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abstract: ' Looking back over 55 years of higher homotopy structures, I reminisce as I recall the early days and ponder how they developed and how I now see them. From the history of Å-structures and later of -structures and their progeny, I hope to highlight some old results which seem not to have garnered the attention they deserve as well as some tantalizing new connections.'
author:
- Jim Stasheff
title: ' and Å-structures: then and now '
---
2ex Dedicated to the memory of Masahiro Sugawara and John Coleman Moore
Introduction
============
Looking back over 55 years of higher homotopy structures, I reminisce as I recall the early days and ponder how they developed and how I now see them. From the history of Å-structures and later of -structures and their progeny, I emphasize my *homotopy* perspective on how they morphed and intertwined in homotopy theory with applications to geometry and physics. A recurring theme is the relation of higher algebraic structures with higher topological, geometric or physical structures [@mss]. There also important higher algebraic structures without homotopy, where ‘higher’ here means generalizations Lie brackets to $n$-ary brackets for $n>2$. I will touch on these only briefly (see Section \[n-ary\]); a very thorough survey is provided by de Azcarraga and Izquierdo . Since 1931 (Dirac’s magnetic monopole), but especially in the last six decades, there has been increased use of cohomological and even homotopy theoretical techniques in mathematical physics. It all began with Gauss in 1833, if not sooner with Kirchof’s laws. The cohomology referred to in Gauss was that of differential forms, div, grad, curl and especially Stokes Theorem (the de Rham complex). I’ll mention some of the more ‘sophisticated’ tools now being used. As I tried to being this survey reasonably up to date, one thing led to another, reinforcing my earlier image not of a tree but of a spider web. I finally had to quit pursuit before I became trapped! My apologies if your favorite strand is not mentioned. I have included bits of history with dates which are often for the published work, not for the earlier arXiv post or samizdat. Acknowledgements: I am grateful to the editors for this opportunity, to all my coauthors and to the many who have responded to earlier versions. The remaining gaps, of which there are several, are mine.
Once upon a time: Å-spaces, algebras, maps
==========================================
For me, the study of associativity began as an undergraduate at Michigan. I was privileged to have a course in classical projective geometry from the eminent relativist George Yuri Rainich[^1]. A later course “The theory of invariants” emphasized the invariants of Maxwell’s equations. Rainich included secondary invariants, preparing me well when I later encountered secondary cohomology operations.
Å-spaces
--------
5ex The history of Å-structures begins, implicitly, in 1957 with the work of Masahiro Sugawara [@sugawara:h; @sugawara:g]. He showed that, with a generalized notion of fibration, the Spanier-Whitehead condition for a space $F$ to be an $H$-space:
*The existence of a fibration with fibre contractible in the total space*
is necessary and sufficient. Sugawara goes on to obtain similar criteria for $F$ to be a homotopy associative $H$-space or a loop space. When I was a graduate student at Princeton, John Moore suggested I look at when a primitive cohomology class $u\in H^n(X,\pi)$ of a topological group or loop space $X$ was the suspension of a class in $H^{n+1}(BX,\pi)$. In other words, when was an H-map $X\to K(\pi,n)$ induced as the loops on a map $BX\to K(\pi,n+1)$. Here, for a topological group or monoid, $BX$ refers to a “classifying space”. In part inspired by the work of Sugawara, I attacked the problem in terms of a filtration of the classifying bundle $EX\to BX$ by the projective spaces $XP(n)$. Sugawara’s work was for $XP(3)$ and $XP(\infty)$. His criteria consisted of an infinite sequence of conditions, generalizations of homotopy associativity. He included conditions involving homotopy inverses which I was able to avoid via mild restrictions.
Any H-space has a projective ‘plane’ $XP(2)$ and homotopy associativity implies the existence of $XP(3)$. Contrast this with classical projective geometry where the existence of a projective 3-space implies strict associativity.
To systematize the higher homotopies, I defined:
An $A_n$ *space* $X$ consists of a space $X$ together with a coherent set of maps $$m_k: K_k \times X^k\to X\ \ for \ \ 2\leq k\leq n$$ where $K_k$ is the (by now) well known $(k-2)$-dimensional associahedron.
My original realization had little symmetry, being based on parameterizations, following Sugawara.
{height="1.2in"}
The *“so-called”* Stasheff polytope was in fact constructed by Tamari in 1951 [@tamari:thesis; @tamari-1962; @tamari-huguet:polyedrale], a full decade before my version.
![from Tamari’s thesis [@tamari:thesis]](St_Tamari-thesis.jpg){height="2in"}
Many other realizations are now popular and have been collected by Forcey [@forcey:zoo] along with other relevant “hedra”. Tamari’s point of view was much different from mine, but just as inspiring for later work, primarily in combinatorics. The book [@tamari:book] has a wealth of offspring.
My multi-indexing of the cells of the associahedra was awkward, but the technology of those years made indexing by trees unavailable. Boardman and Vogt use spaces of binary trees with interior edges given a length in $[0,1]$, producing a cubical subdivision of the associahedra [@BoVo]. Alternatively, the assocciahedron $K_n$ can be realized as the compactification of the space of $n$ distinct points in $[0,1]$ [@mss] Example 4.36, [@jds:config] Appx B. For $n$ distinct points on the circle, the compactification is of the form $S^1\times W_n$, where $W_n$ is called the cyclohedron (see Section \[quest\]). Just as one construction of a classifying space $BG$ for a group $G$ is as a quotient of the disjoint union of pieces $\Delta^n\times G^n$ where $\Delta^n$ is the n-simplex, I constructed $BX$ as a quotient of the disjoint union of pieces $K_n \times X^n$. I was still working on Moore’s problem when I went to Oxford as a Marshall Scholar. I was automatically assigned to J.H.C. Whitehead as supervisor, then transferred to Michael Barratt whom I knew from Princeton. When Barratt moved to Manchester, Ioan James took on my supervision. Though Frank Adams was not at Oxford, he did visit and had a significant impact on my research. Ultimately, this resulted in my theses for Oxford and Princeton. I needed to refer to what would be my Princeton thesis as a prequel to the Oxfod one. Oxford had a strange rule that I could not include anything I *had submitted* elsewhere for a degree, even with attribution, so I made sure to submit to Oxford before returning to the US and finishing the formalities at Princeton. For the record, the 1963 published versions *Homotopy Associativity of H-spaces I and II* [@jds:hahI; @jds:hahII] correspond respectively to the topology of the Princeton thesis and the homological algebra of the Oxford one.
Å-algebras
----------
Thinking of the cellular structure of the associahedra led to:
Given a graded vector space $A= \{A_n\}$, an Å-algebra structure on $A$ is a coherent set of maps $m_k:A^{\otimes k}\to A$ of degree $k-2$.
Here *coherence* refers to the relations 0 = \_[i+j=n+1]{}\_[k=1]{}m\_i (a\_1a\_[k-1]{}m\_j( a\_ka\_[k+j-1]{}) a\_[k+j]{}a\_n) For an ordinary dga $(A,d,m)$, Massey constructed secondary operations now called *Massey products* (Massey’s $d$ was cohomological, i.e. of degree +1). These products generalize easily to an Å-algebra, using $m_3$, etc. For an ordinary associative algebra $(A,m)$, there is the *bar constuction* $BA$, a differential graded coalgeba with differential determined by the multiplication $m$. For an Å-algebra, there is a completely analogous construction using all the $m_k$. The most effective defintion of an Å-morphism from $A_1$ to $A_2$ is as a morphism of dg coalgebras $BA_1\to BA_2$. There is an important but subtle relation between the differentials of the *bar construction spectral sequence* and (higher) Massey products (see [@buijsetal:massey]).
Å-morphisms
-----------
Morphisms of Å-spaces are considerably more subtle. Again the way was led by Sugawara [@sugawara:hc], who presented *strongly homotopy multiplicative maps* of strictly associative $H$-spaces.
\[shmap\] A *strongly homotopy multiplicative map* $X\to Y$ of associative $H$-spaces consists of a coherent family of maps $I^n\times X^n\to Y$.
The analog for dg associative algebras is straightforward. When $X$ and $Y$ are Å-spaces, the parameter spaces for the higher homotopies, now known as the *multiplihedra* (often denoted ${\mathcal J}_n$ or $J_n$), are considerably more complicated and realization as convex polytopes took much longer to appear. An excellent illustrated collection of these and other related polyhera is given by Forcey [@forcey:multiplihedra], see also [@Kawa]. My early picture for $n = 4$ was not really a convex polytope though my drawing of $\cal{J}_4$ appears as a subdivided pentagonal cylinder. Iwase and Mimura in [@iwase-mimura:hha] gave the first detailed definition of the multiplihedra and describe their combinatorial properties. If the range $Y$ is strictly associative, then the multiplihedron ${\mathcal J}_n$ collapses to the associahedron $K_{n+1}$ [@jim:book]. On the other hand, Forcey [@forcey:quotient] observed that if the domain $X$ is strictly associative, then the multiplihedron denoted ${\mathcal J}_n$ collapses to a *composihedron* he created and denotes ${\mathcal CK}(n)$, new for $n\geq 4$. The analog for Å-algebras is much easier to write down, convexity not being an issue. The study of $A_{\infty}$ spaces and algebras continues. There are interesting questions about the extension of $A_n$-maps, as in [@hemmi:review] and about the transfer of $A_{\infty}$ structure through these maps, as in [@markl:tr].
Iterated loop spaces and operads {#iterated}
================================
In terms of the development of higher homotopy structures, perhaps my most important result was this characterization of spaces of the homotopy type of loop spaces in terms of an Å-structure:
A ‘nice’ connected space $X$ has the homotopy type of a based loop space $\Omega Y$ for some $Y$ if and only if $X$ admits the structure of an Å-space.
Here ‘nice’ means of the homotopy type of a CW-complex with a non-degenerate base point. For the standard description of $\Omega Y$ in terms of loops parameterized by $[0,1]$, the maps are generalization of the usual one for homotopy associativity. An alternative is to use the strictly associative *Moore space of loops* $\MY$ with loops parameterized by intervals $[0,r]$ for $r\geq 0$ [@moore:loops]\[mooreloops\]. Apparently Moore never formally published this major contribution, but, thanks to the internet, his seminar of 1955/56 is available at http://faculty.tcu.edu/gfriedman/notes/aht23.pdf. The homotopy equivalences of $X$ and $\Omega Y$ and $\MY$ are indeed Å-morphisms. From characterizing loop spaces, I went on to characterizing loop spaces on H-spaces by constructing a multiplication on a classifying space $BX$. This led to issues of homotopy commutativity with formulas getting out of hand [@jim:hc]. Soon after, Clark [@clark:hc] investigated the subtleties of comparing ${\boldsymbol \Omega}(X\times X)$ with $\MX\times\MX$. In 1967 at the University of Chicago, prompted by a visit from Frank Adams, a seminar [*Iterated homotopies and the bar construction*]{} was organized by Adams and Mac Lane. Thanks to Rainer Vogt we have a great record of that seminar [@vogt:memories]. The audience was exceptional; see Vogt’s listing as well as his recollection of the lectures. He writes that the seminar had a great influence on his work with Mike Boardman, as it did on several of the participants, myself included. In 1968 Boardman and Vogt [@BoVo:bull; @BoVo] (1973), motivated by the many infinite loop spaces then of interest thanks to Bott periodicity and that seminar, emphasized the point of view of *homotopy invariant algebraic structures* to characterize such infinite loop spaces. A key idea was the passage from a strict algebraic structure to one that was homotopy invariant but still of the same homotopy type, suitably interpreted. Since retiring from UNC, I have been a long time guest at U Penn, where the language of algebraic geometry is dominant, puzzling me as to what ‘derived’ referred. To paraphrase Monsieur Jourdain, I was happy to discover that I had been speaking ‘derived’ all my life; i.e. speaking in a related homopy category (see Section \[derived\] for an alternate meaning). Then around 1970, Peter May came along and blew the subject wide open with his development of operads [@may:geom] to handle the complex of homotopy symmetries to characterize iterated loop spaces of any level. Of particular importance early on for Boardman and Vogt and for May was the little $n$-cubes operad.Somewhat later, Getzler [@getzler:bvand2D] introduced the little $n$-disks operad and its framed version. Since then there has been a proliferation of operads with additional structure as well as many generalizations of the concept, [@loday-vallette; @mss:opbook]. Operads were crucial for studying an important issue in the $\infty$-version of commutative algebras: whether to relax the commutativity up to homotopy or to keep the strict symmetry but relax the associativity or relax both. As emphasized by Kontsevich [@kont:trium], the triumvirate of Å-, - and $C_\infty$-algebras play a dominant role. By $C_\infty$-algebra is meant what is also known as a balanced Å-algebra, that is, a strictly graded commutative Å-algebra defined in terms of a coherent set of n-ary products which vanish on shuffles. $C_\infty$-algebras and -algebras are in an adjoint relationship just as are strict associative commutative algebras and Lie algebras. The next most prominent might be $E_\infty$-algebras, dubbed *homotopy everything*. That is a bit of a misnomer, though they are very important for the study of infinite loop spaces. As readers of this journal are well aware, there has been a proliferation of strong homotopy or $\infty$-structure versions of classical algebras such as Gerstenhaber, Batalin-Vilkovisky, Froebenius and on and on (see Sections \[derived\] and \[bvalg\]) as well $\infty$ ‘spaces’ and even $\infty$ ‘group/oids’. Those occur in attempting ‘integration’ as in integration of a Lie algebra to a Lie group [@henriques:int] regarded as a simplicial set or Kan complex. There is also *derived Å* for use over a ring rather than a field [@sagave:derived; @livernet:derivedAoo].
-algebras {#Loo}
=========
In contrast to homotopy associativity, which was considered long before the higher homotopies were recognized, algebras with just ‘Jacobi up to homotopy’ appeared only after the full set of higher Jacobi homotopies were incorporated in -algebras (aka strong homotopy Lie algebras or sh-Lie algebras), which in turn had waited many years to be introduced for lack of applications. However, ‘Jacobi up to homotopy’ was implicit in the many proofs of the graded Lie algebra structure of the Whitehead product since, as Massey said [@Haring],:
> This question was ‘in the air’ among homotopy theorists in the early 1950’s, I don’t believe you can point to any one person and say that he or she raised this.
Retakh and Allday were the first to define Lie-Massey operations, both in 1977: [@retakh:massey] in Russian and [@allday:massey]. A preferable name might be *Massey brackets*. For Retakh, they appeared as obstructions to deformations of complex singularities. (Compare [@douady; @retakh:douady] and see Section \[defs\].) Retakh’s $n$-homotopy multiplicative maps are a special case of -morphisms. Both Retakh and Allday emphasize applications to the Quillen spectral sequence and to rational Whitehead products. For higher Whitehead products, see also [@belchietal:hiwhitehead]. Clear exposition of the Massey brackets and their connection to rational homotopy theory is in chapter V of [@tanre:modeles]. Higher homotopies for Jacobi led to:
[@ls; @lada-markl] An -structure on a graded vector space $V$ is a collection of skew graded symmetric linear brackets $l_n:\bigotimes ^nV \longrightarrow V$ $$\ell_n = [\ ,\dots,\ ]:\Lambda^n V\to V$$ for $n\geq 1$ of degree $2-n$ (for cochain complexes, and $n-2$ for chain complexes) such that \_[i+j=n+1]{}l\_i (l\_j(v\_[(1)]{}…v\_[(j)]{}) v\_[(j+1)]{} …v\_[(n)]{})=0 where $\sigma$ runs through all $(j,n-j)$ unshuffles and for which ‘there exists a set of signs’ (folk saying); in this case, the sign of the unshuffle in the graded sense.
Equivalently, an *-algebra* is a graded vector space $L=\{L_i\}$ with a coderivation differential of degree $\pm 1$ on the graded symmetric coalgebra $C(L)$ on the shift $sL$. For an ordinary Lie algebra, this is the classical Chevalley-Eilenberg [*chain*]{} complex. The map $l_1$ is a differential: $(l_1)^2=0$ and $l_2$ may be compared to an ordinary (graded) Lie bracket $[\ , \ ].$ When $l_1$ or $l_3=0$, the definition yields the usual (graded) Jacobi identity. In general, $l_3$ is a homotopy between the Jacobi expression and $0$ while the other $l_n$’s are known as *higher homotopies* or *higher brackets*. If $l_3=0$, there still may be non-trivial higher $l_n$, e.g. on the homology of a dg Lie algebra (see Section\[qht\]). Physicists like to say products, but we have consistently used *‘brackets’*. There are alternative notations: $$l_n(v_1,\cdots,v_n) = [v_1,\cdots,v_n] \ \text(math)\ \ = [v_1\cdots v_n] \ \text(physics).$$
Note the ambiguity as to the degree $\pm 1$ of $d$ in defining an -algebra. The binary operation is always of degree 0; sometimes the ‘manifest’ grading in examples is $ \bold{not}$ the right one; see examples below. The shift of the bracket now has the same degree as the shift of $\ell_1$. Notice also this bracket extends to an action of the degree 0 piece on the piece of degree 1 (or -1 respectively), as for a module over an algebra (contrast BBvD structures in Section \[BBvD\]).
Belatedly, Sullivan’s models for rational homotopy types were recognized as being of the form $C(L)$ on the shift $sL$ of the -algebra of rational homotopy groups [@sullivan:inf]. The 2-bracket corresponds to the Whitehead or Samelson product and higher order brackets to higher order Whitehead products [@belchietal:hiwhitehead].
structures in physics {#LLphysics}
---------------------
Giovanni Felder was all too briefly my colleague at UNC. Later he joined the club, remarking
*“the $\infty$-virus had a long incubation time and the outbreak came after I left the infection zone.”*
Since then, it has expanded to epidemic proportions in the field theoretic physics community.
In 1982, -algebras appeared in disguise in gravitational physics in work of D’Auria and Fré. Unfortunately they referred to their algebras as free differential algebras; to be precise, their FDA is a dgca (free as a gca ignoring the differential). Around 1984, Gerett Burgers visited Henk Van Dam at UNC and we discussed parts of his thesis; again it looked like an $L_\infty$ structure was lurking there. Later this was confirmed by . The essential idea in [@burgers:diss; @BBvD:probs] was a novel attack on particles of higher spin by letting the gauge parameters act in a *field dependent* way. In 1987, the formulas of the BRST operator in the construction of for constrained Hamiltonian systems [@BF; @FF; @FV] could be recognized as corresponding to an -algebra [@jds:shrep; @jds:ghost; @jds:hrcpa] as did the corresponding Lagrangian formulas [@jds:moskva]. In 1989, the $L_\infty$ structure of CSFT (Closed String Field Theory) was first identified when Zwiebach fortuitously gave a talk in Chapel Hill at the last GUT (Grand Unification Theory) Workshop [@z:csft; @z:book]. By 1993, it seemed appropriate to provide an *Introduction to sh Lie algebras for physicists* [@ls]. In 1998, Roytenberg and Weinstein, building on Roytenberg’s thesis, showed that Courant algebroids give rise to (small) -algebras (see Section \[def:quasi-algebroid\] and [@zambon:hicourant]). -algebras are continuing to be useful (and popular !) in physics[^2] in two ways:
- Solution of a physical problem leads to a structure which later is recognized as that of an -algebra.
- Solution of a physical problem is attacked using knowledge of -algebras.
There are some famous ‘no go’ theorems that rule out certain physical models, e.g. for higher spin particles (see Section \[BBvD\]). What are ruled out are only models in terms of representations of strict Lie algebras (compare Section \[ruths\]).
‘Small’ -algebras
-----------------
CSFT requires the full panoply of higher brackets of all orders, but many other examples of -algebras with at most 3 pieces in the grading have appeared in physics. More generally, the name *Lie $n$-algebra* uses the $n$-categorical language and refers to an -algebra $L$ with $L_n$ = 0 for $n<0$ and for $n> n-1$ or for $n>0$ and $n> -n+1$. Some authors refer to these as ‘truncated’ -algebras.
Zwiebach and other physicists had asked about small examples of -algebras, (physicists’ ‘toy’ models) leading to work of Tom Lada and his student Marilyn Daily. She classified all 3-dimensional -algebras: 2-graded with one 1-dimensional component and one 2-dimensional component; 3-graded where each component is 1-dimensional [@daily].
[**Warning!!**]{} There is a real problem of nomenclature. There are also notions of $n$-Lie algebra which have a $k$-ary bracket *only* for $k=n$ (see Section \[n-ary\]).
In 2017, [@hz] introduced a wide class of field theories they call *standard-form*, which included, by assumption, a (usually small) $L_\infty$-structure; again field dependence was crucial. They went beyond by including a space of ‘field equations’ in their -algebras. Together with Andreas Deser, Irina Kogan and Tom Lada, we are adding such field equations to the approach to show that again field dependence implies the structure. Following , Blumenhagen, Brunner, Kupriyanov and Lüst [@bbkl] attacked the existence of an appropriate $L_\infty$ structure with given initial terms by what they call a bootstrap approach. By this they mean an inductive argument for the $n$-bracket by *solving* the equations for the L$_\infty$ relations. They succeed for non-commutative Chern-Simons and for non-commutative Yang-Mills using properties of the *star product* and various string field theories combining Chern-Simons and Yang-Mills by careful computation (see Section \[morphs\] for uniqueness of their approach). Contrast the inductive argument for BFV and BV (see Section \[bfv\]) where a solution follows from the auxiliary acyclic resolution.
-morphisms {#morphs}
----------
-morphisms $V \to W$ can most efficiently be described as morphisms of dgcas $C(V)\to C(W)$. They are particularly important in Kontsevich’s proof of the Formality Conjecture in relation to deformation quantization of Poisson manifolds [@kont:defquant-pub]. Here the crucial -morphism was from the dg Lie algebra of multivector fields to the dg Lie of multidifferential operators. Since his initial success, there have been many formality theorems upgraded to other settings, for example, for BV algebras [@campos:BVformality] or for a cyclic version [@ward:cyclicMC]. A special case is comprised of the Seiberg-Witten maps (SW maps) [@seiberg-witten] between “non-commutative” gauge field theories, compatible with their gauge structures; in particular, between non-commutative versions of theories with a gauge freedom. When these theories exist, they are consistent deformations of their commutative counterparts.
Blumenhagen, Brinkmann, Kupriyanov and Traube [@bbkt] establish uniqueness of their results (up to gauge equivalence) by constructing SW maps. Aschieri and Deser [@deser-aschieri:SW] construct specific examples in terms of $U(n)$-vector bundles on two-dimensional tori given by globally defined Seiberg-Witten maps (induced from the plane to the torus).
BBvD -algebras {#BBvD}
--------------
In contrast to Lie $n$-algebras as defined above, it is possible to have an -algebra concentrated in degrees $0$ to $n-1$ with $d$ of degree $+1$ as for the higher spin algebras of : They start with a given space of ‘fields’ $\PP$ which is a module over a Lie algebra $\XX$ of gauge symmetries.
By a field dependent gauge transformation of $\XX$ on $\PP,$ they mean a polynomial (or power series) map $\Xi\otimes\LP\to \Phi$: \_() = T(,)=\_[i0]{} T\_i(,) where $T_i$ is linear in $\xi$ and polynomial of homogeneous degree $i$ in $\pp.$ Note the operation $T_0:\Xi \to \PP$ from ‘algebra’ to ‘module’, in contrast to Lie $n$-algebras.
They have a corresponding field dependent generalization of a Lie algebra structure on $\XX$: a polynomial (or power series) map $\Xi\otimes\Xi\otimes\LP\to \Xi$ $$[\xi,\eta](\pp) = C(\xi,\eta,\pp)=\Sigma_{i\geq 0} C_i(\xi,\eta,\pp)$$ where $C_i$ is bilinear in $\xi$ and $\eta$ and of homogeneous degree $i$ in $\pp.$ These operations obey consistency relations which Fulp, Lada and I identified as structure relations of an $L_\infty$ algebra [@fls].
Note the BBvD structure gives rise to an $L_\infty$-algebra structure on the direct sum of the space of fields and the space of gauge parameters, [**not**]{} of the form of an -algebra and its module, rather a Lie 2-algebra. This is similar to what occurs in the BFV and BV formalisms.
The BFV and BV dg algebra formalisms {#bfv}
------------------------------------
For the related but separate notion of a BV algebra, see Definition \[bvalg\]. Work of Batalin-Fradkin-Fradkina-Vilkovisky (BFV) from 1975-1985 [@BF; @FF; @FV] concerned reduction of constrained Hamiltonian systems. The constraints formed a Lie algebra and generated a commutative ideal. Their construction combined a Koszul-Tate resolution with a Chevalley-Eilenberg complex. The sum of the two differentials no longer squared to 0 but required terms of higher order. These were shown to exist using the acyclicity of the Koszul-Tate resolution. In 1988 [@jds:bull], I was able to understand those terms in the context of homological perturbation theory (HPT, a common technique in $\infty$-theories) and representations up to (strong) homotopy (RUTHs) (see Section \[ruths\]). For the Lagrangian version, Batalin and Vilkovisky [@bv:closure; @BV3; @bv:anti] constructed the analogous dg algebra by a similar method [@jds:moskva] . These constructions suggested an -algebra and module, but things were not so simple. My student Lars Kjeseth in his UNC thesis [@lars:hha1; @lars:hha2] showed that the appropriate structure was that of a *strong homotopy* version of a Lie-Rinehart algebra [@rinehart]. This concept then lay dormant until resurrected around 2013 in the work of Johannes Huebschmann [@jh:quasiLR] and of Luca Vitagliano [@luca:shLR]. More generally, [@bfls:gauge] constructed an -algebra on any homological resolution of a Lie algebra.
Some special -algebras {#dft}
----------------------
Special -algebras arise from Courant algebroids, Double Field Theory (DFT) and multisymplectic manifolds. In the last two decades, what is called T-duality in string theory and supergravity led to a formulation of differential geometry on a generalized tangent bundle: $$\label{eq:Edef}
0 \longrightarrow T^*M \longrightarrow E
\stackrel{\pi}{\longrightarrow} TM \longrightarrow 0 .$$ Locally, the bundle $E$ looks like $TM\oplus T^*M$. 10ex$\mathbf{Courant\ algebroids}$ Courant algebroids are structures which include as examples the doubles of Lie bialgebras and bundles $TM\oplus T^*M.$ They are named for T. Courant [@courant] who introduced them in his study of Dirac structures. Given a bilinear skew-symmetric operation $[\ ,\ ]$ on a vector space $ V $, its *Jacobiator* $ J $ is the trilinear operator on $ V $: $$J(e_{1},e_{2},e_{3})=[[e_{1},e_{2}],e_{3}]+[[e_{2},e_{3}],e_{1}]+[[e_{3},e_{1}],e_{2}],$$ $ e_{1},e_{2},e_{3}\in V $. The Jacobiator is obviously skew-symmetric. Of course, in a Lie algebra $ J\equiv 0 $.
\[def:Liealgebroid\] A *Lie algebroid* is a sequence $E\overset{\rho}{\to} TM \overset{p}{\to} M$ of vector bundle maps with a Lie bracket on the space of sections $\Gamma(E)$ such that = (X)(f)Y +f\[X,Y\] for $X,Y\in\Gamma(E), f\in C^\infty(M). $ (It follows that $\rho[X,Y]=[\rho(X),\rho(Y)].$)
\[rinehart\] The purely algebraic analog of a Lie algebroid is a *Lie-Rinehart algebra* over a commutative algebra more general than $C^\infty(M)$ [@rinehart].
\[def:quasi-algebroid\] (Approximate Definition [@roytenberg:thesis; @wein-royt:lmp]) A *Courant algebroid* is an algebroid with anchor $ \rho :E\to TM $ equipped with a nondegenerate symmetric bilinear form $\langle {\cdot }\,\ {\cdot } \rangle$ on the bundle $E,$ a skew-symmetric *Courant* bracket $[\ ,\ ]$ on $ \Gamma (E) $ and a map $ {\mathcal{D}}:{C^{\infty}}(M){\longrightarrow }\Gamma (E) $ satisfying many properties of which the most relevant is that the Courant brackets on $\Gamma(E)$ satisfy the Jacobi identity up to a $\DD$-exact term: For any $e_{1},e_{2},e_{3}\in \Gamma (E) $, J(e\_[1]{},e\_[2]{},e\_[3]{})=T(e\_[1]{},e\_[2]{},e\_[3]{}) where $ T(e_{1},e_{2},e_{3}) $ is the function on the base $ M $ defined by: $$\label{eq:T0}
T(e_{1},e_{2},e_{3})= 1/6 \underset{cyclic}{\sum}\langle [e_{1},e_{2}],e_{3}\rangle.$$
A related structure is due to Dorfman [@dorfman:dirac] for which the bracket is not skew-symmetric but satisfies the Loday (also called Leibniz) version of the Jacobi identity [@loday] which is expressed as a derivation from the left (see Definition \[loday\]). The name *Dorfman bracket* appears in [@royt:dorfman] which discussed Courant-Dorfman structures. The Courant bracket is the skew symmetrization of the Dorfman bracket [@yks:nij].
Roytenberg showed in his PhD thesis [@roytenberg:thesis] the equivalence of this structure with a specific $L_\infty$ algebra $X$, in which $l_1$ is determined by the de Rham differential, $l_2$ by the Lie bracket and the operation $l_3$ contains “flux”-degrees of freedom (e.g. a three-form known as $H$-flux). Here $X$ is a resolution of ${\mathcal H}=coker\DD$: $$\label{eqn:res}
X_2=\ker\DD\stackrel{d_2}{\lon}X_1={C^{\infty}}(M)\stackrel{d_1}{\lon}X_0=\Gamma(E)\lon{ \mathcal H}{\longrightarrow }0,$$ where with $d_1=\DD$ and $d_2$ is the inclusion $\iota:\ker\DD\hookrightarrow{C^{\infty}}(M)$. The Courant brackets on $ \mathcal H$ come from Courant brackets on $\Gamma(E)$. Roytenberg and Weinstein [@wein-royt:lmp] use this to extend the Courant bracket to an -structure on all of their resolution $X$, manifestly a Lie 3-algebra.
Further generalizations known as *higher Courant algebroids* [@zambon:hicourant] are locally $TM\oplus \Lambda^k(T^*M)$. 10ex $\mathbf{Double\ Field\ Theory (DFT)}$ Aldazabal, Marqués and Núñez write in *Double Field Theory: A Pedagogical Review* [@Aldazabal-Nunez]:
> Double Field Theory (DFT) is a proposal to incorporate T-duality, a distinctive symmetry of string theory, as a symmetry of a field theory defined on a *double configuration space*.
Indeed, as originally introduced in physics, DFT (Double Field Theory) refers *not* to a doubling of fields but rather to doubling of an underlying structure such as double vector bundles and Drinfel’d doubles. The original Drinfel’d double occurred in the contexts of quantum groups and of Lie bialgebras. Following Roytenberg [@roytenberg:thesis], Deser and I [@deser-jds] interpret the gauge algebra of DFT in terms of Poisson brackets on a suitable generalized Drinfel’d double. In DFT, it is the coordinates that are doubled and ‘double fields’ refer to fields which depend on both sets of coordinates. Thus we also refer to double functions, double vector fields, double forms, etc. In the physics literature, reference is made to a *C-bracket* to distinguish it from a Poisson bracket, but we showed that the C-bracket is essentially the Poisson bracket on $T^*[1]TM$. (Here $[1]$ denotes the shift in degree of the *fibre* coordinates of $TM$.)
Similarly, Deser and S" amann in [@deser-saemann:ERGI] used the analog of a Poisson bracket on $T^*[n]T[1]M$ to define an action of a Lie $n$-algebra on the algebra of functions $\mathcal{F}:= \mathcal{C}^\infty(T^*[n]T[1]M)$. In particular, what is called “section condition” in the physics of DFT was identified as the requirement for a specific Lie-2 algebra to act on $\mathcal F$ in a well-defined way. As a bundle over $TM$ with a local trivialization with respect to a covering $\mathcal U =\{U_\aa\}$, we have transition functions $a_{\alpha\beta}\in GL(2d,\mathbf R)$ satisfying the usual cocycle condition, but, in terms of the local splitting $TM\oplus T^*M$, there is a higher order ‘twist’ $\tau$ depending on a 2-form $\omega$. Now the cocycle condition fails, the failure depending on $\omega$: $$g{_{\alpha\beta}}\ g{_{\beta\gamma}}\neq g{_{\alpha\gamma}}\quad\mbox{on}\quad U{_{\alpha}}\cap U{_{\beta}}\cap U{_{\gamma}}.$$ This is often described as a failure of associativity, but it is more accurately failure to correspond to a representation. Since $g{_{\alpha\beta}},\ g{_{\beta\gamma}}$ and $g{_{\alpha\gamma}}$ can be expressed in terms of 1-forms, it can be that the difference is an exact form $d\lambda_{\aa\bb\gg}$ for some function $\lambda_{\aa\bb\gg}$ on $U{_{\alpha}}\cap U{_{\beta}}\cap U{_{\gamma}}.$ One would then say the transition functions form a representation up to homotopy (RUTH) or one can speak in terms of ‘gerbes’. (See Section \[wirth\] for higher order generalizations.) 10ex $\mathbf{Multisymplectic\ manifolds}$
[@rogers:multisymplectic] A *multisymplectic* or, more specifically *$n$-plectic*, manifold is one equipped with a closed nondegenerate differential form of degree $n+1$.
As shown by Chris Rogers, [@rogers:multisymplectic], just as a symplectic manifold gives rise to a Poisson algebra of functions, any $n$-plectic manifold gives rise to a Lie $n$-algebra of differential forms with multi-brackets specified via the $n$-plectic structure. The underlying graded vector space consists of a subspace of $(n-1)$-forms he calls *Hamiltonian* together with all $p$-forms for $0 \leq p \leq
n-2:$ $$\label{complex}
{C^{\infty}}(M) \stackrel{d}{\to} \Omega^{1}(M) \stackrel{d}{\to} \cdots \stackrel{d}{\to}
\Omega^{n-2}(M) \stackrel{d}{\to} \Omega^{n-1}_{Ham}.$$
Generalized Jacobi identities and Nambu-Poisson algebras
========================================================
Identities {#n-ary}
----------
There are two important ways to generalize to $n$-variables the Jacobi identity for a Lie algebra written as a left derivation (see Definition \[loday\]): = \[\[a,b\],c\] + \[b,\[a,c\]\]. In older literature, these generalized Jacobi relations are referred to as *fundamental identities*, but, as suggested by de Azcarraga and Izquierdo , a better name might be *characteristic identities*., One identity is the corresponding -relation for a bracket of just $n$ variables: l\_n(l\_n(v\_[(1)]{}…v\_[(n]{}) v\_[(j+1)]{}…v\_[(2n-1)]{})=0.
It has been studied quite independently of my work and of each other by Hanlon and Wachs [@hanlon-wachs] (combinatorial algebraists), by Gnedbaye [@gned] (of Loday’s school) and by de Azcarraga and Bueno [@az-bu] (physicists). On the other hand, the characteristic identity for a *Filippov algebra* [@filippov]) says $ [X_1, X_2, \dots,X_{n-1}, \quad]$ acts as a left derivation.
$$\label{n-der}
[X_1, X_2, \dots,X_{n-1}, [ Y_1, Y_2, \dots, Y_n]] =
\ee
\be
[[X_1,X_2,\dots, X_{n-1}, Y_1],Y_2,\dots,Y_n] +
\dots + [Y_1,\dots, Y_{n-1},[X_1,\dots,X_{n-1},Y_n]].
$$
That identity was known also to Sahoo and Valsakumar [@sahoo]. Unfortunately, both versions are called $n$-Lie algebras, hence my attempt (probably futile) to rename his as Filippov’s. As I recall, I first learned of this other (Filippov) identity from Alexander Vinogradov when we met at the Conference on Secondary Calculus and Cohomological Physics, Moscow, August 1997. (See A. and M. Vinogradov’s [@mmv] for a comparison of these two distinct generalizations of the ordinary Jacobi identity to $n$-ary brackets.) The article by de Azcarraga and Izquierdo is a very thorough survey of even more $n$-ary algebras. All these algebras are important in geometry and in physics where the corresponding structures are on vector bundles over a smooth manifold (see [@wade] and references there in).
Nambu-Poisson $n$-Hamiltonian mechanics
---------------------------------------
Nambu’s original work [@nambu] was a generalization to an $n$-ary bracket of Hamiltonian mechanics with its binary Poisson bracket. Often the literature refers to Nambu-Poisson structures, which emphasizes the setting of $C^\infty$ functions on a smooth manifold as in traditional Hamiltonian mechanics. Just as the latter can be extended to sections of a Lie algebroid, Nambu-Poisson structures on manifolds can be extended to the context of Lie algebroids [@wade]. The Nambu bracket satisfies the Filippov identity [@takh:nambu]. One expects there to be “$\infty$”-versions with the full panoply of applications as for -structures, e.g. homological reduction of constrained Nambu-Poisson algebras (cf. Section \[bfv\]).
Derived bracket and brace and BV algebras
=========================================
with major assistance from Fusun Akman, Yvette Kosmann-Schwarzbach, and Thedia Voronov
Differential operators
----------------------
An increasingly popular approach to -structure is the use of *derived brackets* (see Definition \[derivedbracket\]) introduced by Kosmann-Schwarzbach in [@yks:PtoG], inspired by unpublished notes by Jean-Louis Koszul [@koszul:notes]. She traces their origin to work of Buttin and of A. Vinogradov in the context of unification of brackets in differential geometry, though physics (of integrable systems for instance) was not far behind, especially in work of Irina Dorfman [@dorfman:dirac; @dorfman:book]. See [@yks:derived] for an excellent survey and history. Koszul’s point of view was that of an algebraic characterization of the order of a differential operator. Multilinear operators with arguments that are left multiplication operators $\ell_{a}$ rather than elements $a$ of an algebra may have appeared first in various definitions of the order of a differential operator. For example, the *order* of a differential operator $\D$ can be defined as
[@akman:BV] A linear operator $\D$ on an algebra $A$ is a *differential operator of order $\leq r$* if an inductively defined $(r+1)$-linear form $\Phi_{\D}^{r+1}$ with values in $A$ is identically zero.
or, more simply: *An operator $\D$ on an algebra $A$ is of order at most $n$ if, for all left multipliers $\ell_a$ for $\in A$, the commutator $[\D,\ell_{a}]$ is of order at most $n-1$.*
This approach can be traced back to 1967: Grothendieck (Ch.IV, EGA4), then developed by Koszul [@koszul:notes]. This was carried forward in 1997 by Akman [@akman:BV] who defines higher order differential operators on a general *noncommutative, nonassociative* graded algebra ${{\mathcal{A}}}$. Akman and Ionescu [@akman:hiderived] compare and show equivalence of several definitions of order when the underlying algebra $A$ is classical, i.e., graded commutative and associative. The brackets that are now called “higher derived” (as defined below) can be represented in a form similar to that for defining “order”, but are of interest for providing -structure.
-algebras {#algebras}
---------
Not only -algebras are relevant, but also **-algebras and even -algebras. Loday (1946 - 2012) originally called them *Leibniz algebras*, but he deserves the credit. Recall:
\[loday\] A (left) *-algebra* $(V,[\ ,\ ])$ consists of a vector space $V$ with a *Loday bracket* $[\ ,\ ]$ satisfying the left Leibnitz identity: \[a,\[b,c\]\]= \[\[a,b\],c\] .
Throughout this section, “there exists a set of signs’; here for the graded case.
[@loday:coprod] A (left) *-algebra* $(V,\pi)$ consists of a graded vector space $V$ with a sequence $\pi=(\pi_1,\pi_2,\dots)$ of multivariable coderivations $\pi_i$ on the free graded coalgebra cogenerated by $V$ shifted with a rather unusual coproduct using unshuffles that preserve the (last) element. The sequence $\pi$ is to satisfy $[\pi,\pi]=0$.
Apparently this was defined first by Livernet [@livernet:lodayinfty] for right -algebras, later independently in a different setting by Ammar and Poncin [@ammar-poncin:lodayinfty] with further work by Peddie [@peddie:lodayinfty].
Derived brackets {#derived}
----------------
Kosmann-Schwarzbach defined *derived brackets* in 1996 [@yks:PtoG]:
\[derivedbracket\][@yks:PtoG] If $(V, [~,~], D)$ is a graded differential Lie or Loday algebra with a bracket $[\ ,\ ]$of degree $n$, the [*derived bracket*]{} of $[~,~]$ by $D$ is denoted $$[~,~]_D: V \otimes V \to V$$ and defined by \_D= (-1)\^[|a|+n+1]{} \[Da,b\] , for $a$ and $b \in V$.
Often $D$ is itself given as $D=[\D,\ \ ]$ for an element $\D\in V$. In this generality, the derived bracket is graded Loday and respected by the differential. To obtain a differential graded Lie bracket, additional actions are necessary.
Continuing the iteration of Lie brackets led in 1996 to Bering defining *higher BV anti-brackets* in physics [@beringetal:hibrack]. Somewhat later, independently in math, T. Voronov defined *higher derived brackets* [@tv:hiderived]. This was followed quickly by [@bering:brackets] and further proliferation. Apparently the membrane between math and physics is osmotic: information travels one way faster than the other. The definition closest to my interests is:
*Higher derived brackets* $$B^r_{\Delta}(a_1,\dots,a_r)$$ are defined as $$[\dots [\,[\Delta,a_1],a_2],\dots,a_r]$$ where $(L,[-,-])$ is a Lie algebra with $\D, a_1,\dots,a_r\in L$.
In terms of operators $D$:
\[Alternative\] *Higher derived brackets* $$B^r_{D}(a_1,\dots,a_r)$$ are defined as $$[\dots [Da_1],a_2],\dots,a_r]$$ where $(L,[-,-])$ is a Lie algebra with $ a_1,\dots,a_r\in L$ and $D:L\to L$.
The major uses of higher derived brackets are for describing/constructing -algebras and other $\infty$-algebras. An important case is the derived bracket built with an odd, square-zero differential operator $\D$ of order $\leq 2$, which Akman uses to define a *generalized Batalin-Vilkovisky algebra* structure on an algebra $A$ (see Section \[bvalg\]). The emphasis on order $\leq 2$ is historical, dating back to and their applications in physics.. T. Voronov’s *higher derived brackets* [@tv:hiderived] first occurred in the context of a Lie algebra $L$ with an abelian sub-algebra $A$. The higher derived brackets on $A$ were derived on $L$ and then projected back to $A$. More general higher derived brackets are introduced by Bandiera [@bandiera:hiderived]. Derived (binary and higher) brackets are also important for describing and understanding infinitesimal symmetry actions relevant in physics. The Roytenberg-Weinstein -structure can be expressed and generalized in terms of derived brackets [@roytenberg:thesis], Most recently, Deser and Sämann [@deser-saemann] show binary derived brackets underlie the symmetries of Double Field Theory (Section \[dft\]). They suggest adopting as a guiding principle:
> Whenever we are seeking an infinitesimal action of a fundamental symmetry, we try to find the corresponding derived bracket description.
One should also seek higher derived brackets as in [@tv:hiderived] and hence -structures as called for in [@hz] (see Section \[LLphysics\]).
Braces
------
Brace algebras, like operads, are all about composition. As Akman says: it all depends on what you want to achieve and how far you want to go.
A *brace algebra* is a graded vector space with a collection of braces $x\{x_1, \dots, x_n\}$ of degree $\pm n$ satisfying the identities $$\begin{gathered}
\label{higher}
x\{x_1, \dots, x_m\} \{ y_{1}, \dots , y_{n}\} \\
\begin{split}
= \sum
\pm x\{ y_1, \cdots,
y_{i_1}, x_1 \{ y_{i_1+1}, \cdots,
y_{j_1}\}, y_{j_1+1}, \cdots \hskip10ex \\
\cdots,
y_{i_{m}}, x_m \{ y_{i_{m}+1} ,
\dots , y_{j_m}\}, y_{j_m+1}, \dots, y_n \}
\end{split}\end{gathered}$$ where the sum is over $0 \le i_1 \le \dots \le i_m \le n$ and the sign is due to the $x_i$’s passing through the $y_j$’s in the shuffle.
Thus brace algebras are flexible enough to accommodate insertion of several arguments at one time, but not necessarily filling all slots (an operad does fill all slots). Braces first occurred in work of Kadeishvili [@kad:braces] who called them higher $\smile_1$ products. In later but independent work, Gerstenhaber and A. Voronov named them *braces*, which is now the common term used by several authors. There is a corresponding *brace operad* with subtle use of trees; [@dolgushev-willwacher:braceoperad] explains its relation to Deligne’s ‘conjecture’ for Å-algebras. (Higher) derived brackets can be defined in terms of braces (special compositions of maps).
BV algebras {#bvalg}
-----------
[**Warning!!**]{} Batalin and Vilkovisky have made two *distinct* contributions to gauge field theory:
- the formalism of the Lagrangian dg construction with antifields and ghosts etc. (see \[bfv\]).
- graded algebras with a special differential operator of order 2.
Unfortunately both are sometimes refered to as the BV *formalism*, which preferably should be used for their dg construction.
Recall a *Gerstenhaber algebra* $(A,\cdot,[\ ,\ ])$ is the ‘odd’ analog of a graded Poisson algebra, $[\ , \ ]$ being called a Gerstenhaber bracket. The first example occured in his paper on Hochschild cohomology of an associative algebra [@gerst:coh].
A *Batalin-Vilkovisky algebra*, or *BV algebra*, is a Gerstenhaber algebra where the bracket $[ \; , \; ]$ is obtained from an odd, square zero, second order differential operator $\D$: \[a,b\] =(-1)\^[|a|]{}(ab)-(-1)\^[|a|]{}(a)b-a(b)
Batalin and Vilkovisky introduced the algebras now named in their honor in their study of conformal field theroies and quantization [@bv:quant]. There is a notion of *generalized BV algebra* due to Akman [@akman:BV] in which the bracket $[ \; , \; ]$ is obtained from an odd, square zero, second order differential operator $\D$ as above, though the algebra doesn’t have to be associative nor commutative, but the bracket still measures the deviation of $\D$ from being a derivation. She also considers *differential BV algebras*. For Akman, a major motivation is the work of Lian-Zuckerman [@lz:new] related to VOAs (vertex operator algebras). The restriction to ‘second order’ is just the classical case.
Of course, there is a ‘higher’ notion of $BV_\infty$ algebra. The article by Gálvez-Carrillo, Tonks and Vallette [@vallette-etal:homotopyBV] should be the canonical reference. They point out that several authors call a “homotopy BV-algebra” what is a *commutative* homotopy BV-algebra, that is one for which the product is commutative (compare remarks at the end of Section \[iterated\]). In addition to giving an operadic description (see also [@drummond-cole-vallette]), they give four equivalent definitions of a homotopy BV-algebra and provide applications in four different categories. In particular, they develop the deformation theory and homotopy theory of BV-algebras and of $BV_\infty$ algebras.
Deligne’s question and cyclicity {#quest}
================================
In his seminal work [@gerst:coh], Gerstenhaber showed the Hochschild cohomology $HH^\bullet(A)$ of an associative algebra carries what is now known as a Gerstenhaber algebra structure. Moreover, he constructed homotopies on the Hochschild cochains $CH^\bullet(A)$ to yield the relevant identities on $HH^\bullet(A)$. Thirty years later in 1993 in a letter to several of us, Deligne *asked*: Does the Hochschild cochain complex of an associative ring have a natural action by chains of the small squares operad? This lead to *many* ‘higher structure’ papers resolving this ‘conjecture’ and generalizations, in many of which the little disks operad was invoked instead. In some, braces play a key role. The chains involved were various singular chains on the topological operad or PROP or cellular chains on a related CW complex. The generalization to the Hochschild cochains when $A$ is an Å-algebra followed soon after. To follow this development further means entering a labyrinth and trying to choose a path (see [@mcclure-smith; @kaufmann:spineless] among others). That of [@MandY:defoveroperad] emphasizes relations to deformation theory (see Section \[defs\]). Let me instead pay attention to the cyclic Deligne ‘conjecture’ which, on the one hand, refers to the Hochschild cochain complex for an algebra $A$ with an invariant inner product and, on the other hand, to the *framed little disks operad* [@getz:bv2d]. Solutions involve *cacti* [@sasha:cacti; @kaufmann:cacti], *spineless cacti* [@kaufmann:spineless] and Sullivan chord diagrams [@tz:aso; @tz:cyclic] and of course compactified moduli spaces. The solution is developed further in an Å context and treated particularly well by Ben Ward [@ward:cyclic]. Although the latter clearly involves associativity and cyclicity, it is intriguing to see the approach of Kaufmann and Schwell [@kaufmann-schwell] involves both the associahedron and the cyclohedron, which was introduces by Bott and Taubes [@BT] though only later named in [@jds:config]. Progressing one step further, Tradler considers an $\infty$-inner product [@tradler:inftyinner].
in deformation theory {#defs}
=======================
Deformations of complex structure go back to Riemann, but just for one complex dimension. In higher dimension, even a proper definition of infinitesimal deformation was a problem, resolved by Nijenhuis and Fr" olicher [@fn] by identifying it as an class in the cohomology $H^1$ of the sheaf of germs of holomorphic tangent vectors. Kodaira and Spencer took over from there . The primary obstruction to extending the infinitesimal $\delta$ was a class in $H^2$ given by the product $\delta\smile\delta$. Higher obstructions drew the attention of Douady [@douady], who related them to Massey ‘products’ . Unnoticed for a while, this approach was carried further by Retakh [@retakh:massey]).
What is deformation theory?
---------------------------
Based on the history for complex structures, Gerstenhaber offered the first general description:
> a deformation theory seems to have at least the following aspects:
>
> - A definition of the class of objects within which deformation takes place, and identification of the infinitesimal deformations of a given object with the elements of a suitable cohomology group.
>
> - A theory of the obstructions to the integration of an infinitesimal deformation.
>
> - A parameterization of the set of objects obtainable by deformation from a fixed one, and the construction of a fiber space over this space, the fibers of which are the objects.
>
> - A determination of the natural automorphisms of the parameter space (the modular group of the theory) and determination of the rigid objects. In some cases almost all points of a parameter space will represent the same rigid object, degenerating in various ways to objects admitting proper deformations.
>
By analogy with Riemann’s original work, the associated parameter spaces are referred to as *moduli spaces*. In contrast to classical results, the moduli space need not be in the class of objects being studied. From 1963 to 1968, Gerstenhaber studied deformations of associative algebras in terms of Hochschild cochains and cohomology [@gerst:defs-ra]. This had major impacts: for deformation theory, for physics (see Section \[quant\]) and for higher homotopy theory (see Section \[quest\]). His initial work was soon followed by work of Nijenhuis and Richardson ,[@nij-rich:algdef; @nij-rich:Liedef]. They were the first to articulate something close to the current “metatheorem”, adding dg Lie algebra as a crucial ingredient: *The deformation theory of any mathematical object, e.g., an associative algebra, a complex manifold, etc., can be described starting from a certain differential graded (dg) Lie algebra associated to the mathematical object in question.*\[meta\] This provides a natural setting in which to pursue the obstruction method for trying to integrate “infinitesimal deformations”. The deformation equation is known as the Master Equation in the physics and physics inspired literature and now most commonly as the Maurer-Cartan equation. It is also the equation for a *twisting cochain* used in describing twisted tensor products [@gugjds; @jds:twisted]. As in the above examples, some authors refer to the corresponding *cohomology* as controlling the deformations. In 1986, inspired instead by Goldman-Millson [@goldman-millson], Deligne stated this as a philosophy in a letter to Millson [@deligne:gm]. Meanwhile in 1967, Lichtenbsum and Schlessinger [@schlessinger-lichtenbaum] expressed deformation theory in terms of a *cotangent complex*; think differential forms on the moduli space with the *tangent complex* corresponding to a Lie algebra of derivations. This was carried further in Mike’s thesis [@schlessinger], written from the viewpoint of representability of certain functors. The extension of deformation theory to a variety of mathematical objects is well surveyed by Gerstenhaber and Schack [@gerstenhaber-schack:survey].
The (not quite) metatheorem {#quite}
---------------------------
In his MR review of [@MandY:defoveroperad]. A. Voronov writes with respect to the ‘metatheorem’:
> The uncomfortable generality of the statement might have prevented anybody from making it a theorem and proving it. A route chosen by many was usually the following one: given a mathematical object $A$, construct a dg Lie algebra $L$ and answer as many questions about the deformation theory of $A$ in terms of $L$ as you can.
There have been several theorems of great but not complete generality. Markl [@markl:cotan] develops a deformation theory phrased as a controlling cohomology theory for $k$-algebras over $k$-linear equationally given category, e.g. over operads and PROPs - for bialgebras.. He invites comparison to traditional cotangent cohomology of a commutative algebra (in characteristic zero) based on the free differential graded algebra resolution of the algebra under consideration [@Sjds; @schlessinger]. In [@markl:JHRS10], he provides explicit constructions in terms of ${L_\infty}$-deformation theory, enhanced further in [@markletal:algdiags]; see an alternative [@merkulov-valletteI] for representations of prop(erad)s. Others are expressed in ‘languages the muse did not sing at my cradle’. Kontsevich and Soibleman [@MandY:defoveroperad], develop the existence of deformation theory for an algebra over any (colored) operad, but in a more geometric language of formal dg manifolds. This refers to cofree cocommutative coalgebras with differential in disguise. Thus again -algebras are doing the controlling. They give further insight in the comparison of the algebriac and geometric version of Å-structures in [@kont-soi:notes].The one closest to my native language e.g. of -algebras, is that of Pridham [@pridham:unifying; @pridham:corrig], which is the most general, going beyond characteristic 0. Closer to (derived) algebraic geometry is that of Lurie [@lurie].
Deformation quantization {#quant}
------------------------
Especially important was the eponymous *Gerstenhaber bracket* [@gerst:coh] which was later identified by interpreting the Hochschild complex in terms of coderivations [@jds:intrinsic; @akman:chicken]. His work led to an algebraic description of *deformation quantization* [@bfflsI; @bfflsII], a term derived from physics. Given a Poisson algebra $(A, \{\ ,\ \})$, a deformation quantization is an associative unital $\star$ product on the algebra of formal power series $A[[\hbar]]$ subject to the following two axioms: $$\begin{aligned}
f\star g &=fg+\mathcal{O}(\hbar)\\
f\star g-g\star f & =\hbar\{f,g\}+\mathcal{O}(\hbar^2).\end{aligned}$$
In the physics context, the Poisson bracket is given in terms of differential operators as are the sought after terms of higher order (in which the devil resides!).
-algebras in rational homotopy theory {#qht}
-------------------------------------
-algebras arose by 1977 in my work with Mike Schlesinger [@Sjds; @SS; @jds-ms] on deformation theory of rational homotopy types. Mike and I extended the yoga of control by a dg Lie algebra (See\[meta\]) to similar control by an -algebra, for example, on the homology of a huge strict dg Lie algebra.
Let ${\mathcal H}$ be a simply connected graded commutative algebra of finite type and $(\Lambda Z,d) \to {\mathcal H}$ a filtered model. Differential graded Lie algebras provide a natural setting in which to pursue the obstruction method for trying to integrate “infinitesimal deformations”, elements of $H^1({Der} \Lambda Z)$, to full perturbations. In that regard, $H^*({Der} \Lambda Z)$ appears not only as a graded Lie algebra (in the obvious way) but also as an -algebra. Our main result compares the *moduli space* set of augmented homotopy types of dgca’s $(A,i : {\mathcal H}\approx H(A))$ with the path components of $C(L)$ where $L$ is a sub Lie algebra of $ {Der}\,\Lambda Z$ consisting of the weight decreasing derivations. We were inspired by the work of Kadeishvili on the Å-structure on the homology of a dg associative algebra. This again used the ‘higher structure’ machinery of HPT (Homological Perturbation Theory).
Independently, in 1988, correspondence between Drinfel’d and Schechtman [@Drinfeld:letter; @Toen:drinfeld] develops -algebra under the name *Sugawara - Lie algebra* for the needs of deformations theory. As with ‘Jacobi up to homotopy’, the need for -algebras was ‘in the air’ (See Section \[Loo\]). “A letter from Kharkov to Moscow” [@Drinfeld:letter] deserves further attention. The disparity in the dates recalls the lack of communication between the fSU and the West. The situation was much better two years later when Gerstenhaber and I were able to participate in person at the Euler International Mathematical Institute’s Workshop on [Q]{}uantum [G]{}roups, Deformation Theory, and Representation Theory [@leningrad1990]. Interactions there were to prove very fruitful (see Section \[fields\]).
Representations up to homotopy/RUTHs {#ruths}
====================================
A good bit of group theory has been carried over to Å-algebras, but far from all. Most recently, the analog of Sylow theorems appeared [@prasma:sylow]. It was late in the game before higher homotopy representations received much attention. Although the language is slightly different, an ordinary representation of a group or algebra is equivalent to a morphism to the endomorphisms of another object. This appeared early on in homotopy theory (I learned it as a grad student from Hilton’s *Introduction to Homotopy Theory* [@hilton:intro] - the earliest textbook on the topic) in terms of the action of the based loop space $\Omega B$ on the fiber $F$ of a fibration $F\to E\to B$: $$\Omega B \times F\to F \ \ or \ \ \Omega B \to Haut(F)$$ where $Haut(F)$ denotes the monoid of self homotopy equivalences $F\to F$. The existence of $\Omega B \times F\to F$ follows from the covering homotopy property, but with uniqueness only ‘up to homotopy’. Initially, this was referred to as a *homotopy action*, meaning only that $(f \lambda)\mu$ was homotopic to $f(\lambda)f(\mu),$ with no higher structure. The map $\Omega B \to Haut(F)$ is only an H-map; the full equivalence between such fibrations and Å-maps $\Omega B \to Haut(F)$ had to wait until such maps were available. The corresponding terminology is that of (strong or $\infty$) homotopy action, which has further variants under a variety of names. The definition is simplified if we use the Moore space of loops $\MB$ [@moore:loops] (see Section \[mooreloops\]).
\[Thetas\] A *representation up to homotopy* of $\MB$ on a fibration $E\to B$ is an $A_\infty$-morphism (or shm-morphism [@sugawara:hc]) from $\MB$ to $End_B(E)$ (see Definition \[shmap\]).
In 1971, Nowlan [@nowlan:action] considered fibrations $F\to E\to B$ with associative H-space F as fibre, acting fibrewise on E, but the action being associative only up to higher homotopies. His main result shows that a fibration $\Omega Y \to E \to B$ is fibre homotopy equivalent to one induced by a map of $B \to Y$ if and only if $E$ admits an Å-action of $\Omega Y$. Å-actions occur also for relative loop spaces [@hoefel:Aooaction; @vieira:rel]. In the case of a smooth vector bundle $E \to B$, the corresponding notion is parallel transport, usually determined by a *choice* of connection, hence unique. In physics, various action functionals for quantum field theories correspond to *higher parallel transport* in bundles with graded vector space fibers, corresponding to the higher homotopies of the strong homotopy action. Switching to algebra, there is the corresponding notion of *representation up to homotopy* of an associative dg algebra on a dg vector space $V$. This seems to have first occurred in [@jds:hrcpa] in the context of a Poisson algebra $(P, \{ \ ,\ \})$ with a commutative ideal $I$ closed under $ \{ \ ,\ \}$. Such a structure arises in physics with $A = C^\infty(W)$ for some symplectic manifold, $W$ (see Section \[bfv\] [@BF; @FF; @FV]). There are analogous notions of RUTHs of Lie structures. In particular, in open-closed string field theory (see Section \[ocha\]), the CSFT acts up to homotopy in the strong sense on the OSFT and similalry for OCHAs with the -algebra acting on the Å-algebra by $\infty$-homotopy derivations.. In 2011, RUTHs of Lie algebroids was developed by Abad and Crainic [@abad-crainic:shreps][^3]. In particular, they use representations up to homotopy to define the adjoint representation of a Lie algebroid to control deformations of structure (see Section \[defs\]).
Å-functors, Å-categories and $\infty$-geometry {#wirth}
==============================================
Since associativity is a key property of categories, it is not surprising that Å-categories were eventually defined. In 1993, Fukaya [@fukaya] defined them to handle Morse theoretic homology. Just as one considers Å-morphisms of Å-algebras, one can consider Å-functors (also known as *homotopy coherent functors*) between Å-categories. Such functors were first considered for ordinary strict but topological categories in the context of classification of fibre spaces. For fibrations which are locally homotopy trivial with respect to a good open cover $\{ U{_{\alpha}}\}$ of the base, one can define transition functions $$g{_{\alpha\beta}}: U{_{\alpha}}\cap U{_{\beta}}\to Haut(F),$$ but instead of the cocycle condition for fibre bundles, one obtains only that $g{_{\alpha\beta}}g{_{\beta\gamma}}$ is homotopic to $g{_{\alpha\gamma}}$ as a map of $U{_{\alpha}}\cap
U{_{\beta}}\cap U{_{\gamma}}$ into $Haut(F)$. In 1965, Wirth [@wirth:diss; @jw-jds] showed how a set of coherent higher homotopies arise on multiple intersections. He calls that set a *homotopy transition cocycle.* The disjoint union $\coprod U{_{\alpha}}$ can be given a rather innocuous structure of a topological category $U$, i.e., ${\operatorname{Ob}\,}U = \coprod U{_{\alpha}}$ and ${\operatorname{Mor}\,}U=\coprod U{_{\alpha}}\cap U{_{\beta}}$. Regarding $Haut(F)$ as a category with one object in the standard way, Wirth shows the transition cocycle web of higher homotopies is precisely equivalent to a homotopy coherent functor. For ‘good’ spaces, the usual classification of such fibrations is effected by the realization of this functor via a map $BU\to BHaut(F).$ Having come this far in the Åworld, what about *‘$\infty$-geometry’*? Recall Roytenberg and others define a *dg manifold* as a locally ringed space $ (M, \mathcal{O}_M)$ (in dg commutative algebras over $\mathbb R$), which is locally isomorphic to $(U, \mathcal{O}_U )$, where $ \mathcal{O}_U = C^\infty(U) \otimes S(V^\bullet)$ where $\{U\}$ is an open cover of $M$, $V^\bullet$ is a dg vector space and $S(V^\bullet)$ is the free graded commutative algebra. Again the transition functions $g{_{\alpha\beta}}: U{_{\alpha}}\cap U{_{\beta}}\to Aut S(V^\bullet)$ with respect to an open cover $\{ U{_{\alpha}}\}$ of $M$ satisfy the cocycle condition. Notice that ‘manifold’ is irrelevant, a hold-over from the early days, but useful intuition for those trained in differential geometry; at most a topological space with a good open cover is needed. The coherent homotopy generalization of the definition of a dg-manifold is straightforward, but requires a coherent homotopy cocycle condition, as proposed here: 2ex
A *dg $\infty$-manifold* or *sh-manifold* is a locally ringed space $ (M, \mathcal{O}_M)$ (in dg commutative algebras over $\mathbb R$), which is *locally* homotopy equivalent (as dcga’s) to ($U, \mathcal{O}_U )$, where $ \mathcal{O}_U = C^\infty(U) \otimes S(V^\bullet)$ with $\{U\}$ an open cover of $M$ and $(S(V^\bullet),d)$ is free as graded commutative algebra.
The higher analogs of classical transition functions with a cocycle condition are exactly Wirth’s homotopy transition cocycles.
Å-structures and -structures in physics {#aall}
=======================================
There has not been as much presence of Å-structures in physics as of as of -structure, though Zwiebach followed with Å-algebras for open string field theory in 1997 , a few years after his -algebra for closed string field theory. Then, for open-closed string field theory, Åand were combined. On the other hand, discovery by physicists of ‘mirror symmetry’ among Hodge numbers of dual Calabi-Yau manifolds led Kontsevich to propose *homological mirror symmetry* [@kont:hms] in terms of Å-categories such as were presented by Fukaya [@fukaya] (see Section \[wirth\]). His suggestions included extended moduli spaces and deformations of Å-categories.
String algebra
--------------
In OSFT, strings are considered as paths in a manifold with interactions handled by “joining” two strings to form a third. This is often pictured in one of three ways (see [@jds:almost] for graphics):
- E: endpoint interaction: (most familiar in mathematics as far back as the study of the fundamental group) occurs only when the end of one string agrees with the beginning of the other and the parameterization is adjusted appropriately,
- M: midpoint or half overlap interaction: occurs only when the last half of one agrees with the first half of the other as parameterized,
- V: variable overlap interaction: occurs only when, as parametrized, a last portion of one agrees with the corresponding first portion of the other.
The endpoint case E comes in two flavors:
- with paths parameterized by a fixed interval $[0,1]$ in math or $[0,\pi]$ or $[0,2\pi]$ in physics,
- with paths parameterized by intervals $[0,r]$ for $r\geq 0$ (see Section \[mooreloops\]).
The midpoint case M was considered by Lashof [@lashof:loops] and later by Witten [@witten:noncommsft]; it has the advantage of being associative when defined. Note for the iterated composite of three strings the middle one disappears in the composite! For Lashof and topologists generally, the interaction was regarded as $a+b=c$ whereas for Witten and physicists generally treat the interaction of three strings more symmetrically, cf. by reversing the orientation of the "composite”: $a+b= -c$ or $a+b+c=0$. One could call this a ‘cyclic associative algebra’. According to Kontsevich, his reading of [@jds:almost] led to defining *cyclic Å-algebras*. The variable case V seems to have occurred first in physics in the work of Kaku [@kaku]. Surprisingly, it is associative only up to homotopy but the pentagon relation holds on the nose! In physspeak, that is said as 3-string and 4-string vertices suffice [@hikko]. The images of closed string interaction and open- closed string interaction are much more subtle.
String field theories {#fields}
---------------------
But those are for strings; string fields are functions or forms on the space of strings and they form an algebra under the convolution product where the comultiplication on a string is the set of *decompositions* at arbitrary points in the parameterizing interval. (An excellent and extensive ‘bilingual’ (math and physics) survey is given by Kajiura [@kajiura:survey2010].) One of the striking aspects of Å- and -algebras in physics is the use of an inner product $<\ ,\ >$ for the *action functional*, an integral of a real or complex valued function of the fields. I first noticed this in Zwiebach’s CSFT [@z:csft] for the classical (genus 0) action (\_0, \_1,,\_n) where the $\phi$’s are string fields and the integrand is cyclically symmetric (up to sign). The corresponding -structure is determined by = (v\_0, v\_1,,v\_n). Cyclic -algebras were formalized by Penkava [@penkava:cyclic].
As best I can determine, it was Kontsevich who first considered Å-algebras with an invariant inner product, know as *cyclic Å-algebras*. Retakh recalls he and Feigin discussed one of the talks at the Euler Workshop on [Q]{}uantum [G]{}roups, Deformation Theory, and Representation Theory and a text associated with the talk and showed it to Kontsevich or as Kontsevich says:
> B. Feigin, V. Retakh and I had tried to understand a remark of J. Stasheff on open string theory and higher associative algebras.
This led to his [@kont:trium]. “There’s an operad for that”; see [@ward:cyclic].
Open-Closed Homotopy Algebra and string field theory {#ocha}
----------------------------------------------------
Having considered both Å- and -algebras, plain and fancy, we come to the combination known as *OCHA* for Open-Closed Homotopy Algebra [@hiro-jds:inspired; @hiro-jds:mp]. Inspired by open-closed string field theories [@zwiebach:mixed], OCHAs involve an -algebra acting by derivations (up to strong homotopy) on an Å-algebra but having an additional piece of structure corresponding to a closed string opening to an open string. The relevant operad (rather a colored operad with two colors-one for open and one for closed) is known as the “Swiss cheese operad” (see graphics in [@sasha:cheese] for explanation of the name). The details are quite complicated in the original papers, but, just as other “$\infty$” algebras can be characterized by a single coderivation on an appropriate dgc coalgebra, the same has been achieved for OCHAs by Hoefel [@hoefel:coalg]. A small example appears in [@lada-kade:ocha]. Carqueville has called to my attention the rich class of examples provided by Landau-Ginzburg models , In addition, string theory and string field theory have inspired both string topology, initiated by Chas and Sullivan [@chas-sullivan] and a further variety of $\infty$-algebras.
Scattering amplitudes {#scat}
---------------------
Scattering amplitudes in gauge theories are important ‘observables’. Arkani-Hamed and his colleagues [@arkani:amplituhedron; @arkani:positiveamplituhedron; @arkani:scattering] were led to yet another polyhdedron, the *amplituhedron*, which is a generalization of the ‘positive Grassmannian’. As we have seen, trees are often a starting point leading to more general structures and tree scattering amplitudes are a good place to start. In [@arkani:scattering], they present a “novel construction of the associahedron in kinematic space”.
Now and future
==============
That brings us only somewhat up to date; there is much work in progress “as we come on the air” even in this small part of the space of higher structures. I find in Manin’s *Mathematics, Art, Civilization* [@manin:mac]:
> With the advent of polycategories, enriched categories, Å-categories, and similar structures, we are beginning to speak a language….
Now I find this delightfully ironic since, when I first submitted my theses for publication in AJM, they were deemed too narrow and essentially of no relation to other parts of math! Perhaps Heraclitus was right: All is flux, nothing stays still. : -)
[^1]: Born Yuri Germanovich Rabinovich $https://en.wikipedia.org/wiki/George_Yuri_Rainich$
[^2]: See https://ncatlab.org/nlab/print/L-infinity+algebras+in+physics for an extensive, annotated chronological list - thanks to Urs Schrieber- but beware the linguistic problems.
[^3]: As of this writing, the wiki is not up to date.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The lace expansion has been a powerful tool for investigating mean-field behavior for various stochastic-geometrical models, such as self-avoiding walk and percolation, above their respective upper-critical dimension. In this paper, we prove the lace expansion for the Ising model that is valid for any spin-spin coupling. For the ferromagnetic case, we also prove that the expansion coefficients obey certain diagrammatic bounds that are similar to the diagrammatic bounds on the lace-expansion coefficients for self-avoiding walk. As a result, we obtain Gaussian asymptotics of the critical two-point function for the nearest-neighbor model with $d\gg4$ and for the spread-out model with $d>4$ and $L\gg1$, without assuming reflection positivity.'
author:
- 'Akira Sakai[^1]'
date: 'October 26, 2005[^2]'
title: Lace expansion for the Ising model
---
Introduction and results
========================
Model and the motivation {#ss:model}
------------------------
The Ising model is a statistical-mechanical model that was first introduced in [@i25] as a model for magnets. Consider the $d$-dimensional integer lattice ${{\mathbb Z}^d}$, and let $\Lambda$ be a finite subset of ${{\mathbb Z}^d}$ containing the origin $o\in{{\mathbb Z}^d}$. For example, $\Lambda$ is a $d$-dimensional hypercube centered at the origin. At each site $x\in\Lambda$, there is a spin variable $\varphi_x$ that takes values either $+1$ or $-1$. The Hamiltonian represents the energy of the system, and is defined by $$\begin{aligned}
{\label{eq:hamilton}}
H^h_\Lambda(\varphi)=-\sum_{\{x,y\}\subset\Lambda}J_{x,y}\varphi_x\varphi_y
-h\sum_{x\in\Lambda}\varphi_x,\end{aligned}$$ where $\varphi\equiv\{\varphi_x\}_{x\in\Lambda}$ is a spin configuration, $\{J_{x,y}\}_{x,y\in{{\mathbb Z}^d}}$ is a collection of spin-spin couplings, and $h\in{{\mathbb R}}$ represents the strength of an external magnetic field uniformly imposed on $\Lambda$. We say that the model is ferromagnetic if $J_{x,y}\ge0$ for all pairs $\{x,y\}$; in this case, the Hamiltonian becomes lower as more spins align. The partition function $Z_{p,h;\Lambda}$ at the inverse temperature $p\ge0$ is the expectation of the Boltzmann factor $e^{-pH^h_\Lambda(\varphi)}$ with respect to the product measure $\prod_{x\in\Lambda}(\frac12{\mathbbm{1}{\scriptstyle\{\varphi_x=+1\}}}+\frac12{\mathbbm{1}{\scriptstyle\{\varphi_x=-1\}}})$: $$\begin{aligned}
{\label{eq:ZL-def}}
Z_{p,h;\Lambda}=2^{-|\Lambda|}\sum_{\varphi\in\{\pm1\}^\Lambda}e^{-pH^h_\Lambda
(\varphi)}.\end{aligned}$$ Then, we denote the thermal average of a function $f=f(\varphi)$ by $$\begin{aligned}
{\label{eq:faver}}
{{\langle f \rangle}}_{p,h;\Lambda}=\frac{2^{-|\Lambda|}}{Z_{p,h;\Lambda}}\sum_{\varphi\in
\{\pm1\}^\Lambda}f(\varphi)\,e^{-pH^h_\Lambda(\varphi)}.\end{aligned}$$
Suppose that the spin-spin coupling is translation-invariant, ${{\mathbb Z}^d}$-symmetric and finite-range (i.e., there exists an $L<\infty$ such that $J_{o,x}=0$ if $\|x\|_\infty>L$) and that $J_{o,x}\ge0$ for any $x\in{{\mathbb Z}^d}$ and $h\ge0$. Then, there exist monotone infinite-volume limits of ${{\langle \varphi_x \rangle}}_{p,h;\Lambda}$ and ${{\langle \varphi_x\varphi_y \rangle}}_{p,h;\Lambda}$. Let $$\begin{aligned}
M_{p,h}=\lim_{\Lambda\uparrow{{\mathbb Z}^d}}{{\langle \varphi_o \rangle}}_{p,h;\Lambda},&&
G_p(x)=\lim_{\Lambda\uparrow{{\mathbb Z}^d}}{{\langle \varphi_o\varphi_x \rangle}}_{p,h=0;
\Lambda},&&
\chi_p=\sum_{x\in{{\mathbb Z}^d}}G_p(x).\end{aligned}$$ When $d\ge2$, there exists a unique critical inverse temperature ${p_\text{c}}\in(0,\infty)$ such that the spontaneous magnetization $M^+_p\equiv\lim_{h\downarrow0}M_{p,h}$ equals zero, $G_p(x)$ decays exponentially as $|x|\uparrow\infty$ (we refer, e.g., to [@civ03] for a sharper Ornstein-Zernike result) and thus the magnetic susceptibility $\chi_p$ is finite if $p<{p_\text{c}}$, while $M^+_p>0$ and $\chi_p=\infty$ if $p>{p_\text{c}}$ (see [@abf87] and references therein). We should also refer to [@b05b] for recent results on the phase transition for the Ising model.
We are interested in the behavior of these observables around $p={p_\text{c}}$. The susceptibility $\chi_p$ is known to diverge as $p\uparrow{p_\text{c}}$ [@a82; @ag83]. It is generally expected that $\lim_{p\downarrow{p_\text{c}}}M^+_p=\lim_{h\downarrow0}M_{{p_\text{c}},h}=0$. We believe that there are so-called critical exponents $\gamma=\gamma(d)$, $\beta=\beta(d)$ and $\delta=\delta(d)$, which are insensitive to the precise definition of $J_{o,x}\ge0$ (universality), such that (we use below the limit notation “$\approx$” in some appropriate sense) $$\begin{aligned}
M^+_p\stackrel{p\downarrow{p_\text{c}}}{\approx}(p-{p_\text{c}})^\beta,&&
\chi_p\stackrel{p\uparrow{p_\text{c}}}{\approx}({p_\text{c}}-p)^{-\gamma},&&
M_{{p_\text{c}},h}\stackrel{h\downarrow0}{\approx}h^{1/\delta}.\end{aligned}$$ These exponents (if they exist) are known to obey the mean-field bounds: $\beta\leq1/2$, $\gamma\ge1$ and $\delta\ge3$. For example, $\beta=1/8$, $\gamma=7/4$ and $\delta=15$ for the nearest-neighbor model on ${{\mathbb Z}}^2$ [@o44]. Our ultimate goal is to identify the values of the critical exponents in other dimensions and to understand the universality for the Ising model.
There is a sufficient condition, the so-called bubble condition, for the above critical exponents to take on their respective mean-field values. Namely, the finiteness of $\sum_{x\in{{\mathbb Z}^d}}G_{{p_\text{c}}}(x)^2$ (or the finiteness of $\sum_{x\in{{\mathbb Z}^d}}G_p(x)^2$ uniformly in $p<{p_\text{c}}$) implies that $\beta=1/2$, $\gamma=1$ and $\delta=3$ [@a82; @abf87; @af86; @ag83]. It is therefore crucial to know how fast $G_{{p_\text{c}}}(x)$ (or $G_p(x)$ near $p={p_\text{c}}$) decays as $|x|\uparrow\infty$. We note that the bubble condition holds for $d>4$ if the anomalous dimension $\eta$ takes on its mean-field value $\eta=0$, where the anomalous dimension is another critical exponent formally defined as $$\begin{aligned}
{\label{eq:eta-formal}}
G_{{p_\text{c}}}(x)\stackrel{|x|\uparrow\infty}{\approx}|x|^{-(d-2+\eta)}.\end{aligned}$$
Let $\hat J_k=\sum_{x\in{{\mathbb Z}^d}}J_{o,x}\,e^{ik\cdot x}$ and $\hat G_p(k)=\sum_{x\in{{\mathbb Z}^d}}G_p(x)\,e^{ik\cdot x}$ for $p<{p_\text{c}}$. For a class of models that satisfy the so-called reflection positivity [@fss76], the following infrared bound[^3] holds: $$\begin{aligned}
{\label{eq:IRbd-so}}
0\leq\hat G_p(k)\leq\frac{\text{const.}}{\hat J_0-\hat J_k}\qquad
\text{uniformly in }p<{p_\text{c}},\end{aligned}$$ where $d$ is supposed to be large enough to ensure integrability of the upper bound. For finite-range models, $d$ has to be bigger than 2, since $\hat J_0-\hat J_k\asymp|k|^2$, where “$f\asymp g$” means that $f/g$ is bounded away from zero and infinity. By Parseval’s identity, the infrared bound [(\[eq:IRbd-so\])]{} implies the bubble condition for finite-range reflection-positive models above four dimensions, and therefore $$\begin{aligned}
{\label{eq:MFbehavior}}
M^+_p\stackrel{p\downarrow{p_\text{c}}}{\asymp}(p-{p_\text{c}})^{1/2},&&
\chi_p\stackrel{p\uparrow{p_\text{c}}}{\asymp}({p_\text{c}}-p)^{-1},&&
M_{{p_\text{c}},h}\stackrel{h\downarrow0}{\asymp}h^{1/3}.\end{aligned}$$ The class of reflection-positive models includes the nearest-neighbor model, a variant of the next-nearest-neighbor model, Yukawa potentials, power-law decaying interactions, and their combinations [@bcc05]. For the nearest-neighbor model, we further obtain the following $x$-space Gaussian bound [@s82]: for $x\ne o$, $$\begin{aligned}
{\label{eq:IRbd-sokal}}
G_p(x)\leq\frac{\text{const.}}{|x|^{d-2}}\qquad\text{uniformly in }p<{p_\text{c}}.\end{aligned}$$
The problem in this approach to investigate critical behavior is that, since general finite-range models do not always satisfy reflection positivity, their mean-field behavior cannot necessarily be established, even in high dimensions. If we believe in universality, we expect that finite-range models exhibit the same mean-field behavior as soon as $d>4$. Therefore, it has been desirable to have approaches that do not assume reflection positivity.
The lace expansion has been used successfully to investigate mean-field behavior for self-avoiding walk, percolation, lattice trees/animals and the contact process, above the upper-critical dimension: 4, 6 (4 for oriented percolation), 8 and 4, respectively (see, e.g., [@s04]). One of the advantages in the application of the lace expansion is that we do not have to require reflection positivity to prove a Gaussian infrared bound and mean-field behavior. Another advantage is the possibility to show an asymptotic result for the decay of correlation. Our goal in this paper is to prove the lace-expansion results for the Ising model.
Main results
------------
From now on, we fix $h=0$ and abbreviate, e.g., ${{\langle \varphi_o\varphi_x \rangle}}_{p,h=0;\Lambda}$ to ${{\langle \varphi_o\varphi_x \rangle}}_{p;\Lambda}$. In this paper, we prove the following lace expansion for the two-point function, in which we use the notation $$\begin{aligned}
\tau_{x,y}=\tanh(pJ_{x,y}).\end{aligned}$$
\[prp:Ising-lace\] For any $p\ge0$ and any $\Lambda\subset{{\mathbb Z}^d}$, there exist $\pi_{p;\Lambda}^{{\scriptscriptstyle}(j)}(x)$ and $R_{p;\Lambda}^{{\scriptscriptstyle}(j+1)}(x)$ for $x\in\Lambda$ and $j\ge0$ such that $$\begin{aligned}
{\label{eq:Ising-lace}}
{{\langle \varphi_o\varphi_x \rangle}}_{p;\Lambda}=\Pi_{p;\Lambda}^{{\scriptscriptstyle}(j)}(x)
+\sum_{u,v}\Pi_{p;\Lambda}^{{\scriptscriptstyle}(j)}(u)\,\tau_{u,v}{{\langle
\varphi_v\varphi_x \rangle}}_{p;\Lambda}+(-1)^{j+1}R_{p;\Lambda}^{{\scriptscriptstyle}(j
+1)}(x),\end{aligned}$$ where $$\begin{aligned}
{\label{eq:Pij-def}}
\Pi_{p;\Lambda}^{{\scriptscriptstyle}(j)}(x)&=\sum_{i=0}^j(-1)^i\,\pi_{p;
\Lambda}^{{\scriptscriptstyle}(i)}(x).\end{aligned}$$ For the ferromagnetic case, we have the bounds $$\begin{aligned}
{\label{eq:pij-Rj-naivebd}}
\pi_{p;\Lambda}^{{\scriptscriptstyle}(j)}(x)\ge\delta_{j,0}\delta_{o,x},&&
0\leq R_{p;\Lambda}^{{\scriptscriptstyle}(j+1)}(x)\leq\sum_{u,v}\pi_{p;\Lambda}
^{{\scriptscriptstyle}(j)}(u)\,\tau_{u,v}{{\langle \varphi_v\varphi_x \rangle}}_{p;\Lambda}.\end{aligned}$$
We defer the display of precise expressions of $\pi_{p;\Lambda}^{{\scriptscriptstyle}(i)}(x)$ and $R_{p;\Lambda}^{{\scriptscriptstyle}(j+1)}(x)$ to Section \[sss:complexp\], since we need a certain representation to describe these functions. We introduce this representation in Section \[ss:RCrepr\] and complete the proof of Proposition \[prp:Ising-lace\] in Section \[ss:derivation\].
It is worth emphasizing that the above proposition holds independently of the properties of the spin-spin coupling: $J_{u,v}$ does not have to be translation-invariant or ${{\mathbb Z}^d}$-symmetric. In particular, the identity [(\[eq:Ising-lace\])]{} holds independently of the sign of the spin-spin coupling. A spin glass, whose spin-spin coupling is randomly negative, is an extreme example for which [(\[eq:Ising-lace\])]{} holds.
Whether or not the lace expansion [(\[eq:Ising-lace\])]{} is useful depends on the possibility of good control on the expansion coefficients and the remainder. As explained below, it is indeed possible to have optimal bounds on the expansion coefficients for the nearest-neighbor interaction (i.e., $J_{o,x}={\mathbbm{1}{\scriptstyle\{\|x\|_1=1\}}}$) and for the following spread-out interaction: $$\begin{aligned}
{\label{eq:J-def}}
J_{o,x}=L^{-d}\mu(L^{-1}x)\qquad(1\leq L<\infty),\end{aligned}$$ where $\mu:[-1,1]^d\setminus\{o\}\mapsto[0,\infty)$ is a bounded probability distribution, which is symmetric under rotations by $\pi/2$ and reflections in coordinate hyperplanes, and piecewise continuous so that the Riemann sum $L^{-d}\sum_{x\in{{\mathbb Z}^d}}\mu(L^{-1}x)$ approximates $\int_{{{\mathbb R}^d}}d^dx\;\mu(x)\equiv1$. One of the simplest examples would be $$\begin{aligned}
{\label{eq:Juniform-def}}
J_{o,x}=\frac{{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{0<\|x\|_\infty\leq L\}$}}}}{\sum_{z\in{{\mathbb Z}^d}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{0<\|z
\|_\infty\leq L\}$}}}}=O(L^{-d})\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{0<\|L^{-1}x\|_\infty\leq1\}$}}}.\end{aligned}$$
\[prp:Pij-Rj-bd\] Let $\rho=2(d-4)>0$. For the nearest-neighbor model with $d\gg1$ and for the spread-out model with $L\gg1$, there are finite constants $\theta$ and $\lambda$ such that $$\begin{aligned}
{\label{eq:prp-bds}}
|\Pi_{p;\Lambda}^{{\scriptscriptstyle}(j)}(x)-\delta_{o,x}|&\leq\theta\delta_{o,x}
+\frac{\lambda(1-\delta_{o,x})}{|x|^{d+2+\rho}}\quad(j\ge0),&&
|R_{p;\Lambda}^{{\scriptscriptstyle}(j)}(x)|\to0\quad(j\uparrow\infty),\end{aligned}$$ for any $p\leq{p_\text{c}}$, any $\Lambda\subset{{\mathbb Z}^d}$ and any $x\in\Lambda$.
The proof of Proposition \[prp:Pij-Rj-bd\] depends on certain bounds on the expansion coefficients in terms of two-point functions. These diagrammatic bounds arise from counting the number of “disjoint connections”, corresponding to applications of the BK inequality in percolation (e.g., [@bk85]). We prove these bounds in Section \[s:bounds\], and in anticipation of this, in Section \[s:reduction\] we explain how we use their implication to prove Proposition \[prp:Pij-Rj-bd\], with $\theta=O(d^{-1})$ and $\lambda=O(1)$ for the nearest-neighbor model, and $\theta=O(L^{-2+{\epsilon}})$ and $\lambda=O(\theta^2)$ with a small ${\epsilon}>0$ for the spread-out model.
Let $$\begin{aligned}
\tau\equiv\tau(p)=\sum_x\tau_{o,x},&&
D(x)=\frac{\tau_{o,x}}{\tau},&&
\sigma^2=\sum_x|x|^2D(x).\end{aligned}$$ Due to [(\[eq:prp-bds\])]{} uniformly in $\Lambda\subset{{\mathbb Z}^d}$, there is a limit $\Pi_p(x)\equiv\lim_{\Lambda\uparrow{{\mathbb Z}^d}}
\lim_{j\uparrow\infty}\Pi_{p;\Lambda}^{{\scriptscriptstyle}(j)}(x)$ such that $$\begin{aligned}
{\label{eq:Ising-lace-Zdlim}}
G_p(x)=\Pi_p(x)+(\Pi_p*\tau D*G_p)(x),&&
|\Pi_p(x)-\delta_{o,x}|\leq\theta\delta_{o,x}+\frac{\lambda(1
-\delta_{o,x})}{|x|^{d+2+\rho}},\end{aligned}$$ for any $p\leq{p_\text{c}}$ and any $x\in{{\mathbb Z}^d}$, where $(f*g)(x)=\sum_{y\in{{\mathbb Z}^d}}f(y)\,g(x-y)$. We note that the identity in [(\[eq:Ising-lace-Zdlim\])]{} is similar to the recursion equation for the random-walk Green’s function: $$\begin{aligned}
S_r(x)\equiv\sum_{i=0}^\infty r^iD^{*i}(x)=\delta_{o,x}+(rD*S_r)(x)
\qquad(|r|<1),\end{aligned}$$ where $f^{*i}(x)=(f^{*(i-1)}*f)(x)$, with $f^{*0}(x)=\delta_{o,x}$ by convention. The leading asymptotics of $S_1(x)$ for $d>2$ is known as $\frac{a_d}{\sigma^2}|x|^{-(d-2)}$, where $a_d=\frac{d}2\pi^{-d/2}\Gamma(\frac{d}2-1)$ (e.g., [@h05; @hhs03]). Following the model-independent analysis of the lace expansion in [@h05; @hhs03], we obtain the following asymptotics of the critical two-point function:
\[thm:x-asy\] Let $\rho=2(d-4)>0$ and fix any small ${\epsilon}>0$. For the nearest-neighbor model with $d\gg1$ and for the spread-out model with $L\gg1$, we have that, for $x\ne o$, $$\begin{aligned}
{\label{eq:thm-asy}}
G_{{p_\text{c}}}(x)=\frac{A}{\tau({p_\text{c}})}\,\frac{a_d}{\sigma^2|x|^{d-2}}
\times\begin{cases}
\big(1+O(|x|^{-\frac{(\rho-{\epsilon})\wedge2}d})\big)&(\text{NN model}),\\
\big(1+O(|x|^{-\rho\wedge2+{\epsilon}})\big)
&(\text{SO model}),
\end{cases}\end{aligned}$$ where constants in the error terms may vary depending on ${\epsilon}$, and $$\begin{aligned}
{\label{eq:constants}}
\tau({p_\text{c}})=\bigg(\sum_x\Pi_{{p_\text{c}}}(x)\bigg)^{-1},&&
A=\bigg(1+\frac{\tau({p_\text{c}})}{\sigma^2}\sum_x|x|^2\Pi_{{p_\text{c}}}(x)\bigg)^{-1}.\end{aligned}$$ Consequently, [(\[eq:MFbehavior\])]{} holds and $\eta=0$.
In this paper, we restrict ourselves to the nearest-neighbor model for $d\gg4$ and to the spread-out model for $d>4$ with $L\gg1$. However, it is strongly expected that our method can show the same asymptotics of the critical two-point function for *any* translation-invariant, ${{\mathbb Z}^d}$-symmetric finite-range model above four dimensions, by taking the coordination number sufficiently large.
Organization
------------
In the rest of this paper, we focus our attention on the model-dependent ingredients: the lace expansion for the Ising model (Proposition \[prp:Ising-lace\]) and the bounds on (the alternating sum of) the expansion coefficients for the ferromagnetic models (Proposition \[prp:Pij-Rj-bd\]). In Section \[s:laceexp\], we prove Proposition \[prp:Ising-lace\]. In Section \[s:reduction\], we reduce Proposition \[prp:Pij-Rj-bd\] to a few other propositions, which are then results of the aforementioned diagrammatic bounds on the expansion coefficients. We prove these diagrammatic bounds in Section \[s:bounds\]. As soon as the composition of the diagrams in terms of two-point functions is understood, it is not so hard to establish key elements of the above reduced propositions. We will prove these elements in Section \[ss:proof-so\] for the spread-out model and in Section \[ss:proof-nn\] for the nearest-neighbor model.
Lace expansion for the Ising model {#s:laceexp}
==================================
The lace expansion was initiated by Brydges and Spencer [@bs85] to investigate weakly self-avoiding walk for $d>4$. Later, it was developed for various stochastic-geometrical models, such as strictly self-avoiding walk for $d>4$ (e.g., [@hs92]), lattice trees/animals for $d>8$ (e.g., [@hs90]), unoriented percolation for $d>6$ (e.g., [@hs90']), oriented percolation for $d>4$ (e.g., [@ny93]) and the contact process for $d>4$ (e.g., [@s01]). See [@s04] for an extensive list of references. This is the first lace-expansion paper for the Ising model.
In this section, we prove the lace expansion [(\[eq:Ising-lace\])]{} for the Ising model. From now on, we fix $p\ge0$ and abbreviate, e.g., $\pi_{p;\Lambda}^{{\scriptscriptstyle}(i)}(x)$ to $\pi_\Lambda^{{\scriptscriptstyle}(i)}(x)$.
There may be several ways to derive the lace expansion for ${{\langle \varphi_o\varphi_x \rangle}}_\Lambda$, using, e.g., the high-temperature expansion, the random-walk representation (e.g., [@ffs92]) or the FK random-cluster representation (e.g., [@fk72]). In this paper, we use the random-current representation (Section \[ss:RCrepr\]), which applies to models in the Griffiths-Simon class (e.g., [@a82; @ag83]). This representation is similar in philosophy to the high-temperature expansion, but it turned out to be more efficient in investigating the critical phenomena [@a82; @abf87; @af86; @ag83]. The main advantage in this representation is the source-switching lemma (Lemma \[lmm:switching\] below in Section \[sss:2ndexp\]) by which we have an identity for ${{\langle \varphi_o\varphi_x \rangle}}_\Lambda
-{{\langle \varphi_o\varphi_x \rangle}}_{{\cal A}}$ with “${{\cal A}}\subset\Lambda$” (the meaning will be explained in Section \[ss:RCrepr\]). We will repeatedly apply this identity to complete the lace expansion for ${{\langle \varphi_o\varphi_x \rangle}}_\Lambda$ in Section \[sss:complexp\].
Random-current representation {#ss:RCrepr}
-----------------------------
In this subsection, we describe the random-current representation and introduce some notation that will be essential in the derivation of the lace expansion.
First we introduce some notions and notation. We call a pair of sites $b=\{u,v\}$ with $J_b\ne0$ a *bond*. So far we have used the notation $\Lambda\subset{{\mathbb Z}^d}$ for a site set. However, we will often abuse this notation to describe a *graph* that consists of sites of $\Lambda$ and are equipped with a certain bond set, which we denote by ${{\mathbb B}}_\Lambda$. Note that “$\{u,v\}\in{{\mathbb B}}_\Lambda$” always implies “$u,v\in\Lambda$”, but the latter does not necessarily imply the former. If we regard ${{\cal A}}$ and $\Lambda$ as graphs, then “${{\cal A}}\subset\Lambda$” means that ${{\cal A}}$ is a subset of $\Lambda$ as a site set, and that ${{\mathbb B}}_{{\cal A}}\subset{{\mathbb B}}_\Lambda$.
Now we consider the partition function $Z_{{\cal A}}$ on ${{\cal A}}\subset\Lambda$. By expanding the Boltzmann factor in [(\[eq:ZL-def\])]{}, we obtain $$\begin{aligned}
{\label{eq:ZA-rewr}}
Z_{{\cal A}}&=2^{-|{{\cal A}}|}\sum_{\varphi\in\{\pm1\}^{{\cal A}}}\,\prod_{\{u,v\}\in{{\mathbb B}}_{{\cal A}}}\,
\bigg(\sum_{n_{u,v}\in{{\mathbb Z}_+}}\frac{(p J_{u,v})^{n_{u,v}}}{n_{u,v}!}
\,\varphi_u^{n_{u,v}}\varphi_v^{n_{u,v}}\bigg){\nonumber}\\
&=\sum_{{{\bf n}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_{{\cal A}}}}\bigg(\prod_{b\in{{\mathbb B}}_{{\cal A}}}\frac{(p J_b)
^{n_b}}{n_b!}\bigg)\prod_{v\in{{\cal A}}}\bigg(\frac12\sum_{\varphi_v=\pm1}
\varphi_v^{\sum_{b\ni v}n_b}\bigg),\end{aligned}$$ where we call ${{\bf n}}=\{n_b\}_{b\in{{\mathbb B}}_{{\cal A}}}$ a *current configuration*. Note that the single-spin average in the last line equals 1 if $\sum_{b\ni v}n_b$ is an even integer, and 0 otherwise. Denoting by ${\partial}{{\bf n}}$ the set of *sources* $v\in\Lambda$ at which $\sum_{b\ni v}n_b$ is an *odd* integer, and defining $$\begin{aligned}
{\label{eq:weight}}
w_{{\cal A}}({{\bf n}})=\prod_{b\in{{\mathbb B}}_{{\cal A}}}\frac{(p J_b)^{n_b}}{n_b!}\qquad({{\bf n}}\in
{{\mathbb Z}_+}^{{{\mathbb B}}_{{\cal A}}}),\end{aligned}$$ we obtain $$\begin{aligned}
{\label{eq:ZA-RCrepr1}}
Z_{{\cal A}}=\sum_{{{\bf n}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_{{\cal A}}}}w_{{\cal A}}({{\bf n}})\,\prod_{v\in{{\cal A}}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{
\sum_{b\ni v}n_b\text{ even}\}$}}}=\sum_{{\partial}{{\bf n}}={\varnothing}}w_{{\cal A}}({{\bf n}}).\end{aligned}$$
The partition function $Z_{{\cal A}}$ equals the partition function on $\Lambda$ with $J_b=0$ for all $b\in{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{\cal A}}$. We can also think of $Z_{{\cal A}}$ as the sum of $w_\Lambda({{\bf n}})$ over ${{\bf n}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_\Lambda}$ satisfying ${{\bf n}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}}}\equiv0$, where ${{\bf n}}|_{{\mathbb B}}$ is a projection of ${{\bf n}}$ over the bonds in a bond set ${{\mathbb B}}$, i.e., ${{\bf n}}|_{{\mathbb B}}=\{n_b:b\in{{\mathbb B}}\}$. By this observation, we can rewrite [(\[eq:ZA-RCrepr1\])]{} as $$\begin{aligned}
{\label{eq:ZA-RCrepr2}}
Z_{{\cal A}}=\sum_{\substack{{\partial}{{\bf n}}={\varnothing}\\ {{\bf n}}|_{{{\mathbb B}}_\Lambda\setminus
{{\mathbb B}}_{{{\cal A}}}} \equiv0}}w_\Lambda({{\bf n}}).\end{aligned}$$
Following the same calculation, we can rewrite $Z_{{\cal A}}{{\langle \varphi_x\varphi_y \rangle}}_{{\cal A}}$ for $x,y\in{{\cal A}}$ as $$\begin{aligned}
{\label{eq:2pt-rewr}}
Z_{{\cal A}}{{\langle \varphi_x\varphi_y \rangle}}_{{\cal A}}&=\sum_{{{\bf n}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_{{\cal A}}}}\bigg(\prod_{b\in{{\mathbb B}}_{{\cal A}}}\frac{(p J_b)
^{n_b}}{n_b!}\bigg)\prod_{v\in{{\cal A}}}\bigg(\frac12\sum_{\varphi_v=\pm1}
\varphi_v^{{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle {\scriptscriptstyle}\{v\in x{\vartriangle}y\}$}}}+\sum_{b\ni v}n_b}\bigg){\nonumber}\\
&=\sum_{{\partial}{{\bf n}}=x{\vartriangle}y}w_{{\cal A}}({{\bf n}})=\sum_{\substack{{\partial}{{\bf n}}=x{\vartriangle}y\\
{{\bf n}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}}}\equiv0}}w_\Lambda({{\bf n}}),\end{aligned}$$ where $x{\vartriangle}y$ is an abbreviation for the symmetric difference $\{x\}{\,\triangle\,}\{y\}$: $$\begin{aligned}
{\label{eq:symmdiff}}
x{\vartriangle}y\equiv\{x\}{\,\triangle\,}\{y\}=\begin{cases}
{\varnothing}&\text{if }x=y,\\
\{x,y\}&\text{otherwise}.
\end{cases}\end{aligned}$$ If $x$ or $y$ is in ${{\cal A}}{^{\rm c}}\equiv\Lambda\setminus{{\cal A}}$, then we define both sides of [(\[eq:2pt-rewr\])]{} to be zero. This is consistent with the above representation when $x\ne y$, since, for example, if $x\in{{\cal A}}{^{\rm c}}$, then the leftmost expression of [(\[eq:2pt-rewr\])]{} is a multiple of $\frac12\sum_{\varphi_x=\pm1}\varphi_x=0$, while the last expression in [(\[eq:2pt-rewr\])]{} is also zero because there is no way of connecting $x$ and $y$ on a current configuration ${{\bf n}}$ with ${{\bf n}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}}}\equiv0$.
The key observation in the representation [(\[eq:2pt-rewr\])]{} is that the right-hand side is nonzero only when $x$ and $y$ are connected by a chain of bonds with *odd* currents (see Figure \[fig:RCrepr\]).
![\[fig:RCrepr\]A current configuration with sources at $x$ and $y$. The thick-solid segments represent bonds with odd currents, while the thin-solid segments represent bonds with positive even currents, which cannot be seen in the high-temperature expansion.](RCrepr)
We will exploit this peculiar underlying percolation picture to derive the lace expansion for the two-point function.
Derivation of the lace expansion {#ss:derivation}
--------------------------------
In this subsection, we derive the lace expansion for ${{\langle \varphi_o\varphi_x \rangle}}_\Lambda$ using the random-current representation. In Section \[sss:1stexp\], we introduce some definitions and perform the first stage of the expansion, namely [(\[eq:Ising-lace\])]{} for $j=0$, simply using inclusion-exclusion. In Section \[sss:2ndexp\], we perform the second stage of the expansion, where the source-switching lemma (Lemma \[lmm:switching\]) plays a significant role to carry on the expansion indefinitely. Finally, in Section \[sss:complexp\], we complete the proof of Proposition \[prp:Ising-lace\].
### The first stage of the expansion {#sss:1stexp}
As mentioned in Section \[ss:RCrepr\], the underlying picture in the random-current representation is quite similar to percolation. We exploit this similarity to obtain the lace expansion.
First, we introduce some notions and notation.
\[defn:perc\]
(i) Given ${{\bf n}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_\Lambda}$ and ${{\cal A}}\subset\Lambda$, we say that $x$ is ${{\bf n}}$-connected to $y$ in (the graph) ${{\cal A}}$, and simply write $x{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}y$ *in* ${{\cal A}}$, if either $x=y\in{{\cal A}}$ or there is a self-avoiding path (or we simply call it a path) from $x$ to $y$ consisting of bonds $b\in{{\mathbb B}}_{{\cal A}}$ with $n_b>0$. If ${{\bf n}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_{{\cal A}}}$, we omit “in ${{\cal A}}$” and simply write $x{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}y$. We also define $$\begin{aligned}
{\label{eq:incl/excl}}
\{x{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}y\}=\{x{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}y\}\setminus\{x{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}y\text{ in }
{{\cal A}}{^{\rm c}}\},\end{aligned}$$ and say that $x$ is ${{\bf n}}$-connected to $y$ *through* ${{\cal A}}$.
(ii) Given an event $E$ (i.e., a set of current configurations) and a bond $b$, we define $\{E$ off $b\}$ to be the set of current configurations ${{\bf n}}\in E$ such that changing $n_b$ results in a configuration that is also in $E$. Let ${{\cal C}}_{{\bf n}}^b(x)=\{y:x{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}y\text{ off }b\}$.
(iii) For a *directed* bond $b=(u,v)$, we write ${\underline{b}}=u$ and ${\overline{b}}=v$. We say that a directed bond $b$ is *pivotal* for $x{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}y$ from $x$, if $\{x{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}{\underline{b}}$ off $b\}\cap\{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}y$ in ${{\cal C}}_{{\bf n}}^b(x){^{\rm c}}\}$ occurs. If $\{x{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}y\}$ occurs with no pivotal bonds, we say that $x$ is *${{\bf n}}$-doubly connected to* $y$, and write $x{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}y$.
We begin with the first stage of the lace expansion. First, by using the above percolation language, the two-point function can be written as $$\begin{aligned}
{\label{eq:2pt-perclang}}
{{\langle \varphi_o\varphi_x \rangle}}_\Lambda=\sum_{{\partial}{{\bf n}}=o{\vartriangle}x}\frac{w_\Lambda
({{\bf n}})}{Z_\Lambda}\equiv\sum_{{\partial}{{\bf n}}=o{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}
{Z_\Lambda}\,{{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\}$}}}}.\end{aligned}$$ We decompose the indicator on the right-hand side into two parts depending on whether or not there is a pivotal bond for $o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x$ from $o$; if there is, we take the *first* bond among them. Then, we have $$\begin{aligned}
{\label{eq:0th-ind-fact}}
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\}$}}}={\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}$}}}+\sum_{b\in{{\mathbb B}}_\Lambda}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}
{\underline{b}}\text{ off }b\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{n_b>0\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal C}}_{{\bf n}}^b(o)
{^{\rm c}}\}$}}}.\end{aligned}$$ Let $$\begin{aligned}
{\label{eq:pi0-def}}
\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)=\sum_{{\partial}{{\bf n}}=o{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}
{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}$}}}.\end{aligned}$$ Substituting [(\[eq:0th-ind-fact\])]{} into [(\[eq:2pt-perclang\])]{}, we obtain (see Figure \[fig:1stpiv\])
$$\raisebox{0.2pc}{\includegraphics[scale=0.17]{1stpiv1}}~~~=~~~
\raisebox{-0.5pc}{\includegraphics[scale=0.17]{1stpiv2}}~~~+~~
\sum_b~\raisebox{-1.7pc}{\includegraphics[scale=0.17]{1stpiv3}}$$
$$\begin{aligned}
{\label{eq:pre-1st-exp}}
{{\langle \varphi_o\varphi_x \rangle}}_\Lambda=\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)+\sum_{b\in
{{\mathbb B}}_\Lambda}\;\sum_{{\partial}{{\bf n}}=o{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}{\underline{b}}\text{ off }b\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{n_b>0\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x
\text{ in }{{\cal C}}_{{\bf n}}^b(o){^{\rm c}}\}$}}}.\end{aligned}$$
Next, we consider the sum over ${{\bf n}}$ in [(\[eq:pre-1st-exp\])]{}. Since $b$ is pivotal for $o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x$ from $o\,(\ne x$, due to the last indicator) and ${\partial}{{\bf n}}=o{\vartriangle}x$, in fact $n_b$ is an *odd* integer. We alternate the parity of $n_b$ by changing the source constraint into $o{\vartriangle}b{\vartriangle}x\equiv\{o\}{\,\triangle\,}\{{\underline{b}},{\overline{b}}\}{\,\triangle\,}\{x\}$ and multiplying by $$\begin{aligned}
\frac{\sum_{n\text{ odd}}(p J_b)^n/n!}{\sum_{n\text{ even}}(p
J_b)^n/n!}=\tanh(p J_b)\equiv\tau_b.\end{aligned}$$ Then, the sum over ${{\bf n}}$ in [(\[eq:pre-1st-exp\])]{} equals $$\begin{aligned}
{\label{eq:0th-summand1}}
\sum_{{\partial}{{\bf n}}=o{\vartriangle}b{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}{\underline{b}}\text{ off }b\}$}}}\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{n_b\text{ even}\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal C}}_{{\bf n}}^b(o){^{\rm c}}\}$}}}.\end{aligned}$$ Note that, except for $b$, there are no positive currents on the boundary bonds of ${{\cal C}}_{{\bf n}}^b(o)$.
Now, we condition on ${{\cal C}}_{{\bf n}}^b(o)={{\cal A}}$ and decouple events occurring on ${{\mathbb B}}_{{{\cal A}}{^{\rm c}}}$ from events occurring on ${{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}$, by using the following notation: $$\begin{aligned}
{\label{eq:tildew-def}}
\tilde w_{\Lambda,{{\cal A}}}({{\bf k}})=\prod_{b\in{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\frac{(pJ_b)^{k_b}}{k_b!}\qquad({{\bf k}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_\Lambda
\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}).\end{aligned}$$ Conditioning on ${{\cal C}}_{{\bf n}}^b(o)={{\cal A}}$, multiplying $Z_{{{\cal A}}{^{\rm c}}}/Z_{{{\cal A}}{^{\rm c}}}\equiv1$ (and using the notation ${{\bf k}}={{\bf n}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}$ and ${{\bf m}}={{\bf n}}|_{{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}$) and then summing over ${{\cal A}}\subset\Lambda$, we have $$\begin{aligned}
{\label{eq:0th-summand2}}
{(\ref{eq:0th-summand1})}&=\sum_{{{\cal A}}\subset\Lambda}\,\sum_{\substack{{\partial}{{\bf k}}=o{\vartriangle}{\underline{b}}\\ {\partial}{{\bf m}}={\overline{b}}{\vartriangle}x}}\frac{\tilde w_{\Lambda,{{\cal A}}}({{\bf k}})\,
Z_{{{\cal A}}{^{\rm c}}}}{Z_\Lambda}\,\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}}
{\overset{}{\Longleftrightarrow}}}{\underline{b}}\text{ off }b\\}$}}}\,\cap\,\{{{\cal C}}_{{{\bf k}}}^b(o)={{\cal A}}}\,
\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{k_b\text{ even}\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}$}}
{\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal A}}{^{\rm c}}\}$}}}{\nonumber}\\
&=\sum_{{{\cal A}}\subset\Lambda}\;\sum_{{\partial}{{\bf n}}=o{\vartriangle}{\underline{b}}}\frac{w_\Lambda({{\bf n}})}
{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}{\underline{b}}\text{ off }b\\}$}}}\,\cap\,\{{{\cal C}}_{{\bf n}}^b(o)={{\cal A}}}
\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{n_b\text{ even}\}$}}}\underbrace{\sum_{{\partial}{{\bf m}}={\overline{b}}{\vartriangle}x}\frac{
w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}$}}
{\overset{}{\longleftrightarrow}}}x\text{ (in }{{\cal A}}{^{\rm c}})\}$}}}}_{=\;{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{{{\cal A}}{^{\rm c}}}}{\nonumber}\\
&=\sum_{{\partial}{{\bf n}}=o{\vartriangle}{\underline{b}}}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}{\underline{b}}\text{ off }b\}$}}}\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{n_b\text{ even}\}$}}}\,{{\langle \varphi_{{\overline{b}}}
\varphi_x \rangle}}_{{{\cal C}}_{{\bf n}}^b(o){^{\rm c}}}.\end{aligned}$$ Furthermore, “off $b$” and ${\mathbbm{1}{\scriptstyle\{n_b\text{ even}\}}}$ in the last line can be omitted, since $\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}{\underline{b}}\}\setminus\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}{\underline{b}}$ off $b\}$ and $\{{\partial}{{\bf n}}=o{\vartriangle}{\underline{b}}\}\cap\{n_b$ odd} are subsets of $\{{\overline{b}}\in{{\cal C}}_{{\bf n}}^b(o)\}$, on which ${{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{{{\cal C}}_{{\bf n}}^b(o){^{\rm c}}}=0$. As a result, $$\begin{aligned}
{\label{eq:0th-summand3}}
{(\ref{eq:0th-summand2})}~=\sum_{{\partial}{{\bf n}}=o{\vartriangle}{\underline{b}}}\frac{w_\Lambda({{\bf n}})}
{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}{\underline{b}}\}$}}}\,\tau_b\,{{\langle \varphi_{{\overline{b}}}
\varphi_x \rangle}}_{{{\cal C}}_{{\bf n}}^b(o){^{\rm c}}}.\end{aligned}$$
By [(\[eq:pre-1st-exp\])]{} and [(\[eq:0th-summand3\])]{}, we arrive at $$\begin{aligned}
{\label{eq:1st-exp}}
{{\langle \varphi_o\varphi_x \rangle}}_\Lambda=\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)+\sum_{b\in
{{\mathbb B}}_\Lambda}\pi_\Lambda^{{\scriptscriptstyle}(0)}({\underline{b}})\,\tau_b\,{{\langle \varphi_{{\overline{b}}}
\varphi_x \rangle}}_\Lambda-R_\Lambda^{{\scriptscriptstyle}(1)}(x),\end{aligned}$$ where $$\begin{aligned}
{\label{eq:R1-def}}
R_\Lambda^{{\scriptscriptstyle}(1)}(x)=\sum_{b\in{{\mathbb B}}_\Lambda}\;\sum_{{\partial}{{\bf n}}=o{\vartriangle}{\underline{b}}}
\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}{\underline{b}}\}$}}}\,\tau_b\Big(
{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_\Lambda-{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{
{{\cal C}}_{{\bf n}}^b(o){^{\rm c}}}\Big).\end{aligned}$$ This completes the proof of [(\[eq:Ising-lace\])]{} for $j=0$, with $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$ and $R_\Lambda^{{\scriptscriptstyle}(1)}(x)$ being defined in [(\[eq:pi0-def\])]{} and [(\[eq:R1-def\])]{}, respectively.
### The second stage of the expansion {#sss:2ndexp}
In the next stage of the lace expansion, we further expand $R_\Lambda^{{\scriptscriptstyle}(1)}(x)$ in [(\[eq:1st-exp\])]{}. To do so, we investigate the difference ${{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_\Lambda-{{\langle \varphi_{{\overline{b}}}
\varphi_x \rangle}}_{{{\cal C}}_{{\bf n}}^b(o){^{\rm c}}}$ in [(\[eq:R1-def\])]{}. First, we prove the following key proposition[^4]:
\[prp:through\] For $v,x\in\Lambda$ and ${{\cal A}}\subset\Lambda$, we have $$\begin{aligned}
{\label{eq:lmm-through}}
{{\langle \varphi_v\varphi_x \rangle}}_\Lambda-{{\langle \varphi_v\varphi_x \rangle}}_{{{\cal A}}{^{\rm c}}}
=\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=v{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}
({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}$}}}.\end{aligned}$$ Therefore, ${{\langle \varphi_v\varphi_x \rangle}}_{{{\cal A}}{^{\rm c}}}\leq
{{\langle \varphi_v\varphi_x \rangle}}_\Lambda$ for the ferromagnetic case.
Since both sides of [(\[eq:lmm-through\])]{} are equal to ${\mathbbm{1}{\scriptstyle\{x\in{{\cal A}}\}}}$ when $v=x$ (see below [(\[eq:symmdiff\])]{}), it suffices to prove [(\[eq:lmm-through\])]{} for $v\ne x$.
First, by using [(\[eq:ZA-RCrepr1\])]{}–[(\[eq:2pt-rewr\])]{}, we obtain $$\begin{aligned}
{\label{eq:WZ-num}}
Z_\Lambda Z_{{{\cal A}}{^{\rm c}}}\Big({{\langle \varphi_v\varphi_x \rangle}}_\Lambda-{{\langle \varphi_v
\varphi_x \rangle}}_{{{\cal A}}{^{\rm c}}}\Big)=\sum_{{\partial}{{\bf n}}=\{v,x\}}Z_{{{\cal A}}{^{\rm c}}}\,w_\Lambda
({{\bf n}})-\sum_{{\partial}{{\bf m}}=\{v,x\}}w_{{{\cal A}}{^{\rm c}}}({{\bf m}})\,Z_\Lambda{\nonumber}\\
=\sum_{\substack{{\partial}{{\bf m}}={\varnothing},\,{\partial}{{\bf n}}=\{v,x\}\\ {{\bf m}}|_{{{\mathbb B}}_\Lambda\setminus
{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\equiv0}}w_\Lambda({{\bf m}})\,w_\Lambda({{\bf n}})-\sum_{\substack{
{\partial}{{\bf m}}=\{v,x\},\,{\partial}{{\bf n}}={\varnothing}\\ {{\bf m}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}
\equiv0}}w_\Lambda({{\bf m}})\,w_\Lambda({{\bf n}}).\end{aligned}$$ Note that the second term is equivalent to the first term if the source constraints for ${{\bf m}}$ and ${{\bf n}}$ are exchanged.
Next, we consider the second term of [(\[eq:WZ-num\])]{}, whose exact expression is $$\begin{gathered}
{\label{eq:2ndterm-expl}}
\sum_{\substack{{\partial}{{\bf m}}=\{v,x\},\,{\partial}{{\bf n}}={\varnothing}\\ {{\bf m}}|_{{{\mathbb B}}_\Lambda\setminus
{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\equiv0}}\bigg(\prod_{b\in{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\frac{(pJ_b)^{n_b}}{n_b!}\bigg)\prod_{b\in{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\frac{
(pJ_b)^{m_b+n_b}}{m_b!\,n_b!}=\sum_{{\partial}{{\bf N}}=\{v,x\}}w_\Lambda({{\bf N}})\sum_{
\substack{{\partial}{{\bf m}}=\{v,x\}\\ {{\bf m}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}
\equiv0}}\prod_{b\in{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\binom{N_b}{m_b}.\end{gathered}$$ The following is a variant of the source-switching lemma [@a82; @ghs70] and allows us to change the source constraints in [(\[eq:2ndterm-expl\])]{}.
\[lmm:switching\] $$\begin{aligned}
{\label{eq:switching}}
\sum_{\substack{{\partial}{{\bf m}}=\{v,x\}\\ {{\bf m}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\equiv0}}\,\prod_{b\in{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\binom{N_b}{m_b}={\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}}
{\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal A}}{^{\rm c}}\}$}}}\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {{\bf m}}|_{
{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\equiv0}}\,\prod_{b\in{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\binom{N_b}{m_b}.\end{aligned}$$
The idea of the proof of [(\[eq:switching\])]{} can easily be extended to more general cases, in which the source constraint in the left-hand side of [(\[eq:switching\])]{} is replaced by ${\partial}{{\bf m}}={{\cal V}}$ for some ${{\cal V}}\subset\Lambda$ and that in the right-hand side is replaced by ${\partial}{{\bf m}}={{\cal V}}{\,\triangle\,}\{v,x\}$ (e.g., [@a82]). We will explain the proof of [(\[eq:switching\])]{} after completing the proof of Proposition \[prp:through\].
We continue with the proof of Proposition \[prp:through\]. Substituting [(\[eq:switching\])]{} into [(\[eq:2ndterm-expl\])]{}, we obtain $$\begin{aligned}
{\label{eq:switching-appl}}
{(\ref{eq:2ndterm-expl})}&=\sum_{{\partial}{{\bf N}}=\{v,x\}}w_\Lambda({{\bf N}})\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}}
{\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal A}}{^{\rm c}}\}$}}}\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\
{{\bf m}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\equiv0}}\,\prod_{b\in
{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\binom{N_b}{m_b}{\nonumber}\\
&=\sum_{\substack{{\partial}{{\bf m}}={\varnothing},\,{\partial}{{\bf n}}=\{v,x\}\\ {{\bf m}}|_{{{\mathbb B}}_\Lambda
\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\equiv0}}w_\Lambda({{\bf m}})\,w_\Lambda({{\bf n}})\,
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal A}}{^{\rm c}}\}$}}}.\end{aligned}$$ Note that the source constraints for ${{\bf m}}$ and ${{\bf n}}$ in the last line are identical to those in the first term of [(\[eq:WZ-num\])]{}, under which ${\mathbbm{1}{\scriptstyle\{v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\}}}$ is always 1. By [(\[eq:incl/excl\])]{}, we can rewrite [(\[eq:WZ-num\])]{} as $$\begin{aligned}
{\label{eq:through}}
{{\langle \varphi_v\varphi_x \rangle}}_\Lambda-{{\langle \varphi_v\varphi_x \rangle}}_{{{\cal A}}{^{\rm c}}}
&=\sum_{\substack{{\partial}{{\bf m}}={\varnothing},\,{\partial}{{\bf n}}=\{v,x\}\\ {{\bf m}}|_{{{\mathbb B}}_\Lambda
\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\equiv0}}\frac{w_\Lambda({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}
\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}$}}}.\end{aligned}$$ Using [(\[eq:ZA-RCrepr1\])]{}–[(\[eq:ZA-RCrepr2\])]{} to omit “${{\bf m}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}\equiv0$” and replace $w_\Lambda({{\bf m}})$ by $w_{{{\cal A}}{^{\rm c}}}({{\bf m}})$, we arrive at [(\[eq:lmm-through\])]{}. This completes the proof of Proposition \[prp:through\].
We explain the meaning of the identity [(\[eq:switching\])]{} and the idea of its proof. Given ${{\bf N}}=\{N_b\}_{b\in{{\mathbb B}}_\Lambda}$, we denote by ${{\mathbb G}}_{{\bf N}}$ the graph consisting of $N_b$ *labeled* edges between ${\underline{b}}$ and ${\overline{b}}$ for every $b\in{{\mathbb B}}_\Lambda$ (see Figure \[fig:switching\]).
$$\begin{aligned}
{{\bf N}}~:&\qquad\includegraphics[scale=0.33]{switching1}\\[5pt]
{{\mathbb G}}_{{\bf N}}~:&\qquad\raisebox{-1.8pc}{\includegraphics[scale=0.33]
{switching2}}\\[1pc]
{{\mathbb S}}~:&\qquad\raisebox{-1.8pc}{\includegraphics[scale=0.33]
{switching3}}\\[7pt]
{{\mathbb S}}{\,\triangle\,}\omega~:&\qquad\raisebox{-1.8pc}{\includegraphics[scale=0.33]
{switching4}}\end{aligned}$$
For a subgraph ${{\mathbb S}}\subset{{\mathbb G}}_{{\bf N}}$, we denote by ${\partial}{{\mathbb S}}$ the set of vertices at which the number of incident edges in ${{\mathbb S}}$ is *odd*, and let ${{\mathbb S}}_{{\cal A}}={{\mathbb S}}\cap{{\mathbb G}}_{{{\bf N}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}}}$. Then, the left-hand side of [(\[eq:switching\])]{} equals the cardinality $|{\mathfrak{S}}|$ of $$\begin{aligned}
{\label{eq:Sbefore}}
{\mathfrak{S}}=\{{{\mathbb S}}\subset{{\mathbb G}}_{{\bf N}}:{\partial}{{\mathbb S}}=\{v,x\},~{{\mathbb S}}_{{\cal A}}={\varnothing}\},\end{aligned}$$ and the sum in the right-hand side of [(\[eq:switching\])]{} equals the cardinality $|{\mathfrak{S}}'|$ of $$\begin{aligned}
{\mathfrak{S}}'=\{{{\mathbb S}}\subset{{\mathbb G}}_{{\bf N}}:{\partial}{{\mathbb S}}={\varnothing},~{{\mathbb S}}_{{\cal A}}={\varnothing}\}.\end{aligned}$$ We note that $|{\mathfrak{S}}|$ is zero when there are no paths on ${{\mathbb G}}_{{\bf N}}$ between $v$ and $x$ consisting of edges whose endvertices are both in ${{\cal A}}{^{\rm c}}$, while $|{\mathfrak{S}}'|$ may not be zero. The identity [(\[eq:switching\])]{} reads that $|{\mathfrak{S}}|$ equals $|{\mathfrak{S}}'|$ if we compensate for this discrepancy.
Suppose that there is a path (i.e., a ) $\omega$ from $v$ to $x$ consisting of edges in ${{\mathbb G}}_{{\bf N}}$ whose endvertices are both in ${{\cal A}}{^{\rm c}}$. Then, the map $$\begin{aligned}
{\label{eq:bijection}}
{{\mathbb S}}\in{\mathfrak{S}}~\mapsto~{{\mathbb S}}{\,\triangle\,}\omega\in{\mathfrak{S}}',\end{aligned}$$ is a bijection [@a82; @ghs70], and therefore $|{\mathfrak{S}}|=|{\mathfrak{S}}'|$. Here and in the rest of the paper, the symmetric difference between graphs is only in terms of *edges*. For example, ${{\mathbb S}}{\,\triangle\,}\omega$ is the result of adding or deleting edges (not vertices) contained in $\omega$. This completes the proof of [(\[eq:switching\])]{}.
We now start with the second stage of the expansion by using Proposition \[prp:through\] and applying inclusion-exclusion as in the first stage of the expansion in Section \[sss:1stexp\]. First, we decompose the indicator in [(\[eq:lmm-through\])]{} into two parts depending on whether or not there is a pivotal bond $b$ for $v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x$ from $v$ such that $v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}{\underline{b}}$. Let $$\begin{aligned}
{\label{eq:E-def}}
E_{{\bf N}}(v,x;{{\cal A}})=\{v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}\cap\{\nexists
\text{ pivotal bond }b\text{ for }v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}}
{\overset{}{\longleftrightarrow}}}x
\text{ from $v$ such that }v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}{\underline{b}}\}.\end{aligned}$$ On the event $\{v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}\setminus
E_{{{\bf m}}+{{\bf n}}}(v,x;{{\cal A}})$, we take the *first* pivotal bond $b$ for $v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x$ from $v$ satisfying $v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}{\underline{b}}$. Then, we have (cf., [(\[eq:0th-ind-fact\])]{}) $$\begin{aligned}
{\label{eq:1st-ind-fact}}
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}$}}}={\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E_{{{\bf m}}+{{\bf n}}}(v,x;{{\cal A}})$}}}+\sum_{b\in
{{\mathbb B}}_\Lambda}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E_{{{\bf m}}+{{\bf n}}}(v,{\underline{b}};{{\cal A}})\text{ off }b\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b+n_b>0\}$}}}
\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v){^{\rm c}}\}$}}}.\end{aligned}$$ Let $$\begin{aligned}
{\label{eq:Theta-def}}
\Theta_{v,x;{{\cal A}}}[X]=\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=v{\vartriangle}x}}\frac{
w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E_{{{\bf m}}+{{\bf n}}}(v,x;{{\cal A}})$}}}\,X({{\bf m}}+{{\bf n}}),&&
\Theta_{v,x;{{\cal A}}}=\Theta_{v,x;{{\cal A}}}[1].\end{aligned}$$ Substituting [(\[eq:1st-ind-fact\])]{} into [(\[eq:lmm-through\])]{}, we obtain (see Figure \[fig:through\])
$$\raisebox{-1.3pc}{\includegraphics[scale=0.17]{through1}}~~~=~~~
\raisebox{-1.3pc}{\includegraphics[scale=0.17]{through2}}~~~+~~
\sum_b~\raisebox{-1.3pc}{\includegraphics[scale=0.17]{through3}}$$
$$\begin{aligned}
{\label{eq:2nd-ind-fact}}
&{{\langle \varphi_v\varphi_x \rangle}}_\Lambda-{{\langle \varphi_v\varphi_x \rangle}}_{{{\cal A}}{^{\rm c}}}\\
&=\Theta_{v,x;{{\cal A}}}+\sum_{b\in{{\mathbb B}}_\Lambda}\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\
{\partial}{{\bf n}}=v{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda
({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E_{{{\bf m}}+{{\bf n}}}(v,{\underline{b}};{{\cal A}})\text{ off }b\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b
\text{ even, }n_b\text{ odd}\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal C}}_{
{{\bf m}}+{{\bf n}}}^b(v){^{\rm c}}\}$}}},{\nonumber}\end{aligned}$$
where we have replaced “$m_b+n_b>0$” in [(\[eq:1st-ind-fact\])]{} by “$m_b$ even, $n_b$ odd” that is the only possible combination consistent with the source constraints and the conditions in the indicators. As in [(\[eq:0th-summand1\])]{}, we alternate the parity of $n_b$ by changing the source constraint from ${\partial}{{\bf n}}=v{\vartriangle}x$ to ${\partial}{{\bf n}}=v{\vartriangle}b{\vartriangle}x$ and multiplying by $\tau_b$. Then, the sum over ${{\bf m}}$ and ${{\bf n}}$ in [(\[eq:2nd-ind-fact\])]{} equals $$\begin{aligned}
{\label{eq:3rd-ind-ppfact}}
\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=v{\vartriangle}b{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E_{{{\bf m}}+{{\bf n}}}(v,{\underline{b}};{{\cal A}})\text{ off }b\}$}}}\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b,n_b
\text{ even}\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b
(v){^{\rm c}}\}$}}}.\end{aligned}$$ Then, as in [(\[eq:0th-summand2\])]{}, we condition on ${{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v)={{\cal B}}$ and decouple events occurring on ${{\mathbb B}}_{{{\cal B}}{^{\rm c}}}$ from events occurring on ${{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal B}}{^{\rm c}}}$. Let ${{\bf m}}'={{\bf m}}|_{{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}$, ${{\bf m}}''={{\bf m}}|_{{{\mathbb B}}_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}$, ${{\bf n}}'={{\bf n}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal B}}{^{\rm c}}}}$ and ${{\bf n}}''={{\bf n}}|_{{{\mathbb B}}_{{{\cal B}}{^{\rm c}}}}$. Note that ${\partial}{{\bf m}}'={\partial}{{\bf m}}''={\varnothing}$, ${\partial}{{\bf n}}'=v{\vartriangle}{\underline{b}}$ and ${\partial}{{\bf n}}''={\overline{b}}{\vartriangle}x$. Multiplying [(\[eq:3rd-ind-ppfact\])]{} by $(Z_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}/Z_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}})(Z_{{{\cal B}}{^{\rm c}}}/Z_{{{\cal B}}{^{\rm c}}})\equiv1$ and using the notation [(\[eq:tildew-def\])]{}, we obtain $$\begin{aligned}
{\label{eq:3rd-ind-prefact}}
{(\ref{eq:3rd-ind-ppfact})}&=\sum_{{{\cal B}}\subset\Lambda}\sum_{\substack{{\partial}{{\bf m}}'={\varnothing}\\{\partial}{{\bf n}}'=v{\vartriangle}{\underline{b}}}}\frac{\tilde w_{{{\cal A}}{^{\rm c}},{{\cal B}}}({{\bf m}}')
\,Z_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{\tilde w_{\Lambda,
{{\cal B}}}({{\bf n}}')\,Z_{{{\cal B}}{^{\rm c}}}}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E_{{{\bf m}}'+{{\bf n}}'}(v,{\underline{b}};{{\cal A}})
\text{ off }b\\}$}}}\,\cap\,\{{{\cal C}}_{{{\bf m}}'+{{\bf n}}'}^b(v)={{\cal B}}}\,{\nonumber}\\
&\qquad\qquad\times\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m'_b,n'_b\text{ even}\}$}}}\sum_{\substack{
{\partial}{{\bf m}}''={\varnothing}\\ {\partial}{{\bf n}}''={\overline{b}}{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}
({{\bf m}}'')}{Z_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}\,\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf n}}'')}
{Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}''+{{\bf n}}''$}}
{\overset{}{\longleftrightarrow}}}x\text{ in }{{\cal B}}{^{\rm c}}\}$}}}{\nonumber}\\
&=\sum_{{{\cal B}}\subset\Lambda}\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=v{\vartriangle}{\underline{b}}}}
\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}
\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E_{{{\bf m}}+{{\bf n}}}(v,{\underline{b}};{{\cal A}})\text{ off }b\\}$}}}\,\cap\,\{{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v)=
{{\cal B}}}\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b,n_b\text{ even}\}$}}}\,{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{{{\cal B}}{^{\rm c}}}{\nonumber}\\
&=\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=v{\vartriangle}{\underline{b}}}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}
{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E_{{{\bf m}}+{{\bf n}}}(v,{\underline{b}};
{{\cal A}})\text{ off }b\}$}}}\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b,n_b\text{ even}\}$}}}\,{{\langle \varphi_{{\overline{b}}}
\varphi_x \rangle}}_{{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v){^{\rm c}}},\end{aligned}$$ where we have been able to perform the sum over ${{\bf m}}''$ and ${{\bf n}}''$ independently, due to the fact that ${\mathbbm{1}{\scriptstyle\{{\overline{b}}\,{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}''+{{\bf n}}''$}}
{\overset{}{\longleftrightarrow}}}\,x
\text{ in }{{\cal B}}{^{\rm c}}\}}}\equiv1$ for any ${{\bf n}}''\in{{\mathbb Z}_+}^{{{\mathbb B}}_{{{\cal B}}{^{\rm c}}}}$ with ${\partial}{{\bf n}}''={\overline{b}}{\vartriangle}x$. As in the derivation of [(\[eq:0th-summand3\])]{} from [(\[eq:0th-summand2\])]{}, we can omit “off $b$” and ${\mathbbm{1}{\scriptstyle\{m_b,n_b
\text{ even}\}}}$ in [(\[eq:3rd-ind-prefact\])]{} using the source constraints and the fact that ${{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v){^{\rm c}}}=0$ whenever ${\overline{b}}\in{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v)$. Therefore, $$\begin{aligned}
{\label{eq:3rd-ind-fact}}
{(\ref{eq:3rd-ind-prefact})}~=\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=v{\vartriangle}{\underline{b}}}}
\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}
\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E_{{{\bf m}}+{{\bf n}}}(v,{\underline{b}};{{\cal A}})$}}}\,\tau_b\,{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{
{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v){^{\rm c}}}.\end{aligned}$$ By [(\[eq:Theta-def\])]{}–[(\[eq:3rd-ind-fact\])]{}, we arrive at $$\begin{aligned}
{\label{eq:2nd-exp}}
{{\langle \varphi_v\varphi_x \rangle}}_\Lambda-{{\langle \varphi_v\varphi_x \rangle}}_{{{\cal A}}{^{\rm c}}}=
\Theta_{v,x;{{\cal A}}}&+\sum_{b\in{{\mathbb B}}_\Lambda}\Theta_{v,{\underline{b}};{{\cal A}}}\,\tau_b\,
{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_\Lambda{\nonumber}\\
&-\sum_{b\in{{\mathbb B}}_\Lambda}\Theta_{v,{\underline{b}};{{\cal A}}}\Big[\tau_b\Big({{\langle \varphi_{{\overline{b}}}
\varphi_x \rangle}}_\Lambda-{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{{{\cal C}}^b(v)
{^{\rm c}}}\Big)\Big],\end{aligned}$$ where ${{\cal C}}^b(v)\equiv{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(v)$ is a variable for the operation $\Theta_{v,{\underline{b}};{{\cal A}}}$. This completes the second stage of the expansion.
### Completion of the lace expansion {#sss:complexp}
For notational convenience, we define $w_{\varnothing}({{\bf m}})/Z_{\varnothing}={\mathbbm{1}{\scriptstyle\{{{\bf m}}\equiv0\}}}$. Since $E_{{\bf n}}(o,x;\Lambda)=\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}$ (cf., [(\[eq:E-def\])]{}), we can write $$\begin{aligned}
{\label{eq:pi0-rewr}}
\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)=\Theta_{o,x;\Lambda}.\end{aligned}$$ Also, we can write $R_\Lambda^{{\scriptscriptstyle}(1)}(x)$ in [(\[eq:R1-def\])]{} as $$\begin{aligned}
{\label{eq:R1-rewr}}
R_\Lambda^{{\scriptscriptstyle}(1)}(x)=\sum_b\Theta_{o,{\underline{b}};\Lambda}\Big[\tau_b\Big({{\langle
\varphi_{{\overline{b}}}\varphi_x \rangle}}_\Lambda-{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_{{{\cal C}}^b
(o){^{\rm c}}}\Big)\Big].\end{aligned}$$ Using [(\[eq:2nd-exp\])]{}, we obtain $$\begin{aligned}
{\label{eq:R1R2}}
R_\Lambda^{{\scriptscriptstyle}(1)}(x)=\sum_b\bigg(&\Theta_{o,{\underline{b}};\Lambda}\Big[\tau_b\,
\Theta_{{\overline{b}},x;{{\cal C}}^b(o)}\Big]+\sum_{b'}\Theta_{o,{\underline{b}};\Lambda}\Big[
\tau_b\,\Theta_{{\overline{b}},{\underline{b}}';{{\cal C}}^b(o)}\Big]\,\tau_{b'}{{\langle \varphi_{
{{\overline{b}}^{\raisebox{-2pt}{$\scriptscriptstyle\prime$}}}}\varphi_x \rangle}}_\Lambda{\nonumber}\\
&-\sum_{b'}\Theta_{o,{\underline{b}};\Lambda}\Big[\tau_b\,\Theta_{{\overline{b}},{\underline{b}}';{{\cal C}}^b
(o)}\Big[\tau_{b'}\Big({{\langle \varphi_{{{\overline{b}}^{\raisebox{-2pt}{$\scriptscriptstyle\prime$}}}}\varphi_x \rangle}}_\Lambda-{{\langle
\varphi_{{{\overline{b}}^{\raisebox{-2pt}{$\scriptscriptstyle\prime$}}}}\varphi_x \rangle}}_{{{\cal C}}^{b'}({\overline{b}}){^{\rm c}}}\Big)\Big]
\Big]\bigg),\end{aligned}$$ where ${{\cal C}}^b(o)\equiv{{\cal C}}_{{\bf n}}^b(o)$ is a variable for the outer operation $\Theta_{o,{\underline{b}};\Lambda}$, and ${{\cal C}}^{b'}({\overline{b}})\equiv{{\cal C}}_{{{\bf m}}'+{{\bf n}}'}^{b'}({\overline{b}})$ is a variable for the inner operation $\Theta_{{\overline{b}},{\underline{b}}';{{\cal C}}^b(o)}$. For $j\ge1$, we define $$\begin{aligned}
\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)&=\sum_{b_1,\dots,b_j}\Theta^{{\scriptscriptstyle}(0)}_{o,{\underline{b}}_1;
\Lambda}\Big[\tau_{b_1}\Theta^{{\scriptscriptstyle}(1)}_{{\overline{b}}_1,{\underline{b}}_2;\tilde{{\cal C}}_0}\Big[
\cdots\tau_{b_{j-1}}\Theta^{{\scriptscriptstyle}(j-1)}_{{\overline{b}}_{j-1},{\underline{b}}_j;\tilde{{\cal C}}_{j-2}}
\Big[\tau_{b_j}\Theta^{{\scriptscriptstyle}(j)}_{{\overline{b}}_j,x;\tilde{{\cal C}}_{j-1}}\Big]\cdots\Big]
\Big],{\label{eq:pij-def}}\\
R_\Lambda^{{\scriptscriptstyle}(j)}(x)&=\sum_{b_1,\dots,b_j}\Theta^{{\scriptscriptstyle}(0)}_{o,{\underline{b}}_1;
\Lambda}\Big[\tau_{b_1}\Theta^{{\scriptscriptstyle}(1)}_{{\overline{b}}_1,{\underline{b}}_2;\tilde{{\cal C}}_0}\Big[
\cdots\tau_{b_{j-1}}\Theta^{{\scriptscriptstyle}(j-1)}_{{\overline{b}}_{j-1},{\underline{b}}_j;\tilde{{\cal C}}_{j-2}}
\Big[\tau_{b_j}\Big({{\langle \varphi_{{\overline{b}}_j}\varphi_x \rangle}}_\Lambda-{{\langle \varphi_{
{\overline{b}}_j}\varphi_x \rangle}}_{\tilde{{\cal C}}_{j-1}{^{\rm c}}}\Big)\Big]\cdots\Big]\Big],
{\label{eq:Rj-def}}\end{aligned}$$ where the operation $\Theta^{{\scriptscriptstyle}(i)}$ determines the variable $\tilde{{\cal C}}_i={{\cal C}}_{{{\bf m}}_i+{{\bf n}}_i}^{b_{i+1}}({\overline{b}}_i)$ (provided that ${\overline{b}}_0=o$). Then, we can rewrite [(\[eq:R1R2\])]{} as $$\begin{aligned}
{\label{eq:R1R2-rewr}}
R_\Lambda^{{\scriptscriptstyle}(1)}(x)=\pi_\Lambda^{{\scriptscriptstyle}(1)}(x)+\sum_{b'}\pi_\Lambda
^{{\scriptscriptstyle}(1)}({\underline{b}}')\,\tau_{b'}{{\langle \varphi_{{{\overline{b}}^{\raisebox{-2pt}{$\scriptscriptstyle\prime$}}}}\varphi_x \rangle}}_\Lambda
-R_\Lambda^{{\scriptscriptstyle}(2)}(x).\end{aligned}$$ As a result, $$\begin{aligned}
{{\langle \varphi_o\varphi_x \rangle}}_\Lambda=\big(\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)-\pi_\Lambda
^{{\scriptscriptstyle}(1)}(x)\big)+\sum_b\big(\pi_\Lambda^{{\scriptscriptstyle}(0)}({\underline{b}})-\pi_\Lambda^{{\scriptscriptstyle}(1)}({\underline{b}})\big)\,\tau_b\,{{\langle \varphi_{{\overline{b}}}\varphi_x \rangle}}_\Lambda+R_\Lambda^{
{\scriptscriptstyle}(2)}(x).\end{aligned}$$ By repeated applications of [(\[eq:2nd-exp\])]{} to the remainder $R_\Lambda^{{\scriptscriptstyle}(j)}(x)$, we obtain [(\[eq:Ising-lace\])]{}–[(\[eq:Pij-def\])]{} in Proposition \[prp:Ising-lace\].
For the ferromagnetic case, $\tau_b$ and $w_{{{\cal A}}}({{\bf n}})$ for any ${{\cal A}}\subset\Lambda$ and ${{\bf n}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_{{\cal A}}}$ are nonnegative. This proves the first inequality in [(\[eq:pij-Rj-naivebd\])]{} and, with the help of Proposition \[prp:through\], the nonnegativity of $R_\Lambda^{{\scriptscriptstyle}(j+1)}(x)$ . To prove the upper bound on $R_\Lambda^{{\scriptscriptstyle}(j+1)}(x)$, we simply ignore ${{\langle \varphi_{{\overline{b}}_j}\varphi_x \rangle}}_{\tilde{{\cal C}}_{j-1}{^{\rm c}}}$ in [(\[eq:Rj-def\])]{} and replace $j$ by $j+1$, where $b_{j+1}=\{u,v\}$. This completes the proof of Proposition \[prp:Ising-lace\].
Comparison to percolation {#ss:percolation}
-------------------------
Since we have exploited the underlying percolation picture to derive the lace expansion [(\[eq:Ising-lace\])]{} for the Ising model, it is not so surprising that the expansion coefficients [(\[eq:pi0-rewr\])]{} and [(\[eq:pij-def\])]{} (also recall [(\[eq:Theta-def\])]{}) are quite similar to the lace-expansion coefficients for unoriented bond-percolation (cf., [@hs90']): $$\begin{aligned}
{\label{eq:pij-perc}}
\pi_p^{{\scriptscriptstyle}(j)}(x)=
\begin{cases}
~{\displaystyle}{{\mathbb E}}_p^{{\scriptscriptstyle}(0)}\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}_0$}}
{\overset{}{\Longleftrightarrow}}}x\}$}}}\big]\equiv{{\mathbb P}}_p(o{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}}
{\overset{}{\Longleftrightarrow}}}x)
&(j=0),\\[1pc]
{\displaystyle}\sum_{b_1,\dots,b_j}{{\mathbb E}}_p^{{\scriptscriptstyle}(0)}\Big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}_0$}}
{\overset{}{\Longleftrightarrow}}}{\underline{b}}_1\}$}}}\,
p_{b_1}{{\mathbb E}}_p^{{\scriptscriptstyle}(1)}\Big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E_{{{\bf n}}_1}({\overline{b}}_1,{\underline{b}}_2;\tilde{{\cal C}}_0)$}}}
\cdots p_{b_j}{{\mathbb E}}_p^{{\scriptscriptstyle}(j)}\Big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E_{{{\bf n}}_j}({\overline{b}}_j,x;\tilde
{{\cal C}}_{j-1})$}}}\Big]\cdots\Big]\Big]&(j\ge1),
\end{cases}\end{aligned}$$ where $p\equiv\sum_xp_{o,x}$ is the bond-occupation parameter, and each ${{\mathbb E}}_p^{{\scriptscriptstyle}(i)}$ denotes the expectation with respect to the product measure $\prod_b(p_b{\mathbbm{1}{\scriptstyle\{{{\bf n}}_i|_b=1\}}}+(1-p_b){\mathbbm{1}{\scriptstyle\{{{\bf n}}_i|_b=0\}}})$. In particular, the events involved in [(\[eq:pi0-rewr\])]{} and [(\[eq:pij-def\])]{} are identical to those in [(\[eq:pij-perc\])]{}.
Hoever, there are significant differences between these two models. The major differences are the following:
(a) Each current configuration must satisfy not only the conditions in the indicators, but also its source constraint that is absent in percolation.
(b) An operation $\Theta$ is not an expectation, since the source constraints in the numerator and denominator of $\Theta$ in [(\[eq:Theta-def\])]{} are different.
(c) In each $\Theta^{{\scriptscriptstyle}(i)}$ for $i\ge1$, the sum ${{\bf m}}_i+{{\bf n}}_i$ of two current configurations is coupled with ${{\bf m}}_{i-1}+{{\bf n}}_{i-1}$ via the cluster $\tilde{{\cal C}}_{i-1}$ determined by ${{\bf m}}_{i-1}+{{\bf n}}_{i-1}$. By contrast, in each ${{\mathbb E}}_p^{{\scriptscriptstyle}(i)}$ in [(\[eq:pij-perc\])]{}, a single percolation configuration ${{\bf n}}_i$ is coupled with ${{\bf n}}_{i-1}$ via $\tilde{{\cal C}}_{i-1}={{\cal C}}_{{{\bf n}}_{i-1}}^{b_i}({\overline{b}}_{i-1})$. In addition, ${{\bf m}}_i$ is nonzero only on bonds in ${{\mathbb B}}_{\tilde{{\cal C}}_{i-1}{^{\rm c}}}$, while the current configuration ${{\bf n}}_i$ has no such restriction.
These elements are responsible for the difference in the method of bounding diagrams for the expansion coefficients. Take the $0^\text{th}$-expansion coefficient for example. For percolation, the BK inequality simply tells us that $$\begin{aligned}
{\label{eq:pi0perc-comp}}
\pi_p^{{\scriptscriptstyle}(0)}(x)\leq{{\mathbb P}}_p(o{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}}
{\overset{}{\longleftrightarrow}}}x)^2.\end{aligned}$$ For the ferromagnetic Ising model, on the other hand, we first recall [(\[eq:pi0-def\])]{}, i.e., $$\begin{aligned}
\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)=\sum_{{\partial}{{\bf n}}=o{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}
{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}$}}},\end{aligned}$$ where $w_\Lambda({{\bf n}})/Z_\Lambda\ge0$. Due to the indicator, every current configuration ${{\bf n}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_\Lambda}$ that gives nonzero contribution has at least *two bond-disjoint* paths $\zeta_1,\zeta_2$ from $o$ to $x$ such that $n_b>0$ for all $b\in\zeta_1{\:\Dot{\cup}\:}\zeta_2$. Also, due to the source constraint, there should be at least one path $\zeta$ from $o$ to $x$ such that $n_b$ is odd for all $b\in\zeta$. Suppose, for example, that $\zeta=\zeta_1$ and that $n_b$ for $b\in\zeta_2$ are all positive-even. Since a positive-even integer can split into two odd integers, on the labeled graph ${{\mathbb G}}_{{\bf n}}$ with ${\partial}{{\mathbb G}}_{{\bf n}}=o{\vartriangle}x$ (recall the notation introduced above [(\[eq:Sbefore\])]{}) there are at least *three edge-disjoint* paths from $o$ to $x$. This observation naturally leads us to expect that $$\begin{aligned}
{\label{eq:pi0-comp}}
\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)\leq{{\langle \varphi_o\varphi_x \rangle}}_\Lambda^3\end{aligned}$$ holds for the ferromagnetic Ising model. This naive argument to justify [(\[eq:pi0-comp\])]{} will be made rigorous in Section \[s:bounds\] by taking account of partition functions.
The higher-order expansion coefficients are more involved, due to the above item (c). This will also be explained in detail in Section \[s:bounds\].
Bounds on $\Pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for the ferromagnetic models {#s:reduction}
=================================================================================
From now on, we restrict ourselves to the ferromagnetic models. In this section, we explain how to prove Proposition \[prp:Pij-Rj-bd\] assuming a few other propositions (Propositions \[prp:GimpliesPix\]–\[prp:exp-bootstrap\] below). These propositions are results of diagrammatic bounds on the expansion coefficients in terms of two-point functions. We will show these diagrammatic bounds in Section \[s:bounds\].
The strategy to prove Proposition \[prp:Pij-Rj-bd\] is model-independent, and we follow the strategy in [@h05] for the nearest-neighbor model and that in [@hhs03] for the spread-out model. Since the latter is simpler, we first explain the strategy for the spread-out model. In the rest of this paper, we will frequently use the notation $$\begin{aligned}
{\vbx{|\!|\!|}}=|x|\vee1.\end{aligned}$$ We also emphasize that constants in the $O$-notation used below (e.g., $O(\theta_0)$ in [(\[eq:pi-bd\])]{}) are independent of $\Lambda\subset{{\mathbb Z}^d}$.
Strategy for the spread-out model
---------------------------------
Using the diagrammatic bounds below in Section \[s:bounds\], we will prove in detail in Section \[ss:proof-so\] that the following proposition holds for the spread-out model:
\[prp:GimpliesPix\] Let $J_{o,x}$ be the spread-out interaction. Suppose that $$\begin{aligned}
{\label{eq:IR-xbd}}
\tau\leq2,&&
G(x)\leq\delta_{o,x}+\theta_0{\vbx{|\!|\!|}}^{-q}\end{aligned}$$ hold for some $\theta_0\in(0,\infty)$ and $q\in(\frac{d}2,d)$. Then, for sufficiently small $\theta_0$ (with $\theta_0L^{d-q}$ being bounded away from zero) and any $\Lambda\subset{{\mathbb Z}^d}$, we have $$\begin{aligned}
{\label{eq:pi-bd}}
\pi_\Lambda^{{\scriptscriptstyle}(i)}(x)\leq
\begin{cases}
O(\theta_0)^i\delta_{o,x}+O(\theta_0^3){\vbx{|\!|\!|}}^{-3q}&(i=0,1),\\
O(\theta_0)^i{\vbx{|\!|\!|}}^{-3q}&(i\ge2).
\end{cases}\end{aligned}$$
The exact value of the assumed upper bound on $\tau$ in [(\[eq:IR-xbd\])]{} is unimportant and can be any finite number, as long as it is independent of $\theta_0$ and bigger than the mean-field critical point 1. We note that the exponent $3q$ in [(\[eq:pi-bd\])]{} is due to [(\[eq:pi0-comp\])]{} (and diagrammatic bounds on the higher-expansion coefficients), and is replaced by $2q$ with $q\in(\frac{2d}3,d)$ for percolation, due to, e.g., [(\[eq:pi0perc-comp\])]{}.
We will show below that, at $p={p_\text{c}}$, $$\begin{aligned}
{\label{eq:IR-xbd-so}}
\tau\leq2,&& G(x)\leq\delta_{o,x}+O(L^{-2+\epsilon}){\vbx{|\!|\!|}}^{-(d-2)},\end{aligned}$$ for some small ${\epsilon}>0$. Since $\tau$ and $G(x)$ are nondecreasing and continuous in $p\leq{p_\text{c}}$ for the ferromagnetic models, these bounds imply [(\[eq:IR-xbd\])]{} for all $p\leq{p_\text{c}}$, with $\theta_0=cL^{-2+{\epsilon}}>0$ and $q=d-2$, where $q\in(\frac{d}2,d)$ if $d>4$ and $\theta_0L^{d-q}=cL^{\epsilon}>0$. Then, by Proposition \[prp:GimpliesPix\], the bound [(\[eq:pi-bd\])]{} with $\theta_0=O(L^{-2+{\epsilon}})$ and $q=d-2$ holds for $d>4$ and $\theta_0\ll1$ (thus $L\gg1$). Therefore, by [(\[eq:pij-Rj-naivebd\])]{} with ${{\langle \varphi_v\varphi_x \rangle}}_\Lambda\leq1$, $$\begin{aligned}
{\label{eq:Rj-optSO}}
0\leq R_\Lambda^{{\scriptscriptstyle}(j+1)}(x)\leq\tau\sum_u\pi_\Lambda^{{\scriptscriptstyle}(j)}(u)
=O(\theta_0)^j\to0\qquad(j\uparrow\infty),\end{aligned}$$ and by [(\[eq:Pij-def\])]{} for $j\ge0$, $$\begin{aligned}
{\label{eq:Pij-optSO}}
|\Pi_\Lambda^{{\scriptscriptstyle}(j)}(x)-\delta_{o,x}|\leq O(\theta_0)\delta_{o,x}
+\frac{O(\theta_0^2)}{{\vbx{|\!|\!|}}^{3(d-2)}}=O(\theta_0)\delta_{o,x}+
\frac{O(\theta_0^2)(1-\delta_{o,x})}{|x|^{d+2+\rho}},\end{aligned}$$ where $\rho=2(d-4)$. This completes the proof of Proposition \[prp:Pij-Rj-bd\] for the spread-out model, assuming [(\[eq:IR-xbd-so\])]{} at $p={p_\text{c}}$.
It thus remains to show the bounds in [(\[eq:IR-xbd-so\])]{} at $p={p_\text{c}}$. These bounds are proved by adapting the model-independent bootstrapping argument in [@hhs03] (see the proof of [@hhs03 Proposition 2.2] for self-avoiding walk and percolation), together with the fact that $G(x)$ decays exponentially as $|x|\uparrow\infty$ for every $p<{p_\text{c}}$ [@l80; @s80] so that $\sup_xG(x)$ is continuous in $p<{p_\text{c}}$ [@s05]. We complete the proof.
Strategy for the nearest-neighbor model
---------------------------------------
Since $\sigma^2=O(1)$ for short-range models, we cannot expect that $\theta_0$ in [(\[eq:IR-xbd\])]{} is small, or that Proposition \[prp:GimpliesPix\] is applicable to bound the expansion coefficients in this setting.
Under this circumstance, we follow the strategy in [@h05]. The following is the key proposition, whose proof will be explained in Section \[ss:proof-nn\]:
\[prp:GimpliesPik\] Let $J_{o,x}$ be the nearest-neighbor or spread-out interaction, and suppose that $$\begin{aligned}
{\label{eq:IR-kbd}}
\tau-1\leq\theta_0,&& \sup_x(D*G^{*2})(x)\leq\theta_0,&&
\sup_{\substack{x\equiv(x_1,\dots,x_d)\ne o\\ l=1,\dots,d}}
\bigg(\frac{x_l^2}{\sigma^2}\vee1\bigg)G(x)\leq\theta_0\end{aligned}$$ hold for some $\theta_0\in(0,\infty)$. Then, for sufficiently small $\theta_0$ and any $\Lambda\subset{{\mathbb Z}^d}$, we have $$\begin{aligned}
{\label{eq:pi-sumbd}}
\sum_x\pi_\Lambda^{{\scriptscriptstyle}(i)}(x)\leq
\begin{cases}
1+O(\theta_0^2)&(i=0),\\ O(\theta_0)^i&(i\ge1),
\end{cases}&&
\sum_x|x|^2\pi_\Lambda^{{\scriptscriptstyle}(i)}(x)\leq d\sigma^2(i+1)^2O(\theta_0)^{i
\vee2}.\end{aligned}$$ Furthermore, in addition to [(\[eq:IR-kbd\])]{} with $\theta_0\ll1$, if $$\begin{aligned}
{\label{eq:IR-xbdNN}}
G(x)\leq\lambda_0{\vbx{|\!|\!|}}^{-q}\end{aligned}$$ holds for some $\lambda_0\in[1,\infty)$ and $q\in(0,d)$, then we have for $i\ge0$ $$\begin{aligned}
{\label{eq:pi-kbd}}
\pi_\Lambda^{{\scriptscriptstyle}(i)}(x)\leq O(\theta_0)^i\delta_{o,x}+\frac{\lambda_0^3
(i+1)^{3q+2}O(\theta_0)^{(i-2)\vee0}}{|x|^{3q}}(1-\delta_{o,x}).\end{aligned}$$
First we claim that the assumed bounds in [(\[eq:IR-kbd\])]{} indeed hold for any $p\leq{p_\text{c}}$ if $d>4$ and $\theta_0\ll1$, where $\theta_0=O(d^{-1})$ for the nearest-neighbor model and $\theta_0=O(L^{-d})$ for the spread-out model. The proof is based on the orthodox model-independent bootstrapping argument in, e.g., [@ms93] (see also [@hs02] for improved random-walk estimates; bootstrapping assumptions that are different from, but philosophically similar to, [(\[eq:IR-kbd\])]{} are used in [@hhs?]). Therefore, [(\[eq:pi-sumbd\])]{} holds for $p\leq{p_\text{c}}$ and hence ensures the existence of an infinite-volume limit $\Pi(x)=\lim_{\Lambda\uparrow{{\mathbb Z}^d}}\lim_{j\uparrow\infty}\Pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ that satisfies $$\begin{aligned}
{\label{eq:Pi-bdNN}}
\sum_x|\Pi(x)|=1+O(\theta_0),&& \sum_x|x|^2|\Pi(x)|=d\sigma^2O(\theta_0^2).\end{aligned}$$ As a byproduct, we obtain the identity in [(\[eq:constants\])]{} for $\tau({p_\text{c}})$ for both models. Suppose that $$\begin{aligned}
{\label{eq:IR-xbd-nn}}
G(x)\leq\lambda_0{\vbx{|\!|\!|}}^{-(d-2)}\end{aligned}$$ holds at $p={p_\text{c}}$. Then, by Proposition \[prp:GimpliesPik\], we obtain [(\[eq:pi-kbd\])]{} with $q=d-2$. Using this in [(\[eq:Rj-optSO\])]{}–[(\[eq:Pij-optSO\])]{}, we can prove Proposition \[prp:Pij-Rj-bd\].
To complete the proof, it thus remains to show [(\[eq:IR-xbd-nn\])]{} at $p={p_\text{c}}$. To show this, we use the following proposition:
\[prp:exp-bootstrap\] Let $$\begin{aligned}
{\label{eq:GbarWbar}}
\bar G^{{\scriptscriptstyle}(s)}=\sup_x|x|^sG(x),&& \bar
W^{{\scriptscriptstyle}(t)}=\sup_x\sum_y|y|^tG(y)\,G(x-y),\end{aligned}$$ and suppose that the bounds in [(\[eq:IR-kbd\])]{} hold with $\theta_0\ll1$.
(i) If $\sum_x\Pi(x)=\tau^{-1}$ and $|\Pi(x)|\leq O({\vbx{|\!|\!|}}^{-(d+2)})$, then we have $$\begin{aligned}
{\label{eq:prp-asy}}
G(x)\sim\frac{\sum_x\Pi(x)}{\tau\sum_x|x|^2(D*\Pi)(x)}\,\frac{a_d}
{|x|^{d-2}}\qquad\text{as }|x|\uparrow\infty.\end{aligned}$$
(ii) If $\sum_x|x|^r|\Pi(x)|<\infty$ for some $r>0$, then, for $s,t>0$ which are not odd integers, we have $$\begin{aligned}
\begin{cases}
\bar G^{{\scriptscriptstyle}(s)}<\infty&\text{if}~~~s\leq r~~\text{and}~~s<d-2,\\
\bar W^{{\scriptscriptstyle}(t)}<\infty&\text{if}~~~t\leq\lfloor r\rfloor~~\text{and}
~~t<d-4.
\end{cases}\end{aligned}$$
(iii) If $\bar W^{{\scriptscriptstyle}(t)}<\infty$ for some $t\ge0$, then $\sum_x|x|^{t+2}|\Pi(x)|<\infty$.
The above proposition is a summary of key elements in [@h05 Proposition 1.3 and Lemmas 1.5–1.6] that are sufficient to prove [(\[eq:IR-xbd-nn\])]{} in the current setting. The proofs of Propositions \[prp:exp-bootstrap\](i) and \[prp:exp-bootstrap\](ii) are model-independent and can be found in [@h05 Sections 2 and 4], respectively. The proof of Proposition \[prp:exp-bootstrap\](iii) is similar to that of the first statement of Proposition \[prp:GimpliesPik\]: [(\[eq:IR-kbd\])]{} implies [(\[eq:pi-sumbd\])]{}. We will explain this in Section \[ss:proof-nn\].
Now we continue with the proof of [(\[eq:IR-xbd-nn\])]{}. Fix $p={p_\text{c}}$. Since the asymptotic behavior [(\[eq:prp-asy\])]{} is good enough for the bound [(\[eq:IR-xbd-nn\])]{}, it suffices to check the assumptions of Proposition \[prp:exp-bootstrap\](i). The first assumption on the sum of $\Pi(x)$ is satisfied at $p={p_\text{c}}$, as mentioned below [(\[eq:Pi-bdNN\])]{}. The second assumption is also satisfied if $\bar G^{({{\scriptscriptstyle}\frac{d+2}3})}<\infty$, because of the second statement of Proposition \[prp:GimpliesPik\]: [(\[eq:IR-xbdNN\])]{} implies [(\[eq:pi-kbd\])]{}. By Proposition \[prp:exp-bootstrap\](ii), it thus suffices to show that $\sum_x|x|^{{\scriptscriptstyle}\frac{d+2}3}|\Pi(x)|$ is finite if $d>4$.
To show this, we let $$\begin{aligned}
r_0=2,&& r_{i+1}=\Big((d-2)\wedge\big(\lfloor r_i\rfloor+2\big)
\Big)-{\epsilon},\end{aligned}$$ where $0<{\epsilon}\leq\frac{2}3(d-4)$. Note that, by this definition, $r_i$ for $i\ge1$ equals $((d-2)\wedge(i+3))-{\epsilon}$ and increases until it reaches $d-2-{\epsilon}$. We prove below by induction that $\sum_x|x|^{r_i}|\Pi(x)|$ is finite for all $i\ge0$. This is sufficient for the finiteness of $\sum_x|x|^{{\scriptscriptstyle}\frac{d+2}3}|\Pi(x)|$, since $$\begin{aligned}
\lim_{i\uparrow\infty}r_i=d-2-{\epsilon}\ge d-2-\tfrac{2}3(d-4)
=\tfrac{d+2}3.\end{aligned}$$
Note that, by [(\[eq:Pi-bdNN\])]{}, $\sum_x|x|^{r_0}|\Pi(x)|<\infty$. Suppose $\sum_x|x|^{r_i}|\Pi(x)|<\infty$ for some $i\ge0$. Then, by Proposition \[prp:exp-bootstrap\](ii), $\bar W^{{\scriptscriptstyle}(t)}$ is finite for $t\in(0,\lfloor r_i\rfloor]\cap(0,d-4)$. Since $\lfloor r_0\rfloor=2$ and $\lfloor
r_i\rfloor=(d-3)\wedge(i+2)$ for $i\ge1$, $\bar W^{{\scriptscriptstyle}(T)}$ with $T=(i+2)\wedge(d-4-{\epsilon})$ is finite. Then, by Proposition \[prp:exp-bootstrap\](iii), $\sum_x|x|^{T+2}|\Pi(x)|$ is finite. Since $$\begin{aligned}
T+2=(i+4)\wedge(d-2-{\epsilon})\ge\big((d-2)\wedge(i+4)\big)-{\epsilon}=r_{i+1},\end{aligned}$$ we obtain that $\sum_x|x|^{r_{i+1}}|\Pi(x)|<\infty$. This completes the induction and the proof of [(\[eq:IR-xbd-nn\])]{}. The proof of Proposition \[prp:Pij-Rj-bd\] is now completed.
Diagrammatic bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ {#s:bounds}
=================================================================
In this section, we prove diagrammatic bounds on the expansion coefficients. In Section \[ss:diagram\], we construct diagrams in terms of two-point functions and state the bounds. In Section \[ss:pi0bd\], we prove a key lemma for the diagrammatic bounds and show how to apply this lemma to prove the bound on $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$. In Section \[ss:pijbd\], we prove the bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge1$.
Construction of diagrams {#ss:diagram}
------------------------
To state bounds on the expansion coefficients (as in Proposition \[prp:diagram-bd\] below), we first define diagrammatic functions consisting of two-point functions. Let $$\begin{aligned}
{\label{eq:tildeG-def}}
\tilde G_\Lambda(y,x)
=\sum_{b:{\overline{b}}=x}{{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_\Lambda\tau_b,\end{aligned}$$ which satisfies[^5] $$\begin{aligned}
{\label{eq:G-delta-bd}}
{{\langle \varphi_y\varphi_x \rangle}}_\Lambda\leq\delta_{y,x}+\sum_{b:{\overline{b}}=x}\,
\sum_{\substack{{\partial}{{\bf n}}=y{\vartriangle}x\\ n_b\text{ odd}}}\frac{w_\Lambda
({{\bf n}})}{Z_\Lambda}=\delta_{y,x}+\sum_{b:{\overline{b}}=x}\tau_b\sum_{\substack{
{\partial}{{\bf n}}=y{\vartriangle}{\underline{b}}\\ n_b\text{ even}}}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}
\leq\delta_{y,x}+\tilde G_\Lambda(y,x).\end{aligned}$$ Let $$\begin{aligned}
{\label{eq:psi-def}}
\psi_\Lambda(y,x)=\sum_{j=0}^\infty\big(\tilde G_\Lambda^2\big)^{*j}(y,x)
&\equiv\delta_{y,x}+\sum_{j=1}^\infty\sum_{\substack{u_0,\dots,u_j\\ u_0=
y,\;u_j=x}}\prod_{l=1}^j\tilde G_\Lambda(u_{l-1},u_l)^2,\end{aligned}$$ and define (see the first line in Figure \[fig:P-def\]) $$\begin{aligned}
P_\Lambda^{{\scriptscriptstyle}(1)}(v_1,v'_1)&=2\big(\psi_\Lambda(v_1,v'_1)-
\delta_{v_1,v'_1}\big)\,{{\langle \varphi_{v_1}\varphi_{v'_1} \rangle}}_\Lambda,
{\label{eq:P1-def}}\\[5pt]
P_\Lambda^{{\scriptscriptstyle}(j)}(v_1,v'_j)&=\sum_{\substack{v_2,\dots,v_j\\
v'_1,\dots,v'_{j-1}}}\bigg(\prod_{i=1}^j\big(\psi_\Lambda(v_i,
v'_i)-\delta_{v_i,v'_i}\big)\bigg){{\langle \varphi_{v_1}\varphi_{
v_2} \rangle}}_\Lambda{{\langle \varphi_{v_2}\varphi_{v'_1} \rangle}}_\Lambda{\nonumber}\\
&\qquad\qquad\times\bigg(\prod_{i=2}^{j-1}{{\langle \varphi_{v'_{i
-1}}\varphi_{v_{i+1}} \rangle}}_\Lambda{{\langle \varphi_{v_{i+1}}\varphi_{
v'_i} \rangle}}_\Lambda\bigg){{\langle \varphi_{v'_{j-1}}\varphi_{v'_j} \rangle}}
_\Lambda\qquad(j\ge2),{\label{eq:Pj-def}}\end{aligned}$$ where the empty product for $j=2$ is regarded as 1.
$$\begin{gathered}
P_\Lambda^{{\scriptscriptstyle}(1)}(v_1,v'_1)=\raisebox{-7pt}{\includegraphics[scale=.1]
{P0}}\qquad
P_\Lambda^{{\scriptscriptstyle}(2)}(v_1,v'_2)=\raisebox{-9pt}{\includegraphics[scale=.1]
{P1}}\qquad
P_\Lambda^{{\scriptscriptstyle}(3)}(v_1,v'_3)=\raisebox{-15pt}{\includegraphics[scale=.1]
{P2}}\\[5pt]
P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(1)}}(v_1,v'_1)=\raisebox{-7pt}{\includegraphics
[scale=.1]{P0p}}\hspace{5pc}
P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(1)}}(v_1,v'_1)=\raisebox{-18pt}
{\includegraphics[scale=.1]{P0pp}}+~
\raisebox{-18pt}{\includegraphics[scale=.1]{P0pp2}}\\[5pt]
P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(0)}}(y,x)=\raisebox{-12pt}{\includegraphics
[scale=.1]{P0prime}}\hspace{5pc}
P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(0)}}(y,x)=\raisebox{-18pt}
{\includegraphics[scale=.1]{P0primeprime}}\end{gathered}$$
Next, we define $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(j)}}(v_1,v'_j)$ by replacing one of the $2j-1$ two-point functions on the right-hand side of [(\[eq:P1-def\])]{}–[(\[eq:Pj-def\])]{} by the product of *two* two-point functions, such as replacing ${{\langle \varphi_z\varphi_{z'} \rangle}}_\Lambda$ by ${{\langle \varphi_z\varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_{z'} \rangle}}_\Lambda$, and then summing over all $2j-1$ choices of this replacement. For example, we define (see the second line in Figure \[fig:P-def\]) $$\begin{aligned}
{\label{eq:P'1-def}}
P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(1)}}(v_1,v'_1)=2\big(\psi_\Lambda(v_1,v'_1)-
\delta_{v_1,v'_1}\big){{\langle \varphi_{v_1}\varphi_u \rangle}}_\Lambda{{\langle \varphi_u
\varphi_{v'_1} \rangle}}_\Lambda,\end{aligned}$$ and $$\begin{aligned}
P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(2)}}(v_1,v'_2)=\sum_{v_2,v'_1}\bigg(\prod_{i
=1}^2\big(\psi_\Lambda(v_i,v'_i)-\delta_{v_i,v'_i}\big)\bigg)\Big({{\langle
\varphi_{v_1}\varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_{v_2} \rangle}}_\Lambda
{{\langle \varphi_{v_2}\varphi_{v'_1} \rangle}}_\Lambda{{\langle \varphi_{v'_1}\varphi_{
v'_2} \rangle}}_\Lambda&{\nonumber}\\
+{{\langle \varphi_{v_1}\varphi_{v_2} \rangle}}_\Lambda{{\langle \varphi_{v_2}\varphi_u
\rangle}}_\Lambda{{\langle \varphi_u\varphi_{v'_1} \rangle}}_\Lambda{{\langle \varphi_{v'_1}
\varphi_{v'_2} \rangle}}_\Lambda&{\nonumber}\\[7pt]
+{{\langle \varphi_{v_1}\varphi_{v_2} \rangle}}_\Lambda{{\langle \varphi_{v_2}\varphi_{
v'_1} \rangle}}_\Lambda{{\langle \varphi_{v'_1}\varphi_u \rangle}}_\Lambda{{\langle \varphi_u
\varphi_{v'_2} \rangle}}_\Lambda&\Big).\end{aligned}$$
We define $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(j)}}(v_1,v'_j)$ similarly as follows. First we take *two* two-point functions in $P_\Lambda^{{\scriptscriptstyle}(j)}(v_1,v'_j)$, one of which (say, ${{\langle \varphi_{z_1}\varphi_{z'_1} \rangle}}_\Lambda$ for some $z_1,z'_1$) is among the aforementioned $2j-1$ two-point functions, and the other (say, $\tilde G_\Lambda(z_2,z'_2)$ for some $z_2,z'_2$) is among those of which $\psi_\Lambda(v_i,v'_i)-\delta_{v_i,v'_i}$ for $i=1,\dots,j$ are composed. The product ${{\langle \varphi_{z_1}\varphi_{z'_1} \rangle}}_\Lambda\tilde
G_\Lambda(z_2,z'_2)$ is then replaced by $$\begin{aligned}
&\bigg(\sum_{v'}{{\langle \varphi_{z_1}\varphi_{v'} \rangle}}_\Lambda{{\langle \varphi_{v'}
\varphi_{z'_1} \rangle}}_\Lambda\,\psi_\Lambda(v',v)\bigg)\Big({{\langle \varphi_{z_2}
\varphi_u \rangle}}_\Lambda\tilde G_\Lambda(u,z'_2)+\tilde G_\Lambda(z_2,z'_2)
\,\delta_{u,z'_2}\Big){\nonumber}\\
&+{{\langle \varphi_{z_1}\varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_{z'_1}
\rangle}}_\Lambda\sum_{v'}\Big({{\langle \varphi_{z_2}\varphi_{v'} \rangle}}_\Lambda\tilde
G_\Lambda(v',z'_2)+\tilde G_\Lambda(z_2,z'_2)\,\delta_{v',z'_2}\Big)
\,\psi_\Lambda(v',v).\end{aligned}$$ Finally, we define $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(j)}}(v_1,v'_j)$ by taking account of all possible combinations of ${{\langle \varphi_{z_1} \varphi_{z'_1} \rangle}}_\Lambda$ and $\tilde
G_\Lambda(z_2,z'_2)$. For example, we define $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(1)}}(v_1,v'_1)$ as (see Figure \[fig:P-def\]) $$\begin{aligned}
{\label{eq:P''1-def}}
P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(1)}}(v_1,v'_1){\nonumber}\\
=\sum_{u',u'',v'}\bigg(&2\psi_\Lambda(v_1,u')\,\tilde G_\Lambda(u',u'')
\Big({{\langle \varphi_{u'}\varphi_u \rangle}}_\Lambda\tilde G_\Lambda(u,u'')+\tilde
G_\Lambda(u',u'')\,\delta_{u,u''}\Big)\,\psi_\Lambda(u'',v'_1){\nonumber}\\
&\times{{\langle \varphi_{v_1}\varphi_{v'} \rangle}}_\Lambda{{\langle \varphi_{v'}\varphi_{
v'_1} \rangle}}_\Lambda\psi_\Lambda(v',v)+(\text{permutation of $u$ and }v')
\bigg),\end{aligned}$$ where the permutation term corresponds to the second term for $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(1)}}(v_1,v'_1)$ in Figure \[fig:P-def\].
In addition to the above quantities, we define (see the third line in Figure \[fig:P-def\]) $$\begin{aligned}
P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(0)}}(y,x)&={{\langle \varphi_y\varphi_x \rangle}}_\Lambda^2
{{\langle \varphi_y\varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_x \rangle}}_\Lambda,
{\label{eq:P'0-def}}\\[5pt]
P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(0)}}(y,x)&={{\langle \varphi_y\varphi_x \rangle}}
_\Lambda{{\langle \varphi_y\varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_x \rangle}}
_\Lambda\sum_{v'}{{\langle \varphi_y\varphi_{v'} \rangle}}_\Lambda{{\langle \varphi_{v'}
\varphi_x \rangle}}_\Lambda\,\psi_\Lambda(v',v),{\label{eq:P''0-def}}\end{aligned}$$ and let $$\begin{aligned}
{\label{eq:P'P''-def}}
P'_{\Lambda;u}(y,x)=\sum_{j\ge0}P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(j)}}(y,x),&&
P''_{\Lambda;u,v}(y,x)&=\sum_{j\ge0}P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(j)}}(y,x),\end{aligned}$$ where $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(0)}}(y,x)$ and $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(0)}}(y,x)$ are the leading contributions to $P'_{\Lambda;u}(y,x)$ and $P''_{\Lambda;u,v}(y,x)$, respectively.
Finally, we define $$\begin{aligned}
Q'_{\Lambda;u}(y,x)&=\sum_z\big(\delta_{y,z}+\tilde G_\Lambda(y,z)\big)
P'_{\Lambda;u}(z,x),{\label{eq:Q'-def}}\\
Q''_{\Lambda;u,v}(y,x)&=\sum_z\big(\delta_{y,z}+\tilde G_\Lambda(y,z)
\big)P''_{\Lambda;u,v}(z,x){\nonumber}\\
&\quad+\sum_{v',z}\big(\delta_{y,v'}+\tilde G_\Lambda(y,v')\big)\,\tilde
G_\Lambda(v',z)\,P'_{\Lambda;u}(z,x)\,\psi_\Lambda(v',v).{\label{eq:Q''-def}}\end{aligned}$$
The following are the diagrammatic bounds on the expansion coefficients (see Figure \[fig:piN-bd\]):
\[prp:diagram-bd\] For the ferromagnetic Ising model, we have $$\begin{aligned}
\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)\leq
\begin{cases}{\label{eq:piNbd}}
P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(0)}}(o,x)\equiv{{\langle \varphi_o\varphi_x \rangle}}_\Lambda^3
&(j=0),\\[5pt]
{\displaystyle}\sum_{\substack{b_1,\dots,b_j\\ v_1,\dots,v_j}}P_{\Lambda;v_1}^{\prime
{{\scriptscriptstyle}(0)}}(o,{\underline{b}}_1)\,\bigg(\prod_{i=1}^{j-1}\tau_{b_i}Q''_{\Lambda;v_i,v_{
i+1}}({\overline{b}}_i,{\underline{b}}_{i+1})\bigg)\,\tau_{b_j}Q'_{\Lambda;v_j}({\overline{b}}_j,x)&(j\ge1),
\end{cases}\end{aligned}$$ where, as well as in the rest of the paper, the empty product is regarded as 1 by convention.
$$\begin{aligned}
\pi^{{\scriptscriptstyle}(1)}_\Lambda(x)\lesssim\raisebox{-11pt}{\includegraphics[scale=
0.18]{pi1}}\qquad
\pi^{{\scriptscriptstyle}(2)}_\Lambda(x)\lesssim\raisebox{-20pt}{\includegraphics[scale=
0.18]{pi21}}+\raisebox{-20pt}{\includegraphics[scale=0.18]{pi22}}\end{aligned}$$
Bound on $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$ {#ss:pi0bd}
---------------------------------------------------
The key ingredient of the proof of Proposition \[prp:diagram-bd\] is Lemma \[lmm:GHS-BK\] below, which is an extension of the GHS idea used in the proof of Lemma \[lmm:switching\]. In this subsection, we demonstrate how this extension works to prove the bound on $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$ and the inequality $$\begin{aligned}
{\label{eq:pi0'-bd}}
\sum_{{\partial}{{\bf n}}=o{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}
x\\}$}}}\,\cap\,\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}y}\leq P_{\Lambda;y}^{\prime{{\scriptscriptstyle}(0)}}(o,x),\end{aligned}$$ which will be used in Section \[ss:pijbd\] to obtain the bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge1$.
Since the inequality is trivial if $x=o$, we restrict our attention to the case of $x\ne o$.
First we note that, for each current configuration ${{\bf n}}$ with ${\partial}{{\bf n}}=\{o,x\}$ and ${\mathbbm{1}{\scriptstyle\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}}}=1$, there are at least *three edge-disjoint* paths on ${{\mathbb G}}_{{\bf n}}$ between $o$ and $x$. See, for example, the first term on the right-hand side in Figure \[fig:1stpiv\]. Suppose that the thick line in that picture, referred to as $\zeta_1$ and split into $\zeta_{11}{\:\Dot{\cup}\:}\zeta_{12}{\:\Dot{\cup}\:}\zeta_{13}$ from $o$ to $x$, consists of bonds $b$ with $n_b=1$, and that the thin lines, referred to as $\zeta_2$ and $\zeta_3$ that terminate at $o$ and $x$ respectively, consist of bonds $b'$ with $n_{b'}=2$. Let $\zeta'_i$, for $i=2,3$, be the duplication of $\zeta_i$. Then, the three paths $\zeta_2{\:\Dot{\cup}\:}\zeta_{13}$, $\zeta'_2{\:\Dot{\cup}\:}\zeta_{12}{\:\Dot{\cup}\:}\zeta_3$ and $\zeta_{11}{\:\Dot{\cup}\:}\zeta'_3$ are edge-disjoint.
Then, by multiplying $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$ by *two* dummies $(Z_\Lambda/Z_\Lambda)^2\,(\equiv1)$, we obtain $$\begin{aligned}
{\label{eq:pi0*Z2}}
\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)&=\sum_{\substack{{\partial}{{\bf n}}=\{o,x\}\\ {\partial}{{\bf m}}'={\partial}{{\bf m}}''
={\varnothing}}}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,\frac{w_\Lambda({{\bf m}}')}{Z_\Lambda}
\,\frac{w_\Lambda({{\bf m}}'')}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}$}}}{\nonumber}\\
&=\sum_{{\partial}{{\bf N}}=\{o,x\}}\frac{w_\Lambda({{\bf N}})}{Z_\Lambda^3}\sum_{\substack{
{\partial}{{\bf n}}=\{o,x\}\\ {\partial}{{\bf m}}'={\partial}{{\bf m}}''={\varnothing}\\ {{\bf N}}\equiv{{\bf n}}+{{\bf m}}'+{{\bf m}}''}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}$}}}\prod_b\frac{N_b!}{n_b!\;m'_b!\;m''_b!},\end{aligned}$$ where the sum over ${{\bf n}},{{\bf m}}',{{\bf m}}''$ in the second line equals the cardinality of the following set of partitions: $$\begin{aligned}
{\label{eq:Ssubset}}
{\mathfrak{S}}_0=\bigg\{({{\mathbb S}}_0,{{\mathbb S}}_1,{{\mathbb S}}_2):{{\mathbb G}}_{{\bf N}}={\mathop{\Dot{\bigcup}}}_{i=0,1,2}{{\mathbb S}}_i,\;
{\partial}{{\mathbb S}}_0=\{o,x\},\;{\partial}{{\mathbb S}}_1={\partial}{{\mathbb S}}_2={\varnothing},\;o{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}}
{\overset{}{\Longleftrightarrow}}}x\text{ in }
{{\mathbb S}}_0\bigg\},\end{aligned}$$ where “$o{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}}
{\overset{}{\Longleftrightarrow}}}x$ in ${{\mathbb S}}_0$” means that there are at least two *bond*-disjoint paths in ${{\mathbb S}}_0$. We will show $|{\mathfrak{S}}_0|\leq|{\mathfrak{S}}'_0|$, where $$\begin{aligned}
{\label{eq:Ssupset}}
{\mathfrak{S}}'_0=\bigg\{({{\mathbb S}}_0,{{\mathbb S}}_1,{{\mathbb S}}_2):{{\mathbb G}}_{{\bf N}}={\mathop{\Dot{\bigcup}}}_{i=0,1,2}{{\mathbb S}}_i,\;
{\partial}{{\mathbb S}}_0={\partial}{{\mathbb S}}_1={\partial}{{\mathbb S}}_2=\{o,x\}\bigg\}.\end{aligned}$$ This implies [(\[eq:piNbd\])]{} for $j=0$, because $$\begin{aligned}
|{\mathfrak{S}}'_0|=\sum_{\substack{{\partial}{{\bf n}}={\partial}{{\bf m}}'={\partial}{{\bf m}}''=\{o,x\}\\ {{\bf N}}\equiv
{{\bf n}}+{{\bf m}}'+{{\bf m}}''}}\prod_b\frac{N_b!}{n_b!\,m'_b!\,
m''_b!},\end{aligned}$$ and $$\begin{aligned}
\sum_{{\partial}{{\bf N}}=\{o,x\}}\frac{w_\Lambda({{\bf N}})}{Z_\Lambda^3}\sum_{\substack{
{\partial}{{\bf n}}={\partial}{{\bf m}}'={\partial}{{\bf m}}''=\{o,x\}\\ {{\bf N}}\equiv{{\bf n}}+{{\bf m}}'+{{\bf m}}''}}\prod_b
\frac{N_b!}{n_b!\;m'_b!\;m''_b!}=\bigg(\sum_{{\partial}{{\bf n}}=\{
o,x\}}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\bigg)^3.\end{aligned}$$
It remains to show $|{\mathfrak{S}}_0|\leq|{\mathfrak{S}}'_0|$. To do so, we use the following lemma, in which we denote by $\Omega_{z\to z'}^{{{\bf N}}}$ the set of paths on ${{\mathbb G}}_{{\bf N}}$ from $z$ to $z'$ and write $\omega\cap\omega'={\varnothing}$ to mean that $\omega$ and $\omega'$ are *edge*-disjoint (not necessarily *bond*-disjoint).
\[lmm:GHS-BK\] Given a current configuration ${{\bf N}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_\Lambda}$, $k\ge1$, ${{\cal V}}\subset\Lambda$ and $z_i\ne z'_i\in\Lambda$ for $i=1,\dots,k$, we let $$\begin{aligned}
{\label{eq:fS-gen}}
{\mathfrak{S}}=\left\{({{\mathbb S}}_0,{{\mathbb S}}_1,\dots,{{\mathbb S}}_k):
\begin{array}{r}
{{\mathbb G}}_{{\bf N}}={\mathop{\Dot{\bigcup}}}_{i=0}^{\raisebox{-3pt}{$\scriptstyle k$}}
{{\mathbb S}}_i,\;{\partial}{{\mathbb S}}_0={{\cal V}},\;{\partial}{{\mathbb S}}_i={\varnothing}~(i=1,\dots,k),\;\\
{{}^\exists}\omega_i\in\Omega^{{\bf N}}_{z_i\to z'_i}~(i=1,\dots,k)~
\text{\rm such that }\omega_i\subset{{\mathbb S}}_0{\:\Dot{\cup}\:}{{\mathbb S}}_i\\
\text{\rm and }\omega_i\cap\omega_j={\varnothing}~(i\ne j)
\end{array}\right\},\end{aligned}$$ and define ${\mathfrak{S}}'$ to be the right-hand side of [(\[eq:fS-gen\])]{} with “${\partial}{{\mathbb S}}_0={{\cal V}}$, ${\partial}{{\mathbb S}}_i={\varnothing}$” being replaced by “${\partial}{{\mathbb S}}_0={{\cal V}}{\,\triangle\,}\{z_1,z'_1\}{\,\triangle\,}\cdots{\,\triangle\,}\{z_k,z'_k\}$, ${\partial}{{\mathbb S}}_i=\{z_i,z'_i\}$”. Then, $|{\mathfrak{S}}|=|{\mathfrak{S}}'|$.
We will prove this lemma at the end of this subsection.
Now we use Lemma \[lmm:GHS-BK\] with $k=2$ and ${{\cal V}}=\{z_1,z'_1\}=\{z_2,z'_2\}=\{o,x\}$. Note that ${\mathfrak{S}}_0$ in [(\[eq:Ssubset\])]{} is a subset of ${\mathfrak{S}}$, since ${\mathfrak{S}}$ includes partitions $({{\mathbb S}}_0,{{\mathbb S}}_1,{{\mathbb S}}_2)$ in which there does not exist two *bond*-disjoint paths on ${{\mathbb S}}_0$. In addition, ${\mathfrak{S}}'$ is trivially a subset of ${\mathfrak{S}}'_0$ in [(\[eq:Ssupset\])]{}. Therefore, we have $|{\mathfrak{S}}_0|\leq|{\mathfrak{S}}'_0|$. This completes the proof of [(\[eq:piNbd\])]{} for $j=0$.
Here, we summarize the basic steps that we have followed to bound $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$ and which we generalize to prove [(\[eq:pi0’-bd\])]{} below and the bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge1$ in Section \[sss:dbconn\].
(i) Count the (minimum) number, say, $k+1$, of *edge-disjoint* paths on ${{\mathbb G}}_{{\bf n}}$ that satisfy the source constraint (as well as other additional conditions, if there are) of the considered function $f(x)$. For example, $k=2$ for $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)\equiv\frac1{Z_\Lambda}\sum_{{\partial}{{\bf n}}=\{o,x\}}
w_\Lambda({{\bf n}})\,{\mathbbm{1}{\scriptstyle\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}}}$.
(ii) Multiply $f(x)$ by $(\frac{Z_\Lambda}{Z_\Lambda})^k=\prod_{i=1}^k
(\frac1{Z_\Lambda}\sum_{{\partial}{{\bf m}}_i={\varnothing}}w_\Lambda({{\bf m}}_i)) \,(\equiv1)$ and then overlap the $k$ dummies ${{\bf m}}_1,\dots,{{\bf m}}_k$ on the original current configuration ${{\bf n}}$. Choose $k$ paths $\omega_1,\dots,\omega_k$ among $k+1$ edge-disjoint paths on ${{\mathbb G}}_{{{\bf n}}+\sum_{i=1}^k{{\bf m}}_i}$.
(iii) Use Lemma \[lmm:GHS-BK\] to exchange the occupation status of edges on $\omega_i$ between ${{\mathbb G}}_{{\bf n}}$ and ${{\mathbb G}}_{{{\bf m}}_i}$ for every $i=1,\dots,k$. The current configurations after the mapping, denoted by $\tilde{{\bf n}},\tilde{{\bf m}}_1,\dots,\tilde{{\bf m}}_k$, satisfy ${\partial}\tilde{{\bf n}}={\partial}{{\bf n}}{\vartriangle}{\partial}\omega_1{\vartriangle}\cdots{\vartriangle}{\partial}\omega_k$ and ${\partial}\tilde{{\bf m}}_i={\partial}\omega_i$ for $i=1,\dots,k$.
If $y=o$ or $x$, then [(\[eq:pi0’-bd\])]{} is reduced to the inequality for $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$. Also, if $y\ne o=x$, then the left-hand side of [(\[eq:pi0’-bd\])]{} multiplied by $Z_\Lambda/Z_\Lambda=\sum_{{\partial}{{\bf m}}={\varnothing}}w_\Lambda({{\bf m}})/Z_\Lambda\equiv1$ equals $$\begin{aligned}
{\label{eq:dbbd}}
\sum_{{\partial}{{\bf n}}={\partial}{{\bf m}}={\varnothing}}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,\frac{w_\Lambda
({{\bf m}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}y\}$}}}&\leq\sum_{{\partial}{{\bf n}}={\partial}{{\bf m}}={\varnothing}}\frac{
w_\Lambda({{\bf n}})}{Z_\Lambda}\,\frac{w_\Lambda({{\bf m}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}+{{\bf m}}$}}
{\overset{}{\longleftrightarrow}}}y\}$}}}{\nonumber}\\
&=\sum_{{\partial}{{\bf n}}={\partial}{{\bf m}}=\{o,y\}}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,\frac{
w_\Lambda({{\bf m}})}{Z_\Lambda}\;={{\langle \varphi_o\varphi_y \rangle}}_\Lambda^2,\end{aligned}$$ where the first equality is due to Lemma \[lmm:switching\]. Therefore, we can assume $o\ne x\ne y\ne o$.
We follow the three steps described above.
\(i) Since $y\notin{\partial}{{\bf n}}=\{o,x\}$ and ${\mathbbm{1}{\scriptstyle\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\\}}}\,\cap\,\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}y}=1$, it is not hard to see that there is an *edge*-disjoint cycle (closed path) $o\to
y\to x\to o$. Since a cycle does not have a source, there must be another edge-disjoint connection from $o$ to $x$, due to the source constraint ${\partial}{{\bf n}}=\{o,x\}$. Therefore, there are at least $4\,(=k+1)$ edge-disjoint paths on ${{\mathbb G}}_{{\bf n}}$: one is between $o$ and $y$, another is between $y$ and $x$, and the other two are between $o$ and $x$.
\(ii) Multiplying both sides of [(\[eq:pi0’-bd\])]{} by $(Z_\Lambda/Z_\Lambda)^3$ is equivalent to $$\begin{aligned}
{\label{eq:pi0'-equiv}}
&\sum_{{\partial}{{\bf N}}=\{o,x\}}\frac{w_\Lambda({{\bf N}})}{Z_\Lambda^4}\sum_{\substack{
{\partial}{{\bf n}}=\{o,x\}\\ {\partial}{{\bf m}}_i={\varnothing}~{{}^\forall}i=1,2,3\\ {{\bf N}}={{\bf n}}+\sum_{i=1}^3
{{\bf m}}_i}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\\}$}}}\,\cap\,\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}y}\prod_b
\frac{N_b!}{n_b!\;m^{{\scriptscriptstyle}(1)}_b!\;m^{{\scriptscriptstyle}(2)}_b!\;m^{{\scriptscriptstyle}(3)}_b!}{\nonumber}\\
&\quad\leq\sum_{{\partial}{{\bf N}}=\{o,x\}}\frac{w_\Lambda({{\bf N}})}{Z_\Lambda^4}\sum_{
\substack{{\partial}{{\bf n}}={\partial}{{\bf m}}_3=\{o,x\}\\ {\partial}{{\bf m}}_1=\{o,y\},~{\partial}{{\bf m}}_2=\{y,x\}\\
{{\bf N}}={{\bf n}}+\sum_{i=1}^3{{\bf m}}_i}}\prod_b\frac{N_b!}{n_b!\;
m^{{\scriptscriptstyle}(1)}_b!\;m^{{\scriptscriptstyle}(2)}_b!\;m^{{\scriptscriptstyle}(3)}_b!},\end{aligned}$$ where we have used the notation $m_b^{{\scriptscriptstyle}(i)}={{\bf m}}_i|_b$. Note that the second sum on the left-hand side equals the cardinality of $$\begin{aligned}
{\label{eq:S03sub}}
\bigg\{({{\mathbb S}}_0,{{\mathbb S}}_1,{{\mathbb S}}_2,{{\mathbb S}}_3):
\begin{array}{c}
{{\mathbb G}}_{{\bf N}}={\mathop{\Dot{\bigcup}}}_{i=0}^{\raisebox{-3pt}{$\scriptstyle3$}}
{{\mathbb S}}_i,\;{\partial}{{\mathbb S}}_0=\{o,x\},\;{\partial}{{\mathbb S}}_1={\partial}{{\mathbb S}}_2={\partial}{{\mathbb S}}_3={\varnothing}\\
o{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}}
{\overset{}{\Longleftrightarrow}}}x\text{ in }{{\mathbb S}}_0,~o{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}}
{\overset{}{\longleftrightarrow}}}y\text{ in }{{\mathbb S}}_0
\end{array}\bigg\},\end{aligned}$$ and the second sum on the right-hand side of [(\[eq:pi0’-equiv\])]{} equals the cardinality of $$\begin{aligned}
{\label{eq:S03sup}}
\textstyle\Big\{({{\mathbb S}}_0,{{\mathbb S}}_1,{{\mathbb S}}_2,{{\mathbb S}}_3):{{\mathbb G}}_{{\bf N}}={\mathop{\Dot{\bigcup}}}_{i=0}^{
\raisebox{-3pt}{$\scriptstyle3$}}{{\mathbb S}}_i,\;{\partial}{{\mathbb S}}_0={\partial}{{\mathbb S}}_3=\{o,x\},\;{\partial}{{\mathbb S}}_1=\{o,y\},\;{\partial}{{\mathbb S}}_2=\{y,x\}\Big\}.\end{aligned}$$ Therefore, to prove [(\[eq:pi0’-equiv\])]{}, it is sufficient to show that the cardinality of [(\[eq:S03sub\])]{} is not bigger than that of [(\[eq:S03sup\])]{}.
\(iii) Now we use Lemma \[lmm:GHS-BK\] with $k=3$ and ${{\cal V}}=\{z_3,z'_3\}=\{o,x\}$, $\{z_1,z'_1\}=\{o,y\}$ and $\{z_2,z'_2\}=\{y,x\}$. Since [(\[eq:S03sub\])]{} is a subset of ${\mathfrak{S}}$ in the current setting, while ${\mathfrak{S}}'$ is a subset of [(\[eq:S03sup\])]{}, we obtain [(\[eq:pi0’-equiv\])]{}. This completes the proof of [(\[eq:pi0’-bd\])]{}.
We prove Lemma \[lmm:GHS-BK\] by decomposing ${\mathfrak{S}}^{(\prime)}$ into ${\mathop{\Dot{\bigcup}}}_{\vec\omega_k}{\mathfrak{S}}_{\vec\omega_k}^{(\prime)}$ (described in detail below) and then constructing a bijection from ${\mathfrak{S}}_{\vec\omega_k}$ to ${\mathfrak{S}}'_{\vec\omega_k}$ for every $\vec\omega_k$. To do so, we first introduce some notation.
1. For every $i=1,\dots,k$, we introduce an arbitrarily fixed order among elements in $\Omega_{z_i\to z'_i}^{{\bf N}}$. For $\omega,\omega'\in\Omega_{z_i\to z'_i}^{{\bf N}}$, we write $\omega\prec\omega'$ if $\omega$ is earlier than $\omega'$ in this order. Let $\tilde\Omega_{z_1\to z'_1}^{{\bf N}}$ be the set of paths $\zeta\in\Omega_{z_1\to z'_1}^{{\bf N}}$ such that there are $k-1$ edge-disjoint paths on ${{\mathbb G}}_{{{\bf N}}}\setminus\zeta$ (= the resulting graph by removing the edges in $\zeta$) each of which connects $z_i$ and $z'_i$ for every $i=2,\dots,k$.
2. Then, for $\omega_1\in\tilde\Omega_{z_1\to z'_1}^{{\bf N}}$, we define $\Xi_{z_2\to z'_2}^{{{\bf N}};\omega_1}$ to be the set of paths $\zeta\in\Omega_{z_2\to z'_2}^{{\bf N}}$ on ${{\mathbb G}}_{{\bf N}}\setminus\omega_1$ such that $\zeta\not\supset\xi$ for any $\xi\in\tilde\Omega_{z_1\to
z'_1}^{{\bf N}}$ earlier than $\omega_1$. Then, we define $\tilde\Omega_{z_2\to z'_2}^{{{\bf N}};\omega_1}$ to be the set of paths $\zeta\in\Xi_{z_2\to z'_2}^{{{\bf N}};\omega_1}$ such that there are $k-2$ edge-disjoint paths on ${{\mathbb G}}_{{{\bf N}}}\setminus(\omega_1{\:\Dot{\cup}\:}\zeta)$ each of which is from $z_i$ to $z'_i$ for $i=3,\dots,k$.
3. More generally, for $l<k$ and $\vec\omega_l=(\omega_1,
\dots,\omega_l)$ with $\omega_1\in\tilde\Omega_{z_1\to z'_1}^{{\bf N}}$, $\omega_2\in\tilde\Omega_{z_2\to z'_2}^{{{\bf N}};\omega_1},\dots$, $\omega_l\in\tilde\Omega_{z_l\to z'_l}^{{{\bf N}};\vec\omega_{l-1}}$, we define $\Xi_{z_{l+1}\to z'_{l+1}}^{{{\bf N}};\vec\omega_l}$ to be the set of paths $\zeta\in\Omega_{z_{l+1}\to z'_{l+1}}^{{\bf N}}$ on ${{\mathbb G}}_{{\bf N}}\setminus{\mathop{\Dot{\bigcup}}}_{i=1}^{\raisebox{-3pt}{$\scriptstyle l$}}
\omega_i$ such that $\zeta\not\supset\xi$ for any $\xi\in\tilde\Omega_{z_i\to z'_i}^{{{\bf N}};\vec\omega_{i-1}}$ earlier than $\omega_i$, for every $i=1,\dots,l$. Then, we define $\tilde\Omega_{z_{l+1}\to z'_{l+1}}^{{{\bf N}};\vec\omega_l}$ to be the set of paths $\zeta\in\Xi_{z_{l+1}\to z'_{l+1}}^{{{\bf N}};\vec\omega_l}$ such that there are $k-(l+1)$ edge-disjoint paths on ${{\mathbb G}}_{
{{\bf N}}}\setminus({\mathop{\Dot{\bigcup}}}_{i=1}^{\raisebox{-3pt}{$\scriptstyle
l$}}\omega_i {\:\Dot{\cup}\:}\zeta)$ each of which is from $z_i$ to $z'_i$ for $i=l+2,\dots,k$.
4. If $l=k-1$, then we simply define $\tilde\Omega_{z_k\to z'_k}^{{{\bf N}};\vec
\omega_{k-1}}=\Xi_{z_k\to z'_k}^{{{\bf N}};\vec\omega_{k-1}}$. We will also abuse the notation to denote $\tilde\Omega_{z_1\to z'_1}^{{\bf N}}$ by $\tilde\Omega_{z_1\to z'_1}^{{{\bf N}};\vec\omega_0}$.
Using the above notation, we can decompose ${\mathfrak{S}}^{(\prime)}$ disjointly as follows. For a collection $\omega_i\in\tilde\Omega_{z_i\to z'_i}^{{{\bf N}};\vec\omega_{i-1}}$ for $i=1,\dots,k$, we denote by ${\mathfrak{S}}_{\vec\omega_k}^{(\prime)}$ the set of partitions $\vec{{\mathbb S}}_k\equiv({{\mathbb S}}_0,{{\mathbb S}}_1,\dots,{{\mathbb S}}_k)\in{\mathfrak{S}}^{(\prime)}$ such that, for every $i=1,\dots,k$, the earliest element of $\tilde\Omega_{z_i\to
z'_i}^{{{\bf N}};\vec\omega_{i-1}}$ contained in ${{\mathbb S}}_0{\:\Dot{\cup}\:}{{\mathbb S}}_i$ is $\omega_i$. Then, ${\mathfrak{S}}^{(\prime)}$ is decomposed as $$\begin{aligned}
{\label{eq:SS'-dec}}
{\mathfrak{S}}^{(\prime)}={\mathop{\Dot{\bigcup}}}_{\omega_1\in\tilde\Omega_{z_1\to z'_1}^{{\bf N}}}\,
{\mathop{\Dot{\bigcup}}}_{\omega_2\in\tilde\Omega_{z_2\to z'_2}^{{{\bf N}};\omega_1}}\cdots
{\mathop{\Dot{\bigcup}}}_{\omega_k\in\tilde\Omega_{z_k\to z'_k}^{{{\bf N}};\vec\omega_{k-1}}}
{\mathfrak{S}}_{\vec\omega_k}^{(\prime)}.\end{aligned}$$
To complete the proof of Lemma \[lmm:GHS-BK\], it suffices to construct a bijection from ${\mathfrak{S}}_{\vec\omega_k}$ to ${\mathfrak{S}}'_{\vec\omega_k}$ for every $\vec\omega_k$. For $\vec{{\mathbb S}}_k\in{\mathfrak{S}}_{\vec\omega_k}$, we define $$\begin{aligned}
{\label{eq:Fdef}}
\textstyle\vec F_{\vec\omega_k}(\vec{{\mathbb S}}_k)\equiv\big(F_{\vec\omega_k}^{
{\scriptscriptstyle}(0)}({{\mathbb S}}_0),\dots,F_{\vec\omega_k}^{{\scriptscriptstyle}(k)}({{\mathbb S}}_k)\big)=\Big({{\mathbb S}}_0
{\,\triangle\,}{\mathop{\Dot{\bigcup}}}_{i=1}^{\raisebox{-3pt}{$\scriptstyle k$}}\omega_i,\,{{\mathbb S}}_1
{\,\triangle\,}\omega_1,\dots,{{\mathbb S}}_k{\,\triangle\,}\omega_k\Big),\end{aligned}$$ where ${\partial}F_{\vec\omega_k}^{{\scriptscriptstyle}(0)}({{\mathbb S}}_0)={{\cal V}}{\,\triangle\,}\{z_1,z'_1\}
{\,\triangle\,}\cdots{\,\triangle\,}\{z_k,z'_k\}$ and ${\partial}F_{\vec\omega_k}^{{\scriptscriptstyle}(i)}({{\mathbb S}}_i)= \{z_i,z'_i\}$ for $i=1,\dots,k$. Note that, by definition using symmetric difference, we have $\vec
F_{ \vec\omega_k}(\vec F_{\vec\omega_k}(\vec{{\mathbb S}}_k))=\vec{{\mathbb S}}_k$. Also, by simple combinatorics using $\omega_i\cap\omega_j={{\mathbb S}}_i\cap{{\mathbb S}}_j={\varnothing}$ and $\omega_j\subset{{\mathbb S}}_0{\:\Dot{\cup}\:}{{\mathbb S}}_j$ for $1\leq j\leq k$ and $i\ne j$, we have $$\begin{aligned}
{\label{eq:F0DcupFi}}
F_{\vec\omega_k}^{{\scriptscriptstyle}(i)}({{\mathbb S}}_i)\cap F_{\vec\omega_k}^{{\scriptscriptstyle}(j)}({{\mathbb S}}_j)=
{\varnothing},&&
F_{\vec\omega_k}^{{\scriptscriptstyle}(0)}({{\mathbb S}}_0){\:\Dot{\cup}\:}F_{\vec\omega_k}^{{\scriptscriptstyle}(j)}
({{\mathbb S}}_j)\textstyle=\Big({{\mathbb S}}_0{\,\triangle\,}{\mathop{\Dot{\bigcup}}}_{i\ne j}\omega_i\Big){\:\Dot{\cup}\:}{{\mathbb S}}_j.\end{aligned}$$ Since $\omega_j\subset{{\mathbb S}}_0{\:\Dot{\cup}\:}{{\mathbb S}}_j$ and $\omega_j\cap{\mathop{\Dot{\bigcup}}}_{i\ne j}\omega_i={\varnothing}$, we have $\omega_j\subset F_{\vec\omega_k}^{{\scriptscriptstyle}(0)}({{\mathbb S}}_0){\:\Dot{\cup}\:}F_{\vec\omega_k}^{{\scriptscriptstyle}(j)} ({{\mathbb S}}_j)$.
It remains to show that $\omega_j$ is the earliest element of $\tilde\Omega_{z_j\to
z'_j}^{{{\bf N}};\vec\omega_{j-1}}$ in $F_{\vec\omega_k}^{{\scriptscriptstyle}(0)}({{\mathbb S}}_0){\:\Dot{\cup}\:}F_{\vec\omega_k}^{{\scriptscriptstyle}(j)} ({{\mathbb S}}_j)$. To see this, we first recall that $\tilde\Omega_{z_j\to z'_j}^{{{\bf N}};\vec \omega_{j-1}}$ is a set of paths on ${{\mathbb G}}_{{\bf N}}\setminus{\mathop{\Dot{\bigcup}}}_{i<j} \omega_i$, so that its earliest element contained in $({{\mathbb S}}_0{\,\triangle\,}{\mathop{\Dot{\bigcup}}}_{i<j}\omega_i){\:\Dot{\cup}\:}{{\mathbb S}}_j$ is still $\omega_j$. Furthermore, since each $\tilde\Omega_{z_i\to z'_i}^{{{\bf N}};\vec \omega_{i-1}}$ for $i>j$ is a set of paths that do not fully contain $\omega_j$ or any earlier element of $\tilde\Omega_{z_j\to
z'_j}^{{{\bf N}};\vec\omega_{j-1}}$ as a subset, $\omega_j$ is still the earliest element of $$\begin{aligned}
\bigg(\textstyle\Big({{\mathbb S}}_0{\,\triangle\,}{\mathop{\Dot{\bigcup}}}_{i<j}\omega_i\Big){\:\Dot{\cup}\:}{{\mathbb S}}_j\bigg)
{\,\triangle\,}\Big({\mathop{\Dot{\bigcup}}}_{i>j}\omega_i\Big)=\Big({{\mathbb S}}_0{\,\triangle\,}{\mathop{\Dot{\bigcup}}}_{i\ne j}
\omega_i\Big){\:\Dot{\cup}\:}{{\mathbb S}}_j\equiv F_{\vec\omega_k}^{{\scriptscriptstyle}(0)}({{\mathbb S}}_0){\:\Dot{\cup}\:}F_{\vec\omega_k}^{{\scriptscriptstyle}(j)}({{\mathbb S}}_j).\end{aligned}$$ Therefore, $\vec F_{\vec\omega_k}$ is a bijection from ${\mathfrak{S}}_{\vec\omega_k}$ to ${\mathfrak{S}}'_{\vec\omega_k}$. This completes the proof of Lemma \[lmm:GHS-BK\].
Bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge1$ {#ss:pijbd}
----------------------------------------------------------------
First we prove [(\[eq:piNbd\])]{} for $j\ge1$ assuming the following two lemmas, in which we recall [(\[eq:Theta-def\])]{} and use $$\begin{aligned}
&E'_{{\bf N}}(z,x;{{\cal A}})=\{z{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}\cap\{z{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}}
{\overset{}{\Longleftrightarrow}}}x\},&
&E''_{{\bf N}}(z,x,v;{{\cal A}})=E'_{{\bf N}}(z,x;{{\cal A}})\cap\{z{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf N}}$}}
{\overset{}{\longleftrightarrow}}}v\},{\label{eq:E'E''-def}}\\
&\Theta'_{z,x;{{\cal A}}}=\!\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=z{\vartriangle}x}}\!\!
\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}
\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E'_{{{\bf m}}+{{\bf n}}}(z,x;{{\cal A}})$}}},\quad&
&\Theta''_{z,x,v;{{\cal A}}}=\!\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=z{\vartriangle}x}}\!\!
\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}
\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E''_{{{\bf m}}+{{\bf n}}}(z,x,v;{{\cal A}})$}}}.{\label{eq:Theta'Theta''-def}}\end{aligned}$$
\[lmm:Thetabds\] For the ferromagnetic Ising model, we have $$\begin{aligned}
\Theta_{y,x;{{\cal A}}}&\leq\sum_z\big(\delta_{y,z}+\tilde G_\Lambda(y,z)\big)
\,\Theta'_{z,x;{{\cal A}}},{\label{eq:Theta[1]-bd}}\\[5pt]
\Theta_{y,x;{{\cal A}}}\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}}
{\overset{}{\longleftrightarrow}}}v\}$}}}\big]&\leq\sum_z\big(\delta_{y,z}
+\tilde G_\Lambda(y,z)\big)\,\Theta''_{z,x,v;{{\cal A}}}{\nonumber}\\
&\quad+\sum_{v',z}\big(\delta_{y,v'}+\tilde G_\Lambda(y,v')\big)\,\tilde G_\Lambda(v',z)\,\Theta'_{z,x;
{{\cal A}}}\,\psi_\Lambda(v',v).{\label{eq:Theta[I]-bd}}\end{aligned}$$
\[lmm:Theta’Theta”bd\] For the ferromagnetic Ising model, we have $$\begin{aligned}
{\label{eq:Theta'Theta''bd}}
\Theta'_{y,x;{{\cal A}}}\leq\sum_{u\in{{\cal A}}}P'_{\Lambda;u}(y,x),&&
\Theta''_{y,x,v;{{\cal A}}}\leq\sum_{u\in{{\cal A}}}P''_{\Lambda;u,v}(y,x).\end{aligned}$$
We prove Lemma \[lmm:Thetabds\] in Section \[sss:chopping-off\], and Lemma \[lmm:Theta’Theta”bd\] in Section \[sss:dbconn\].
Recall [(\[eq:pij-def\])]{}. By [(\[eq:Theta\[1\]-bd\])]{}, [(\[eq:Theta’Theta”bd\])]{} and [(\[eq:Q’-def\])]{}, we obtain $$\begin{aligned}
{\label{eq:nest-diagbd}}
\Theta^{{\scriptscriptstyle}(j-1)}_{{\overline{b}}_{j-1},{\underline{b}}_j;\tilde{{\cal C}}_{j-2}}\Big[\tau_{b_j}
\Theta^{{\scriptscriptstyle}(j)}_{{\overline{b}}_j,x;\tilde{{\cal C}}_{j-1}}\Big]&\leq\Theta^{
{\scriptscriptstyle}(j-1)}_{{\overline{b}}_{j-1},{\underline{b}}_j;\tilde{{\cal C}}_{j-2}}\bigg[\sum_z\tau_{b_j}\big(
\delta_{{\overline{b}}_j,z}+\tilde G_\Lambda({\overline{b}}_j,z)\big)\sum_{v_j\in
\tilde{{\cal C}}_{j-1}}P'_{\Lambda;v_j}(z,x)\bigg]{\nonumber}\\
&\leq\sum_{v_j}\Theta^{{\scriptscriptstyle}(j-1)}_{{\overline{b}}_{j-1},{\underline{b}}_j;\tilde{{\cal C}}_{j-2}}
\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}_{j-1}{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}}
{\overset{}{\longleftrightarrow}}}v_j\}$}}}\big]\,\tau_{b_j}Q'_{\Lambda;v_j}
({\overline{b}}_j,x).\end{aligned}$$ For $j=1$, we use [(\[eq:pi0’-bd\])]{} and [(\[eq:nest-diagbd\])]{} to obtain $$\begin{aligned}
{\label{eq:pi0'-bd-appl}}
&\pi_\Lambda^{{\scriptscriptstyle}(1)}(x)\equiv\sum_{b_1}\Theta^{{\scriptscriptstyle}(0)}_{o,{\underline{b}}_1;\Lambda}
\Big[\tau_{b_1}\,\Theta^{{\scriptscriptstyle}(1)}_{{\overline{b}}_1,x;\tilde{{\cal C}}_0}\Big]\leq\sum_{b_1,
v_1}\Theta^{{\scriptscriptstyle}(0)}_{o,{\underline{b}}_1;\Lambda}\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}}
{\overset{}{\longleftrightarrow}}}v_1\}$}}}\big]\,\tau_{
b_1}Q'_{\Lambda;v_1}({\overline{b}}_1,x)\\
&~=\sum_{b_1,v_1}\bigg(\sum_{{\partial}{{\bf n}}=o{\vartriangle}{\underline{b}}_1}\frac{w_\Lambda({{\bf n}})}
{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}{\underline{b}}_1\\}$}}}\,\cap\,\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v_1}\bigg)\tau_{
b_1}Q'_{\Lambda;v_1}({\overline{b}}_1,x)\leq\sum_{b_1,v_1}P_{\Lambda;v_1}^{\prime{
{\scriptscriptstyle}(0)}}(o,{\underline{b}}_1)\,\tau_{b_1}Q'_{\Lambda;v_1}({\overline{b}}_1,x).{\nonumber}\end{aligned}$$ For $j\ge2$, we use [(\[eq:Theta\[I\]-bd\])]{}–[(\[eq:Theta’Theta”bd\])]{} and then [(\[eq:Q’-def\])]{}–[(\[eq:Q”-def\])]{} to obtain $$\begin{aligned}
{\label{eq:Theta-bd-appl}}
&\Theta^{{\scriptscriptstyle}(j-1)}_{{\overline{b}}_{j-1},{\underline{b}}_j;\tilde{{\cal C}}_{j-2}}\Big[\tau_{b_j}\Theta^{
{\scriptscriptstyle}(j)}_{{\overline{b}}_j,x;\tilde{{\cal C}}_{j-1}}\Big]\leq\sum_{v_j}\Theta^{{\scriptscriptstyle}(j-1)}_{
{\overline{b}}_{j-1},{\underline{b}}_j;\tilde{{\cal C}}_{j-2}}\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{\overline{b}}_{j-1}{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}}
{\overset{}{\longleftrightarrow}}}v_j\}$}}}\big]\,\tau_{
b_j}Q'_{\Lambda;v_j}({\overline{b}}_j,x){\nonumber}\\
&\quad\leq\sum_{v_j}\tau_{b_j}Q'_{\Lambda;v_j}({\overline{b}}_j,x)\Bigg(\sum_z\big(
\delta_{{\overline{b}}_{j-1},z}+\tilde G_\Lambda({\overline{b}}_{j-1},z)\big)\sum_{v_{j-1}\in
\tilde{{\cal C}}_{j-2}}P''_{\Lambda;v_{j-1},v_j}(z,{\underline{b}}_j){\nonumber}\\
&\qquad\qquad+\sum_{v',z}\big(\delta_{{\overline{b}}_{j-1},v'}+\tilde G_\Lambda({\overline{b}}_{
j-1},v')\big)\,\tilde G_\Lambda(v',z)\,\sum_{v_{j-1}\in\tilde{{\cal C}}_{j-2}}
P'_{\Lambda;v_{j-1}}(z,{\underline{b}}_j)\,\psi_\Lambda(v',v_j)\Bigg){\nonumber}\\
&\quad\leq\sum_{v_{j-1},v_j}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v_{j-1}\in\tilde{{\cal C}}_{j-2}\}$}}}\,Q''_{\Lambda;
v_{j-1},v_j}({\overline{b}}_{j-1},{\underline{b}}_j)\,\tau_{b_j}Q'_{\Lambda;v_j}({\overline{b}}_j,x).\end{aligned}$$ We repeatedly use [(\[eq:Theta\[I\]-bd\])]{}–[(\[eq:Theta’Theta”bd\])]{} to bound $\Theta^{{\scriptscriptstyle}(i)}_{{\overline{b}}_i,{\underline{b}}_{i+1};\tilde{{\cal C}}_{i-1}}
[{\mathbbm{1}{\scriptstyle\{{\overline{b}}_i {\underset{\raisebox{5pt}{${\scriptscriptstyle}$}}
{\overset{}{\longleftrightarrow}}}v_{i+1}\}}}]$ for $i=j-2,\dots,1$ as in [(\[eq:Theta-bd-appl\])]{}, and then at the end we apply [(\[eq:pi0’-bd\])]{} as in [(\[eq:pi0’-bd-appl\])]{} to obtain [(\[eq:piNbd\])]{}. This completes the proof.
### Proof of Lemma \[lmm:Thetabds\] {#sss:chopping-off}
Recall [(\[eq:Theta-def\])]{} and [(\[eq:Theta’Theta”-def\])]{}. Then, to prove [(\[eq:Theta\[1\]-bd\])]{}, it suffices to bound the contribution from ${\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}})\setminus E'_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}})$}}}$ by $\sum_z\tilde G_\Lambda(y,z)\,\Theta'_{z,x;{{\cal A}}}$.
First we recall [(\[eq:E-def\])]{} and [(\[eq:E’E”-def\])]{}. Then, we have $$\begin{aligned}
E_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}})\setminus E'_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}})=E_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}})
\cap\big\{\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}\big\}.\end{aligned}$$ On $\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}$, there is at least one pivotal bond for $y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x$ from $y$. Let $b$ be the last pivotal bond among them. Then, we have ${\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\text{ off }b$, $m_b+n_b>0$, and $y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}{\underline{b}}$ in ${{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x){^{\rm c}}$. Moreover, on the event $E_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}})$, we have that $y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}{\underline{b}}$ in ${{\cal A}}{^{\rm c}}$ and ${\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x$. Since $\{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\text{ off }b\}\cap\{{\overline{b}}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}=\{E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})\text{ off }b\}$ on the event that $b$ is pivotal for $y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x$ from $y$, we have $$\begin{aligned}
{\label{eq:EE'-dec}}
&E_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}})\setminus E'_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}}){\nonumber}\\
&={\mathop{\Dot{\bigcup}}}_b\Big\{\{E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})\text{ off }b\}\cap\{m_b+n_b>0
\}\cap\big\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}{\underline{b}}\text{ in }{{\cal A}}{^{\rm c}}\cap{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x)
{^{\rm c}}\big\}\Big\}.\end{aligned}$$ Therefore, we obtain $$\begin{aligned}
{\label{eq:Theta[1]-rewr}}
&\Theta_{y,x;{{\cal A}}}-\Theta'_{y,x;{{\cal A}}}{\nonumber}\\
&=\sum_b\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=y{\vartriangle}x}}\frac{
w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}
{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})\text{ off }b\}$}}}\,
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b+n_b>0\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}{\underline{b}}\text{ in }{{\cal A}}{^{\rm c}}\cap{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x){^{\rm c}}\}$}}}.\end{aligned}$$
It remains to bound the right-hand side of [(\[eq:Theta\[1\]-rewr\])]{}, which is nonzero only if $m_b$ is even and $n_b$ is odd, due to the source constraints and the conditions in the indicators. First, as in [(\[eq:2nd-ind-fact\])]{}, we alternate the parity of $n_b$ by changing the source constraint into ${\partial}{{\bf n}}=y{\vartriangle}b{\vartriangle}x$ and multiplying by $\tau_b$. Then, by conditioning on ${{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x)$ as in [(\[eq:3rd-ind-prefact\])]{} (i.e., conditioning on ${{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x)={{\cal B}}$, letting ${{\bf m}}'={{\bf m}}|_{{{\mathbb B}}_{{{\cal A}}{^{\rm c}}}\setminus{{\mathbb B}}_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}$, ${{\bf m}}''={{\bf m}}|_{{{\mathbb B}}_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}$, ${{\bf n}}'={{\bf n}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal B}}{^{\rm c}}}}$ and ${{\bf n}}''={{\bf n}}|_{{{\mathbb B}}_{{{\cal B}}{^{\rm c}}}}$, and then summing over ${{\cal B}}\subset\Lambda$), we obtain $$\begin{aligned}
{\label{eq:2ndind-contr}}
\sum_{{{\cal B}}\subset\Lambda}\sum_{\substack{{\partial}{{\bf m}}'={\varnothing}\\ {\partial}{{\bf n}}'=
{\overline{b}}{\vartriangle}x}}\frac{\tilde w_{{{\cal A}}{^{\rm c}},{{\cal B}}}({{\bf m}}')\,Z_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{\tilde w_{\Lambda,{{\cal B}}}
({{\bf n}}')\,Z_{{{\cal B}}{^{\rm c}}}}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E'_{{{\bf m}}'+{{\bf n}}'}({\overline{b}},x;
{{\cal A}})\text{ off }b\\}$}}}\,\cap\,\{{{\cal C}}^b_{{{\bf m}}'+{{\bf n}}'}(x)={{\cal B}}}{\nonumber}\\
\times\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m'_b,n'_b\text{ even}\}$}}}\underbrace{\sum_{
\substack{{\partial}{{\bf m}}''={\varnothing}\\ {\partial}{{\bf n}}''=y{\vartriangle}{\underline{b}}}}\frac{w_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}({{\bf m}}'')}{Z_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}\,
\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf n}}'')}{Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}''+
{{\bf n}}''$}}
{\overset{}{\longleftrightarrow}}}{\underline{b}}~\text{ in }{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}\}$}}}}_{\stackrel{
\because{(\ref{eq:switching-appl})}\;}={{\langle \varphi_y\varphi_{{\underline{b}}}
\rangle}}_{{{\cal A}}{^{\rm c}}\cap\,{{\cal B}}{^{\rm c}}}}{\nonumber}\\
=\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}={\overline{b}}{\vartriangle}x}}\frac{
w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}
{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})\text{ off }b\}$}}}\,
\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b,n_b\text{ even}\}$}}}\,{{\langle \varphi_y\varphi_{{\underline{b}}}
\rangle}}_{{{\cal A}}{^{\rm c}}\cap\,{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x){^{\rm c}}}.\end{aligned}$$ Since ${{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_{{{\cal A}}{^{\rm c}}\cap\,{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x){^{\rm c}}}=0$ on $E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})\setminus\{E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})$ off $b\}\subset\{{\underline{b}}\in{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x)\}$ and on the event that $m_b$ or $n_b$ is odd (see below [(\[eq:0th-summand2\])]{} or above [(\[eq:3rd-ind-fact\])]{}), we can omit “off $b$” and ${\mathbbm{1}{\scriptstyle\{m_b,n_b\text{ even}\}}}$ in [(\[eq:2ndind-contr\])]{}. Since ${{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_{{{\cal A}}{^{\rm c}}\cap\,{{\cal C}}_{{{\bf m}}+ {{\bf n}}}^b(x)
{^{\rm c}}}\leq{{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_\Lambda$ due to Proposition \[prp:through\], we have $$\begin{aligned}
{\label{eq:2ndind-contrbd}}
{(\ref{eq:2ndind-contr})}\leq{{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_\Lambda\tau_b
\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}={\overline{b}}{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})$}}}={{\langle \varphi_y\varphi_{{\underline{b}}}
\rangle}}_\Lambda\tau_b\,\Theta'_{{\overline{b}},x;{{\cal A}}}.\end{aligned}$$ Therefore, [(\[eq:Theta\[1\]-rewr\])]{} is bounded by $\sum_b{{\langle \varphi_y
\varphi_{{\underline{b}}} \rangle}}_\Lambda\tau_b\,\Theta'_{{\overline{b}},x;{{\cal A}}}\equiv\sum_z\tilde
G_\Lambda(y,z)\,\Theta'_{z,x;{{\cal A}}}$. This completes the proof of [(\[eq:Theta\[1\]-bd\])]{}.
Recall [(\[eq:Theta-def\])]{} and [(\[eq:Theta’Theta”-def\])]{}. To prove [(\[eq:Theta\[I\]-bd\])]{}, we investigate $$\begin{aligned}
L&\equiv\big\{E_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}})\cap\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v\}\big\}
\setminus E''_{{{\bf m}}+{{\bf n}}}(y,x,v;{{\cal A}}){\nonumber}\\
&=\{E_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}})\setminus E'_{{{\bf m}}+{{\bf n}}}(y,x;{{\cal A}})\}\cap
\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v\},\end{aligned}$$ where $\Theta_{y,x;{{\cal A}}}[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle L$}}}]=\Theta_{y,x;{{\cal A}}}[{\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}}
{\overset{}{\longleftrightarrow}}}
v\}}}]-\Theta''_{y,x,v;{{\cal A}}}$.
First we recall [(\[eq:EE’-dec\])]{}, in which $b$ is the last pivotal bond for $y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x$ from $y$, and define $$\begin{aligned}
R_1(b)&=\{E''_{{{\bf m}}+{{\bf n}}}({\overline{b}},x,v;{{\cal A}})\text{ off }b\}\cap\{m_b
+n_b>0\}\cap\big\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}{\underline{b}}\text{ in }{{\cal A}}{^{\rm c}}\cap
{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x){^{\rm c}}\big\},{\label{eq:R1b-def}}\\
R_2(b)&=\{E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})\text{ off }b\}
\cap\{m_b+n_b>0\}\cap\big\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}{\underline{b}}\text{ in }{{\cal A}}{^{\rm c}}\cap{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x){^{\rm c}},\;y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v\big\},
{\label{eq:R2b-def}}\end{aligned}$$ where $v\in{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x)$ on $R_1(b)$, while $v\in{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(y)$ on $R_2(b)$. Since $$\begin{aligned}
{\label{eq:EE'E''-predec}}
L={\mathop{\Dot{\bigcup}}}_b\{R_1(b){\:\Dot{\cup}\:}R_2(b)\},\end{aligned}$$ we have $$\begin{aligned}
{\label{eq:EE'E''predec2}}
\Theta_{y,x;{{\cal A}}}[{\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}}
{\overset{}{\longleftrightarrow}}} v\}}}]-\Theta''_{y,x,v;{{\cal A}}}=\sum_b
\Big(\Theta_{y,x;{{\cal A}}}\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle R_1(b)$}}}\big]+\Theta_{y,x;{{\cal A}}}
\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle R_2(b)$}}}\big]\Big).\end{aligned}$$ Following the same argument as in [(\[eq:2ndind-contr\])]{}–[(\[eq:2ndind-contrbd\])]{}, we easily obtain $$\begin{aligned}
{\label{eq:EE'E''predec3}}
\Theta_{y,x;{{\cal A}}}\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle R_1(b)$}}}\big]&=\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}={\overline{b}}{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{
{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E''_{{{\bf m}}+{{\bf n}}}({\overline{b}},x,v;{{\cal A}})\text{ off }b\}$}}}\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b,n_b
\text{ even}\}$}}}\,{{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_{{{\cal A}}{^{\rm c}}\cap\,
{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x){^{\rm c}}}{\nonumber}\\
&\leq{{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_\Lambda\tau_b\sum_{\substack{
{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}={\overline{b}}{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{
{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E''_{{{\bf m}}+{{\bf n}}}({\overline{b}},x,v;{{\cal A}})$}}}={{\langle \varphi_y\varphi_{{\underline{b}}} \rangle}}_\Lambda\tau_b
\,\Theta''_{{\overline{b}},x,v;{{\cal A}}}.\end{aligned}$$ Similarly, we have $$\begin{aligned}
{\label{eq:EE'E''decpre3}}
\Theta_{y,x;{{\cal A}}}\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle R_2(b)$}}}\big]&=\sum_{{{\cal B}}\subset
\Lambda}\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}={\overline{b}}{\vartriangle}x}}
\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda
({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})\text{ off }
b\\}$}}}\,\cap\,\{{{\cal C}}^b_{{{\bf m}}+{{\bf n}}}(x)={{\cal B}}}\,\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b,n_b
\text{ even}\}$}}}{\nonumber}\\
&\qquad\qquad\times\sum_{\substack{{\partial}{{\bf h}}={\varnothing}\\ {\partial}{{\bf k}}=y{\vartriangle}{\underline{b}}}}
\frac{w_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}({{\bf h}})}{Z_{{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}}}}\,\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})}{Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}{\underline{b}}\text{ in }{{\cal A}}{^{\rm c}}\cap{{\cal B}}{^{\rm c}},~y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\text{ (in }{{\cal B}}{^{\rm c}})\}$}}}{\nonumber}\\
&\leq\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}={\overline{b}}{\vartriangle}x}}\frac{
w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}
{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})\text{ off }b\}$}}}\,
\tau_b{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{m_b,n_b\text{ even}\}$}}}\,\Psi_{y,{\underline{b}},v;{{\cal A}},{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x)},\end{aligned}$$ where $$\begin{aligned}
{\label{eq:Psi-def}}
\Psi_{y,z,v;{{\cal A}},{{\cal B}}}=\sum_{\substack{{\partial}{{\bf h}}={\varnothing}\\ {\partial}{{\bf k}}=
y{\vartriangle}z}}\frac{w_{{{\cal A}}{^{\rm c}}\cap\,{{\cal B}}{^{\rm c}}}({{\bf h}})}{Z_{{{\cal A}}{^{\rm c}}\cap\,{{\cal B}}{^{\rm c}}}}\,\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})}{Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\}$}}}.\end{aligned}$$ We note that, by ignoring the indicator in [(\[eq:Psi-def\])]{}, we have $0\leq\Psi_{y,z,v;{{\cal A}},{{\cal B}}}\leq{{\langle \varphi_y\varphi_z \rangle}}_{{{\cal B}}{^{\rm c}}}$, which is zero whenever $z\in{{\cal B}}$. Therefore, we can omit “off $b$” and ${\mathbbm{1}{\scriptstyle\{m_b,n_b\text{ even}\}}}$ in [(\[eq:EE’E”decpre3\])]{} to obtain $$\begin{aligned}
{\label{eq:EE'E''dec3}}
\Theta_{y,x;{{\cal A}}}\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle R_2(b)$}}}\big]\leq\sum_{\substack{{\partial}{{\bf m}}=
{\varnothing}\\ {\partial}{{\bf n}}={\overline{b}}{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}
\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})$}}}
\,\tau_b\,\Psi_{y,{\underline{b}},v;{{\cal A}},{{\cal C}}_{{{\bf m}}+{{\bf n}}}^b(x)}.\end{aligned}$$ Substituting [(\[eq:EE’E”predec3\])]{} and [(\[eq:EE’E”dec3\])]{} to [(\[eq:EE’E”predec2\])]{}, we arrive at $$\begin{aligned}
{\label{eq:EE'E''dec2}}
\Theta_{y,x;{{\cal A}}}\big[{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}$}}
{\overset{}{\longleftrightarrow}}}v\}$}}}\big]&\leq\sum_z\big(\delta_{y,
z}+\tilde G_\Lambda(y,z)\big)\,\Theta''_{z,x,v;{{\cal A}}}{\nonumber}\\
&\quad+\sum_b\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}={\overline{b}}{\vartriangle}x}}\frac{
w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}
\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle E'_{{{\bf m}}+{{\bf n}}}({\overline{b}},x;{{\cal A}})$}}}\,\tau_b\,\Psi_{y,{\underline{b}},v;{{\cal A}},{{\cal C}}_{
{{\bf m}}+{{\bf n}}}^b(x)}.\end{aligned}$$ The proof of [(\[eq:Theta\[I\]-bd\])]{} is completed by using $$\begin{aligned}
{\label{eq:Psi-bd}}
\Psi_{y,z,v;{{\cal A}},{{\cal B}}}\leq\sum_{v'}{{\langle \varphi_y\varphi_{v'} \rangle}}_\Lambda
{{\langle \varphi_{v'}\varphi_z \rangle}}_\Lambda\,\psi_\Lambda(v',v),\end{aligned}$$ and replacing ${{\langle \varphi_y\varphi_{v'} \rangle}}_\Lambda$ in [(\[eq:Psi-bd\])]{} by $\delta_{y,v'}+\tilde G_\Lambda(y,v')$, due to [(\[eq:G-delta-bd\])]{}.
To complete the proof of [(\[eq:Theta\[I\]-bd\])]{}, it thus remains to show [(\[eq:Psi-bd\])]{}. First we note that, if ${{\cal A}}\subset{{\cal B}}$, then by Lemma \[lmm:switching\] we have $$\begin{aligned}
\Psi_{y,z,v;{{\cal A}},{{\cal B}}}=\sum_{\substack{{\partial}{{\bf h}}={\varnothing}\\ {\partial}{{\bf k}}=y{\vartriangle}z}}
\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf h}})}{Z_{{{\cal B}}{^{\rm c}}}}\,\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})}
{Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\}$}}}={{\langle \varphi_y\varphi_v \rangle}}_{{{\cal B}}{^{\rm c}}}{{\langle \varphi_v\varphi_z \rangle}}_{{{\cal B}}{^{\rm c}}}\leq{{\langle \varphi_y
\varphi_v \rangle}}_\Lambda{{\langle \varphi_v\varphi_z \rangle}}_\Lambda.\end{aligned}$$ However, to prove [(\[eq:Psi-bd\])]{} for a general ${{\cal A}}$ that does not necessarily satisfy ${{\cal A}}\subset{{\cal B}}$, we use $$\begin{aligned}
{\label{eq:Psi-ind-dec}}
\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\}=\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\}{\:\Dot{\cup}\:}\big\{\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\}
\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\}\big\},\end{aligned}$$ and consider the two events on the right-hand side separately. The contribution to $\Psi_{y,z,v;{{\cal A}},{{\cal B}}}$ from $\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\}$ is easily bounded, similarly to [(\[eq:dbbd\])]{}, as $$\begin{aligned}
{\label{eq:psi-delta}}
\sum_{{\partial}{{\bf k}}=y{\vartriangle}z}\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})}{Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\}$}}}\leq\sum_{\substack{{\partial}{{\bf k}}=y{\vartriangle}z\\ {\partial}{{\bf k}}'={\varnothing}}}\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})}{Z_{{{\cal B}}{^{\rm c}}}}\,\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}}')}{Z_{{{\cal B}}{^{\rm c}}}}\,
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}+{{\bf k}}'$}}
{\overset{}{\longleftrightarrow}}}v\}$}}}&={{\langle \varphi_y\varphi_v \rangle}}_{{{\cal B}}{^{\rm c}}}{{\langle
\varphi_v\varphi_z \rangle}}_{{{\cal B}}{^{\rm c}}}{\nonumber}\\
&\leq{{\langle \varphi_y\varphi_v \rangle}}_\Lambda{{\langle \varphi_v\varphi_z \rangle}}_\Lambda.\end{aligned}$$
Next we consider the contribution to $\Psi_{y,z,v;{{\cal A}},{{\cal B}}}$ from $\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\}$ in [(\[eq:Psi-ind-dec\])]{}. We denote by ${{\cal C}}_{{\bf k}}(y)$ the set of sites ${{\bf k}}$-connected from $y$. Since $v\in{{\cal C}}_{{{\bf h}}+{{\bf k}}}(y)\setminus{{\cal C}}_{{\bf k}}(y)$, there is a *nonzero* alternating chain of mutually-disjoint ${{\bf h}}$-connected clusters and mutually-disjoint ${{\bf k}}$-connected clusters, from some $u_0\in{{\cal C}}_{{\bf k}}(y)$ to $v$. Therefore, we have $$\begin{aligned}
{\label{eq:ind-bd}}
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\\}$}}}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v}\leq\sum_{j=1}^\infty
\sum_{\substack{u_0,\dots,u_j\\ u_l\ne u_{l'}\,{{}^\forall}l\ne l'\\ u_j
=v}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}u_0\}$}}}\bigg(\prod_{l\ge0}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{u_{2l}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}$}}
{\overset{}{\longleftrightarrow}}}u_{2l
+1}\}$}}}\bigg)\bigg(\prod_{l\ge1}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{u_{2l-1}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}u_{2l}\}$}}}\bigg){\nonumber}\\
\times\bigg(\prod_{\substack{l,l'\ge0\\ l\ne l'}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{\bf h}}(u_{2l})
\,\cap\,{{\cal C}}_{{\bf h}}(u_{2l'})={\varnothing}\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{\bf k}}(u_{2l})\,\cap\,{{\cal C}}_{{\bf k}}(u_{2l'})={\varnothing}\}$}}}\bigg),\end{aligned}$$ where we regard an empty product as 1. Using this bound, we can perform the sums over ${{\bf h}}$ and ${{\bf k}}$ in [(\[eq:Psi-def\])]{} independently.
For $j=1$ and given $u_0\ne u_1=v$, the summand of [(\[eq:ind-bd\])]{} equals ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}u_0\}}}{\mathbbm{1}{\scriptstyle\{u_0{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}$}}
{\overset{}{\longleftrightarrow}}}v\}}}$, which is simply equal to ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}$}}
{\overset{}{\longleftrightarrow}}}v\}}}$ if $u_0=y$. Then, by [(\[eq:psi-delta\])]{} and [(\[eq:G-delta-bd\])]{}, the contribution from this to $\Psi_{y,z,v;{{\cal A}},{{\cal B}}}$ is $$\begin{aligned}
{\label{eq:psi-delta-G2}}
\sum_{{\partial}{{\bf k}}=y{\vartriangle}z}\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})}{Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}u_0\}$}}}\sum_{{\partial}{{\bf h}}={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}\cap\,{{\cal B}}{^{\rm c}}}({{\bf h}})}
{Z_{{{\cal A}}{^{\rm c}}\cap\,{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{u_0{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}$}}
{\overset{}{\longleftrightarrow}}}v\}$}}}\leq{{\langle \varphi_y
\varphi_{u_0} \rangle}}_\Lambda{{\langle \varphi_{u_0}\varphi_z \rangle}}_\Lambda\,\tilde
G_\Lambda(u_0,v)^2.\end{aligned}$$
Fix $j\ge2$ and a sequence of distinct sites $u_0,\dots,u_j\,(=v)$, and first consider the contribution to the sum over ${{\bf k}}$ in [(\[eq:Psi-def\])]{} from the relevant indicators in the right-hand side of [(\[eq:ind-bd\])]{}, which is $$\begin{aligned}
{\label{eq:nsum-0thbd}}
&\sum_{{\partial}{{\bf k}}=y{\vartriangle}z}\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})}{Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}u_0\}$}}}\bigg(\prod_{l\ge1}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{u_{2l-1}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}u_{2l}\}$}}}\bigg)
\prod_{\substack{l,l'\ge0\\ l\ne l'}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{\bf k}}(u_{2l})\,\cap\,
{{\cal C}}_{{\bf k}}(u_{2l'})={\varnothing}\}$}}}\\
&=\sum_{{\partial}{{\bf k}}=y{\vartriangle}z}\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})}{Z_{{{\cal B}}{^{\rm c}}}}
\bigg(\prod_{l\ge1}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{u_{2l-1}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}u_{2l}\}$}}}\bigg)\bigg(
\prod_{\substack{l,l'\ge1\\ l\ne l'}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{\bf k}}(u_{2l})\,\cap\,
{{\cal C}}_{{\bf k}}(u_{2l'})={\varnothing}\}$}}}\bigg){\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}u_0\\}$}}}\,\cap\,\{{{\cal C}}_{{\bf k}}(u_0)\,\cap\,{{\cal U}}_{{{\bf k}};1}={\varnothing}}{\nonumber},\end{aligned}$$ where ${{\cal U}}_{{{\bf k}};1}={\mathop{\Dot{\bigcup}}}_{l\ge1}{{\cal C}}_{{\bf k}}(u_{2l})$. Conditioning on ${{\cal U}}_{{{\bf k}};1}$, we obtain that $$\begin{gathered}
{(\ref{eq:nsum-0thbd})}=\sum_{{\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})}
{Z_{{{\cal B}}{^{\rm c}}}}\bigg(\prod_{l\ge1}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{u_{2l-1}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}u_{2l}\}$}}}
\bigg)\bigg(\prod_{\substack{l,l'\ge1\\ l\ne l'}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{\bf k}}(u_{2l})\,\cap\,{{\cal C}}_{{\bf k}}(u_{2l'})={\varnothing}\}$}}}\bigg){\nonumber}\\
\times\underbrace{\sum_{{\partial}{{\bf k}}'=y{\vartriangle}z}\frac{w_{{{\cal B}}{^{\rm c}}\cap\,
{{\cal U}}_{{{\bf k}};1}{^{\rm c}}}({{\bf k}}')}{Z_{{{\cal B}}{^{\rm c}}\cap\,{{\cal U}}_{{{\bf k}};1}{^{\rm c}}}}
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}'$}}
{\overset{}{\longleftrightarrow}}}u_0\}$}}}}_{\stackrel{\because{(\ref{eq:psi-delta})}\;}\leq
{{\langle \varphi_y\varphi_{u_0} \rangle}}_\Lambda{{\langle \varphi_{u_0}\varphi_z
\rangle}}_\Lambda}.{\label{eq:nsum-1stbd}}\end{gathered}$$ Then, by conditioning on ${{\cal U}}_{{{\bf k}};2}\equiv{\mathop{\Dot{\bigcup}}}_{l\ge2}{{\cal C}}_{{\bf k}}(u_{2l})$, following the same computation as above and using [(\[eq:G-delta-bd\])]{}, we further obtain that $$\begin{gathered}
{(\ref{eq:nsum-0thbd})}\leq{{\langle \varphi_y\varphi_{u_0} \rangle}}_\Lambda{{\langle
\varphi_{u_0}\varphi_z \rangle}}_\Lambda\sum_{{\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal B}}{^{\rm c}}}
({{\bf k}})}{Z_{{{\cal B}}{^{\rm c}}}}\bigg(\prod_{l\ge2}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{u_{2l-1}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}u_{2
l}\}$}}}\bigg)\bigg(\prod_{\substack{l,l'\ge2\\ l\ne l'}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{\bf k}}(u_{2l})\,\cap\,{{\cal C}}_{{\bf k}}(u_{2l'})={\varnothing}\}$}}}\bigg){\nonumber}\\
\times\underbrace{\sum_{{\partial}{{\bf k}}'={\varnothing}}\frac{w_{{{\cal B}}{^{\rm c}}\cap\,{{\cal U}}_{{{\bf k}};2}{^{\rm c}}}({{\bf k}}')}{Z_{{{\cal B}}{^{\rm c}}\cap\,{{\cal U}}_{{{\bf k}};2}{^{\rm c}}}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{u_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf k}}'$}}
{\overset{}{\longleftrightarrow}}}u_2\}$}}}}_{\leq\;\tilde G_\Lambda(u_1,u_2)^2}.{\label{eq:nsum-2ndbd}}\end{gathered}$$ We repeat this computation until all indicators for ${{\bf k}}$ are used up. We also apply the same argument to the sum over ${{\bf h}}$ in [(\[eq:Psi-def\])]{}. Summarizing these bounds with [(\[eq:psi-delta\])]{} and [(\[eq:psi-delta-G2\])]{}, and replacing $u_0$ in [(\[eq:ind-bd\])]{}–[(\[eq:nsum-1stbd\])]{} by $v'$, we obtain [(\[eq:Psi-bd\])]{}. This completes the proof of [(\[eq:Theta\[I\]-bd\])]{}.
### Proof of Lemma \[lmm:Theta’Theta”bd\] {#sss:dbconn}
We note that the common factor ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}}}$ in $\Theta'_{y,x;{{\cal A}}}$ and $\Theta''_{y,x,v;{{\cal A}}}$ can be decomposed as $$\begin{aligned}
{\label{eq:Theta'-evdec}}
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}$}}}={\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}$}}}+{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\\}$}}}
\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x}.\end{aligned}$$ We estimate the contributions from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}}}$ to $\Theta'_{y,x;{{\cal A}}}$ and $\Theta''_{y,x,v;{{\cal A}}}$ in the following paragraphs (a) and (b), respectively. Then, in the paragraphs (c) and (d) below, we will estimate the contributions from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\\}}}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x}$ in [(\[eq:Theta’-evdec\])]{} to $\Theta'_{y,x;{{\cal A}}}$ and $\Theta''_{y,x,v;{{\cal A}}}$, respectively.
**(a)** First we investigate the contribution to $\Theta'_{y,x;{{\cal A}}}$ from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}}}$: $$\begin{aligned}
{\label{eq:contr-(a)}}
\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=y{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}
({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x}.\end{aligned}$$ For a set of events $E_1,\dots,E_N$, we define $E_1\circ\cdots\circ E_N$ to be the event that $E_1,\dots,E_N$ occur *bond*-disjointly. Then, we have $$\begin{aligned}
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x}\leq{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x}\leq\sum_{u\in{{\cal A}}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}u
\\}$}}}\,\circ\,\{u{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\}\,\circ\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x},\end{aligned}$$ where the right-hand side does not depend on ${{\bf m}}$. Therefore, the contribution to $\Theta'_{y,x;{{\cal A}}}$ is bounded by $$\begin{aligned}
{\label{eq:Theta'-bd1stbd}}
{(\ref{eq:contr-(a)})}\leq\sum_{u\in{{\cal A}}}\,\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{
w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}u\\}$}}}\,\circ\,\{u{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\}\,\circ\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x}\leq\sum_{u\in{{\cal A}}}P_{\Lambda;
u}^{\prime{{\scriptscriptstyle}(0)}}(y,x),\end{aligned}$$ where we have applied the same argument as in the proof of [(\[eq:pi0’-bd\])]{}, which is around [(\[eq:dbbd\])]{}–[(\[eq:S03sup\])]{}.
**(b)** Next we investigate the contribution to $\Theta''_{y,x,v;{{\cal A}}}$ from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}}}$ in [(\[eq:Theta’-evdec\])]{}: $$\begin{aligned}
{\label{eq:contr-(b)}}
\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=y{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}
({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+
{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v}.\end{aligned}$$ Note that, by using [(\[eq:Psi-ind-dec\])]{} and ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}}}\leq{\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}}}$, we have $$\begin{aligned}
{\label{eq:Theta''-1stindbd}}
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}
v}\leq{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x}\,\Big({\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}
v\}$}}}+{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v\\}$}}}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v}\Big).\end{aligned}$$ We investigate the contributions from the two indicators in the parentheses separately.
We begin with the contribution from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v\}}}$, which is independent of ${{\bf m}}$. Since $$\begin{aligned}
\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}\cap\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}\cap\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v\}&\subset\{y
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}\circ\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x,~y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v\},\\
\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}&\subset\bigcup_{u\in{{\cal A}}}\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}u\}\circ\{u
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\},{\label{eq:Theta''-1stind1st-pcontr}}\end{aligned}$$ the contribution to [(\[eq:contr-(b)\])]{} from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v\}}}$ in [(\[eq:Theta”-1stindbd\])]{} is bounded by $$\begin{aligned}
{\label{eq:Theta''-1stind1stcontr}}
\sum_{u\in{{\cal A}}}\,\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}u\\}$}}}\,\circ\,\{u{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\}\,\circ\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x,~y
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v}.\end{aligned}$$ We follow Steps (i)–(iii) described above [(\[eq:dbbd\])]{} in Section \[ss:pi0bd\]. Without loss of generality, we can assume that $y,u,x$ and $v$ are all different; otherwise, the following argument can be simplified. (i) Since $y$ and $x$ are sources, but $u$ and $v$ are not, there is an edge-disjoint cycle $y\to u\to x\to
v\to y$, with an extra edge-disjoint path from $y$ to $x$. Therefore, we have in total at least $5\,(=4+1)$ edge-disjoint paths. (ii) Multiplying by $(Z_\Lambda/Z_\Lambda)^4$, we have $$\begin{aligned}
{\label{eq:Theta''-bd1stprebd}}
{(\ref{eq:Theta''-1stind1stcontr})}=\sum_{u\in{{\cal A}}}\,\sum_{{\partial}{{\bf N}}=y{\vartriangle}x}\frac{w_\Lambda({{\bf N}})}
{Z_\Lambda^5}\sum_{\substack{{\partial}{{\bf n}}=y{\vartriangle}x\\ {\partial}{{\bf m}}_i={\varnothing}~
{{}^\forall}i=1,\dots,4\\ {{\bf N}}={{\bf n}}+\sum_{i=1}^4{{\bf m}}_i}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}u\\}$}}}\,\circ\,\{u{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\}\,\circ\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x,~y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v}\prod_b\frac{N_b!}{n_b!\prod_{i=1}^4m^{{\scriptscriptstyle}(i)}_b!},\end{aligned}$$ where we have used the notation $m_b^{{\scriptscriptstyle}(i)}={{\bf m}}_i|_b$. (iii) The sum over ${{\bf n}},{{\bf m}}_1,\dots,{{\bf m}}_4$ in [(\[eq:Theta”-bd1stprebd\])]{} is bounded by the cardinality of ${\mathfrak{S}}$ in Lemma \[lmm:GHS-BK\] with $k=4$, ${{\cal V}}=\{y,x\}$, $\{z_1,z'_1\}=\{y,u\}$, $\{z_2,z'_2\}=\{u,x\}$, $\{z_3,z'_3\}=\{y,v\}$ and $\{z_4,z'_4\}=\{v,x\}$. Bounding the cardinality of ${\mathfrak{S}}'$ in Lemma \[lmm:GHS-BK\] for this setting, we obtain $$\begin{aligned}
{\label{eq:Theta''-bd1stbd1}}
{(\ref{eq:Theta''-bd1stprebd})}&\leq\sum_{u\in{{\cal A}}}\,\sum_{{\partial}{{\bf N}}=y{\vartriangle}x}\frac{w_\Lambda({{\bf N}})}{Z_\Lambda^5}\sum_{\substack{{\partial}{{\bf n}}=y{\vartriangle}x\\ {\partial}{{\bf m}}_1=y{\vartriangle}u,~{\partial}{{\bf m}}_2=u{\vartriangle}x\\ {\partial}{{\bf m}}_3=y{\vartriangle}v,~{\partial}{{\bf m}}_4=v{\vartriangle}x\\ {{\bf N}}={{\bf n}}+\sum_{i=1}^4{{\bf m}}_i}}\prod_b\frac{N_b!}{n_b!
\prod_{i=1}^4m^{{\scriptscriptstyle}(i)}_b!}{\nonumber}\\
&\leq\sum_{u\in{{\cal A}}}{{\langle \varphi_y\varphi_x \rangle}}_\Lambda{{\langle \varphi_y
\varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_x \rangle}}_\Lambda{{\langle \varphi_y
\varphi_v \rangle}}_\Lambda{{\langle \varphi_v\varphi_x \rangle}}_\Lambda.\end{aligned}$$
Next we investigate the contribution to [(\[eq:contr-(b)\])]{} from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v\\}}}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v}$ in [(\[eq:Theta”-1stindbd\])]{}. On the event $\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}\cap\{\{
y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v\}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v\}\}$, there exists a $v_0\ne v$ such that $\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\}\circ\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x,~y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v_0\}$ occurs and that $v_0$ and $v$ are connected via a nonzero alternating chain of mutually-disjoint ${{\bf m}}$-connected clusters and mutually-disjoint ${{\bf n}}$-connected clusters. Therefore, by [(\[eq:ind-bd\])]{} and [(\[eq:Theta”-1stind1st-pcontr\])]{} (see also [(\[eq:Theta”-1stind1stcontr\])]{}), we obtain $$\begin{aligned}
{\label{eq:Theta''-1stindbd2}}
&{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}\,\cap\,\{\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v
\}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v\}}{\nonumber}\\
&\leq\sum_{u\in{{\cal A}}}\,\sum_{j\ge1}\sum_{\substack{v_0,\dots,v_j\\ v_l\ne
v_{l'}\,{{}^\forall}l\ne l'\\ v_j=v}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}u\\}$}}}\,\circ\,\{u{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}
x\}\,\circ\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x,~y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v_0}\,\bigg(\prod_{l\ge0}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v_{2
l}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}$}}
{\overset{}{\longleftrightarrow}}}v_{2l+1}\}$}}}\bigg){\nonumber}\\
&\qquad\times\bigg(\prod_{l\ge1}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v_{2l-1}{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v_{2l}\}$}}}\bigg)\bigg(
\prod_{\substack{l,l'\ge0\\ l\ne l'}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{\bf m}}(v_{2l})\,\cap\,{{\cal C}}_{{\bf m}}(v_{2l'})={\varnothing}\}$}}}\;{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{\bf n}}(v_{2l})\,\cap\,{{\cal C}}_{{\bf n}}(v_{2l'})={\varnothing}\}$}}}\bigg).\end{aligned}$$ For the three products of indicators, we repeate the same argument as in [(\[eq:psi-delta-G2\])]{}–[(\[eq:nsum-2ndbd\])]{} to derive the factor $\psi_\Lambda(v_0,v)-\delta_{v_0,v}$. As a result, we have $$\begin{aligned}
{\label{eq:Theta''-prebd1stbd2}}
&\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=y{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}
{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}
\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}\,\cap\,\{\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v\}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v\}}{\nonumber}\\
&~\leq\sum_{v_0}\big(\psi_\Lambda(v_0,v)-\delta_{v_0,v}\big)\sum_{u\in{{\cal A}}}
\,\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}
u\\}$}}}\,\circ\,\{u{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\}\,\circ\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x,~y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v_0}.\end{aligned}$$ Following the same argument as in [(\[eq:Theta”-1stind1stcontr\])]{}–[(\[eq:Theta”-bd1stbd1\])]{}, we obtain $$\begin{aligned}
{\label{eq:Theta''-bd1stbd2}}
{(\ref{eq:Theta''-prebd1stbd2})}&\leq\sum_{u\in{{\cal A}},\;v_0}\big(\psi_\Lambda(v_0,
v)-\delta_{v_0,v}\big)\,{{\langle \varphi_y\varphi_x \rangle}}_\Lambda{{\langle \varphi_y
\varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_x \rangle}}_\Lambda{{\langle \varphi_y
\varphi_{v_0} \rangle}}_\Lambda{{\langle \varphi_{v_0}\varphi_x \rangle}}_\Lambda{\nonumber}\\
&\leq\sum_{u\in{{\cal A}}}\Big(P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(0)}}(y,x)-
{{\langle \varphi_y\varphi_x \rangle}}_\Lambda{{\langle \varphi_y\varphi_u \rangle}}_\Lambda
{{\langle \varphi_u\varphi_x \rangle}}_\Lambda{{\langle \varphi_y\varphi_v \rangle}}_\Lambda
{{\langle \varphi_v\varphi_x \rangle}}_\Lambda\Big).\end{aligned}$$
Summarizing [(\[eq:Theta”-1stindbd\])]{}, [(\[eq:Theta”-bd1stbd1\])]{} and [(\[eq:Theta”-bd1stbd2\])]{}, we arrive at $$\begin{aligned}
{\label{eq:Theta''-0bdfin}}
{(\ref{eq:contr-(b)})}\leq\sum_{u\in{{\cal A}}}P_{\Lambda;u,v}^{\prime\prime
{{\scriptscriptstyle}(0)}}(y,x).\end{aligned}$$ This completes the bound on the contribution to $\Theta''_{y,x,v;{{\cal A}}}$ from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}}}$ in [(\[eq:Theta’-evdec\])]{}.
**(c)** The contribution to $\Theta'_{y,x;{{\cal A}}}$ from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\\}}}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x}$ in [(\[eq:Theta’-evdec\])]{} equals $$\begin{aligned}
{\label{eq:contr-(c)}}
\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=y{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}
{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}}.\end{aligned}$$ Note that, if ${\mathbbm{1}{\scriptstyle\{{\partial}{{\bf n}}=y{\vartriangle}x\\}}}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x}=1$, then $y$ is ${{\bf n}}$-connected, but not ${{\bf n}}$-doubly connected, to $x$, and therefore there exists at least one pivotal bond for $y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x$. Given an ordered set of bonds $\vec
b_T=(b_1,\dots,b_T)$, we define $$\begin{aligned}
{\label{eq:H-def}}
H_{{{\bf n}};\vec b_T}(y,x)=\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}{\underline{b}}_1\}\cap\bigcap_{i=1}^T\Big\{\{{\overline{b}}_i
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}{\underline{b}}_{i+1}\}\cap\big\{n_{b_i}>0,~b_i\text{ is pivotal for }y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\big\}\Big\},\end{aligned}$$ where, by convention, ${\underline{b}}_{T+1}=x$. Then, by ${\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}$}}}\leq{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}$}}}$, we obtain $$\begin{aligned}
{\label{eq:Theta'-2ndindbd2}}
{(\ref{eq:contr-(c)})}&=\sum_{T\ge1}\sum_{\vec b_T}\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\
{\partial}{{\bf n}}=y{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda
({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,H_{{{\bf n}};\vec b_T}(y,x)
\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x}{\nonumber}\\
&\leq\sum_{T\ge1}\sum_{\vec b_T}\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=y
{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}
{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,H_{{{\bf n}};\vec b_T}(y,x)\,\cap\,
\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x}.\end{aligned}$$
On the event $H_{{{\bf n}};\vec b_T}(y,x)$, we denote the ${{\bf n}}$-double connections between the pivotal bonds $b_1,\dots,b_T$ by $$\begin{aligned}
{{\cal D}}_{{{\bf n}};i}=\begin{cases}
{{\cal C}}_{{\bf n}}^{b_1}(y)&(i=0),\\
{{\cal C}}_{{\bf n}}^{b_{i+1}}(y)\setminus{{\cal C}}_{{\bf n}}^{b_i}(y)&(i=1,\dots,T-1),\\
{{\cal C}}_{{\bf n}}(y)\setminus{{\cal C}}_{{\bf n}}^{b_T}(y)&(i=T).
\end{cases}\end{aligned}$$ As in Figure \[fig:lace-edges\], we can think of ${{\cal C}}_{{\bf n}}(y)$ as the interval $[0,T]$, where each integer $i\in[0,T]$ corresponds to ${{\cal D}}_{{{\bf n}};i}$ and the unit interval $(i-1,i)\subset[0,T]$ corresponds to the pivotal bond $b_i$. Since $y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x$, we see that, for every $b_i$, there must be an $({{\bf m}}+{{\bf n}})$-bypath (i.e., an $({{\bf m}}+{{\bf n}})$-connection that does not go through $b_i$) from some $z\in{{\cal D}}_{{{\bf n}};s}$ with $s<i$ to some $z'\in{{\cal D}}_{{{\bf n}};t}$ with $t\ge i$. We abbreviate $\{s,t\}$ to $st$ if there is no confusion. Let ${{\cal L}}_{[0,T]}^{{\scriptscriptstyle}(1)}=\{\{0T\}\}$, ${{\cal L}}_{[0,T]}^{{\scriptscriptstyle}(2)}=\{\{0t_1,s_2T\}:0<s_2\leq t_1<T\}$ and generally for $j\leq T$ (see Figure \[fig:lace-edges\]),
![\[fig:lace-edges\]An element in ${{\cal L}}_{[0,8]}^{{\scriptscriptstyle}(4)}$, which consists of $s_1t_1=\{0,3\}$, $s_2t_2=\{2,4\}$, $s_3t_3=\{4,6\}$ and $s_4t_4=\{5,8\}$.](lace-edges)
$$\begin{aligned}
{{\cal L}}_{[0,T]}^{{\scriptscriptstyle}(j)}=\big\{\{s_it_i\}_{i=1}^j:0=s_1<s_2\leq t_1<s_3
\leq\cdots\leq t_{j-2}<s_j\leq t_{j-1}<t_j=T\big\}.\end{aligned}$$
For every $j\in\{1,\dots,T\}$, we have $\bigcup_{st\in\Gamma}[s,t]=
[0,T]$ for any $\Gamma\in{{\cal L}}_{[0,T]}^{{\scriptscriptstyle}(j)}$, which implies double connection. Conditioning on ${{\cal C}}_{{\bf n}}(y)\equiv\bigcup_{i=0}^{
\raisebox{-3pt}{$\scriptstyle T$}}{{\cal D}}_{{{\bf n}};i}={{\cal B}}$ (and denoting ${{\bf k}}={{\bf n}}|_{{{\mathbb B}}_{{{\cal B}}{^{\rm c}}}}$, ${{\bf h}}={{\bf n}}|_{{{\mathbb B}}_\Lambda\setminus{{\mathbb B}}_{{{\cal B}}{^{\rm c}}}}$ and ${{\cal D}}_{{{\bf n}};i}\equiv{{\cal D}}_{{{\bf h}};i}={{\cal B}}_i$) and multiplying by $Z_{{{\cal B}}{^{\rm c}}}/Z_{{{\cal B}}{^{\rm c}}}$, we obtain $$\begin{aligned}
{\label{eq:Theta'-2ndindbd2.2}}
{(\ref{eq:Theta'-2ndindbd2})}=\sum_{{{\cal B}}\subset\Lambda}\,\sum_{T\ge1}
\sum_{\vec b_T}\sum_{\substack{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}\\ {\partial}{{\bf h}}=y{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{\tilde w_{\Lambda,
{{\cal B}}}({{\bf h}})\,Z_{{{\cal B}}{^{\rm c}}}}{Z_\Lambda}\,\frac{w_{{{\cal B}}{^{\rm c}}}({{\bf k}})}
{Z_{{{\cal B}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf h}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,H_{{{\bf h}};\vec b_T}(y,x)
\,\cap\,\{{{\cal C}}_{{\bf h}}(y)={{\cal B}}}{\nonumber}\\
\times\sum_{j=1}^T\sum_{\{s_it_i\}_{i=1}^j\in{{\cal L}}_{[0,T]}^{(j)}}
\,\sum_{\substack{z_1,\dots,z_j\\ z'_1,\dots,z'_j}}\bigg(\prod_{i=1}^j
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i\in{{\cal B}}_{s_i},~z'_i\in{{\cal B}}_{t_i}\\}$}}}\,\cap\,\{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}
z'_i}\bigg)\prod_{i\ne l}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\,\cap\,{{\cal C}}_{{{\bf m}}+{{\bf k}}}
(z_l)={\varnothing}\}$}}}.\end{aligned}$$ Reorganizing this expression and then summing over ${{\cal B}}\subset{{\cal A}}$, we obtain $$\begin{aligned}
{\label{eq:Theta'-2ndindbd3}}
{(\ref{eq:Theta'-2ndindbd2.2})}&=\sum_{T\ge1}\sum_{\vec
b_T}\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{w_\Lambda
({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}\,\cap\,H_{{{\bf n}};\vec b_T}
(y,x)$}}}{\nonumber}\\
&\quad\times\sum_{j=1}^T\sum_{\{s_it_i\}_{i=1}^j\in{{\cal L}}_{[0,T]}^{(j)}}\,
\sum_{\substack{z_1,\dots,z_j\\ z'_1,\dots,z'_j}}\bigg(\prod_{i=1}^j
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i\in{{\cal D}}_{{{\bf n}};s_i},~z'_i\in{{\cal D}}_{{{\bf n}};t_i}\}$}}}\bigg){\nonumber}\\
&\quad\times\sum_{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}
{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}
\bigg(\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}z'_i\}$}}}\bigg)\prod_{i\ne l}
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\,\cap\,{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_l)={\varnothing}\}$}}},\end{aligned}$$ where we have denoted ${{\cal C}}_{{\bf n}}(y)$ by $\tilde{{\cal D}}$. In the rightmost expression, the first line determines $\tilde{{\cal D}}$ that contains vertices $z_i,z'_i$ for all $i=1,\dots,j$ in a specific manner, while the second line determines the bypaths ${{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)$ joining $z_i$ and $z'_i$ for every $i=1,\dots,j$. We first derive ${{\bf n}}$-independent bounds on these bypaths in the following paragraph (c-1). Then, in (c-2) below, we will bound the first two lines of the rightmost expression in [(\[eq:Theta’-2ndindbd3\])]{}.
**(c-1)** For $j=1$, the last line of the rightmost expression in [(\[eq:Theta’-2ndindbd3\])]{} simply equals $$\begin{aligned}
{\label{eq:Theta'-2ndindbd3:j=1}}
\sum_{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,
\frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_1
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}z'_1\}$}}}.\end{aligned}$$ Since $z_1,z'_1\in\tilde{{\cal D}}$ and $z_1\ne z'_1$, these two vertices are connected via a nonzero alternating chain of mutually-disjoint ${{\bf m}}$-connected clusters and mutually-disjoint ${{\bf k}}$-connected clusters. Moreover, since $z_1,z'_1\in\tilde{{\cal D}}$ and ${{\bf k}}\in{{\mathbb Z}_+}^{{{\mathbb B}}_{\tilde{{\cal D}}{^{\rm c}}}}$, this chain of bubbles starts and ends with ${{\bf m}}$-connected clusters (possibly with a single ${{\bf m}}$-connected cluster), not with ${{\bf k}}$-connected clusters. Therefore, by following the argument around [(\[eq:ind-bd\])]{}–[(\[eq:nsum-2ndbd\])]{}, we can easily show $$\begin{aligned}
{\label{eq:Theta'-2ndindbd3:j=1bd}}
{(\ref{eq:Theta'-2ndindbd3:j=1})}\leq\sum_{l\ge1}\big(\tilde G_\Lambda^2
\big)^{*(2l-1)}(z_1,z'_1).\end{aligned}$$
For $j\ge2$, since ${{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)$ for $i=1,\dots,j$ are mutually-disjoint due to the last product of the indicators in [(\[eq:Theta’-2ndindbd3\])]{}, we can treat each bypath separately by the conditioning-on-clusters argument. By conditioning on ${{\cal V}}_{{{\bf m}}+{{\bf k}}}\equiv{\mathop{\Dot{\bigcup}}}_{i\ge2}{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)$, the last line in the rightmost expression of [(\[eq:Theta’-2ndindbd3\])]{} equals $$\begin{gathered}
\sum_{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,
\frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}\bigg(\prod_{i=
2}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}z'_i\}$}}}\bigg)\bigg(\prod_{\substack{i,l\ge
2\\ i\ne l}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\,\cap\,{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_l)={\varnothing}\}$}}}
\bigg){\nonumber}\\
\times\sum_{{\partial}{{\bf m}}'={\partial}{{\bf k}}'={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+
{{\bf k}}}{^{\rm c}}}({{\bf m}}')}{Z_{{{\cal A}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}}\,\frac{
w_{\tilde{{\cal D}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}({{\bf k}}')}{Z_{\tilde{{\cal D}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}'+{{\bf k}}'$}}
{\overset{}{\longleftrightarrow}}}z'_1\}$}}}.
{\label{eq:lace-edges}}\end{gathered}$$ By using [(\[eq:Theta’-2ndindbd3:j=1bd\])]{} (and replacing ${{\cal A}}{^{\rm c}}$ and $\tilde{{\cal D}}{^{\rm c}}$ in [(\[eq:Theta’-2ndindbd3:j=1\])]{} by ${{\cal A}}{^{\rm c}}\cap{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}$ and $\tilde{{\cal D}}{^{\rm c}}\cap{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}$, respectively), the second line of [(\[eq:lace-edges\])]{} is bounded by $\sum_{l\ge1}(\tilde G_\Lambda^2)^{*(2l-1)}(z_1,z'_1)$. Repeating the same argument until the remaining products of the indicators are used up, we obtain $$\begin{aligned}
{\label{eq:lace-edgesbd}}
{(\ref{eq:lace-edges})}&\leq\prod_{i=1}^j\sum_{l\ge1}\big(\tilde
G_\Lambda^2\big)^{*(2l-1)}(z_i,z'_i).\end{aligned}$$
We have proved that $$\begin{aligned}
{\label{eq:Theta'-2ndindbd4}}
{(\ref{eq:Theta'-2ndindbd3})}\leq\sum_{j\ge1}\sum_{\substack{z_1,\dots,
z_j\\ z'_1,\dots,z'_j}}\bigg(\prod_{i=1}^j\sum_{l\ge1}\big(\tilde
G_\Lambda^2\big)^{*(2l-1)}(z_i,z'_i)\bigg)\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}
\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}$}}}{\nonumber}\\
\times\sum_{T\ge j}\sum_{\vec b_T}\sum_{\{s_it_i\}_{i=1}^j\in{{\cal L}}_{[0,
T]}^{(j)}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle H_{{{\bf n}};\vec b_T}(y,x)$}}}\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i\in{{\cal D}}_{
{{\bf n}};s_i},\,z'_i\in{{\cal D}}_{{{\bf n}};t_i}\}$}}}.\end{aligned}$$
**(c-2)** Since [(\[eq:Theta’-2ndindbd4\])]{} depends only on a single current configuration, we may use Lemma \[lmm:GHS-BK\] to obtain an upper bound. To do so, we first simplify the second line of [(\[eq:Theta’-2ndindbd4\])]{}, which is, by definition, equal to the indicator of the disjoint union $$\begin{aligned}
{\label{eq:fin-ind}}
&{\mathop{\Dot{\bigcup}}}_{T\ge j}\,{\mathop{\Dot{\bigcup}}}_{\vec b_T}{\mathop{\Dot{\bigcup}}}_{\{s_it_i\}_{i=1}^j\in
{{\cal L}}_{[0,T]}^{{\scriptscriptstyle}(j)}}\bigg\{H_{{{\bf n}};\vec b_T}(y,x)\cap\bigcap_{i=1}^j
\big\{z_i\in{{\cal D}}_{{{\bf n}};s_i},\,z'_i\in{{\cal D}}_{{{\bf n}};t_i}\big\}\bigg\}\\
&={\mathop{\Dot{\bigcup}}}_{e_1,\dots,e_j}\Bigg\{{\mathop{\Dot{\bigcup}}}_{T\ge j}\,{\mathop{\Dot{\bigcup}}}_{\vec b_T}
{\mathop{\Dot{\bigcup}}}_{\substack{\{s_it_i\}_{i=1}^j\in{{\cal L}}_{[0,T]}^{{\scriptscriptstyle}(j)}\\ b_{t_i
+1}=e_{i+1}\;{{}^\forall}i=0,\dots,j-1}}\bigg\{H_{{{\bf n}};\vec b_T}(y,x)\cap
\bigcap_{i=1}^j\big\{z_i\in{{\cal D}}_{{{\bf n}};s_i},\,z'_i\in{{\cal D}}_{{{\bf n}};t_i}\big\}
\bigg\}\Bigg\},{\nonumber}\end{aligned}$$ where $t_0=0$ by convention. On the left-hand side of [(\[eq:fin-ind\])]{}, the first two unions identify the number and location of the pivotal bonds for $y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x$, and the third union identifies the indices of double connections associated with the bypaths between $z_i$ and $z'_i$, for every $i=1,\dots,j$. The union over $e_1,\dots,e_j$ on the right-hand side identifies some of the pivotal bonds $b_1,\dots,b_T$ that are essential to decompose the chain of double connections $H_{{{\bf n}};\vec b_T}(y,x)$ into the following building blocks (see Figure \[fig:I-def\]):
$$\begin{gathered}
I_1(y,z,x)=~~\raisebox{-12pt}{\includegraphics[scale=0.12]{I1}}\hspace{7pc}
I_2(y,z',x)=~~\raisebox{-12pt}{\includegraphics[scale=0.12]{I2}}\\[1pc]
I_3(y,z,z',x)=~~\raisebox{-12pt}{\includegraphics[scale=0.12]{I31}}\quad~
\cup\quad~\raisebox{-12pt}{\includegraphics[scale=0.12]{I32}}\end{gathered}$$
$$\begin{gathered}
I_1(y,z,x)=\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x,~y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}z\},\qquad
I_2(y,z',x)=\bigcup_u\big\{\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}u\}\circ I_1(u,z',x)\big\},
{\label{eq:I12-def}}\\
I_3(y,z,z',x)=\bigcup_u\Big\{\{I_2(y,z,u)\circ I_2(u,z',x)\}\cup\big\{
\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}u\}\circ\{I_1(u,z,x)\cap I_1(u,z',x)\}\big\}\Big\}.
{\label{eq:I3-def}}\end{gathered}$$
For example, since ${{\cal L}}_{[0,T]}^{{\scriptscriptstyle}(1)}=\{\{0T\}\}$, we have $$\begin{aligned}
{\label{eq:fin-ind:=1}}
&({(\ref{eq:fin-ind})}\text{ for }j=1)={\mathop{\Dot{\bigcup}}}_{e_1}{\mathop{\Dot{\bigcup}}}_{T\ge1}\,
{\mathop{\Dot{\bigcup}}}_{\vec b_T:b_1=e_1}\Big\{H_{{{\bf n}};\vec b_T}(y,x)\cap\big\{
z_1\in{{\cal D}}_{{{\bf n}};0},\,z'_1\in{{\cal D}}_{{{\bf n}};T}\big\}\Big\}{\nonumber}\\
&\qquad\subset{\mathop{\Dot{\bigcup}}}_{e_1}\Big\{\big\{I_1(y,z_1,{\underline{e}}_1)\circ I_2
({\overline{e}}_1,z'_1,x)\big\}\cap\big\{n_{e_1}>0,~e_1\text{ is pivotal for }
y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\big\}\Big\}.\end{aligned}$$ It is not hard to see in general that $$\begin{aligned}
{\label{eq:fin-ind:geq2}}
&({(\ref{eq:fin-ind})}\text{ for }j\ge2){\nonumber}\\
&\quad\subset{\mathop{\Dot{\bigcup}}}_{e_1,\dots,e_j}\bigg\{\Big\{I_1(y,z_1,{\underline{e}}_1)
\circ I_3({\overline{e}}_1,z_2,z'_1,{\underline{e}}_2)\circ\cdots\circ I_3({\overline{e}}_{j-1},
z_j,z'_{j-1},{\underline{e}}_j)\circ I_2({\overline{e}}_j,z'_j,x)\Big\}{\nonumber}\\
&\hspace{5pc}\cap\bigcap_{i=1}^j\big\{n_{e_i}>0,~e_i
\text{ is pivotal for }y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x\big\}\bigg\}.\end{aligned}$$
To bound [(\[eq:Theta’-2ndindbd4\])]{} using Lemma \[lmm:GHS-BK\], we further consider an event that includes [(\[eq:fin-ind:=1\])]{}–[(\[eq:fin-ind:geq2\])]{} as subsets. Without losing generality, we can assume that $y\ne{\underline{e}}_1$, ${\overline{e}}_{i-1}\ne{\underline{e}}_i$ for $i=2,\dots,j$, and ${\overline{e}}_j\ne x$; otherwise, the following argument can be simplified. We consider each event $I_i$ in [(\[eq:fin-ind:=1\])]{}–[(\[eq:fin-ind:geq2\])]{} individually, and to do so, we assume that $y$ and ${\underline{e}}_1$ are the only sources for $I_1(y,z_1,{\underline{e}}_1)$, that ${\overline{e}}_{i-1}$ and ${\underline{e}}_i$ are the only sources for $I_3({\overline{e}}_{i-1},z_i,z'_{i-1},{\underline{e}}_i)$ for every $i=2,\dots,j$, and that ${\overline{e}}_j$ and $x$ are the only sources for $I_2({\overline{e}}_j,z'_j,x)$. This is because $y$ and $x$ are the only sources for the entire event [(\[eq:fin-ind:geq2\])]{}, and every $e_i$ is pivotal for $y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}x$.
On $I_1(y,z,x)$ with $y,x$ being the only sources, according to the observation in Step (i) described below [(\[eq:dbbd\])]{}, we have two edge-disjoint connections from $y$ to $z$, one of which may go through $x$, and another edge-disjoint connection from $y$ to $x$ (cf., $I_1(y,z,x)$ in Figure \[fig:I-def\]). Therefore, $$\begin{aligned}
{\label{eq:I1-supset}}
I_1(y,z,x)\subset\big\{{{}^\exists}\omega_1,\omega_2\in\Omega_{y\to z}^{{\bf n}}\,{{}^\exists}\omega_3\in\Omega_{y\to x}^{{\bf n}}\text{ such that }~\omega_i
\cap\omega_l={\varnothing}~(i\ne l)\big\}.\end{aligned}$$ Similarly, for $I_2(y,z',x)$ with $y,x$ being the only sources (cf., $I_2(y,z',x)$ in Figure \[fig:I-def\]), $$\begin{aligned}
{\label{eq:I2-supset}}
I_2(y,z',x)\subset\big\{{{}^\exists}\omega_1,\omega_2\in\Omega_{x\to
z'}^{{\bf n}}\,{{}^\exists}\omega_3\in\Omega_{y\to x}^{{\bf n}}\text{ such that }
~\omega_i\cap\omega_l={\varnothing}~(i\ne l)\big\}.\end{aligned}$$
On $I_3(y,z,z',x)$ with $y,x$ being the only sources, there are at least three edge-disjoint paths, one from $y$ to $z$, another one from $z$ to $z'$, and another one from $z'$ to $x$. It is not hard to see this from $\bigcup_u\{I_2(y,z,u)\circ I_2(u,z',x)\}$ in [(\[eq:I3-def\])]{}, which corresponds to the first event depicted in Figure \[fig:I-def\]. It is also possible to extract such three edge-disjoint paths from the remaining event in [(\[eq:I3-def\])]{}. See the second event depicted in Figure \[fig:I-def\] for one of the worst topological situations. Since there are at least three edge-disjoint paths between $u$ and $x$, say, $\zeta_1,\zeta_2$ and $\zeta_3$, we can go from $y$ to $z$ via $\zeta_1$ and a part of $\zeta_2$, and go from $z$ to $z'$ via the middle part of $\zeta_2$, and then go from $z'$ to $x$ via the remaining part of $\zeta_2$ and $\zeta_3$. The other cases can be dealt with similarly. As a result, we have $$\begin{aligned}
{\label{eq:I3-supset}}
I_3(y,z,z',x)\subset\big\{{{}^\exists}\omega_1\in\Omega_{y\to z}^{{\bf n}}\,{{}^\exists}\omega_2\in\Omega_{z\to z'}^{{\bf n}}\,{{}^\exists}\omega_3\in\Omega_{z'\to x}^{{\bf n}}\text{ such that }\omega_i\cap\omega_l={\varnothing}~(i\ne l)\big\}.\end{aligned}$$
Since $$\begin{aligned}
\bigcup_e\Big\{\big\{\{{{}^\exists}\omega\in\Omega_{z\to{\underline{e}}}^{{\bf n}}\}\circ\{{{}^\exists}\omega\in\Omega_{{\overline{e}}\to z'}^{{\bf n}}\}\big\}\cap\{n_e>0\}\Big\}\subset\{{{}^\exists}\omega\in\Omega_{z\to z'}^{{\bf n}}\},\end{aligned}$$ we see that [(\[eq:fin-ind:=1\])]{} is a subset of $$\begin{aligned}
{\label{eq:tildeI-def:=1}}
\tilde I_{z_1,z'_1}^{{\scriptscriptstyle}(1)}(y,x)=\left\{\!
\begin{array}{c}
{{}^\exists}\omega_1,\omega_2\in\Omega_{z_1\to y}^{{\bf n}}\;{{}^\exists}\omega_3\in
\Omega_{y\to x}^{{\bf n}}\;{{}^\exists}\omega_4,\omega_5\in\Omega_{x\to z'_1}^{{\bf n}}\\
\text{such that }~\omega_i\cap\omega_l={\varnothing}~(i\ne l)
\end{array}\!\right\},\end{aligned}$$ and that [(\[eq:fin-ind:geq2\])]{} is a subset of (see Figure \[fig:eventI\]) $$\begin{aligned}
{\label{eq:tildeI-def:geq2}}
\tilde I_{\vec z_j,\vec z'_j}^{{\scriptscriptstyle}(j)}(y,x)=\left\{\!
\begin{array}{c}
{{}^\exists}\omega_1,\omega_2\in\Omega_{z_1\to y}^{{\bf n}}\;{{}^\exists}\omega_3\in
\Omega_{y\to z_2}^{{\bf n}}\;{{}^\exists}\omega_4\in\Omega_{z_2\to z'_1}^{{\bf n}}\;
{{}^\exists}\omega_5\in\Omega_{z'_1\to z_3}^{{\bf n}}\cdots\\
\cdots{{}^\exists}\omega_{2j}\in\Omega_{z_j\to z'_{j-1}}^{{\bf n}}\,{{}^\exists}\omega_{
2j+1}\in\Omega_{z'_{j-1}\to x}^{{\bf n}}\;{{}^\exists}\omega_{2j+2},\omega_{2j+3}
\in\Omega_{x\to z'_j}^{{\bf n}}\\
\text{such that }~\omega_i\cap\omega_l={\varnothing}~(i\ne l)
\end{array}\!\right\},\end{aligned}$$
![\[fig:eventI\]A schematic representation of $\tilde I_{\vec z_j,\vec z'_j}^{{\scriptscriptstyle}(j)}(y,x)$ for $j\ge2$ consisting of $2j+3$ edge-disjoint paths on ${{\mathbb G}}_{{\bf n}}$.](eventI)
where $\vec z_j^{(\prime)}=(z_1^{(\prime)},\dots,z_j^{(\prime)})$. Therefore, $$\begin{aligned}
{\label{eq:Theta'-2ndindbd5}}
{(\ref{eq:Theta'-2ndindbd4})}\leq\sum_{j\ge1}\sum_{\substack{z_1,\dots,z_j\\
z'_1,\dots,z'_j}}\bigg(\prod_{i=1}^j\sum_{l\ge1}\big(\tilde G_\Lambda^2
\big)^{*(2l-1)}(z_i,z'_i)\bigg)\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{w_\Lambda
({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \tilde I_{\vec z_j,
\vec z'_j}^{{\scriptscriptstyle}(j)}(y,x)$}}}.\end{aligned}$$
Now we apply Lemma \[lmm:GHS-BK\] to bound [(\[eq:Theta’-2ndindbd5\])]{}. To clearly understand how it is applied, for now we ignore ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}}}$ in [(\[eq:Theta’-2ndindbd5\])]{} and only consider the contribution from ${\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \tilde I_{\vec z_j, \vec z'_j}^{{\scriptscriptstyle}(j)}(y,x)$}}}$. Without losing generality, we assume that $y,x,z_i,z'_i$ for $i=1,\dots,j$ are all different. Since there are $2j+3$ edge-disjoint paths on ${{\mathbb G}}_{{\bf n}}$ as in [(\[eq:tildeI-def:=1\])]{}–[(\[eq:tildeI-def:geq2\])]{} (see also Figure \[fig:eventI\]), we multiply [(\[eq:Theta’-2ndindbd5\])]{} by $(Z_\Lambda/Z_\Lambda)^{2j+2}$, following Step (ii) of the strategy described in Section \[ss:pi0bd\]. Overlapping the $2j+3$ current configurations and using Lemma \[lmm:GHS-BK\] with ${{\cal V}}=\{y,x\}$ and $k=2j+2$, we obtain $$\begin{aligned}
{\label{eq:Theta'-2ndindbd6}}
\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}&\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \tilde
I_{\vec z_j,\vec z'_j}^{{\scriptscriptstyle}(j)}(y,x)$}}}\leq{{\langle \varphi_{z_1}\varphi_y
\rangle}}_\Lambda^2{{\langle \varphi_x\varphi_{z'_j} \rangle}}_\Lambda^2\\
&\times\begin{cases}
{\displaystyle}{{\langle \varphi_y\varphi_x \rangle}}_\Lambda&(j=1),\\
{\displaystyle}{{\langle \varphi_y\varphi_{z_2} \rangle}}_\Lambda{{\langle \varphi_{z_2}\varphi_{
z'_1} \rangle}}_\Lambda\bigg(\prod_{i=2}^{j-1}{{\langle \varphi_{z'_{i-1}}\varphi_{
z_{i+1}} \rangle}}_\Lambda{{\langle \varphi_{z_{i+1}}\varphi_{z'_i} \rangle}}_\Lambda\bigg)
{{\langle \varphi_{z'_{j-1}}\varphi_x \rangle}}_\Lambda&(j\ge2).
\end{cases}{\nonumber}\end{aligned}$$ Note that, by [(\[eq:G-delta-bd\])]{}, we have $$\begin{aligned}
\left.\begin{array}{r}
\sum_{l\ge1}(\tilde G_\Lambda^2)^{*(2l-1)}(y,x)\\[5pt]
\sum_z{{\langle \varphi_z\varphi_y \rangle}}_\Lambda^2\sum_{l\ge1}(\tilde
G_\Lambda^2)^{*(2l-1)}(z,x)\\[5pt]
\sum_{z'}{{\langle \varphi_x\varphi_{z'} \rangle}}_\Lambda^2\sum_{l\ge1}
(\tilde G_\Lambda^2)^{*(2l-1)}(y,z')
\end{array}\right\}&\leq\psi_\Lambda(y,x)-\delta_{y,x},\\[5pt]
\sum_{z,z'}{{\langle \varphi_z\varphi_y \rangle}}_\Lambda^2{{\langle \varphi_x
\varphi_{z'} \rangle}}_\Lambda^2\sum_{l\ge1}\big(\tilde G_\Lambda^2
\big)^{*(2l-1)}(z,z')&\leq2\big(\psi_\Lambda(y,x)-\delta_{y,x}\big).\end{aligned}$$ Therefore, [(\[eq:Theta’-2ndindbd5\])]{} without ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}}}$ is bounded by $$\begin{aligned}
{\label{eq:Theta'-2ndindbd7}}
&{{\langle \varphi_y\varphi_x \rangle}}_\Lambda\sum_{z_1,z'_1}{{\langle \varphi_{z_1}
\varphi_y \rangle}}_\Lambda^2{{\langle \varphi_x\varphi_{z'_1} \rangle}}_\Lambda^2\sum_{l
\ge1}\big(\tilde G_\Lambda^2\big)^{*(2l-1)}(z_1,z'_1){\nonumber}\\
&+\sum_{j\ge2}\sum_{\substack{z_2,\dots,z_j\\ z'_1,\dots,z'_{j-1}}}
\bigg(\prod_{i=2}^{j-1}\big(\psi_\Lambda(z_i,z'_i)-\delta_{z_i,z'_i}
\big)\bigg)\bigg(\sum_{z_1}{{\langle \varphi_y\varphi_{z_1} \rangle}}_\Lambda^2
\sum_{l\ge1}\big(\tilde G_\Lambda^2\big)^{*(2l-1)}(z_1,z'_1)\bigg)
{\nonumber}\\
&\hspace{4pc}\times\bigg(\sum_{z'_j}{{\langle \varphi_x\varphi_{z'_j}
\rangle}}_\Lambda^2\sum_{l\ge1}\big(\tilde G_\Lambda^2\big)^{*(2l-1)}(z_j,
z'_j)\bigg){{\langle \varphi_y\varphi_{z_2} \rangle}}_\Lambda{{\langle \varphi_{z_2}
\varphi_{z'_1} \rangle}}_\Lambda{\nonumber}\\
&\hspace{4pc}\times\bigg(\prod_{i=2}^{j-1}{{\langle \varphi_{z'_{i-1}}
\varphi_{z_{i+1}} \rangle}}_\Lambda{{\langle \varphi_{z_{i+1}}\varphi_{z'_i} \rangle}}_\Lambda
\bigg){{\langle \varphi_{z'_{j-1}}\varphi_x \rangle}}_\Lambda\leq\sum_{j\ge1}
P_\Lambda^{{\scriptscriptstyle}(j)}(y,x).\end{aligned}$$
If ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}}}$ is present in the above argument, then at least one of the paths $\omega_i$ for $i=3,\dots,2j+1$ has to go through ${{\cal A}}$. For example, if $\omega_3~(\in\Omega_{y\to
z_2}^{{\bf n}})$ goes through ${{\cal A}}$, then we can split it into two edge-disjoint paths at some $u\in{{\cal A}}$, such as $\omega'_3\in\Omega_{y\to u}^{{\bf n}}$ and $\omega''_3\in\Omega_{u\to
z_2}^{{\bf n}}$. The contribution from this case is bounded, by following the same argument as above, by [(\[eq:Theta’-2ndindbd6\])]{} with ${{\langle \varphi_y\varphi_{z_2} \rangle}}_\Lambda$ being replaced by $\sum_{u\in{{\cal A}}}
{{\langle \varphi_y\varphi_u \rangle}}_\Lambda{{\langle \varphi_u\varphi_{z_2} \rangle}}_\Lambda$. Bounding the other $2j-2$ cases similarly and summing these bounds over $j\ge1$, we obtain $$\begin{aligned}
{\label{eq:Theta'-2ndindbd8}}
{(\ref{eq:Theta'-2ndindbd5})}\leq\sum_{u\in{{\cal A}}}\sum_{j\ge1}P_{\Lambda;
u}^{\prime{{\scriptscriptstyle}(j)}}(y,x).\end{aligned}$$
This together with [(\[eq:Theta’-bd1stbd\])]{} in the above paragraph (a) complete the proof of the bound on $\Theta'_{y,x;{{\cal A}}}$ in [(\[eq:Theta’Theta”bd\])]{}.
**(d)** Finally, we investigate the contribution to $\Theta''_{y,x,v;{{\cal A}}}$ from ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\\}}}\setminus
\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x}$ in [(\[eq:Theta’-evdec\])]{}: $$\begin{aligned}
{\label{eq:contr-(d)}}
\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf n}}=y{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}
{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,\{\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}\setminus\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}\}\,\cap
\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v}.\end{aligned}$$ Using $H_{{{\bf n}};\vec b_T}(y,x)$ defined in [(\[eq:H-def\])]{}, we can write [(\[eq:contr-(d)\])]{} as (cf., [(\[eq:Theta’-2ndindbd2\])]{}) $$\begin{aligned}
{\label{eq:Theta''-2ndindrewr}}
{(\ref{eq:contr-(d)})}=\sum_{T\ge1}\sum_{\vec b_T}\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\
{\partial}{{\bf n}}=y{\vartriangle}x}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_\Lambda
({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\\}$}}}\,\cap\,H_{{{\bf n}};\vec b_T}(y,x)
\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\Longleftrightarrow}}}x\}\,\cap\,\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{}{\longleftrightarrow}}}v}.\end{aligned}$$ To bound this, we will also use a similar expression to [(\[eq:Theta’-2ndindbd3\])]{}, in which ${{\bf k}}={{\bf n}}|_{{{\mathbb B}}_{\tilde{{\cal D}}{^{\rm c}}}}$ with $\tilde{{\cal D}}={{\cal C}}_{{\bf n}}^b(y)$. We investigate [(\[eq:Theta”-2ndindrewr\])]{} separately (in the following paragraphs (d-1) and (d-2)) depending on whether or not there is a bypath ${{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)$ for some $i\in\{1,\dots,j\}$ containing $v$.
**(d-1)** If there is such a bypath, then we use ${\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}}}\leq{\mathbbm{1}{\scriptstyle\{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}}}$ as in [(\[eq:Theta’-2ndindbd2\])]{} to bound the contribution from this case to [(\[eq:Theta”-2ndindrewr\])]{} by $$\begin{aligned}
{\label{eq:Theta''-2ndindbd1}}
\sum_{T\ge1}\sum_{\vec b_T}\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}
{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}\,\cap\,H_{{{\bf n}};\vec b_T}(y,x)$}}}
\sum_{j=1}^T\,\sum_{\{s_it_i\}_{i=1}^j\in{{\cal L}}_{[0,T]}^{(j)}}\,\sum_{
\substack{z_1,\dots,z_j\\ z'_1,\dots,z'_j}}\bigg(\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i
\in{{\cal D}}_{{{\bf n}};s_i},~z'_i\in{{\cal D}}_{{{\bf n}};t_i}\}$}}}\bigg){\nonumber}\\
\times\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf k}}={\varnothing}}}\frac{w_{{{\cal A}}{^{\rm c}}}
({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}\bigg(\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}z'_i\}$}}}\bigg)\bigg(
\prod_{i\ne l}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\,\cap\,{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_l)={\varnothing}\}$}}}
\bigg)\sum_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v\in{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\}$}}}.\end{aligned}$$ Note that the last sum of the indicators is the only difference from [(\[eq:Theta’-2ndindbd3\])]{}.
When $j=1$, the second line of [(\[eq:Theta”-2ndindbd1\])]{} equals $$\begin{aligned}
{\label{eq:Theta''-2ndindbd1:j=1}}
\sum_{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,
\frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_1
{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}z'_1\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\}$}}}.\end{aligned}$$ As described in [(\[eq:Theta’-2ndindbd3:j=1\])]{}–[(\[eq:Theta’-2ndindbd3:j=1bd\])]{}, we can bound [(\[eq:Theta”-2ndindbd1:j=1\])]{} without ${\mathbbm{1}{\scriptstyle\{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\}}}$ by a chain of bubbles $\sum_{l\ge1}(\tilde G_\Lambda^2)^{*(2l-1)}(z_1,z'_1)$. If ${\mathbbm{1}{\scriptstyle\{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\}}}=1$, then, by the argument around [(\[eq:ind-bd\])]{}–[(\[eq:nsum-2ndbd\])]{}, one of the bubbles has an extra vertex $v'$ that is further connected to $v$ with another chain of bubbles $\psi_\Lambda(v',v)$. That is, the effect of ${\mathbbm{1}{\scriptstyle\{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\}}}$ is to replace one of the $\tilde G_\Lambda$’s in the chain of bubbles, say, $\tilde G_\Lambda(a,a')$, by $\sum_{v'}({{\langle \varphi_{a}
\varphi_{v'} \rangle}}_\Lambda\tilde G_\Lambda(v',a')+\tilde G_\Lambda
(a,a')\delta_{v',a'})\,\psi_\Lambda(v',v)$. Let $$\begin{aligned}
{\label{eq:g-def}}
g_{\Lambda;y}(z,z')=\sum_{l\ge1}\sum_{i=1}^{2l-1}\sum_{a,a'}\big(
\tilde G_\Lambda^2\big)^{*(i-1)}(z,a)\,\tilde G_\Lambda(a,a')\,
\big(\tilde G_\Lambda^2\big)^{*(2l-1-i)}(a',z'){\nonumber}\\
\times\Big({{\langle \varphi_a\varphi_y \rangle}}_\Lambda\tilde G_\Lambda(y,a')
+\tilde G_\Lambda(a,a')\,\delta_{y,a'}\Big).\end{aligned}$$ Then, we have $$\begin{aligned}
{\label{eq:Theta''-2ndindbd1:j=1bd}}
{(\ref{eq:Theta''-2ndindbd1:j=1})}\leq\sum_{v'}g_{\Lambda;v'}(z_1,z'_1)
\,\psi_\Lambda(v',v).\end{aligned}$$
Let $j\ge2$ and consider the contribution to [(\[eq:Theta”-2ndindbd1\])]{} from ${\mathbbm{1}{\scriptstyle\{v\in{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_1)\}}}$; the contribution from ${\mathbbm{1}{\scriptstyle\{v\in{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\}}}$ with $i\ne1$ can be estimated in the same way. By conditioning on ${{\cal V}}_{{{\bf m}}+{{\bf k}}}\equiv{\mathop{\Dot{\bigcup}}}_{i\ge2}{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)$ as in [(\[eq:lace-edges\])]{}, the contribution to the second line of [(\[eq:Theta”-2ndindbd1\])]{} from ${\mathbbm{1}{\scriptstyle\{v\in{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_1)\}}}\equiv{\mathbbm{1}{\scriptstyle\{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\}}}$ equals $$\begin{aligned}
{\label{eq:Theta''-2ndindbd2}}
&\sum_{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,
\frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}\bigg(\prod_{i=
2}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}z'_i\}$}}}\bigg)\bigg(\prod_{\substack{i,i'\ge
2\\ i\ne i'}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\,\cap\,{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_{i'})=
{\varnothing}\}$}}}\bigg){\nonumber}\\
&\qquad\times\sum_{{\partial}{{\bf m}}'={\partial}{{\bf k}}'={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}\cap\,{{\cal V}}_{
{{\bf m}}+{{\bf k}}}{^{\rm c}}}({{\bf m}}')}{Z_{{{\cal A}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}}\,
\frac{w_{\tilde{{\cal D}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}({{\bf k}}')}{Z_{\tilde
{{\cal D}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}'+{{\bf k}}'$}}
{\overset{}{\longleftrightarrow}}}z'_1\}$}}}
\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}'+{{\bf k}}'$}}
{\overset{}{\longleftrightarrow}}}v\}$}}},\end{aligned}$$ where the second line is bounded by [(\[eq:Theta”-2ndindbd1:j=1bd\])]{} for $j=1$, and then the first line is bounded by $\prod_{i=2}^j\sum_{l\ge1}(\tilde G_\Lambda^2)^{*(2l-1)}(z_i,z'_i)$, due to [(\[eq:lace-edges\])]{}–[(\[eq:lace-edgesbd\])]{}.
Summarizing the above bounds, we have (cf., [(\[eq:Theta’-2ndindbd5\])]{}) $$\begin{aligned}
{\label{eq:Theta''-2ndindbd2.2}}
{(\ref{eq:Theta''-2ndindbd1})}\leq\sum_{j\ge1}\sum_{\substack{z_1,\dots,
z_j\\ z'_1,\dots,z'_j}}\bigg(&\sum_{h=1}^j\sum_{v'}g_{\Lambda;v'}
(z_h,z'_h)\,\psi_\Lambda(v',v)\prod_{i\ne h}\sum_{l\ge1}\big(
\tilde G_\Lambda^2\big)^{*(2l-1)}(z_i,z'_i)\bigg){\nonumber}\\
&\times\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{
y{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}x\}$}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \tilde I_{\vec z_j,\vec z'_j}^{{\scriptscriptstyle}(j)}(y,
x)$}}},\end{aligned}$$ to which we can apply the bound discussed between [(\[eq:Theta’-2ndindbd2\])]{} and [(\[eq:Theta’-2ndindbd8\])]{}.
**(d-2)** If $v\notin{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)$ for any $i=1,\dots,j$, then there exists a $v'\in{{\cal D}}_{{{\bf n}};l}$ for some $l\in\{0,\dots,T\}$ such that $v'{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v$ and ${{\cal C}}_{{{\bf m}}+{{\bf k}}}(v')\cap{{\cal C}}_{{{\bf m}}+{{\bf k}}} (z_i)={\varnothing}$ for any $i$. In addition, since all connections from $y$ to $x$ on the graph $\tilde{{\cal D}}\cup{\mathop{\Dot{\bigcup}}}_{i=1}^{\raisebox{-2pt} {$\scriptstyle
j$}}{{\cal C}}_{{{\bf m}}+{{\bf k}}} (z_i)$ have to go through ${{\cal A}}$, there is an $h\in\{1,\dots,j\}$ such that $z_h{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}z'_h$. Therefore, the contribution from this case to [(\[eq:Theta”-2ndindrewr\])]{} is bounded by $$\begin{gathered}
\sum_{T\ge1}\sum_{\vec b_T}\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\!\frac{w_\Lambda({{\bf n}})}
{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle H_{{{\bf n}};\vec b_T}(y,x)$}}}\sum_{j=1}^T\sum_{\{s_it_i\}_{i
=1}^j\in{{\cal L}}_{[0,T]}^{(j)}}\sum_{\substack{v',z_1,\dots,z_j\\ z'_1,\dots,
z'_j}}\!\bigg(\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i\in{{\cal D}}_{{{\bf n}};s_i},\;z'_i\in{{\cal D}}_{{{\bf n}};
t_i}\}$}}}\bigg)\sum_{l=0}^T{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v'\in{{\cal D}}_{{{\bf n}};l}\}$}}}{\nonumber}\\
\times\sum_{\substack{{\partial}{{\bf m}}={\varnothing}\\ {\partial}{{\bf k}}={\varnothing}}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}
{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}
\bigg(\sum_{h=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_h{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}z'_h\}$}}}
\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}z'_i\}$}}}\bigg)\bigg(\prod_{i\ne i'}
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\,\cap\,{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_{i'})={\varnothing}\}$}}}\bigg){\nonumber}\\
\times{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v'{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}v\}$}}}\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(v')\,\cap\,
{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)={\varnothing}\}$}}},{\label{eq:Theta''-2ndindbd3}}\end{gathered}$$ where, by conditioning on ${{\cal S}}_{{{\bf m}}+{{\bf k}}}\equiv{\mathop{\Dot{\bigcup}}}_{i=1}^{\raisebox
{-2pt}{$\scriptstyle j$}}{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)$, the last two lines are (see below [(\[eq:lace-edges\])]{}) $$\begin{gathered}
\sum_{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,
\frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}\bigg(\sum_{h=1}^j
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_h{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}z'_h\}$}}}\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}z'_i\}$}}}
\bigg)\bigg(\prod_{i\ne i'}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\,\cap\,{{\cal C}}_{{{\bf m}}+{{\bf k}}}
(z_{i'})={\varnothing}\}$}}}\bigg){\nonumber}\\
\times\underbrace{\sum_{{\partial}{{\bf m}}''={\partial}{{\bf k}}''={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}\cap\,
{{\cal S}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}({{\bf m}}'')}{Z_{{{\cal A}}{^{\rm c}}\cap\,{{\cal S}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}}\,
\frac{w_{\tilde{{\cal D}}{^{\rm c}}\cap\,{{\cal S}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}({{\bf k}}'')}{Z_{\tilde{{\cal D}}{^{\rm c}}\cap\,{{\cal S}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v'{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}''+{{\bf k}}''$}}
{\overset{}{\longleftrightarrow}}}v\}$}}}}_{\leq\;
\psi_\Lambda(v',v)}.{\label{eq:Theta''-2ndindbd3-l2,3}}\end{gathered}$$
When $j=1$, we have $$\begin{aligned}
{\label{eq:Theta''-2ndindbd3-l2,3:j=1}}
({(\ref{eq:Theta''-2ndindbd3-l2,3})}\text{ for }j=1)\leq\psi_\Lambda
(v',v)\sum_{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,\frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}\,
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}z'_1\}$}}}.\end{aligned}$$ If we ignore the “through ${{\cal A}}$”-condition in the last indicator, then the sum is bounded, as in [(\[eq:Theta’-2ndindbd3:j=1bd\])]{}, by a chain of bubbles $\sum_{l\ge1}(\tilde
G_\Lambda^2)^{*(2l-1)}(z_1,z'_1)$. However, because of this condition, one of the $\tilde G_\Lambda$’s in the bound, say, $\tilde G_\Lambda(a,a')$, is replaced by $\sum_{u\in{{\cal A}}}({{\langle \varphi_{a}\varphi_u \rangle}}_\Lambda\tilde
G_\Lambda(u,a') +\tilde G_\Lambda(a,a')\delta_{u,a'})$. Using [(\[eq:g-def\])]{}, we have $$\begin{aligned}
{\label{eq:Theta''-2ndindbd3-l2,3:j=1bd}}
{(\ref{eq:Theta''-2ndindbd3-l2,3:j=1})}\leq\psi_\Lambda(v',v)\sum_{y\in{{\cal A}}}
g_{\Lambda;y}(z_1,z'_1).\end{aligned}$$
Let $j\ge2$ and consider the contribution to [(\[eq:Theta”-2ndindbd3-l2,3\])]{} from ${\mathbbm{1}{\scriptstyle\{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}z'_1\}}}$; the contributions from ${\mathbbm{1}{\scriptstyle\{z_h{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}z'_h\}}}$ with $h\ne1$ can be estimated similarly. By conditioning on ${{\cal V}}_{{{\bf m}}+{{\bf k}}}\equiv{\mathop{\Dot{\bigcup}}}_{i\ge2}{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)$, the contribution to [(\[eq:Theta”-2ndindbd3-l2,3\])]{} from ${\mathbbm{1}{\scriptstyle\{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}z'_1\}}}$ equals $$\begin{aligned}
{\label{eq:Theta''-2ndindbd4}}
\sum_{{\partial}{{\bf m}}={\partial}{{\bf k}}={\varnothing}}&\frac{w_{{{\cal A}}{^{\rm c}}}({{\bf m}})}{Z_{{{\cal A}}{^{\rm c}}}}\,
\frac{w_{\tilde{{\cal D}}{^{\rm c}}}({{\bf k}})}{Z_{\tilde{{\cal D}}{^{\rm c}}}}\bigg(\prod_{i=
2}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}+{{\bf k}}$}}
{\overset{}{\longleftrightarrow}}}z'_i\}$}}}\bigg)\bigg(\prod_{\substack{i,i'\ge
2\\ i\ne i'}}{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_i)\,\cap\,{{\cal C}}_{{{\bf m}}+{{\bf k}}}(z_{i'})=
{\varnothing}\}$}}}\bigg){\nonumber}\\
&\times\psi_\Lambda(v',v)\sum_{{\partial}{{\bf m}}'={\partial}{{\bf k}}'={\varnothing}}\frac{w_{{{\cal A}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}({{\bf m}}')}{Z_{{{\cal A}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}
{^{\rm c}}}}\,\frac{w_{\tilde{{\cal D}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}({{\bf k}}')}
{Z_{\tilde{{\cal D}}{^{\rm c}}\cap\,{{\cal V}}_{{{\bf m}}+{{\bf k}}}{^{\rm c}}}}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_1{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf m}}'+
{{\bf k}}'$}}
{\overset{{{\cal A}}}{\longleftrightarrow}}}z'_1\}$}}},\end{aligned}$$ where the second line is bounded by [(\[eq:Theta”-2ndindbd3-l2,3:j=1bd\])]{} for $j=1$, and then the first line is bounded by $\prod_{i=2}^j\sum_{l\ge1}(\tilde G_\Lambda^2)^{
*(2l-1)}(z_i,z'_i)$, as described below [(\[eq:Theta”-2ndindbd2\])]{}.
As a result, [(\[eq:Theta”-2ndindbd3\])]{} is bounded by $$\begin{aligned}
{\label{eq:Theta''-2ndindbd5}}
&\sum_{j\ge1}\sum_{\substack{v'\!,z_1,\dots,z_j\\ z'_1,\dots,z'_j}}
\psi_\Lambda(v',v)\bigg(\sum_{h=1}^j\sum_{y\in{{\cal A}}}g_{\Lambda;y}(z_h,
z'_h)\prod_{i\ne h}\sum_{l\ge1}\big(\tilde G_\Lambda^2\big)^{*(2l-
1)}(z_i,z'_i)\bigg){\nonumber}\\
&\times\sum_{{\partial}{{\bf n}}=y{\vartriangle}x}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\sum_{T
\ge j}\sum_{\vec b_T}\sum_{\{s_it_i\}_{i=1}^j\in{{\cal L}}_{[0,T]}^{(j)}}
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle H_{{{\bf n}};\vec b_T}(y,x)$}}}\bigg(\prod_{i=1}^j{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{z_i\in{{\cal D}}_{{{\bf n}};
s_i},\,z'_i\in{{\cal D}}_{{{\bf n}};t_i}\}$}}}\bigg)\sum_{l=0}^T{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{v'\in{{\cal D}}_{{{\bf n}};l}\}$}}}.\end{aligned}$$ The second line can be bounded by following the argument between [(\[eq:Theta’-2ndindbd4\])]{} and [(\[eq:Theta’-2ndindbd7\])]{}; note that the sum of the indicators in [(\[eq:Theta”-2ndindbd5\])]{}, except for the last factor $\sum_{l=0}^T{\mathbbm{1}{\scriptstyle\{v'\in{{\cal D}}_{{{\bf n}};l}\}}}$, is identical to that in [(\[eq:Theta’-2ndindbd4\])]{}. First, we rewrite the sum of the indicators in [(\[eq:Theta”-2ndindbd5\])]{} as a single indicator of an event ${{\cal E}}$ similar to [(\[eq:fin-ind\])]{}. Then, we construct another event similar to $\tilde I^{{\scriptscriptstyle}(j)}_{\vec z_j,\vec z'_j}(y,x)$ in [(\[eq:tildeI-def:=1\])]{}–[(\[eq:tildeI-def:geq2\])]{}, of which ${{\cal E}}$ is a subset. Due to $\sum_{l=0}^T{\mathbbm{1}{\scriptstyle\{v'\in{{\cal D}}_{{{\bf n}};l}\}}}$ in [(\[eq:Theta”-2ndindbd5\])]{}, one of the paths in the definition of $\tilde I^{{\scriptscriptstyle}(j)}_{\vec z_j,\vec z'_j}(y,x)$, say, $\omega_i\in\Omega^{{{\bf n}}}_{a\to a'}$ for some $a,a'$ (depending on $i$) is split into two edge-disjoint paths $\omega'_i\in\Omega^{{{\bf n}}}_{a\to v'}$ and $\omega''_i\in\Omega^{{{\bf n}}}_{v'\to a'}$, followed by the summation over $i=3,\dots,2j+1$ (cf., Figure \[fig:eventI\]). Finally, we apply Lemma \[lmm:GHS-BK\] to obtain the desired bound on the last line of [(\[eq:Theta”-2ndindbd5\])]{}.
Summarizing the above (d-1) and (d-2), we obtain $$\begin{aligned}
{(\ref{eq:Theta''-2ndindrewr})}\leq\sum_{j\ge1}\sum_{u\in{{\cal A}}}P_{
\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(j)}}(y,x).\end{aligned}$$
This together with [(\[eq:Theta”-0bdfin\])]{} in the above paragraph (b) complete the proof of the bound on $\Theta''_{y,x,v;{{\cal A}}}$ in [(\[eq:Theta’Theta”bd\])]{}.
Bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ assuming the decay of $G(x)$
=================================================================================
Using the diagrammatic bounds proved in the previous section, we prove Proposition \[prp:GimpliesPix\] in Section \[ss:proof-so\], and Propositions \[prp:GimpliesPik\] and \[prp:exp-bootstrap\](iii) in Section \[ss:proof-nn\].
Bounds for the spread-out model {#ss:proof-so}
-------------------------------
We prove Proposition \[prp:GimpliesPix\] for the spread-out model using the following convolution bounds:
\[prp:conv-star\]
(i) Let $a\ge b>0$ and $a+b>d$. There is a $C=C(a,b,d)$ such that $$\begin{aligned}
{\label{eq:conv}}
\sum_y\frac1{{\vby-v{|\!|\!|}}^a}\,\frac1{{\vbx-y{|\!|\!|}}^b}\leq\frac{C}
{{\vbx-v{|\!|\!|}}^{(a\wedge d+b)-d}}.\end{aligned}$$
(ii) Let $q\in(\frac{d}2,d)$. There is a $C'=C'(d,q)$ such that $$\begin{aligned}
\sum_z\frac1{{\vbx-z{|\!|\!|}}^q}\,\frac1{{\vbx'-z{|\!|\!|}}^q}\,\frac1{{\vbz-y{|\!|\!|}}^q}\,
\frac1{{\vbz-y'{|\!|\!|}}^q}\leq\frac{C'}{{\vbx-y{|\!|\!|}}^q{\vbx'-y'{|\!|\!|}}^q}.{\label{eq:star}}\end{aligned}$$
The inequality [(\[eq:conv\])]{} is identical to [@hhs03 Proposition 1.7(i)]. We use this to prove [(\[eq:star\])]{}. By the triangle inequality, we have $\frac12{\vbx-y{|\!|\!|}}\leq{\vbx-z{|\!|\!|}}\vee{\vbz-y{|\!|\!|}}$ and $\frac12{\vbx'-y'{|\!|\!|}}\leq{\vbx'-z{|\!|\!|}}\vee{\vbz-y'{|\!|\!|}}$. Suppose that ${\vbx-z{|\!|\!|}}\leq{\vbz-y{|\!|\!|}}$ and ${\vbx'-z{|\!|\!|}}\leq{\vbz-y'{|\!|\!|}}$. Then, by [(\[eq:conv\])]{} with $a=b=q$, the contribution from this case is bounded by $$\begin{aligned}
\frac{2^{2q}}{{\vbx-y{|\!|\!|}}^q{\vbx'-y'{|\!|\!|}}^q}\sum_z\frac1{{\vbx-z{|\!|\!|}}^q}\,
\frac1{{\vbx'-z{|\!|\!|}}^q}\leq\frac{2^{2q}c{\vbx-x'{|\!|\!|}}^{d-2q}}{{\vbx-y{|\!|\!|}}^q
{\vbx'-y'{|\!|\!|}}^q},\end{aligned}$$ for some $c<\infty$, where we note that ${\vbx-x'{|\!|\!|}}^{d-2q}\leq1$ because of $\frac12d<q$. The other three possible cases can be estimated similarly (see Figure \[fig:star\](a)). This completes the proof of Proposition \[prp:conv-star\].
$$\begin{aligned}
\begin{array}{cc}
\text{(a)}&{\displaystyle}\sum_z~~\raisebox{-1.4pc}{\includegraphics[scale=0.2]
{star1}}~~~~\lesssim~~~~\raisebox{-1.4pc}{\includegraphics[scale=0.2]
{star2}}\\[2pc]
\text{(b)}&\qquad{\displaystyle}\sum_{u_j,v_j}~~\raisebox{-21pt}{\includegraphics
[scale=0.2]{fish1}}~~~~\lesssim~~~~\sum_{v_j}~~\raisebox{-14pt}{
\includegraphics[scale=0.2]{fish2}}~~~~\lesssim~~~\raisebox{-14pt}{
\includegraphics[scale=0.2]{fish3}}
\end{array}\end{aligned}$$
Before going into the proof of Proposition \[prp:GimpliesPix\], we summarize prerequisites. Recall that [(\[eq:Q’-def\])]{}–[(\[eq:Q”-def\])]{} involve $\tilde G_\Lambda$, and note that, by [(\[eq:G-delta-bd\])]{}, $$\begin{aligned}
{\label{eq:pi0-1stbd}}
{{\langle \varphi_o\varphi_x \rangle}}_\Lambda^3\leq\delta_{o,x}+\tilde G_\Lambda(o,x)^3.\end{aligned}$$ We first show that $$\begin{aligned}
{\label{eq:tildeG-bd}}
\tilde G_\Lambda(o,x)\leq\frac{O(\theta_0)}{{\vbx{|\!|\!|}}^q},&&
\sum_{b:{\underline{b}}=o}\tau_b\big(\delta_{{\overline{b}},x}+\tilde G_\Lambda({\overline{b}},x)\big)
\leq\frac{O(\theta_0)}{{\vbx{|\!|\!|}}^q}\end{aligned}$$ hold assuming the bounds in [(\[eq:IR-xbd\])]{}.
By the assumed bound $\tau\leq2$ in [(\[eq:IR-xbd\])]{}, we have $$\begin{aligned}
{\label{eq:tildeG-1stbd}}
\tilde G_\Lambda(o,x)=\tau D(x)+\sum_{y\ne x}\tau D(y)\,{{\langle \varphi_y
\varphi_x \rangle}}_\Lambda\leq2D(x)+\sum_{y\ne x}2D(y)\,G(x-y),\end{aligned}$$ where, and from now on without stating explicitly, we use the translation invariance of $G(x)$ and the fact that $G(x-y)$ is an increasing limit of ${{\langle \varphi_y\varphi_x \rangle}}_\Lambda$ as $\Lambda\uparrow{{\mathbb Z}^d}$. By [(\[eq:J-def\])]{} and the assumption in Proposition \[prp:GimpliesPix\] that $\theta_0L^{d-q}$, with $q<d$, is bounded away from zero, we obtain $$\begin{aligned}
{\label{eq:Dbd}}
D(x)\leq O(L^{-d}){\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{0<\|x\|_\infty\leq L\}$}}}\leq\frac{O(L^{-d+q})}
{{\vbx{|\!|\!|}}^q}\leq\frac{O(\theta_0)}{{\vbx{|\!|\!|}}^q}.\end{aligned}$$ For the last term in [(\[eq:tildeG-1stbd\])]{}, we consider the cases for $|x|\leq2\sqrt{d}L$ and $|x|\ge2\sqrt{d}L$ separately.
When $|x|\leq2\sqrt{d}L$, we use [(\[eq:Dbd\])]{}, [(\[eq:IR-xbd\])]{} and [(\[eq:conv\])]{} with $\frac12d<q<d$ to obtain $$\begin{aligned}
\sum_{y\ne x}D(y)\,G(x-y)\leq\sum_y\frac{O(L^{-d+q})}{{\vby{|\!|\!|}}^q}\,
\frac{\theta_0}{{\vbx-y{|\!|\!|}}^q}\leq\frac{O(\theta_0L^{-d+q})}
{{\vbx{|\!|\!|}}^{2q-d}}\leq\frac{O(\theta_0)}{{\vbx{|\!|\!|}}^q}.\end{aligned}$$
When $|x|\ge2\sqrt{d}L$, we use the triangle inequality $|x-y|\ge|x|-|y|$ and the fact that $D(y)$ is nonzero only when $0<\|y\|_\infty\leq L$ (so that $|y|\leq\sqrt{d}\|y\|_\infty\leq\sqrt{d}L\leq\frac12|x|$). Then, we obtain $$\begin{aligned}
\sum_{y\ne x}D(y)\,G(x-y)\leq\sum_yD(y)\,\frac{2^q\theta_0}{{\vbx{|\!|\!|}}^q}
=\frac{2^q\theta_0}{{\vbx{|\!|\!|}}^q}.\end{aligned}$$
This completes the proof of the first inequality in [(\[eq:tildeG-bd\])]{}. The second inequality can be proved similarly.
By repeated use of [(\[eq:tildeG-bd\])]{} and Proposition \[prp:conv-star\](i) with $a=b=2q$ (or Proposition \[prp:conv-star\](ii) with $x=x'$ and $y=y'$), we obtain $$\begin{aligned}
{\label{eq:psi-bd}}
\psi_\Lambda(v',v)\leq\delta_{v',v}+\frac{O(\theta_0^2)}{{\vbv-v'{|\!|\!|}}^{2q}}.\end{aligned}$$ Together with the naive bound $G(x)\leq O(1){\vbx{|\!|\!|}}^{-q}$ (cf., [(\[eq:IR-xbd\])]{}) as well as Proposition \[prp:conv-star\](ii) (with $x=x'$ or $y=y'$), we also obtain $$\begin{aligned}
{\label{eq:GGpsi-bd}}
\sum_{v'}G(v'-y)\,G(z-v')\,\psi_\Lambda(v',v)&\leq G(v-y)\,G(z-v)+
\sum_{v'}\frac{O(\theta_0^2)}{{\vbv'-y{|\!|\!|}}^q{\vbz-v'{|\!|\!|}}^q{\vbv-v'{|\!|\!|}}^{2q}}
{\nonumber}\\
&\leq\frac{O(1)}{{\vbv-y{|\!|\!|}}^q{\vbz-v{|\!|\!|}}^q}.\end{aligned}$$ The $O(1)$ term in the right-hand side is replaced by $O(\theta_0)$ or $O(\theta_0^2)$ depending on the number of $G$’s on the left being replaced by $\tilde G_\Lambda$’s.
Since [(\[eq:pi0-1stbd\])]{}–[(\[eq:tildeG-bd\])]{} immediately imply the bound on $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$, it suffices to prove the bounds on $\pi_\Lambda^{{\scriptscriptstyle}(i)}(x)$ for $i\ge1$. To do so, we first estimate the building blocks of the diagrammatic bound [(\[eq:piNbd\])]{}: $\sum_{b:{\underline{b}}=y}\tau_b\,Q'_{\Lambda;u}({\overline{b}},x)$ and $\sum_{b:{\underline{b}}=y}\tau_b\,Q''_{\Lambda;u,v}({\overline{b}},x)$.
Recall [(\[eq:P’0-def\])]{}–[(\[eq:Q”-def\])]{}. First, by using $G(x)\leq O(1){\vbx{|\!|\!|}}^{-q}$ and [(\[eq:GGpsi-bd\])]{}, we obtain $$\begin{aligned}
P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(0)}}(y,x)&\leq\frac{O(1)}{{\vbx-y{|\!|\!|}}^{2q}
{\vbu-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q},{\label{eq:P'0-bd}}\\
P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(0)}}(y,x)&\leq\frac{O(1)}{{\vbx
-y{|\!|\!|}}^q{\vbu-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q{\vbv-y{|\!|\!|}}^q{\vbx-v{|\!|\!|}}^q}.{\label{eq:P''0-bd}}\end{aligned}$$ We will show at the end of this subsection that, for $j\ge1$, $$\begin{aligned}
P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(j)}}(y,x)&\leq\frac{O(j)\,O(\theta_0^2)^j}
{{\vbx-y{|\!|\!|}}^{2q}{\vbu-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q},{\label{eq:P'j-bd}}\\
P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(j)}}(y,x)&\leq\frac{O(j^2)\,O
(\theta_0^2)^j}{{\vbx-y{|\!|\!|}}^q{\vbu-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q{\vbv-y{|\!|\!|}}^q
{\vbx-v{|\!|\!|}}^q}.{\label{eq:P''j-bd}}\end{aligned}$$ As a result, $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(0)}}(y,x)$ (resp., $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(0)}}(y,x)$) is the leading term of $P'_{\Lambda;u}(y,x)$ (resp., $P''_{\Lambda;u,v}(y,x)$), which thus obeys the same bound as in [(\[eq:P’0-bd\])]{} (resp., [(\[eq:P”0-bd\])]{}), with a different constant in $O(1)$. Combining these bounds with [(\[eq:tildeG-bd\])]{} and [(\[eq:GGpsi-bd\])]{} (with both $G$ in the left-hand side being replace by $\tilde G_\Lambda$) and then using Proposition \[prp:conv-star\](ii), we obtain $$\begin{aligned}
{\label{eq:bb1-bd}}
\sum_{b:{\underline{b}}=y}\tau_b\,Q'_{\Lambda;u}({\overline{b}},x)&\leq\sum_z\frac{O(\theta_0)}
{{\vbz-y{|\!|\!|}}^q}\,\frac1{{\vbx-z{|\!|\!|}}^{2q}{\vbu-z{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q}\leq
\frac{O(\theta_0)}{{\vbx-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^{2q}},\end{aligned}$$ and $$\begin{aligned}
{\label{eq:bb2-bd}}
\sum_{b:{\underline{b}}=y}\tau_b\,Q''_{\Lambda;u,v}({\overline{b}},x)&\leq\sum_z\frac{O(\theta_0)}
{{\vbz-y{|\!|\!|}}^q}\,\frac1{{\vbx-z{|\!|\!|}}^q{\vbu-z{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q{\vbv-z{|\!|\!|}}^q
{\vbx-v{|\!|\!|}}^q}{\nonumber}\\
&\quad+\sum_z\frac{O(\theta_0)}{{\vbv-y{|\!|\!|}}^q}\,\frac{O(\theta_0)}{{\vbz-
v{|\!|\!|}}^q}\,\frac1{{\vbx-z{|\!|\!|}}^{2q}{\vbu-z{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q}{\nonumber}\\
&\leq\frac{O(\theta_0)}{{\vbv-y{|\!|\!|}}^q{\vbx-v{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^{2q}}.\end{aligned}$$ This completes bounding the building blocks.
Now we prove the bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge1$. For the bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge2$, we simply apply [(\[eq:P’0-bd\])]{} and [(\[eq:bb1-bd\])]{}–[(\[eq:bb2-bd\])]{} to the diagrammatic bound [(\[eq:piNbd\])]{}. Then, we obtain $$\begin{aligned}
{\label{eq:piNgeq2-prebd}}
\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)\leq\sum_{\substack{u_1,\dots,u_j\\ v_1,\dots,
v_j}}\frac{O(1)}{{\vbu_1{|\!|\!|}}^{2q}{\vbv_1{|\!|\!|}}^q{\vbu_1-v_1{|\!|\!|}}^q}&\bigg(
\prod_{i=1}^{j-1}\frac{O(\theta_0)}{{\vbv_{i+1}-u_i{|\!|\!|}}^q{\vbu_{i+1}-
v_{i+1}{|\!|\!|}}^q{\vbu_{i+1}-v_i{|\!|\!|}}^{2q}}\bigg){\nonumber}\\
&\times\frac{O(\theta_0)}{{\vbx-u_j{|\!|\!|}}^q{\vbx-v_j{|\!|\!|}}^{2q}}\qquad(j\ge2).\end{aligned}$$ First, we consider the sum over $u_j$ and $v_j$. By successive applications of Proposition \[prp:conv-star\](ii) (with $x=x'$ or $y=y'$), we obtain (see Figure \[fig:star\](b)) $$\begin{aligned}
{\label{eq:succ-appl}}
&\sum_{v_j}\sum_{u_j}\frac{O(\theta_0)}{{\vbv_j-u_{j-1}{|\!|\!|}}^q{\vbu_j
-v_j{|\!|\!|}}^q{\vbu_j-v_{j-1}{|\!|\!|}}^{2q}}\,\frac{O(\theta_0)}{{\vbx-u_j{|\!|\!|}}^q
{\vbx-v_j{|\!|\!|}}^{2q}}\\
&\leq\sum_{v_j}\frac{O(\theta_0)^2}{{\vbv_j-u_{j-1}{|\!|\!|}}^q{\vbv_{j-1}
-v_j{|\!|\!|}}^q{\vbx-v_{j-1}{|\!|\!|}}^q{\vbx-v_j{|\!|\!|}}^{2q}}\leq\frac{O(\theta_0)^2}
{{\vbx-u_{j-1}{|\!|\!|}}^q{\vbx-v_{j-1}{|\!|\!|}}^{2q}},{\nonumber}\end{aligned}$$ and thus $$\begin{aligned}
\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)\leq\sum_{\substack{u_1,\dots,u_{j-1}\\ v_1,
\dots,v_{j-1}}}\frac{O(1)}{{\vbu_1{|\!|\!|}}^{2q}{\vbv_1{|\!|\!|}}^q{\vbu_1-v_1{|\!|\!|}}^q}
&\bigg(\prod_{i=1}^{j-2}\frac{O(\theta_0)}{{\vbv_{i+1}-u_i{|\!|\!|}}^q{\vbu_{i
+1}-v_{i+1}{|\!|\!|}}^q{\vbu_{i+1}-v_i{|\!|\!|}}^{2q}}\bigg){\nonumber}\\
&\times\frac{O(\theta_0)^2}{{\vbx-u_{j-1}{|\!|\!|}}^q{\vbx-v_{j-1}{|\!|\!|}}^{2q}}.\end{aligned}$$ Repeating the application of Proposition \[prp:conv-star\](ii) as in [(\[eq:succ-appl\])]{}, we end up with $$\begin{aligned}
{\label{eq:piNgeq2-bd}}
\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)&\leq\sum_{u_1,v_1}\frac{O(1)}{{\vbu_1{|\!|\!|}}^{2
q}{\vbv_1{|\!|\!|}}^q{\vbu_1-v_1{|\!|\!|}}^q}\,\frac{O(\theta_0)^j}{{\vbx-u_1{|\!|\!|}}^q
{\vbx-v_1{|\!|\!|}}^{2q}}\leq\frac{O(\theta_0)^j}{{\vbx{|\!|\!|}}^{3q}}.\end{aligned}$$
For the bound on $\pi_\Lambda^{{\scriptscriptstyle}(1)}(x)$, we use the following bound, instead of [(\[eq:P’0-bd\])]{}: $$\begin{aligned}
{\label{eq:P'0-dec}}
P_{\Lambda;v}^{\prime{{\scriptscriptstyle}(0)}}(o,u)=\delta_{o,u}\delta_{o,v}+(1-\delta_{
o,u}\delta_{o,v})\,P_{\Lambda;v}^{\prime{{\scriptscriptstyle}(0)}}(o,u)\leq\delta_{o,u}
\delta_{o,v}+\frac{O(\theta_0^2)}{{\vbu{|\!|\!|}}^{2q}{\vbv{|\!|\!|}}^q{\vbu-v{|\!|\!|}}^q}.\end{aligned}$$ In addition, instead of using [(\[eq:bb1-bd\])]{}, we use $$\begin{aligned}
{\label{eq:bb1-dec}}
\sum_{b:{\underline{b}}=u}\tau_b\,Q'_{\Lambda;v}({\overline{b}},x)&\leq\sum_z\frac{O(\theta_0)}
{{\vbz-u{|\!|\!|}}^q}\bigg(\delta_{z,v}\delta_{z,x}+(1-\delta_{z,x}\delta_{z,
v})\,P_{\Lambda;v}^{\prime{{\scriptscriptstyle}(0)}}(z,x)+\sum_{j\ge1}P_{\Lambda;v}^{
\prime{{\scriptscriptstyle}(j)}}(z,x)\bigg){\nonumber}\\
&\leq\frac{O(\theta_0)}{{\vbx-u{|\!|\!|}}^q}\,\delta_{v,x}+\sum_z\frac{O(
\theta_0^3)}{{\vbz-u{|\!|\!|}}^q{\vbx-z{|\!|\!|}}^{2q}{\vbv-z{|\!|\!|}}^q{\vbx-v{|\!|\!|}}^q}{\nonumber}\\
&\leq\frac{O(\theta_0)}{{\vbx-u{|\!|\!|}}^q}\,\delta_{v,x}+\frac{O(\theta_0^3)}
{{\vbx-u{|\!|\!|}}^q{\vbx-v{|\!|\!|}}^{2q}},\end{aligned}$$ due to [(\[eq:tildeG-bd\])]{}, [(\[eq:P’j-bd\])]{} and [(\[eq:P’0-dec\])]{}. Applying [(\[eq:P’0-dec\])]{}–[(\[eq:bb1-dec\])]{} to [(\[eq:piNbd\])]{} for $j=1$ and then using Proposition \[prp:conv-star\](ii), we end up with $$\begin{aligned}
\pi_\Lambda^{{\scriptscriptstyle}(1)}(x)&\leq O(\theta_0)\,\delta_{o,x}+\frac{O(\theta_0^3)}
{{\vbx{|\!|\!|}}^{3q}}+\sum_{u,v}\frac{O(\theta_0^2)}{{\vbu{|\!|\!|}}^{2q}{\vbv{|\!|\!|}}^q
{\vbu-v{|\!|\!|}}^q}\bigg(\frac{O(\theta_0)\,\delta_{v,x}}{{\vbx-u{|\!|\!|}}^q}+\frac{
O(\theta_0^3)}{{\vbx-u{|\!|\!|}}^q{\vbx-v{|\!|\!|}}^{2q}}\bigg){\nonumber}\\
&\leq O(\theta_0)\,\delta_{o,x}+\frac{O(\theta_0^3)}{{\vbx{|\!|\!|}}^{3q}}.\end{aligned}$$
To complete the proof of Proposition \[prp:GimpliesPix\], it thus remains to show [(\[eq:P’j-bd\])]{}–[(\[eq:P”j-bd\])]{}. The inequality [(\[eq:P’j-bd\])]{} for $j=1$ immediately follows from the definition [(\[eq:P’1-def\])]{} of $P_{\Lambda;u}^{\prime{\scriptscriptstyle}(1)}$ (see also Figure \[fig:P-def\]) and the bound [(\[eq:psi-bd\])]{} on $\psi_\Lambda-\delta$. To prove [(\[eq:P”j-bd\])]{} for $j=1$, we first recall the definition [(\[eq:P”1-def\])]{} of $P_{\Lambda;u,v}^{\prime\prime{\scriptscriptstyle}(1)}$ (and Figure \[fig:P-def\]). Note that, by [(\[eq:GGpsi-bd\])]{}, $\sum_{v'}G(v'-y)\,G(z-v')\,\psi_\Lambda(v',v)$ obeys the same bound on $\sum_{v'}G(v'-y)\,G(z-v')$ (with a different $O(1)$ term). That is, the effect of an additional $\psi_\Lambda$ is not significant. Therefore, the bound on $P_{\Lambda;u,v}^{\prime\prime{\scriptscriptstyle}(1)}$ is identical, with a possible modification of the $O(1)$ multiple, to the bound on $P_{\Lambda;u}^{\prime{\scriptscriptstyle}(1)}$ (or $P_{\Lambda;v}^{\prime{\scriptscriptstyle}(1)}$) with $v$ (resp., $u$) “being embedded” in one of the bubbles consisting of $\psi_\Lambda-\delta$. By [(\[eq:psi-bd\])]{}, $\psi_\Lambda(y,x)-\delta_{y,x}$ with $v$ being embedded in one of its bubbles is bounded as $$\begin{aligned}
{\label{eq:psipsi-bd}}
&\sum_{k=1}^\infty\sum_{l=1}^k\sum_{y',x'}\big(\tilde G_\Lambda^2
\big)^{*(l-1)}(y,y')\,\tilde G_\Lambda(y',x')\Big({{\langle \varphi_{y'}
\varphi_v \rangle}}_\Lambda\tilde G_\Lambda(v,x')+\tilde G_\Lambda(y',x')\,
\delta_{v,x'}\Big)\big(\tilde G_\Lambda^2\big)^{*(k-l)}
(x',x){\nonumber}\\
&=\sum_{y',x'}\psi_\Lambda(y,y')\,\tilde G_\Lambda(y',x')\Big(
{{\langle \varphi_{y'}\varphi_v \rangle}}_\Lambda\tilde G_\Lambda(v,x')+\tilde
G_\Lambda(y',x')\,\delta_{v,x'}\Big)\psi_\Lambda(x',x){\nonumber}\\
&\leq\sum_{y',x'}\frac{O(1)}{{\vby'-y{|\!|\!|}}^{2q}}\,\frac{O(\theta_0)}
{{\vbx'-y'{|\!|\!|}}^q}\,\frac{O(\theta_0)}{{\vbv-y'{|\!|\!|}}^q{\vbx'-v{|\!|\!|}}^q}\,
\frac{O(1)}{{\vbx-x'{|\!|\!|}}^{2q}}\leq\frac{O(\theta_0^2)}
{{\vbx-y{|\!|\!|}}^q{\vbv-y{|\!|\!|}}^q{\vbx-v{|\!|\!|}}^q}.\end{aligned}$$ By this observation and using [(\[eq:IR-xbd\])]{} to bound the remaining two two-point functions consisting of $P_{\Lambda;u,v}^{\prime\prime{\scriptscriptstyle}(1)}$ (recall [(\[eq:P”1-def\])]{}), we obtain [(\[eq:P”j-bd\])]{} for $j=1$.
For [(\[eq:P’j-bd\])]{}–[(\[eq:P”j-bd\])]{} with $j\ge2$, we first note that, by applying [(\[eq:IR-xbd\])]{} and [(\[eq:psi-bd\])]{} to the definition [(\[eq:Pj-def\])]{} of $P_\Lambda^{{\scriptscriptstyle}(j)}(y,x)$, we have $$\begin{gathered}
P_\Lambda^{{\scriptscriptstyle}(j)}(y,x)\leq\sum_{\substack{v_2,\dots,v_j\\ v'_1,\dots,
v'_{j-1}}}\frac{O(\theta_0^2)}{{\vbv'_1-y{|\!|\!|}}^{2q}{\vbv_2-y{|\!|\!|}}^q{\vbv'_1
-v_2{|\!|\!|}}^q}\prod_{i=2}^{j-1}\frac{O(\theta_0^2)}{{\vbv'_i-v_i{|\!|\!|}}^{2q}{{|\!|\!|}v_{i+1}-v'_{i-1}{|\!|\!|}}^q{\vbv'_i-v_{i+1}{|\!|\!|}}^q}{\nonumber}\\
\times\frac{O(\theta_0^2)}{{\vbx-v_j{|\!|\!|}}^{2q}{\vbx-v'_{j-1}{|\!|\!|}}^q}.{\label{eq:Pj-bd}}\end{gathered}$$ By definition, the bound on $P_{\Lambda;u}^{\prime{\scriptscriptstyle}(j)}(y,x)$ is obtained by “embedding $u$” in one of the $2j-1$ factors of ${{|\!|\!|}\cdots{|\!|\!|}}^q$ (not ${{|\!|\!|}\cdots{|\!|\!|}}^{2q}$) and then summing over all these $2j-1$ choices. For example, the contribution from the case in which ${\vbv_2-y{|\!|\!|}}^q$ is replaced by ${\vbu-y{|\!|\!|}}^q{\vbv_2-u{|\!|\!|}}^q$ is bounded, similarly to [(\[eq:piNgeq2-bd\])]{}, by $$\begin{aligned}
&\sum_{v_2,v'_1}\frac{O(\theta_0^2)}{{\vbv'_1-y{|\!|\!|}}^{2q}{\vbu-y{|\!|\!|}}^q{\vbv_2
-u{|\!|\!|}}^q{\vbv'_1-v_2{|\!|\!|}}^q}\,\frac{O(\theta_0^2)^{j-1}}{{\vbx-v'_1{|\!|\!|}}^q{\vbx
-v_2{|\!|\!|}}^{2q}}{\nonumber}\\
&\leq\sum_{v'_1}\frac{O(\theta_0^2)^j}{{\vbv'_1-y{|\!|\!|}}^{2q}{\vbu-y{|\!|\!|}}^q{{|\!|\!|}x-u{|\!|\!|}}^q{\vbx-v'_1{|\!|\!|}}^{2q}}\leq\frac{O(\theta_0^2)^j}{{\vbx-y{|\!|\!|}}^{2q}{\vbu
-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q}.\end{aligned}$$ The other $2j-2$ contributions can be estimated in a similar way, with the same form of the bound. This completes the proof of [(\[eq:P’j-bd\])]{}.
By [(\[eq:psipsi-bd\])]{}, the bound on $P_{\Lambda;u,v}^{\prime\prime{\scriptscriptstyle}(j)}(y,x)$ is also obtained by “embedding $u$ and $v$” in one of the $2j-1$ factors of ${{|\!|\!|}\cdots{|\!|\!|}}^q$ and one of the $j$ factors of ${{|\!|\!|}\cdots{|\!|\!|}}^{2q}$ in [(\[eq:Pj-bd\])]{}, and then summing over all these combinations. For example, the contribution from the case in which ${\vbv_2-y{|\!|\!|}}^q$ and ${\vbv'_1-y{|\!|\!|}}^{2q}$ in [(\[eq:Pj-bd\])]{} are replaced, respectively, by ${\vbu-y{|\!|\!|}}^q{\vbv_2-u{|\!|\!|}}^q$ and ${\vbv'_1-y{|\!|\!|}}^q{\vbv-y{|\!|\!|}}^q{\vbv'_1-v{|\!|\!|}}^q$, is bounded by $$\begin{aligned}
&\sum_{v_2,v'_1}\frac{O(\theta_0^2)}{{\vbv'_1-y{|\!|\!|}}^q{\vbv-y{|\!|\!|}}^q{\vbv'_1
-v{|\!|\!|}}^q{\vbu-y{|\!|\!|}}^q{\vbv_2-u{|\!|\!|}}^q{\vbv'_1-v_2{|\!|\!|}}^q}\,\frac{O(\theta_0^2)^{j
-1}}{{\vbx-v'_1{|\!|\!|}}^q{\vbx-v_2{|\!|\!|}}^{2q}}{\nonumber}\\
&\leq\sum_{v'_1}\frac{O(\theta_0^2)^j}{{\vbv'_1-y{|\!|\!|}}^q{\vbv-y{|\!|\!|}}^q{\vbv'_1
-v{|\!|\!|}}^q{\vbu-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q{\vbx-v'_1{|\!|\!|}}^{2q}}{\nonumber}\\
&\leq\frac{O(\theta_0^2)^j}{{\vbx-y{|\!|\!|}}^q{\vbu-y{|\!|\!|}}^q{\vbx-u{|\!|\!|}}^q
{\vbv-y{|\!|\!|}}^q{\vbx-v{|\!|\!|}}^q}.\end{aligned}$$ The other $(2j-1)j-1$ contributions can be estimated similarly, with the same form of the bound. This completes the proof of [(\[eq:P”j-bd\])]{} and thus Proposition \[prp:GimpliesPix\].
Bounds for finite-range models {#ss:proof-nn}
------------------------------
First, we prove [(\[eq:pi-sumbd\])]{} and Proposition \[prp:exp-bootstrap\](iii) assuming [(\[eq:IR-kbd\])]{}. Then, we prove [(\[eq:pi-kbd\])]{} assuming [(\[eq:IR-kbd\])]{} and [(\[eq:IR-xbdNN\])]{} to complete the proof of Propositions \[prp:GimpliesPik\].
By applying [(\[eq:G-delta-bd\])]{} to the bound [(\[eq:piNbd\])]{} on $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$, it is easy to show that, for $r=0,2$, $$\begin{aligned}
{\label{eq:pi0-rthmombd}}
\sum_x|x|^r\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)\leq\delta_{r,0}+\sum_{x\ne o}
|x|^r{{\langle \varphi_o\varphi_x \rangle}}_\Lambda^3&\leq\delta_{r,0}+\Big(
\sup_{x\ne o}|x|^rG(x)\Big)\sum_{x\ne o}(\tau D*G)(x)\,G(x){\nonumber}\\
&\leq\delta_{r,0}+(d\sigma^2)^{\delta_{r,2}}O(\theta_0)^2.\end{aligned}$$
For $i\ge1$, by using the diagrammatic bound [(\[eq:piNbd\])]{} and translation invariance, we have $$\begin{aligned}
{\label{eq:dec-bd}}
\sum_x\pi_\Lambda^{{\scriptscriptstyle}(i)}(x)\leq\bigg(\sum_{v,x}P_{\Lambda;v}
^{\prime{{\scriptscriptstyle}(0)}}(o,x)\bigg)\bigg(\sup_y\sum_{z,v,x}\tau_{y,z}
Q''_{\Lambda;o,v}(z,x)\bigg)^{i-1}\bigg(\sup_y\sum_{z,x}\tau_{y,
z}Q'_{\Lambda;o}(z,x)\bigg).\end{aligned}$$ The proof of the bound on $\sum_x\pi_\Lambda^{{\scriptscriptstyle}(i)}(x)$ for $i\ge1$ is completed by showing that $$\begin{aligned}
{\label{eq:block-sumbd}}
\bigg(\sum_{v,x}P_{\Lambda;v}^{\prime{{\scriptscriptstyle}(0)}}(o,x)-1\bigg)\vee
\bigg(\sup_y\sum_{z,v,x}\tau_{y,z}Q''_{\Lambda;o,v}(z,x)\bigg)
\vee\bigg(\sup_y\sum_{z,x}\tau_{y,z}Q'_{\Lambda;o}(z,x)\bigg)
=O(\theta_0).\end{aligned}$$ The key idea to obtain this estimate is that the bounding diagrams for the Ising model are similar to those for self-avoiding walk (cf., Figure \[fig:piN-bd\]). The diagrams for self-avoiding walk are known to be bounded by products of bubble diagrams (see, e.g., [@ms93]), and we can apply the same method to bound the diagrams for the Ising model by products of bubbles.
For example, consider $$\begin{aligned}
{\label{eq:tau*Q'-rewr}}
\sum_{z,x}\tau_{y,z}Q'_{\Lambda;o}(z,x)=\sum_{z',x}\bigg(\sum_z
\tau_{y,z}\big(\delta_{z,z'}+\tilde G_\Lambda(z,z')\big)\bigg)
P'_{\Lambda;o}(z',x).\end{aligned}$$ The factor of $\theta_0$ is due to the nonzero line segment $\sum_z\tau_{y,z}(\delta_{z,z'}+\tilde G_\Lambda(z,z'))$, because $$\begin{gathered}
\sum_z\tau_{o,z}\big(\delta_{z,x}+\tilde G_\Lambda(z,x)\big)=\tau
D(x)+\tau\sum_zD(z)\,\tilde G_\Lambda(z,x)\leq O(\theta_0)+\tau
\sup_x\tilde G_\Lambda(o,x),{\label{eq:tau*delta+G-bd}}\\
\tilde G_\Lambda(o,x)\leq\tau D(x)+\tau\sum_{y\ne o}G(y)\,D(x-y)\leq
O(\theta_0)+\tau\sup_{y\ne o}G(y)=O(\theta_0),{\label{eq:tildeG-bdnn}}\end{gathered}$$ where we have used translation invariance, [(\[eq:IR-kbd\])]{} and $\sup_xD(x)=O(\theta_0)$. By [(\[eq:P’P”-def\])]{}, $$\begin{aligned}
{\label{eq:tau*Q'-rewrbd}}
{(\ref{eq:tau*Q'-rewr})}\leq O(\theta_0)\sum_{z',x}P'_{\Lambda;o}(z',x)
=O(\theta_0)\sum_{z',x}\bigg(P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(0)}}(z',x)
+\sum_{j\ge1}P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(j)}}(z',x)\bigg).\end{aligned}$$ Similarly to [(\[eq:pi0-rthmombd\])]{} for $r=0$, the sum of $P_{\Lambda;v}^{\prime{{\scriptscriptstyle}(0)}}(z',x)$ is easily estimated as $1+O(\theta_0)$. We claim that the sum of $P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(j)}}(z',x)$ for $j\geq1$ is $(2j-1)\,O(\theta_0)^j$, since $P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(j)}}(z',x)$ is a sum of $2j-1$ terms, each of which contains $j$ chains of nonzero bubbles; each chain is $\psi_\Lambda(v,v')-\delta_{v,v'}$ for some $v,v'$ and satisfies $$\begin{aligned}
\sum_{v'}\big(\psi_\Lambda(v,v')-\delta_{v,v'}\big)\leq\sum_{l\ge1}
\Big(\tau^2\big(D*(D*G^{*2})\big)(o)\Big)^l=\sum_{l\ge1}
O(\theta_0)^l=O(\theta_0).\end{aligned}$$ For example, $$\begin{aligned}
\sum_{z',x}P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(4)}}(z',x)&=\raisebox{-1.2pc}
{\includegraphics[scale=0.15]{Pprime4}}~+6\text{ other possibilities},\end{aligned}$$ which can be estimated, by translation invariance, as $$\begin{aligned}
\raisebox{-1.5pc}{\includegraphics[scale=0.15]{Pprime4}}&\leq
~\raisebox{-1.8pc}{\includegraphics[scale=0.15]{Pprime4dec}}{\nonumber}\\[5pt]
&\leq\bigg(\sum_y\big(\psi_\Lambda(o,y)-\delta_{o,y}\big)\bigg)^4\big(
\bar W^{{\scriptscriptstyle}(0)}\big)^4=O(\theta_0)^4,\end{aligned}$$ where $\bar W^{{\scriptscriptstyle}(t)}$ is given by [(\[eq:GbarWbar\])]{}.
The sum of $\tau_{y,z}Q''_{\Lambda;o,v}(z,x)$ in [(\[eq:block-sumbd\])]{} is estimated similarly [@sNN]. We complete the proof of the bound on $\sum_x\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge1$.
To estimate $\sum_x|x|^2\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge1$, we recall that, in each bounding diagram, there are at least three distinct paths between $o$ and $x$: the uppermost path (i.e., $o\to
b_1\to v_2\to b_3\to\cdots\to x$ in [(\[eq:piNbd\])]{}; see also Figure \[fig:piN-bd\]), the lowermost path (i.e., $o\to v_1\to
b_2\to v_3\to\cdots\to x$) and a middle zigzag path. We use the lowermost path to bound $|x|^2$ as $$\begin{aligned}
{\label{eq:x2-bd}}
|x|^2=\sum_{n=0}^j|a_n|^2+2\sum_{0\leq m<n\leq j}a_m\cdot a_n
\leq(j+1)\sum_{n=0}^j|a_n|^2,\end{aligned}$$ where $a_0=v_1$, $a_1={\underline{b}}_2-v_1$ ,$a_2=v_3-{\underline{b}}_2,\dots$, and $a_j=x-v_j$ or $x-{\underline{b}}_j$ depending on the parity of $j$.
$$\begin{gathered}
\includegraphics[scale=0.16]{pi3dec}\\[1pc]
\text{(i)}\quad\raisebox{-1.2pc}{\includegraphics[scale=0.12]
{pi3dec4}}\qquad\qquad
\text{(ii)}\quad\raisebox{-1.2pc}{\includegraphics[scale=0.12]
{pi3dec1}}\\[5pt]
\text{(iii)}\quad\raisebox{-1.2pc}{\includegraphics[scale=0.12]
{pi3dec2}}\qquad~~~\&~~\qquad\raisebox{-1.2pc}{\includegraphics
[scale=0.12]{pi3dec3}}\end{gathered}$$
We discuss the contributions to $\sum_x|x|^2\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ from (i) $|a_j|^2$, (ii) $|a_0|^2$ and (iii) $|a_n|^2$ for $n\ne0,j$, separately (cf., Figure \[fig:pi3-dec\]).
\(i) The contribution from $|a_j|^2$ is bounded by $$\begin{aligned}
{\label{eq:2nddec-bd:n=j}}
&\bigg(\sum_{v,y}P_{\Lambda;v}^{\prime{{\scriptscriptstyle}(0)}}(o,y)\bigg)\bigg(\sup_y
\sum_{\substack{b,v,z\\ {\underline{b}}=y}}\tau_bQ''_{\Lambda;o,v}({\overline{b}},z)\bigg)^{j
-1}{\nonumber}\\
&\qquad\qquad\times\bigg(\sup_y\sum_{\substack{b,x\\ {\underline{b}}=y}}\Big(|x|^2
{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{j\text{ odd}\}$}}}+|x-{\underline{b}}|^2{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{j\text{ even}\}$}}}\Big)\,\tau_b
Q'_{\Lambda;o}({\overline{b}},x)\bigg){\nonumber}\\
&\leq O(\theta_0)^{j-1}\sup_y\sum_{z,z',x}\Big(|x|^2{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{j\text{ odd}\}$}}}
+|x-y|^2{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{j\text{ even}\}$}}}\Big)\,\tau_{y,z}\big(\delta_{z,z'}+\tilde
G_\Lambda(z,z')\big)P'_{\Lambda;o}(z',x).\end{aligned}$$ By [(\[eq:P’0-def\])]{}, the leading contribution from $P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(0)}}(z',x)$ for an odd $j$ can be estimated as $$\begin{aligned}
{\label{eq:2nddec-bd:n=jbd}}
&\sup_y\sum_{z,z',x}|x|^2\tau_{y,z}\big(\delta_{z,z'}+\tilde
G_\Lambda(z,z')\big)P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(0)}}(z',x){\nonumber}\\
&=\sup_y\sum_{z,z',x}\tau_{y,z}\big(\delta_{z,z'}+\tilde G_\Lambda(
z,z')\big)\,{{\langle \varphi_{z'}\varphi_o \rangle}}_\Lambda\,{{\langle \varphi_{z'}
\varphi_x \rangle}}_\Lambda^2\,|x|^2{{\langle \varphi_o\varphi_x \rangle}}_\Lambda{\nonumber}\\
&\leq\sup_y\Big((\tau D*G)(y)+(\tau D*G)^{*2}(y)\Big)\,G^{*2}(o)\,
\bar G^{{\scriptscriptstyle}(2)}=d\sigma^2O(\theta_0)^2,\end{aligned}$$ where $\bar G^{{\scriptscriptstyle}(s)}$ is given by [(\[eq:GbarWbar\])]{}. The other contributions from $P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(i)}}(z',x)$ for $i\ge1$ and from the even-$j$ case can be estimated similarly; if $j$ is even, then, by using $|x-y|^2\leq2|z'-y|^2+2|x-z'|^2$ and estimating the contributions from $|z'-y|^2$ and $|x-z'|^2$ separately, we obtain that the supremum in [(\[eq:2nddec-bd:n=j\])]{} is $d\sigma^2O(\theta_0)$. Consequently, [(\[eq:2nddec-bd:n=j\])]{} is $d\sigma^2O(\theta_0)^{2\lfloor{{\scriptscriptstyle}\frac{j+1}2}\rfloor}$.
\(ii) To bound the contributions to $\sum_x|x|^2\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ from $|a_n|^2$ for $n<j$, we define (cf., Figure \[fig:tildeQ”\]) $$\begin{aligned}
{\label{eq:tildeQ''-def}}
\tilde Q''_{\Lambda;u,v}(y,x)=\sum_b\bigg(P''_{\Lambda;u,v}(y,
{\underline{b}})+\sum_{y'}\tilde G_\Lambda(y,y')\,P'_{\Lambda;u}(y',{\underline{b}})
\,\psi_\Lambda(y,v)\bigg)\,\tau_b\big(\delta_{{\overline{b}},x}+\tilde
G_\Lambda({\overline{b}},x)\big).\end{aligned}$$ By translation invariance and a similar argument to show [(\[eq:block-sumbd\])]{}, we can easily prove $$\begin{aligned}
{\label{eq:tildeQ''-bd}}
\sup_z\sum_{y,v}\tilde Q''_{\Lambda;o,v}(y,v+z)=\sum_{y,v}\tilde
Q''_{\Lambda;v,o}(y,z)=O(\theta_0).\end{aligned}$$ Therefore, the contribution from $|a_0|^2$ to $\sum_x|x|^2
\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ is bounded by $$\begin{aligned}
{\label{eq:2nddec-bd:n=0}}
&\bigg(\sup_y\sum_{v,b}|v|^2P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(0)}}(v,{\underline{b}})\,
\tau_b\big(\delta_{{\overline{b}},y}+\tilde G_\Lambda({\overline{b}},y)\big)\bigg)\bigg(
\sup_z\sum_{y,v}\tilde Q''_{\Lambda;v,o}(y,z)\bigg)^{j-1}\bigg(
\sum_{z,x}P'_{\Lambda;o}(z,x)\bigg){\nonumber}\\
&\quad\leq d\sigma^2O(\theta_0)^{j+1}.\end{aligned}$$
![\[fig:tildeQ”\]The leading diagrams of $\tilde
Q''_{\Lambda;u,v}(y,x)$, due to $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(0)}}(y,{\underline{b}})$ and $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(0)}}(y',{\underline{b}})$ in [(\[eq:tildeQ”-def\])]{}, respectively.](tildeQpp1 "fig:") ![\[fig:tildeQ”\]The leading diagrams of $\tilde
Q''_{\Lambda;u,v}(y,x)$, due to $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(0)}}(y,{\underline{b}})$ and $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(0)}}(y',{\underline{b}})$ in [(\[eq:tildeQ”-def\])]{}, respectively.](tildeQpp2 "fig:")
\(iii) By translation invariance and [(\[eq:tildeQ”-def\])]{}–[(\[eq:tildeQ”-bd\])]{}, the contribution from $|a_n|^2$ for an $n\ne0,j$ is bounded by $$\begin{aligned}
{\label{eq:2nddec-bd:0<n<j}}
&\bigg(\sum_{v,y}P_{\Lambda;v}^{\prime{{\scriptscriptstyle}(0)}}(o,y)\bigg)\bigg(\sup_y
\sum_{\substack{b,v,z\\ {\underline{b}}=y}}\tau_bQ''_{\Lambda;o,v}({\overline{b}},z)\bigg)^{n
-1}\bigg(\sup_z\sum_{y,v}\tilde Q''_{\Lambda;v,o}(y,z)\bigg)^{j-1-n}
\bigg(\sum_{z,x}P'_{\Lambda;o}(z,x)\bigg){\nonumber}\\
&\times\bigg(\sup_{y,z}\sum_{\substack{b,b',v\\ {\underline{b}}=y}}\Big(|{\underline{b}}'
|^2{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{n\text{ odd}\}$}}}+|v-{\underline{b}}|^2{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{n\text{ even}\}$}}}\Big)\,\tau_b
Q''_{\Lambda;o,v}({\overline{b}},{\underline{b}}')\,\tau_{b'}\big(\delta_{{{\overline{b}}^{\raisebox{-2pt}{$\scriptscriptstyle\prime$}}},v+z}
+\tilde G_\Lambda({{\overline{b}}^{\raisebox{-2pt}{$\scriptstyle\prime$}}},v+z)\big)\bigg),\end{aligned}$$ where the first line is $O(\theta_0)^{j-2}$. The leading contribution to the second line from $P_{\Lambda;o,v}^{\prime\prime{{\scriptscriptstyle}(0)}}$ and $P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(0)}}$ in $Q''_{\Lambda;o,v}$ for an odd $n$ is bounded, due to translation invariance, by $$\begin{aligned}
{\label{eq:2nddec-bd:0<n<jbd}}
&\bar G^{{\scriptscriptstyle}(2)}\sup_{y,z}\Bigg(~\raisebox{-1.4pc}{\includegraphics
[scale=0.14]{Pprime0Wdec24}}~+\raisebox{-1.4pc}{\includegraphics
[scale=0.14]{Pprime0Wdec25}}~\Bigg){\nonumber}\\
&\leq d\sigma^2O(\theta_0)^2\sup_z\Bigg(~\raisebox{-1.4pc}
{\includegraphics[scale=0.14]{Pprime0Wdec26}}~+~\raisebox{-1.4pc}
{\includegraphics[scale=0.14]{Pprime0Wdec27}}~\Bigg)\leq d\sigma^2
O(\theta_0)^3.\end{aligned}$$ The other contributions from $P_{\Lambda;o,v}^{\prime\prime{{\scriptscriptstyle}(i)}}$ and $P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(i)}}$ for $i\ge1$ and from the even-$n$ case can be estimated similarly; if $n$ is even, then the second supremum in [(\[eq:2nddec-bd:0<n<jbd\])]{} is $O(\theta_0)$. Therefore, [(\[eq:2nddec-bd:0<n<j\])]{} is $d\sigma^2O(\theta_0)^{2\lfloor{{\scriptscriptstyle}\frac{j+1}2}\rfloor}$.
Summarizing the above (i)–(iii) and using $2\lfloor{{\scriptscriptstyle}\frac{j+1}2}\rfloor\geq j\vee2$ for $j\ge1$, we have $$\begin{aligned}
\frac1{j+1}\sum_x|x|^2\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)\leq d\sigma^2\Big(
jO(\theta_0)^{2\lfloor{{\scriptscriptstyle}\frac{j+1}2}\rfloor}+O(\theta_0)^{j+1}
\Big)\leq d\sigma^2(j+1)\,O(\theta_0)^{j\vee2}.\end{aligned}$$ This together with [(\[eq:pi0-rthmombd\])]{} complete the proof of [(\[eq:pi-sumbd\])]{}.
It is easy to see that $$\begin{aligned}
{\label{eq:pi0-t+2ndmombd}}
\sum_x|x|^{t+2}\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)\leq\sum_x|x|^{t+2}G(x)^3\leq
\bar G^{{\scriptscriptstyle}(2)}\sum_x|x|^tG(x)^2\leq d\sigma^2\theta_0\bar W^{{\scriptscriptstyle}(t)}.\end{aligned}$$ We show below that, for $j\ge1$, $$\begin{aligned}
{\label{eq:pij-t+2ndmombd}}
\sum_x|x|^{t+2}\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)\leq d\sigma^2\bar W^{{\scriptscriptstyle}(t)}
(j+1)^{t+3}O(\theta_0)^{j\vee2-1},\end{aligned}$$ where the bound is independent of $\Lambda$. Due to these uniform bounds, we conclude that the sum of $|x|^{t+2}|\Pi(x)|$ is finite if $\theta_0\ll1$.
Now we explain the main idea of the proof of [(\[eq:pij-t+2ndmombd\])]{}. First we recall that, in the proof of the bound on $\sum_x|x|^2\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$, we distribute $|x|^2$ along the lowermost path of each bounding diagram. To bound $\sum_x|x|^{t+2}\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$, we again use the lowermost path in the same way to distribute $|x|^2$, and use the uppermost path to distribute the remaining $|x|^t$. More precisely, we use $$\begin{aligned}
{\label{eq:|x|-max}}
|x|\leq(j+1)\max_{n=0,1,\dots,j}|a'_n|,\end{aligned}$$ where $a'_0,a'_1,\dots,a'_j$ are the displacements along the uppermost path: $a'_0={\underline{b}}_1$, $a'_1=v_2-{\underline{b}}_1$, $a'_2={\underline{b}}_3-v_2,\dots$, and $a'_j=x-v_j$ or $x-{\underline{b}}_j$ depending on the parity of $j$. Let $m$ be such that $|a'_m|=\max_n|a'_n|$.
For the contribution to $\sum_x|x|^{t+2}\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ from $|a_n|^2$ in [(\[eq:x2-bd\])]{} for $n\ne m$, we simply follow the same strategy as explained above in the paragraphs (i)–(iii) to prove the bound on $\sum_x|x|^2\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$. The only difference is that one of the bubbles $\bar W^{{\scriptscriptstyle}(0)}$ contained in the bound on the $m^\text{th}$ block is now replaced by $\bar W^{{\scriptscriptstyle}(t)}$.
The contribution from $|a_m|^2$ in [(\[eq:x2-bd\])]{} can be estimated in a similar way, except for a few complicated cases, due to $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(i)}}$ and $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(i)}}$ for $i\ge1$ contained in the $m^\text{th}$ block. For example, let $j$ be even and let $m=j$ (cf., the second line of [(\[eq:2nddec-bd:n=j\])]{}). The following are two possibile diagrams in the contribution from $P_{\Lambda;o}^{\prime{{\scriptscriptstyle}(4)}}(f,x)$ to $\sum_{z,x}|x-y|^2|x|^t\tau_{y,z}Q'_{\Lambda;o}(z,x)$: $$\begin{aligned}
{\label{eq:IRSchwarz}}
\text{(i)}\quad\raisebox{-1.5pc}{\includegraphics[scale=0.14]
{IRnonSchwarz}}\hspace{5pc}
\text{(ii)}\quad\raisebox{-1.5pc}{\includegraphics[scale=0.14]
{IRSchwarz}}\end{aligned}$$ where, for simplicity, $\psi_\Lambda(f,g)-\delta_{f,g}$ and $\psi_\Lambda(u,z)-\delta_{u,z}$ are reduced to $\tilde
G_\Lambda(f,g)^2$ and $\tilde G_\Lambda(f,g)^2$, respectively. We suppose that $|v|$ is bigger than $|w-v|$ and $|x-w|$ along the lowermost path from $o$ to $x$ through $v$ and $w$, so that $|x|^t$ is bounded by $3^t|v|^t$. We also suppose that $|z-u|$ in (\[eq:IRSchwarz\].i) (resp., $|g-f|$ in (\[eq:IRSchwarz\].ii)) is bigger than the end-to-end distance of any of the other four segments along the uppermost path from $y$ to $x$ through $f,g,u$ and $z$. Therefore, we can bound $|x-y|^2$ by $5^2|z-u|^2$ in (\[eq:IRSchwarz\].i) (resp., $5^2|g-f|^2$ in (\[eq:IRSchwarz\].ii)) and bound the weighted arc between $u$ and $z$ (resp., between $f$ and $g$) by $5^2\bar G^{{\scriptscriptstyle}(2)}$. By translation invariance, the remaining diagram of (\[eq:IRSchwarz\].i) is easily bounded as $$\begin{aligned}
{\label{eq:IRSchwarz-bdi}}
\sum_{f',g,u',v}\!\raisebox{-1.4pc}{\includegraphics[scale=0.14]
{IRnonSchwarzdec1}}\;=\sup_{f',g,u'}\,\raisebox{-1.4pc}
{\includegraphics[scale=0.14]{IRnonSchwarzdec2}}\leq\bar
W^{{\scriptscriptstyle}(t)}\,O(\theta_0)^4,\end{aligned}$$ where the power 4 (not 3) is due to the fact that the segment from $u'$ in the last block is nonzero.
To bound the remaining diagram of (\[eq:IRSchwarz\].ii) is a little trickier. We note that at least one of $|u|,|z-u|,|w-z|$ and $|v-w|$ along the path from $o$ to $v$ through $u,z,w$ is bigger than $\frac14|v|$. Suppose $|v-w|\ge\frac14|v|$, so that $|v|^t\leq2^t|v-w|^{t/2}|v|^{t/2}$. Then, by using the Schwarz inequality, we obtain $$\begin{aligned}
{\label{eq:IRSchwarz-bdii}}
\raisebox{-1.3pc}{\includegraphics[scale=0.13]{IRSchwarzdec1}}~~\leq~~
\Bigg(~~\raisebox{-2pc}{\includegraphics[scale=0.12]{IRSchwarzdec2}}
~~\Bigg)^{1/2}~\Bigg(~~\raisebox{-1.5pc}{\includegraphics[scale=0.12]
{IRSchwarzdec3}}~~\Bigg)^{1/2},\end{aligned}$$ where the two weighted arcs between $o$ and $v$ in the second term is $|v|^tG(v)^2\equiv(|v|^{t/2}G(v))^2$. By translation invariance and the fact that the north-east and north-west segments from $g$ in the first term are nonzero, we obtain $$\begin{aligned}
\raisebox{-2pc}{\includegraphics[scale=0.13]{IRSchwarzdec2}}~
&\leq\bigg(\sup_z\raisebox{-0.9pc}{\includegraphics[scale=0.12]
{IRSchwarzdec4}}~\bigg)\Big(\sup_{g'}\tau(D*G^{*2})(g')\Big)^2\bar
W^{{\scriptscriptstyle}(0)}\bigg(\sum_v\big(\psi_\lambda(o,v)-\delta_{o,v}\big)
\bigg)^2{\nonumber}\\
&\leq O(\theta_0)^5.\end{aligned}$$ With the help of $(\bar W^{{\scriptscriptstyle}(t/2)})^2\leq\bar W^{{\scriptscriptstyle}(0)}\bar W^{{\scriptscriptstyle}(t)}$ (due to the Schwarz inequality), we also obtain $$\begin{aligned}
\raisebox{-1.8pc}{\includegraphics[scale=0.13]{IRSchwarzdec3}}~~\leq
\,\bar W^{{\scriptscriptstyle}(0)}\,\bar W^{{\scriptscriptstyle}(t)}\Bigg(\sup_v\raisebox{-1.1pc}
{\includegraphics[scale=0.13]{IRSchwarzdec5}}~\Bigg)^2\leq\big(\bar
W^{{\scriptscriptstyle}(t)}\big)^2O(\theta_0)^4.\end{aligned}$$ Therefore, [(\[eq:IRSchwarz-bdii\])]{} is bounded by $\bar W^{{\scriptscriptstyle}(t)}O(\theta_0)^{9/2}$.
The other cases can be estimated similarly [@sNN]. As a result, we obtain $$\begin{aligned}
\sum_x|x|^{t+2}\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)\leq\sum_{m=0}^jd\sigma^2
\bar W^{{\scriptscriptstyle}(t)}(j+1)^{t+2}O(\theta_0)^{j\vee2-1},\end{aligned}$$ which implies [(\[eq:pij-t+2ndmombd\])]{}. This completes the proof of Proposition \[prp:exp-bootstrap\](iii).
If $x=o$, then we simply use the bound on the sum in [(\[eq:pi-sumbd\])]{} to obtain $\pi_\Lambda^{{\scriptscriptstyle}(i)}(o)\leq
O(\theta_0)^i$ for any $i\ge0$. It is also easy to see that $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$ with $x\ne o$ obeys [(\[eq:pi-kbd\])]{}, due to [(\[eq:IR-xbdNN\])]{} and the diagrammatic bound [(\[eq:piNbd\])]{}. It thus remains to show [(\[eq:pi-kbd\])]{} for $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ with $x\ne o$ and $j\ge1$.
The idea of the proof is somewhat similar to that of Proposition \[prp:exp-bootstrap\](iii) explained above. First, we take $|a_m|\equiv\max_n|a_n|$ from the lowermost path and $|a'_l|\equiv\max_n|a'_n|$ from the uppermost path of a bounding diagram. Note that, by [(\[eq:|x|-max\])]{}, $|a_m|$ and $|a'_l|$ are both bigger than $\frac1{j+1}|x|$. That is, $|a_m|^{-q}$ and $|a'_l|^{-q}$ are both bounded from above by $(j+1)^q|x|^{-q}$. If the path corresponding to $a_m$ in the $m^\text{th}$ block consists of $N$ segments, we take the “longest” segment whose end-to-end distance is therefore bigger than $\frac1{N(j+1)}|x|$. That is, the corresponding two-point function is bounded by $\lambda_0
N^q(j+1)^q|x|^{-q}$. Here, $N$ depends on the parity of $m$, as well as on $i\ge0$ for $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(i)}}$ (or $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(i)}}$ if $m=0$ or $j$) and the location of $u,v$ in each diagram, and is at most $N\leq O(i+1)$. However, the number of nonzero chains of bubbles contained in each diagram of $P_{\Lambda;u}^{\prime{{\scriptscriptstyle}(i)}}$ and $P_{\Lambda;u,v}^{\prime\prime{{\scriptscriptstyle}(i)}}$ is $O(i)$, and hence their contribution would be $O(\theta_0)^{O(i)}$. This compensates the growing factor of $N^q$, and therefore we will not have to take the effect of $N$ seriously. The same is true for $a'_l$, and we refrain from repeating the same argument.
Next, we take the “longest” segment, denoted $a''$, among those which together with $a'_l$ (or a part of it) form a “loop”; a similar observation was used to obtain [(\[eq:IRSchwarz-bdii\])]{}. The loop consists of segments contained in the $l^\text{th}$ block and possibly in the $(l-1)^\text{st}$ block, and hence the number of choices for $a''$ is at most $O(i_{l-1}+i_l+1)$, where $i_l$ is the index of $P_{\Lambda}^{\prime{{\scriptscriptstyle}(i_l)}}$ or $P_{\Lambda}^{\prime\prime{{\scriptscriptstyle}(i_l)}}$ in the $l^\text{th}$ block ($i_{-1}=0$ by convention). By [(\[eq:|x|-max\])]{}, we have $|a''|\ge
O(i_{l-1}+i_l+1)^{-1}|a'_l|$, and the corresponding two-point function is bounded by $\lambda_0O(i_{l-1}+i_l+1)^q
(j+1)^q|x|^{-q}$. As explained above, the effect of $O(i_{l-1}+i_l+1)^q$ would not be significant after summing over $i_{l-1}$ and $i_l$.
We have explained how to extract three “long” segments from each bounding diagram, which provide the factor $\lambda_0^3(j+1)^{3q}|x|^{-3q}$ in [(\[eq:pi-kbd\])]{}; the extra factor of $(j+1)^2$ in [(\[eq:pi-kbd\])]{} is due to the number of choices of $m,l\in\{0,1,\dots,j\}$. Therefore, the remaining task is to control the rest of the diagram.
Suppose, for example, $0<m<l<j$ (so that $j\ge3$). Using $\tilde Q''_\Lambda$ defined in [(\[eq:tildeQ”-def\])]{}, we can reorganize the diagrammatic bound [(\[eq:piNbd\])]{} on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ as (cf., [(\[eq:2nddec-bd:0<n<j\])]{}) $$\begin{aligned}
{\label{eq:diagbd-reorg}}
\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)&\leq\sum_{\substack{b_m,v_m\\ y_{l+1},
v_{l+1}}}\bigg(\sum_{\substack{b_1,\dots,b_{m-1}\\ v_1,\dots,
v_{m-1}}}P_{\Lambda;v_1}^{\prime{{\scriptscriptstyle}(0)}}(o,{\underline{b}}_1)\prod_{i=
1}^{m-1}\tau_{b_i}Q''_{\Lambda;v_i,v_{i+1}}({\overline{b}}_i,{\underline{b}}_{i+1})
\bigg){\nonumber}\\
&\qquad\times\sum_{b_{l+1}}\bigg(\sum_{\substack{b_{m+1},\dots,
b_l\\ v_{m+1},\dots,v_l}}\prod_{i=m}^l\tau_{b_i}Q''_{\Lambda;
v_i,v_{i+1}}({\overline{b}}_i,{\underline{b}}_{i+1})\bigg)\tau_{b_{l+1}}\big(\delta_{
{\overline{b}}_{l+1},y_{l+1}}+\tilde G_\Lambda({\overline{b}}_{l+1},y_{l+1})\big){\nonumber}\\
&\qquad\times\Bigg(\sum_{\substack{y_{l+2},\dots,y_j\\ v_{l+2},
\dots,v_j}}\bigg(\prod_{i=l+1}^{j-1}\tilde Q''_{\Lambda;v_i,
v_{i+1}}(y_i,y_{i+1})\bigg)P'_{\Lambda;v_j}(y_j,x)\Bigg).\end{aligned}$$ As explained above, we bound three “long” two-point functions contained in the second line of [(\[eq:diagbd-reorg\])]{}; let $Y_{m,l}$ be the supremum of what remains in the second line over $b_m,v_m,y_{l+1},v_{l+1}$. Then we can perform the sum of the first line over $b_m,v_m$ and the sum of the third line over $y_{l+1},v_{l+1}$ independently; the former is $O(\theta_0)^{m-1}$ and the latter is $O(\theta_0)^{j-1-l}$, due to [(\[eq:block-sumbd\])]{} and [(\[eq:tildeQ”-bd\])]{}, respectively. Finally, we can bound $Y_{m,l}$ using the Schwarz inequality by $O(\theta_0)^{l-m}$, where $l-m$ is the number of nonzero segments in the second line of [(\[eq:diagbd-reorg\])]{} (i.e., $\sum_{b_i}\tau_{b_i}(\delta_{{\overline{b}}_i,y_i}+\tilde
G_\Lambda({\overline{b}}_i,y_i))$ for some $y_m,\dots,y_{l+1}$) minus 2 (= the maximum number of those along the uppermost and lowermost paths that are extracted to obtain the aformentioned $|x|$-decaying term). For example, one of the leading contributions to $Y_{m,m+4}$ is bounded, by using translation invariance and the Schwarz inequality, as $$\begin{aligned}
\sup_{u,v,y}\raisebox{-1pc}{\includegraphics[scale=0.14]{Yml1}}~\leq
O(\theta_0)~\sup_{u,z}\raisebox{-1pc}{\includegraphics[scale=0.14]
{Yml2}}\\
\leq O(\theta_0)^{3/2}\left(\raisebox{-1.9pc}{\includegraphics[scale
=0.14]{Yml3}}\right)^{1/2}\leq O(\theta_0)^2~\sup_{s'}\raisebox{-1pc}
{\includegraphics[scale=0.14]{Yml4}}~&\leq O(\theta_0)^4.{\nonumber}\end{aligned}$$
The other cases can be estimated similarly [@sNN]. This completes the proof of [(\[eq:pi-kbd\])]{}.
Acknowledgements {#acknowledgements .unnumbered}
================
First of all, I am grateful to Masao Ohno for having drawn my attention to the subject of this paper. I would like to thank Takashi Hara for stimulating discussions and his hospitality during my visit to Kyushu University in December 2004 and April 2005. I would also like to thank Aernout van Enter for useful discussions on reflection positivity. Special thanks go to Mark Holmes and John Imbrie for continual encouragement and valuable comments to the former versions of the manuscript, and Remco van der Hofstad for his constant support in various aspects. This work was supported in part by the Postdoctoral Fellowship of EURANDOM, and in part by the Netherlands Organization for Scientific Research (NWO).
[99]{}
M. Aizenman. Geometric analysis of $\phi^4$ fields and Ising models. *Commun. Math. Phys.* [**86**]{} (1982): 1–48.
M. Aizenman, D.J. Barsky and R. Fernández. The phase transition in a general class of Ising-type models is sharp. *J. Statist. Phys.* [**47**]{} (1987): 343–374.
M. Aizenman and R. Fernández. On the critical behavior of the magnetization in high-dimensional Ising models. *J. Statist. Phys.* [**44**]{} (1986): 393–454.
M. Aizenman and R. Graham. On the renormalized coupling constant and the susceptibility in $\phi_4^4$ field theory and the Ising model in four dimensions. *Nucl. Phys. B* [**225**]{} (1983): 261–288.
J. van den Berg and H. Kesten. Inequalities with applications to percolation and reliability. (1985): 556–569.
M. Biskup, L. Chayes and N. Crawford. Mean-field driven first-order phase transitions in systems with long-range interactions. *Preprint* (2005).
T. Bodineau. Translation invariant Gibbs states for the Ising model. *Preprint* (2005). To appear in *Probab. Th. Rel. Fields*.
D. Brydges and T. Spencer. Self-avoiding walk in 5 or more dimensions. *Commum. Math. Phys.* [**97**]{} (1985): 125–148.
M. Campanino, D. Ioffe and Y. Velenik. Ornstein-Zernike theory for finite range Ising models above $T_c$. *Probab. Th. Rel. Fields* [**125**]{} (2003): 305–349.
R. Fernández, J. Fröhlich and A.D. Sokal. *Random walks, critical phenomena, and triviality in quantum field theory*. Springer, Berlin (1992).
C.M. Fortuin and P.W. Kasteleyn. On the random-cluster model I. Introduction and relation to other models. *Physica* [**57**]{} (1972): 536–564.
J. Fröhlich, B. Simon and T. Spencer. Infrared bounds, phase transitions and continuous symmetry breaking. *Commun. Math. Phys.* [**50**]{} (1976): 79–95.
R.B. Griffiths, C.A. Hurst and S. Sherman. Concavity of magnetization of an Ising ferromagnet in a positive external field. *J. Math. Phys.* [**11**]{} (1970): 790–795.
T. Hara. Decay of correlations in nearest-neighbour self-avoiding walk, percolation, lattice trees and animals. *Preprint* (2005).
T. Hara, R. van der Hofstad and G. Slade. Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. *Ann. Probab.* [**31**]{} (2003): 349–408.
T. Hara and G. Slade. On the upper critical dimension of lattice trees and lattice animals. *J. Statist. Phys.* [**59**]{} (1990): 1469–1510.
T. Hara and G. Slade. Mean-field critical behaviour for percolation in high dimensions. *Commun. Math. Phys.* [**128**]{} (1990): 333–391.
T. Hara and G. Slade. Self-avoiding walk in five or more dimensions. I. The critical behaviour. *Commun. Math. Phys.* [**147**]{} (1992): 101–136.
T. Hara and G. Slade. Mean-field behaviour and the lace expansion. *Probability and Phase Transition* (ed., G. R. Grimmett). Kluwer, Dordrecht (1994): 87–122.
M. Heydenreich, R. van der Hofstad and A. Sakai. Mean-field behavior for long- and finite-range Ising model, percolation and self-avoiding walk. *In preparation*.
R. van der Hofstad and G. Slade. A generalised inductive approach to the lace expansion. *Probab. Th. Rel. Fields* [**122**]{} (2002): 389–430.
E. Ising. Beitrag zur Theorie des Ferromagnetismus. *Zeitschrift für Physik* [**31**]{} (1925): 253–258.
E.H. Lieb. A refinement of Simon’s correlation inequality. (1980): 127–135.
N. Madras and G. Slade. *The Self-Avoiding Walk*. Birkhäuser, Boston (1993).
B.G. Nguyen and W-S. Yang. Triangle condition for oriented percolation in high dimensions. *Ann. Probab.* [**21**]{} (1993): 1809–1844.
L. Onsager. Crystal statistics. I. A two-dimensional model with an order-disorder transition. (1944): 117–149.
A. Sakai. Mean-field critical behavior for the contact process. *J. Statist. Phys.* [**104**]{} (2001): 111–143.
A. Sakai. Asymptotic behavior of the critical two-point function for spread-out Ising ferromagnets above four dimensions. *Unpublished manuscript* (2005).
A. Sakai. Diagrammatic estimates of the lace-expansion coefficients for finite-range Ising ferromagnets. *Unpublished notes* (2006).
B. Simon. Correlation inequalities and the decay of correlations in ferromagnets. (1980): 111–126.
G. Slade. The lace expansion and its applications. Springer Lecture Notes in Mathematics [**1879**]{} (2006).
A.D. Sokal. An alternate constructive approach to the $\varphi_3^4$ quantum field theory, and a possible destructive approach to $\varphi_4^4$. *Ann. Inst. Henri Poincaré Phys. Théorique* [**37**]{} (1982): 317–398.
[^1]: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK. [[email protected]]{}
[^2]: Updated: November 13, 2006
[^3]: In [(\[eq:IRbd-so\])]{} and [(\[eq:IRbd-sokal\])]{}, we also use the fact that, for $p<{p_\text{c}}$, our $G_p$ (i.e., the infinite-volume limit of the two-point function under the free-boundary condition) is equal to the infinite-volume limit of the two-point function under the periodic-boundary condition.
[^4]: The mean-field results in [@a82; @abf87; @af86; @ag83] are based on a couple of differential inequalities for $M_{p,h}$ and $\chi_p$ (under the periodic-boundary condition) using a certain random-walk representation. We can simplify the proof of the same differential inequalities (under the free-boundary condition as well) using Proposition \[prp:through\].
[^5]: Repeated applications of [(\[eq:G-delta-bd\])]{} to the translation-invariant models result in the random-walk bound: ${{\langle \varphi_o\varphi_x \rangle}}_\Lambda\leq S_\tau(x)$ for $\Lambda\subset{{\mathbb Z}^d}$ and $\tau\leq1$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
A high-performance shooting algorithm is developed to compute time-periodic solutions of the free-surface Euler equations with spectral accuracy in double and quadruple precision. The method is used to study resonance and its effect on standing water waves. We identify new nucleation mechanisms in which isolated large-amplitude solutions, and closed loops of such solutions, suddenly exist for depths below a critical threshold. We also study degenerate and secondary bifurcations related to Wilton’s ripples in the traveling case, and explore the breakdown of self-similarity at the crests of extreme standing waves. In shallow water, we find that standing waves take the form of counter-propagating solitary waves that repeatedly collide quasi-elastically. In deep water with surface tension, we find that standing waves resemble counter-propagating depression waves. We also discuss existence and non-uniqueness of solutions, and smooth versus erratic dependence of Fourier modes on wave amplitude and fluid depth.
In the numerical method, robustness is achieved by posing the problem as an overdetermined nonlinear system and using either adjoint-based minimization techniques or a quadratically convergent trust-region method to minimize the objective function. Efficiency is achieved in the trust-region approach by parallelizing the Jacobian computation so the setup cost of computing the Dirichlet-to-Neumann operator in the variational equation is not repeated for each column. Updates of the Jacobian are also delayed until the previous Jacobian ceases to be useful. Accuracy is maintained using spectral collocation with optional mesh refinement in space, a high order Runge-Kutta or spectral deferred correction method in time, and quadruple-precision for improved navigation of delicate regions of parameter space as well as validation of double-precision results. Implementation issues for GPU acceleration are briefly discussed, and the performance of the algorithm is tested for a number of hardware configurations.
address: 'Department of Mathematics, University of California, Berkeley'
author:
- Jon Wilkening
- Jia Yu
title: |
Overdetermined Shooting Methods for Computing\
Standing Water Waves with Spectral Accuracy
---
=1
water waves ,standing waves , resonance ,bifurcation ,Wilton’s ripples , trust-region shooting method ,boundary integral method ,spectral deferred correction ,GPU acceleration , quadruple precision
Introduction
============
Time-periodic solutions of the free-surface Euler equations serve as an excellent benchmark for the design and implementation of numerical algorithms for two-point boundary value problems governed by nonlinear partial differential equations. In particular, there is a large body of existing work on numerical methods for computing standing waves [@schwartz:81; @mercer:92; @mercer:94; @smith:roberts:99; @tsai:jeng:94; @bryant:stiassnie:94; @schultz; @okamura:03] and short-crested waves [@roberts83b; @Marchant87b; @bridges01; @Ioualalen06] for performance comparison. Moreover, many of these previous studies reach contradictory scientific conclusions that warrant further investigation, especially concerning extreme waves and the formation of a corner or cusp. Penney and Price [@penney:52] predicted a 90 degree corner, which was verified experimentally by G. I. Taylor [@taylor:53], who was nevertheless skeptical of their analysis. Grant [@grant] and Okamura [@okamura:98] gave theoretical arguments supporting the 90 degree corner. Schwartz and Whitney [@schwartz:81] and Okamura [@okamura:03] performed numerical experiments that backed the 90 degree conjecture. Mercer and Roberts [@mercer:92] predicted a somewhat sharper angle and mentioned 60 degrees as a possibility. Schultz *et al.* [@schultz] obtained results similar to Mercer and Roberts, and proposed that a cusp may actually form rather than a corner. Wilkening [@breakdown] showed that extreme waves do not approach a limiting wave at all due to fine scale structure that emerges at the surface of very large amplitude waves and prevents the wave crest from sharpening in a self-similar manner. This raises many new questions about the behavior of large-amplitude standing waves, which we will explore in Section \[sec:breakdown\].
On the theoretical side, it has long been known [@concus:64; @vanden:broeck:book; @iooss05] that standing water waves suffer from a small-divisor problem that obstructs convergence of the perturbation expansions developed by Rayleigh [@rayleigh1876], Penney and Price [@penney:52], Tadjbakhsh and Keller [@tadjbakhsh], Concus [@concus:62], Schwartz and Whitney [@schwartz:81], and others. Penney and Price [@penney:52] went so far as to state, “there seems little likelihood that a proof of the existence of the stationary waves will ever be given.” Remarkably, Plotnikov and Toland [@plotnikov01], together with Iooss [@iooss05], have recently established existence of small-amplitude standing waves using a Nash-Moser iteration. As often happens in small-divisor problems [@craig:wayne; @bourgain99], solutions could only be proved to exist for values of an amplitude parameter in a totally disconnected Cantor set. No assertion is made about parameter values outside of this set. This raises intriguing new questions about whether resonance really causes a complete loss of smoothness in the dependence of solutions on amplitude, or if these results are an artifact of the use of Nash-Moser theory to prove existence. While a complete answer can only come through further analysis, insight can be gained by studying high precision numerical solutions.
In previous numerical studies, the most effective methods for computing standing water waves have been Fourier collocation in space and time [@vandenBroeck:81; @tsai:jeng:94; @okamura:03; @ioualalen:03; @okamura:10], semi-analytic series expansions [@schwartz:81; @amick:87], and shooting methods [@mercer:92; @mercer:94; @schultz; @smith:roberts:99]. In Fourier collocation, time-periodicity is built into the basis, and the equations of motion are imposed at collocation points to obtain a large nonlinear system of equations. This is the usual approach taken in analysis to prove existence of time-periodic solutions, e.g. of nonlinear wave equations [@craig:wayne] or nonlinear Schrödinger equations [@bourgain99]. The drawback as a numerical method is that the number of unknowns in the nonlinear system grows like $(\Delta
x\,\Delta t)^{-1}$ rather than $\Delta x^{-1}$ for a shooting method, which limits the resolution one can achieve. Orthogonal collocation, as implemented in the software package AUTO [@doedel91], would be less efficient than Fourier collocation as more timesteps will be required to achieve the same accuracy.
The semi-analytic series expansions of Schwartz and Whitney [@schwartz:81; @amick:87] are a significant improvement over previous perturbation methods [@rayleigh1876; @penney:52; @tadjbakhsh; @concus:62] in that the authors show how to compute an arbitrary number of terms rather than stopping at 3rd or 5th order. They also used conformal mapping to flatten the boundary, which leads to a more promising representation of the solution of Laplace’s equation. As a numerical method, the coefficients of the expansion are expensive to compute, which limits the number of terms one can obtain in practice. (Schwartz and Whitney stopped at 25th order). It may also be that the resulting series is an asymptotic series rather than a convergent series. Nevertheless, these series expansions play an essential role in the proof of existence of standing waves on deep water by Plotnikov, Toland and Iooss [@iooss05].
In a shooting method, one augments the known boundary values at one endpoint with additional prescribed data to make the initial value problem well posed, then looks for values of the new data to satisfy the boundary conditions at the other endpoint. For ordinary differential equations, this normally leads to a system of equations with the same number of equations as unknowns. The same is true of multi-shooting methods [@keller:68; @stoer:bulirsch; @guckenheimer:00]. When the boundary value problem is governed by a system of partial differential equations, it is customary to discretize the PDE to obtain an ODE, then proceed as described above. However, because of aliasing errors, quadrature errors, filtering errors, and amplification by the derivative operator, discretization causes larger errors in high-frequency modes than low-frequency modes when the solution is evolved in time. These errors can cause the shooting method to be too aggressive in its search for initial conditions, and to explore regions of parameter space (the space of initial conditions) where either the numerical solution is inaccurate, or the physical solution becomes singular before reaching the other endpoint. Even if safeguards are put in place to penalize high-frequency modes in the search for initial conditions, the Jacobian is often poorly conditioned due to these discretization errors.
We have found that posing boundary value problems governed by PDEs as overdetermined, nonlinear least squares problems can dramatically improve the robustness of shooting methods in two critical ways. First, we improve accuracy by padding the initial condition with high-frequency modes that are constrained to be zero. With enough padding, all the degrees of freedom controlled by the shooting method can be resolved sufficiently to compute a reliable Jacobian. Second, adding more rows to the Jacobian increases its smallest singular values, often improving the condition number by several orders of magnitude. The extra rows come from including the high-frequency modes of the boundary conditions in the system of equations, even though they are not included in the list of augmented initial conditions. As a rule of thumb, it is usually sufficient to set the top 1/3 to 1/2 of the Fourier spectrum to zero initially; additional zero-padding has little effect on the numerical solution or the condition number. Validation of accuracy by monitoring energy conservation and decay rates of Fourier modes will be discussed in Section \[sec:breakdown\], along with mesh refinement studies and comparison with quadruple precision calculations.
In this paper, we present two methods of solving the nonlinear least squares problem that arises in the overdetermined shooting framework. The first is the adjoint continuation method (ACM) of Ambrose and Wilkening [@benj1; @benj2; @vtxs1; @lasers], in which the gradient of the objective function with respect to initial conditions is computed by solving an adjoint PDE, and the BFGS algorithm [@bfgs; @nocedal] is used for the minimization. This was the approach used by one of the authors in her dissertation [@jia:thesis] to obtain the results of Sections \[sec:unit\] and \[sec:surf\]. In the second approach, we exploit an opportunity for parallelism that makes computing the entire Jacobian feasible. Once this is done, a variant of the Levenberg-Marquardt method (with less frequent Jacobian updates) is used to rapidly converge to the solution. The main challenge here is organizing the computation to maximize re-use of setup costs in solving the variational equation with multiple right-hand sides, to minimize communication between threads or with the GPU device, and to ensure that most of the linear algebra occurs at level 3 BLAS speed. The performance of the algorithms on various platforms is reported in Section \[sec:perform\].
The scientific focus of the present work is on resonance and its effect on existence, non-uniqueness, and physical behavior of standing water waves. A summary of our main results is given in the abstract, and in more detail at the beginning of Section \[sec:results\]. We mention here that resonant modes generally take the form of higher-frequency, secondary standing waves oscillating at the surface of larger-scale, primary standing waves. Because the equations are nonlinear, only certain combinations of amplitude and phase can occur for each component wave. This leads to non-uniqueness through multiple branches of solutions. In shallow water, bifurcation curves of high-frequency Fourier modes behave erratically and contain many gaps where solutions do not appear to exist. This is expected on theoretical grounds. However, these bifurcation “curves” become smoother, or “heal,” as fluid depth increases. In infinite depth, such resonant effects are largely invisible, which we quantify and discuss in Section \[sec:conclude\].
In future work [@water:stable], the methods of this paper will be used to study other families of time-periodic solutions of the free-surface Euler equations with less symmetry than is assumed here, e.g. traveling-standing waves, unidirectional solitary wave interactions, and collisions of gravity-capillary solitary waves. The stability of these solutions will also be analyzed in [@water:stable] using Floquet methods.
Equations of motion and time-stepping {#sec:eqs}
=====================================
The effectiveness of a shooting algorithm for solving two-point boundary value problems is limited by the accuracy of the time-stepper. In this section, we describe a boundary integral formulation of the water wave problem that is spectrally accurate in space and arbitrary order in time. We also describe how to implement the method in double and quadruple precision using a GPU, and discuss symmetries of the problem that can be exploited to reduce the work of computing standing waves by a factor of 4. The method is similar to other boundary integral formulations [@lh76; @baker:82; @krasny:86; @mercer:92; @mercer:94; @smith:roberts:99; @baker10], but is simpler to implement than the angle–arclength formulation used in [@hls94; @ceniceros:99; @HLS01; @vtxs1], and avoids issues of identifying two curves that are equal “up to reparametrization” when the $x$ and $y$ coordinates of the interface are both evolved (in non-symmetric problems). Our approach also avoids sawtooth instabilities that sometimes occur when using Lagrangian markers [@lh76; @mercer:92]. This is consistent with the results of Baker and Nachbin [@baker:nachbin:98], who found that sawtooth instabilities can be controlled without filtering using the correct combination of spectral differentiation and interpolation schemes. While conformal mapping methods [@dyachenko:1996; @nachbin:04; @milewski:11] are more efficient than boundary integral methods in many situations, they are not suitable for modeling extreme waves as the spacing between grid points expands severely in regions where wave crests form, which is the opposite of what is needed for an efficient representation of the solution via mesh refinement.
Equations of motion {#sec:motion}
-------------------
We consider a two-dimensional irrotational ideal fluid [@whitham74; @johnson97; @craik04; @craik05] bounded below by a flat wall and above by an evolving surface, $\eta(x,t)$. Because the flow is irrotational, there is a velocity potential $\phi$ such that $\mathbf{u}=\nabla\phi$. The restriction of $\phi$ to the free surface is denoted $\varphi(x,t)=\phi(x,\eta(x,t),t)$. The equations of motion governing $\eta(x,t)$ and $\varphi(x,t)$ are
\[eq:ww\] $$\begin{aligned}
\label{eq:ww:1}
\eta_t &= \phi_y - \eta_x\phi_x, \\[-3pt]
\label{eq:ww:2}
\varphi_t &= P\left[\phi_y\eta_t - \frac{1}{2}\phi_x^2 -
\frac{1}{2}\phi_y^2 - g\eta +
\frac{\sigma}{\rho}\partial_x\left(\frac{\eta_x}{\sqrt{1+\eta_x^2}}
\right)\right].
\end{aligned}$$
Here $g$ is the acceleration of gravity, $\rho$ is the fluid density, $\sigma\ge0$ is the surface tension (possibly zero), and $P$ is the $L^2$ projection to zero mean that annihilates constant functions, $$P = {\operatorname}{id} - P_0, \qquad
P_0 f = \frac{1}{2\pi}\int_0^{2\pi} f(x)\,dx.$$ This projection is not standard in (\[eq:ww:2\]), but yields a convenient convention for selecting the arbitrary additive constant in the potential. In fact, if the fluid has infinite depth and the mean surface height is zero, $P$ has no effect in (\[eq:ww:2\]) at the PDE level, ignoring roundoff and discretization errors. The velocity components $u=\phi_x$, $v=\phi_y$ on the right hand side of (\[eq:ww\]) are evaluated at the free surface to determine $\eta_t$ and $\varphi_t$. The system is closed by relating $\phi$ in the fluid to $\eta$ and $\varphi$ on the surface as the solution of Laplace’s equation
\[eq:dno\] $$\begin{aligned}
\label{eq:dno:1}
\phi_{xx} + \phi_{yy} &= 0, & -{h}&< y < \eta, \\
\label{eq:dno:2}
\phi_y &= 0, & y &= -{h}, \\
\label{eq:dno:3}
\phi &= \varphi, & y &= \eta,\end{aligned}$$
where ${h}$ is the mean fluid depth (possibly infinite). We assume $\eta(x,t)$ and $\mathbf{u}(x,y,t)$ are $2\pi$-periodic in $x$. Applying a horizontal Galilean transformation if necessary, we may also assume $\phi$ is $2\pi$-periodic in $x$. We generally assume ${h}=0$ in the finite depth case and absorb the mean fluid depth into $\eta$ itself. This causes $-\eta(x)$ to be a reflection of the free surface across the bottom boundary, which simplifies many formulas in the boundary integral formulation below. The same strategy can also be applied in the presence of a more general bottom topography [@nachbin:04].
Equation (\[eq:ww:1\]) is a kinematic condition requiring that particles on the surface remain there. Equation (\[eq:ww:2\]) comes from $\varphi_t = \phi_y\eta_t + \phi_t$ and the unsteady Bernoulli equation, $\phi_t + \frac{1}{2}|\nabla\phi|^2 + gy + \frac{p}{\rho} =
c(t)$, where $c(t)$ is constant in space but otherwise arbitrary. At the free surface, we assume the pressure jump across the interface due to surface tension is proportional to curvature, $p_0-p\vert_{y=\eta} = \sigma\kappa$. The ambient pressure $p_0$ is absorbed into the arbitrary function $c(t)$, which is chosen to preserve the mean of $\varphi(x,t)$: $$\label{eq:c:def}
c(t) = \frac{p_0}{\rho} + P_0\left[
\eta_x\phi_x\phi_y + \frac{1}{2}\phi_x^2 - \frac{1}{2}\phi_y^2
+ g\eta - \frac{\sigma}{\rho}\partial_x\left(\frac{\eta_x}{
\sqrt{1+\eta_x^2}}\right)
\right].$$ The advantage of this construction is that $\mathbf{u}=\nabla\phi$ is time-periodic with period $T$ if and only if $\eta$ and $\varphi$ are time-periodic with the same period. Otherwise, $\varphi(x,T)$ could differ from $\varphi(x,0)$ by a constant function without affecting the periodicity of $\mathbf{u}$.
Details of our boundary integral formulation are given in Appendix \[sec:BI\]. Briefly, we identify $\mathbb{R}^2$ with $\mathbb{C}$ and parametrize the free surface by $$\label{eq:zeta}
\zeta(\alpha) = \xi(\alpha) + i\eta(\xi(\alpha)),$$ where the change of variables $x=\xi(\alpha)$ allows for smooth mesh refinement in regions of high curvature, and $t$ has been suppressed in the notation. We compute the Dirichlet-Neumann operator [@craig:sulem:93], $$\label{eq:DNO:def}
{\mathcal{G}}\varphi(x) = \sqrt{1+\eta'(x)^2}\,\, {\frac{\partial \phi}{\partial n}}(x+i\eta(x)),$$ which appears implicitly in the right hand side of (\[eq:ww\]) through $\phi_x$ and $\phi_y$, in three steps. First, we solve the integral equation $$\label{eq:fred}
\frac{1}{2}\mu(\alpha) + \frac{1}{2\pi}\int_0^{2\pi}
[K_1(\alpha,\beta) + K_2(\alpha,\beta)]\mu(\beta)\,d\beta =
\varphi(\xi(\alpha))$$ for the dipole density, $\mu(\alpha)$, in terms of the (known) Dirichlet data $\varphi(\xi(\alpha))$. Formulas for $K_1$ and $K_2$ are given in (\[eq:K1K2\]) below. These kernels are smooth functions (even at $\alpha=\beta$), so the integral is not singular; see Appendix \[sec:BI\]. Second, we differentiate $\mu(\alpha)$ to obtain the vortex sheet strength, $\gamma(\alpha)=\mu'(\alpha)$. Finally, we evaluate the normal derivative of $\phi$ at the free surface via $$\label{eq:G}
{\mathcal{G}}\varphi(\xi(\alpha)) = \frac{1}{|\xi'(\alpha)|}
\left[\frac{1}{2}H\gamma(\alpha) + \frac{1}{2\pi}\int_0^{2\pi}
[G_1(\alpha,\beta) + G_2(\alpha,\beta)]\gamma(\beta)\,d\beta
\right].$$ $G_1$ and $G_2$ are defined in (\[eq:K1K2\]) below, and $H$ is the Hilbert transform, which is diagonal in Fourier space with symbol $\hat H_k=-i{\operatorname}{sgn}(k)$. The only unbounded operation in this procedure is the second step, in which $\gamma(\alpha)$ is obtained from $\mu(\alpha)$ by taking a derivative.
Once ${\mathcal{G}}\varphi(x)$ is known, we compute $\phi_x$ and $\phi_y$ on the boundary using $$\label{eq:uv:from:G}
\begin{pmatrix} \phi_x \\ \phi_y \end{pmatrix} =
\frac{1}{1+\eta'(x)^2}\begin{pmatrix}
1 & -\eta'(x) \\ \eta'(x) & 1 \end{pmatrix}
\begin{pmatrix}
\varphi'(x) \\
{\mathcal{G}}\varphi(x)
\end{pmatrix},$$ which allows us to evaluate (\[eq:ww:1\]) and (\[eq:ww:2\]) for $\eta_t$ and $\varphi_t$. Alternatively, one can write the right hand side of (\[eq:ww\]) directly in terms of $\varphi'(x)$ and ${\mathcal{G}}\varphi(x)$.
GPU-accelerated time-stepping and quadruple precision
-----------------------------------------------------
![\[fig:spacing\] Dependence of mesh spacing on the parameter $\rho$ (dropping the subscript $l$) in (\[eq:xi\]), with mesh refinement near $x=\pi$. (left) Plots of $x=\xi(\alpha)$ for $\rho=0.0$, $0.02$, $0.04$, $0.08$, $0.25$, $0.6$ and $1.0$. (center) $E(\alpha)=\partial\xi/\partial\alpha$ represents the grid spacing relative to uniform spacing. Comparison of $E(\alpha)$ and $E(\xi^{-1}(x))$ shows how the grid points are re-distributed. (right) A magnified view near $\alpha=\pi$ shows that when $\rho$ reaches 0, $\xi(\alpha)$ ceases to be a diffeomorphism and $E(\xi^{-1}(x))$ forms a cusp. ](figs/spacing){width=".8\linewidth"}
Next we turn to the question of discretization. Because we are interested in studying large amplitude standing waves that develop relatively sharp wave crests for brief periods of time, we discretize space and time adaptively. Time is divided into $\nu$ segments $\theta_lT$, where $\theta_1 + \cdots + \theta_\nu = 1$ and $T$ is the simulation time, usually an estimate of the period or quarter-period. In the simulations reported here, $\nu$ ranges from 1 to 5 and each $\theta_l$ was close to $1/\nu$ (within a factor of two). On segment $l$, we fix the number of (uniform) timesteps, $N_l$, the number of spatial grid points, $M_l$, and the function $$\label{eq:xi}
\xi_l(\alpha) = \int_0^\alpha E_l(\beta)\,d\beta, \quad
E_l(\alpha) = \left\{\!\!\begin{array}{rl}
1 - P\big[A_l\sin^4(\alpha/2)\big], & \text{to refine near } x=\pi \\[2pt]
1 - P\big[A_l\cos^4(\alpha/2)\big], & \text{to refine near } x=0
\end{array}\!\!\right\}, \quad
A_l = \frac{8(1-\rho_l)}{5+3\rho_l},$$ which controls the grid spacing in the change of variables $x=\xi_l(\alpha)$; see Figure \[fig:spacing\]. As before, $P$ projects out the mean. The parameter $\rho_l$ lies in the range $0<\rho_l\le1$ and satisfies $$\label{eq:rho:def}
\rho_l = \frac{\min\{E_l(0), E_l(\pi)\}}{\max\{E_l(0), E_l(\pi)\}}, \qquad
\min\{E_l(0), E_l(\pi)\} = \frac{8\rho_l}{5+3\rho_l}, \qquad
\max\{E_l(0), E_l(\pi)\} = \frac{8}{5+3\rho_l}.$$ Note that $\rho_l=1$ corresponds to uniform spacing while $\rho_l=0$ corresponds to the singular limit where $\xi_l$ ceases to be a diffeomorphism at one point. This approach takes advantage of the fact that we can arrange in advance that the wave crests will form at $x=0$ and $x=\pi$, alternating between the two in time. A more automated approach would be to have the grid spacing evolve with the wave profile, perhaps as a function of curvature, rather than asking the user to specify the change of variables. We did not experiment with this idea since our approach also allows the number of grid points to increase in time, which would be complicated in an automated approach. We always set $\rho_1=1$ so that $x=\alpha$ on the first segment. Respacing the grid from segment $l$ to $l+1$ boils down to interpolating $\eta$ and $\varphi$ to obtain values on the new mesh, e.g. $\eta\circ\xi_{l+1}(\alpha_j) = \eta\circ\xi_l(
\xi_l^{-1}\circ\xi_{l+1}(\alpha_j))$, $\alpha_j=2\pi j/M_l$, which is straightforward by Newton’s method. To be safe, we avoid refining the mesh in one region at the expense of another; thus, if $\rho_{l+1}<
\rho_l$, we also require $(M_{l+1}/M_l)\ge(5+3\rho_l)/(5+3\rho_{l+1})$ so that the grid spacing decreases throughout the interval, but more so in the region where the wave crest is forming.
Since the evolution equations are not stiff unless the surface tension is large, high order explicit time-stepping schemes work well. For each Runge-Kutta stage within a timestep on a given segment $l$, the integral equation (\[eq:fred\]) is solved by collocation using uniformly spaced grid points $\alpha_j=2\pi j/M_l$ and the (spectrally accurate) trapezoidal rule, $$\frac{1}{2\pi}\int_0^{2\pi}K(\alpha_i,\beta)\mu(\beta)\,d\beta \approx
\frac{1}{M_l} \sum_{j=0}^{M_l-1} K(\alpha_i,\alpha_j)\mu(\alpha_j).$$ The matrices $K_{ij}=K(\alpha_i,\alpha_j)/M_l$ and $G_{ij}=G(\alpha_i,\alpha_j)/M_l$ that represent the discretized integral operators in (\[eq:fred\]) and (\[eq:G\]) are computed simultaneously and in parallel. The formulas are $K(\alpha,\beta)=K_1(\alpha,\beta)+K_2(\alpha,\beta)$ and $G(\alpha,\beta)=G_1(\alpha,\beta)+G_2(\alpha,\beta)$ with $$\label{eq:K1K2}
\begin{aligned}
&K_1 = {\operatorname{Im}}\left\{\frac{\zeta'(\beta)}{2}
\cot\frac{\zeta(\alpha) - \zeta(\beta)}{2}
- \frac{1}{2}\cot\frac{\alpha-\beta}{2}\right\}, \quad
K_2 = {\operatorname{Im}}\left\{\frac{\bar\zeta'(\beta)}{2}\cot
\frac{\zeta(\alpha)-\bar\zeta(\beta)}{2}\right\}, \\
& G_1 = {\operatorname{Re}}\left\{\frac{\zeta'(\alpha)}{2}
\cot\frac{\zeta(\alpha) - \zeta(\beta)}{2}
- \frac{1}{2}\cot\frac{\alpha-\beta}{2}\right\}, \quad
G_2 = {\operatorname{Re}}\left\{\frac{\zeta'(\alpha)}{2}\cot
\frac{\zeta(\alpha)-\bar\zeta(\beta)}{2}\right\}.
\end{aligned}$$ As explained in Appendix \[sec:BI\], these kernels have been regularized. Indeed, $K_1(\alpha,\beta)$ and $G_1(\alpha,\beta)$ are continuous at $\beta=\alpha$ if we define $K_1 =
-{\operatorname{Im}}\{\zeta''(\alpha)/[2\zeta'(\alpha)]\}$ and $G_1={\operatorname{Re}}\{\zeta''(\alpha)/[2\zeta'(\alpha)]\}$. These formulas are used when computing the diagonal entries $K_{ii}$ and $G_{ii}$. The terms $\cot((\alpha_i-\alpha_j)/2)$ in (\[eq:K1K2\]) are computed once and for all at the start. If the fluid depth is infinite, $K_2$ and $G_2$ are omitted. GMRES is used to solve (\[eq:fred\]) for $\mu$, which consistently takes 4-30 iterations to reach machine precision (independent of problem size). In quadruple precision, the typical range is 9-36 GMRES iterations. The FFT is used to compute $\mu'$ and $H\gamma$ in (\[eq:G\]), as well as $\zeta'$, $\zeta''$, $\eta'$, and $\varphi'$.
We wrote 3 versions of the code, which differ only in how the matrices $K$ and $G$ are computed. The simplest version uses openMP *parallel for* loops to distribute the work among all available threads. The most complicated version is parallelized using MPI and scalapack. In this case, the matrices $K$ and $G$ are stored in block-cyclic layout [@Lim96] across the processors, and each processor computes only the matrix entries it is responsible for. The fastest version of the code is parallelized on a GPU in the cuda programming language. First, the CPU sends the GPU the vector $\zeta(\alpha_j)$, which holds $M_l$ complex numbers. Next, the GPU computes the matrices $K$ and $G$ and stores them in device memory. Finally, in the GMRES iteration, Krylov vectors are sent to the GPU, which applies the matrix $K$ and returns the result as a vector. After the last Krylov iteration, the device also applies $G$ to $\mu$ to help compute ${\mathcal{G}}\varphi$ in (\[eq:G\]). Thus, communication with the GPU involves passing vectors of length $M_l$, while $O(M_l^2)$ flops must be performed on each vector passed in. As a result, communication does not pose a computational bottleneck, and the device operates at near 100% efficiency. We remark that the formula $$\cot\frac{x+iy}{2} = \begin{cases}
[\cos(x)+\cosh(y)]\big/[\sin(x)+i\sinh(y)], & \cos(x)\ge 0, \\
[\sin(x)-i\sinh(y)]\big/[\cosh(y)-\cos(x)], & \cos(x)<0
\end{cases}$$ is relatively expensive to evaluate. Thus, it pays to compute $K$ and $G$ simultaneously (to re-use $\sin$, $\cos$, $\sinh$, $\cosh$ results), and to actually store the matrices in device memory rather than re-compute the matrix entries each time a matrix-vector product is required.
In double-precision, we evolve (\[eq:ww\]) using Dormand and Prince’s DOP853 scheme [@hairer:I]. This is a 13 stage, 8th order, “first same as last” Runge-Kutta method, so the effective cost of each step is 12 function evaluations. We apply the 36th order filter described in [@hou:li:07] to the right hand side of (1e) and (1f) each time they are evaluated in the Runge-Kutta procedure, and to the solution itself at the end of each time-step. This filter consists of multiplying the $k$th Fourier mode by $$\label{eq:filter}
\exp\left[-36\big(|k|/k_\text{max}\big)^{36}\right], \qquad
k_\text{max} = M/2,
$$ which allows the highest-frequency Fourier modes to remain non-zero (to help resolve the solution) while still suppressing aliasing errors. To achieve truncation errors of order $10^{-30}$ in quadruple-precision, the 8th order method requires too many timesteps. Through trial and error, we found that a 15th order spectral deferred correction (SDC) method [@dutt; @huang; @minion] is the most efficient scheme for achieving this level of accuracy. Our GPU implementation of quadruple precision arithmetic will be discussed briefly in Section \[sec:trust\]. The variant of SDC that we use in this paper employs eight Radau IIa quadrature nodes [@hairer:I]. The initial values at the nodes are obtained via fourth order Runge-Kutta. Ten correction sweeps are then performed to improve the solution to $O(h^{15})$ accuracy at the quadrature nodes. We use pure Picard corrections instead of the more standard forward-Euler corrections as they have slightly better stability properties. The final integration step yields a local truncation error of $O(h^{16})$; hence, the method is 15th order. See [@sdc:picard] for more information about this variant of the SDC method and its properties. If one wished to go beyond quadruple-precision arithmetic, it is straightforward to increase the order of the time-stepping scheme accordingly. We did not investigate the use of symplectic integrators since our approach already conserves energy to 12-16 digits of accuracy in double precision, and 24-32 digits in quadruple precision.
Translational and time-reversal symmetry {#sec:sym}
----------------------------------------
In this paper, we restrict attention to symmetric standing waves of the type studied in [@rayleigh1876; @penney:52; @tadjbakhsh; @concus:62; @mercer:92; @mercer:94; @schultz; @ioualalen:03]. For these waves, it is only necessary to evolve the solution over a quarter period. Indeed, if at some time $T/4$ the fluid comes to rest ($\varphi\equiv0$), a time-reversal argument shows that the solution will evolve back to the initial state at $T/2$ with the sign of $\varphi$ reversed. More precisely, the condition $\varphi(x,T/4)=0$ implies that $\eta(x,T/2)=\eta(x,0)$ and $\varphi(x,T/2)=
-\varphi(x,0)$. Now suppose that, upon translation by $\pi$, $\eta(x,0)$ remains invariant while $\varphi(x,0)$ changes sign. Then we see that $\eta_1(x,t)=\eta(x+\pi,T/2+t)$ and $\varphi_1(x,t)=\varphi(x+\pi,T/2+t)$ are solutions of (\[eq:ww\]) with initial conditions $$\begin{aligned}
\eta_1(x,0)&=\eta(x+\pi,T/2)=\eta(x+\pi,0)=\eta(x,0),\\
\varphi_1(x,0)&=\varphi(x+\pi,T/2)=-\varphi(x+\pi,0)=\varphi(x,0).\end{aligned}$$ Therefore, $\eta_1=\eta$, $\varphi_1=\varphi$, and $$\begin{aligned}
\eta(x,T) &= \eta_1(x-\pi,T/2) = \eta(x-\pi,T/2) = \eta(x-\pi,0) = \eta(x,0), \\
\varphi(x,T) &= \varphi_1(x-\pi,T/2) = \varphi(x-\pi,T/2) =
-\varphi(x-\pi,0) = \varphi(x,0).\end{aligned}$$ Hence, $\eta$ and $\varphi$ are time-periodic with period $T$. It is natural to expect standing waves to have even symmetry when the origin is placed at a crest or trough and the fluid comes to rest. This assumption implies that $\eta$ and $\varphi$ will remain even functions for all time since $\eta_t$ and $\varphi_t$ in (\[eq:ww\]) are even whenever $\eta$ and $\varphi$ are. Under all these assumptions, the evolution of $\eta$ and $\varphi$ from $T/2$ to $T$ is a mirror image (about $x=\frac{\pi}{2}$ or $x=\frac{3\pi}{2}$) of the evolution from $0$ to $T/2$.
Once the initial conditions and period are found using symmetry to accelerate the search for time-periodic solutions, we double-check that the numerical solution evolved from $0$ to $T$ is indeed time-periodic. Mercer and Roberts exploited similar symmetries in their numerical computations [@mercer:92; @mercer:94].
Overdetermined shooting methods {#sec:od:shoot}
===============================
As discussed in the introduction, two-point boundary value problems governed by partial differential equations must be discretized before solving them numerically. However, truncation errors lead to loss of accuracy in the highest-frequency modes of the numerical solution, which can cause difficulty for the convergence of shooting methods. We will see below that robustness can be achieved by posing these problems as overdetermined nonlinear systems.
In Section \[sec:n:ls\], we define two objective functions with the property that driving them to zero is equivalent to finding a time-periodic standing wave. One of the objective functions exploits the symmetry discussed above to reduce the simulation time by a factor of 4. The other is more robust as it naturally penalizes high-frequency Fourier modes of the initial conditions. Both objective functions use symmetry to reduce the number of unknowns and eliminate phase shifts of the standing waves in space and time. The problem is overdetermined because the highest-frequency Fourier modes are constrained to be zero initially but not at the final time. Also, because $T/4$ often corresponds to a sharply crested wave profile, there are more active Fourier modes in the solution at that time than at $t=0$. By refining the mesh adaptively, we include all of these active modes in the objective functions, making them more overdetermined. The idea that the underlying dynamics of standing water waves is lower-dimensional than predicted by counting active Fourier modes has recently been explored by Williams, *et al.* [@pod].
In Sections \[sec:acm\] and \[sec:trust\], we describe two methods for solving the resulting nonlinear least squares problem. The first is the Adjoint Continuation Method [@benj1; @benj2; @vtxs1; @lasers], in which the gradient of the objective function is computed by solving an adjoint PDE and the BFGS algorithm [@bfgs; @nocedal] is used for the minimization. The second is a trust-region approach in which the Jacobian is computed by solving the variational equation in parallel with multiple right-hand sides. This allows the work of computing the Dirichlet-Neumann operator to be shared across all the columns of the Jacobian. We also discuss implementation issues in quadruple precision on a GPU.
Nonlinear least squares formulation {#sec:n:ls}
-----------------------------------
In the symmetric standing wave case considered here, we assume the initial conditions are even functions satisfying $\eta(x+\pi,0)=\eta(x,0)$ and $\varphi(x+\pi,0)=-\varphi(x,0)$. In Fourier space, they take the form $$\label{eq:init:stand}
\begin{aligned}
\hat\eta_k(0) &= c_{|k|}, \qquad (k=\pm2,\pm4,\pm6,\dots\;;\; |k|\le n), \\
\hat\varphi_k(0) &= c_{|k|}, \qquad (k=\pm1,\pm3,\pm5,\dots\;;\; |k|\le n),
\end{aligned}$$ where $c_1,\dots,c_n$ are real numbers, and all other Fourier modes of the initial conditions are set to zero. (In the finite depth case, we also set $\hat\eta_0={h}$, the mean fluid depth.) Here $n$ is taken to be somewhat smaller than $M_1$, e.g. $n\approx
\frac{1}{3}M_1$, where $M_1$ is the number of spatial grid points used during the first $N_1$ timesteps. (Recall that subscripts on $M$ and $N$ refer to mesh refinement sub-intervals.) Note that high-frequency Fourier modes of the initial condition are zero-padded to improve resolution of the first $n$ Fourier modes.
In addition to the Fourier modes of the initial condition, the period of the solution is unknown. We add a zeroth component to $c$ to represent the period: $$\label{eq:T:c0}
T = c_0.$$ Our goal is to find $c\in\mathbb{R}^{n+1}$ such that $\varphi(x,T/4)=0$. We therefore define the objective function $$\label{eq:f:phi}
f(c) = \frac{1}{2}r(c)^Tr(c) \approx
\frac{1}{4\pi}\int_0^{2\pi}
\varphi(x,T/4)^2\,dx,
\qquad
r_i = \varphi(\xi_\nu(\alpha_i),T/4)\sqrt{E_\nu(\alpha_i)/M_\nu},$$ where $\nu$ is the index of the final sub-interval in the mesh refinement strategy and the square root is a quadrature weight to approximate the integral via the trapezoidal rule after the change of variables $x=\xi_\nu(\alpha)$, $dx=E_\nu(\alpha)\,d\alpha$. Note that $r\in\mathbb{R}^m$ with $m=M_\nu$, which is usually several times larger than $n$, the number of non-zero initial conditions. The numerical solution is not sensitive to the choice of $m$ and $n$ as long as enough zero-padding is included in the initial condition to resolve the highest frequency Fourier modes. This will be confirmed in Section \[sec:breakdown\] through mesh-refinement studies and comparison with quadruple-precision computations.
One can also use an objective function that measures deviation from time-periodicity directly: $$\label{eq:f:naive}
f(c) \approx
\frac{1}{4\pi}\int_0^{2\pi}
\big[ \eta(x,T) - \eta(x,0) \big]^2 +
\big[ \varphi(x,T) - \varphi(x,0) \big]^2\,dx.$$ When the underlying PDE is stiff (e.g. for the Benjamin-Ono [@benj1; @benj2] or KdV equations), an objective function of the form (\[eq:f:naive\]) has a key advantage over (\[eq:f:phi\]). For stiff problems, semi-implicit time-stepping methods are used in order to take reasonably large time-steps. Such methods damp high-frequency modes of the initial condition. This causes these modes to have little effect on an objective function of the form (\[eq:f:phi\]); thus, the Jacobian $J_{ij}=\partial r_i/\partial
c_j$ can be poorly conditioned if the shooting method attempts to solve for too many modes. By contrast, when implemented via (\[eq:f:naive\]), the initial conditions of high-frequency modes are heavily penalized for deviating from the damped values at time $T$. As a result, the Jacobian does not suffer from rank deficiency, and high-frequency modes do not drift far from zero unless doing so is helpful. Since the water wave is not stiff, we use explicit schemes that do not significantly damp high-frequency modes; therefore, the computational advantage of evolving over a quarter-period outweigh any robustness advantage of using (\[eq:f:naive\]).
We used symmetry to reduce the number of unknown initial conditions in (\[eq:init:stand\]). This has the added benefit of selecting the spatial and temporal phase of each solution in a systematic manner. In problems where the symmetries of the solution are not known in advance, or to search for symmetry-breaking bifurcations, one can revert to the approach described in [@benj1], where both real and imaginary parts of the leading Fourier modes of the initial condition were computed in the search for time-periodic solutions. To eliminate spatial and temporal phase shifts, one of the Fourier modes was constrained to be real and its time derivative was required to be imaginary. Constraining the time-derivative of a mode is most easily done with a penalty function [@benj1]. Alternatively, if two modes are constrained to be real and their time-derivatives are left arbitrary, it is easier to remove their imaginary parts from the search space than to use a penalty function.
Once phase shifts have been eliminated, the families of time-periodic solutions we have found appear to sweep out two-parameter families of solutions. To compute a solution in a family, we specify the mean depth and the value of one of the $c_k$ in (\[eq:init:stand\]) or (\[eq:T:c0\]) and solve for the other $c_j$ to minimize the objective function. If $f$ is reduced below a specified threshold (typically $10^{-26}$ in double-precision or $10^{-52}$ in quadruple precision), we consider the solution to be time-periodic. If $f$ reaches a local minimum that is higher than the specified threshold, we either (1) refine the mesh, increase $n$, and try again; (2) choose a different value of $c_k$ closer to the previous successful value; or (3) change the index $k$ specifying which Fourier mode is used as a bifurcation parameter. Switching to a different $k$ is often useful when tracking a fold in the bifurcation curve. Since $c\in\mathbb{R}^{n+1}$ and one parameter has been frozen, $f$ is effectively a function of $n$ variables.
We note that once $n$ and the mesh parameters $\nu$, $\theta_l$, $A_l$, $M_l$ and $N_l$ are chosen, $f(c)$ is a smooth function that can be minimized using a variety of optimization techniques. Small divisors come into play when deciding whether $f$ would really converge to zero in the mesh refinement limit (with ever increasing numerical precision). The answer may depend on whether the bifurcation parameters ($\hat\eta_0$ and either $\eta(a,0)$ or one of the $c_k$) are allowed to vary within the tolerance of the current roundoff threshold each time the mesh is refined and the floating point precision is increased. While it is likely that small divisors prevent the existence of smooth families of exact solutions, exceedingly accurate approximate solutions do appear to sweep out smooth families, with occasional disconnections in the bifurcation curves due to resonance.
Adjoint continuation method {#sec:acm}
---------------------------
Having recast the shooting method as an overdetermined nonlinear least squares problem, we must now minimize the functional $f$ in (\[eq:f:phi\]) or (\[eq:f:naive\]). The first approach we tried was the adjoint continuation method (ACM) developed by Ambrose and Wilkening to study time-periodic solutions of the Benjamin-Ono equation [@benj1; @benj2] and the vortex sheet with surface tension [@vtxs1]. The method has also been used by Williams *et al.* to study the stability transition from single-pulse to multi-pulse dynamics in a mode-locked laser system [@lasers].
The idea of the ACM is to compute the gradient of $f$ with respect to the initial conditions by solving an adjoint PDE, and then minimize $f$ using the BFGS method [@bfgs; @nocedal]. BFGS is a quasi-Newton algorithm that builds an approximate (inverse) Hessian matrix from the sequence of gradient vectors it encounters on successive line-searches. In more detail, let $q=(\eta,\varphi)$ and denote the system (\[eq:ww\]) abstractly by $$\label{eq:qt}
q_t = F(q), \qquad q(x,0)=q_0(x).$$ We define the inner product $$\label{eq:inner:prod}
\langle q_1,q_2\rangle=\frac{1}{2\pi}\int_0^{2\pi}
\left[\eta_1(x)\eta_2(x)+\varphi_1(x)\varphi_2(x)\right]dx$$ so that $f$ in (\[eq:f:phi\]), written now as a function of the initial conditions and proposed period, which themselves depend on $c$ via (\[eq:init:stand\]) and (\[eq:T:c0\]), takes the form $$\label{eq:f:phi:again}
f(q_0,T) = \frac{1}{2}\|\,\big(0,\varphi(\cdot,T/4)\big)\,\|^2 =
\frac{1}{4\pi} \int_0^{2\pi} \varphi(x,T/4)^2\,dx,$$ where $q=(\eta,\varphi)$ solves (\[eq:qt\]). The case with $f$ of the form (\[eq:f:naive\]) is similar, so we omit details here. In the course of minimizing $f$, the BFGS algorithm will repeatedly query the user to evaluate both $f(c)$ and its gradient $\nabla_c
f(c)$ at a sequence of points $c\in\mathbb{R}^{n+1}$. The $T$ derivative, $\partial f/\partial c_0$, is easily obtained by evaluating $$\label{eq:dfdT}
{\frac{\partial f}{\partial T}} = \frac{1}{8\pi}\int_0^{2\pi}
\varphi(x,T/4)\,\varphi_t(x,T/4)\,dx$$ using the trapezoidal rule after changing variables, $x=\xi_\nu(\alpha)$, $dx=E_\nu(\alpha)\,d\alpha$. Note that $\varphi(\cdot,T/4)$ and $\varphi_t(\cdot,T/4)$ are already known by solving (\[eq:ww\]). One way to compute the other components of $\nabla_c f$, say $\partial
f/\partial c_k$, would be to solve the variational equation, (written abstractly here and explicitly in Appendix \[sec:adjoint\]) $$\label{eq:dot}
\dot{q}_t=D F(q(\cdot,t))\dot{q}, \qquad
\dot q(x,0) = \dot q_0(x)$$ with initial conditions $$\label{eq:dot:ic}
\dot q_0(x) = \begin{cases}
(e^{ikx}+e^{-ikx},0), & k=1,3,5,\dots \\
(0,e^{ikx}+e^{-ikx}), & k=2,4,6,\dots
\end{cases}$$ to obtain $$\label{eq:f:dot1}
{\frac{\partial f}{\partial c_k}} =
\dot{f} = \left.\frac{d}{d{\varepsilon}}\right|_{{\varepsilon}=0}
f(q_0+{\varepsilon}\dot{q}_0,T)=
\big\langle\, \big(0,\varphi(\cdot,T/4)\big)\,,
\big(0,\dot{\varphi}(\cdot,T/4)\big)\,\big\rangle.$$ Note that a dot denotes a directional derivative with respect to the initial condition, not a time-derivative. To avoid the expense of solving (\[eq:dot\]) repeatedly for each value of $k$, we solve a single adjoint PDE to find $\delta f/\delta q_0$ such that $\dot f = \langle\, \delta f/\delta q_0\,,\,\dot q_0 \,\rangle$. From (\[eq:f:dot1\]), we have $$\label{eq:f:dot2}
\dot{f} = \big\langle\,\big(0,\varphi(\cdot,T/4)\big)\,,
\big( \dot\eta(\cdot,T/4),\dot{\varphi}(\cdot,T/4)\big)\,
\big\rangle = \langle\, \tilde q_0\,,\dot q(\cdot,T/4)\,\rangle,$$ where we have defined $\tilde q_0 = (\tilde\eta_0,\tilde\varphi_0)$ with $\tilde\eta_0 = 0$ and $\tilde\varphi_0 = \varphi(\cdot,T/4)$. Note that replacing 0 by $\dot\eta(\cdot,T/4)$ did not affect the inner product. Next we observe that the solution $\tilde q(x,s)$ of the adjoint equation $$\label{eq:adj}
\tilde{q}_s=D F(q(\cdot,T/4-s))^*\tilde{q}, \qquad
\tilde q(\cdot,0) = \tilde q_0,$$ which evolves backward in time ($s=T/4-t$), has the property that $$\langle\tilde{q}(\cdot,T/4-t),\dot{q}(\cdot,t)\rangle=\text{const}.$$ Setting $t=T/4$ shows that this constant is actually $\dot
f$. Setting $t=0$ gives the form we want: $$\dot f = \langle\, \delta f/\delta q_0\,,\,\dot q_0 \,\rangle, \qquad
{\frac{\delta f}{\delta q_0}} = \tilde q(\cdot,T/4).$$ From (\[eq:dot:ic\]), we obtain $$\label{eq:nabla:f}
{\frac{\partial f}{\partial c_k}} = \left\{\begin{array}{cc}
2{\operatorname{Re}}\big\{{{\tilde\eta}^{\scriptscriptstyle\bm\wedge}}_k(T/4)\big\}, & k = 1,3,5,\dots \\
2{\operatorname{Re}}\big\{{{\tilde\varphi}^{\scriptscriptstyle\bm\wedge}}_k(T/4)\big\}, & k = 2,4,6,\dots
\end{array}\right\}.$$ Together with (\[eq:dfdT\]), this gives all the components of $\nabla_c f$ at once. Explicit formulas for the linearized and adjoint equations (\[eq:dot\]) and (\[eq:adj\]) are derived in Appendix \[sec:adjoint\].
Like (\[eq:dot\]), the adjoint equation (\[eq:adj\]) is linear, but non-autonomous, due to the presence of the solution $q(t)$ of (\[eq:qt\]) in the equation. In the BFGS method, the gradient is always called immediately after computing the function value; thus, if $q(t)$ and $q_t(t)$ are stored in memory at each timestep in the forward solve, they are available in the adjoint solve at intermediate Runge-Kutta steps through cubic Hermite interpolation. We actually use dense output formulas [@shampine86; @hairer:I] for the 5th and 8th order Dormand-Prince schemes since cubic Hermite interpolation limits the accuracy of the adjoint solve to 4th order, but the idea is the same. If there is insufficient memory to store the solution at every timestep, we store the solution at equally spaced mile-markers and re-compute $q$ between them when $\tilde q$ reaches that region. Thus, $\nabla f$ can be computed in approximately the same amount of time as $f$ itself, or twice the time if mile-markers are used.
It is worth mentioning that, when discretized, the values of $\eta$ and $\varphi$ are stored on a non-uniformly spaced grid for each segment $l\in\{2,\dots,\nu\}$ in the mesh-refinement strategy. The adjoint variables $\tilde\eta$, $\tilde\varphi$ are stored at the same mesh points, and are initialized by $$\tilde\eta_0\circ\xi_\nu(\alpha_i)=0, \qquad
\tilde\varphi_0\circ\xi_\nu(\alpha_i) = \varphi(\xi_\nu(\alpha_i),T/4),$$ with no additional weight factors needed. This works because the inner product (\[eq:inner:prod\]) is defined with respect to $x$ rather than $\alpha$, and the change of variables has been accounted for by the factor $\sqrt{E_\nu(\alpha_i)/M_\nu}$ in the formula (\[eq:f:phi\]) for $f$.
Trust-region shooting method {#sec:trust}
----------------------------
While the ACM method gives an efficient way of computing the gradient of $f$, it takes many line-search iterations to build up an accurate approximation of the Hessian of $f$. This misses a key opportunity for parallelism and re-use of data that can be exploited if we switch from the BFGS framework to a Levenberg-Marquardt approach [@nocedal]. Instead of solving the adjoint equation (\[eq:adj\]) to compute $\nabla f$ efficiently, we solve the variational equation (\[eq:dot\]) with multiple right-hand sides to compute all the columns of the Jacobian simultaneously. From (\[eq:f:phi\]), we see that $$\label{eq:Jik}
J_{ik} = {\frac{\partial r_i}{\partial c_k}} = \begin{cases}
\varphi_t(\xi_\nu(\alpha_i),T/4)\sqrt{E_\nu(\alpha_i)/M_\nu}, & k=0, \\
\dot\varphi(\xi_\nu(\alpha_i),T/4)\sqrt{E_\nu(\alpha_i)/M_\nu}, & k\ge1,
\end{cases}$$ where $\dot q_0$ is initialized as in (\[eq:dot:ic\]) for $k\ge1$. We avoid the need to store $q$ at every timestep (or at mile-markers) by evolving $q$ along with $\dot q$ rather than interpolating $q$: $$\label{eq:q:qdot}
{\frac{\partial }{\partial t}}
\begin{pmatrix} q \\ \dot q \end{pmatrix} =
\begin{pmatrix} F(q) \\ DF(q)\dot q \end{pmatrix}, \qquad
\begin{aligned}
q(0) &= q_0 = (\eta_0,\varphi_0), \\
\dot q(0) &= \dot q_0 = \partial q_0/\partial c_k.
\end{aligned}$$ In practice, we replace $\dot q$ in (\[eq:q:qdot\]) by the matrix $\dot Q=[\dot q_{(k=1)},\dots,\dot q_{(k=n)}]$ to compute all the columns of $J$ (besides $k=0$) at once. The linearized equations (\[eq:lin2\]) involve the same Dirichlet-to-Neumann operator as the nonlinear equations (\[eq:ww\]), so the matrices $K$ and $G$ in (\[eq:fred\]) and (\[eq:G\]) only have to be computed once to evolve the entire matrix $\dot Q$ through a Runge-Kutta stage. Moreover, the linear algebra involved can be implemented at level 3 BLAS speed. For large problems, we perform an LU-factorization of $K$, the cost of which is made up for many times over by replacing GMRES iterations with a single back-solve stage for each right-hand side. In the GPU version of the code, all the linear algebra involving $K$ and $G$ is performed on the device (using the CULA library). As before, communication with the device is minimal in comparison to the computational work performed there.
We emphasize that the main advantage of solving linearized equations is that the same DNO operator is used for each column of $\dot Q$ in a given Runge-Kutta stage. This opportunity is lost in the simpler approach of approximating $J$ through finite differences by evolving (\[eq:qt\]) repeatedly, with initial conditions perturbed in each coordinate direction: $$\label{eq:J:fd}
J_{ik}\approx\frac{r_i(c+{\varepsilon}e_k)-r_i(c)}{{\varepsilon}},
\qquad e_k = (0,\dots,0,1,0,\dots,0)^T\in\mathbb{R}^{n+1}.$$ Thus, while finite differences can also be parallelized efficiently by evolving these solutions independently, the matrices $K$ and $G$ will be computed $n$ times more often in the finite difference approach, and most of the linear algebra will drop from running at level 3 BLAS speed to level 2. Details of our Levenberg-Marquardt implementation are given in Appendix \[sec:levmar\], where we discuss how to re-use the Jacobian several times rather than re-computing it each time a step is accepted.
The CULA and LAPACK libraries could not be used for quadruple precision calculations, and we did not try FFTW in that mode. Instead, we used custom FFT and linear algebra libraries (written by Wilkening) for this purpose. However, for the GPU, we did not have any previous code to build on. Our solution was to write a block version of matrix-matrix multiplication in CUDA to compute residuals in quadruple precision, then use iterative refinement to solve for the corrections in double-precision, using the CULA library. Although quadruple precision is not native on any current GPU, we found M. Lu’s *gqd* package [@gqd], which is a CUDA implementation of Bailey’s *qd* package [@qd], to be quite fast. Our code is written so that the floating point type can be changed through a simple *typedef* in a header file. This is possible in C++ by overloading function names and operators to call the appropriate versions of routines based on the argument types.
Numerical results {#sec:results}
=================
This section is organized as follows: In Section \[sec:unit\], we use the Adjoint Continuation Method to study standing waves of wavelength $2\pi$ in water of uniform depth $h=1$. Several disconnections in the bifurcation curves are encountered, which are shown to correspond physically to higher-frequency standing waves superposed (nonlinearly) on the low-frequency carrier wave. In Section \[sec:nuc\], we use the trust-region approach to study a nucleation event in which isolated large-amplitude solutions, and closed loops of such solutions, suddenly exist for depths below a threshold value. This gives a new mechanism for the creation of additional branches of solutions (besides harmonic resonance [@mercer:94; @smith:roberts:99]). In Section \[sec:bif:degen\], we study a “Wilton ripple” phenomenon [@vandenBroeck:84; @chen:saffman:79; @bryant:stiassnie:94; @bridges:standing:3d; @akers:wilton] in which a pair of “mixed mode” solutions bifurcate along side the “pure mode” solutions at a critical depth. Our numerical solutions are accurate enough to identify the leading terms in the asymptotic expansion of these mixed mode solutions. Following the mixed-mode branches via numerical continuation reveals that they meet up with the pure mode branches again at large amplitude. We also study how this degenerate bifurcation splits when the fluid depth is perturbed. In Section \[sec:breakdown\], we study what goes wrong in the Penney and Price conjecture, which predicts that the limiting standing wave of extreme form will develop sharp 90 degree corner angles at the wave crests. We also discuss energy conservation, decay of Fourier modes, and validation of accuracy. In Section \[sec:shallow\], we study collisions of counter-propagating solitary water waves that are elastic in the sense that the background radiation is identical before and after the collision. In Section \[sec:surf\], we study time-periodic gravity-capillary waves of the type studied by Concus [@concus:62] and Vanden-Broeck [@vandenBroeck:84] using perturbation theory. Finally, in Section \[sec:perform\], we compare the performance of the algorithms on a variety of parallel machines, using MINPACK as a benchmark for solving nonlinear least squares problems.
Standing waves of unit depth {#sec:unit}
----------------------------
We begin by computing a family of symmetric standing waves with mean fluid depth $\hat\eta_0 = {h}= 1$ and zero surface tension. The linearized equations about a flat rest state are $$\label{eq:lin}
\dot\eta_t = {\mathcal{G}}\dot\varphi, \qquad
\dot\varphi_t = P[-g\dot\eta], \qquad
\Big({\mathcal{G}}\big[e^{ikx}\big] = \big[k\tanh k{h}\big]e^{ikx}\Big).$$ Thus, the linearized problem has standing wave solutions of the form $$\label{eq:lin:soln}
\dot\eta = A\sin\omega t \cos kx, \quad
\dot\varphi = B\cos\omega t \cos kx, \quad
\omega^2 = kg\tanh k{h}, \quad
A/B = \sqrt{(k/g)\tanh kh}.$$ Setting $h=1$, $g=1$, $k=1$, these solutions have period $T=2\pi/\omega \approx 7.19976$. Here $B$ is a free parameter controlling the amplitude, and $A$ is determined by $A/B=\sqrt{\tanh 1}$.
To find time-periodic solutions of the nonlinear problem, we start with a small amplitude linearized solution as an initial guess. Holding $c_1=\hat\varphi_1(0)$ constant, we solve for the other $c_k$ in (\[eq:init:stand\]) using the ACM method of Section \[sec:acm\]. Note that $c_1=B$ in the linearized regime. We then repeat this procedure for another value of $c_1$ to obtain a second small-amplitude solution of the nonlinear problem. The particular choices we made were $c_1=-0.001$ and $c_1=-0.002$. We then varied $c_1$ in increments of $-0.001$, using linear extrapolation from the previous two solutions for the initial guess. The results are shown in Fig. \[fig:bif10\]. The two representative solutions labeled A and B show that the amplitude of the wave increases and the crest sharpens as the magnitude of $c_1$ increases. We chose $c_1$ to be negative so the peak at $T/4$ would occur at $x=\pi$ rather than $x=0$. An identical bifurcation curve (reflected about the $T$-axis) would be obtained by increasing $c_1$ from 0 to positive values. The solutions A and B would then be shifted by $\pi$ in space.
![\[fig:bif10\] A family of standing water waves of unit depth (${h}=1$) bifurcates from the stationary solution at $T=2\pi/\sqrt{\tanh{1}}\approx7.200$. We used the ACM method to track the family out of the linearized regime via numerical continuation. The period initially decreases with amplitude, but later increases to surpass the period of the linearized standing waves. A resonance near solution $A$ causes the 9th Fourier mode of $\varphi$ to jump discontinuously as the period increases. This resonance has little effect on the first Fourier mode.](figs/bif_water_10_csd){width="\linewidth"}
For most values of $c_1$ between $0.0$ and $-0.23$, the ACM method has no difficulty finding time-periodic solutions to an accuracy of $f<10^{-26}$. However, at $c_1=-0.201$, the minimum value of $f$ exceeds this target. On further investigation, we found there was a small gap, $c_1\in(-0.20113,-0.20124)$, where we were unable to compute time-periodic solutions even after increasing $M$ from 256 to 512 and decreasing the continuation stepsize to $\Delta c_1 =
1.0\times 10^{-5}$. By plotting other Fourier modes of the initial conditions versus the period, we noticed that the 9th mode jumps discontinuously when $c_1$ crosses this gap. A similar disconnection appears to be developing near solution B.
Studying the results of Fig. \[fig:bif10\], we suspected we could find additional solutions by back-tracking from B to the region of the bifurcation curve around $c_9=-7.0\times 10^{-6}$ and performing a large extrapolation step to $c_9\approx-1.0\times 10^{-5}$, hoping to jump over the disconnection at B. This worked as expected, causing us to land on the branch that terminates at G in Fig \[fig:bif100\]. We used the same technique to jump from this branch to a solution between E and D. We were unable to find any new branches beyond C by extrapolation from earlier consecutive pairs of solutions.
![\[fig:bif100\] Several branches of standing waves were found by extrapolation across disconnections in the bifurcation curves. These disconnections are caused by resonant modes that may be interpreted physically as high-frequency standing waves superposed (nonlinearly) on the low-frequency carrier wave. ](figs/bif_water_100_csd){width="\linewidth"}
Next we track each solution branch as far as possible in each direction. This requires switching among the $c_k$ as bifurcation parameters when traversing different regions of the solution space. The period, $c_0=T$, is one of the options. We also experimented with pseudo-arclength continuation [@keller:68; @doedel91; @smith:roberts:99], but found that it is necessary to re-scale the Fourier modes to successfully traverse folds in the bifurcation diagram. This requires just as much human intervention as switching among the $c_k$, so we abandoned the approach. The disconnections at A and B turn out to meet each other, so that B is part of a closed loop and A is connected to the branch containing G. We stopped at G, F, C because the computations became too expensive to continue further with the desired accuracy of $f<10^{-26}$ using the adjoint continuation method.
The use of Fourier modes of the initial conditions in the bifurcation diagrams is unconventional, but yields insight about the effect of resonance on the dynamics of standing waves. We observe experimentally that disconnections in the bifurcation curves correspond to higher-frequency standing waves appearing at the surface of lower-frequency carrier waves. Because the equations are nonlinear, only certain combinations of amplitude and phase can occur. We generally see two possible solutions, one in which the high and low-frequency component waves are in phase with each other, and another where they are out of phase. For example, solutions F and G in Fig. \[fig:bif100\] can both be described as a $k=7$ wave-number standing wave oscillating on top of a $k=1$ carrier wave, but the smaller wave sharpens the crest at F and flattens it at G, being 180 degrees out of phase at F versus G when the composite wave comes to rest. (All the standing waves of this paper reach a rest state at $t=T/4$, by construction. Other types of solutions will be considered in future work [@water:stable].) In Section \[sec:bif:degen\], we show that this disconnection between branches F and G is caused by a $(3,7)$ harmonic resonance at fluid depth $h=1.0397$, where the period of the $k=1$ mode is equal to 3 times the period of the $k=7$ mode for small-amplitude waves [@smith:roberts:99; @ioualalen:03].
In Fig. \[fig:bif100A\], we plot $c_7=\hat\varphi_7(0)$ versus $T$, along with the evolution of $\varphi(x,t)$ for several solutions over time. Note that the scale on the $y$-axis is 20 times larger here (with $c_7$) than in Fig. \[fig:bif100\] (with $c_9$). This is why the secondary standing waves in the plots of solutions F and G appear to have wave number $k=7$. We also note that the disconnections at A and B are nearly invisible in the plot of $c_7$ vs $T$. This is because the dominant wave number of these branches is $k=9$. Similarly, it is difficult to observe any of the side branches in the plot of $c_1$ vs $T$ in Fig. \[fig:bif100\] since they all sweep back and forth along nearly the same curve. We will return to this point in Section \[sec:bif:degen\].
![\[fig:bif100A\] Bifurcation diagram showing $c_7=\hat\varphi_7(0)$ versus $T$ for standing waves of unit depth, along with snapshots of the evolution of $\varphi(x,t)$ for three of these solutions. A secondary standing wave with wave number $k=7$ can be seen visibly superposed on $\varphi(x,0)$ in solution F, which corresponds to the large value of $c_7$ at F in the diagram.](figs/bif_water_100A_csd){width="\linewidth"}
![\[fig:bif105\] If the mean depth, $h$, is increased from 1.0 to 1.05, the loop structure between A and B in Fig. \[fig:bif100\] disappears, and branches F and G meet each other a second time at another imperfect bifurcation. As $h$ increases further, these loops shrink, disappearing completely by the time $h=1.09$.](figs/csd_105_trans7){width="\linewidth"}
Nucleation of imperfect bifurcations {#sec:nuc}
------------------------------------
We next consider the effect of fluid depth on these bifurcation curves. We found the ACM method was too slow to perform this study effectively, which partly motivated us to develop the trust region shooting algorithm. As shown in Fig. \[fig:bif105\], if the fluid depth is increased from $h=1.0$ to $h=1.05$, it becomes possible to track branches F and G to completion. The large amplitude oscillations in the 7th Fourier mode eventually die back down when these branches are followed past the folds at $c_7\approx \pm4\times 10^{-3}$ in Fig. \[fig:bif105\]. The branches eventually meet each other at an imperfect bifurcation close to the initial bifurcation from the zero-amplitude state to the $k=1$ standing wave solutions. This imperfect bifurcation was not present at $h=1$. Its nucleation will be investigated in greater detail in Section \[sec:bif:degen\]. The small bifurcation loops at A and B in Fig. \[fig:bif100\] have disappeared by the time $h=1.05$. If we continue to increase $h$ to $1.07$, the top wing of the S-shaped bifurcation loop breaks free from the bottom wing and forms a closed loop. This loop disappears by the time $h$ reaches $1.08$. By $h=1.09$, the $k=7$ resonance has all but disappeared.
In Fig. \[fig:bif10\], we saw that the period of standing waves of unit depth decreases to a local minimum before increasing with wave amplitude. Two of the plots of Fig. \[fig:bif105\] show that this remains true for $h=1.05$, but not for $h=1.07$. In the latter case, the period begins increasing immediately rather than first decreasing to a minimum. This is consistent with the asymptotic analysis of Tadjbakhsh and Keller [@tadjbakhsh], which predicts that $$\label{eq:omega:correction}
{\textstyle}\omega = \omega_0 + \frac{1}{2}\epsilon^2\omega_2 + O(\epsilon^3),
\qquad \omega_0^2 = \tanh h, \qquad
\omega_2 = \frac{1}{32}(9\omega_0^{-7} - 12\omega_0^{-3}
-3 \omega_0 - 2\omega_0^5),$$ where $\epsilon$ controls the wave amplitude, and agrees with $A$ in (\[eq:lin:soln\]) to linear order. The correction term $\omega_2$ is positive for $h<1.0581$ and negative for $h>1.0581$.
We will see in Section \[sec:bif:degen\] that the nucleation of bifurcation branches between $h=1.09$ and $h=1.0$ is partly caused by a $(3,7)$ harmonic resonance (defined below) at fluid depth $h=1.0397$. As this mechanism is complicated, we also looked for simpler examples in deeper water. The simplest case we found is shown in Fig. \[fig:bif205\]. For fluid depth $h=2$, we noticed a pair of disconnections in the bifurcation curves that were not present for $h=2.1$. The 23rd Fourier mode of the initial condition exhibits the largest deviation from 0 on the side branches of these disconnections. However, as discussed in the next section, this is not caused by a harmonic resonance of type $(m,23)$ for some integer $m$. To investigate the formation of these side branches, we swept through the region $6.64\le T\le 6.68$ with slightly different values of $h$, using $c_0=T$ as the bifurcation parameter. As shown in Fig. \[fig:bif205\], when $h=2.0455$, the bifurcation curve bulges slightly but does not break. As $h$ is decreased to $2.045$, a pair of disconnections appear and spread apart from each other. We selected $h=2.0453$ as a good starting point to follow the side branches. As we hoped would happen, the two red side branches in the second panel of the figure met up with each other (at $c_{23}\approx
7\times 10^{-5}$), as did the two black branches (at $c_{23}\approx
-7\times 10^{-5}$). We switched between $T$ and $c_{23}$ as bifurcation parameters to follow these curves. We then computed two paths (not shown) in which $c_{23}=\pm 4\times 10^{-5}$ was held fixed as $h$ was increased. We selected 4 of these solutions to serve as starting points to track the remaining curves in Figure \[fig:bif205\], which have fluid depths $h_2$ through $h_5$ given in the figure. We adjusted $h_2$ to achieve a near three-way bifurcation. This bifurcation is quite difficult to compute as the Hessian of $f$ becomes nearly singular; for this reason, some of the solutions had to be computed in quadruple precision to avoid falling off the curves. Finally, to find the points A and B where a single, isolated solution exists at a critical depth, we computed $h$ as a function of $(T,c_{23})$ on a small $10\times 10$ grid patch near A and B, and maximized the polynomial interpolant using Mathematica.
![\[fig:bif205\] A pair of imperfect bifurcations were found to coalesce as fluid depth increases, leaving behind two closed loops and a smooth bifurcation curve running between them. The loops each shrink to a point and disappear as fluid depth continues to increase. Reversing the process shows that isolated solutions can nucleate new branches of solutions as fluid depth decreases. (right) The nucleated solutions A and B are nearly identical on large scales, but contain secondary, high-frequency standing waves at smaller scales that are out of phase with each other. These small oscillations become visible when the slope of the wave profile is plotted. ](figs/bif205){width="\linewidth"}
Degenerate and secondary bifurcations due to harmonic resonance {#sec:bif:degen}
---------------------------------------------------------------
In this section, we explore the source of the resonance between the $k=1$ and $k=7$ modes in water of depth $h$ close to 1. While harmonic resonances such as this have long been known to cause imperfect bifurcations [@mercer:94; @smith:roberts:99; @ioualalen:03], we are unaware that anyone has been able to track the side-branches all the way back to the origin, where they meet up with mixed-mode solutions of the type studied asymptotically by Vanden-Broeck [@vandenBroeck:84] and numerically by Bryant and Stiassnie [@bryant:stiassnie:94]. In the traveling wave case, such mixed-mode solutions are known as Wilton’s ripples [@chen:saffman:79; @akers:wilton]. When the fluid depth is perturbed, we find that the degenerate bifurcation splits into a primary bifurcation and two secondary bifurcations [@bauer:keller]. This is consistent with Bridges’ work on perturbation of degenerate bifurcations in three-dimensional standing water waves in the weakly nonlinear regime [@bridges:standing:3d].
We begin by observing that the ratio of the periods of two small-amplitude standing waves is $$m = \frac{T_1}{T_2} = \frac{\omega_2}{\omega_1} =
\sqrt{\frac{k_2\tanh k_2{h}}{k_1\tanh k_1{h}}}.$$ If we require $m$ to be an integer and set $k_1=1$, we obtain $$\label{eq:harm:res}
k_2\tanh k_2h = m^2\tanh h.$$ Following [@mercer:94; @smith:roberts:99; @ioualalen:03], we say there is a harmonic resonance of order $(m,k_2)$ if $h$ satisfies (\[eq:harm:res\]). At this depth, linearized standing waves of wave number $k=1$ have a period exactly $m$ times larger than standing waves of wave number $k=k_2$. This nomenclature comes from the short-crested waves literature [@ioualalen:93; @ioualalen:96]; a more general framework can be imagined in which $k_1$ is not assumed equal to 1 and $m$ is allowed to be rational, but we do not need such generality.
We remark that the nucleation event discussed in the previous section does not appear to be connected to a harmonic resonance. In that example, the fundamental mode must have a fairly large amplitude before the secondary wave becomes active, and the secondary wave is not a clean $k=23$ mode. Also, no integer $m$ causes the fluid depth of an $(m,23)$ resonance to be close to $2.045$. The situation is simply that at a certain amplitude, the $k=1$ standing wave excites a higher-frequency, smaller amplitude standing wave that oscillates at its surface. It is not possible to decrease both of their amplitudes to zero without destroying the resonant interaction in this case.
We now restrict attention to the $(3,7)$ harmonic resonance. Setting $m=3$ and $k_2=7$ in (\[eq:harm:res\]) yields $$7\tanh 7h = 9\tanh h, \quad
h>0 \qquad \Rightarrow \qquad
h = h_\text{crit} \approx 1.0397189.$$ In the nonlinear problem, when the fluid depth has this critical value, we find that the $k=7$ and $k=1$ branches persist as if the other were not present. Indeed, the former can be computed as a family of $k=1$ solutions on a fluid of depth $7h$. The latter can be computed by taking a pure $k=1$ solution of the linearized problem as a starting guess and solving for the other Fourier modes of the initial conditions, as before. When this is done, after setting $\epsilon = \hat\varphi_1(0)$, we find that $\hat\varphi_3(0)=O(\epsilon^3)$, $\hat\varphi_5(0)=O(\epsilon^5)$, $\hat\varphi_7(0)=O(\epsilon^5)$, and $\hat\varphi_9(0)=O(\epsilon^7)$. To obtain these numbers, we used 10 values of $\epsilon$ between $10^{-4}$ and $10^{-3}$ and computed the slope of a log-log plot. The calculations were done in quadruple precision with a 32 digit estimate of $h_\text{crit}$ to avoid corruption by roundoff error. If we repeat this procedure with $h=1.0$, we find instead that $\hat\varphi_7(0)=O(\epsilon^7)$, $\hat\varphi_9(0)=O(\epsilon^9)$. Thus, the degeneracy of the bifurcation at $h_\text{crit}$ appears to slow the decay rate of the 7th and higher modes, but not enough to affect the behavior at linear order.
We were surprised to discover that two additional branches also bifurcate from the stationary solution when $h=h_\text{crit}$. For these branches, we find that $\hat\varphi_k(0) = O(\epsilon^p)$, where the first several values of $p$ are $$\begin{array}{r|r}
k & p \\ \hline
1 & 1 \\
3 & 3 \\
5 & 3
\end{array} \qquad\quad
\begin{array}{r|r}
k & p \\ \hline
7 & 1 \\
9 & 3 \\
11 & 5
\end{array} \qquad\quad
\begin{array}{r|r}
k & p \\ \hline
13 & 3 \\
15 & 3 \\
17 & 5
\end{array} \qquad\quad
\begin{array}{r|r}
k & p \\ \hline
19 & 5 \\
21 & 3 \\
23 & 5
\end{array} \qquad\quad
\begin{array}{r|r}
k & p \\ \hline
25 & 7 \\
27 & 5 \\
29 & 5
\end{array}$$ These numbers were computed as described above, with $\epsilon$ ranging between $10^{-4}$ and $10^{-3}$. To get a clean integer for $\hat\varphi_{21}(0)$, $\hat\varphi_{27}(0)$ and $\hat\varphi_{29}(0)$, we had to drop down to the range $10^{-5}\le{\varepsilon}\le10^{-4}$. Using the Aitken-Neville algorithm [@deuflhard] to extrapolate $\hat\varphi_k(0)/{\varepsilon}^p$ to ${\varepsilon}=0$, we obtain the leading coefficients for the two branches: $$\begin{array}{r|l}
k & \qquad\qquad\hat\varphi_k(0) \\ \hline
1 & {\phantom{-}}\epsilon \\
7 & {\phantom{-}}0.034152137008 \epsilon + O(\epsilon^3) \\
3 & -0.376330285335 \epsilon^3 + O(\epsilon^5) \\
5 & {\phantom{-}}0.065341882841 \epsilon^3 + O(\epsilon^5) \\
9 & -0.172818320378 \epsilon^3 + O(\epsilon^5) \\
13 & {\phantom{-}}0.019277463225 \epsilon^3 + O(\epsilon^5) \\
15 & -0.011062972892 \epsilon^3 + O(\epsilon^5) \\
21 & -1.303045\times 10^{-8} \epsilon^3 + O(\epsilon^5)
\end{array}
\qquad\qquad
\begin{array}{r|l}
k & \qquad\qquad\hat\varphi_k(0) \\ \hline
1 & {\phantom{-}}\epsilon \\
7 & -0.034152137008 \epsilon + O(\epsilon^3) \\
3 & -0.376330285335 \epsilon^3 + O(\epsilon^5) \\
5 & -0.065341882841 \epsilon^3 + O(\epsilon^5) \\
9 & {\phantom{-}}0.172818320378 \epsilon^3 + O(\epsilon^5) \\
13 & {\phantom{-}}0.019277463225 \epsilon^3 + O(\epsilon^5) \\
15 & -0.011062972892 \epsilon^3 + O(\epsilon^5) \\
21 & {\phantom{-}}1.303045\times 10^{-8} \epsilon^3 + O(\epsilon^5)
\end{array}$$ In summary, there are four families of solutions of the nonlinear problem that bifurcate from the stationary solution. In the small amplitude limit, they approach a pure $k=1$ mode, a pure $k=7$ mode, and two mixed modes involving both $k=1$ and $k=7$ wave numbers. For convenience, we will refer to these branches as “pure” and “mixed” based on their limiting behavior in the linearized regime. The mixed mode solutions are examples of the Wilton’s ripple phenomenon [@vandenBroeck:84; @chen:saffman:79; @akers:wilton] in which multiple wavelengths are present in the leading order asymptotics.
![\[fig:spider\] Perturbation of this degenerate bifurcation causes a pair of imperfect ($h>h_\text{crit}$) or perfect ($h<h_\text{crit}$) secondary bifurcations to form. Red markers are the solutions actually computed, while blue markers correspond to the same solutions, phase-shifted in space by $\pi$. ](figs/spider){width="\linewidth"}
When these four branches are tracked in both directions, we end up with eight rays of solutions emanating from the equilibrium configuration, labeled a–h in Figure \[fig:spider\]. Rays a and b consist of pure $k=1$ mode solutions, with negative and positive amplitude, respectively, where amplitude refers to $\hat\varphi_1(0)$. Rays e and h are the pure $k=7$ mode solutions, and rays c,d,f,g are the mixed mode solutions. It is remarkable that rays a and f, as well as b and c, are globally connected to each other by a large loop in the bifurcation diagram. By contrast, for the Benjamin-Ono equation [@benj3], additional branches of solutions that emanate from a degenerate bifurcation belong to different levels of the hierarchy of time-periodic solutions than the main branches; thus, solutions on the additional branches have a different number of phase parameters, and cannot meet up with one of the main branches without another bifurcation.
We now investigate what happens to these rays when the fluid depth is perturbed. When $h$ increases from $h_\text{crit}$ to $1.04$, rays e and h (the pure $k=7$ solutions) break free from the other 6 rays. An imperfect bifurcation forms on rays a and b, linking the former to f and d, and the latter to c and g. Aside from this local reshuffling of branch connections near the stationary solution, the global bifurcation structure of $h=1.04$ is similar to $h=h_\text{crit}$. In the other direction, when $h=1.03<h_\text{crit}$, rays a and b (the pure $k=1$ solutions) disconnect from the other rays. Instead of forming imperfect bifurcations as before, rays c and d separate from the $\hat\varphi_7=0$ axis, but remain connected to ray e through a perfect bifurcation. The same is true of rays f, g and h. Thus, we have identified a case where perturbing a degenerate bifurcation causes it to break up into a primary bifurcation and two secondary bifurcations [@bauer:keller], either perfect ($h<h_\text{crit}$) or imperfect ($h>h_\text{crit}$).
The reason one is perfect and the other is not can be explained heuristically as follows. All the solutions on the pure $k=7$ branch have Fourier modes $\hat\varphi_k(t)$, with $k$ not divisible by 7, exactly equal to zero. These modes can be eliminated from the nonlinear system of equations by reformulating the problem as a $k=1$ solution on a fluid of depth $7h$. This reformulation removes the resonant interaction by restricting the $k=1$ mode (in the original formulation) to remain zero. The simplest way for this mode to become non-zero, i.e. deviate from rays e,h in Fig. \[fig:spider\], is through a subharmonic bifurcation (with $k=7$ as the fundamental wavelength) in which the Jacobian $J$ in (\[eq:Jik\]) develops a non-trivial kernel containing a null vector $c\in l^2(\mathbb{N})$ with $c_1=\hat\varphi_1(0)\ne0$. Here $c$ contains the even modes of $\eta$ and the odd modes of $\varphi$ at $t=0$, as in (\[eq:init:stand\]), but with $n=\infty$. If such a kernel exists, one expects to be able to perturb the solution in this direction, positively or negatively, to obtain a pitchfork bifurcation. By contrast, solutions on the $k=1$ branch develop non-zero higher-frequency modes through non-linear mode interactions. So while $c_1=0$ on the $k=7$ branch, $c_7\ne0$ on the $k=1$ branch. Since there is no way to control the influence of the 7th mode, e.g. by constraining it to be zero, there really is no “pure” $k=1$ branch to bifurcate from, and the result is an imperfect bifurcation.
![\[fig:104plots\] Solutions labeled F, G, H and K in Fig. \[fig:spider\] show the transition from standing waves that form crests at $x=\pi$ to those that form crests at $x=0$ when $t=T/4$. This transition occurs where branch f meets branch g in the $h<h_\text{crit}$ case. For $h>h_\text{crit}$, branch f meets branch d at an imperfect bifurcation, and the solution at the end of path d would resemble solution K, shifted in space by $\pi$. ](figs/csd104plots){width="\linewidth"}
The fact that ray f is connected to g for $h<h_\text{crit}$, and to d for $h>h_\text{crit}$, has a curious effect on the form of the numerical solution at the end of the red branch, the branch of solutions actually computed, in Fig. \[fig:spider\]. In the former case, $c_1$ changes sign from branch f to g, and we end up at solution K in Fig. \[fig:104plots\], which forms a wave crest at $x=0$ at $t=T/4$. In the latter case, $c_1$ remains negative from branch f to d (or branch a to d if the imperfect bifurcation is traversed without branch jumping) and we end up at a solution similar to K, but phase shifted, so that a wave crest forms at $x=\pi$ at $t=T/4$. Note that the sign of $c_1$ determines whether the fluid starts out flowing toward $x=\pi$ and away from $x=0$, or vice-versa.
The transition from wave crests at $x=\pi$ to wave crests at $x=0$ when $t=T/4$ is shown in Fig. \[fig:104plots\]. Solutions F, G and H may all be described as $k=7$ standing waves superposed on $k=1$ standing waves. Note that solution F bulges upward at $x=\pi$ when $t=T/4$, while solution G bulges downward there. A striking feature of these plots is that the $k=7$ modes of $\varphi$ and $\eta$ nearly vanish at $t=T/12$ and $t=T/6$, respectively. This occurs because the $k=7$ mode oscillates 3 times faster than the $k=1$ mode. Solution H is a pure $k=7$ solution, which means $\varphi(x,t)$ vanishes identically at $t=\frac{2m+1}{4}\left(\frac{T}{3}\right)$, $m\ge0$, while $\hat\eta_7(t)$ passes through zero at $t=\frac{m}{2}\left(\frac{T}{3}\right)$, $m\ge0$. Since solutions F and G are close to solution H, $\hat\varphi_7(t)$ and $\hat\eta_7(t)$ pass close to zero at these times, leading to smoother solutions dominated by the first Fourier mode at these times.
![\[fig:bif104\] Solutions O and P, the bifurcation points from the equilibrium state to the pure $k=1$ and $k=7$ standing waves, respectively, separate from each other through a change in period as fluid depth varies from $h=h_\text{crit}$. The imperfect bifurcation can occur arbitrarily close to solution O by taking $h\searrow h_\text{crit}$. Turning points in $\hat\varphi_7(0)$ and $T$ occur at solution H for $h<h_\text{crit}$. ](figs/csd_bif_104){width="\linewidth"}
Figure \[fig:bif104\] shows how the period varies along each of these solution branches. The period varies with fluid depth more rapidly for solutions on the $k=1$ branch than on the $k=7$ branch since the slope of $\tanh h$ is 18000 times larger than that of $\tanh
7h$ when $h\approx 1$. As a result, bifurcation point O (to the $k=1$ branch) moves visibly when $h$ changes from $1.03$ to $1.04$, while bifurcation point P (to the $k=7$ branch) hardly moves at all. We also see that the period increases with amplitude on branches e and h. By contrast, on branches a and b, it decreases to a local minimum before increasing with amplitude. This is consistent with the asymptotic analysis of Tadjbakhsh and Keller discussed previously; see (\[eq:omega:correction\]) above. Finally, we note that both $T$ and $c_7$ have a turning point at solution H, the bifurcation point connecting branches f and g to branch h. This causes paths f and g to lie nearly on top of each other for much of the bifurcation diagram. Other examples of distinct bifurcation curves tracing back and forth over nearly the same paths are present (but difficult to discern) in Figures \[fig:spider\] and \[fig:bif104\] when $h=1.03$, and will be discussed further in Section \[sec:conclude\].
Breakdown of self-similarity and the Penney and Price conjecture {#sec:breakdown}
----------------------------------------------------------------
In Sections \[sec:nuc\] and \[sec:bif:degen\] above, we have seen that increasing the fluid depth causes disconnections in the bifurcation diagrams to “heal,” and it is natural to ask if any will persist to the infinite depth limit. The answer turns out to be yes, which is not surprising from a theoretical point of view since infinite depth standing waves are completely resonant [@iooss05], involving state transition operators with infinite dimensional kernels and a small-divisor problem on the complement of this kernel. Nevertheless, examples of such disconnecitons have only recently been observed in numerical simulations [@breakdown], and show no evidence of being densely distributed along bifurcation curves. In this section, we expand on the results of [@breakdown], filling in essential details and providing new material not discussed there. The scarcity of observable disconnections will be discussed further in the conclusion section.
The main question addressed in [@breakdown] is whether standing waves of extreme form approach a limiting wave profile with a geometric singularity at the wave crest when the bifurcation curve terminates. This type of question has a long history, starting with Stokes [@stokes:1880; @craik:05], who predicted that the periodic traveling wave of greatest height would feature wave crests with sharp, $120^\circ$ interior crest angles. While there are some surprises concerning oscillatory asymptotic behavior at the crest of the almost highest traveling wave [@lhf:77; @lhf:78; @breakdown], it has been confirmed both theoretically [@amick:82] and numerically [@maklakov; @gandzha:07] that a limiting extreme traveling wave does exist, and possesses a sharp $120^\circ$ wave crest. For standing waves, a similar conjecture was made by Penney and Price in 1952 [@penney:52], who predicted that the limiting extreme wave would develop sharp, 90 degree interior crest angles each time the fluid comes to rest. As discussed in the introduction, numerous experimental, theoretical and numerical studies [@taylor:53; @grant; @okamura:98; @okamura:03; @okamura:10; @mercer:92; @schultz] have reached contradictory conclusions on the limiting behavior at the crests of extreme standing waves.
![\[fig:bif:00\] Snapshots of several standing waves over a quarter period in water of infinite depth, along with bifurcation diagrams showing where they fit in. (center) Conventional bifurcation diagram showing wave height versus crest acceleration. The turning point at A was discovered by Mercer and Roberts [@mercer:92] while the turning point at C and subsequent bifurcation structure were discovered by Wilkening [@breakdown]. The wave height $h$ eventually exceeds the local maximum at A. (right) The continuation parameters actually used were $c_1$, $c_5$, $c_{60}$ and $T$ rather than $h$ or $A_c$. ](figs/bif00){width="\linewidth"}
Penney and Price expected wave height, defined as half the maximum crest-to-trough height, to increase monotonically from the zero-amplitude equilibrium wave to the extreme wave. Mercer and Roberts found that wave height reaches a turning point, achieving a local maximum of $h=0.62017$ at $A_c=0.92631$, where $A_c$ is the downward acceleration of a fluid particle at the wave crest at the instant the fluid comes to rest (assuming $g=1$ in (\[eq:ww\])). Since $h$ is not a monotonic function, they proposed using crest acceleration as a bifurcation parameter instead. However, as shown in Fig. \[fig:bif:00\], crest acceleration also fails to be a monotonic function. The bifurcation curve that was supposed to terminate at the extreme wave when $A_c$ reaches 1 becomes fragmented for $0.99<A_c<1$. Just as in the finite depth case, this fragmentation is due to resonant interactions between the large-scale carrier wave and various smaller-scale, secondary standing waves that appear at the surface of the primary wave. Figure \[fig:tip\] shows several examples of the oscillatory structures that are excited by resonance, both in space and in time. The amplitude of the higher-frequency oscillations are small enough in each case that the vertical position of a particle traveling from trough to crest increases monotonically in time; however, plotting differences of solutions on nearby branches as a function of time reveals the temporal behavior of the secondary standing waves. We note that the frequency of oscillation of the secondary waves (near $x=\pi$) decreases as $t$ approaches $T/4$. This seems reasonable as fluid particles at the crest are nearly in free-fall at this time when $A_c$ is close to 1; thus, the driving force of the secondary oscillations is low there. In shallow water, the secondary oscillations are often strong enough to lead to non-monotonic particle trajectories from trough to crest (see Section \[sec:shallow\]).
![\[fig:tip\] Oscillatory structures near the wave crest, and time-evolution of a particle from trough to crest, for several extreme standing waves. Labels correspond to the bifurcation diagrams in Figure \[fig:bif:00\]. These solutions take the form of higher-frequency standing waves superimposed nonlinearly on lower-frequency carrier waves. The higher-frequency oscillations occur both in space and time. ](figs/tip){width="\linewidth"}
If a limiting wave profile does not materialize as $A_c$ approaches 1, a natural question arises as to what will terminate the bifurcation curves. In [@breakdown], it was emphasized that oscillations at the crest tip prevent self-similar sharpening to a corner, as happens in the traveling case. We note here that the entire wave profile, not just the crest tip, develops high-frequency oscillations on small scales toward the end of each bifurcation curve. This is illustrated in Fig. \[fig:breakdown\], and suggests that if these bifurcation curves do end somewhere, without looping back to merge with another disconnection, it may be due to solutions becoming increasingly rough, with some Sobolev norm diverging in the limit. We also remark that since many of these standing waves come close to forming a $90^\circ$ corner, there may well exist nearly time-periodic solutions that do pass through a singular state. Taylor’s thought experiment [@taylor:53] in which water is piled up in a crested configuration and released from rest could be applied to a sharply crested perturbation of the rest state of one of our standing waves. However, like Taylor, we see no reason that $90^\circ$ would be the only allowable crest angle. It is conceivable that $90^\circ$ is the only angle for which smooth solutions can propagate forward from a singular initial condition, but we know of no such results.
![\[fig:breakdown\] Evolution of surface height and its derivatives over a quarter period for solution O in Figure \[fig:bif:00\]. The plots at right of $\eta_{xx}$ are shown only at time $T/4$ for clarity. In the far right panel, we also plotted solution E for comparison. Each solution that terminates a branch in the bifurcation diagram is highly oscillatory; we followed the branches to the point that the computations became too expensive to continue further. ](figs/breakdown){width="\linewidth"}
The increase in roughness of the solutions as crest acceleration approaches $1$ may also be observed by plotting Fourier mode amplitudes for various solutions along the bifurcation curve. In Fig. \[fig:fourier\], we compare the Fourier spectrum of $\eta$ at $t=0$ and $t=T/4$ for solutions O and A. In both of these simulations, we parametrized the curve non-uniformly, as in (\[eq:xi\]), to increase resolution near the crest tip. Thus, a distinction must be made between computing Fourier modes with respect to $x$ versus $\alpha$: $$\label{eq:fourier}
\hat\eta_k(t) = \frac{1}{2\pi}\int_0^{2\pi} \eta(x,t)e^{-ikx}\,dx, \qquad
\text{ or } \qquad
\hat\eta_k(t) = \frac{1}{2\pi}\int_0^{2\pi}
\eta(\xi_{l(t)}(\alpha),t)e^{-ik\alpha}\,d\alpha.$$ In this figure, we use the latter convention, since $\eta\circ\xi_l$ and $\varphi\circ\xi_l$ are the quantities actually evolved in time, and the decay rate of Fourier modes is faster with respect to $\alpha$. At $t=0$, the two formulas in (\[eq:fourier\]) agree since we require $\xi_1(\alpha)=\alpha$. For solution O, the Fourier modes of the initial conditions decay to $|c_k|<10^{-12}$ for $2k\ge3000$. At $t=T/4$ we have $\max(|\hat\eta_k|,|\hat\varphi_k|)<10^{-12}$ for $2k\ge6500$ with $\rho_\nu=0.09$. For solution A, we have $|c_k|<10^{-29}$ for $2k\ge
400$ and $\max(|\hat\eta_k|,|\hat\varphi_k|)<10^{-29}$ for $2k\ge 800$ at $t=T/4$ with $\rho_\nu=0.4$. Here $T=1.629324$ for solution O and $T=1.634989$ for solution A. The mesh parameters used in these simulations are listed in Table \[tab:param:OA\]. We remark that for solutions such as O with fairly sharp wave crests, decreasing $\rho_\nu$ generally leads to faster decay of Fourier modes, but also amplifies roundoff errors due to closer grid spacing near the crest. Further decrease of $\rho_\nu$ in the double-precision calculation does more harm than good.
$$\begin{array}{l||c||c|c|c|c||c|c|c|c||c||c|c|c|c||c|c|c|c}
\text{\;\;solution} & \nu & \theta_1 & \theta_2 & \theta_3 & \theta_4 & \rho_1 & \rho_2 & \rho_3 & \rho_4 & n & M_1 & M_2 & M_3 & M_4 & N_1 & N_2 & N_3 & N_4 \\ \hline
\text{A (quad)} & {\scriptstyle}2 & {\scriptstyle}0.2 & {\scriptstyle}0.8 & {\scriptstyle}- & {\scriptstyle}- & {\scriptstyle}1.0 & {\scriptstyle}0.4 & {\scriptstyle}- & {\scriptstyle}- & {\scriptstyle}200 & {\scriptstyle}768 & {\scriptstyle}1024 & {\scriptstyle}- & {\scriptstyle}- & {\scriptstyle}24 & {\scriptstyle}144 & {\scriptstyle}- & {\scriptstyle}- \\
\text{O (double)} & {\scriptstyle}4 & {\scriptstyle}0.2 & {\scriptstyle}0.3 & {\scriptstyle}0.3 & {\scriptstyle}0.2 & {\scriptstyle}1.0 & {\scriptstyle}0.4 & {\scriptstyle}0.1 & {\scriptstyle}0.09 & {\scriptstyle}1500 & {\scriptstyle}4608 & {\scriptstyle}6144 & {\scriptstyle}6912 & {\scriptstyle}8192 & {\scriptstyle}192 & {\scriptstyle}432 & {\scriptstyle}576 & {\scriptstyle}480 \\
\text{O (quad)} & {\scriptstyle}4 & {\scriptstyle}0.1 & {\scriptstyle}0.3 & {\scriptstyle}0.4 & {\scriptstyle}0.2 & {\scriptstyle}1.0 & {\scriptstyle}0.25 & {\scriptstyle}0.08 & {\scriptstyle}0.05 & {\scriptstyle}1500 & {\scriptstyle}6144 & {\scriptstyle}7500 & {\scriptstyle}8192 & {\scriptstyle}9216 & {\scriptstyle}60 & {\scriptstyle}216 & {\scriptstyle}384 & {\scriptstyle}240
\end{array}$$
The effect of roundoff-error and the 36th order filter can both be seen in the second panel of Fig. \[fig:fourier\]. In exact arithmetic, $\eta(x,t)$ and $\varphi(x,t)$ would remain even functions for all time. However, in numerical simulations, the imaginary parts of $\hat\eta_k(t)$ and $\hat\varphi_k(t)$ drift away from zero, giving a useful indicator of how much the solution has been corrupted by roundoff error. The filter (\[eq:filter\]) has little effect on the first 70 percent of the Fourier modes, but strongly damps out the last 15 percent. By monitoring the decay of Fourier modes through plots like this, one can ensure that the simulations are fully resolved, and that filtering does not introduce more error than is already introduced by roundoff error. We also monitor energy conservation, $$E(t) = \frac{1}{2} \int_0^{2\pi} \left[\varphi(x,t){\mathcal{G}}\varphi(x,t) +
g\eta(x,t)^2\right]\,dx,$$ choosing time-steps small enough that $E$ remains constant to as many digits as possible, typically 14 in double-precision and 29 in quadruple precision. Note that ${\mathcal{G}}$ depends on time through $\eta$.
![\[fig:fourier\] The Fourier modes of $\eta$ (shown) and $\varphi$ (not shown) are monitored to decide how many grid points are needed to resolve the solution. The parameter $\rho_l$ controls the nonuniform spacing of gridpoints via (\[eq:xi\]). The real (black) and imaginary (grey) parts of $\hat{\eta}_k(t)$ are plotted in positions $2k$ and $2k+1$, respectively. (Left) the minimization was performed in double precision to obtain these initial conditions. The result was checked in quadruple precision to eliminate roundoff error. (Right) the minimization was performed in quadruple precision, yielding $f=2.1\times10^{-60}$. ](figs/UerrFourier){width="\linewidth"}
![\[fig:evol:PhiOA\] Plots of the residual $\varphi(x,T/4)$ for solutions O and A. (left) When minimized in double-precision arithmetic, we obtain $f=1.3\times 10^{-26}$. Shown here is a re-computation of the solution in quadruple precision on a finer mesh using the same initial conditions. This yields $f=8.6\times 10^{-27}$, which is even smaller than predicted in double-precision. Re-spacing the grid maps the curve $E_\nu(\alpha)$ to $E_\nu(\xi_\nu^{-1}(x))$, improving the resolution that can be achieved with 9216 gridpoints. Note that the oscillations in $\varphi(x,T/4)$ are fully resolved. (right) The minimization was performed in quadruple-precision arithmetic, yielding $f=2.1\times 10^{-60}$. The velocity potential is nearly 30 orders of magnitude smaller at $t=T/4$ than at $t=0$. ](figs/evolPhiOA){width="\linewidth"}
Because solution A remains smoother and involves many fewer Fourier modes than solution O, it was possible for us to perform the entire computation in quadruple precision arithmetic. This allowed us to reduce $f$ in (\[eq:f:phi\]) to $2.1\times 10^{-60}$. As shown in Fig. \[fig:evol:PhiOA\], the velocity potential of this solution drops from $O(1)$ at $t=0$ to less than $3\times 10^{-29}$ at $t=T/4$, in the uniform norm. While it was not possible to perform the minimization for solution O in quadruple precision arithmetic (due to memory limitations of the GPU device), we were able to check the double-precision result in quadruple precision to confirm that $f$ is not under-predicted by the minimization procedure. Because the DOPRI8 and SDC15 methods involve 12 and 99 internal Runge-Kutta stages per time-step, respectively, more function evaluations were involved in advancing the quadruple-precision calculations through time even though $N_l$ is larger in the double-precision calculations. As shown in Fig. \[fig:evol:PhiOA\], the oscillations in $\varphi(x,T/4)$ remain fully resolved in the quadruple precision calculation; thus, $f=8.6\times 10^{-27}$ is an accurate measure of the squared error. The predicted value of $f$ in double-precision (obtained by minimizing $f$) was $f=1.3\times
10^{-26}$. Since minimizing $f$ entails eliminating as many significant digits of $\varphi(x,T/4)$ as possible, the resulting value of $f$ is not expected to be highly accurate. What is important is that minimizing $f$ in double-precision does not grossly underestimate its minimum value. In fact, its value is often over-estimated, as occurred here. This robustness is a major benefit of posing the problem as an *overdetermined* non-linear system. The only way to achieve a small value of $f$ is to accurately track a solution of the PDE for which the exact $f$ is small. Roundoff errors and truncation errors will cause the components of $r$ in (\[eq:f:phi\]) to drift away from zero, leading to an incompatible system of equations with minimum residual of the order of the accumulated errors.
There is a big advantage to choosing $t=0$ to occur at the midpoint between rest states rather than at a rest state. The reason is that many more Fourier modes are needed to represent $\eta$ when the wave crest is relatively sharp, which for us occurs when $t$ is near $T/4$. For example, solutions O and A in Fig. \[fig:fourier\] have more than twice as many active Fourier modes at $t=T/4$ as they did at $t=0$, even using a non-uniform grid to better resolve the crested region. Setting up the problem this way leads to fewer Fourier modes of the initial condition to solve for, and increases the number of non-linear equations. Thus, the system is more overdetermined, adding robustness to the computation.
We conclude this section by mentioning that “branch jumping” is very easy to accomplish (and hard to avoid) in the numerical continuation algorithm. For strong disconnections such as at solution B in Fig. \[fig:bif10\], it is sometimes necessary to backtrack away from the disconnection and then take a big step, hoping to land beyond the gap. Since we measure the residual error in an overdetermined fashion, it is obvious if we land in a gap where there is no time-periodic solution — the minimum value of $f$ remains large in that case. However, most disconnections can be traversed without backtracking, or even knowing in advance of their presence. The disconnections in Figure \[fig:bif:00\] were all discovered by accident in this way. Once a disconnection is observed, we can go back and follow side branches to look for global re-connections or new families of time-periodic solutions.
Counter-propagating solitary waves in shallow water {#sec:shallow}
---------------------------------------------------
In previous sections, we saw that decreasing the fluid depth causes nucleation of loop structures in the bifurcation curves that nearly (or actually) meet at imperfect (or perfect) bifurcations. Some of these disconnections persist in the infinite depth limit. We now consider the other extreme of standing waves in very shallow water.
In Figure \[fig:bif:005\], we track the $k=1$ family of standing waves out of the linear regime for water of depth $h=0.05$ and spatial period $2\pi$. The period of the solutions in the linear regime is $T=2\pi/\sqrt{\tanh 0.05}=28.1110$, compared to $T=7.19976$ when $h=1$ and $T=2\pi=6.28319$ when $h=\infty$. Thus, the waves travel much slower in shallow water. We also see that $T$ decreases with amplitude as the waves leave the linear regime, consistent with Tadjbakhsh and Keller’s result, Equation (\[eq:omega:correction\]) above, that the sign of the quadratic correction term in angular frequency is positive for $h<1.0581$. Many more disconnections have appeared in the bifurcation diagrams at this depth than were observed in the cases $h\approx 1.0$ and $h=\infty$ considered above. It was not possible to track all the side branches that have emerged to see if they reconnect with each other. However, each time we detected that the minimization algorithm had jumped from one branch to another, we did backtrack to fill in enough points to observe which modes were excited by the resonance. In general, higher-frequency Fourier modes of the initial condition possess more disconnections, even though all the bifurcation curves describe the same family of solutions. For example, in Fig. \[fig:bif:005\], we see that $\hat\varphi_{17}(0)$ has much stronger disconnections than $\hat\varphi_1(0)$, and those of $\hat\eta_{36}(0)$ are stronger still. This suggests that high-frequency resonances have little effect on the dynamics of lower-frequency modes. Nevertheless, there is *some* effect, since even $\hat\varphi_1(0)$ exhibits visible disconnections, and in fact has some gaps in $T$ where solutions could not be found. We interpret these gaps as numerical manifestations of the Cantor-like structures that arise in analytical studies of standing water waves due to small divisors [@plotnikov01; @iooss05]. This will be discussed further in the conclusion section.
![\[fig:bif:005\] Bifurcation diagrams showing the dependence of $\hat\varphi_1(0)$, $\hat\varphi_{17}(0)$, and $\hat\eta_{36}(0)$ on $T$ for a family of standing water waves in shallow ($h=0.05$) water. Many more disconnections are visible at this depth than were observed in the $h=1.0$ and $h=\infty$ cases above. ](figs/bif005){width="\linewidth"}
![\[fig:evol:005\] Standing waves in shallow water take the form of counter-propagating solitary waves that interact elastically. The low-amplitude radiation normally associated with inelastic collisions is already present before the interaction, and does not increase as a result of the interaction. In solutions A and B, this radiation consists of small-amplitude, high-frequency standing waves over which the solitary waves travel. In solution C, this radiation is a chaotic mix of standing and counter-propagating traveling waves of different wave numbers. ](figs/evol005){width="\linewidth"}
In Figure \[fig:evol:005\], we show time-elapsed snapshots of the standing wave solutions labeled A–C in Figure \[fig:bif:005\]. These standing waves no longer lead to large scale sloshing modes in which the fluid rushes from center to sides and back in bulk. Instead, a pair of counter-propagating solitary waves travel back and forth across the domain, alternately colliding at $x=\pi$ and $x=0$ at times $t=T/4+(T/2)\mathbb{Z}$. In the unit depth case above, we observed in Figure \[fig:bif100\] that disconnections in the bifurcation curve correspond to secondary standing waves appearing with one of two phases at the surface of a primary carrier wave. The same is true of these solitary wave interactions. While it is difficult to observe in a static image, movies of solutions A and B in Figure \[fig:evol:005\] reveal that the primary solitary waves travel over smaller standing waves with higher wave number and angular frequency. As a result, a fluid particle at $x=\pi$ will oscillate up and down with the secondary standing wave until the solitary waves collide, pushing the particle upward a great distance. By contrast, in the infinte-depth case, we saw in Figure \[fig:tip\] that $\eta(\pi,t)$ increases monotonically from trough to crest in spite of the secondary waves. Each time a disconnection in Figure \[fig:bif:005\] is crossed, the background standing wave (or some of its component waves) change phase by $180^\circ$. Solution B is positioned near the center of a bifurcation branch, far from major disconnections in the bifurcation curves. As a result, the water surface over which the solitary waves travel remains particularly calm for solution B. By contrast the background waves of solution C are quite large in amplitude, with many active wave numbers. A Floquet stability analysis, presented elsewhere [@water:stable], shows that solutions A and B are linearly stable to harmonic perturbations while solution C is unstable.
Gravity-capillary standing waves {#sec:surf}
--------------------------------
In this section, we consider the effect of surface tension on the dynamics of standing water waves. We restrict attention to waves of the type considered by Concus [@concus:62] and Vanden-Broeck [@vandenBroeck:84], leaving collisions of gravity-capillary waves [@milewski:11] for future work [@water:stable]. The only change in the linearized equations (\[eq:lin\]) when surface tension is included is that $\dot\varphi_t=P\big[-g\dot\eta+(\sigma/\rho)\dot\eta_{xx}\big]$. The standing wave solutions of the linearized problem continue to have the form (\[eq:lin:soln\]), but with $$\omega^2 = \left(g + \frac{\sigma}{\rho}k^2\right)k\tanh kh, \qquad
A/B = \sqrt{k\tanh kh\Big/\big[g + (\sigma/\rho)k^2\big]}.$$ We choose length and time-scales so that $g=1$ and $\sigma/\rho=1$. For simplicity, we consider only the $k=1$ bifurcation in the infinite depth case, and continue to assume all functions are $2\pi$-periodic in space. In this configuration, the period of the linearized standing waves is $T=2\pi/\sqrt{2}\approx 4.443$. For real water (assuming $\sigma=72 \text{ dyne}/\text{cm}$), 4.443 units of dimensionless time corresponds to $0.0739$ seconds, and $2\pi$ spatial units corresponds to $1.7$ cm.
![\[fig:bif:surf\] Bifurcation diagrams showing the dependence of various Fourier modes of the initial conditions on the period for standing water waves with surface tension. The bifurcation curve splits into several disjoint branches between solutions B and C as resonant waves appear on the fluid surface. The curve labeled ‘quadratic correction’ is given in Equation (\[eq:concus\]). ](figs/bifSurf){width="\linewidth"}
![\[fig:evol:surf\] Time-elapsed snapshots of four standing waves over a quarter-period. At $t=0$, a pair of counter-propagating depression waves move away from each other as the fluid flows to the center. Solutions B, C and D exhibit higher-frequency standing waves oscillating on the surface of the low-frequency carrier wave. All of the solutions reach a rest state where $\varphi\equiv0$ at $t=T/4$. ](figs/evolSurf){width="\linewidth"}
The results are summarized in Figure \[fig:bif:surf\]. As the bifurcation parameter, $c_1=\hat\varphi_1(0)$, increases in magnitude beyond the realm of linear theory, the period increases, just as in the zero surface tension case for $h=\infty$. Quantitatively, our results agree with Concus’ prediction [@concus:62] that $$\label{eq:concus}
T = \sqrt{2}\,\pi\left(1 + \frac{197}{320}c_1^2\right), \qquad
c_1 = \hat\varphi_1(0)$$ in the infinite depth case when the surface tension parameter $\delta:=\sigma k^2/(\sigma k^2+\rho g)$ is equal to 1/2. Equation (\[eq:concus\]) is plotted in the left panel of Figure \[fig:bif:surf\] for comparison. Shortly after solution B ($c_1=-0.464$, $T=5.10$), a complicated sequence of imperfect bifurcations occurs in which several disjoint families of solutions pass near each other. Comparison of solutions B, C and D in Figure \[fig:evol:surf\] suggests that these disconnections are due to the excitation of different patterns of smaller-scale capillary waves oscillating on the free surface. An interesting difference between these standing waves and their zero surface-tension counterparts (e.g. in Figure \[fig:bif10\]) is that the “solitary” waves that appear in the transition periods between rest states of maximum amplitude are inverted. Thus, we can think of these solutions as counter-propagating depression waves [@crapper; @vandenBroeck:crapper] that are tuned to be time-periodic, just as the zero surface-tension case leads to counter-propagating Stokes waves. The depression waves travel outward as fluid flows to the center, whereas the Stokes waves travel inward, carrying the fluid with them. Figure \[fig:evol:surf:P\] shows snapshots of particle trajectories for solution C, color coded by pressure. The methodology for computing this pressure is given at the end of Appendix \[sec:BI\]. Negative pressure (relative to the ambient air pressure $p_0$ in (\[eq:c:def\]), which is set to zero for convenience) arises beneath the depression waves as they pass, which leads to larger pressure gradients, faster wave speeds, and shorter periods than were seen in previous sections.
![\[fig:evol:surf:P\] A more detailed view of solution C from Figure \[fig:evol:surf\] at times $t=0$ and $t=T/4$ showing regions of negative pressure beneath depression waves. These images are taken from movies in which passively advected particles have been added to the fluid for visualization, color coded by pressure using (\[eq:pressure\]). A secondary standing wave leads to visible variations in curvature and pressure at time $T/4$. ](figs/evolSurfP){width="\linewidth"}
Performance comparison {#sec:perform}
----------------------
We conclude our results with a comparison of running times for the various algorithms and machines used to generate the data reported above. Our machines consist of a laptop, a desktop, a server, a GPU device, and a supercluster. The laptop is a Macbook Pro, 2.53 GHz Intel Core i5 machine. The desktop is a Mac Pro with two quad-core 2.8 GHz Intel Nehalem processors. The rackmount server has two six-core 3.33 GHz Intel Westmere processors and an NVidia M2050 GPU, and is running Ubuntu Linux. The cluster is the Lawrencium cluster (LR1) at Lawrence Berkeley National Laboratory. Each node of the cluster contains two quad-core 2.66 GHz Intel Harpertown processors. Intel’s math kernel library and scalapack library were used for the linear algebra on Lawrencium.
$$\begin{array}{c|c|c|c|c|c|c|c|c|c}
\text{index range} & n & M & N & N_\text{quad} & \text{bif par} & \text{start} & \text{end} & T_\text{start} & T_\text{end} \\ \hline
\text{101--150} & 20 & 128 & 60 & 24 & \hat\varphi_1(0) & -0.004 & -0.2 & 1.5708 & 1.6034 \\
\text{150--174} & 32 & 192 & 96 & 36 & \hat\varphi_1(0) & -0.2 & -0.26 & 1.6034 & 1.6265 \\
\text{174--184} & 50 & 256 & 96 & 48 & \hat\varphi_1(0) & -0.26 & -0.275 & 1.6265 & 1.6332 \\
\text{184--194} & 54 & 384 & 120 & 60 & \hat\varphi_5(0) & 0.001071 & 0.001856 & 1.6332 & 1.6359 \\
\text{194--200} & 64 & 512 & 144 & 72 & \hat\varphi_5(0) & 0.001856 & 0.002117 & 1.6359 & 1.6358 \\
\text{200--210} & 75 & 768 & 180 & 96 & \hat\varphi_5(0) & 0.002117 & 0.002515 & 1.6358 & 1.6348 \\
\text{210--220} & 96 & 1024 & 240 & 120 & \hat\varphi_5(0) & 0.002515 & 0.002981 & 1.6348 & 1.6326
\end{array}$$
Our first test consists of computing the first 120 deep-water standing wave solutions reported in Figure \[fig:bif:00\] (up through solution B). The running times increase with amplitude due to an increase in the number of gridpoints ($M$), timesteps ($N$), and unknown Fourier modes of the initial conditions ($n$). The parameters used in this test are given in Table \[tab:param:small\], with running times reported in the left panel of Figure \[fig:times\]. For each index range, we computed the average time required to reduce $f$ below $10^{-25}$ (or $10^{-50}$ in quadruple precision), using linear extrapolation from the previous two solutions as a starting guess. In the Adjoint Continuation Method, the first solution in each range (with index 101, 150, 174, etc.) takes much longer than subsequent minimizations. This is because we re-build the inverse Hessian from scratch when the problem size changes, but not from one solution to the next in a given index range. This is illustrated in the figure by plotting the maximum and median number of seconds required to find a solution in a given index range, along with the average.
For these smaller problems, the DOPRI5 and DOPRI8 schemes are of comparable efficiency for double-precision accuracy. We used the former for this particular test. In quadruple precision, we switched to the SDC15 scheme, which is more efficient than DOPRI5 and DOPRI8 in reducing $f$ below $10^{-50}$. We also doubled $M$ and $n$ in the quadruple-precision runs. The MINPACK benchmark results were optimized as much as possible (using the GPU with Error Correcting Code (ECC) turned off) to give as fair a comparison as possible. For the benchmark, the Jacobian was computed via forward differences, as in (\[eq:J:fd\]). The ACM method works well on small problems, but starts to slow down relative to the benchmark around $M=1024$. The trust region shooting method is much faster than the ACM (and the benchmark) due to the fact that all the columns of the Jacobian employ the same Dirichlet to Neumann operator at each timestep. Thus, we save a factor of $n$ in setup costs by computing $n$ columns of the Jacobian simultaneously. Moreover, most of the work can be organized to run at level 3 BLAS speed. Note that the GPU is slower than the multi-core CPU in double-precision for small problems, but eventually wins out as the opportunity for parallelism increases. In quadruple-precision, the GPU is substantially faster than the CPU for all problem sizes tested as there is more arithmetic to be done relative to communication costs.
![\[fig:times\] Performance of the algorithms on various architectures. (left) Each data point is the average running time (in seconds per solution) for solutions in each index range listed in Table \[tab:param:small\]. For the Adjoint Continuation Method (ACM), which takes much longer for the first solution than subsequent solutions due to re-use of the Hessian information, we also report the longest running time and the median running time. (right) Time taken in each segment of the mesh-refinement strategy to evolve solution O in Figure \[fig:breakdown\] through $1/60$th of a quarter-period. The parameters for each segment are given in Table \[tab:param:large\]. The times listed for the Jacobian are the cost of evolving all 1200 columns through time $T/240$. ](figs/run_time){width="\linewidth"}
$$\begin{array}{r||c||c|c|c|c||c|c|c|c||c||c|c|c|c||c|c|c|c||c}
& \nu & \theta_1 & \theta_2 & \theta_3 & \theta_4 & \rho_1 & \rho_2 & \rho_3 & \rho_4 & n & M_1 & M_2 & M_3 & M_4 & d_1 & d_2 & d_3 & d_4 & \text{scheme} \\ \hline
\text{double} & {\scriptstyle}4 & {\scriptstyle}0.2 & {\scriptstyle}0.2 & {\scriptstyle}0.2 & {\scriptstyle}0.4 & {\scriptstyle}1.0 & {\scriptstyle}0.4 & {\scriptstyle}0.12 & {\scriptstyle}0.09 & {\scriptstyle}1200 & {\scriptstyle}3456 & {\scriptstyle}4608 & {\scriptstyle}5184 & {\scriptstyle}6144 & {\scriptstyle}10 & {\scriptstyle}20 & {\scriptstyle}30 & {\scriptstyle}40 & \text{\footnotesize DOPRI8} \\
\text{quad} & {\scriptstyle}4 & {\scriptstyle}0.1 & {\scriptstyle}0.3 & {\scriptstyle}0.4 & {\scriptstyle}0.2 & {\scriptstyle}1.0 & {\scriptstyle}0.25 & {\scriptstyle}0.08 & {\scriptstyle}0.05 & {\scriptstyle}1500 & {\scriptstyle}6144 & {\scriptstyle}7500 & {\scriptstyle}8192 & {\scriptstyle}9216 & {\scriptstyle}10 & {\scriptstyle}12 & {\scriptstyle}16 & {\scriptstyle}20 & \text{\footnotesize SDC15}
\end{array}$$
Our second test consists of timing each phase of the computation of solution O in Figure \[fig:breakdown\]. The parameters used for the performance comparison are given in Table \[tab:param:large\]. For the double-precision calculation, we later refined the mesh to the values listed in Table \[tab:param:OA\] in Section \[sec:breakdown\]; however, this was done on one machine only. (The value of $f$ here is $3.9\times 10^{-23}$ versus $1.3\times10^{-26}$ in Section \[sec:breakdown\].) The quantities $d_l$ in Table \[tab:param:large\] are the number of timesteps between mile-markers where the energy and plots of the solution were recorded. In this case, we recorded 60 slices of the solution between $t=0$ and $t=T/4$. The running times in the right panel of Figure \[fig:times\] report the time to advance from one mile-marker to the next. It was not possible to solve this problem via the ACM method or MINPACK, so this test compares running times of the trust region method on several machines. In quadruple precision, we evolved the solution but did not compute the Jacobian. Two of the jobs on the cluster (1 node and 2 nodes) were terminated early due to insufficient available wall-clock time. When using the GPU, there is little improvement in performance in also running openMP on the CPU. For example, switching from 12 threads (shown in the figure) to one thread (not shown) slows the computation of the Jacobian by about 10 percent, but speeds up the computation of the solution by about 1 percent. When evolving the solution on a large problem, the GPU is fully utilized; however, when evolving the Jacobian, the GPU is idle about 60 percent of the time. Thus, we can run 2-3 jobs simultaneously to improve the effective performance of the GPU by another factor of 2 over what is plotted in the figures. This is also true of the Lawrencium cluster — while using more nodes to solve a single problem stops paying off around 8 nodes, we can run multiple jobs independently. Most of the large-amplitude solutions in Figure \[fig:bif:00\] were computed in this way on the Lawrencium cluster, before we acquired the GPU device.
Conclusion {#sec:conclude}
==========
We have shown how to compute time-periodic solutions of the free-surface Euler equations with improved resolution, accuracy and robustness by formulating the shooting method as an overdetermined nonlinear least squares problem and exploiting parallelism in the Jacobian calculation. This made it possible to resolve a long-standing open question, posed by Penney and Price in 1952, on whether the most extreme standing wave develops wave crests with sharp 90 degree corners each time the fluid comes to rest. Previous numerical studies reached different conclusions about the form of the limiting wave, but none were able to resolve the fine-scale oscillations that develop due to resonant effects. While we cannot say for certain that no standing wave exists that forms sharp corners at periodic time-intervals, we can say that such a wave does not lie at the end of a family of increasingly sharp standing waves parametrized by crest acceleration, $A_c$. Indeed, crest acceleration is not a monotonic function, and the bifurcation curve becomes fragmented as $A_c\rightarrow1$, with different branches corresponding to different fine-scale oscillation patterns that emerge at the surface of the wave. Following any of these branches in either direction leads to increasingly oscillatory solutions with curvature that appears to blow up throughout the interval $[0,2\pi]$, not just at the crest tip.
Small-amplitude standing waves have been proved to exist in finite depth by Plotnikov and Toland [@plotnikov01], and in infinite depth by Plotnikov, Toland and Iooss [@iooss05]. However, the proofs rely on a Nash-Moser iteration that only guarantees existence for values of the amplitude in a totally disconnected Cantor set [@craig:wayne; @bourgain99]. In shallow water, with $h=0.05$, we do see evidence that solutions do not come in smooth families. For example, in Figure \[fig:bif:005\], the number of visible disconnections in the bifurcation diagrams increases dramatically from $\hat\varphi_1(0)$ to $\hat\varphi_{17}(0)$ to $\hat\eta_{36}(0)$. There are also a few gaps along the $T$-axis where the numerical method failed to find a solution, i.e. the minimum value of $f$ did not decrease below the target of $10^{-26}$ regardless of how many additional Fourier modes were included in the simulation. It is easy to imagine that removing all the gaps that arise in this fashion as the mesh is refined and the numerical precision is increased could lead to a Cantor-like set of allowed periods.
Our numerical method measures success by how small the objective function $f$ and residual $r$ become. It will succeed if it can find initial conditions that are close enough to those of an exactly time-periodic solution, or at least of a solution that is time-periodic up to roundoff error tolerances. For the residual to be small, the bifurcation parameter must nearly belong to the Cantor set of allowed values, but membership need not be exact. If the Cantor set is fat enough (i.e. has nearly full measure), then most values of the bifurcation parameter will be close to some element of the set — roundoff error fills in the smallest gaps. While it is possible that our numerical method would report a false positive, this seems unlikely. The residuals of our solutions are not under-predicted by the minimization algorithm due to formulation of the problem as an overdetermined system. Indeed, we saw in Figure \[fig:evol:PhiOA\] that $f$ decreases from $1.3\times10^{-26}$ to $8.6\times10^{-27}$ for solution O when the initial conditions are evolved on a finer mesh in quadruple precision, and decreases from $1.9\times10^{-28}$ to $2.1\times10^{-60}$ for solution A when the minimization is repeated in quadruple precision. This latter test is particularly convincing that the method is converging to an exactly time-periodic solution.
![\[fig:quad:00\] Study of resonance in deep water standing waves. (left) Switching from double- to quadruple-precision arithmetic reveals only one additional disconnection in the bifurcation curves. The inset graph shows how the disconnections of Figure \[fig:bif:00\] look when $c_{47}$ is plotted rather than $c_1$, $c_5$ and $c_{60}$. Solutions B and D are the points where $|c_{47}/c_1|\approx 3\times 10^{-16}$, just barely above the roundoff threshold. The gap in crest acceleration between these solutions is $A_c(D)-A_c(B)=1.4\times 10^{-9}$; thus, it is extremely unlikely in a parameter study that one would land in this gap. Outside of this gap, resonant effects from this disconnection are smaller than the roundoff threshold. (center and right) Resonance causes bursts of growth in the Fourier spectrum, but the modes continue to decay exponentially in the long run. ](figs/quad00){width="\linewidth"}
If standing waves on water of infinite depth do not come in smooth families, as suggested by the analysis of [@iooss05], they are remarkably well approximated by them. Prior to our work, no numerical evidence of disconnections in the bifurcation curves had been observed. Wilkening [@breakdown] found several disconnections for values of crest acceleration $A_c>0.99$, but none at smaller values. As shown in Figure \[fig:quad:00\], there is one additional disconnection around $A_c=0.947$ that can be observed in double-precision that was missed in [@breakdown]. However, the points at which resonance is supposed to cause difficulty are expected to be dense over the whole range $0<A_c<1$. We re-computed the solutions up to $A_c=0.8907$ in quadruple precision, expecting several new disconnections to emerge in high-frequency Fourier modes. Surprisingly, we could only find one, at $A_c=0.658621$. Using a bisection algorithm to zoom in on the disconnection in the 47th Fourier mode from both sides (using $c_5$ as the bifurcation parameter), we were able to extend the side branches from $c_{47}\approx \pm 10^{-27}$ to $c_{47}\approx \pm 10^{-12}$. These side branches become observable in double-precision at points B and D in Figure \[fig:quad:00\]. However, the gap in crest acceleration between solutions B and D is only $1.4\times 10^{-9}$ units wide. Thus, it is extremely unlikely that this resonance could be detected in double-precision without knowing where to look. Presumably the same issue prevents us from seeing additional disconnections in quadruple-precision. This suggests that the Cantor-like set of allowed values of the amplitude parameter is very fat, with gaps decaying to zero rapidly with the wave number of the resonant mode.
In finite depth, with $h\approx 1$, a connection can be seen between resonance and non-uniqueness of solutions. The main difference from the $h=0.05$ and $h=\infty$ cases is that for $h\approx1$, the disconnections lead to side-branches that can be tracked a great distance via numerical continuation, and are often found to be globally connected to one another. Traversing these side branches causes high-frequency modes to sweep out small-amplitude loop-shaped structures. These loops are “long and thin” in the sense that low-frequency modes trace back over the previously swept out bifurcation curves while traversing the loop, with little deviation in the lateral direction. For example, in Figure \[fig:bif103w\], the 27th Fourier mode executes a number of excursions in which it grows to around $10^{-6}$, causing the period and first Fourier modes to sweep back and forth over much larger ranges, $7.167<T<7.229$ and $-0.200>\hat\varphi_1(0)>-0.262$. These loops are plotted in Figures \[fig:spider\] and \[fig:bif104\] as well, but the curves are indistinguishable from one another at this resolution since the lateral deviations are so small. Looking at the third panel of Figure \[fig:bif103w\], one might ask, “how many solutions are there with period $T=7.2$.” If we had not noticed any of the disconnections (note the exponential scaling of the axis), we would have answered 1. If we had only tracked the outer wings, we would have answered 3. Having tracked all the branches shown, the answer appears to be 5. But of course there are probably infinitely many disconnections in higher-frequency Fourier modes that we did not resolve or track, and some of these may lead to additional solutions with $T=1.2$. Physically, all these crossings of $T=1.2$ correspond to a hierarchy of “standing waves on standing waves,” with different mode amplitudes and phases working together to create a globally time-periodic solution with this period. The fact that the low-frequency bifurcation curves sweep back and forth over nearly the same graph reinforces the physically reasonable idea that high-frequency, low amplitude waves oscillating on the surface of low-frequency, large amplitude waves will not significantly change the large-scale behavior.
![\[fig:bif103w\] A closer look at the bifurcation structure in Figure \[fig:spider\] in the $h=1.03$ case reveals a number of additional side-branches that trace back and forth over nearly the same curves when low-frequency modes are plotted (left), but become well-separated when high frequency modes are plotted (right center, right). A small gap near $T=7.175$ has formed on one of the wings in the plots of $\hat\varphi_{27}(0)$ and $\hat\varphi_{69}(0)$ vs $T$. ](figs/bif103w){width="\linewidth"}
In summary, time-periodic water waves occur in abundance in numerical simulations, and appear to be highly non-unique, partly due to the Wilton’s ripple phenomenon of mixed-mode solutions co-existing with pure-mode solutions near a degenerate bifurcation, and also due to a tendency of the bifurcation curves to fold back on themselves each time a resonant mode is excited. Proofs of existence based on Nash-Moser iteration must somehow select among these multiple solutions, and it would be interesting to know whether the Cantor-like structure in the analysis is caused by a true lack of existence for parameter values outside of this set, or is partly caused by non-uniqueness. Finally, we note that most of the disconnections in the numerically computed bifurcation curves disappear in the infinite depth limit, and remarkably small residuals can be achieved with smooth families of approximate solutions. This calls for further investigation of the extent to which the obstacles to proving smooth dependence of solutions on amplitude can be overcome or quantified.
**Acknowledgments**
This research was supported in part by the Director, Office of Science, Computational and Technology Research, U.S. Department of Energy under Contract No. DE-AC02-05CH11231, and by the National Science Foundation through grant DMS-0955078. Some of the computations were performed on the Lawrencium supercluster at LBNL.
Boundary integral formulation {#sec:BI}
=============================
While many numerical methods exist to evolve irrotational flow problems [@lh76; @baker:82; @krasny:86; @mercer:92; @dias:bridges; @HLS01; @vtxs1; @dyachenko:1996; @milewski:11; @baker10], we have found that a direct boundary integral implementation of (\[eq:ww\]) is the simplest and most effective approach for problems where $\eta$ remains single valued, i.e. the interface does not overturn. Suppressing $t$ in the notation, we represent the complex velocity potential $\Phi(z)=\phi(z)+i\psi(z)$ as a Cauchy integral [@muskh] $$\label{eq:cauchy}
\Phi(z) = \frac{1}{2\pi i}PV\!\!\int_{-\infty}^\infty
\frac{-\zeta'(\alpha)}{\zeta(\alpha)-z}\mu(\alpha)\,d\alpha, \qquad
\zeta(\alpha) = \xi(\alpha) + i\eta(\xi(\alpha)),$$ where $z$ is a field point in the fluid, $\mu(\alpha)$ is the (real-valued) dipole density, $\zeta(\alpha)$ parametrizes the free surface, $PV$ indicates a principal value integral, $\eta(x)$ retains its meaning from equation (\[eq:ww\]), and the change of variables $x=\xi(\alpha)$ will be used to smoothly refine the mesh in regions of high curvature. The minus sign in (\[eq:cauchy\]) accounts for the fact that Cauchy integrals are usually parametrized counter-clockwise, but we have parametrized the curve so the fluid region lies to the right of $\zeta(\alpha)$. When the fluid depth is finite, we impose the tangential flow condition using an identical double-layer potential on the mirror image surface, $\bar\zeta(\alpha)$. This assumes we have set $h=0$ in (\[eq:dno\]), absorbing the mean fluid depth into $\eta$ itself. We also use $\frac{1}{2}\cot\frac{z}{2} = PV\sum_k\frac{1}{z+2\pi k}$ to sum (\[eq:cauchy\]) over periodic images. The result is $$\label{eq:Phi}
\Phi(z) = \frac{1}{2\pi i}\int_0^{2\pi}
\left[ \frac{\zeta'(\alpha)}{2}\cot\left(\frac{z-\zeta(\alpha)}{2}
\right) -
\frac{\bar\zeta'(\alpha)}{2}\cot\left(\frac{z-\bar\zeta(\alpha)}{2}
\right)\right]\mu(\alpha)\,d\alpha.$$ Note that $\Phi$ is real-valued on the $x$-axis, indicating that the stream function $\psi$ is zero (and therefore constant) along the bottom boundary.
As $z$ approaches $\zeta(\alpha)$ from above ($+$) or below ($-$), the Plemelj formula [@muskh] gives $$\label{eq:plemelj:1}
\Phi\big(\zeta(\alpha)^\pm\big) = \mp\frac{1}{2}\mu(\alpha) +
\frac{PV}{2\pi i}\int_0^{2\pi}
\left[
\frac{\zeta'(\beta)}{2}\cot\frac{\zeta(\alpha)-
\zeta(\beta)}{2} -
\frac{\bar\zeta'(\beta)}{2}\cot\frac{\zeta(\alpha)-
\bar\zeta(\beta)}{2}
\right]
\mu(\beta)\,d\beta.$$ We regularize the principal value integral by subtracting and adding $\frac{1}{2}\cot\left(\frac{\alpha-\beta}{2}\right)$ from the first term in brackets [@vande:vooren:80; @pullin:82; @baker:nachbin:98]. The result is $$\label{eq:Phi:formula}
\Phi\big(\zeta(\alpha)^\pm\big) = \mp\frac{1}{2}\mu(\alpha)
- \frac{i}{2}H\mu(\alpha) + \frac{1}{2\pi i}\int_0^{2\pi}
[{\widetilde{K}}_1(\alpha,\beta) + {\widetilde{K}}_2(\alpha,\beta)]
\mu(\beta)\,d\beta,$$ where $Hf(\alpha)=\frac{1}{\pi}PV\!\int_{-\infty}^\infty
\frac{f(\beta)}{\alpha-\beta}\,d\alpha = \frac{1}{\pi}PV\!
\int_0^{2\pi}\frac{f(\beta)}{2}\cot\left(
\frac{\alpha-\beta}{2}\right)d\beta$ is the Hilbert transform and $$\label{eq:K1K2:til}
{\widetilde{K}}_1(\alpha,\beta) = \frac{\zeta'(\beta)}{2}
\cot\frac{\zeta(\alpha) - \zeta(\beta)}{2}
- \frac{1}{2}\cot\frac{\alpha-\beta}{2}, \qquad
{\widetilde{K}}_2(\alpha,\beta) = \frac{\bar\zeta'(\beta)}{2}\cot
\frac{\zeta(\alpha)-\bar\zeta(\beta)}{2}.$$ We note that ${\widetilde{K}}_1(\alpha,\beta)$ is continuous at $\beta=\alpha$ if we define ${\widetilde{K}}_1(\alpha,\alpha)=-\zeta''(\alpha)/[2\zeta'(\alpha)]$. Taking the real part of (\[eq:Phi:formula\]) at $z=\zeta(\alpha)^-$ yields a second-kind Fredholm integral equation for $\mu(\alpha)$ in terms of $\varphi(\xi(\alpha))$, $$\label{eq:fred2}
\frac{1}{2}\mu(\alpha) + \frac{1}{2\pi}\int_0^{2\pi}
[K_1(\alpha,\beta) + K_2(\alpha,\beta)]\mu(\beta)\,d\beta =
\varphi(\xi(\alpha)),$$ where $K_j(\alpha,\beta) = {\operatorname{Im}}\{ {\widetilde{K}}_j(\alpha,\beta)\}$. Once $\mu(\alpha)$ is known, it follows from (\[eq:Phi\]) that $$\label{eq:dPhi1}
\Phi'(z) = u(z)-iv(z) = \frac{1}{2\pi i}\int_0^{2\pi}
\left[\frac{1}{2}\cot\left(\frac{z-\zeta(\alpha)}{2}\right) -
\frac{1}{2}\cot\left(\frac{z-\bar{\zeta}(\alpha)}{2}\right)\right]
\gamma(\alpha)\,d\alpha,$$ where $\gamma(\alpha)=\mu'(\alpha)$ is the (normalized) vortex sheet strength. As $z$ approaches $\zeta(\alpha)$ from above or below, one may show [@jia:thesis] that $$\label{eq:plemelj:2}
\zeta'(\alpha)\Phi'\big(\zeta(\alpha)^\pm\big) =
\mp\frac{1}{2}\gamma(\alpha) + \frac{PV}{2\pi i}\int_0^{2\pi}
\left[
\frac{\zeta'(\alpha)}{2}\cot\frac{\zeta(\alpha) - \zeta(\beta)}{2}
- \frac{\zeta'(\alpha)}{2}\cot\frac{\zeta(\alpha) - \bar\zeta(\beta)}{2}
\right]\gamma(\beta)\,d\beta.$$ Note that $\zeta'$ is evaluated at $\beta$ in (\[eq:plemelj:1\]) and at $\alpha$ in (\[eq:plemelj:2\]) inside the integral. We regularize the principal value integral using the same technique as before to obtain $$\label{eq:dPhi2}
\zeta'(\alpha)\Phi'\big(\zeta(\alpha)^\pm\big) =
\mp\frac{1}{2}\gamma(\alpha) - \frac{i}{2}H\gamma(\alpha) +
\frac{1}{2\pi i}\int_0^{2\pi} [{\widetilde{G}}_1(\alpha,\beta)
+ {\widetilde{G}}_2(\alpha,\beta)]\gamma(\beta)\,d\beta,$$ where $$\label{eq:G1G2:til}
{\widetilde{G}}_1(\alpha,\beta) = \frac{\zeta'(\alpha)}{2}
\cot\frac{\zeta(\alpha) - \zeta(\beta)}{2}
- \frac{1}{2}\cot\frac{\alpha-\beta}{2}, \qquad
{\widetilde{G}}_2(\alpha,\beta) = \frac{\zeta'(\alpha)}{2}\cot
\frac{\zeta(\alpha)-\bar\zeta(\beta)}{2}.$$ ${\widetilde{G}}_1(\alpha,\beta)$ is continuous at $\beta=\alpha$ if we define ${\widetilde{G}}_1(\alpha,\alpha) = \zeta''(\alpha)/[2\zeta'(\alpha)]$. We could read off $u=\phi_x$ and $v=\phi_y$ from (\[eq:dPhi2\]) for use in the right hand side of (\[eq:ww\]). Instead, as an intermediate step, we compute the output of the Dirichlet-Neumann operator defined in (\[eq:DNO:def\]), $$\label{eq:DNO:formula}
\begin{aligned}
|\xi'(\alpha)|{\mathcal{G}}\varphi(\xi(\alpha)) &=
|\zeta'(\alpha)|{\frac{\partial \phi}{\partial n}}(\zeta(\alpha))
= \lim_{z\rightarrow \zeta(\alpha)^-}
{\operatorname{Re}}\left\{i\zeta'(\alpha)[u(z)-iv(z)]\right\} \\
&= \frac{1}{2}H\gamma(\alpha) + \frac{1}{2\pi}\int_0^{2\pi}
[G_1(\alpha,\beta) + G_2(\alpha,\beta)]\gamma(\beta)\,d\beta.
\end{aligned}$$ Here $G_j(\alpha,\beta) = {\operatorname{Re}}\{{\widetilde{G}}_j(\alpha,\beta)\}$ and $i\zeta'(\alpha)/|\zeta'(\alpha)|$ represents the normal vector to the curve. Note that the dot product of two complex numbers $z$ and $w$ (thought of as vectors in $\mathbb{R}^2$) is ${\operatorname{Re}}\{z\bar w\}$. Once ${\mathcal{G}}\varphi(x)$ is known, we can evaluate the right hand side of (\[eq:ww\]) using (\[eq:uv:from:G\]).
For visualization, it is often useful to evaluate the velocity and pressure inside the fluid. The velocity was already given in terms of the vortex sheet strength in (\[eq:dPhi1\]) above. For pressure, we use the unsteady Bernoulli equation $$\label{eq:pressure}
\phi_t + \frac{1}{2}|\nabla\phi|^2 + gy + \frac{p}{\rho} = c(t),$$ where $c(t)$ was given in (\[eq:c:def\]). One option for computing $\phi_t$ is to differentiate (\[eq:fred2\]) with respect to time to obtain an integral equation for $\mu_t$ (see [@water:pod]), then express $\phi_t$ in terms of $\mu_t$ by differentiating (\[eq:Phi\]). A simpler approach is to differentiate the Laplace equation (\[eq:dno\]) with respect to time. The value of $\phi_t$ on the upper boundary is $\varphi_t - \phi_y\eta_t$, which is known. Since the real part of (\[eq:Phi\]) gives the solution $\phi(z)$ of Laplace’s equation with boundary condition $\varphi$ on the upper surface, we can replace $\varphi$ with $\varphi_t - \phi_y\eta_t$ in (\[eq:fred2\]) to convert (\[eq:Phi\]) into a formula for $\phi_t(z)$ instead.
Linearized and adjoint equations for the water wave {#sec:adjoint}
===================================================
In this section we derive explicit formulas for the variational and adjoint equations of Sections \[sec:acm\] and \[sec:trust\]. A dot will be used to denote a directional derivative with respect to the initial conditions. The equation $\dot q_t = DF(q)\dot q$ of (\[eq:dot\]) is simply
\[eq:lin2\] $$\begin{aligned}
\dot\eta(x,0)=\dot\eta_0(x),\quad \dot\varphi(x,0) =\dot\varphi_0(x),&&
t& =0,\label{eq:l1}\\
\dot\phi_{xx}+\dot\phi_{yy} =0,&&
-h& <y<\eta,\label{eq:l2}\\
\dot\phi_y =0,&&
y& =-h,\label{eq:l3}\\
\dot\phi+\phi_y\dot\eta =\dot\varphi,&&
y& =\eta,\label{eq:l4}\\
\dot\eta_t+\dot\eta_x\phi_x+\eta_x\dot\phi_x+
\eta_x\phi_{xy}\dot\eta =\dot\phi_y+\phi_{yy}\dot\eta,&&
y& =\eta,\label{eq:l5}\\
\dot\varphi_t
= P\left[-\bigg(\eta_x\phi_x\phi_y + \frac{1}{2}\phi_x^2 -
\frac{1}{2}\phi_y^2\bigg)^\text{\large .} - g\dot\eta +
\frac{\sigma}{\rho}\partial_x\left(\frac{\dot\eta_x}{(1+\eta_x^2)^{3/2}}\right)\right],&&
y&=\eta.\label{eq:l6}
\end{aligned}$$
Note that evaluation of $\dot\phi(x,y,t)$ on the free surface gives $\big[\dot\varphi(x)-\phi_y(x,\eta(x),t)\dot\eta(x)\big]$ rather than $\dot\varphi(x)$ due to the boundary perturbation. Making use of $\phi_{yy}=-\phi_{xx}$, (\[eq:l5\]) can be simplified to $$\label{eq:l5a}
\dot\eta_t = \big(\dot\phi_y - \eta_x\dot\phi_x\big) -
\big(\dot\eta\phi_x\big)',$$ where a prime indicates an $x$-derivative along the free surface, e.g. $f':=\frac{d}{dx}f(x,\eta(x),t)=f_x+\eta_x f_y$. Equation (\[eq:l6\]) may also be simplified, using $$\begin{aligned}
\notag
\bigg(\eta_x\phi_x\phi_y + \frac{1}{2}\phi_x^2-\frac{1}{2}\phi_y^2
\bigg)^\text{\large .} &=
\dot\eta_x\phi_x\phi_y + \eta_x\dot\phi_x\phi_y +
\eta_x\phi_{xy}\dot\eta\phi_y + \eta_x\phi_x\dot\phi_y +
\eta_x\phi_x\phi_{yy}\dot\eta + \phi_x\dot\phi_x + \phi_x\phi_{xy}\dot\eta
-\phi_y\dot\phi_y - \phi_y\phi_{yy}\dot\eta \\
\label{eq:l6a}
&= \big(\dot\eta\phi_x\phi_y\big)' + \phi_x\dot\phi' -
\phi_y\big(\dot\phi_y - \eta_x\dot\phi_x\big).\end{aligned}$$ The equation $\tilde{q}_s=D F(q)^*\tilde{q}$ is obtained from $$\label{eq:tilde1}
\begin{split}
\langle\dot{q},\tilde{q}_s\rangle=\langle\dot{q}_t,\tilde{q}\rangle&=
\frac{1}{2\pi}\int_0^{2\pi}\left[\underline{(\dot\phi_y-\eta_x\dot\phi_x)}
- (\dot\eta\phi_x)'\right]
\tilde\eta\,dx\\
&\qquad+\frac{1}{2\pi}\int_0^{2\pi}P\left[-(\dot\eta\phi_x\phi_y)'-
\phi_x\dot\phi'+\phi_y\underline{(\dot\phi_y-\eta_x\dot\phi_x)}
-g\dot\eta+\frac{\sigma}{\rho}\partial_x\left(
\frac{\dot\eta_x}{(1+\eta_x^2)^{3/2}}\right)\right]\tilde\varphi\,dx.
\end{split}$$ The right-hand side must now be re-organized so we can identify $\tilde q_s$. $P$ is self-adjoint, so it can be transferred from the bracketed term to $\tilde\varphi$. The underlined terms may be written ${\mathcal{G}}\dot\phi$, where $\dot\phi$ is evaluated on the free surface. Green’s identity shows that ${\mathcal{G}}$ is self-adjoint. Indeed, if $\dot\phi$ and $\chi$ satisfy Laplace’s equation with Neumann conditions on the bottom boundary, then $$0 = \iint(\chi\Delta\dot\phi-\dot\phi\Delta\chi)d A =
\int\chi\frac{\partial\dot\phi}{\partial n}-
\dot\phi\frac{\partial\chi}{\partial n} ds =
\int\chi{\mathcal{G}}\dot\phi\,dx - \int\dot\phi{\mathcal{G}}\chi\,dx.$$ Thus, from (\[eq:tilde1\]), we obtain $$\langle\dot{q},\tilde{q}_s\rangle = \frac{1}{2\pi}\int_0^{2\pi}
\left[ \dot\phi{\mathcal{G}}\chi +
\dot\eta\phi_x\tilde\eta' + \dot\eta\phi_x\phi_y (P\tilde\varphi)'
+ \dot\phi(\phi_x P\tilde\varphi)' - g\dot\eta P\tilde\varphi
+ \frac{\sigma}{\rho}\dot\eta \partial_x\left( \frac{
\tilde\varphi_x}{(1+\eta_x^2)^{3/2}} \right)\right]dx,$$ where $\chi$ is an auxiliary solution of Laplace’s equation defined to be $\big(\tilde\eta + \phi_y P\tilde\varphi\big)$ on the free surface. Finally, we substitute $\dot\phi = \dot\varphi - \phi_y\dot\eta$ and match terms to arrive at the adjoint system
\[eq:adj2\] $$\begin{aligned}
\tilde\eta(x,0)=0,\quad
\tilde\varphi(x,0) =\varphi(x,T/4),&&
s&=0,\label{eq:a1}\\
\chi_{xx}+\chi_{yy} =0,&&
-h&<y<\eta,\label{eq:a2}\\
\chi_y =0,&&
y&=-h,\label{eq:a3}\\
\chi =\tilde\eta+\phi_y P \tilde\varphi,&&
y&=\eta,\label{eq:a4}\\
\tilde\varphi_s= (\chi_y - \eta_x\chi_x) +
(\phi_x P\tilde\varphi)',&&
y&=\eta,\label{eq:a5}\\
\tilde\eta_s = -\phi_y(\chi_y-\eta_x\chi_x)
+ \phi_x\tilde\eta_x - \phi_y\phi_x'P\tilde\varphi
- g P\tilde\varphi+\frac{\sigma}{\rho}\partial_x
\left(\frac{\tilde\varphi_x}{(1+\eta_x^2)^{3/2}}\right),&&
y&=\eta.\label{eq:a6}\end{aligned}$$
The initial conditions (\[eq:a1\]) are specific to the objective function (\[eq:f:phi:again\]), but are easily modified to handle the alternative objective function (\[eq:f:naive\]). Note that the adjoint problem has the same structure as the forward and linearized problems, with a Dirichlet to Neumann map appearing in the evolution equations for $\tilde\eta$ and $\tilde\varphi$. We use the boundary integral method described in Appendix \[sec:BI\] to compute ${\mathcal{G}}\chi$, and employ a dense output formula to interpolate $\eta$ and $\varphi$ between timesteps at intermediate Runge-Kutta stages of the adjoint problem, as explained in Section \[sec:acm\].
Levenberg-Marquardt implementation with delayed Jacobian updates {#sec:levmar}
================================================================
Since minimizing $f$ in (\[eq:f:phi\]) is a small-residual nonlinear least squares problem, the Levenberg-Marquardt method [@nocedal] is quadratically convergent. Our goal in this section is to discuss modifications of the algorithm in which re-computation of the Jacobian is delayed until the previously computed Jacobian ceases to be useful. By appropriately adjusting the step size in the numerical continuation algorithm, it is usually only necessary to compute the Jacobian once per solution. Briefly, the Levenberg-Marquardt method works by minimizing the quadratic function $$f_\text{approx}(p) = f(c) + g^Tp + \frac{1}{2}p^TBp,
\qquad g = \nabla f(c) = J^T(c)r(c), \qquad B = J(c)^TJ(c)$$ over the trust region $\|p\|\le\Delta$. The true Hessian of $f$ at $c$ satisfies $H-B=\sum_i r_i\nabla^2r_i$, which is small if $r$ is small. The solution of this constrained quadratic minimization problem is the same as the solution of a linear least-squares problem with an unknown parameter $\lambda$: $$\min_p\left\|\begin{pmatrix} J \\ \sqrt{\lambda}\,I \end{pmatrix} p
+ \begin{pmatrix} r \\ 0 \end{pmatrix} \right\|, \qquad
\lambda\ge 0, \qquad (\|p\|-\Delta)\lambda=0.$$ Formulating the problem this way (instead of solving $(B+\lambda
I)p=-g$) avoids squaring the condition number of $J$. Rather than use the MINPACK algorithm [@nocedal] to find the Lagrange multiplier $\lambda$, we compute the (thin) SVD of $J$, and define $$J = USV^T, \qquad S={\operatorname}{diag}\{\sigma\}, \qquad
\tilde p=V^Tp, \qquad \tilde r = U^T r, \qquad
\tilde g = S^T\tilde r.$$ Here $U$ is $m\times n$ and $S=S^T$ is $n\times n$. This leads to an equivalent problem $$\label{eq:trust:tilde}
\min_{\tilde p}\left\|\begin{pmatrix} S \\ \sqrt{\lambda}\,I \end{pmatrix}
\tilde p
+ \begin{pmatrix} \tilde r \\ 0 \end{pmatrix} \right\|, \qquad
\lambda\ge 0, \qquad (\|\tilde p\|-\Delta)\lambda=0,$$ which can be solved in $O(n)$ time by performing a Newton iteration on $\tau(\lambda)$, defined as $$\tau(\lambda)=\frac{1}{\|\tilde p\|}-\frac{1}{\Delta}, \qquad
\tilde p = \arg\min\left\|\begin{pmatrix} S \\
\sqrt\lambda\,I \end{pmatrix} \tilde p +
\begin{pmatrix} \tilde r \\ 0 \end{pmatrix} \right\|.$$ It is easy to show that $\tau$ is an increasing, concave down function for $\lambda\ge0$ (assuming $S$ is non-singular); thus, if $\tau(0)<0$, the Newton iteration starting at $\lambda^{{(0)}}=0$ will increase monotonically to the solution of (\[eq:trust:tilde\]) with $\tau(\lambda^{{(l)}})$ increasing to zero. This Newton iteration is equivalent to
We use $\text{tol}=10^{-12}$ in double-precision and $10^{-24}$ in quadruple precision. It is not critical that $\lambda$ be computed to such high accuracy, but as the Newton iteration is inexpensive once the SVD of $J$ is known, there is no reason not to iterate to convergence. At the end, we set $p=V\tilde p$.
We remark that it is more common to compute $\lambda$ by a sequence of QR factorizations of $[J;\sqrt{\lambda^{{(l)}}}\,I]$, as is done in MINPACK. However, the SVD approach is simpler, and similar in speed, since several QR factorizations have to be performed to compute $\lambda$ while only one SVD must be computed. Moreover, we can re-use $J$ several times instead of re-computing it each time a step is accepted. When this is done, it pays to have factored $J=USV^T$ up front.
Delaying the computation of $J$ requires a modified strategy for updating the trust region radius, as well as a means of deciding when the minimization is complete, and when to re-compute $J$. Our design decisions are summarized as follows:
1\. The algorithm terminates if $f=0$, or if $c$ is unchanged from the previous iteration (i.e. $c+p$ equals $c$ in floating point arithmetic), or if the algorithm reaches the *roundoff\_regime* phase, and then a step is rejected or *stepsJ* reaches *max\_stepsJ*. Here *stepsJ* counts accepted steps since $J$ was last evaluated, and the *roundoff\_regime* phase begins if $f<f_\text{tol}$ or $\Delta<g_\text{tol}$, where the tolerances and *max\_stepsJ* are specified by the user. If the Jacobian has just been computed (i.e. $\text{\emph{stepsJ}}=0$), we also check if $\|g\|<g_\text{tol}$ or $|df|/f<df_\text{tol}$ to trigger *roundoff\_regime*. Here $df=f_\text{approx}(c+p)-f(c)$ is the predicted change in $f$ when minimizing the quadratic model $f_\text{approx}$ over the trust region, and $df_\text{tol}$ is specified by the user. We used $$f_\text{tol}=10^{-26}, \qquad g_\text{tol}=10^{-13}, \qquad
\text{\emph{max\_stepsJ}}=10, \qquad df_\text{tol} = 10^{-5}.$$ The idea of *roundoff\_regime* is to try to improve $f$ through a few additional residual calculations without recomputing $J$.
2\. Steps are accepted if $\rho=[f(c+p)-f(c)]/df>0$; otherwise they are rejected. Note that $\rho$ is the ratio of the actual change to the predicted change, the latter being negative. We also use $\rho$ to adjust $\Delta$. If $\rho<\rho_0=1/4$, we replace $\Delta$ by $\|p\|$ times $\alpha_0=3/8$. If $\rho>\rho_1=0.85$ and $\|p\|>0.9\Delta$, we multiply $\Delta$ by $\alpha_1=1.875$. Otherwise we leave $\Delta$ alone. So far this agrees with the standard trust region mechanism [@nocedal] for adjusting $\Delta$, with slightly different parameters. What we do differently is define a parameter *delta\_trigger* to be a prescribed fraction, namely $\alpha_2=0.2$, of *delta\_first\_rejected*, the first rejected radius after (or coinciding with) an accepted step. Note that the radius is rejected ($\rho<\rho_0$), not necessarily the step ($\rho\le0$). The reason to wait for an accepted step is to let the trust region shrink normally several times in a row if the Jacobian is freshly computed ($\text{\emph{stepsJ}}=0$).
3\. The Jacobian is re-computed if *roundoff\_regime* has not occurred, and either *stepsJ* reaches *max\_stepsJ*, or $\text{\emph{stepsJ}}>0$ and $\Delta$ drops below *delta\_trigger*, or $\text{\emph{stepsJ}}>0$ and $|df|/f<df_\text{tol}$. This last test avoids iterating on an old Jacobian if the new residual is nearly orthogonal to its columns — there is little point in continuing if $f_\text{approx}$ cannot be decreased significantly. The parameters $\alpha_i$ were chosen so that $$\max(\alpha_0^2,\alpha_0^3\alpha_1^2)<\alpha_2<
\min(\alpha_0,\alpha_0^2\alpha_1),$$ which triggers the re-computation of $J$ if two radii are rejected in a row, or on a reject-accept-reject-accept-reject sequence, assuming $\|p\|=\Delta$ on each rejection. Before computing $J$, if *delta\_first\_rejected* has been defined since $J$ was last computed, we reset $\Delta$ to $$\Delta = \text{\emph{delta\_first\_rejected}}/\alpha_1.$$ This makes up for the decreases in $\Delta$ that occur due to using an old Jacobian.
4\. We compute $r$ but not $J$ if a step is rejected on a freshly computed Jacobian, or if a step is accepted or rejected without triggering one of the conditions mentioned above for computing $J$.
**References**
[10]{} url \#1[`#1`]{}urlprefixhref \#1\#2[\#2]{} \#1[\#1]{}
L. W. Schwartz, A. K. Whitney, A semi-analytic solution for nonlinear standing waves in deep water, J. Fluid Mech. 107 (1981) 147–171.
G. N. Mercer, A. J. Roberts, [Standing waves in deep water: Their stability and extreme form]{}, Phys. Fluids A 4 (2) (1992) 259–269.
G. N. Mercer, A. J. Roberts, The form of standing waves on finite depth water, Wave Motion 19 (1994) 233–244.
D. H. Smith, A. J. Roberts, Branching behavior of standing waves — the signatures of resonance, Phys. Fluids 11 (1999) 1051–1064.
C. P. Tsai, D. S. Jeng, Numerical [Fourier]{} solutions of standing waves in finite water depth, Appl. Ocean Res. 16 (1994) 185–193.
P. J. Bryant, M. Stiassnie, Different forms for nonlinear standing waves in deep water, J. Fluid Mech. 272 (1994) 135–156.
W. W. Schultz, J. M. Vanden-Broeck, L. Jiang, M. Perlin, [Highly nonlinear standing water waves with small capillary effect]{}, J. Fluid Mech. 369 (1998) 253–272.
M. Okamura, [Standing gravity waves of large amplitude on deep water]{}, Wave Motion 37 (2003) 173–182.
A. J. Roberts, Highly nonlinear short-crested water waves, J. Fluid Mech. 135 (1983) 301–321.
T. R. Marchant, A. J. Roberts, Properties of short-crested waves in water of finite depth, J. Austral. Math. Soc. B 29 (1987) 103–125.
T. J. Bridges, F. Dias, D. Menasce, Steady three-dimensional water-wave patterns on finite-depth fluid, J. Fluid Mech. 436 (2001) 145–175.
M. Ioualalen, M. Okamura, S. Cornier, C. Kharif, A. J. Roberts, Computation of short-crested deep-water waves, J. Waterway, Port, Coast and Ocean Engrg 132 (3) (2006) 157–165.
W. G. Penney, A. T. Price, Finite periodic stationary gravity waves in a perfect liquid, part [II]{}, Phil. Trans. R. Soc. London A 244 (1952) 254–284.
G. I. Taylor, [An experimental study of standing waves]{}, Proc. Roy. Soc. A 218 (1953) 44–59.
M. A. Grant, Standing [Stokes]{} waves of maximum height, J. Fluid Mech. 60 (1973) 593–604.
M. Okamura, On the enclosed crest angle of the limiting profile of standing waves, Wave Motion 28 (1998) 79–87.
J. Wilkening, Breakdown of self-similarity at the crests of large amplitude standing water waves, Phys. Rev. Lett 107 (2011) 184501.
P. Concus, Standing capillary–gravity waves of finite amplitude: Corrigendum, J. Fluid Mech. 19 (1964) 264–266.
J.-M. Vanden-Broeck, Gravity–Capillary Free–Surface Flows, Cambridge University Press, Cambridge, 2010.
G. Iooss, P. I. Plotnikov, J. F. Toland, Standing waves on an infinitely deep perfect fluid under gravity, Arch. Rat. Mech. Anal. 177 (2005) 367–478.
B. Rayleigh, On waves, Philos. Mag. 1 (1876) 257–279.
I. Tadjbakhsh, J. B. Keller, Standing surface waves of finite amplitude, J. Fluid Mech. 8 (1960) 442–451.
P. Concus, Standing capillary–gravity waves of finite amplitude, J. Fluid Mech. 14 (1962) 568–576.
P. Plotnikov, J. Toland, Nash-moser theory for standing water waves, Arch. Rat. Mech. Anal. 159 (2001) 1–83.
W. Craig, C. E. Wayne, Newton’s method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math 46 (1993) 1409–1498.
J. Bourgain, Nonlinear Schr¬odinger equations. In Hyperbolic equations and frequency interactions, American Mathematical Society, Providence, 1999.
J.-M. Vanden-Broeck, L. W. Schwartz, Numerical calculation of standing waves in water of arbitrary uniform depth, Phys. Fluids 24 (5) (1981) 812–815.
M. Okamura, M. Ioualalen, C. Kharif, Standing waves on water of uniform depth: on their resonances and matching with short-crested waves, J. Fluid Mech. 495 (2003) 145–156.
M. Okamura, Almost limiting short-crested gravity waves in deep water, J. Fluid Mech. 646 (2010) 481–503.
C. J. Amick, J. F. Toland, The semi-analytic theory of standing waves, Proc. Roy. Soc. Lond. A 411 (1987) 123–138.
E. J. Doedel, H. B. Keller, J. P. Kernévez, Numerical analysis and control of bifurcation problems: ([II]{}) [Bifurcation]{} in infinite dimensions, Int. J. Bifurcation and Chaos 1 (1991) 745–772.
H. B. Keller, Numerical Methods for Two-Point Boundary-Value Problems, Blaisdell, New York, 1968.
J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, 3rd Edition, Springer, New York, 2002.
J. Guckenheimer, B. Meloon, Computing periodic orbits and their bifurcations with automatic differentiation, SIAM J. Sci. Comput. 22 (3) (2000) 951–985.
D. M. Ambrose, J. Wilkening, Computation of time-periodic solutions of the [Benjamin]{}–[Ono]{} equation, J. Nonlinear Sci. 20 (3) (2010) 277–308.
D. M. Ambrose, J. Wilkening, Global paths of time-periodic solutions of the [Benjamin]{}–[Ono]{} equation connecting pairs of traveling waves, Comm. App. Math. and Comp. Sci. 4 (1) (2009) 177–215.
D. M. Ambrose, J. Wilkening, Computation of symmetric, time-periodic solutions of the vortex sheet with surface tension, Proc. Nat. Acad. Sci. 107 (8) (2010) 3361–3366.
M. O. Williams, J. Wilkening, E. Shlizerman, J. N. Kutz, Continuation of periodic solutions in the waveguide array mode-locked laser, Physica D 240 (22) (2011) 1791–1804.
C. G. Broyden, The convergence of a class of double-rank minimization algorithms, [Parts I and II]{}, J. Inst Maths Applics 6 (1970) 76–90, 222–231.
J. Nocedal, S. J. Wright, Numerical Optimization, Springer, New York, 1999.
J. Yu, A local construction of the [Smith]{} normal form of a matrix polynomial, and time-periodic gravity-driven water waves, Ph.D. thesis, University of California, Berkeley (May 2010).
J. Wilkening, Stability of solitary water wave collisions and standing waves, (in preparation).
M. S. Longuet-Higgins, E. D. Cokelet, The deformation of steep surface waves on water. [I]{}. a numerical method of computation, Proc. Royal Soc. A 350 (1976) 1–26.
G. R. Baker, D. I. Meiron, S. A. Orszag, Generalized vortex methods for free-surface flow problems, J. Fluid Mech. 123 (1982) 477–501.
R. Krasny, Desingularization of periodic vortex sheet roll-up, J. Comput. Phys. 65 (1986) 292–313.
G. R. Baker, C. Xie, Singularities in the complex physical plane for deep water waves, J. Fluid Mech. 685 (2011) 83–116.
T. Y. Hou, J. S. Lowengrub, M. J. Shelley, Removing the stiffness from interfacial flows with surface tension, J. Comput. Phys. 114 (1994) 312–338.
H. D. Ceniceros, T. Y. Hou, Dynamic generation of capillary waves, Phys. Fluids 11 (5) (1999) 1042–1050.
T. Y. Hou, J. S. Lowengrub, M. J. Shelley, Boundary integral methods for multicomponent fluids and multiphase materials, J. Comput. Phys. 169 (2001) 302–362.
G. Baker, A. Nachbin, Stable methods for vortex sheet motion in the presence of surface tension, SIAM J. Sci. Comput. 19 (5) (1998) 1737–1766.
A. L. Dyachenko, V. E. Zakharov, E. A. Kuznetsov, Nonlinear dynamics on the free surface of an ideal fluid, Plasma Phys. Rep. 22 (1996) 916–928.
W. Artiles, A. Nachbin, Nonlinear evolution of surface gravity waves over highly variable depth, Phys. Rev. Lett. 93 (2004) 234501.
P. A. Milewski, J.-M. Vanden-Broeck, Z. Wang, Dynamics of steep two-dimensional gravity–capillary solitary waves, J. Fluid Mech. 664 (2010) 466–477.
G. B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.
R. S. Johnson, A modern introduction to the mathematical theory of water waves, Cambridge University Press, Cambridge, UK, 1997.
A. D. D. Craik, The origins of water wave theory, Ann. Rev. Fluid Mech. 36 (2004) 1–28.
A. D. D. Craik, George [Gabriel]{} [Stokes]{} on water wave theory, Ann. Rev. Fluid Mech. 37 (2005) 23–42.
W. Craig, C. Sulem, Numerical simulation of gravity waves, J. Comput. Phys. 108 (1993) 73–83.
Y. W. Lim, P. B. Bhat, V. K. Prasanna, Efficient algorithms for block-cyclic redistribution of arrays, in: IEEE Symposium on Parallel and Distributed Processing, 1996, pp. 74–83.
E. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations [I]{}: Nonstiff Problems, 2nd Edition, Springer, Berlin, 2000.
T. Y. Hou, R. Li, Computing nearly singular solutions using pseudo-spectral methods, J. Comput. Phys. 226 (2007) 379–397.
A. Dutt, L. Greengard, V. Rokhlin, Spectral deferred correction methods for ordinary differential equations, BIT 40 (2) (2000) 241–266.
J. Huang, J. Jia, M. Minion, Accelerating the convergence of spectral deferred correction methods, J. Comput. Phys. 214 (2006) 633–656.
A. T. Layton, M. L. Minion, Implications of the choice of quadrature nodes for [Picard]{} integral deferred corrections methods for ordinary differential equations, BIT Numerical Mathematics 45 (2005) 341–373.
J. Wilkening, Hybrid spectral deferred correction methods that crossover to pure [Picard]{} iteration, Tech. rep., Lawrence Berkeley National Laboratory, (in preparation) (2012).
M. O. Williams, E. Shlizerman, J. Wilkening, J. N. Kutz, The low dimensionality of time-periodic standing waves in water of finite and infinite depth, SIAM J. Appl. Dyn. Syst.(accepted).
L. F. Shampine, Some practical [Runge]{}–[Kutta]{} formulas, Mathematics of Computation 46 (1986) 135–150.
M. Lu, B. He, Q. Luo, Supporting extended precision on graphics processors, Proceedings of the Sixth International Workshop on Data Management on New Hardware (2010) 19–26.
Y. Hida, X. S. Li, D. H. Bailey, The qd package for double-double and quad-double arithmetic, `http://crd-legacy.lbl.gov/~dhbailey/mpdist/` (2003–2012).
J.-M. Vanden-Broeck, Nonlinear gravity-capillary standing waves in water of arbitrary uniform depth, J. Fluid Mech. 139 (1984) 97–104.
B. Chen, P. G. Saffman, Steady gravity-capillary waves on deep water–[I]{}. weakly nonlinear waves, Studies in Appl. Math. 60 (1979) 183–210.
T. J. Bridges, Secondary bifurcation and change of type for three-dimensional standing waves in finite depth, J. Fluid Mech. 179 (1987) 137–153.
B. F. Akers, W. Gao, Wilton ripples in weakly nonlinear model equations, Comm. Math. Sci. 10 (3) (2012) 1015–1024.
L. Bauer, H. B. Keller, E. L. Raiss, Multiple eigenvalues lead to secondary bifurcation, SIAM Review 17 (1) (1975) 101–122.
M. Ioualalen, C. Kharif, Stability of three-dimensional progressive gravity waves on deep water to superharmonic disturbances, Eur. J. Mech. B 12 (1993) 401–414.
M. Ioualalen, A. J. Roberts, C. Kharif, On the observability of finite-depth short-crested water waves, J. Fluid Mech. 322 (1996) 1–19.
P. Deuflhard, A. Hohmann, Numerical Analysis in Modern Scientific Computing, An Introduction, 2nd Edition, Springer, New York, 2003.
J. Wilkening, An infinite branching hierarchy of time-periodic solutions of the [Benjamin]{}–[Ono]{} equationArXiv:0811.4209.
G. G. Stokes, Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change of form, in: Mathematical and physical papers, Vol. 1, Cambridge University Press, 1880, pp. 225–228.
A. D. D. Craik, George [Gabriel]{} [Stokes]{} on water wave theory, Annual Rev. Fluid Mech. 37 (2005) 23–42.
M. S. Longuet-Higgins, M. J. H. Fox, Theory of the almost-highest wave: the inner solution, J. Fluid Mech. 80 (4) (1977) 721–741.
M. S. Longuet-Higgins, M. J. H. Fox, Theory of the almost-highest wave. part 2. matching and analytic extension, J. Fluid Mech. 85 (4) (1978) 769–786.
C. J. Amick, L. E. Fraenkel, J. F. Toland, On the [Stokes]{} conjecture for the wave of extreme form, Acta Math. 148 (1) (1982) 193–214.
D. V. Maklakov, Almost-highest gravity waves on water of finite depth, Euro. J. Appl. Math. 13 (2002) 67–93.
I. S. Gandzha, V. P. Lukomsky, On water waves with a corner at the crest, Proc. R. Soc. A 463 (2007) 1597–1614.
D. G. Crapper, An exact solution for progressive capillary waves of arbitrary amplitude, J. Fluid Mech. 2 (1957) 532–540.
J.-M. Vanden-Broeck, A new family of capillary waves, J. Fluid Mech. 98 (1) (1980) 161–169.
F. Dias, T. J. Bridges, The numerical computation of freely propagating time-dependent irrotational water waves, Fluid Dynamics Research 38 (2006) 803–830.
N. I. Muskhelishvili, Singular Integral Equations, 2nd Edition, Dover, New York, 1992.
A. I. [Van de Vooren]{}, A numerical investigation of the rolling up of vortex sheets, Proc. Royal Soc. London Ser. A 373 (1980) 67–91.
D. I. Pullin, Numerical studies of surface-tension effects in nonlinear [Kelvin]{}–[Helmholtz]{} and [Rayleyigh]{}–[Taylor]{} instabilities, J. Fluid Mech. 119 (1982) 507–532.
M. O. Williams, E. Shlizerman, J. Wilkening, J. N. Kutz, The low dimensionality of time-periodic standing waves in water of finite and infinite depth, SIAM J. Appl. Dyn. Syst.(accepted).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The evidence for a new singlet scalar particle from the 750 GeV diphoton excess, and the absence of any other signal of new physics at the LHC so far, suggest the existence of new coloured scalars. To study this possibility, we propose a supersymmetry inspired simplified model, extending the Standard Model with a singlet scalar and with heavy scalar fields carrying both colour and electric charges – the ‘squarks’. To allow the latter to decay, and to generate the dark matter of the Universe, we also add a neutral fermion to the particle content. We show that this model provides a two-parameter fit to the observed diphoton excess consistently with cosmology, while the allowed parameter space is bounded by the consistency of the model. In the context of our simplified model this implies the existence of other supersymmetric particles accessible at the LHC, rendering this scenario falsifiable. If this excess persists, it will imply a paradigm shift in assessing supersymmetry breaking and the role of scalars in low scale physics.'
address:
- 'Dipart. di Fisica Teorica, Università di Trieste, Strada Costiera 11, I-34151 Trieste, Italy and INFN, Sezione di Trieste, Via Valerio 2, I-34127 Trieste, Italy'
- 'NICPB, Rävala 10, Tallinn 10143, Estonia'
- |
INFN, Sezione di Roma, c/o Dipart. di Fisica, Università di Roma “La Sapienza",\
Piazzale Aldo Moro 2, I-00185 Rome, Italy
- 'Institute of Physics, University of Tartu, Estonia'
author:
- 'E. Gabrielli'
- 'K. Kannike'
- 'B. Mele'
- 'M. Raidal'
- 'C. Spethmann'
- 'H. Veermäe'
bibliography:
- '750GeVdiphoton\_v3.bib'
title: A SUSY Inspired Simplified Model for the 750 GeV Diphoton Excess
---
Introduction
============
The discovery of the Higgs boson at the LHC [@Aad:2012tfa; @Chatrchyan:2012xdj] completed the observation of all fundamental degrees of freedom predicted by the Standard Model (SM) of particle interactions. Nevertheless, it is widely believed that the SM suffers from a series of shortcomings, related to the stability of the electroweak scale and the absence of a candidate for the dark matter (DM) of the Universe, for instance. Solutions to these problems require extending the present theoretical framework to include new degrees of freedom, possibly relevant at the energy scales probed by colliders in the near future.
The LHC Run 2 at 13 TeV collision energy provides the potential to probe physics at shorter distances compared to LHC Run 1 at 7 TeV and 8 TeV. The exploration has just started with about 4 fb$^{-1}$ of integrated luminosity delivered to the ATLAS and CMS experiments, beginning in June 2015. Searching for two-particle resonances is an especially adequate way to look for new physics manifestations when new thresholds in collision energies are reached. Both ATLAS and CMS are presently analysing the new data sets, and trying to get the most out of the very first run at 13 TeV.
In a recent CERN seminar both ATLAS [@ATLAS] and CMS [@CMS] presented a photon pair excess with an invariant mass at about 750 GeV, with a [*local*]{} significance varying (depending on the narrow- or wide-width assumption) in the range 2.6 to 3.9 $\sigma$. The signal also exhibits some compatibility with the photon-pair studies of Run 1 data by both ATLAS and CMS. Assuming that the observed diphoton excess is due to a new resonance, CMS provides a combination of Run 1 plus Run 2 data for its production cross section times branching fraction into photons to be $4.5\pm 1.9$ fb [@CMS]. The corresponding ATLAS result for 13 TeV was estimated to be $10.6\pm2.9$ fb [@DiChiara:2015vdm]. While further data will be needed in order to clarify whether the observed excess is robust, it is exciting to assume that the di-photon excess is really pointing to the existence of new physics below a scale of 1 TeV, and to try to determine which kind of SM extension can predict such an effect. Presently, no anomaly in any other final state has been detected [@ATLAS; @CMS], which severely restricts any realistic explanation of the excess.
The most natural interpretation of the observed diphoton excess is due to the decays of a singlet scalar $S$ into photons, $S\to\gamma\gamma$ [@DiChiara:2015vdm; @Franceschini:2015kwy; @Knapen:2015dap; @Buttazzo:2015txu; @Backovic:2015fnp; @Mambrini:2015wyu; @Gupta:2015zzs; @Ellis:2015oso; @McDermott:2015sck; @Dutta:2015wqh; @Cao:2015pto; @Kobakhidze:2015ldh; @Martinez:2015kmn; @Bian:2015kjt; @Chakrabortty:2015hff; @Falkowski:2015swt; @Bai:2015nbs].[^1] The existence of light scalars much below the cut-off scale of the SM (such as the Planck scale) requires some mechanism to protect their masses against radiative corrections from the cut-off scale. By far the most popular solution to the hierarchy problem is supersymmetry. However, in recent years supersymmetry has lost some of its appeal because of the severe experimental constraints from the LHC [@ATLAS; @CMS].
In the context of the diphoton excess the conventional supersymmetric models, such as the Minimal Supersymmetric Standard Model (MSSM), have several shortcomings. Firstly, the excess cannot be explained within the MSSM alone [@DiChiara:2015vdm; @Buttazzo:2015txu; @Angelescu:2015uiz; @Gupta:2015zzs], and the framework must be extended to accommodate the new signal. This points into the direction of the Next-to-Minimal Supersymmetric Standard Model (NMSSM) rather than the MSSM. Secondly, most of the diphoton excess studies so far have assumed the existence of heavy coloured vectorlike fermions that, at one loop, induce singlet scalar couplings to gluons and photons [@Franceschini:2015kwy; @Knapen:2015dap; @Pilaftsis:2015ycr; @Buttazzo:2015txu; @Angelescu:2015uiz; @Gupta:2015zzs; @Ellis:2015oso; @McDermott:2015sck; @Kobakhidze:2015ldh; @Martinez:2015kmn; @No:2015bsn; @Chao:2015ttq; @Fichet:2015vvy; @Curtin:2015jcv; @Falkowski:2015swt]. Such coloured fermions, however, do not exist in the particle content of any supersymmetric extension of the SM. In addition, new coloured fermions are severely constrained by LHC searches and must be very heavy [@ATLAS; @CMS]. Therefore, extending the non-minimal supersymmetric models further with charged and coloured vectorlike fermions implies that the model that is supersymmetrised is not the SM. The model becomes unnecessarily complicated without any obvious need for these specific new particles.
In this work we argue that the diphoton excess hints at the existence of relatively light coloured and charged scalars. First, these particles, [*the squarks*]{}, do exist in any supersymmetric extension of the SM and there is no need to extend the model with new [*ad hoc*]{} particles. Second, the LHC constraints on coloured scalar masses are much less stringent than on coloured fermions, such as gluinos.[^2] These arguments allow for relatively light squarks in the loops generating $gg\to S\to\gamma\gamma$ that are potentially observable at the LHC in coming years, rendering this scenario directly testable. Third, one of the favourable feature of supersymmetric models is the existence of dark matter (DM) that comes for free as the lightest neutral superpartner of gauge bosons and scalars. We use these arguments to address the LHC diphoton excess.
Motivated by the above mentioned good features of supersymmetric theories, we propose a supersymmetry inspired simplified model that is able to explain the diphoton excess consistently with all other LHC results and with the existence of DM. Although minimal by construction, and therefore not supersymmetric by itself, this model uses the particle content of the NMSSM and can be embedded into the latter.[^3] Therefore, the mass spectrum of this type of supersymmetric models must be very different from the ones predicted by simple supersymmetry breaking scenarios.
We study this effective model carefully and show that the requirement of a physical, charge and colour conserving vacuum restricts the allowed mass parameters to be constrained from above, rendering the model testable or falsifiable by collider experiments. In the context of the simplified model this statement means that the effective theory breaks down and the new supersymmetric degrees of freedom must appear to cure the model. Therefore, if verified, our framework [*predicts*]{} the existence of new supersymmetric particles at the reach of next collider runs. Thus the di-photon excess may change our present understanding of the supersymmetry breaking patterns and the role of scalars in supersymmetric models.
![Leading order contributions to the main decay modes of $S$.[]{data-label="fig:Sdec"}](Stofinal)
The SUSY Inspired Simplified Model
==================================
We construct a supersymmetry inspired simplified model that produces a narrow scalar resonance $gg \to S \to \gamma \gamma$. This resonance is necessarily a singlet under the SM gauge group. As shown in Fig. \[fig:Sdec\], its interactions with photons and gluons are therefore induced at loop level by another scalar field $\tilde{Q}$ that is coloured and carries hypercharge, which we assume to be $q_{\tilde{Q}} = 2/3$. $\tilde{Q}$ is therefore identical to the well known right-handed up-type squark that transforms in the fundamental representation of $SU(3)$ but is a singlet under $SU(2)$. To avoid any conflict with LHC phenomenology and cosmological and astrophysical observations, the squark $\tilde{Q}$ is required to be unstable. As in the supersymmetric extension of the SM, we take it to decay into a quark and a neutralino-like fermion $\chi_{0}$, which is the dark matter candidate in our scenario.
Thus we consider a minimal extension of the SM with the real singlet scalar field $S$, three generations of ‘squarks’ $\tilde{Q_{i}}$ and the ‘neutralino’ $\chi_{0}$. Obviously, the model with this particle-content is by itself not supersymmetric, but requires embedding into a supersymmetric theory. The general Lagrangian for the given particle sector contains the following terms $$\begin{aligned}
{\cal L}_{\rm kin} & = |D_{\mu} \tilde{Q}_{i}|^{2} + \frac{1}{2}(\partial_{\mu} S) (\partial^{\mu} S)
\\
& - M^2_{\tilde{Q}} |\tilde{Q_{i}}|^{2} - \frac{1}{2} M^2_{S0} S^{2}
+ \frac{1}{2} \bar{\chi}_{0}(\slashed{\partial} - m_{\chi_{0}})\chi_{0}, \notag
\\
{\cal L}_{\rm dec}
& = \frac{1}{2} y_{\chi} S \chi^{T}_{0}\chi_{0}
+ (y_{i} \tilde{Q_{i}}^{\dagger}\chi^{T}_{0} U_{iR} + \text{h.c.}),
\\
\label{Lag_scalar}
{\cal L}_{\rm scalar}
& =
- \mu_{\tilde{Q}} S |\tilde{Q_{i}}|^{2}
- \frac{\mu_{S}}{3} S^{3}
- \frac{\lambda_{S}}{4} S^{4}
\\
&
- \lambda_{S\tilde{Q}} S^{2} |\tilde{Q_{i}}|^{2}
- \lambda_{\tilde{Q}} |\tilde{Q_{i}}|^{2} |\tilde{Q_{j}}|^{2} \notag
\\
& - \lambda'_{\tilde{Q}} (\tilde{Q}_{i}^{\dag} \tilde{Q}_{j}) (\tilde{Q}_{j}^{\dag} \tilde{Q}_{i}), \notag
\\
{\cal L}_{H}
& =
- (\mu_{H} S + \lambda_{HS} S^{2}) H^{\dag} H \\
& - \lambda_{\tilde{HQ}} (\tilde{Q_{i}}^{\dag} \tilde{Q_{i}}) (H^{\dag} H), \notag\end{aligned}$$ with the covariant derivative $D_{\mu}=\partial_{\mu}+g_sT^a G^a_{\mu} +e q_{\tilde{Q}} A_{\mu}$, where $G^a_{\mu}$ is the gluon and $A_{\mu}$ is the photon field, and we sum over the generation indices $i$ for $\tilde{Q}_{i}$. We assume a flavour symmetry to forbid any other terms involving $\tilde{Q}_{i}$. Eq. contains the interactions among the two scalars, most importantly the first term with the coupling $\mu_{\tilde{Q}}$ which has dimensions of mass.
We require that $M_{\chi}, M_{\tilde{Q}} > M_{S}/2 $ to forbid tree level decays of the $S$ resonance. Also, instability of $\tilde{Q}$ dictates that $M_{\tilde{Q}} > M_{\chi}$. This choice has the benefit of providing a dark matter candidate – the neutralino $\chi_{0}$. It has recently been shown [@Backovic:2015fnp; @Mambrini:2015wyu] that in such a setup the observed amount of DM can be produced from thermal freeze-out analogously to the MSSM. Thus the DM in our scenario is a thermal relic in the form of the stable neutralino $\chi^0.$ To satisfy the observation $\Omega_{\rm DM} h\sim 0.1,$ the neutralino mass must be ${\cal O}(300)$ GeV [@Backovic:2015fnp; @Mambrini:2015wyu] , implying a somewhat compressed spectrum. The latter implies that the model is not severely constrained by the LHC searches for squark pair production for the final state of two jets and missing energy [@Aad:2015iea].
As we have already commented, this model does not fit into the MSSM but requires some extended supersymmetric model, the NMSSM being the simplest of them. We note that the mass spectrum of such a model must feature light scalars while the gluino must be heavy to comply with the LHC bound. Since our study is phenomenological, we just assume this supersymmetry breaking pattern.
Conditions for a Physical Vacuum
================================
We consider the conditions for the vacuum of the model not to break colour and electric charge. We need to ensure the following:
1. The potential is bounded from below in the limit of large field values.
2. The squarks $\tilde{Q}_{i}$ do not get VEVs, which would break colour and electric charge. The true vacuum should be at $S = 0$ and $\tilde{Q} = 0$, therefore the potential has to be positive everywhere else,[^4] $$V(S \neq 0, \tilde{Q} \neq 0) > 0.$$
3. $S$ does not get a VEV: a non-zero VEV for $S$ would shift the mass of $S$ away from $M_{S}$.[^5]
The potential must be bounded from below in order for a finite minimum of potential energy to exist. In the limit of large field values, we can ignore the dimensionful terms in the scalar potential. The full bounded below conditions can be found via co-positivity constraints on the quartic part of the scalar potential [@Kannike:2012pe]: $$\begin{aligned}
& \lambda_{S} > 0, \quad \lambda_{\tilde{Q}} + \theta(-\lambda'_{\tilde{Q}}) \lambda'_{\tilde{Q}} > 0, \quad \lambda_{H} > 0, \\
& \bar{\lambda}_{SQ} \equiv 2 \sqrt{\lambda_{S} [\lambda_{\tilde{Q}} + \theta(-\lambda'_{\tilde{Q}}) \lambda'_{\tilde{Q}}]} + \lambda_{S\tilde{Q}} > 0, \\
& \bar{\lambda}_{HQ} \equiv 2 \sqrt{\lambda_{H} [\lambda_{\tilde{Q}} + \theta(-\lambda'_{\tilde{Q}}) \lambda'_{\tilde{Q}}]} + \lambda_{H\tilde{Q}} > 0, \\
& \bar{\lambda}_{HS} \equiv 2 \sqrt{\lambda_{H} \lambda_{S}} + \lambda_{HS} > 0, \\
& \lambda_{HS} \sqrt{\lambda_{\tilde{Q}} + \theta(-\lambda'_{\tilde{Q}}) \lambda'_{\tilde{Q}}} + \lambda_{H\tilde{Q}} \sqrt{\lambda_{S}} + \lambda_{S\tilde{Q}} \sqrt{\lambda_{H}}
\notag \\
& + 2 \sqrt{[\lambda_{\tilde{Q}} + \theta(-\lambda'_{\tilde{Q}}) \lambda'_{\tilde{Q}}] \lambda_{S} \lambda_{H}}
\\
& + \sqrt{\bar{\lambda}_{SQ} \bar{\lambda}_{HQ} \bar{\lambda}_{HS}} > 0, \notag\end{aligned}$$ where $\theta$ is the Heaviside step function. The conditions can be satisfied by taking $\lambda_{S} \geq 0$, $\lambda_{\tilde{Q}} \geq 0$, $\lambda'_{\tilde{Q}} \geq 0$, $\lambda_{S\tilde{Q}} \geq 0$, $\lambda_{HS} \geq 0$, $\lambda_{H\tilde{Q}} \geq 0$. The stationary point equations for the new particles are then $$\begin{aligned}
0 &= \mu_{\tilde{Q}} |\tilde{Q}_{i}|^{2} + (M_{S}^{2} + 2 \lambda_{S\tilde{Q}} |\tilde{Q}_{i}|^{2}) S \label{eq:min:S} \\
& + \mu_{S} S^{2} + \lambda_{S} S^{3}, \notag \\
0 &= |\tilde{Q}_{i}| (M_{\tilde{Q}}^{2} + 2 \lambda_{\tilde{Q}} |\tilde{Q}_{i}|^{2} + \mu_{\tilde{Q}} S + \lambda_{S\tilde{Q}} S^{2}). \label{eq:min:Q}
\end{aligned}$$ If $\tilde{Q}_{i} = 0$, we need $$\mu_{S}^{2} < 4 \lambda_{S} M_{S}^{2}$$ for $S$ not to get a VEV. We will take $\mu_{S} \simeq 0$ to get the largest allowed parameter space for the diphoton signal. However, we note that a small but non-zero $\mu_{S}$ could always be generated at two loops.
$S$ and $\tilde{Q}_{i}$ could also get non-zero VEVs simultaneously. We need to forbid this to prevent a coloured vacuum. The forbidden part of the parameter space is found by requiring that the vacua where $S$ and $\tilde{Q}_{i}$ have non-zero VEV – if they exist – are local minima of the potential, that is, $V > 0$.
Note that the bound does not depend on the number of flavours, which cancels out in the minimisation equations. Especially, the $\lambda'_{\tilde{Q}}$ term does not affect the result, since its minimal value is zero. Also, it is plausible that the bound will not be weakened by much if the flavor symmetry is abandoned.
To fit the diphoton signal we need a large $\mu_{\tilde{Q}}$ that tends to destabilise the SM vacuum. This effect can be countered with large quartic couplings. In Fig. \[fig:XSecPlot\] we show the forbidden region on $\mu_{\tilde{Q}}$ vs. $M_{\tilde{Q}}^{2}$ plane with gray colour for the least constraining choice $\lambda_{\tilde{Q}} = \lambda_{S} = \lambda_{S\tilde{Q}} = 4 \pi$. In the context of this effective model, that must be embedded into a supersymmetric model, the appearance of non-perturbative couplings signals the break-down of the effective model. This implies that the supersymmetric particles of the full model must appear below the scale given by this constraint.
![$\sigma(pp \to S) \times \text{BR}(S \to \gamma \gamma)$ at the 13 TeV LHC. The colored regions correspond to 4.5$\pm$1.9 fb (inner region) and 4.5$\pm$ 3.8 fb (outer region) corresponding to the $1\sigma$ and $2\sigma$ regions for $N_f = 3$ degenerate squark generations. The horizontal axis shows the mass of the colored scalar particle $\tilde{Q}$ and the vertical axis the trilinear $S\tilde{Q}\tilde{Q}$ coupling. The grey shaded region is forbidden by the presence of colour symmetry breaking assuming $\lambda_{\tilde{Q}} = \lambda_{S} = \lambda_{S\tilde{Q}} = 4 \pi$.[]{data-label="fig:XSecPlot"}](XsecPlot){width="45.00000%"}
Event Rates
===========
We choose the mass of the singlet to be on the resonance, $M_{S} = 750$ GeV. At this energy scale $\alpha_s(M_{S}) = 0.0894(31)$ [@Chatrchyan:2013txa], whereas for $\alpha = 1/137$ we use the zero momentum value. From the CMS [@CMS] we know that $\sigma (pp \to S \to \gamma \gamma) \simeq 4.5~\text{fb}$.
The partial decay widths of the singlet $S$ into two photons and into two gluons are [@Gunion:1989we; @Djouadi:2005gj] $$\begin{aligned}
\label{eq:partial_G}
\Gamma (S \to \gamma \gamma)
&= \frac{ \alpha^{2} M_{S}^{3}\mu_{\tilde{Q}}^{2}}{1024 \pi^{3}M_{\tilde{Q}}^{4}} N_{f}^{2} N_{c}^{2} q_{\tilde{Q}}^{4} \left| A_{0}(\tau) \right|^{2},
\\
\Gamma (S \to g g)
&= \frac{ \alpha_{s}^{2} M_{S}^{3} \mu_{\tilde{Q}}^{2}}{512 \pi^{3} M_{\tilde{Q}}^{4}} N_{f}^{2} \left| A_{0}(\tau) \right|^{2},\end{aligned}$$ respectively. $N_c = 3$ denotes the dimension of the representation for the squarks and $N_f$ the number of squark flavors. The scalar loop function is given by [@Gunion:1989we] $$A_{0}(\tau) = \tau(1 - \tau f(\tau)),$$ with $\tau = 4 M_{\tilde{Q}}^{2}/M_{S}^{2}$, and the universal scaling function is $$\! f(\tau) =
\begin{cases}
\arcsin^{2} \sqrt{1/\tau}
& \tau \geq 1 ,
\\
-\left( {\rm arccosh} \sqrt{1/\tau} - i \pi/2 \right)^{2}
& \tau < 1.
\end{cases}$$
The cross section for producing the diphoton signal via the decay of $S$ in the narrow width approximation is $$\sigma(gg \to S \to \gamma \gamma) = \sigma (gg \to S) \, \text{BR}(S \to \gamma \gamma),$$ where the production cross section is related to the decay width into gluons by $$\label{eq:partonicGGF}
\sigma(gg \to S) = \frac{\pi^{2}}{8 M_{S}} \Gamma (S \to g g) \delta(\hat s - M_{S}^{2}).$$ Taking into account that $\Gamma (S \to \gamma \gamma) \ll \Gamma (S \to g g) \simeq \Gamma_{S}$, the branching ratio reads $$\text{BR}(S \to \gamma \gamma)
\simeq \frac{1}{2} \frac{\alpha^{2}}{\alpha_{s}^{2}} N_{c}^{2}q_{\tilde{Q}}^{4}
\simeq 0.58\%, $$ where we used $\alpha_s(M_{S}) \simeq 0.09$, $\alpha \simeq 1/137$ and assumed the up type squark with the charge $q_{\tilde{Q}} = 2/3$. We remark that if the dominant decay mode of $S$ is $S \to g g$ as assumed here, the cross section for the resonant production of diphotons by gluon-gluon fusion is approximately independent of the details of the strong interaction since $$\label{eq:partonicGGFapprox}
\sigma(gg \to S \to \gamma \gamma)
\simeq \frac{\pi^{2}}{8 M_{S}} \Gamma (S \to \gamma\gamma)\delta(\hat s - M_{S}^{2}).$$ At the level of precision considered here, we assume that this cancellation also holds if higher orders corrections in $\alpha_s$ are taken into account.
To calculate the $S$ resonance production cross section at the LHC, we integrate Eq. numerically using the MSTW parton distribution function (pdf) set [@Martin:2009iq] $$\begin{aligned}
\sigma(gg \to S \to \gamma \gamma) = \frac{\pi^{2}}{8 M_{S}^3} I_{\rm pdf} \Gamma (S \to \gamma\gamma) ,\end{aligned}$$ where $\sqrt{s} = 13\,{\rm{\,TeV}}$ is the center of mass energy of LHC proton-proton collisions, and $$\begin{aligned}
I_{\rm pdf}
= \int_{M_S^2/s}^1 \; \frac{\mathrm{d}x}{x} \; \bar{g}(x) \; \bar{g}\left(\frac{M_S^2}{sx}\right)
\approx 5.8,\end{aligned}$$ is the dimensionless pdf integral evaluated at $\sqrt{s} = 13\,{\rm{\,TeV}}$. Here $g(x, M_S) = \bar{g}(x, M_S)/x$ is the pdf of the gluon at momentum fraction $x$ evaluated at the scale $M_S=750$ GeV.
To reproduce the observed signal, we find that the partial decay width to photons is $$\Gamma(S \to \gamma \gamma) \approx (0.68 \pm 0.28) \mbox{ MeV}.$$ The parameter space that reproduces the observed decay width for $N_f = 3$ generations is depicted in Fig. \[fig:XSecPlot\]. Accounting for unitarity and preserving color and charge symmetries, it follows, that within the $1\sigma$ band the data favors $N_f \geq 2$ generations of light squarks with masses below $M_{\tilde{Q}} \lesssim 800{\rm{\,GeV}}$ and a relatively large coupling to the scalar $S$ of $\mu_{\tilde{Q}} \gtrsim 2{\rm{\,TeV}}$. As was also noted in [@Gupta:2015zzs] in the context of a different model, we similarly find that the signal can not be reproduced by a single generation of light squarks within $1\sigma$.
The most important result evident in Fig. \[fig:XSecPlot\] is that the allowed parameter space of this effective model is bounded to a small region by the di-photon excess and by the consistency of the effective model. This implies that new particles must be present in Nature at the scale ${\cal O}(1)$ TeV.
Effective Field Theory Approach
===============================
We turn to analyse our scenario in terms of the effective Lagrangian approach. In the case the squark is heavier than the singlet $S$, the latter can acquire effective couplings with photons and gluons by integrating out the squark field. For generic squarks this corresponds to an effective Lagrangian $${\mathcal{L}}_{\rm eff}
= \frac{1}{\Lambda_{\gamma}} S \,F^{\mu \nu}F_{\mu \nu} + \frac{1}{\Lambda_{G}} S \,G^{a \mu \nu}G^a_{\mu \nu},
\label{Leff}$$ where $F_{\mu \nu}$ and $G^a_{\mu \nu}$ are the field strengths of the SM gauge fields, while $\Lambda_{i}$ denote the effective scale of the non-renormalizable interaction. In the simplified model considered above, the condition $M_{\tilde{Q}} \gg M_S$ cannot hold for physically allowed parameters, see Fig. \[fig:XSecPlot\]. Thus the rates obtained by using the effective Lagrangian need to explicitly account for the loop function $A_{0}$ (i.e. non-trivial scaling of $\Lambda_{i}$) to get accurate results, even if the expansion $E/\Lambda_{i}$ naively seems to be well defined.
Nevertheless, in this context the formalism of effective Lagrangian approach is very useful, since it allows to capture in a model-independent way the crucial information concerning the underlying dynamics responsible of generating the effective coupling. If the effective operator is generated perturbatively by integrating out particles running in the loops, its coefficient has the general form $$\begin{aligned}
\label{eff_scale}
\frac{1}{\Lambda_{i}} = \frac{\alpha_{i}}{4\pi} \frac{N_{e} g_{\tilde{Q}S}}{m_{\tilde{Q}}} C_{i},\end{aligned}$$ where $C_{i}$ is an $\mathcal{O}(1)$ factor originating from loop integrals and $g_{S}$ denotes an effective coupling between $S$ and the mediators and $N_{e}$ is the effective number of degrees of freedom running in the loops. The cross section obtained from the effective Lagrangian is roughly $$\sigma(pp \to S \to \gamma \gamma)
\approx \frac{\alpha^2 N_e^{2}g_{S}^{2}}{512\, m_{\tilde{Q}}^{2}}.$$ Fixing its value to $5~\text{fb}$ suggests that in order to reproduce the required phenomenology of the observed diphoton excess, the effective coupling defined by Eq. should satisfy $$\begin{aligned}
N_e g_{S} \approx 70 \times \frac{m_{\tilde{Q}}}{M_{S}}\, .\end{aligned}$$ As we can see, this would require necessarily a $g_{S} \simeq \mathcal{O}(10)$ if $N_e\sim {\cal O}(1)$. Then, from these results one can naively guess that this large number of $g_{S}$ points towards either strong dynamics or a relatively large number of degrees of freedom in the loops. This is, indeed, justified conclusion if one considers vector-like fermions running in the loop [@Franceschini:2015kwy; @Knapen:2015dap; @Pilaftsis:2015ycr; @Buttazzo:2015txu; @Angelescu:2015uiz; @Gupta:2015zzs; @Ellis:2015oso; @McDermott:2015sck; @Kobakhidze:2015ldh; @Martinez:2015kmn; @No:2015bsn; @Chao:2015ttq; @Fichet:2015vvy; @Curtin:2015jcv; @Falkowski:2015swt; @Aloni:2015mxa], where $g_S$ coincides with the corresponding fermion Yukawa coupling to the scalar resonance. In this respect, we qualitatively agree with the effective model approach conclusions of Ref. [@Aloni:2015mxa] on the large production rates.
However, as we have shown with the present simplified model, when scalar fields are propagating in the loop, the above conclusions do not hold anymore. The coupling $g_S$ can be made naturally (and consistently) very large, even in the framework of weakly coupled field theories, being related to the ratio $g_{S} = \mu_{\tilde{Q}}/m_{\tilde{Q}} \sim\mathcal{O}(10)$. This is the advantage of having [*soft*]{} coupling $\mu_{\tilde{Q}}$ in theories with scalars. However, this requires that the scalar resonance should not get a VEV or equivalently it should not be a Higgs-like particle. We have seen that constraints from the colour-charge breaking minima could limit the ratio $\mu_{\tilde{Q}}/m_{\tilde{Q}}$. This implies that the present simplified model breaks down and Eq. does not correspond to the scale in Eq. any more. The correct interpretation would require the knowledge of the full supersymmetric theory.
Discussion and Conclusions
==========================
We have shown that the recently claimed evidence for the 750 GeV diphoton excess at the LHC can actually, contrary to the general opinion, favour supersymmetry in Nature. However, the corresponding supersymmetric theory must contain a singlet in addition to the SM particle content, and the mass spectrum of the sparticles must be rather unusual featuring several light scalars while the gluino must be heavy to satisfy the LHC constraints.
To study the diphoton excess we have presented a simplified model that captures the required properties of the supersymmetric theory it is to be embedded in. As the result, we have shown that the NMSSM-like particle content is sufficient to generate large enough $gg\to S$ and $S\to \gamma\gamma$ processes at loop level to explain the observations. In particular, the coloured scalars in the loops have an advantage over the fermions to produce the needed large signal because of the possibly large dimensionful coupling $\mu_{\tilde{Q}}.$ We have also shown that the requirement of a colour and charge conserving vacuum constrains the parameter space of this scenario so that the model is testable. In the context of the simplified model, that by itself is not supersymmetric, this implies that the model breaks down at rather low energy where new superpartners of the complete supersymmetric model must appear to save physics. The concrete prediction of our scenario is the existence of relatively light squarks which should be searched for at the LHC.
We conclude that, if this scenario will turn out to be the explanation of the diphoton excess, supersymmetry, indeed, was ‘just around the corner.’ However, to study the full the model and its precise properties would require more discoveries at the LHC or at the future 100 TeV collider.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thank Luca Marzola and Stefano Di Chiara for useful discussions. This work was supported by the grants IUT23-6, PUT716, PUT799 and by the EU through the ERDF CoE program.
References {#references .unnumbered}
==========
[^1]: Solutions with pseudoscalars have also been considered in [@DiChiara:2015vdm; @Pilaftsis:2015ycr; @Low:2015qep; @Higaki:2015jag; @Molinaro:2015cwg; @Becirevic:2015fmu].
[^2]: The only exception are strongly coupled scalar diquarks [@Ma:1998pi], exotic scalars coupled to two valence quarks, that should produce quark-quark resonances at the LHC and which masses are, therefore, constrained to be above 6 TeV [@Khachatryan:2015dcf] The models with coloured scalars in the loop presented in Ref. [@Knapen:2015dap] are based on diquarks that are not superpartners of quarks. Also, their model does not contain dark matter candidates.
[^3]: Alternatively, the singlet could be a sgoldstino [@Petersson:2015mkr; @Bellazzini:2015nxw; @Demidov:2015zqn], the superpartner of supersymmetry breaking goldstone.
[^4]: The Higgs portal couplings are already strongly constrained [@Falkowski:2015swt] and we neglect them for phenomenological reasons. We also assume that $\mu_{H} \simeq 0$ to prevent a large decay width of $S$ into Higgs bosons.
[^5]: A VEV for $S$ would also generate large contribution to the mass of the squark, which would need to be fine-tuned.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We consider non-selfadjoint operator algebras ${{\mathfrak{L}(G,\lambda)}}$ generated by weighted creation operators on the Fock-Hilbert spaces of countable directed graphs $G$. These algebras may be viewed as noncommutative generalizations of weighted Bergman space algebras, or as weighted versions of the free semigroupoid algebras of directed graphs. A complete description of the commutant is obtained together with broad conditions that ensure the double commutant property. It is also shown that the double commutant property may fail for ${{\mathfrak{L}(G,\lambda)}}$ in the case of the single vertex graph with two edges and a suitable choice of left weight function $\lambda$.'
address:
- 'Department of Mathematics & Statistics, University of Guelph, Guelph, ON, Canada N1G 2W1'
- 'School of Mathematics & Statistics, University College Dublin, Belfield, Dublin 4, Ireland'
- 'Department of Mathematics & Statistics, Lancaster University, Lancaster, U.K., LA1 4YF'
author:
- 'David W. Kribs'
- 'Rupert H. Levene'
- 'Stephen C. Power'
bibliography:
- 'myrefs.bib'
title: Commutants of Weighted Shift Directed Graph Operator Algebras
---
Introduction
============
For over two decades, operator algebras associated with directed graphs and their generalizations have received intense interest in the operator algebra and mathematics community. This class of algebras includes many interesting examples, often with connections to different areas, such as dynamical systems, and at the same time is sufficiently broad that results for them have given insights to the general theory of operator algebras. The most fundamental non-selfadjoint algebras in this class are the tensor algebras [@muhlysolel; @popescu] and free semigroupoid algebras of directed graphs [@katsouliskribs; @kribspower; @kribssolel], including free semigroup algebras [@davidsonpitts; @kennedy]. Each of these is generated by creation operators on a Fock-type Hilbert space defined by the graph, and there is now an extensive body of work for these algebras. In this paper we consider weighted creation operator generalizations, in the weak operator topology (WOT) closed setting, and we investigate their algebraic structure. The resulting [*weighted shift directed graph algebras*]{} ${{\mathfrak{L}(G,\lambda)}}$ may be viewed as the minimal generalization of two different classes of non-selfadjoint algebras: the free semigroupoid algebras of directed graphs on the one hand, and on the other, the classical unilateral weighted shift algebras associated with single variable weighted Bergman spaces.
The paper is organized as follows. In the next section we introduce the notation $\lambda$, $\rho$, for certain left and right weight functions for the path semigroupoid of a directed graph $G$, and define their associated weighted creation operators (which need not be bounded) and their respective operator algebras, ${{\mathfrak{L}(G,\lambda)}}$ and ${{\mathfrak{R}(G,\rho)}}$, on the Fock space $\H_G$. In the subsequent section we investigate the structure of the commutant algebra ${{\mathfrak{L}(G,\lambda)}}'$ and obtain its characterization under the natural condition (left-boundedness of $\lambda$) that all the weighted left creation operators are bounded. In the proof we identify a simple commuting square condition that relates the left weight $\lambda$ to a particular right weight $\rho$ which is relevant to the commutant, and we exploit this to show that ${{\mathfrak{L}(G,\lambda)}}' ={{\mathfrak{R}(G,\rho)}}$ for this right weight. In the fourth section we investigate the double commutant ${{\mathfrak{L}(G,\lambda)}}''$ and obtain broad conditions which ensure the double commutant property ${{\mathfrak{L}(G,\lambda)}}''={{\mathfrak{L}(G,\lambda)}}$.
A range of illuminating examples is also given. In particular, for the single vertex graph with two edges it is shown that there exist left-bounded weights $\lambda$ for which ${{\mathfrak{L}(G,\lambda)}} '' = \B(\H_G)$. On the other hand, for the directed $2$-cycle graph, with two vertices and two edges, necessary and sufficient conditions are obtained for the double commutant property.
Our focus here is on the analysis of generalized weighted shifts and the non-selfadjoint operator algebras they generate, in a setting that embraces both commutative and non-commutative versions, and is built upon the contemporary directed graph operator algebra framework. In fact the first foray in this direction for single vertex directed graphs gave sufficient conditions for the determination of the commutant and for reflexivity [@kribs04], the basic general goals being to extend results from the single variable commutative case and to expose new phenomena in the non-commutative directed graph setting. Our concern in the present paper is to characterize commutants for the left and right algebras by identifying explicit conditions at the level of weighted graphs. It would be interesting to connect this double commutant investigation with the recent work [@marcouxmastnak] on a general double commutant theorem for non-selfadjoint algebras, and with recent work on weighted Hardy algebras of correspondences [@dor-on; @muhlysolel-weighted].
We leave the natural problems of invariant subspace structure and reflexivity for these algebras for investigation elsewhere. It should be possible to identify a large class of these algebras as being reflexive, and in doing so, extend results from the case of weighted Bergman spaces [@Shields74] and partial results from the weighted free semigroupoid algebra case [@kribs04]. Additionally, non-reflexive examples have not yet been constructed in the non-commutative case. This should also be possible with extended notions of strictly cyclic weighted shifts to our setting.
Weighted Shift Directed Graph Algebras
======================================
Let $G$ be a countable directed graph with edge set $E(G)=\{e,f,\ldots\}$ and vertex set $V(G)=\{x,y,\ldots\}$. We will write $G^+=\{ u,v,w,\ldots \}$ for the set of finite paths in $G$, including the vertices regarded as paths of length $0$. Note that if $G$ is finite (by which we mean that both $V(G)$ and $E(G)$ are finite), then the set $\{w\in G^+\colon |w|< k\}$ is finite for each $k\ge1$, where $|w|$ denotes the length of a path $w$. We write $s(w)$ and $r(w)$ for the source and range vertices of a path $w$; in particular, if $x\in V(G)$, then $r(x)=x=s(x)$. We will also take a right to left orientation for path products, so that $w = r(w) w s(w)$ for all $w\in G^+$, and for $v,w\in G^+$ we have $wv\in G^+$ if and only if $s(w)=r(v)$.
To each such graph $G$ we associate the Hilbert space $\H_G=\ell^2(G^+)$, called the Fock space of $G$, with canonical orthonormal basis $\{ \xi_v : v\in G^+ \}$. The vectors $\xi_x$ for $x\in V(G)$ are called vacuum vectors. The left creation operators on $\H_G$ are the partial isometries defined as follows: $L_w \xi_v = \xi_{wv}$ whenever $wv\in G^+$, and $L_w \xi_v = 0$ otherwise. (These operators may also be viewed as generated by the left regular representation of the path semigroupoid of the graph.) Pictorially, as an accompaniment to the directed graph, one can view the actions of the generators $L_e$ as tracing out downward tree structures that lay out the basis vectors for $\H_G$. One tree is present for each vertex $x$ in $G$, with the top tree vertex in each tree corresponding to a vacuum vector $\xi_x$, and the tree edges corresponding to the basis pairs $(\xi_w, \xi_{ew})$.
We call a function $\lambda\colon G^+\times G^+\to [0,\infty)$ a *left weight* on $G$ if
1. $\lambda(v,w)>0\iff wv\in G^+$; and
2. $\lambda$ satisfies the *(left) cocycle condition*: $$\lambda(v,w_2w_1) = \lambda(w_1v,w_2)\lambda(v,w_1)$$ for all $v,w_1,w_2\in G^+$ with $w_2w_1v\in G^+$.
Note that if $v\in G^+$, then $r(v)\in V(G)$ satisfies $r(v)=r(v)^2$, hence $\lambda(v,r(v))=\lambda(v,r(v))^2$ and so $\lambda(v,r(v))=1$. In particular, for $x\in V(G)$ we have $x=s(x)=r(x)$ and so $\lambda(s(x),x)=\lambda(x,r(x))=1$. Note also that the edge weights $\lambda(v,e)$ (where $ev\in G^+$ and $e\in E(G)$) determine the entire function $\lambda$ through the left cocycle condition. Indeed, if we attach the weight $\lambda(v,e)$ to the edge in the Fock space tree corresponding to the move $\xi_v\mapsto \xi_{ev}$ defined by $L_e\xi_v = \xi_{ev}$, then we can view $\lambda(v,w)$ (when non-zero) as the product of the individual weights one crosses when moving from $\xi_v$ to $\xi_{wv}$ in that tree. See subsequent sections for more discussion on this “forest” perspective.
Given such a left weight $\lambda$, we define (by a mild abuse of notation) $$\lambda(w)=\sup_{v\in G^+} \lambda(v,w)\in [0,\infty]$$ for each $w\in G^+$. We say that $\lambda$ is *left-bounded at $w$* if $\lambda(w) < \infty$, and that $\lambda$ is *left-bounded* if this condition holds for all $w\in G^+$. The cocycle condition gives $\lambda(w_2w_1)\leq \lambda(w_2)\lambda(w_1)$ whenever $w_2w_1\in G^+$, so $\lambda$ is left-bounded if and only if $\lambda$ is left-bounded at every edge $e\in E(G)$.
If $\lambda$ is left-bounded at $w$, then we define the weighted left shift operator $L_{\lambda,w}\in \B(\H_G)$ to be the continuous linear extension of $$L_{\lambda,w} \, \xi_v =
\begin{cases}
\lambda(v,w) \,\xi_{wv} &\text{if $wv\in G^+$}\\
0 &\text{otherwise.}
\end{cases}$$ Since $\lambda(w)<\infty$, it is easy to see that this gives a well-defined operator with $\|L_{\lambda,w}\|=\lambda(w)$. Moreover, if $\lambda$ is left-bounded, then by the cocycle condition we see that $w\mapsto L_{\lambda,w}$ is a semigroupoid homomorphism: $$L_{\lambda,w_2w_1}=L_{\lambda,w_2}L_{\lambda,w_1}\quad \text{whenever $w_2w_1\in G^+$}.$$
We remark that one could also consider complex-valued left weight functions rather than weights taking non-negative values only. However, the corresponding weighted left shift operators would be jointly unitarily equivalent to weighted shift operators defined by a non-negative weight function. To see this, consider a complex-valued left weight $\mu\colon G^+\times G^+\to \bC$, by which we mean that $\mu(v,w)\ne0\iff wv\in G^+$ and $\mu$ satisfies the left cocycle condition. Define corresponding weighted left shifts $L_{\mu,w}$ exactly as above, let $\lambda\colon G^+\times G^+\to [0,\infty)$ be the non-negative left weight $\lambda(v,w)=|\mu(v,w)|$ and consider $\beta\colon G^+\times G^+\to \bT$, $\beta(v,w)=\tfrac{\lambda(v,w)}{\mu(v,w)}$ when $wv\in G^+$, and $\beta(v,w)=1$ otherwise. Note that $\beta$ then satisfies the left cocycle condition, so in particular we have $\beta(s(v),wv)=\beta(v,w)\beta(s(v),v)$ whenever $wv\in G^+$. The unitary operator $U_\beta$ mapping $\xi_v$ to $\beta(s(v),v)\xi_v$ satisfies $$U_\beta L_{\mu,w}\xi_v = \mu(v,w)\beta(s(v),wv)\xi_{wv}=\beta(s(v),v)\lambda(v,w)\xi_{wv} = L_{\lambda,w}U_\beta\xi_v$$ whenever $wv\in G^+$, hence $U_\beta L_{\mu,w}=L_{\lambda,w}U_\beta$, i.e., $L_{\mu,w}=U_\beta^*L_{\lambda,w}U_\beta$.
Observe also that the requirement that $\lambda(v,w) \ne 0$ when $wv\in G^+$ is equivalent to requiring $L_{\lambda,w}$ to be injective on the set $\{\xi_v : wv\in G^+\}$, and is thus an assumption we build into the weights to avoid degeneracies in the analysis. Finally note that each operator $L_{\lambda,w}$ factors as a product of $L_w$ and a diagonal (with respect to the standard basis) weight operator, just as in the single variable case of [@Shields74] which is recovered when $G$ consists of a single vertex with a single loop edge.
We now define the algebras ${{\mathfrak{L}(G,\lambda)}}$ that we shall consider in the paper. In the case of the single vertex, single loop edge graph, these algebras include classical unilateral weighted shift algebras such as those associated with weighted Bergman spaces; see the survey article [@Shields74] for an entrance point into the literature. The case of a single vertex graph and multiple loop edges was first considered along with some reflexivity type problems in [@kribs04].
If ${\lambda}$ is a left weight on a directed graph $G$, then we write ${{\mathfrak{L}(G,\lambda)}}$ for the WOT-closed unital operator algebra generated by the family of weighted left shift operators $\{ L_{\lambda,w} : w \in G^+,\ \lambda(w)<\infty\}$.
If (as is often the case below) $\lambda$ is left-bounded, then the set $$\{L_{\lambda,w}\colon w\in V(G)\cup E(G)\}$$ also generates ${{\mathfrak{L}(G,\lambda)}}$ as a WOT-closed unital operator algebra.
Let us call a strictly positive function $\alpha\colon G^+\to (0,\infty)$ with $\alpha(x)=1$ for all $x\in V(G)$ a *path weight* on $G$. For any such $\alpha$, there is a corresponding left weight $\lambda_\alpha$ on $G$ given by $$\lambda_\alpha(v,w)=
\begin{cases}
\frac{\alpha(wv)}{\alpha(v)}&\text{if $wv\in G^+$}\\
0&\text{otherwise.}
\end{cases}$$ Conversely, from any left weight $\lambda$, we obtain a corresponding path weight $\alpha_\lambda\colon v\mapsto
\lambda(s(v),v)$, and these correspondences are inverses of one another. This observation allows us to easily construct examples of left weights.
The left-handed notions above have right-handed counterparts which will play an important role in describing commutants. A *right weight* on $G$ is a function $\rho\colon G^+\times G^+\to [0,\infty)$ satisfying $\rho(v,u)>0\iff vu\in G^+$ and the *(right) cocycle condition* $$\rho(v,u_1u_2)=\rho(vu_1,u_2)\rho(v,u_1)$$ for all $v,u_1,u_2\in
G^+$ with $vu_1u_2\in G^+$. We then have $\rho(v,s(v))=1$ for all $v\in
G^+$. We write $\rho(u)=\sup_v\rho(v,u)$, and say $\rho$ is *right-bounded at $u$* if $\rho(u)<\infty$. We may then consider the weighted right shift operator $R_{\rho,u}\in \B(\H_G)$ (with $\|R_{\rho,u}\|=\rho(u)$) which satisfies the defining equation $$R_{\rho,u}\xi_v=
\begin{cases}
\rho(v,u) \,\xi_{vu} &\text{if $vu\in G^+$}\\
0 &\text{otherwise.}
\end{cases}$$ We have $\rho(u_1u_2)\leq \rho(u_2)\rho(u_1)$, and $R_{\rho,u_1u_2}=R_{\rho,u_2}R_{\rho,u_1}$ whenever $u_1u_2\in G^+$ and $\rho$ is right-bounded at $u_1$ and at $u_2$.
We write ${{\mathfrak{R}(G,\rho)}}$ for the WOT-closed unital operator algebra generated by $
\{R_{\rho,u}\colon u\in G^+,\ \rho(u)<\infty\}
$.
A right weight is *right-bounded* if it is right-bounded at every $u\in G^+$. Finally, we observe that $$\rho_\alpha(v,u)=
\begin{cases}
\frac{\alpha(vu)}{\alpha(v)}&\text{if $vu\in G^+$}\\
0&\text{otherwise}
\end{cases}$$ defines a one-to-one correspondence between path weights $\alpha$ and right weights $\rho=\rho_\alpha$.
Each of these right-handed definitions may be derived by applying the corresponding left-handed definition to the opposite graph of $G$, and making appropriate identifications. Note that the suprema defining $\lambda(u)$ and $\rho(u)$ are taken over the first argument, in $\lambda(\cdot,u)$ and $\rho(\cdot,u)$, and so in particular the notion of right-boundedness for a left weight function does not arise. A path weight $\alpha$, on the other hand, may be said to be left (resp. right) bounded if the associated map $\lambda_\alpha$ (resp. $\rho_\alpha$) is left-bounded (resp. right-bounded).
The weighted shift creation operators $L_{\lambda, e}$ for edges of $e$, and also sums of these operators, are in fact special cases of a wide class of weighted shift operators defined on general countable trees, rather than our graph generated trees. The single operator theory for these general shifts, such as conditions for hyponormality and $p$-hyponormality, is developed in the recent book of Jablo´nski, Jung, and Stochel [@jjs-12].
Muhly and Solel have recently defined weighted shift versions of the Hardy algebras $H^\infty(E)$ [@muhlysolel-weighted] that can be associated with a correspondence $E$ (a self-dual right Hilbert $C^*$-module) over a $W^*$-algebra $M$. The Hardy algebras $\A=H^\infty(E)$ in fact provide generalizations of the free semigroupoid graph algebras in which the self-adjoint (diagonal) subalgebra $\A \cap \A^*$ is no longer commutative. At the expense of a much higher level of technicality, the weighted shift versions of these Hardy algebras similarly extend the weighted shift directed graph algebras ${{\mathfrak{L}(G,\lambda)}}$.
Commutant Structure
===================
Let $\lambda$ be a left weight on $G$, and let $\rho$ be a right weight on $G$. We say that the pair $(\lambda,\rho)$ satisfies the *commuting square condition* at $(w,u)\in G^+\times G^+$ if $$\rho(wv,u)\lambda(v,w)=\lambda(vu,w)\rho(v,u)$$ for every $v\in G^+$ with $wvu\in G^+$. If $\lambda$ is left-bounded at $w$ and $\rho$ is right-bounded at $u$, then a simple computation shows that this condition holds if and only if $$R_{\rho,u}L_{\lambda,w}=L_{\lambda,w}R_{\rho,u}.$$
Recall that associated with the left weight $\lambda$ is a forest graph whose vertices are labelled by the elements of $G^+$ and whose edges $(v,ev)$ are labelled by the individual weights $\lambda(v,e)$ for $e\in E(G)$. We may now augment this ${\lambda}$-labelled forest by additional ${\rho}$-edges $(v,ve)$, which are labelled by the individual nonzero weights ${\rho}(v,e)$. The resulting labelled graph is the union of two labelled edge-disjoint forests which share the same vertex set. The commuting square condition can be viewed as a commuting square within this labelled graph, for the weights indicated in Figure \[f:weights\].
![The commuting square condition. Solid lines are paths made of edges labelled by $\lambda$-weights, and dashed lines are paths of edges labelled by $\rho$-weights.[]{data-label="f:weights"}](diagram)
Let $\lambda$ be a left weight on $G$. A right weight $\rho$ on $G$ is a *right companion* to $\lambda$ if $(\lambda,\rho)$ satisfies the commuting square condition at every $(w,u)\in
G^+\times G^+$. We call a right companion $\rho$ to $\lambda$ *canonical* if $\rho(r(e),e)=\lambda(s(e),e)$ for all $e\in E(G)$.
\[prop:canonical\] For any left weight $\lambda$ on $G$, there is a unique canonical right companion $\rho$ to $\lambda$, namely $\rho=\rho_\alpha$ where $\alpha$ is the path weight with $\lambda=\lambda_\alpha$. Moreover, if $\rho_1$ and $\rho_2$ are both right companions to $\lambda$, then ${{\mathfrak{R}(G,\rho_1)}}={{\mathfrak{R}(G,\rho_2)}}$.
Let $\alpha=\alpha_\lambda$ be the path weight given by $\alpha(v)=\lambda(s(v),v)$. Then $\lambda=\lambda_\alpha$ and by an easy calculation, the right weight $\rho_\alpha$ (defined in the previous section) is a canonical right companion to $\lambda$.
If $\rho_1$ and $\rho_2$ are both right companions to $\lambda$, then applying the commuting square condition for $\rho_1$ and $\rho_2$ with $v=r(u)=s(w)$ shows that $q(u):=\frac{\rho_2(r(u),u)}{\rho_1(r(u),u)}>0$ satisfies $\rho_2(w,u)=q(u)\rho_1(w,u)$ for any $w,u$ with $wu\in G^+$. So $\rho_1$ is right-bounded at $u$ if and only if $\rho_2$ is right-bounded at $u$, and in this case $R_{\rho_1,u}=q(u)R_{\rho_2,u}$; hence ${{\mathfrak{R}(G,\rho_1)}}={{\mathfrak{R}(G,\rho_2)}}$.
If $\rho_1$ and $\rho_2$ are both canonical right companions to $\lambda$, then $q(e)=1$ for all $e\in E(G)$. Applying the cocycle condition for $\rho_1$ and $\rho_2$ to the relation $\rho_2=q\cdot \rho_1$ shows that $q(u_1u_2)=q(u_2)q(u_1)$ whenever $u_1u_2\in G^+$; hence $q(u)=1$ for all $u\in G^+$, so $\rho_1=\rho_2$.
For $k\geq0$, let $Q_k$ be the orthogonal projection of $\H_G$ onto the closed linear span of $\{ \xi_v : |v|=k \}$. For $j\in\bZ$, define a complete contraction $\Phi_j\colon \B(\H_G)\to \B(\H_G)$ by $$\Phi_j(X) = \sum_{m\geq \max\{ 0,-j\}} Q_m X Q_{m+j}.$$ Also for $k\in\bN$, define $\Sigma_k\colon \B(\H_G)\to \B(\H_G)$ via the Cesaro-type sums $$\Sigma_k(X) = \sum_{|j|<k} \Big( 1 - \frac{|j|}{k} \Big) \Phi_j(X).$$
The following lemma is well-known. For completeness, we include a short proof.
\[lem:Sigma-k\] For $k\ge0$ and $X\in \B(\H_G)$, we have $\|\Sigma_k(X)\|\leq
\|X\|$, and $\Sigma_k(X)$ converges to $X$ in the strong operator topology as $k\to \infty$.
Let $z\in \bT$ and let $U_z$ be the diagonal unitary operator on $\H_G$ for which $U_z\xi_v = z^{|v|}\xi_v$ for each $v \in
G^+$. Then $U_zQ_sXQ_tU_z^* = z^{s-t}Q_sXQ_t$ for any $s,t\ge0$. Since $\sum_{\ell\ge0}Q_\ell=I$, it follows that for $j\in \bZ$, we have $$\Phi_j(X) = \int _{|z|=1}z^{j}U_zXU_z^*dz.$$ Writing $F_k(z)=\sum_{j=-k}^k(1-\frac{|j|}{k+1})z^j$ for the usual Fejér kernel, we see that$$\Sigma_k(X) = \int_{|z|=1}F_{k-1}(z)U_zXU_z^*dz.$$ Considering the scalars $\langle \Sigma_k(X)\xi, \zeta \rangle$, for $\xi, \zeta \in \H_G$, and the fact that $\|F_{k-1}\|_{L^1(\bT)}=1$, it follows that $\|\Sigma_k(X)\| \leq \|X\|$ for all $k$.
Let $\xi \in \H_G$. Then $$\|(X-\Sigma_k(X))\xi\| \leq \int_{|z|=1}F_{k-1}(z)\|(X - U_zXU_z^*)\xi\|dz.$$ The operators $U_z, U_z^*$ converge to the identity operator in the strong operator topology, as $z$ tends to $1$, and $F_k$ tends weak star to the unit point mass measure at $z=1$ as $k\to \infty$. It follows that $\Sigma_k(X)\xi \to X\xi$ as $k\to \infty$, and so $\Sigma_k(X){\stackrel{\textsc{sot}}{\to}}X$.
\[lem:Xf\] Let $\rho$ be a right weight on $G$ and suppose that $f\colon G^+\to
\bC$ has the property that if $f(u)\ne0$, then $\rho(u)<\infty$. Let $\H_0$ be the dense subspace of $\H_G$ spanned by $\{\xi_v\colon
v\in G^+\}$, and consider the sesquilinear form $A_f\colon
\H_0\times\H_0\to \bC$ with $$A_f(\xi_v,\xi_w)=
\begin{cases}
f(u)\rho(v,u)&\text{if $w=vu$ for some $u\in G^+$}\\
0&\text{otherwise}.
\end{cases}$$ If $A_f$ is bounded on $\H_0\times \H_0$, then the operator $X_f\in
\B(\H_G)$ implementing the continuous extension of $A_f$ to $\H_G\times\H_G$ satisfies $X_f\in {{\mathfrak{R}(G,\rho)}}$.
Let $(G_1,G_2,\dots)$ be a sequence of finite subgraphs of $G$ which increases to $G$; that is, $V(G_n)$ and $E(G_n)$ are finite sets for each $n$, and $V(G_n)\uparrow V(G)$ and $E(G_n)\uparrow E(G)$. For $n\ge1$, let $P_n$ be the projection onto the closure of the subspace of $\H_G$ spanned by $\{\xi_v\colon v\in G_n^+\}$. For $v,w\in
G^+$, a calculation shows that $\langle
P_n\Phi_j(X_f)P_n\xi_v,\xi_w\rangle=0$ unless $v,w\in G_n^+$ with $w=vu$ for some $u\in G_n^+$ with $|u|=-j$ and $\rho(u)<\infty$; and that in the latter case, $$\langle
P_n\Phi_j(X_f)P_n\xi_v,\xi_w\rangle=A_f(\xi_v,\xi_w)=f(u)\rho(v,u).$$ It follows that $P_n\Phi_j(X_f)P_n= P_nF_{j,n} P_n$ where $$F_{j,n}=
\displaystyle\sum_{\substack{u\in G_n^+,\,|u|=-j,\\\rho(u)<\infty}} f(u)R_{\rho,u}.$$ (We have $F_{j,n}=0$ if $j>0$.) Since $V(G_n)$ and $E(G_n)$ are finite, $F_{j,n}$ is a finite linear combination of operators $R_{\rho,u}$, so $F_{j,n}\in {{\mathfrak{R}(G,\rho)}}$. Now $P_n{\stackrel{\textsc{sot}}{\to}}I$, so $$\Phi_j(X_f)=\operatorname*{{\mbox{\scshape sot}}-lim}_{n\to \infty} P_n F_{j,n} P_n.$$ In fact, we will shortly see that $$\Phi_j(X_f)=\operatorname*{{\mbox{\scshape wot}}-lim}_{n\to \infty}
F_{j,n}.$$ From this, it follows that $\Phi_j(X_f)\in {{\mathfrak{R}(G,\rho)}}$, so $\Sigma_k(X_f)\in {{\mathfrak{R}(G,\rho)}}$ for all $k\ge1$, allowing us to conclude, by Lemma \[lem:Sigma-k\], that $X_f=\operatorname*{{\mbox{\scshape sot}}-lim}_{k\to\infty}\Sigma_k(X_f)\in {{\mathfrak{R}(G,\rho)}}$ as desired.
To see this, we will first show that $\{\|F_{j,n}\|\colon n\ge
1\}$ is bounded. Note that for any $v\in G^+$, we have the norm-convergent sums $$X_f\xi_v=\sum_{w\in G^+} \langle X_f\xi_v,\xi_w\rangle \xi_w = \sum_{w\in G^+}A_f(\xi_v,\xi_w)\xi_w = \sum_{u\in G^+}f(u)\rho(v,u)\xi_{vu}.$$ Moreover, $$F_{j,n}\xi_v=\sum_{\substack{u\in G_n^+,\,|u|=-j,\\\rho(u)<\infty}}f(u)\rho(v,u)\xi_{vu}$$ so $\|F_{j,n}\xi_v\|\leq \|X_f\|$. For $i=1,2$, if $v_iu_i\in G^+$ and $|u_1|=|u_2|=-j$, then $v_1\ne v_2\implies v_1u_1\ne v_2u_2$. It follows that $\{ F_{j,n}\xi_v\colon v\in G^+\}$ is a pairwise orthogonal family of vectors for each $n\ge1$, hence $F_{j,n}=\sum^{\oplus}_{v\in G^+} F_{j,n}\xi_v\xi_v^*$ and so $$\|F_{j,n}\|=\sup_{v\in G^+} \|F_{j,n} \xi_v\|\leq \|X_f\|.$$ Now $P_n^\perp:=I-P_n{\stackrel{\textsc{sot}}{\to}}0$ as $n\to \infty$, so $P_nF_{j,n}P_n^\perp{\stackrel{\textsc{sot}}{\to}}0$ and $P_n^\perp F_{j,n}{\stackrel{\textsc{wot}}{\to}}0$ as $n\to \infty$. Hence $$F_{j,n}=P_nF_{j,n}P_n+P_nF_{j,n}P_n^\perp+P_n^\perp F_{j,n}{\stackrel{\textsc{wot}}{\to}}\Phi_j(X_f) \quad\text{as $n\to \infty$},$$ which completes the proof.
It is not difficult to see that if the function $f$ in Lemma \[lem:Xf\] has finite support, then $$X_f= \sum_{\substack{u\in G^+,\\\rho(u)<\infty}}f(u)R_{\rho,u}.$$ Heuristically, it is useful to think of $X_f$ as the formal series given by this formula even when the support of $f$ is infinite.
\[lem:ker\] Let $\lambda$ be a left-bounded left weight on $G$. If $K\in {{\mathfrak{L}(G,\lambda)}}'$ and $K\xi_x=0$ for all $x\in V(G)$, then $K=0$.
Given $w\in G^+$, consider $x=s(w)$. We have $$K\xi_w=\lambda(x,w)^{-1}KL_{\lambda,w}\xi_x=\lambda(x,w)^{-1}L_{\lambda,x}K\xi_x=0,$$ so $K=0$.
We are now ready to prove our main result.
\[commutant\_thm\] If $\lambda$ is a left-bounded left weight on $G$ and $\rho$ is its canonical right companion, then the commutant of ${{\mathfrak{L}(G,\lambda)}}$ coincides with ${{\mathfrak{R}(G,\rho)}}$.
The observations at the start of this section show that $L_{\lambda,w}$ commutes with $R_{\rho,u}$ whenever $w,u\in G^+$ and $\rho(u)<\infty$. Hence ${{\mathfrak{L}(G,\lambda)}}^\prime$ contains ${{\mathfrak{R}(G,\rho)}}$.
To prove the other inclusion, begin by fixing $S \in {{\mathfrak{L}(G,\lambda)}}^\prime$. For $u\in G^+$, consider the coefficients $a_u\in \bC$ defined by $$a_u=\langle S\xi_{r(u)},\xi_u\rangle.$$ Observe that for any $x\in V(G)$, the operator $L_{\lambda,x}=L_x$ is a projection with range spanned by $\{\xi_u\colon u\in r^{-1}(x)\}$, and $L_{x} S\xi_x=SL_{x}\xi_x=S\xi_x$. Hence $$S\xi_x=\sum_{u\in G^+} \langle S\xi_x,\xi_u\rangle \xi_u=\sum_{u\in r^{-1}(x)} a_u\xi_u$$ with convergence in norm.
If $v,w\in G^+$, then $\xi_v=\lambda(s(v),v)^{-1}L_{\lambda,v}
\xi_{s(v)}$ and $[S,L_{\lambda,v}]=0$, and $L_{\lambda,v}^*\xi_w=0$ unless $w=vu$ for some $u\in G^+$, and $L_{\lambda,v}^*\xi_{vu}=\lambda(u,v)\xi_u$. Now $\frac{\lambda(u,v)}{\lambda(s(v),v)}=\frac{\rho(v,u)}{\rho(r(u),u)}$ by the commuting square condition, and it follows that $$\begin{aligned}
\langle S\xi_v,\xi_w\rangle &=
\begin{cases}
\frac{\rho(v,u)}{\rho(r(u),u)}a_u &\text{if $w=vu$ for some $u\in G^+$}\\0&\text{otherwise}.
\end{cases}\end{aligned}$$ In particular, if $a_u\ne 0$, then $\rho(u)<\infty$ since $$\|S\|\ge\sup_{\{v\in G^+\colon vu\in G^+\}}|\langle S\xi_v,\xi_{vu}\rangle | = \sup_{v\in G^+}\frac{\rho(v,u)}{\rho(r(u),u)}|a_u|=\frac{\rho(u)}{\rho(r(u),u)}|a_u|.$$
In view of this, if we define $f\colon G^+\to \bC$ by $f(u)=a_u
\rho(r(u),u)^{-1}$, then we may legitimately consider the bilinear form $A_{f}\colon \H_0\times \H_0\to \bC$, defined as in Lemma \[lem:Xf\]. Consider the operators $\Sigma_{k}(S)$. If $v,w\in G^+$ and $\big||w|-|v|\big|<k$, then $$\begin{aligned}
\langle \Sigma_{k}(S)\xi_v,\xi_w\rangle
&= \sum_{|j|<k}\left(1-\frac{|j|}k\right)\sum_{m\geq \max\{0,-j\}}\langle SQ_{m+j}\xi_v,Q_m\xi_w\rangle\\
&= \left(1-\frac{\big||w|-|v|\big|}k\right)\langle S\xi_v,\xi_w\rangle\\
&=
\begin{cases}
\left(1-\frac{|w|-|v|}k\right)f(u)\rho(v,u)&\text{if $w=vu$ for some $u\in G^+$}\\0&\text{otherwise}
\end{cases}\\
&= \left(1-\frac{|w|-|v|}k\right)A_f(\xi_v,\xi_w).\end{aligned}$$ Hence for any $\xi,\eta\in \H_0$, we have $A_f(\xi,\eta)=\lim_{k\to \infty}\langle \Sigma_k(S)\xi,\eta\rangle$, so by Lemma \[lem:Sigma-k\], $$|A_f(\xi,\eta)|\leq \sup_{k\ge1}\|\Sigma_k(S)\|\,\|\xi\|\,\|\eta\|\leq \|S\|\,\|\xi\|\,\|\eta\|.$$ Thus $A_f$ is bounded on $\H_0\times \H_0$. By Lemma \[lem:Xf\], the bounded linear operator $X=X_f$ implementing $A_f$ is in ${{\mathfrak{R}(G,\rho)}}$. So $X\in
{{\mathfrak{L}(G,\lambda)}}'$, and (as above) we conclude that for $x\in V(G)$, the vector $X\xi_x$ is in the closed subspace spanned by $\{\xi_u\colon
u\in r^{-1}(x)\}$. Moreover, for any $u\in r^{-1}(x)$ we have $$\langle X\xi_x,\xi_u\rangle = A_f(\xi_x,\xi_u)=f(u)\rho(x,u)=a_u=\langle S\xi_x,\xi_u\rangle.$$ So $X\xi_x=S\xi_x$ for all $x\in V(G)$. Since $X,S\in {{\mathfrak{L}(G,\lambda)}}'$, we have $K=X-S\in {{\mathfrak{L}(G,\lambda)}}'$ and $K\xi_x=0$ for all $x\in V(G)$. Hence $S=X$ by Lemma \[lem:ker\], and so $S\in {{\mathfrak{R}(G,\rho)}}$, which completes the proof.
This result generalizes and improves on a few previous results. The case of the single vertex and single edge graph yields classical single variable weighted shift operators, and there, the notion of right-boundedness simply corresponds to the weight sequence being bounded below. Hence this result generalizes the fundamental commutant theorem for weighted Bergman spaces $H^\infty(\beta)$ [@Shields74]. In the case of a single vertex graph with $n$ edges and unit weights this result captures the commutant theorem for free semigroup algebras $\L_n$ [@davidsonpitts2], and it improves on the commutant result of [@kribs04], which established a special case of the theorem in the single vertex multi-edged weighted shift case. Finally, this result generalizes the commutant theorem for free semigroupoid algebras [@kribspower] which are determined by general unweighted directed graphs.
Double Commutant Theorems
=========================
When $\rho$ is right-bounded, we obtain the following mirror image of Theorem \[commutant\_thm\] which may be established with a flipped version of the preceding proof. For brevity, we will instead pass to the opposite graph ${{{G}^t}}$ of $G$, which is essentially “$G$ with the edges reversed”. More formally, we set $V({{{G}^t}})=V(G)$, $E({{{G}^t}})=E(G)$ and $({{{G}^t}})^+=\{{{{v}^t}}\colon v\in G^+\}$, where ${{{(wv)}^t}}={{{v}^t}}{{{w}^t}}$ for $wv\in G^+$, and ${{{u}^t}}=u$ for $u\in V(G)\cup E(G)$; the source and range maps for ${{{G}^t}}$ are given by $ {{{s}^t}}({{{v}^t}})=r(v)$ and ${{{r}^t}}({{{v}^t}})=s(v)$.
If $\lambda$ is a left weight on $G$ whose canonical right companion $\rho$ is right-bounded, then the commutant of ${{\mathfrak{R}(G,\rho)}}$ coincides with ${{\mathfrak{L}(G,\lambda)}}$.
Let ${{{G}^t}}$ be the opposite graph of $G$ and let ${{{\rho}^t}}({{{v}^t}},{{{u}^t}})={\rho}(v,u)$ for $v,u\in G^+$; since $\rho$ is a right-bounded right weight on $G$, it follows that ${{{\rho}^t}}$ is a left-bounded left weight on $G^t$. A calculation using the path weight associated with $\lambda$ and $\rho$ shows that the canonical right companion to ${{{\rho}^t}}$ is the right weight ${{{\lambda}^t}}$ on $G^t$ given by ${{{\lambda}^t}}(v^t,w^t)=\lambda(v,w)$. Let $U\colon
\H_{{{{G}^t}}}\to \H_G$ be the unitary with $U\xi_{{{{v}^t}}}=\xi_v$. For $u,w\in G^+$ with $\lambda(w)<\infty$, by checking values on basis vectors we see that $$UL_{\rho^t,{{{u}^t}}}U^*=R_{\rho,u} {{\quad\text{and}\quad}}UR_{\lambda^t,{{{w}^t}}}U^*=L_{\lambda,w},$$ so $$U{{\mathfrak{L}({{{G}^t}},{{{{\rho}}^t}})}} U^*={{\mathfrak{R}(G,\rho)}} {{\quad\text{and}\quad}}U {{\mathfrak{R}({{{G}^t}},{{{{\lambda}}^t}})}}U^*={{\mathfrak{L}(G,\lambda)}}.$$ By Theorem \[commutant\_thm\], ${{\mathfrak{L}({{{G}^t}},{{{{\rho}}^t}})}}'={{\mathfrak{R}({{{G}^t}},{{{\lambda}^t}})}}$, hence $$\begin{aligned}
{{\mathfrak{R}(G,\rho)}}'=(U{{\mathfrak{L}({{{G}^t}},{{{{\rho}}^t}})}}U^*)'&=U{{\mathfrak{L}({{{G}^t}},{{{\rho}^t}})}}'U^*\\&=U{{\mathfrak{R}({{{G}^t}},{{{{\lambda}}^t}})}}U^*={{\mathfrak{L}(G,\lambda)}}.\qedhere
\end{aligned}$$
Combining the previous two results leads us to the following double commutant theorem.
\[preconj\] If $\lambda$ is a left-bounded left weight on $G$ whose canonical right companion $\rho$ is right-bounded, then ${{\mathfrak{L}(G,\lambda)}}$ coincides with its double commutant: $${{\mathfrak{L}(G,\lambda)}}'' = {{\mathfrak{L}(G,\lambda)}}.$$
If $\alpha\colon G^+\to (0,\infty)$ is any path weight with $\sup_v\alpha(v)<\infty$ and $\inf_v\alpha(v)>0$, then plainly $\lambda_\alpha$ is left-bounded and its canonical right companion $\rho_\alpha$ is right-bounded, giving a large class of weights satisfying the hypotheses of this result. In particular, if $|G^+|<\infty$ (i.e., if $G$ is a finite *acyclic* directed graph), then for any left weight $\lambda$ on $G$, we see that ${{\mathfrak{L}(G,\lambda)}}$ is an algebra of $n\times n$ matrices with ${{\mathfrak{L}(G,\lambda)}}''={{\mathfrak{L}(G,\lambda)}}$, where $n=|G^+|$.
On the other hand, there are many weights which satisfy the hypotheses of Theorem \[preconj\] but violate these boundedness conditions for the path weight $\alpha$. For example, if $G^+$ contains paths of arbitrary length and $\alpha(v)=f(|v|)$ where $f\colon \bN_0\to (0,\infty)$ is any decreasing function with $f(0)=1$ and $f(k)\to 0$ as $k\to \infty$, then $\lambda_\alpha$ is left-bounded and $\rho_\alpha$ is right-bounded, but $\inf_v\alpha(v)=0$.
We now show one way to weaken the hypotheses in Theorem \[preconj\], at least if $G$ is a finite directed graph. We first require a technical lemma.
\[lem:phi\]
Let $\rho$ be a right weight on $G$ and let $u\in G^+$ with $\rho(u)<\infty$. Let $k\ge0$ and let $X\in \B(\H_G)$. If $[X,R_{\rho,u}]=0$, then $[\Sigma_{k}(X),R_{\rho,u}]=0$.
By calculating values on canonical basis vectors, we observe that $$Q_mR_{\rho,u}=
\begin{cases}
R_{\rho,u}Q_{m-|w|}&\text{if $m\ge|u|$}\\
0&\text{if $0\leq m<|u|$.}
\end{cases}$$ So if $[X,R_{\rho,u}]=0$, then $$\Phi_j(X)R_{\rho,u}=\sum_{m\geq
\max\{|u|,|u|-j\}}R_{\rho,u}Q_{m-|u|}XQ_{m+j-|u|}=R_{\rho,u}\Phi_j(X).$$ Since $\Sigma_{k}(X)$ is a linear combination of the operators $\Phi_j(X)$, which all commute with $R_{\rho,u}$, we see that $\Sigma_{k}(X)$ commutes with $R_{\rho,u}$.
For any right weight ${\rho}$ on $G$, let us write $$G^+_{\rho}=\{u\in G^+\colon {\rho}(u)<\infty\}.$$ Since $\rho(x)=1$ for $x\in V(G)$, we have $V(G)\subseteq G^+_\rho$. Moreover, since $\rho(vw)\leq \rho(v)\rho(w)$ for any $vw\in G^+$, we see that $G^+_\rho$ is a subsemigroupoid in $G^+$.
\[thm:tails\] Let ${\lambda}$ be a left-bounded left weight on a finite directed graph $G$, with canonical right companion $\rho$. If $$\label{eq:tails}
\forall\,v\in G^+\ \exists\,u_v\in G^+_\rho\colon vu_v\in G^+_\rho,$$ then ${{\mathfrak{L}(G,\lambda)}}''={{\mathfrak{L}(G,\lambda)}}$.
It suffices to show that ${{\mathfrak{L}(G,\lambda)}}''\subseteq {{\mathfrak{L}(G,\lambda)}}$; equivalently (by Theorem \[commutant\_thm\]) that ${{\mathfrak{R}(G,\rho)}}'\subseteq {{\mathfrak{L}(G,\lambda)}}$. Suppose that $T\in{{\mathfrak{R}(G,\rho)}}'$. If $v\in G^+$, then $R_{\rho,s(v)}=R_{s(v)}$ is a projection in ${{\mathfrak{R}(G,\rho)}}$ with $\xi_v=R_{s(v)}\xi_v$ and $[T,R_{s(v)}]=0$, from which it follows that $T\xi_v\in R_{s(v)}\H_G$. Moreover, if $u\in
G^+_\rho$, then the restriction of $R_{\rho,u}^*R_{\rho,u}$ to $R_{r(u)}\H_G$ is an injective diagonal operator since it maps $\xi_v$ to $\rho(v,u)^2\xi_v$ if $s(v)=r(u)$.
Now suppose $K\in {{\mathfrak{R}(G,\rho)}}'$ and $K\xi_x=0$ for all $x\in V(G)$; we claim that we necessarily have $K=0$. To see this, let $v\in G^+$, let $u_v$ be as in Eq. (\[eq:tails\]) and note that $r(u_v)=s(v)$, since $vu_v\in G^+$. Now $$\begin{aligned}
R_{\rho,u_v}^*R_{\rho,u_v}K\xi_v&= R_{\rho,u_v}^*KR_{\rho,u_v}\xi_v =\rho(v,u_v)R^*_{\rho,u_v}K\xi_{vu_v} \\&= \rho(v,u_v)\rho(r(v),vu_v)^{-1}R_{\rho,u_v}^*R_{\rho,vu_v}K\xi_{r(v)}=0,
\end{aligned}$$ so $K\xi_v=0$ by the observations of the previous paragraph, establishing the claim.
Now let $T\in {{\mathfrak{R}(G,\rho)}}'$ be arbitrary. Since $G$ is finite, for each $k\in \bN$ the set $\{w\in G^+\colon |w|<k\}$ is finite and we may consider the operator $$p_k(T)=\sum_{\{w\in G^+\colon |w|<k\}} \left(1-\frac{|w|}k\right) a_w
\lambda(s(w),w)^{-1}L_{\lambda,w}$$ where $a_w=\langle
T\xi_{s(w)},\xi_w\rangle$ for $w\in G^+$. Clearly, $p_k(T)\in
{{\mathfrak{L}(G,\lambda)}}$. As observed above, we have $T\xi_x\in R_x\H_G$, so $$T\xi_x=\sum_{w\in s^{-1}(x)} a_w\xi_w.$$ By Lemma \[lem:phi\], the operators $\Sigma_k(T)$ are in ${{\mathfrak{R}(G,\rho)}}'$. Hence $K=\Sigma_k(T)-p_k(T)\in {{\mathfrak{R}(G,\rho)}}'$, and a calculation gives $K\xi_x=0$ for all $x\in V(G)$. Hence $\Sigma_k(T)=p_k(T)\in {{\mathfrak{L}(G,\lambda)}}$. Since ${{\mathfrak{L}(G,\lambda)}}$ is strongly closed and $\Sigma_k(T)\to T$ strongly, we obtain $T\in {{\mathfrak{L}(G,\lambda)}}$. Hence ${{\mathfrak{R}(G,\rho)}}'\subseteq{{\mathfrak{L}(G,\lambda)}}$ which completes the proof.
We now give some examples illustrating this result. If $u\in G^+$, it will be useful to write $$G^+u=\{vu\colon v\in G^+ \text{ and } vu\in G^+\}.$$
![The path weight $\alpha$ considered in Example \[exa:rho(v)\][]{data-label="f:eg47a"}](eg45-alpha)
![The path weight $\alpha$ considered in Example \[exa:rho(v)\][]{data-label="f:eg47a"}](eg47-alpha)
\[exa:rho(v)1\] Let $G$ be the directed graph with a single vertex $\phi$ and two loop edges, $e$ and $f$, so that $G^+=\{\phi,e,f,ee=e^2,ef,fe,f^2,eee=e^3,\dots\}$ and $s(w)=r(w)=\phi$ for every $w\in G^+$ (see Figure \[f:eg45\]). As indicated in Figure \[f:eg45a\], we consider the path weight $\alpha\colon G^+\to (0,\infty)$ given by $$\alpha(v)=
\begin{cases}
2^{-|v|}&\text{if $v\in G^+e$}\\
1&\text{otherwise},
\end{cases}$$ and let $\lambda=\lambda_\alpha$. It is easy to check that the left weight $\lambda$ is left-bounded (in fact $\lambda(w)\leq 1$ for all $w\in G^+$); the canonical right companion of $\lambda$ is $\rho=\rho_\alpha$ by Proposition \[prop:canonical\].
We claim that while $\rho$ is not right-bounded, we have $$G^+_{\rho}=\{\phi\}\cup G^+e = G^+\setminus G^+f,$$ so that Eq. (\[eq:tails\]) holds with $u_v=e$ for all $v\in G^+$, hence ${{\mathfrak{L}(G,\lambda)}}''={{\mathfrak{L}(G,\lambda)}}$.
Let us check that $G^+_{\rho}$ is indeed of this form. We have ${\rho}(v,u)=\frac{\alpha(vu)}{\alpha(v)}$ for any $ v,u\in G^+$. In particular, if $u\in G^+f$, then $\alpha(vu)=1$ and so $\rho(v,u)=2^{|v|}$ for $v\in G^+e$, so ${\rho}(u)=\infty$. On the other hand, if $u\in
G^+\setminus G^+f$, then $\alpha(vu)=2^{-|vu|}\leq \alpha(v)$ for all $v\in G^+$, so $\rho(u)\leq 1$.
\[exa:twovertex\] Let $G$ be the directed $2$-cycle, so that $V(G)=\{x,y\}$ and $E(G)=\{e,f\}$ where $s(e)=r(f)=x$ and $s(f)=r(e)=y$ (see Figure \[f:eg46\]). For this particular graph $G$, we will show that if $\lambda$ is any left-bounded left weight on $G$ with right companion $\rho$, then ${{\mathfrak{L}(G,\lambda)}}''={{\mathfrak{L}(G,\lambda)}}$ if and only if $G_\rho^+$ satisfies Eq. (\[eq:tails\]).
Note that the edges in any path in $G^+$ must alternate: $$G^+=\{x,y,e,f,ef,fe,efe,fef,efef,fefe,\dots\}.$$ We first show that $ef\in G^+_\rho$. Let $v\in G^+\setminus V(G)$ with $vef\in G^+$. Either $v=(ef)^{k}$ or $v=f(ef)^{k-1}=(fe)^{k-1}f$ for some $k\ge1$. Let $\alpha\colon G^+\to (0,\infty)$ be the path weight with $\lambda=\lambda_\alpha$ and $\rho=\rho_\alpha$. Since $(ef)^kef=(ef)^{k+1}=ef(ef)^k$, we have $$\rho((ef)^k,ef)=\frac{\alpha((ef)^{k+1})}{\alpha((ef)^k)} = \lambda((ef)^k,ef)\leq \lambda(ef)<\infty$$ and since $f(ef)^{k-1}ef=f(ef)^k=(fe)^kf$, we have $$\rho(f(ef)^{k-1},ef)=\frac{\alpha((fe)^{k}f)}{\alpha((fe)^{k-1}f)}=\lambda((fe)^{k-1}f,fe)\leq \lambda(fe)<\infty,$$ so $\rho(ef)<\infty$, i.e., $ef\in G^+_\rho$. Similarly, $fe\in
G^+_\rho$. Since $G_\rho^+$ is a semigroupoid, we have $\langle ef,fe\rangle \subseteq G^+_\rho$ where $\langle
ef,fe\rangle := V(G)\cup \{ (ef)^k,(fe)^k\colon k\ge1\}$. If $G_\rho^+\supsetneq\langle ef,fe\rangle$, then $G_\rho^+$ contains an element of odd length. By symmetry, we may assume this is of the form $e(fe)^n$ for some $n\ge0$. We then also have $e(fe)^m=e(fe)^n(fe)^{m-n}\in G_\rho^+$ for any $m> n$, so if we define $u_v$ for $v\in G^+$ by $$u_v=
\begin{cases}
s(v)&\text{if $v\in \langle ef,fe\rangle$}\\
(fe)^n&\text{if $v=e(fe)^k$ for some $k\ge0$}\\
e(fe)^n&\text{if $v=f(ef)^k$ for some $k\ge0$,}
\end{cases}$$ then $u_v\in G^+_\rho$ and $vu_v\in G^+_\rho$ for all $v\in G^+$, so Eq. (\[eq:tails\]) holds and so ${{\mathfrak{L}(G,\lambda)}}''={{\mathfrak{L}(G,\lambda)}}$ by Theorem \[thm:tails\].
On the other hand, if $G^+_\rho=\langle ef,fe\rangle$, then Eq. (\[eq:tails\]) does not hold, since if $|v|$ is odd and $u\in
G^+_\rho$ with $vu\in G^+$, then $|vu|$ is also odd, so $vu\not\in
G_\rho^+$. In this case, it is not difficult to see that the orthogonal projection $P$ onto the closed linear span of $\{\xi_v\colon \text{$v=x$ or $v=(fe)^k$, $k\ge1$}\}$ commutes with $R_{\rho,u}$ for $u\in \{x,y,ef,fe\}$, hence $P\in
{{\mathfrak{R}(G,\rho)}}'={{\mathfrak{L}(G,\lambda)}}''$. If $T\in {{\mathfrak{L}(G,\lambda)}}$, then $\langle
T\xi_x,\xi_x\rangle = \langle T\xi_f,\xi_f\rangle$. Since $P\xi_x=\xi_x$ and $P\xi_f=0$, we have $P\not\in {{\mathfrak{L}(G,\lambda)}}$. So ${{\mathfrak{L}(G,\lambda)}}''\ne {{\mathfrak{L}(G,\lambda)}}$.
We note that it is indeed possible for Eq. (\[eq:tails\]) to fail for this graph $G$. For example, let $\alpha\colon G^+\to
(0,\infty)$ with $\alpha(v)=1$ for all $v\in s^{-1}(x)$, and $\alpha(v)=2^{f(|v|)}$ for $v\in s^{-1}(y)$ where $f\colon \bN_0\to
\bZ$ is a function with $f(0)=0$, and $f(n+1)\in \{f(n)-1,f(n)+1\}$ for all $n\in \bN_0$ and with $\sup_n f(n)=\infty$ and $\inf_n
f(n)=-\infty$. One may then check that the left weight $\lambda_\alpha$ is left-bounded, and that its canonical right companion $\rho$ satisfies $G_\rho^+=\langle ef,fe\rangle$, so Eq. (\[eq:tails\]) fails.
\[exa:rho(v)\] For a general left-bounded weight $\lambda$, the double commutant property for ${{\mathfrak{L}(G,\lambda)}}$ can fail very badly. For example, let $G$ again be the directed graph with a single vertex $\phi$ and two loop edges $e$ and $f$, and let us now define a path weight $\alpha\colon
G^+\to (0,\infty)$ recursively by setting $\alpha(\phi)=\alpha(e)=\alpha(f)=1$, and $$\begin{aligned}
\alpha(ewe)&=\tfrac12 \alpha(we),&\alpha(fwf)&=\tfrac 12 \alpha(wf),\\
\alpha(ewf)&=\alpha(wf), &\alpha(fwe)&=\alpha(we).
\end{aligned}$$ This is illustrated in Figures \[f:eg45\] and \[f:eg47a\]. Take $\lambda=\lambda_\alpha$ and $\rho=\rho_\alpha$. Observe that ${\lambda}$ is a left-bounded left weight since $\alpha(wv)\leq \alpha(v)$ for all $w,v\in G^+$. For any $k\in
\bN$ and $w\in G^+$, $${\rho}(we)\geq
{\rho}(f^k,we)=\frac{\alpha(f^kwe)}{\alpha(f^k)}=2^{k-1}
\alpha(we)\to \infty\text{ as~$k\to \infty$,}$$ hence ${\rho}(we)=\infty$; by symmetry, ${\rho}(wf)=\infty$. Hence $
G^+_{\rho}=\{\phi\}$. Since $R_{\rho,\phi}=R_\phi=I$, we have ${{\mathfrak{R}(G,\rho)}}=\bC I$ and so ${{\mathfrak{L}(G,\lambda)}}''={{\mathfrak{R}(G,\rho)}}'=\B(\H_G)\ne {{\mathfrak{L}(G,\lambda)}}$.
The commutant result yields other structural results on the algebras, such as the following.
Let ${\lambda}$ be a left-bounded left weight on $G$ with canonical right companion $\rho$. If either $\rho$ is right-bounded, or $G$ is finite and $G^+_{\rho}$ satisfies Eq. (\[eq:tails\]), then ${{\mathfrak{L}(G,\lambda)}}$ is inverse closed.
This is a well-known property of commutants: if $A\in{{\mathfrak{L}(G,\lambda)}}
= {{\mathfrak{R}(G,\rho)}}'$ is invertible in $\B(\H_G)$, then for all $R\in{{\mathfrak{R}(G,\rho)}}$, $A^{-1}R = A^{-1} RAA^{-1} = RA^{-1}$, and hence $A^{-1}\in{{\mathfrak{R}(G,\rho)}}' = {{\mathfrak{L}(G,\lambda)}}$.
If ${\lambda}$ is a left-bounded left weight on $G$ whose canonical right companion $\rho$ is right-bounded, then every normal element of ${{\mathfrak{L}(G,\lambda)}}$ lies in the SOT-closure of the linear span of the projections $L_{\lambda,x}$ for $x\in V(G)$.
Let $N$ be a normal element of ${{\mathfrak{L}(G,\lambda)}}$. Set $a_x = \langle N\xi_x,\xi_x\rangle$ for $x\in V(G)$ and let $M$ be the SOT-convergent sum $M=\sum_{x\in V(G)}a_xL_{\lambda,x}$. Observe that each $\xi_x$ is an eigenvector for ${{\mathfrak{L}(G,\lambda)}}^*$, as for all $u\in G^+\setminus\{x\}$ and $A\in {{\mathfrak{L}(G,\lambda)}}$, we have $$\langle A^*\xi_x, \xi_u\rangle =\tfrac1{{\rho}(r(u),u)} \langle\xi_x, A R_{\rho,u} \xi_{r(u)}\rangle=
\tfrac1{{\rho}(r(u),u)}\langle R_{\rho,u}^*\xi_x,A\xi_{r(u)}\rangle=0.$$ Thus $N^* \xi_x = \overline{a_x} \xi_x$, and by normality $N\xi_x = a_x \xi_x$. Hence for all $u\in G^+$, $$N\xi_u = \tfrac1{{\rho}(r(u),u)}N R_{\rho,u} \xi_{r(u)} = \tfrac1{{\rho}(r(u),u)}R_{\rho,u} N\xi_{r(u)} = a_{r(u)} \xi_u=M\xi_u,$$ so $N=M$.
Building on Theorems \[preconj\] and \[thm:tails\], a natural open problem is to determine weighted graph conditions that fully characterize when the algebra ${{\mathfrak{L}(G,\lambda)}}$ and its double commutant ${{\mathfrak{L}(G,\lambda)}}''$ coincide.
This work was partly supported by a London Mathematical Society travel grant. The first named author was partly supported by NSERC Discovery Grant 400160 and a University Research Chair at Guelph. We are grateful to the referee for helpful suggestions.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we formulate and investigate a generalized consensus algorithm which makes an attempt to unify distributed averaging and maximizing algorithms considered in the literature. Each node iteratively updates its state as a time-varying weighted average of its own state, the minimal state, and the maximal state of its neighbors. We prove that finite-time consensus is almost impossible for averaging under this uniform model. Both time-dependent and state-dependent graphs are considered, and various necessary and/or sufficient conditions are presented on the consensus convergence. For time-dependent graphs, we show that quasi-strong connectivity is critical for averaging, as is strong connectivity for maximizing. For state-dependent graphs defined by a $\mu$-nearest-neighbor rule, where each node interacts with its $\mu$ nearest smaller neighbors and the $\mu$ nearest larger neighbors, we show that $\mu+1$ is a critical threshold on the total number of nodes for the transit from finite-time to asymptotic convergence for averaging, in the absence of node self-confidence. The threshold is $2\mu$ if each node chooses to connect only to neighbors with unique values. Numerical examples illustrate the tightness of the conditions. The results characterize some fundamental similarities and differences between distributed averaging and maximizing algorithms.'
author:
- 'Guodong Shi and Karl Henrik Johansson[^1]'
title: |
**Finite-time and Asymptotic Convergence of\
Distributed Averaging and Maximizing Algorithms[^2]**
---
[**Keywords:**]{} Averaging algorithms, Max-consensus, Finite-time convergence, State-dependent connections
Introduction
============
Distributed averaging algorithms, where each node iteratively averages its neighbors’ states, have been extensively studied in the literature, due to its wide applicability in engineering [@tsi; @jad03; @cortes], computer science [@cs2; @cs3], and social science [@degroot; @social1; @golub]. Recently also the max-consensus algorithms have attracted attention. These algorithms compute the maximal value among the nodes, and have been used for leader election, network size estimation, and various applications in wireless networks [@cortes; @julien2].
The convergence to a consensus is central in the study of averaging and maximizing algorithms but can be hard to analyze, especially when the node interactions are carried out over a switching graph. The most convenient way of modeling the switching node interactions is just to assume the communication graphs are defined by a sequence of time-dependent graphs over the node set. The connectivity of this sequence of graphs plays an important role for the network to reach consensus. Joint connectivity, i.e., connectivity of the union graph over time intervals, has been considered, and various convergence conditions have been established [@tsi; @vb2; @jad03; @saber04; @ren; @mor; @caoming1; @caoming2; @shi09; @mor]. The switching topology can be dependent on the node states. For instance, in Krause’s model, each node is connected only to nodes within a certain distance [@krause]. Vicsec’s model has a similar setting but with higher-order node dynamics [@vic95]. Because the node dynamics is coupled with the graph dynamics for state-dependent graphs, the convergence analysis is quite challenging. Deterministic consensus algorithms with state-dependent graph were studied in [@vb1; @julien], and convergence results under probabilistic models were established in [@tang; @liuguo].
Few studies have discussed the fundamental similarities and differences between distributed averaging and maximizing. Averaging and maximizing consensus algorithms are both distributed information processing over graphs, where nodes communicate and exchange information with its neighbors in the aim of collective convergence. Average consensus algorithms in the literature are based on two standing assumptions: local cohesion and node self-confidence. The node states iteratively update as a weighted average of its neighbors’ states, with a positive lower bound for the weight corresponding to its own state [@tsi; @vb2; @jad03; @ren; @caoming1; @caoming2; @tsi11]. Average consensus algorithms can also be viewed as the equivalent state evolution process where each node updates its state as a weighted average of its own state, and the minimum and maximum states of its neighbors. Maximizing (or minimizing) consensus algorithms are simply based on that each node updates its state to the maximal (minimal) state among its neighbors [@max1; @max2]. Asymptotic convergence is common in the study of averaging consensus algorithms [@caoming1; @ren; @tsi11; @jad03], while it has been shown that maximizing algorithms converge in general in finite time [@max1; @max2]. Finite-time convergence of averaging algorithms was investigated in [@cortes] for a continuous-time model, and recently finite-time consensus in discrete time was discussed in [@it] for a special case of gossiping [@boyd].
In this paper, we make the simple observation that averaging and maximizing algorithms can be viewed as instances of a more general distributed processing model. Using this model the transition of the consensus convergence can be studied for the two classes of distributed algorithms in a unified way. Each node iteratively updates its state as a weighted average of its own state together with the minimum and maximum states of its neighbors. By special cases for the weight parameters, averaging and maximizing algorithms can be analyzed. Under this uniform model, we prove for averaging that finite-time consensus is impossible in general, and asymptotical consensus is possible only when the node self-confidence satisfies a divergence condition. Both time-dependent and state-dependent graphs are considered, and various necessary and/or sufficient conditions are presented on the consensus convergence. For time-dependent graphs, we show that quasi-strong connectivity (existence of a spanning tree) is critical for averaging, as is strong connectivity for maximizing. We use a $\mu$-nearest-neighbor rule to generate state-dependent graphs, in which each node interacts with its $\mu$ nearest smaller neighbors ($\mu$ neighbors with smaller state values), and the nearest $\mu$ larger neighbors. This model is motivated from recent studies of collective bird behavior [@pnas]. For averaging algorithms without node self-confidence under such state-dependent graphs, we show that $\mu+1$ is a critical value for the total number of nodes: finite-time consensus is achieved globally if the number of nodes is no larger than $\mu+1$, and finite-time consensus fails for almost all initial conditions if the number of nodes is larger than $\mu+1$. Moreover, it is shown that this critical number of nodes is instead $2\mu$ if each node chooses to connect only to neighbors with distinct values in the neighbor rule.
The rest of the paper is organized as follows. In Section 2 we introduce the considered network model, the uniform processing algorithm, and the consensus problem. The impossibilities of finite-time or asymptotic consensus are studied in Section 3. Consensus convergence is studied for time-dependent and state-dependent graphs in Sections 4 and 5, respectively. We show numerical examples in Section 6 and finally some concluding remarks are given in Section 7.
Problem Definition
==================
In this section, we introduce the network model, the considered algorithm, and define the problem of interest.
Network
-------
We first recall some concepts and notations in graph theory [@god]. A directed graph (digraph) $\mathcal
{G}=(\mathcal {V}, \mathcal {E})$ consists of a finite set $\mathcal{V}$ of nodes and an arc set $\mathcal {E}\subseteq \mathcal{V}\times\mathcal{V}$. An element $e=(i,j)\in\mathcal {E}$ is called an [*arc*]{} from node $i\in \mathcal{V}$ to $j\in\mathcal{V}$. If the arcs are pairwise distinct in an alternating sequence $ v_{0}e_{1}v_1e_{2}v_{2}\dots e_{k}v_{k}$ of nodes $v_{i}\in\mathcal{V}$ and arcs $e_{i}=(v_{i-1},v_{i})\in\mathcal {E}$ for $i=1,2,\dots,k$, the sequence is called a (directed) [*path*]{} with [*length*]{} $k$. If there exists a path from node $i$ to node $j$, then node $j$ is said to be reachable from node $i$. Each node is thought to be reachable by itself. A node $v$ from which any other node is reachable is called a [*center*]{} (or a [*root*]{}) of $\mathcal {G}$. A digraph $\mathcal
{G}$ is said to be [*strongly connected*]{} if node $i$ is reachable from $j$ for any two nodes $i,j\in\mathcal{V}$; [*quasi-strongly connected*]{} if $\mathcal {G}$ has a center [@ber]. The [*distance*]{} from $i$ to $j$ in a digraph $\mathcal{G}$, $d(i,j)$, is the length of a shortest simple path $i \rightarrow j$ if $j$ is reachable from $i$, and the [*diameter*]{} of $\mathcal
{G}$ is $\rm{diam}(\mathcal{G})$$=\max\{d(i,j)|i,j \in\mathcal
{V},\ j\mbox{ is reachable from}\ i\}$. The union of two digraphs with the same node set $\mathcal {G}_1=(\mathcal {V},\mathcal {E}_1)$ and $\mathcal
{G}_2=(\mathcal {V},\mathcal {E}_2)$ is defined as $\mathcal {G}_1\cup\mathcal
{G}_2=(\mathcal {V},\mathcal {E}_1\cup\mathcal {E}_2)$. A digraph $\mathcal {G}$ is said to be bidirectional if for every two nodes $i$ and $j$, $(i,j)\in \mathcal{E}$ if and only if $(j,i)\in \mathcal{E}$ . A bidirectional graph $\mathcal{G}$ is said to be [*connected*]{} if there is a path between any two nodes.
Consider a network with node set $\mathcal{V}=\{1,2,\dots,n\}$, $n\geq 3$. Time is slotted. Denote the state of node $i$ at time $k\geq 0$ as $x_i(k)\in\mathds{R}$. Then $x(k)=\big(x_1(k) \dots x_n(k)\big)^T$ represents the network state. For time-varying graphs, we use the following definition.
The interactions among the nodes are determined by a given sequence of digraphs with node set $\mathcal{V}$, denoted as $\mathcal{G}_k=
(\mathcal{V}, \mathcal{E}_k)$, $k=0,1,\dots$.
Throughout this paper, we call node $j$ a [*neighbor*]{} of node $i$ if there is an arc from $j$ to $i$ in the graph. Each node is supposed to always be a neighbor of itself. Let $\mathcal{N}_i(k)$ represent the neighbor set of node $i$ at time $k$.
Algorithm
---------
The classical average consensus algorithm in the literature is given by $$\begin{aligned}
\label{r1}
x_i(k+1)=\sum_{j\in\mathcal{N}_i(k)}a_{ij}(k)x_j(k),\ \ i=1\dots,n.\end{aligned}$$ Two standing assumptions are fundamental in determining the nature of its dynamics:
- [*(Local Cohesion)*]{} $\sum_{j\in\mathcal{N}_i(k)}a_{ij}(k)=1$ for all $i$ and $k$;
- [*(Self-confidence)*]{} There exists a constant $\eta>0$ such that $a_{ii}(k)\geq \eta$ for all $i$ and $k$.
These assumptions are widely imposed in the existing works, e.g., [@jad03; @tsi; @tsi2; @tsi11; @caoming1; @caoming2; @ren; @vb2]. With A1 and A2, we have $$\begin{aligned}
\label{102}
\sum_{j\in\mathcal{N}_i(k)}a_{ij}(k)x_j(k)=\eta x_i(k) + \big(a_{ii}(k)-\eta\big)x_i(k) +\sum_{j\in\mathcal{N}_i(k), j\neq i}a_{ij}(k)x_j(k)
\end{aligned}$$ and $$\begin{aligned}
\label{100}
\big(1-\eta\big)\min_{j\in \mathcal{N}_i(k)}x_j(k)\leq \big(a_{ii}(k)-\eta\big)x_i(k) +\sum_{j\in\mathcal{N}_i(k), j\neq i}a_{ij}(k)x_j(k)\leq \big(1-\eta\big)\max_{j\in \mathcal{N}_i(k)}x_j(k).
\end{aligned}$$
Noting the fact that for any $c\in[a,b]$ there exists a unique $\lambda\in[0,1]$ satisfying $c=\lambda a+(1-\lambda)b$, we see from (\[100\]) that for every $i$ and $k$, there exists $\beta_k^{\langle i\rangle}\in [0,1]$ such that $$\begin{aligned}
\label{101}
&\big(a_{ii}(k)-\eta\big)x_i(k) +\sum_{j\in\mathcal{N}_i(k), j\neq i}a_{ij}(k)x_j(k)\nonumber\\
&= \beta_k^{\langle i\rangle}\big(1-\eta\big)\min_{j\in \mathcal{N}_i(k)}x_j(k)+ \big(1-\beta_k^{\langle i\rangle}\big)\big(1-\eta\big)\max_{j\in \mathcal{N}_i(k)}x_j(k)\nonumber\\
&= \alpha^{\langle i\rangle}_k\min_{j\in \mathcal{N}_i(k)}x_j(k)+\big(1-\eta-\alpha^{\langle i\rangle}_k\big)\max_{j\in \mathcal{N}_i(k)} x_j(k),
\end{aligned}$$ where $\alpha^{\langle i\rangle}_k=\beta_k^{\langle i\rangle}(1-\eta)\in[0,1-\eta]$.
Therefore, in light of (\[102\]) and (\[101\]), based on assumptions A1 and A2, we can always write the average consensus algorithm (\[r1\]) into the following equivalent form: $$\begin{aligned}
\label{r2}
x_i(k+1)=\eta x_i(k)+\alpha^{\langle i\rangle}_k\min_{j\in \mathcal{N}_i(k)}x_j(k)+\big(1-\eta-\alpha^{\langle i\rangle}_k\big)\max_{j\in \mathcal{N}_i(k)} x_j(k),\end{aligned}$$ where $\alpha_k^{\langle i\rangle}\in[0,1-\eta]$ for all $i$ and $k$. Thus, the information processing principle behind distributed averaging is that each node iteratively takes a weighted average of its current state and the minimum and maximum states of its neighbor set.
The standard maximizing algorithm [@max1; @max2] is defined by $$\begin{aligned}
x_i(k+1)=\max_{j\in \mathcal{N}_i(k)} x_j(k),\end{aligned}$$ so distributed maximizing is each node interacting with its neighbors and simply taking the maximal state within its neighbor set.
In this paper, we aim to present a model under which we can discuss fundamental differences of some distributed information processing mechanisms. We consider the following algorithm for the node updates: $$\begin{aligned}
\label{9}
x_i(k+1)=\eta_k x_i(k)+\alpha_k \min_{j\in \mathcal{N}_i(k)}x_j(k)+\big(1-\eta_k-\alpha_k\big)\max_{j\in \mathcal{N}_i(k)} x_j(k),\end{aligned}$$ where $\alpha_k, \eta_k\geq 0$ and $\alpha_k+\eta_k \leq 1$. We denote the set of all algorithms of the form (\[9\]) by $\mathcal{A}$, when the parameter $(\alpha_k, \eta_k)$ takes value as $ \eta_k\in[0,1], \alpha_k\in[0,1-\eta_k].$ This model is a special case of (\[r2\]) as the parameter $\alpha_k$ is not depending on the node index $i$ in (\[9\]).
Note that $\mathcal{A}$ represents a uniform model for distributed averaging and maximizing algorithms. Obeying the cohesion and self-confidence assumptions, the set of (weighted) averaging algorithms, $\mathcal{A}_{\rm ave}$, consists of algorithms in the form of (\[9\]) with parameters $\eta_k\in(0,1], \alpha_k\in[0,1-\eta_k].$ The set of maximizing algorithms, $\mathcal{A}_{\rm max}$, is given by the parameter set $ \eta_k\equiv 0\ {\rm and}\ \alpha_k \equiv 0 .$
Algorithm (\[9\]) is more restrictive than (\[r2\]) in the sense that the averaging weight $\alpha^{\langle i\rangle}_k$ in (\[r2\]) might vary for different nodes. Hence, (\[9\]) cannot in general capture the averaging algorithm (\[r1\]). Except for this difference, the standing assumptions A1 and A2 of average consensus algorithms are still fulfilled for algorithm (\[9\]). Moreover, note that for (\[9\]) no lower bound is imposed on the node self-confidence.
In Algorithm (\[9\]) each node’s update only depends on the states of the minimum and maximum neighbor states at every time step. In other words, not all links are active explicitly in the iterations. Therefore, the existing convergence results on averaging algorithms cannot be applied directly, since these results rely on the connectivity of the communication graph.
If besides A1 we impose a double stochasticity assumption [@tsi11; @tsi2; @vb2] on the arc weights $a_{ij}(k)$, i.e., $\sum_{i\in\mathcal{N}_j(k)} a_{ji}(k)=1$ for all $i$ and $k$, each node’s state will converge to the average of all initial values during the evolution of Algorithm (\[r1\]) as long as consensus is reached. In the absence of double stochasticity assumption, if there is a consensus under time-varying communication, we know that the consensus limit of (\[r1\]) still is a convex combination of the initial values with coefficients invariant with respect to initial conditions [@caoming1; @caoming2; @tsi11]. However, neither of these conclusions holds for Algorithm (\[9\]). It is straightforward to see that the convergence limit is a convex combination of the initial values if consensus is reached. But due to the state-dependent node update in (\[9\]), the coefficients in the convex combination of the consensus limit indeed depend on the initial condition (even with fixed communication graph).
Problem
-------
Let $\big\{x(k;x^0)=\big(x_1(k;x^0) \dots x_n(k;x^0)\big)^T\big\}_0^\infty$ be the sequence generated by (\[9\]) for initial time $k_0$ and initial value $x^0=x(k_0)=\big(x_1(k_0)\dots x_n(k_0)\big)^T \in\mathds{R}^n$. We will identify $x(k;x^0)$ as $x(k)$ in the following discussions. We introduce the following definition on the convergence of the considered algorithm.
\(i) Asymptotic consensus is achieved for Algorithm (\[9\]) for initial condition $x(k_0)=x^0\in\mathds{R}^n$ if there exists $z_\ast(x^0)\in\mathds{R}$ such that $$\lim_{k\rightarrow \infty}x_i(k)=z_\ast,\ \ i=1,\dots,n.$$ Global asymptotic consensus is achieved if asymptotic consensus is achieved for all $k_0\geq 0$ and $x^0\in \mathds{R}^n$.
\(ii) Finite-time consensus is achieved for Algorithm (\[9\]) for initial condition $x(k_0)=x^0\in\mathds{R}^n$ if there exist $z_\ast(x^0)\in\mathds{R}$ and an integer $T_\ast(x^0)>0$ such that $$x_i(T_\ast)=z_\ast,\ \ i=1,\dots,n.$$ Global finite-time consensus is achieved if finite-time consensus is achieved for all $k_0\geq 0$ and $x^0\in \mathds{R}^n$.
The algorithm reaching consensus is equivalent with that $x(k)$ converges to the manifold $$\mathrm{C}=\Big\{x=(x_1 \dots x_n)^T:\ x_1=\dots=x_n\Big\}.$$ We call $\mathrm{C}$ the consensus manifold. Its dimension is one.
In the following, we focus on the impossibilities and possibilities of asymptotic or finite-time consensus. We will show that the convergence properties drastically change when Algorithm (\[9\]) transits from averaging to maximizing.
Convergence Impossibilities
===========================
In this section, we discuss the impossibilities of asymptotic or finite-time convergence for the averaging algorithms in $\mathcal{A}_{\rm ave}$. One theorem for each case is proven.
\[thmr1\] For every averaging algorithm in $\mathcal{A}_{\rm ave}$, finite-time consensus fails for all initial values in $\mathds{R}^n$ except for initial values on the consensus manifold.
[*Proof.* ]{} We define $$h(k)=\min_{i\in\mathcal{V}} x_i(k);\ \ H(k)=\max_{i\in\mathcal{V}} x_i(k).$$ Introduce $\Phi(k)=H(k)-h(k)$. Then clearly asymptotic consensus is achieved if and only if $\lim_{k\rightarrow \infty} \Phi(k)=0$.
Take a node $i$ satisfying $x_i(k)=h(k)$. We have $$\begin{aligned}
\label{11}
x_i(k+1)&=\eta_{k} x_i(k)+\alpha_{k} \min_{j\in \mathcal{N}_i(k)}x_j(k)+\big(1-\eta_{k}-\alpha_{k}\big)\max_{j\in \mathcal{N}_i(k)} x_j(k)\nonumber\\
&\leq (\alpha_{k}+\eta_{k})h(k)+(1-\eta_{k}-\alpha_{k})H(k).\end{aligned}$$ Similarly, taking another node $m$ satisfying $x_m(k)=H(k)$, we obtain $$\begin{aligned}
\label{12}
x_m(k+1)&=\eta_{k} x_j(k)+\alpha_{k} \min_{m\in \mathcal{N}_i(k)}x_j(k)+\big(1-\eta_{k}-\alpha_{k}\big)\max_{m\in \mathcal{N}_i(k)} x_j(k)\nonumber\\
&\geq \alpha_{k}h(k)+(1-\alpha_{k})H(k).\end{aligned}$$
With (\[11\]) and (\[12\]), we conclude that $$\begin{aligned}
\label{14}
\Phi (k+1)&= \max_{i\in\mathcal{V}} x_i(k)- \min_{i\in\mathcal{V}} x_i(k)\geq x_m(k+1)-x_i(k+1)\geq \eta_k \Phi(k).\end{aligned}$$ Therefore, since (\[14\]) holds for all $k$, we immediately obtain that for every algorithm in the averaging set $\mathcal{A}_{\rm ave}$, $$\begin{aligned}
\label{15}
\Phi(K) \geq \Phi(k_0)\prod_{k=k_0}^{K-1} \eta_k >0\end{aligned}$$ for all $K\geq k_0$ as long as $\Phi(k_0)>0$. Noticing that the initial values satisfying $\Phi(k_0)=0$ are on the consensus manifold, the desired conclusion follows. $\square$
Since the consensus manifold is a one-dimensional manifold in $\mathds{R}^n$, Theorem \[thmr1\] indicates that finite-time convergence is almost impossible for average consensus algorithms. This partially explains why finite-time convergence results are rare for averaging algorithms in the literature.
Next, we discuss the impossibility of asymptotic consensus. The following lemma is well-known.
\[lemr1\] Let $\{b_k\}_0^\infty$ be a sequence of real numbers with $b_k\in[0,1)$ for all $k$. Then $\sum_{k=0}^\infty b_k=\infty$ if and only if $\prod_{k=0}^{\infty}(1-b_k)=0$.
The following theorem on asymptotic convergence holds.
\[thmr2\] For every averaging algorithm in $\mathcal{A}_{\rm ave}$, asymptotic consensus fails for all initial values in $\mathds{R}^n$ except for initial values on the consensus manifold, if $\sum_{k=0}^\infty \big(1-\eta_k\big)<\infty$.
[*Proof.*]{} In light of Lemma \[lemr1\] and (\[15\]), we see that for every algorithm in the averaging set $\mathcal{A}_{\rm ave}$, $$\begin{aligned}
\lim_{K\rightarrow \infty}\Phi(K) \geq \Phi(k_0)\prod_{k=k_0}^{\infty} \eta_k >0\end{aligned}$$ if $\sum_{k=0}^\infty \big(1-\eta_k\big)<\infty$ for all initial values satisfying $\Phi(k_0)>0$. The desired conclusion thus follows. $\square$
Theorem \[thmr2\] indicates that $\sum_{k=0}^\infty \big(1-\eta_k\big)=\infty$ is a necessary condition for average algorithms to reach asymptotic consensus. Note that $\eta_k$ measures node self-confidence. Thus, the condition $\sum_{k=0}^\infty \big(1-\eta_k\big)=\infty$ characterizes the maximal self-confidence that nodes can hold and still reach consensus.
It is worth pointing out that Theorems \[thmr1\] and \[thmr2\] hold for any communication graph. In the following discussions, we turn to the possibilities for consensus. Then, however, the communication graph plays an important role. Time-dependent and state-dependent graphs will be studied, respectively, in the following two sections.
Time-dependent Graphs
=====================
In this section, we focus on time-dependent graphs. We first discuss a special case when the network topology is fixed, and the required connectivity for average and max-min algorithms will be treated. Next, time-varying communications will be discussed, and a series of conditions will be presented on the asymptotic or finite-time convergence of the considered algorithm under jointly connected graphs.
Fixed Graph
-----------
For fixed communication graphs, we present the following result.
\[thmt1\] Suppose $\mathcal{G}_k\equiv\mathcal{G}_\ast$ is a fixed digraph for all $k$.
\(i) For every algorithm in $\mathcal{A}_{\rm ave}$, global asymptotic consensus can be achieved only if $\mathcal{G}_\ast$ is quasi-strongly connected.
\(ii) For every algorithm in $\mathcal{A}_{\rm max}$, global finite-time consensus is achieved if and only if $\mathcal{G}_\ast$ is strongly connected.
[*Proof.*]{} (i) If $\mathcal{G}_\ast$ is not quasi-strongly connected, there exist two distinct nodes $i$ and $j$ such that $\mathcal {V}_1\cap \mathcal {V}_2=\emptyset$, where $\mathcal {V}_1=\{\mbox{nodes\ from\ which\ $i$\ is\ reachable\ in}\ \mathcal
{G}_\ast\}$ and $\mathcal {V}_2=\{$nodes from which $j$ is reachable in $\mathcal
{G}_\ast\}$. Consequently, nodes in $\mathcal{V}_1$ never receive information from nodes in $\mathcal{V}_2$. Take $x_i(k_0)=0$ for $i\in \mathcal{V}_1$ and $x_i(k_0)=1$ for $i\in \mathcal{V}_2$. Obviously, consensus cannot be achieved under this initial condition. The conclusion holds.
\(ii) (Sufficiency.) Let $v_0$ be a node with the maximal value initially. Then after one step all the nodes for which $v_0$ is a neighbor will reach the maximal value. Proceeding the analysis we see that the whole network will converge to the initial maximum in finite time.
(Necessity.) Assume that $\mathcal{G}_\ast$ is not strongly connected. There will be two different nodes $i_{\ast}$ and $j_{\ast}$ such that $j_\ast$ is not reachable from $i_\ast$. Introduce $\mathcal{V}_\ast=\{j:$ $j$ is reachable from $i_\ast\}$. Then $\mathcal{V}_\ast\neq \mathcal{V}$ because $j_\ast\notin \mathcal{V}_\ast$. Moreover, the definition of $\mathcal{V}_\ast$ guarantees that all the nodes in $\mathcal{V}\setminus \mathcal{V}_\ast$ will never be influenced by the nodes in $\mathcal{V}_\ast$. Therefore, letting the initial maximum be unique and reached by some node in $\mathcal{V}_\ast$, consensus will not be reached.
The proof is complete.$\square$
As will be shown in the following discussions, quasi-strong connectivity is not only necessary, but also sufficient to guarantee global asymptotic consensus for the algorithms in the averaging set $\mathcal{A}_{\rm ave}$ under some mild conditions on the parameters $(\alpha_k,\eta_k)$. Therefore, Theorem \[thmt1\] clearly states that quasi-strong connectivity is critical for averaging, as is strong connectivity for maximizing.
Time-varying Graph
------------------
We now turn to time-varying graphs. Denote the joint graph of $\mathcal
{G}_k$ over time interval $[k_1,k_2]$ as $\mathcal {G}\big([k_1,k_2]\big)=(\mathcal {V},\cup_{k\in[k_1,k_2]}\mathcal
{E}(k))$, where $0\leq k_1\leq k_2\leq +\infty$. We introduce the following definitions on the joint connectivity of time-varying graphs.
\(i) $\mathcal
{G}_k$ is [*uniformly jointly quasi-strongly connected*]{} (strongly connected) if there exists an integer $B\geq1$ such that $
\mathcal {G}\big([k,k+B-1]\big)$ is quasi-strongly connected (strongly connected) for all $k\geq0$.
\(ii) $\mathcal
{G}_k$ is [*infinitely jointly strongly connected*]{} if $
\mathcal {G}\big([k,\infty)\big)$ is strongly connected for all $k\geq0$.
\(iii) Suppose $\mathcal {G}_k$ is bidirectional for all $k$. Then $\mathcal
{G}_k$ is [*infinitely jointly connected*]{} if $
\mathcal {G}\big([k,\infty)\big)$ is connected for all $k\geq0$.
The uniformly joint connectivity, which requires the union graph to be connected over each bounded interval, has been extensively studied in the literature, e.g., [@tsi; @jad03; @caoming1; @caoming2; @ren]. The infinitely joint connectivity is a more general case since it does not impose an upper bound for the length of the interval where connectivity is guaranteed for the union graph. Convergence results for consensus algorithms based on infinitely joint connectivity are given in [@mor; @shi09; @shi11].
The following conclusion holds for uniformly jointly quasi-strongly connected graphs.
\[thmt2\] Suppose $\mathcal{G}_k$ is uniformly jointly quasi-strongly connected. Algorithms in the averaging set $\mathcal{A}_{\rm ave}$ achieve global asymptotic consensus if either $$\begin{aligned}
\label{20}
\Huge{\sum}_{s=0}^\infty \bigg[ \prod_{k= s(n-1)^2B}^{(s+1)(n-1)^2B-1}\alpha_k \bigg]=\infty\end{aligned}$$ or $$\begin{aligned}
\label{21}
\Huge{\sum}_{s=0}^\infty \bigg[\prod_{k= s(n-1)^2B}^{(s+1)(n-1)^2B-1}\big(1-\alpha_k -\eta_k\big)\bigg] =\infty.\end{aligned}$$
Theorem \[thmt2\] hence states that divergence of certain products of the algorithm parameters guarantees global asymptotic consensus.
It is straightforward to see that for a non-negative sequence $\{b_k\}$ with $b_k\geq b_{k+1}$ for all $k$, $\sum_{s=0}^\infty \prod_{k= s(n-1)^2B}^{(s+1)(n-1)^2-1}b_k =\infty$ if and only if $\sum_{k=0}^\infty b_k^{(n-1)^2B} =\infty$. Thus, the following corollary follows from Theorem \[thmt2\].
Suppose $\mathcal{G}_k$ is uniformly jointly quasi-strongly connected.
\(i) Assume that $\alpha_k\geq \alpha_{k+1}$ for all $k$. Algorithms in the averaging set $\mathcal{A}_{\rm ave}$ achieve global asymptotic consensus if $\sum_{k=0}^\infty \alpha_k^{(n-1)^2B} =\infty$.
\(ii) Assume that $\alpha_k+ \eta_k \leq \alpha_{k+1}+\eta_{k+1}$ for all $k$. Algorithms in the averaging set $\mathcal{A}_{\rm ave}$ achieve global asymptotic consensus if $\sum_{k=0}^\infty \big(1-\alpha_k-\eta_k\big)^{(n-1)^2B} =\infty$.
For uniformly jointly strongly connected graphs, it turns out that consensus can be achieved under weaker conditions on $(\alpha_k,\eta_k)$.
\[thmr4\] Suppose $\mathcal{G}_k$ is uniformly jointly strongly connected. Algorithms in the averaging set $\mathcal{A}_{\rm ave}$ achieve global asymptotic consensus if either $$\begin{aligned}
\Huge{\sum}_{s=0}^\infty\bigg[ \prod_{k= s(n-1)B}^{(s+1)(n-1)B-1}\alpha_k \bigg]=\infty\end{aligned}$$ or $$\begin{aligned}
\Huge{\sum}_{s=0}^\infty \bigg[ \prod_{k= s(n-1)B}^{(s+1)(n-1)B-1}\big(1-\alpha_k-\eta_k\big)\bigg] =\infty.\end{aligned}$$
Similarly, Theorem \[thmr4\] leads to the following corollary.
Suppose $\mathcal{G}_k$ is uniformly jointly strongly connected.
\(i) Assume that $\alpha_k\geq \alpha_{k+1}$ for all $k$. Averaging algorithms in the set $\mathcal{A}_{\rm ave}$ achieve global asymptotic consensus if $\sum_{k=0}^\infty \alpha_k^{(n-1)B} =\infty$.
\(ii) Assume that $\alpha_k+ \eta_k \leq \alpha_{k+1}+\eta_{k+1}$ for all $k$. Averaging algorithms in the set $\mathcal{A}_{\rm ave}$ achieve global asymptotic consensus if $\sum_{k=0}^\infty \big(1-\alpha_k-\eta_k\big)^{(n-1)B} =\infty$.
For bidirectional graphs, the conditions are much simpler to state. We present the following result.
\[thmr3\] Suppose $\mathcal {G}_k$ is bidirectional for all $k$ and $\mathcal{G}_k$ is infinitely jointly connected. Averaging algorithms in the set $\mathcal{A}_{\rm ave}$ achieves achieve global asymptotic consensus if there exists a constant $\alpha_\ast \in (0,1)$ such that either $\alpha_k\geq \alpha_\ast$ or $1-\alpha_k-\eta_k\geq \alpha_\ast$ for all $k$.
The convergence of algorithms in the maximizing set $\mathcal{A}_{\rm max}$ is stated as follows.
\[thmt3\] Maximizing algorithms in the set $\mathcal{A}_{\rm max}$ achieve global finite-time consensus if $\mathcal{G}_k$ is infinitely jointly strongly connected.
Theorems \[thmt2\]–\[thmt3\] together provide a comprehensive understanding of the convergence conditions for the considered model (\[9\]) under time-varying graphs. Infinitely jointly strong connectivity is sufficient for global finite-time consensus for algorithms in $\mathcal{A}_{\rm max}$ according to Theorem \[thmt3\], while infinitely joint connectivity cannot ensure global asymptotic consensus for algorithms in $\mathcal{A}_{\rm ave}$ in general. Thus, in this sense algorithms in $\mathcal{A}_{\rm ave}$ and $\mathcal{A}_{\rm max}$ are fundamentally different under infinitely jointly connected graphs.
The rest of this section contains the proofs of Theorems \[thmt2\]–\[thmt3\].
### **Proof of Theorem \[thmt2\]**
We continue to use the following notations: $$h(k)=\min_{i\in\mathcal{V}}x_i(k), \quad H(k)=\max_{i \in\mathcal{V}}x_i(k),$$ and $\Phi(k)=H(k)-h(k)$. Following any solution of (\[9\]), it is obvious to see that $h(k)$ is non-decreasing and $H(k)$ is non-increasing.
Note that if (\[20\]) guarantees asymptotic consensus for algorithm (\[9\]), replacing the node states $x_i(k)$ with $-x_i(k)$ leads to that (\[21\]) guarantees asymptotic consensus of algorithm (\[9\]) for $-x_i(k), i=1,\dots,n$. Since consensus for $x_i(k), i=1,\dots,n$ is equivalent with consensus for $-x_i(k), i=1,\dots,n$, (\[20\]) and (\[21\]) are equivalent in terms of consensus convergence. Thus, we just need to show that (\[20\]) is a sufficient condition for asymptotic consensus.
Take $k_\ast\geq0$ as any moment in the iterative algorithm. Take $(n-1)^2$ intervals $[k_\ast,k_\ast+B-1]$, $[k_\ast+B,k_\ast+2B-1], \dots$, $[k_\ast+(n^2-2n)B,k_\ast+(n-1)^2B-1]$. Since $\mathcal{G}_k$ is uniformly jointly quasi-strongly connected, the union graph on each of these intervals has at least one center node. Consequently, there must be a node $v_0$ and $n-1$ integers $0\leq b_1<b_2<\dots<b_{n-1}\leq n^2-2n$ such that $v_0$ is a center of $\mathcal{G}\big([k_\ast+b_iB,k_\ast+(b_i+1)B-1]\big),i=1,\dots,n-1$. Assume that $$\begin{aligned}
x_{v_0}(k_\ast)\leq \frac{1}{2} h(k_\ast)+ \frac{1}{2} H(k_\ast).\end{aligned}$$
We first bound $x_{v_0}(k)$ for $k\in[k_\ast,k_\ast+(n-1)^2B]$. It is not hard to see that $$\begin{aligned}
x_{v_0}(k_\ast+1)&=\eta_{k_\ast}x_{v_0}(k_\ast)+ \alpha_{k_\ast} \min_{j\in \mathcal{N}_{v_0}(k_\ast)}x_j(k_\ast)+\big(1-\alpha_{k_\ast}-\eta_{k_\ast}\big)\max_{j\in \mathcal{N}_{v_0}(k_\ast)} x_j(k_\ast)\nonumber\\
&\leq \big(\alpha_{k_\ast}+\eta_{k_\ast}\big) \Big(\frac{1}{2} h(k_\ast)+ \frac{1}{2} H(k_\ast)\Big)+\big(1-\alpha_{k_\ast}-\eta_{k_\ast}\big)H(k_\ast)\nonumber\\
&\leq \alpha_{k_\ast} \Big(\frac{1}{2} h(k_\ast)+ \frac{1}{2} H(k_\ast)\Big)+\big(1-\alpha_{k_\ast}\big)H(k_\ast)\nonumber\\
&= \frac{\alpha_{k_\ast}}{2}h(k_\ast)+\big(1-\frac{\alpha_{k_\ast}}{2}\big)H(k_\ast).\end{aligned}$$ Proceeding, we obtain $$\begin{aligned}
\label{3}
x_{v_0}(k_\ast+m)\leq \frac{\prod_{k=k_\ast} ^{k_\ast +m-1}\alpha_{k} }{2}h(k_\ast)+\big(1-\frac{\prod_{k=k_\ast} ^{k_\ast +m-1}\alpha_{k} }{2}\big)H(k_\ast),\ \ m=0,1,\dots.\end{aligned}$$
Since $v_0$ is a center of $\mathcal{G}\big([k_\ast+b_1B,k_\ast+(b_1+1)B-1]\big)$, there exists another node $v_1$ such that $v_0$ is a neighbor of $v_1$ for some $k_1\in [k_\ast+b_1B,k_\ast+(b_1+1)B-1]$. As a result, based on (\[3\]), we have $$\begin{aligned}
x_{v_1}({k}_1+1)& =\eta_{k_1}x_{v_1}(k_1)+\alpha_{k_1}\min_{j\in \mathcal{N}_{v_1}(k_1)}x_j(k_1)+\big(1-\alpha_{k_1}-\eta_{k_1}\big)\max_{j\in \mathcal{N}_{v_1}(k_1)} x_j(k_1)\nonumber\\
&\leq \alpha_{k_1} x_{v_0}(k_1)+\big(1-\alpha_{k_1}\big)H(k_1)\nonumber\\
&\leq \alpha_{k_1} \Big(\frac{\prod_{k=k_\ast} ^{k_1-1}\alpha_{k} }{2}h(k_\ast)+\big(1-\frac{\prod_{k=k_\ast} ^{k_1-1}\alpha_{k}}{2}\big)H(k_\ast)\Big)+\big(1-\alpha_{k_1}\big)H(k_\ast)\nonumber\\
&= \frac{\prod_{k=k_\ast} ^{k_1}\alpha_{k} }{2}h(k_\ast)+\big(1-\frac{\prod_{k=k_\ast} ^{k_1}\alpha_{k}}{2}\big)H(k_\ast).\end{aligned}$$ Proceeding, we have $$\begin{aligned}
x_{v_0}(k_\ast+m)\leq \frac{\prod_{k=k_\ast} ^{k_\ast +m-1}\alpha_{k} }{2}h(k_\ast)+\big(1-\frac{\prod_{k=k_\ast} ^{k_\ast +m-1}\alpha_{k} }{2}\big)H(k_\ast),\ \ m=(b_1+1)B,\dots.\end{aligned}$$
Continuing the analysis on time intervals $[k_\ast+b_iB,k_\ast+(b_i+1)B-1]$ for $i=2,\dots,n-1$ and nodes $v_2,v_3,\dots,v_{n-1}$, similar upper bounds for each node can be obtained: $$\begin{aligned}
x_{v_i}(k_\ast+m)\leq \frac{\prod_{k=k_\ast} ^{k_\ast +m-1}\alpha_{k} }{2}h(k_\ast)+\big(1-\frac{\prod_{k=k_\ast} ^{k_\ast +m-1}\alpha_{k} }{2}\big)H(k_\ast),\ \ m=(b_i+1)B,\dots.\end{aligned}$$ This immediately leads to $$\begin{aligned}
x_{v_i}(k_\ast+(n-1)^2B)\leq \frac{\prod_{k=k_\ast} ^{k_\ast +(n-1)^2B-1}\alpha_{k} }{2}h(k_\ast)+\big(1-\frac{\prod_{k=k_\ast} ^{k_\ast +(n-1)^2B-1}\alpha_{k} }{2}\big)H(k_\ast),\ i=0,1,\dots,n\end{aligned}$$ which implies $$\begin{aligned}
H\big(k_\ast+(n-1)^2B\big)\leq \frac{\prod_{k=k_\ast} ^{k_\ast +(n-1)^2B-1}\alpha_{k} }{2}h(k_\ast)+\big(1-\frac{\prod_{k=k_\ast} ^{k_\ast +(n-1)^2B-1}\alpha_{k} }{2}\big)H(k_\ast).\end{aligned}$$ Thus, we have $$\begin{aligned}
\label{6}
\Phi\big(k_\ast+(n-1)^2B\big)&= {H}\big(k_\ast+(n-1)^2B\big)-{h}\big(k_\ast+(n-1)^2B\big)\nonumber\\
&\leq \frac{\prod_{k=k_\ast} ^{k_\ast +(n-1)^2B-1}\alpha_{k} }{2}h(k_\ast)+\big(1-\frac{\prod_{k=k_\ast} ^{k_\ast +(n-1)^2B-1}\alpha_{k} }{2}\big)H(k_\ast)-h(k_\ast)\nonumber\\
&=\Big(1- \frac{\prod_{k=k_\ast} ^{k_\ast +(n-1)^2B-1}\alpha_{k} }{2}\Big)\Phi(k_\ast).\end{aligned}$$
From a symmetric analysis by bounding $h(k_\ast+(n-1)^2B)$ from below, we know that (\[6\]) also holds for the other condition with $x_{v_0}(k_\ast)\geq \frac{1}{2} h(k_\ast)+ \frac{1}{2} H(k_\ast)$. Therefore, since $k_\ast$ is selected arbitrarily, we can assume the initial time is $k_0=0$, without loss of generality, and then conclude that $$\begin{aligned}
\Phi\big(K(n-1)^2B\big)
&\leq \Phi(0) \prod_{s=0} ^{K-1}\Bigg(1- \frac{1 }{2}\prod_{k=s(n-1)^2B} ^{(s+1)(n-1)^2B-1}\alpha_{k}\Bigg).\end{aligned}$$
The desired conclusion follows immediately from Lemma \[lemr1\].
### Proof of Theorem \[thmr4\]
Notice that in a strongly connected graph, every node is a center node. Therefore, when $\mathcal{G}_k$ is uniformly jointly strongly connected, taking $k_\ast\geq0$ as any moment in the iteration and $n-1$ intervals $[k_\ast,k_\ast+B-1]$, $[k_\ast+B,k_\ast+2B-1], \dots$, $[k_\ast+(n-2)B,k_\ast+(n-1)B-1]$, any node $i\in\mathcal{V}$ is a center node for the union graph over each of these intervals. As a result, the desired conclusion follows repeating the analysis used in the proof of Theorem \[thmt2\].
### Proof of Theorem \[thmr3\]
Similar to the proof of Theorem \[thmt2\], we only need to show that the existence of a constant $\alpha_\ast \in (0,1)$ such that $\alpha_k\geq \alpha_\ast$ is sufficient for asymptotic consensus.
Take $k^\ast_1\geq0$ as an arbitrary moment in the iterative algorithm. Take a node $u_0\in\mathcal{V}$ satisfying $x_{u_0}(k^\ast_1)=h(k^\ast_1)$. We define $$\begin{aligned}
k_1= \inf\big\{k\geq k^\ast_1:\mbox{there exists another node connecting $u_0$ at time}\ k\big\}\end{aligned}$$ and then $$\begin{aligned}
\mathcal{V}_1= \big\{k\geq k^\ast_1:\mbox{nodes which are connected to $u_0$ at time}\ k_1\big\}.\end{aligned}$$ Thus, we have $$\begin{aligned}
x_{u_0}(k_1+1)&=\eta_{k_1}x_{u_0}(k_1)+\alpha_{k_1}\min_{j\in \mathcal{N}_{u_0}(k_1)}x_j(k_1)+\big(1-\alpha_{k_1} -\eta_{k_1}\big)\max_{j\in \mathcal{N}_{u_0}(k_1)} x_j(k_1)\nonumber\\
&\leq (\alpha_{k_1}+\eta_{k_1}) x_{u_0}(k_1)+\big(1-\alpha_{k_1}-\eta_{k_1}\big)H(k_1)\nonumber\\
&\leq (\alpha_{k_1}+\eta_{k_1}) h(k_1^\ast)+\big(1-\alpha_{k_1}-\eta_{k_1}\big)H(k_1^\ast)\nonumber\\
&\leq \alpha_{k_1} h(k_1^\ast)+\big(1-\alpha_{k_1}\big)H(k_1^\ast)\nonumber\\
&\leq \alpha_{\ast} h(k_1^\ast)+\big(1-\alpha_{\ast}\big)H(k_1^\ast)\end{aligned}$$ and $$\begin{aligned}
x_{i}(k_1+1)&=\eta_{k_1}x_{i}(k_1)+\alpha_{k_1}\min_{j\in \mathcal{N}_{i}(k_1)}x_j(k_1)+\big(1-\alpha_{k_1}\big)\max_{j\in \mathcal{N}_{i}(k_1)} x_j(k_1)\nonumber\\
&\leq \alpha_{k_1} x_{u_0}(k_1)+\big(1-\alpha_{k_1}\big)H(k_1)\nonumber\\
&\leq \alpha_{k_1} h(k^\ast_1)+\big(1-\alpha_{k_1}\big)H(k^\ast_1)\nonumber\\
&\leq \alpha_{\ast} h(k_1^\ast)+\big(1-\alpha_{\ast}\big)H(k_1^\ast)\end{aligned}$$ for all $i\in \mathcal{V}_1$.
Note that if nodes in $\{u_0\}\cup\mathcal{V}_1$ are not connected with other nodes in $\mathcal{V}\setminus (\{u_0\}\cup\mathcal{V}_1)$ during $[k_1+1,k_1+s]$, $s\geq 1$, we have that for all $i\in\{u_0\}\cup \mathcal{V}_1$, $$\begin{aligned}
x_{i}(k_1+m)\leq \alpha_{\ast} h(k_1^\ast)+\big(1-\alpha_{\ast}\big)H(k_1^\ast),\ \ m=1,\dots,s+1.\end{aligned}$$
Continuing the estimate, $k_2,\dots,k_d$ and $\mathcal{V}_2,\dots,\mathcal{V}_d$ can be defined correspondingly until $\mathcal{V}=\{u_0\}\cup(\cup_{i=1}^d \mathcal{V}_i)$, so eventually we have $$\begin{aligned}
x_{i}(k_d+1)\leq \alpha^d_\ast h(k^\ast_1)+\big(1-\alpha^d_\ast\big)H(k^\ast_1), \ \ i=1,\dots,n,\end{aligned}$$ which implies $$\begin{aligned}
\label{5}
{H}(k_d+1)\leq \alpha^d_\ast h(k^\ast_1)+\big(1-\alpha^d_\ast\big)H(k^\ast_1).\end{aligned}$$ We denote $k^\ast_2=k_d+1$. Because it holds that $d\leq n-1$, we see from (\[5\]) that $$\begin{aligned}
\Phi(k^\ast_2)\leq \big(1-\alpha^{n-1}_\ast\big)\Phi(k^\ast_1).\end{aligned}$$ Since $\mathcal{G}_k$ is infinitely jointly connected, this process can be carried on for an infinite sequence $k_1^\ast<k_2^\ast<\dots$. Thus, asymptotic consensus is achieved for all initial conditions. This completes the proof.
### Proof of Theorem \[thmt3\]
Let $v_0$ be a node with the maximal value initially. Because $\mathcal{G}_k$ is infinitely jointly strongly connected, we can define $$\begin{aligned}
k_1= \inf\big\{k\geq k^\ast_1:\mbox{there exists another node for which $v_0$ is a neighbor at time}\ k\big\}\end{aligned}$$ and then $$\begin{aligned}
\mathcal{V}_1= \big\{k\geq k^\ast_1:\mbox{nodes for which $v_0$ is a neighbor at time}\ k_1\big\}.\end{aligned}$$ Then at time ${k}_1+1$ all the nodes in $\mathcal{V}_1$ will reach the maximal value. Proceeding the analysis we know that the whole network will converge to the initial maximum in finite time.$\square$
State-dependent Graphs
======================
In this section, we investigate the convergence of Algorithm (\[9\]) for state-dependent graphs. We are interested in a particular set of averaging algorithms, $\mathcal{A}_{\rm ave}^\ast$, where $(\alpha_k, \eta_k)$ takes value $ \eta_k \equiv 0$, $\alpha_k\in(0,1) $. Algorithms in $\mathcal{A}_{\rm ave}^\ast$ correspond to the case when the self-confidence assumption A2 does not hold, and are of the form $$\begin{aligned}
\label{r100}
x_i(k+1)=\alpha_k\min_{j\in \mathcal{N}_i(k)}x_j(k)+\big(1-\alpha_k\big)\max_{j\in \mathcal{N}_i(k)} x_j(k).\end{aligned}$$ Algorithms in $\mathcal{A}_{\rm ave}^\ast$ still have local cohesion. Hence, they fulfill Assumption A1 but not A2. In fact, averaging algorithms without self-confidence have been investigated in classical works on the convergence of product of stochastic matrices, e.g., [@wolf; @haj; @degroot].
State-dependent Communication
-----------------------------
In both Krause’s [@krause] and Vicsek’s [@vic95] models, nodes interact with neighbors whose distance is within a certain communication range. Convergence analysis for consensus algorithms under such models can be found in [@vb1; @julien; @tang; @liuguo]. Recently, it was discovered through empirical data that in a bird flock each bird seems to interact with a fixed number of nearest neighbors, rather than with all neighbors within a fixed metric distance [@pnas]. Nearest-neighbor model has been studied under a probabilistic setting on the graph connectivity for wireless communication networks [@kumar]. From a social network point of view, the evolution of opinions may result from similar models since members tend to exchange information with a fixed number of other members who hold a similar opinion as themselves [@degroot; @julien]. In this section, we consider a network model in which nodes interact only with other nodes having a close state value. Consider the following nearest-neighbor rule.
For a positive integer $\mu$ and any node $i\in\mathcal{V}$, there is a link entering $i$ from each node in the set $\mathcal{N}_i^-(k)\cup\mathcal{N}_i^+(k)$, where $$\mbox{$\mathcal{N}_i^-(k)=\big\{$nearest $\mu$ neighbors from $\{j\in\mathcal{V}: x_{j}(k)$$<x_i(k)\}\big\}$}$$ denotes the nearest smaller neighbor set, and $$\mbox{$\mathcal{N}_i^+(k)=\big\{$nearest $\mu$ neighbors from $\{j\in\mathcal{V}: x_{j}(k)$$>x_i(k)\}\big\}$}$$ denotes the nearest larger neighbor set. The graph defined by this nearest neighbor rule is denoted as $\mathcal{G}^{\mu}_{x(k)}$, $k=0,1,\dots$.
Naturally, if there are less than $\mu$ nodes with states smaller than $x_i(k)$, $\mathcal{N}_i^-(k)$ has less that $\mu$ elements. Similar condition holds for $\mathcal{N}_i^+(k)$. Hence, the number of neighbor nodes is not necessarily fixed in the nearest-neighbor graph.
Note that, at each time $k$, the nearest-neighbor graph is uniquely determined by the node states. The node interactions are indeed determined by the distance between the node states. In this sense, the nearest-neighbor graph shares similar structure with Krause’s model [@krause; @vb1], where each node communicates with the nodes within certain radius. This nearest-neighbor graph also fulfills the interaction structure in the bird flock model discussed in [@pnas] since each node communicates with an almost fixed number of neighbors, nearest from above and below.
Note that in the definition of the nearest-neighbor graph, nodes may have neighbors with the same state values. We consider the following nearest-value graph, where each node considers only neighbors with different state values.
[*(Nearest-value Graph)*]{} For a positive integer $\mu$ and any node $i\in\mathcal{V}$, there is a link entering $i$ from each node in the set $\mathcal{N}_i^-(k)\cup\mathcal{N}_i^+(k)$, where $$\mbox{$\mathcal{N}_i^-(k)=\big\{$nearest $\mu$ neighbors with different values from $\{j\in\mathcal{V}: x_{j}(k)$$<x_i(k)\}\big\}$}$$ denotes the nearest smaller neighbor set, and $$\mbox{$\mathcal{N}_i^+(k)=\big\{$nearest $\mu$ neighbors with different values from $\{j\in\mathcal{V}: x_{j}(k)$$>x_i(k)\}\big\}$}$$ denotes the nearest larger neighbor set. The graph defined by this nearest neighbor rule is denoted as $\mathcal{G}^{\mu v}_{x(k)}$, $k=0,1,\dots$.
An illustration of nearest-neighbor and nearest-value graphs at a specific time instance $k$ is shown in Figure \[diff\] for $n=4$ nodes and $\mu=2$.
![Examples of nearest-neighbor graph $\mathcal{G}^{\mu}_{x(k)}$ and nearest-value graph $\mathcal{G}^{\mu v}_{x(k)}$ for $\mu=2$. Note that for a given set of states, these graphs are in general not unique. []{data-label="diff"}](neighborgraph.eps){width="\textwidth"}
![Examples of nearest-neighbor graph $\mathcal{G}^{\mu}_{x(k)}$ and nearest-value graph $\mathcal{G}^{\mu v}_{x(k)}$ for $\mu=2$. Note that for a given set of states, these graphs are in general not unique. []{data-label="diff"}](valuegraph.eps){width="\textwidth"}
Basic Lemmas
------------
We first establish two useful lemmas for the analysis of nearest-neighbor and nearest-value graphs. The following lemma indicates that the order of node states is preserved.
\[lem1\] For any two nodes $u,v\in\mathcal{V}$ and every algorithm in $\mathcal{A}$, under either the nearest-neighbor graph $\mathcal{G}^{\mu}_{x(k)}$ or the nearest-value graph $\mathcal{G}^{\mu v}_{x(k)}$, we have
\(i) $x_u(k+1)=x_v(k+1)$ if $x_u(k)=x_v(k)$;
\(ii) $x_u(k+1)\leq x_v(k+1)$ if $x_u(k)< x_v(k)$.
[*Proof.*]{} When $x_u(k)=x_v(k)$, we have $\{j: x_{j}(k)<x_u(k)\}$=$\{j: x_{j}(k)<x_v(k)\}$ and $\{j: x_{j}(k)>x_u(k)\}$=$\{j: x_{j}(k)>x_v(k)\}$. Thus, for either $\mathcal{G}^{\mu}_{x(k)}$ or $\mathcal{G}^{\mu v}_{x(k)}$, both $$\min_{j\in \mathcal{N}_u(k)} x_j(k) =\min_{j\in \mathcal{N}_v(k)} x_j(k) \ \ \mbox{and}\ \ \max_{j\in \mathcal{N}_u(k)} x_j(k) =\max_{j\in \mathcal{N}_v(k)} x_j(k)$$ hold. Then (i) follows straightforwardly.
If $x_u(k)<x_v(k)$, it is easy to see that $$\min_{j\in \mathcal{N}_u(k)} x_j(k) \leq \min_{j\in \mathcal{N}_v(k)} x_j(k) ;\quad \max_{j\in \mathcal{N}_u(k)} x_j(k) \leq \max_{j\in \mathcal{N}_v(k)} x_j(k)$$ according to the definition of neighbor sets, which implies (ii) immediately. $\square$
Define $$\Upsilon_k=\Big| \big\{x_1(k),\dots,x_n(k)\big\}\Big|$$ as the number of distinct node states at time $k$, where $\big|S\big|$ for a set $S$ represents its cardinality. Then Lemma \[lem1\] implies that $\Upsilon_{k+1}\leq \Upsilon_k$ for all $k\geq 0$. This point plays an important role in the convergence analysis.
Moreover, for both the nearest-neighbor graph $\mathcal{G}^\mu_{x(k)}$ and the nearest-value graph $\mathcal{G}^{\mu v}_{x(k)}$, in order to distinguish the node states under different values of neighbors, we denote $x_i^\mu(k)$ as the state of node $i$ when the number of larger or smaller neighbors is $\mu$. Correspondingly, we denote $$h^{\mu}(k)=\min_{i\in\mathcal{V}} x_i^\mu (k),\ \ H^\mu(k)=\max_{i\in\mathcal{V}} x_i^\mu(k).$$ and $\Phi^\mu(k)=H^\mu(k)-h^\mu(k)$. We give another lemma indicating that the convergence speed increases as the number of neighbors increases, which is quite intuitive because apparently graph connectivity increases as the number of neighbors increases.
\[lem2\] Consider either the nearest-neighbor graph $\mathcal{G}^\mu_{x(k)}$ or the nearest-value graph $\mathcal{G}^{\mu v}_{x(k)}$. Given two integers $1\leq\mu_1\leq \mu_2$. For every algorithm in $\mathcal{A}$ and every initial value, we have $\Phi^{\mu_1}(k)\geq \Phi^{\mu_2}(k)$ for all $k$.
[*Proof.*]{} Fix the initial condition at time $k_0$. Let $m\in\mathcal{V}$ be a node satisfying $x_m^{\mu_1}(k_0)=h^{\mu_1}(k_0)$ and $x_m^{\mu_2}(k_0)=h^{\mu_2}(k_0)$. The order preservation property given by Lemma \[lem1\] guarantees that $x_m^{\mu_1}(k)=h^{\mu_1}(k)$ and $x_m^{\mu_2}(k)=h^{\mu_2}(k)$ for all $k\geq k_0$. It is straightforward to see that $x_m^{\mu_1}(k_0+1)\leq x_m^{\mu_2}(k_0+1)$ if $\mu_1 \leq \mu_2$, and continuing we know that $x_m^{\mu_1}(k_0+s)\leq x_m^{\mu_2}(k_0+s)$ for all $s\geq 2$. Thus, we have $h^{\mu_1}(k) \leq h^{\mu_2}(k)$ for all $k\geq k_0$. A symmetric analysis leads to $H^{\mu_1}(k) \geq H^{\mu_2}(k)$ for all $k$ and the desired conclusion thus follows. $\square$
Convergence for Nearest-neighbor Graph
--------------------------------------
For algorithms in the set $\mathcal{A}_{\rm ave}^\ast$, we present the following result under nearest-neighbor graph.
\[thm8\] Consider the nearest-neighbor graph $\mathcal{G}^\mu_{x(k)}$.
\(i) When $n\leq \mu+1$, each algorithm in $\mathcal{A}_{\rm ave}^\ast$ achieves global finite-time consensus;
\(ii) When $n> \mu+1$, each algorithm in $\mathcal{A}_{\rm ave}^\ast$ fails to achieve finite-time consensus for almost all initial values;
\(iii) When $n> \mu+1$, each algorithm in $\mathcal{A}_{\rm ave}^\ast$ achieves global asymptotic consensus if $\{\alpha_k\}$ is monotone.
[*Proof.*]{} (i) When $n\leq \mu+1$, the communication graph is the complete graph. Thus, consensus will be achieved in one step following (\[9\]) for every algorithm in $\mathcal{A}_{\rm ave}^\ast$.
\(ii) Let $n> \mu+1$. We define two index set $$\mathcal{I}_k^-=\big\{i:\ x_i(k)=h(k)=\min_{i\in\mathcal{V}} x_i (k)\big\};\ \mathcal{I}_k^+=\big\{i:\ x_i(k)=H(k)=\max_{i\in\mathcal{V}} x_i (k)\big\}.$$
[*Claim.*]{} Suppose both $\mathcal{I}_k^-$ and $\mathcal{I}_k^+$ contain one node only. Then so do $\mathcal{I}_{k+1}^-$ and $\mathcal{I}_{k+1}^+$.
Let $u$ and $v$ be the unique element in $\mathcal{I}_k^-$ and $\mathcal{I}_k^+$, respectively. Take $m\in\mathcal{V}\setminus\{u\}$. Noting the fact that $x_m(k)>x_u (k)$ and $\mu\leq n-2$, we have $$\min_{j\in \mathcal{N}_u(k) } x_j(k) \leq \min_{j\in \mathcal{N}_m(k) } x_j(k),\ \max_{j\in \mathcal{N}_u(k) } x_j(k) <\max_{j\in \mathcal{N}_m(k) } x_j(k).$$ This leads to $x_{m}(k+1)>x_u(k+1)$. Therefore, $u$ is still the unique element in $\mathcal{I}_{k+1}^-$. Similarly we can prove that $v$ is still the unique element in $\mathcal{I}_{k+1}^+$. The claim holds.
Now observe that $$\Delta\doteq\bigcup_{u\neq v} \big\{x=(x_1\dots x_n)^T:\ x_u <\min_{m\in\mathcal{V}\setminus\{u\}} x_m\ {\rm and}\ x_v>\max_{m\in\mathcal{V}\setminus\{v\}} x_m,\big\}$$ has measure zero with respect to the standard Lebesgue measure on $\mathds{R}^n$. For any initial value not in $\Delta$, we have both $\mathcal{I}_k^-$ and $\mathcal{I}_k^+$ contain one unique element, and thus finite-time consensus is impossible. The desired conclusion follows.
\(iii) Recall that $$\Upsilon_k=\Big| \big\{x_1(k),\dots,x_n(k)\big\}\Big|.$$ Since $\Upsilon_{k+1}\leq \Upsilon_k$ holds for all $k$ according to Lemma \[lem1\], there exists two integers $0\leq m\leq n$ and $T\geq0$ such that $$\begin{aligned}
\Upsilon_k=m,\end{aligned}$$ for all $k\geq T$. Thus, we can sort the possible node states for all $k\geq T$ as $$y_1(k)<y_2(k)<\dots< y_m(k).$$
Apparently $m\neq 1, 2$ since otherwise the graph is complete for time $\ell$ with $\Upsilon_{\ell}=1,2$ and consensus is reached after one step. We assume $m\geq 3$ in the following discussions.
Algorithm (\[r100\]) can be equivalently transformed to the dynamics on $\{y_1(k),\dots,y_m(k)\}$. Moreover, based on Lemma \[lem2\], we only need to prove asymptotic consensus for the case $\mu=1$.
Let $\mu=1$ and $k\geq T$. For algorithms in $\mathcal{A}_{\rm ave}^\ast$, the dynamics of $\{y_1(k),\dots,y_m(k)\}$ can be written: $$\label{22}
\begin{cases}
y_1(k+1)=\alpha_k y_1(k)+(1-\alpha_k ) y_{2}(k); \\
y_2(k+1)=\alpha_k y_{1}(k)+(1-\alpha_k) y_{3}(k);\\
\quad \quad \quad \vdots \\
y_{m-1}(k+1)=\alpha_k y_{m-2}(k)+(1-\alpha_k) y_{m}(k);\\
y_m(k+1)=\alpha_k y_{m-1}(k)+(1-\alpha_k) y_{m}(k).
\end{cases}$$
Now let $\{\alpha_k\}$ be monotone, say, non-decreasing. Then we have $\alpha_k \geq \alpha_T>0$. Therefore, for all $k\geq T$, we have $$\begin{aligned}
y_1(k+1)=\alpha_k y_1(k)+(1-\alpha_k ) y_{2}(k)\leq \alpha_T y_1(k)+(1-\alpha_T) y_{m}(k),\end{aligned}$$ and continuing we know that $$\begin{aligned}
y_1(k+s)\leq \alpha_T^s y_1(k)+(1-\alpha_T^s) y_{m}(k), s\geq 1.\end{aligned}$$ Similarly for $y_2(k)$, we have $$\begin{aligned}
y_2(k+2)&=\alpha_{k+1}y_{1}(k+1)+(1-\alpha_{k+1}) y_{3}(k+1) \nonumber\\
&\leq \alpha_T \big( \alpha_T y_1(k)+(1-\alpha_T) y_{m}(k) \big)+(1-\alpha_T) y_{m}(k)\nonumber\\
&\leq \alpha_T^2 y_1(k)+(1-\alpha_T^2) y_{m}(k)\end{aligned}$$ and $$\begin{aligned}
y_2(k+s)\leq \alpha_T^s y_1(k)+(1-\alpha_T^s) y_{m}(k), \ s\geq 2.\end{aligned}$$
Proceeding the analysis, eventually we arrive at $$\begin{aligned}
y_i(k+n-1)\leq \alpha_T^{n-1} y_1(k)+(1-\alpha_T^{n-1}) y_{m}(k), \ i=1,\dots,n,\end{aligned}$$ which yields $$\begin{aligned}
\Phi(k+n-1)&=y_m(k+n-1)-y_1(k+n-1)\nonumber\\
&\leq\alpha_T^{n-1} y_1(k)+(1-\alpha_T^{n-1}) y_{m}(k)-y_1(k)\nonumber\\
&= (1-\alpha_T^{n-1})\Phi(k).\end{aligned}$$ Thus, global asymptotic consensus is achieved. The other case with $\{\alpha_k\}$ being non-increasing can be proved using a symmetric argument. The desired conclusion follows.
This completes the proof of the theorem.$\square$
In Theorem \[thm8\], the asymptotic consensus statement relies on the condition that $\{\alpha_k\}$ is monotone. From the proof of Theorem \[thm8\] we see that this condition can be replaced by that there exists a constant $\varepsilon \in (0,1)$ such that either $\alpha_k\geq \varepsilon$ or $\alpha_k\leq1- \varepsilon$ for all $k$. In fact, we conjecture that the asymptotic consensus statement of Theorem \[thm8\] holds true for all $\{\alpha_k\}$, i.e., we believe that asymptotic consensus is achieved for all algorithms in $\mathcal{A}_{\rm ave}^\ast$ under nearest-neighbor graphs.
Theorem \[thm8\] indicates that $\mu+1$ is a critical number of nodes for nearest-neighbor graphs: for algorithms in $\mathcal{A}_{\rm ave}^\ast$, finite-time consensus holds globally if $n\leq \mu+1$, and fails almost globally if $n>\mu+1$. Note that $n\leq \mu+1$ implies that the communication graph is the complete graph, which is rare in general. Recalling that Theorem \[thmr1\] showed that finite-time consensus fails almost globally for algorithms in $\mathcal{A}_{\rm ave}$, we conclude that finite-time consensus is generally rare for averaging algorithms in $\mathcal{A}$, no matter with ($\mathcal{A}_{\rm ave}$) or without ($\mathcal{A}_{\rm ave}^\ast$) the self-confidence assumption.
For algorithms in $\mathcal{A}_{\rm max}$, we present the following result.
Consider the nearest-neighbor graph $\mathcal{G}^\mu_{x(k)}$. Algorithms in $\mathcal{A}_{\rm max}$ achieve global finite-time consensus in no more than $\lceil \frac{n}{\mu}\rceil$ steps, where $\lceil z\rceil$ represents the smallest integer no smaller than $z$.
[*Proof.*]{} Without loss of generality, we assume that $x_1(0),\dots,x_n(0)$ are mutually different. We sort the initial values of the nodes as $$x_{i_1}(0)<x_{i_2}(0)<\dots< x_{i_n}(0).$$ Here $i_m$ denotes node with the $m$’th largest value initially.
Observing that $i_n$ is a right-hand side neighbor of nodes $i_{n-\mu},i_{n-\mu+1},\dots,i_{n-1}$, we have $$x_{i_\tau}(1)=x_{i_n}(0), \ \tau=n-\mu,\dots,n.$$ This leads to $\Upsilon_1= \Upsilon_0-\mu$ where $
\Upsilon_k=\big| \big\{x_1(k),\dots,x_n(k)\big\}\big|$. Proceeding the same analysis we know that consensus is achieved in no more than $\lceil \frac{n}{\mu}\rceil$ steps. The desired conclusion follows. $\square$
Convergence for Nearest-value Graph
-----------------------------------
In this subsection, we study the convergence for nearest-value graphs. Since nearest-value graph $\mathcal{G}^{\mu v}_{x(k)}$ indeed increases the connectivity of $\mathcal{G}^{\mu}_{x(k)}$, the asymptotic consensus statement of Theorem \[thm8\] also holds for $\mathcal{G}^{\mu v}_{x(k)}$. The main result for finite-time consensus of nearest-value graphs is presented as follows. It turns out that the critical number of nodes for nearest-value graphs is $2\mu$.
\[thm10\]Consider the nearest-value graph $\mathcal{G}^{\mu v}_{x(k)}$.
\(i) When $n\leq 2\mu$, algorithms in $\mathcal{A}_{\rm ave}^\ast$ achieve global finite-time consensus in no more than $\lceil \log_2(2\mu+1)\rceil$ steps;
\(ii) When $n> 2\mu$, algorithms in $\mathcal{A}_{\rm ave}^\ast$ fail to achieve finite-time consensus for almost all initial conditions.
[*Proof.*]{} (i) Suppose $n\leq 2\mu$. Based on Lemma \[lem2\], without loss of generality, we assume $n=2\mu$ and the initial values of the nodes are mutually different. Now we have $\Upsilon_0=\big| \{x_1(0),\dots,x_n(0)\}\big|=2\mu$. We fist show the following claim.
[*Claim.*]{} If $\Upsilon_k=2\mu-A$ with $A\geq 0$ an integer, then $\Upsilon_{k+1}\leq \Upsilon_k-A-1$.
We order the node states at time $k$ and denote them as $$Y_1<Y_2<\dots< Y_{\Upsilon_k}.$$ When $\Upsilon_k=2\mu-A$, it is not hard to find that the for all $m=\mu-A,\dots,\mu+1$, each node with value $Y_{\Upsilon_m}$ will connect to some node with value $Y_1$, and some other node with value $Y_{\Upsilon_k}$. Therefore, the nodes with value $Y_{\Upsilon_m}, m=\mu-A,\dots,\mu+1$ will reach the same state after the $k$’th update. The claim holds.
Therefore, by induction we have $\Upsilon_{k}=\max\{0, \Upsilon_0-\sum_{m=0}^{k-1} 2^m\}=\max\{0, 2\mu-(2^k-1)\}$. The conclusion (i) follows straightforwardly.
\(ii) Suppose $n>2\mu$. Let $x_1(0),\dots,x_n(0)$ be mutually different. Then it is not hard to see that for any two nodes $u$ and $v$ with $x_u(0)<x_v(0)$, at least one of $$\min_{j\in \mathcal{N}_u(0) } x_j(0) <\min_{j\in \mathcal{N}_v(0) } x_j(0)$$ or $$\max_{j\in \mathcal{N}_u(0) } x_j(0) <\max_{j\in \mathcal{N}_v(0) } x_j(0)$$ holds. This immediately leads to $x_u(1)<x_v(1)$. Because $u$ and $v$ are arbitrarily chosen, we can conclude that $\Upsilon_1=\Upsilon_0$. By an induction argument we see that $\Upsilon_k=\Upsilon_0=n$ for all $k\geq0$, or equivalently, consensus cannot be achieved in finite time. Now observe that $$\bigcup_{i\geq j} \big\{x=(x_1\dots x_n)^T:x_i=x_j\big\}$$ has measure zero with respect to the standard Lebesgue measure on $\mathds{R}^n$. The desired conclusion thus follows. $\square$
Numerical Examples
==================
In this section, we present two numerical examples.
[**Example 1.**]{} Consider a network with six nodes $\mathcal{V}=\{1,\dots,6\}$. The communication graph is fixed and directed, as indicated in Figure \[graph\]. Initial values for each node are $x_i(0)=i-1, i=1,\dots,6$.
Let $\eta_k\equiv 0$ and $\alpha_k\equiv\alpha$ in Algorithm (\[9\]). Consider eleven values in the interval $[0,1/2]$ of the parameter $\alpha$. Recall that $\Phi(k)=\max_{i\in\mathcal{V}}x_i(k) - \min_{i\in\mathcal{V}}x_i(k) $ is the consensus measure. The trajectory corresponding to each value of $\alpha$ is shown in Figure \[traj\]. We see that finite-time convergence is achieved only when $\alpha$ is zero, i.e., only for the maximizing case of Algorithm (\[9\]).
[**Example 2.**]{} Consider a network with $n=128$ nodes. Take initial value for the $i$’th node as $x_i(0)=i$. Let $\eta_k\equiv 0$ and $\alpha_k\equiv0.5$ in Algorithm (\[9\]). The communication graph is given by the nearest-neighbor or nearest-value rule. The evolution of the convergence properties depending on $\mu$ is shown in Figure \[neighborgraph\] and \[valuegraph\]. In Figure \[neighborgraph\], we illustrate the conclusion in Theorem \[thm8\] about that finite-time convergence does not hold under nearest-neighbor graphs $\mathcal{G}^{\mu }_{x(k)}$ for $\mu<127$. In Figure \[valuegraph\], we illustrate the conclusion in Theorem \[thm10\] about that finite-time convergence does not hold under nearest-value graphs $\mathcal{G}^{\mu v}_{x(k)}$ for $\mu<64$.
In Figure \[rate\], we show the dependence of the convergence rates on the number of neighbors for nearest-value graphs. We see that there is a sharp increase of the convergence rate when $\mu$ is larger than about five. This is consistent with the convergence speed for consensus algorithm on Cayley graphs, where the convergence rate increases very fast as the graph degree increases [@speranzon].
Conclusions
===========
This paper focused on a uniform model for distributed averaging and maximizing. Each node iteratively updated its state as a weighted average of its own state, the minimal state, and maximal state among its neighbors. We proved that finite-time consensus is almost impossible for averaging under the uniform model. The communication graph was time-dependent or state-dependent. Necessary and sufficient conditions were established on the graph to ensure a global consensus. For time-dependent graphs, we showed that quasi-strong connectivity is critical for averaging algorithms, as is strong connectivity for maximizing algorithms. For state-dependent graphs defined by a $\mu$-nearest-neighbor rule, where each node interacts with its $\mu$ nearest smaller neighbors and the $\mu$ nearest larger neighbors, we showed that $\mu+1$ is a critical number of nodes when consensus transits from finite time to asymptotic convergence in the absence of node self-confidence: finite-time consensus disappears suddenly when the number of nodes is larger than $\mu+1$. This critical number of nodes turned out to be $2\mu$ if each node chooses to connect to nodes with different values. The results revealed the fundamental connection and difference between distributed averaging and maximizing, but more challenges still lie in the principles underlying the two types of algorithms, such as their convergence rates.
[Acknowledgment ]{}
The authors would like to thank Dr. Kin Cheong Sou for his generous help on the numerical examples.
[99]{}
C. Godsil and G. Royle. New York: Springer-Verlag, 2001.
C. Berge and A. Ghouila-Houri. . John Wiley and Sons, New York, 1965.
J. Wolfowitz, “Products of indecomposable, aperiodic, stochastic matrices," [*Proc. Amer. Math. Soc.*]{}, vol. 15, pp. 733-736, 1963.
J. Hajnal, “Weak ergodicity in non-homogeneous Markov chains," [*Proc. Cambridge Philos. Soc.*]{}, no. 54, pp. 233-246, 1958.
M. H. DeGroot, “Reaching a consensus," [*Journal of the American Statistical Association*]{}, vol. 69, no. 345, pp. 118-121, 1974. P. M. DeMarzo, D. Vayanos, J. Zwiebel, “Persuasion bias, social influence, and unidimensional opinions," [*Quarterly Journal of Economics*]{}, vol. 118, no. 3, pp. 909-968, 2003.
B. Golub and M. O. Jackson, “Naïve learning in social networks and the wisdom of crowds," [*American Economic Journal: Microeconomics*]{}, vol. 2, no. 1, pp. 112-149, 2007. S. Muthukrishnan, B. Ghosh, and M. Schultz, “First and second order diffusive methods for rapid, coarse, distributed load balancing," [*Theory of Computing Systems*]{}, vol. 31, pp. 331-354, 1998.
R. Diekmann, A. Frommer, and B. Monien, “Efficient schemes for nearest neighbor load balancing," [*Parallel Computing*]{}, vol. 25, pp. 789-812, 1999.
N. Malpani, J. L. Welch and N. Vaidya, “Leader election algorithms for mobile ad hoc networks," [*Proc. 4th international workshop on Discrete algorithms and methods for mobile computing and communications*]{}, pp. 96-104, 2000.
J. Tsitsiklis, D. Bertsekas, and M. Athans, “Distributed asynchronous deterministic and stochastic gradient optimization algorithms," [*IEEE Trans. Autom. Control*]{}, vol. 31, pp. 803-812, 1986.
A. Jadbabaie, J. Lin, and A. S. Morse, “Coordination of groups of mobile autonomous agents using nearest neighbor rules, [*IEEE Trans. Autom. Control*]{}, vol. 48, no. 6, pp. 988-1001, 2003.
R. Olfati-Saber and R. Murray, “Consensus problems in the networks of agents with switching topology and time dealys," , vol. 49, no. 9, pp. 1520-1533, 2004.
M. Cao, A. S. Morse and B. D. O. Anderson, “Reaching a consensus in a dynamically changing environment: a graphical approach," , vol. 47, no. 2, 575-600, 2008.
M. Cao, A. S. Morse and B. D. O. Anderson, “Reaching a consensus in a dynamically changing environment: convergence rates, measurement delays, and asynchronous events," , vol. 47, no. 2, 601-623, 2008.
W. Ren and R. Beard, “Consensus seeking in multi-agent systems under dynamically changing interaction topologies," [*IEEE Trans. Autom. Control*]{}, vol. 50, no. 5, pp. 655-661, 2005.
L. Moreau, “Stability of multi-agent systems with time-dependent communication links," [*IEEE Trans. Autom. Control*]{}, vol. 50, pp. 169-182, 2005.
G. Shi and Y. Hong, “Global target aggregation and state agreement of nonlinear multi-agent systems with switching topologies," [*Automatica*]{}, vol. 45, no. 5, pp. 1165-1175, 2009.
G. Shi, Y. Hong and K. H. Johansson, “Connectivity and set tracking of multi-agent systems guided by multiple moving leaders," [*IEEE Trans. Autom. Control*]{}, vol. 57, no. 3, pp. 663-676, 2012.
A. Nedić, A. Olshevsky, A. Ozdaglar, and J. N. Tsitsiklis, “On distributed averaging algorithms and qantization effects," [*IEEE Trans. Autom. Control*]{}, vol. 54, no. 11, pp. 2506-2517, 2009.
A. Olshevsky and J. N. Tsitsiklis, “Convergence speed in distributed consensus and averaging," [*SIAM Review*]{}, vol. 53, no. 4, pp. 747-772, 2011.
T. Vicsek, A. Czirok, E. B. Jacob, I. Cohen, and O. Schochet. Novel type of phase transitions in a system of self-driven particles. , vol. 75, 1226-1229, 1995.
U. Krause, “Soziale dynamiken mit vielen interakteuren eine problemskizze," In [*Modellierung und Simulation von Dynamiken mit vielen interagierenden Akteuren*]{}, pp. 37-51, 1997.
V. Blondel, J. M. Hendrickx and J. Tsitsiklis, “On Krause’s consensus formation model with state-dependent connectivity," [*IEEE Transactions on Automatic Control*]{}, vol. 54, no. 11, pp. 2506-2517, 2009
V. Blondel, J. M. Hendrickx, A. Olshevsky and J. Tsitsiklis, “Convergence in multiagent coordination, consensus, and flocking," [*IEEE Conf. Decision and Control*]{}, pp. 2996-3000, 2005.
G. Tang and L. Guo, “Convergence of a class of multi-agent systems in probabilistic framework," [*Journal of Systems Science and Complexity*]{}, 20(2), pp. 173-197, 2007.
Z. Liu and L. Guo, “Synchronization of multi-agent systems without connectivity assumptions," [*Automatica*]{}, vol. 45, no. 12, pp. 2744-2753, 2009.
V. D. Blondel, J. M. Hendrickx, and J. N. Tsitsiklis, “Continuous-time average-preserving opinion dynamics with opinion-dependent communications," [*SIAM Journal on Control and Optimization*]{}, vol. 48, no. 8, pp. 5214-5240, 2010.
M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, “Interaction ruling animal collective behavior depends on topological rather than metric distance: evidence from a field study," [*PNAS*]{}, vol. 105, no. 4, pp. 1232-1237, 2008.
F. Xue and P. R. Kumar, “On the $\theta$-coverage and connectivity of large random networks," [*IEEE Trans. Infom. Theory*]{}, vol. 52, no. 6, pp. 2289-2299, 2006.
J. M. Hendrickx, A. Olshevsky, and J. N. Tsitsiklis, “Distributed anonymous discrete function computation," [*IEEE Trans. Autom. Control*]{}, vol. 56, no. 10, pp. 2276-2289, 2011.
B. M. Nejad, S. A. Attia and J. Raisch, “Max-consensus in a max-plus algebraic setting: The case of fixed communication topologies," in [*International Symposium on Information, Communication and Automation Technologies*]{}, pp.1-7, 2009.
J. Cortés, “Finite-time convergent gradient flows with applications to network consensus," [*Automatica*]{}, vol. 42, no.11, pp. 1993-2000, 2006.
J. Cortés, “Analysis and design of distributed algorithms for $\chi$-consensus," [*IEEE Conference on Decision and Control*]{}, pp. 3363-3368, 2006.
F. Iutzeler, P. Ciblat, and J. Jakubowicz, “Analysis of max-consensus algorithms in wireless channels," [*IEEE Transactions on Signal Processing*]{}, 2012 (submitted).
D. Varagnolo, G. Pillonetto, and L. Schenato, “Distributed statistical estimation of the number of nodes in sensor networks,” in [*IEEE Conference on Decision and Control*]{}, 2011
R. Carli, F. Fagnani, A. Speranzon and S. Zampieri, “Communication constraints in the average consensus problem," [*Automatica*]{}, vol.44, no. 3, pp. 671-684, 2008.
S. Boyd, A. Ghosh, B. Prabhakar and D. Shah, “Randomized gossip algorithms," [*IEEE Trans. Information Theory*]{}, vol. 52, no. 6, pp. 2508-2530, 2006.
G. Shi, M. Johansson and K. H. Johansson, “When do gossip algorithms converge in finite time?" avaiable at [http://arxiv.org/pdf/1206.0992v1.pdf]{}.
[^1]: The authors are with ACCESS Linnaeus Centre, School of Electrical Engineering, Royal Institute of Technology, Stockholm 10044, Sweden. Email: [$\{$guodongs, kallej$\}[email protected]]{}
[^2]: This work has been supported in part by the Knut and Alice Wallenberg Foundation, the Swedish Research Council and KTH SRA TNG.
| {
"pile_set_name": "ArXiv"
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---
author:
- 'J. Aleksić'
- 'S. Ansoldi'
- 'L. A. Antonelli'
- 'P. Antoranz'
- 'A. Babic'
- 'P. Bangale'
- 'J. A. Barrio'
- 'J. Becerra González'
- 'W. Bednarek'
- 'E. Bernardini'
- 'B. Biasuzzi'
- 'A. Biland'
- 'O. Blanch'
- 'S. Bonnefoy'
- 'G. Bonnoli'
- 'F. Borracci'
- 'T. Bretz'
- 'E. Carmona$^*$'
- 'A. Carosi'
- 'P. Colin'
- 'E. Colombo'
- 'J. L. Contreras'
- 'J. Cortina'
- 'S. Covino'
- 'P. Da Vela'
- 'F. Dazzi'
- 'A. De Angelis'
- 'G. De Caneva'
- 'B. De Lotto'
- 'E. de Oña Wilhelmi'
- 'C. Delgado Mendez'
- 'D. Dominis Prester'
- 'D. Dorner'
- 'M. Doro'
- 'S. Einecke'
- 'D. Eisenacher'
- 'D. Elsaesser'
- 'M. V. Fonseca'
- 'L. Font'
- 'K. Frantzen'
- 'C. Fruck'
- 'D. Galindo'
- 'R. J. García López'
- 'M. Garczarczyk'
- 'D. Garrido Terrats'
- 'M. Gaug'
- 'N. Godinović'
- 'A. González Muñoz'
- 'S. R. Gozzini'
- 'D. Hadasch'
- 'Y. Hanabata'
- 'M. Hayashida'
- 'J. Herrera'
- 'D. Hildebrand'
- 'J. Hose'
- 'D. Hrupec'
- 'W. Idec'
- 'V. Kadenius'
- 'H. Kellermann'
- 'K. Kodani'
- 'Y. Konno'
- 'J. Krause'
- 'H. Kubo'
- 'J. Kushida'
- 'A. La Barbera'
- 'D. Lelas'
- 'N. Lewandowska'
- 'E. Lindfors'
- 'S. Lombardi'
- 'M. López'
- 'R. López-Coto[^1]'
- 'A. López-Oramas'
- 'E. Lorenz'
- 'I. Lozano'
- 'M. Makariev'
- 'K. Mallot'
- 'G. Maneva'
- 'N. Mankuzhiyil'
- 'K. Mannheim'
- 'L. Maraschi'
- 'B. Marcote'
- 'M. Mariotti'
- 'M. Martínez'
- 'D. Mazin'
- 'U. Menzel'
- 'J. M. Miranda'
- 'R. Mirzoyan'
- 'A. Moralejo'
- 'P. Munar-Adrover'
- 'D. Nakajima'
- 'A. Niedzwiecki'
- 'K. Nilsson'
- 'K. Nishijima'
- 'K. Noda'
- 'R. Orito'
- 'A. Overkemping'
- 'S. Paiano'
- 'M. Palatiello'
- 'D. Paneque'
- 'R. Paoletti'
- 'J. M. Paredes'
- 'X. Paredes-Fortuny'
- 'M. Persic'
- 'P. G. Prada Moroni'
- 'E. Prandini'
- 'I. Puljak'
- 'R. Reinthal'
- 'W. Rhode'
- 'M. Ribó'
- 'J. Rico'
- 'J. Rodriguez Garcia'
- 'S. Rügamer'
- 'T. Saito'
- 'K. Saito'
- 'K. Satalecka'
- 'V. Scalzotto'
- 'V. Scapin'
- 'C. Schultz'
- 'T. Schweizer'
- 'S. N. Shore'
- 'A. Sillanpää'
- 'J. Sitarek'
- 'I. Snidaric'
- 'D. Sobczynska'
- 'F. Spanier'
- 'V. Stamatescu'
- 'A. Stamerra'
- 'T. Steinbring'
- 'J. Storz'
- 'M. Strzys'
- 'L. Takalo'
- 'H. Takami'
- 'F. Tavecchio'
- 'P. Temnikov'
- 'T. Terzić'
- 'D. Tescaro'
- 'M. Teshima'
- 'J. Thaele'
- 'O. Tibolla'
- 'D. F. Torres'
- 'T. Toyama'
- 'A. Treves'
- 'M. Uellenbeck'
- 'P. Vogler'
- 'R. Zanin'
- '(the MAGIC Collaboration) and J. Martín , M.A. Pérez-Torres'
bibliography:
- './aa.bib'
date: 'Received ... / Accepted ... Draft version '
title: 'Discovery of TeV gamma-ray emission from the pulsar wind nebula 3C 58 by MAGIC'
---
[The pulsar wind nebula (PWN) 3C 58 is one of the historical very high-energy (VHE; $E>$100 GeV) $\gamma$-ray source candidates. It is energized by one of the highest spin-down power pulsars known (5% of Crab pulsar) and it has been compared with the Crab nebula because of their morphological similarities. This object was previously observed by imaging atmospheric Cherenkov telescopes (Whipple, VERITAS and MAGIC), although it was not detected, with an upper limit of 2.3 % Crab unit (C.U.) at VHE. It was detected by the [*Fermi*]{} Large Area Telescope (LAT) with a spectrum extending beyond 100 GeV.]{} [We aim to extend the spectrum of 3C 58 beyond the energies reported by the Fermi Collaboration and probe acceleration of particles in the PWN up to energies of a few tens of TeV.]{} [We analyzed 81 hours of 3C 58 data taken in the period between August 2013 and January 2014 with the MAGIC telescopes. ]{} [We detected VHE $\gamma$-ray emission from 3C 58 with a significance of $5.7\sigma$ and an integral flux of 0.65% C.U. above 1 TeV. According to our results, 3C 58 is the least luminous VHE $\gamma$-ray PWN ever detected at VHE and has the lowest flux at VHE to date. The differential energy spectrum between 400 GeV and 10 TeV is well described by a power-law function d$\phi$/d$E$=$f_0$($E$/1 TeV)$^{-\Gamma}$ with $f_{0} = (2.0 \pm 0.4_{\text{stat}} \pm 0.6_{\text{sys}}){\times10^{-13}} \text{cm}^{-2} \text{s}^{-1} \text{TeV}^{-1}$ and $ \Gamma = 2.4 \pm 0.2_{\text{stat}}\pm 0.2_{\text{sys}}$. The skymap is compatible with an unresolved source.]{} [We report the first significant detection of PWN 3C 58 at TeV energies. We compare our results with the expectations of time-dependent models in which electrons upscatter photon fields. The best representation favors a distance to the PWN of 2 kpc and far-infrared (FIR) values similar to cosmic microwave background photon fields. If we consider an unexpectedly high FIR density, the data can also be reproduced by models assuming a 3.2 kpc distance. A low magnetic field, far from equipartition, is required to explain the VHE data. Hadronic contribution from the hosting supernova remnant (SNR) requires an unrealistic energy budget given the density of the medium, disfavoring cosmic-ray acceleration in the SNR as origin of the VHE $\gamma$-ray emission.]{}
Introduction {#intro}
============
The supernova remnant 3C 58 (SNR G130.7+3.1) has a flat radio spectrum and is brightest near the center, therefore it was classified as a pulsar wind nebula [PWN; @PWN_classification]. It is centered on PSR J0205+6449, a pulsar discovered in 2002 with the [*Chandra*]{} X-ray observatory [@Discovery_X-rays]. It is widely assumed that 3C 58 is located at a distance of 3.2 kpc [@Distance_old], but a new H I measurement suggests a distance of 2 kpc [@Kothes2013]. The age of the system is estimated to be $\sim$ 2.5 kyr [@Chevalier] from the PWN evolution and energetics. However, the historical association of the PWN with the supernova of 1181 [@SN1181] and different measurements such as the velocity of the optical knots [@Optical_knots], neutron star cooling models [proposing that an exotic cooling mechanism must operate in this pulsar; @Slane2002], the proper motion of the pulsar [@Pulsar_proper_motion], and radio expansion of the nebula [@7kyr] derive ages ranging from 0.8 kyr up to 7 kyr. The pulsar has one of the highest spin-down powers known ($\dot E$ = 2.7$\times$10$^{37}$erg s$^{-1}$). The PWN has a size of 9$^{\prime}$$\times$6$^{\prime}$ in radio, infrared (IR), and X-rays [@Size; @Size_X-rays; @Slane2004; @IR_Mag_field]. Its luminosity is $L_{\text{ 0.5 -- 10 keV}}=2.4\times 10^{34}$ erg s$^{-1}$ in the X-ray band, which is more than 3 orders of magnitude lower than that of the Crab nebula [@X_Luminosity]. 3C 58 has been compared with the Crab because the jet-torus structure is similar [@Slane2004]. Because of these morphological similarities with the Crab nebula and its high spin-down power (5% of Crab), 3C 58 has historically been considered one of the PWNe most likely to emit $\gamma$ rays.
The pulsar J0205+6449 has a period $P$=65.68 ms, a spin-down rate $\dot P=1.93\times10^{-13}$s s$^{-1}$, and a characteristic age of 5.38 kyr [@Discovery_X-rays]. It was discovered by the [*Fermi*]{}-LAT in pulsed $\gamma$ rays. The measured energy flux is $F_{\gamma>0.1 \rm \text{GeV}}$=(5.4$\pm$0.2)$\times$10$^{-11}$ erg cm$^{-2}$s$^{-1}$ with a luminosity of $L_{\gamma>0.1 \rm \text{GeV}}$=(2.4$\pm$0.1)$\times$10$^{34}$ erg s$^{-1}$, assuming a distance for the pulsar of 1.95 kpc [@Distance_pulsar]. The spectrum is well described by a power-law with an exponential cutoff at E$_{\text{cutoff}}$=1.6 GeV [@SecondPulsarCatalog]. No pulsed emission was detected at energies above 10 GeV [@FermiAbove10]. In the off-peak region, defined as the region between the two $\gamma$-ray pulsed peaks (off-peak phase interval $\phi$=0.64–0.99), the Fermi Collaboration reported the detection of emission from 3C 58 [@SecondPulsarCatalog]. The reported energy flux is (1.75$\pm$0.68)$\times$10$^{-11}$erg cm$^{-2}$s$^{-1}$ and the differential energy spectrum between 100 MeV and 316 GeV is well described by a power-law with photon index $\Gamma=1.61\pm0.21$. No hint of spatial extension was reported at those energies. The association of the high-energy unpulsed steady emission with the PWN is favored, although an hadronic origin related to the associated SNR can not be ruled out. 3C 58 was tagged as a potential TeV $\gamma$-ray source by the Fermi Collaboration [@FermiAbove10].
The PWN 3C 58 was previously observed in the very high-energy (VHE; E $>$ 100 GeV) range by several imaging atmospheric Cherenkov telescopes. The Whipple telescope reported an integral flux upper limit of 1.31$\times$10$^{-11}$ cm$^{-2}$s$^{-1}$ [$\sim$ 19 % C.U. at an energy threshold of 500 GeV; @WhippleULs], and VERITAS established upper limits at the level of 2.3 % C.U. above an energy of 300 GeV [@3C58_Aliu_VERITAS]. MAGIC-I observed the source in 2005 and established integral upper limits above 110 GeV at the level of 7.7$\times$10$^{-12}$ cm$^{-2}$s$^{-1}$ [$\sim$4 % C.U.; @MAGICULs]. The improved sensitivity of the MAGIC telescopes with respect to previous observations and the [*Fermi*]{}-LAT results motivated us to perform deep VHE observations of the source.
Observations
============
MAGIC is a stereoscopic system of two imaging atmospheric Cherenkov telescopes situated in the Canary island of La Palma, Spain (28.8$^\circ$N, 17.9$^\circ$ W at 2225 m above sea level). During 2011 and 2012 it underwent a major upgrade of the digital trigger, readout systems, and one of the cameras [@Upgrade]. The system achieves a sensitivity of (0.71 $\pm$ 0.02)% of the Crab nebula flux above 250 GeV in 50 hours at low zenith angles [@Performance2013].
MAGIC observed 3C 58 in the period between 4 August 2013 to 5 January 2014 for 99 hours, and after quality cuts, 81 hours of the data were used for the analysis. The source was observed at zenith angles between 36$^\circ$ and 52$^\circ$. The data were taken in *wobble-mode* pointing at four different positions situated 0.4$^\circ$ away from the source to evaluate the background simultaneously with 3C 58 observations [@Wobble].
Analysis and results {#results}
====================
The data were analyzed using the MARS analysis framework [@MARS]. For every event, we calculated the size of the image (number of photoelectrons (phe) in the image of the shower) and the angular distance $\theta$ between the reconstructed arrival direction of the gamma-ray and the source position. We determined the arrival direction combining of the individual telescope information using the Disp method [@MARS]. A random-forest algorithm was used to estimate a global variable, the hadronness, which was used for background rejection [@RF]. The energy of a given event was estimated from the image sizes of the shower, its reconstructed impact parameter, and the zenith angle with Monte Carlo (MC) filled look-up tables. We optimized the cut parameters for detecting a 1% C.U. point-like source on an independent Crab nebula data sample by maximizing the Li & Ma significance [@LiMa Eq. 17, hereafter LiMa]. We selected events with a $\theta^2$ angle $<$ 0.01 deg$^2$, hadronness $<$ 0.18, and size in both telescopes $>$ 300 phe. For each pointing we used five off-source regions to estimate the background. We used MC $\gamma$ rays to calculate the energy threshold of the analysis, defined here as the peak of the true energy distribution. To calculate the flux and spectral energy distribution (SED), the effective area of the telescopes was estimated using the MC simulation of $\gamma$-rays. The SED was finally unfolded using the Schmelling method to account for the energy resolution and energy bias of the instrument [@Unfolding].
The applied cuts and the zenith angle of the observations yield an energy threshold of 420 GeV. The significance of the signal, calculated with the LiMa formula, is $5.7\sigma$, which establishes 3C 58 as a $\gamma$-ray source. The $\theta^2$ distribution is shown in Figure \[Theta2\]. As the five OFF positions were taken for each of the wobble positions, the OFF histograms were re-weighted depending on the time taken on each wobble position.
![Distribution of squared angular distance, $\theta^2$, between the reconstructed arrival directions of gamma-ray candidate events and the position of PSR 0205+6449 ([*red points*]{}). The distribution of $\theta^2$ for the OFF positions is also shown ([*gray filled histogram*]{}). The vertical dashed line defines the signal region ($\theta^2_{\text{cut}}$=0.01 deg$^2$), $N_{\text{on}}$ is the number of events in the source region, $N_{\text{off}}$ is the number of background events, estimated from the background regions and $N_{\text{ex}}$=$N_{\text{on}}$-$N_{\text{off}}$ is the number of excess events.[]{data-label="Theta2"}](Figures/Theta2.pdf){width="50.00000%"}
We show in Figure \[Skymaps\] the relative flux (excess/background) skymap, produced using the same cuts as for the $\theta^2$ calculation. The test statistics (TS) significance, which is the LiMa significance applied on a smoothed and modeled background estimate, is higher than 6 at the position of the pulsar PSR J0205+6449. The excess of the VHE skymap was fit with a Gaussian function. The best-fit position is RA(J2000) = 2 h 05 m 31(09)$_\text{stat}$(11)$_\text{sys}$ s ; DEC (J2000) = $64^\circ$ 51$^{\prime}$(1)$_\text{stat}$(1)$_\text{sys}$. This position is statistically deviant by $2\sigma$ from the position of the pulsar, but is compatible with it if systematic errors are taken into account. In the bottom left of the image we show the point spread function (PSF) of the smeared map at the corresponding energies, which is the result of the sum in quadrature of the instrumental angular resolution and the applied smearing (4.7$^{\prime}$ radius, at the analysis energy threshold). The extension of the VHE source is compatible with the instrument PSF. The VLA contours are coincident with the detected $\gamma$-ray excess.
![Relative flux (excess/background) map for MAGIC observations. The cyan circle indicates the position of PSR J0205+6449 and the black cross shows the fitted centroid of the MAGIC image with its statistical uncertainty. In green we plot the contour levels for the TS starting at 4 and increasing in steps of 1. The magenta contours represent the VLA flux at 1.4 GHz [@VLA], starting at 0.25 Jy and increasing in steps of 0.25 Jy.[]{data-label="Skymaps"}](Figures/FR_legend.png){width="50.00000%"}
\[SED\]
Figure \[SED\] shows the energy spectrum for the MAGIC data, together with published predictions for the gamma-ray emission from several authors, and two spectra obtained with three years of [*Fermi*]{}-LAT data, which were retrieved from the [*Fermi*]{}-LAT second pulsar-catalog [2PC, @SecondPulsarCatalog] and the [*Fermi*]{} high-energy LAT catalog [1FHL, @FermiAbove10]. The 1FHL catalog used events from the [*Pass 7 Clean class*]{}, which provides a substantial reduction of residual cosmic-ray background above 10 GeV, at the expense of a slightly smaller collection area, compared with the [*Pass 7 Source class*]{} that was adopted for 2PC [@Clean_class]. The two $\gamma$-ray spectra from 3C58 reported in the 2PC and 1FHL catalogs agree within statistical uncertainties. The differential energy spectrum of the source is well fit by a single power-law function d$\phi$/d$E$=$f_0$($E/$1 TeV)$^{-\Gamma}$ with $f_{0} = (2.0 \pm 0.4_{\text{stat}} \pm 0.6_{\text{sys}}){\times10^{-13}} \text{cm}^{-2} \text{s}^{-1} \text{TeV}^{-1}$, $ \Gamma = 2.4 \pm 0.2_{\text{stat}}\pm 0.2_{\text{sys}}$ and $\chi^2$=0.04/2. The systematic errors were estimated from the MAGIC performance paper [@Performance2012] including the upgraded telescope performances. The integral flux above 1 TeV is $F_{E>1\text{ TeV}}=1.4 {\times10^{-13}} \text{cm}^{-2} \text{s}^{-1}$. Taking into account a distance of 2 kpc, the luminosity of the source above 1 TeV is $L_{\gamma, E>1 \text{ TeV}}=(3.0\pm1.1)\times$10$^{32}$$d^2_{2}$ erg s$^{-1}$, where $d_{2}$ is the distance normalized to 2 kpc.
![3C 58 spectral energy distribution in the range between 0.1 GeV and 20 TeV. Red circles are the VHE points reported in this work. The best-fit function is drawn in red and the systematic uncertainty is represented by the yellow shaded area. Black squares and black arrows are taken from the [*Fermi*]{}-LAT second pulsar-catalog results [@SecondPulsarCatalog]. Blue squares are taken from the [*Fermi*]{} high-energy LAT catalog [@FermiAbove10]. The magenta line is the SED prediction for 3C 58 taken from Figure 10 of [@Wlodek2003]. The clear green dashed-dotted line is the SED predicted by [@Tanaka13], assuming an age of 1 kyr, and the dark green dotted line is the prediction from the same paper, assuming an age of 2.5 kyr. The blue dashed line represents the SED predicted by [@3C58_Diego] assuming that the Galactic FIR background is high enough to reach a flux detectable by the MAGIC sensitivity in 50h.[]{data-label="SED"}](Figures/SED_unfolded.pdf){width="50.00000%"}
Discussion
==========
Several models have been proposed that predict the VHE $\gamma$-ray emission of PWN 3C 58. [@Bucciantini] presented a one zone model of the spectral evolution of PWNe and applied it to 3C 58 using a distance of 3.2 kpc. The VHE emission from this model consists of inverse Compton (IC) scattering of CMB photons and optical-to-IR photons, and also of pion decay. The flux of $\gamma$ rays above 400 GeV predicted by this model is about an order of magnitude lower than the observation.
[@Wlodek2003] proposed a time-dependent model in which positrons gain energy in the process of resonant scattering by heavy nuclei. The VHE emission is produced by IC scattering of leptons off CMB, IR, and synchrotron photons and by the decay of pions due to the interaction of nuclei with the matter of the nebula. The age of 3C 58 is assumed to be 5 kyr, using a distance of 3.2 kpc and an expansion velocity of 1000 km s$^{-1}$. According to this model, the predicted integral flux above 400 GeV is $\sim$10$^{-13}$ cm$^{-2}$s$^{-1}$, while the integral flux above 420 GeV measured here is 5$\times$10$^{-13}$cm$^{-2}$s$^{-1}$. Calculations by [@Wlodek2005], using the same model with an initial expansion velocity of 2000 km s$^{-1}$ and considering IC scattering only from the CMB, are consistent with the observed spectrum. However, the magnetic field derived in this case is $B\sim$14$\mu$G and it underestimates the radio emission of the nebula, although a more complex spectral shape might account for the radio nebula emission.
[@Tanaka10] developed a time-dependent model of the spectral evolution of PWNe including synchrotron emission, synchrotron self-Compton, and IC. They evolved the electron energy distribution using an advective differential equation. To calculate the observability of 3C 58 at TeV energies they assumed a distance of 2 kpc and two different ages: 2.5 kyr and 1 kyr [@Tanaka13]. For the 2.5 kyr age, they obtained a magnetic field B$\geq17 \mu$G, while for an age of 1 kyr, the magnetic field obtained is B=40 $\mu$G. The emission predicted by this model is closer to the [*Fermi*]{} result for an age of 2.5 kyr.
[@Jonatan] presented a different time-dependent leptonic diffusion-loss equation model without approximations, including synchrotron emission, synchrotron self-Compton, IC, and bremsstrahlung. They assumed a distance of 3.2 kpc and an age of 2.5 kyr to calculate the observability of 3C 58 at high energies [@3C58_Diego]. The predicted emission, without considering any additional photon source other than the CMB, is more than an order of magnitude lower than the flux reported here. It predicts VHE emission detectable by MAGIC in 50 hours for an FIR-dominated photon background with an energy density of 5 eV/cm$^3$. This would be more than one order of magnitude higher than the local IR density in the Galactic background radiation model used in GALPROP [$\sim$0.2 eV cm$^{-3}$; @GALPROP]. The magnetic field derived from this model is 35 $\mu$G. To reproduce the observations, a large FIR background or a revised distance to the PWN of 2 kpc are required. In the first case, a nearby star or the SNR itself might provide the necessary FIR targets, although no detection of an enhancement has been found in the direction of the PWN. As we mentioned in Sec. \[intro\], a distance of 2 kpc has recently been proposed by [@Kothes2013] based on the recent H I measurements of the Canadian Galactic Plane Survey. At this distance, a lower photon density is required to fit the VHE data (Torres. 2014, priv. comm.).
We have shown different time-dependent models in this section that predict the VHE emission of 3C 58. The SEDs predicted by them are shown in Figure \[SED\]. They use different assumptions for the evolution of the PWN and its emission. [@Bucciantini] divided the evolution of the SNR into phases and modeled the PWN evolution inside it. In [@Wlodek2003] model, nuclei play an important role in accelerating particles inside the PWN. [@3C58_Diego] and [@Tanaka10] modeled the evolution of the particle distribution by solving the diffusion-loss equation. [@3C58_Diego] fully solved the diffusion-loss equation, while [@Tanaka10] neglected an escape term in the equation as an approximation. Another difference between these latter two models is that [@Tanaka10] took synchrotron emission, synchrotron self-Compton and IC into account, while [@3C58_Diego] also consider the bremsstrahlung. The models that fit the $\gamma$-ray data derived a low magnetic field, far from equipartition, very low for a young PWN, but comparable with the value derived by [@IR_Mag_field] using other data.
We also evaluated the possibility that the VHE $\gamma$ rays might be produced by hadronic emission in the SNR that is associated with 3C 58. For the calculations, we assumed a distance of 2 kpc, a density of the medium of 0.38 $d^{-1/2}_{2}$ cm$^{-3}$ [@Slane2004], and an initial energy of the supernova explosion of 10$^{51}$ erg. We calculated the expected flux above 1 TeV following the approach reported in [@Fluxes_SNRs]. We find that the cosmic-ray nuclei have to be accelerated with an efficiency above 100% to account for our observed flux, thus we conclude that cosmic rays from the SNR are unlikely to be the origin of the TeV emission. Note, however, that dense clumps embedded within the remnant may change these conclusions [@Clumps].
Conclusions
===========
We have for the first time detected VHE $\gamma$ rays up to TeV energies from the PWN 3C 58. The measured luminosity and flux make 3C 58 the least-luminous VHE $\gamma$-ray PWN known and the object with the lowest flux at VHE to date. Only a closer distance of 2 kpc or a high local FIR photon density can qualitatively reproduce the multiwavelength data of this object in the published models. Since the high FIR density is unexpected, the closer distance with FIR photon density comparable with the averaged value in the Galaxy is favored. The models that fit the $\gamma$-ray data derived magnetic fields which are very far from equipartition. Following the assumptions in [@Fluxes_SNRs], it is highly unlikely that the measured flux comes from hadronic emission of the SNR. The VHE spectral results we presented will help to explain the broadband emission of this source within the available theoretical scenarios.
We would like to thank the Instituto de Astrofísica de Canarias for the excellent working conditions at the Observatorio del Roque de los Muchachos in La Palma. The support of the German BMBF and MPG, the Italian INFN, the Swiss National Fund SNF, and the Spanish MINECO is gratefully acknowledged. This work was also supported by the CPAN CSD2007-00042 and MultiDark CSD2009-00064 projects of the Spanish Consolider-Ingenio 2010 programme, by grant 127740 of the Academy of Finland, by the DFG Cluster of Excellence “Origin and Structure of the Universe”, by the Croatian Science Foundation (HrZZ) Project 09/176, by the DFG Collaborative Research Centers SFB823/C4 and SFB876/C3, and by the Polish MNiSzW grant 745/N-HESS-MAGIC/2010/0. We would like to thank S. J. Tanaka for providing us useful information about his model.
[^1]: Corresponding authors: R. López-Coto, , E. Carmona,
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'D. Streich'
- 'R. S. de Jong'
- 'J. Bailin'
- 'P. Goudfrooij'
- 'D. Radburn-Smith'
- 'M. Vlajic'
bibliography:
- '../../bibliography.bib'
date: 'Received .../ Accepted ...'
title: 'On the relation between metallicity and RGB color in HST/ACS data'
---
Introduction
============
Measuring the metallicity and its gradients in galaxies is a key issue for understanding galaxy formation and evolution.
Outside the Local Group, spectroscopic metallicity determination of (resolved) stars is not feasible at the moment, except for the few very bright supergiants [@kudritzki12]. At the same time, the number of available color magnitude diagrams (CMDs) of nearby galaxies is increasing rapidly, e.g. from the GHOSTS [@radburn11] and ANGST [@dalcanton09] surveys. These CMDs can be used to derive metallicities.
The color of the red giant branch (RGB) has long been known to depend on the metallicity [@hoyle55; @sandage66; @demarque82]. This has been extensively used to measure metallicities of old populations.
There are many ways to convert the color to a metallicity: some authors define indices of an observed population, e.g. the color of the RGB at a given magnitude or the slope of the RGB, [as defined for example in @dacosta90; @lee93; @saviane00; @valenti04], while others measured metallicities on a star by star basis by interpolating between either globular cluster fiducial lines [e.g. @tanaka10; @tiede04] or analytic RGB functions [calibrated with globular cluster data, e.g. @zoccali03; @gullieuszik07; @held10] or stellar evolution models [e.g. @richardson09; @babusiaux05], and therefore generating metallicity distribution functions for a population.
Uncertainties that arise from a specific calibration or a given isochrone or cluster template set are typically not well studied. Furthermore, an observational relation for the widely used HST ACS filters F606W-F814W is still missing in the literature[^1]. Here we aim to address these shortcomings.
This paper is organized as follows: After an introduction to the data and isochrones we use in chapter \[data\], an observational color metallicity relation is derived in chapter \[results\]. A discussion and summary follow in chapters \[discussion\] and \[summary\].
Data and Isochrones {#data}
===================
In this work we use the data of 71 globular clusters observed as part of the ACS Globular Cluster Survey [ACSGCS; @sarajedini07] and its extension [@dotter11]. These data contain photometry in the F606W and F814W filters and is publicly available at the homepage of the ACSGCS team[^2]. For the determination of photometric uncertainties and completeness, the results from artificial star tests are also available. A detailed description of the data reduction is given in @anderson08.
To compare the different clusters, it is necessary to transform the apparent magnitudes into absolute, reddening-free magnitudes. For this purpose, we use the distance modulus and color excess from the GC database of W. Harris [@harris96; @harris10] and the extinction ratios for the ACS filters given by @sirianni05 [Table 14]. Metallicities, metallicity uncertainties and $\alpha$-abundances are taken from @carretta09 [@carretta10], if not stated otherwise.
In order to measure the color of the clusters RGBs they must have a sufficient number of stars in the RGB region. We selected therefore only those clusters for our study, which have more than five stars brighter than $M_{F814W}=-2$ and least one star brighter than $M_{F814W}=-3$. A list of the clusters used is given in Appendix \[app:properties\] (Table \[GC\_literature\]).
For comparison with theoretical models, we use four sets of isochrones: the new PARSEC isochrones [@bressan12] and their predecessors, the (old) Padua isochrones[^3] [@girardi10; @marigo08 and references therein], the BaSTI isochrones[^4] [@pietrinferni06; @pietrinferni04] and the Dartmouth isochrones[^5] [@dotter07].
Results
=======
Color measurement {#sec:colormeasurement}
-----------------
We use two indices to define the color of the RGB: $C_{-3.0}=(F606W-F814W)_{M=-3.0}$ and $C_{-3.5}=(F606W-F814W)_{M=-3.5}$, i.e. the color of the RGB at an absolute F814W magnitude of -3.0 and -3.5, respectively (see Fig. \[CMD\_diverse\] for some typical CMDs). Equivalent indices for the Johnson-Cousins filter system were already used by @dacosta90, @lee93 and also by @saviane00. These indices have the advantage of only depending on relatively bright stars and can therefore be measured in distant galaxies, as well. We use also a third index, the S-index, which is the slope of the RGB [@saviane00; @hartwick68]. This slope is measured between two points of the RGB, one at the level of the horizontal branch and the other two magnitudes brighter. While this index needs deeper data, and therefore its usage in extragalactic systems is limited, it has the advantage of being independent of extinction and distance errors.
In order to provide a robust measurement of the color at a given magnitude we interpolated the RGB with a hyperbola of the form: $$M=a+b\cdot \textrm{color}+c/(\textrm{color}+d)$$ Such a function was already used by @saviane00 to find a one-parameter representation of the RGB; they defined the parameters $a$, $b$, $c$, and $d$ as a quadratic function of metallicity. Here, we are only interested in a good interpolation in sparse parts of the RGB and can therefore use $a$, $b$, $c$, and $d$ as free parameters for each cluster. In order to reduce problems due to contamination, we define a region of probable RGB stars, which also excludes the horizontal branch/red clump part of the CMD. Note that we fit the curve directly to the color/magnitude points of the stars and not to the ridge line of the RGB [in contrast to @saviane00]. More details of the fitting process and some example plots with the exclusion region are shown in the Appendix.
To calculate the S-index, we first determined the horizontal branch magnitude of each system by visual inspection of the associated CMDs. This was typically F606W$\approx$0.40mag, with a 1-sigma variation of 0.10mag. We measured the color at this magnitude (and at 2 magnitudes brighter) from the fitted RGB used previously, and calculated the S-index as the slope between these points
Metallicity determination
-------------------------
The iron abundance \[Fe/H\] is often used synonymously with metallicity. However, from the theoretical point of view of stellar evolution, all elements are important in determining the properties of stellar atmospheres. Therefore the color of red giants is expected to depend on the overall metallicity \[M/H\] rather than on \[Fe/H\]. Unfortunately, there are very few measurements of the abundances of other elements in globular clusters.
We use here the abundances given in @carretta10, who have measured \[Fe/H\] for all GCs in our sample and have compiled \[$\alpha$/Fe\] values for many of them. According to @salaris93, these two measurements can be combined to get the overall metallicity with the formula $$[M/H] = [Fe/H] + \log_{10}(0.638*10^{[\alpha/Fe]}+0.362) .$$ For clusters that have no individual $\alpha$ measurement, we had to estimate its $\alpha$ abundance. Since the spread of \[$\alpha$/Fe\] among globular clusters is rather small, such an estimate will only introduce small errors. In Fig. \[alphas\], \[$\alpha$/Fe\] is plotted against \[Fe/H\], where we have assumed an uncertainty of $0.05$ in the $\alpha$ abundance. The straight line is a linear regression, which we use for the estimation of \[$\alpha$/Fe\], where it is not available. The scatter around this regression line is 0.1dex, which we adopt as the individual uncertainty in the estimated \[$\alpha$/Fe\].
![Alpha abundance as a function of \[Fe/H\] for all clusters in @carretta10. The text in the lower left corner gives the formula of the regression line and the scatter around this line. We used these for estimating the \[$\alpha$/Fe\] and its uncertainty for clusters without individual alpha measurement.[]{data-label="alphas"}](alpha-FeH_carretta){width="0.99\columnwidth"}
Uncertainties
-------------
To determine the uncertainties of our color measurements, we performed a bootstrap analysis. The uncertainty in the fit is derived by fitting the RGB of 500 samples that are drawn randomly from the original data. Each re-sample has the same number of stars as the original sample, but may contain some stars multiple times while others are absent.
We also incorporated in the bootstraps a shift due to the uncertainties in extinction and distance. According to @harris10, the uncertainty in extinction is of the order 10% in E(B-V), but is at least 0.01mag, while the uncertainty in distance modulus is 0.1mag.
The uncertainty in distance is important because we measure the color at a given absolute magnitude. This is particularly significant for the metal-rich clusters where the color of the RGB is strongly dependent on magnitude, as opposed to metal-poor clusters where the RGB is nearly vertical on a CMD. The resulting uncertainty in $C_{-3.5}$ ranges from approximately 0.01mag at \[M/H\]=-2 to approximately 0.1mag at \[M/H\]=-0.2.
The uncertainties in the metallicity are the sum of the uncertainties in \[Fe/H\] @carretta09, and in \[$\alpha$/Fe\], which we adopt as 0.05dex for clusters with individual alpha-abundance measurements and 0.1dex for clusters with estimated values.
Color metallicity relation
--------------------------
Using the colors and metallicities described above, we can now look at the color-metallicity relations.
The results are shown in Fig. \[metalfitindices\]. There is a clear relation between RGB color and spectroscopic metallicity. This relation can be parametrized with the function $F606W-F814W=a_0\exp(\mathrm{[M/H]}/a_1)+a_2.$ Using the orthogonal distance regression (ODR) algorithm [@boggs87; @boggs92][^6], we determined the three parameters, that are shown in Table \[fitparams\]. The ODR uses the uncertainties on both variables to determine the best fit. Hence, both the uncertainties in color and metallicity, as described above, are considered during the fit and their effects are included in the final uncertainties of the resulting fit parameters. The residual varinaces for both relations are $\sigma_{res}^2<1$, so the adopted uncertainties can explain the observed scatter in the relations.
$a_0$ $a_1$ $a_2$
------------ ----------------- ----------------- -----------------
$C_{-3.5}$ $0.95\pm0.11$ $0.602\pm0.069$ $0.920\pm0.015$
$C_{-3.0}$ $0.567\pm0.056$ $0.75\pm0.12$ $0.845\pm0.018$
S-index $3.67\pm0.76$ $-9.3\pm1.2$ $-2.08\pm0.44$
: Fit parameters of the color–metallicity relations. For $C_{-3.5}$ and $C_{-3.0}$ the relation is exponential: $C_{i}=a_0\exp(\mathrm{[M/H]}/a_1)+a_2$, for the S-index it is linear $S=a_0+a_1\mathrm{[M/H]}$. []{data-label="fitparams"}
{width="99.00000%"}
The S-index
-----------
The slope of the RGB as a function of metallicity can be seen in Fig. \[s\_index\].
![*top panel:* The RGB slope as a function of metallicity. The solid black line is the best-fit quadratic function, as given in the equation in the bottom left. Grey lines give the $1\sigma$, $2\sigma$ and $3\sigma$ confidence ranges of the the best fit relation. *Bottom panel:* weighted orthogonal residuals, i.e. the orthogonal distances to the best fit line divided by the respective uncertainties. []{data-label="s_index"}](s_index){width="0.99\columnwidth"}
The reported uncertainties of the S-index are a combination of the uncertainties of the RGB fit (determined through a bootstrap analysis as described above) and the uncertainty in the determination of the HB level, which we set here to $\sigma_{HBmag}=0.1$mag.
As expected, the slope of the RGB gets smaller with increasing metallicity, while at the low-metallicity end the RGB slope is insensitive to metallicity. We have fitted a quadratic function to the data, which is shown in Fig. \[s\_index\] together with the associated best-fit parameters. The choice of a quadratic function for the fit proves to be appropriate as no trends are seen in the residuals. Moreover, the variance of the residuals is only $\sigma_{res}^2=1.17$, i.e. the residuals are only slightly larger than expected from the individual measurement uncertainties. The maximum of the parabola is at \[M/H\]=-2.14, which is beyond the metallicity range of the observed clusters.
Discussion
==========
Analyzing residuals
-------------------
We examine the residuals to look for a possible second parameter that influences the color or slope of the RGB and could produce some scatter in a simple color-metallicity relation. Figures \[residuals1\] and \[residuals2\] show the residuals of the fit of the color-metallicity relation, that is shown in Fig. \[metalfitindices\], and Figures \[S\_residuals1\] and \[S\_residuals2\] the residuals of the fit to the slope metallicity relation, that is shown in Fig. \[s\_index\].
![Residuals of the fit of the color-metallicity relation as function of metallicity (upper panel), iron abundance (middle panel) and alpha enhancement (lower panel). Symbols and colors are as in Fig. \[metalfitindices\].[]{data-label="residuals1"}](residuals_metals){width="0.99\columnwidth"}
![Residuals of the fit of the color-metallicity relation as function of age [upper panel, @marinfranch09], extinction (middle panel) and galactic latitude (lower panel). Symbols and colors are as in Fig. \[metalfitindices\].[]{data-label="residuals2"}](residuals_age_pos){width="0.99\columnwidth"}
![Residuals of the fit of the slope-metallicity relation as function of metallicity (upper panel), iron abundance (middle panel) and alpha enhancement (lower panel).[]{data-label="S_residuals1"}](S_residuals_metals){width="0.99\columnwidth"}
![Residuals of the fit of the slope-metallicity relation as function of age [upper panel, @marinfranch09], extinction (middle panel) and galactic latitude (lower panel).[]{data-label="S_residuals2"}](S_residuals_agepos){width="0.99\columnwidth"}
The residuals as a function of metallicity, \[Fe/H\] and \[$\alpha$/Fe\], are shown in Fig. \[residuals1\] and Fig. \[S\_residuals1\]. There is no trend with any of these parameters, neither in the color- nor slope-metallicity relations.
From theoretical studies, the age is known to have an effect on the color of the RGB. In fact, a weak trend of the residuals with age can be seen in Figure \[residuals2\] (upper panel), with older clusters being slightly redder than younger clusters. The slope of the regression lines shown there is $1.94\pm2.13$ for $C_{-3.5}$ and $3.00\pm1.85$ for $C_{-3.0}$, which makes the trend significant for the $C_{-3.0}$ index. In order to quantify the effects of age on the CMD, we analyse the residuals in color space[^7] in Figure \[color\_residuals\].
![Color residuals of the fit of the color-metallicity relation as function of age. The text gives the regression line formulas for both indices.[]{data-label="color_residuals"}](color_residuals){width="0.99\columnwidth"}
Assuming a typical age for globular clusters of 12.8Gyr (as @marinfranch09 do using the isochrones of @dotter07) we can transform the slopes of the regression lines in Figure \[color\_residuals\] to actual color changes. These are $0.0062\pm0.0041$mag/Gyr for $C_{-3.5}$ and $0.0078\pm0.0026$mag/Gyr for $C_{-3.0}$. While this is a very small effect for the age range observed in our globular clusters (10Gyr to 14Gyr), it can make significant differences when extrapolated to younger populations; e.g. an 8Gyr old population would be bluer than predicted by our relation by about 0.03mag. Note also that the S index does not show any systematic trends with age.
To test for problems with the extinction values, we looked for trends with E(B-V) and galactic latitude (Fig. \[residuals2\] and Fig. \[S\_residuals2\], middle and lower panel). We do not find any systematics here.
Comparison
----------
We can compare our relations to those derived from stellar evolution models, and to relations from ground-based data transformed to the HST/ACS filter systems.
For comparison with theoretical relations, we use the isochrone set from the Padua, Dartmouth and BaSTI groups. For all isochrone sets we used ages of 8Gyr, 10Gyr and 13Gyr. For Dartmouth, we use $\alpha$-enhancements of $[\alpha/Fe]=\{0.0,0.2,0.4\}$ and for BaSTI models $[\alpha/Fe]=\{0.0,0.4\}$. The PARSEC[^8] and Padua isochrones are available only with solar scaled abundances.
{width="99.00000%"} {width="99.00000%"} {width="99.00000%"}
{width="99.00000%"} {width="99.00000%"} {width="99.00000%"}
All these isochrone sets show a qualitatively similar behavior. The RGB gets redder (Fig. \[relations\]) and shallower (Fig. \[relations\_sindex\]) with increasing metallicity. A higher age also leads to a redder RGB, but this effect is relatively small. An age difference of 5Gyr causes the same color difference as a metallicity difference of only 0.1dex (see Fig. \[relations\], top row). At a given total metallicity \[M/H\], the $\alpha$-abundance has almost no effect on the RGB color (Fig. \[relations\], middle row). This supports the assumption that the color of the RGB is mainly influenced by \[M/H\] and not \[Fe/H\].
The increasing curvature of the RGB with increasing metallicity prevents the RGB of some metal rich clusters from reaching F814W$=-3.5$mag, but bend down at fainter magnitudes. In the Dartmouth models this applies for isochrones with \[M/H\]$>-0.4$, in Padua models isochrones with \[M/H\]$>-0.3$. However, the BaSTI RGB isochrones all reach F814W$=-3.5$mag, even at super-solar metallicities. Among our clusters, NGC6838 (\[M/H\]$=-0.53$; it also has very few stars in the RGB) and NGC6441 (\[M/H\]$=-0.29$) are affected by this.
In order to quantify the agreement between our relations and other relations, we have performed a Monte Carlo resampling of our relations by drawing random parameter sets $a_i$ from a multivariate Gaussian distribution with the mean and covariance matrix as given by the best fit. The 68.3%, 95.5%, and 99.7% confidence interval are shown as contours in Figures \[metalfitindices\], \[s\_index\], \[relations\], and \[relations\_sindex\].
![Comparison of the observed S-index metallicity relation with isochrones of varying age. The gray contours show the 1$\sigma$, 2$\sigma$, and 3$\sigma$ confidence levels of the fit.[]{data-label="relations_sindex"}](isochrone-age_comparison_s-index){width="0.99\columnwidth"}
As can be seen in Figures \[relations\] and \[relations\_sindex\], BaSTI isochrones show good agreement with our observational result. At most metallicities the $\alpha$-enhanced BaSTI isochrone falls within the the 1$\sigma$ confidence range of our observational relation; only for \[M/H\]$>$-0.4 are the isochrones significantly redder ($>3\sigma$) and shallower than our relation. The Dartmouth isochrones agree well at very low metallicities, but tend to predict slightly redder colors and shallower slopes at intermediate and higher metallicities. In contrast, results from the Padua isochrones are bluer by almost 0.15mag and much steeper at lower metallicities, and redder and shallower at the high metallicity end. Determining the reason for this offset is beyond the scope of this paper, but this problem has been known to lead to higher metallicity estimates, when Padua isochrones are used [@lejeune99].
####
Existing color-metallicity relations are given in the standard Johnson-Cousin filters. Thus to compare these with our analysis we use the transformations to the HST/ACS filter set described in @sirianni05. Two such transformations are provided, one observationally based and the other synthetic. The former uses observations of horizontal branch and RGB stars in the metal-poor (\[Fe/H\]=$-2.15$) globular cluster NGC2419. This cluster does not contain stars with $(V-I) > 1.3$, hence the transformation at these redder colors are extrapolated and should be used with caution. The transformation is:
$$F606W - F814W = -0.055 + 0.762(V - I)$$
For the synthetic transformation, stellar models with $(V-I) < 1.8$ were used. Hence, for redder colors, the extrapolation should again be treated with caution. The transformation is given as:
$$F606W - F814W = 0.062 + 0.646(V - I) + 0.053(V - I)^2$$
We use both these transformations on the color-metallicity relations of @saviane00, who determined relations for the indices $(V - I)_{-3.0}$ and $(V - I)_{-3.5}$, and for @dacosta90 [for $(V - I)_{-3.0}$] and @lee93 [for $(V - I)_{-3.5}$]. To shift these transformations, which are defined for \[Fe/H\], to the \[M/H\] scale, we used the same \[Fe/H\]-\[$\alpha$/Fe\] relation as for the data.
Note that these two transformations have a relative offset of of about 0.05mag, which can be seen in Fig. \[relations\] as the two almost parallel lines in the lower panel.[^9]
From Fig. \[relations\] it can be seen that these transformed relations are always bluer at the low metallicity end and have a steeper slope than our relations.[^10]
Part of this discrepancy can be explained by the different metallicity scales used for the various relations. While we use the metallicity scale of @carretta09 [,C+09], earlier relations were determined either in the Zinn & West scale [@zinn84 ZW84] or the Carretta & Gratton scale [@carretta97 CG97]. The adopted C+09 scale is comparable to the ZW84 scale; however, the CG97 scale yields higher metallicities for \[Fe/H\]$\lesssim-1$ and lower metallicities for \[Fe/H\]$\gtrsim-1$ (see \[fehscales\]).
![Comparison of different metallicity scales. On the x-axis the C+09 scale, that is adopted in this paper, is shown. Blue crosses show the clusters from @carretta97, red crosses from @zinn84. The plus symbols show the metallicities determined in @rutledge97 [RHS] based on the Ca triplet, calibrated to both scales.[]{data-label="fehscales"}](FeH_scales){width="0.99\columnwidth"}
Hence, using the CG97 scale will lead to a steeper color-metallicty relation than found from our measurements (see the bottom panel of Fig. \[relations\]).
Inverting the relation
----------------------
The main purpose of the color metallicity relation is to estimate metallicities of old stellar population. The uncertainties arising from the inverted relation are highly nonlinear. In Fig. \[inverted\_residuals\] we plot the difference between the spectroscopic metallicities and the metallicities derived with our relation. It is apparent that for bluer colors (i.e. lower metallicities) the difference can be very large. If the color is near the pole of the metallicity-color function, the formal uncertainties can be infinite. Then only an upper limit on the metallicity can be derived. For all clusters with $C_{-3.5}<1.2$ (or $C_{-3.0}<1.0$) the scatter in the metallicity differences is about 0.3dex. We suggest using this as a minimum uncertainty for metallicities derived from our relation in that color range. For redder colors, the uncertainty drops in half.
![Error distribution of the metallicity determination using the inverted color metallicity relations. Lines with errorbars are the running mean and standard deviation which are computed using a bin width of 0.3mag for $C_{-3.5}$ and 0.15mag for $C_{-3.0}$. Symbols and colors are as in Fig. \[metalfitindices\]. Note the different scales on the x-axis for the two distributions.[]{data-label="inverted_residuals"}](Z_residuals){width="0.99\columnwidth"}
Conclusions and summary {#summary}
=======================
In this paper, we derived relations between the colors and the slope of the RGB and metallicity using data from globular clusters. The details of the relations are summarized in Table \[fitparams\]. When using these relations for determining metallicities of old resolved stellar populations, the following points should be kept in mind:
- The color changes very little with metallicity for $\mbox{[M/H]}\lesssim -1.0$, the slope changes little below $[M/H]\lesssim-1.5$. Therefore, inverting the relation in this regime introduces large uncertainties. This makes a photometric metallicity determination rather inaccurate in this metallicity range.
- Our relation agrees well with the prediction from BaSTI isochrones. Dartmouth isochrones are slightly redder, Padua isochrones bluer than our data. Thus, for the purpose of determining metallicities of old populations we recommend the use of BaSTI isochrones.
- A comparison with other color-metallicity-relations from the literature, both empirical and theoretical, shows some scatter between these relations. Therefore a comparison of metallicities derived from different methods/relations will introduce systematic offsets. This should be kept in mind whenever the use of a homogenous method is not possible.
We thank our referee, Ivo Saviane, for the careful reading of our manuscript, the detailed look into the data, and the comments that helped to improve this paper. We also thank Benne Holwerda and Antonela Monachesi for their comments on an earlier version of this paper that improved its final quality.
DS gratefully acknowledges the support from DLR via grant 50OR1012 and a scholarship from the Cusanuswerk.
Description of the fit of the RGB
=================================
We parametrized the RGB with the function $$M=a+b\cdot \mbox{color}+c/(\mbox{color}+d) ,$$ as given in @saviane00. Since the data do not only contain RGB stars, but also the horizontal branch, blue stragglers and foreground stars, we have defined a region to guarantee a high fraction of RGB stars in our fit sample. The extent of this region can be seen as the red frame in Fig. \[CMD\_diverse\]. Note that this region excludes also the red clump.
In some clusters, there is a distinct asymptotic giant branch (AGB) visible, which lies on the blue side of the RGB (it is mostly seen at F814W magnitudes between -1 and -2). As in @saviane00, we have removed these AGB stars by excluding all detections that lie blue wards of a reference line with the same slope for all clusters (denoted in Figures \[CMD\_diverse\] through \[CMDs\_diverse\_last\] by a dashed red line). The horizontal position of the reference line was set to be 0.05mag blue wards (at F814W=-0.5) of a first fit of all stars in the RGB region and then excluding. The fit including all stars is shown in the CMDs as black dashed line, while the final fit after the AGB removal is shown as black solid line.
For the actual fit we used the python package scipy.odr. This routine performs an orthogonal distance regression, i.e. it minimizes the orthogonal distance between the curve and the data points. The distance of each data point is weighted with its measurement uncertainty. This method is a variation of the typical $\chi^2$ minimization, now generalized for data with uncertainties on both variables.
The ACSGCS team reports photometric uncertainties for each individual star, which are typically quite small; the median uncertainty in F814W is only 0.003mag. This is much smaller than both the observed scatter in the RGB and the errors that are found in the artificial star test at a level of F814W$\approx0$. (There are no artificial star tests at brighter magnitudes.) The mean measurement error estimated from the difference of the input and recovered magnitudes in the artificial star test are 0.06mag in F814W and 0.03mag in color. These estimated are added in quadrature to the reported uncertainties of each star. The smaller error in color is due to the fact that errors in both bands are correlated. Thus the uncertainty of the difference of both bands is smaller than the uncertainty in each band.
Finally, we visually inspected each CMD with its fit, to check for any residual problems of our clusters. After this inspection we excluded four more clusters from the sample: NGC6838 and NGC6441, because their RGB fits do not reach the $M_I=-3.5$ level; NGC6388 (and again NGC6441), because their red clumps seem to be to faint [@bellini13 also found problems with differential reddening and multiple populations in these two clusters]; and NGC6715, because it has a clear and strong second RGB.
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Properties of the Globular clusters {#app:properties}
===================================
The properties of the globular clusters, from the literature and determined in this work, are summarized in Table \[GC\_literature\] and Table \[GC\_results\] respectively.
---------- ------------ ------------ ----------- --------- ------------------ ------------------- ------------------- ----------------- ------
name RA DEC (m-M$)_V$ E(B-V) \[Fe/H\]$_{H10}$ \[Fe/H\]$_{C+10}$ $\sigma_{[Fe/H]}$ \[$\alpha$/Fe\] age
\[$\deg$\] \[$\deg$\] \[mag\] \[mag\] \[dex\] \[dex\] \[dex\] \[dex\]
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
Arp 2 292.1838 -29.6444 17.59 0.10 -1.75 -1.74 0.08 0.34 0.85
IC4499 225.0769 -81.7863 17.08 0.23 -1.53 -1.62 0.09 ... ...
Lynga 7 242.7652 -54.6822 16.78 0.73 -1.01 -0.68 0.06 ... 1.13
NGC104 6.0236 -71.9187 13.37 0.04 -0.72 -0.76 0.02 0.42 1.02
NGC362 15.8094 -69.1512 14.83 0.05 -1.26 -1.30 0.04 0.30 0.81
NGC1261 48.0675 -54.7838 16.09 0.01 -1.27 -1.27 0.08 ... 0.80
NGC1851 78.5282 -39.9534 15.47 0.02 -1.18 -1.18 0.08 0.38 0.78
NGC2298 102.2475 -35.9947 15.60 0.14 -1.92 -1.96 0.04 0.50 0.99
NGC2808 138.0129 -63.1365 15.59 0.22 -1.14 -1.18 0.04 0.33 0.85
NGC3201 154.4034 -45.5875 14.20 0.24 -1.59 -1.51 0.02 0.33 0.80
NGC4590 189.8666 -25.2559 15.21 0.05 -2.23 -2.27 0.04 0.35 0.90
NGC4833 194.8913 -69.1235 15.08 0.32 -1.85 -1.89 0.05 ... 0.98
NGC5024 198.2302 18.1682 16.32 0.02 -2.10 -2.06 0.09 ... 0.99
NGC5139 201.6968 -46.5204 13.94 0.12 -1.53 -1.64 0.09 ... 0.90
NGC5272 205.5484 28.3773 15.07 0.01 -1.50 -1.50 0.05 0.34 0.89
NGC5286 206.6117 -50.6257 16.08 0.24 -1.69 -1.70 0.07 ... 0.98
NGC5904 229.6384 2.0810 14.46 0.03 -1.29 -1.33 0.02 0.38 0.83
NGC5927 232.0029 -49.3270 15.82 0.45 -0.49 -0.29 0.07 ... 0.99
NGC5986 236.5125 -36.2136 15.96 0.28 -1.59 -1.63 0.08 ... 0.95
NGC6093 244.2600 -21.0239 15.56 0.18 -1.75 -1.75 0.08 0.24 0.98
NGC6101 246.4505 -71.7978 16.10 0.05 -1.98 -1.98 0.07 ... 0.98
NGC6121 245.8968 -25.4743 12.82 0.35 -1.16 -1.18 0.02 0.51 0.98
NGC6144 246.8077 -25.9765 15.86 0.36 -1.76 -1.82 0.05 ... 1.08
NGC6171 248.1328 -12.9462 15.05 0.33 -1.02 -1.03 0.02 0.49 1.09
NGC6205 250.4218 36.4599 14.33 0.02 -1.53 -1.58 0.04 0.31 0.91
NGC6218 251.8091 -0.0515 14.01 0.19 -1.37 -1.43 0.02 0.41 0.99
NGC6254 254.2877 -3.8997 14.08 0.28 -1.56 -1.57 0.02 0.37 0.89
NGC6304 258.6344 -28.5380 15.52 0.54 -0.45 -0.37 0.07 ... 1.06
NGC6341 259.2808 43.1359 14.65 0.02 -2.31 -2.35 0.05 0.46 1.03
NGC6362 262.9791 -66.9517 14.68 0.09 -0.99 -1.07 0.05 ... 1.06
NGC6426 266.2277 3.1701 17.68 0.36 -2.15 -2.36 0.06 ... ...
NGC6496 269.7653 -43.7341 15.74 0.15 -0.46 -0.46 0.07 ... 0.97
NGC6541 272.0098 -42.2851 14.82 0.14 -1.81 -1.82 0.08 0.43 1.01
NGC6584 274.6567 -51.7842 15.96 0.10 -1.50 -1.50 0.09 ... 0.88
NGC6624 275.9188 -29.6390 15.36 0.28 -0.44 -0.42 0.07 ... 0.98
NGC6637 277.8462 -31.6519 15.28 0.18 -0.64 -0.59 0.07 0.31 1.02
NGC6652 278.9401 -31.0093 15.28 0.09 -0.81 -0.76 0.14 ... 1.01
NGC6656 279.0998 -22.0953 13.60 0.34 -1.70 -1.70 0.08 0.38 0.99
NGC6681 280.8032 -31.7079 14.99 0.07 -1.62 -1.62 0.08 ... 1.00
NGC6717 283.7752 -21.2985 14.94 0.22 -1.26 -1.26 0.07 ... 1.03
NGC6723 284.8881 -35.3678 14.84 0.05 -1.10 -1.10 0.07 0.50 1.02
NGC6752 287.7171 -58.0154 13.13 0.04 -1.54 -1.55 0.01 0.43 0.92
NGC6779 289.1482 30.1835 15.68 0.26 -1.98 -2.00 0.09 ... 1.07
NGC6809 294.9988 -29.0353 13.89 0.08 -1.94 -1.93 0.02 0.42 0.96
NGC6934 308.5474 7.4045 16.28 0.10 -1.47 -1.56 0.09 ... 0.87
NGC6981 313.3654 -11.4627 16.31 0.05 -1.42 -1.48 0.07 ... 0.85
NGC7006 315.3724 16.1873 18.23 0.05 -1.52 -1.46 0.06 0.28 ...
NGC7078 322.4930 12.1670 15.39 0.10 -2.37 -2.33 0.02 0.40 1.01
NGC7089 323.3626 0.8233 15.50 0.06 -1.65 -1.66 0.07 0.41 0.92
NGC7099 325.0922 -22.8201 14.64 0.03 -2.27 -2.33 0.02 0.37 1.01
Pal 2 71.5246 31.3815 21.01 1.24 -1.42 -1.29 0.09 ... ...
Rup 106 189.6675 -50.8497 17.25 0.20 -1.68 -1.78 0.08 -0.03 ...
Terzan 8 295.4350 -32.0005 17.47 0.12 -2.16 -2.02 0.06 0.45 0.95
---------- ------------ ------------ ----------- --------- ------------------ ------------------- ------------------- ----------------- ------
\[GC\_literature\]
---------- ---------------- ----------------- ---------------- ----------------- -------- ------------ -------------
name (V-I$)_{-3.5}$ $\sigma_{-3.5}$ (V-I$)_{-3.0}$ $\sigma_{-3.0}$ $S$ $\sigma_S$ $M_{V}(HB)$
\[mag\] \[mag\] \[mag\] \[mag\] \[mag\]
(1) (2) (3) (4) (5) (6) (7) (8)
Arp 2 0.999 0.027 0.920 0.011 13.415 0.826 0.45
IC4499 0.976 0.016 0.902 0.007 13.460 0.620 0.30
Lynga 7 1.321 0.042 1.158 0.016 7.829 0.587 0.40
NGC104 1.397 0.017 1.162 0.006 6.860 0.540 0.44
NGC362 1.069 0.009 0.953 0.005 11.578 0.568 0.52
NGC1261 1.092 0.024 0.977 0.007 11.190 0.622 0.53
NGC1851 1.110 0.007 0.999 0.005 11.059 0.545 0.53
NGC2298 1.054 0.019 0.969 0.010 13.736 0.762 0.35
NGC2808 1.077 0.008 0.967 0.004 11.215 0.461 0.56
NGC3201 1.048 0.012 0.973 0.005 11.307 0.512 0.30
NGC4590 0.950 0.012 0.896 0.006 13.559 0.505 0.30
NGC4833 0.976 0.004 0.916 0.003 13.888 0.476 0.30
NGC5024 0.975 0.002 0.897 0.002 12.324 0.525 0.20
NGC5139 1.044 0.011 0.960 0.006 11.553 0.521 0.40
NGC5272 1.018 0.003 0.929 0.002 12.281 0.528 0.40
NGC5286 0.985 0.005 0.913 0.003 14.017 0.554 0.40
NGC5904 1.057 0.005 0.962 0.003 11.818 0.523 0.50
NGC5927 1.883 0.091 1.353 0.025 4.849 0.680 0.51
NGC5986 1.019 0.006 0.939 0.004 12.747 0.553 0.40
NGC6093 1.053 0.006 0.968 0.006 12.131 0.589 0.40
NGC6101 1.024 0.018 0.956 0.007 13.104 0.698 0.30
NGC6121 1.179 0.028 1.068 0.015 10.401 0.849 0.30
NGC6144 1.028 0.014 0.961 0.009 14.346 0.962 0.25
NGC6171 1.231 0.017 1.108 0.017 9.498 0.622 0.42
NGC6205 1.038 0.008 0.948 0.003 11.467 0.509 0.40
NGC6218 1.119 0.015 1.003 0.007 9.887 0.583 0.30
NGC6254 1.061 0.021 0.972 0.009 11.391 0.524 0.50
NGC6304 2.051 0.176 1.392 0.043 4.257 0.848 0.49
NGC6341 0.956 0.003 0.889 0.003 13.381 0.491 0.30
NGC6362 1.186 0.023 1.034 0.009 9.554 0.571 0.54
NGC6426 1.008 0.027 0.946 0.026 13.152 1.967 0.40
NGC6496 1.587 0.078 1.308 0.036 5.976 0.772 0.53
NGC6541 0.954 0.004 0.883 0.003 12.937 0.523 0.30
NGC6584 1.015 0.005 0.925 0.003 12.120 0.557 0.40
NGC6624 1.514 0.096 1.215 0.040 6.412 0.742 0.47
NGC6637 1.427 0.023 1.159 0.009 7.243 0.675 0.46
NGC6652 1.321 0.021 1.147 0.014 7.829 0.564 0.47
NGC6656 1.014 0.012 0.949 0.005 13.284 0.523 0.50
NGC6681 1.087 0.010 0.983 0.005 12.476 0.579 0.60
NGC6717 1.091 0.062 0.995 0.041 11.462 1.647 0.60
NGC6723 1.149 0.022 1.033 0.010 10.487 0.574 0.49
NGC6752 1.075 0.025 0.980 0.010 10.063 0.564 0.25
NGC6779 0.960 0.005 0.888 0.002 13.579 0.519 0.40
NGC6809 1.011 0.007 0.938 0.006 13.041 0.638 0.40
NGC6934 1.050 0.018 0.962 0.015 10.874 1.048 0.40
NGC6981 1.015 0.009 0.933 0.004 12.173 0.559 0.40
NGC7006 1.035 0.014 0.955 0.006 12.705 0.575 0.40
NGC7078 0.927 0.002 0.861 0.002 13.877 0.520 0.30
NGC7089 0.996 0.005 0.909 0.006 12.324 0.615 0.30
NGC7099 0.978 0.005 0.908 0.003 13.211 0.475 0.40
Pal 2 1.019 0.015 0.962 0.010 11.582 0.558 0.30
Rup 106 0.965 0.014 0.890 0.009 13.509 0.873 0.30
Terzan 8 0.990 0.023 0.905 0.011 13.440 0.872 0.30
---------- ---------------- ----------------- ---------------- ----------------- -------- ------------ -------------
\[GC\_results\]
[^1]: @mormany05 actually have found such a relation, but they used only three clusters and did not publish the details.
[^2]: <http://www.astro.ufl.edu/~ata/public_hstgc/>
[^3]: <http://stev.oapd.inaf.it/cgi-bin/cmd>
[^4]: <http://albione.oa-teramo.inaf.it/>
[^5]: <http://stellar.dartmouth.edu/~models/index.html>
[^6]: We used the Python implementation of this algorithm that is part of the Scipy library: <http://docs.scipy.org/doc/scipy/reference/odr.html>
[^7]: i.e. we ignore uncertainties in metallicity and only look at the color offset between the data and the best fit relation
[^8]: Actually, @bressan12 write about $\alpha$-enhanced PARSEC isochrones, but these are not (yet) publicly available.
[^9]: The offset can already be seen in @sirianni05 [Fig. 21] as an offset in plot of (V-I) versus V-F606W.
[^10]: Strictly speaking, we compare slightly different things here: The transformed relations measure the color at constant I-band magnitude, while in this work we have measured the color at constant F814W magnitude. We can ignore this difference here because the difference between I-band and F814W is small. According to the transformations given above, the differences between F814W and I are always smaller than 0.05mag and the resulting error in the color measurement of the RGB is always smaller than 0.01mag (except for the two reddest clusters, for which it can reach 0.06 mag). Therefore the effect on the total color-metallicity relation is negligible.
| {
"pile_set_name": "ArXiv"
} |
{
"pile_set_name": "ArXiv"
} |
|
---
abstract: 'The entanglement entropy (EE) has emerged as an important window into the structure of complex quantum states of matter. We analyze the universal part of the EE for gapless systems put on tori in 2d/3d, denoted by ${\chi}$. Focusing on scale invariant systems, we derive general non-perturbative properties for the shape dependence of $\chi$, and reveal surprising relations to the EE associated with corners in the entangling surface. We obtain closed-form expressions for $\chi$ in 2d/3d within a model that arises in the study of conformal field theories (CFTs), and use them to obtain ansatzes without fitting parameters for the 2d/3d free boson CFTs. Our numerical lattice calculations show that the ansatzes are highly accurate. Finally, we discuss how the torus EE [can act as a fingerprint of exotic states such as]{} gapless quantum spin liquids, e.g. Kitaev’s honeycomb model.'
author:
- 'William Witczak-Krempa'
- 'Lauren E. Hayward Sierens'
- 'Roger G. Melko'
bibliography:
- 'Biblo.bib'
title: Cornering gapless quantum states via their torus entanglement
---
Measures of quantum entanglement have emerged as powerful tools to characterize complex many-body systems[@Calabrese1; @Casini1; @Fradkin_book; @Wen_book2; @laflorencie], such as phases with topological order, gapless spin liquids and quantum critical states lacking long-lived excitations. The entanglement entropy (EE) and its Rényi relatives have proven especially useful. The EE of a spatial region $A$, heuristically, measures the amount of entanglement between the inside of $A$ and the outside. Different regions will reveal different properties about the physical state. Generally, a convenient choice is to work on a space that is periodic in at least one direction, [[*i.e.*]{}]{}a cylinder or, particularly in the case of finite-size lattice calculations, a torus. In this setting, region $A$ is often chosen to wrap around at least one cycle, making it topologically non-trivial. In a large class of topologically ordered systems in 2 spatial dimensions (2d), the EE of the groundstate on a cylinder or torus reveals a wealth of information[@Dong08; @Zhang12; @Cincio] about the fractionalized excitations (anyons). Furthermore, these EEs have proved to be useful diagnostics in the search for such exotic phases[@isakov; @Wang11; @jiang12; @depenbrock]. In contrast, for gapless states, analytical[@Fradkin_book; @Fradkin06; @Stephan09; @Hsu09; @Max09; @Hsu10; @Oshikawa10; @Max11; @Swingle12; @fradkin; @Pretko] and numerical[@Stephan09; @Ju_2012; @Inglis_2013; @Bohdan; @Helmes2014; @Luitz; @Laflorencie2015] studies have revealed that the situation is more intricate and numerous open questions remain.
In this work, we analyze the universal torus EE of gapless theories in 2d/3d. We focus our attention on scale invariant systems such as conformal field theories (CFTs) and Lifshitz quantum critical theories ($z\neq 1$), thus excluding the extra complexity due to Fermi surfaces. We derive general properties of the torus EE in 2d/3d using strong subadditivity[@Lieb73] and other considerations. We then make new connections between the shape dependence of the torus EE, and the EE associated with sharp corners[@Casini1], as shown in [Fig.\[fig:torus\]]{}d. The comparison is natural because both quantities are expressed in terms of an angle variable. Surprisingly, we find that the angle dependence of both the torus and corner functions are nearly equal when properly normalized, [Fig.\[fig:comparison\]]{}. This is illustrated using free CFTs, and strongly coupled ones. To gain more intuition about the shape dependence of universal term, we derive a closed-form expression for the torus EE in 2d/3d using a CFT construction. This allows us to make approximate predictions for the free boson CFT, *without any fitting parameters*. Our numerical analysis shows that these predictions work accurately. We then discuss how the torus EE can be used to reveal both the topological and geometrical degrees of freedom of gapless spin liquids, using the Kitaev model as an example.
![[**a & b)**]{} 2d space with a torus topology. We study the EE of a cylindrical region $A$. [**c)**]{} Constraints on the torus EE function ${\chi}({\theta})$ result from dividing $A$ into 3 parts, and applying strong subadditivity. [**d)**]{} Region with a sharp corner.[]{data-label="fig:torus"}](torus-corner.pdf)
[**Fundamentals of torus entanglement:**]{} We consider a system on a flat torus, [Fig.\[fig:torus\]]{}, [[*i.e.*]{}]{}we identify the coordinate $r_i$ with $r_i\!+\! L_i$, $i=x,y$. Given the corresponding groundstate, we study its EE associated with a cylindrical region $A$ of length $L_A$, $S(A)=-{\rm tr}(\rho_A\ln \rho_A)$; $\rho_A$ is the reduced density matrix of $A$. The EE scales as $$\begin{aligned}
\label{SA}
S(A)= \mathcal B\, 2L_y/{\delta}-{\chi}+O({\delta}/L_y),\end{aligned}$$ in the limit where $L_i,L_A$ far exceed the microscopic (UV) scale $\delta$, which can be taken to be the lattice spacing. The first term corresponds to the “area law”, with a non-universal prefactor $\mathcal B$. Our interest lies in the ${\delta}$-independent term, $-\chi$, because it is *universal*. It remains constant with growing $L_y$, at fixed ratios $L_A/L_i$, but in general depends non-trivially on both ratios. $\chi$ thus constitutes a non-trivial measure of the low-energy degrees of freedom of the system, and as we shall see, acts as fingerprint of the state.
We now obtain non-perturbative properties of the torus function ${\chi}({\theta};{b})$, where we have defined the [natural]{} angular variable ${\theta}\!=\!2\pi L_A/L_x$, and the aspect ratio ${b}\!=\!L_x/L_y$ (we shall often keep the $b$-dependence implicit). First, since we are dealing with pure states, the EE of $A$ must equal that of its complement, [[*i.e.*]{}]{}$\chi({\theta})=\chi(2\pi- {\theta})$; we shall henceforth restrict ourselves to $0\!<\!{\theta}\!\leq\!\pi$, as in [Fig.\[fig:comparison\]]{}. Further, since the limit where $A$ approaches half the torus is not singular, ${\chi}$ will be analytic about $\pi$: $$\begin{aligned}
\label{exp}
{\chi}({\theta}\approx\pi) = \sum_{\ell=0} c_\ell \cdot(\pi-{\theta})^{2\ell},\end{aligned}$$ where only even powers appear due to the aforementioned reflection symmetry about $\pi$. The $c_\ell$ depend on the aspect ratio ${b}$, and it would be interesting to understand which properties of the state they encode. To derive further constraints on ${\chi}$, we invoke an important property of the EE, namely its strong subadditivity[@Lieb73] (SSA), which implies the following inequality for 3 non-overlapping regions: $S(A_1\!\cup\! A_2\!\cup\! A_3)+S(A_2)\leq S(A_1\!\cup\! A_2)+S(A_2\cup A_3)$. The key idea is to divide $A$ into 3 regions as in [Fig.\[fig:torus\]]{}c, with angles ${\theta}_i$, and apply SSA. Substituting [Eq.(\[SA\])]{} into the SSA inequality, we find that the boundary law contributions cancel and we are left with ${\chi}({\theta}_1+{\theta}_2 +{\theta}_3)+{\chi}({\theta}_2)\geq{\chi}({\theta}_1\!+\!{\theta}_2)+{\chi}({\theta}_2\!+\!{\theta}_3)$. From this, we can derive $$\begin{aligned}
\label{ineq}
{\chi}'({\theta})\leq 0 \,, \qquad {\chi}''({\theta}) \geq 0,\end{aligned}$$ for $0\!<\!{\theta}\!\leq\!\pi$, [[*i.e.*]{}]{}the torus function ${\chi}({\theta})$ is convex decreasing on that interval. As a direct consequence of the inequalities (\[ineq\]), the second expansion coefficient in [Eq.(\[exp\])]{} satisfies $c_1\geq 0$ (for all aspect ratios).
![Comparing the universal torus and corner EE of various CFTs in 2d.[]{data-label="fig:comparison"}](EE_2D_torusVsCorner_PRL.pdf)
[We now examine the limit ${\theta}\!\to\!0$ with $L_{x,y}$ fixed, in which case the EE reduces to that of a (periodic) thin strip of width $L_A\!\to\!0$ and length $L_y\!\gg\! L_A$. We argue that the periodicity in the $x,y$-directions and the associated boundary conditions do not influence ${\chi}$ in this limit since the EE is dominated by degrees of freedom that do not exceed length scales $\sim\! L_A\ll L_{x,y}$. The total $\chi$ can be obtained by adding the contributions from these local patches, and will be proportional to $L_y/L_A$. We can thus relate the thin slice limit on the torus to the EE of a thin strip in *infinite* space. For scale invariant systems, this reads[@Casini1] $S_{\rm strip}=\mathcal B 2L/{\delta}- \kappa L/L_A$, where $L_A$ is the strip’s width. $L$ is the long-distance regulator of the infinite strip; alternatively, we can define the EE per unit length, $S_{\rm strip}/L$. ${\chi}$ will thus have [the same $\kappa L_y/L_A$]{} divergence in the thin slice limit:]{} $$\begin{aligned}
\label{thin}
{\chi}({\theta}\to 0) = \kappa\,\frac{L_y}{L_A} = \frac{2\pi\kappa}{b\,{\theta}}.\end{aligned}$$ [Further, by virtue of (\[ineq\]), $\kappa\geq 0$.]{} This means that in the small-$\theta$ limit the full EE, [Eq.(\[SA\])]{}, decreases since the universal contribution $\chi$ appears with a negative sign. This is consistent since when $A$ vanishes, $S\!=\!0$. [The universal constant $\kappa$ has been computed for certain critical theories[@Casini_rev]; it will play a central role in our discussion.]{}
[**Relation to corner entanglement:**]{} The above properties share striking similarities with the EE associated with sharp corners, as we now explain. Given a region $A$ [in the infinite plane]{} that contains a corner with opening angle ${\vartheta}$, [Fig.\[fig:torus\]]{}d, the EE scales as $$\begin{aligned}
S(A)= B\,L/{\delta}- a({\vartheta}) \ln(L/{\delta}) + \dotsb, \end{aligned}$$ where $B$ is the area law prefactor, and $a({\vartheta})$ is a *universal* coefficient arising from the corner[@Casini3; @Fradkin06; @Hirata07; @Casini_rev; @Kallin13; @Kallin14]. It encodes rich low-energy information about the state[@Fradkin06; @corner-prl; @corner-twist; @Kallin13; @Gustainis; @Helmes2014; @Helmes2; @free-corners; @laflorencie], but in contrast to $\chi$, it vanishes for gapped systems and is thus blind to purely topological degrees of freedom. $a({\vartheta})$ is also symmetric about $\pi$ (at which point the corner disappears), and can be expanded as in [Eq.(\[exp\])]{}. For CFTs, the leading term in the expansion is[@corner-prl; @Bueno2; @faulkner15] $(\pi^2C_T/24)(\pi-{\vartheta})^2$, where $C_T$ determines the 2-point function of the stress tensor (and thus of the energy density) in the groundstate. [Fig.\[fig:comparison\]]{} shows $a({\vartheta})$ for the free scalar/Dirac fermion[@Casini_rev] and holographic CFTs[@Hirata07]. Further, $a({\vartheta})$ obeys the [same]{} monotonicity and convexity conditions[@Hirata07] (\[ineq\]). Finally, in the sharp corner limit ${\vartheta}\to 0$, the corner function shows a $1/{\vartheta}$ divergence[@Casini_rev] just as $\chi$: $a({\vartheta}\!\to\! 0)\!=\! \kappa_c/{\vartheta}$. For CFTs, $\kappa_c\!=\!\kappa$ is exactly the same universal constant that controls the divergence of $\chi({\theta}\!\to\! 0)$, [Eq.(\[thin\])]{}. This holds because the sharp corner geometry can be conformally mapped to that of a thin strip[@Bueno2], which controls ${\chi}({\theta}\!\to\! 0)$ as discussed above. [It would be interesting to see if non-conformal critical theories ($z\!\neq\! 1$) have the same relation between their sharp-corner $\kappa_c$ and thin-slice coefficients $\kappa$.]{}
Given the similar asymptotics of ${\chi}({\theta})$ and $a({\vartheta})$, one can wonder how they compare at intermediate angles. [Fig.\[fig:comparison\]]{} shows the torus and corner functions of various CFTs. For a meaningful comparison, we normalize them by the thin-slice/sharp-corner coefficient $\kappa$. Surprisingly, all curves nearly overlap in the entire range of angles. What makes the collapse more remarkable is that the curves for the holographic CFTs[@fradkin] and the Extensive Mutual Information model[@Casini05; @Casini08] (defined below) hold for *all* aspect ratios $b\leq 1$ [(App. \[ap:properties\_EMI\])]{}, a non-trivial fact in itself. The same $b$-independence of $b{\chi}$ approximately holds for the massless scalar, as we illustrate with numerical data at $b\!=\! 1,\tfrac{1}{2},\tfrac{1}{4}$, taken from [Fig.\[fig:scalar\_2d\]]{}. The reason for the collapse constitutes an open question beyond the scope of this work, [but it suggests a deeper relation between wavefunctions on spaces with different topologies/geometries.]{} [**Ansatz from extensive mutual information:**]{} To gain further intuition about the EE on tori, we derive a closed-form ansatz for ${\chi}({\theta})$ that can be meaningfully compared with a large class of gapless states, particularly CFTs. To do so we use the Extensive Mutual Information model (EMI)[@Casini05; @Casini08; @Swingle10], which has proven useful in the analysis of the EE of CFTs in various dimensions[@Casini05; @Casini08; @Swingle10; @corner-prl; @corner-twist]. The EMI is not defined through a Hamiltonian, but instead allows for a simple geometric computation of the EE within the bounds of conformal symmetry, and has passed numerous non-trivial tests[@Casini05; @Swingle10; @corner-prl; @corner-twist]. The resulting EE of the EMI can be interpreted[@Swingle10] in terms of an ansatz for twist (or swap) operators used to compute Rényi and entanglement entropies. The designation EMI comes from the fact that its mutual information $I(A,B)=S(A)+S(B)-S(A\cup B)$ is extensive: $I(A,B\cup C)=I(A,B)+I(A,C)$. In infinite flat space, the EE of a region $A$ can be computed as follows within the EMI: $$\begin{aligned}
\label{emi}
S(A) = \int_{\partial A}\! d {{\bf r}}_1 \int_{\partial A} \! d{{\bf r}}_2 \,\,\hat n_1\cdot \hat n_2 \, C({{\bf r}}_1-{{\bf r}}_2),
$$ where $\hat{{{\bf n}}}$ denotes the unit normal to the boundary $\partial A$, and $C({{\bf r}})=s_1/|{{\bf r}}|^{2(d-1)}$. The coordinates ${{\bf r}}_{1,2}$ live on $\partial A$, and $s_1$ is a positive constant. In order to apply the prescription (\[emi\]) to the torus, we need to account for the periodicity when determining the function $C$. [However, contrary to the infinite plane, conformal invariance and the extensivity of the mutual information do not suffice to fix $C$ on the torus, and one is left with a richer set of possibilities. A simple choice for $C$ is described in App.\[ap:EMI\]; the resulting torus EE reads:]{} $$\begin{aligned}
\label{tor-emi}
{\chi_{\rm \scriptscriptstyle EMI}}({\theta})\! = \!4\kappa\! \left[ \frac{\cot{^{-1}}\!\left(\tfrac{{b}}{\pi}{\theta}\right)}{{b}\,{\theta}} \! + \!
\frac{\cot{^{-1}}\!\left(\tfrac{{b}}{\pi}(2\pi-{\theta})\right)}{{b}\,(2\pi-{\theta})} \right]\!
+ 2{\gamma}\! \end{aligned}$$ where $\cot{^{-1}}z$ is the inverse cotangent, and ${\gamma}$ is a constant. We have normalized the first term of (\[tor-emi\]) using $\kappa$ so as to reproduce the expected small ${\theta}$ divergence, [Eq.(\[thin\])]{}. ${\chi_{\rm \scriptscriptstyle EMI}}$ is thus non-negative for all angles and aspect ratios. Our result is naturally symmetric and analytic about ${\theta}=\pi$, as in [Eq.(\[exp\])]{}, and obeys the constraints (\[ineq\]) from SSA. [Eq.(\[tor-emi\])]{} thus provides a closed-form candidate function to analyze the EE of strongly interacting states, especially CFTs, on tori. This is a powerful tool since virtually no other analytic results exist in this case. In an important development, a semi-analytical result was obtained[@fradkin] for ${\chi}$ in special CFTs using the holographic AdS/CFT correspondence. However, singular behavior was found as the aspect ratio goes through $b\!=\!1$. Such non-analycities are not expected for generic CFTs, as in the quantum critical Ising model, and are indeed absent in [Eq.(\[tor-emi\])]{} and in the free boson CFT ([Fig.\[fig:scalar\_2d\]]{}). Nevertheless, as noted above, striking similarities exist for $b\leq 1$ between the EMI and AdS functions, [Fig.\[fig:comparison\]]{}. In the latter case $b\cdot({\chi}-{\chi}(\pi))$ is exactly independent of $b$[@fradkin], while for the EMI, this holds to excellent accuracy and is not a priori obvious from [Eq.(\[tor-emi\])]{}, see App.\[ap:properties\_EMI\] for more details.
A useful limit to consider is the thin torus: ${b}\!\to\!\infty$ with ${\theta}$ fixed, in which case [Eq.(\[tor-emi\])]{} reduces to $2{\gamma}+ O(b^{-2})$. Namely, [the universal EE]{} approaches a pure constant independent of $L_A,L_i$, [which is twice the universal EE associated with a semi-infinite cylindrical bipartition of an infinite cylinder (App. \[ap:EMI\]).]{} This is consistent with the expectation that a generic CFT will not contain gapless modes in the 1d limit [because the contracting $y$-direction leads to a large $\sim\! 1/L_y$ gap.]{} Otherwise, [the EE]{} would scale as $\sim\ln[\tfrac{L_x}{\pi\delta}\sin(\tfrac{\pi L_A}{L_x})]$ [when $L_A$ is changed, corresponding to the behavior of a critical 1d system on a circle]{}[@Calabrese1]. The [absence of such critical scaling in the thin torus limit]{} is verified[@fradkin] in the strongly coupled holographic CFTs mentioned above. Exceptions do occur, [[*e.g.*]{}]{}for non-interacting CFTs with *periodic* boundary conditions due to zero [energy]{} modes, but one can twist the boundary conditions to [gap them out]{} (see below). [**3d torus:**]{} We now explore the largely uncharted territory of torus entanglement in gapless 3d theories. We take the subregion $A$ to be a hyper-cylinder of length $L_A$ aligned along $x$, [Fig.\[fig:scalar\_3d\]]{}. The corresponding angle variable is again ${\theta}=2\pi L_A/L_x$. The analog of [Eq.(\[SA\])]{} in 3d reads: $$\begin{aligned}
\label{SA3d}
S^{3d}(A)= \mathcal B\frac{2 L_yL_z}{\delta^2}-{\chi^{3d}}+O(\delta/L_{y,z}), \end{aligned}$$ where ${\chi^{3d}}({\theta};{b}_y,{b}_z)$ now depends on the 2 aspect ratios, $b_{y,z}=L_x/L_{y,z}$. The general properties obtained above for the 2d torus function ${\chi}$ can be adapted *mutatis mutandis* to the 3d case. In particular, ${\chi^{3d}}$ will be convex decreasing for $0\!<\!{\theta}\!\leq\! \pi$, as in [Eq.(\[ineq\])]{}, and will be analytic about ${\theta}\!=\!\pi$. Further, in the small-${\theta}$ limit we find $$\begin{aligned}
\label{small_3d}
{\chi^{3d}}({\theta}\to 0) = {\kappa^{3d}}\, \frac{L_y L_z}{L_A^2} = \frac{(2\pi)^2{\kappa^{3d}}}{b_y b_z\, \theta^2} \end{aligned}$$ since the EE effectively becomes that of an infinite thin slab with thickness $L_A$. Our 2d argument given above can be generalized to argue that the system is insensitive to the periodicity of the $x,y,z$ directions in this limit. ${\kappa^{3d}}\geq 0$ is a universal constant characterizing the theory[@Casini_rev], and it is the 3d analog of the 2d $\kappa$ encountered above. As we have done in 2d, we can use the EMI to obtain a closed-form torus function ${\chi^{3d}_{\rm \scriptscriptstyle EMI}}({\theta})$. [[Fig.\[fig:scalar\_3d\]]{} shows the result for different aspect ratios]{}; the full answer is given in App.\[ap:EMI\].
![[**Main**]{}: Torus function ${\chi}$ for the massless scalar in 2d for various aspect ratios ${b}=L_x/L_y$; it has been vertically offset for clarity. The points are numerical data, and the lines are the predictions obtained using ${\chi_{\rm \scriptscriptstyle EMI}}$, *without any fitting parameters*. [**Inset**]{}: Dirac fermion data[@fradkin] at $b\!=\!1$, the corresponding ${\chi_{\rm \scriptscriptstyle EMI}}$, and the complex scalar data for comparison. The axes represent the same quantities as in the main plot.[]{data-label="fig:scalar_2d"}](EE_2D_PBCx_APBCy_mass0-0_PRL.pdf)
![[**Main**]{}: Torus function ${\chi^{3d}}$ for the massless free boson in 3d [for various aspect ratios $b_y\!=\! b_z\!=\! b$]{}. The points are the numerical data and the line is the prediction obtained using the ansatz ${\chi^{3d}_{\rm \scriptscriptstyle EMI}}$, *without any fitting parameters*. [**Inset**]{}: Opposing faces of the box are identified to give a 3-torus topology. Region $A$ is a hyper-cylinder extending along $x$.[]{data-label="fig:scalar_3d"}](torus3dp2.pdf)
[**Torus EE for lattice bosons:**]{} We numerically evaluate the torus EE of a free and massless relativistic boson (a CFT) in 2d/3d using the square/cubic lattice realization of the Hamiltonian $H\! =\! \int \! d^d x [ \tfrac{1}{2}\pi^2 + \tfrac{1}{2}( \nabla \phi)^2 ]$, where $\phi$ is the 1-component boson and $\pi$ its conjugate momentum. This theory corresponds to the Gaussian fixed point of the interacting quantum critical Ising model in 2d/3d, and constitutes a key benchmark system. We obtain the torus EE [by directly evaluating the reduced density matrix of $A$]{} from the two-point vacuum correlation functions $\left\langle \phi_\mathbf{x} \phi_\mathbf{x'} \right\rangle$ and $\left\langle \pi_\mathbf{x} \pi_\mathbf{x'} \right\rangle$ for lattice sites $\mathbf{x},\mathbf{x'}\!\in\! A$ [@Peschel]; details are given in App.\[ap:numerics\]. We perform our 2d calculations on lattices of size [$L_x=500$, and our 3d calculations on lattices of size $L_x = 100, 140, 200, 288, 456$ for aspect ratios $b _y\!=\! b_z \!= \tfrac{1}{4}, \tfrac{1}{2},1,2,4$ (respectively).]{} Each lattice has antiperiodic boundary conditions (APBC) in the $y$-direction and PBC along the remaining directions. We use the former to avoid the $\mathbf{k}\!=\!0$ zero mode present for PBC. The numerical results in 2d/3d are shown in Figs.\[fig:scalar\_2d\],\[fig:scalar\_3d\], respectively. The solid lines in both figures correspond to the EMI candidate functions, Eq.(\[tor-emi\]) in 2d, while the 3d one is given in the appendix. Crucially, *no fitting* to the data has been performed. Instead, to generate the lines we have relied on two facts: First, the EMI torus functions relative to their value at $\theta\!=\!\pi$, ${\chi_{\rm \scriptscriptstyle EMI}}^{3d}(\theta)-{\chi_{\rm \scriptscriptstyle EMI}}^{3d}(\pi)$, depend on a single universal constant, $\kappa^{3d}$. Second, this constant has been computed in a *different* context for the massless scalar in 2d/3d[@Casini_rev]: $\kappa_{\rm sc} = 0.0397$, ${\kappa^{3d}}_{\rm sc} =5.54\! \times\! 10^{-3}$. The resulting ansatz curves and the data agree with each other exceptionally well, which is surprising since we have not done any fitting. The agreement in [2d/3d]{} extends over a wide range of aspect ratios, meaning the ansatz even captures the $b$-dependence [without any fitting]{}! We note that since the EMI does not describe a free boson CFT, we expect that some of the deviations are intrinsic.
The inset of [Fig.\[fig:scalar\_2d\]]{} shows the data for a massless free Dirac fermion (another CFT) obtained numerically in Ref. with (A)PBC along $(y)x$. In this case, we again know the value of the small-${\theta}$ constant[@Casini_rev], $\kappa_{\rm Dirac}=0.0722$, which allows us to fix the ${\chi_{\rm \scriptscriptstyle EMI}}$ ansatz; the result is the line in the inset of [Fig.\[fig:scalar\_2d\]]{}. We have also shown the data for a complex scalar, which overlaps almost exactly with that of the Dirac fermion. Part of the agreement can be explained from the fact that the complex scalar has $\kappa\!=\!2\kappa_{\rm sc}\!=\!0.0794$, which is close to the Dirac value.
[**Summary & outlook:**]{} We have seen that the universal EE of cylindrical regions on tori reveals non-trivial information about scale invariant quantum systems, like conformal field theories, in 2d/3d. Our findings range from general non-perturbative properties to concrete examples involving bosons on a lattice. We note that many of these results can be extended to the Rényi entropies $S_n$. In particular, in the thin slice limit ${\chi}_n$ will show the same divergence as in Eqs.(\[thin\],\[small\_3d\]), but with ${\kappa}_n$. A torus function was previously derived[@Stephan13] at $n\!\geq \! 2$ for a family of 2d Lifshitz quantum critical points[@Ardonne04], and it was successfully compared with the von Neumann case in various theories. Many of our results apply to that function[@will-prep].
[Since the torus EE can also capture topological information about the excitations[@Dong08; @Zhang12; @Cincio] (relating to anyons, say), it will be interesting to use it to obtain fingerprints for gapless spin liquids or deconfined quantum critical points.]{} In this vein, a simple example where ${\chi}$ encodes both topological and geometrical degrees of freedom is Kitaev’s gapless spin liquid on the honeycomb lattice[@Kitaev06]. In this frustrated spin model, the emergent long-distance degrees of freedom are 2 massless Majorana fermions coupled to a $Z_2$ gauge field. We expect the universal EE to be ${\chi}_{\rm f}({\theta})+{\chi}_{\rm top}$, owing to the factorization of the fermions and $Z_2$ contributions[@Yao10]. ${\chi}_{\rm top}$ is purely topological and comes from the $Z_2$ gauge theory[@Zhang12], while the fermions yield the shape dependent ${\chi}_{\rm f}({\theta})$. Inspired by this capability of $\chi$ to capture both topological and gapless degrees of freedom, we ask ask whether the torus EE can yield a RG monotone, in the same spirit as the disk EE[@sinha10; @CH12]?
*Acknowledgments*—We are thankful to P. Bueno, X. Chen, E. Fradkin, A. Lucas for useful discussions. WWK is grateful for the hospitality of Perimeter Institute, where this work was initiated. WWK was funded by a fellowship from NSERC, and by MURI grant W911NF-14-1-0003 from ARO. LHS was partially funded by the Ontario Graduate Scholarship. RM is supported by NSERC of Canada, the Canada Research Chair Program and the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research & Innovation.
[**Supplementary Information**]{}
Torus entanglement of the Extensive Mutual Information model {#ap:EMI}
============================================================
![a) Semi-infinite subregion $A$ of a space with the topology of an infinite cylinder. b) Finite subregion $A$ of the infinite cylinder. c) Finite subregion $A$ of a space with a torus topology. Each green arrow denotes a normal vector, $\hat{{{\bf n}}}$, to the entangling surface. []{data-label="fig:cyl"}](cylinder_plots.pdf)
Let us first recall the calculation of the EE within the Extensive Mutual Information model (EMI) for a region $A$ in infinite flat space[@Casini05; @Casini08; @Swingle10]. One needs to evaluate the following double integral over two copies of the entangling surface $\partial A$: $$\begin{aligned}
\label{s-emi}
S(A) &= \int_{\partial A}\! d {{\bf r}}_1\! \int_{\partial A} \! d {{\bf r}}_2 \, \hat n_1\cdot \hat n_2 \, C({{\bf r}}_1-{{\bf r}}_2), \end{aligned}$$ with $$\begin{aligned}
C({{\bf r}}) &=\frac{s_1}{|{{\bf r}}|^{2(d-1)}}, \label{emi2}\end{aligned}$$ where $d$ is the spatial dimension, $\hat{{{\bf n}}}$ the vector normal to $\partial A$, and ${{\bf r}}_{12}={{\bf r}}_1-{{\bf r}}_2$ is the separation vector.
The prescription given in [Eq.(\[emi2\])]{} cannot be applied to the torus because it does not account for the periodicity of space. On the torus we require $C({{\bf r}})$ to be periodic along all the spatial directions. Further, at distances much shorter than the linear dimensions of the torus, $L_i$, the $C$-function needs to reduce to [Eq.(\[emi2\])]{}, i.e. $$\begin{aligned}
C(r_i\ll L_i ) = s_1/|{{\bf r}}|^{2(d-1)}\,,\end{aligned}$$ because in this limit the boundary conditions of space should not affect the EE. This scaling naturally follows from the scale invariance of the system on the infinite plane[@Casini05; @Casini08; @Swingle10]. We note that under such general conditions, $S(A)$ defined in [Eq.(\[s-emi\])]{} will have an extensive mutual information $I$ for non-intersecting regions $A,B,C$ living on the torus: $$\begin{aligned}
I(A,B\cup C) = I(A,B) + I(A,C)\end{aligned}$$ where $I(A,B)=S(A)+S(B) - S(A\cup B)$. We are thus led to the conclusion that on the torus, the form of $C({{\bf r}})$ is not entirely constrained by symmetry and extensivity of the mutual information. This is not very surprising because conformal symmetry on the torus is much less powerful than in infinite space $\mathbb R^d$. For instance, conformal symmetry in 1+1 Euclidean spacetime dimensions is not sufficiently powerful to fix the 2-point functions of local operators in a CFT on the torus; they will thus depend on many details of each given theory. In contrast, such correlation functions are fixed by symmetry on the infinite plane.
We thus see that the structure of the EMI on the torus is richer than on the plane. In order to obtain a viable EMI ansatz, we shall construct a $C$-function that extends [Eq.(\[emi2\])]{} to the torus in a simple way. In Section \[sec:new-emi\], we construct a second EMI ansatz using a different $C({{\bf r}})$, which has very similar properties to the first one. This fact, together with the excellent agreement with the numerical data for the boson and Dirac fermion on the lattice, show the robustness and utility of the EMI construction.
Infinite cylinder
-----------------
Let us begin with the simpler case where the entire space takes the topology of an infinite cylinder with circumference $L_y$, [Fig.\[fig:cyl\]]{}a.
*a. Semi-infinite region $A$*
We first take the region $A$ to be a semi-infinite cylinder which ends at $x=0$. The entangling surface is a circle, and constitutes the domain of integration in the EMI calculation. The expression for the EE entropy reads $$\begin{aligned}
S &= \int_0^{L_y}\! dy_1 \int_0^{L_y}\! dy_2 \, \hat n_1\cdot \hat n_2 \, C({{\bf r}}_1-{{\bf r}}_2) \\
&= \int_0^{L_y}\! dy_1 \int_0^{L_y}\! dy_2 \, s_1 \frac{\hat x\cdot\hat x}{\tilde y_{12}^2}, \label{cyl1}\end{aligned}$$ where we have defined $y_{12}=y_1-y_2$, and $$\begin{aligned}
\label{ytilde}
|\tilde y| =
\begin{cases}
|y|, & \mbox{if }\, |y|\leq L_y/2 \\
L_y - |y|, & \mbox{if }\, L_y/2 < |y|\leq L_y,
\end{cases} \end{aligned}$$ [Eq.(\[ytilde\])]{} defines a rather simple distance function on a line segment with periodic boundary conditions, [[*i.e.*]{}]{}a (“flat”) circle. This constitutes our minimal prescription in modifying [Eq.(\[emi2\])]{} to account for the periodicity in the $y$-direction. [We shall see that this ansatz yields sensible and transparent answers for the EE.]{} In fact, our results for the EE on the cylinder and torus topologies will be shown to satisfy all the known requirements (see main text). Further, they will provide very accurate candidate functions to compare with numerical data for non-interacting bosons and fermions.
In obtaining [Eq.(\[cyl1\])]{}, we have used the fact that both normal vectors are $\hat{{{\bf n}}}_{1,2}=\hat x$, and $x_{1,2}=0$ on the entangling surface. We change variables to $Y=(y_1+y_2)/2$ and $y_{12}$, and perform the integral over $Y$: $$\begin{aligned}
S &= s_1L_y \int_{-L_y}^{L_y} \frac{dy_{12}}{\tilde y_{12}^2}
= s_1L_y\, 4\!\int_{{\delta}}^{L_y/2} \frac{dy_{12}}{y_{12}^2},\end{aligned}$$ where in the last equality we have used the definition of $\tilde y_{12}$, [Eq.(\[ytilde\])]{}. We also introduced a UV cutoff ${\delta}$ to make the integral finite. The final result for the EE in the limit $L_y\gg {\delta}$ reads $$\begin{aligned}
\label{semi-inf}
S = \mathcal B \frac{L_y}{{\delta}} - {\gamma}+ \dotsb,\end{aligned}$$ where $\mathcal B=4s_1$, and ${\gamma}=8 s_1$. Both constants are positive since $s_1>0$. We have thus recovered the boundary law term, and a universal (negative) contribution $-{\gamma}$.
*b. Finite region $A$*
Let us now consider the more interesting case where the subregion $A$ is a cylinder of finite length $L_A$, as shown in [Fig.\[fig:cyl\]]{}b. The boundary of region $A$ is now composed of 2 disjoint circles (left and right): $\partial A=L\cup R$. The EE computed within the EMI will thus be composed of 4 contributions, depending on whether ${{\bf r}}_i$ lies on the left or right circle: $$\begin{aligned}
\label{LR}
S &= S^{LL}+ S^{RR} + S^{LR} + S^{RL} {\nonumber \\}&= 2S^{RR} + 2S^{RL}.\end{aligned}$$ In the last equality we have used the translation symmetry along the $x$-direction. Now, $S^{RR}$ is exactly given by the semi-infinite cylinder answer, [Eq.(\[semi-inf\])]{}. It remains to compute $S^{RL}$: $$\begin{aligned}
S^{RL} = \int_0^{L_y}\! dy_1 \int_0^{L_y}\! dy_2 \, \frac{-s_1}{L_A^2 + \tilde y_{12}^2}, \end{aligned}$$ where we have used $\hat{{{\bf n}}}_1=-\hat{{{\bf n}}}_2$ when ${{\bf r}}_{1,2}$ do not lie on the same circle (disconnected component of $\partial A$). By again changing variables to $Y$ and $y_{12}$, and performing the integrals we obtain $$\begin{aligned}
S^{RL} = - s_1\, \frac{4L_y}{L_A} \cot{^{-1}}\! \left( \frac{2L_A}{L_y} \right).\end{aligned}$$ Note that this result is entirely independent of the cutoff ${\delta}$. The final answer for the EE of $A$ thus reads: $$\begin{aligned}
S = \mathcal B \frac{2L_y}{{\delta}} - {\chi^{\rm cyl}}+ \dotsb ,\end{aligned}$$ where we have defined the cylinder function ${\chi^{\rm cyl}}(L_A/L_y)$, $$\begin{aligned}
\label{cyl-emi}
{\chi^{\rm cyl}}&= 2{\gamma}+ \frac{2\kappa}{\pi} \frac{L_y}{L_A}\cot{^{-1}}\! \left( \frac{2L_A}{L_y} \right), \end{aligned}$$ which depends on the single dimensionless ratio, $L_A/L_y$. Here $\kappa=4\pi s_1$ is the “thin strip” constant: It determines the subleading term in the EE of a thin strip of width $L_A$ in the infinite plane[@Casini_rev], $$\begin{aligned}
S_{\rm strip} = \mathcal B \frac{2L}{{\delta}} - \kappa \frac{L}{L_A} +\dotsb ,\end{aligned}$$ where $L$ is a long-distance regulator for region $A$. $\mathcal B$ and ${\gamma}$ are as above, in particular ${\gamma}=8s_1$.
![Full ${\chi_{\rm \scriptscriptstyle EMI}}$ (no vertical offset) for different aspect ratios (using $s_1=1/8$). From bottom to top: $L_x/L_y=4,1,1/2,1/4$. The horizontal dashed line corresponds to $L_x/L_y=\infty$, in which case ${\chi_{\rm \scriptscriptstyle EMI}}=2{\gamma}=2$. []{data-label="fig:EMI"}](chiEMI_2D.pdf)
Torus in 2d
-----------
We can now tackle the torus topology in 2d, [Fig.\[fig:cyl\]]{}c. The extra complication compared with the infinite cylinder is that the space is now periodic in the $x$-direction. As a result the final answer must be the same under the exchange $L_A\leftrightarrow L_x-L_A$ (by purity). The structure of the EE within in the EMI will still decompose into 2 terms, as in [Eq.(\[LR\])]{}. $S_{RR}$ will be the same as in the cylinder calculation above because it is not sensitive to the $x$-cycle. In contrast, $S_{RL}$ does know about the $x$-cycle. The simplest way to accommodate for this is to add a “mirror” contribution to [Eq.(\[emi2\])]{}, $C^{\rm mirror}=s_1 /[(L_x\!-\!L_A)^2+\tilde{y}_{12}^2]$, in the calculation of $S^{RL}=S^{LR}$. Performing the calculation with the additional mirror term, we finally obtain $$\begin{aligned}
S = \mathcal B \frac{L_y}{{\delta}} - {\chi_{\rm \scriptscriptstyle EMI}}+ \dotsb,\end{aligned}$$ where the torus EE function is $$\begin{aligned}
{\chi_{\rm \scriptscriptstyle EMI}}= 2{\gamma}+ \frac{2\kappa L_y}{\pi}\! \left[ \frac{\cot{^{-1}}\!\left( \frac{{2L_A}_{}}{L_y}\right) }{L_A}
+ \frac{\cot{^{-1}}\!\left( \frac{2(L_x-L_A{)}_{} }{L_y}\right) }{L_x-L_A} \right]\!\end{aligned}$$ and $\mathcal B=4 s_1$, ${\gamma}=8s_1$ and $\kappa=4\pi s_1$ are the same constants as in the infinite cylinder calculations above. $\mathcal B$ is independent of the aspect ratio of the torus. Further, ${\chi_{\rm \scriptscriptstyle EMI}}$ is non-negative. [Fig.\[fig:EMI\]]{} shows the $b,{\theta}$-dependence of ${\chi_{\rm \scriptscriptstyle EMI}}$.
Torus in 3d
-----------
We now turn to the torus topology in 3d, see [Fig.\[fig:scalar\_3d\]]{}. We compactify the spatial dimensions such that the $i$th coordinate $r_i$ is identified with $r_i+L_i$, $i=x,y,z$. The entangling surface consists of 2 disconnected parts, each of which is a 2-torus. As a result, the EE within the EMI again decomposes into 2 terms as in [Eq.(\[LR\])]{}. The first term, $S^{RR}$, comes from having ${{\bf r}}_1$ and ${{\bf r}}_2$ both on the right 2-torus $R$: $$\begin{aligned}
S^{RR} = \int dy_1 dz_1 \int dy_2 dz_2 \frac{s_1}{\left(\tilde y_{12}^2 + \tilde z_{12}^2\right)^2}, \end{aligned}$$ where $\tilde y_{12}$ is as defined in [Eq.(\[ytilde\])]{}, $\tilde z_{12}$ is analogously defined but with $L_z$ instead of $L_y$. We change integration variables to $Y=(y_1+y_2)/2$, $y_{12}$, and $Z=(z_1+z_2)/2$, $z_{12}$, and perform the integrals over $Y,Z$: $$\begin{aligned}
S^{RR}
= 16L_yL_z\! \int_{0}^{\frac{{L_y} }{2}}\!\! dy_{12}\int_{0}^{\frac{{L_z}_{} }{2}}\!\! dz_{12}
\frac{s_1}{\left(y_{12}^2 + z_{12}^2\right)^2}, \end{aligned}$$ where we were able to remove the tildes by restricting the domain of integration. Performing both integrals we get $$\begin{gathered}
S^{RR} = 2\pi s_1 \frac{L_yL_z}{{\delta}^2} - 16s_1\left[ 1+\frac{L_y}{L_z}\tan{^{-1}}\!\left(\frac{L_y}{L_z}\right)\right. \\ \left.
+\frac{L_z}{L_y}\tan{^{-1}}\!\left(\frac{L_z}{L_y}\right) \right] + O({\delta}/L_{y,z}),\end{gathered}$$ which is symmetric under $L_y\leftrightarrow L_z$, as expected. The term in brackets is independent of the UV cutoff ${\delta}$, and will contribute to the torus function ${\chi^{3d}_{\rm \scriptscriptstyle EMI}}$.
Next, we turn to the $S^{RL}$ term in [Eq.(\[LR\])]{}. As in the 2d torus calculation, $S^{RL}$ will receive a contribution from a term with $|x_{12}|=L_A$, and from a “mirror” term with $|x_{12}|=L_x-L_A$, in order to account for the periodicity along $x$. The first contribution reads $$\begin{aligned}
S^{RL}_{(1)}= \int dy_1 dz_1 \int dy_2 dz_2 \frac{-s_1}{\left(L_A^2+ \tilde y_{12}^2 + \tilde z_{12}^2\right)^2} ,\end{aligned}$$ where we have used $\hat{{{\bf n}}}_1=-\hat{{{\bf n}}}_2$. Again changing variables to center-of-mass and relative coordinates, and performing the integration over the former we get $$\begin{aligned}
S^{RL}_{(1)} &= \! \int_0^{\frac{L_y}{2}} \!\! dy_{12}\int_0^{\frac{L_{z_{\!}}}{2}} \!\! dz_{12}
\frac{-16L_yL_zs_1}{\left(L_A^2 + y_{12}^2 + z_{12}^2\right)^2} \\
&=-\frac{16L_y L_z s_1}{2L_A^2}\! \left[ \frac{L_y \tan{^{-1}}\!\Big( \frac{ {L_z}_{} }{ \sqrt{4L_A^2+L_y^2} } \Big) }{\sqrt{4L_A^2+L_y^2}} + y\leftrightarrow z \right]\!.\nonumber
\end{aligned}$$ Thus the final answer for the EE is $$\begin{aligned}
S=\mathcal B\frac{2L_y L_z}{{\delta}^2} - {\chi^{3d}_{\rm \scriptscriptstyle EMI}}+\dotsb ,\end{aligned}$$ where
$$\begin{gathered}
\label{torEMI-3d-full}
{\chi^{3d}_{\rm \scriptscriptstyle EMI}}= 2{\gamma}^{3d} + \frac{4{\kappa^{3d}}}{\pi}\!\left[ \frac{L_y}{L_z}\tan{^{-1}}\!\left(\frac{L_y}{L_z} \right) +
\frac{L_y L_z}{2L_A^2}\frac{L_y \tan{^{-1}}\!\Big(\frac{{L_z}_{}}{\sqrt{4L_A^2+L_y^2}} \Big) }{\sqrt{4L_A^2+L_y^2}} \right. \\
\left. + \, \frac{L_y L_z}{2 (L_x \!-\!L_A)^2 } \frac{L_y \tan{^{-1}}\!\Big(\frac{{L_z}_{}}{\sqrt{4(L_x-L_A)^2+L_y^2}} \Big) }{\sqrt{4(L_x\! -\! L_A)^2+L_y^2}}
+ y\leftrightarrow z \right]
\end{gathered}$$
with ${\gamma}^{3d}=16s_1$ and ${\kappa^{3d}}=8\pi s_1$. We note that the square brackets contain 6 terms due to the contributions with $y$ and $z$ interchanged. Thus, ${\chi^{3d}_{\rm \scriptscriptstyle EMI}}$ is fully symmetric under the exchange $L_y\leftrightarrow L_z$, as it should be. We have defined ${\kappa^{3d}}$ as the thin slab coefficient: $$\begin{aligned}
\lim_{L_A\to 0} {\chi_{\rm \scriptscriptstyle EMI}}= {\kappa^{3d}}\frac{L_y L_z}{L_A^2},\end{aligned}$$ which is readily obtained from [Eq.(\[torEMI-3d-full\])]{} by using the identity $\tan{^{-1}}z +\tan{^{-1}}(1/z)=\pi/2$, with $z>0$. [The result simplifies for $b_y=b_z=b$ to give $$\begin{gathered}
\label{tort-emi}
{\chi^{3d}_{\rm \scriptscriptstyle EMI}}({\theta}) = 16\pi {\kappa^{3d}}\Bigg[\frac{1}{8\pi} + \frac{\cot{^{-1}}\! \sqrt{1+(b{\theta}/\pi)^2}}{b^2{\theta}^2 \sqrt{1+(b{\theta}/\pi)^2}} \\
+ \frac{\cot{^{-1}}\!\sqrt{1+b^2(2\pi-{\theta})^2/\pi^2} }{b^2(2\pi-{\theta})^2 \sqrt{1+b^2(2\pi-{\theta})^2/\pi^2}} \Bigg] +
2{\gamma}^{3d}.\end{gathered}$$ This is the case we study numerically in [Fig.\[fig:scalar\_3d\]]{} of the main text for the non-interacting gapless boson. We note that ${\chi^{3d}_{\rm \scriptscriptstyle EMI}}$ has a similar structure to the 2d result, [[*e.g.*]{}]{}the appearance of $\cot{^{-1}}$.]{}
Properties in $2d$ {#ap:properties_EMI}
------------------
### Special scaling for $b \leq 1$
We discuss the special scaling encountered for $b\cdot{\chi_{\rm \scriptscriptstyle EMI}}({\theta})$, relative to its value at ${\theta}=\pi$: $$\begin{aligned}
{\widetilde{{\chi}}_{\rm \scriptscriptstyle EMI}}(\theta) &= \frac{b}{2\pi} \left[ {\chi_{\rm \scriptscriptstyle EMI}}({\theta};b)-{\chi_{\rm \scriptscriptstyle EMI}}(\pi;b) \right] \\
&= \frac{4\kappa}{\pi} \left( \frac{\cot{^{-1}}(b{\theta}/\pi)}{{\theta}} + \frac{\cot{^{-1}}(b(2\pi- {\theta})/\pi)}{2\pi-{\theta}} \right. {\nonumber \\}& \qquad \qquad \left. -\frac{2}{\pi}\cot{^{-1}}b\right) .\end{aligned}$$ As was mentioned in the main text, ${\widetilde{{\chi}}_{\rm \scriptscriptstyle EMI}}$ is approximately independent of the aspect ratio $b$ in the range $b\leq 1$. [Fig.\[fig:chi-tilde\]]{} demonstrates that this statement holds very accurately.
We note that in the $b\to 0$ limit $$\begin{aligned}
{\widetilde{{\chi}}_{\rm \scriptscriptstyle EMI}}({\theta}; 0) = \frac{2\kappa}{\pi} \, \frac{(\pi-{\theta})^2}{{\theta}(2\pi-{\theta})}.\end{aligned}$$ We recognize this as precisely the corner EE function $a({\theta})$ of the family of Lifshitz quantum critical points with conformal wavefunctions[@Fradkin06]! For that family of $z=2$ theories[@Ardonne04], $\kappa=\pi c/24$[@Fradkin06; @WKB], where $c$ is the Virasoro central charge of the parent 1d CFT that describes the equal-time correlations of the theory. At present, we do not have an explanation for the appearance of this corner function in the torus EE of the EMI. We note that the latter has a different corner function $a_{\rm EMI}(\theta)$.[@Casini08; @Swingle10]
### Smooth and thin-torus limit expansions
We here give the leading terms in the smooth ${\theta}\approx\pi$ and thin slice ${\theta}\approx 0$ expansions for ${\chi_{\rm \scriptscriptstyle EMI}}$ in 2d. In the smooth limit we get $$\begin{gathered}
{\chi_{\rm \scriptscriptstyle EMI}}({\theta}\approx\pi)=
\left(2{\gamma}+ \frac{8 \kappa \cot ^{-1}b}{\pi b} \right) \\
+ \frac{8 \kappa \left(b+2 b^3+\left(b^2+1\right)^2 \cot ^{-1}b\right)}{\pi ^3 b \left(b^2+1\right)^2}(\theta -\pi )^2 + O({\theta}-\pi)^4\end{gathered}$$ while in the thin slice limit we get $$\begin{gathered}
{\chi_{\rm \scriptscriptstyle EMI}}({\theta}\approx 0)= \frac{2 \pi \kappa }{b \theta } + \left(2{\gamma}+ \frac{2 \kappa \cot ^{-1}(2 b)}{\pi b} -\frac{4 \kappa }{\pi }\right)
+O({\theta}).\end{gathered}$$ We see that the first term matches the thin strip contribution, as described above and in the main text.
Another ansatz {#sec:new-emi}
--------------
The reader might wonder how generic is the ansatz described in the above sections? Indeed, one could have chosen another ansatz for the $C({{\bf r}})$ function in [Eq.(\[s-emi\])]{}. Here, we introduce another ansatz, and show that it has very similar properties to the first one. We focus on $d=2$, where the new ansatz for the $C$-function reads: $$\begin{aligned}
\label{new-emi}
C({{\bf r}}) = \frac{s_1}{\operatorname{ch}(x)^2 + \operatorname{ch}(y)^2}\end{aligned}$$ where $$\begin{aligned}
\label{chord}
\operatorname{ch}(r_i) = \frac{L_i}{\pi} \sin\left(\frac{\pi r_i}{L_i}\right) \end{aligned}$$ is the chord length on a circle of circumference $L_i$. [Eq.(\[new-emi\])]{} is manifestly periodic under $r_i\to r_i+L_i$, and reduces to [Eq.(\[emi2\])]{} at short distances. The resulting torus function $\chi$ can be easily obtained using [Eq.(\[s-emi\])]{}: $$\begin{aligned}
\chi(\theta;b) = \frac{\pi\kappa}{b\sin(\theta/2) \sqrt{1+b^2\sin^2(\theta/2)} }\,. \end{aligned}$$ This new ansatz, just like the original one, obeys all the known physical constraints required for $\chi$, such as being convex decreasing on $[0,\pi)$, and the reflection property $\chi(2\pi-\theta)=\chi(\theta)$. Further, in [Fig.\[fig:new-emi\]]{} we quantitatively compare the new ansatz with the old one, using $\chi(\theta)-\chi(\pi)$ normalized by $\kappa$. This is the same quantity that is compared with the boson and fermion numerical data in the main text. We see that the deviations are very small.
![We compare the new EMI ansatz (dashed) with the old one (black and solid) for 3 different aspect ratios. Top to bottom: $b=1/4,1,4$.[]{data-label="fig:new-emi"}](comparing_EMI_ansatz.pdf)
Numerical calculations {#ap:numerics}
======================
We calculate the EE for the lattice Hamiltonian of a free relativistic boson in the massless limit, which is given by $$\begin{aligned}
H = \frac{1}{2} \sum_\mathbf{x}
& \Big[ \pi_{\mathbf{x}}^2 + \left( \phi_{x_1+1, x_2, \ldots, x_d} - \phi_{\mathbf{x}} \right)^2 \nonumber \\
\phantom{\bigg(} & {} + \ldots + \left( \phi_{x_1, x_2, \ldots, x_d+1} - \phi_{\mathbf{x}} \right)^2 \Big], \label{eq:Hamiltonian_lattice}\end{aligned}$$ where $d$ is the spatial dimension of the lattice, $\mathbf{x} = (x_1, x_2, \ldots, x_d)$ represents the spatial lattice coordinates, each $x_i$ is summed from 1 to $L_i$, and $L_i$ is the lattice length along the $i^\mathrm{th}$ dimension.
For translationally invariant boundary conditions, the two-point vacuum correlation functions corresponding to this Hamiltonian are given by $$\begin{aligned}
\left\langle \phi_0 \phi_\mathbf{x} \right\rangle &=
\frac{1}{2N} \sum_\mathbf{k}
\frac{1}{\omega_\mathbf{k}} \cos \left( k_1 x_1 \right) \cos \left( k_2 x_2 \right) \cdots \cos \left( k_d x_d \right)
, \nonumber \\
\left\langle \pi_0 \pi_\mathbf{x} \right\rangle &= \frac{1}{2N} \sum_\mathbf{k}
\omega_\mathbf{k}
\cos \left( k_1 x_1 \right) \cos \left( k_2 x_2 \right) \cdots \cos \left( k_d x_d \right), \label{eq:correlators_PBC}\end{aligned}$$ where $N = L_1 L_2 \cdots L_d$ is the total number of lattice sites and $$\omega_\mathbf{k} = 2 \sqrt{ \sin^2\left( k_1/2 \right) + \sin^2\left( k_2/2 \right) + \ldots + \sin^2\left( k_d/2 \right)}. \label{eq:omega}$$ The values of the momenta $\mathbf{k}$ are quantized such that $k_i = 2 n_i \pi /L_i$ when the lattice has PBC along the $i^\mathrm{th}$ lattice direction and similarly $k_i = (2 n_i +1) \pi /L_i$ for APBC (where $n_i = 0, 1, \ldots L_i-1$).
Note that, for a fully periodic lattice, the correlator $\left\langle \phi_{0} \phi_\mathbf{x} \right\rangle$ (and, as we will see, the EE) diverges since $\omega_\mathbf{k}=0$ for the zero mode $\mathbf{k} = 0$. In order to avoid this divergence, our calculations impose APBC along the $y$-direction ([[*i.e.*]{}]{}the $x_2$-direction) and PBC along the remaining lattice directions. In doing so, we have $\omega_\mathbf{k} \neq 0$ for all allowed values of $\mathbf{k}$.
These two-point correlators define the [$N \times N$]{} matrices $X_{ab} = \left\langle \phi_{\mathbf{x}_a} \phi_{\mathbf{x}_b} \right\rangle =
\left\langle \phi_0 \phi_{\mathbf{x}_b-\mathbf{x}_a} \right\rangle$ and $P_{ab} = \left\langle \pi_0 \pi_{\mathbf{x}_b-\mathbf{x}_a} \right\rangle$, where $a,b$ label lattice sites. To get $S(A)$ we only need to know the [smaller $N_A \times N_A$]{} matrices $X_A$ and $P_A$, which are the sections of $X$ and $P$ (respectively) with indices $i,j$ restricted to [the $N_A$ sites of]{} region $A$ [@Peschel]. The EE is given in terms of the eigenvalues $\nu_\ell$ of $\sqrt{X_A P_A}$ as[@Casini:2009] $$\begin{aligned}
S(A) = \sum_{\ell{=1}}^{{N_A}} & \left[ \left( \nu_\ell + \frac{1}{2} \right) \log \left( \nu_\ell + \frac{1}{2} \right) \right. \nonumber \\
& \;{} - \left. \left( \nu_\ell - \frac{1}{2} \right) \log \left( \nu_\ell - \frac{1}{2} \right) \right].\end{aligned}$$
[In order to access the EE on larger lattices, for the case of the torus geometry we employ an extension of the above methods as given in Ref. that takes advantage of the translational symmetry along $(d-1)$ spatial lattice directions. In this modified method, we map the $(d + 1)$-dimensional model to an effective model consisting of $L_2 \times L_3 \times ... \times L_d$ separate $(1+ 1)$-dimensional chains. ]{}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Dirac equation written on the boundary of the Nutku helicoid space consists of a system of ordinary differential equations. We tried to analyze this system and we found that it has a higher singularity than those of the Heun’s equations which give the solutions of the Dirac equation in the bulk. We also lose an independent integral of motion on the boundary. This facts explain why we could not find the solution of the system on the boundary in terms of known functions. We make the stability analysis of the helicoid and catenoid cases and end up with an appendix which gives a new example where one encounters a form of the Heun equation.'
author:
- |
T.Birkandan$^{\ast }$ and M. Hortaçsu\* $^{1}$\
[$^{\ast }$[ Istanbul Technical University, Department of Physics, Istanbul, Turkey]{}. ]{}
title: Singularity Structure and Stability Analysis of the Dirac Equation on the Boundary of the Nutku Helicoid Solution
---
PACS: 04.62.+v, 02.30.Hq
Introduction
============
Although one usually needs only different forms of the hypergeometric equation or its confluent forms to describe many different phenomena in theoretical physics, functions with higher singularity structure are seen more and more in the literature \[1-12\]. A common form is the Heun function [@heun], which is studied extensively in the books by Ronveaux and Slavyanov et al [@ronveaux][@slavyanov], the seminal book by Ince [@Ince] as well as in several articles \[17-20\]. Although for linear equations both Ince and Slavyanov et al end their singularity analysis with Heun type functions, sometimes equations with even more singularities are needed for relatively simple situations, which lack some symmetries.
As an example of such a case, here we study the singularity structure of the Dirac equations, written in the background of the Nutku helicoid solution[@yavuz1], restricted to the boundary of the helicoid. This metric was formerly studied by Lorenz-Petzold [@lp]. One can study the scalar field in this background and obtain the propagator in a closed form [@yavuz2], thanks to four integrals of motion allowed by the metric. One can also write the Dirac equation and obtain the solutions in terms of Mathieu functions \[24-26\]. One needs to study the eigenvalue problem on the boundary to impose the boundary conditions in this problem. The similar problem in the bulk, using partial differential equations, can be solved in terms of known functions. On the boundary, we get a system of ordinary differential equations. At first glance, this system seems to be easier to analyze. When we investigate the system further, we find that this system has higher form of singularities and fewer integrals of motion. These turn out to be the reasons why we cannot express the solution in terms of the functions cited in Ince’s book[@Ince].
In this note, we comment on the symmetries of the new problem and study the singularity structure of the new system. We show that one gets an equation of higher form of singularity. Since we do not have closed solutions, we then use stability analysis to see if this system describes a stable system. To our surprise we find that although the answer is not affirmative for the helicoid case, we get a limit cycle for the related catenoid solution.
Below we summarize our results. In an appendix, we show a new example where one encounters a form of the Heun equation. This is the solution of the laplacian in the background of the Eguchi-Hanson solution [@eguchi], trivially extended to five dimensions.
Our Analysis
============
The Nutku helicoid metric is given as
$$\begin{aligned}
{\normalsize ds}^{2} &=&\frac{1}{\sqrt{1+\frac{a^{2}}{r^{2}}}}%
[dr^{2}+(r^{2}+a^{2})d\theta ^{2}+\left( 1+\frac{a^{2}}{r^{2}}\sin
^{2}\theta \right) dy^{2} \notag \\
&&-\frac{a^{2}}{r^{2}}\sin 2\theta dydz{\normalsize +}\left( 1+\frac{a^{2}}{%
r^{2}}\cos ^{2}\theta \right) {\normalsize dz}^{2}{\normalsize ]}.\end{aligned}$$
where $0<r<\infty $, $0\leq \theta \leq 2\pi $, $y$ and $z$ are along the Killing directions and will be taken to be periodic coordinates on a 2-torus [@yavuz2]. This is an example of a multi-center metric. This metric reduces to the flat metric if we take $a=0$. $$ds^{2}=dr^{2}+r^{2}d\theta ^{2}+dy^{2}+dz^{2}.$$If we make the following transformation $$r=a\sinh x,$$the metric is written as $$\begin{aligned}
ds^{2} &=&\frac{a^{2}}{2}\sinh 2x(dx^{2}+d\theta ^{2}) \notag \\
&&+\frac{2}{\sinh 2x}[(\sinh ^{2}x+\sin ^{2}\theta )dy^{2} \\
&&-\sin 2\theta dydz+(\sinh ^{2}x+\cos ^{2}\theta )dz^{2}]. \notag\end{aligned}$$
The solutions of the Dirac equation, written in the background of the Nutku helicoid metric, can be expressed as a special form of Heun functions [@tolga1][@tolga2]. We could reduce the double confluent Heun function obtained for the radial equation to the Mathieu function with coordinate transformations. Mathieu function is a related but much more studied function with similar singularity structure. In the work cited above, [@tolga1][@tolga2], we tried to get the solution of the little Dirac equation, the name used for the equation restricted to a boundary of the helicoid. We needed this solution also to be able to calculate the index of the differential operator. For a fixed value of the radial coordinate we had only a coupled system of ordinary differential equations, which, in general, should be much simpler to solve than the coupled system of partial differential equations obtained for the full Dirac equation, written for the bulk. We were successful to obtain the solution in this latter case in terms of Mathieu functions. We were not able to identify the solutions for the little Dirac equation , though.
The first thing we check is whether we lose any of the three Killing vectors and one Killing tensor. For the metric in question, one has three integrals of motion, [@yavuz2] namely $p_{y},p_{z}$ and $g^{\mu
\nu }p_{\mu }p_{\nu }={\mu^{2}}$, namely $$\left( \frac{dS_{x}}{dx}\right) ^{2}+\left( \frac{dS_{\theta }}{d\theta }%
\right) ^{2}+a^{2}\,\left( p_{y}^{2}+p_{z}^{2}\right) \,\sinh ^{2}x-\frac{%
\mu ^{2}\,a^{2}}{2}\,\sinh 2x+\,a^{2}\,\left( \cos \theta \,p_{y}\,+\,\sin
\theta p_{z}\right) ^{2}=0,$$ and an extra integral of motion, the Killing tensor [@yavuz2] , $$K=-p_{\theta }^{2}-a^{2}(cos\theta p_{y}+sin\theta p_{z})^{2}$$ which gives us the angular equation for a fixed value of the constant $%
\lambda $ . See eq.s (46) and (47) of [@yavuz2]. When one restricts the solution to a fixed value of the radial coordinate, the value of $g^{\mu \nu
}p_{\mu }p_{\nu }$ is not an independent constant of motion from the other two Killing vectors and the Killing tensor.
Trying to see the effect of this result on our problem, we investigate the singularity structure of the equation we get for the little Dirac operator. Here we study the simplest case, where the eigenvalue $%
\lambda $ is equal to zero, since the answer to our problem is already apparent here. For this case, instead of getting a system of four coupled equations we get coupling only between two of them at a time. When we analyze the system by reducing them to a second order equation for a single dependent variable, we find an operator with two irregular and one regular singularities, which is one more than allowed for the equations considered among the Heun functions. The double confluent Heun function, the solution one obtains for the full Dirac equation, has two irregular singularities, missing the extra regular singularity of the case studied here.
To make our discussion concrete we explicitly perform the calculation in the next section.
Singularities
-------------
The Dirac equation written in the background of the Nutku helicoids metric is written as $$(\partial _{x}+i\partial _{\theta })\Psi _{3}\ +iak[cos(\theta -\phi
+ix)]\Psi _{4}=0,$$$$(\partial _{x}-i\partial _{\theta })\Psi _{4}\ -iak[cos(\theta -\phi
-ix)]\Psi _{3}=0,$$$$(-\partial _{x}+i\partial _{\theta })f_{1}\ +iak[cos(\theta -\phi +ix)]f_{2}{%
=0},$$$$(-\partial _{x}-i\partial _{\theta })f_{2}\ -iak[cos(\theta -\phi
-ix)]f_{1}=0.$$
These equations have simple solutions [@Nuri] which can also be expanded in terms of products of radial and angular Mathieu functions [@Chaos][@tolga1]. Problem arises when these solutions are restricted to boundary [@tolga2].
To impose these boundary conditions we need to write the little Dirac equation, the Dirac equation restricted to the boundary, where the variable $x$ takes a fixed value $x_{0}$. We choose to write the equations in the form, $${\frac{\sqrt{2}}{{a}}}\{i\frac{d}{d\theta }\Psi _{3}\ +ikacos(\theta -\phi
+ix_{0})\Psi _{4}\}=\lambda f_{1},$$$${\frac{\sqrt{2}}{{a}}}\{-i\frac{d}{d\theta }\Psi _{4}\ -iakcos(\theta -\phi
-ix_{0})\Psi _{3}\}=\lambda f_{2},$$$${\frac{\sqrt{2}}{{a}}}\{-i\frac{d}{d\theta }f_{1}\ -iakcos(\theta -\phi
+ix_{0})f_{2}\}=\lambda \Psi _{3},$$$${\frac{\sqrt{2}}{{a}}}\{i\frac{d}{d\theta }f_{2}\ +iakcos(\theta -\phi
-ix_{0})f_{1}\}=\lambda \Psi _{4}.$$Here $\lambda $ is the eigenvalue of the little Dirac equation. We take $\lambda =0$ as the simplest case. The transformation
$$\Theta =\theta -\phi -ix_{0}$$
can be used. Then we solve $f_{1\text{ }}$ in the latter two equations in terms of $f_{2}$:$$-\frac{d^{2}}{d\Theta ^{2}}f_{2}-\tan \Theta \frac{d}{d\Theta }f_{2}+\frac{%
(ak)^{2}}{2}[\cos (2\Theta )\cosh (2x_{0})-i\sin (2\Theta )\sinh
(2x_{0})+\cosh (2x_{0})]f_{2}=0$$
When we make the transformation$$u=e^{2i\Theta },$$
the equation reads,$$\{4(u+1)u[u\frac{d^{2}}{du^{2}}+\frac{d}{du}]-2iu(u-1)\frac{d}{du}+\frac{%
(ak)^{2}}{2}(u+1)[ue^{-2x_{0}}+\frac{1}{u}e^{2x_{0}}+\cosh
(2x_{0})]\}f_{2}=0. \label{denklem}$$
This equation has irregular singularities at $u=0$ and $\infty $ and a regular singularity at $u=-1$. If we try a solution in the form $\overset{\infty }{\underset{n=-\infty }{\sum }}a_{n}u^{n}$ around the irregular singularity $u=0$ we end up with a four-term recursion relation as
$$\begin{aligned}
&&a_{n-1}[4(n^{2}-2n+1)-2i(n-1)+\frac{(ak)^{2}}{2}(\frac{3}{2}e^{-2x_{0}}+%
\frac{1}{2}e^{2x_{0}})] \notag \\
&&+a_{n}[4n^{2}+2in+\frac{(ak)^{2}}{2}(\frac{3}{2}e^{-2x_{0}}+\frac{1}{2}%
e^{2x_{0}})]+ \\
&&a_{n-2}[\frac{(ak)^{2}}{2}e^{-2x_{0}}]+a_{n+1}[\frac{(ak)^{2}}{2}%
e^{2x_{0}}] \notag\end{aligned}$$
As it is known, in the Heun equation case, this kind of series solution gives a three-term relation [@r155].
If we search for a solution of the Thomé type we may try a solution of the form $f_{2}=e^{\frac{A}{\sqrt{u}}}g(u)$. This form does not allow us to get a Taylor series expansion around the irregular point $u=0$ [@slav115] [@olver].
If we try a series solution around the regular singularity at $%
u=-1 $ as
$\overset{\infty }{\underset{n=0}{\sum }}a_{n}(u+1)^{n+\alpha }$ we find a relation between five consecutive coefficients for the solution. Therefore, we may conclude that the solution of this equation cannot be written in terms of Heun functions or simplier special functions.
To check this further, we first set the coefficient of $\frac{1}{u}
$ term in equation \[denklem\] equal to zero to change our irregular singularity at zero to a regular one. Then we keep this term and discard the $ue^{-2x_{0}}$ term to reduce the singularity structure of infinity. In both cases one can check that the solution can be expressed in terms of confluent Heun functions. This shows that reducing one of the singularities yields a Heun function. Thus, we conclude that the full equation \[denklem\] is not one of the better known equations in the literature, which are included in the computer packages like Maple, cited in the seminal book by Ince [Ince]{}.
To investigate the type of our equation we try to get a confluent form of a new equation,$$y^{\prime \prime }(z)+(\frac{1-\mu _{0}}{z}+\frac{1-\mu _{1}}{z+1}+\frac{%
1-\mu _{2}}{z-a})y^{\prime }(z)+\frac{\beta _{0}+\beta _{1}z+\beta _{2}z^{2}%
}{z^{2}(z-a)}y(z)=0 \label{bizim}$$
with regular singularities at $0$, $-1$ and $a$ and an irregular singularity at infinity. This equation differs from the generalized Heun equation [@schmid][@schafke]:$$y^{\prime \prime }(z)+(\frac{1-\mu _{0}}{z}+\frac{1-\mu _{1}}{z+1}+\frac{%
1-\mu _{2}}{z-a}-\alpha )y^{\prime }(z)+\frac{\beta _{0}+\beta _{1}z+\beta
_{2}z^{2}}{z(z+1)(z-a)}y(z)=0 \label{genheun1}$$
which also has regular singularities at $0$, $-1$ and $a$ and an irregular singularity at infinity. These two equations both have four-term recursion relations. They, however, have different singularity ranks according to the classification given in [@ronveaux293]. When we put $a=0
$ in equation \[bizim\], we get$$y^{\prime \prime }(z)+(\frac{2-\mu _{0}-\mu _{2}}{z}+\frac{1-\mu _{1}}{z+1}%
)y^{\prime }(z)+\frac{\beta _{0}+\beta _{1}z+\beta _{2}z^{2}}{z^{3}}y(z)=0.
\label{baskabizim}$$
We get a singularity structure as a regular singularity at $-1$ and two irregular singularities at zero and infinity like the equation [denklem]{}.
Both equations \[denklem\] and \[baskabizim\] have four-term recursion relations when a Laurent power series solution is attempted. We may name this equation as the confluent form of the equation \[bizim\]. It is in the same form as our original equation rewritten as $$\{\frac{d^{2}}{du^{2}}+[\frac{1}{u}(1+\frac{i}{2})+\frac{i}{u+1}]\frac{d}{du}%
+\frac{(ak)^{2}}{2}[\frac{e^{-2x_{0}}}{u}+\frac{e^{2x_{0}}}{u^{3}}+\frac{%
\cosh (2x_{0})}{u^{2}}]\}f_{2}=0.$$
Both of these equations have s-rank multisymbols $\{1,\frac{3}{2},%
\frac{3}{2}\}$ referring to the singularities at $\{-1,0,\infty \}$ [ronveaux293]{}.
We could not obtain a confluent equation similar to the equation \[denklem\] from the generalized Heun equation \[genheun1\]. If we simply put $a=0$ in this equation we get the confluent Heun solution. We can obtain an equation with the same singularity structure as our equation only if we write the equation,$$y^{\prime \prime }(z)+(\frac{1-\mu _{0}}{z}+\frac{1-\mu _{1}}{z+1}+\frac{%
1-\mu _{2}}{z-a})y^{\prime }(z)+\frac{\frac{\beta _{-1}}{z}+\beta _{0}+\beta
_{1}z+\beta _{2}z^{2}}{z(z+1)(z-a)}y(z)=0, \label{beta3}$$
and make $a$ approach zero. Then we end up with an equation having the same singularity structure as in equation \[denklem\].
If we want to compare equation \[denklem\] with equation [genheun1]{} we have to form a confluent form of the latter equation. To coalesce the singularities at zero, we make a detour and then use standart techiques [@filip]. We first translate the singularity at zero to a singularity at $b$,$$y^{\prime \prime }(z)+(\frac{1-\mu _{0}}{z-b}+\frac{1-\mu _{1}}{z+1}+\frac{%
1-\mu _{2}}{z-a}-\alpha )y^{\prime }(z)+\frac{\beta _{0}+\beta _{1}z+\beta
_{2}z^{2}}{(z-b)(z+1)(z-a)}y(z)=0 \label{genheun}$$
This equation has regular singularities at $z=-1,a,b$ and an irregular singularity at infinity. We make the transformation $z=\frac{1}{v}$. Then, the equation \[genheun\] becomes,$$\begin{aligned}
&&y^{\prime \prime }(v)+(\frac{2-(1-\mu _{0})-(1-\mu _{1})-(1-\mu _{2})}{v}+%
\frac{\alpha }{v^{2}} \notag \\
&&-\frac{1-\mu _{0}}{\frac{1}{b}-v}-\frac{1-\mu _{1}}{v+1}-\frac{1-\mu _{2}}{%
\frac{1}{a}-v})y^{\prime }(v) \notag \\
&&+\frac{\frac{\beta _{0}}{v}+\frac{\beta _{1}}{v^{2}}+\frac{\beta _{2}}{%
v^{3}}}{(1-bv)(1+v)(1-av)}y(v)=0\end{aligned}$$
We set $\mu _{0}=-\mu _{1}=1/b=2/a=\epsilon $ and take the limit $%
\epsilon \rightarrow \infty .$ Then we transform back to the original variables using $v=\frac{1}{z}$ to obtain$$y^{\prime \prime }(z)+(\frac{3-\mu _{1}}{z}-\alpha -\frac{\mu _{1}-1}{z(z+1)}%
+\frac{1}{z^{2}})y^{\prime }(z)+\frac{\beta _{0}+\beta _{1}z+\beta _{2}z^{2}%
}{z^{2}(z+1)}y(z)=0 \label{onda}$$
This equation is a confluent form of the equation \[genheun\]. It has a regular singularity at $-1$ and two irregular singularities at zero and infinity and a four-way recursion relation when expanded around zero using a Laurent expansion. This equation has s-rank multisymbols $\{1,2,2\}$ referring to singularities at $\{-1,0,\infty \}$ [@ronveaux293]. Here we get the rank-2 irregular singularities at zero and infinity only from the coefficient of the first derivative whereas in equation \[denklem\] and in equation \[baskabizim\] the coefficient of the term without derivatives gives us these singularities. They also have different ranks. Even if we set $\alpha =0$ in equation \[onda\] then we get rank=$\{1,2,\frac{3}{2}\}$ which is different from the rank of our original equation. Hence, our equation \[denklem\] may be a confluent form only of a variation of the equation \[genheun1\], like equation \[beta3\].
Stability analysis for the little Dirac Equation
------------------------------------------------
The little Dirac equation is a system of linear differential equations with periodic coefficients. Then we can write the system as [bellman]{}[@codlev]:
$$\partial _{\theta }\psi =P(\theta )\psi$$
Here $P(\theta +\tau )=P(\theta )$ and $\tau $ is the period of the coefficients. $(\tau \neq 0)$. According to Bellman, if $\psi (0)=I$ , we can write $$\psi (\theta )=Q(\theta )e^{B\theta. }$$
The matrix $Q(\theta )$ is also periodic with the period $\tau $. Then we have$$\begin{aligned}
\psi (\theta +\tau ) &=&Q(\theta +\tau )e^{B\theta }e^{B\tau } \notag \\
&=&Q(\theta )e^{B\theta }e^{B\tau } \\
&=&\psi (\theta )e^{B\tau }. \notag\end{aligned}$$
We can define $e^{B\tau }\equiv C$ and use
$$C=\psi (\theta )^{-1}\psi (\theta +\tau )$$
to obtain $C$. The Jordan normal form of $C$, $T$ being the transformation matrix,
$$C=T%
\begin{pmatrix}
L_{1} & & \\
& \ddots & \\
& & L_{r}%
\end{pmatrix}%
T^{-1}$$
gives us the $B$ matrix:
$$B=\frac{1}{\tau }\ln L$$
The eigenvalues of $B$ give us the characteristic roots. We will use these characteristic roots with different parameters in our stability analysis. The characteristic roots are given by $\alpha _{i}$, ($i=1..4$).
Table 1. The change in the characteristic roots with respect to parameters (helicoid case)
[|c|c|c|c|c|c|]{} $a$ & $k$ & $x_{0}$ & $\lambda $ &
---------------------------------------
real parts of the
characteristic roots ($\times 2\pi $)
---------------------------------------
&
----------------------
signs of the
characteristic roots
----------------------
\
$1$ & $1$ & $1$ & $1$ & $\alpha _{1,2,3,4}=9.25029$ & $++--$\
$0.5$ & $"$ & $"$ & $"$ & $\alpha _{1,2,3,4}=4.03926$ & $"$\
$0.8$ & $"$ & $"$ & $"$ & $\alpha _{1,2,3,4}=7.2163$ & $"$\
$1.1$ & $"$ & $"$ & $"$ & $\alpha _{1,2,3,4}=10.2542$ & $"$\
$1.5$ & $"$ & $"$ & $"$ & $\alpha _{1,2,3,4}=14.2215$ & $"$\
$1$ & $0.5$ & $1$ & $1$ & $\alpha _{1,2,3,4}=5.8327$ & $"$\
$"$ & $0.8$ & $"$ & $"$ & $\alpha _{1,2,3,4}=7.75421$ & $"$\
$"$ & $1.1$ & $"$ & $"$ & $\alpha _{1,2,3,4}=10.0319$ & $"$\
$"$ & $1.5$ & $"$ & $"$ & $\alpha _{1,2,3,4}=13.2745$ & $"$\
$1$ & $1$ & $0.5$ & $1$ & $\alpha _{1,2,3,4}=6.56206$ & $"$\
$"$ & $"$ & $0.8$ & $"$ & $\alpha _{1,2,3,4}=7.91586$ & $"$\
$"$ & $"$ & $1.1$ & $"$ & $\alpha _{1,2,3,4}=10.0632$ & $"$\
$"$ & $"$ & $1.5$ & $"$ & $\alpha _{1,2,3,4}=14.4648$ & $"$\
$1$ & $1$ & $1$ & $0.5$ & $\alpha _{1,2,3,4}=8.30164$ & $"$\
$"$ & $"$ & $"$ & $0.8$ & $\alpha _{1,2,3,4}=8.81342$ & $"$\
$"$ & $"$ & $"$ & $1.1$ & $\alpha _{1,2,3,4}=9.49249$ & $"$\
$"$ & $"$ & $"$ & $1.5$ & $\alpha _{1,2,3,4}=10.5888$ & $"$\
Our calculations indicate that $f_{1}$ and $f_{2}$ solutions are not stable (positive characteristic root) while, $\Psi _{3}$ and $\Psi _{4} $ solutions are stable (negative characteristic root). As it is seen in Table 1, when we keep all the other parameters constant and vary only $a$, the value of the roots are influenced most, whereas the effect of the variation in the value of $\lambda $ changes the value of the roots least. We also find that when these parameters exceed unity in absolute value, we encounter inconsistencies in the numerical values. The separation between consecutive roots increase and some negative roots go to positive values for large values of the parameters. If we keep the values of the parameters in the range $[-1,1] $, we seem to have no such problems.
The periodicity of the defined $Q$ can be checked using numerical means.
We use $$\psi (\theta +\tau )=Q(\theta +\tau )e^{B\theta }C \\$$
and for $\theta =0$, $\psi (\theta )=Q(\theta )e^{B\theta }$ to give,
$$\psi (\tau )C^{-1}=Q(\tau )=Q(0)=\psi (0)=I .$$
We check numerically that this equation is satisfied; hence, $Q$ is periodic.
For the catenoid case, one replaces $a$ with $ia$ in the metric. The same stability procedure is performed for this case and we find the characteristic roots given in the Table 2.
Table 2. The change in the characteristic roots with respect to parameters (catenoid case)
[|c|c|c|c|c|c|]{} $a$ ($\times $ $i$) & $k$ & $x_{0}$ & $\lambda $ & characteristic roots ($%
\times 2\pi $) &
----------------------
signs of the
characteristic roots
----------------------
\
$1$ & $1$ & $1$ & $1$ & $%
\begin{tabular}[b]{l}
$\_[1,2]{}=-5.8454310\^[-8]{}+2.08426i$ \\
$\_[3,4]{}=-6.4096710\^[-8]{}+0.72266i$%
\end{tabular}%
$ & $+-+-$\
$0.5$ & $"$ & $"$ & $"$ & $%
\begin{tabular}[b]{l}
$\_[1,2]{}=-2.4055310\^[-7]{}-1.19724i$ \\
$\_[3,4]{}=-2.4922510\^[-7]{}-2.3721i$%
\end{tabular}%
$ & $-+-+$\
$0.8$ & $"$ & $"$ & $"$ & $%
\begin{tabular}[b]{l}
$\_[1,2]{}=-5.6304610\^[-7]{}-0.233835i$ \\
$\_[3,4]{}=-5.7817810\^[-7]{}-2.42696i$%
\end{tabular}%
$ & $-+-+$\
$1.1$ & $"$ & $"$ & $"$ & $%
\begin{tabular}[b]{l}
$\_[1,2]{}=-7.8285610\^[-7]{}+2.9982i$ \\
$\_[3,4]{}=-7.8835810\^[-7]{}-0.186969i$%
\end{tabular}%
$ & $+-+-$\
$1.5$ & $"$ & $"$ & $"$ & $%
\begin{tabular}[b]{l}
$\_[1,2]{}=-7.5353910\^[-7]{}-0.526163i$ \\
$\_[3,4]{}=-7.725510\^[-7]{}-2.35978i$%
\end{tabular}%
$ & $-+-+$\
$1$ & $0.5$ & $1$ & $1$ & $%
\begin{tabular}[b]{l}
$\_[1,2]{}=3.4908110\^[-8]{}+0.661356i$ \\
$\_[3,4]{}=2.0279410\^[-8]{}-1.31726i$%
\end{tabular}%
$ & $+--+$\
$"$ & $0.8$ & $"$ & $"$ & $%
\begin{tabular}[b]{l}
$\_[1,2]{}=-4.0495710\^[-7]{}+0.864554i$ \\
$\_[3,4]{}=-4.2282310\^[-7]{}+2.3638i$%
\end{tabular}%
$ & $+-+-$\
$"$ & $1.1$ & $"$ & $"$ & $%
\begin{tabular}[b]{l}
$\_[1,2]{}=-2.257710\^[-7]{}+2.7328i$ \\
$\_[3,4]{}=-2.3092910\^[-7]{}+0.120058i$%
\end{tabular}%
$ & $+-+-$\
$"$ & $1.5$ & $"$ & $"$ & $%
\begin{tabular}[b]{l}
$\_[1,2]{}=-1.3391610\^[-7]{}+0.715427i$ \\
$\_[3,4]{}=-1.3890410\^[-7]{}-2.75007i$%
\end{tabular}%
$ & $+--+$\
$1$ & $1$ & $0.5$ & $1$ & $%
\begin{tabular}[b]{l}
$\_[1,2]{}=-4.6862510\^[-8]{}-0.232525i$ \\
$\_[3,4]{}=-5.3477110\^[-8]{}-2.02431i$%
\end{tabular}%
$ & $-+-+$\
$"$ & $"$ & $0.8$ & $"$ & $%
\begin{tabular}[b]{l}
$\_[1,2]{}=-4.6990110\^[-7]{}-1.06977i$ \\
$\_[3,4]{}=-4.7298610\^[-7]{}-2.11709i$%
\end{tabular}%
$ & $-+-+$\
$"$ & $"$ & $1.1$ & $"$ & $%
\begin{tabular}[b]{l}
$\_[1,2]{}=-6.0642510\^[-7]{}-2.74153i$ \\
$\_[3,4]{}=-6.1418810\^[-7]{}-0.143986i$%
\end{tabular}%
$ & $-+-+$\
$"$ & $"$ & $1.5$ & $"$ & $%
\begin{tabular}[b]{l}
$\_[1,2]{}=-4.2637510\^[-7]{}+0.299098i$ \\
$\_[3,4]{}=-4.421410\^[-7]{}+1.55822i$%
\end{tabular}%
$ & $+-+-$\
$1$ & $1$ & $1$ & $0.5$ & $%
\begin{tabular}[b]{l}
$\_[1,2]{}=-4.6777310\^[-7]{}+0.761149i$ \\
$\_[3,4]{}=-4.6962710\^[-7]{}+0.741915i$%
\end{tabular}%
$ & $+-+-$\
$"$ & $"$ & $"$ & $0.8$ & $%
\begin{tabular}[b]{l}
$\_[1,2]{}=-2.8225610\^[-7]{}-1.5208i$ \\
$\_[3,4]{}=-2.8657110\^[-7]{}-0.79892i$%
\end{tabular}%
$ & $-+-+$\
$"$ & $"$ & $"$ & $1.1$ & $%
\begin{tabular}[b]{l}
$\_[1,2]{}=-4.1447910\^[-7]{}+2.38047i$ \\
$\_[3,4]{}=-4.198510\^[-7]{}+0.650337i$%
\end{tabular}%
$ & $+-+-$\
$"$ & $"$ & $"$ & $1.5$ & $%
\begin{tabular}[b]{l}
$\_[1,2]{}=-3.3391410\^[-7]{}-2.62503i$ \\
$\_[3,4]{}=-3.4206710\^[-7]{}-0.145628i$%
\end{tabular}%
$ & $-+-+$\
\
We see that the real parts of these roots are compatible with assigning to value zero within numerical errors. This corresponds to a limit cycle$\cite{hurewicz}\cite{lefschetz}$.
Conclusion
==========
Here we performed a systematic analysis of the Dirac equation restricted to the boundary when it is written in the background of the Nutku helicoid solution [@yavuz2] We find that the resulting system of ordinary differential equations has a singularity which is higher than those of the Heun functions which are solutions for the bulk. We also lose an independent integral of motion. This fact explains why we could not obtain the solution of the system on the boundary in terms of well known functions.
The stability analysis we performed shows that although this system is not stable, a related system, the catenoid solution is. We can, thus, give a meaning to its solutions, although we can not get explicit solutions for the little Dirac equation obtained from it too.
Appendix: Scalar field in the background of the extended Eguchi-Hanson solution
===============================================================================
To go to five dimensions, we can add a time component to the Eguchi-Hanson metric [@eguchi] so that we have $$ds^{2}=-dt^{2}+{\frac{{1}}{{1-{\frac{{a^{4}}}{{r^{4}}}}}}}%
dr^{2}+r^{2}(\sigma _{x}^{2}+\sigma _{y}^{2})+r^{2}(1-{\frac{{a^{4}}}{{r^{4}}%
}})\sigma _{z}^{2}$$where $$\sigma _{x}={\frac{{1}}{{2}}}(-\cos \xi d\theta -\sin \theta \sin \xi d\phi )$$$$\sigma _{y}={\frac{{1}}{{2}}}(\sin \xi d\theta -\sin \theta \cos \xi d\phi )$$$$\sigma _{z}={\frac{{1}}{{2}}}(-d\xi -\cos \theta d\phi ).~$$This is a vacuum solution.
If we take $$\Phi =e^{ikt}e^{in\phi }e^{i(m+{\frac{{1}}{{2}}})\xi }\varphi (r,\theta ),$$we find the scalar equation as
$$\begin{aligned}
H\varphi (r,\theta ) &=&({\frac{{r^{4}-a^{4}}}{{r^{2}}}}\partial _{rr}+{%
\frac{{3r^{4}+a^{4}}}{{r^{3}}}}\partial _{r}+k^{2}r^{2}+{\frac{{4a^{4}m^{2}}%
}{{a^{4}-r^{4}}}}+ \notag \\
&&4\partial _{\theta \theta }+4\cot \theta \partial _{\theta }+{\frac{{%
8mn\cos \theta -4(m^{2}+n^{2})}}{{\sin ^{2}\theta }})}\varphi (r,\theta ).\end{aligned}$$
If we take $\varphi (r,\theta )=f(r)g(\theta )$, the solution of the radial part is expressed in terms of confluent Heun ($\mathit{H}_{C}$) functions.
$$f\left( r\right) =\left( -a^{4}+r^{4}\right) ^{{\frac{{1}}{{2}}}\,m}\
\mathit{H}_{C}\left( 0,m,m,{\frac{{1}}{{2}}}\,{k}^{2}{a}^{2},{\frac{{1}}{{2}}%
}\,{m}^{2}-{\frac{{1}}{{4}}}\,\lambda -{\frac{{1}}{{4}}}\,{k}^{2}{a}^{2},\,{{%
\frac{{{a}^{2}+{r}^{2}}}{{2{a}^{2}}}}}\right) \$$
$$+\left( {a}^{2}+{r}^{2}\right) ^{-{\frac{{1}}{{2}}}\,m}\left(
r^{2}-a^{2}\right) ^{{\frac{{1}}{{2}}}\,m}\mathit{H}_{C}\left( 0,-m,m,{\frac{%
{1}}{{2}}}\,{k}^{2}{a}^{2},{\frac{{1}}{{2}}}\,{m}^{2}-{\frac{{1}}{{4}}}%
\,\lambda -{\frac{{1}}{{4}}}\,{k}^{2}{a}^{2},{{\frac{{{a}^{2}+{r}^{2}}}{{2{a}%
^{2}}}}}\right)$$
The angular solution is in terms of hypergeometric solutions.
$$g\left( \theta \right) ={\frac{{1}}{{\sin \theta }}}{\{\sqrt{2-2\,\cos
\left( \theta \right) }\left( {\frac{{1}}{{2}}}\,\cos \left( \theta \right) -%
{\frac{{1}}{{2}}}\right) ^{{\frac{{1}}{{2}}}\,m}\left( {\frac{{1}}{{2}}}%
\,\cos \left( \theta \right) -{\frac{{1}}{{2}}}\right) ^{-{\frac{{1}}{{2}}}%
\,n}}\$$
$${[\left( 2\,\cos \left( \theta \right) +2\right) ^{{\frac{{1}}{{2}}}-{\frac{{%
1}}{{2}}}\,n-{\frac{{1}}{{2}}}\,m}}\$$
$$\times {\mathit{_{2}F_{1}}}({[-n+{\frac{{1}}{{2}}}\,\sqrt{\lambda +1}+{\frac{%
{1}}{{2}}},}\ {-n-{\frac{{1}}{{2}}}\,\sqrt{\lambda +1}+{\frac{{1}}{{2}}}%
],[1-n-m],{\frac{{1}}{{2}}}\,\cos \left( \theta \right) +{\frac{{1}}{{2}}})}%
\$$
$${+\left( 2\,\cos \left( \theta \right) +2\right) ^{{\frac{{1}}{{2}}}+{\frac{{%
1}}{{2}}}\,n+{\frac{{1}}{{2}}}\,m}}\$$
$$\times {\mathit{_{2}F_{1}}([m+{\frac{{1}}{{2}}}\,\sqrt{\lambda +1}+{\frac{{1}%
}{{2}}},m-{\frac{{1}}{{2}}}\,\sqrt{\lambda +1}+{\frac{{1}}{{2}}}],}\ {%
[1+n+m],{\frac{{1}}{{2}}}\,\cos \left( \theta \right) +{\frac{{1}}{{2}}})]\}}$$
If the variable transformation $r=a\sqrt{\cosh x}$ is made, the solution can be expressed as
$$f\left( x\right) ={\frac{{1}}{{\sinh x}}}{\{\left( \sinh \left( x\right)
\right) ^{m+1}}\ \mathit{H}_{C}{\left( 0,m,m,{\frac{{1}}{{2}}}\,{k}^{2}{a}%
^{2},{\frac{{1}}{{2}}}\,{m}^{2}-{\frac{{1}}{{4}}}\,\lambda -{\frac{{1}}{{4}}}%
\,{k}^{2}{a}^{2},{\frac{{1}}{{2}}}\,\cosh \left( x\right) +{\frac{{1}}{{2}}}%
\right) }\$$
$${+\left( 2\,\cosh \left( x\right) +2\right) ^{-{\frac{{1}}{{2}}}\,m+{\frac{{1%
}}{{2}}}}\left( 2\,\cosh \left( x\right) -2\right) ^{{\frac{{1}}{{2}}}\,m+{%
\frac{{1}}{{2}}}}}$$
$$\times \mathit{H}_{C}{\left( 0,-m,m,{\frac{{1}}{{2}}}\,{k}^{2}{a}^{2},{\frac{%
{1}}{{2}}}\,{m}^{2}-{\frac{{1}}{{4}}}\,\lambda -{\frac{{1}}{{4}}}\,{k}^{2}{a}%
^{2},{\frac{{1}}{{2}}}\,\cosh \left( x\right) +{\frac{{1}}{{2}}}\right) \}}.$$
We tried to express the equation for the radial part in terms of $%
u={\frac{{a^{2}+r^{2}}}{{2a^{2}}}}$ to see the singularity structure more clearly. Then the radial differential operator reads $$4{\frac{{d^{2}}}{{du^{2}}}}+4\left( {\frac{{1}}{{u-1}}}+{\frac{{1}}{{u}}}%
\right) {\frac{{d}}{{du}}}+k^{2}a^{2}\left( {\frac{{1}}{{u-1}}}+{\frac{{1}}{{%
u}}}\right) +{\frac{{m^{2}}}{{u^{2}(1-u)^{2}}}}.$$
This operator has two regular singularities at zero and one, and an irregular singularity at infinity, the singularity structure of the confluent Heun equation. This is different from the hypergeometric equation, which has regular singularities at zero, one and infinity.
**Acknowledgement**: We would like to thank Prof. Ayşe Bilge for correspondence and discussions. This work is supported by TÜBİTAK, the Scientific and Technological Council of Turkey. The work of M.H. is also supported by TÜBA, the Academy of Sciences of Turkey.
[99]{} S.A. Teukolsky, Physical Review Letters **29**, 1114 (1972)
E.W. Leaver, Proceedings of Royal Society London A **402**, 285 (1985)
A.H. Wilson, Proceedings of Royal Society **A118**, 617-647 (1928)
B.D.B. Figueiredo, math-phys/0509013, also Journal of Mathematical Physics **46** (11), 113503 (2005)
S.Y. Slavyanov, *Asymptotic Solutions of the One-dimensional Schrodinger Equation* (Leningrad University Press) (in Russian) (1991), Translation into English: S.Y. Slavyanov, *Asymptotic Solutions of the One-dimensional Schrodinger Equation* (Amer. Math. Soc. Trans. of Math. Monographs) 151 (1996)
G. Siopsis, hep-th/0407157, Nuclear Physics **B715**, 483-498 (2005)
D. Batic, H. Schmid, M. Winklmeier, gr-qc/0607017, Journal of Physics A: Math. General **39**, 12559 (2006)
P.P. Fiziev, Classical and Quantum Gravity **23**, 2447 (2006), gr-qc/0509123
P.P. Fiziev, gr-qc/0603003
D. Batic, H. Schmid, Journal of Mathematical Physics **48**, 042502 (2007)
R. Manvelyan et al., hep-th/0001179, Nuclear Physics **B579**, 177-208 (2000)
T. Oota, Y. Yasui, Nuclear Physics **B742**, 275 (2006)
K. Heun, Mathematische Annalen **33**, 161 (1889)
A. Ronveaux (ed.), *Heun’s Differential Equations* (Oxford University Press) (1995)
S.Y. Slavyanov, W. Lay, *Special Functions, A Unified Theory Based on Singularities*(Oxford University Press) (2000)
E.L. Ince, *Ordinary Differential Equations* (Dover Publications) (1926,1956)
R. Schafke, D. Schmidt, SIAM Journal of Mathematical Analysis **11**, 848 (1980)
R.S. Maier, math.CA/0408317
R.S. Maier, math.CA/0203264, also Journal of Differential Equations **213**, 171-203 (2005)
N. Gurappa, P.K. Panigrahi, Journal of Physics A: Math. General **37** L605-L608 (2004)
Y. Nutku, Physical Review Letters **77**, 4702 (1996)
D. Lorenz-Petzold, Journal of Mathematical Physics **24**, 2632 (1983)
A.N. Aliev, M. Hortaçsu, J. Kalayci , Y. Nutku, Classical and Quantum Gravity **16**, 631 (1999)
Y. Sucu, N. Ünal, Classical and Quantum Gravity **21**, 1443 (2004)
T. Birkandan, M. Hortaçsu, Journal of Physics A: Math. Theor. **40**, 1 (2007)
T. Birkandan, M. Hortaçsu, Journal of Mathematical Physics **48**, 092301 (2007)
T. Eguchi, A.J. Hanson, Physics Letters **74B**, 249 (1978)
L. Chaos-Cador, E. Ley-Koo, Revista Mexicana de Fisica **48**, 67 (2002)
A. Ronveaux (ed.), *Heun’s Differential Equations* (Oxford University Press) p.155 (1995)
F.W.J. Olver, *Asyptotics and Special Functions* (Academic Press) (1974)
A. Ronveaux (ed.), *Heun’s Differential Equations* (Oxford University Press) p.293 (1995)
P. Dennery, A. Krzywicki, *Mathematics for Physicists* (Dover Publications) (1967)
R. Bellman, *Stability Theory of Differential Equations* (McGraw-Hill) (1953)
E.A. Coddington, N. Levinson, *Theory of Ordinary Differential Equations* (Krieger Pub Co.) (1984)
W. Hurewicz, *Lectures on Ordinary Differential Equations* (Dover Publications) (2002)
S. Lefschetz, *Differential Equations: Geometric Theory* (Dover Publications) (2005)
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We show that a generic principally polarized abelian variety is uniquely determined by its gradient theta-hyperplanes, the non-projectivized version of those studied in [@cap1], [@cap2], [@cap4], which in a sense are a generalization to ppavs of bitangents of plane curves. More precisely, we show that, generically, the set of gradients of all odd theta functions at the point zero uniquely determines a ppav with level (4,8) structure. We also show that our map is an immersion of the moduli space of ppavs.'
address: 'Mathematics Department, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544, USADipartimento di Matematica, Università di Roma, Piazzale Aldo Moro, 2, I-00185 Roma, Italy'
author:
- Samuel Grushevsky
- Riccardo Salvati Manni
date: 'May 11, 2003.'
title: Gradients of odd theta functions
---
Definitions and notations
=========================
We denote by $\H_g$ the [*Siegel upper half-space*]{} — the space of complex symmetric $g\times g$ matrices with positive definite imaginary part. An element $\tau\in\H_g$ is called a [*period matrix*]{}, and defines the complex abelian variety $X_\tau:=\C^g/\Z^g+\tau \Z^g$. The group $\Gamma_g:={\rm
Sp}(2g,\Z)$ acts on $\H_g$ by automorphisms: for $\gamma:=\pmatrix
a&b\\ c&d\endpmatrix\in{\rm Sp}(2g,\Z)$ the action is $\gamma\tau:=(a\tau+b)(c\tau+d)^{-1}$. The quotient of $\H_g$ by the action of the symplectic group is the moduli space of principally polarized abelian varieties (ppavs): $\A_g:=\H_g/{\rm
Sp}(2g,\Z)$. A ppav is called irreducible if it is not a direct product of two lower-dimensional ppavs, i.e. if its period matrix $\tau$ is not conjugate by the action of $\Gamma_g$ to a matrix that splits as $\tau_1\oplus\tau_2$ for two lower-dimensional period matrices. For us the case $g=1$ is special and in the following we will always assume $g>1$.
We define the [*level*]{} subgroups of the symplectic group to be $$\Gamma_g(n):=\left\lbrace\gamma=\pmatrix a&b\\ c&d\endpmatrix
\in\Gamma_g\, |\, \gamma\equiv\pmatrix 1&0\\
0&1\endpmatrix\ {\rm mod}\ n\right\rbrace$$ $$\Gamma_g(n,2n):=\left\lbrace\gamma\in\Gamma_g(n)\, |\, {\rm diag}
(a^tb)\equiv{\rm diag}(c^td)\equiv0\ {\rm mod}\ 2n\right\rbrace.$$ The corresponding [*level moduli spaces of ppavs*]{} are denoted $\A_g^n$ and $\A_g^{n,2n}$, respectively.
A function $F:\H_g\to\C$ is called a [*modular form of weight $k$ with respect to $\Gamma\subset\Gamma_g$*]{} if $$F(\gamma\tau)=\det(c\tau+d)^kF(\tau),\quad \forall \gamma=
\pmatrix a&b\\ c&d\endpmatrix\in\Gamma,\ \forall \tau\in\H_g$$
More generally, let $\rho:{\rm GL}(g,\C)\to\operatorname{End} V$ be some representation. Then a map $F:\H_g\to V$ is called a [*$\rho$- or $V$-valued modular form*]{}, or simply a [*vector-valued modular form*]{}, if the choice of $\rho$ is clear, with respect to $\Gamma\subset\Gamma_g$ if $$F(\gamma\tau)=\rho(c\tau+d)F(\tau),\quad \forall \gamma=
\pmatrix a&b\\ c&d\endpmatrix\in\Gamma,\
\forall \tau\in\H_g.$$
For $\e,\de\in \Z_2^g$, thought of as vectors of zeros and ones, and $z\in \C^g$ we define the [*theta function with characteristic $[\e,\de]$*]{} to be $$\tt\e\de(\tau,z):=\sum\limits_{m\in\Z^g} \exp \pi i \left[\left(
m+\frac{\e}{2},\tau(m+\frac{\e}{2})\right)+2\left(m+\frac{\e}{2},z+
\frac{\de}{2}\right)\right].$$ A [*characteristic*]{} $[\e,\de]$ is called [*even*]{} or [*odd*]{} depending on whether the scalar product $\e\cdot\de\in\Z_2$ is zero or one, and the corresponding theta function is even or odd in $z$, respectively. The number of even (resp. odd) characteristics is $2^{g-1}(2^g+1)$ (resp. $2^{g-1}(2^g-1)$).
For $\e\in\Z_2^g$ we also define the [*second order theta function*]{} with characteristic $\e$ to be $$\T[\e](\tau,z):=\tt{\e}{0}(2\tau,2z).$$ The group $\Gamma_g$ acts on the set of characteristics as follows: $$\gamma \pmatrix\e\\ \de\endpmatrix :=\pmatrix d & -c \\
-b & a \endpmatrix \pmatrix \e\\ \de \endpmatrix +\frac{1}{2}
\pmatrix \diag(cd^t)\cr \diag(ab^t) \endpmatrix,$$ where the resulting characteristics is taken modulo 2. This action is not transitive, in fact the parity of the characteristics is an invariant. The transformation law for theta functions under the action of the symplectic group is (see [@ig1]): $$\label{translaw} \theta\left[\gamma \pmatrix\e\\ \de \endpmatrix
\right](\gamma\tau,(c\tau+d)^{-t}
z)=\phi(\e,\de,\gamma,\tau,z)\det (c\tau+d)^{1/2}\tt\e\de(\tau,z),$$ where $\phi$ is some complicated explicit function. It is further known (see [@ig1], [@sm1]) that $\phi|_{z=0} $ does not depend on $\tau$, and that for $\gamma\in\Gamma_g(4,8)$ we have $\phi|_{z=0}=1$, while $\gamma\in\Gamma_g(4,8)$ acts trivially on the characteristics $[\e,\de]$. Thus the values of theta functions at $z=0$, called [*theta constants*]{}, are modular forms of weight one half with respect to $\Gamma_g(4,8)$. Similarly it is known that the theta constants of second order are modular forms of weight one half with respect to $\Gamma_g(2,4)$. The action of $\Gamma_g(2)/ \Gamma_g(4,8)$ on the set of theta constants with characteristics is by certain characters whose values are fourth roots of the unity, and is well understood — see [@sm1]. The action of $\Gamma_g/\Gamma_g(2)$ on the set of characteristics is by permutations.
All odd theta constants with characteristics vanish identically, as the corresponding theta functions are odd functions of $z$, and thus there are $2^{g-1}(2^g+1)$ non-trivial theta constants with characteristics, and $2^g$ theta constants of the second order.
Differentiating the theta transformation law above with respect to $z_i$ and then evaluating at $z=0$, we see that for $\gamma\in
\Gamma_g(4,8)$ and $[\e,\de]$ odd $$\frac{\p}{\p z_i}\tt\e\de(\tau,z)|_{z=0}=\det(c\tau+d)^{1/2}
\sum\limits_j (c\tau+d)_{ij}\frac{\p}{\p
z_j}\tt\e\de(\gamma\tau,z)|_{z=0};$$ in other words the gradient vector $\lbrace\frac{\p}{\p
z_i}\tt\e\de(\tau,0)\rbrace_{{\rm all}\ i}$ is a $\C^g$-valued modular form with respect to $\Gamma_g(4,8)$ under the representation $\rho(M):=(\det M)^{1/2}\cdot M$.
The set of all even theta constants with characteristics defines the map $$\P Th:\A_g^{4,8}\to \P^{2^{ g-1}(2^g+1)-1},\quad \P Th(\tau):=\lbrace
\tt\e\de(\tau,0) \rbrace_{{\rm all\ even}\ [\e,\de]}$$ Theta constants of the second order similarly define the map $$\P Th_2:\A_g^{2,4}\to \P^{2^g-1},\quad \P Th_2(\tau):=\lbrace\T
[\e](\tau,0) \rbrace_{{\rm all}\ \e}.$$ Considering the set of gradients of all odd theta functions at zero gives the map $$grTh:\H_g\to (\C^g)^{\times 2^{g-1}(2^g-1)}\qquad grTh(\tau):=\left\lbrace
\vec{\rm grad}_z\tt\e\de\right\rbrace_{{\rm all\ odd}\ [\e,\de]},$$ which due to modular properties descends to the quotient map $$\P grTh:\A_g^{4,8}\to (\C^g)^{\times 2^{g-1}(2^g-1)}/\rho({\rm GL}(g,\C)),$$ where ${\rm GL}(g,\C)$ acts simultaneously on all $\C^g$’s in the product by $\rho$.
Because of Lefschetz theorem for abelian varieties for any $\tau$ the rank of the $2^{g-1}(2^g-1)\times g$ matrix of derivatives $\frac{\p}{\p_{z_i}}\tt \e\de(\tau,0)$ is always $g$ (see [@sm3]). Thus if we think of this matrix as a $g$-tuple of vectors in $\C^{2^{g-1}(2^g-1)}$, it is always non-degenerate. Thus the image of $\P grTh$ in fact lies in the grassmannian, $$\P grTh:\A_g^{4,8}\to{\rm Gr}_\C(g,\,2^{g-1}(2^g-1))$$ of $g$-dimensional subspaces in $\C^{2^{g-1}(2^g-1)}$. The Plücker’s coordinates of this map are modular forms of weight $\frac{g}{2}+1$ and have been extensively studied — see [@fr885], [@fay], [@ig2], [@sm3].
It is known that the map $\P Th$ is an embedding — see [@ig1] and references therein. In [@sm1] it is shown that $\P Th_2$ is also injective. However, there appears to be a small gap in the proof there, so at the moment we can only say that the map is injective for $g \leq 3$ and generically injective for $g\ge 4$. In this work we will avoid using this result except for the case $g=2$. We remark that, because of the transformation formula of theta functions, all the above maps are $\Gamma_g$-equivariant. Here we prove the following properties of $\P grTh$:
For $g\geq 3$ the map $\P grTh$ is generically injective on $\A_g^{4,8}$. For genus 2 the map is finite of degree 16.
In the course of the proof we give explicitly an open set in $\A_g^{4,8}$ where the map is injective, and obtain, denoting by $\J_g\subset\A_g$ the locus of Jacobians,
For $g\geq 3$ the map $\P grTh$ is also generically injective on $\J_g^{4,8}$.
We will also show
For $g\geq 2$ the map $\P grTh$ is injective on tangent spaces.
The map $\P grTh$ is closely related to the indicated in [@cap2] generalization to abelian varieties of the map obtained by sending a Jacobian to the collection of the hyperplanes tangent to the canonical curve at $g-1$ points, considered in [@cap1], [@cap2], [@cap4]. This map itself is in fact the generalization of the map of a plane quartic to the set of its bitangents. It was shown by Aronhold in [@aronh] that some subset of bitangents (i.e. their directions), now called an Aronhold system, together with the points of their tangency serves to recover the curve, but until [@cap2] it was not known that the bitangent directions themselves determine a (generic) curve.
Indeed, suppose we have a hyperplane tangent to the canonical curve at $g-1$ points. The reduced divisor that it cuts on the curve is then the square root of the canonical, and thus a theta characteristic. However, it is an effective theta characteristic, which for a generic curve means it is an odd theta characteristic. On the other hand, for each odd theta characteristics we get a $(g-1)$-tangent hyperplane to the canonical curve, and analytically it is clear what the direction of this hyperplane is: it is given by the gradient of the corresponding odd theta function with characteristics at zero.
In our terms the map to $(g-1)$-tangent hyperplanes is thus the map $$gr\P Th:\J_g\to S^{2^{g-1}(2^g-1)}(\P^{g-1})/{\rm PGL}(g,\C)$$ of the locus of Jacobians $\J_g$, obtained by sending $\tau$ to the set of gradients of all odd theta constants, but [*each*]{} projectivized independently, and considered as a point in $\P^{g-1}$ and not in $\C^g$ (we have the symmetric power of $\P^{g-1}$ here instead of the direct power because we are forgetting the level structure, and thus the characteristics may be permuted).
This map $gr\P Th$ considered by Caporaso and Sernesi is quite different from our $\P grTh$. As explained in [@cap2] and [@cap4], $gr\P Th$ is not defined on all of $\J_g$ or $\A_g$ — the problem occurs for those $\tau$ for which one of the components of $grTh(\tau)$ is equal to zero. Where it is defined, it factors through $\P grTh$: $$\begin{matrix}
&\A_g&\mathop{\longrightarrow}\limits^{\P grTh}\quad&{\rm
Gr}_\C(g,\,2^{g-1}(2^g-1))\ \\
&&\quad\searrow \hbox{\xxx gr\yyy P\xxx Th}\quad&\downarrow\pi\\
&&&(\P^{g-1})^{2^{g-1}(2^g-1)}/{\rm PGL}(g,\C)\ ,
\end{matrix}$$ where $\pi$ is the natural projection, which is not always defined.
In [@cap2] and [@cap4] it is shown that when restricted to its domain on the Jacobian locus, $gr\P Th$ is generically injective. While our corollary 2 states that $\P grTh|_{\J_g}$ is generically injective, it does not serve to reproduce the result of Caporaso and Sernesi, as we do not handle the projectivization map $\pi$, which might collapse images different points.
In the following, we will often omit the arguments $\tau$ and $z=0$ for theta functions and their derivatives, and will write $\p_{z_i}$ instead of $\frac{\p}{\p z_i}$. The letters of the Latin alphabet will denote coordinates for vectors in $\C^g$, i.e. will range from 1 to $g$. The letters of the Greek alphabet denote characteristics for $g$-dimensional theta functions, i.e. lie in $\Z_2^g$.
$\t$’s and $\T$’s
=================
The fundamental relation between theta functions with characteristics and theta functions of the second order comes from the fact that the squares of theta functions with characteristics are sections of the bundle on the abelian variety for which the theta functions of thesecond order form a basis for the space of sections. The relationship and others more general are special cases of Riemann’s addition theorem for theta functions (see, for example, [@ig1]): $$\label{tT}
\begin{matrix}
\tt\a\b(2\tau,2z)\tt{\a+\e}\b(2\tau,2x)\\
=\frac{1}{2^g}\sum\limits_{\s\in\Z_2^g}(-
1)^{\a\cdot\s}\tt\e{\b+\s}(\tau,z+x)\tt\e\s (\tau,z-x),
\end{matrix}$$ which is valid for all $\tau$, $z$, $x$ and $\a,\b,\e$. Let us denote $$\label{defC}
\begin{matrix}
C_{ij\,\e\de}^{\b}(\tau):=\p_{z_i}\tt\e{\b+\de}(\tau,0)\p_{z_j}
\tt\e\de(\tau,0)\\
+\p_{z_j}\tt\e{\b+\de}(\tau,0)\p_{z_i}\tt\e\de(\tau,0),
\end{matrix}$$ which is zero unless $[\e,\b+\de]$ and $[\e,\de]$ are odd, and $$\begin{matrix}
A_{ij\,\e\de}^{\b}(\tau):=\p_{z_i}\p_{z_j}\tt\de\b(2\tau,0)\tt\e\b(2\tau,0)-
\\
\tt\de\b(2\tau,0)\p_{z_i}\p_{z_j}\tt\e\b(2\tau,0),
\end{matrix}$$ which is zero unless $[\de,\b]$ and $[\e,\b]$ are both even characteristics. Note also that by the heat equation we have $\p_{z_i}\p_{z_j}\t=\p_{ \tau_{ij}}\t$ up to a constant that is not important to us.
We then have the following relation
If $[\e,\de]$ and $[\e,\b+\de]$ are odd characteristics, $$C_{ij\,\e\de}^{\b}= \frac{1}{2}\sum\limits_{\a\in\Z_2^g}(-1)^{
\a\cdot\de }A_{ij\, {\e+\a} \a}^{\b}.$$
Let us take the sum of the equations (\[tT\]) for different $\a$, each with coefficient $(-1)^{\a\cdot\de}$, where $\de$ is some characteristic. We get $$\label{tT2}
\begin{matrix}
\sum\limits_{\a\in\Z_2^g}
(-1)^{\a\cdot\de}\tt\a\b(2\tau,2z)\tt{\a+\e}\b(2\tau,0)\\
={1\over 2^g}\sum\limits_{\a,\s\in\Z_2^g}(-1)^{
\a\cdot(\s+\de)}\tt\e{\b+\s}(\tau,z)\tt\e\s (\tau,z)\\
=\tt\e{\b+\de}(\tau,z)\tt\e\de (\tau,z).
\end{matrix}$$ Now differentiate this relation twice with respect to $z_i$ and $z_j$ and then evaluate at $z=0$ to prove the lemma.
Moreover, the expression of $C$’s in terms of $A$’s is invertible:
$$\label{Adef}
A_{ij\,{\a+\e}\a}^{\b}=\frac{1}{2^{g-1}}\sum_{\lbrace\s|[\e,\s]
{\rm\ odd}\rbrace}(-1)^{\a\cdot\s}C_{ij\,\e\s}^\b$$
In (\[tT\]), we assume that $[\a,\b] $ and $[\a+\e,\b]$ are even characteristics. Differentiating, we get $$\p_{z_i}\p_{z_j}\tt\a\b(2\tau,0) \tt{\a+\e}\b(2\tau,0)$$ $$=\frac{1}{2^{g}}\p_{z_i}\p_{z_j}\left.\left(\sum\limits_{\s\in\Z_2^g}(-1)^{\a\cdot\s}
\tt\e{\b+\s}( \tau, z) \tt\e\s (\tau,z)\right)\right|_{z=0}.$$ Similarly, switching $\a$ and $\a+\e$ we have $$\tt\a \b ( 2\tau,
0)\p_{z_i}\p_{z_j} \tt{\a+\e}\b (2\tau,0)$$ $$=\frac{1}{2^{g}}\p_{z_i}\p_{z_j}\left.\left(\sum\limits_{\s\in\Z_2^g}(-
1)^{(\a+\e)\cdot\s} \tt\e{\b+\s}( \tau, z) \tt\e\s
(\tau,z))\right)\right|_{z=0}.$$ Subtracting and computing separately for the cases of $[\e,\s]$ odd and even, we get the statement of the lemma.
Recovering $\P Th_2(\tau)$ from $\P grTh(\tau)$
===============================================
The following identity holds for all $i,j,\e,\de,\s,\b$, and all $\tau\in\H_g$: \[propline\] $$\label{syseq}
A_{ij,\e\de}^{\b}(\tau)\tt\s\b(2\tau)+A_{ij,\de\s}^{\b}(\tau)\tt\e\b
(2\tau)+A_{ij,\s\e}^{\b}(\tau)\tt\de\b(2\tau)=0.$$
From the definition of $A$’s it follows that the above expression is the determinant of the matrix $\left(\begin{matrix}
\tt\e\b&\tt\e\b&\p_{\tau_{ij}}\tt\e\b\\
\tt\de\b&\tt\de\b&\p_{\tau_{ij}}\tt\de\b\\
\tt\s\b&\tt\s\b&\p_{\tau_{ij}}\tt\s\b
\end{matrix}\right)$.
\[propminor\] For any $i,j,I,J,\e,\de,\s,\tau$ we have $$\label{betaalpha}
\begin{matrix}
\left(A_{ij,\e\de}^\b(\tau) A_{IJ,\s\de}^\b(\tau)-A_{IJ,\e\de}^\b(\tau)
A_{ij,\s\de}^\b(\tau)\right)\tt\e\b(2\tau)\\
=\left(A_{ij,\de\e}^\b(\tau) A_{IJ,\s\e}^\b(\tau)-A_{IJ,\de\e}^\b(\tau)
A_{ij,\s\e}^\b(\tau)\right)\tt\de\b(2\tau).
\end{matrix}$$
The identity is straightforward if we substitute the definition of $A$’s in terms of $\t$’s and their derivatives. Indeed $$\label{minors}
A_{ij,\e\de}^\b A_{IJ,\s\de}^\b-A_{IJ,\e\de}^\b A_{ij,\s\de}^\b=
\tt\de\b\det\pmatrix
\tt\e\b & \p_{\tau_{ij}}\tt\e\b & \p_{\tau_{IJ}}\tt\e\b\\
\tt\de\b & \p_{\tau_{ij}}\tt\de\b & \p_{\tau_{IJ}}\tt\de\b\\
\tt\s\b & \p_{\tau_{ij}}\tt\s\b & \p_{\tau_{IJ}}\tt\s\b
\endpmatrix$$ evaluated at $2\tau$. Another proof would be to use the previous proposition for $\e,\de,\s$ with $i,j$ and then with $I,J$ to express $\tt\s\b$ in two different ways, and then equate these two expressions.
Let us consider the case $\b=0$ for the propositions above. Then the $A_{ij,\e\de}^{0}$ are $2\times 2$ minors of the $2^g\times
\left(\frac{g(g+1)}{2}+1\right)$ matrix $M$ with columns $(\T[\e],\lbrace\p_{\tau_{ij}}\T[\e]\rbrace_{ i\le j})_{\rm all\
i,j}$ (i.e. in each column there are the values either of $\T[\e]$ or of its derivative, for all $\e\in\Z_2^g$). This has maximal rank, equal to $\frac{g(g+1)}{2}+1$, for irreducible abelian varieties — see [@sasaki] and [@sm1]. Inductively for reducible abelian varieties we see that for $g\ge 2$ the rank of $M$ is at least three.
For $\b=0$ the system of equations (\[propline\]) has a unique projective solution for $\lbrace\T[\e](\tau)\rbrace_{{\rm all}\
\e}$ for fixed $A$’s. Since $A$’s are expressible in terms of $C$’s, which are combinations of gradients of odd theta functions, this means that $\P Th_2(\tau)$ is determined uniquely by $\P
grTh(\tau)$. \[getTh2\]
This is a consequence of the fact that the matrix $M$ above has at least three, and that the solutions of the system (\[propline\]) are invariant under the action of ${\rm GL}(g,\C)$. Basically we need to show that the system has maximal rank. Suppose we are given $\P grTh(\tau)$, i.e. all $A^0_{ij,\e\de}$’s. Let us pick a representative $\tau\in\H_g$ and think of $grTh(\tau)$ — we will deal with the action of ${\rm GL}(g,\C)$ later.
Since the matrix $M$ described above has rank at least 3, we can pick a non-degenerate $3\times 3$ minor in it. For irreducible $\tau$ the matrix $M$ is of maximal rank, and thus this minor can be chosen to contain the first column. From the fact that the theta constants of reducible abelian varieties are products of lower-dimensional theta constants it follows that such a choice is also possible for reducible abelian varieties.
Suppose now that this non-degenerate minor is $$\det\pmatrix
\T[\e] &\p_{\tau_{ij}}\T[\e] &\p_{\tau_{IJ}}\T[\e]\\
\T[\de] &\p_{\tau_{ij}}\T[\de] &\p_{\tau_{IJ}}\T[\de]\\
\T[\s] &\p_{\tau_{ij}}\T[\s] &\p_{\tau_{IJ}}\T[\s]
\endpmatrix\ne 0.$$ Then at least one of $\T[\e],\T[\de],\T[\s]$ must also be non-zero — by renaming let it be $\T[\e]$. Then the combination of $A$’s in the right-hand-side of the formula (\[minors\]) is non-zero, and thus we can use proposition \[propminor\] to express $\frac{\T[\de]}{\T[\e]}$ and $\frac{\T[\s]}{\T[\e]}$ in terms of $A$’s, i.e. in terms of $grTh(\tau)$.
Furthermore, since $\T[\e]$ and the $3\times 3$ minor above are non-zero, some $2\times 2$ subminor containing $\T[\e]$ must also be non-zero. By renaming let it be $$\det\pmatrix
\T[\e] &\p_{\tau_{ij}}\T[\e]\\
\T[\de] &\p_{\tau_{ij}}\T[\de]\\
\endpmatrix=A_{ij,\e\de}^0\ne 0.$$ Then we can use proposition \[propline\] with $\e,\de,i,j$ and any $\s$ to express all $\frac{\T[\s]}{\T[\e]}$ in terms of $A$’s. Thus from $grTh(\tau)$ we can recover $\P Th_2(\tau)$ uniquely.
Now we have to deal with the action of ${\rm GL}(g,\C)$ to finish the proof. However, the system (\[syseq\]) is acted upon by the adjoint action of ${\rm GL}(g,\C)$ (if we consider each $A_{ij\,\e\de}^0$ as a matrix labeled by $i,j$), and thus transformed into an equivalent system. This equivalent system will have the same solutions, and as we have shown the solution to be unique, it has the same solution.
Generic injectivity of $\P grTh$
================================
If it were known that $\P Th_2$ is injective, we would be already done, and could conclude the injectivity of $\P grTh$ at level (2,4). But, as this is not yet known, we need to do extra work.
The genus $g=2$ case is rather special: in this case it is known that $\P Th_2$ is injective; however, there are only six odd characteristics, while $$\vert \Gamma_2(2,4)/\Gamma_2(4,8)\vert=2^{10},$$ Thus in genus two the map $\P grTh(\tau)$ factors over some subgroup $\Gamma$ such that $\Gamma_2(4,8)\subset \Gamma\subset
\Gamma_2(2,4)$, and the injectivity of $\P grTh$ holds only on $\A_2^\Gamma$.
In general, to recover $\P Th(\tau)$ from $\P Th_2(\tau)$, we just need to know which sign to choose for each $\tt\e\de$, since their squares are already expressible in terms of theta constants of the second order by (\[tT2\]).
To get more control over the signs, let us consider the case $\b\neq 0$ of the equations (\[tT\]). Then $A_{ij,\e\de}^{\b}$ are $2\times 2$ minors of the $2^g\times
\left(\frac{g(g+1)}{2}+1\right)$ matrix $M^{\b}$ consisting of $(\tt\a\b(2\tau,0),\lbrace\p_{\tau_{ij}} \tt \a\b (2\tau,0)
\rbrace_{i\le j})_{{\rm all\ }\a}$, which, unlike the $M$ considered above, has $2^{g- 1}$ null rows corresponding to those $\a$ for which $[\a,\b]$ is odd. To use an argument similar to the one in the previous section relating $\P grTh$ and $\P Th_2$, we need
For $g\geq 3$ the matrix $M^\b$ has rank at least three for all $\tau$.
Fix an irreducible period matrix $\tau$ and consider the abelian variety $X:=X_{\tau}$. Let $\cal L$ denote the symmetric line bundle inducing a principal polarization on $X$, for which $\tt 0
0 (\tau,z)$ is the basis for sections. We denote by $\Theta$ the associated divisor. For any $x\in X$, let $t_x:X\to X$ be the traslation by $x$. Setting $x:=\b/4$, we consider the line bundle $\cal N:=t_{x}^{*}\cal L^2$. A basis for the sections of $\cal N$ are the theta functions $\tt \a\b (2\tau,2z)$. As a consequence of (\[tT\]), we see that for any $x\in X$, all $$\tt 0\b(\tau,z+x)\tt 00(\tau,z-x)$$ are linear combinations of $\tt\a\b(2\tau,2z)$. We recall that if $\tau$ is irreducibile, the Gauss maps $G_x:t_{x}^*\Theta\to
\P^{g-1}$ are dominant.
Suppose $M^\b$ is not of maximal rank. Then for some $\lambda$ and $c$’s we have $$\sum_{i\leq j} c_{ij}\p_{\tau_{ij}} \tt \a\b (2\tau,0)=\lambda \tt
\a\b (2\tau,0).$$ Thus we have $$\sum\limits_{i\leq j} c_{ij}\p_{z_{i}} \p_{z_{j}}\left.\left(\tt
0\b (\tau, z-x)\tt 00(\tau,z+x)\right)\right|_{z=0}$$ $$=\lambda\tt 0\b (\tau,x)\tt 00 (\tau,x)$$ If $x$ does not belong to $\Theta\cup t_{\b/2}^*\Theta$, the coefficient of $\lambda$ in the above relation is not zero. Vice versa, assuming that $x\in \Theta\cap t_{\b/2}^*\Theta$, we have $$\sum\limits_{i\leq j} c_{ij}\p_{z_{i}}\tt 0\b (\tau,x)
\p_{z_{j}}\tt 00(\tau,x)+\p_{z_{j}} \tt 0\b (\tau,x)
\p_{z_{i}}\tt 00(\tau,x)=0.$$ Now, since $\tau$ is irreducible, the singular locus of $\Theta$ has codimension at least two in $\Theta$, see [@el], so there exists such an $x$ in the smooth part of $\Theta$ and $t_{\b/2}^*\Theta$. Moreover, for $g\ge 3$ by a linear transformation we can find $x_1$ and $x_2$ such that $$\p_{z_{j}}\tt 00(\tau,x_1)=\de_{j}^1{\rm\ and\ }\p_{z_{j}}\tt
00(\tau,x_2) =\de_{j}^2,$$ where $\de_i^j$ is Krönecker’s delta, and either $\p_{z_{j}}\tt
0\b(\tau,x_2)$ is not proportional to $\p_{z_{j}}\tt
00(\tau,x_1)$ or $\p_{z_{j}}\tt 0\b(\tau,x_1)$ is not proportional to $\p_{z_{j}}\tt 00(\tau, x_2)$. These properties impose three linearly independent conditions on the coefficients $c_{ij},\,\lambda$. Hence, for irreducible $\tau$ the matrix $M^{\b}$ has at least three linearly independent columns. If the point $\tau$ is reducible, we can use similar facts about the Gauss map, do directly the genus three case and use some inductive argument to finish the proof.
Thus we get a generalization of proposition \[getTh2\] in the same way:
Given $\P grTh(\tau)$, the equations (\[syseq\]) for $\b\neq 0$ have a unique projective solution for $\lbrace\tt\a\b(2\tau,0)
\rbrace_{\rm all\ \a}$.
From the above proposition and formula (\[tT2\]) it follows that
All products of the type $$\tt\e{\b+\de}(\tau,0)\tt\e\de (\tau,0)$$ are determined by $\P grTh$ uniquely up to a multiplicative constant $t_{[0, \b]}$.
Obviously the above statement is true for every point $\tau$, thus, using the $\Gamma_g$-equivariance of the map $\P grTh$ and observing that the homogenuos action of $\Gamma_g$ on the set of characteristics is transitive on the set of characteristics different from $[0,0]$, by using the above corollary stated for points $\gamma\tau$ for all $\gamma$ we prove
\[cor\] All products of the type $$\tt\a\b(\tau,0)\tt{\a+\e}{\b+\de}(\tau,0)$$ are determined by $\P grTh(\tau)$ uniquely up to a multiplicative constant $t_{\e, \de}$.
Now we are able to prove our main theorem.
We will show that generically $\P grTh(\tau)$ determines $\P
Th(\tau)$ uniquely. Assume that there are two points $\tau$ and $\tau'$ for which $\P grTh(\tau)=\P grTh(\tau').$ Since $\P
Th_2(\tau)=\P Th_2(\tau')$ by proposition \[getTh2\], $$\tt\a\b(\tau,0)^2=c^2\tt\a\b(\tau',0)^2\qquad \forall [\a,\b],$$ where $c$ is a constant independent of $\a,\b$. Hence $$\tt\a\b(\tau,0)=c s_{\a,\b}\tt\a\b(\tau',0),$$ where $s_{\a, \b}$ is a sign depending on $\a,\b$. Replacing $\tau$ by $\gamma\tau$ and $c$ by $-c$ if necessary, we can assume $\tt 0 0 (\tau,0)\neq 0$ and $s_{0, 0}=1$. Now from the two previous corollaries it follows that $$\label{ok} t_{\a, \b} = t_{0,0} s_{\a ,\b },\qquad
s_{\a,\b}s_{\e,\de}=s_{\a+\e,\b+\de}\qquad \forall\a, \e,\de,\b$$ whenever $$\label{okif}
\tt\a\b(\tau,0)\cdot\tt\e\de(\tau,0)\cdot\tt{\a+\e}{\b+\de}(\tau,0)\ne0.$$
We would like to prove that all $s_{\a ,\b }=1$. Using (\[tT\]) we can easily check that for all $[\e, \de]$, among the products $\tt\a\b(\tau,0)\tt{\a+\e}{\b+\de}(\tau,0)$ appearing in Corollary \[cor\] there is at least one different from $0$. Hence there is a linear basis $[\e_1,\de_1],\dots,[\e_{2g},\de_{2g}]$ for the set of characteristics such that the associated theta constants do not vanish at $\tau$. We denote by $A_1(\tau)$ this set of characteristics. Now recall that an element $\gamma=\pmatrix a&b\\
c&d\endpmatrix\in \Gamma_g(4)$ acts on theta constants by the character $$\label{char}
\phi(\e,\,\de,\gamma)=(-1)^{{\rm diag(b)\cdot\e}\over
4}(-1)^{{\rm diag(c)\cdot\de}\over 4}$$ by the theta transformation law (\[translaw\]). We observe that on one side this action is compatible with the statement of Corollary \[cor\]. Thus we can choose $\gamma\in\Gamma_g(4)$ such that $$\tt{\e_j}{\de_j}(\tau,0)=c\,\tt{\e_j}{\de_j}(\gamma\tau ',0),$$ i.e. $s_{\e_j,\de_j }=1$ for all $j=1,\dots, 2g$.
Now let $A(\tau)$ be the set of characteristics whose associated theta constants do not vanish at $\tau$. We denote by $A_2(\tau)$ the subset of $A(\tau)$ consisting of the elements of $A_1(\tau)$ and the sums of two elements of $A_1(\tau)$ that are still in $A(\tau)$. Since the condition (\[okif\]) holds for characteristics in $A_2(\tau)$, (\[ok\]) is satisfied and thus $s_{\e,\de }=1$ since this is true for $A_1(\tau)$.
Furthermore, let us denote by $A_3(\tau)$ the subset of $A$ whose elements are either in $A_2(\tau)$ or sums of two elements of $A_2(\tau)$. Iterating the process, we get a certain subset $B(\tau)$. We would like to have $B(\tau)=A(\tau)$. Obviously this holds and we have no trouble if we assume the non-vanishing of all theta constants with characteristic $[\e,0]$ and $[0,\de ]$: then whenever $\tt\e\de(\tau,0)\ne0$, we can use $$\tt\e0(\tau,0)\tt0\de(\tau,0)\tt\e\de(\tau,0)\ne0$$ for the condition (\[okif\]) and by formula (\[ok\]) we are done. So if we restrict to the open set determined by this non-vanishing, we have $\P Th(\tau)=\P Th(\gamma \tau')$ for some $\gamma\in\Gamma_g$, and thus from the injectivity of $\P Th$ follows that $\tau$ and $\gamma\tau'$ are $\Gamma_g(4,8)$–conjugate, so that also $\P grTh(\tau)=\P
grTh(\gamma \tau')$. Now it is left to show that $\gamma\in\Gamma_g(4,8)$.
Assume the contrary: $\gamma\in\Gamma_g(4)\setminus\Gamma_g(4,8)$. We claim that then $\gamma$ acts non-trivially on $\P grTh(\tau)$, so that we would have $\P grTh(\tau)=\P grTh(\gamma\tau')\not=\P
grTh(\tau')$, which is a contradiction. Indeed, $\gamma$ acts on each gradient by multiplication by a sign $\phi(\e,\de,\gamma)$. Consider all odd $[\e,\de]$ such that $\phi(\e,\de, \gamma)=-1$: if for at least one of those and one of the remaining (otherwise there is multiplication by $-1$) the associated gradient $\vec{\rm
grad}_z \tt\e\de$ is not the zero vector, we are done. Up to conjugating by some element of $\Gamma_g$, we can assume that the $b$ in $\gamma$ is such that ${\rm diag}(b)\equiv 0{\rm\ mod\ }8$. Since all level subgroups are normal, a conjugate of $\gamma$ lies exactly in the level subgroups in which $\gamma$ lies.
From formula (\[char\]) we see that if ${\rm diag}\,
c\cdot\de\equiv 4{\rm\ mod\ }8$ for some fixed $\de$ (resp. $\equiv 0$), then $\phi(\e, \de,\gamma)=-1$ (resp. $=1$) for all $\e$. But since the map $X_\tau\to \P^{2^g-1}$ defined by $z\to\tt\e0(\tau,z)$ is of maximal rank at the point $\de/2$ (recall that $\tt\e0(\tau,z+\de/2)=\tt\e\de(\tau,z)$), all gradients at zero of odd theta functions of the form $\tt\e\de$ cannot vanish simultaneously for all $\e$.
As a consequence of the above proof, if we have $B(\tau)=A(\tau)$ for all $\tau$, the map $\P grTh$ is injective.
To the best of our knowledge this is always true. In particular it easy to check that even when $\tau$ is $\Gamma_g$-conjugate to a diagonal matrix, i.e. is the period matrix of a product of elliptic curves and thus has the maximal possible number of vanishing theta constants.
Thus we can say that $\P grTh$ is injective, if for all $\tau$ the corresponding subset $A(\tau)$ contains $A(\tilde\tau)$, with $\tilde\tau$ being $\Gamma_g$-conjugate to a diagonal matrix. For all examples of abelian varieties that we know the set of characteristics for which the associated theta constants vanish is always contained in such a set for a diagonal period matrix, up to conjugation. Thus in all the examples that we know we do have $A(\tau)\supset A(\tilde\tau)$ and the map $\P grTh$ is injective at $\tau$.
Since for $g=3,\, 4$ the combinatorics of the possible vanishing of theta constants is known well, and the worst cases are the reducible and the hyperelliptic, which can be treated by hand, we have
For $g=3,\, 4$ the map $\P grTh$ is injective on $\A_g^{4,8}$.
We also prove the injectivity of $\P grTh$ on generic Jacobians: notice that it does not directly follow from theorem 1 a priori as the Jacobians may not be the “generic” abelian varieties.
From the proof of theorem 1 we see that the map $\P grTh$ is injective at $\tau$ if none of the theta constants at $\tau$ vanish. However, it is known classically from the works of Riemann that no theta constant vanishes identically on $\J_g$ Since $\J_g$ is irreducible, the subset of $\J_g$ where no theta constants vanish is Zariski open, and there the injectivity of $\P
grTh$ holds.
Injectivity on the tangent space
================================
Now let us show that $\P grTh$ is smooth. Let us study the situation in general terms first.
Suppose $\P f:X\to Gr(k,N)$ is an analytic map of a complex variety to the grassmannian, locally near some $p\in X$ given by $f:x_1\ldots x_M\to f_1(x), \ldots f_k(x)$, where $x_i$ are the local coordinates near $p$, and each $f_i$ is a vector in $\C^N$. Then $\dd \P f|_p$ is injective if and only if for all $v\in\C^M\setminus\lbrace 0\rbrace$ at least for one $I$ the vector $\p_v f_I(0)$ is linearly independent with $(f_1(0),\ldots,
f_k(0))$. In particular it is injective if the vectors $(f_1(0),\ldots, f_k(0),\p_{x_1}f_I(0),\ldots, \p_{x_M}f_I(0))$ are linearly independent for some $I$. \[diffinj\]
Indeed let us consider the linearization of $f$ near $p$; for $v\in \C^M$ infinitesimally small we have $f(v)=(f_1(0)+\p_vf_1(0),\ldots, f_N(0)+\p_vf_N(0))$. $\dd \P
f|_p$ is non-degenerate iff it does not map any tangent vector $v$ to zero, i.e. if $f(v)$ represents a point in the grassmannian different from $f(0)$ for all $v$. This is equivalent to saying that for any $v$ at least one of the vectors making up $f(v)$ does not lie in $\P f(0)$, i.e. that at least one $\p_vf_I(0)$ is linearly independent with $(f_1(0),\ldots, f_k(0))$.
Now if $(f_1(0),\ldots, f_k(0),\p_{x_1}f_I(0),\ldots, \p_{x_M}
f_I(0))$ are linearly independent for some $I$, then it implies that any linear combination $$\p_vf_I(0)=\sum v_j\p_{x_j}f_i(0)$$ is linearly independent with $(f_1(0),\ldots, f_k(0))$ for all $v$.
We use the above lemma for $f=grTh$, $k=g$, $M=g(g+1)/2$ and $N=2^{g-1}(2^g-1)$. In [@sm2] it is stated (Theorem 2b) that $\P grTh$ is an immersion away from $\Gamma_g$-conjugates of points that are reducible as a product of a one-dimensional and a $g-1$-dimensional abelian variety. The proof there proceeds by showing that the rank of the matrix $(grTh(\tau),\p_{\tau_{ij}}grTh(\tau))$, (i.e. of $(f_i(0),\p_{x_j}f_i(0))_{\rm all\ i,j}$ in the notations of the lemma \[diffinj\]) is maximal exactly for points that are not one-reducible. The maximality of this rank implies the maximality of the rank of any submatrix, in particular the one for which we need linear independence to apply lemma \[diffinj\]. However, the converse, implicitly assumed in [@sm2], is in fact not true: it would be requiring in addition to the lemma that $(\p_{x_j}f_i(0))$ are linearly independent, which is not necessary.
Thus the argument in [@sm2] shows that $\P grTh$ is an immersion away from $\H_1\times \H_{g-1}$, and we only need to deal with $\tau=\pmatrix \lambda&0\\ 0&\tau'\endpmatrix$. In this case write $[\e,\de]=[\e_1\,\e',\de_1\,\de']$, and let indices $i,j,I$ always be greater than one. Then $$\p_{z_1}\tt\e\de(\tau)=\p_{z_1}\tt{\e_1}{\de_1}(\lambda)\tt{\e'}{\de'}
(\tau');
\ \p_{z_I}\tt\e\de(\tau)=\tt{\e_1}{\de_1}(\lambda)\p_{z_I}\tt{\e'}{\de'}
(\tau')$$ Using the heat equation, for $\dd\, \p_{z_1}\tt\e\de|_\tau$ we have $$\p_{\tau_{ij}}\p_{z_1}\tt\e\de=\p_{z_1}\tt{\e_1}{\de_1}\p_{\tau_{ij}}
\tt{\e'}{\de'};\ \p_{\tau_{11}}\p_{z_1}\tt\e\de=\p_{z_1}^3
\tt{\e_1}{\de_1}\tt{\e'}{\de'}$$ $$\p_{\tau_{1i}}\p_{z_1}\tt\e\de=\p_{\tau_{11}}\tt{\e_1}{\de_1}\p_{z_i}
\tt{\e'}{\de'}.$$ What are the possible linear relations among these vectors? Arrange the vectors into a matrix and split this matrix into two corresponding to whether $[\e',\de']$ is odd or even. We notice that all derivatives above are non-zero only for one parity of $[\e',\de']$ (i.e. every column of the matrix has non-zero elements only in one of the two submatrices), and thus the matrix of vectors is in $2\times 2$ submatrix block form, and we can compute the rank by adding the ranks of the blocks.
For $[\e',\de']$ odd, the only non-zero elements of the corresponding row are $\p_{z_i}\tt\e\de$ and $\p_{\tau_{1i}}\p_{z_1}\tt\e\de$, which are independent because all $\p_{z_i}\tt{\e'} {\de'}$ are non-collinear (as $grTh(\tau')$) and the matrix $(\tt{\e_1}{\de_1},\p_{\tau_{11}}
\tt{\e_1}{\de_1})$ has maximal rank, two.
On the other hand, if $[\e',\de']$ is even, then $\p_{z_1}\tt\e\de$ and $\p_{\tau_{ij}}\p_{z_1}\tt\e\de$ are independent because the matrix $(\tt{\e'}{\de'},
\p_{\tau_{ij}}\tt{\e'}{\de'})$ has maximal rank, while $\p_{z_1}^3\tt\e\de$ is proportional to $\p_{z_1}\tt\e\de$.
Thus using the above lemma we see that if $v\not\in\C\p_{\tau_{
11}}$, the vectors $(f_1(0),f_i(0), \p_vf_1(0))$ are linearly independent. By the lemma, to prove the injectivity of $\dd \P f$ we then need to show that for $v=\p_{\tau_{11}}$ the vectors $(f_1(0),f_i(0),\p_vf_I(0))$ are linearly independent. Indeed we compute $$\p_{\tau_{11}}\p_{z_I}\tt\e\de(\tau)=
\p_{\tau_{11}}\tt{\e_1}{\de_1}(\lambda) \p_{z_I}\tt{\e'}{\de'},$$ which is linearly independent with $\p_{z_I}\tt\e\de$ because $(\tt{\e_1}{\de_1},\p_{\tau_{11}}\tt{\e_1}{\de_1})$ has maximal rank, and with $\p_{z_1}\tt\e\de$ because one is zero for $[\e',\de']$ odd, and the other — for $[\e',\de']$ even.
Acknowledgements {#acknowledgements .unnumbered}
================
We are very grateful to Lucia Caporaso and Edoardo Sernesi for bringing the subject to our attention and explaining to us their work on theta-hyperplanes, which inspired this article. We would also like to thank the organizers of the Complex Analysis meeting in Oberwolfach in August 2002 for bringing professor Caporaso and the first author to the same place at the same time, and thus making this work begin.
[Fr885]{}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study an extension of optimal transport techniques to stationary Markov chains from a computational perspective. In this context, naively applying optimal transport to the stationary distributions of the Markov chains of interest would not capture the Markovian dynamics. Instead, we study a new problem, called the optimal transition coupling problem, in which the optimal transport problem is constrained to the set of stationary Markovian couplings satisfying a certain transition matrix condition. After drawing a connection between this problem and Markov decision processes, we prove that solutions can be obtained via the policy iteration algorithm. For settings with large state spaces, we also define a regularized problem, propose a faster, approximate algorithm, and provide bounds on the computational complexity of each iteration. Finally, we validate our theoretical results empirically, demonstrating that the approximate algorithm exhibits faster overall runtime with low error in a simulation study.'
author:
- |
Kevin O’Connor\
UNC-Chapel Hill\
`[email protected]`\
Kevin McGoff\
UNC-Charlotte\
`[email protected]`\
Andrew B. Nobel\
UNC-Chapel Hill\
`[email protected]`
bibliography:
- 'references.bib'
title: |
Optimal Transport for Stationary Markov Chains\
via Policy Iteration
---
Introduction {#sec:intro}
============
The application and computation of optimal transport plans has recently received a great deal of attention within the machine learning community. Recent applications of optimal transport include a variety of problems in generative modeling [@arjovsky2017wasserstein; @genevay2017learning; @salimans2018improving; @kolouri2018sliced; @deshpande2018generative] and supervised learning [@frogner2015learning; @janati2019wasserstein; @Luise2018DifferentialPO]. In this paper, we study the optimal transport (OT) problem in the case where the objects of interest are stationary Markov chains. In particular, we provide algorithms for computing solutions to a constrained form of the OT problem by combining existing ideas from computational OT with techniques from Markov decision processes.
[The principled extension of computational OT techniques to distributions capturing stochastic structure, for example processes with serial dependence, is an important problem in computational OT.]{} Indeed, several recent applications of OT, including modeling the growth of cell populations over time [@schiebinger2019optimal] and embedding natural language [@xu2018distilled], fit naturally into the framework of dependent processes. Furthermore, such extensions of OT open the door to applications in climate science, finance, epidemiology and other fields where observations possess temporal structure. The case of stationary Markov chains considered in this paper constitutes a step towards rigorous extensions of computational OT to dependent [processes]{}.
#### Main contributions
\(1) We propose the optimal transport problem for stationary Markov chains (OTC) in terms of transition couplings. (2) [We recast the OTC problem for Markov chains as a multichain, average-cost Markov decision process.]{} (3) We prove that the standard policy iteration algorithm converges to a global solution of this problem. (4) We leverage regularization techniques for computational optimal transport to derive a faster, approximate algorithm. (5) We provide initial validation for our theoretical results through a simulation study.
#### Notation
Let ${\mathbb{R}}_+$ be the non-negative reals and $\Delta_n = \{u \in \mathbb{R}^n_+ | \sum_{i=1}^n u_i = 1\}$ denote the probability simplex in ${\mathbb{R}}^n$. Given a metric space ${\mathcal{U}}$, let ${\mathcal{M}}({\mathcal{U}})$ denote the set of Borel probability measures on ${\mathcal{U}}$. For a vector $u \in {\mathbb{R}}^n$, let $\|u\|_\infty = \max_i |u_i|$ and $\|u\|_1 = \sum_i |u_i|$. [Occasionally we will treat matrices in ${\mathbb{R}}^{n\times n}$ as vectors in ${\mathbb{R}}^{n^2}$.]{}
Constraining the optimal transport problem {#sec:background_on_otc}
==========================================
The optimal transport problem is defined in terms of couplings. Given probability measures $\mu \in {\mathcal{M}}({\mathcal{U}})$ and $\nu \in {\mathcal{M}}({\mathcal{V}})$ defined on [metric spaces]{} ${\mathcal{U}}$ and ${\mathcal{V}}$, a *coupling* of $\mu$ and $\nu$ is defined as a probability measure $\pi \in {\mathcal{M}}({\mathcal{U}}\times{\mathcal{V}})$ such that $\pi(A\times {\mathcal{V}}) = \mu(A)$ and $\pi({\mathcal{U}}\times B) = \nu(B)$ for every measurable $A \subset {\mathcal{U}}$ and $B \subset {\mathcal{V}}$. Letting $\Pi(\mu, \nu)$ denote the set of couplings of $\mu$ and $\nu$, the optimal transport problem with respect to a cost function $\tilde{c}: {\mathcal{U}}\times{\mathcal{V}}\rightarrow {\mathbb{R}}$ is defined as $$\label{eq:ot_problem}
\mbox{minimize} \,\, \int \tilde{c}\, d\pi \,\, \mbox{subject to} \,\, \pi \in \Pi(\mu, \nu).$$ [As stated, the problem makes no particular assumptions about what the measures $\mu$ and $\nu$ describe. In most existing applications, $\mu$ and $\nu$ represent static quantities such as images or measurements of gene expression. However, in other application areas, the measures $\mu$ and $\nu$ may represent dependent processes. For example, $\mu$ and $\nu$ may correspond to sequences of words, the symptoms of a patient over time, or daily high temperatures. In these cases, additional care is needed in order to study .]{}
As a step toward computational OT for general dependent processes, we consider the case when $\mu$ and $\nu$ represent stationary Markov chains $X = (X_0, X_1, ...)$ and $Y = (Y_0, Y_1, ...)$ taking values in finite spaces ${\mathcal{X}}$ and ${\mathcal{Y}}$, respectively. Unlike general processes, which may exhibit infinitely long-range dependence, stationary Markov chains have a relatively simple dependence structure and are thus especially conducive to computation. Without loss of generality, we assume that ${\mathcal{X}}$ and ${\mathcal{Y}}$ both contain $d$ points. [Let $P, Q \in {\mathbb{R}}_+^{d \times d}$ be the respective transition matrices, and let $p, q \in \Delta_d$ be the respective stationary distributions, of the chains $X$ and $Y$.]{} For a review of basic Markov chain theory, we refer the reader to [@levin2017markov]. We also suppose that a non-negative cost function $c: {\mathcal{X}}\times {\mathcal{Y}}\rightarrow {\mathbb{R}}_+$ has been specified. We remark that this setting mirrors that of standard OT, where a cost or metric is defined *a priori* on the sample space.
[One may naively apply the standard formulation of the optimal transport problem in this setting, by taking ${\mathcal{U}}= {\mathcal{X}}$, ${\mathcal{V}}= {\mathcal{Y}}$, $\tilde{c}(x, y) = c(x,y)$, and considering the optimal coupling of the stationary distributions $p$ and $q$. Note that this marginal approach does not capture the dependence structure of $X$ and $Y$, and can lead to misleading conclusions. Consider for example the case when ${\mathcal{X}}= {\mathcal{Y}}= \{0, 1\}$, $c(x,y) = \delta(x \neq y)$, $$P= \kbordermatrix{&0&1 \\ 0 & \nicefrac{1}{2} & \nicefrac{1}{2} \\ 1 & \nicefrac{1}{2} & \nicefrac{1}{2}}, \quad\mbox{and}\quad Q = \kbordermatrix{&0&1 \\ 0 & 0& 1 \\ 1 & 1 & 0}.$$ Even though $X$ is iid and $Y$ is deterministic, their optimal transport distance is zero since $p = q$. Furthermore, the optimal coupling only specifies a distribution on ${\mathcal{X}}\times {\mathcal{Y}}$: it does not provide a means of generating a joint process having $X$ and $Y$ as marginals. We seek a variation of that captures and preserves the stochastic structure (stationarity, Markovity) of $X$ and $Y$.]{}
[As an alternative to the marginal approach, one may consider instead the full measures ${\mathbb{P}}\in {\mathcal{M}}({\mathcal{X}}^{\mathbb{N}})$ and ${\mathbb{Q}}\in {\mathcal{M}}({\mathcal{Y}}^{\mathbb{N}})$ of the processes $X$ and $Y$.]{} In particular, ${\mathbb{P}}$ is the unique probability measure on ${\mathcal{X}}^{\mathbb{N}}$ such that for any cylinder set $[a_i^j] := \{{\mathbf{x}}\in {\mathcal{X}}^{\mathbb{N}}: x_k = a_k, i \leq k \leq j\}$, $${\mathbb{P}}([a_i^j]) := p(a_i) \prod\limits_{k=i+1}^j P(a_{k-1}, a_k),$$ and ${\mathbb{Q}}$ is defined similarly in terms of $q$ and $Q$. Then one may let ${\mathcal{U}}= {\mathcal{X}}^{\mathbb{N}}$, ${\mathcal{V}}= {\mathcal{Y}}^{\mathbb{N}}$, ${\mathbf{x}}= (x_0, x_1, ...)$, ${\mathbf{y}}= (y_0, y_1, ...)$, $\tilde{c}({\mathbf{x}}, {\mathbf{y}}) = c(x_0, y_0)$ and couple ${\mathbb{P}}$ and ${\mathbb{Q}}$, obtaining a probability measure on the joint sequence space $({\mathcal{X}}\times{\mathcal{Y}})^{\mathbb{N}}$. However, such a coupling need not be stationary or Markovian. [In order to capture the dynamics of $X$ and $Y$, we might restrict the feasible set to couplings of ${\mathbb{P}}$ and ${\mathbb{Q}}$ that have the same dependence structure as $X$ and $Y$, namely the family $\Pi_M({\mathbb{P}}, {\mathbb{Q}})$ of stationary Markovian couplings. While this is a natural choice, the minimum expected cost over this set may violate the triangle inequality even when the cost is a metric [@ellis1976thedj; @ellis1978distances]. Moreover, the family $\Pi_M({\mathbb{P}}, {\mathbb{Q}})$ is not characterized by a simple set of constraints [@boyle2009hidden].]{} For the sake of computational tractability, we require a subset of $\Pi_M({\mathbb{P}}, {\mathbb{Q}})$ which admits a sufficiently simple, computationally tractable representation. To alleviate these issues, we further constrain the set of couplings to the subset of $\Pi_M({\mathbb{P}}, {\mathbb{Q}})$ whose transition distributions are couplings of those of $X$ and $Y$. Note that, to reduce notation when considering vectors and matrices indexed by elements of ${\mathcal{X}}\times{\mathcal{Y}}$, we will indicate only the cardinality of the index set and adopt an indexing convention whereby a vector $u \in \mathbb{R}^{d^2}$ is indexed as $u(x, y)$ and a matrix $R \in {\mathbb{R}}^{d^2\times d^2}$ is indexed as $R((x,y),(x',y'))$ for $(x,y),(x',y') \in {\mathcal{X}}\times{\mathcal{Y}}$. Note further that we regard vectors of the form $R((x,y),\cdot)$ as row vectors.
\[def:transitioncouplings\] A paired chain in $\Pi_M({\mathbb{P}}, {\mathbb{Q}})$ with transition matrix $R$ is called a **transition coupling** of $X$ and $Y$ if, for every $(x,y) \in {\mathcal{X}}\times {\mathcal{Y}}$, $R((x,y),\cdot) \in \Pi(P(x,\cdot), Q(y,\cdot))$. We denote the set of transition couplings of $X$ and $Y$ by ${\Pi_{\tiny TC}({\mathbb{P}}, {\mathbb{Q}})}$ and, abusing notation slightly, the set of transition matrices satisfying the condition above by ${\Pi(P, Q)}$.
The couplings defined in Definition \[def:transitioncouplings\] are referred to as “Markovian couplings” in the literature [@levin2017markov] and have been used, for example, to study diffusions [@banerjee2018coupling; @banerjee2016coupling; @banerjee2017rigidity]. We refer to such couplings as “transition couplings” in order to distinguish them from elements of $\Pi_M({\mathbb{P}}, {\mathbb{Q}})$. Note that ${\Pi_{\tiny TC}({\mathbb{P}}, {\mathbb{Q}})}\neq \emptyset$ since it contains the independent coupling, that is, the distribution obtained by coupling $X$ and $Y$ independently.
A key advantage of considering ${\Pi_{\tiny TC}({\mathbb{P}}, {\mathbb{Q}})}$ over $\Pi_M({\mathbb{P}}, {\mathbb{Q}})$ is that the constraints defining this set are linear and thus computationally tractable. In the case that $X$ and $Y$ are irreducible, this set of transition matrices actually characterizes the set of transition couplings.
[prop]{}[transmatchar]{} \[prop:transmat\_char\] Let $X$ and $Y$ be irreducible stationary Markov chains with transition matrices $P$ and $Q$, respectively. Then any stationary Markov chain with a transition matrix contained in ${\Pi(P, Q)}$ is a transition coupling of $X$ and $Y$.
For brevity, we will also use “transition couplings” to refer to elements of ${\Pi(P, Q)}$. Defining $\tilde{c}: ({\mathcal{X}}\times {\mathcal{Y}})^{\mathbb{N}}\rightarrow {\mathbb{R}}_+$ such that $\tilde{c}({\mathbf{x}}, {\mathbf{y}}) = c(x_0, y_0)$, we define the *optimal transition coupling (OTC) problem* by $$\label{eq:otc_problem_rv}
\mbox{minimize}\,\, \int \tilde{c} \, d\pi \,\, \mbox{subject to} \,\, \pi \in {\Pi_{\tiny TC}({\mathbb{P}}, {\mathbb{Q}})}.$$ As shown in Appendix \[app:existence\], the minimum in is achieved by an element of ${\Pi_{\tiny TC}({\mathbb{P}}, {\mathbb{Q}})}$ under our assumptions. Problem involves the minimization of a linear objective over the non-convex set ${\Pi_{\tiny TC}({\mathbb{P}}, {\mathbb{Q}})}$ and thus poses a significant computational challenge. However, Proposition \[prop:transmat\_char\] allows us to optimize instead over the convex polyhedron ${\Pi(P, Q)}$. Informally, can be restated as the minimization of ${\mathbb{E}}c(X_0', Y'_0)$ over $R \in {\Pi(P, Q)}$, where $(X', Y')$ is a stationary Markov chain generated by $R$. However, this reformulation of has a non-convex objective, so some care is needed in order to obtain global solutions.
Related Work {#sec:related_work}
------------
Stationary couplings of stationary processes, known as a *joinings*, were first studied in [@furstenberg1967disjointness]. Distances between processes based on joinings have been proposed in the ergodic theory literature [@ornstein1973application; @gray1975generalization], but have been explored primarily as a theoretical tool: no tractable algorithms have been proposed for computing such distances exactly. In the context of Markov chains, coupling methods have been widely used as a tool to establish rates of convergence [@griffeath1976coupling; @lindvall2002lectures]. Optimal Markovian couplings of Markov processes are studied in [@ellis1976thedj; @ellis1978distances; @ellis1980conditions; @ellis1980kamae]. Despite the theoretical progress, little work has been done to develop tractable algorithms for computing optimal couplings of Markov chains. In [@mufa1994optimal; @zhang2000existence; @song2016measuring] the authors consider a different, computationally simpler, form of the optimal transition coupling problem studied here, in which one minimizes the expected cost of the next step. We also remark that the optimal transition coupling problem appears in an unpublished manuscript of Aldous and Diaconis.
Other work has considered modifications of the Wasserstein distance for time series. The work [@cazelles2019wasserstein] studies the Wasserstein-Fourier distance, which is the Wasserstein distance between normalized power spectral densities, while [@muskulus2011wasserstein] suggest using the optimal transport cost between the $k$-block empirical measures constructed from observed samples. For general observed sequences, [@su2018order] consider only couplings that do not disturb the ordering of the two sequences too much, as quantified by the inverse difference moment. In contrast to these approaches, we seek a more direct modification of the optimal transport problem itself that best captures the Markovian dynamics.
Connection to Markov decision processes {#sec:connection_to_mdp}
=======================================
In the remainder of the paper, we focus on developing tractable algorithms for solving the OTC problem . We begin by making a connection between the OTC problem and Markov decision processes (MDP), which allows us to build upon existing techniques and algorithms in the MDP literature.
A Markov decision process is characterized by a 4-tuple $({\mathcal{S}}, {\mathcal{A}}, {\mathcal{P}}, c')$ consisting of a state space $\mathcal{S}$, an action space ${\mathcal{A}}= \bigcup_s {\mathcal{A}}_s$, a set of transition distributions ${\mathcal{P}}= \{p(\cdot | s, a): s \in {\mathcal{S}}, a \in {\mathcal{A}}\}$, and a cost function $c': \mathcal{S}\times {\mathcal{A}}\rightarrow\mathbb{R}$. At each time step the process occupies a state $s \in \mathcal{S}$ and an agent chooses an action $a \in {\mathcal{A}}_s$; the process then moves to a new state according to the distribution $p(\cdot | s, a)$, incurring a cost $c'(s, a)$. Informally, the goal of the agent is to choose actions in order to incur minimum average cost. The behavior of an agent is described by a family $\gamma = \{ \gamma_s(\cdot) : s \in \mathcal{S} \}$ of distributions $\gamma_s(\cdot) \in {\mathcal{M}}(\mathcal{A}_s)$ on the set of admissible actions, which is known as a [*policy*]{}. An agent following policy $\gamma$ chooses her next action according to $\gamma_s(\cdot)$ whenever the system is in state $s$, independently of her previous actions.
It is easy to see that, in conjunction with the transition distributions ${\mathcal{P}}$, every policy $\gamma$ induces a stationary Markov chain on the state space $\mathcal{S}$ of the MDP. In the average-cost MDP problem the goal is to identify a policy for which the induced Markov chain minimizes the limiting average cost, namely a policy $\gamma$ minimizing $$\label{eq:mdp_average_cost}
\overline{c}_\gamma(s) := \lim\limits_{T \rightarrow\infty} \frac{1}{T}\sum\limits_{t=1}^T \mathbb{E}_\gamma \left[c'(s_t, a_t) \bigg| s_0 = s\right],$$ for each $s \in {\mathcal{S}}$. Note that the expectation in is taken with respect to the Markov chain induced by $\gamma$, and that the limit exists by the ergodic theorem. In general, the limiting average cost $\overline{c}_\gamma(s)$ will depend on the initial state $s$, but if $\gamma$ induces an ergodic chain then the average cost will be constant. If all policies induce ergodic Markov chains, the MDP is referred to as “unichain”; otherwise the MDP is classified as “multichain”. We refer the reader to [@puterman2005markov] for more details on MDP’s.
The OTC problem may readily be recast as an MDP. In detail, let the state space $\mathcal{S} = {\mathcal{X}}\times {\mathcal{Y}}$. Furthermore, letting $s = (x,y)$ denote an element of $\mathcal{S}$, define the set of admissible actions in $s$ as ${\mathcal{A}}_s = {\Pi(P(x,\cdot), Q(y,\cdot))}$, the transition distributions $p(\cdot | s, r_s) := r_s(\cdot)$ for $r_s \in {\mathcal{A}}_s$, and the cost function $c'(s, r_s) = c(x,y)$. Note that the cost function $c'$ is independent of the action $r_s$. We refer to this MDP as TC-MDP. Any policy $\gamma$ for TC-MDP specifies distributions over ${\Pi(P(x,\cdot), Q(y,\cdot))}$ for each $(x,y) \in {\mathcal{X}}\times{\mathcal{Y}}$ and thus corresponds to a single distribution over ${\Pi(P, Q)}$ that governs the random actions of the agent. In TC-MDP it suffices to consider only deterministic policies $\gamma$, namely those such that for each state $s = (x,y)$ the distribution $\gamma_s(\cdot)$ is a point mass at unique element of ${\mathcal{A}}_{s} = {\Pi(P(x,\cdot), Q(y,\cdot))}$.
[prop]{}[deterministicpolicy]{} \[prop:deterministic\_policy\] Let $\gamma$ be a policy for TC-MDP. Then there exists a deterministic policy $\tilde{\gamma}$ such that $\overline{c}_\gamma(s) = \overline{c}_{\tilde{\gamma}}(s)$ for every $s \in {\mathcal{S}}$.
As such, optimization over deterministic policies and over ${\Pi(P, Q)}$ are equivalent. Going forward, we refer to $R \in {\Pi(P, Q)}$ directly instead of the equivalent deterministic policy $\tilde{\gamma}$ in our notation. We briefly note that, even when $X$ and $Y$ are ergodic, the same may not be true of the stationary Markov chain induced by a transition coupling matrix $R \in {\Pi(P, Q)}$ (see Appendix \[sec:redicible\_tc\]). Specifically, a single element of ${\Pi(P, Q)}$ may have multiple stationary distributions and thus give rise to multiple stationary Markov chains depending on the initial state $s \in {\mathcal{S}}$. Thus TC-MDP is classified as multichain.
Finally, supposing that $X$ and $Y$ are irreducible, note the equivalence of the objective functions in TC-MDP and : For every $R \in {\Pi(P, Q)}$ and $s \in {\mathcal{S}}$, let $\pi_R^s \in {\mathcal{M}}(({\mathcal{X}}\times{\mathcal{Y}})^{\mathbb{N}})$ denote the measure of the stationary Markov chain generated by $R$ with stationary distribution arising from the initial state $s$. Note that by Proposition \[prop:transmat\_char\], $\pi_R^s \in {\Pi_{\tiny TC}({\mathbb{P}}, {\mathbb{Q}})}$. Then for every $s \in {\mathcal{S}}$, $$\overline{c}_R(s) = \lim\limits_{T\rightarrow\infty} \frac{1}{T} \sum\limits_{t=1}^T \sum\limits_{s_t} R^t(s,s_t) c(s_t) = \int \tilde{c} \, d\pi_R^s.$$ If $R^* \in {\Pi(P, Q)}$ is optimal in TC-MDP, then $\overline{c}_{R^*}(s) = \min_{R \in {\Pi(P, Q)}} \overline{c}_{R}(s)$ for every $s \in {\mathcal{S}}$. Letting $s^* \in \operatorname*{argmin}_{s \in {\mathcal{S}}} \overline{c}_{R^*}(s)$, $\pi_{R^*}^{s^*}$ is thus an optimal transition coupling of $X$ and $Y$. So any solution to TC-MDP necessarily yields a solution to .
Policy iteration for optimal transition couplings {#sec:policy_iteration}
=================================================
Now that we have shown that the OTC problem can be viewed as an MDP, we can leverage existing algorithms for MDP’s to obtain solutions. To this end, we propose to use the policy iteration algorithm [@howard1960dynamic] because of its favorable convergence properties and ease of implementation. To facilitate our discussion, in what follows, we regard the cost function $c$ and limiting average cost $\overline{c}_R$ as vectors in ${\mathbb{R}}_+^{d^2}$. For each $R \in {\Pi(P, Q)}$, standard results [@puterman2005markov] guarantee that the limit $\overline{R} := \lim_{T\rightarrow\infty} \nicefrac{1}{T} \sum_{t=0}^{T-1} R^t$ exists. When $R$ is aperiodic and irreducible, the Perron-Frobenius theorem implies that $\overline{R} = \lim_{T\rightarrow\infty} R^T$ and the rows of $\overline{R}$ are equal to the stationary distributions of $R$.
[r]{}[0.50]{}
\[alg:pia\] $R_0 \leftarrow P \otimes Q$, $n \leftarrow 0$
The policy iteration algorithm repeatedly [evaluates]{} and [improves]{} policies. For a given policy $R \in {\Pi(P, Q)}$ the evaluation step computes the average cost (*gain*) vector $g = \overline{R} \, c$ and the total extra cost (*bias*) vector $h = \sum_{t=0}^\infty R^t(c - g)$. In practice, $g$ and $h$ may be obtained by solving a linear system of equations rather than evaluating infinite sums (see Algorithm \[alg:exact\_pe\]). The policy improvement step selects a new transition coupling matrix $R'$ that minimizes $R' \, g$ or $R' \, h$ element-wise (see Algorithm \[alg:exact\_pi\]). In particular, one may select $R'$ such that for each $(x,y)$ the row $r = R'((x,y),\cdot)$ minimizes $r g$ or $r h$ over $r \in {\Pi(P(x,\cdot), Q(y,\cdot))}$ . Once a fixed point in the evaluation and improvement process is reached, the procedure terminates. The resulting policy iteration algorithm will be referred to as `ExactOTC` (see Algorithm \[alg:pia\]). We initialize Algorithm \[alg:pia\] to the independent transition coupling $P \otimes Q$, which satisfies $P\otimes Q((x,y),(x',y')) = P(x,x')Q(y,y')$.
\[alg:exact\_pe\] Solve for $(g, h, w)$ such that $$\left[\begin{array}{ccc} I - R & 0 &0 \\ I & I - R & 0 \\ 0 & I & I - R \end{array}\right] \left[\begin{array}{c} g \\ h \\ w \end{array}\right] = \left[\begin{array}{c} 0 \\ c \\ 0 \end{array}\right]$$
\[alg:exact\_pi\] $R' \leftarrow \operatorname*{argmin}_{R \in \Pi} R g$
For finite state and action spaces, policy iteration is known to yield an optimal policy for the average-cost MDP in a finite number of steps [@puterman2005markov]. While policy iteration may fail to converge for general compact action spaces [@dekker1987counter; @schweitzer1985undiscounted; @puterman2005markov], as is the case for TC-MDP, we may exploit the polyhedral structure of ${\Pi(P, Q)}$ to establish the following convergence result.
[thm]{}[convergenceofpia]{} \[thm:convergence\_of\_pia\] Algorithm \[alg:pia\] converges to a solution $(g^*, h^*, R^*)$ of TC-MDP in a finite number of iterations. Moreover, if $X$ and $Y$ are irreducible, $R^*$ is an optimal transition coupling of $X$ and $Y$.
Recall from the discussion in Section \[sec:connection\_to\_mdp\] that a solution to TC-MDP necessarily yields a solution to . Thus Theorem \[thm:convergence\_of\_pia\] ensures that a solution to the OTC problem can be obtained from Algorithm \[alg:pia\] in a finite number of iteration. A proof of this result can be found in Appendix \[app:pia\_convergence\].
Fast approximate policy iteration {#sec:fast_approx_pia}
=================================
The simplicity of Algorithm \[alg:pia\] along with Theorem \[thm:convergence\_of\_pia\] make it an appealing method for solving the OTC problem when $d$ is small. However, each call to Algorithm \[alg:exact\_pe\] involves solving a system of $3d^2$ linear equations, requiring a total of ${\mathcal{O}}(d^6)$ operations. Furthermore, each call to Algorithm \[alg:exact\_pi\] entails solving $d^2$ linear programs each with ${\mathcal{O}}(d)$ constraints, which can be accomplished in a total time of ${\mathcal{O}}(d^5 \log d)$. For even moderate $d$, this may be too slow for practical use. A similar dependence on the dimension of each coupling is observed in exact OT algorithms, such as the Network Simplex Algorithm [@peyre2019computational]. To alleviate the poor scaling with $d$, one may use entropic regularization, whereby a negative entropy term is added to the OT objective. Cuturi [@cuturi2013sinkhorn] showed that solutions to the regularized OT problem may be obtained efficiently via Sinkhorn’s algorithm [@sinkhorn1967diagonal]. More recently, [@altschuler2017near] proved that Sinkhorn’s algorithm yields an approximation of the OT cost with error bounded by $\varepsilon$ in near-linear time with respect to the dimension of the couplings under consideration. This represents the state of the art in terms of dependence on dimension for arbitrary discrete measures. Analogously, one may hope that a similar dependence on the size of elements of ${\Pi(P, Q)}$ may be achievable for each policy iteration when solving the OTC problem.
In this section, we explore the extension of entropic regularization techniques to the OTC problem and provide an approximate algorithm that runs in $\tilde{{\mathcal{O}}}(d^4)$ time per iteration, where $\tilde{{\mathcal{O}}}(\cdot)$ hides poly-logarithmic factors. This complexity is nearly-linear in the dimension $d^4$ of the transition couplings. Mirroring the derivation of entropic OT, we first introduce a constrained OTC problem in which we consider transition couplings that are close to the independent transition coupling. We show that this type of constraint induces beneficial regularity properties among the transition couplings in the constrained set. We also propose a truncation-based approximation of the `ExactPE` algorithm, which we call `ApproxPE`. Using the regularity of the constrained set, we show that one can obtain approximations of the gain and bias from `ApproxPE` with error bounded by $\varepsilon$ in $\tilde{{\mathcal{O}}}(d^4 \log \varepsilon^{-1})$ time. Furthermore, we propose an entropy-regularized approximation to the `ExactPI` algorithm, called `ApproxPI`. We perform a new analysis of the Sinkhorn algorithm to show that `ApproxPI` yields an improved transition coupling with error bounded by $\varepsilon$ in $\tilde{{\mathcal{O}}}(d^4 \varepsilon^{-4})$ time. Combining these two algorithms, we obtain the `FastEntropicOTC` algorithm, which runs in $\tilde{{\mathcal{O}}}(d^4 \varepsilon^{-4})$ time per iteration. Finally, we provide empirical support for our theoretical results through a simulation study. We find that the improved efficiency at each iteration of `FastEntropicOTC` leads to a much faster runtime in practice as compared to `ExactOTC`. Our experiments also show that `FastEntropicOTC` yields an expected cost that closely approximates the unregularized OTC cost.
Constrained Optimal Transition Coupling Problem
-----------------------------------------------
We begin by defining a constrained set of transition couplings. Let ${\mathcal{K}}(\cdot \| \cdot)$ be the Kullback-Leibler (KL) divergence, where for any $u, v \in \Delta_{d^2}$ with $u \ll v$, ${\mathcal{K}}(u \| v) = \sum_{s} u(s) \log(u(s) / v(s))$ (letting $0 \log(0/0) = 0$). Then for every $\eta > 0$ and $(x,y) \in {\mathcal{X}}\times {\mathcal{Y}}$, define the set $${\Pi_\eta(P(x,\cdot), Q(y,\cdot))}= \left\{ r \in {\Pi(P(x,\cdot), Q(y,\cdot))}: \, {\mathcal{K}}\big(r \| P\otimes Q((x,y),\cdot)\big) \leq \eta\right\},$$ and the subset of transition coupling matrices ${\Pi_\eta(P,Q)}= \{ R \in {\Pi(P, Q)}: R((x,y),\cdot) \in {\Pi_\eta(P(x,\cdot), Q(y,\cdot))}, \, \forall (x,y) \in {\mathcal{X}}\times {\mathcal{Y}}\}$. Thus, elements of ${\Pi_\eta(P,Q)}$ have rows that are close in KL-divergence to the rows of the independent transition coupling $P \otimes Q$. When $P$ and $Q$ are aperiodic and irreducible, we find that $P\otimes Q$ is as well. We prove that, for appropriate $\eta$, the proximity of each element in ${\Pi_\eta(P,Q)}$ to $P\otimes Q$ in KL-divergence is enough to establish two beneficial regularity properties for the entire set.
[prop]{}[propertiesofentropictc]{} \[prop:properties\_of\_entropic\_tc\] Let $P$ and $Q$ be aperiodic and irreducible. Then for $\eta$ small enough, every $R \in {\Pi_\eta(P,Q)}$ is aperiodic and irreducible and thus has a unique stationary distribution $\lambda_R \in \Delta_{d^2}$. Moreover, there exist constants $M < \infty$ and $\alpha \in (0, 1)$ such that for any $t \geq 1$, $$\max\limits_{R\in {\Pi_\eta(P,Q)}} \max\limits_{s \in {\mathcal{X}}\times{\mathcal{Y}}} \|R^t(s, \cdot) - \lambda_{R}\|_1 \leq M \alpha^t.$$
We give an explicit choice of $\eta$ in the proof of Proposition \[prop:properties\_of\_entropic\_tc\]. Now, let ${\Pi_{\tiny TC}^\eta({\mathbb{P}}, {\mathbb{Q}})}$ be the set of transition couplings with transition matrices in ${\Pi_\eta(P,Q)}$ and define the constrained OTC problem, $$\label{eq:entropic_otc}
\mbox{minimize} \,\, \int \tilde{c} \, d\pi \,\, \mbox{subject to} \,\, \pi \in {\Pi_{\tiny TC}^\eta({\mathbb{P}}, {\mathbb{Q}})}.$$ For completeness, we prove that a solution to exists in Appendix \[app:existence\]. Since the function $R \mapsto {\mathcal{K}}(R(s,\cdot) \| P\otimes Q(s,\cdot))$ is uniformly bounded over $R \in {\Pi(P, Q)}$ for each $s \in {\mathcal{X}}\times {\mathcal{Y}}$, coincides with the unconstrained OTC problem for large enough $\eta$. As such, we expect that, for large $\eta$, a solution to will be close in some sense to a solution of . While we do not provide a proof of this fact, we give some empirical evidence that this is the case in Section \[sec:experiments\]. Finally, note that corresponds to an MDP in the same way that does but with a constrained set of policies. In the rest of the section, we seek fast approximations of Algorithms \[alg:exact\_pe\] and \[alg:exact\_pi\] for this constrained MDP. From now on, we assume that $P$ and $Q$ are aperiodic and irreducible and fix $\eta > 0$ such that Proposition \[prop:properties\_of\_entropic\_tc\] holds.
Fast Approximate Policy Evaluation
----------------------------------
[r]{}[0.28]{}
\[alg:fast\_pe\] $\tilde{g} \leftarrow (\nicefrac{1}{d^2} (R^L c)^\top \mathbbm{1})\mathbbm{1}$ $\tilde{h} \leftarrow \sum_{t=0}^T R^t(c - \tilde{g})$
Next, we propose a fast approximation of Algorithm \[alg:exact\_pe\]. By our choice of $\eta$ and Proposition \[prop:properties\_of\_entropic\_tc\], all elements of ${\Pi_\eta(P,Q)}$ are aperiodic and irreducible. Thus, the gain vector is constant and may be written as $g = g_0 \mathbbm{1}$ for a scalar $g_0$. Fixing $R \in {\Pi_\eta(P,Q)}$ and $L, T \geq 1$, we approximate the gain $g$ by $\tilde{g} := (\nicefrac{1}{d^2} (R^L c)^\top \mathbbm{1})\mathbbm{1}$ and the bias vector $h$ as $\tilde{h} := \sum_{t=0}^T R^t(c - \tilde{g})$. The resulting algorithm, which we refer to as `ApproxPE`, is detailed in Algorithm \[alg:fast\_pe\]. Note that $\tilde{g}$ and $\tilde{h}$ can be computed in $\mathcal{O}(L d^4)$ and ${\mathcal{O}}(T d^4)$ time, respectively. Since $g$ and $h$ are equal to the limits of $\tilde{g}$ and $\tilde{h}$ as $L, T \rightarrow \infty$, we expect that larger $L$ and $T$ will yield better approximations. One must ensure that the $L$ and $T$ that are required for a good approximation do not grow too quickly with $d$. Using Proposition \[prop:properties\_of\_entropic\_tc\], we show that this is the case.
[prop]{}[policyevaluationcomplexity]{} \[prop:policy\_evaluation\_complexity\] Let $P$ and $Q$ be aperiodic and irreducible transition matrices, $R \in {\Pi_\eta(P,Q)}$ and $\varepsilon > 0$. Furthermore, let $g \in {\mathbb{R}}^{d^2}$ and $h \in {\mathbb{R}}^{d^2}$ be the gain and bias of $R$, respectively. Then for appropriate choice of $L$ and $T$, $\emph{\texttt{ApproxPE}}(R, L, T)$ yields $(\tilde{g}, \tilde{h})$ such that $\|\tilde{g} - g\|_\infty \leq \varepsilon$ and $\|\tilde{h} - h\|_1 \leq \varepsilon$ in $\tilde{\mathcal{O}}(d^4 \log \varepsilon^{-1})$ time.
In particular, `ApproxPE` does approximate `ExactPE` in time scaling like $\tilde{{\mathcal{O}}}(d^4)$. Explicit choices of $L$ and $T$ are given in the proof of Proposition \[prop:policy\_evaluation\_complexity\], which may be found in Appendix \[app:complexity\].
Entropic Policy Improvement {#sec:entropic_regularization}
---------------------------
Next we describe a means of approximating Algorithm \[alg:exact\_pi\]. Note that since the gain vector for any element of ${\Pi_\eta(P,Q)}$ is constant, we need only improve policies with respect to the bias vector. For the constrained MDP, exact policy improvement can be performed by calling `ExactPI` with $\Pi = {\Pi_\eta(P,Q)}$. However, no computation time is saved by doing this. Instead, we settle for an algorithm that yields approximately improved transition couplings with better efficiency. To find such an approximation, we reconsider the linear optimization problems that comprise the policy improvement step. Namely, for each $s = (x,y) \in {\mathcal{X}}\times {\mathcal{Y}}$, $$\label{eq:ent_policy_improvement}
\mbox{minimize} \,\, \sum\limits_{s'} r(s') h(s') \,\, \mbox{subject to} \,\, r \in {\Pi_\eta(P(x,\cdot), Q(y,\cdot))}.$$ Recognizing that is in fact a constrained OT problem, it is equivalent to $$\label{eq:penalized_ot}
\mbox{minimize} \,\, \sum\limits_{s'} r(s') h(s') + \frac{1}{\xi_s} \sum_{s'} r(s') \log r(s') \,\, \mbox{subject to} \,\, r\in {\Pi(P(x,\cdot), Q(y,\cdot))},$$
[r]{}[0.52]{}
\[alg:approx\_entropic\_pi\]
for some $\xi_s < \infty$ depending on $\eta$ and $h$ [@dessein2018regularized]. Problem is an instance of an entropy-regularized OT problem. In order to solve , we use the `ApproxOT` algorithm of [@altschuler2017near], detailed in Appendix \[app:complexity\]. Using `ApproxOT` instead of solving exactly, we obtain the `ApproxPI` algorithm detailed in Algorithm \[alg:approx\_entropic\_pi\]. It was shown in [@altschuler2017near] that `ApproxOT` yields an approximation of the OT cost in near-linear time with respect to the size of the couplings of interest. However, in order to control the the approximation error of `ApproxPI`, we rely on a different analysis showing that one can obtain an approximation of the regularized optimal coupling in near-linear time. To the best of our knowledge, this result does not exist in the literature, so we provide a proof in Appendix \[app:complexity\]. Using this result, we show the following complexity bound.
[prop]{}[policyimprovementcomplexity]{} \[prop:policy\_improvement\_complexity\] Let $P$ and $Q$ be aperiodic and irreducible, $h \in {\mathbb{R}}^{d^2}$ and $\varepsilon > 0$. Then there exist finite constants $(\xi_s)$ such that $\emph{\texttt{ApproxPI}}(h, (\xi_{s}), \varepsilon)$ returns $\hat{R} \in {\Pi(P, Q)}$ with $\max_s \|\hat{R}(s,\cdot) - R^*(s,\cdot)\|_1 \leq \varepsilon$ for some $R^* \in \operatorname*{argmin}_{R' \in {\Pi_\eta(P,Q)}} R'h$ in $\tilde{{\mathcal{O}}}(d^4 \varepsilon^{-4})$ time.
To summarize, this result states that `ApproxPI` approximates `ExactPI` in $\tilde{{\mathcal{O}}}(d^4)$ time rather than $\tilde{{\mathcal{O}}}(d^5)$ as previously discussed. In practice, further speedups are possible by utilizing the fact that the $d^2$ entropic OT problems to be solved are decoupled and thus may be computed in parallel.
`FastEntropicPIA`
-----------------
[r]{}[0.48]{}
\[alg:fastentropic\_pia\] $n \leftarrow 0$
Finally, using Algorithms \[alg:fast\_pe\] and \[alg:approx\_entropic\_pi\], we define the `FastEntropicOTC` algorithm, detailed in Algorithm \[alg:fastentropic\_pia\]. Essentially, `FastEntropicOTC` is defined by taking `ExactOTC` for ${\Pi_\eta(P,Q)}$ and replacing `ExactPE` and `ExactPI` by their approximations, `ApproxPE` and `ApproxPI`. In practice, `ApproxPI` returns transition couplings in the relative interior of ${\Pi(P, Q)}$, so the iterates of `FastEntropicOTC` are not restricted to the finite set of extreme points of ${\Pi(P, Q)}$. Thus, convergence for Algorithm \[alg:fastentropic\_pia\] must be assessed differently than in Algorithm \[alg:pia\]. In our simulations we found that the element-wise inequality $\tilde{g}_{n+1} \geq \tilde{g}_n$ works well as an indicator of convergence.
[0.48]{} ![A comparison of total runtimes between `ExactOTC` and `FastEntropicOTC` and approximation errors for a range of $d$ and $\xi$. Error bars show the maximum and minimum values over five simulations. []{data-label="fig:exp_results"}](runtime_diff_plot_05-29-20_09-42-41.png "fig:")
[0.48]{} ![A comparison of total runtimes between `ExactOTC` and `FastEntropicOTC` and approximation errors for a range of $d$ and $\xi$. Error bars show the maximum and minimum values over five simulations. []{data-label="fig:exp_results"}](error_plot_05-29-20_09-42-41.png "fig:")
Simulation study {#sec:experiments}
================
In order to validate the use of Algorithm \[alg:fastentropic\_pia\] as a fast approximation of Algorithm \[alg:pia\], we performed a simulation study to compare the two. In each simulation, we generated random $P$, $Q$ and $c$ and ran both `ExactOTC` and `FastEntropicOTC` until convergence. We used a range of parameters, letting $d \in \{10, 20, ... 100\}$, $L = 100$, $T = 1000$, and $\xi \in \{75, 100, 200\}$, where $\xi_s = \xi$ for all $s \in {\mathcal{X}}\times {\mathcal{Y}}$. Complete implementation details may be found in Appendix \[app:experimental\_details\]. The resulting runtimes and errors are reported in Figure \[fig:exp\_results\]. In our simulations, we found that the time savings at each iteration from `FastEntropicOTC` resulted in substantial time savings over the entire runtime of the algorithm without substantial loss of accuracy. Moreover, weakening the regularization by increasing $\xi$ reduces the error of `FastEntropicOTC` with little additional runtime. This suggests that `FastEntropicOTC` may be a more efficient alternative to `ExactOTC` when $d$ is large.
Broader impact {#broader-impact .unnumbered}
==============
Optimal transport has been used in several applications including image analysis and generation, domain adaptation, modeling cell development, and embedding natural language. Our work extends the optimal transport problem to a setting in which the objects of interest are stationary Markov chains Markov chains can be used for modeling a variety of phenomena including population dynamics, gene sequences, text, and music. We anticipate that our work can be applied to problems arising in these areas.
The authors would like to thank Quoc Tran-Dinh for helpful discussions and Jason Altschuler for contributions to the proof of Lemma \[lemma:bound\_on\_sinkhorn\_error\]. K.O. and A.N were supported in part by NIH Grant R01 HG009125-01. K.M. gratefully acknowledges the support of NSF Grant DMS-1847144. K.M. and A.N. were supported in part by NSF Grant DMS-1613261. A.N. was supported in part by NSF Grant DMS-1613072.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We extend a result of Ahlgren and Ono [@ao] on congruences for traces of singular moduli of level $1$ to traces defined in terms of Hauptmodul associated to certain groups of genus 0 of higher levels.'
address: 'Department of Mathematics $\&$ Statistics, Queen’s University, Kingston, Ontario, Canada K7L 3N6'
author:
- Robert Osburn
title: Congruences for Traces of Singular moduli
---
Introduction
============
Let $j(z)$ denote the usual elliptic modular function on $\operatorname{SL}_2(\mathbb Z)$ with $q$-expansion ($q:=e^{2\pi i
z}$)
$j(z)=q^{-1}+744+196884q+21493760q^2+\cdots.$
The values of $j(z)$ at imaginary quadratic arguments in the upper half of the complex plane are known as singular moduli. Singular moduli are important algebraic integers which generate ring class field extensions of imaginary quadratic fields (Theorem 11.1 in [@cox]), are related to supersingular elliptic curves ([@ao]), and to Borcherds products of modular forms ([@b1], [@b2]).
Let $d$ denote a positive integer congruent to 0 or 3 modulo 4 so that $-d$ is the discriminant of an order in an imaginary quadratic field. Denote by $\mathcal{Q}_d$ the set of positive definite integral binary quadratic forms $$Q(x,y)=ax^2+bxy+cy^2$$ with discriminant $-d=b^2-4ac$. To each $Q \in \mathcal{Q}_d$, let $\alpha_Q$ be the unique complex number in the upper half plane which is a root of $Q(x,1)$; the singular modulus $j(\alpha_Q)$ depends only on the equivalence class of $Q$ under the action of $\Gamma:= PSL_2(\mathbb Z)$. Define $\omega_Q\in \{1, 2, 3\}$ by $$\omega_Q:
=\left \{ \begin{array}{l}
2 \quad \mbox{if $Q \sim_{\Gamma} [a,0,a]$},\\
3 \quad \mbox{if $Q \sim_{\Gamma} [a,a,a]$},\\
1 \quad \mbox{otherwise.}
\end{array}
\right. \\$$ Let $J(z)$ be the Hauptmodul $$J(z):= j(z)-744=q^{-1}+196884q+21493760q^2+\cdots.$$ Zagier [@z] defined the trace of the singular moduli of discriminant $-d$ as $$t(d):=\sum_{Q\in \mathcal{Q}_d/\Gamma}
\frac{J(\alpha_Q)}{\omega_Q}=\sum_{Q\in \mathcal{Q}_d/\Gamma}
\frac{j(\alpha_Q)-744}{\omega_Q} \in \mathbb{Z}.$$
Zagier has shown that $t(d)$ has some interesting properties. Namely, the following result (see Theorem 1 in [@z]) shows that the $t(d)$’s are Fourier coefficients of a half-integral weight modular form.
Let $\theta_1(z)$ and $E_4(z)$ be defined by $$\begin{aligned}
&E_4(z):=1+240\sum_{n=1}^{\infty}\frac{n^3q^n}{1-q^n},\\
&\theta_1(z):=
\frac{\eta^2(z)}{\eta(2z)}
=\sum_{n=-\infty}^{\infty}(-1)^nq^{n^2}=1-2q+2q^4-2q^9+\cdots.\\
\end{aligned}$$ and let $g(z)$ be defined by $$\begin{aligned}
g(z): &=-q^{-1}+2+\sum_{0<
d\equiv 0, 3\pmod 4}t(d)q^d \\
\end{aligned}$$
Then
$$\begin{aligned}
g(z) &=-\frac{\theta_1(z)E_4(4z)}{\eta^6(4z)} \\
&=-q^{-1}+2-248q^3+492q^4-4119q^7\cdots \\
\end{aligned}$$
i.e., $g(z)$ is a modular form of weight $\frac{3}{2}$ on $\Gamma_{0}(4)$, holomorphic on the upper half plane and meromorphic at the cusps.
Now what about divisibility properties of $t(d)$ as $d$ varies? In this direction, Ahlgren and Ono [@ao] recently proved the following result which shows that these traces $t(d)$ satisfy congruences based on the factorization of primes in certain imaginary quadratic fields.
If $d$ is a positive integer for which an odd prime $l$ splits in $\mathbb Q(\sqrt{-d})$, then $$t(l^2d)\equiv 0\pmod l.$$
Recently, Kim [@kim2] and Zagier [@z] defined an analogous trace of singular moduli by replacing the $j$-function by a modular function of higher level, in particular by the Hauptmodul associated to other groups of genus 0. Let $\Gamma_0(N)^{*}$ be the group generated by $\Gamma_0(N)$ and all Atkin-Lehner involutions $W_e$ for $e||N$, i.e., $e$ is a positive divisor of $N$ for which gcd$(e, N/e)=1$. There are only finitely many values of $N$ for which $\Gamma_0(N)^{*}$ is of genus 0 (see [@fricke1], [@fricke2], or [@ogg]). In particular, there are only finitely many prime values of $N$. For such a prime $p$, let $j_{p}^{*}$ be the corresponding Hauptmodul. For these primes $p$, Kim and Zagier define a trace $t^{(p)}(d)$ (see Section 3 below) in terms of singular values of $j_{p}^{*}$. The goal of this paper is to prove that the same type of congruence holds for $t^{(p)}(d)$, namely
Let $p$ be a prime for which $\Gamma_0(p)^{*}$ is of genus 0. If $d$ is a positive integer such that $-d$ is congruent to a square modulo $4p$ and for which an odd prime $l \neq p$ splits in $\mathbb
Q(\sqrt{-d})$, then $$t^{(p)}(l^2d)\equiv 0\bmod l.$$
Preliminaries on Modular and Jacobi forms
=========================================
We first recall some facts about half-integral weight modular forms (see [@kob], [@koh]). If $f(z)$ is a function of the upper half-plane, $\lambda \in \frac{1}{2}\mathbb Z$, and $\left(\begin{matrix} a & b \\
c & d \\ \end{matrix} \right) \in GL_{2}^{+}(\mathbb R)$, then we define the slash operator by
$\displaystyle f(z)|_{\lambda}\left(\begin{matrix} a & b \\
c & d \\ \end{matrix} \right) :=
(ad-bc)^{\frac{\lambda}{2}}(cz+d)^{-\lambda}f\Big(\frac{az+b}{cz+d}\Big)$
Here we take the branch of the square root having non-negative real part. If $\gamma=\left(\begin{matrix} a & b \\
c & d \\ \end{matrix} \right) \in \Gamma_0(4)$, then define
$\displaystyle j(\gamma, z):=\Big(\frac{c}{d}\Big)\epsilon_d^{-1}\sqrt{cz+d}$,
where
$$\epsilon:=
\left \{ \begin{array}{l}
1 \quad \mbox{if $d \equiv 1 \pmod 4$},\\
i \quad \mbox{if $d \equiv -1 \pmod 4$}.
\end{array}
\right. \\$$
If $k$ is an integer and $N$ is an odd positive integer, then let $\mathcal{M}_{k+{\frac{1}{2}}}(\Gamma_0(4N))$ denote the infinite dimensional vector space of nearly holomorphic modular forms of weight $k+\frac{1}{2}$ on $\Gamma_0(4N)$. These are functions $f(z)$ which are holomorphic on the upper half-plane, meromorphic at the cusps, and which satisfy
$$f({\gamma}z)=j(\gamma, z)^{2k+1}f(z)$$
for all $\gamma \in \Gamma_0(4N)$. Denote by $\mathcal{M}^{+}_{k+{\frac{1}{2}}}(\Gamma_0(4N))$ the “Kohnen plus-spaces” (see [@koh]) of nearly holomorphic forms which transform according to (1) and which have a Fourier expansion of the form
$\displaystyle \sum_{(-1)^{k}n \equiv 0,1 \pmod 4} a(n)q^n$.
We recall some properties of Hecke operators on $\mathcal{M}^{+}_{k+{\frac{1}{2}}}(\Gamma_0(4N))$. If $l$ is a prime such that $l \nmid N$, then the Hecke operator $T_{k+\frac{1}{2}, 4N}(l^2)$ on a modular form
$f(z):= \displaystyle \sum_{(-1)^{k}n \equiv 0,1 \pmod 4} a(n)q^n
\in \mathcal{M}^{+}_{k+{\frac{1}{2}}}(\Gamma_0(4N))$
is given by
$f(z)|T_{k+\frac{1}{2}, 4N}(l^2):= \displaystyle \sum_{(-1)^{k}n
\equiv 0,1 \pmod 4} \Bigl ( a(l^{2}n) +
\Biggl(\frac{(-1)^{k}n}{l}\Biggr)l^{k-1}a(n) + l^{2k-1}a(n/l^2)
\Bigr ) q^n$
where $\Bigl(\frac{*}{l}\Bigr)$ is a Legendre symbol. Let us now recall some facts about Jacobi forms (see [@ez]). A Jacobi form on $\operatorname{SL}_2(\mathbb Z)$ is a holomorphic function
$\phi: \frak{H} \times \mathbb C \to \mathbb C$
satisfying
$\displaystyle \phi\Big(\frac{a{\tau}+b}{c{\tau}+d}, \frac{z}{c{\tau} + d}\Big)=
(c{\tau}+d)^{k} e^{2{\pi}iN\frac{cz^2}{c{\tau}+ d}} \phi(\tau, z)$
$\displaystyle \phi({\tau}, z+{\lambda}{\tau}+ {\mu})=
e^{-2{\pi}iN({\lambda}^2{\tau}+2{\lambda}z)} \phi(\tau, z)$
for all $\left( \begin{matrix} a & b \\
c & d \\ \end{matrix} \right) \in \operatorname{SL}_2(\mathbb Z)$ and $(\lambda,\mu) \in {\mathbb Z}^2$, and having a Fourier expansion of the form ($q=e^{2{\pi}i{\tau}}$, $\zeta=e^{2{\pi}iz}$)
$\phi(\tau, z) = \displaystyle \sum_{n=0}^{\infty}
\sum_{\substack{r\in \mathbb Z \\ r^2 \leq 4Nn}}
c(n,r)q^{n}{\zeta}^{r}$.
Here $k$ and $N$ are the weight and index of $\phi$, respectively. Let $J_{k,N}$ denote the space of Jacobi forms of weight $k$ and index $N$ on $\operatorname{SL}_2(\mathbb Z)$. By Theorem 2.2 in [@ez], the coefficient $c(n,r)$ depends only on $4Nn - r^2$ and $r \bmod 2N$. By definition $c(n,r)=0$ unless $4Nn
- r^2 \geq 0$. If we drop the condition $4Nn - r^2 \geq 0$, we obtain a nearly holomorphic Jacobi form. Let $J^{!}_{k,N}$ be the space of nearly holomorphic Jacobi forms of weight $k$ and index $N$.
Traces
======
Let $\Gamma_0(N)^{*}$ be the group generated by $\Gamma_0(N)$ and all Atkin-Lehner involutions $W_e$ for $e||N$, that is, $e$ is a positive divisor of $N$ for which gcd$(e, N/e)=1$. $W_e$ can be represented by a matrix of the form $\frac{1}{\sqrt{e}}\left( \begin{matrix} ex & y \\
Nz & ew \\ \end{matrix} \right)$ with $x$, $y$, $z$, $w \in
\mathbb Z$ and $xwe-yzN/e=1$. There are only finitely many values of $N$ for which $\Gamma_0(N)^{*}$ is of genus 0 (see [@fricke1], [@fricke2], or [@ogg]). In particular, if we let $\frak{S}$ denote the set of prime values for such $N$, then
$\frak{S}=\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71
\}.$
For $p \in \mathfrak{S}$, let $j_{p}^{*}$ be the corresponding Hauptmodul with Fourier expansion
$q^{-1} + 0 + a_{1}q + a_{2}q^2 + \dots$
Let us now define a trace in terms of the $j_{p}^{*}$’s. Let $d$ be a positive integer such that $-d$ is congruent to a square modulo $4p$. Choose an integer $\beta \bmod 2p$ such that ${\beta}^2 \equiv -d \bmod 4p$ and consider the set
$\mathcal{Q}_{d, p, \beta}= \{[a,b,c] \in \mathcal{Q}_d : a \equiv
0 \bmod p$, $b \equiv \beta \bmod 2p \}$.
Note that $\Gamma_0(p)$ acts on $\mathcal{Q}_{d, p, \beta}$. Assume that $d$ is not divisible as a discriminant by the square of any prime dividing $p$, i.e. not divisible by $p^2$. Then we have a bijection via the natural map between
$\mathcal{Q}_{d, p, \beta} \diagup \Gamma_0(p)$
and
$\mathcal{Q}_d \diagup \Gamma$
as the image of the root $\alpha_{Q}$, $Q \in
\mathcal{Q}_{d, p, \beta}$, in $\Gamma_0(p) \diagup \mathfrak{H}$ corresponds to a Heegner point. We could then define a trace $t^{(p, \beta)}(d)$ as the sum of the values of $j_{p}^{*}$ with $Q$ running over a set of representatives for $\mathcal{Q}_{d, p,
\beta} \diagup \Gamma_0(p)$. As $t^{(p, \beta)}(d)$ is independent of $\beta$, we define the trace $t^{(p)}(d)$ (see Section 8 of [@z] or Section 1 of [@kim2])
$t^{(p)}(d) = \displaystyle \sum_{Q}
\frac{j_{p}^{*}(\alpha_{Q})}{\omega_{Q}} \in \mathbb{Z}$
where the sum is over $\Gamma_0(p)^{*}$ representatives of forms $Q=[a,b,c]$ satisfying $a \equiv 0 \bmod p$.
For $p=2$, we have $t^{(2)}(4)=\frac{1}{2}
j_{2}^{*}(\frac{1+i}{2})=-52$, $t^{(2)}(7)=
j_{2}^{*}(\frac{1+\sqrt{-7}}{4})=-23$, $t^{(2)}(8)=
j_{2}^{*}(\frac{\sqrt{-2}}{2})=152$. For $p=3$, we have $t^{(3)}(3)=\frac{1}{3} j_{3}^{*}(\frac{-3+\sqrt{-3}}{6})=-14$, $t^{(3)}(11)= j_{3}^{*}(\frac{1+\sqrt{-11}}{6})=22$. Moreover by the table in Section 8 of [@z], we have:
[c|ccc]{}\
$d$ & $t^{(2)}(d)$ & $t^{(3)}(d)$ & $t^{(5)}(d)$\
3 & & $-14$ &\
4 & $-52$ & & $-8$\
7 & $-23$ & &\
8 & 152 & $-34$ &\
11 & & 22 & $-12$\
12 & $-496$ & 52 &\
15 & $-1$ & $-138$ & $-38$\
16 & 1036 & & $-6$\
19 & & & 20\
20 & $-2256$ & $-116$ & 12\
23 & $-94$ & 115 &\
24 & 4400 & 348 & $-44$\
27 & & $-482$ &\
28 & $-8192$ & &\
The empty entries correspond to $-d$ which are not congruent to squares modulo $4p$.
By the discussion in Section 8 of [@z] or Section 2.2 in [@kim2], there exist forms $\phi_{p} \in J^{!}_{2,p}$ uniquely characterized by the condition that their Fourier coefficients $c(n,r)=B(4pn-r^2)$ depend only on $r^2-4pn$ and where $B(-1)=1$, $B(d)=0$ if $d=4pn-r^2<0$, $\neq -1$ and $B(0)=-2$. Define $g_p(z)$ as
$g_p(z):= q^{-1} + \displaystyle \sum_{d \geq 0} B(d)q^d$.
By the correspondence between Jacobi forms and half-integral weight forms (Theorem 5.6 in [@ez]), $g_p(z) \in
\mathcal{M}^{+}_{\frac{3}{2}}(\Gamma_0(4p))$. As the dimension of $J_{2,p}$ is zero, we have that for every integer $d \geq 0$ such that $-d$ is congruent to a square modulo $4p$, there exists a unique $f_{d,p} \in \mathcal{M}^{+}_{\frac{1}{2}}(\Gamma_0(4p))$ with Fourier expansion
$f_{d,p}(z)= q^{-d} + \displaystyle \sum_{0<D\equiv 0,1 \pmod 4}
A(D,d) q^D$.
An explicit construction of $f_{d,p}$ can be found in the appendix of [@kim1] and the uniqueness of $f_{d,p}$ follows from the discussion at the end of Section 2 in [@kim1]. The following result relates the Fourier coefficients $A(1,d)$ and $B(d)$ and shows that the traces $t^{(p)}(d)$ are Fourier coefficients of a nearly holomorphic Jacobi form of weight 2 and index $p$ (see Theorem 8 in [@z] or Lemma 3.5 and Corollary 3.6 in [@kim2]).
Let $p$ be a prime for which $\Gamma_0(p)^{*}$ is of genus 0.\
(i) Let $d=4pn-r^2$ for some integers $n$ and $r$. Let $A(1,d)$ be the coefficient of $q$ in $f_{d,p}$ and $B(d)$ be the coefficient of ${q^n}{\zeta^{r}}$ in $\phi_{p}$. Then $A(1,d)=-B(d)$.\
(ii) For each natural number $d$ which is congruent to a square modulo $4p$, let $t^{(p)}(d)$ be defined as above. We also put $t^{(p)}(-1)=-1, t^{(p)}(d)=0$ for $d<-1$. Then $t^{(p)}(d)=-B(d)$.
Proof of Theorem 1.3
====================
The proof requires the study of Hecke operators $T_{k+\frac{1}{2},
4p}(l^2)$ on the forms $g_p(z)$ and $f_{d,p}(z)$. Define integers $A_{l}(d)$ and $B_{l}(d)$ by
$A_{l}(d):=$ the coefficient of $q$ in $f_{d,p}|T_{\frac{1}{2},
4p}(l^2)$,
$B_{l}(d):=$ the coefficient of $q^{d}$ in $g_p(z)|T_{\frac{3}{2},
4p}(l^2)$.
From equation (19) of [@z], we have
$A_{l}(d)=A(1,d) + lA(l^2,d)$.
Also note that we have
$g_p(z)|T_{\frac{3}{2}, 4p}(l^2)= q^{-1} + lq^{-{l^2}} +
\displaystyle \sum_{0<d \equiv 0, 3 \pmod 4} \Bigl ( B(l^{2}d) +
\Biggl(\frac{-d}{l}\Biggr)B(d) + lB(d/l^2) \Bigr ) q^d$
and so $B_{l}(d) = B(l^{2}d) +
\Bigl(\frac{-d}{l}\Bigr)B(d) + lB(d/l^2)$. Now suppose $p$ is in $\frak{S}$ and $d$ is a positive integer such that $-d$ is a square modulo $4p$ and for which an odd prime $l\neq p$ splits in $\mathbb Q(\sqrt{-d})$. Then $\Bigl(\frac{-d}{l}\Bigr)=1$. By Theorem 3.2 and the above calculations, we have $$\begin{aligned}
t^{(p)}(l^{2}d)&= -B(l^{2}d) \\
&= -B_{l}(d) + \Bigl(\frac{-d}{l}\Bigr)B(d) + lB(d/l^{2}) \\
&\equiv -B_{l}(d) + B(d) \bmod l \\
&\equiv A_{l}(d) + B(d) \bmod l \\
&\equiv A(1,d) + lA(l^{2},d) + B(d) \bmod l \\
&\equiv -B(d) + B(d) \bmod l \\
&\equiv 0 \bmod l.\\
\end{aligned}$$
We now illustrate Theorem 1.3. If $p=2$ and $l=3$, then for every non-negative integer $s$, we have
$t^{(2)}(3^2(24s+23)) \equiv 0 \bmod 3$.
In particular, if we want to compute $t^{(2)}(207)$, then we are interested in $\phi_{2} \in J^{!}_{2,2}$. By Theorem 9.3 in [@ez] and the discussion preceding Table 8 in [@z], $J^{!}_{2,2}$ is the free polynomial algebra over
$\mathbb C[E_4(\tau), E_6(\tau),
{\Delta}^{-1}]\diagup({E_4(\tau)}^{3} - {E_6(\tau)}^{2})$
on two generators $a$ and $b$ where $\displaystyle \Delta=\frac{{E_4(\tau)}^{3} - {E_6(\tau)}^{2}}{1728}$. The Fourier expansions of $a$ and $b$ begin
$$\begin{aligned}
a&=(\zeta - 2 + {\zeta}^{-1}) + (-2{\zeta}^{2} + 8{\zeta} - 12 +
8{\zeta}^{-1} - 2{\zeta}^{-2})q + ({\zeta}^{3} -12{\zeta}^2 +
39{\zeta} - 56 + \cdots)q^2 \\ &+ (8{\zeta}^3 - 56{\zeta}^2 +
152{\zeta} - 208 + \cdots)q^3 + \cdots.
\end{aligned}$$
$$\begin{aligned}
b&=(\zeta + 10 + {\zeta}^-1) + (10{\zeta}^2 -64{\zeta} + 108 -
64{\zeta} + 10{\zeta}^2)q + ({\zeta}^3 + 108{\zeta}^2 - 513{\zeta}
\\ &+ 808 - \cdots)q^2 + (-64{\zeta}^3 + 808{\zeta}^2 - 2752\zeta +
4016 - \cdots)q^3 + \cdots.
\end{aligned}$$
The coefficients for $a$ and $b$ can be obtained using Table 1 or the recursion formulas on page 39 of [@ez]. The representation of $\phi_{2}$ in terms of $a$ and $b$ is:
$\displaystyle \phi_{2}=\frac{1}{12}a(E_4(\tau)b-E_6(\tau)a)$.
By Theorem 3.2, we have $t^{(2)}(207)=-B(207)$. As $8n-r^2=207$ has a solution $n=29$ and $r=5$, then $B(207)$ is the coefficient of $q^{29}{\zeta}^5$ which is $-113643$. Thus
$t^{(2)}(207)=113643 \equiv 0 \bmod 3$.
\(1) Zagier actually defined $t^{(N)}(d)$ and proved part (ii) of Theorem 3.2 for all $N$ such that $\Gamma_0(N)^{*}$ is of genus 0 (see Section 8 in [@z]). One might be able to prove part (i) of Theorem 3.2 in the case $N$ is squarefree. If so, then a congruence, similar to Theorem 1.3, should hold for $t^{(N)}(d)$, $N$ squarefree. If $N$ is not squarefree, then C. Kim has kindly pointed out part (i) of Theorem 3.2 does not hold. For example, if $N=4$ and $d=7$, one can construct $f_{7,4}$ (see the appendix in [@kim2]) and compute that
$f_{7,4}=q^{-7} -55q + 0q^{4} + 220q^{9} + \cdots.$
Thus $A(1,7)=-55$. But $B(7)=23$ (see Remark 3.1).
\(2) We should note that Theorem 1.3 is an extension of the simplest case of Theorem 1 in [@ao]. Ono and Ahlgren have also proven congruences for $t(d)$ which involve ramified or inert primes in quadratic fields. In fact, they prove that a positive proportion of primes yield congruences for $t(d)$ (see parts (2) and (3) of Theorem 1 in [@ao]). It would be interesting to see if such congruences hold for $t^{(p)}(d)$ or $t^{(N)}(d)$.
\(3) The Monster $\mathbb{M}$ is the largest of the sporadic simple groups of order
$2^{46}3^{20}5^{9}7^{6}11^{2}13^{3}17\cdot19\cdot23\cdot29\cdot31\cdot41\cdot47\cdot59\cdot71$
Ogg [@ogg] noticed that the primes dividing the order of $\mathbb{M}$ are exactly those in the set $\frak{S}$. The monster $\mathbb{M}$ acts on a graded vector algebra $V=V_{-1}
\bigoplus_{n\geq 1} V_n$ (see Frenkel, Lepowsky, and Meurman [@flm] for the construction). For any element $g \in \mathbb{M}$, let $Tr(g|V_n)$ denote the trace of $g$ acting on $V_n$ for each $n$. Then $Tr(g|V_{-1})=1$ and $Tr(g|V_n) \in \mathbb Z$ for every $n \geq 1$. The Thompson series is defined by:
$T_{g}(z)= q^{-1} + \displaystyle \sum_{n \geq 1} Tr(g|V_n)q^n$.
The authors in [@cy] study Thompson series evaluated at imaginary quadratic arguments, i.e. “singular moduli” of Thompson series. It is possible to define a trace of singular moduli of Thompson series. A natural question is “do we have congruences for these traces?”
Acknowledgments {#acknowledgments .unnumbered}
===============
The author would like to thank Imin Chen, Chang Heon Kim, Ken Ono, and Noriko Yui for their valuable comments.
[10]{}
S. Ahlgren, K. Ono, [*Arithmetic of Singular Moduli and Class Equations*]{}, to appear in Compositio Math.
R. E. Borcherds, [*Automorphic forms on $\mathbb O_{s+2,2}(\mathbb R)$ and infinite products*]{}, Invent. Math. Vol 120, (1995), 161–213.
R. E. Borcherds, [*Automorphic forms on $\mathbb O_{s+2,2}(\mathbb R)^{+}$ and generalized Kac-Moody algebras*]{}, Proc. Int. Congress of Mathematicians (Zürich, 1994) (1995), 744–752.
I. Chen, N. Yui, *Singular values of Thompson series*, in “Groups, Difference Sets and Monster” (K. T. Arau et al., Eds.), pp. 255–326, de Gruyter, Berlin, 1995.
D. Cox, [*Primes of the Form $x^2+ny^2$*]{}, John Wiley $\&$ Sons, Inc, New York, 1989.
M. Eichler, D. Zagier, [*The Theory of Jacobi forms*]{}, Progress in Math. **55**, Birkhäuser-Verlag, 1985.
I. Frenkel, J. Leopowsky, A. Meurman, [*Vertex Operator Algebras and the Monster*]{}, Academic Press, New York 1988.
R. Fricke, [*Die Elliptische Funktionen und Ihre Anwendungen*]{}, 2-ter Teil, Teubner, Leipzig 1922.
R. Fricke, [*Lehrbuch der Algebra III (Algebraische Zahlen)*]{}, Vieweg, Braunschweig 1928.
C.H. Kim, *Borcherds products associated with certain Thompson series*, Compositio Math. **140** (2004), 541–551.
C.H. Kim, *Traces of singular values and Borcherds products*, preprint, 2003.
N. Koblitz, *Introduction to elliptic curves and modular forms*, Springer-Verlag, 1984.
W. Kohnen, [*Newforms of half-integral weight*]{}, J. reine angew. Math **333**, (1982), 32-72.
A. Ogg, *Automorphismes de courbes modulaires*, S[é]{}minaire Delange-Pisot-Poitou (Th[é]{}orie des nombres) 16e ann[é]{}e (1974/75), No. 7, 7-01-7-08.
D. Zagier, [*Traces of singular moduli*]{}, Motives, Polylogarithms and Hodge Theory, Part I. International Press Lecture Series, editors F. Bogomolov and L. Katzarkov, International Press, Somerville (2002), 211–244.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Solar flares involve the sudden release of magnetic energy in the solar corona. Accelerated nonthermal electrons have often been invoked as the primary means for transporting the bulk of the released energy to the lower solar atmosphere. However, significant challenges remain for this scenario, especially in accounting for the large number of accelerated electrons inferred from observations. Propagating magnetohydrodynamics (MHD) waves, particularly those with subsecond/second-scale periods, have been proposed as an alternative means for transporting the released flare energy, 1[likely alongside the electron beams]{}, while observational evidence remains elusive. Here we report a possible detection of such waves in the 1[late]{} impulsive phase of a two-ribbon flare. This is based on ultrahigh cadence dynamic imaging spectroscopic observations of a peculiar type of decimetric radio bursts obtained by the Karl G. Jansky Very Large Array. Radio imaging at each time and frequency pixel allows us to trace the spatiotemporal motion of the source, which agrees with the implications of the frequency drift pattern in the dynamic spectrum. The radio source1[, propagating at 1000–2000 km s$^{-1}$ in projection, shows close spatial and temporal association with transient brightenings on the flare ribbon]{}. In addition, multitudes of subsecond-period oscillations are present in the radio emission. We interpret the observed radio bursts as 1[short-period]{} MHD wave packets propagating along newly reconnected magnetic flux tubes linking to the flare ribbon. The estimated energy flux carried by the waves is comparable to that needed to account for the plasma heating 1[during the late impulsive phase of this flare]{}.'
author:
- 'Sijie Yu ()'
- 'Bin Chen ()'
bibliography:
- 'Yu2018.bib'
title: 'Possible Detection of Subsecond-period Propagating Magnetohydrodynamics Waves in Post-reconnection Magnetic Loops during a Two-ribbon Solar Flare'
---
Introduction {#sec:intro}
============
An outstanding question in solar flare studies is how a large amount of magnetic energy released in a flare (up to 10$^{33}$ erg) is converted into other forms of energy in accelerated particles, heated plasma, waves/turbulence, and bulk motions, and transported throughout the flare region. The collisional thick-target model (CTTM; @1971SoPh...18..489B), along with the framework of the standard CSHKP flare scenario [@1964NASSP..50..451C; @1966Natur.211..695S; @1974SoPh...34..323H; @1976SoPh...50...85K], assumes that 1[a considerable fraction]{} of the magnetic energy released via reconnection goes into acceleration of charged electrons and ions to nonthermal energies in the solar corona 1[[@2004JGRA..10910104E; @2005JGRA..11011103E; @2012ApJ...759...71E]]{}. The downward-propagating electrons along the reconnected, close field lines slam into the dense chromosphere and lose most of their energy through Coulomb collisions. This sudden energy loss results in the 1[intense]{} heating of the chromospheric material within a confined region at the footpoints of the closed arcades, driving hot and dense material upward and filling the arcades — a process known as “chromospheric evaporation.” The arcades, in turn, accumulate a large emission measure at high temperatures, thereby appearing particularly bright in extreme ultraviolet (EUV) and soft X-ray (SXR) wavelengths (see, e.g., a recent review by @2017LRSP...14....2B).
The CTTM model has been successful in accounting for a variety of flare phenomena, most notably the “Neupert effect”: The high-energy, hard X-ray (HXR) emission tends to coincide temporally with the rate of the rising lower-energy, SXR emission during the primary phase of energy release (also known as the “impulsive phase”) of a flare . Other outstanding examples include the decreasing height and area of HXR footpoint sources with increasing energy. However, significant challenges remain for the CTTM model (see, e.g., and references therein). One challenge is the so-called “number problem”: the total number of nonthermal electrons required to account for the observed HXR, (E)UV, or white light (WL) footpoint sources or flare ribbons can be very large compared to that available in the corona [e.g., @2007ApJ...656.1187F; @2011ApJ...739...96K]. Similar implications have been argued based on observations of coronal HXR sources — the inferred number density of nonthermal electrons is a large fraction of, or in some cases, nearly equal to, the total electron density available in the corona [@2007ApJ...669L..49K; @2008ApJ...678L..63K; @2010ApJ...714.1108K]. 1[This requires electrons to replenished the corona at the same rate as nonthermal electrons precipitate from it, otherwise the coronal acceleration region would be quickly evacuated. A scenario that invokes return currents, which involve electrons flowing up from the chromosphere into the corona to neutralize the depletion of the coronal electrons, has been suggested to alleviate the difficulty ]{}. 1[Nevertheless, these considerations have led various authors to suggest alternative scenarios that invoke electron (re)acceleration in the lower, denser solar atmosphere .]{} Other mechanisms have also been proposed for heating the chromospheric plasma, such as thermal conduction or magnetohydrodynamics (MHD) waves . In all cases, alternative means, 1[possibly operating alongside accelerated electrons]{} as in the CTTM model, are postulated to transport a sizable portion of the released flare energy from the reconnection region, presumably located in the corona, downward to spatially confined regions in the lower solar atmosphere.
One excellent way to provide such focused energy transport other than electron beams is via propagating plasma waves within reconnected flare arcades . A variety of plasma waves, including Alfvén waves and fast-mode and slow-mode magnetosonic waves, can arise as a natural consequence of the flare energy being released in an impulsive fashion (see, e.g., recent studies by @2017ApJ...847....1T and @2018ApJ...860..138P). As argued by @2008ApJ...675.1645F and @2013ApJ...765...81R, plasma waves are capable of carrying a significant amount of flare energy, which may be comparable to that needed to power the radiative emissions of a flare. 1[An intriguing recent numerical study by @2016ApJ...818L..20R demonstrated that the waves can drive chromospheric evaporation in a strikingly similar fashion to the way electron beams do. Their results were then confirmed by @2016ApJ...827..101K, who further showed that the detailed shapes of certain chromospheric lines could be used as a potential observational test to distinguish between the wave- and electron-beam heating scenarios.]{}
Observationally, flare-associated quasi-periodic pulsations (QPPs) with different periods ranging from $<$1 s to tens of minutes have been detected at virtually all wavelengths. One of the main origins for the QPPs is thought to be MHD oscillations or waves (see, e.g., @2009SSRv..149..119N for a review). Observational evidence for large-scale wave-like phenomena associated with flares has also frequently been reported using spatially-resolved imaging data (see reviews by e.g., @2012SoPh..281..187P [@2014SoPh..289.3233L; @2015LRSP...12....3W; @2017SoPh..292....7L], a study of a large sample of such events in @2013ApJ...776...58N, and a most recent observation of the 2017 September 10 X8.2 flare in @2018ApJ...864L..24L). Observational evidence that links the response in the lower solar atmosphere to downward-propagating MHD waves, however, is rather rare. One outstanding example was from @2016NatCo...713104L, who found a sudden sunspot rotation during the impulsive phase of a flare based on observations from the Goode Solar Telescope of the Big Bear Solar Observatory (GST/BBSO), possibly triggered by downward-propagating waves generated by the release of flare energy. Another interesting study by @2015ApJ...810....4B reported long-period ($\sim$140 s), slow ($\sim$20 km s$^{-1}$) oscillating flare ribbons based on observations by the *Interface Region Imaging Spectrograph*, although the authors interpreted the oscillating phenomenon in terms of instabilities in the reconnection current sheet rather than MHD waves. It is worthwhile to point out that, in the Earth’s magnetosphere, direct evidence for Alfvén waves propagating along the outer boundary of the “plasma sheet” (which is analogous to newly reconnected flare loops) has been reported based on *in situ* measurements. These waves have been argued to be responsible in transporting a significant amount of energy flux (in the form of Poynting flux) from the site of energy release in the magnetotail toward the Earth, which, in turn, powers the auroral emission that is analogous to flare ribbons on the Sun [@2000JGR...10518675W; @2002JGRA..107.1201W; @2000GeoRL..27.3169K].
Recently, numerical and analytical models have been developed to investigate 1[energy transport and deposition]{} from the corona to the low solar atmosphere by MHD waves [@2013ApJ...765...81R; @2016ApJ...827..101K; @2016ApJ...818L..20R; @2017ApJ...847....1T; @2018ApJ...853..101R]. An important finding is that short-period MHD waves, especially those having periods of about one second or less, carry a significant amount of energy [@2017ApJ...847....1T], suffer much less energy loss when propagating out from the corona to the lower solar atmosphere [@2013ApJ...765...81R; @2018ApJ...860..138P], and are much more efficient in dissipating the energy in the upper chromosphere than their long-period counterparts [@2008ApJ...675.1645F; @2013ApJ...765...81R; @2016ApJ...818L..20R]. Therefore, these short-period MHD waves are thought to be a potential candidate for an alternative carrier for energy released in flares. Subsecond-period ($P<1$ s) QPPs have frequently been reported in radio and X-ray light curves and/or dynamic spectra (e.g., ). 1[However, most of the large-scale wave-like phenomena detected on the basis of imaging data fall into the long-period regime ($>$10 s, e.g., @2009SSRv..149..119N), with some rare exceptions from eclipse observations [e.g., @2002SoPh..207..241P].]{} This is mainly due to the limitation on temporal cadence of current WL/EUV imaging instrumentation, or the lack of radio/X-ray imaging capability at high temporal cadence with sufficient dynamic range or counting statistics.
Here we report ultrahigh cadence (0.05 s) spectroscopic imaging of a peculiar type of radio bursts in the decimetric wavelength range (“[dm-$\lambda$]{}” hereafter) that is likely associated with subsecond-period MHD waves propagating along flaring arcades. The bursts were recorded by the Karl G. Jansky Very Large Array (VLA) in a *GOES*-class C7.2 flare that is associated with a failed filament eruption and large-scale coronal EUV waves. We further show that these MHD waves may carry a significant amount of energy flux that is comparable to the average energy flux needed for driving the plasma heating at the flare ribbons. In Section \[sec-obs\], we present VLA dynamic imaging spectroscopic observations of the radio bursts, supported by complementary magnetic, EUV, and X-ray data. In Section \[sec-discussion\], we interpret the observations within a physical scenario that involves propagating short-period MHD wave packets and discuss their energetics. We briefly summarize our findings in Section \[sec-conclusion\].
Observations {#sec-obs}
============
Event overview {#sec-overview}
--------------
The VLA is a general-purpose radio interferometer operating at $<$1–50 GHz. It has completed a major upgrade [@2011ApJ...739L...1P] and was partially commissioned for solar observation in late 2011 [@2013ApJ...763L..21C]. It is capable of making broadband radio imaging spectroscopic observations in more than one thousand spectral channels with ultrahigh time resolution of tens of millisecond-scale. Recent studies with the VLA have demonstrated its unique power in using coherent solar radio bursts to diagnose the production and transport of energetic electrons in solar flares by utilizing its imaging capabilities with spectrometer-like time and spectral resolution [@2013ApJ...763L..21C; @2014ApJ...794..149C; @2015Sci...350.1238C; @2018ApJ...866...62C; @2017ApJ...848...77W].
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The event under investigation occurred on 2014 November 11 in NOAA active region (AR) $12201$, located at $44\degr$ east from the central meridian. It is a *GOES*-class C7.2 solar flare (flare identifier “SOL2014-11-01T16:39:00L085C095” following the IAU convention suggested by @2010SoPh..263....1L). This event was well observed by the Atmospheric Imaging Assembly (AIA; @2012SoPh..275...17L) and the Helioseismic and Magnetic Imager (HMI; @2012SoPh..275..207S) aboard the *Solar Dynamics Observatory* (*SDO*). The impulsive phase of the flare started from $\sim$16:39 UT and was partially covered by *RHESSI* [@2002SoPh..210....3L] until 16:42 UT, when the spacecraft entered the South Atlantic Anomaly (SAA). The VLA was used to observe the Sun from 16:30:10 UT to 20:40:09 UT and captured the entire flare. The observations were made in frequency bands between 1 and 2 GHz with 50 ms cadence and 2 MHz spectral resolution in dual circular polarizations. The 27-antenna array was in the C configuration (maximum baseline length 3 km), yielding an intrinsic angular resolution of $35''.7\times 16''.3$ at $\nu=1$ GHz at the time of the observation (and this scales linearly with $1/\nu$). The deconvolved synthesis images are restored with a $30''$ circular beam.
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Figures \[fig-overview\] and \[fig-euv\] show an overview of the time history and general context of the flare event. The *GOES* 1–8 Å SXR flux starts to rise at 16:39 UT and peaks at around 16:46 UT, during which time a filament is seen to erupt (green arrows in Figure \[fig-euv\]) but it does not fully detach from the surface and forms a coronal mass ejection—a phenomenon known as a “failed eruption.” During this period, both the HXR light curve (blue curve in Figure \[fig-overview\](D)) and the SXR derivative (red curve in Figure \[fig-overview\](D)) display multiple bursty features, which is characteristic of the flare’s impulsive phase, during which the primary energy release occurs. Precipitating nonthermal electrons lose most of their energy in the dense chromosphere, resulting in HXR sources at the footpoints of the reconnected flare arcades via bremsstrahlung radiation (contours in Figure \[fig-euv\](B)). Bright flare ribbons, visible in UV/EUV passbands (shown in Figure \[fig-euv\] in purple, which is mostly contributed by AIA 304 Å), are formed due to heating of the chromospheric/photospheric material by precipitated nonthermal electrons or by other means. The evaporated chromospheric material fills the flare arcades and forms bright coronal loops, best seen in EUV passbands that are sensitive to relatively high coronal temperatures (green and blue colors in Figure \[fig-euv\], which show AIA 211 and 94 Å bands that correspond to plasma temperatures of 2 MK and 7 MK, respectively). Many of the impulsive peaks in the SXR derivative have counterparts in the light curves from the Radio Solar Telescope Network (RSTN) (Figure \[fig-overview\](B) and (C)), which are also visible in the VLA 1–2 GHz dynamic spectrum as short-duration radio bursts (Figure \[fig-overview\](A)), suggesting that they are both closely associated with accelerated nonthermal electrons. The [dm-$\lambda$]{} bursts have complex fine spectrotemporal structures, especially in the lower-frequency portion of the radio dynamic spectrum.
The radio bursts under study appear during the late impulsive phase (shaded area in Figure \[fig-overview\](A–C) demarcated with vertical dashed lines). Two main episodes can be distinguished in the dynamic spectrum, each of which lasts for $\sim$10–20 seconds (referred to as “Burst 1” and “Burst 2” hereafter). An enlarged view of these bursts is available in Figures \[fig-ribbonbrightenings\](A) and \[fig-radio-spec-imaging\](A). From the imaging data, the bursts have a peak brightness temperature $T_B$ of $\sim1.1 \times 10^7$ K. 1[The total flux density is $\sim1$ sfu (solar flux unit; 1 sfu $= 10^4$ Jy).]{} In addition, the bursts are nearly 100% polarized with left-hand circular polarization (LCP). These properties are consistent with radio emission associated with a coherent radiation mechanism. In the dynamic spectrum, the bursts appear as arch-shaped emission lanes, which display a low-high-low frequency drift pattern. The frequency drift rate $d\nu/dt$ is between 60 and 200 MHz/s (or a relative drift rate of $\dot{\nu}/\nu\approx0.04$–0.2), which is about one order of magnitude lower than type III radio bursts emitted by beams of fast electrons, but similar to fiber bursts and lace bursts in the same frequency range . Such bursts with an intermediate frequency drift rate are sometimes referred to as the “intermediate drift bursts” . The multiple episodes of positive- and negative-drifting features resemble to some extent the “lace bursts” in the literature . However, the emission lanes of these bursts appear to be much smoother, while the lace bursts, at least from the few cases reported in the literature, have a much more fragmentary and chaotic appearance.
Radio imaging of the bursts places the burst source (red contours in Figures \[fig-euv\](E) and (F)) near the northern flare ribbon. The location of the radio bursts is also very close to the *RHESSI* 12–25 keV HXR footpoint source, shown in Figure \[fig-euv\](B) as white contours, albeit the latter is obtained several minutes earlier (at 16:40 UT) before the spacecraft enters the SAA. 1[A more detailed investigation reveals a close temporal and spatial association between the radio bursts and the transient (E)UV brightenings at the northern flare ribbon. Figure \[fig-ribbonbrightenings\](B) shows an AIA 304 Å background-detrended image sequence during the time interval of the radio dynamic spectrum shown in Figure \[fig-ribbonbrightenings\](A). During this period, the northern ribbon features the appearance of two transient EUV brightenings during radio Bursts 1 and 2, and the location of the brightenings is very close to the radio source (red).]{} The appearance of the radio source during the flare’s impulsive phase, as well as its close spatial and temporal association with the ribbon brightenings, suggests that the radio source is intimately related to the release and transport of the flare energy. More detailed discussions of the spectral, temporal, and spatial features of the bursts based on radio dynamic imaging spectroscopy will be presented in the next subsection.
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Another interesting feature of this event is that it is accompanied by large-scale, 1[fast-propagating]{} disturbances 1[(“PDs” hereafter), observed in EUV,]{} during the impulsive and gradual phase of the flare; they are usually interpreted as propagating MHD waves in the corona [@2012SoPh..281..187P; @2013ApJ...776...58N; @2014SoPh..289.3233L; @2015LRSP...12....3W; @2017SoPh..292....7L; @2018ApJ...864L..24L]. Using AIA 171, 193, and 211 Å running-ratio images (ratio of current frame to the previous frame), a large-scale 1[PD feature]{} (denoted as 1[“PD1”]{} in Figure \[fig-LSWave\]) is present in the area between AR 12201 and AR 12200. In addition, another large-scale 1[PD]{} appears to move outward above the limb (denoted 1[“PD2”]{} in Figure \[fig-LSWave\]). The temporal evolution of the two 1[PDs]{} is displayed in the time-distance plots in Figure \[fig-LSWave\](G) and (H), made along two slices labeled “S1” and “S2” in Figure \[fig-LSWave\](A), respectively. The initialization of the large-scale 1[PDs]{} coincides with the onset of the flare, demonstrating their close association with the flare energy release. The large-scale 1[PDs]{} propagate at a speed of 400–500 km s$^{-1}$, with 1[PD1]{} clearly experiencing multiple deflections by magnetic structures of the ARs. We note that the radio bursts are observed during the period when 1[PD1]{} remains in the flaring region (Figure \[fig-LSWave\](C)). This is a strong indication of the presence of ubiquitous MHD disturbances in and around the flaring region during the time of the radio bursts.
Radio Dynamic Spectroscopic Imaging {#sec-spec-imaging}
-----------------------------------
The capability of simultaneous imaging and dynamic spectroscopy offered by the VLA allows each pixel in the dynamic spectrum to form a radio image. As an example, Figure \[fig-radio-spec-imaging\](B) shows a three-dimensional (3D) rendering of a VLA spectral image cube taken for Burst 2 within a 100 ms integration (at 16:46:18.2 UT; the timing is shown as the vertical dotted line in panel (A)). The two horizontal slices in Figure \[fig-radio-spec-imaging\] (B) indicate the radio images at the peak frequencies of the two emission lanes at that time (circles in panel (A)). The same two radio images are shown in Figures \[fig-radio-spec-imaging\](C) and (D) as green and blue contours overlaid on the AIA EUV 304 Å image and the HMI photospheric line-of-sight (LOS) magnetogram respectively. As discussed in the previous subsection, the radio sources are located near the northern flare ribbon. In the magnetogram, this flare ribbon corresponds to a region with a positive magnetic polarity. As the bursts are 100% LCP, they are likely polarized in the sense of $o$ mode.
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We produce an independent 3D spectral image cube for each time pixel when the radio burst of interest is present in the radio dynamic spectrum, thereby creating a four-dimensional (4D) spectrotemporal image cube. From the 4D cube we are able to derive the spectrotemporal variation intrinsic to this radio source of interest by isolating its flux from all other sources present on the solar disk in the spatial domain, resulting in a spatial resolved, or “vector” radio dynamic spectrum of the source (Figure \[fig-radio-spec-imaging\](E) and (F)). This technique was first introduced by @2015Sci...350.1238C based on VLA dynamic spectroscopic imaging data, and was subsequently applied in a number of recent studies with VLA data [@2017ApJ...848...77W; @2018ApJ...866...62C]. A similar approach is discussed in a recent study by @2017SoPh..292..168M based on data from the Murchison Widefield Array. The resulting vector dynamic spectra show clearer features of the radio bursts than the cross-power dynamic spectra obtained at short baselines (which are a proxy for the total-power dynamic spectra; Figure \[fig-radio-spec-imaging\](A)). The improvement, however, is not substantial, which is consistent with the imaging results in which this burst source is shown as the dominant emission on the solar disk. To highlight the fine structure of the bursts, we further enhance the vector dynamic spectrum by using the contrast-limited adaptive histogram equalization technique [@Pizer1987], shown in Figures \[fig-radio-spec-imaging\](G) and (H).
Bursts 1 and 2 share similar spectrotemporal features. They contain at least one emission lane that starts with a positive drift rate toward higher frequency ($d\nu/dt>0$, sometimes referred to in the literature as “reverse drift” because “normal drift” bursts show negative frequency drifts). It then turns over rather smoothly at the highest frequency point and drifts toward lower frequency with a negative frequency drift rate ($d\nu/dt<0$). The total frequency variation $\Delta \nu_{\rm tot}/\nu$ can be up to 30%. Burst 2 undergoes two repeated cycles of positive-to-negative frequency drift. At least three distinct emission lanes are clearly visible (denoted as “L1”, “L2”, and “L3” in Figure \[fig-radio-spec-imaging\](F)) with two additional faint lanes that can only been distinguished in the enhanced dynamic spectrum (arrows in Figure \[fig-radio-spec-imaging\](H)). Although the three bright emission lanes of Burst 2 occur close together in time, they differ in their intensity, peak emission frequency, frequency drift rate, and frequency turnover time $t_{\rm o}$ (defined as the time when the emission frequency reaches the highest value and the frequency drift rate $\dot{\nu}$ goes to nearly zero; it is indicated by orange arrows in Figure \[fig-radio-spec-imaging\](F) and (H)). The average instantaneous frequency bandwidth $\Delta \nu$ of the emission lanes is about 60–100 MHz, corresponding to a relative frequency bandwidth $\Delta \nu/\nu\approx 6\%$.
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More detailed inspection of the dynamic spectral features of the stronger burst (Burst 2; based on the full 50 ms cadence data) reveals multitudes of very short, subsecond-scale fine structures on each emission lane (Figure \[fig-wavelet\](A)). Figures \[fig-wavelet\](B–E) provide an enlarged view of four segments of the emission lanes for Burst 2 (labelled as “S1”, “S2”, “S3”, “S4” in Figure \[fig-wavelet\](A)), which have been detrended to remove their overall frequency drift pattern. The bursts appear to oscillate quasi-periodically in their emission frequency around the central “ridge” of the emission lane. 1[We use a damped oscillation profile $$\delta\nu(t) = \delta\nu_0\exp(-t/\tau_A)\sin\bigg[\frac{2\pi t}{P/(1-t/\tau_P)^3}\bigg]$$ to fit the four segments (Figure \[fig-wavelet\](B–E)). The oscillations have an amplitude of $\delta\nu_0\approx10$–$30\ \rm{MHz}$ (or a relative amplitude of $\delta\nu/\nu$ of $\sim$1–2$\%$), period of $P\approx0.3$–1.0 s, and damping times of $\tau_A\approx0.5$–5 s in amplitude and $\tau_P \gtrsim$ 30 s in period.]{} Wavelet analysis of such oscillation patterns in emission frequency confirms that the oscillations display very short, subsecond-scale periods ranging from $\sim$0.3–1.0 s (Figure \[fig-wavelet\](F–I)).
Radio imaging of each pixel in the dynamic spectrum where the bursts are found provides key information on the spatial variation of the radio source as a function of time and frequency. For each image at a given frequency $\nu$ and time $t$, we fit the source with a 2D Gaussian function and determine the source centroid $I_{\rm pk}(\theta, \phi, \nu, t$), where $I_{\rm pk}$ is the peak intensity, and $\theta$ and $\phi$ are the centroid position in helioprojective longitude and latitude. As shown in several previous studies, the uncertainty of the centroid location for unresolved, point-like sources (which is likely the case for the coherent bursts under study) is determined by $\sigma\approx\theta_{\rm FWHM}/\mathrm{S/N}\sqrt{8\ln 2}$, where $\theta_{\rm FWHM}$ is the FWHM beamwidth and S/N is the ratio of the peak flux to the root-mean-square noise of the image [@1988ApJ...330..809R; @1997PASP..109..166C; @2018ApJ...866...62C]. In our data, typical values are $\theta_{\rm FWHM}\approx 30''$ and $\mathrm{S/N}\gtrsim20$, which give $\sigma\lesssim0.6''$. However, as discussed later in Section \[sec-discussion\], the bursts are likely associated with fundamental plasma radiation, which is known to be prone to scattering effects as the radiation propagates through the inhomogeneous corona toward the observer [@1994ApJ...426..774B; @2017NatCo...8.1515K; @2018ApJ...856...73C; @2018SoPh..293..132M]. Therefore, the estimate of uncertainty given above should only be considered as a lower limit. In fact, by obtaining the centroid locations of all frequency-time pixels on the emission lane within a small time period ($\sim$0.5 s) and frequency range ($\sim$50 MHz), we find that they are distributed rather randomly within an area of a FWHM size of $\sim 2''\times2''$. Hence we estimate the actual position uncertainty of the centroids as $\sigma\approx1''$.
We focus on Burst 2 for detailed investigations of the spatial, temporal, and spectral variation of the source centroid since it has the best S/N. For each emission lane, we first extract all time and frequency pixels where the intensity exceeds 50% of its peak intensity. An example for such a selection for lane L1 of Burst 2 is shown in Figure \[fig-L1-3d\](A) enclosed by the white contour. Figure \[fig-L1-3d\](B) shows the resulting centroid positions as a function of frequency (colored dots from blue to red in increasing frequency) for emission lane L1. To further improve positional accuracy and reduce cluttering in the figure, each dot in the plot represents the average position for centroids at all frequency pixels across the emission lane (that have an intensity above 50% of the peak) for a given time $t$, with the color representing their mean frequency. The background of Figure \[fig-L1-3d\](B) is the HMI photospheric magnetogram shown in grayscale, overlaid with the AIA 1600 Å image. The latter clearly shows the double flare ribbons in red.
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Figure \[fig-L1\_time\](A) shows the same distribution of radio centroids derived from emission lane L1 as in Figure \[fig-L1-3d\](B), but colored in time instead. It displays an evident motion in projection: the radio source first moves toward the flare ribbon as frequency increases (blue to red color in Figure \[fig-L1-3d\](B) and blue to white color in Figure \[fig-L1\_time\](A)) until it reaches the maximum frequency at the lowest height, and then bounces back in the opposite direction away from the ribbon as frequency decreases (red to blue color in Figure \[fig-L1-3d\](B) and white to red color in Figure \[fig-L1\_time\](A)). The average speed in projection is $\sim$1000–2000 km s$^{-1}$, which is typical for propagating Alfvén or fast-mode magnetosonic waves in the low corona [e.g., @2013ApJ...776...58N; @2018ApJ...864L..24L].This is a strong indication that the radio emission is associated with a propagating Alfvén or fast-mode MHD disturbance in a magnetic tube in the close vicinity of the flare ribbon. As discussed in Section \[sec-overview\], the presence of ubiquitous MHD disturbances in the flaring region is strongly implicated by the observation of large-scale, fast 1[PDs]{} observed by *SDO*/AIA at about the same time.
Motion of Radio source motion in 3D {#sec-3d}
-----------------------------------
In order to place the location of the radio centroids into the physical context of the flare, we perform potential field extrapolation based on the *SDO*/HMI LOS photospheric data right after the flare peak at 17:00 UT [@2014SoPh..289.3549B; @2014SoPh..289.3483H] to derive the coronal magnetic field. Selected magnetic field lines from the extrapolation results are shown in Figure \[fig-L1-3d\](B) for regions around the location of the radio burst centroids and the postflare arcades. It is shown that the spatial distribution of the positions of the radio centroids at different frequencies tends to follow the magnetic field lines (yellow) rooted around the northern flare ribbon, with its higher-frequency end located closer to the ribbon. This is consistent with the expectation for plasma radiation, in which emission occurs at a higher emission frequency in regions with higher plasma density, which are typically located at lower coronal heights ($\nu\approx s\nu_{pe}\approx 8980s\sqrt{n_e}$ Hz, where $s$ is the harmonic number, $\nu_{\rm pe}$ is the electron plasma frequency, and $n_e$ is the local electron density in cm$^{-3}$).
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Since the emission is highly polarized, it is reasonable to assume fundamental plasma radiation as the emission mechanism responsible (i.e., harmonic number $s$=1 and $\nu \approx \nu_{\rm pe}$). In this case, the emission frequency $\nu$ of the radio source centroid $I_{\rm pk}(\theta, \phi, \nu, t$) can be directly translated into the plasma density of the source $n_e$. By further assuming a coronal density model $n_e(h)$ where $h$ is the coronal height, we can thus map the measured centroid locations in 2D projection to three dimensional (3D) locations in the corona, i.e., from $I_{\rm pk}(\theta, \phi, \nu, t$) to $I_{\rm pk}(\theta, \phi, h, t$). A similar practice has been used in , and more recently @2017ApJ...848...77W, for deriving 3D trajectories of [dm-$\lambda$]{} fiber bursts in the corona. Here we adopt a barometric density model with an exponential form $$n_\mathrm{e}(h)=n_{e0}\,\mathrm{exp}\left(-\frac{h-h_0}{L_n}\right)
\label{equ-density-model}$$ where $h$ is the height above the solar surface, $L_n$ is the density scale height, and $n_{e0}$ is the density at a reference height $h_0$. Such a density model describes the variation in density for an isothermal, plane-parallel atmosphere under hydrostatic equilibrium (e.g., @2005psci.book.....A), and has been widely used in the literature as a zero-order approximation for estimating the coronal heights of various solar coherent radio bursts . For simplicity, we fix the parameters $n_{e0}$ and $h_0$ to be $\sim 3\times 10^{10}\ \mathrm{cm^{-3}}$ and $\sim$2000 km at the top of the chromosphere according to the VAL model [@1981ApJS...45..635V], and investigate the effect of different choices of $L_n$ on the resulting 3D distribution of the radio source centroids.
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Figures \[fig-L1-3d\](C) and (D) demonstrate the inferred 3D distributions of the radio source centroids with different choices of $L_n$ from 10 Mm to 70 Mm, viewed from the eastern and northern sides of the AR respectively. Each set of 3D centroid positions at a given $L_n$ is shown as dots of the same color (from red to blue in increasing $L_n$). It is obvious from the figure that the choice of a greater value of $L_n$ yields a more stretched distribution of the radio centroids in height, and vice versa. Such a proportionality between the vertical extent $h_{\rm tot}$ of the radio sources and $L_n$ is straightforward to find if we combine the barometric density model (Eq. \[equ-density-model\]) with the frequency–density relation for plasma radiation $\nu \propto \sqrt{n_e}$, which gives $h_{\rm tot} \approx 2L_n\Delta\nu_{\rm tot}/\nu$, where $\Delta\nu_{\rm tot}$ is the total frequency width of the radio burst determined from the dynamic spectrum. More importantly, different choices of $L_n$ affect how the radio source centroids are distributed with regard to the extrapolated magnetic field lines in 3D. For small $L_n$ values, the centroids tend to be distributed across the field lines within a small range of vertical heights, while for $L_n$ values in the intermediate range ($\sim$35–50 Mm), the spatial extent of the centroids tends to agree with the direction of the extrapolated field lines. As discussed earlier, the temporal evolution of a radio source (1000–2000 km s$^{-1}$ in projection) is consistent with a physical motion of the emission source at Alfvénic or fast-mode magnetosonic speed. Because the radio source appears to be closely associated with the flare ribbon both spatially and temporally (see Section \[sec-overview\]), we assume that the radio source moves along (or within a small angle with respect to) the reconnected magnetic loops that link to the flare ribbon. In this case, the corresponding $L_n$ values fall into the $\sim$35–50 Mm range. For subsequent discussions, we will adopt $L_n=40$ Mm, with the understanding that this parameter is not very well determined due to the inherent limitations of magnetic field extrapolation the uncertainty on the exact direction of propagation of the radio source in 3D, and it may vary from burst to burst.
Figure \[fig-L1\_time\](B)–(D) shows the inferred 3D spatial and temporal evolution of the radio centroids of emission lane L1 after adopting the coronal density model with $L_n=40$ Mm, viewing from the east side of the AR. It is clear that the radio source first moves downward along a converging magnetic field tube (panels B and C) and then bounces backward after it reaches the lowest altitude (or highest frequency). We also extend the same analysis to emission lanes L2 and L3 of Burst 2. The results show a similar spatiotemporal evolution of the radio source centroids as lane L1 (Figures \[fig-L2\] and \[fig-L3\]). 1[We caution that the absolute height of the radio source as well as the point of reflection, however, depends strongly on the selection of parameters in the coronal density and magnetic field model adopted here]{}, which may very well be different for radio bursts propagating along different flare loops. Therefore, the 3D source evolution shown in Figures \[fig-L1\_time\]–\[fig-L3\] should only be considered as a qualitative representation.
DISCUSSIONS {#sec-discussion}
===========
We briefly summarize the observational results in the previous section as follows.
1. The radio bursts of interest appeared during the late impulsive phase of a C7.2 two-ribbon solar flare that was associated with a failed filament eruption, when large-scale, fast-propagating EUV disturbances were observed throughout the flare region.
2. The location of the radio source coincides with the northern flare ribbon and HXR footpoints. 1[In addition, the radio source appears to show close spatial and temporal association with transient (E)UV brightenings on the ribbon.]{}
3. The bursts have a high brightness temperature of $>10^7$ K and are completely polarized in the sense of $o$ mode.
4. The bursts consist of multiple emission lanes that exhibit a low–high–low frequency drift pattern in the radio dynamic spectrum with a moderate relative frequency drift rate of $\dot{\nu}/\nu \lesssim 0.2 \rm{s}^{-1}$, which is typical for intermediate drift bursts in the decimetric wavelength range.
5. Imaging at all time and frequency pixels where the bursts are present shows that the radio source propagates at a speed of 1–2 Mm s$^{-1}$ in projection. The low–high–low frequency drift pattern corresponds to the source firstly moving downward along the flaring loop before it reaching the lowest point and bouncing back upward.
6. Some of the emission lanes consist of multitudes of subsecond-period oscillations in emission frequency with an amplitude of 1[$\delta\nu/\nu\approx$1–2$\%$]{}.
What is the nature of the propagating radio source that is reflected at or near the flare ribbon? First, it is most likely associated with fundamental plasma radiation, which is due to the nonlinear conversion from plasma Langmuir waves induced by the presence of nonthermal electrons. This is because that the bursts have a narrow frequency bandwidth ($\delta \nu/\nu \approx 6\%$) and fast temporally varying features, and are nearly 100% polarized. Second, the propagation speed of the emission source (1–2 Mm s$^{-1}$ in projection) is too slow for electron beams emitting type III bursts (which usually propagate at 0.1$c$–0.5$c$, see, e.g., @2013ApJ...763L..21C [@2018ApJ...866...62C; @2017ApJ...851..151M]), but 1[likely]{} too fast for slow-mode magnetosonic waves, 1[unless the temperature in the source reaches over $\sim$50 MK]{}. The most probable candidate for the radio-emission-carrying disturbance is thus Alfvénic or fast-mode magnetosonic waves, which propagate at $\sim$1–4 Mm s$^{-1}$ under typical coronal conditions. The Alfvénic or fast-mode waves can be excited by a broadband driver, such as the impulsive flare energy release, and propagate outward from the site of energy release. For fast-mode waves to achieve focused, field-aligned energy transport, an overdense magnetic tube would be required to act as a waveguide , which, in our case, can be the freshly reconnected flaring loops that connect to the flare ribbons. The observed reflection of the waves at or near the flare ribbon may be due to sharp gradients at and/or below the transition region . However, this is less clear from our observations regarding the physical connection between the nonthermal electrons (responsible for the production of Langmuir waves) and the MHD waves: the energetic electrons could be accelerated locally within the waves by a variety of means, including acceleration by parallel electric field, turbulence, or a first-order Fermi process with the wavefront acting as a moving mirror (e.g., @2008ApJ...675.1645F), or they could originate from an acceleration site elsewhere (e.g., at the reconnection site or the top of the flare loops) but be trapped with the propagating MHD waves.
It is particularly intriguing that some of the emission lanes show fast, subsecond-scale quasi-periodic oscillations in the emission frequency with an amplitude of $\delta\nu/\nu\approx$1[1–2$\%$]{}. Under the plasma radiation scenario, $\delta\nu/\nu$ can be directly translated into small density perturbations of $\delta n_e/n_e\approx 2\delta\nu/\nu\approx $1[2–4$\%$]{}. 1[If these small-amplitude oscillations in frequency can be interpreted as weak density perturbations associated with the propagating waves, the scenario of fast-mode magnetosonic mode scenario would be more probable, because pure Alfven modes are incompressible.]{} We note that such small density disturbances are hardly detectable by current EUV or SXR imaging instrumentation, mainly because the resulting small fluctuation level in the EUV/SXR intensity $\delta I/I\lesssim 2\delta n_e/n_e\approx 4\%$ is very difficult to detect against the background. In addition, the subsecond periodicity of the density perturbations is at least an order of magnitude below the time cadence of the current EUV/SXR imaging instrumentation (e.g., 12 s for *SDO*/AIA). We note, however, that subsecond-scale oscillations in the solar corona have been reported in the literature based on non-imaging radio or X-ray light curves or dynamic spectra during flares . @1987SoPh..111..113A summarized the possible mechanisms into three categories: (1) quasi-periodic injections of nonthermal electrons, (2) fast cyclic self-organizing systems of plasma instabilities associated with the wave–particle or wave–wave interaction processes, and (3) MHD oscillations. While we cannot completely rule out the other possibilities, the observed oscillations in radio emission frequency (or plasma density), combined with the fast-moving radio source with a speed characteristic of Alfvénic or fast-mode waves, are a strong indication of a weakly compressible, propagating MHD wave packets in the flaring loops that cause localized quasi-periodic modulations of the plasma density along their way.
The spatial scale of the radio-emitting fast-wave packages can be inferred from the instantaneous frequency bandwidth $\Delta \nu/\nu$ of individual emission lanes based on the plasma radiation scenario: $\Delta L = 2L_n(\Delta \nu/\nu)$, where $L_n=n_e/(dn_e/dl)$ is the density scale height. For a magnetic loop under hydrostatic equilibrium, the density gradient is along the vertical direction $z$, and the density scale height is $L_n=n_e/(dn_e/dl)=2k_BT/(\mu m_Hg)\approx46 T_{\rm MK}$ Mm, where $g$ is the gravitational acceleration near the solar surface, $m_H$ is the mass of the hydrogen atom, $T_{\rm MK}$ is the coronal temperature in megakelvin, and $\mu\approx 1.27$ is the mean molecular weight for typical coronal conditions [@2005psci.book.....A]. In this case, a frequency bandwidth of $\Delta \nu/\nu \approx 6\%$ implies a vertical extent of the source of $\Delta L_z \approx 5.5T_{\rm MK}$ Mm. Such an estimate of the source size is not inconsistent with the distribution on a small scale of a few megameters of the radio source centroids across all frequencies on the emission lane at a given time in the plane of the sky $\Delta L_{\parallel}$, although the latter is complicated by the scattering of the radio waves due to coronal inhomogeneities (see discussions in Section \[sec-spec-imaging\]). It is interesting to note that this size estimation is about an order of magnitude smaller than the apparent size of each radio image (with a half-power-full-maximum size of $\sim$30–50 Mm; see Figure \[fig-radio-spec-imaging\](D)). Such an extended radio image can likely be attributed to the angular broadening of the radio source caused by random scattering of the radio waves traversing the inhomogeneous corona [@1994ApJ...426..774B]. Indeed, @1994ApJ...426..774B estimated an angular broadening of a few tens of arcseconds at our observing frequency and source longitude, which is of the same order of magnitude as our apparent source size.
The wavelength associated with the subsecond-period oscillations can be estimated via $\lambda\approx v_{\rm p}P$, where $v_{\rm p}$ is the phase speed of the waves, taken to be of the same order of magnitude as the observed wave speed $\sim$3 Mm s$^{-1}$ (after assuming an inclination angle of 60$^\circ$ inferred from the extrapolation of the magnetic field, see Section \[sec-3d\]) that presumably represents the group speed of the wave packet $v_{\rm g}$ (see, e.g., @1984ApJ...279..857R for discussions regarding the relation between $v_{\rm p}$ and $v_{\rm g}$), and $P$ is the wave period, taken to be $\sim$0.5 s from the observed periods of the density fluctuations (see Figure \[fig-wavelet\]). Therefore, the wavelength of the oscillations is estimated as $\lambda\approx 1.5$ Mm, much smaller than the size of the propagating radio source ($\Delta L > \Delta L_z \approx 5.5T_{\rm MK} $ Mm). We therefore argue that each radio source is likely a propagating MHD wave packet that consists of multiple short-period oscillations.
During each burst period, multiple emission lanes are present in the radio dynamic spectrum with almost synchronous frequency drift behavior (which is particularly clear for Burst 2; see Figure \[fig-radio-spec-imaging\](H)). Imaging results of the different emission lanes suggest that they are all located at the same site and share very similar spatiotemporal behavior in projection, but show subtle differences (see Figures \[fig-L1\_time\](A), \[fig-L2\](C), and \[fig-L3\](C)). Their different emission frequencies, however, imply that the corresponding propagating disturbances have different plasma densities. Some other types of solar [dm-$\lambda$]{} bursts, in particular, zebra-pattern bursts (ZBs), also display multiple emission lanes. One leading theory for ZBs attributes the observed multiple lanes to radio emission at the plasma upper-hybrid frequency $\nu_{\rm uh}$ that coincides with harmonics of the electron gyrofrequency $\nu_{\mathrm{ce}}$, i.e., $\nu\approx\nu_{\rm uh}\approx (\nu_{\mathrm{pe}}^2+\nu_{\mathrm{ce}}^2)^{1/2}\approx s\nu_{\mathrm{ce}}$ . However, unlike the ZBs, the frequency spacing between different emission lanes in this burst is irregular and varies in time. Moreover, although the frequency turnover times of different emission lanes $t_{\rm o}$ are very close to each other, they differ by $\sim$0.5–0.8 s (orange arrows in Figure \[fig-radio-spec-imaging\](F) and (H)) and does not show a systematic lag in frequency as is usually present in ZBs [@2007SoPh..246..431C; @2007SoPh..241..127K; @2013ApJ...777..159Y]. Therefore, we argue that the different emission lanes are not due to harmonics of a particular plasma wave mode. Instead, they are related to different wave packets, which are triggered by the same impulsive energy release event, propagating in magnetic flux tubes with different plasma properties.
{width="\textwidth"}
The schematic in Figure \[fig-cartoon\] summarizes our interpretation of the observed radio bursts in terms of propagating MHD wave packets that contain multiple subsecond-period oscillations within the context of the filament eruption and two-ribbon flare. As introduced in Section \[sec:intro\], subsecond-period MHD waves may 1[be a viable mechanism]{} responsible for transporting a substantial amount of the magnetic energy released in the corona downward to the lower atmosphere, resulting in intense plasma heating and/or particle acceleration. Let us consider the scenario of fast-mode MHD waves guided by dense magnetic flux tubes as an example . The kinetic energy flux associated with propagating MHD waves can be estimated as $F_K\approx\frac{1}{2}\rho\delta v^2 v_g$ [@2007Sci...317.1192T; @2014ApJ...795...18V], where $\rho\approx m_Hn_e$ is the mass density, $\delta v$ is the amplitude of the velocity perturbation, and $v_g$ is the group speed of the propagating MHD wave. Estimates for both $\rho$ and $v_g$ can be conveniently obtained from our observations of the radio emission frequency and the radio source motion. Although the velocity perturbation $\delta v$ is not directly measured by our observations, it is intimately related to the observed density perturbation amplitude $\delta\rho \approx m_H \delta n_e$ through the continuity equation in the regime of small perturbation: $$\frac{d(\delta\rho)}{dt}=-\rho_0 \nabla \cdot \delta v,$$ It is beyond the scope of the current study to examine the detailed relation for all possible wave modes propagating in coronal loops with different density profiles. Nevertheless, under typical coronal conditions, it has been shown by previous studies that, under typical coronal conditions, $\delta v/v_g$ is of the same order of magnitude as $\delta n_e/n_e$ for fast-mode MHD waves propagating along dense coronal loops . The latter is found to be $\delta n_e/n_e\approx2\delta\nu/\nu\approx$1[2–4$\%$]{}. Following these assumptions, we estimate the energy flux as 1[(2–8)$\times 10^8$]{} erg s$^{-1}$ cm$^{-2}$, with $n_e\approx 2\times 10^{10}$ cm$^{-3}$, $\delta v/v_g \approx $1[2–4$\%$]{}, and $v_g\approx 3$ Mm s$^{-1}$.
{width="\textwidth"}
Is the estimated energy flux carried by the MHD disturbances energetically important in this flare? The energy flux required to power flares can be inferred using a variety of observational diagnostic methods including broadband imaging of flare ribbons in WL and UV [@2007ApJ...656.1187F; @2012ApJ...752..124Q; @2013ApJ...770..111L], as well as HXR spectroscopic and imaging observations of flare footpoints [@2007ApJ...656.1187F]. Here we adopt the method developed by @2012ApJ...752..124Q to estimate the energy flux 1[needed to account for flare heating]{} based on *SDO*/AIA 1600 Å UV observations of the flare ribbons. The energy flux $F_i(t)$ of flare heating is related to UV 1600 Å ribbon brightening at pixel $i$ as $$F_i(t)=\lambda I_i^{\rm pk} \exp\left[-\frac{(t-t_i^{\rm pk})^2}{2\tau_i^2}\right] \mathrm{erg\ s^{-1}\ cm^{-2}},$$\[eq-energy\]
where the exponential term is the Gaussian function used to fit the rise phase of the light curve of the UV count rate that has a characteristic rise time $\tau_i$ and peaks at $t_i^{\rm pk}$, and $\lambda$ is the scaling factor that converts the observed peak UV count rate $I_i^{\rm pk}$ at pixel $i$ (in DN s$^{-1}$ pixel$^{-1}$, where DN is data numbers) to 1[the estimated energy flux responsible for the flare heating (erg s$^{-1}$ cm$^{-2}$), which depends not only on the mechanism of UV radiation upon heating in the lower atmosphere, but also on the instrument response. @2012ApJ...752..124Q and @2013ApJ...770..111L performed detailed modeling studies of loop heating of two flares, and found that $\lambda$ generally lies in the range (2-3)$\times10^5\ \mathrm{erg\ DN^{-1}\ pixel/cm^{-2}}$ to best match the model-computed *GOES* SXR light curves with the observations. Here we take $\lambda \approx 2.7\times10^5\ \mathrm{erg\ DN^{-1}\ pixel/cm^{-2}}$ quoted in @2012ApJ...752..124Q for our order-of-magnitude estimate]{}. We have traced all pixels in AIA 1600 Å UV images that show flare ribbon brightenings, which are shown in Figure \[fig-ribbon\](A) colored by their peak time $t_i^{\rm pk}$ from purple to red. The flare ribbons show an evident separating motion during the impulsive phase of the flare, which is characteristic of two-ribbon flares and has been considered as one of the primary evidence for magnetic-reconnection-driven flare energy release [@2002ApJ...565.1335Q]. The corresponding light curves of the UV count rate for all ribbon pixels are shown in Figure \[fig-ribbon\](B), again colored by their peak time (only the rising portion of the light curve is shown). The UV ribbon brightenings agree very well in time with the *GOES* SXR derivative (thick red curve in Figure \[fig-ribbon\](C)), suggesting that heating of the flare loops is mainly driven by the “evaporation” of the heated chromospheric plasma. The estimated energy flux averaged over all ribbon pixels $\overline{F}(t)$ based on Eq. \[eq-energy\] is shown as the blue curve in Figure \[fig-ribbon\](C). 1[Also shown is the average $\overline{F}(t)$ estimated using only pixels of the northern ribbon (dashed blue curve), with which the radio bursts appear to be associated temporally and spatially (see Figure \[fig-ribbonbrightenings\]).]{} The values are in the range of $10^8$–$10^9$ $\mathrm{erg\ s^{-1}\ cm^{-2}}$, which is typical for *GOES* C-class flares. At the time of the radio burst, the average $\overline{F}(t)$ 1[at the northern ribbon]{} is about 1[$4\times 10^8\ \mathrm{erg\ s^{-1}\ cm^{-2}}$]{}, which is comparable to the estimated energy flux carried by the observed subsecond-period MHD wave packets.
We note, however, taht such coherent-burst-emitting waves can only be observed when the following conditions are met. 1[(1) Flare-accelerated electrons are present in the vicinity of the MHD waves. (2) Conditions are satisfied for inducing nonlinear growth of Langmuir waves and the subsequent conversion to transverse radio waves. (3) The radio waves are emitted within the bandwidth of the instrument (1–2 GHz in our case). (4) The instrument is sensitive enough to distinguish the radio bursts from the background active region and flare emission—the flux density of the bursts is only $\sim 1$ sfu (1 sfu = 10$^4$ Jansky) in our case, which is barely above the noise level of most non-imaging solar radio spectrometers.]{} For these reasons, the radio bursts appear relatively rare, and thus their volume filling factor in the entire flaring region is essentially unknown. 1[Moreover, although possible signatures of wave damping seem to be present in some bursts that we observe (see Figure \[fig-wavelet\](B–E)), which may be due to energy loss during their propagation, the fraction of total energy deposited to the lower solar atmosphere from the waves remains undetermined in this study.]{} However, considering the presence of ubiquitous large-scale fast EUV waves throughout the active region around the same time, it is reasonable to postulate that these short-period waves are also ubiquitously present in the flaring region. If this is the case, these waves may 1[play a role in transporting the released flare energy during the late impulsive phase of this flare, likely alongside the accelerated electrons, and the subsequent heating of the flare ribbons and arcades]{}.
Conclusion {#sec-conclusion}
==========
Here we report radio imaging of propagating MHD waves along post-reconnection flare loops during the late impulsive phase of a two-ribbon flare. This is based on observations of a peculiar type of [dm-$\lambda$]{} radio bursts recorded by the VLA. In the radio dynamic spectrum, the bursts show a low–high–low frequency drift pattern with a moderate frequency drift rate of $\dot{\nu}/\nu \lesssim 0.2$. VLA’s unique capability of imaging with spectrometer-like temporal and spectral resolution (50 ms and 2 MHz) allows us to image the radio source at every pixel in the dynamic spectrum where the burst is present. In accordance with its low–high–low frequency drift behavior, we find that the radio source firstly moves downward toward a flare ribbon before it reaches the lowest height and turns upward. The measured speed in projection is $\sim$1–2 Mm/s, which is characteristic of Alfvénic or fast-mode MHD waves in the low corona. Furthermore, we find that the bursts consist of many subsecond, quasi-periodic oscillations in emission frequency, interpreted as fast oscillations within propagating MHD wave packets. As illustrated in Figure \[fig-cartoon\], these wave packets are likely triggered by the impulsive flare energy release, and subsequently propagate downward along the newly reconnected field lines down to the flare ribbons. From the observed density oscillations and the source motion, we estimate that these wave packets carry an energy flux of 1[(2–8)$\times 10^8$ erg s$^{-1}$ cm$^{-2}$]{}, which is comparable to the average energy flux required for driving the flare heating 1[during the late impulsive phase of the flare estimated from the UV ribbon brightenings. In addition, the radio source seems to show a close spatial and temporal association with the transient brightenings on the flare ribbon]{}. As introduced in Section \[sec:intro\], such subsecond-period MHD waves have long been postulated as an alternative or complementary means for transporting the bulk of energy released in flares alongside electron beams, resulting in strong plasma heating and/or particle acceleration. Here we provide, to the best of our knowledge, the first possible observational evidence for these subsecond-period MHD waves propagating in post-reconnection magnetic loops derived from imaging and spectroscopy data, and demonstrate 1[their possible role in driving plasma heating during the late impulsive phase of this flare event. Future studies are required to, first of all, investigate their presence in other flare events, and moreover, establish whether or not they are energetically important in transporting the released flare energy during different flare phases]{}.
We thank Sophie Musset for her help in producing the *RHESSI* X-ray image. We also thank Tim Bastian, John Wygant, Lindsay Glesener, Kathy Reeves, and Dale Gary for helpful discussions, 1[as well as an anonymous referee who provided constructive comments to improve the paper]{}. The National Radio Astronomy Observatory is a facility of the National Science Foundation (NSF) operated under cooperative agreement by Associated Universities, Inc. This work made use of open-source software packages including CASA [@2007ASPC..376..127M], SunPy , and Astropy . B.C. and S.Y. are supported by NASA grant NNX17AB82G and NSF grant AGS-1654382 to the New Jersey Institute of Technology.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this article we study combinatorial degenerations of minimal surfaces of Kodaira dimension 0 over local fields, and in particular show that the ‘type’ of the degeneration can be read off from the monodromy operator acting on a suitable cohomology group. This can be viewed as an arithmetic analogue of results of Persson and Kulikov on degenerations of complex surfaces, and extends various particular cases studied by Matsumoto, Liedtke and Matsumoto and Hernández Mada. We also study ‘maximally unipotent’ degenerations of Calabi–Yau threefolds, following Kollár and Xu, showing in this case that the dual intersection graph is a 3-sphere.'
address:
- |
Università degli Studi di Padova\
Dipartimento di Matematica “Tullio Levi-Civita”\
Via Trieste, 63\
35121 Padova\
Italia
- |
Università degli Studi di Padova\
Dipartimento di Matematica “Tullio Levi-Civita”\
Via Trieste, 63\
35121 Padova\
Italia
author:
- Bruno Chiarellotto
- Christopher Lazda
bibliography:
- '/Users/Chris/Dropbox/LaTeX/lib.bib'
title: 'Combinatorial degenerations of surfaces and Calabi–Yau threefolds'
---
Introduction
============
Fix a complete discrete valuation ring $R$ with perfect residue field $k$ of characteristic $p>3$ and fraction field $F$. Let $\pi$ be a uniformiser for $R$, and let $X$ be a smooth and projective scheme over $F$. Let $\overline{F}$ be a separable closure of $F$.
\[defmod\] A *model* of $X$ over $R$ is a regular algebraic space $\cur{X}$, proper and flat over $\cur{X}$ over $R$, whose generic fibre is isomorphic to $X$, and whose special fibre is a scheme. We say that a model is semistable if it is étale locally smooth over $R[x_1,\ldots,x_d](x_1\ldots x_r-\pi)$, and strictly semistable if furthermore the irreducible components of the special fibre $Y$ are smooth over $k$.
A major question in arithmetic geometry is that of determining criteria under which $X$ has good or semistable reduction over $F$, i.e. admits a model $\cur{X}$ which is smooth and proper over $R$, or semistable over $R$. In general the question of determining good reduction criteria comes in two flavours.
1. Does there exists a model $\cur{X}$ of $X$ which is smooth over $R$?
2. Given a semistable model $\cur{X}$ of $X$, can we tell whether or not $\cur{X}$ is smooth?
We will refer to the first of these as the problem of ‘abstract’ good reduction, and the second as the problem of ‘concrete’ good reduction. The sorts of criteria we expect are those that can be expressed in certain homological or homotopical invariants of the variety in question. In this article we will mainly concentrate on these problems for minimal smooth projective surfaces over $F$ of Kodaira dimension $0$. These naturally fall into four classes:
- K3 surfaces;
- Enriques surfaces;
- abelian surfaces;
- bielliptic surfaces,
and in each case we have both the abstract and concrete good reduction problem. Note that for this article we will generally use ‘abelian surface’ to mean a surface over $F$ that is geometrically an abelian surface, i.e. we do not necessarily assume the existence of an $F$-rational point (or thus of a group law).
In the analogous complex analytic situation (i.e. that of a semistable, projective degeneration $X \rightarrow \Delta$ over the open unit disc with general fibre $X_t$ a minimal complex algebraic surface with $\kappa=0$) it was shown by Persson [@Per77] and Kulikov [@Kul77] that, under a certain (reasonably strong) hypothesis on the total space $X$ one could quite explicitly describe the ‘shape’ of the special fibre, and that these shapes naturally fall into three ‘types’ depending on the nilpotency index of the logarithm of the monodromy on a suitable cohomology group. Our main result here is an analogue of this result in an ‘arithmetic’ context, namely classifying the special fibre of a strictly semistable scheme over $R$ whose generic fibre is a surface of one of the above types, in terms of the monodromy operator on a suitable cohomology group. The exact form of the theorem is somewhat tricky to state simply, so here we content ourselves with providing a rough outline and refer to the body of the article for more detailed statements.
\[vague\]Let $X/F$ be a minimal surface with $\kappa=0$, and let $\ell$ be a prime (possibly equal to $p$). Let $\mathscr{X}/R$ be a ‘minimal’ model of $X$ in the sense of Definition \[minmod\]. Then the special fibre $Y$ of $\mathscr{X}$ is ‘combinatorial’, and moreover there exists an ‘$\ell$-adic local system’ $V_\ell$ on $X$ such that $Y$ is of Type I, II or III as the nilpotency index of a certain monodromy operator on $H^i(X,V_\ell)$ is $1$, $2$ or $3$ respectively.
1. We will not give the definition of ‘combinatorial’ surfaces here, see Definitions \[crdk\], \[crde\], \[crda\] and \[crdb\].
2. When $\mathrm{char}(F)=0$ or $\mathrm{char}(F)=p\neq \ell$ then the local system $V_\ell$ is a $\Q_\ell$-étale sheaf on $X$, and the corresponding cohomology group is $H^i_{\et}(X_{\overline{F}},V_{\ell})$. This is an $\ell$-adic representation of $G_F$, de Rham when $\ell=p$ and $\mathrm{char}(F)=0$, and hence has a monodromy operator attached to it.
3. When $\mathrm{char}(F)=p=\ell$ then the local system $V_\ell=V_p$ is an overconvergent $F$-isocrystal, and the corresponding cohomology group is a certain form of rigid cohomology $H^i_\rig(X/\rk,V_p)$. This is a $\pn$-module over the Robba ring $\rk$ and hence has a monodromy operator by the $p$-adic local monodromy theorem. For more details on $p$-adic cohomology in equicharacteristic $p$ case see §\[rpad\].
Certain types of results of this sort have been studied before, for example by Matsumoto in [@Mat15] (for $\mathrm{char}(F)\neq \ell$ and $X$ a K3 surface), Liedtke and Matsumoto in [@LM14] ($\mathrm{char}(F)=0$, $\ell\neq p$ and $X$ K3 or Enriques), Hernández-Mada in [@HM15] ($\mathrm{char}(F)=0$, $\ell=p$ and $X$ K3 or Enriques), and Pérez Buendía in [@PB14] ($\mathrm{char}(F)=0$, $\ell=p$ and $X$ K3), and our purpose here is partly to unify these existing results into a broader picture, and partly to fill in various gaps, for example allowing $\ell=p=\mathrm{char}(F)$ in the case of K3 surfaces. It is perhaps worth noting that even treating the case of abelian surfaces is not quite as irrelevant as it may seem (given the rather well-known results on good reduction criteria for abelian varieties) since our result describes the possible shape of the special fibre of a *proper*, but not necessarily smooth model. We also relate these shapes to the more classical description of the special fibre of the Néron model, at least after a finite base change (Proposition \[typerank\]).
In each case (K3, Enriques, abelian, bielliptic) the proof of the theorem is in two parts. The first consists of showing that the special fibre $Y$ is combinatorial, this uses coherent cohomology and some basic (logarithmic) algebraic geometry. The second then divides the possible shapes into types depending on the nilpotency index of a certain monodromy operator $N$, this uses the weight spectral sequence and the weight monodromy conjecture (which in all cases is known for dimensions $\leq2$). Although we do not use it explicitly, constantly lurking in the background here is a Clemens–Schmid type exact sequence of the sort considered in [@CT14]. Unfortunately, while the structure of the argument in all 4 cases is similar, we were not able to provide a single argument to cover all of them, hence parts of this article may seem somewhat repetitive.
The major hypothesis in the theorem is ‘minimality’ of the model $\mathscr{X}$, which is more or less the assumption that the canonical divisor $K_{\mathscr{X}}$ of $\mathscr{X}$ is numerically trivial. For K3 surfaces one expects that such models exist (at least after a finite base change), and Matsumoto in [@Kul77] showed that this is true if the semistable reduction conjecture is true for K3 surfaces. For abelian surfaces, this argument adapts to show that one does always have such a model after a finite base change (Theorem \[concab\]), however, for Enriques surfaces there are counterexamples to the existence of such models (see [@LM14]) and it seems likely that the same true for bielliptic surfaces. Unfortunately, the methods used by Persson, Kulikov et al. to describe the special fibre when one does not necessarily have these ‘minimal models’ do not seem to be at all adaptable to the arithmetic situation.
Finally, we turn towards addressing similar questions in higher dimensions by looking at certain ‘maximally unipotent’ degenerations of Calabi–Yau threefolds. The inspiration here is the recent work of Kollár and Xu in [@KX15] on log Calabi–Yau pairs, using recently proved results on the Minimal Model Program for threefolds in positive characteristic (in particular the existence of Mori fibre spaces from [@BW14]). The main result we obtain (Theorem \[CY3\]) is only part of the story, unfortunately, proceeding any further (at least using the methods of this article) will require knowing that the weight monodromy conjecture holds in the given situation, so is only likely to be currently possible in equicharacteristic. A key part of the proof uses a certain description of the homotopy type (in particular the fundamental group) of Berkovich spaces, which forces us to restrict to models $\mathscr{X}/R$ which are schemes, rather than algebraic spaces. As the example of K3 surfaces shows, however, any result concerning the ‘abstract’ good reduction problem is likely to involve algebraic spaces, and will therefore require methods to handle this case.
Notation and conventions {#notation-and-conventions .unnumbered}
------------------------
Throughout $k$ will be a perfect field of characteristic $p>3$, $R$ will be a complete DVR with residue field $k$ and fraction field $F$, which may be of characteristic $0$ or $p$. We will choose a uniformiser $\pi$ for $F$, and let $\overline{F}$ denote a separable closure. We will denote by $q$ some fixed power of $p$ such that $\F_q\subset k$.
A variety over a field will be a separated scheme of finite type, and when $X$ is proper and $\mathscr{F}$ is a coherent sheaf on $X$ we will write $$h^i(X,\mathscr{F})=\dim H^i(X,\mathscr{F})\;\;\;\;\text{and}\;\;\;\;\chi(X,\mathscr{F})=\sum_i (-1)^i h^i(X,\mathscr{F}).$$ We will also write $\chi(X)=\chi(X,\mathcal{O}_X)$, since we always mean coherent Euler-Poincaré characteristics (rather than topological ones) this should not cause confusion.
Unless otherwise mentioned, a surface over any field will always mean a smooth, projective and geometrically connected surface. A ruled surface of genus $g$ is a surface $X$ together with a morphism $f:X\rightarrow C$ to a smooth projective surface $C$ of genus $g$, whose generic fibre is isomorphic to $\P^1$. If we let $F$ denote a smooth fibre of $f$ then an $n$-*ruling* of $f$ (for some $n\geq 1$) will be a smooth curve $D \subset X$ such $D\cdot F=n$, a $1$-ruling will be referred to simply as a ruling.
Review of $p$-adic cohomology in equicharacteristic {#rpad}
===================================================
In this section we will briefly review some of the material from [@LP16] on $p$-adic cohomology when $\mathrm{char}(F)=p$, and explain some of the facts alluded to in the introduction, in particular the existence of monodromy operators. We will therefore let $W=W(k)$ denote the ring of Witt vectors of $k$, $K$ its fraction field, and $\sigma$ the $q$-power Frobenius on $W$ and $K$. In this situation, we have an isomorphism $F\cong k\lser{\pi}$ where $\pi$ is our choice of uniformiser. We will let $\rk$ denote the Robba ring over $K$, that is the ring of series $\sum_i a_it^i$ with $a_i\in K$ such that:
- for all $\rho<1$, $\norm{a_i}\rho^i\rightarrow 0$ as $i\rightarrow \infty$;
- for some $\eta<1$, $\norm{a_i}\eta^i\rightarrow 0$ as $i\rightarrow -\infty$.
In other words, it is the ring of functions convergent on some semi-open annulus $\eta\leq \norm{t}<1$. The ring of integral elements $\rk^{\mathrm{int}}$ (i.e. those with $a_i\in W$) is therefore a lift of $F$ to characteristic $0$, in the sense that mapping $t\mapsto \pi$ induces $\rk^{\mathrm{int}}/(p)\cong F$. We will denote by $\sigma$ a Frobenius on $\rk$, i.e. a continuous $\sigma$-linear endomorphism preserving $\rk^{\mathrm{int}}$ and lifting the absolute $q$-power Frobenius on $F$, we will moreover assume that $\sigma(t)=ut^q$ for some $u\in (W\pow{t}\otimes_W K)^\times$. The reader is welcome to assume that $\sigma(\sum_i a_it^i)=\sum_i \sigma(a_i)t^{iq}$. Let $\partial_t:\rk\rightarrow \rk$ denote the derivation given by differentiation with respect to $t$.
\[pnm1\] A $\pn$-module over $\rk$ is a finite free $\rk$-module $M$ together with:
- a connection, that is a $K$-linear map $\nabla:M\rightarrow M$ such that $$\nabla(rm)=\partial_t(r)m+r\nabla(m)\;\;\;\;\text{for all}\;\; r\in \rk \;\; \text{and} \;\;m\in M;$$
- a horizontal Frobenius $\varphi:\sigma^*M:=M\otimes_{\rk,\sigma} \rk \isomto M$.
Then $\pn$-modules over $\rk$ should be considered as $p$-adic analogues of Galois representations, for example, they satisfy a local monodromy theorem (see [@Ked04a]) and hence have a canonical monodromy operator $N$ attached to them (see [@Mar08]). More specifically, the connection $\nabla$ should be viewed as an analogue of the action of the inertia subgroup $I_F$ and the Frobenius $\varphi$ the action of some Frobenius lift in $G_F$. The analogue for $\pn$-modules of inertia acting unipotently (on an $\ell$-adic representation for $\ell\neq p$) or of a $p$-adic Galois representation being semistable (when $\mathrm{char}(F)=0$) is therefore the connection acting unipotently, i.e. there being a basis $m_1,\ldots,m_n$ such that $\nabla(m_i)\in \rk m_1+\ldots+\rk m_{i-1}$ for all $i$. The analogue of being unramified or crystalline for a $\pn$-module $M$ is therefore the connection acting trivially, or in other words $M$ admitting a basis of horizontal sections. We call such $\pn$-modules $M$ solvable.
Let $\mathcal{E}_K^\dagger\subset \rk$ denote the bounded Robba ring, that is the subring consisting of series $\sum_i a_it^i$ such that $\norm{a_i}$ is bounded, we therefore have the notion of a $\pn$-module over $\ekd$, as in Definition \[pnm1\]. The main purpose of the book [@LP16] was to define cohomology groups $$X\mapsto H^i_\rig(X/\ekd)$$ for $i\geq0$ associated to any $k\lser{\pi}$-variety $X$ (i.e. separated $k\lser{\pi}$-scheme of finite type), as well as versions with compact support $H^i_{c,\rig}(X/\ekd)$ or support in a closed subscheme $Z\subset X$, $H^i_{Z,\rig}(X/\ekd)$. These are $\pn$-modules over $\ekd$ and enjoy all the same formal properties as $\ell$-adic étale cohomology for $\ell\neq p$. Here we list a few of them.
1. If $X$ is of dimension $d$ then $H^i_\rig(X/\ekd)=H^i_{c,\rig}(X/\ekd)=H^i_{Z,\rig}(X/\ekd)=0$ for $i$ outside the range $0\leq i\leq 2d$.
2. (Künneth formula) For any $X,Y$ over $k\lser{\pi}$ we have $$H^n_{c,\rig}(X\times Y/\ekd) \cong \bigoplus_{i+j=n} H^i_{c,\rig}(X/\ekd)\otimes_{\ekd} H^j_{c,\rig}(Y/\ekd)$$ and if $X$ and $Y$ are smooth over $k\lser{\pi}$ we also have $$H^n_{\rig}(X\times Y/\ekd) \cong \bigoplus_{i+j=n} H^i_{\rig}(X/\ekd)\otimes_{\ekd} H^j_{\rig}(Y/\ekd).$$
3. (Poincaré duality) For any $X$ smooth over $k\lser{\pi}$ of equidimension $d$ we have a perfect pairing $$H^i_\rig(X/\ekd) \times H^{2d-i}_{c,\rig}(X/\ekd) \rightarrow H^{2d}_{c,\rig}(X/\ekd)\cong \ekd(-d)$$ where $(-d)$ is the Tate twist which multiplies the Frobenius structure on the constant $\pn$-module $\ekd$ by $q^d$.
4. (Excision) For any closed $Z\subset X$ with complement $U\subset X$ we have long exact sequences $$\ldots \rightarrow H^i_{Z,\rig}(X/\ekd) \rightarrow H^i_\rig(X/\ekd) \rightarrow H^i_\rig(U/\ekd) \rightarrow \ldots$$ and $$\ldots \rightarrow H^i_{c,\rig}(U/\ekd) \rightarrow H^i_{c,\rig}(X/\ekd) \rightarrow H^i_{c,\rig}(Z/\ekd) \rightarrow \ldots.$$
5. (Gysin) For any closed immersion $Z\hookrightarrow X$ of smooth schemes over $k\lser{\pi}$, of constant codimension $c$ there is a Gysin isomorphism $$H^i_{Z,\rig}(X/\ekd)\cong H^{i-2c}_\rig(Z/\ekd)(-c).$$
6. There is a ‘forget supports’ map $H^i_{c,\rig}(X/\ekd)\rightarrow H^i_\rig(X/\ekd)$ which is an isomorphism whenever $X$ is proper over $k\lser{\pi}$.
7. Let $U\subset C$ be an open subcurve of a smooth projective curve $C$ of genus $g$, with complementary divisor $D$ of degree $d$. Then $$\dim_{\ekd} H^1_\rig(U/\ekd)= \begin{cases} 2g-1+d & \text{if } d\geq 1, \\
2g & \text{if }d=0.
\end{cases}$$
8. Let $A$ be an abelian variety over $k\lser{\pi}$ of dimension $g$. Then $H^1_\rig(A/\ekd)$ is (more or less) isomorphic to the contravariant Dieudonné module of the $p$-divisible group $A[p^\infty]$ of $A$, has dimension $2g$, and $$H^i_\rig(A/\ekd)\cong \textstyle{\bigwedge^i} H^1_\rig(A/\ekd).$$
All of these properties were proved in [@LP16]. We may therefore define, for any variety $X/k\lser{\pi}$ $$H^i_\rig(X/\rk) := H^i_\rig(X/\ekd)\otimes_{\ekd}\rk$$ as $\pn$-modules over $\rk$. That the property of a $\pn$-module being solvable (resp. unipotent) really is the correct analogue of a Galois representation being unramified or crystalline (resp. unipotent or semistable) is suggested by the following result.
Let $X/k\lser{\pi}$ be smooth and proper. Then if $X$ has good (resp. semistable reduction) then $H^i_\rig(X/\rk)$ is solvable (resp. unipotent) for all $i\geq0$. If moreover $X$ is an abelian variety, then the converse also holds.
In [@LP16] was also shown an equicharacteristic analogue of the $C_{\mathrm{st}}$-conjecture, namely that when $\mathcal{X}/R$ is proper and semistable, the cohomology $H^i_\rig(X/\rk)$ of the generic fibre can be recovered from the log-crystalline cohomology $H^i_{\log\text{-}\mathrm{cris}}(Y^{\log}/W^{\log})\otimes_W K$ of the special fibre. Our task for the remainder of this section is to generalise this result to algebraic spaces (with fibres that are schemes).
So fix a smooth and proper variety $X/F$ and a semistable model $\mathcal{X}/R$ (see Definition \[defmod\]) for $X$. Let $Y^{\log}$ denote the special fibre of $\mathcal{X}$ with its induced log structure, and let $W^{\log}$ denote $W$ with the log structure defined by $1\mapsto 0$. Then the log-crystalline cohomology $H^i_{\log\text{-}\mathrm{cris}}(Y^{\log}/W^{\log})\otimes_W K$ is a $(\varphi,N)$-module over $K$, i.e. a vector space with semilinear Frobenius $\varphi$ and nilpotent monodromy operator $N$ satisfying $N\varphi=q\varphi N$, and the rigid cohomology $H^i_\rig(X/\rk)$ is a $\pn$-module over $\rk$. There is a fully faithful functor $$(-)\otimes_K \rk: \underline{\mathbf{M}\Phi}^N_K\rightarrow \underline{\mathbf{M}\Phi}^\nabla_{\rk}$$ from the category $\underline{\mathbf{M}\Phi}^N_K$ of $(\varphi,N)$-modules over $K$ to that of $\pn$-modules over $\rk$, whose essential image consists exactly of the unipotent $\pn$-modules, i.e. those which are iterated extensions of constant ones. The analogue of Fontaine’s $C_{\mathrm{st}}$ conjecture in the equicharacteristic world is then the following.
\[cstpas\] There is an isomorphism $$\left(H^i_{\log\text{-}\mathrm{cris}}(Y^{\log}/W^{\log})\otimes_W K\right)\otimes_K \rk \cong H^i_\rig(X/\rk)$$ in $ \underline{\mathbf{M}\Phi}^\nabla_{\rk} $.
Thanks to the extension of logarithmic crystalline cohomology and Hyodo–Kato cohomology to algebraic stacks by Olsson in [@Ols07a], in particular base change (Theorem 2.6.2) and the construction of the monodromy operator (§6.5), the same proof as given in the scheme case (see Chapter 5 of [@LP16]) works for algebraic spaces as well.
In [@LP16] was defined the notion of an overconvergent $F$-isocrystal on $X$, relative to $K$. These play the role in the $p$-adic theory of lisse $\ell$-adic sheaves in $\ell$-adic cohomology. Classically, i.e. over $k$, one can associate these objects to $p$-adic representations of the fundamental group, and we will need to do this also over Laurent series fields. We only need this for representations $\rho$ with finite image, and in this case the construction is simple. So let $\rho:\pi_1^{\et}(X,\overline{x})\rightarrow G$ be a finite quotient of the étale fundamental group of a smooth and proper variety over $F$, then this corresponds to a finite, étale, Galois cover $f:X'\rightarrow X$, and hence from results of [@LP16] we have a pushforward functor $$f_*:F\text{-}\mathrm{Isoc}^\dagger(X'/K)\rightarrow F\text{-}\mathrm{Isoc}^\dagger(X/K)$$ from overconvergent $F$-isocrystals on $X'$ to those on $X$. We may therefore define $V_\rho\in F\text{-}\mathrm{Isoc}^\dagger(X/K)$ to be the pushforward $f_*\mathcal{O}^\dagger_{X'/K}$ of the constant isocrystal on $X'$.
SNCL varieties {#snclv}
==============
In this section, following F. Kato in §11 of [@Kat96], we will introduce the key notion of a simple normal crossings log variety over $k$, or SNCL variety for short.
We say a geometrically connected variety $Y/k$ is a normal crossings variety over $k$ if it is étale locally étale over $k[x_0,\ldots,x_d]/(x_0\cdots x_r)$.
Let $Y$ denote a normal crossings variety over $k$, and let $M_Y$ be a log structure on $Y$. Then we say that $M_Y$ is of embedding type if étale locally on $Y$ it is (isomorphic to) the log structure associated to the homomorphism of monoids $$\N^{r+1} \rightarrow \frac{k[x_0,\ldots,x_d]}{(x_0\cdots x_r)}$$ sending the $i$th basis element of $\N^{r+1}$ to $x_i$.
Note that the existence of such a log structure imposes conditions on $Y$, and the log structure $M_Y$ is *not* determined by the geometry of the underlying scheme $Y$. In fact, one can show that such a log structure exists if and only if, denoting by $D$ the singular locus of $Y$, there exists a line bundle $\mathcal{L}$ on $Y$ such that $\mathcal{E}\mathit{xt}^1(\Omega^1_{Y/k},\mathcal{O}_Y) \cong \mathcal{L}\otimes \mathcal{O}_D$ (see for example Theorem 11.7 of [@Kat96]).
We say that a log scheme $Y^{\log}$ of embedding type is of semistable type if there exists a log smooth morphism $Y^{\log}\rightarrow \spec{k}^{\log}$ where the latter is endowed with the log structure of the punctured point.
Again, the existence of such a morphism implies conditions on $Y$, namely that $\mathcal{E}\mathit{xt}^1(\Omega^1_{Y/k},\mathcal{O}_Y) \cong \mathcal{O}_D$ (where again $D$ is the singular locus).
A SNCL variety over $k$ is a smooth log scheme $Y^{\log}$ over $k^{\log}$ of semistable type, such that the irreducible components of $Y$ are all smooth.
Any SNCL variety $Y^{\log}$ is log smooth over $k^{\log}$ by definition, and for all $p\geq0$ we will let $\Lambda^p_{Y^{\log}/k^{\log}}$ denote the locally free sheaf of logarithmic $p$-forms on $Y$. We will also let $\omega_Y=\Lambda^{\dim Y}_{Y^{\log}/k^{\log}}$ denote the line bundle of top degree differential forms.
\[dualsheaf\] The sheaf $\omega_Y$ is a dualising sheaf for $Y$.
Follows immediately from Proposition 2.14 and Theorem 2.21 of [@Tsu99a].
We will also need a spectral sequence for the cohomology of semistable varieties. This should be well-known, but we could not find a suitable reference.
\[sslc\] Let $Y^{\log}$ be a SNCL variety over $k$ of dimension $n$, with smooth components $Y_1,\ldots,Y_N$. For each $0\leq s\leq n$ write $$Y^{(s)}=\CMcoprod_{\substack{ I\subset \{1,\ldots,N\}\\ \norm{I}=s+1}} \bigcap_{i\in I} Y_i,$$ and let $i_s:Y^{(s)}\rightarrow Y$ denote the natural map. For $1\leq t\leq s+1$ let $$\partial^s_t: Y^{(s+1)}\rightarrow Y^{(s)}$$ be the canonical map induced by the natural inclusion $Y_{\{i_1,\ldots,i_{s+1}\}}\rightarrow Y_{\{i_1,\ldots,\hat{i}_t,\ldots,i_{s+1}\}}$. Then the there exists an exact sequence $$0 \rightarrow \cur{O}_Y\overset{d^{-1}}{\rightarrow} i_{0*} \cur{O}_{Y^{(0)}}\overset{d^0}{\rightarrow} \ldots \overset{d^{n-1}}{\rightarrow} i_{n*}\cur{O}_Y^{(n)} \rightarrow 0$$ of sheaves on $Y$ where $d^{-1}=i_0^*$ and $$d^s = \sum_{t=1}^{s+1} (-1)^t \partial_t^{s*}$$ for $s\geq 0$.
We define a complex $$0 \rightarrow \cur{O}_Y\rightarrow i_{0*} \cur{O}_{Y^{(0)}}\rightarrow \ldots \rightarrow i_{n*}\cur{O}_Y^{(n)} \rightarrow 0$$ using the formulae in the statement of the lemma, to check it is in fact exact (or indeed, to check that it is even a complex) we may work locally, and hence assume that $Y$ is smooth over $\spec{\frac{k[x_1,\ldots,x_{d}]}{(x_1\ldots x_r)}}$. But now we can just use flat base change to reduce to the case where $Y=\spec{\frac{k[x_1,\ldots,x_{d}]}{(x_1\ldots x_r)}}$, which follows from a straightforward computation.
In the above situation, there exists a spectral sequence $$E^{s,t}_1:=H^t(Y^{(s)},\cur{O}_{Y^{(s)}})\Rightarrow H^{s+t}(Y,\cur{O}_Y).$$
Some useful results
===================
In this section we prove three lemmas that will come in handy later on. The first characterises surfaces with effective anticanonical divisor of a certain form, analogous to Lemma 3.3.7 of [@Per77] in the complex case.
\[paac\] Let $k$ be an algebraically closed field, and $V$ a surface with canonical divisor $K_V$. Let $\{C_i\}$ be a non-empty family of smooth curves $C_i$ on $V$, such that the divisor $D=\sum_iC_i$ is a simple normal crossings divisor, and we have $K_V+D=0$ in $\mathrm{Pic}(V)$. Then one of the following must happen.
1. $V$ is an elliptic ruled surface, and $D=E_1+E_2$ is a sum of disjoint elliptic curves, which are rulings on $V$.
2. $V$ is an elliptic ruled surface, and $D=E$ is a single elliptic curve, which is a 2-ruling on $V$,
3. $V$ is rational, and $D=E$ is an elliptic curve.
4. $V$ is rational, and $D=\sum_{i=1}^d C_i$ is a cycle of rational curves on $V$, i.e. either $d=2$ and $C_1\cdot C_2=2$, or $d>2$ and $C_1\cdot C_2 = C_2\cdot C_3 =\ldots = C_d \cdot C_1=1$, with all other intersection numbers $0$.
The point is that since the classification of surfaces is essentially the same in characteristic $p$ as characteristic $0$, Persson’s original proof carries over verbatim. We reproduce it here for the reader’s benefit.
The hypotheses imply that $V$ is of Kodaira dimension $-\infty$, and hence is either rational or ruled. For each curve $C_i$, let $T_{C_i}$ denote the number of double points on $C_i$, that is $\sum_{j\neq i} C_i\cdot C_j$. By the genus formula we have $$2g(C_i)-2 = C_i\cdot (C_i+K_V ) = -T_{C_i}$$ (here $K_V$ is the canonical divisor) and hence either $T_{C_i}=0$ and $g(C_i)=1$ or $T_{C_i}=2$ and $g(C_i)=0$. Hence $D$ is a disjoint sum of elliptic curves and cycles of rational curves.
Let $\pi:V\rightarrow V_0$ be a map onto a minimal model. For any $i$ such that $\pi$ does not contract $C_i$, let $C_{0i}=\pi(C_i)$, and let $D_0:=\pi(D)$. Any exceptional curve $E$ has to either be a component of a rational cycle or meet exactly one component of $D$ in exactly one point (because $D\cdot E=-K_V\cdot E = 1$). It then follows that $D_0$ has the same form as $D$ (i.e. is a disjoint union of elliptic curves and cycles of rational curves) except that it might also contain nodal rational curves, not meeting any other components. If $V_0\cong \P^2$, then the only possibilities for $D_0$ are a triangle of lines, a conic plus a line, a single elliptic curve or a nodal cubic. Therefore $(V,D)$ has the form claimed.
Otherwise, $V_0$ is a $\P^1$ bundle over a smooth projective curve, let $F \subset V_0$ be a fibre intersecting all $C_{0i}$ properly. Applying the genus formula again gives $K_{V_0}\cdot F=-2$, hence $D_0\cdot F=2=\sum_i C_{0i} \cdot F$. Each connected component of $D_0$ is either a rational cycle, a nodal rational curve or an elliptic curve, and the first two kinds of components have to intersect $F$ with multiplicity $\geq 2$ (in the second case this is because it cannot be either a fibre or a degree 1 cover of the base). Hence if some $C_{0i}$ is an elliptic curve $E_1$, then either $E_1\cdot F=2$, in which case $D_0=E_1$, or $E_1\cdot F=1$, in which case we must have $D_0=E_1+E_2$ for some other elliptic curve $E_2$. In the first case $V_0$ can be elliptic ruled, in which case $E_1$ is a 2-ruling, or rational. In the second case $V_0$ must be elliptic ruled, and both $E_1$ and $E_2$ are rulings. Otherwise, each $C_{0i}$ is a rational curve, $V_0$ must be rational and $D_0$ is either a single cycle of smooth rational curves or a single nodal rational curve. Again, this implies that $(V,D)$ has the form claimed.
We will also need the following cohomological computation.
\[betti\]
1. Let $V$ be an elliptic ruled surface over $k$, and let $\ell$ be a prime number $\neq p$. Then $\dim_{\Q_\ell}H^1_\et(V_{\overline{k}},\Q_\ell)=\dim_K H^1_\rig(V/K)=2$.
2. Let $V$ be a rational surface over $k$, and let $\ell$ be a prime number $\neq p$. Then $\dim_{\Q_\ell}H^1_\et(V_{\overline{k}},\Q_\ell)=\dim_K H^1_\rig(V/K)=0$.
One may use the excision exact sequence in either rigid or $\ell$-adic étale cohomology to see that the first Betti number of a smooth projective surface is unchanged under monoidal transformations, and is hence a birational invariant. We may therefore reduce to the case of $E\times\P^1$ or $\P^1\times\P^1$, which follows from the Künneth formula.
Finally, we have the following (well known) result.
\[gcggcs\] Let $\mathcal{X}/R$ be proper and flat. Assume that the generic fibre $X$ is geometrically connected. Then so is the special fibre $Y$.
Since $\mathcal{X}$ is proper and flat over $R$, the zeroth cohomology $H^0(\mathcal{X},\mathcal{O}_{\mathcal{X}})$ is torsion free and finitely generated over $R$, hence it is free. Since the generic fibre is geometrically connected, it is of rank 1, and the natural map $R\rightarrow H^0(\mathcal{X},\mathcal{O}_{\mathcal{X}})$ is an isomorphism. Since this also holds after any finite flat base change $R\rightarrow R'$, it follows from Zariski’s Main Theorem [@stacks [Tag 0A1C](http://stacks.math.columbia.edu/tag/0A1C)] that $Y$ must in fact be geometrically connected.
Minimal models, logarithmic surfaces and combinatorial reduction
================================================================
The purpose of this section is to introduce the notion of a minimal model of a surface of Kodaira dimension $0$, as well as the corresponding logarithmic and combinatorial versions of these surfaces. The basic idea in all cases is that we have $$\text{minimal}\Rightarrow \text{logarithmic}\Rightarrow \text{combinatorial}$$ and although the general form that the picture takes is the same in all 4 cases, there are enough differences to merit describing how it works separately in each case. This unfortunately means that the next few sections are somewhat repetitive.
Let $X/F$ be a smooth, projective, geometrically connected minimal surface of Kodaira dimension 0, and denote the canonical sheaf by $\omega_X$. Then $X$ falls into one of the following four cases.
1. $\omega_X\cong \mathcal{O}_X$ and $h^1(X,\mathcal{O}_X)=0$. Then $X$ is a K3 surface.
2. $h^0(X,\omega_X)=0$ and $h^1(X,\mathcal{O}_X)=0$. Then $X$ is an Enriques surface.
3. $\omega_X\cong \mathcal{O}_X$ and $h^1(X,\mathcal{O}_X)=2$. Then $X$ is an abelian surface.
4. $h^0(X,\omega_X)=0$ and $h^1(X,\mathcal{O}_X)=1$. Then $X$ is a bielliptic surface.
Note that if $X$ is an Enriques surface we have $\omega_X^{\otimes 2}\cong \mathcal{O}_X$ and if $X$ is a bielliptic surface we have $\omega_X^{\otimes m}\cong\mathcal{O}_X$ for $m=2,3,4$ or $6$. Also note that since $p>3$ the classification of such surfaces is the same over $k$ as over $F$ (i.e. we do not have to consider the ‘extraordinary’ Enriques or bielliptic surfaces). In all cases we may therefore define an integer $m$ as the smallest positive integer such that $\omega_X^{\otimes m}\cong \mathcal{O}_X$. If $\mathscr{X}/R$ is a semistable model for $X$ then we will let $\mathscr{X}^{\log}$ denote the log scheme with log structure induced by the special fibre, this is log smooth over $R^{\log}$, where the log structure is again induced by the special fibre $\pi=0$. We will let $\omega_\mathscr{X}=\Lambda^2_{\mathcal{X}^{\log}/R^{\log}}$ denote the line bundle of logarithmic 2-forms on $\mathscr{X}$. We will also let $Y$ denote the special fibre, and $Y^{\log}/k^{\log}$ the smooth log scheme whose log structure is the one pulled back from that on $\mathscr{X}$.
\[minmod\] Let $\mathscr{X}/R$ be a semistable model for $X$. Then we say that $\mathscr{X}$ is minimal if it is strictly semistable and $\omega_{\mathscr{X}}^{\otimes m}\cong \mathcal{O}_{\mathscr{X}}$.
When $X$ is an Enriques surface, there are counter-examples to the existence of such minimal models, even allowing for finite extensions of $R$.
The first stage is in passing from minimal models to logarithmic surfaces of Kodaira dimension $0$, the latter being defined by logarithmic analogues of the above criteria.
\[logs\] Let $Y^{\log}/k^{\log}$ be a proper SNCL scheme over $k$, of dimension $2$, and let $\omega_Y=\Lambda^2_{Y^{\log}/k^{\log}}$ be its canonical sheaf. Then we say that $Y^{\log}$ is a:
1. logarithmic K3 surface if $\omega_Y\cong \mathcal{O}_Y$ and $h^1(Y,\mathcal{O}_Y)=0$;
2. logarithmic Enriques surface if $\omega_Y$ is torsion in $\mathrm{Pic}(Y)$, $h^0(Y,\omega_Y)=0$ and $h^1(Y,\mathcal{O}_Y)=0$;
3. logarithmic abelian surface if $\omega_Y\cong \mathcal{O}_Y$ and $h^1(Y,\mathcal{O}_Y)=2$;
4. logarithmic bielliptic surface if $\omega_Y$ is torsion in $\mathrm{Pic}(Y)$, $h^0(Y,\omega_Y)=0$ and $h^1(Y,\mathcal{O}_Y)=1$;
Let $X/F$ be a minimal surface of Kodaira dimension $0$, and $\mathscr{X}/R$ a minimal model. Then $Y^{\log}$ is a logarithmic K3 (resp. Enriques, abelain, bielliptic) surface if $X$ is K3 (resp. Enriques, abelian, bielliptic).
Note that the only obstruction to $Y^{\log}/k^{\log}$ being an SNCL variety is geometric connectedness, which follows from Lemma \[gcggcs\]. The conditions on the canonical sheaf $\omega_Y$ in Definition \[logs\] follow from the definition of minimality, it therefore suffices to verify the required dimensions of the coherent cohomology groups on $Y$. We divide into the four cases.
First assume that $X$ is a K3 surface. Then we have $\chi(X,\mathcal{O}_X)=2$, and hence by local constancy of $\chi$ under a flat map (see Chapter III, Theorem 9.9 of [@Har77]) we must also have that $\chi(Y,\mathcal{O}_Y)=2$. Since $Y$ is geometrically connected by Lemma \[gcggcs\], we have $h^0(Y,\mathcal{O}_Y)=1$, and therefore $h^2(Y,\mathcal{O}_Y)-h^1(Y,\mathcal{O}_Y)=1$. But by Proposition \[dualsheaf\] we must have $h^2(Y,\mathcal{O}_Y)=h^0(Y,\omega_Y)$, and by definition of minimality we know that $\omega_Y\cong \mathcal{O}_Y$. Hence $h^2(Y,\mathcal{O}_Y)=1$ and therefore $h^1(Y,\mathcal{O}_Y)=0$. Hence $Y^{\log}$ is a logarithmic K3 surface.
Next assume that $X$ is Enriques. Then as above, we have that $h^0(Y,\mathcal{O}_Y)=1$ and hence by local constancy of $\chi$, that $h^1(Y,\mathcal{O}_Y)=h^2(Y,\mathcal{O}_Y)$. Let $\pi:\widetilde{\mathscr{X}}\rightarrow \mathscr{X}$ denote the canonical double cover coming from the 2-torsion element $\omega_\mathscr{X}\in \mathrm{Pic}(\mathscr{X})$, with generic fibre $\widetilde{X}\rightarrow X$ and special fibre $\widetilde{Y}\rightarrow Y$. Then $\widetilde{\mathscr{X}}$ is a minimal model of the K3 surface $\widetilde{X}$, and hence $\widetilde{Y}^{\log}$ is a logarithmic K3 surface. Hence $h^1(\widetilde{Y},\mathcal{O}_{\widetilde{Y}})=0$, and since $\mathcal{O}_Y\subset \pi_*\mathcal{O}_{\widetilde{Y}}$ is a direct summand, we must have $h^1(Y,\mathcal{O}_Y) =0$, and therefore $h^0(Y,\omega_Y)=h^2(Y,\mathcal{O}_Y)=0$. Thus $Y^{\log}$ is a logarithmic Enriques surface.
The case of abelian surfaces is handled entirely similarly to that of K3 surfaces, and the case of bielliptic surfaces is then deduced as Enriques is deduced from K3.
The next notion is that of combinatorial versions of the above four cases.
\[crdk\] Let $Y$ be a proper surface over $k$ (not necessarily smooth). We say that $Y$ is a combinatorial K3 surface if, geometrically (i.e. over $\overline{k}$), one of the following situations occurs:
- (Type I) $Y$ is a smooth K3 surface.
- (Type II) $Y=Y_1\cup \ldots \cup Y_N$ is a chain with $Y_1,Y_N$ smooth rational surfaces and all other $Y_i$ elliptic ruled surfaces, with each double curve on each ‘inner’ component a ruling. The dual graph of $Y_{\overline{k}}$ is a straight line with endpoints $Y_1$ and $Y_N$.
- (Type III) $Y$ is a union of smooth rational surfaces, the double curves on each component form a cycle of rational curves, and the dual graph of $Y_{\overline{k}}$ is a triangulation of $S^2$.
\[crde\] Let $Y$ be a proper surface over $k$ (not necessarily smooth). We say that $Y$ is a combinatorial Enriques surface if, geometrically, one of the following situations occurs:
- (Type I) $Y$ is a smooth Enriques surface.
- (Type II) $Y=Y_1\cup \ldots \cup Y_N$ is a chain of surfaces, with $Y_1$ rational and all others elliptic ruled, with each double curve on each ‘inner’ component a ruling and the double curve on $Y_N$ a 2-ruling. The dual graph of $Y_{\overline{k}}$ is a straight line with endpoints $Y_1$ and $Y_N$.
- (Type III) $Y$ is a union of smooth rational surfaces, the double curves on each component form a cycle of rational curves, and the dual graph of $Y_{\overline{k}}$ is a triangulation of $\P^2(\R)$.
\[crda\] Let $Y$ be a proper surface over $k$ (not necessarily smooth). We say that $Y$ is a combinatorial abelian surface if, geometrically, one of the following situations occurs:
- (Type I) $Y$ is a smooth abelian surface.
- (Type II) $Y=Y_1\cup \ldots \cup Y_N$ is a cycle of elliptic ruled surfaces, with each double curve a ruling. The dual graph of $Y_{\overline{k}}$ is a circle.
- (Type III) $Y$ is a union of smooth rational surfaces, the double curves on each component form a cycle of rational curves, and the dual graph of $Y_{\overline{k}}$ is a triangulation of the torus $S^1\times S^1$.
\[crdb\] Let $Y$ be a proper surface over $k$ (not necessarily smooth). We say that $Y$ is a combinatorial bielliptic surface if, geometrically, one of the following situations occurs:
- (Type I) $Y$ is a smooth bielliptic surface.
- (Type II) $Y=Y_1\cup \ldots \cup Y_N$ is either a cycle or chain of elliptic ruled surfaces, with each double curve either a ruling (cycles or ‘inner’ components of a chain) or a 2-ruling (‘end’ components of a chain). The dual graph of $Y_{\overline{k}}$ is either a circle or a line segment.
- (Type III) $Y$ is a union of smooth rational surfaces, the double curves on each component form a cycle of rational curves, and the dual graph of $Y_{\overline{k}}$ is a triangulation of the Klein bottle.
Of course, in each case logarithmic surfaces will turn out to be combinatorial, this has been proved by Nakkajima for K3 and Enriques surfaces, and we will show it during the course of this article for abelian (Theorem \[crass\]) and bielliptic (Theorem \[hst\]) surfaces.
K3 surfaces
===========
In this section, we will properly state and prove Theorem \[vague\] for K3 surfaces. The case when $\mathrm{char}(F)=0$ and $\ell=p$ is due to Hernández-Mada in [@HM15], and Perez Buendía in [@PB14] and the case $\ell\neq p$ should be well-known (and at least part of it is implicitly proved in [@Kul77]), however, we could not find a reference in the literature so we include a proof here for completeness. We begin with a result of Nakkajima.
Let $Y^{\log}$ be a logarithmic K3 surface over $k$. Then the underlying scheme $Y$ is a combinatorial K3 surface.
A proof of this result given entirely in terms of coherent cohomology can be given as in Theorem \[crass\] below.
Let $\mathscr{X}/R$ be a minimal semistable model of a K3 surface $X/F$. Then the special fibre $Y$ is a combinatorial K3 surface.
For a K3 surface $X/K$, and for all $\ell\neq p$, the second cohomology group $H^2_\et(X_{\overline{F}},{\Q_\ell})$ is a finite dimensional $\Q_\ell$ vector space with a continuous Galois action, which is quasi-unipotent. If $\ell=p$ and $\mathrm{char}(F)=0$ then $H^2_\et(X_{\overline{F}},{\Q_p})$ is a de Rham representation of $G_F$, and if $\mathrm{char}(F)=p$ then $H^2_\rig(X/\rk)$ is a $\pn$-module over $\rk$.
If we therefore let $H^2(X)$ stand for:
- $H^2_\et(X_{\overline{F}},{\Q_\ell})$ if $\ell\neq p$;
- $H^2_\et(X_{\overline{F}},{\Q_p})$ if $\ell=p$ and $\mathrm{char}(F)=0$;
- $H^2_\rig(X/\rk)$ if $\ell=p$ and $\mathrm{char}(F)=p$;
then in all cases we get a monodromy operator $N$ on $H^2(X)$.
\[k3main\] Let $\cur{X}/R$ be a minimal semistable model of a K3 surface $X$, and $Y$ its special fibre, which is a combinatorial K3 surface. Then $Y$ is of Type I,II or III respectively as the nilpotency index of $N$ on $H^2(X)$ is $1$, $2$ or $3$.
The case $\ell=p$ and $\mathrm{char}(F)=0$ is due to Hernández Mada, and in fact the case $\ell=\mathrm{char}(F)=p$ also follows from his result by applying the results in Chapter 5 of [@LP16].
To deal with the case $\ell\neq \mathrm{char}(k)$, we use the weight spectral sequence (for algebraic spaces this is Proposition 2.3 of [@Kul77]). Let $Y=Y_1\cup \ldots \cup Y_N$ be the components of $Y$, $C_{ij}=Y_i\cap Y_j$ the double curves and $$Y^{(0)}=\CMcoprod_i Y_i,\;\;Y^{(1)}=\CMcoprod_{i<j}C_{ij},\;\;Y^{(2)}=\CMcoprod_{i<j<k}Y_i\cap Y_j\cap Y_k.$$ We consider the weight spectral sequence $$E_1^{s,t} = \bigoplus_{j\geq \max\{0,-s\}} H^{t-2j}_\et(Y^{(s+2j)}_{\overline{k}},\Q_\ell)(-j)\Rightarrow H^{s+t}_\et(X_{\overline{F}},\Q_\ell)$$ which degenerates at $E_2$ and is compatible with monodromy in the sense that there exists a morphism $N:E_r^{s,t}\rightarrow E_r^{s+2,t-2}$ of spectral sequences abutting to the monodromy operator on $H^{s+t}_\et(X_{\overline{K}},\Q_\ell)$. Moreover, by the weight-monodromy conjecture (see Remark 6.8(1) of [@Nak06]) we know that $N^r$ induces an isomorphism $E_2^{-r,w+r}\isomto E_2^{r,w-r}$. Hence we can characterise the three cases where $N$ has nilpotency index $1,2$ or $3$ in terms of the weight spectral sequence as follows.
1. $N=0$ if and only if $E_2^{1,1}=E_2^{2,0}=0$.
2. $N\neq 0$, $N^2=0$ if and only if $E_2^{1,1}\neq0$ and $E_2^{2,0}=0$.
3. $N^2\neq0$, $N^3=0$ if and only if $E_2^{1,1},E_2^{2,0}\neq 0$.
Hence it suffices to show the following.
1. If $Y$ is of Type I, then $E_2^{1,1}=0$.
2. If $Y$ is of Type II, then $E_2^{1,1}\neq0$ and $E_2^{2,0}=0$.
3. If $Y$ is of Type III, then $E_2^{2,0}\neq 0$.
The first of these is clear, and in both the Type II and III cases the term $$E_2^{2,0}= \mathrm{coker}\left( H^0(Y^{(1)}_{\overline{k}},\Q_\ell) \rightarrow H^0(Y^{(2)}_{\overline{k}},\Q_\ell) \right)$$ is simply the second singular cohomology $H^2_\mathrm{sing}(\Gamma,\Q_\ell)$ of the dual graph $\Gamma$. For Type II this is $0$, and for Type III this is $1$-dimensional over $\Q_\ell$, hence it suffices to show that if $Y$ is of Type II, then $E_2^{1,1}\neq0$.
But we know that $$\dim_{\Q_\ell} E_2^{-1,2}+\dim_{\Q_\ell} E_2^{0,1}+\dim_{\Q_\ell} E_2^{1,0}=\dim_{\Q_\ell} H^1_\et(X,\Q_\ell)=0$$ and hence $\dim_{\Q_\ell}E_2^{0,1}=0$. Therefore we have $$\dim_{\Q_\ell} E_2^{1,1}= \dim_{\Q_\ell}H^1_\et(Y^{(1)},\Q_\ell)-\dim_{\Q_\ell} H^1_\et(Y^{(0)},\Q_\ell)$$ which using Lemma \[betti\] we can check to be equal to $2(N-1)-2(N-2)=2$. Hence $E_2^{1,1}\neq0$ as required.
Enriques surfaces {#en}
=================
To deal with the case of Enriques surfaces, we again start with a result of Nakkajima, analogous to the one quoted above.
Let $Y^{\log}$ be a logarithmic Enriques surface over $k$. Then the underlying scheme $Y$ is a combinatorial Enriques surface.
Again, it is possible to prove this only using coherent cohomology as in Theorem \[hst\] below.
Let $\mathscr{X}/R$ be a minimal semistable model of an Enriques surface $X/F$. Then the special fibre $Y$ is a combinatorial Enriques surface.
If $X/F$ is an Enriques surface, then for all $\ell\neq p$ the second homotopy group $\pi_2^\et(X_{\overline{F}})_{\Q_\ell}$ (for the definition of the higher homotopy groups of algebraic varieties, see [@AM69]) is a finite dimensional $\Q_\ell$ vector space with a continuous Galois action, which is quasi-unipotent. If $\ell=p$ and $\mathrm{char}(F)=0$ then $\pi_2^\et(X_{\overline{F}})_{\Q_p}$ is a de Rham representation of $G_F$. If $\mathrm{char}(F)=p$ there is (currently!) no general theory of higher homotopy groups, so instead we cheat somewhat and use the known properties of the higher étale homotopy groups to justify making the following definition.
We define $\pi_2^\rig(X/\rk):=H^2_\rig(\widetilde{X}/\rk)^\vee$, where $\widetilde{X}\rightarrow X$ is the canonical double cover of $X$.
Thus $\pi_2^\rig(X/\rk)$ is a $\pn$-module over $\rk$. Again, if we let $\pi_2(X)$ stand for any of $\pi_2^\et(X_{\overline{F}})_{\Q_\ell}$, $\pi_2^\et(X_{\overline{F}})_{\Q_p}$ or $\pi_2^\rig(X/\rk)$, then in all cases we have a monodromy operator $N$ associated to $\pi_2(X)$.
\[cgre\] Let $\cur{X}/R$ be a minimal semistable model of an Enriques surface $X$, and $Y$ its special fibre, which is a combinatorial Enriques surface. Then $Y$ is of Type I,II or III respectively as the nilpotency index of $N$ on $\pi_2(X)$ is $1$, $2$ or $3$.
1. As noted in the introduction, a result very similar to this was proved in [@HM15].
2. The result as stated here is slightly different to Theorem \[vague\]. There are in fact two ways of stating it, one using the second homotopy group $\pi_2$ and one using the cohomology of a certain rank 2 local system $V$ on $X$, given by pushing forward the constant sheaf on the K3 double cover of $X$.
If we let $\widetilde{\cur{X}}$ denote the canonical double cover of $\cur{X}$, with special fibre $\widetilde{Y}$ and generic fibre $\widetilde{X}$, then as remarked above, $\widetilde{X}$ is a smooth K3 surface over $K$, and $\widetilde{\cur{X}}$ is a minimal semistable model for $\widetilde{X}$. Hence $\widetilde{Y}$ is a combinatorial K3 surface, whose type can be deduced from the nilpotency index of the monodromy operator $N$ on $H^2_\et(\widetilde{X}_{\overline{F}},\Q_\ell)$.
Now note that since $\widetilde{X}$ is simply connected, we have $$\pi_2^\et(X_{\overline{F}})_{\Q_\ell} \cong \pi_2^\et(\widetilde{X}_{\overline{F}})_{\Q_\ell} \cong H_2^\et(\widetilde{X}_{\overline{F}},\Q_\ell) \cong H^2_\et(\widetilde{X}_{\overline{F}},\Q_\ell)^\vee$$ for all $\ell$ (including $\ell=p$ when $\mathrm{char}(F)=0$), and the corresponding isomorphism holds by definition for $\pi_2^\rig(X/\rk)$. Hence $\widetilde{Y}$ is of Type I,II or III respectively as the nilpotency index of $N$ on $\pi_2(X)$ is 1,2 or 3. It therefore suffices to show that the type of $\widetilde{Y}$ is the same as that of $Y$.
Note that we have a finite étale map $f:\widetilde{Y}\rightarrow Y$, therefore if $\widetilde{Y}$ is of Type I, that is a smooth K3 surface, then we must also have that $Y$ is smooth, hence of Type I. If $Y$ is not smooth, then let the components of $Y$ be $Y_1,\ldots,Y_N$, and the components of $\widetilde{Y}$ be $\widetilde{Y}_1,\ldots,\widetilde{Y}_M$. After pulling back $f$ to each component $Y_i$, one of two things can occur:
1. $f^{-1}(Y_i)$ is irreducible, and we get a non-trivial 2-cover $\widetilde{Y}_j\rightarrow Y_i$;
2. $f^{-1}(Y_i)$ splits into 2 disjoint components $\widetilde{Y}_j,\widetilde{Y}_{j'}$, each mapping isomorphically onto $Y$.
If $\widetilde{Y}$ is of Type III, then each component $\widetilde{Y}_j$ is rational, hence, since rational varieties are simply connected each component of $Y$ is also rational, and $Y$ is of Type III. If $\widetilde{Y}$ is of Type II, then one of two things can happen.
1. $M>2$ and there exists a component of $\widetilde{Y}$ which is an elliptic ruled surface.
2. $M=2$ and $\widetilde{Y}=\widetilde{Y}_1\cup\widetilde{Y}_2$ consists of $2$ rational surfaces meeting along an elliptic curve.
In the first case, one verifies that $Y$ must also have a component isomorphic to an elliptic ruled surface (since a rational surface cannot be an unramified cover of an elliptic ruled surface), and is therefore of Type II. In the second case, $Y$ must also have 2 components, (since otherwise $Y$, and therefore $\widetilde{Y}$, would be smooth), and each component of $\widetilde{Y}$ would be a non-trivial double cover of a component of $Y$. But since the components of $Y$ are either rational or elliptic ruled, this cannot happen.
Abelian surfaces
================
In order to deal with abelian surfaces, we need the following analogue of Nakkajima’s result,
\[crass\] Let $Y^{\log}$ be a logarithmic abelian surface over $k$. Then the underlying scheme $Y$ is a combinatorial abelian surface.
We may assume that $k=\overline{k}$. We adapt the proof of Theorem II of [@Kul77]. Let $Y_1,\ldots,Y_N$ denote the components of $Y$, $C_{ij}=Y_i\cap Y_j$ for $i\neq j$ the double curves, and $T_{C_{ij}}$ the number of triple points on each curve $C_{ij}$. We may assume that $N>1$.
Note that $\omega_Y|_{Y_i} \cong \Omega^2_{Y_i/k}(\log \sum_{j\neq i} C_{ij})\cong \cur{O}_{Y_i}$ and hence the divisor $K_{Y_i} + \sum_{j\neq i} C_{ij}$ on $Y_i$ is principal, where $K_{Y_i}$ is a canonical divisor on $Y_i$. Write $D_i=\sum_{j\neq i} C_{ij}$. Now applying Lemma \[paac\] gives us the following possibilities for each $(Y_i,D_i)$:
1. $Y_i$ is an elliptic ruled surface, and either:
1. $D_i=E_1+E_2$ where $E_1,E_2$ are 2 non-intersecting rulings;
2. a $D_i=E$ is a single 2-ruling.
2. $Y_i$ is a rational surface, and either:
1. $D_i=E$ is an elliptic curve inside $Y_i$;
2. $D_i=C_1+\ldots +C_d$ is a cycle of rational curves on $Y_i$.
First suppose that there is some $i$ such that case (2)(b) happens. Then this must also occur on each neighbour of $Y_i$, and since $Y$ is connected, it follows that this occurs on each component. The dual graph $\Gamma$ is therefore a triangulation of a compact surface without border.
Write $$Y^{(0)}=\CMcoprod_i Y_i,\;\;Y^{(1)}=\CMcoprod_{i<j}C_{ij},\;\;Y^{(2)}=\CMcoprod_{i<j<k}Y_i\cap Y_j\cap Y_k,$$ and consider the spectral sequence $H^t(Y^{(s)},\cur{O}_{Y^{(s)}})\Rightarrow H^{s+t}(Y,\cur{O}_Y) $ constructed in §\[snclv\]. Since the components $Y_i$ and the curves $C_{ij}$ are rational, it follows that $H^t(Y^{(s)},\cur{O}_{Y^{(s)}})=0$ for $t>0$ (see for example, Theorem 1 of [@CR11]), and therefore that the coherent cohomology $H^i(Y,\cur{O}_Y)$ of $Y$ is the same as the $k$-valued singular cohomology $H^i_\mathrm{sing}(\Gamma,k)$ of $\Gamma$. But since $p\neq 2$, the $k$-Betti numbers $\dim_k H^i_\mathrm{sing}(\Gamma,k)$ are the same as the $\Q$-Betti numbers $\dim_{\Q}H^i_\mathrm{sing}(\Gamma,\Q)$, the latter must therefore be $1,2,1$ and by the classification of closed 2-manifolds we can deduce that $\Gamma$ is a torus.
Finally let us suppose that all the double curves $C_{ij}$ are elliptic curves, so that each $T_{C_{ij}}=0$ (see the proof of Lemma \[paac\]). Again examining the spectral sequence $H^t(Y^{(s)},\cur{O}_{Y^{(s)}})\Rightarrow H^{s+t}(Y,\cur{O}_Y)$ and using the fact that $\chi(E)=0$ for an elliptic curve, we can see that $0=\chi(Y)=\chi(Y^{(0)})=\bigoplus_i\chi(Y_i)$. Since each $Y_i$ is either rational ($\chi=1$) or elliptic ruled ($\chi=0$), it follows that each $Y_i$ must be elliptic ruled, and we are in the case (1) above. The dual graph $\Gamma$ is one dimensional, and since each component has on it at most two double curves, $\Gamma$ is either a line segment or a circle.
If $\Gamma$ were a line segment, then $Y=Y_1\cup_{E_1}\ldots \cup_{E_{N-1}} Y_N$ would be a chain. Then birational invariance of coherent cohomology would imply that the maps $$\begin{aligned}
H^0(Y_i,\mathcal{O}_{Y_i}) &\rightarrow H^0(E_i,\mathcal{O}_{E_i}) \\
H^0(Y_{i+1},\mathcal{O}_{Y_{i+1}}) &\rightarrow H^0(E_i,\mathcal{O}_{E_i}) \\
H^1(Y_i,\mathcal{O}_{Y_i}) &\rightarrow H^1(E_i,\mathcal{O}_{E_i}) \\
H^1(Y_{i+1},\mathcal{O}_{Y_{i+1}}) &\rightarrow H^1(E_i,\mathcal{O}_{E_i}) \end{aligned}$$ would be isomorphisms, and hence some basic linear algebra would imply surjectivity of the maps $$\begin{aligned}
H^0(Y^{(0)},\mathcal{O}_{Y^{(0)}}) &\rightarrow H^0(Y^{(1)},\mathcal{O}_{Y^{(1)}}) \\
H^1(Y^{(0)},\mathcal{O}_{Y^{(0)}}) &\rightarrow H^1(Y^{(1)},\mathcal{O}_{Y^{(1)}}).\end{aligned}$$ Also, we would have $\dim_k H^1(Y^{(0)},\mathcal{O}_{Y^{(0)}})={N}$ and $\dim_k H^1(Y^{(1)},\mathcal{O}_{Y^{(1)}})={N-1}$, and hence again examining the spectral sequence $H^t(Y^{(s)},\cur{O}_{Y^{(s)}})\Rightarrow H^{s+t}(Y,\cur{O}_Y)$ would imply that $\dim_k H^1(Y,\mathcal{O}_Y)=1$. Since we know that in fact $\dim_kH^1(Y,\mathcal{O}_Y)=2$ (by the definition of a logarithmic abelian surface), this cannot happen. Hence $\Gamma$ must be a circle and $Y$ is of Type II.
Let $\mathscr{X}/R$ be a minimal semistable model of an abelian surface $X/F$. Then the special fibre $Y$ is a combinatorial abelian surface.
If $X/F$ is an abelian surface, then for any prime $\ell\neq p$ we consider the quasi-unipotent $G_F$-representation $H^2_\et(X_{\overline{K}},\Q_\ell)$. For $\ell= p$ and $\mathrm{char}(F)=0$ we may also consider the de Rham representation $H^2_\et(X_{\overline{K}},\Q_p)$, and when $\mathrm{char}(F)=p=\ell$ the $\pn$-module $H^2_\rig(X/\rk)$. Again letting $H^2(X)$ stand for any of the above second cohomology groups then, in each case, we have a nilpotent monodromy operator $N$ associated to $H^2(X)$.
\[abmain\] Let $\cur{X}/R$ be a minimal semistable model for $X$, with special fibre $Y$. Then $Y$ is combinatorial of Type I, II or III respectively as the nilpotency index of $N$ on $H^2(X)$ is $1$, $2$ or $3$.
We will treat the case $\ell\neq p$ and $\mathrm{char}(F)=0$, the other cases are handled entirely similarly. Let $Y_1,\ldots , Y_N$ be the smooth components of the special fibre $Y$. For any $I=\{ i_1\ldots,i_n\}$ write $Y_I=\cap_{i\in I} Y_i$ and for any $s\geq 0$ write $Y^{(s)}=\CMcoprod_{\norm{I}=s+1}Y_I$, these are all smooth over $k$ and empty if $s>2$.
As in the proof of Theorem \[k3main\] we consider the weight spectral sequence $$E_1^{s,t} = \bigoplus_{j\geq \max\{0,-s\}} H^{t-2j}_\et(Y^{(s+2j)}_{\overline{k}},\Q_\ell)(-j)\Rightarrow H^{s+t}_\et(X_{\overline{F}},\Q_\ell).$$ As before it suffices to show the following.
1. If $Y$ is of Type I, then $E_2^{1,1}=0$.
2. If $Y$ is of Type II, then $E_2^{1,1}\neq0$ and $E_2^{2,0}=0$.
3. If $Y$ is of Type III, then $E_2^{2,0}\neq 0$.
Again, the first of these is trivial, and in both the Type II and III cases the term $E_2^{2,0}$ is the second singular cohomology $H^2_\mathrm{sing}(\Gamma,\Q_\ell)$ of the dual graph $\Gamma$. It therefore suffices to show that if $Y$ is of Type II, then $E_2^{1,1}\neq0$.
To show this, note that we have $\dim_{\Q_\ell} E_2^{i,0}=\dim_{\Q_\ell} H^i_\mathrm{sing}(\Gamma,\Q_\ell)$, which is $1$ for $i=0,1$ and zero otherwise. Hence by the fact that $E_2^{-r,w+r}\isomto E_2^{r,w-r}$ we may deduce that $\dim_{\Q_\ell}E_2^{-1,2}=1$, and hence from the fact that $$\dim_{\Q_\ell}E_2^{-1,2}+\dim_{\Q_\ell}E_2^{0,1}+\dim_{\Q_\ell}E_2^{1,0}=\dim_{\Q_\ell} H^1_\et(X_{\overline{F}},\Q_\ell)=4$$ that $\dim_{\Q_\ell}E_2^{0,1}=2$. If we write $Y=Y_1\cup \ldots \cup Y_N$ as a union of $N$ elliptic ruled surfaces, then $Y^{(1)}$ is a disjoint union of $N$ elliptic curves. Hence by Lemma \[betti\] we must have $$\dim_{\Q_\ell}H^1_\et(Y^{(0)}_{\overline{k}},\Q_\ell)= \dim_{\Q_\ell}H^1_\et(Y^{(1)}_{\overline{k}},\Q_\ell)=2N.$$ Hence $\dim_{\Q_\ell}E_2^{1,1}=\dim_{\Q_\ell}E_2^{0,1}=2$ and therefore $E_2^{1,1}\neq0$.
When $\ell=p$, the $\ell$-adic weight spectral should be replaced by the $p$-adic one constructed by Mokrane in [@Mok93]. That this abuts to the $p$-adic étale cohomology when $\mathrm{char}(F)=0$ follows from Matsumoto’s extension of Fontaine’s $C_{\mathrm{st}}$ conjecture to algebraic spaces in [@Mat15], and that it abuts to the $\rk$-valued rigid cohomology when $\mathrm{char}(F)=p$ follows from Proposition \[cstpas\].
Bielliptic surfaces
===================
We can now complete our treatment of minimal models of surfaces of Kodaira dimension $0$ by investigating what happens for bielliptic surfaces.
\[hst\] Let $Y^{\log}$ be a logarithmic bielliptic surface over $k$. Then the underlying scheme $Y$ is a combinatorial bielliptic surface.
We may assume $k=\overline{k}$. Let $\pi:\widetilde{Y}^{\log}\rightarrow Y^{\log}$ be the canonical $m$-cover associated to $\omega_{Y^{\log}}$. Then one easily checks that $\widetilde{Y}^{\log}$ is a logarithmic abelian surface over $k$, and hence is combinatorial of Type I, II or III. If $\widetilde{Y}$ is of Type I, then $\widetilde{Y}$, and therefore $Y$, must be smooth over $k$, and hence $Y$ is a smooth bielliptic surface over $k$, i.e. of Type I.
So assume that $\widetilde{Y}$ is of Type II or III. Let $\widetilde{Y}_1,\ldots,\widetilde{Y}_M$ denote the components of $\widetilde{Y}$ and $Y_1,\ldots,Y_N$ those of $Y$. Note that as in the proof of Theorem \[crass\] we have $$m(K_{Y_i}+\sum_{j\neq i}C_{ij})=0$$ in $\mathrm{Pic}(Y_i)$, where $C_{ij}$ are the double curves.
Suppose that $\widetilde{Y}$ is of Type II. Note that each component of $\widetilde{Y}$ is finite étale over some component of $Y$, and hence each component of $Y$ is an elliptic ruled surface. For each $Y_i$ choose some $\widetilde{Y}_l \rightarrow Y_i$ finite étale, and let $\widetilde{C}_{lj}$ be the inverse image of the double curves. Then we have $$K_{\widetilde{Y}_l}+\sum_j\widetilde{C}_{lj}=0$$ in $\mathrm{Pic}(\widetilde{Y}_l)$. Applying Lemma \[paac\] we can see that $\sum_j\widetilde{C}_{il}$ is either a single elliptic curve $E$, which is a 2-ruling on $\widetilde{Y}_l$, or two disjoint rulings $E_1,E_2$. Hence the same is true for $\sum_j C_{ij}$ on $Y_i$, and therefore $Y$ is of Type II.
Finally, suppose that $\widetilde{Y}$ is of Type III. Then again, each component of $\widetilde{Y}$ is finite étale over some component of $Y$, hence all of the latter are rational. Since the Picard group of a rational surface is torsion free, it follows that we must have $$K_{Y_i}+\sum_{j\neq i}C_{ij} =0$$ on each $Y_i$. Hence applying Lemma \[paac\] as in the proof of Theorem \[crass\] it suffices to show that the dual graph $\Gamma$ of $Y$ is a triangulation of the Klein bottle. But now examining the spectral sequence $$E^{s,t}_1:=H^t(Y^{(s)},\cur{O}_{Y^{(s)}})\Rightarrow H^{s+t}(Y,\cur{O}_Y)$$ (where $Y^{(s)}$ is defined similarly to before), and using the fact that $\mathrm{char}(k)>2$, we can see that the Betti numbers of $\Gamma$ are the same as the dimensions of the coherent cohomology of $Y$, and therefore $Y$ is of Type III.
To formulate the analogue of Theorem \[abmain\] for bielliptic surfaces, we will need to construct a family of canonical local systems on our bielliptic surface $X$. Note that the torsion element $\omega_X\in \mathrm{Pic}(X)[m]\in H^1(X,\mu_m)$ gives rise to a $\mu_m$-torsor over $X$, and hence a canonical $\Q$-valued permutation representation $\rho$ of the fundamental group $\pi_1^\et(X,\bar{x})$, and we can use this to construct canonical $\ell$- or $p$-adic local systems on $X$. When $\ell\neq p$ we obtain a continuous representation $\rho\otimes_{\Q} \Q_\ell$ of $\pi_1^{\et}(X,\bar{x})$ and hence a lisse $\ell$-adic sheaf $V_\ell$ on $X$, and when $\ell=p$ and $\mathrm{char}(F)=0$ we may do the same to obtain a lisse $p$-adic sheaf $V_p$ on $X$, and when $\ell=\mathrm{char}(F)=p$ we obtain an overconvergent $F$-isocrystal $V_p$ on $X/K$ using the construction of §\[rpad\].
Then the local systems $V_\ell,V_p$ do not depend on the choice of point $\bar{x}$, and the $G_F$-representations $H^2_\et(X_{\overline{F}},V_\ell)$ and $H^2_\et(X_{\overline{F}},V_p)$ when $\mathrm{char}(F)=0$ are quasi-unipotent and de Rham respectively, we may also consider the $\pn$-module $$H^2_\rig(X/\rk,V_p):=H^2_\rig(X/\ekd,V_p)\otimes_{\ekd}\rk$$ over $\rk$. Letting $H^2(X,V)$ stand for any of $H^2_\et(X_{\overline{F}},V_\ell)$, $H^2_\et(X_{\overline{F}},V_p)$ or $H^2_\et(X/\cur{R}_K,V_p)$, in all cases we obtain monodromy operators $N$ associated to $H^2(X,V)$.
This construction might seem a little laboured, since what we are really constructing is simply the pushforward of the constant sheaf via the canonical abelian cover of $X$. The point of describing it in the above way is to emphasise the fact that the local systems $V_\ell,V_p$ are entirely intrinsic to $X$.
\[bimain\] Let $\cur{X}/R$ be a minimal semistable model for $X$, with special fibre $Y$. Then $Y$ is combinatorial of Type I, II or III respectively as the nilpotency index of $N$ on $H^2(X,V)$ is $1$, $2$ or $3$.
The local systems $V_\ell,V_p$ are by construction such that there exists a finite étale cover $\widetilde{\cur{X}}\rightarrow \cur{X}$ Galois with group $G$, such that $\widetilde{\cur{X}}$ is a minimal model of an abelian surface $\widetilde{X}$ and $H^2(X,V)\cong H^2(\widetilde{X})$. The special fibre $\widetilde{Y}$ is therefore a finite étale cover of $Y$, also Galois with group $G$, and is a combinatorial abelian surface of Type I, II or III according to the nilpotency index of $N$ on $H^2(X,V)$. Hence we must show that $\widetilde{Y}$ and $Y$ have the same type; this was shown during the course of the proof of Theorem \[hst\].
Existence of models and abstract good reduction
===============================================
As explained in the introduction, our results so far are essentially ‘one half’ of the good reduction problem for surfaces with $\kappa=0$, the other half consists of trying to actually find models nice enough to be able to apply the above methods.
Let $X/F$ be a minimal surface of Kodaira dimension $0$. Then we say that $X$ admits potentially combinatorial reduction if after replacing $F$ by a finite separable extension, there exists a minimal model $\mathcal{X}/R$ of $X$.
Then thanks to the results of the previous sections, for surfaces with potentially combinatorial reduction, we can describe the ‘type’ of the reduction in terms of the nilpotency index of the monodromy operator on a suitable cohomology or homotopy group of $X$ (either $\ell$-adic or $p$-adic). We can therefore answer questions of ‘abstract reduction’ type by establishing whether or not surfaces have potentially combinatorial reduction. The strongest result one might hope for is that every such surface has potentially combinatorial reduction. Unfortunately, this is not the case.
There exist Enriques surfaces over $\Q_p$ which do not admit potentially combinatorial reduction.
This can in fact be seen already in the complex analytic case of a degenerating family $\mathscr{X}\rightarrow \Delta$ of Kähler manifolds over a disc (see Appendix 2 of [@Per77]). In Proposition 2.1 of [@LM14] it is shown that if a K3 surface over $F$ admits potentially strictly semistable reduction, then it admits potentially combinatorial reduction. Again, while the former is always conjectured, it can only be proved under certain conditions, see Corollary 2.2 of *loc. cit*. Since we know that abelian surfaces admit potentially strictly semistable reduction, we can use their argument to prove the following.
\[concab\] Abelian surfaces $X/F$ admit potentially combinatorial reduction.
By Theorem 4.6 of [@Kun98], after replacing $F$ by a finite separable extension, we may assume that there exists a strictly semistable scheme model $\mathcal{X}/R$ of $X$. By applying the Minimal Model Program of [@Kaw94] there exists another scheme model $\mathcal{X}'$ for $X$ such that:
1. the components of the special fibre of $\mathcal{X}'$ are geometrically normal and integral $\Q$-Cartier divisors on $\mathcal{X}'$;
2. $\mathscr{X}'$ is regular away from a finite set $\Sigma$ of closed points on its special fibre, and $\mathcal{X}'$ has only terminal singularities at these points;
3. the special fibre is a normal crossings divisor on $\mathcal{X}\setminus \Sigma$;
4. the relative canonical Weil divisor $K_{\mathcal{X}'/R}$ is $\Q$-Cartier and n.e.f. relative to $R$.
Now, since the canonical divisor $K_{X}$ on the generic fibre is trivial, it follows that we may write $K_{\mathscr{X}'/R}$ as a linear combination $\sum_i a_iV_i$ of the components of the special fibre $Y'$ of $\mathcal{X}'$. Moreover since $\sum_i V_i=0$ we may in fact assume that $a_i\leq 0$ for all $i$ and $a_i=0$ for some $i$. Since $K_{\mathscr{X}'/\Q}$ is n.e.f. relative to $R$, arguing as in Lemma 4.7 of [@Mau14] shows that in fact we must have $a_i=0$ for all $i$, and hence $K_{\mathscr{X}'/R}=0$. In particular it is Cartier (not just $\Q$-Cartier) and therefore applying Theorem 4.4 of [@Kaw94] we can see that in fact $\mathcal{X}'$ is strictly semistable away from a finite set of isolated rational double points on components of $Y'$.
Finally, applying Theorem 2.9.2 of [@Sai04] and Theorem 2 of [@Art74] we may, after replacing $F$ by a finite separable extension, find a strictly semistable algebraic space model $\mathscr{X}''/R$ for $X$ and a birational morphism $\mathscr{X}''\rightarrow \mathscr{X}'$ which is an isomorphism outside a closed subset of each special fibre, of codimension $\geq 2$ in the total space. Since we know that $K_{\mathscr{X}'/R}=0$, it follows that $K_{\mathscr{X}''/R}=0$, and therefore $\mathscr{X}''$ is a minimal model in the sense of Definition \[minmod\].
Of course, this begs the question as to whether or not bielliptic surfaces admit potentially combinatorial reduction, we are not sure whether to expect this or not.
Finally, we would like to relate the ‘type’ of combinatorial reduction for abelian (and hence bielliptic) surfaces to the more traditional invariants associated to abelian varieties with semi-abelian reduction. So suppose that we have an abelian surface $X/F$. Then after a finite separable extension, we may assume that $X$ admits the structure of an abelian variety over $F$, let us therefore call it $A$ instead. After making a further extension, we may assume that $A$ has semi-abelian reduction, i.e. there exist a semi-abelian scheme over $R$ whose generic fibre is $A$. In this situation we have a ‘uniformisation cross’ for $A$ (see for example §2 of [@CI99]), which is a diagram $$\xymatrix{ & T\ar[d] & \\ \Gamma\ar[r] & G\ar[r]^{\pi} \ar[d] & A \\ & B }$$ where $T$ is a torus over $F$, $B$ is an abelian variety with good reduction, $G$ is an extension of $B$ by $T$ and $\Gamma$ is a discrete group. Fixing a prime $\ell\neq p$, the monodromy operator on $H^1_\et(A_{\overline{F}},\Q_\ell)$ can be defined as follows. We have an exact sequence $$0\rightarrow \mathrm{Hom}(\Gamma,\Q_\ell)\rightarrow H^1_\et(A_{\overline{F}},\Q_\ell)\rightarrow H^1_\et(G_{\overline{F}},\Q_\ell)\rightarrow 0$$ and a non-degenerate pairing $$\Gamma\times \mathrm{Hom}(T,\G_m)\rightarrow \Q$$ and the monodromy operator on $H^1_\et(A_{\overline{F}},\Q_\ell)$ is then the composition $$H^1_\et(A_{\overline{F}},\Q_\ell)\rightarrow H^1_\et(T_{\overline{F}},\Q_\ell)\rightarrow \mathrm{Hom}(T,\G_m)\otimes_{\Z}\Q_\ell\rightarrow \mathrm{Hom}(\Gamma,\Q_\ell)\rightarrow H^1_\et(A_{\overline{F}},\Q_\ell)$$ (see for example [@CI99]). Since the first map is surjective, the last injective, and all others are isomorphisms, we have that the dimension of the image of monodromy on $H^1_\et(A_{\overline{F}},\Q_\ell)$ is equal to the dimension of $H^1_\et(T_{\overline{F}},\Q_\ell)$, and therefore to the rank of $T$. Using some simple linear algebra, one can therefore give the nilpotency index of $N$ on $H^2_\et(A_{\overline{F}},\Q_\ell)=\bigwedge^2 H^1_\et(A_{\overline{F}},\Q_\ell)$ as follows:
1. $\mathrm{rank}(T)=0\Rightarrow N=0$ on $ H^2_\et(A_{\overline{F}},\Q_\ell)$;
2. $\mathrm{rank}(T)=1 \Rightarrow N\neq 0$, $N^2=0$ on $ H^2_\et(A_{\overline{F}},\Q_\ell)$;
3. $\mathrm{rank}(T)=2 \Rightarrow N^2\neq 0$, $N^3=0$ on $ H^2_\et(A_{\overline{F}},\Q_\ell)$.
Hence we have the following.
\[typerank\] $A$ has potentially combinatorial reduction of Type I,II or III as $\mathrm{rank}(T)$ is $0,1$ or $2$ respectively.
Towards higher dimensions
=========================
In this final section of the article, we begin to investigate the shape of degenerations in higher dimensions, in particular looking at Calabi–Yau threefolds and concentrating on the ‘maximal intersection case’, analogous to the Type III degeneration of K3 surfaces. In characteristic 0 some fairly general results in this direction are proved in [@KX15], and the approach there provides much of the inspiration for the main result of this section, Theorem \[CY3\], as well as some of the key ingredients of its proof. Many of the proofs there rely on results from the log minimal model program (LMMP), which happily has recently been solved for threefolds in characteristics $>5$ by Hacon–Xu [@HX15], Birkar [@Bir13] and Birkar–Waldron [@BW14]. Given these results, many of our proofs consist of working through special low dimensional cases of [@KX15] explicitly (and gaining slightly more information than given there), although there are certain places where specifically characteristic $p$ arguments are needed.
Since we will need to use the LMMP for threefolds, we will assume throughout that $p>5$. Unfortunately, since we will also need to know results on the homotopy type of Berkovich spaces, we will also need to assume that our models are in fact schemes, rather than algebraic spaces.
A Calabi–Yau variety over $F$ is a smooth, projective, geometrically connected variety $X/F$ such that:
- the canonical sheaf $\omega_X=\Omega^{\dim X}_{X/F}$ is trivial, i.e. $\omega_X\cong \mathcal{O}_X$;
- $X$ is geometrically simply connected, i.e. $\pi_1^{\et}(X_{\overline{F}},x)=\{1\}$ for any $x\in X(\overline{F})$;
- $H^i(X,\mathcal{O}_X)=0$ for all $0<i<\dim X$.
In dimension 2 these are exactly the K3 surfaces, and we will be interested in what we can say about degenerations of Calabi–Yau varieties in dimension 3. Here one expects to be able to divide ‘suitably nice’ semistable degenerations into 4 ‘types’ depending on the nilpotency index of $N$ acting on $H^3(X)$ (for some suitable Weil cohomology theory). In this section we will treat the ‘Type IV’ situation.
We say that a morphism $f:X\rightarrow S$ of algebraic varieties (over an algebraically closed field) is a Mori fibre space if it is projective with connected fibres, and the anticanonical divisor $-K_X$ is $f$-ample, i.e. ample on all fibres of $f$.
\[ccyt4\] Let $Y=\cup_i V_i$ be a simple normal crossings variety over $k$ of dimension 3. We say that $Y$ is a combinatorial Calabi–Yau of Type IV if geometrically (i.e. over $\overline{k}$) we have:
- each component $V_i$ is birational to a Mori fibre space over a unirational base;
- each connected component of every double surface $S_{ij}$ is rational;
- each connected component of every triple curve $C_{ijk}$ is rational;
- the dual graph $\Gamma$ of $Y$ is a triangulation of the 3-sphere $S^3$.
1. It is worth noting that in characteristic $0$ these conditions imply that $V_i$ is rationally connected, and the analogue of the condition in dimension $2$ implies rationality, even in characteristic $p$.
2. We may in fact assume that we have the above shape after a finite extension of $k$.
Let $H^3(X)$ stand for either $H^3_\et(X_{\overline{F}},\Q_\ell)$ if $\mathrm{char}(F)=0$ or $\ell \neq p$, or $H^3_\rig(X/\rk)$ if $\mathrm{char}(F)=p$. In all cases, we have a natural monodromy operator $N$ acting on $H^3(X)$, such that $N^4=0$. As a first step in the study of Calabi–Yau degenerations in dimension 3, the main result of this section is the following.
\[CY3\] Let $\mathscr{X}$ be a strictly semistable $R$-scheme with generic fibre $X$ a Calabi–Yau threefold. Assume moreover that the sheaf of logarithmic $3$-forms $\omega_\mathscr{X}$ on $\mathscr{X}$ relative to $R$ is trivial, and that $N^3\neq 0$ on $H^3(X)$. Then the special fibre $Y$ of $\mathscr{X}$ is a combinatorial Calabi–Yau of Type IV.
As before, we will only treat the case $\mathrm{char}(F)=0$ and $\ell\neq p$, the others are handled identically. We may also assume that $k=\overline{k}$. Let $V_i$ denote the components of $Y$, $S_{ij}$ the double surfaces, $C_{ijk}$ the triple curves and $P_{ijkl}$ the quadruple points. Write $Y^{(0)}=\CMcoprod_i V_i$, $Y^{(1)}=\CMcoprod_{ij} S_{ij}$ et cetera. The only point where the hypothesis on the nilpotency index of $N$ is used is to prove the following lemma.
Suppose that $N^3\neq 0$. Then $Y$ has ‘maximal intersection’, i.e. there exists a quadruple point $P_{ijkl}$.
If there is no quadruple point $P_{ijkl}$ then $Y^{(3)}=\emptyset$. Let $W_n$ denote the weight filtration on $H^3_\et(X_{\overline{F}},\Q_\ell)$, so that $W_{-1}=0$ and $W_6=H^3_\et(X_{\overline{F}},\Q_\ell)$. The monodromy operator $N^3$ sends $W_i$ into $W_{i-6}$, in particular $N^3(H^3_\et(X_{\overline{F}},\Q_\ell))\subset W_0$. But $Y^{(3)}=\emptyset$ implies that $W_0=0$ and hence $N^3=0$.
Note that we do not need to know the weight-monodromy conjecture in order for the lemma to hold, we simply need to know compatibility of $N$ with the weight filtration.
For each $i$ we will let $D_i=\sum_{j\neq i} S_{ij}$, so that by the assumption $\omega_\mathcal{X}\cong \mathcal{O}_{\mathcal{X}}$ and the adjunction formula we have $-K_{V_i}=D_i$ for all $i$. Similarly setting $E_{ij}=\sum_{k\neq i,j} C_{ijk}$ we obtain $-K_{S_{ij}}=E_{ij}$ and setting $F_{ijk}=\sum_{l\neq ijk} P_{ijkl}$ we can see that $-K_{C_{ijk}}=F_{ijk}$. The lemma shows that there exists some $V_{i}$ containing a quadruple point, and the first key step in proving Theorem \[CY3\] is showing that this is actually true for every $i$. The main ingredient in this is the following.
\[divcon\] Let $(V,D)$ be a pair consisting of a smooth projective threefold $V$ over $\overline{k}$ and a non-empty strict normal crossings divisor $D\subset V$. Assume that $K_V+D=0$, and that $D$ is disconnected. Then $D$ consists of two disjoint irreducible components $D_1$ and $D_2$.
The corresponding result for surfaces follows from Lemma \[paac\].
The characteristic 0 version of this result is Proposition 4.37 of [@Kol13]. However, thanks to the proof of the Minimal Model Program for threefolds in characteristic $p>5$, in particular the connectedness principle and the existence of Mori fibre spaces in [@Bir13; @BW14] the same proof works here. So we will run the MMP on the smooth 3-fold $V$. It follows from Theorem 1.7 of [@BW14] that this terminates in a Mori fibre space $p:V^*\rightarrow S$, and by the connectedness principle (Theorem 1.8 of [@Bir13]) it suffices to prove that the strict transform $D^*\subset V^*$ consists of 2 irreducible components. Now we simply follow the proof of Proposition 4.37 of [@Kol13], which goes as follows.
We know that there exists some component $D_1^*\subset D^*$ which positively intersects the ray contracted by $p$. Choose another component $D_2^*\subset D^*$ disjoint from $D^*_1$, and choose some fibre $F_s$ of $p$ meeting $D_2^*$. Since $D_2^*$ is disjoint from $D_1^*$, it follows that it cannot contain $F_s$, and hence intersects $F_s$ positively. Hence both $D_1^*$ and $D_2^*$ are $p$-ample, intersecting the contracted ray positively. Hence the generic fibre of $p$ is of dimension $1$, and is a regular (not necessarily smooth) Fano curve. It then follows that if we choose a general fibre $F_g$ of $p$, then $D_i^*\cdot F_g=1$ for $i=1,2$ and all other components of $D^*$ are $p$-vertical, hence trivial as claimed.
Every component of $Y$ contains a quadruple point.
By connectedness of $Y$ it suffices to show that each neighbour of $V_{i}$ also contains a quadruple point. Note that by Proposition \[divcon\] the divisor $D_{i}$ is connected, by hypothesis there exists a double surface $S_{ij}$ in $D_i$ containing a quadruple point, and hence it suffices to show that each double surface $S_{ik}$ meeting $S_{ij}$ contains a quadruple point. But if not, then $C_{ijk}$ would form a connected component of $E_{ij}$ and hence again applying Lemma \[paac\] we would see that $S_{ij}$ could not contain a quadruple point. Therefore $S_{ik}$ must contain a quadruple point, and we are done.
Of course this also shows that each double surface $S_{ij}$ contains a quadruple point, hence by repeatedly applying Lemma \[paac\] we can conclude that each surface $S_{ij}$ and each curve $C_{ijk}$ is rational. We may therefore see as in the proof of Theorem \[crass\] that the dual graph of each $D_i$ is a closed 2-manifold. Moreover, applying the MMP to each $V_i$ produces a Mori fibre space $W_i\rightarrow Z_i$, such that the divisor $D_i=\sum_{j\neq i}S_{ij}$ dominates $Z_i$. Therefore $V_i$ has the form described in Definition \[ccyt4\].
Finally, to show that the dual graph $\Gamma$ is a 3-sphere, we consider, for every vertex $\gamma$, corresponding to a component $V_i$ of $Y$, the ‘star’ of $\gamma$, i.e. the subcomplex of $\Gamma$ consisting of those cells meeting $\gamma$. This is a cone over the dual graph of $D_i$, hence $\Gamma$ is a closed 3-manifold.
The dual graph $\Gamma$ is simply connected.
Let $\C_p$ denote the completion of the algebraic closure of $F$, and $\mathcal{O}_{\C_p}$ its ring of integers. Let $\frak{X}$ denote the base change to $\mathcal{O}_{\C_p}$ of the $\pi$-adic completion of $\mathscr{X}$, this $\frak{X}$ is polystable over $\mathcal{O}_{\C_p}$ in the sense of Definition 1.2 of [@Ber99]. Let $X_{\C_p}^\mathrm{an}$ denote the generic fibre of $\frak{X}$, considered as a Berkovich space, or in other words the analytification of the base change of $X$ to $\C_p$.
Let $\pi_1^\et(X^\mathrm{an}_{\C_p})$ denote the étale fundamental group of $X_{\C_p}^\mathrm{an}$ in the sense of [@dJ95a], and by $\pi_1^\mathrm{top}(X^\mathrm{an}_{\C_p})$ the fundamental group of the underlying topological space of $X_{\C_p}^\mathrm{an}$. Theorem 2.10(iii) of [@dJ95a] together with rigid analytic GAGA shows that the profinite completion of $\pi_1^\et(X^\mathrm{an}_{\C_p})$ is trivial, since it is isomorphic to the algebraic étale fundamental group $\pi_1^\et(X_{\C_p})$ of $X_{\C_p}$, and $X$ is Calabi–Yau. Next, by Remark 2.11 of [@dJ95a] together with Theorem 9.1 of [@Ber99] we have a surjection $\pi_1^\et(X^\mathrm{an}_{\C_p})\rightarrow \pi_1^\mathrm{top}(X^\mathrm{an}_{\C_p})$ and hence the profinite completion of $\pi_1^\mathrm{top}(X^\mathrm{an}_{\C_p})$ is trivial.
Now by Theorem 8.2 of [@Ber99] we have $\pi_1(\Gamma)\cong \pi_1^\mathrm{top}(X^\mathrm{an}_{\C_p})$ and hence the profinite completion of $\pi_1(\Gamma)$ is trivial. Since $\Gamma$ is a 3-manifold, we may finally apply [@Hem87] to conclude that $\pi_1(\Gamma)$ is trivial as claimed.
We may now conclude the proof of Theorem \[CY3\] using the Poincaré conjecture. In fact, if we know that the weight monodromy conjecture holds, then we have the following converse.
Let $\mathscr{X}$ be a strictly semistable $R$-scheme whose generic fibre is a Calabi–Yau threefold $X$, such that $\omega_\mathscr{X}\cong \mathcal{O}_{\mathscr{X}}$. Assume that the special fibre $Y$ is a combinatorial Calabi–Yau of Type IV. If the weight monodromy conjecture holds for $H^3(X)$, then $N^3\neq 0$.
Again, we assume that $\ell\neq p$, the other cases are handled similarly. Consider the weight spectral sequence $E_r^{p,q}$ for $\mathscr{X}$. The hypotheses imply that $N^3$ induces an isomorphism $$N^3:E_2^{-3,6} \rightarrow E_2^{3,0}$$ and to show that $N^3\neq 0$ it therefore suffices to show that $E_2^{3,0}\neq 0$. Writing out the weight spectral sequence explicitly we see that we have an isomorphism $$E_2^{3,0}\cong H^3_\mathrm{sing}(\Gamma,\Q_\ell)$$ where $\Gamma\simeq S^3$ is the dual graph of $Y$, and hence the claim follows.
This is in particular the case if $\mathrm{char}(F)=p$ (when $\ell\neq p$ this is [@Ito05], when $\ell=p$ it is Chapter 5 of [@LP16]) or $\mathrm{char}(F)=0$, $\ell\neq p$ and $X$ is a complete intersection in some projective space (which follows from [@Sch12]).
Acknowledgements {#acknowledgements .unnumbered}
================
B. Chiarellotto was supported by the grant MIUR-PRIN 2010-11 “Arithmetic Algebraic Geometry and Number Theory”. C. Lazda was supported by a Marie Curie fellowship of the Istituto Nazionale di Alta Matematica. Both authors would like to thank the anonymous referee for a careful reading of the paper, and for suggesting several important improvements.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We consider the behaviour of current fluctuations in the one-dimensional partially asymmetric zero-range process with open boundaries. Significantly, we find that the distribution of large current fluctuations does not satisfy the Gallavotti-Cohen symmetry and that such a breakdown can generally occur in systems with unbounded state space. We also discuss the dependence of the asymptotic current distribution on the initial state of the system.'
author:
- 'R. J. Harris[^1]'
- 'A. R[á]{}kos'
- 'G. M. Sch[ü]{}tz'
bibliography:
- 'allref.bib'
title: 'Breakdown of [G]{}allavotti-[C]{}ohen symmetry for stochastic dynamics'
---
Substantial progress in the understanding of nonequilibrium systems has been achieved recently through so-called fluctuation theorems [@Evans02b]. Specifically, the Gallavotti-Cohen fluctuation theorem (GCFT) can be loosely written as $$\frac{p(-\sigma,t)}{p(\sigma,t)} \sim e^{-\sigma t} \label{e:GCFT}$$ where $p(\sigma,t)$ is the probability to observe an average value $\sigma$ for the entropy production in time interval $t$ and $\sim$ denotes the limiting behaviour for large $t$. This theorem was first derived for deterministic systems [@Gallavotti95] (motivated by computer simulations of sheared fluids [@Evans93]) and subsequently for stochastic dynamics [@Kurchan98; @Lebowitz99]. From [@Ciliberto98] onwards there have been successful attempts at experimental verification, including for simple random processes such as the driven two-level system in [@Schuler05]. Strictly the GCFT is a property of non-equilibrium steady states but, for systems with a unique stationary state it is usually also expected to hold for arbitrary initial states (see, e.g., [@Cohen99; @Searles99] for discussion on this point). We will refer to this more general property of the large deviation function as “GC symmetry”. Some related issues have previously been discussed for Langevin dynamics [@Kurchan98; @Farago02; @Baiesi06]; we consider the more general case of stochastic *many-particle* systems.
Specifically, we explore the GC symmetry in the context of a paradigmatic non-equilibrium model—the zero-range process [@Spitzer70]. For certain parameter values, this interacting particle system exhibits a condensation phenomenon [@Evans00; @Jeon00b] in which a macroscopic proportion of particles pile up on a single site. Condensation transitions are well-known in colloidal and granular systems [@Shim04] and also occur in a variety of other physical and nonphysical contexts [@Evans05]. In [@Me05] it was argued that current fluctuations in the asymmetric zero-range process with open boundary conditions can become spatially-inhomogeneous for large fluctuations—a precursor of the condensation which occurs for strong boundary driving. Here, for a specialized case, we explicitly calculate the current distribution in this large-fluctuation regime and thus prove a breakdown of the symmetry relation . Significantly, we argue that our analytical approach predicts that this effect also occurs for more general models. Fianlly, we discuss the relation of our results to GCFT breakdowns found in some other works [@Bonetto05; @vanZon03; @vanZon04].
Let us begin by defining our model—the partially asymmetric zero-range process (PAZRP) on an open one-dimensional lattice of $L$ sites [@Levine04c]. Each site can contain any integer number of particles, the topmost of which hops randomly to a neighbouring site after an exponentially distributed waiting time. In the bulk particles move to the right (left) with rate $p w_n$ ($q w_n$) where $w_n$ depends only on the occupation number $n$ of the departure site. Particles are injected onto site 1 ($L$) with rate $\alpha$ ($\delta$) and removed with rate $\gamma w_n$ ($\beta w_n$). If the partition function has a finite radius of convergence (i.e, $\lim_{n\to\infty} w_n$ is finite) then for strong boundary driving a growing condensate occurs at one or both of the boundary sites [@Levine04c].
We are interested in the probability distribution of integrated current $J_l(t)$, i.e., the net number of particle jumps between sites $l$ and $l+1$ in time interval $[0,t]$. The long-time asymptotic behaviour of this distribution is characterized by the limit of the generating function $$e_l(\lambda) =\lim_{t \rightarrow \infty} - \frac{1}{t} \ln {\langle e^{-\lambda J_l(t)} \rangle}. \label{e:e_l}$$ which implies [@Lebowitz99] a large deviation property for the asymptotic probability distribution, $p_l(j,t)=\mathrm{Prob}(j_l=j,t)$, of the observed “average” current $j_l=J_l/t$ $$p_l(j,t) \sim e^{-t\hat{e}_l(j)} \label{e:pj}$$ where $\hat{e}_l(j)$ is the Legendre transformation of $e_l(\lambda)$, i.e., $
\hat{e}_l(j)=\max_{\lambda}\{e_l(\lambda)-\lambda j \}. \label{e:lang}
$ To calculate the current distribution we employ the quantum Hamiltonian formalism [@Schutz01] where the master equation for the probability vector $|P_t\rangle$ resembles a Schrödinger equation with Hamiltonian $H$ (see [@Levine04c] for details). The generating function $\langle e^{-\lambda J_l(t)} \rangle$ can then be written as $\langle s | e^{-\tilde{H}_l t} |P_0\rangle$ where $\tilde{H}_l$ is a modified Hamiltonian in which the terms in $H$ giving a unit increase/decrease in $J_l$ are multiplied by $e^{\mp\lambda}$ [@Me05]. Here $|P_0\rangle$ is the initial probability distribution and $\langle s|$ is a summation vector giving the average value over all configurations. For the current into the system from the left (which can be positive or negative) we consider $\tilde{H}_0$ with lowest eigenvalue $\tilde{e}_{0}(\lambda)$ and corresponding eigenvector $|\tilde{0}\rangle$. In the case where $\langle s | \tilde{0} \rangle$ and $\langle \tilde{0} | P_0 \rangle$ are finite, the long-time limiting behaviour is given by $$\langle e^{-\lambda J_0(t)} \rangle \sim \langle s |\tilde{0} \rangle \langle \tilde{0} | P_0\rangle e^{-\tilde{e}_0(\lambda) t} \label{e:pre}$$ In this case we have $e_0(\lambda)=\tilde{e}_0(\lambda)$ and the form of $\tilde{H}_0$ imposes the GC symmetry relation $$e_0(\lambda)=e_0(2E-\lambda) \label{e:GCFTe}$$ which leads, via , to the relationship with $\sigma=2Ej$ and effective field $E$ given by $e^{2E}= (\alpha \beta / \gamma\delta ) ( p/q )^{L-1}$. The field $E$ can be related to a force $F=2Ek_BT$.
While the ground-state eigenvalue calculated in [@Me05] is independent of $w_n$, the latter determine the form of the eigenvectors $\langle \tilde{0}|$ and $|\tilde{0} \rangle$. If $\lim_{n\to\infty} w_n$ is finite then $\langle s | \tilde{0} \rangle$ diverges for some values of $\lambda$. For a fixed initial particle configuration $\langle \tilde{0} | P_0 \rangle$ is always finite. However, for a normalized distribution over initial configurations (e.g., the steady-state) $\langle \tilde{0} | P_0 \rangle$ can also diverge (again in the case where $w_n$ is bounded) meaning that *the asymptotic current distribution retains a dependence on the initial state.* This has important consequences for measurement of the current fluctuations in simulation (or equivalent experiments). Suppose we start from a fixed initial particle configuration, e.g., the empty lattice, wait for some time $T_1$ and then measure the current over a time interval $T_2$. These are two noncommuting timescales—if we take $T_2 \to \infty$ faster than $T_1 \to \infty$ we will measure the asymptotic distribution of current fluctuations corresponding to the fixed initial condition which may differ from the asymptotic behaviour of steady-state current fluctuations obtained by taking $T_1 \to \infty$ before $T_2 \to \infty$.
We first specialize to the case of the single-site PAZRP, i.e, one site with “input” (left) and “output” (right) bonds. In this model explicit calculation of the matrix element $\langle s | e^{-\tilde{H}_0 t} |P_0 \rangle$ is possible. For simplicity we consider here $w_n=1$, anticipating qualitatively the same effects for any bounded $w_n$. We take the case $\alpha-\gamma<\beta-\delta$ in order to ensure a well-defined steady state and assume an initial Boltzmann distribution $$|P_0\rangle = \sum_{n=0}^\infty x^n (1-x) |n\rangle$$ where $|n\rangle$ denotes the state with site occupied by $n$ particles and the fugacity $x=e^{-\beta \mu}<1$. The steady state is $ x=(\alpha+\delta)/(\beta+\gamma)$ while $x \to 0$ gives the empty site. By ergodicity this gives the same asymptotic current distribution as any fixed initial particle number.
Explicit computations yield an integral form for the generating function of input current $$\begin{gathered}
\langle s | e^{-\tilde{H}_0 t} |P_0 \rangle = \frac{ x-1}{2\pi i} \biggl\{ \oint_{C_1} e^{-\varepsilon(z)t} \frac{1}{(z-1)(z- x)}\, dz \\
+ \oint_{C_2} e^{-\varepsilon(z)t} \frac{ x^{-1} [u_\lambda/v_\lambda - z u_\lambda/(\beta + \gamma)]}{(z-1)[z- x^{-1}u_\lambda/v_\lambda][z-u_\lambda/(\beta + \gamma)]}\, dz \biggr\} \label{e:intinb}\end{gathered}$$ with $$\varepsilon(z)=\alpha+\beta+\gamma+\delta-v_\lambda z- u_\lambda z^{-1}.$$ Here, for notational brevity we write $
u_\lambda \equiv {\alpha e^{-\lambda} +\delta}
$, $
v_\lambda \equiv {\beta + \gamma e^\lambda}.
$ and for later use also define the parameter combination $
\eta=\sqrt{[(\beta+\gamma)^2-\beta\delta-\alpha\gamma]^2-4\alpha\beta\gamma\delta}.
$ The contour $C_1$ ($C_2$) is an anti-clockwise circle of radius $ x+\epsilon$ ($\epsilon$) around the origin of the complex plane with $\epsilon \to 0$.
In order to extract the large-time behaviour from this integral representation we use a saddle-point method, taking careful account of the contributions from residues when the saddle-point contour is deformed through poles in the integrand. This yields changes in behaviour at the values of $\lambda$ given in Table \[t:lam\].
[ll]{} Values of $\lambda$ & Corresponding values of $j$\
$e^{\lambda_1} \equiv \frac{\alpha}{\beta+\gamma-\delta}$ & $j_a \equiv \frac{(\beta+\gamma-\delta)^2-\alpha\gamma}{\beta+\gamma-\delta}$, $j_b \equiv \frac{\beta(\beta+\gamma-\delta)^2-\alpha\gamma\delta}{(\beta+\gamma)(\beta+\gamma-\delta)}$\
$e^{\lambda_2} \equiv \frac{(\beta+\gamma)^2-\alpha\gamma-\beta\delta+\eta}{2\gamma\delta}$ & $j_c \equiv -\frac{\eta}{\beta+\gamma}$\
$e^{\lambda_3} \equiv \frac{\delta-\beta x^2+\sqrt{(\delta-\beta x^2)^2+4\alpha\gamma x^2}}{2\gamma x^2} $ & $j_d\equiv \frac{-(\delta-\beta x^2)}{ x}$\
$e^{\lambda_4} \equiv \frac{\beta(1- x)+\gamma}{\gamma x} $ & $j_e \equiv \frac{\alpha\beta\gamma x^2-\delta\left[\beta(1- x)+\gamma \right]^2}{ x(\beta+\gamma)\left[\beta(1- x)+\gamma \right]} $, $j_f \equiv \frac{\alpha\gamma-\left[\beta(1- x)+\gamma \right]^2}{\beta(1- x)+\gamma}$\
For $$x< x_c\equiv\frac{-\eta+(\beta+\gamma)^2-\alpha\gamma+\beta\delta}{2\beta(\beta+\gamma)}$$ we find $$e_0(\lambda)=
\begin{cases}
\alpha(1-e^{-\lambda})+\gamma(1-e^\lambda) & \lambda < \lambda_1 \\
\alpha+\delta-\frac{u_\lambda v_\lambda}{\beta+\gamma} & \lambda_1<\lambda<\lambda_2 \\
\alpha+\beta+\gamma+\delta-2\sqrt{u_\lambda v_\lambda} & \lambda_2<\lambda<\lambda_3 \\
\alpha+\beta+\gamma+\delta-v_\lambda x - u_\lambda x^{-1} & \lambda_3<\lambda
\end{cases}$$ whereas for $ x> x_c$ we get $$e_0(\lambda)=
\begin{cases}
\alpha(1-e^{-\lambda})+\gamma(1-e^\lambda) & \lambda < \lambda_1 \\
\alpha+\delta-\frac{u_\lambda v_\lambda}{\beta+\gamma} & \lambda_1 < \lambda < \lambda_4 \\
\alpha+\beta+\gamma+\delta-v_\lambda x - u_\lambda x^{-1} & \lambda_4 < \lambda.
\end{cases}$$ We note that the form of $e_0(\lambda)$ seen in the regime $\lambda_1<\lambda<\lambda_3$ ($\lambda_4$) is the groundstate eigenvalue of $\tilde{H}_0$ [@Me05]. At $\lambda=\lambda_2$ the spectrum of $\tilde{H}_0$ becomes gapless. The changes at $\lambda_1$ and $\lambda_3$ ($\lambda_4$) correspond to the divergence of $\langle s | \tilde{0} \rangle$ and $\langle \tilde{0} | P_0 \rangle$ respectively. One immediately sees that the symmetry relation is only obeyed for a limited range of $\lambda$.
Via Legendre transformation we obtain the large deviation behaviour of $j_0=J_0/t$. The resulting “phase diagram” is shown in Fig. \[f:pdj\] where
\[Cr\]\[Cl\][$j\quad$]{} \[Tc\]\[Bc\][$ x$]{} \[\]\[\][III]{} \[\]\[\][I]{} \[\]\[\][II]{} \[\]\[\][VI]{} \[\]\[\][V]{} \[\]\[\][ IV]{} \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\]
$\hat{e}_0(j)$ has the following forms in the different regions: $$\hat{e}_0(j)=
\begin{cases}
f_j(\alpha,\gamma) & \text{I} \\
g_j\!\!\left(\frac{(\alpha-\beta-\gamma+\delta)(\beta-\delta)}{\beta+\gamma-\delta},\frac{\beta+\gamma-\delta}{\alpha}\right) & \text{II} \\
f_j\!\!\left(\frac{\alpha\beta}{\beta+\gamma},\frac{\gamma\delta}{\beta+\gamma}\right) & \text{III} \\
f_j(\alpha,\gamma)+f_j(\beta,\delta) & \text{IV} \\
f_j(\alpha,\gamma)+g_j(\beta(1- x)+\delta(1- x^{-1}), x) & \text{V} \\
g_j\!\!\left(\frac{(1- x)\left\{\alpha\beta x-\delta\left[\beta(1- x)+\gamma\right]\right\}}{ x\left[\beta(1- x)+\gamma\right]},\frac{\gamma x}{\beta(1- x)+\gamma} \right) & \text{VI} \\
\end{cases} \label{e:ejres} \\$$ with $$\begin{aligned}
f_j(a,b)&=a+b-\sqrt{j^2+4ab}+j \ln \frac{j+\sqrt{j^2+4ab}}{2a} \\
g_j(a,b)&=a+j \ln b.\end{aligned}$$ The function $f_j(a,b)$ is the “random walk” current distribution of a single bond with Poissonian jumps of rate $a$ to the right and $b$ to the left. The straightline function $g_j(a,b)$ gives an exponential decay of $p_0(j,t)$ with increasing $j$. We now give some brief remarks on the physical interpretation of these behaviours.
In region III, the current across the input bond is dependent on the current across the output bond, resulting in a distribution with mean $(\alpha\beta-\gamma\delta)/(\beta+\gamma)$ and diffusion constant $(\alpha\beta+\gamma\delta)/(\beta+\gamma)$. In IV ($j$ large and negative) there is a temporary build-up of particles on the site (an “instantaneous condensate”[@Me05]) and so to see $j_0=j$, requires a current of $j$ across both bonds independently. In I ($j$ large and positive) the piling-up of particles on the site means the input bond does not feel the presence of the output bond. The $ x$-dependence in region V arises from the possibility of an arbitrarily large initial occupation. II and VI are transition regimes involving linear combinations of two different behaviours. They correspond to values of $\lambda$ where $e_0(\lambda)$ has a discontinuous derivative (cf. a first order phase transition). Analogous results for $e_1(\lambda)$ and $\hat{e}_1(j)$ which characterize the distribution of outgoing current are obtained by the replacements $\alpha \leftrightarrow \delta$, $\beta \leftrightarrow \gamma$, $p \leftrightarrow q$, $\lambda \leftrightarrow -\lambda$, $j \leftrightarrow -j$.
\[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Tc\]\[Bc\][$j$]{} \[Bc\]\[Tc\][$\hat{e}(-j)-\hat{e}(j)$]{} \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\]
The GC symmetry states that, $\hat{e}(-j)-\hat{e}(j)$ should be a straight line (of slope $\log[(\alpha\beta)/(\gamma\delta)]$ in this single-site case) but the results imply that this only holds for small $j$ (specifically in the shaded region of Fig. \[f:pdj\]). In Fig. \[f:GCFT\] we test this prediction against simulation for both input and output bonds. The Monte Carlo simulation results were obtained using an efficient event-driven (continuous time) algorithm; for steady-state results the number of histories with each initial occupation was weighted according to the known steady-state distribution [@Levine04c]. For increasing measurement times the simulation data converges towards the long-time limits predicted by our theory rather than the straight line predicted by GC symmetry.
Unfortunately, since for increasing times it becomes exponentially more unlikely to measure a current fluctuation away from the mean, it is difficult to get long-time simulation data for a large range of $j$. A further check is provided by numerical evaluation of the integral followed by numerical Fourier transform to give the finite time distribution of $p(j,t)$—for small $t$ this gives excellent agreement with the simulation data; for larger $t$ the integrals converge too slowly for the method to be useful.
We now turn to numerical results for a larger system with a different choice of bounded $w_n$, see Fig. \[f:bnd\].
\[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Tc\]\[Tc\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Cr\]\[Cr\] \[Tc\]\[Bc\][$j$]{} \[Bc\]\[Tc\][$\hat{e}(-j)-\hat{e}(j)$]{} \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\] \[Cl\]\[Cl\]
In the finite-time simulation regime one again sees indications of violation of GC symmetry with bond-dependent form. Physically, we argue that the inhomogeneity of the fluctuations across the two different bonds in the single-site PAZRP and the associated violation of the GC symmetry is a result of the temporary build-up of particles on the site. In general, this possibility is expected to occur in any open-boundary zero-range process with $\lim_{n\to\infty} w_n$ finite (even when the boundary parameters are chosen so that there is a well-defined steady state, i.e., no permanent condensation).
Mathematically, the observed breakdown of the GC symmetry results from the divergence of $\langle s | \tilde{0} \rangle$ and $\langle \tilde{0} | P_0 \rangle$. For models where the number of particle configurations $N$ is limited, these quantities are finite and the relation holds for any initial state. However, the limit $N \to \infty$ does not necessarily commute with the $t\to\infty$ limit taken (implicitly) in and (explicitly) in . This non-commutation of limits leads in some cases to the violation of *even for steady-state initial conditions*. This and the initial state dependence (due to non-commuting timescales) are the main issues highlighted by our work.
We now give a more general explanation of this GC breakdown and highlight some connections to previous works. Firstly, consider the observed bond dependence. For systems with bounded state space, currents across different bonds differ by finite boundary terms which vanish in the long-time limit so any combination of currents has the same large deviation behaviour. In contrast, for systems with unbounded state space, current fluctuations can be spatially inhomogeneous and the boundary terms non-vanishing. However, there is always a specific weighted sum of currents for which these boundary terms cancel, giving an action functional analogous to heat production (see [@Lebowitz99]). For the choice $w_n=1$ this is $
W=2\sum_{l=0}^L E_l J_l
$ where $E_l$ is the effective field across each bond, e.g., for the single-site PAZRP we have $e^{2E_0}=\alpha/\gamma$ and $e^{2E_1}=\delta/\beta$.
However, it can readily be seen that the large deviations of $W$ still do not satisfy the GC symmetry. This is due to the presence of further non-vanishing boundary terms. Consider instead the modified action functional (again for $w_n=1$) $$W'=2\sum_{l=0}^L E_l J_l - \ln \frac{P_0(\{n\}(t))}{P_0(\{n\}(0))}$$ where $\{n\}(t)$ represents the configuration of particles at time $t$ and $P_0$ is the initial distribution. The fluctuations of this quantity do satisfy the relationship *even for finite times*—this is a statement of the transient fluctuation theorem of Evans and Searles [@Evans94; @Searles99] (see also [@Carberry04; @Wang05c] for recent experimental tests). Only for bounded state space (finite potentials) do the boundary terms containing the initial distribution vanish in the long-time limit leading to recovery of the GC symmetry and the steady-state theorem.
Note that if one measures only a single current (e.g., $J_0$ or $J_1$) but starts with an initial distribution corresponding to detailed balance across that bond, the boundary terms cancel and the GC symmetry *is* observed. A particular example is the zero-current case $\alpha\beta=\gamma\delta$ with an initial equilibrium distribution, $ x=(\alpha+\delta)/(\beta+\gamma)$—the current fluctuations across both bonds become symmetric $\hat{e}(j)=\hat{e}(-j)$ as predicted by the GCFT with $E \to 0$. This also implies the usual Green-Kubo formula and Onsager reciprocity relations [@Lebowitz99]. For other values of $x$ a breakdown of is still predicted in the $E \to 0$ limit (despite the system’s ergodicity).
An analogous apparent breakdown of the GCFT in models with *deterministic* dynamics and unbounded potentials was discussed by Bonnetto et al. [@Bonetto05]. They argue for the restoration of the symmetry by removal of the “unphysical” singular terms An earlier study of a model with both deterministic and stochastic forces [@vanZon03; @vanZon04] (see [@Garnier05] for experimental realization) found a modified form of heat fluctuation theorem for large fluctuations. In contrast to [@Bonetto05; @vanZon03; @vanZon04], we do not find a constant value for the ratio of probabilities for large forward and backward fluctuations.
A. Rákos acknowledges financial support from the Israel Science Foundation.
[^1]: E-mail:
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Since AlN has emerged as an important piezoelectric material for a wide variety of applications, efforts have been made to increase its piezoelectric response via alloying with transition metals that can substitute for Al in the wurtzite lattice. Herein, we report density functional theory calculations of structure and properties of the Cr-AlN system for Cr concentrations ranging from zero to beyond the wurtzite-rocksalt transition point. By studying the different contributions to the longitudinal piezoelectric coefficient, we propose that the physical origin of the enhanced piezoelectricity in Cr$_x$Al$_{1-x}$N alloys is the increase of the internal parameter $u$ of the wurtzite structure upon substitution of Al with the larger Cr ions. Among a set of wurtzite-structured materials, we have found that Cr-AlN has the most sensitive piezoelectric coefficient with respect to alloying concentration. Based on these results, we propose that Cr-AlN is a viable piezoelectric material whose properties can be tuned via Cr composition. We support this proposal by combinatorial synthesis experiments, which show that Cr can be incorporated in the AlN lattice up to 30% before a detectable transition to rocksalt occurs. At this Cr content, the piezoelectric modulus $d_{33}$ is approximately four times larger than that of pure AlN. This finding, combined with the relative ease of synthesis under non-equilibrium conditions, may propel Cr-AlN as a prime piezoelectric material for applications such as resonators and acoustic wave generators.'
author:
- 'Sukriti Manna$^1$, Kevin R. Talley$^{2,3}$, Prashun Gorai$^{2,3}$, John Mangum$^2$, Andriy Zakutayev$^3$, Geoff L. Brennecka$^2$, Vladan Stevanović$^{2,3}$, and Cristian V. Ciobanu$^1$[^1]'
title: Enhanced piezoelectric response of AlN via CrN alloying
---
Introduction
============
Aluminum nitride has emerged as an important material for micro-electromechanical (MEMS) based systems[@fu2017advances; @muralt2008recent] such as surface and bulk acoustic resonators,[@fu2017advances; @loebl2003piezoelectric] atomic force microscopy (AFM) cantilevers,[@fu2017advances] accelerometers,[@gerfers2007sub; @wang2017mems] oscillators,[@zuo20101] resonators for energy harvesting,[@wang2017aln; @elfrink2009vibration] and band-pass filters.[@yang2003highly] The advantages of using AlN in MEMS devices include metal$-$oxide$-$semiconductor (CMOS) compatibility, high thermal conductivity, and high temperature stability. In addition, its low permittivity and high mechanical stiffness are particularly important for resonantor applications.[@fu2017advances; @muralt2017aln] However, the piezoelectric constants of AlN thin films are lower than those of other commonly used piezoelectric materials. For example, the out-of-plane piezoelectric strain modulus[@nomenclature] $d_{33}$ of reactively sputtered AlN films is reported to be 5.5 pC/N, whereas $d_{33}$ for ZnO can be at least twice as large,[@kang2017enhanced] and PZT films can be over 100 pC/N.[@muralt2008recent]
It is therefore desirable to find ways to increase the piezoelectric response of AlN in order to integrate AlN-based devices into existing and new systems. A common way to engineer piezoelectric properties of AlN is by alloying with transition metal nitrides (Sc, Y, others), which can lead to a several-fold increase in the field-induced strain via increases in the longitudinal piezoelectric coefficient $e_{33}$ and simultaneous decreases in the longitudinal elastic stiffness $C_{33}$.[@akiyama2009enhancement; @caro2015piezoelectric; @manna2017tuning; @tasnadi2010origin] In the case of ScN alloying, the origins of this response have been studied,[@tasnadi2010origin] and it is presumed that other such systems which also involve AlN alloyed with rocksalt-structured end members are similar: as the content of the rocksalt end member in the alloy increases, the accompanying structural frustration enables a greater piezoelectric response. This structural frustration, however, is also accompanied by thermodynamic driving forces for phase separation[@hoglund2010wurtzite] which, with increased alloy concentration, lead to the destruction of the piezoelectric response upon transition to the (centrosymmetric, cubic) rocksalt structure. The experimental realization of large alloy contents without phase separation or severe degradation of film texture and crystalline quality can be quite difficult,[@hoglund2010wurtzite; @mayrhofer2015microstructure] even when using non-equilibrium deposition processes such as sputtering. Thus, it is desirable to find alloy systems for which the structural transition from wurtzite to rocksalt occurs at low alloying concentrations since these may be more easily synthesized and more stable, while also (hypothetically) providing comparable property enhancements as those observed in the more-studied Sc-AlN alloy system. Among the AlN-based systems presently accessible experimentally, Cr-AlN has the lowest transition composition between the wurtzite and rocksalt structures, occurring at approximately 25% CrN concentration.[@mayrhofer2008structure; @holec2010pressure] This motivates the investigation of the piezoelectric properties of the Cr$_x$Al$_{1-x}$N system, which we also refer to, for simplicity, in terms of Cr substitution for Al.
In this article, we study Cr-substituted AlN using density functional theory (DFT) calculations of structural, mechanical, and piezoelectric properties. Given that Cr has unpaired $d$ electrons, a challenge to overcome in these calculations is the simulation of a truly representative random distribution of the spins of Cr ions, whose placement in the AlN lattice involves not only chemical disordering, but spin disordering as well. Among a set of wurtzite-based materials, we have found that Cr-doped AlN is the alloy whose piezoelectric stress coefficient $e_{33}$ is the most sensitive to alloying concentration and also has the lowest wurtzite-to-rocksalt transition composition. The key factor leading to the enhanced piezoelectricity in Cr$_x$Al$_{1-x}$N alloys is the ionic contribution to the coefficient $e_{33}$; this ionic contribution is increased through the internal $u$ parameter of the wurtzite structure when alloyed with the (larger) Cr ions. Therefore, we propose Cr$_x$Al$_{1-x}$N as a viable piezoelectric material with properties that can be tuned via Cr composition. To further support this proposal, we have performed combinatorial synthesis and subsequent characterization of Cr$_x$Al$_{1-x}$N films, and have showed that Cr can be incorporated in the AlN lattice up to 30% before a detectable transition to rocksalt occurs. At this Cr content, the piezoelectric modulus $d_{33}$ is four times larger than that of AlN. Pending future device fabrication and accurate measurements of properties and device performance, this significant increase in $d_{33}$ can propel Cr-AlN to be the choice material for applications such as resonators, GHz telecommunications, or acoustic wave generators.
Methods
=======
Paramagnetic Representation of Cr-AlN Alloys
--------------------------------------------
Starting with a computational supercell of wurtzite AlN, any desired Cr concentration is realized by substituting a corresponding number of Al ions with Cr ions in the cation sub-lattice. In order to realistically simulate the chemical disorder of actual Cr-AlN alloys while maintaining a tractable size for the computational cell, we use special quasirandom structures (SQS).[@zunger1990special; @van2009multicomponent; @van2013efficient] The Cr$^{3+}$ ions have unpaired $d$ electrons, which require spin-polarized DFT calculations. Another important aspect of the calculations is that the Cr$_x$Al$_{1-x}$N alloys are paramagnetic,[@mayrhofer2008structure; @endo2007crystal; @endo2005magnetic] and this state has to be captured explicitly in the DFT calculations. Therefore, in addition to the configurational disorder simulated via SQS, the paramagnetic state requires truly random configurations for the spins associated with the Cr$^{3+}$ ions.[@alling2010effect; @abrikosov2016recent] However, as shown by Abrikosov [*et al.*]{},[@abrikosov2016recent] the paramagnetic state can be approximated by using disordered, collinear, static spins because such state yields zero spin-spin correlation functions. To represent the paramagnetic state of Cr$_x$Al$_{1-x}$N, for a given alloy structure with $n$ Cr sites, we performed a minimum of $n \choose 2$ and maximum 20 calculations. In these calculations, the spins on Cr sites are randomly initialized subject to the restriction of zero total spin for each concentration and each SQS structure. An example of such a random distribution of initial spins is illustrated in Figure \[schematics-spin\] for $x$ = 25% Cr concentration.
![Schematics of cation sublattice of Cr$_{x}$Al$_{1-x}$N alloy. Al (Cr) sites are shown as gray (green) spheres. At a given Cr concentration, the Cr sites of each configuration have a different and random spin initialization with zero total spin in order to capture the paramagnetic state.[]{data-label="schematics-spin"}](fig1-spindisorder){width="7cm"}
Details of the DFT Calculations
-------------------------------
Structural optimizations and calculations of piezoelectric and elastic constants were carried out using the Vienna Ab-initio Simulation Package (VASP),[@kresse1996efficiency] with projector augmented waves (PAW) in the generalized gradient approximation using the Perdew-Burke-Ernzerhof (PBE) exchange-correlation function[@perdew1996generalized] and an on-site Hubbard term[@dudarev1998electron] $U$ for the Cr $3d$ states. The plane wave cutoff energy was set to 540 eV in all calculations. For the wurtzite structures, we have used $4\times4\times 2$ (128 atoms) and $2\times2\times2$ (32 atoms) SQS supercells; for the rocksalt structures, the computations were carried out on $2\times 2 \times 2$ (64 atoms) SQS supercells. Brillouin zone sampling was performed by employing $1\times 1\times 1$ and $2 \times2\times 2$ Monkhorst-Pack[@monkhorst1976special] $k$-point meshes for the wurzite and rocksalt structures, respectively, with the origin set at the $\Gamma$ point in each case. Piezoelectric coefficients were calculated using density functional perturbation theory, and the elastic constants were computed by finite differences.[@gonze1997dynamical; @wu2005systematic] The on-site Coulomb interaction for Cr atoms was set at 3 eV, through a Dudarev approach.[@dudarev1998electron] Before performing the calculations for elastic and piezoelectric constants, we performed cell shape, volume, and ionic relaxations in order to obtain the equilibrium lattice parameters and ionic positions at each particular Cr concentration and SQS alloy.
Experimental Procedures
-----------------------
Combinatorial synthesis of Cr$_x$Al$_{1-x}$N films was performed through reactive physical vapor deposition (PVD). Two inch diameter circular aluminum (99.9999%) and chromium (99.999%) metallic targets were arranged at 45$^{\rm o}$ angles measured from the normal to a plasma-cleaned Si(100) substrate inside a custom vacuum system with a base pressure of $5\times 10^{-6}$ torr. Magnetron RF sputtering with a power of 60 W for aluminum targets and 40 W for the chromium targets was performed at a deposition pressure of $3\times 10^{-3}$ torr, with 8 sccm of argon and 4 sccm of nitrogen, and a substrate temperature of 400 $^{\rm o}$C. Aluminum glow discharges were oriented opposite to each other, with the chromium target perpendicular to both, resulting in a film library with a compositional range in one direction.[@config1; @config2] Each sample library was subdivided into eleven regions across the composition gradient, which were subsequently characterized by x-ray diffraction (XRD) and x-ray fluorescence (XRF), performed on a Bruker D8 Discovery diffractometer with a 2D area detector in a theta-2theta configuration and a Fischer XUV vacuum x-ray spectrometer, respectively.
Results and Discussion
======================
Enthalpy of Mixing
------------------
The enthalpy of mixing as a function of the Cr concentration $x$, at zero pressure, is defined with respect to the pure wurzite-AlN and rocksalt-CrN phases via $$\Delta H_{\text{mix}}(x) = E_{\text{Cr}_{x}\text{Al}_{1-x}\text{N}} - xE_{\text{rs}\text{-CrN}} -(1-x)E_{\text{w}\text{-AlN}},$$ where $E_{\text{Cr}_{x}\text{Al}_{1-x}\text{N}}$, $E_{\text{rs}-\text{CrN}}$, and $E_{\text{w}-\text{AlN}}$ are the total energies per atom of the SQS alloy, pure AlN phase, and pure CrN phase, respectively . The DFT calculated mixing enthalpies for the wurtzite and rocksalt phases of Cr$_{x}$Al$_{1-x}$N are shown in Figure \[enthalpy\](a). The wurzite phase is found to be favorable up to $x=0.25$, beyond which rocksalt alloys are stable; this wurzite to rocksalt phase transition point is consistent with previous experimental observations and other theoretical predictions.[@mayrhofer2008structure; @holec2010pressure]
We have compared the mixing enthalpy of the Cr$_x$Al$_{1-x}$N alloys with that of several other common wurzite-based nitrides,[@akiyama2009enhancement; @tholander2016ab; @vzukauskaite2012yxal1; @manna2017tuning; @mayrhofer2015microstructure] Sc$_{x}$Al$_{1-x}$N, Y$_{x}$Al$_{1-x}$N, and Y$_{x}$In$_{1-x}$N, with the results shown in Figure \[enthalpy\](b). The mixing enthalpies are positive for all cases, meaning that the alloying of AlN or InN with their respective end members is an endothermic process. In practice, these alloys are formed as disordered solid solutions obtained using physical vapor deposition techniques operating at relatively low substrate temperatures because of the energetic plasmas involved.[@hoglund2010wurtzite; @luo2009influence] Figure \[enthalpy\](b) shows that the mixing enthalpy in Cr$_x$Al$_{1-x}$N lies between values corresponding to other systems synthesized experimentally, hence Cr$_x$Al$_{1-x}$N is no more difficult to synthesize than the others. More importantly, the enthalpy calculations show that the transition to rocksalt occurs at the lowest alloy concentration across the wurtzite systems considered, which is important for achieving maximum piezoresponse-enhancing structural frustration with a minimum of dopant concentration in order to retain the single-phase wurtzite.
![(a) DFT-calculated mixing enthalpies of the wurtzite and rocksalt phases of Cr$_{x}$Al$_{1-x}$N as functions of Cr concentration. (b) Calculated mixing enthalpies for several wurzite-based nitride alloys grown experimentally.[]{data-label="enthalpy"}](fig2-enthalpy){width="7cm"}
{width="13cm"}
{width="10cm"}
![(a) Variation of internal parameter, u with Cr addition. (b) Crystal structure of wurzite AlN.[]{data-label="tetrahedra"}](fig5-u_plot){width="5cm"}
Piezoelectric Stress Coefficients
---------------------------------
The piezoelectric coefficients $e_{ij}$ for different spin configurations in SQS supercells with the same Cr content are shown in Figure \[all-e\]. For clarity, the panels in Figure \[all-e\] are arranged in the same fashion as the piezoelectric tensor when represented as a matrix in Voigt notation. The vertical scale is the same for all coefficients except $e_{33}, e_{31}$, and $e_{32}$. The scatter in the results corresponds to different SQS supercells at each Cr concentration; this is an effect of the finite size of the system, in which local distortions around Cr atoms lead to small variations of the lattice constants and angles. It is for this reason that we average the SQS results at each Cr concentration, thereby obtaining smoother variations of the piezoelectric coefficients. At 25% Cr, the value of $e_{33}$ becomes $\sim$1.7 times larger than that corresponding to pure AlN.
The piezoelectric coefficient $e_{33}$ of wurzite Cr$_x$Al$_{1-x}$N is shown in Figure \[e33-details\](a) as a function of Cr concentration, and can be written as[@bernardini1997spontaneous] $$e_{33}(x) = e_{33}^{\text{clamped}}(x) + e_{33}^{\text{non-clamped}}(x),
\label{eqe33partition}$$ in which $e_{33}^{\text{clamped}}(x)$ describes the electronic response to strain and is evaluated by freezing the internal atomic coordinates at their equilibrium positions. The term $e_{33}^{\text{non-clamped}}(x)$ is due to changes in internal coordinates, and is given by $$e_{33}^{\text{non-clamped}}(x) = \frac{4eZ_{33}^{*}(x)}{\sqrt{3} a(x)^{2}}\frac{du(x)}{d\epsilon}
\label{nclamped}$$ where $e$ is the (positive) electron charge, $a(x)$ is the equilibrium lattice constant, $u(x)$ is the internal parameter of the wurtzite, $Z_{33}^{*}(x)$ is the dynamical Born charge in units of $e$, and $\epsilon$ is the macroscopic applied strain. $e_{33}^{\text{non-clamped}}(x)$ describes the piezoelectric response coming from the displacements of internal atomic coordinates produced by the macroscopic strain. Based on Eqs. (\[eqe33partition\]) and (\[nclamped\]), panels (b) through (f) in Figure \[e33-details\] show the different relevant quantities contributing to $e_{33}$ in order to identify the main factors responsible for the increase of piezoelectric response with Cr addition. Direct inspection of Figures \[e33-details\](a-c) indicates that the main contribution to the increase of $e_{33}$ comes from the non-clamped ionic part, Figure \[e33-details\](c). Since the Born charge $Z_{33}^{*}$ \[Figure \[e33-details\](d)\] is practically constant, the key factor that leads to increasing the piezoelectric coefficient is the strain sensitivity $du/d\epsilon$ of the internal parameter $u$ \[Figure \[e33-details\](e)\].
Although the internal parameter $u$ is an average value across the entire supercell, the individual average $u$ parameters can also be determined separately for AlN and CrN tetrahedra \[Figure \[tetrahedra\](a,b)\]. The internal parameter $u$ of AlN tetrahedra \[Figure \[tetrahedra\](b)\] does not change significantly, while that of the CrN tetrahedra grows approximately linearly with Cr concentration \[Figure \[tetrahedra\](a)\]. In an alloy system where AlN tetrahedra are the majority, this variation can be understood based on (i) the fact that the ionic radius of Cr is about 10% larger than that of Al, and (b) the increase in Cr concentration will lead to average $u$ parameters mimicking the variation of the $u$ parameter corresponding to CrN tetrahedra.
Comparison with Other Wurtzite-Based Alloys
-------------------------------------------
![Comparison of change in $e_{33}$ with addition of different transition metals for $x\leq$ 25% regime.[]{data-label="e33-compare"}](fig6-e33-compare){width="7cm"}
The results from calculations of the piezoelectric properties of Cr$_x$Al$_{1-x}$N with $x$ from 0 to 25% Cr are plotted in Figure \[e33-compare\], together with the calculated values for Sc$_x$Al$_{1-x}$N, Y$_x$Al$_{1-x}$N, and Y$_x$In$_{1-x}$N. In Cr$_x$Al$_{1-x}$N, $e_{33}$ increases rapidly from 1.46 to 2.40 C/m$^{2}$ for Cr concentrations from 0 to 25%. For all other alloys considered, the increase is smaller in the same interval of solute concentration: for Sc$_x$Al$_{1-x}$N, Y$_x$Al$_{1-x}$N, and Y$_x$In$_{1-x}$N, $e_{33}$ increases, respectively, from 1.55 to 1.9 C/m$^{2}$, 1.55 to 1.7 C/m$^{2}$, and 0.9 to 1.2 C/m$^{2}$. Within the $x\leq$ 25% range, Cr is more effective than any of the other studied transition elements in improving piezoelectric response of AlN-based alloys.
![Variation of (a) $C_{33}$ and (b) $d_{33}$ for several nitride-based wurtzite alloys.[]{data-label="C33d33"}](fig7-C33d33){width="6cm"}
The experimentally measurable property is $d_{33}$, which is commonly known as piezoelectric strain modulus and relates the electric polarization vector with stress. The relationship between the piezoelectric strain and stress moduli is[@nye1985physical] $$d_{ij} = \sum_{k=1}^{6} e_{ik}(C^{-1})_{kj},
\label{eC}$$ where $C_{ij}$ are the elements of the stiffness tensor in Voigt notation. The variation of the elastic constant $C_{33}$ in Cr$_x$Al$_{1-x}$N with $x$ is shown in Figure \[C33d33\](a), along with the other systems considered here. For all of these wurtzite-based piezoelectrics, the increase in piezoelectric response with alloying element concentration is accompanied by mechanical softening (decrease in $C_{33}$). From Eq. (\[eC\]), it follows that the increase in $e_{33}$ (Figure \[e33-compare\]) and the mechanical softening \[Figure \[C33d33\](a)\] cooperate to lead to the increase of $d_{33}$ values with alloy concentration $x$. Our calculated $d_{33}$ values for Cr$_x$Al$_{1-x}$N are in good agreement with experimental data from Ref. \[Figure \[C33d33\](b)\] for Cr concentrations up to 6.3%. Beyond this concentration, Luo et al.[@luo2009influence] report a drop in the $d_{33}$ values of their films, which is attributed to changes in film texture. We have also extended our calculations of piezoelectric coefficients beyond 25% Cr composition in wurtzite structures. Figure \[e33-25\] shows that $e_{33}$ continues to increase at least up to 37.5% Cr. The calculations done at 50% Cr, which start with wurtzite SQS configurations, evolve into rocksalt configurations during relaxation, which explains the decrease of $e_{33}$ to zero in Figure \[e33-25\].
![Variation of $e_{33}$ as function of Cr concentration for wurtzite phase alloys up to 50%.[]{data-label="e33-25"}](fig8-e33-boundary){width="7.5cm"}
Material $e_{33}$ (C/m$^2)$ $d_{33}$ (pC/N) Refs.
------------------ -------------------- ----------------- ------- --
AlN 1.55 4.5-5.3
Sc-AlN, 10% Sc 1.61 7.8
Y-AlN, 6% Y 1.5 4.0
Y-InN, 14% Y 1.1 5.1
[*this work:*]{}
Cr-AlN, 12.5% Cr 1.84 9.86
Cr-AlN, 25.0% Cr 2.35 16.45
Cr-AlN, 30.0% Cr 2.59 19.52
: Piezoelectric properties of AlN and a few wurtzite alloys for piezoelectric device applications.[]{data-label="table:properties"}
It is worthwhile to compare the performance of several AlN wurzite-based materials for their use in applications. These applications, which are mainly resonators, ultrasound wave generators, GHz telecommunications, FBAR devices, bulk or surface acoustic generators, and biosensors, lead to a multitude of application-specific figures of merit for different utilization modes of the piezoelectric material. However, most figures of merit rely on the piezoelectric properties $e_{33}$ and $d_{33}$, both of which in general should be as large as possible for increased piezoelectric device sensitivity. The most used wurtzite material for these applications is AlN, although there are several other options as well (refer to Table I). Alloying with ScN is promising in that it offers an increased $d_{33}$ for about 10% Sc concentration; larger Sc concentrations are possible, but the growth process becomes more costly and the material is likely to lose texture with increased Sc content. Options such as alloying with YN offer marginal improvement at 6% Y content, and YN-doped InN (14% Y content) fares similarly (Table I). Our results indicate that CrN alloying of AlN can reach superior values for the piezoelectric properties, nearly quadrupling the value of $d_{33}$ (Table I) with respect to AlN. The fact that the transition point is the lowest (Figure \[enthalpy\]) of all wurtzite-based materials relevant for the technologies mentioned above, makes the CrN alloying easier compared with the other materials (which require higher alloy content) and hence renders Cr-AlN a prime candidate for synthesis of new, CrN-alloyed piezoelectrics for resonators and acoustic generators. As we shall see in Sec. III.D, the non-equilibrium growth techniques can bring Cr content past the transition point without significant formation of the (non-piezoelectric) rocksalt phase. Consequently, the piezoelectric properties are expected to be significantly better than those of AlN, especially $d_{33}$ (refer to Table \[table:properties\]). Indeed, this is born out in experiments (data points in Figure \[C33d33\]b). Measurements of figures of merit for specific device configurations will be needed in the future, as those require not only combinations of elastic and piezoelectric properties, but dielectric properties as well.[@manna2017tuning]
Experimental Results
--------------------
![ Representative transmission electron microscopy (TEM) image (a) and an energy dispersive spectroscopy (EDS) line scan (b) of an (Al$_{1-x}$Cr$_x$)N film cross section containing $\sim$7% Cr, confirming the incorporation of Cr into wurtzite solid solution. EDS data were collected along the dashed black line shown in panel (a).[]{data-label="TEM"}](fig9-TEMPresentation1){width="8.5cm"}
![X-ray diffraction patterns of the thin film combinatorial libraries plotted against the film composition, with comparison to the patterns for wurtzite (WZ) [@eddine1977etude] and rocksalt(RS)[@wyckoff1960crystal] structures. For alloying content $x<30\%$, the films grow predominantly with the wurtzite structure. At higher Cr concentrations, $x>30\%$, both rocksalt and wurtzite phases are detected, and the wurtzite exhibits degraded texture. No films were produced with compositions in regions where no intensity is shown. []{data-label="kevin"}](fig10-kevinXRD){width="8.5cm"}
To bring experimental support to our proposal that the Cr-AlN system can become a key piezoelectric material to replace AlN and perhaps even ZnO for future applications, we have to ensure that the texture obtained during growth is stable for sufficiently high CrN concentrations. After synthesizing Cr-AlN alloys through reactive PVD, we have performed transmission electron microscopy (TEM) analysis of the films grown in order to check for textural integrity (i.e., grains oriented primarily with the $c$ axis close to the surface normal) and for the onset of the rocksalt phase. At CrN content below 25% \[the theoretical boundary shown in Figure \[enthalpy\](a)\], our films display no significant texture variations. For example, Figure 9(a) shows a typical TEM micrograph wherein texture is preserved over the film thickness. Additionally, our energy dispersive spectrocopy (EDS) characterization shows nearly constant Cr content through the sample \[Figure \[TEM\](b)\]. Further characterization by XRD was performed for all CrN compositions in the combinatorially synthesized films. Figure \[kevin\] shows the XRD results for the 88 discrete Cr$_x$Al$_{1-x}$N compositions produced in an effort to test the possibilities of synthesizing alloys in a wide range of concentrations, including alloys beyond the wurtzite-to-rocksalt transition point. At low alloying levels, the films grow exclusively with the wurtzite structure and a $(002)$ preferred orientation, as indicated by the dominant presence of the wurtzite $(002)$ diffraction peak (Figure \[kevin\], left side). Films grown by reaction PVD under the conditions used here accept chromium into the wurtzite lattice and grow primarily with the ideal $(002)$ orientation. With increased CrN content, the wurtzite $(012)$ and $(010)$ peaks appear, indicating some deviations from the original, and still predominant $(002)$ orientation of the film. The metastability of this alloy is overcome at an approximate composition of $x \simeq 30$%, where the polycrystalline rocksalt phase appears, as revealed by the rocksalt $(002)$ and $(111)$ peaks (Figure \[kevin\], right hand side). These experimental results show that wurtzite Cr$_x$Al$_{1-x}$N solid solutions can be synthesized without observable phase separation up to concentrations of 30% Cr. Wurtzite material still exists at global compositions beyond 30%, but in a wurtzite-rocksalt phase mixture, which will diminish the piezoelectric properties because of the presence of a significant amount of centrosymmetric rocksalt phase in the mixture.
There are few studies of Cr alloyed into wurtzite AlN,[@luo2009influence; @felmetsger2011; @endo2005magnetic] reporting Cr-doped alloys grown by magnetron sputtering. The Cr concentration previously attained is below 10%, although the limits of Cr alloying were not actually tested in the previous reports.[@luo2009influence; @felmetsger2011; @endo2005magnetic] Our combinatorial synthesis results show that Cr can be doped into the wurtzite lattice up to 11% before the predominant (002) film texture starts to change, and up to 30% before the rocksalt phase appears.
Concluding Remarks
==================
By using a physical representation of the paramagnetic state of substitutional Cr in a wurtzite AlN matrix and performing the necessary averaging over spin configurations at each Cr concentration, we computed the structural, mechanical, and piezoelectric properties of Cr-AlN alloys. Our combinatorial synthesis experiments showed that Cr-AlN are relatively easy to synthesize, and also showed that the reactive PVD procedure resulted in Cr-AlN alloys retaining the wurtzite structure for alloying concentrations up to 30% Cr. Remarkably, our DFT calculations of piezoelectric properties revealed that for 12.5% Cr $d_{33}$ is twice that of pure AlN, and for 30% Cr this modulus is about four times larger than that of AlN.
From a technological standpoint, this finding should make Cr-AlN the prime candidate to replace the current-wurtzite based materials in resonators and acoustic wave generators. The larger piezoelectric response (than AlN) may lead to smaller power consumption and perhaps even to avenues to further miniaturize various devices. While the substitutional alloying with Cr would improve the piezoelectric response for every type of device in which currently AlN is being used, one may wonder why not alloying with other trivalent metals, such as Y or Sc. In particular, Sc has been shown to significantly increase the piezoelectric modulus as well.[@tasnadi2010origin] Even though Sc-AlN has more exciting properties[@caro2015piezoelectric; @tasnadi2010origin] than Cr-AlN, the reason why ScN alloys have not taken over the resonator market so far is that the outstanding enhancements in piezoelectric properties occur at very high Sc concentration (Fig. 2, $x>55$%), at which the stability of the wurtzite phase is rather poor. Cr-AlN has a low wurtzite-to-rocksalt transition concentration, and therefore can offer certain piezoelectric enhancements at alloying levels that are easier to stabilize during the synthesis.
In order to ensure significant impact of Cr-AlN alloys as materials to outperform and replace the established piezoelectrics AlN and ZnO, two avenues should be pursued in the near future. First, to benefit from the $300$% increase in $d_{33}$ at 30% Cr content, it is not sufficient that the rocksalt phase does not form up to that Cr concentration: we also have to avoid the formation of (012) and (010)-oriented grains during growth, which would downgrade (simply through directional averaging) the piezoelectric enhancements associated with the (002)-oriented grains. To that end, we envision changing substrates so as to enable better lattice matching with Cr-alloys with over 25% Cr. This can effectively prevent the (012) and (010) textures from emerging, therefore creating the conditions to take advantage of the large increase in $d_{33}$ reported here. Second, future experimental efforts should measure device performance especially to understand the additional aspect of how Cr content in wurtzite affects the bandgap and whether there would be deleterious leakage effects at larger Cr concentrations. Assuming a worst case scenario, these effects can be mitigated by co-alloying with a non-metalic atomic species (e.g. boron).
Pursuing the two directions above can make Cr-AlN suitable for simultaneous optical and mechanical resonators,[@yale1; @yale2] which are relatively new applications that currently exploit multi-physics aspects of AlN. At present, the characterization of Cr-AlN for these multifunctional applications that require simultaneous engineering of the photonic and acoustic band structure is rather incipient, and only few relevant properties of the Cr-AlN alloys are known: for example, for a Cr concentration of about 2%, the bandgap is virtually unchanged, while the adsorption band decreases from 6 to 3.5 eV.[@highly2] Future theoretical and experimental work to investigate, e.g., photoelastic effect and optical attenuation, is necessary in order to fully uncover the potential of Cr-doped AlN for these applications. For now, we surmise that the technological reason for which one would replace AlN with Cr-AlN for use in multifunctional resonators is the trade-off between the increase in vibrational amplitude and decrease in frequency: while low amounts of Cr may lower the frequency somewhat, the oscillation amplitude would increase due to larger piezoelectric response. The decrease in frequency can be mitigated by co-doping with a small trivalent element (boron), as shown for other doped AlN alloys.[@manna2017tuning] Last but not least, it is worth noting that doping with Cr could enable magnetic polarizaton of the Cr ions in the wurtzite lattice and/or of the minority carriers: these effects are non-existent in pure AlN, and could be pursued for spintronic applications or for low-hysteresis magnets.[@endo2005magnetic]
The significant increase of piezoelectric modulus reported here provides significant drive to pursue the two directions identified above, and overcome routine barriers towards establishing Cr-AlN as a replacement for AlN with large performance enhancements.
[*Acknowledgments.*]{} The authors gratefully acknowledge the support of the National Science Foundation through Grant No. DMREF-1534503. The DFT calculations were performed using the high-performance computing facilities at Colorado School of Mines (Golden Energy Computing Organization) and at National Renewable Energy Laboratory (NREL). Synthesis and characterization facilities at NREL were supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, as part of the Energy Frontier Research Center “Center for Next Generation of Materials by Design: Incorporating Metastability” under contract No. DE-AC36-08GO28308.
[99]{}
Y. Q. Fu, J. K. Luo, N.T. Nguyen, A. J. Walton, A. J. Flewitt, X.T. Zu, Y. Li, G. McHale, A. Matthews, E. Iborra, H. Du, and W. I. Milne, Advances in Piezoelectric Thin Films for Acoustic Biosensors, Acoustofluidics and Lab-on-Chip Applications, Prog. Mater. Sci. [**89**]{}, 31 (2017).
P. Muralt, Recent Progress in Materials Issues for Piezoelectric MEMS, J. Am. Ceram. Soc. [**91**]{}, 1385 (2008).
H. Loebl, M. Klee, C. Metzmacher, W. Brand, R. Milsom, and P. Lok, Piezoelectric Thin AlN Films for Bulk Acoustic Wave (BAW) Resonators, Mater. Chem. Phys. [**79**]{}, 143 (2003).
F. Gerfers, M. Kohlstadt, H. Bar, M.-Y. He, Y. Manoli, and L.-P. Wang, Sub-$\mu$g Ultra-Low-Noise MEMS Accelerometers Based on CMOS-Compatible Piezoelectric AlN Thin Films, pp. 1191–1194 in [*Solid-State Sensors, Actuators and Microsystems Conference, (TRANSDUCERS 2007)*]{} (2017).
Y. Wang, H. Ding, X. Le, W. Wang, and J. Xie, A MEMS Piezoelectric In-Plane Resonant Accelerometer Based on Aluminum Nitride with Two-Stage Microleverage Mechanism, Sens. Actuators. A. Phys. [**254**]{}, 126 (2017).
C. Zuo, J. Van der Spiegel, and G. Piazza, 1.05-GHz CMOS Oscillator Based on Lateral- Field-Excited Piezoelectric AlN Contour- Mode MEMS Resonators, IEEE Trans. Ultrason., Ferroelect., Freq. Control. [**57**]{}, 82 (2010).
N. Wang, C. Sun, L. Y. Siow, H. Ji, P. Chang, Q. Zhang, and Y. Gu, AlN Wideband Energy Harvesters with Wafer-Level Vacuum Packaging Utilizing Three-Wafer Bonding, pp. 841–844 in [*IEEE 30th International Conference on Micro Electro Mechanical Systems*]{} (2017).
R. Elfrink, T. Kamel, M. Goedbloed, S. Matova, D. Hohlfeld, Y. Van Andel, and R. Van Schaijk, Vibration Energy Harvesting with Aluminum Nitride-Based Piezoelectric Devices, J. Micromech. Microeng. [**19**]{}, 094005 (2009).
C.-M. Yang, K. Uehara, S.-K. Kim, S. Kameda, H. Nakase, and K. Tsubouchi, Highly $c$-Axis-Oriented AIN Film using MOCVD for 5GHz-Band FBAR Filter, pp. 170–173 in [*IEEE Symposium on Ultrasonics*]{}, Vol. [**1**]{} (2003).
P. Muralt, AlN Thin Film Processing and Basic Properties, pp. 3–37 in [*Piezoelectric MEMS Resonators*]{} (Springer, International Publishing, Switzerland, 2017)
By convention, we refer to $e_{ij}$, which couple strain and dielectric displacement (or electric field and stress), as piezoelectric coeffcients, while $d_{ij}$, which couple strain with electric field (or stress with dielectric displacement), are referred to as piezoelectric moduli. In both cases, we express the individual matrix elements in standard reduced Voigt notation.
X. Kang, S. Shetty, L. Garten, J. F. Ihlefeld, S. Trolier- McKinstry, and J.-P. Maria, Enhanced Dielectric and Piezoelectric Responses in Zn$_{1-x}$Mg$_x$O Thin Films near the Phase Separation Boundary, [Appl. Phys. Lett. ]{}[**110**]{}, 042903 (2017).
M. Akiyama, T. Kamohara, K. Kano, A. Teshigahara, Y. Takeuchi, and N. Kawahara, Enhancement of Piezoelectric Response in Scandium Aluminum Nitride Alloy Thin Films Prepared by Dual Reactive Cosputtering, [Adv. Mater.]{} [**21**]{}, 593 (2009).
M. A. Caro, S. Zhang, T. Riekkinen, M. Ylilammi, M. A. Moram, O. Lopez-Acevedo, J. Molarius, and T. Laurila, Piezoelectric Coefficients and Spontaneous Polarization of ScAlN, J. Phys. Condens. Matter. [**27**]{}, 245901 (2015).
S. Manna, G. L. Brennecka, V. Stevanović, and C. V. Ciobanu, Tuning the Piezoelectric and Mechanical Properties of the AlN System via Alloying with YN and BN, [J. Appl. Phys.]{} [**122**]{}, 105101 (2017).
F. Tasnádi, B. Alling, C. Höglund, G. Wingqvist, J. Birch, L. Hultman, and I. A. Abrikosov, Origin of the Anomalous Piezoelectric Response in Wurtzite Sc$_x$Al$_{1−x}$N Alloys, [Phys. Rev. Lett.]{} [**104**]{}, 137601 (2010).
C. Höglund, J. Birch, B. Alling, J. Bareño, Z. Czigány, P. O. Persson, G. Wingqvist, A. Žukauskaitė, and L. Hultman, Wurtzite Structure Sc$_{1−x}$Al$_x$N Solid Solution Films Grown by Reactive Magnetron Sputter Epitaxy: Structural Characterization and First-Principles alculations, [J. Appl. Phys.]{} [**107**]{}, 123515 (2010).
P. M. Mayrhofer, H. Riedl, H. Euchner, M. Stöger-Pollach, P. H. Mayrhofer, A. Bittner, and U. Schmid, Microstructure and Piezoelectric Response of Y$_x$Al$_{1− x}$N Thin Films, [Acta Mat.]{} [**100**]{}, 81 (2015).
P. Mayrhofer, D. Music, T. Reeswinkel, H.-G. Fu, and J. Schneider, Structure, Elastic Properties and Phase Stability of Cr$_{1-x}$Al$_x$N, [Acta Mat.]{} [**56**]{}, 2469 (2008).
D. Holec, F. Rovere, P. H. Mayrhofer, and P. B. Barna, Pressure-Dependent Stability of Cubic and Wurtzite Phases within the TiN-AlN and CrN-AlN Systems, [Scripta Mat.]{} [**62**]{}, 349 (2010).
A. Zunger, S.-H. Wei, L. Ferreira, and J. E. Bernard, Special Quasirandom Structures, [Phys. Rev. Lett.]{} [**65**]{}, 353 (1990).
A. van de Walle, Multicomponent Multisublattice Alloys, Nonconfigurational Entropy and Other Additions to the Alloy Theoretic Automated Toolkit, Calphad [**33**]{}, 266 (2009).
A. Van de Walle, P. Tiwary, M. De Jong, D. Olmsted, M. Asta, A. Dick, D. Shin, Y. Wang, L.-Q. Chen, and Z.-K. Liu, Efficient Stochastic Generation of Special Quasirandom Structures, [Calphad]{} [**42**]{}, 13 (2013).
Y. Endo, T. Sato, Y. Kawamura, and M. Yamamoto, Crystal Structure and Magnetic Properties of Cr-doped AlN Films with Various Cr Concentrations, [Mater. Trans.]{} [**48**]{}, 465 (2007).
Y. Endo, T. Sato, A. Takita, Y. Kawamura, and M. Yamamoto, Magnetic, Electrical Properties, and Structure of Cr-AlN and Mn-AlN Thin Films Grown on Si Substrates, IEEE Trans. Magn. [**41**]{}, 2718 (2005).
B. Alling, T. Marten, and I. Abrikosov, Effect of Magnetic Disorder and Strong Electron Correlations on the Thermodynamics of CrN, [Phys. Rev. B]{} [**82**]{}, 184430 (2010).
I. Abrikosov, A. Ponomareva, P. Steneteg, S. Barannikova, and B. Alling, Recent Progress in Simulations of the Paramagnetic State of Magnetic Materials, [Curr. Opin. Solid State Mater. Sci.]{} [**20**]{}, 85 (2016).
G. Kresse and J. Furthmüller, Efficiency of Ab-Initio Total Energy Calculations for Metals and Semiconductors Using a Plane-Wave Basis Set, [Comp. Mat. Sci.]{} [**6**]{}, 15 (1996).
J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized Gradient Approximation Made Simple, [Phys. Rev. Lett. ]{} [**77**]{}, 3865 (1996).
H. J. Monkhorst and J. D. Pack, Special Points for Brillouin-Zone Integrations, [Phys. Rev. B ]{}[**13**]{}, 5188 (1976).
X. Gonze and C. Lee, Dynamical Matrices, Born Effective Charges, Dielectric Permittivity Tensors, and Interatomic Force Constants from Density-Functional Perturbation Theory, Phys. Rev. B [**55**]{}, 10355 (1997).
X. Wu, D. Vanderbilt, and D. Hamann, Systematic Treatment of Displacements, Strains, and Electric Fields in Density-Functional Perturbation Theory, Phys. Rev. B [**72**]{}, 035105 (2005).
S. Dudarev, G. Botton, S. Savrasov, C. Humphreys, and A. Sutton, Electron-Energy-Loss Spectra and the Structural Stability of Nickel Oxide: An LSDA+U Study, [Phys. Rev. B]{} [**57**]{}, 1505 (1998).
A. W. Welch, L. L. Baranowski, P. Zawadzki, S. Lany, C. A. Wolden, A. Zakutayev, CuSbSe$_2$ Photovoltaic Devices with 3% Efficiency [Appl. Phys. Express]{} [**8**]{}, 082301 (2015).
A. W. Welch, L. L. Baranowski, H. Peng, H. Hempel, R. Eichberger, T. Unold, S. Lany, C. A. Wolden, A. Zakutayev, Trade-Offs in Thin Film Solar Cells with Layered Chalcostibite Photovoltaic Absorbers, [Adv. Energy Mater.]{} [**7**]{}, 1601935 (2017).
C. Tholander, J. Birch, F. Tasnádi, L. Hultman, J. Palisaitis, P. O. [Å]{}. Persson, J. Jensen, P. Sandström, B. Alling, and A. Žukauskaitė, Ab Initio Calculations and Experimental Study of Piezoelectric Y$_x$In$_{1-x}$N Thin Films Deposited Using Reactive Magnetron Sputter Epitaxy, [Acta Mat.]{} [**105**]{}, 199 (2016).
A. Žukauskaitė, C. Tholander, J. Palisaitis, P. O. [Å]{}. Persson, V. Darakchieva, N. B. Sedrine, F. Tasnádi, B. Alling, J. Birch, and L. Hultman, Y$_x$Al$_{1-x}$N Thin Films, [J. Phys. D.]{} [**45**]{}, 422001 (2012).
J. Luo, B. Fan, F. Zeng, and F. Pan, Influence of Cr-Doping on Microstructure and Piezoelectric Response of AlN Films, [J. Phys. D.]{} [**42**]{}, 235406 (2009).
F. Bernardini, V. Fiorentini, and D. Vanderbilt, Spontaneous Polarization and Piezoelectric Constants of III-V Nitrides, Phys. Rev. B [**56**]{}, R10024 (1997).
J. F. Nye, [*Physical Properties of Crystals: Their Representation by Tensors and Matrices*]{} (Oxford University Press, 1985).
M. N. Eddine, E. Bertaut, M. Roubin, and J. Paris, Etude Cristallographique de Cr$_{1-x}$V$_x$N à Basse Temperature, [Acta Crystallogr. B.]{} [**33**]{}, 3010 (1977).
R. W. G. Wyckoff and R. W. Wyckoff, [*Crystal structures*]{}, Vol. [**2**]{} (Interscience New York, 1960).
V. V. Felmetsger and M. K. Mikhov, Reactive Magnetron Sputtering of Piezoelectric Cr-Doped AlN Thin Films, pp. 835-839 in [*IEEE International Ultrasonics Symposium*]{} (2011).
R. Matloub, A. Artieda, C. Sandu, E. Milyutin, and P. Muralt, Electromechanical Properties of Al$_{0.9}$Sc$_{0.1}$N Thin Films Evaluated at 2.5 GHz Film Bulk Acoustic Resonators, [Appl. Phys. Lett. ]{} [ **99**]{}, 092903 (2011).
G. Wingqvist, F. Tasnádi, A. Žukauskaitė, J. Birch, H. Arvin, and L. Hultman, Increased Electromechanical Coupling in w-Sc$_x$Al$_{1-x}$N, [Appl. Phys. Lett. ]{} [ **97**]{}, 112902 (2010).
L. R. Fan, X. K. Sun, C. Xiong, C. Schuck, H. X. Tang, Aluminum Nitride Piezo-Acousto-Photonic Crystal Nanocavity with High Quality Factors, Appl. Phys. Lett. [**102**]{}, 153507 (2013).
C. Xiong, L. R. Fan, X. K. Sun, H. X. Tang, Cavity Piezooptomechanics: Piezoelectrically Excited, Optically Transduced Optomechanical Resonators, Appl. Phys. Lett. [**102**]{}, 021110 (2013).
A. Y. Polyakov, N. B. Smirnov, A. V. Govorkov, R. M. Frazier, J. Y. Liefer, G. T. Thaler, C. R. Abernathy, S. J. Pearton, and J. M. Zavada, Properties of Highly Cr-Doped AlN, Appl. Phys. Lett [**85**]{}, 4067 (2004).
[^1]: Corresponding author, email: [email protected]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Let $W$ be a right-angled Coxeter group corresponding to a finite non-discrete graph $\mathcal{G}$ with at least $3$ vertices. Our main theorem says that $\mathcal{G}^c$ is connected if and only if for any infinite index quasiconvex subgroup $H$ of $W$ and any finite subset $\{ \gamma_1, \ldots , \gamma_n \} \subset W \setminus H$ there is a surjection $f$ from $W$ to a finite alternating group such that $f (\gamma_i) \notin f (H)$. A corollary is that a right-angled Artin group splits as a direct product of cyclic groups and groups with many alternating quotients in the above sense.
Similarly, finitely generated subgroups of closed, orientable, hyperbolic surface groups can be separated from finitely many elements in an alternating quotient, answering positively the conjecture of Wilton [@wilton2012alternating].
author:
- Michal Buran
bibliography:
- '../mybib.bib'
title: 'Alternating quotients of right-angled Coxeter groups'
---
Introduction
============
It is often fruitful to study an infinite discrete group via its finite quotients. For this reason, conditions that guarantee many finite quotients can be useful.
One such notion is residual finiteness. A group $G$ is said to be *residually finite* if for every $g \in G \setminus \{e \}$, there exists a homomorphism $f: G \rightarrow F$, where $F$ is a finite group and $f(g) \neq e$.
We could try to strengthen this notion by requiring that any finite set of non-trivial elements is not killed by some map to a finite group. But these two notions are equivalent as we could simply take product of maps, which don’t kill the individual elements.
Another way to modify this is to require that the image of $\gamma$ avoids the image of a specified subgroup $H < G$, which does not contain $\gamma$. If this is true for all finitely generated subgroups $H$, we say that $G$ is *subgroup separable*.
Finitely generated free groups are subgroup separable [@hall1949coset Theorem 5.1]. The finite quotient $F$ of a free group could be a priori anything. Wilton proved that (for a free group with at least two generators) we can require $f$ to be a surjection onto a finite alternating group, thus giving us some control over the maps which ‘witness’ subgroup separability [@wilton2012alternating].
Scott showed that closed, orientable, hyperbolic surface groups are subgroup separable [@scott1978subgroups].
Extending and combining methods from both papers, our main theorem shows that even in the case of hyperbolic surface groups, we can require the image to be a finite alternating group.
Let $H$ be a subgroup of a finitely generated group $G$, let $\mathcal{C}$ be a class of groups. We say that $H$ is *$\mathcal{C}$-separable* if for any choice of $\{ \gamma_1, \ldots , \gamma_m \} \subset G \setminus H$ there is a surjection $f$ from $G$ to a group in $\mathcal{C}$ such that $f (\gamma_i) \notin f (H)$ for all $i$.
Note the difference between this terminology and the one above. We talk about subgroups as $\mathcal{C}$-separable in contrast with subgroup separability, which is a property of the entire group.
We will usually take $\mathcal{C}$ to be the class of alternating groups or symmetric groups. We will denote these classes by $\mathcal{A}$ and $\mathcal{S}$, respectively.
In this case, there is a difference between taking a single $\gamma_1$ and multiple group elements as a product of maps surjecting alternating groups is not a map onto an alternating group. In particular, if $G = A_n \times A_m$ then any $\gamma \in G \setminus \{e\}$ does not map to $e$ under at least one of the projections onto factors. However, if we take $\gamma_1, \ldots, \gamma_k$ to be an enumeration of $G \setminus \{e\}$, then the image of any map injective on these elements is isomorphic to $G$ and hence not an alternating group.
The following is our main result.
Let $\mathcal{G}$ be a non-discrete finite simplicial graph of size at least $3$. Every infinite index quasiconvex subgroup of a right-angled Coxeter group $W$ associated to $\mathcal{G}$ is $\mathcal{A}$-separable (and $\mathcal{S}$-separable) if and only if $\mathcal{G}^c$ is connected.
If $\mathcal{G}$ was a discrete graph, there would be difficulties in controlling a permutation parity of the images of generators. It is possible that this can be resolved.
We require infinite index as otherwise the finite quotient by the normal subgroup contained in $H$ could potentially have no alternating quotients.
Quasiconvexity is required as not all finitely generated subgroups of RACG are $\mathcal{C}$-separable, where $\mathcal{C}$ is the set of finite groups [@haglund2008special Example 10.3].
Every finitely generated right-angled Artin group is a direct product of cyclic group and groups whose infinite index quasiconvex subgroups are $\mathcal{A}$-separable.
Infinite index quasiconvex subgroups of closed, orientable, hyperbolic surface groups are $\mathcal{A}$-separable.
Preliminaries
=============
$\mathcal{A}$-separability
--------------------------
We will establish some properties of $\mathcal{A}$-separability.
\[product\] Let $A$ and $B$ be non-trivial finitely generated groups. Then $\{e \} < A \times B$ is not $\mathcal{A}$-separable.
There are only finitely many surjections from $A \times B$ onto $A_2, A_3$ and $A_4$. If $A \times B$ is infinite, then there is a non-identity element $g$ in the kernel of all these maps. Consider elements $g, (e,b), (a,e)$, where $a \neq e$, $b \neq e$. Suppose $f:A \times B \rightarrow A_n$ is a surjection, which does not map these elements to $e$.
By the choice of $g$, we have $n>4$. The group $f(A \times e)$ is a normal subgroup of $A_n$, so it is $e$ or $A_n$. Similarly for $e \times B$. However $A_n$ is not commutative, so one of $A \times e, e \times B$ is mapped to $e$.
If both $A$ and $B$ are finite and $\{e \} < A \times B$ is $\mathcal{A}$-separable, enumerate $A \times B$ as $\gamma_1, \ldots \gamma_m$. Applying the $\mathcal{A}$-separability condition with respect to this set, we get an isomorphism $f : A \times B \rightarrow A_n$. However, $A_n$ is not a direct product, so one of $A, B$ is $A_n$ and the other is trivial.
This implies that passing to a finite degree extension does not in general preserve $\mathcal{A}$-separability of quasiconvex subgroups. However passing to finite-index subgroup does:
\[fi\] Let $G$ be a finitely generated group, let $H$ be a finite-index subgroup of $G$, and let $K$ be an infinite index subgroup of $H$. If $K$ is $\mathcal{A}$-separable in $G$, then it is $\mathcal{A}$-separable in $H$.
We need $K$ to be infinite index in $H$, as otherwise it’s possible that $K = N(H)$ in the notation of the proof below. E.g. take $G = A_n$, $H$ a proper subgroup, $K = \{e\}$.
Suppose $\gamma_1, \ldots, \gamma_n \in H \setminus K$.
Let $N(H) = \bigcap_{g \in G} H^g$ be a normal subgroup contained in $H$. Then $N(H)$ is still finite index and let $M = [G : N(H)]$ be this index. Since $G$ is finitely generated, there are only finitely many surjections $f : G \twoheadrightarrow A_m$ with $m \leq M$. The intersection of preimages of $f(K)$ over such surjections is a finite intersection of finite index subroups, hence a finite index subgroup. So there exists some $\gamma_0 \in G \setminus K$ such that $f(\gamma_0) \in f(K)$ for all $f : G \twoheadrightarrow A_m$ with $m \leq M$.
As $K$ is $\mathcal{A}$-separable in $G$, there exists a surjection $f : G \twoheadrightarrow A_m$, such that $f(\gamma_i) \notin f(K)$ for all $i \in \{0, \ldots n\}$. By the choice of $\gamma_0$ we have $m > M$. But $[A_m:f(N(H))] \leq M$, so $f(N(H)) = A_m$. In particular, $f(H) = A_m$ and $f|_H$ is the desired surjection.
Cube complexes
--------------
For further details of the definitions from this section, the reader is referred to [@haglund2008special].
*An $n$-dimensional cube $C$* is $I^n$, where $I = [-1,1]$. *A face* of a cube is a subset $F = \{\underline{x}: x_i = (-1)^\epsilon \}$, where $1 \leq i \leq n$, $\epsilon =0,1$.
Suppose $\mathcal{C}$ is a set of cubes and $\mathcal{F}$ is a set of maps between these cubes, each of which is an inclusion of a face. Suppose that every face of a cube in $\mathcal{C}$ is an image of at most one inclusion of a face $f \in \mathcal{F}$. Then *the cube complex $X$ associated to $(\mathcal{C},\mathcal{F})$* is $$X = (\bigsqcup_{C \in \mathcal{C}} C) / \sim$$ where $\sim$ is the smallest equivalence relation containing $x \sim f(x)$ for every $f \in \mathcal{F}$, $x \in Dom(f)$.
*A midcube $M$* of a cube $I^n$ is a set of the form $\{\underline{x}: x_i = 0 \}$ for some $ 1 \leq i \leq n$.
If $f: C \rightarrow C'$ is an inclusion of a face and $M$ is a midcube of $C$, then $f(M)$ is contained in unique midcube $M'$ of $C'$. Moreover $f|_M :M \rightarrow M'$ is an inclusion of a face.
Let $X$ be a cube complex associated to $(\mathcal{C},\mathcal{F})$. Let $\mathcal{M}$ be the set of midcubes of cubes of $\mathcal{C}$. Let $\mathcal{F'}$ be the set of restrictions of maps in $\mathcal{F}$ to midcubes.
The pair $(\mathcal{M}, \mathcal{F}')$ satisfies that every face is an image of at most one inclusion of a face, so there is an associated cube complex $X'$. Moreover, inclusions of midcubes descend to a map $\varphi : X' \rightarrow X$. *A hyperplane $H$* is a connected component of $X'$ together with a map $\varphi|_H$.
Hyperplanes are analogous to codimension-1 submanifolds.
Suppose $X$ is a cube complex.
Define a relation of *elementary parallelism* on oriented edges of $X$ by $\overrightarrow{e_1} \sim \overrightarrow{e_2}$ if they form opposite edges of a square. Extend this to the smallest equivalence relation. *The wall $W(\overrightarrow{e})$* is the equivalence class containing $\overrightarrow{e}$. Similarly, we can define an elementary parallelism on unoriented edges and *an unoriented wall $W(e)$*.
We denote by $\overleftarrow{e}$ the edge $\overrightarrow{e}$ with the opposite orientation.
There is a bijective correspondence between unoriented walls and hyperplanes, where $W(e)$ corresponds to $H(e)$, a hyperplane which contains the unique midcube of $e$. We say $H(e)$ is dual to $e$. By abuse of notation, we sometimes identify $H(e)$ with its image.
The following notion was invented by Haglund and Wise and was originally called *A-special* [@haglund2008special Definition 3.2].
A cube complex is *special* if the following holds.
1. For all edges $\overrightarrow{e} \notin W(\overleftarrow{e})$. We say the hyperplanes are $2$-sided.
2. Whenever $\overrightarrow{e_2} \in W(\overrightarrow{e_1})$, then $e_1$ and $e_2$ are not consecutive edges in a square. Equivalently, each hyperplane embeds.
3. Whenever $\overrightarrow{e_2} \in W(\overrightarrow{e_1})$, $\overrightarrow{e_2} \neq \overrightarrow{e_1}$, then the initial point of $\overrightarrow{e_2}$ is not the initial point of $\overrightarrow{e_1}$. We say that no hyperplane directly self-osculates.
4. Whenever $\overrightarrow{e_2} \in W(\overrightarrow{e_1})$ and $\overrightarrow{f_2} \in W(\overrightarrow{f_1})$ and $e_1$ and $f_1$ form two consecutive edges of a square and $\overrightarrow{e_2}$ and $\overrightarrow{f_2}$ start at the same vertex, then $\overleftarrow{e_2}$ and $\overrightarrow{f_2}$ are two consecutive edges in some square. We say that no two hyperplanes inter-osculate.
Haglund and Wise have shown that $CAT(0)$ cube complexes are special [@haglund2008special Example 3.3.(3)]. In this paper, we will only ever use specialness of these complexes.
Every special cube complex is contained in a nonpositively curved complex with the same $2$-skeleton [@haglund2008special Lemma 3.13]. The hyperplane $H(e)$ separates a $CAT(0)$ cube complex $X$ into two connected components.
\[Half-space, [@haglund2008finite]\] Let $X \backslash \backslash H$ be the union of cubes disjoint from $H$. If $X$ is $CAT(0)$, $X \backslash \backslash H$ has two connected components. Call them *half-spaces $H^-$ and $H^+$*
Define *$N(H)$* to be the union of all cubes intersecting $H$. Let *$\partial N(H)$* consist of cubes of $N(H)$ that don’t intersect $H$. In the case of a simply connected special cube complex $\partial N(H)$ has two components; call them *$\partial N(H)^ +$* and *$\partial N(H)^ -$*.
\[Convex subcomplex\] A subcomplex $Y$ of a cube complex $X$ is *(combinatorially geodesically) convex* if any geodesic in $X^{(1)}$ with endpoints in $Y$ is contained in $Y$.
The components of the boundary of a hyperplane $\partial N(H)^ +$, $\partial N(H)^ -$ and half-spaces are combinatorially geodesically convex [@haglund2008finite Lemma 2.10]. Any intersection of half-spaces is convex [@haglund2008finite Corollary 2.16] and a convex subcomplex of a $CAT(0)$ cube complex coincides with the intersection of all half-spaces containing it [@haglund2008finite Proposition 2.17].
\[Bounding hyperplane\] A hyperplane *bounds* a convex cubical subcomplex $Y \subset X$ if it is dual to an edge with endpoints $v \in Y$ and $v' \notin Y$.
Right-angled Coxeter and Artin groups
-------------------------------------
\[Right-angled Coxeter group\] Given a graph $\mathcal{G}$ with vertex set $I$, let $S = \{ s_i : i \in I \}$. *The right-angled Coxeter group* associated to $\mathcal{G}$ is the group $C(\mathcal{G})$ given by the presentation $\langle S \mid s_i^2 = 1 \text{ for } i \in I, [s_i, s_j]=1 \text{ for } (i,j) \in E(\mathcal{G}) \rangle$.
The right-angled Coxeter group $C(\mathcal{G})$ acts on *the Davis–Moussong Complex $DM(\mathcal{G})$* [@haglund2008special]. Througout the paper if we talk about the action of $C(\mathcal{G})$ on a cube complex, we mean this action. The Davis–Moussong complex is similar to Cayley complex, but it doesn’t contain ‘duplicate squares’ and it contains higher dimensional cubes.
Fix a vertex $v_0 \in DM(\mathcal{G})$. There is a bijection between the vertices of $DM(\mathcal{G})$ and the elements of $C(\mathcal{G})$ given by $gv_0 \longleftrightarrow g$. Vertices $g v_0$ and $g s v_0$ are connected by an edge $ge_s$ labelled $s$. If the generators $s_{i_1}, s_{i_2}, \ldots s_{i_n}$ pairwise commute, there is an $n$-cube with the vertex set $\{ g (\Pi_{j \in P} s_{i_j}) v : P \subset \{1, \ldots, n \} \}$.
Note that $g s_i g^{-1}$ acts on the left on $DM(\mathcal{G})$ as a reflection in $H(ge_{s_i})$. There is also a right action of $C(\mathcal{G})$ on $DM(\mathcal{G})^0$, where $s_i$ sends $g v_0$ to $g s_i v_0$ – the vertex to which $g$ is connected by an edge labeled $s_i$. This action does not extend to $DM(\mathcal{G})$.
More generally, if $\Gamma$ is a subgroup of $C(\mathcal{G})$, the action of $C(\mathcal{G})$ on the right cosets of $\Gamma$ can be realized geometrically as an action of $C(\mathcal{G})$ on $\Gamma \backslash DM(\mathcal{G})^0$. This action is given by $ (\Gamma hv_0 ).g= \Gamma hg v_0$. If $\Gamma$ acts on $DM(\mathcal{G})$ co-compactly, this gives a finite permutation action. We will use this to construct maps from $C(\mathcal{G})$ to $S_n$.
If $G$ acts on a cube complex $X$, we say $H < G$ is *quasiconvex* if there is a non-empty convex subcomplex $Y \subset X$, which is invariant under $H$ and moreover $H$ acts on $Y$ cocompactly. We say, that $H$ acts on $X$ with *core* $Y$.
If $G$ is hyperbolic, this coincides with the usual notion of quasiconvexity [@haglund2008finite].
The right-angled Artin group associated to a simplicial graph $\mathcal{G}$ is $A(\mathcal{G}) = \langle g_v: g \in V(\mathcal{G}) \mid g_u g_v = g_v g_u \text{ for } \{u,v\} \in E(\mathcal{G}) \rangle$.
The next lemma relates RAAGs and RACGs.
[@davis2000right] \[RAAGtoRACG\] Given a graph $\mathcal{G}$, define a graph $\mathcal{H}$ as follows:
- $V(\mathcal{H})=V(\mathcal{G}) \times \{0,1\}$
- $(u,1)$ and $(v,1)$ are connected by an edge if $\{u, v\}$ is an edge of $\mathcal{G}$. The $(u,0)$ and $(v,1)$ are connected by an edge if $u$ and $v$ are distinct. Similarly, $(u,0)$ and $(v,0)$ are connected by an edge if $u$ and $v$ are distinct.
The right-angled Artin groups $A(\mathcal{G})$ is a finite-index subgroups of the right-angled Coxeter group $C(\mathcal{H})$ via the inclusion $\iota$ extending $g_u \longrightarrow s_{(u,0)} s_{(u,1)}$.
A right-angled Artin group $A(\mathcal{G})$ acts on Salvetti complex $X=X(\mathcal{G})$, which consists of the following:
- $X^0 = A(\mathcal{G})$
- If generators $g_{u_1}, g_{u_2}, \ldots g_{u_n}$ pairwise commute and $g \in A(\mathcal{G})$, there is a unique $n$-cube with the vertex set $\{ g (\Pi_{j \in P} g_{u_{j}}) : P \subset \{1, \ldots, n \} \}$.
The action of the right-angled group on the vertex set is by the left multiplication and it extends uniquely to the entire cube complex.
For the rest of the paper whenever we talk about the action of a RACG or RAAG on a cube complex, we mean the canonical action on the associated Davis-Moussong Complex or Salvetti complex, respectively.
Jordan’s Theorem
----------------
A subgroup $G < S_n$ is called *primitive* if it acts transitively on $\{1, \ldots ,n \}$ and it does not preserve any nontrivial partition.
If $n$ is a prime and $G$ is transitive, then the action is primitive.
Our main tool is the following.
[@dixon1996permutation Theorem 3.3D] For each $k$ there exists $n$ such that if $G < S_n$ is a primitive subgroup and there exists $\gamma \in G \setminus \{ e \}$, which moves less than $k$ elements, then $G = S_n$ or $A_n$.
The main theorem and its proof
==============================
Our main theorem relates combinatorics of $\mathcal{G}$ and $\mathcal{A}$-separability of $C(\mathcal{G})$.
\[main\] Let $\mathcal{G}$ be a non-discrete finite simplicial graph of size at least $3$. Then all infinite-index quasiconvex subgroups of the right-angled Coxeter group associated to $\mathcal{G}$ are $\mathcal{A}$-separable and $\mathcal{S}$-separable if and only if $\mathcal{G}^c$ is connected.
Recall that here quasiconvex means that it acts cocompactly on a convex subcomplex of Davis-Moussong complex. Similar result holds for RAAGs.
\[AsepRAAG\] Let $\mathcal{G}$ be a finite simplicial graph of size at least $2$. Then all infinite index quasiconvex subgroups of the right-angled Artin group associated to $\mathcal{G}$ are $\mathcal{A}$-separable if and only if $\mathcal{G}^c$ is connected.
Here quasiconvex means that the subgroup acts cocompactly on a convex subcomplex of Salvetti complex. There is another action of the Artin group on a cube complex given by embedding the group in right-angled Coxeter group as described in the Lemma \[RAAGtoRACG\]. We will first show that quasiconvexity with respect to Salvetti complex implies quasiconvexity with respect to Davis-Moussong complex.
\[qcinRACGtoqcinRAAG\] Suppose $\mathcal{G}$ is a simplicial complex, and $K$ a quasiconvex subgroup of $A(\mathcal{G})$ with respect to the action on $X(\mathcal{G})$. Let $\mathcal{H}$ be as in lemma $\ref{RAAGtoRACG}$ and identify $A(\mathcal{G})$ with a subgroup of $C(\mathcal{H})$ in the same lemma. Then $K$ is quasiconvex in $C(\mathcal{H})$ with respect to the action on $DM(\mathcal{H})$.
Recall that $N(H)$ is the union of all cubes intersecting a hyperplane $H$. For a hyperplane $H$ in a $CAT(0)$ cube complex $X$, $N(H) \simeq H \times [0,1]$. We can collapse $N(H)$ onto $H$. Formally, say $(x,t) \sim (x, t')$ for all $x \in H$ and $t,t' \in [0,1]$. *Collapse of neighbourhood of $H$* is the quotient map $X \longrightarrow X / \sim$. We can collapse multiple neighbourhoods simultaneously by quotienting by the smallest equivalence relation, which contains the equivalence relation for each hyperplane.
Let $f: (DM(\mathcal{H}), v_0) \longrightarrow (Y,y)$ be the simultaneous collapse of all hyperplanes labelled by $s_{(v,0)}$ for all $v \in \mathcal{G}$. See Figure \[tree\]. The equivalence relation commutes with the action of $C(\mathcal{H})$, so there is an induced action of $C(\mathcal{H})$ on $Y$.
We collapsed all edges with labels from $\mathcal{G} \times \{0\}$ so for all $s_{(v,0)}$ and all $g \in C(\mathcal{H})$, we have $gs_{(v,0)}.y = g.y$. Let $f': X(\mathcal{G}) \longrightarrow Y$ be defined as follows
- Vertices: Send $g$ to $g.y$.
- Edges: Send the edge between $g$ and $gg_{v}$ to the edge between $g.y$ and $gg_{v}.y$. It is indeed an edge as $g.y = gs_{(v,0)}.y$ and $gs_{(v,0)}s_{(v,1)}.y$
- Squares: Send the square with vertices $g, gg_{v}, gg_u, gg_u g_v$ to the square with vertices $g.y, gg_{v}.y, gg_u.y, gg_u g_v.y$.
- Higher dimensions: Extend analogously.
The map $f'$ is $A(\mathcal{G})$-equivariant cube complex isomorphism.
For any $y \in Y$, we have that $(f')^{-1}(y)$ is contained in a cube of dimension at most $d$, where $d$ is the size of maximal clique in $\mathcal{G}$. Suppose $K$ acts cocompactly on a convex subcomplex $Z$ of $X(\mathcal{G})$. Then $K$ acts cocompactly on $W:= f^{-1} f'(Z) \subset DM(\mathcal{H})$.
It remains to show that $W$ is convex. Let $e$ be an edge in $DM(\mathcal{H})$ with exactly one endpoint in $W$. The edge $e$ is labelled by some $s_{(v,1)}$ as all edges labelled by $s_{(v,0)}$ either lie entirely in $W$ or have an empty intersection with it. Collapsing map sends parallel edges to parallel edges (unless it sends them both to a vertex) and any sequence of elementary parallelisms in codomain lifts to the domain, so $f(H(e)) = H(f(e))$. In particular, if $H(e)$ intersects $W$, then $H(f(e))$ intersects $Z$ and by the convexity of $Z$, $f(e)$ lies entirely in $Z$, which contradicts that $e$ doesn’t lie entirely in $W$.
So a quasiconvex with respect to the action on $X(\mathcal{G})$ implies quasiconvex with respect to the action on $DM(\mathcal{H})$.
(-29.91,-23.18) rectangle (27.36,21.97); (-8,-2)– (10,-2); (-2,4)– (-2,-2); (-8,-2)– (-8,16); (-2,4)– (-8,4); (-8,-2)– (-26,-2); (-14,4)– (-14,-2); (-14,4)– (-8,4); (-2,-8)– (-2,-2); (-8,-2)– (-8,-20); (-2,-8)– (-8,-8); (-8,-2)– (-26,-2); (-14,-8)– (-14,-2); (-14,-8)– (-8,-8); (4,-2)– (10,-2); (6,0)– (6,-2); (4,-2)– (4,4); (6,0)– (4,0); (4,-2)– (-2,-2); (2,0)– (2,-2); (2,0)– (4,0); (6,-4)– (6,-2); (4,-2)– (4,-8); (6,-4)– (4,-4); (4,-2)– (-2,-2); (2,-4)– (2,-2); (2,-4)– (4,-4); (-2,-2)– (10,-2); (4,2)– (6,2); (4.67,2.67)– (4.67,2); (4,2)– (4,4); (4.67,2.67)– (4,2.67); (3.33,2.67)– (3.33,2); (3.33,2.67)– (4,2.67); (4.67,1.33)– (4.67,2); (4,2)– (4,0); (4.67,1.33)– (4,1.33); (3.33,1.33)– (3.33,2); (3.33,1.33)– (4,1.33); (-10,-2)– (8,-2); (2,-2)– (8,-2); (2,2)– (4,2); (8.67,-1.33)– (8.67,-2); (8,-2)– (8,0); (8.67,-1.33)– (8,-1.33); (7.33,-1.33)– (7.33,-2); (7.33,-1.33)– (8,-1.33); (-2,4)– (2,0); (6,0)– (7.33,-1.33); (6,0)– (4.67,1.33); (2,0)– (3.33,1.33); (-8,-2)– (-8,16); (-2,4)– (-8,4); (-2,4)– (-2,-2); (-8,10)– (-8,16); (-6,12)– (-8,12); (-8,10)– (-2,10); (-6,12)– (-6,10); (-8,10)– (-8,4); (-6,8)– (-8,8); (-6,8)– (-6,10); (-10,12)– (-8,12); (-10,12)– (-10,10); (-8,10)– (-8,4); (-10,8)– (-8,8); (-10,8)– (-10,10); (-8,4)– (-8,16); (-4,10)– (-4,12); (-3.33,10.67)– (-4,10.67); (-4,10)– (-2,10); (-3.33,10.67)– (-3.33,10); (-3.33,9.33)– (-4,9.33); (-3.33,9.33)– (-3.33,10); (-4.67,10.67)– (-4,10.67); (-4,10)– (-6,10); (-4.67,10.67)– (-4.67,10); (-4.67,9.33)– (-4,9.33); (-4.67,9.33)– (-4.67,10); (-4,8)– (-4,10); (-7.33,14.67)– (-8,14.67); (-8,14)– (-6,14); (-7.33,14.67)– (-7.33,14); (-7.33,13.33)– (-8,13.33); (-7.33,13.33)– (-7.33,14); (-2,4)– (-6,8); (-6,12)– (-7.33,13.33); (-6,12)– (-4.67,10.67); (-6,8)– (-4.67,9.33); (-8,-2)– (-26,-2); (-14,4)– (-14,-2); (-14,4)– (-8,4); (-20,-2)– (-26,-2); (-22,0)– (-22,-2); (-20,-2)– (-20,4); (-22,0)– (-20,0); (-20,-2)– (-14,-2); (-18,0)– (-18,-2); (-18,0)– (-20,0); (-22,-4)– (-22,-2); (-22,-4)– (-20,-4); (-20,-2)– (-14,-2); (-18,-4)– (-18,-2); (-18,-4)– (-20,-4); (-14,-2)– (-26,-2); (-20,2)– (-22,2); (-20.67,2.67)– (-20.67,2); (-20,2)– (-20,4); (-20.67,2.67)– (-20,2.67); (-19.33,2.67)– (-19.33,2); (-19.33,2.67)– (-20,2.67); (-20.67,1.33)– (-20.67,2); (-20,2)– (-20,0); (-20.67,1.33)– (-20,1.33); (-19.33,1.33)– (-19.33,2); (-19.33,1.33)– (-20,1.33); (-18,2)– (-20,2); (-24.67,-1.33)– (-24.67,-2); (-24,-2)– (-24,0); (-24.67,-1.33)– (-24,-1.33); (-23.33,-1.33)– (-23.33,-2); (-23.33,-1.33)– (-24,-1.33); (-14,4)– (-18,0); (-22,0)– (-23.33,-1.33); (-22,0)– (-20.67,1.33); (-18,0)– (-19.33,1.33); (-8,-2)– (-8,16); (-14,4)– (-8,4); (-14,4)– (-14,-2); (-8,10)– (-8,16); (-10,12)– (-8,12); (-8,10)– (-14,10); (-10,12)– (-10,10); (-8,10)– (-8,4); (-10,8)– (-8,8); (-10,8)– (-10,10); (-6,12)– (-8,12); (-6,12)– (-6,10); (-8,10)– (-8,4); (-6,8)– (-8,8); (-6,8)– (-6,10); (-8,4)– (-8,16); (-12,10)– (-12,12); (-12.67,10.67)– (-12,10.67); (-12,10)– (-14,10); (-12.67,10.67)– (-12.67,10); (-12.67,9.33)– (-12,9.33); (-12.67,9.33)– (-12.67,10); (-11.33,10.67)– (-12,10.67); (-12,10)– (-10,10); (-11.33,10.67)– (-11.33,10); (-11.33,9.33)– (-12,9.33); (-11.33,9.33)– (-11.33,10); (-12,8)– (-12,10); (-8.67,14.67)– (-8,14.67); (-8,14)– (-10,14); (-8.67,14.67)– (-8.67,14); (-8.67,13.33)– (-8,13.33); (-8.67,13.33)– (-8.67,14); (-14,4)– (-10,8); (-10,12)– (-8.67,13.33); (-10,12)– (-11.33,10.67); (-10,8)– (-11.33,9.33); (-2,-8)– (-2,-2); (-8,-2)– (-8,-20); (-2,-8)– (-8,-8); (-8,-2)– (-26,-2); (-14,-8)– (-14,-2); (-14,-8)– (-8,-8); (-2,4)– (-2,-2); (-2,4)– (-8,4); (-8,-2)– (-26,-2); (-14,4)– (-14,-2); (-14,4)– (-8,4); (4,-2)– (10,-2); (6,-4)– (6,-2); (4,-2)– (4,-8); (6,-4)– (4,-4); (4,-2)– (-2,-2); (2,-4)– (2,-2); (2,-4)– (4,-4); (6,0)– (6,-2); (4,-2)– (4,4); (6,0)– (4,0); (4,-2)– (-2,-2); (2,0)– (2,-2); (2,0)– (4,0); (-2,-2)– (10,-2); (4,-6)– (6,-6); (4.67,-6.67)– (4.67,-6); (4,-6)– (4,-8); (4.67,-6.67)– (4,-6.67); (3.33,-6.67)– (3.33,-6); (3.33,-6.67)– (4,-6.67); (4.67,-5.33)– (4.67,-6); (4,-6)– (4,-4); (4.67,-5.33)– (4,-5.33); (3.33,-5.33)– (3.33,-6); (3.33,-5.33)– (4,-5.33); (2,-6)– (4,-6); (8.67,-2.67)– (8.67,-2); (8,-2)– (8,-4); (8.67,-2.67)– (8,-2.67); (7.33,-2.67)– (7.33,-2); (7.33,-2.67)– (8,-2.67); (-2,-8)– (2,-4); (6,-4)– (7.33,-2.67); (6,-4)– (4.67,-5.33); (2,-4)– (3.33,-5.33); (-8,-2)– (-8,-20); (-2,-8)– (-8,-8); (-2,-8)– (-2,-2); (-8,-14)– (-8,-20); (-6,-16)– (-8,-16); (-8,-14)– (-2,-14); (-6,-16)– (-6,-14); (-8,-14)– (-8,-8); (-6,-12)– (-8,-12); (-6,-12)– (-6,-14); (-10,-16)– (-8,-16); (-10,-16)– (-10,-14); (-8,-14)– (-8,-8); (-10,-12)– (-8,-12); (-10,-12)– (-10,-14); (-8,-8)– (-8,-20); (-4,-14)– (-4,-16); (-3.33,-14.67)– (-4,-14.67); (-4,-14)– (-2,-14); (-3.33,-14.67)– (-3.33,-14); (-3.33,-13.33)– (-4,-13.33); (-3.33,-13.33)– (-3.33,-14); (-4.67,-14.67)– (-4,-14.67); (-4,-14)– (-6,-14); (-4.67,-14.67)– (-4.67,-14); (-4.67,-13.33)– (-4,-13.33); (-4.67,-13.33)– (-4.67,-14); (-4,-12)– (-4,-14); (-7.33,-18.67)– (-8,-18.67); (-8,-18)– (-6,-18); (-7.33,-18.67)– (-7.33,-18); (-7.33,-17.33)– (-8,-17.33); (-7.33,-17.33)– (-7.33,-18); (-2,-8)– (-6,-12); (-6,-16)– (-7.33,-17.33); (-6,-16)– (-4.67,-14.67); (-6,-12)– (-4.67,-13.33); (-8,-2)– (-26,-2); (-14,-8)– (-14,-2); (-14,-8)– (-8,-8); (-20,-2)– (-26,-2); (-22,-4)– (-22,-2); (-20,-2)– (-20,-8); (-22,-4)– (-20,-4); (-20,-2)– (-14,-2); (-18,-4)– (-18,-2); (-18,-4)– (-20,-4); (-22,0)– (-22,-2); (-22,0)– (-20,0); (-20,-2)– (-14,-2); (-18,0)– (-18,-2); (-18,0)– (-20,0); (-14,-2)– (-26,-2); (-20,-6)– (-22,-6); (-20.67,-6.67)– (-20.67,-6); (-20,-6)– (-20,-8); (-20.67,-6.67)– (-20,-6.67); (-19.33,-6.67)– (-19.33,-6); (-19.33,-6.67)– (-20,-6.67); (-20.67,-5.33)– (-20.67,-6); (-20,-6)– (-20,-4); (-20.67,-5.33)– (-20,-5.33); (-19.33,-5.33)– (-19.33,-6); (-19.33,-5.33)– (-20,-5.33); (-18,-6)– (-20,-6); (-24.67,-2.67)– (-24.67,-2); (-24,-2)– (-24,-4); (-24.67,-2.67)– (-24,-2.67); (-23.33,-2.67)– (-23.33,-2); (-23.33,-2.67)– (-24,-2.67); (-14,-8)– (-18,-4); (-22,-4)– (-23.33,-2.67); (-22,-4)– (-20.67,-5.33); (-18,-4)– (-19.33,-5.33); (-8,-2)– (-8,-20); (-14,-8)– (-8,-8); (-14,-8)– (-14,-2); (-8,-14)– (-8,-20); (-10,-16)– (-8,-16); (-8,-14)– (-14,-14); (-10,-16)– (-10,-14); (-8,-14)– (-8,-8); (-10,-12)– (-8,-12); (-10,-12)– (-10,-14); (-6,-16)– (-8,-16); (-6,-16)– (-6,-14); (-8,-14)– (-8,-8); (-6,-12)– (-8,-12); (-6,-12)– (-6,-14); (-8,-8)– (-8,-20); (-12,-14)– (-12,-16); (-12.67,-14.67)– (-12,-14.67); (-12,-14)– (-14,-14); (-12.67,-14.67)– (-12.67,-14); (-12.67,-13.33)– (-12,-13.33); (-12.67,-13.33)– (-12.67,-14); (-11.33,-14.67)– (-12,-14.67); (-12,-14)– (-10,-14); (-11.33,-14.67)– (-11.33,-14); (-11.33,-13.33)– (-12,-13.33); (-11.33,-13.33)– (-11.33,-14); (-12,-12)– (-12,-14); (-8.67,-18.67)– (-8,-18.67); (-8,-18)– (-10,-18); (-8.67,-18.67)– (-8.67,-18); (-8.67,-17.33)– (-8,-17.33); (-8.67,-17.33)– (-8.67,-18); (-14,-8)– (-10,-12); (-10,-16)– (-8.67,-17.33); (-10,-16)– (-11.33,-14.67); (-10,-12)– (-11.33,-13.33);
\[Proof of corollary \[AsepRAAG\]\] $\Rightarrow$: If $H,K$ are components of $\mathcal{G}^c$, then $A(\mathcal{G}) = A(H^c) \times A(K^c)$ so by Lemma \[product\] the trivial subgroup $\{e\}$ is not $\mathcal{A}$-separable in $A(\mathcal{G})$.
$\Leftarrow$: Let $\mathcal{H}$ be as in Lemma \[RAAGtoRACG\].
Suppose $U$ is a proper component of $\mathcal{H}^c$. The vertices $(v,0)$ and $(v,1)$ are not connected by an edge in $\mathcal{H}$, so $U^0$ is of the form $V \times \{0,1\}$ for some $V \subsetneq \mathcal{G}^0$. But then looking at $V \times \{1\} \subset \mathcal{G} \times \{1\}$ gives that $V^0$ is a vertex set of a proper component of $\mathcal{G}^c$.
So $\mathcal{G}^c$ connected implies $\mathcal{H}^c$ is connected.
By lemma \[qcinRACGtoqcinRAAG\] the quasiconvexity condition of the Main theorem \[main\] is also satisfied.
(-1.45,-1.32) rectangle (1.69,1.13); (0,0) – (52.63:0.08) arc (52.63:142.63:0.08) – cycle; (0,0) circle (1cm); plot\[domain=1.53:1.978,variable=\]([1\*4.39\*cos(r)+0\*4.39\*sin(r)]{},[0\*4.39\*cos(r)+1\*4.39\*sin(r)]{}); plot\[domain=1.694:2.944,variable=\]([1\*1.39\*cos(r)+0\*1.39\*sin(r)]{},[0\*1.39\*cos(r)+1\*1.39\*sin(r)]{}); plot\[domain=0.482:1.697,variable=\]([1\*1.43\*cos(r)+0\*1.43\*sin(r)]{},[0\*1.43\*cos(r)+1\*1.43\*sin(r)]{}); (0.09,-0.61)– (-0.15,-0.43); (-0.15,-0.43)– (0.02,-0.21); (0.05,-0.62) node\[anchor=north west\] [$y$]{}; (-0.17,-0.47) node\[anchor=north west\] [$e_1$]{}; (-0.15,-0.17) node\[anchor=north west\] [$e_2$]{}; (0.28,-0.18) node\[anchor=north west\] [$H(e_1)$]{}; (0.27,-0.48) node\[anchor=north west\] [$H(e_2)$]{}; (-0.2, 0.05) node\[anchor=north west\] [$L$]{};
(0.09,-0.61) circle (1pt); (-0.15,-0.43) circle (1pt); (0.02,-0.21) circle (1pt); (0.06,-0.41) circle (1pt); (0.06,-0.3) node [$90\textrm{{\ensuremath{^\circ}}}$]{};
[@scott1985correction Correction to the proof of Theorem 3.1] \[fgtoqc\] A closed, orientable, hyperbolic surface group $G$ is a finite index subgroup of $C(C_5)$, where $C_5$ is a cycle of length $5$. Moreover, for a suitable embedding $G \xhookrightarrow{} C(C_5)$, all finitely generated subgroups of $G$ are quasiconvex in $C(C_5)$ with respect to the action on $DM(C_5)$.
Scott uses a different terminology, so it makes sense to summarise the proof. The natural generators of $C(C_5)$ act on the hyperbolic plane by reflections in the sides of a right-angled pentagon. Translates of the pentagon give a tiling of the hyperbolic plane. Dual to this cell complex is a square complex $DM(C_5)$. Under this identification, the geodesic lines bounding the the pentagons of the tiling become hyperplanes of $DM(C_5)$.
Suppose $H$ is a finitely generated subgroup of the surface group $G = \pi_1 (\Sigma)$. Let $\Sigma_H$ be the covering space associated to $H$. By Lemma 1.5 in $\cite{scott1978subgroups}$, there exists a closed, compact, incompressible subsurface $\Sigma' \subset \Sigma_H$ such that the induced map $\pi_1 \Sigma' \longrightarrow \pi_1\Sigma_H$ is surjective. Moreover, by [@scott1985correction Correction to the proof of Theorem 3.1] we can require $\Sigma'$ to have a geodesic boundary.
Let $\widetilde{\Sigma'}$ be the lift of $\Sigma '$ to $\mathbb{H}^2 = DM(C_5)$. Let $Y$ be the intersection of all half-spaces containing $\widetilde{\Sigma'}$. Suppose $y$ lies in $Y$, but not in $N_3(\widetilde{\Sigma'})$ and that $e_1, e_2$ are the first two edges of the combinatorial geodesic from $y$ to $\widetilde{\Sigma'}$. Since $y \in Y$, both $H(e_1)$ and $H(e_2)$ intersect $\widetilde{\Sigma'}$. Consequently, $H(e_1)$ intersects $H(e_2)$ as $H(e_2)$ does not separate $H(e_1)$ from $\widetilde{\Sigma'}$. Call the intersection $y'$. The point $y$ is a centre of a pentagon and $y'$ is a vertex of the same pentagon, so the distance between them doesn’t depend on $y$ (for example by specialness of $DM(\mathcal{G})$). See Figure \[correction\].
The closest boundary component $L$ of $\widetilde{\Sigma'}$ to $y$ is seen from $y'$ at more than the right angle (remember that the hyperplanes are geodesics). But such a point is within distance $\int_{t=0 }^{\pi/4} \frac{1}{\cos(t)}dt$ of $L$. To see this, take $L$ to be the vertical ray through $(0,0)$ in the upper half-plane model to the. Then the set of points with obtuse subtended angle is contained between rays $y = x$ and $y =-x$. Geodesic between these rays and $L$ is an arc of length $\int_{t=0}^{\pi/4} \frac{r \sqrt{\cos^2(t) + \sin^2(t)}}{r \cos(t)} dt = \int_{t=0 }^{\pi/4} \frac{1}{\cos(t)}dt$.
(-0.66,-0.26) rectangle (1.05,1.09); (0.5,0.53) – (0.47,0.53) – (0.47,0.5) – (0.5,0.5) – cycle; (0,0) – (-25.22:0.04) arc (-25.22:90:0.04) – cycle; plot(,[(-0-0\*)/1]{}); (0,0) – (0,1.09); plot\[domain=0:3.14,variable=\]([1\*0.5\*cos(r)+0\*0.5\*sin(r)]{},[0\*0.5\*cos(r)+1\*0.5\*sin(r)]{}); (0.5,0.5) – (0.5,1.09); plot(,[(-0–0.5\*)/0.5]{}); plot(,[(-0–0.5\*)/-0.5]{}); (-0.08,0.65) node\[anchor=north west\] [$L$]{}; plot\[domain=0.79:1.57,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=0:1.13,variable=\]([1\*0.8\*cos(r)+0\*0.8\*sin(r)]{},[0\*0.8\*cos(r)+1\*0.8\*sin(r)]{}); (-0.46,0.73) – (-0.46,1.09); (-0.54,0.76) node\[anchor=north west\] [$$y'$$]{}; (0.21,0.72) node\[anchor=north west\] [a]{};
(0.44,0.53) node [$90\textrm{{\ensuremath{^\circ}}}$]{}; (-0.46,0.73) circle (0.5pt); (-0.35,0.75) node [$>90\textrm{{\ensuremath{^\circ}}}$]{};
Therefore $y'$ (and hence $y$) is at a uniformly bounded distance from $\widetilde{\Sigma'}$ and the action of $H$ on $Y$ is cocompact.
\[surfaces\] All finitely generated infinite index subgroups of closed, orientable, hyperbolic surface group $G$ are $\mathcal{A}$-separable in $G$.
By Lemma \[fgtoqc\], finitely generated subgroups of $G$ are quasiconvex in $C(C_5)$. By the main theorem \[main\] they are $\mathcal{A}$-separable in $C(C_5)$. By Lemma \[fi\], they are $\mathcal{A}$-separable in $G$.
\[Disjoint hyperplanes, bounding hyperplanes, positive half-space\] Let $X$ be a cube complex, $Y$ a convex subcomplex. Let $\mathcal{D}(Y)$ be the set of hyperplanes disjoint from $Y$. Let $\mathcal{B}(Y)$ the set of hyperplanes bounding $Y$.
If $H \in \mathcal{D}(Y)$, denote by *$H^+$* the half-space of $X\backslash \backslash H$ containing $Y$.
Recall that any intersection of half-spaces is convex and conversely any convex subcomplex is an intersection of the half-spaces containing it. Hence it is equivalent to specify a convex subcomplex or the half-spaces in which it is contained (or the set of disjoint hyperplanes if there can be no confusion about the choice of half-spaces, e.g. if only one choice gives a non-empty intersection).
Suppose $G$ acts on a cube complex $X$ with core $Y$. Define *deletion* as removing a bounding hyperplane $H_0$ and all its $G$-translates from $\mathcal{D}(Y)$. That is the result of deletion of $H_0$ is $Y' = \cap_{H \in \mathcal{D}(Y) \setminus G. \{H_0 \} } H^+$.
The cube complex $V = H_0^- \cap Y'$ is called a *vertebra*. See Figures \[fig:pentagons\] and \[vertebra\].
(-4.37,-3.17) rectangle (6.57,5.42); (0,0)– (0,1); (0,0)– (1,0); (1,1)– (1,0); (1,1)– (1,1.56); (0.71,1.71)– (0.85,1.71); plot\[domain=3.93:4.71,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([-1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); (2,0)– (2,1); (2,0)– (1,0); (1,1)– (1,1.56); (1.29,1.71)– (1.15,1.71); plot\[domain=3.93:4.71,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([-1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); (4,0)– (4,1); (4,0)– (3,0); (3,1)– (3,0); (3,1)– (3,1.56); (3.29,1.71)– (3.15,1.71); (2,0)– (3,0); (3,1)– (3,1.56); (2.71,1.71)– (2.85,1.71); plot\[domain=3.93:4.71,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([-1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([-1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); (8,0)– (8,1); (8,0)– (7,0); (7,1)– (7,0); (7,1)– (7,1.56); (7.29,1.71)– (7.15,1.71); (6,0)– (6,1); (6,0)– (7,0); (7,1)– (7,1.56); (6.71,1.71)– (6.85,1.71); (4,0)– (5,0); (5,1)– (5,0); (5,1)– (5,1.56); (4.71,1.71)– (4.85,1.71); (6,0)– (5,0); (5,1)– (5,1.56); (5.29,1.71)– (5.15,1.71); plot\[domain=3.93:4.71,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([-1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([-1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([-1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([-1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([-1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); plot\[domain=3.93:4.71,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=4.71:5.5,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=-0.79:0,variable=\]([1\*0.71\*cos(r)+0\*0.71\*sin(r)]{},[0\*0.71\*cos(r)+1\*0.71\*sin(r)]{}); plot\[domain=3.142:4.71,variable=\]([1\*0.15\*cos(r)+0\*0.15\*sin(r)]{},[0\*0.15\*cos(r)+1\*0.15\*sin(r)]{}); (0,0)– (-1,0); (-1,1)– (-1,0); (-1,1)– (-1,1.56); (-0.71,1.71)– (-0.85,1.71); (-2,0)– (-2,1); (-2,0)– (-1,0); (-1,1)– (-1,1.56); (-1.29,1.71)– (-1.15,1.71); (-4,0)– (-4,1); (-4,0)– (-3,0); (-3,1)– (-3,0); (-3,1)– (-3,1.56); (-3.29,1.71)– (-3.15,1.71); (-2,0)– (-3,0); (-3,1)– (-3,1.56); (-2.71,1.71)– (-2.85,1.71); (-8,0)– (-8,1); (-8,0)– (-7,0); (-7,1)– (-7,0); (-7,1)– (-7,1.56); (-7.29,1.71)– (-7.15,1.71); (-6,0)– (-6,1); (-6,0)– (-7,0); (-7,1)– (-7,1.56); (-6.71,1.71)– (-6.85,1.71); (-4,0)– (-5,0); (-5,1)– (-5,0); (-5,1)– (-5,1.56); (-4.71,1.71)– (-4.85,1.71); (-6,0)– (-5,0); (-5,1)– (-5,1.56); (-5.29,1.71)– (-5.15,1.71); (-1.1,0.65) node\[anchor=north west\] [$ s_1 $]{}; (-0.10,0.65) node\[anchor=north west\] [$ s_4 $]{}; (-0.90,1.13) node\[anchor=north west\] [$ s_2 $]{};
A vertebra is an intersection of two combinatorially geodesically convex sets, so it also is combinatorially geodesically convex. In particular, it is connected.
We say $G$ acts *without self-intersections* on a cube complex $X$, if $N(gH) \cap N(H) \neq \emptyset$ implies $gH = H$ for all hyperplanes $H$ of $X$ and $g \in G$.
\[scale=2\] (1,0) – (0,0); (1,0) node\[anchor=north\] [$V$]{}; (0,0) – ($ (-1/2,{sqrt(3)/2}) $); (0,0) – ($ (-1/2,{-sqrt(3)/2}) $); ($ (-1/2,{sqrt(3)/2}) $) – ($ (-1/2,{sqrt(3)/2}) +(1/2,{sqrt(3)/2})$); ($ (-1/2,-{sqrt(3)/2}) $) – ($ (-1/2,-{sqrt(3)/2}) +(1/2,{-sqrt(3)/2})$); ($ (-1/2,{sqrt(3)/2}) $) – ($ (-1/2,{sqrt(3)/2}) -(1,0)$); ($ (-1/2,{-sqrt(3)/2}) $) – ($ (-1/2,{-sqrt(3)/2}) -(1,0)$); (1/2,0.05) – (1/2,-0.05) node\[midway,anchor=north\] [$H_0$]{}; (0,-2) – (-3/2,-2) node\[midway,anchor=north\] [$Y'$]{}; (1,-2.3) – (-3/2,-2.3) node\[midway,anchor=north\] [$Y$]{};
\[lemmaA\] Suppose that $G$ acts without self-intersections on a locally compact $CAT(0)$ cube complex $X$ with core $Y$. Then the result $Y'$ of deletion of $H_0$ is also a core for $G$. Let $G_{H_0} (:= \{g \in G | g.H_0 = H_0 \}$ be the stabilizer of $H_0$ in $G$. If $C$ is a set of orbit representatives for the action of $G$ on the vertices of $Y$ and $D$ is a set of orbit representatives for the action of $G_{H_0}$ on the vertices of the vertebra $V = H_0^- \cap Y'$, then $C' = C \sqcup D$ is a set of orbit representatives for the action of $G$ on the vertices of $Y'$.
Recall that $CAT(0)$ implies special.
First note that $\mathcal{D}(Y') = \mathcal{D}(Y) \setminus G. \{H_0 \}$ by definition and $\mathcal{B}(Y) \setminus G. \{H_0 \} \subset \mathcal{B}(Y')$ as a bounding hyperplane $Y$ still bounds $Y'$ unless it is a translate of $H_0$.
The set of half-spaces containing $Y$ is invariant under $G$, hence $Y'$ is invariant. The subcomplex $Y'$ is an intersection of half-spaces, hence convex. Suppose $v \in Y' \setminus Y$. Let $v_0, v_1 \ldots v_k$ be a combinatorial geodesic from $v$ to $Y$ of shortest length with edges $e_1, \ldots e_k$ and suppose $k>1$. Let’s $H_i$ be the hyperplane dual to $e_i$. Then as $v_{k-1} \notin Y$, we have $H_k \in G. \{H_0\}$. Since $G$ acts on $X$ without self-intersections $H_{k-1} \notin G. \{H_0\}$. And $H_{k-1} \notin \mathcal{D}(Y')$, because $v_0, v_k \in Y'$ and $Y'$ is convex, so $e_{k-1} \in Y'$
Therefore $H_{k-1} \notin \mathcal{D}(Y)$. It must intersect $Y$, so it is not entirely contained in $H_k^-$ and it intersects $H_k$. Because the cube complex is special, $H_k$ and $H_{k-1}$ don’t interosculate. In particular, there is a square with two consecutive sides $e_{k-1}$ and $e_k$. Let $e'_j$ be the edge opposite $e_j$ in this square. We can now construct a shorter path from $v_0$ to $Y$ with edges $e_1, \ldots, e_{k-2}, e'_{k}$. Contradiction.
So $k \leq 1$ and $Y'$ lies in a $1$-neighbourhood of $Y$ and therefore the action is cocompact.
There is a unique edge connecting $v \in Y' \setminus Y$ to $Y$ as any path of length $2$ is a geodesic or is contained in some square. In the first case by convexity of $Y$, we have $v \in Y$. In the second, $H_0 \notin \mathcal{D}(Y)$.
By invariance of $Y$, the $G$-translates of $V$ don’t intersect $Y$. Suppose $v \in Y' \setminus Y$. There is a unique hyperplane in $G. \{H_0 \}$ dual to an edge $e_1$, which connects $v$ to $Y$. Say $g.H_0$. Then $v$ belongs to a unique translate of $V$, namely $g.V$.
Let $\mathcal{G}$ be a finite simplicial graph. If $K$ is a subgroup of a right-angled Coxeter group $C(\mathcal{G})$ and it acts on the Davis-Moussong complex with core $Y$, then deletion produces another core.
The Davis-Moussong complex $DM(\mathcal{G})$ is a $CAT(0)$ cube complex, hence simply connected special. The action of $C(\mathcal{G})$ on it preserves labels. In any square the consecutive edges have distinct labels, so the action is without self-intersections. The restriction to $K$ is also without self-intersections.
\[intersection\] Suppose $G$ acts on a $CAT(0)$ cube complex $X$ with core $Y$. If $Y' \subset X$ is constructed from $Y$ using a deletion of $H = H(e)$, then each edge in $V = H^- \cap Y'$ is dual to a hyperplane intersecting $H$.
Let $e'$ be an edge in $V$ and $H'$ a hyperplane dual to $e'$. If $H' \cap H = \emptyset$, $H'$ is contained entirely in $H^-$. But then $H'$ is disjoint from $Y$. In particular one of the endpoints of $e'$ is in the opposite half-space of $X \backslash \backslash H'$ than $Y$.
Since $Y'$ is the intersection of all half-spaces containing $Y$ with the exception of the $G$-translates of $H^+$, the hyperplane $H'$ is $gH$ for some $g \in G$.
The subcomplex $Y$ is $G$-invariant and $H$ bounds $Y$, hence $H'$ bounds $Y$. This contradicts $H' \subset H^-$.
\[commutation\] Suppose $G<C(\mathcal{G})$ acts on $DM(\mathcal{G})$ with core $Y$. If $Y' \subset X$ is constructed from $Y$ using a deletion of $H = H(e)$, then each edge in $V = H^- \cap Y'$ has a label which commutes with the label of $e$.
Suppose $Y$ is a subcomplex of $X$ and $p=e_1 e_2 \ldots e_n$ is a path in $X$ and then *deletion of hyperplanes along the path $p$* is the deletion of $H(e_1), H(e_2), \ldots H(e_n)$.
If $v \in X$, and $s_1, s_2, \ldots s_n$ is a sequence of edge labels, then *the deletion with labels $s_1, s_2, \ldots s_n$ at $v$* is the *deletion of hyperplanes along $p$*, where $p$ is a path $e_1, e_2, \ldots e_n$ starting at $v$ with $e_i$ labelled $s_i$.
Suppose $Y_n$ was built from $Y_0$ using a series of deletion of hyperplanes $H_1, \ldots H_n$. We call $T=Y_n \cap H_1^-$ *a tail*. Moreover, if $H_j$ corresponds to $s_{i_j}$ and there is no confusion about the initial vertex of the path, we say that $Y_n$ was built from $Y_0$ with respect to $i_1, \ldots i_n$.
\[reduce\] Suppose $\mathcal{G}$ is a finite simplicial graph. Suppose $\mathcal{G}^c$ is connected, $| \mathcal{G}| >1$ and $H$ acts on $DM(\mathcal{G})$ with a core $Y \subsetneq DM(\mathcal{G})$. Then there exists a core $Y'$ which can be obtained from $Y$ by deletion along a path $e_1, e_2 \ldots e_n$ with the vertebra $Y' \cap H(e_n)^-$ a single vertex.
The hypothesis that $\mathcal{G}^c$ is connected is necessary. Consider the situation when $\mathcal{G}$ is a square. Then $C(\mathcal{G}) = D_\infty \times D_\infty$ and $DM(\mathcal{G})$ is the standard tiling of $\mathbb{R}^2$. Let $H = D_\infty$ be the subgroup generated by two non-commuting generators of $C(\mathcal{G})$. The invariance of the core and cocompactness of the action imply that any core for $H$ is of the form $\mathbb{R} \times [k,l]$ for some $k,l \in \mathbb{Z}$.
Every hyperplane intersecting such a core divides it into two infinite half-spaces.
Since $Y$ is a proper subcomplex, there exists $e_1$ be such that $H(e_1) = H_1$ bounds $Y$. Let $v_0$ be the endpoint of $e_1$, which lies in $Y$. Let $v_1$ be the other endpoint. Say the label of $e_1$ is $s_1$. Let $Y_1$ be a cube complex obtained from $Y$ by deletion of $H_1$.
Let $S_1$ be the set of generators labelling the edges of vertebra $V_1$. Then by corollary $\ref{commutation}$, $s_1$ commutes with all generators in $S_1$.
If $e_2 \notin V_1$ is an edge with endpoint $v_1$, whose label $s_2$ does not commute with $s_1$, we can define $H_2, Y_2, V_2$ and $S_2$ similarly as before. Just as before the generators of $S_2$ commute with $s_2$
The hyperplanes $H_1$ and $H_2$ don’t intersect, so $N(H_2) \subset H_1^-$. There is an inclusion of $V_2$ into $V_1$ given by sending a vertex of $V_2$ to the unique vertex of $V_1$ to which it is connected by an edge labelled $s_2$. Extending this map to edges and cubes is a label preserving map between cube complexes $V_2$ and $V_1$. It follows that $S_2$ is a (not necessarily proper) subset of $S_1$.
We will now show that, by a series of such operations, we can reach a situation where $S_n = \emptyset$. I.e. the vertebra $V_n$ is a single vertex.
Suppose we have already applied deletion $i$ times and $S_i$ is non-empty. We will use a series of deletions to get $S_{k+1} \subsetneq S_{k} \subset S_{k-1} \subset \ldots \subset S_{i+1} \subset S_i$. By an abuse of notation, we’ll identify the vertices of $\mathcal{G}^c$ with the labels and with the generators of the right-angled Coxeter group. (Rather than having a generator $s_v$ for every vertex $v \in V(\mathcal{G})$ and using these as labels.)
Since the group does not split as a product, there exists some $a \in S_i$ and $b \notin S_i$ which don’t commute. Since $\mathcal{G}^c$ is connected, there exists a vertex path $s_{i-1}, \ldots s_k = b$ in $\mathcal{G}^c$ from the vertex $s_{i-1}$, which is the label of the hyperplane we removed last.
Succesive generators in this path don’t commute. Indeed assume that $s_j$ and $s_{j+1}$ commute. Take $v \in DM(\mathcal{G})$, let $e_1, e_2, e_3, e_4$ be edges of the path starting at $v$ with labels $s_j, s_{j+1}, s_j, s_{j+1}$. This is a closed loop, since $s_j$ and $s_{j+1}$ commute. The hyperplane $H(e_1)$ separates $v$ from $s_j s_{j+1} v$, so it has to be dual to one of $e_3$ and $e_4$. Parallel edges have the same labels so $H(e_1) = H(e_3)$. Similarly $H(e_2) = H(e_4)$. The hyperplane $H(e_1)$ separates $e_2$ from $e_4$, so $H(e_1)$ and $H(e_2)$ have to cross. Davis-Moussong complex is special, so there is a square where $e_1$ and $e_2$ are successive edges. By construction of the complex, $s_j$ and $s_{j+1}$ are connected by an edge is $\mathcal{G}$. This contradicts adjacency of $s_j$ and $s_{j+1}$ in $\mathcal{G}^c$.
Apply deletion of hyperplanes labeled $s_i, \ldots , s_k$ starting at some vertex of $v \in V_{i-1}$. Note that the $j$th hyperplane we remove belongs in a subset of $\mathcal{B}(Y_{j-1})$ as $s_i \ldots s_{j-1} v \in V_{j-1}$ and $s_j$ does not commute with $s_{j-1}$. Moreover, $S_j = \{s \in S_{j-1} : ss_j = s_j s\}$. In particular, $S_{k+1} \subset S_i$ and $a$ does not belong to $S_{k+1}$ as $as_k \neq s_ka$. Similarly, the hyperplane $H'$ dual to edge between $v$ and $s_{j+1}$ is dual
Therefore $S_{k+1}$ is a proper subset of $S_i$ and we can continue this process until we get an empty $S_n$.
We can even control the label of the hyperplane which was removed last. Indeed, if the last removed hyperplane had label $s_i$, and $b$ is some other generator, pick a vertex path between $s_i$ and $b$ in $\mathcal{G}^c$. Then remove hyperplanes labelled by vertices on this path, starting at the unique vertex of a vertebra.
By lemma \[lemmaA\] there is a set of orbit representatives $K$ for the action of $G$ on $Y_n$ with $T \subset K$.
Haglund shows the following [@haglund2008finite Proof of Theorem A].
\[Scott\] Suppose $G < C(\mathcal{G})$ acts on $DM(\mathcal{G})$ with a core $Y$ with a set of orbit representatives $K$. Let $\Gamma_0 <C(\mathcal{G})$ be generated by the reflections in the hyperplanes bounding $Y$. Let $\Gamma_1 = \Gamma_1 (Y)= \langle G, \Gamma_0 \rangle$. Then $Y$ is a fundamental domain for the action of $\Gamma_0$ on $X$ and $K$ is a a set of orbit representatives for the action of $\Gamma_1$ on $X$.
Let $C(\mathcal{G})$ act on the right cosets of $\Gamma_1 < C(\mathcal{G})$. We have that $s \in S$ sends $\Gamma_1 g$ to $\Gamma_1 g s = (\Gamma_1 g s g^{-1}) g$. But $gs g^{-1}$ is a reflection in the hyperplane $H(ge_s)$. By definition of $\Gamma_0$ if $H(ge_s)$ bounds $Y$, $gs g^{-1} \in \Gamma_0$ and $\Gamma_1 g$ is fixed by $s$.
Moreover, if $K = \{g_1 v_0, \ldots ,g_n v_0 \}$, then $\{g_0, \ldots g_n\}$ is a set of right coset representatives for $\Gamma_1$.
We will first prove that by a suitable sequence of deletions, we can satisfy the conditions of Jordan’s theorem. It follows that we can construct quotients that are either alternating or symmetric.
If $Y$ is a subset of $X$, then $N_1(Y)$ is union of closed cubes, which have non-empty intersection with $Y$. We define inductively $N_i(Y) = N_1(N_{r-1}(Y))$.
If $Y$ is convex, then so is $N_r(Y)$ (as a neighbourhood is obtained by removing bounding hyperplanes and therefore it is an intersection of convex subcomplexes). And if $H$ acts cocompactly on $Y$, it still acts cocompactly on $N_r(Y)$.
Let $C(\mathcal{G})$ be the right-angled Coxeter group associated to $\mathcal{G}$ a finite simplicial graph, $|\mathcal{G}| >2$ , and $H$ acts with a proper core $Y$. Let $\mathcal{C}$ be the class of symmetric and alternating groups. If $\mathcal{G}^c$ is connected $H$ is $\mathcal{C}$-separable.
As $H$ acts with a proper core, there exists a generator of $C(\mathcal{G})$ not contained in $H$. Say $s_0 \notin H$.
Suppose $\gamma_1, \ldots \gamma_n \notin H$.
Fix $v \in Y$. Without loss of generality, we may assume that $Y$ contains $N(v)$ and $\gamma_i v$ for all $i$ (otherwise replace $Y$ with $N_r(Y)$ for a sufficiently large $r$). Moreover, by lemma \[reduce\] we may assume that there exists a hyperplane $H_0 \notin \mathcal{D}(Y)$ with $|H_0^- \cap Y| = 1$ and by the remark after the proof the label of $H_0$ is $s_0$.
As $\mathcal{G}^c$ is connected, there exists a generator $s_1$ not commuting with $s_0$. Let $v_0$ be the unique vertex of $H_0^- \cap Y$. Delete $k$ hyperplanes labelled alternately by $s_1$ and $s_0$ starting at $v_0$ – delete hyperplanes $H(e_{s_1}), H(s_1 e_{s_2}), H(s_1 s_2 e_{s_3}), \ldots , H(s_1 \ldots s_{k-1} e_{s_k})$ etc. where $k$ is to be specified later and $e_{s_i}$ is the edge labelled $s_i$ starting at $v_0$. Call the resulting core $Y'$.
Let $\Gamma_0$ be the group generated by reflections in hyperplanes bounding $Y'$. Let $\Gamma_1 = \langle \Gamma_0, H \rangle$. Then $[C(\mathcal{G}): \Gamma_1] = |H \setminus Y'|$, where $|H \setminus Y'|$ denotes the number of vertices of $H \setminus Y'$. A suitable choice of $k$ makes this a prime. As $\Gamma_1 \setminus C(\mathcal{G}) \cong H \setminus Y'$ and $\gamma_i v \notin H.v$, we may choose $\gamma_i$ as one of the coset representatives. In particular, $\gamma_i$ does not fix $\Gamma_1$, so it does not act as an element of $H$.
Let $s_3$ be a generator distinct from $s_1$ and $s_2$.
By the remark after lemma \[Scott\], $s_3$ fixes the cosets corresponding to the vertices of the tail. So it moves at most $|H \setminus Y|$ elements. By taking $k$ large enough so that $|H \setminus Y'|$ is still a prime, we may ensure that the conditions of Jordan’s lemma are satisfied.
Changing parity
===============
We shall now prove that we may force the action to be alternating (similarly we can force it to be symmetric).
*The parity of $s_i$ with respect to the core $Y$* is the parity of $s_i$ acting on the right cosets of $\Gamma_1 (Y)$.
We will modify the construction of the tail in order to make each $s_i$ act as an even permutation (or we will make at least one of $s_i$ acts as an odd permutation).
Suppose $g.v_0$ is in the tail. If the edge between $g.v_0$ and $gs.v_0$ is in the tail, then $g.v_0$ and $gs.v_0$ map to distinct vertices in $\Gamma_1 \setminus X$, hence $\Gamma_1 g \neq \Gamma_1 g s$.
If $gs.v_0$ is not in the tail, then the hyperplane dual to this edge bounds $Y$ and the reflection in this hyperplane belongs to $\Gamma_1$. Therefore $\Gamma_1 = \Gamma_1 g s g^{-1}$ or equivalently $\Gamma_1 g = \Gamma_1 g s$.
More precisely, suppose $H$ acts with core $Y$ and $Y'$ is the core resulting from deletion of $H_0, \ldots, H_k$, and the label of $H_i$ is $s_i$. Moreover assume $H_0 \cap Y'$ is a single edge.
Then the parity of $s_1$ with respect to $Y'$ is the sum of the parity of $s_1$ with respect to $Y$ and the number of edges labelled $s_1$ in $H_0^- \cap Y'$. So we can control parity of $s_1$ by changing the number of edges with label $s_1$ in the tail. Suppose that the conditions of Jordan’s theorem are satisfied with a margin $M$ (i.e. the conditions are satisfied even if $s_3$ moves $|H \setminus Y| + M$ elements). Taking $M = (|\mathcal{G}|-2) (2d+1)+16$, where $d$ is the diameter of $\mathcal{G}^c$ will be sufficient.
First let us show that we can deal with parity of all generators other than $s_1$ and $s_2$.
\[parity\] For any $i \in I \setminus \{1,2\}$, if the tail of $Y$ is a path with labels $s_1, s_2, \ldots s_1, s_2, s_1$ of length at least $2d_{\mathcal{G}^c}(v_1,v_i)+1$ starting at vertex $V$, then there exists a core $Y'$ such that in the associated action the parity of $s_i$ changed and the parities of no $s_j$ changed for $j \in I \setminus \{1,2,i\}$. Moreover, $|H \setminus Y| = |H \setminus Y'|$ and $Y'$ contains a tail of the same length as $Y$ and the labels of these two paths are the same with the exception of a subpath labeled $s_1, s_2, \ldots s_1, s_2, s_1$ of length $2d_{\mathcal{G}^c}(v_1,v_i)+1$.
(0,0) – (10,0) ; (2,4) – (-6,4); in [0,4,8]{} (,0) – node\[anchor=east\] [$s_1$]{} (,1); in [2,6,10]{} (,0) – node\[anchor=east\] [$s_2$]{} (, 2); in [2,-2,-6]{} (,4) – node\[anchor=east\] [$s_2$]{} (,2); in [0,-4]{} (,4) – node\[anchor=east\] [$s_1$]{} (, 3);
in [-2,2,6]{} ([-1]{},2) – ([+1]{},2); in [1,5,9]{} (,0) node\[anchor=north\] [$s_3$]{}; in [3,7]{} (,0) node\[anchor=north\] [$s_4$]{}; in [-5.5,-2.5,-1.5]{} (,2) node\[anchor=north\] [$s_5$]{}; in [1.5,2.5,5.5,6.5,9.5]{} (,2) node\[anchor=south\] [$s_5$]{}; in [-3,1]{} (,4) node\[anchor=south\] [$s_3$]{};
(0,2) node\[anchor=south west\] [$s_4$]{}; (0,2) node\[anchor=south east\] [$s_4$]{}; (0,2) node\[anchor=north west\] [$s_4$]{}; (-4,2) node\[anchor=south west\] [$s_4$]{}; (-4,2) node\[anchor=south east\] [$s_4$]{}; (4,2) node\[anchor=north west\] [$s_4$]{}; (4,2) node\[anchor=north east\] [$s_4$]{}; (8,2) node\[anchor=north west\] [$s_4$]{}; (8,2) node\[anchor=north east\] [$s_4$]{};
(-6,2) – (-5,2); (10,2) – (9,2); (-3,2) arc (0:180:1); (0,1) arc (-90:180:1); (3,2) arc (-180:0:1); (7,2) arc (-180:0:1);
Say $v_1 = v_{i_0}, v_{i_1}, \ldots v_{i_d} = v_i$ is a path in $\mathcal{G}^c$ of the shortest length. Let $Y'$ be a subcomplex built using deletions of hyperplanes $s_{i_0}, s_{i_1}, \ldots s_{i_d}, s_{i_{d-1}}, \ldots, s_{i_0}, s_2, s_1, \ldots s_1$ starting at $v$.
Compared to $Y$, the tail of this complex contains two more edges labeled $s_{j_i}$ for $0 < j < d$. It also contains an extra edge labeled $s_{i_d} = s_i$, so the parity of $s_i$ changed and the parity of other generators $s_j $ remains the same for $j \neq 1,2,i$.
Now let’s change the parity of a generator that appears in the tail.
\[squares\] If the tail of $Y$ contains a path with labels $s_1, s_2, \ldots s_1, s_2, s_1$ of length at least $7$, then there exists a core $Y'$ such that in the associated action only the parity of $s_1$ changed. Moreover, $|H \setminus Y| = |H \setminus Y'|$ and $Y'$ is built from the same complex as $Y$ using a sequence of deletions, whose labels agree with that of $Y$ with the exception of $5$ deletions. (We allow a deletion to be replaced by no deletion.)
(0,0) node\[anchor=north\] [$v_1$]{} circle (0.05); (1,0) node\[anchor=north\] [$v_2$]{} circle (0.05); (1/2,[sqrt(3)/2]{}) node\[anchor=south\] [$v_3$]{} circle (0.05); (1,0) – (1/2,[sqrt(3)/2]{});
(2,0) – node\[anchor=south east\] [$s_2$]{} ([2+sqrt(2)]{},[sqrt(2)]{}) – node\[anchor= south west\] [$s_1$]{} ([2+2\*sqrt(2)]{},0) – node\[anchor=south east\] [$s_2$]{} ([2+3\*sqrt(2)]{},[sqrt(2)]{});
(7,0) – node\[anchor=south east\] [$s_2$]{} ([7+sqrt(2)]{},[sqrt(2)]{}) – node\[anchor=south west\] [$s_3$]{} ([7+2\*sqrt(2)]{},0) – node\[anchor=north west\] [$s_2$]{} ([7+sqrt(2)]{},[-sqrt(2)]{}) – node\[anchor=north east\] [$s_3$]{} (7,0);
1. Suppose there is some $s_3$ commuting with $s_2$, but not with $s_1$. Then instead of the deletion of the hyperplanes $s_2, s_1, s_2$, delete the hyperplanes labelled $s_2, s_3$. This creates a square. Continue building the tail starting from one of the vertices of the square using the deletions of the hyperplanes with the same labels as before. The new tail contains the same number of $s_2$ labels, two more of $s_3$ and one fewer $s_1$. Hence only the parity of $s_1$ changed.
To be precise, we need to take the path labelled $s_1, s_2, s_1$ which is a subpath of a path labelled $s_2,s_1,s_2,s_1,s_2$ in the tail, as otherwise deleting a hyperplane labelled $s_3$ could potentially introduce more than just a side of a square. Similarly for the other cases in this proof.
2. Suppose there is some $s_3$ commuting with $s_1$, but not $s_2$. Then instead of the deletion of the hyperplanes labelled $s_1, s_2, s_1, s_2, s_1$, delete the hyperplanes labelled $s_1, s_3$ and then delete the hyperplanes labelled $s_2$ at two of the vertices of the square. This creates a square with two spurs. Continue building the tail starting from the remaining vertex of the square. The new tail contains the same number of $s_2$ labels, two more of $s_3$ and one fewer $s_1$. Hence only the parity of $s_1$ changed.
3. Suppose there is no generator commuting with exactly one of $s_1, s_2$. As the graph $\mathcal{G}$ is non-discrete, there is a generator commuting with both $s_1$ and $s_2$. The component of $\mathcal{G}^c$ containing $v_1$ and $v_2$ is not a proper subgraph so there exist $s_3, s_4$ such that $s_3$ commutes with both $s_1$ and $s_2$, and $s_4$ does not commute with any of the $s_1,s_2,s_3$. Now instead of the deletion of the hyperplanes labelled $s_1, s_2, s_1, s_2, s_1$, delete the hyperplanes labelled $s_4, s_1, s_3$. This creates a square with labels $s_1, s_3, s_1, s_3$. Perform deletion with respect to $s_4$ at one of the vertices. Then continue building the tail.
Let $Y'$ be the new subcomplex. By construction $|H \setminus Y| = |H \setminus Y'|$ and the sequences of labels deleted hyperplanes for the two complexes differ at no more than $5$ places.
Using lemmas $\ref{parity}$ and $\ref{squares}$, we can now modify segments of the tail to make the parity of all elements even. This completes the proof of the main theorem.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The reaction $pp$$\rightarrow$$pp$$\pi$$^{0}$$\pi$$^{0}$ has been investigated at a beam energy of 1.4 GeV using the WASA-at-COSY facility. The total cross section is found to be (324 $\pm$ $21_\text{systematic}$ $\pm$ $58_\text{normalization}$) $\mu$b. In order to study the production mechanism, differential kinematical distributions have been evaluated. The differential distributions indicate that both initial state protons are excited into intermediate $\Delta(1232)$ resonances, each decaying into a proton and a single pion, thereby producing the pion pair in the final state. No significant contribution of the Roper resonance $N^{*}(1440)$ via its decay into a proton and two pions is found.'
address:
- 'Division of Nuclear Physics, Department of Physics and Astronomy, Uppsala University, Box 516, 75120 Uppsala, Sweden'
- 'Physikalisches Institut, Friedrich–Alexander–Universität Erlangen–Nürnberg, Erwin–Rommel-Str. 1, 91058 Erlangen, Germany'
- 'Department of Nuclear Reactions, The Andrzej Soltan Institute for Nuclear Studies, ul. Hoza 69, 00-681, Warsaw, Poland'
- 'Physikalisches Institut, Eberhard–Karls–Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany'
- 'Institute of Physics, Jagiellonian University, ul. Reymonta 4, 30-059 Kraków, Poland'
- 'Institut für Kernphysik, Westfälische Wilhelms–Universität Münster, Wilhelm–Klemm–Str. 9, 48149 Münster, Germany'
- 'High Energy Physics Department, The Andrzej Soltan Institute for Nuclear Studies, ul. Hoza 69, 00-681, Warsaw, Poland'
- 'Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai–400076, Maharashtra, India'
- 'Helmholtz–Institut für Strahlen– und Kernphysik, Rheinische Friedrich–Wilhelms–Universität Bonn, Nu[ß]{}allee 14–16, 53115 Bonn, Germany'
- 'Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany'
- 'Jülich Center for Hadron Physics, Forschungszentrum Jülich, 52425 Jülich, Germany'
- 'Institut für Experimentalphysik I, Ruhr–Universität Bochum, Universitätsstr. 150, 44780 Bochum, Germany'
- 'Zentralinstitut für Elektronik, Forschungszentrum Jülich, 52425 Jülich, Germany'
- 'Institute for Theoretical and Experimental Physics, State Scientific Center of the Russian Federation, Bolshaya Cheremushkinskaya 25, 117218 Moscow, Russia'
- 'II. Physikalisches Institut, Justus–Liebig–Universität Gie[ß]{}en, Heinrich–Buff–Ring 16, 35392 Giessen, Germany'
- 'High Energy Physics Division, Petersburg Nuclear Physics Institute, Orlova Rosha 2, 188300 Gatchina, Russia'
- 'August Che[ł]{}kowski Institute of Physics, University of Silesia, Uniwersytecka 4, 40-007, Katowice, Poland'
- 'The Henryk Niewodnicza[ń]{}ski Institute of Nuclear Physics, Polish Academy of Sciences, 152 Radzikowskiego St, 31-342 Kraków, Poland'
- 'Veksler and Baldin Laboratory of High Energiy Physics, Joint Institute for Nuclear Physics, Joliot–Curie 6, 141980 Dubna, Russia'
- 'Dzhelepov Laboratory of Nuclear Problems, Joint Institute for Nuclear Physics, Joliot–Curie 6, 141980 Dubna, Russia'
- 'Institute of Modern Physics, Chinese Academy of Sciences, 509 Nanchang Rd., 730000 Lanzhou, China'
- 'Department of Cosmic Ray Physics, The Andrzej Soltan Institute for Nuclear Studies, ul. Uniwersytecka 5, 90-950 Lodz, Poland'
author:
- |
The WASA-at-COSY Collaboration\
P. Adlarson
- 'C. Adolph'
- 'W. Augustyniak'
- 'M. Bashkanov'
- 'T. Bednarski'
- 'F.S. Bergmann'
- 'M. Ber[ł]{}owski'
- 'H. Bhatt'
- 'K.–T. Brinkmann'
- 'M. Büscher'
- 'H. Calén'
- 'H. Clement'
- 'D. Coderre'
- 'E. Czerwi[ń]{}ski'
- 'E. Doroshkevich'
- 'R. Engels'
- 'W. Erven'
- 'W. Eyrich'
- 'P. Fedorets'
- 'K. Föhl'
- 'K. Fransson'
- 'F. Goldenbaum'
- 'P. Goslawski'
- 'K. Grigoryev'
- 'C.–O. Gullström'
- 'L. Heijkenskjöld'
- 'J. Heimlich'
- 'V. Hejny'
- 'F. Hinterberger'
- 'M. Hodana'
- 'B. Höistad'
- 'M. Jacewicz'
- 'M. Janusz'
- 'A. Jany'
- 'B.R. Jany'
- 'L. Jarczyk'
- 'T. Johansson'
- 'B. Kamys'
- 'G. Kemmerling'
- 'O. Khakimova'
- 'A. Khoukaz'
- 'S. Kistryn'
- 'J. Klaja'
- 'H. Kleines'
- 'B. K[ł]{}os'
- 'F. Kren'
- 'W. Krzemie[ń]{}'
- 'P. Kulessa'
- 'A. Kupść'
- 'K. Lalwani'
- 'B. Lorentz'
- 'H. Machner'
- 'A. Magiera'
- 'R. Maier'
- 'B. Maria[ń]{}ski'
- 'P. Marciniewski'
- 'M. Mikirtychiants'
- 'H.–P. Morsch'
- 'P. Moskal'
- 'B.K. Nandi'
- 'S. Nied[ź]{}wiecki'
- 'H. Ohm'
- 'A. Passfeld'
- 'C. Pauly'
- 'E. Perez del Rio'
- 'Y. Petukhov'
- 'N. Piskunov'
- 'P. Pluci[ń]{}ski'
- 'P. Podkopa[ł]{}'
- 'A. Povtoreyko'
- 'D. Prasuhn'
- 'A. Pricking'
- 'K. Pysz'
- 'T. Rausmann'
- 'C.F. Redmer'
- 'J. Ritman'
- 'Z. Rudy'
- 'S. Sawant'
- 'S. Schadmand'
- 'A. Schmidt'
- 'T. Sefzick'
- 'V. Serdyuk'
- 'N. Shah'
- 'M. Siemaszko'
- 'T. Skorodko'
- 'M. Skurzok'
- 'J. Smyrski'
- 'V. Sopov'
- 'R. Stassen'
- 'J. Stepaniak'
- 'G. Sterzenbach'
- 'H. Stockhorst'
- 'A. Szczurek'
- 'A. Täschner'
- 'T. Tolba'
- 'A. Trzci[ń]{}ski'
- 'R. Varma'
- 'P. Vlasov'
- 'G.J. Wagner'
- 'W. Wglorz'
- 'U. Wiedner'
- 'A. Winnemöller'
- 'M. Wolke'
- 'A. Wro[ń]{}ska'
- 'P. Wüstner'
- 'P. Wurm'
- 'X. Yuan'
- 'L. Yurev'
- 'J. Zabierowski'
- 'C. Zheng'
- 'M.J. Zieli[ń]{}ski'
- 'W. Zipper'
- 'J. Z[ł]{}oma[ń]{}czuk'
- 'P. [Ż]{}upra[ń]{}ski'
title: '$\pi^{0}\pi^{0}$ Production in Proton-Proton Collisions at $T_{p}$=1.4 GeV'
---
Introduction {#Int}
============
Investigations of the two-pion decay of mesons and baryons have been extensively carried out in pion-induced $\pi$$N$$\rightarrow$$\pi\pi$$N$ [@PhysRevC.69.045202] and photon-induced $\gamma$$N$$\rightarrow$$\pi\pi$$N$ [@Thoma:2007bm] reactions. Double pion production in nucleon-nucleon ($NN$) collisions is of particular interest in view of studying the simultaneous excitation of the two baryons and their subsequent decays. Here, the simplest case is considered: the excitation of the two nucleons into the $\Delta$(1232) resonance.\
Several theoretical models for double pion production have been suggested in the energy range from the production threshold up to several GeV [@Tejedor1994667; @Oset1985584]. A full reaction model describing the double pion production in $NN$ collisions has been developed by Alvarez-Ruso [@AlvarezRuso1998519]. More advanced calculations by Cao, Zou and Xu that include relativistic corrections based on the calculations of Ref. [@AlvarezRuso1998519] have been recently published [@PhysRevC.81.065201]. These models include and study both the resonant and the non-resonant terms of $\pi\pi$-production. The models predict that at energies near threshold the $\pi\pi$ production is dominated by the excitation of one of the nucleons into the Roper resonance $N$$^{*}$(1440)$P$$_{11}$ via $\sigma$-exchange, followed by its s-wave decay $N$$^{*}$$\rightarrow$$N$($\pi$$\pi$)$^{\text{s-wave}}_{I=0}$ (where $I$ indicates the isospin of the $\pi\pi$ system). As the beam energy increases, the p-wave decay $N$$^{*}$$\rightarrow$$\Delta$(1232)$\pi$$\rightarrow$N($\pi$$\pi$) gives an increasing contribution to the cross section. At higher energies the double $\Delta$(1232) excitation (dominantly via ($\pi$-$\rho$)-exchange) is expected to be the dominant reaction mechanism for $\pi\pi$ production.\
Such reactions have been studied experimentally at intermediate and higher energy regions [@Shimizu1982571; @PhysRev.138.B670]. Recently, exclusive high-statistics measurements have become available from near threshold ($T_{p}$=650 MeV) up to $T_{p}$=1.3 GeV from the PROMICE/WASA [@Johanson200275; @PhysRevLett.88.192301; @PhysRevC.67.052202], CELSIUS/WASA [@springerlink:10.1140/epja/i2008-10569-6; @PhysRevLett.102.052301; @Skorodko200930; @Skorodko2011115], COSY-TOF [@springerlink:10.1140/epja/i2008-10637-y] and ANKE [@PhysRevLett.102.192301] experiments. The analysis of the data obtained from these experiments, with the exception of ANKE (where only $pp$ final states in the diproton $^{1}$$S$$_{0}$ quasi-particle state have been considered), indicate that in case of $pp$ collisions (isovector channel) only two $t$-channel reaction mechanisms dominate: the excitation of the Roper resonance $N$$^{*}$(1440) (via $\sigma$-exchange) at energies close to threshold [@PhysRevLett.88.192301; @PhysRevLett.102.052301], and the excitation of the $\Delta\Delta$ system (via ($\pi$-$\rho$)-exchange) at energies $T_{p}$$\geq$1.3 GeV [@Skorodko2011115]. Model predictions are found to be in good agreement with the experimental results at energies close to threshold. At energies $T$$_{p}$$\geq$1 GeV, the Roper resonance contribution is over-predicted in the theoretical calculations. However, the predicted $\Delta\Delta$ excitation is in accordance with the data, if relativistic corrections are taken into account. A detailed description of the relativistic corrections applied to the model calculations of Ref. [@AlvarezRuso1998519] can be found in Refs. [@Skorodko200930; @Skorodko2011115]. On the other hand, the ANKE results reveal dominance of the $\Delta\Delta$ excitation already at $T_{p}$=0.8 GeV.\
In contrast to the exclusive measurements at low and intermediate energies, there is little experimental information on double pion production in $NN$ collisions at higher energies (i.e. $T_{p}$$>$1.3 GeV). Only the total cross sections are provided at $T_{p}$=1.36 GeV [@Koch2004] and $T_{p}$=1.48 GeV [@PhysRev.138.B670].\
Here, we report on the total as well as differential cross section measurements of the $pp$$\rightarrow$$pp$$\pi$$^{0}$$\pi$$^{0}$ reaction at $T$$_{p}$ = 1.4 GeV using the WASA at COSY facility [@WASApro2004]. The beam energy corresponds to a center-of-mass energy of $\sqrt{s}$=2.48 GeV, $i.e.$ twice the $\Delta$ mass, thereby allowing a stringent test of the $t$-channel $\Delta\Delta$ mechanism as expected from the diverse model calculations.
Experimental Setup {#Exp}
==================
The experimental data were collected using the Wide Angle Shower Apparatus (WASA). WASA is an internal target experiment at the COoler SYnchrotron (COSY) of the Forschungszentrum Jülich, Germany. The detection system provides nearly full solid angle coverage for both charged and neutral particles. It allows multi-body final state hadronic interactions to be studied with high efficiency. The WASA facility consists of a central and a forward detector part and a cryogenic microsphere (pellet) target.\
The pellet target generator is located above the central detector. It provides frozen pure hydrogen or deuteron pellets of about 25 $\mu$m diameter (as the targets), thereby minimizing background reactions from other materials.\
The central detector is built around the interaction point and covers polar scattering angles between 20$^{\circ}$–169$^{\circ}$. The innermost detector, the mini drift chamber, is housed within the magnetic field of a superconducting solenoid and is used in determining the momenta of charged particles. The next layer, the plastic scintillator barrel provides fast signal for first level trigger and charged particle identification. As the outermost layer, 1012 CsI(Na) crystals of the calorimeter enable the measurement of the energy deposited by charged particles as well as the reconstruction of electromagnetic showers.\
The forward detection system covers the polar angular range of 3$^{\circ}$–18$^{\circ}$. The multi-plane straw tube detector is implemented for the precise reconstruction of charged particle track coordinates. An arrangement of segmented plastic scintillator layers, the forward range hodoscope, is used to reconstruct kinetic energies of scattered particles by the $\Delta$$E-E$ technique. A three-layered thin hodoscope provides fast charged particle discrimination. The forward detector can provide a tag on meson production via the missing mass of the reconstructed recoil particles. The trigger for the present experiment demanded at least one charged particle candidate to reach the first layer of the forward range hodoscope. For more details about the WASA-at-COSY facility see Ref. [@WASApro2004].
Data Analysis {#DSel}
=============
Recoil protons from the $pp$$\rightarrow$$pp$$\pi$$^{0}$$\pi$$^{0}$ reaction with $T_{p}$=1.4 GeV are detected in the forward detector, while the two neutral pions are reconstructed in the central detector. The main criterion to select the event sample demands 1 or 2 charged tracks in the forward detector and exactly 4 neutral tracks in the central detector. With this selection, the geometrical acceptance of the $pp$$\rightarrow$$pp$$\pi$$^{0}$$\pi$$^{0}$ reaction is found to be $45$$\%$. Two event samples are selected: the first includes events with only one proton detected in the forward detector while the other proton is scattered outside the forward detector. The second contains events when two protons were detected in the forward detector. The combination of both data samples gives a finite acceptance over all of the avaliable phase space, as shown in Fig. 1. Here, as an example, two two-dimensional acceptance distributions of $p\pi$$^{0}$ pairs (left plot) and of $p\pi^{0}\pi^{0}$- versus $p\pi$$^{0}$-invariant masses (right plot) are presented. The Monte Carlo plots are based on equally populated phase space and show that nearly the full phase space is covered.
{width=".23\textwidth"} {width=".23\textwidth"}
{width=".23\textwidth"} {width=".23\textwidth"}
The identification of protons in the forward detector is based on the $\Delta E-E$ method, as shown in Fig. 2 (left plot). The depicted selection criterion selects not only protons that are stopped in the forward range hodoscope but also those that punch through. The selection helps to reject the contribution resulting from hadronic interactions in detector material.\
Neutral pions have been reconstructed from the photon pairs detected in the central detector. The reconstruction procedure is based on the minimum $\chi$$^{2}$ method which is applied to select the two-photon combinations with invariant masses closest to the $\pi$$^{0}$ mass. Figure 2 (right plot), shows the distribution of invariant masses ($M_{\gamma\gamma}$) for all three unique combinations of the 4 photons forming two $\gamma\gamma$ pairs. The figure shows that the $M_{\gamma\gamma}$ distribution peaks at the $\pi$$^{0}$ mass, as expected. The small background at low and high $\gamma\gamma$ masses is due to wrong $\gamma$-pair combinations and due to the finite resolution of the detector.\
Furthermore, a kinematic fit with six constraints, four for total energy-momentum conservation and two for each of the two $\gamma \gamma$ pair masses being equate to the $\pi$$^{0}$ mass, is applied in order to suppress the contribution from background channels and to recover the information of the unmeasured proton, scattered into the central detector or into inactive material. For consistency, the kinematic fit routine is always applied with one unmeasured proton in the final state. Hence, in the case where two protons are registered in the forward detector only one proton is selected and the other one is ignored. The proton with the lower energy is found to have better resolution. Therefore, it is chosen as the measured value in the kinematic fit routine while the higher energy one is treated as the unmeasured variable [@Tolba2010].\
The absolute normalization of the data has been achieved by normalizing to the measured $pp$$\rightarrow$$pp$$\eta$ cross section [@Chiavassa1994270]. Two decay modes of the $\eta$ meson, $\eta$$\rightarrow$$3\pi$$^{0}$ and $\eta$$\rightarrow$$2\gamma$, were chosen because they have similar final state particles as the $pp$$\rightarrow$$pp$$\pi$$^{0}$$\pi$$^{0}$ reaction. These channels have an additional advantage that they are the dominant neutral decay modes of the $\eta$ meson [@PDG2010].\
The data are corrected for the detector efficiency and acceptance by a Monte Carlo simulation using a toy model tuned to match the data. The tuned toy model is constructed by generating a four-body final state phase space distribution of the $pp$$\rightarrow$$pp$$\pi$$^{0}$$\pi$$^{0}$ reaction, employing the GEANT phase space generator, based on the FOWL program [@James77]. Then, the generated event weight is modified to describe the 2$\pi$$^{0}$ production mechanism according to the production of two $\Delta$(1232)P$_{33}$ resonances in the intermediate state, each decaying into $p\pi$$^{0}$. The partial wave amplitude that describes the decay of $\Delta$(1232) into $p\pi$-system is derived in Ref. [@Risser197368]. This amplitude together with correction terms for the measured proton and pion angular distributions in the center-of-mass system, as well as for the $M$$_{\pi^{0}\pi^{0}}$ and $M$$_{p\pi^{0}}$ distributions are multiplied by the generated weights of each event. The Monte Carlo simulations are then compared with the data, and this step is repeated until the data and the simulations are in good agreement. The tuned toy model is explained in detail in Ref. [@Tolba2010].
Results {#Res}
=======
The total cross section of the $pp$$\rightarrow$$pp$$\pi$$^{0}$$\pi$$^{0}$ reaction at $T$$_{p}$ = 1.4 GeV is determined to be $\sigma_\text{tot}$=(324 $\pm$ $21_\text{systematic}$ $\pm$ $58_\text{normalization}$) $\mu$b. The total cross section error is evaluated in terms of statistical and systematical uncertainties. The statistical error is found to be $<$1$\%$ and thus negligible compared to the systimatical contribution. The systematical error is constructed from two terms, systematic effects and normalization. The systematic contribution is estimated by observing the variation of the results with different analysis constraints where the varied parameters are assumed to be independent of each other. The systematic term is calculated from the following main contributions: 1) applying different selection regions to the flat part of the confidence level (probability) distribution of the kinematic fit, the contribution from this term is found to be 5$\%$, 2) the contribution from the correction for the detector acceptance generated by different Monte Carlo models is found to be 4$\%$, and 3) constraining the reconstruction particles to satisfy the geometrical boundaries of the central and the forward detectors, the contribution from this term is found to be 1$\%$. The total error from the systematic term is the square root of the quadratic sum of the individual terms and found to be 6.5$\%$. The normalization term is constructed from two main components: 1) contribution from the $pp$$\rightarrow$$pp$$\eta$ analysis, found to be 14$\%$, and 2) the uncertainty of the cross section value in Ref. [@Chiavassa1994270] which is found to be 11$\%$. The total error contribution from the normalization term is estimated to be 18$\%$.\
Figure 3 compares the cross section from this work (solid circle) with the previous experimental data [@Shimizu1982571; @Johanson200275; @Skorodko200930; @Koch2004; @PhysRev.138.B670] and to the theoretical expectations calculated in Ref. [@AlvarezRuso1998519]. The new data point is compatible with the earlier results [@Shimizu1982571; @PhysRev.138.B670; @Johanson200275; @Skorodko200930; @Koch2004] and corroborates the strongly rising trend of the cross section starting at $\sim$ 1170 MeV. As has been verified in Ref. [@Skorodko200930], the trend of rising total cross sections from threshold up to $T_{p}$$\sim$1 GeV is due to the dominance of the Roper resonance. Above 1 GeV, it levels off and proceeds with only a slight increase up to $T_{p}$$\sim$1170 MeV. The rise in the cross section values at higher energies $T_{p}$$>$1170 MeV is expected to be associated with the dominance of the $\Delta\Delta$ excitation. A similar interpretation with a better description of the excitation function is obtained with the modifications applied in Ref. [@Skorodko2011115]
![Total cross section for the $pp$$\rightarrow$$pp$$\pi$$^{0}$$\pi$$^{0}$ reaction as a function of $T_{p}$. The result of this work (solid circle), at $T$$_{p}$ = 1.4 GeV, is compared to the data from PROMICE/WASA (open triangles) [@Johanson200275], CELSIUS/WASA (filled triangles) [@Skorodko200930], at 1.36 GeV (square) [@Koch2004], bubble chamber results (inverted triangles) [@Shimizu1982571] and (star) [@PhysRev.138.B670] and the theoretical calculations of Ref. [@AlvarezRuso1998519]. Here, the long dashed doted line represents the production via double $\Delta$(1232), the dashed line represents the production via Roper resonance decay $N$$^{*}$(1440)$\rightarrow$$\Delta\pi$$^{0}$, the dashed doted line represents the production via direct decay $N$$^{*}$(1440)$\rightarrow$$p\pi$$^{0}$$\pi$$^{0}$ and the solid black line represents the expected total cross section.](fig5.eps){width=".40\textwidth"}
In order to study the mechanism of the $pp$$\rightarrow$$pp$$\pi$$^{0}$$\pi$$^{0}$ reaction, seven independent kinematic variables are necessary to cover the available phase space of the reaction. Therefore, different kinematical variables describing the system have been investigated after the data have been corrected for the detector efficiency and acceptance using the tuned toy model. The corrected data are compared to an uniformly populated phase space distribution and the models according to Refs. [@AlvarezRuso1998519] and [@Skorodko2011115]. All theoretical models are normalized to the same total cross section as the data. The differential distributions presented here have been chosen because they are sensitive to contributions from intermediate $\Delta$(1232) and/or the $N$$^{*}$(1440) resonances.
![Comparison of data (black dots) to the theoretical expectations calculated from Ref. [@AlvarezRuso1998519] (red-dashed line) and Ref. [@Skorodko2011115] (blue-line), and with uniformly populated phase space (shaded area). Left: differential distribution of the $\pi$$^{0}$$\pi$$^{0}$-invariant mass, $M$$_{\pi^{0}\pi^{0}}$. Right: differential distribution of two pion opening angle in the center-of-mass system, $cos$$\delta$$^\text{CM}_{\pi^{0}\pi^{0}}$.](fig6.eps "fig:"){width=".23\textwidth"} ![Comparison of data (black dots) to the theoretical expectations calculated from Ref. [@AlvarezRuso1998519] (red-dashed line) and Ref. [@Skorodko2011115] (blue-line), and with uniformly populated phase space (shaded area). Left: differential distribution of the $\pi$$^{0}$$\pi$$^{0}$-invariant mass, $M$$_{\pi^{0}\pi^{0}}$. Right: differential distribution of two pion opening angle in the center-of-mass system, $cos$$\delta$$^\text{CM}_{\pi^{0}\pi^{0}}$.](fig7.eps "fig:"){width=".23\textwidth"}
Figure 4 shows that the $\pi$$^{0}$$\pi$$^{0}$-invariant mass ($M$$_{\pi^{0}\pi^{0}}$) distribution (left plot) is similar to the uniformly populated phase space distribution, while it shows a strong deviation from the calculations of Ref. [@AlvarezRuso1998519], where the model calculations predict two large enhancements at lower and higher $M$$_{\pi^{0}\pi^{0}}$ values. The enhancement at higher $M$$_{\pi^{0}\pi^{0}}$ values is due to the dominance of the $\rho$ exchange in the model calculations. In contrast, the data are well described by the assumption of $t$-channel meson exchange leading to $\Delta\Delta$ excitation of Ref. [@Skorodko2011115] (blue line), where the $\rho$-exchange contribution is strongly reduced compared to the original calculations [@AlvarezRuso1998519]. The systematic enhancement at low $M$$_{\pi^{0}\pi^{0}}$ values indicates the tendency of the two pions to be emitted parallel with respect to each other in the center-of-mass frame. This behavior is seen as well in the two pion opening angle distribution $cos$$\delta$$^\text{CM}_{\pi^{0}\pi^{0}}$ (right plot of Fig. 4), where the data is enhanced at $cos$$\delta$$^\text{CM}_{\pi^{0}\pi^{0}}$ = 1, relative to the phase space spectrum. Here, the data are well described by the modified calculations of Ref. [@Skorodko2011115], while a large deviation is observed from the calculations of Ref. [@AlvarezRuso1998519]. The strong peaking of the latter calculations at an opening angle of 180$^{\circ}$ is due to the enhancement at higher values of $M$$_{\pi^{0}\pi^{0}}$ in left frame of Fig. 4.
{width=".23\textwidth"} {width=".23\textwidth"}
{width=".23\textwidth"} {width=".23\textwidth"}
The upper and lower left plots of Fig. 5 show indications for the $\Delta\Delta$ excitation in the correlation of the $M$$_{p\pi^{0}}$ pairs (upper) and in the one-dimensional projection onto the $M$$_{p\pi^{0}}$-axis (lower). Here, evidence for the $\Delta$(1232) resonance can be seen as a strong enhancement at $M$$_{p\pi^{0}}$ $\sim$ $M$$_{\Delta}$ = 1.232 GeV/c$^{2}$. The uniform phase space distribution of the lower plot shows the strongest deviation with respect to the data due to the $\Delta$ excitation in the data points. In contrast, no significant evidence for the presence of the Roper resonance $N^{*}$(1440) is observed in the right-hand plots of Fig. 5. Here, one would expect an enhancement around $M$$_{p\pi^{0}\pi^{0}}$ = 1.44 GeV/c$^{2}$.\
The upper left plot of Fig. 6 shows the $pp$-invariant mass spectrum, $M$$_{pp}$, which peaks slightly to the right with respect to the uniformly populated phase space distribution. The $M$$_{pp}$ spectrum is complementary to that for $M$$_{\pi^{0}\pi^{0}}$ from Fig. 4 that is located to the left relative to the phase space spectrum. The upper right plot shows the angular distribution of the protons in the center-of-mass frame, $cos$$\theta$$^\text{CM}_{p}$. It exhibits an anisotropic behavior, in agreement with the theoretical calculations. The strong forward-backward peaking of the $cos$$\theta$$^\text{CM}_{p}$ spectrum is associated with $\pi$-$\rho$ exchange mediating the $pp$ interaction. The angular distribution of the $p\pi$$^{0}$-system in the center-of-mass frame, cos$\theta$$^\text{CM}_{p\pi^{0}}$, (lower right) shows a forward-backward symmetry. That is similar in shape to the $cos$$\theta$$^\text{CM}_{p}$ distribution as expected from the large p/$\pi$$^{0}$ mass ratio. In the $pp\pi$$^{0}$-invariant mass distribution, $M$$_{pp\pi^{0}}$, (lower left plot) the data peak near the sum of the proton and $\Delta$ masses as expected for $\Delta\Delta$ production at threshold. Here, the modified model [@Skorodko2011115] and the data points are in good agreement, while the calculations of Ref. [@AlvarezRuso1998519] are slightly shifted to lower $M$$_{pp\pi^{0}}$ values and deviate from the data points. The data deviate strongly from the distribution generated according to phase space of the process. This is due to the $\Delta$ excitation, consistent with the observations in the other invariant mass distributions.
{width=".23\textwidth"} {width=".23\textwidth"}
{width=".23\textwidth"} {width=".23\textwidth"}
Conclusion and outlook {#con}
======================
Data for the total cross section of the $pp$$\rightarrow$$pp$$\pi$$^{0}$$\pi$$^{0}$ reaction at $T$$_{p}$ = 1.4 GeV have been obtained. Differential distributions are compared to theoretical predictions. Calculations of Ref. [@AlvarezRuso1998519] show a stronger deviation from the experimental data as compared to the refined calculations of Ref. [@Skorodko2011115]. The latter show small deviations for some differential distributions, e.g. $cos$$\theta$$^\text{CM}_{p\pi^{0}}$ (lower right plot of Figure 6), leaving room for further optimization of the model. From studying the reaction dynamics the $\pi$$^{0}$$\pi$$^{0}$ production at $T$$_{p}$ = 1.4 GeV is found to be dominated by the $t$-channel double $\Delta$(1232) excitation as intermediate state, without a significant contribution from the Roper resonance $N^{*}$(1440).\
The investigation of the production of charged pions ($\pi$$^{+}$$\pi$$^{-}$) is the next step in the study of the double pion production in $NN$ collisions with WASA-at-COSY. This channel is of special interest in order to study the contribution from the vector meson $\rho$$^{0}$(770) which is expected to play an important role in the $\pi$$^{+}$$\pi$$^{-}$ production. Moreover, the extension to higher proton energies will shed light on the role of heavier resonances.
Acknowledgments {#Ack}
===============
This work was in part supported by: the Forschungszentrum Jülich including the COSY-FFE program, the European Community under the FP7-Infrastructure-2008-1, the German-BMBF, the German-Indian DAAD-DST exchange program, VIQCD and the German Research Foundation (DFG), the Wallenberg Foundation, the Swedish Research Council, the Göran Gustafsson Foundation, the Polish Ministry of Science and Higher Education and the Polish National Science Center and Foundation for Polish Science - MPD program.\
We also want to thank the technical and administration staff at the Forschungszentrum Jülich and at the participating institutes.\
This work is part of the PhD thesis of Tamer Tolba.
[24]{}
S. Prakhov, et al., *Phys. Rev. C* **69**, (2004) 045202.
U. Thoma, et al., *Phys. Lett. B* **659**, (2008) 87.
J. Tejedor, E. Oset, *Nucl. Phys. A* **571**, (1994) 667.
E. Oset, M. J. Vicente-Vacas, *Nucl. Phys. A* **446**, (1985) 584.
L. Alvarez-Ruso, et al., *Nucl. Phys. A* **633**, (1998) 519.
Xu. Cao, et al., *Phys. Rev. C* **81**, (2010) 12.
F. Shimizu, et al., *Nucl. Phys. A* **386**, (1982) 571.
A. M. Eisner, et al., *Phys. Rev* **138**, (1965) B670.
J. Johanson et al., *Nucl. Phys. A* **712**, (2002) 75.
W. Brodowski, et al., *Phys. Rev. Lett.* **88**, (2002) 192301.
J. Pätzold, et al., *Phys. Rev. C* **67**, (2003) 052202.
T. Skorodko, et al., *Eur. Phys. J. A* **35**, (2008) 317.
M. Bashkanov, et al., *Phys. Rev. Lett.* **102**, (2009) 052301.
T. Skorodko, et al., *Phys. Lett. B* **679**, (2009) 30.
T. Skorodko, et al., *Phys. Lett. B* **695**, (2011) 115.
S. Abd El-Bary, et al., *Eur. Phys. J. A* **37**, (2008) 267.
S. Dymov , et al., *Phys. Rev. Lett.* **102**, (2009) 192301.
I. Koch, *PhD Thesis, Uppsala University*, (2004).
H. H. Adam, et al., *arXiv:nucl-ex/0411038*, (2004).
T. Tolba, *PhD Thesis, Ruhr-Universität Bochum*, (2010).
E. Chiavassa, et al., *Phys. Lett. B* **322**, (1994) 270.
K. Nakamura, et al., *Particle Data Group), JP G, 075021* **37**, (2010).
F. James, *CERN Program Library Long Writeup W505*, (1977).
T. Risser, M. D. Shuster, *Phys. Lett. B***43**, (1973) 68.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'By introducing in the hydrodynamic model, i.e. in the hydrodynamic equations and the corresponding boundary conditions, the higher order terms in the deviation of the shape, we obtain in the second order the Korteweg de Vries equation (KdV). The same equation is obtained by introducing in the liquid drop model (LDM), i.e. in the kinetic, surface and Coulomb terms, the higher terms in the second order. The KdV equation has the cnoidal waves as steady-state solutions. These waves could describe the small anharmonic vibrations of spherical nuclei up to the solitary waves. The solitons could describe the preformation of clusters on the nuclear surface. We apply this nonlinear liquid drop model to the alpha formation in heavy nuclei. We find an additional minimum in the total energy of such systems, corresponding to the solitons as clusters on the nuclear surface. By introducing the shell effects we choose this minimum to be degenerated with the ground state. The spectroscopic factor is given by the ratio of the square amplitudes in the two minima.'
author:
- |
Andrei Ludu$^1$ , Aureliu Săndulescu$^2$ and Walter Greiner\
\
title: 'Nonlinear Liquid Drop Model. Cnoidal Waves'
---
15.6cm 0.5cm 1.0cm
PACS numbers: 23.60.+e, 21.60.Gx, 24.30.-v, 25.70.ef
Introduction
============
It is well known that liquid drop model, as a collective model of the nucleus describes excelently the spectra of spherical nuclei as small vibrations (harmonic in the linear approximation or anharmonic in higher approximations) around their shape. On the other hand it is known that on the nuclear surface of heavy nuclei close to the magic nuclei ($^{208}$Pb, $^{100}$Sn) a large enhancement of clusters (alpha, carbon, oxigen, neon, magnesium, silicon) exists which leads to the emission of such clusters as natural decays \[1,2\]. It is also clear that traditional collective models \[3\] are not able to give a complete explanation of such natural decays, i.e. they still did not completely answer to the main physical question: why should nucleons join together and spontaneous form an isolated cluster on the nuclear surface ? Only by the introduction in the shell model of many body correlation effects we could form an isolated bump, stable in time. It is possible to describe the formation of such clusters in a collective model \[4,5\] ?
In the present paper by introducing the nonlinearities in the liquid drop model we succeded to give a positive answer to this problem. In a nonlinear liquid drop model we can describe simultaneously, by cnoidal waves the transitions from small vibrations to the formation of solitons. The experimental discovery and the theoretical foundation of solitons \[6\] as non-dispersive localized waves moving uniformly, lead to a powerfull theory of classical field equations with such solutions \[7-9\]. Also it was possible to explain the correspondence between classical soliton solutions and the extended-particle states of the quantized version of the theory. This lead to a generalisation of the semiclassical expansion of quantum mechanics in quantum field theory. In the last case solitons (and breathers or instanton solutions) are non-perturbative. Of course, the above methods work only if the initial physical model allows first the existence of some non-trivial classical localized solutions.
From the mathematical physics point of view solitons, as solutions of non-linear evolution equations, are isolated waves which preserve their shape and have finite and localized energy density. Recently has been shown that solitons may be also relevant in nuclear physics \[4,5\] or particle physics \[10\]. Also it was realized that many field theoretical models for particle interactions, and even for quantum extended particles \[11\], possess soliton or breather solutions and that the solitons ought to be interpreted as additional particle-like structures in theory. The traveling solutions of the KdV equation \[7\] and singular solutions having poles at $\pm \infty$. Recently, in order to describe the quasimolecular spectra, we have introduced in \[5\] a one-dimensional soliton model for the cluster (alpha particle) and the rotator-vibrator model for the nucleus. An excelent agreement with the experimental data was obtained.
We conclude that the nonlinear terms lead to new qualitative picture of the liquid drop model, i.e. the harmonic oscillations grow into anharmonic ones which can lead to a stable soliton configuration. Both the potential picture and the phase space portraits support this behaviour. We stress that all these results are embedded into a Hamiltonian formalism.
In the present paper we first introduce in the hydrodynamical model the higher terms in the deviation of the spherical shape. We have shown that in the third order we obtain the Korteweg de Vries equation (KdV). In the following chapter 3 by introducing in the liquid drop model the third order terms in the deviation of the shape we obtained the same KdV equation as in the nonlinear hydrodynamical model. We should like to stress that these nonlinear equations are Hamiltonian eqautions which describe the total energy of the system. Chapter 4 contains the nonlinear solutions: cnoidal waves and the singular solutions for KdV equation. Last chapter discuss the application of nonlinear liquid drop model to the alpha preformation factors.
The nonlinear hydrodynamic model
================================
There are two possible ways to describe the classical dynamics of a liquid drop: first is the fluid (hydrodynamic) approach based on the continuity, Euler and the equations of state together with boundary conditions and the second is the Hamiltonian approach. However there is a deep connection between these two approaches. The motion of a perfect incompressible liquid in a domain is governed only by the Euler equation, since the continuity and state equations reduce in this case to the Poisson equation. The boundary conditions ask the inner product between the velocity field and the volume differentiable 3-form to be zero (special procedure occures when the boundary itself is variable in time - free surface - and results in nonlinear contributions to the differential equations). The flow of the velocity field belongs to the group of volume preserving transformations of the volume of the drop and hence it could be a geodesic on the manifold of this group. Now, the connection with the second approach comes from the observation that the Euler equation is Hamiltonian in the sense that, between all other possible flows, only those which satisfy Euler equation are geodesics. In the case of the liquid drop model the dominant terms are the volume, the surface and Coulombian. All terms are dependent on the geometry of the surface. Therefore, we expect that the dynamics of the liquid drop is strongly dominated (phenomenologicaly) by the dynamics of the free surface, in both these approaches.
Let us describe the surface of the nucleus in spherical coordinates $(r,\theta ,\phi )$ as a function of the polar angles $\theta $ and $\phi $, by writing the nuclear radius in the form $$\begin{aligned}
r=R_0 (1+\xi (\theta , \phi ,t)),\end{aligned}$$ where $R_0$ is the radius of the spherical nucleus and the shape function $\xi $ is the difference in the radius between the deformed and the spherical one. In the following we search for deformed surfaces which contain stable traveling waves which can lead to a bump. The stability is fulfilled if the function describing the surface arises as a solitary wave solution of one of the classical nonlinear equations like KdV (Korteweg de Vries), whose stability was clearly established in literature \[7,12,13\]. These equations are tractable as dynamical systems in the frame of nonlinear infinite-dimensional Hamiltonian field theories. The cnoidal waves as solutions of KdV equation describe small vibrations up to the solitary waves. In the case of the spherical surface, the localization condition is realised for values of the angular half-width $L$ of the shape function $\xi (\theta ,\phi ,t)$ smaller than $\pi $, at any moment of time \[4,5\]. Without any loss of generality and in the spirit of the above theory, it is convenient to look for a special space-time behaviour of the shape function of the form $$\begin{aligned}
\xi (\theta ,\phi ,t)=g(\theta )\eta (\phi -Vt)\end{aligned}$$ with $g$ an arbitrary bounded, non-vanishing continuous function, $\eta $ a compact supported, or rapidly decreasing function with $V$ defining the tangential velocity of the traveling solution $\eta $ on the surface. This solution represents a stable traveling perturbation ($\eta $) in the $\phi $ direction, having a given transversal profile ($g$) in the $\theta $ direction. This is different from the traditional liquid drop model case when one expands the shape function in spherical harmonics and where we have shown \[5\] one needs to consider more than 10 multipoles to fit such shapes of a localised bump.
We suppose (without any loose of generality, due to the spherical symmetry) that the bumps are situated on a circle $\theta =\pi / 2 $ such that the variable $\theta $ plays only the role of a parameter in the corresponding dynamical equations, and that the bumps travel along the $\phi $ coordinate only. This choice results in the separation of variables in the shape function in eq.(2). Since in the investigation of the dynamics of the surface the coordinate $r$ will be involved only in the (free surface) boundary conditions, then $\phi $ remains the unique free coordinate. Hence, we can reduce the whole 3-dimensional problem to a 1-dimensional formalism.
In the hydrodynamic approach we treat the nucleus is a perfect ideal fluid layer (incompressible, irrotational and without viscosity) described by the field velocity ${\vec v}(r,\theta ,\phi ,t)$ and by the constant mass density $\rho =$const. From the continuity equation we have $div $ ${\vec v}=0$ and due to the irrotationality condition, $\nabla \times {\vec v}=0$, we have a potential flow, described by the velocity potential $\Phi (r,\theta ,\phi ,t)$, and the corresponding Laplace equation $$\begin{aligned}
{\vec v}=\nabla \Phi, \ \ \ \ \ \triangle \Phi =0.\end{aligned}$$ The dynamics of this perfect fluid is described by the Euler equation $$\begin{aligned}
{{\partial {\vec v}} \over {\partial t}}+({\vec v}\cdot \nabla ){\vec v}
=-{{1} \over {\rho }}\nabla P+{{1} \over {\rho }}{\vec f},\end{aligned}$$ where $P$ is the pressure and ${\vec f}$ is the volume density of the forces acting into this fluid, e.g. for the Coulombian one we have $
{\vec f} =-{\rho }_{el}\nabla \Psi$, with $\Psi $ the electrostatic potential and $\rho _{el}$ the charge density, supposed to be constant, too. By using eqs.(3), eq.(4) becomes, in the Coulombian case $$\begin{aligned}
\biggl ( \Phi _{t}+{1 \over 2}|\nabla \Phi |^{2}\biggr ) \bigg | _{\Sigma }
=-{{1} \over {\rho }}P
-{{\rho _{el}} \over {\rho }}\Psi |_{\Sigma }.\end{aligned}$$ To determine uniquely the unknown functions $\Phi $ and $\xi $ we need, in completion of eqs.(3,5), the boundary conditions for the scalar harmonic field $\Phi $, on (maximum) two closed surfaces: the external free surface of the nucleus described by eqs.(1) or (2) and the inner surface (if it exists) of the fluid layer. The latter condition can be expressed in a simpler form if we consider that the motion is limited to only a thin fluid layer characterised by zero radial velocity of the flow on its inner surface. This last condition expresses the existence of a rigid core in the volume of the nucleus. The first boundary condition can be expressed in the most general form of the kinematical constrain of the free surface of the fluid described by eq.(1), \[4,5,7\] $$\begin{aligned}
{{dr} \over {dt}} \biggr | _{\Sigma }=
{\biggl ( {{\partial r} \over {\partial t}}+
{{\partial r} \over {\partial \theta}}{{d\theta } \over {dt}}+
{{\partial r} \over {\partial \phi}}{{d\phi } \over {dt}}
\biggr ) } \biggr | _{\Sigma },\end{aligned}$$ where $r(t,\phi ,\theta )$ in the RHS represents the shape function described in eq.(1) and the label $\Sigma $ means that eq.(6) is taken on the free surface $\Sigma $. This equation allows very general types of movements, including traveling and vibrational waves. Eq.(6) reduces to the form ${{dr} \over {dt}}\biggr | _{\Sigma }=
{{\partial r} \over {\partial t}} \biggr | _{\Sigma }
$ when one considers only its linear approximation \[3,14\], i.e. that one used in the Bohr-Mottelson model. This linearization implies the existence of only collective radial vibrations and does no allow any motion along the tangential direction. Eq.(6) can be written in terms of the derivatives of the potential of the flow and the shape function $\xi $ $$\begin{aligned}
\Phi _{r} \biggr | _{\Sigma }= R_0 {\biggl (
\xi _{t} +
{{\xi _{\theta }} \over {r^2}}\Phi _ {\theta }+
{{\xi _{\phi }} \over {r^2 \sin ^{2} \theta }}\Phi _{\phi }
\biggr ) } \biggr | _{\Sigma }, \end{aligned}$$ where ${{\partial \Phi } \over {\partial r}}=v_r=\dot r$ is the radial velocity and ${{1} \over {r}}
{{\partial \Phi } \over {\partial \theta}}=v_{\theta }=r\dot{\theta } $, ${{1} \over {r \sin \theta}}
{{\partial \Phi } \over {\partial \phi }}=v_{\phi }=r \dot {\phi }\sin
\theta$ are the tangential velocities. We denote here the partial differentiation by suffixes, $\partial \Phi /\partial \phi =\Phi _{\phi }$, etc. The existence of a rigid core of radius $R_0 -h(\theta )>0$, $h(\theta)\ll R_0 $, introduces the second boundary condition for the radial velocity on the surface of this core in the form $$\begin{aligned}
v_r |_{r=R_0 -h}={{\partial \Phi } \over {\partial r}} \biggr | _{r=R_0 -h}=0.\end{aligned}$$ Both eqs.(7,8) are von Neumann type of boundary conditions. The motion of the fluid is described by the Laplace equation eq.(3) for $\Phi $, and by the two boundary conditions, eqs.(7,8), for $\Phi $ and $\xi $. To these equations we have to add the dynamical equation in the form of Euler-Lagrange equation if we use a Lagrangean formalism, or in the form of Hamilton equation if we use a Hamiltonian formalism. In the present paper we consider the contribution of the nonlinear terms in all equations (e.g. the second and third terms of RHS of eqs.(6,7)). The corresponding solutions should reduce to the standard normal modes of vibrations, if we restrict to the linear approximation.
For such typical hydrodynamical problems, like that described by eqs.(3,5,7,8), one generaly uses, in the linear approximation, the expansion in spherical harmonics. Such an expansion is no more appropriate for the nonlinear cases. Hence, we use for the potential of the flow the expansion $$\begin{aligned}
\Phi=\sum_{n=0}^{\infty}\biggl (
{{r-R_0 } \over {R_0 }}
\biggr )^{n}f_{n}(\theta , \phi ,t),\end{aligned}$$ where the functions $f_n$ are not orthogonal on the surface of the sphere and do not form in general a complete system. The convergence of eq.(9) is controlled by the value of the small quantity ${{r-R_0 } \over {R_0 }}
\le max|\xi |=\epsilon$, \[4\]. From the Laplace equation (in spherical coordinates), and the expansions $$\begin{aligned}
{{1} \over {r^n }}={1 \over {R_{0}^{n} }}\sum_{k=0}^{\infty }
(-1)^k ((n-1)k+1)\xi ^{k}, \ \ \ \ k=1,2,\end{aligned}$$ we obtain a system of equations which result in the recurrence relations for the unknown functions $f_n $ $$\begin{aligned}
f_n= {{(-1)^{n-1}(n-1)\triangle _{\Omega }f_{0}-2(n-1)f_{n-1}+
\sum_{k=1}^{n-2}(-1)^{n-k}(2k-(n-k-1)\triangle _{\Omega} f_{k})}
\over {n(n-1)}},\end{aligned}$$ with $n\ge 2$ and where $\triangle _{\Omega }=
{{1} \over {\sin \theta }}{{\partial } \over {\partial \theta}}
\biggl ( \sin \theta {{\partial } \over {\partial \theta}}
\biggr ) +{{1} \over {\sin ^{2} \theta }}{{\partial } \over {\partial \phi }}
$ is the angular part of the Laplacean operator in spherical coordinates. Eq. (11) reduces the unknown functions to only two: $\triangle _{\Omega }f_0 $ and $f_1 $: $$f_2 =-{1 \over 2}(\triangle _{\Omega }f_0 +2f_1 ),$$ $$\begin{aligned}
f_3 ={1 \over 6}(4\triangle _{\Omega }f_0 -
4\triangle _{\Omega }f_1 +4f_1 +2),\end{aligned}$$ $$f_4 ={1 \over 24}({\triangle }^{2}_{\Omega }f_0 -
14\triangle _{\Omega }f_0 +8\triangle _{\Omega }f_1 -8f_1 ) \ \ \dots .$$ If we choose the independent functions $\triangle _{\Omega }f_0 $ and $f_1 $ to be smooth on the sphere, they must be bounded together with all the $f_n $’s (these being linear combinations of higher derivatives of $f_{0}$ and $f_1 $) and hence the series in eq.(9) are indeed controlled by the difference in the radius between the deformed and the spherical one. However, in the following we will use only truncated polynomials of these series.
By introducing eqs.(11-12) in the second boundary condition, eq.(8), we obtain the condition $$\begin{aligned}
\sum_{n=1}^{\infty }n\biggl ( -{{h} \over {R_0 }}
\biggr )^{n-1}f_n =0,\end{aligned}$$ which reads, in the first order in $h/R_0 $ $$\begin{aligned}
f_1 ={{2h} \over R_0 }f_2 .\end{aligned}$$ From eqs.(12,14) the unknown function $f_1 $ is obtained, in the smallest order in $h/R_0 $ $$\begin{aligned}
\triangle _{\Omega }f_0 =-\biggl ( {{R_0 } \over {h}}+2
\biggr ) f_1 .\end{aligned}$$ Concerning the first boundary condition held at the free surface $\Sigma $, eq.(7), we need to calculate the derivatives of the potential of the flow on that surface $$\Phi _{r}|_{\Sigma }=\sum_{n}n{{(r-R_0 )^{n-1}_{\Sigma }} \over {R_{0}^{n}}}
f_{n}
= {{f_{1}} \over {R_0 }}+{{2\xi f_2 } \over {R_{0}}}+{\cal O}_{2}(\xi ),$$ $$\begin{aligned}
\Phi _{\phi }|_{\Sigma }=\sum_{n}\xi ^{n}f_{n,\phi }=
f_{0,\phi }+\xi f_{1,\phi }+{\cal O}_{2}(\xi ),\end{aligned}$$ $$\Phi _{\theta }|_{\Sigma }=\sum_{n}\xi ^{n}f_{n,\theta }=
f_{0,\theta }+\xi f_{1,\theta }+{\cal O}_{2}(\xi ).$$ By introducing the series eqs.(10,16) in eq.(7), for the traveling wave solution eq.(2), we have the equation $$\begin{aligned}
f_1 +2\xi f_2 =
R_{0}^{2} \xi _t+ {{\xi _{\phi }(1-2\xi )} \over {\sin ^{2}\theta }}
(f_{0,\phi }+\xi f_{1,\phi})+
\xi _{\theta }(1-2\xi )(f_{0,\theta }+\xi f_{1,\theta }) .\end{aligned}$$ We keep the nonlinearity of the boundary conditions, eqs.(7,8,13), in the first order (i.e. the first order in the expression of $f_0$ and the second order in the expression of $f_1$). Consequently, in order to be consistent, it is enough to take the linear approximation of the solution for $f_1 $ in eq.(17), like in the case of the normal modes of vibrations $$\begin{aligned}
f_1 =R_{0}^{2}\xi_{t}+{\cal O}_{2}(\xi ).\end{aligned}$$ Hence, by introducing the linear approximation for $f_1 $ (eq.(18)) in eq.(17) $$\begin{aligned}
2\xi f_2 =
{1 \over {\sin ^2 \theta }}\biggl (
-\xi _{\phi }f_{0,\phi }
+\xi \xi _{\phi }(f_{1,\phi }-2f_{0,\phi }) \biggr )
+\xi \xi _{\theta }(f_{1,\theta }-2f_{0,\theta }),\end{aligned}$$ and by taking the expression of $f_2 $ from the recurrence relations, eq.(14) and $\triangle _{\Omega }f_0 $ from eq.(15), we obtain the form of $f_0 $, in the second order in $\xi $ $$\begin{aligned}
f_{0,\phi}=-{{R_{0}^{3}\sin ^2 \theta } \over {h}}{{\xi \xi _{t}} \over {\xi
_{\phi}}}(1+2\xi )-{{\xi _{\theta}f_{0,\theta }} \over {\xi _{\phi }}}
+{\cal O}_{3}(\xi ).\end{aligned}$$ In the case of traveling wave profile of the form $\xi (\theta ,\phi ,t)=g(\theta )\eta (\phi -Vt)$, which introduces the restriction $\xi _{\phi }=-V \xi _{t}$ and vanishes the tangential velocity in the $\theta $-direction, eq.(20) becomes $$\begin{aligned}
f_{0,\phi }={{VR_{0}^{3}\sin ^{2}\theta } \over {h}}\xi(1+2\xi )
+{\cal O}_{3}(\xi ).\end{aligned}$$ Eqs.(18,20,21) describe, in the second order in $\xi$, the connection between the velocity potential (the flow) and the shape function, through the boundary conditions. This fact is a typical feature of nonlinear systems. The dependence of $\Phi |_{\Sigma }$ on the polar angles, in the second order in $\xi $, has the form of a quadrupole in the $\theta $-direction and depends only on $\xi $ and its derivatives in the $\phi $-direction. For traveling wave profiles the tangential velocity in the direction of motion of the perturbation, $v_{\phi }=\Phi _{\phi }/r\sin \theta $ is proportional with $\xi $ in the first order $$\begin{aligned}
v_{\phi }={{2VR_{0}\sin \theta } \over {h}}\xi+{\cal O}_{2}(\xi ),\end{aligned}$$ with the constant of proportionality being exactly the coupling parameter $\chi $ from our previous solitonic model, i.e. eq.(20) in \[4\]. This can be seen also from Figs. 1 and 2a in \[4\], where it is clearly stated the $1/h$ dependence of $\chi $, for fixed velocity $V$, like in eqs.(21,22). Eqs.(20-22) are valid for any traveling wave shape functions, case which includes harmonic or anharmonic oscillations, solitons, breathers, cnoidal waves, etc. \[7-9,13\].
In order to obtain the dynamical equation for the surface $\Sigma $ we follow the usual formalism for the normal vibration of a sphere \[14\], corrected with the corresponding nonlinear terms \[4,5,7,8\], i.e. we solve the Euler equation on the free surface with respect to the potential flow $\Phi $ and the shape function $\xi $, for given pressure, force fields and boundary conditions. The pressure at the free surface $\Sigma $ can be obtained from the surface energy of the deformed nucleus, $U_S$ $$\begin{aligned}
U_S =\sigma R_{0}^{2}\int _{0}^{2\pi }\int _{0}^{\pi}
(1+\xi )\sqrt{(1+\xi )^2 +\xi _{\theta }^{2}
+{{\xi _{\phi }^{2}} \over {\sin ^{2} \theta }}}\sin \theta d\theta d\phi , \end{aligned}$$ where $\sigma $ is the pressure surface coefficient. Indeed, by expanding in series the square root in eq.(23) with respect to $\xi $, up to the third order, we obtain for the first variation of the functional $U_S$ $$\begin{aligned}
\delta U_S =\sigma
R_{0}^{2}\int_{0}^{2\pi }\int _{0}^{\pi }
\biggl (
2+2\xi +\xi ^2 -
\triangle _{\Omega }\xi+
3\xi ^{2}\xi _{\theta } ctg \theta
\biggr )\delta \xi \sin \theta d\theta d\phi +{\cal O}_{4}(\xi ).\end{aligned}$$ Following \[14\] the surface pressure on $\Sigma $ is given by the local curvature radius of the surface, and from the volume conservation we have $$\begin{aligned}
P|_{\Sigma }=\sigma \biggl ( {1 \over {R_1 }}+{1 \over {R_2 }} \biggr )=
\sigma R_{0}^{2}{{\delta a_{\xi}} \over {R_{0}^{3}(1+2\xi)}},\end{aligned}$$ where $R_{1,2}$ are the principal radii of curvature of the surface of the fluid and $\delta a_{\xi }$ is the term in the paranthesis in eq.(24). By introducing eq.(24) in eq.(25) we obtain the expression of the surface pressure as a function of the shape function $$\begin{aligned}
P|_{\Sigma }={{\sigma } \over {R_0 }}
(-2\xi-4\xi ^2 -\triangle _{\Omega }\xi +3\xi \xi ^{2}_{\theta }
ctg \theta )+const.\end{aligned}$$ The terms of order three in $\xi _{\phi ,\theta}, \xi _{\phi ,\phi }$ and $\xi _{\theta ,\theta }$, could be neglected in eq.(25) due to the high localisation of the solution (the relative amplitude of the deformation $\epsilon $ is smaller than its angular halh-width $L$, $\xi \xi _{\phi \phi }/R_{0}^{2}\simeq \epsilon ^{2}/L^2 \ll 1$, etc.).
The Coulombian potential is given by a Poisson equation, $\triangle \Psi =\rho _{el}/\epsilon _{0}$, with $\epsilon _{0}$ the vacuum dielectric constant. By using the same method like for $\Phi $, \[4\], we obtain in the second order for $\xi $, the form $$\begin{aligned}
\Psi |_{\Sigma }={{\rho _{el} R_{0}^{2}}\over {3\epsilon _{0}}}
\biggl ( 1-\xi -{{\xi ^2 }\over {6}}
\biggr ) .\end{aligned}$$
In order to write the Euler equation, eq.(5), for the above restrictions, we take the surface pressure from eq.(26), the velocity potential from eqs.(9,14,18,21), and the Coulombian potential from eq.(27) and we obtain, in the second order in $\xi $ and the first order in its derivatives $$\Phi _{t}|_{\Sigma }+{{V^2 R_{0}^{4}\sin ^2 \theta }\over {2h^2 }}\xi ^2 =
{{\sigma }\over {\rho R_0 }}(2\xi +4\xi ^2+\triangle _{\Omega }
\xi -3\xi ^2 \xi _{\theta }ctg \theta )$$ $$\begin{aligned}
+{{\rho _{el}^{2} R_{0}^{2}}\over {3\epsilon _{0}\rho }}
\biggl ( \xi +{{\xi ^2 }\over {6}} \biggr ) +const.\end{aligned}$$ Neglecting all nonlinear terms in this expression, we obtain $$\begin{aligned}
\Phi _{t}|_{\Sigma }=-Vf_{0,\phi }=
-{{\sigma }\over {\rho R_0 }}(2\xi +\triangle _{\Omega }
\xi )-
{{\rho _{el}^{2} R_{0}^{2}}\over {3\epsilon _{0}\rho }}\xi
+const.,\end{aligned}$$ which, together with the linearised eq.(7), i.e. $\Phi _{r}|_{\Sigma }=R_0 \xi _{t}$, describes the normal modes of vibration of the liquid drop in the presence of the Coulombian field with spherical harmonics solutions for the angular part and complex exponential for the time dependence. This linear approximation represents exactly the reduction of the present model to the traditional liquid drop model.
In Eq.(28) is a nonlinear partial differential equation with respect to $\theta $ and $\phi $. By differentiating it with respect to $\phi $, by using eqs.(15,18) and by re-ordering the terms, we obtain, in the second order $$\begin{aligned}
A(\theta )\eta _{t }+B(\theta )\eta _{\phi }+C(\theta )g(\theta )\eta \eta _{\phi }+
D(\theta )\eta _{\phi \phi \phi }=0, \end{aligned}$$ which is a Korteweg de Vries (KdV) equation with variable coefficients depending on $\theta $, as a parameter $$A={{VR_{0}^{2}(R_0 +2h)\sin ^2 \theta }\over {h}}; \ \ \
B=-{{\sigma }\over {\rho R_0 }}{{(2g+\triangle
g)} \over {g}}-{{\rho _{el^{2}}R_{0}^2 }\over {3\epsilon _{0}\rho }};$$ $$\begin{aligned}
C=8\biggl (
{{V^2R_{0}^{4}\sin ^4 \theta
} \over {8h^2 }}-{{\sigma } \over {\rho R_{0}}} \biggr ) -
{{\rho _{el}^{2}R_{0}^2
}\over {9\epsilon _{0}\rho }} ; \ \ \
D=-{{\sigma }\over {\rho R_{0}\sin ^2 \theta }}, \end{aligned}$$ where by $\triangle g$ we understand only the action over $\theta $. First we want to make a qualitative analysis of the solutions of this nonlinear equation. The solutions can be clasificated by either using the algebraic hierachies and topological arguments \[7-9,12,13\] or by using the phase space portrait of these solutions \[8,12\]. The latter way gives a simpler picture of the classes of admisible solutions and, in addition, characterize their periodic and bounded character. In order to comment on the traveling wave solutions eq.(2), we use the relation $\partial _{t}=-V\partial _{\phi}$, with $V$ a parameter to be determined. We can then integrate the resulting equation once and write it in the form: $\eta _{tt}=(VA-A_0 )V^2 /C\eta -BV^2 /2C\eta ^2$. This is the Newton equation of motion for a one-dimensional particle described by the coordinate $\eta (t)$ in a “potential force” given by the RHS of the above expression. By integrating once this “force” we obtain the effective one-dimensional potential associated with this motion, $U(\eta
,V)$, depending on $V$ as a parameter. For a given value of $V$ we can construct a certain phase space of this type of solutions. All possible solutions of the KdV equation are classified according with all admissible trajectories of this particle, under the action of the effective potential, in the phase space, \[8,12\], Figs.1. In Fig.1a we present the effective potential shape and the associate trajectories in the phase space for the KdV equation. The possible trajectories with constant energy ($E=U(\eta ,V)+{1 \over 2}\eta _{t}^{2}$), are classified through the parameter energy, $E$. If we reduce to the linear approximation, the effective potential is quadratic and the solutions allow only harmonic oscillations, Fig1b. This situation represents the case of the traditional liquid drop model where only the harmonic oscillations are taken into account, and the governing equation becomes the Helmoltz equation, \[3,15\]. The KdV equation has a cubic potential energy, Fig1b. This is an example in which one can see directly the effect of the introduction of the nonlinearity: the harmonic oscillator effective potential gets a pocket and a saddle point, which is responsible for the new soliton solutions. Oscillations still exist and become anharmonic, being described by the cnoidal wave solutions of the KdV equation. At the superior boundary of the pocket, where the energy $E$ equals the potential energy at the saddle, the anharmonic oscillations become aperiodic and describe the soliton solution of the KdV equation. For values of the energy higher than the saddle (the soliton energy), or for smaller than the bottom of the valley, the solutions decay into instable, singular ones, having poles towards infinity. However, these residual solutions are not taken into account here since they are not traveling waves.
Eq.(30) has as solutions the cnoidal waves. The localised solution, solitons have the form $$\begin{aligned}
\eta (\phi ,t)=g(\theta )sech ^2 \biggl ( {{\phi -V(\theta )t} \over {L(\theta )}}
\biggr ),\end{aligned}$$ where $g(\theta )\le \eta _0 =\epsilon R_0 $ is the soliton amplitude, $V(\theta )$ is its angular velocity and $L(\theta )$ is its half-width. Due to the nonlinearity of the equation, these coefficients depend on the coefficients of the KdV equation, through the relations, \[7,8\] $$\begin{aligned}
L(\theta )=\sqrt{{{12D} \over {\eta _0 C}}}; \ \ \ \ \ V(\theta )=
{{g(\theta) C+3A}\over {3B}},\end{aligned}$$ in other words, a higher soliton moves faster and has a larger half-width. The soliton is extremely stable against perturbations in the sense of smooth modifications of the coefficients of the equations, initial conditions, introduction of small additional terms in the equation or when interacting with other solutions of this equation (scattering). Solitons have a constant traveling profile in time and an infinite number of integrals of motion \[7-9,12,13\]. Details concerning the explicit dependence of the soliton parameters, as functions of the coefficients of the KdV equation eq.(30), are given [*[in extenso]{}*]{} in \[4\].
In the following we analyse the stability of the steady-wave solutions of the KdV equation, eq.(30), against the dependence of its coefficients on $\theta $. In order to have a real traveling wave along the $\phi $-direction, the parameters of the soliton solution must fulfil the following restrictions:
1\. The amplitude of the soliton and the depth of the layer must decrease from their maximum values, ($\eta _{0},h$) on the circle $\theta =\pi /2$ towards 0, when $\theta \rightarrow 0,\pi $. Then, the function $g(\theta )$ must increase, when $\theta : 0, \pi \rightarrow
\pi /2, $ from $0$ to $\eta _{0}$. The function $h(\theta )$ must increase when $\theta : 0, \pi \rightarrow
\pi /2, $ from $0$ to $h$. The numbers $\eta _{0}, h(\pi /2)$ (i.e. the maximum amplitude of the soliton and the maximum depth of the layer) are free parameters, as have been shown in \[4,5\].
2\. The (angular) half-width $L$, eq.(33), must be constant: $$\begin{aligned}
D(\theta )={\cal C}_1 C(\theta )g(\theta ), \end{aligned}$$ 3. The (angular) velocity $V$, eq.(33), must be also constant: $$\begin{aligned}
D(\theta )={\cal C}_2 B(\theta ), \end{aligned}$$ where ${\cal C}_{1,2}$ are constants. Taking into account the mutual relations between the parameter of the soliton solution and the coefficients of the KdV equation, eqs.(33) and the restrictions 1-3, we obtain a system of two differential equations, eqs.(34,35), for the two unknown functions $L(\theta ),V(\theta )$, with bilocal Cauchy conditions at the ends of $[0, \pi ]$, given by condition 1. Consequently we have for these functions a well defined boundary condition problem with unique solutions, depending parametricaly on $\eta _{0},h(\pi /2)$. Both differential equations, eqs.(34,35), are well defined in $\theta [0,\pi ]$ and have no poles.
The nonlinear liquid drop model
===============================
The liquid drop model has an infinite-dimensional Hamiltonian structure described by a nonlinear Hamiltonian function. Such systems can be treated in the same way as the finite-dimensional ones, excepting some difficulties related with the differentiability of the flow and with the definition of the symplectic structure, \[12\]. These occure, on one side, due to the fact that the vector fields are only densely defined (since we are dealing with partial differential equations) and, on the other side, due to the fact that the linear Hamiltonian and the nonlinear one could have different symplectic structures (different associated symplectic forms, phase spaces, etc.). If these difficulties are overcome, the theory of Hamilton equations and the invariants and conservation laws may be used, too.
The dynamics governing traveling waves and small (even anharmonic) oscillations in a perfect irrotational fluid, in 1+1 space-time dimensions, can be described by a scalar field $\Phi (x,t)$ and a Hamiltonian $H_{[\Phi ]}=\int _{D}h dx$ with the Hamiltonian density $h$ $$\begin{aligned}
H_{[\Phi ]}=\int _{x_1 }^{x_2 } \biggl ( {1 \over 2} \Phi _{t}^{2}
-{{1} \over {V^2 }}|\Phi _x |^2 +F(\Phi ) \biggr ) dx\end{aligned}$$ where $x_{1,2}=\pm \infty $ for the flow along the line ($D=R$), and $x_{1,2}=0,\pi $ ($x\rightarrow \phi \in D=[0,\pi ]$), for a flow on a closed manifold. The first term in the integrand stays for the kinetic energy ($T$) and the next two for the potential energy ($U$), the last one describing the restoring forces in a general manner. Such potentials occur in quantum theory of self-interacting mesons where the function $F$ governs the nonlinear part of the interaction. The configuration space for this problem is some space of smooth real fields with the associated symplectic form generated by the usual $L^2 $ inner product. This approach can also provide a Lagrangian $L=T-V=\int _{D}{\cal L} dx$ and the equation of motion is $$\begin{aligned}
\Phi _{t}=\Phi _{tt}-V^2
\Phi _{xx}-\partial _{x}{{dF} \over {d\Phi }}.\end{aligned}$$ This formalism provides periodical harmonic solutions in $x$ for $F=0$ and nonlinear waves for general $F$. In order to obtain localised solutions, we have to use another type of Hamiltonian, i.e. a KdV (or modified KdV, etc.) one. This could be provided by the Hamiltonian $$\begin{aligned}
H[\Phi ]=\int _{D}h(\Phi,\Phi _x ,\Phi _{xx}, ...)dx,\end{aligned}$$ where $D$ is again the domain of the generic space coordinate $x$, \[9\]. The Hamiltonian vector field id given by $X_{H[\Phi ]}={{\partial } \over {\partial x}}{{\delta h} \over {\delta \Phi
}}$, where $\delta /\delta \Phi $ is the functional derivative of $h$ with respect to the $\Phi $ $$\begin{aligned}
{{\delta h} \over {\delta \Phi }}=
\sum_{k \ge 0}(-1)^{k+j}\partial _{t}^{j}\partial _{\phi }^{k}
{{\partial h}\over {\partial \Phi _{\phi }^{k}\Phi _{t}^{j}}},\end{aligned}$$ with $\partial _{t}^{j}=\partial ^{j}/ \partial t^{j}$, $\Phi _{\phi }^{k}=\partial ^{k}\Phi / \partial {\phi }^{k}$, etc. The corresponding Hamilton equation becomes $$\begin{aligned}
{{\partial \Phi } \over {\partial t}}=
-{{\partial } \over {\partial x}}{{\delta h} \over {\delta \Phi }},\end{aligned}$$ i.e. $\Phi (x,t)$ are the integral curves of $X_{H_{[\Phi ]}}$. Localised solutions must vanish at infinity for $D=R$, and must be periodical or rapidly decreasing functions for $D=[0,\pi]$. A symplectic structure on these spaces of functions is given by the skew-scalar product, \[12\] of two arbitrary functions $\Phi _1 $ and $\Phi _2 $ $$\begin{aligned}
\omega (\Phi _1 ,\Phi _2 )={1 \over 2}\int _{D}\biggl (
\int_{x_2}^{x}(\Phi _1 (x)\Phi _2 (y) - \Phi _1 (y) \Phi _2 (x) )
dy\biggr ) dx.\end{aligned}$$ However it is a qualitative difference between the two Hamiltonians, eqs.(36,38), occuring from their different symplectic geometries. This difference has consequences in the existence of two different phase spaces, and finaly, in the difficulty of smoothly connecting their characteristic solutions, i.e. linear oscillations and solitary waves.
The liquid drop model consists in the sum of the kinetic and potential energy of the fluid, $E=T+U$. All the terms in $E$ depend on two functions: the shape of the surface $\xi (\theta ,\phi ,t)$ and the potential flow $\Phi (r, \theta ,
\phi ,t)$. In the following we use for the shape function the factorization given in eq.(2) and we consider only one canonical coordinate, $\phi $. In the following we consider only those solutions $\xi (\theta ,\phi ,t)$ in which the coordinate $\theta $ describing the transversal profile of the traveling wave is considered to be decoupled of $\phi $, eq.(2). All the terms dependent of $\theta $ will be absorbed in the coefficients of some integrals, and will become the parameters of the model. In this case the energy becomes a functional of $\eta $ only.
The potential energy $U$ has at least three terms: the surface energy ($U_S$), the Coulombian energy ($E_C$) and the shell energy ($E_{sh}$). The first term describing the surface contribution is $$\begin{aligned}
U_S =\sigma ({\cal A}_{\xi }-{\cal A}_{0}),\end{aligned}$$ where $\sigma $ is the surface pressure coefficient, ${\cal A}_{\xi }$ being the area of the deformed nucleus and ${\cal A}_0$ the area of the spherical nucleus, of radius $R_0$. Both these areas encircle the same volume $V_0=
{{4\pi {R}_{0}^{3}} \over 3}$ of the nucleus. The surface of the deformed nucleus depends only of the shape function $\xi (\theta ,\phi ,t)$, eqs.(1,2,23). It is possible to expand eq.(23) for small deformations $\xi /R_0<<1$, in its Taylor series in all its three arguments $\xi, \xi _{\theta },
\xi _{\phi }$, around 0. By introducing the shape function,eq.(2) in eq.(23) we obtain, in the third order $$U_{S}[\eta ]=\sigma R_{0}^{2}\int_{0}^{2\pi }\biggl [
2S_{1,0}^{1}\eta +(S_{1,0}^{1}+{1 \over 2}S_{0,1}^{1}){\eta }^2$$ $$\begin{aligned}
+{1 \over 2}S_{1,2}^{1}{\eta }^3
+{1 \over 2}S_{2,0}^{-1}{\eta }_{\phi }^{2}\biggr ]d\phi
+{\cal O}_{3}(\eta ,\eta _{\phi }),\end{aligned}$$ where ${\cal O}_3(\eta ,\eta _{\phi })$ represents the contributions of terms involving higher orders than 3 in the function $\xi $ and its derivatives. We introduce the notations $$\begin{aligned}
S_{n,m}^{k}=\int_{0}^{\pi }g^{n}g_{\theta }^{m}{\sin }^{k}\theta
d\theta .\end{aligned}$$ The term proportional with $\eta \eta _{\phi }^{2}$ in eq.(23) is neglected because it belongs to the fourth order, since the second derivative to the square introduced a factor of $\eta _{0}^{3}/L^2 $ which is smaller compared with $\eta _{0}^{3}$, ($\eta _{0}$ being the amplitude of the perturbation).
The volume conservation condition $$\begin{aligned}
V_0=\int _{\Sigma }{{r^{3}(\theta ,\phi )} \over 3}d\Omega=
{{4\pi } \over 3}R_{0}^{3},\end{aligned}$$ leads, according with eqs.(1,2) to the restriction $$\begin{aligned}
\int _{\Sigma }(3\xi +3\xi ^2 +\xi ^3 )d\Omega =
\int_{0}^{2\pi }(3A_{1,0}^{1}\eta +3A_{2,0}^{1}\eta ^2 +A_{3,0}^{1}
\eta ^3 )d\phi =V_{cluster}.\end{aligned}$$
The second term in the potential energy is given by the Coulomb interaction, for constant charge density of the nucleus $\rho _{el}$, in the volume $V_0 $ $$\begin{aligned}
U_C [\eta ] ={{\rho _e} \over 2}\int _{\cal V}^{'}
\int _{\cal V} {1 \over {|{\vec r}-{\vec r}'|}}
d{\cal V}d{\cal V}^{'}.\end{aligned}$$ The Coulomb energy contains three terms: the self energy of the core of radius $R_0$ and of the bump (the deformation) and the interaction energy between these. Adding these terms, subtracting the Coulomb energy of the initial spherical configuration and taking into account the terms up to the third order in $\xi$, we can write the Coulomb energy in the form, \[6\] $$\begin{aligned}
U_{C}=U^{(1)}_{C} \biggl [ 1+(C_{1,0}^{1}+C_{1,1}^{1})\int _{0}^{2\pi }\eta
d\phi
+C_{2,0}^{1}\int_ {0}^{2\pi}\eta ^2 d\phi+
C_{3,1}^{1}\int_ {0}^{2\pi}\eta ^3 d\phi
\biggr ]+U_{C}^{(0)},\end{aligned}$$ where $$U_{C}^{(1)}=3.093(Z_0 -Z_{cl})^2 (\rho {\cal V}_{0})^{-1/3},$$ $$\begin{aligned}
U_{C}^{(0)}=-0.665Z_{0}^{2}(\rho {\cal V}_{0})^{-1/3}+8.6275Z_{cl}^{2}
(\rho {\cal V}_{0})^{-1/3},\end{aligned}$$ are constants and the Coulombian energy is given in MKS units. The numbers $Z_0 ,Z_{cl}$ represent the atomic mass numbers of the parent nucleus and of the corresponding cluster, respectively. The coefficients $C_{ij}$ in eq.(48) are defined in the form $$\begin{aligned}
C_{ij}^{k}={1 \over {R_{0}^i}}\int_{0}^{\pi}h^i g^j \sin ^{k} \theta d\theta .\end{aligned}$$
The shell energy is introduced by considering that the main contribution to the shell effects in cluster decays is due to the final nucleus, close to the double magic nucleus $^{208}$Pb. The spherical core $r\le R_0 -h$ represents the final nucleus, which is also unexcited for the even-even case. We introduce the shell energy like a measure of the overlap between the core and the final nucleus, on one side, and between the final emitted cluster and the bump, on the other hand $$\begin{aligned}
E_{sh}={{ V_{over }}\over {V+[V_0 -(V_{cluster}+V_{layer})]-V_{over}}},\end{aligned}$$ where $V_{over }$ denotes the volume of the overlap between the volumes of the initial $V_0$ and final $V$ nuclei, $V_{cluster}$ is the soliton volume, eq.(46), and $V_{layer}$ is the layer volume on which the soliton is moving (i.e. $r\in [R_0 -h,R_0 ]$). We use this form for the shell energy multiplied with a constant $U_0$, choosen such that the total energy of the system in the state of rezidual nucleus + cluster to be degenerated with the ground state energy.
The kinetic energy is given by $$\begin{aligned}
T={{\rho } \over 2}\int_{\cal V}v^2d{\cal V}=
{{\rho } \over 2}\int_{\cal V}|\nabla \Phi |^2d{\cal V}=
{{\rho } \over 2}\oint_{\Sigma }\Phi \nabla \Phi \cdot d{\vec S},\end{aligned}$$ where $d{\vec S}=R^{2}_{0}(1+\xi _{\theta }^{2}+\xi _{\phi }^{2}
)^{-1/2}(1, -\xi _{\theta }, -\xi _{\phi }
)\sin \theta d\theta d\phi $ is the oriented surface element of $\Sigma $, eq.(1). Explicitly
$$\begin{aligned}
T={{\rho } \over 2}\int_{\Sigma }
{{
\Phi \Phi _r -{{\Phi \Phi _{\theta }\xi _{\theta }} \over {r}}
-{{\Phi \Phi _{\phi }\xi _{\phi } } \over {r\sin \theta }}
} \over {
\sqrt{1+\xi _{\theta }^2 +\xi _{\phi }^2}
}}
dS\end{aligned}$$
where $dS=R^2 \sin \theta d\theta d\phi $ is the scalar surface element. Now we use the first boundary condition. By taking $\Phi _r$ from eq.(16) and introducing it in eq.(53) the dependence of the kinetic energy of the tangential velocity along $\theta $-direction $\Phi _{\theta }$ becomes negligible and we obtain $$\begin{aligned}
T={{R_{0}^{2}\rho } \over 2}\int_{0}^{\pi}\int_{0}^{2\pi}
{{
R_0 \Phi \eta _t \sin \theta +{1 \over {R_0}}
\xi _{\phi } \Phi \Phi _{\phi }(1-\sin \theta )} \over
{\sqrt{1+\xi _{\theta }^{2}+\xi _{\phi }^{2}}}} d\theta d\phi ,\end{aligned}$$ where we have approximated $r|_{\Sigma }\simeq R_0 $ in the numerator, since the corresponding higher corrections will occure in higher order than three in $\xi $. We use now the second boundary condition, which, together with the first, gives the form of the functions $f_n$ in the structure of $\Phi |_{\Sigma
}$. In the following we take the expression of $\Phi $ from eq.(9), with its coefficients $f_n$ obtained in eqs.(12,14,18,21) and using the derivatives of $\Phi $ from eqs.(16), we have for the kinetic energy in eq.(54), in the second order in $\xi $ $$T[\eta ]={{R_{0}^{6}\rho V} \over {2}}\biggl ( C_{-1,2}^{3}\int_{0}^{2\pi}
\eta _{t} \int_{0 }^{\phi } \eta ^2 ({\tilde {\phi}})
d{\tilde {\phi }} d\phi
+2C_{-1,2}^{3}\int_{0}^{2\pi}\eta _{t} \int_{0 }^{\phi } \eta
({\tilde {\phi}}) d{\tilde {\phi }}
d\phi \biggr )$$ $$\begin{aligned}
+{{R_{0}^{7}\rho V^{2}} \over {2}}
\biggl (
(C_{-2,3}^{5}-C_{-2,3}^{6})\int_{0}^{\pi }\eta \eta _{\phi }
\int_{0 }^{\phi } \eta ({\tilde {\phi}}) d{\tilde {\phi }}\biggr )d\phi ,\end{aligned}$$ where $C_{ij}^{k}$ are defined in eq.(50) and the symbol ${\tilde {\phi }}$ is a dummy variable of integration. Since we are looking only for traveling wave solutions (which include any sort of oscillations or moving bumps) we can transform the derivative with respect to time into the derivative with respect to $\phi $, by using eq.(2), i.e. $\partial _t =
-V \partial _{\phi }$. We perform then an integration by parts for those term containing the primitives of $\eta $, and we finaly obtain for the kinetic energy $$\begin{aligned}
T[\eta ]=T^{(1)}\int_{0}^{2\pi }\eta ^2 d\phi +T^{(2)}\int_{0}^{2\pi }\eta ^3 d\phi ,\end{aligned}$$ with the parametric functions $T^{(i)}$ given by $$T^{(1)} ={{R_{0}^{6}\rho V^2 } \over 2}C_{-1,2}^{3},$$ $$\begin{aligned}
T^{(2)}={{R_{0}^{6}\rho V^2 } \over 2}(2C_{-1,2}^{3}R_0 +C_{-2,3}^{5}+
R_0 C_{-2,3}^{6}).\end{aligned}$$
The total energy of the system is the sum of the terms given in eqs.(43,48,51,56). We write it, in the second order, as a functional depending on the unknown functions $\eta (\phi ,t) ,g(\theta ), h(\theta )$ and the free parameter $V$ $$E[\eta ,g(\theta ),h(\theta )]=
\int_{0}^{2\pi }\biggl [
\biggl ( 2\sigma R_{0}^{2}S_{1,0}^{1}
+U^{(1)}_{C}(C_{1,0}^{1}+C_{1,1}^{1})\biggr ) \eta$$ $$+ \biggl ( \sigma R_{0}^{2}\biggl ( S_{1,0}^{1}+{1 \over 2}
S_{0,1}^{1}\biggr )
+U_{C}^{(1)}C_{2,0}^{1}+T^{(1)}
\biggr ){\eta }^2
+\biggl (
+{{\sigma R_{0}^{2}} \over 2}S_{1,2}^{1}+U_{C}^{(1)}C_{3,1}^{1}+T^{(2)}
\biggr )\eta ^{3}$$ $$\begin{aligned}
+{{\sigma R_{0}^{2}} \over 2}S_{2,0}^{-1}{\eta }_{\phi }^{2} \biggr ] d\phi +
E_{sh}[V_{cluster}]+const.\end{aligned}$$ We note that in the deduction of eq.(58) we have implicitely used the boundary conditions, in the expression of $T$. This functional may be interpreted as depending on $\eta $ only, the function $g(\theta )$ being fixed through the conditions 1-3. Generally, the expression $E$ in eq.(58) depends on three functions $\eta (\phi -Vt )$, $g(\theta )$ and $h(\eta )$, in the third order of approximation. In order to formulate eq.(58) as the conservation law of the energy, i.e. the functional $E$ to describe a Hamiltonian functional, we use the abstract definition of a Hamiltonian system as the Lie algebra of the Poisson bracket $$\begin{aligned}
\{ F_1 ,F_2 \} =\int _{0}^{\pi }\int _{0}^{2 \pi }\biggl (
{{\delta F_1 }\over {\delta \xi }}{{\partial }\over {\partial \phi }}
{{\delta F_2 }\over {\delta \xi}}+
{{\delta F_1 }\over {\delta \xi }}{{\partial }\over {\partial \theta }}
{{\delta F_2 }\over {\delta \xi}}
\biggr )\sin \theta d\theta d\phi ,\end{aligned}$$ where $F_i =\int \int f_i (\xi ^{k}(\theta ,\phi ,t))\sin \theta d\theta
d\phi $ are generic functionals and $\xi ^{k}$ represents the set of the function $\xi $ together with its derivatives with respect to $\theta
,\phi $ up to a certain order \[12\]. If we take the energy expression, eq.(58) as a Hamiltonian, $E \rightarrow H[\eta ]$, then the time derivative of any quantity $F[\eta ]$ is given by $$\begin{aligned}
F_t =[F,H].\end{aligned}$$ We analyse eq.(60) for $\xi (\theta ,\phi ,t)=g(\theta )
\eta (\phi -Vt)$, i.e. the case investigated in section 2, eq.(2), with respect to the variation of the function $\eta $ only, and for fixed $g$ and $h$ introduced as parametric functions. We define $F=\int_{0}^{2\pi}
\eta (\phi -Vt)d\phi $ and, since $\delta F /\delta \eta =1$, we get from eqs.(58,60) $$\begin{aligned}
{{dF}\over {dt}}=\int_{0}^{2\pi }\eta _t d\phi =\int_{0}^{2\pi}
\partial _{\phi }{{\delta ({\cal A}\eta +{\cal B}\eta ^2 +{\cal C}\eta ^3 +
{\cal D}\eta _{\phi }^{2})}
\over {\delta \eta }}d\phi ,\end{aligned}$$ where ${\cal A}=2\sigma R_{0}^{2}S_{1,0}^{1}+U_{C}^{1}(C_{1,0}^{1}+
C_{1,1}^{1})$, ${\cal B}=\sigma R_{0}^{2}(S_{1,0}^{1}+{1 \over 2}
S_{0,1}^{1})+U_{C}^{1}C_{2,0}^{1}+T^{(1)}$, ${\cal C}={{\sigma R_{0}^{2}}\over {2}}S_{1,2}^{1}
+U_{C}^{(1)}C_{3,1}^{1}+T^{(2)}$ and ${\cal D}={{\sigma R_{0}^{2}}\over 2}S_{2,0}^{-1}$, from eqs.(58). Eq.(61) gives $$\begin{aligned}
\int_{0}^{2\pi }\eta _t d\phi =\int_{0}^{2\pi }
(2{\cal B}\eta_ {\phi }+6{\cal C}\eta \eta _{\phi }-2{\cal D}
\eta _{\phi \phi \phi })d\phi .\end{aligned}$$ Eq.(62) lead to the KdV equation, similar with the Euler approach in section 2. Therefore we have shown that the energy of the nonlinear liquid drop model, eq.(58), can be interpreted as the Hamiltonian of a of the one-dimensional KdV equation \[5,7-10,13,14\], in agreement with the result obtained in section 2, eq.(30), from the Euler equation approach. The coefficients of the terms in the above expression depend on two arbitrary functions but of different argument, and hence they represent a parametric dependence and are not involved in the Hamiltonian dynamics of the function $\eta (\phi
,t)$ describing the traveling wave profile in the direction of propagation. Similar de-coupling of the coordinates is used in the theory of 2-dimensional solitary wave as in the Kortweg-Petviashvili (KP) equation \[12,13\].
The cnoidal and solitary wave solutions
=======================================
The KdV equation in its most general form $$\begin{aligned}
A\eta _t +B\eta _{\phi } +C\eta \eta _{\phi }+D\eta _{\phi \phi \phi }=0,\end{aligned}$$ has three classes of exact solutions, depending on the three initial conditions, for the same values of its coefficients, $A,B,C,D$. The most general steady-state solution of the KdV eqution has the form of an oscillation that can reduce to the simple-pulse solution in the limit that the oscillations period tends to infinity. This solutions can be classified using the phase space portrait, Fig.1, in cnoidal waves, solitary waves and singular solutions, Fig.2. In the following we denote with $V$ the traveling velocity (phase velocity), with $L$ the half-width and with $\eta _{0}$ the amplitude of the solutions, eqs.(32,33). For small amplitudes $|\eta _0 |\ll 1$, i.e. $C\simeq 0$, the KdV equation with has approximate solutions in the form of stationary small amplitude oscillations, if $sign(CD)=+1$. This is the case when the nonlinear term (the coefficient $C$ in eq.(31)) is neglected, the dispersion law becomes $L=2\pi / V^{3/2}$ and the group velocity is $V_{gr}=3V$. These solutions are instable and fall into cnoidal waves, if their amplitude is increased with the increasing of the energy.
The algebraic convenience is obtained by writing the steady state solutions in the form of eq.(2) with $\eta _{t}=-V\eta _{\phi }$. This allows us to use, in the following, the variable $z=\phi -Vt$. The KdV equation reduces to an ordinary differential equation for which a first quadrature may be immediatley obtained. A second quadrature is possible, after the first one, by multiplication of the resulting equation with $\eta _{z}$. We find that $$\begin{aligned}
\eta _{z}^{2}={{2C}\over {3D}}\eta ^3 +{{AV-B}\over {D}}\eta ^2+ a\eta +b,\end{aligned}$$ where $a,b$ are constants of integration. Eq.(64) factorizes into $$\begin{aligned}
\eta _{z}^{2}=-4(\eta -\alpha _1 )(\eta -\alpha _2 )(\eta -\alpha _3
)=-U[\eta ],\end{aligned}$$ with $\alpha _1 \le \alpha _2 \le \alpha _3 $ real roots and with the same interpretation for $U[\eta ]$ like in eq.(58) and Figs.1, of the kinetic (LHS of eq.(62)), potential ($U$) and total ($E$) energies. Since $\eta _z $ is real, for finite amplitude oscillations $\eta $ must be confined to the range $\alpha _2 \le \eta \le \alpha _3$, Fig.3. In between that range, the potential energy has a valley which characterizes the stability of the solutions. Setting $$\begin{aligned}
\alpha _3 -\eta \biggl ( \sqrt{{(\alpha _3 -\alpha _1 )C
}\over {6D}}z\biggr ) =(\alpha _3 -\alpha _2 )w^2
(z),\end{aligned}$$ we obtain the differential equation $$\begin{aligned}
w_{z}^{2}=(1-w^2 )(1-k^2 w^2 ),\end{aligned}$$ with $m^2 ={{\alpha _3 -\alpha _2 }\over {\alpha _3 -\alpha _1 }}$. This equation defines the Jacobi elliptic functions $sn(z|m)$ of parameter $m$ of amplitude $z$ \[15\] $$\begin{aligned}
sn(z|m)=\sin \beta , \ \ \
z=\int_{0}^{\beta }{{dx}\over {\sqrt{1-m\sin ^2 x}}}, \end{aligned}$$ and evidently $z(\pi /2,m)=K(m), cn^2 (z|m)=1-sn(z|m)$. The square of the cnoidal sinus and cosinus oscillates between 0 and 1, when $m$ takes real values in $[0,1]$, with a period equal with $2K(m)$, where $$\begin{aligned}
K(m)=\int_{0}^{1}{{dx}\over {[(1-x^2 )(1-m^2 x^2 )]^{1/2}}},\end{aligned}$$ represents the Jacobi elliptic integral. The solutions become $$\begin{aligned}
\eta (\phi -Vt)=\alpha _2 +(\alpha _3 -\alpha _2 )cn^2 \biggl ( \sqrt{{{C(\alpha _3
-\alpha _1 )}\over {6D}}}(\phi -Vt)\bigg | m \biggr ) .\end{aligned}$$ The solution $\eta $ oscillates between $\alpha _2$ and $\alpha _3$, with a period $T=2K(m)\sqrt{{6D}\over {(\alpha _3 -\alpha _1 )C}}$, as one can see from the negative potential valley in Fig.3. This solution is not allowed to exist outside of this valley. However, the shape of the valley can be modified by choosing different values for the parameters $\alpha _i $, i.e. one can associate different potential pictures to different sets of initial conditions. The function $cn(z|m)$ has two limiting forms $cn(z|0)=\cos z$, $cn(z|1)=$sech $z$, see Fig.2, too. The potential for both these limits is presented in Fig.3. If $\alpha _2 $ approaches $\alpha _1 $, $m$ approaches 1 and $T,K
\rightarrow \infty$. We obtain in this limit a localised solution $$\begin{aligned}
\eta (\phi -Vt)=\alpha _2 +(\alpha _3 -\alpha _2)cn^2 \biggl (
\sqrt{{{C(\alpha _3 -\alpha _2 )}\over {6D}}}(\phi -Vt)\bigg | 1
\biggr ),\end{aligned}$$ which is a bump of half-width $L=\sqrt{{6D} \over {C(\alpha _3 -\alpha _2 )}}$ and amplitude $\alpha _3 -\alpha _2 $ above a reference level $\alpha _2$. More, since $\alpha _1 +\alpha _2 +\alpha _3 ={{3(B-AV)}\over {2C}}$, we can write, in the limit $\alpha _2 =0$ (so that $\alpha _3 ={{3(B-AV)} \over {2C}}$) $$\begin{aligned}
\eta (\phi -Vt)=\eta _0 sech ^2 \biggl [
\sqrt{{{\eta _0 C}\over {12 D}}}(\phi -Vt)
\biggr ],\end{aligned}$$ which is exactly the soliton solution, i.e. eq.(32). From Fig.3 we see that the solution is defined between the limits $\alpha _1 ,\alpha _3 $ (the maximal domain of definition) and, due to the value zero of the derivative of $\eta $ in $\alpha _1$ (the saddle point), it requests infinite time to approach this limit, therefore it has a high stability. On the other hand, a steady-state solution in the form of a small oscillation is obtained when $\alpha _3 \rightarrow \alpha _2 $ and $m \rightarrow 0 , T\rightarrow \pi
/2$. In this limit we have the solution for small harmonic oscillations in the form $$\begin{aligned}
\eta (\phi -Vt)=\alpha _2 +(\alpha _3 -\alpha _2 )\cos ^2 \biggl (
\sqrt{{{(\alpha _3 -\alpha _1 )C}\over {6D}}}(\phi -Vt)
\biggr ).\end{aligned}$$ In this latter case the range for $\eta $ becomes very small, in principle a point situated in the bottom of the potential valley centered on $\alpha _3 $, Fig.3, (but in fact a small domain, since the KdV equation was deduced in the second order only) In conclusion, within the same equation, only by modifying the initial conditions and the parameter $V$, one can obtain both periodic solutions and localised bumps. A picture showing the smooth deformation from the cosine function towards the soliton is presented in Fig.4. Such different choices introduce different potentials, as in Fig.3. In this way, following a certain path in the space of these parameters, the dynamical evolution can smoothly approach both limits. The solutions depend on three free parameters (constants of integration) subjected to two additional conditions. On one side we have the condition of the volume conservations, requested by the physical solutions, eqs.(45,46). On the other side the solutions must be periodic, i.e. $$\begin{aligned}
K\sqrt{{\alpha _3 -\alpha _2 }\over {\alpha _3 -\alpha _1 }}=
{{\pi}\over {n}}\sqrt{\alpha _3 -\alpha _1 },\end{aligned}$$ for $n=1,2,\dots ,N$ with $N$ always finite ($K(m)$ is bounded from below) such that $N\le 2\sqrt{\alpha _3 -\alpha _1 }$. Consequently the whole model contains only a free parameter which can be choosen to be one of the $\alpha
$’s, $V$ or $\eta _0 $, and we can draw the potential shape $U(r, \eta _0 )$ as a 2-dimensional surface. In Fig.5 we present such a contour levels potential picture where the coordinates are the amplitude of the soliton $\eta _0$ (denoted $r$) and the parameter $\alpha _2 $, (denoted $a$). Each cross section of the potential surface parallel to the $r$-axis represent the potential of the solution, for a certain initial condition (a fixed value of $\alpha _2 $) and a certain fixed volume. In the limit of the soliton wave ($r$ around 3) we have a deep potential hole ($\alpha _2 =\alpha _3 \simeq 1.6$) and in the limit of the small oscillations ($r$ around $2.4$, $\alpha _2 =\alpha _1
\simeq 3.2$) we obtain again a minimum, embedded into a large potential valley. The reference level of the soliton ($r\simeq 1.4$) is smaller than the average reference level of the small oscillations, due to the volume conservation. The evolution of the corresponding solution $\eta (\phi -Vt ;\alpha _2 )$ between these limiting forms is plotted in Fig.6.
The spectroscopic factors
=========================
The spectroscopic factor are defined as the ration between the experimental decay ($\lambda _{exp}$) over the theoretical one-body constant ($\lambda
_{Gamow}$). For the alpha decay, the relative ratios are well reproduced but the absolute values are underestimated at least by two orders. Possible explnations arrise from the fact that the many-body correlations are neglected in these approaches, the correct antisymmetrization of the channel wave function, the impossibility of separation in two factors of the transition element in the integral theory \[1,2\]. In the present paper we consider that we have introduced the many-body effects in a phenomenologic way, based on a collective model, resulting in the possibility of describing of the large enhancement of the alpha cluster on the nucler surface. hence, we try to give an explanation of the spectroscopic factors based on this model. By considering the cnoidal waves as solutions of both the hydrodynamic equations (Euler) and the liquid drop model equations, we obtain new shapes which show that, starting from the harmonic normal modes on the nucler surface and due to the nonlinearities, some nucleons are grouping together to form the emitted cluster. In the cnoidal wave description, the spectroscopic factors may be given by the ratios of the two wave amplitudes in the corresponding wells, evaluated with the help of barrier penetrability between the two minima, Fig.5, \[4,5\]. The spectroscopic factors are given by the penetrabilities of the above barriers $$\begin{aligned}
S=exp\biggl (
-{{2}\over {\hbar }}\int_{0}^{\eta _{0}}(A_{cluster}-E[\eta ])^{1/2}d\eta
\biggr ),\end{aligned}$$ where $\eta _0$ is the final amplitude of the soliton, Fig.6, . The function $E[\eta ]$ is the total energy from eq.(57), with the parameter $U_0 $ fitted such that the second minimum of the potential energy, Fig.7, to be degenerated in energy with the first one. The numerical parameters are identical with that one used in the references \[4,5\]. The result is compared with similar calculations \[4,5,16,17\] and with the experimental preformation factors for $^{208}$Pb, \[s\]: $S_{exp}=0.085$, $S_{ref. [4,5]}=0.095$, $S_{BW}=0.0063$ and $S_{soliton}=0.07$. The first theoretical result was obtained in the frame of a previous form of the present model \[4\]. The second result represents the spectroscopic factor $S_{BW}=(6.3\times 10^{-3})^{{A_{cluster}-1}\over {3}}$, obtained in \[mm\] by using a semiempirical heavy ion potential. However we mention that there exists some ambiguities in the definition of the spectroscopic factors.
Conclusions
===========
In this paper we present a nonlinear hydrodynamic model describing the new large amplitude collective motion in nuclei, suggested by the new exotic alpha (and cluster) decays. From the Euler equation, subjected to nonlinear boundary conditions at the free surface and the inner surface of the fluid layer, we obtain, up to the third order in the deviation of the surface from the spherical shape, a dynamical equation for the radial coordinate in the form of the Korteweg de Vries eqution. The new introduced nonlinear terms lead to a model describing steady-states as knoidal waves on the surface of the nucleus. These solutions are described by the Jacobi elliptic functions and cover continuously all the range between small harmonic oscillations, anharmonic oscillations up to solitary wave. This model approaches the traditional liquid drop model in the linear approximation, when the KdV equation transforms into the Helmholtz eqution and the cnoidal waves approach their periodic limit, i.e. the normal modes of vibration of nuclei. From the total energy of the liquid drop model (surface, Coulombian, kinetic and shell energy), calculated in the same order of approximation, it results a functional depending on the shape functions which is exactly the Hamiltonian of the Korteweg de Vries equation, hence a Hamiltonian formalism applied to the present model lead to the same dynamical equation. The final results for the dynamical equations and the energy are in agreement with the previous (one-dimensional) version of the same model, \[4,5\]. The solitary wave limit of the solutions could describe the preformation of the clusters on the nuclear surface. We investigate the potential energy, depending on three parameters: the amplitude of the deformation, the phase velocity of the deformation, and the depth of the layer, under the restrictions of constant volume of the cluster and periodicity of the solutions on the surface. From the expression of the potential energy we find a minimum associated with the harmonic vibration, (small oscillation limit) and an additional minimum corresponding to the solitonic shapes as clusters on the nuclear surface. By choosing the shell effect contribution such that the two minima are degenerated, we obtain a potential energy profile which could describe the potential barrier associated with the preformation of such clusters. The spectroscopic factors, calculated as the ratio of the square amplitudes in the two minima, are in good agreement with the experiments.
[99]{} A. Sandulescu, W. Greiner,[*[Rep. Prog. Phys.]{}*]{}, [**[55]{}**]{} 1423 (1989)
P. B. Price, [*[Ann. Rev. Nucl. Part. Sci.]{}*]{}, [**[39]{}**]{} 19 (1989)
W. Greiner and J. Eisenberg, [*Nuclear Models*]{} (North-Holland, 1987)
A. Ludu, A. Sandulescu and W. Greiner, [*[Int. J. Modern Phys. E]{}*]{} [**1**]{}, 169 (1992)
A. Sandulescu, A. Ludu and W. Greiner, Proc. Inter. Conf. [*Nucleus and Clusters*]{}, Turku, Finland, June 1991, eds. M. Brenner and F. B. Malik (Springer-Verlag, Berlin 1991), p.262; A. Ludu, A. Sandulescu and W. Greiner, [*[Int. J. Modern Phys. E]{}*]{} [**2**]{}, 4, (1993) 855; [*J. Phys. G: Nucl. Part. Phys.*]{} [**21**]{} 1715 (1995); A. Ludu, A. Sandulescu, W. Greiner, K. M. Källmann, M. Brenner, T. Lönnroth and P. Manngärd, [*[J. Phys. G: Nucl. Part.]{}*]{}, [**[21]{}**]{} L41 (1995); V. G. Kartavenko, K. A. Gridnev and W. Greiner , [*[Int. J. Mod. Phys.]{}*]{}, [**3**]{}, 1219 (1994); V. G. Kartavenko, A. Ludu, A. Sandulescu and W. Greiner, [*Int. J. Mod. Phys. E*]{} [**5**]{} 329 (1996)
D. J. Korteweg, G. de Vries, [*[Phil. Mag.]{}*]{}, [**39**]{} (1895) 422
G.L. Lamb, [*Elements of Soliton Theory*]{}, (John Wiley & Sons, New York, 1980); C. Rebbi and G. Soliani, [*Solitons and Particles*]{}, (World Scientific, Singapore, 1984); S. Novikov, S. V. Manakov, [*Theory of Solitons. The Inverse Scattering Method*]{}, (Consultants Bureau, New York, 1984); R. K. Bullough and P. J. Caudrey, Eds., [*[Solitons]{}*]{}, (Topics in Current Physics, Springer-Verlag, Berlin, 1980)
G. Eilenberger, [*[Solitons. Mathematical Methods for Physicists]{}*]{}, (Springer-Verlag, Berlin, 1981)
R. K. Bullogh and P. J. Caudrey, [*[Solitons]{}*]{}, (Springer-Verlag, Berlin, 1980); P. G. Drazin, R. S. Johnson, [*[Solitons : an Introduction]{}*]{}, (Cambridge University Press, Cambridge, 1989).
C. Rebbi and G. Soliani, [*[Solitons and Particles]{}*]{} (World Scientific, Singapore, 1984)
K. Kreutz, Phys. Rev. D [**12**]{} 3126 (1975); R. Rajaraman, [*Solitons and Instantons. An Introduction to Solitons and Instantons in Quantum Field Theory*]{}, (North - Holland, Amsterdam, 1984)
R. Abraham and J. E. Marsden, [*Foundations of Mechanics*]{}, (The Benjamin/Cummings Publishing Company, Inc., Reading, Massachustes, 1978); V. I. Arnold, [*Mathematical Methods of Classical Mechanics*]{}, (Springer-Verlag, New York, 1978)
F. B. Estabrook and W. D. Wahlquist, [*[J. Math. Phys.]{}*]{}, [**17**]{} 1293 (1976); Y. Nogami, F. M. Toyama and S. Zhao , [*[J. Phys. A: Math. Gen.]{}*]{}, [**28**]{} 1413 (1995); Mohammad and M. Can, J. Phys. A: Math. Gen., [**28**]{} 3223 (1995); H. D. Wahlquist and F. B. Estabrook, [*[Phys. Rev. Lett.]{}*]{} [**31**]{} 1386; (1973); [*[J. Math. Phys.]{}*]{}, [**16**]{} 1 (1975); M. D. Kruskal, R. M. Miura, C. S. Gardner and N. J. Zabusky, [*[J. Math. Phys.]{}*]{}, [**11**]{} 952 (1970); R. M. Miura, [*[SIAM Rev.]{}*]{} [**18**]{} 412 (1976) and R. M. Miura, C. S. Gardner and M. D. Kruskal, [*[J. Math. Phys.]{}*]{}, [**9**]{} 1204 (1968)
L. D. Landau and E. M. Lifschitz, [*[Fluid Mechanics]{}*]{} (Pergamon Press, Oxford, 1987), $\S$ 62
M. Abramowitz and I. A. Stegun, [*[Handbook of Mathematical Functions]{}*]{}, (Dover Publications Inc., New York, 1968)
H. J. Mang, [*[Phys. Rev.]{}*]{} [**[119]{}**]{} 1069 (1960)
R. Blendowske and A. Walliser, [*[Phys. Rev. Lett]{}*]{} [**[61]{}**]{} 1930 (1988).
**[Figure Captions]{}**
Figs.1 0.5cm The potential picture and the phase space trajectories for some traveling wave solutions. We present, in the left side of Fig.1a the effective cubic potential associated with the KdV equation, eq.(30), and in the right side the corresponding phase space portrait with the curves describing motions of constant energy. One associates with the finite potential valley a class of periodic solutions consisting in anharmonic oscillations (a), centerd around the bottom (A) of the valley. In their superior limit these solutions are bounded by a special singular solution (the separatrix - S) of infinite period, hence a localised bump for the function $\eta (\phi )\simeq \eta (t)$. Starting from the right side of the potential valley to the saddle point (the origin of the phase space), the corrsponding motion represents the soliton solution. The right limit of the (S) curve gives the soliton amplitude. The amplitude of the anharmonic oscillations (cnoidal wave, i.e. periodic solutions of the KdV equation) are limited by the soliton height. Eq.(30) has also other classes of singular solutions, (b-c), which have no physical semnification for the present model, i.e. not being traveling waves.
In Fig.1b we present the same picture, for the linear approximation of eq.(30). The effective potential is quadratic and gives periodic solutions, harmonic oscillations. For comparison, we ploted in the same frame the original cubic potential. One can see how the nonlinearity developes and creates the pocket and the saddle.
Fig.2 0.5cm We present the three classes of solutions ($\eta (\phi -Vt)$) of the KdV equation: the soliton solution (one soliton bump of height 12 in the figure), cnoidal waves (large and small amplitude anharmonic oscillations ), and the singular (unphysical) solutions (decreasing to $-\infty $). The cnoidal waves are similar with some periodic copies of the soliton shape, and the small amplitude ones are (in the presented case) just cosine functions.
Fig.3 0.5cm The effective potential associated with the KdV equation is plotted against the coordinate $z$, eq.(65). The general steady-state solution ($cn(z|m)$) has three real zeros, ($\alpha _i $), which controll the range of the amplitude. It has two limits: the soliton ($Sech$) and the harmonic oscillations ($cos$). The soliton realises the deeper potential valley, where from its stability.
Fig.4 0.5cm The cnoidal cosine function $cn(z|m)$ ploted against its argument and the parameter $m$. For $m=1$ the function approaches the $Sech$ limit and for $m=0$ the cnoidal cosine approaches the cos limit.
Fig.5 0.5cm The contour plots of the potential surface $U[\eta ]=U(r,\alpha _2)$ plotted for a fixed volume $V_{cluster}=4$. One can see the initial state characterized by harmonic oscillations, ($r=2.4, \alpha _2 =3.2$), and the final state of minimum energy associated with the soliton, ($r=3, \alpha _2 =1.6$).
Fig.6 0.5cm The shape of the surface associated with the general solution of the KdV equation, plotted against the coordinate $r=R_0 (1+\eta (\theta ))$ and the parameter $V$. The smooth transition from the harmonic oscillations form ($V\simeq 0$) to the final solitonic form ($V\simeq 2$), under the volume conservation restriction, is presented.
Fig.7 0.5cm The potential barrier around a path of minimum energy, in the plane of parameters $\eta _0 , \alpha _2 $, from the small oscillations (first minimum of zero energy) to the solitonic solution (second minimum). The higher curve represents the energy without the term $E_{sh}$. The curve which has the second minimum degenerated in energy with the first one, represents the total energy, with the shell corrections included. Along the path therer are labeled the corresponding values of $\eta _0 $.
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Energy of a spherically symmetric charged dilaton black hole
A. CHAMORRO and K. S. VIRBHADRA
Departamento de Física Teórica, Universidad del País Vasco
Apartado 644, 48080 Bilbao, Spain
In recent years there is considerable interest in obtaining charged dilaton black hole solutions and investigating their properties $[1-6]$. Garfinkle, Horowitz, and Strominger (GHS) $[1]$ considered the action S = d\^4 x and obtained a nice form of static and spherically symmetric charged dilaton black hole solution $[1,2]$, given by the line element ds\^2 = B dt\^2 - B\^[-1]{} dr\^2 - D r\^2 (d\^2 + \^2d\^2), the dilaton field $\Phi$, where e\^[2]{} = \^[(1-)/]{}, and the component of the electromagnetic field tensor F\_[tr]{} = , where B = (1-) (1-)\^, D = (1-)\^[1-]{}, and = . $r_{+}$ and $r_{-}$ are related through & &2M = r\_[+]{} + r\_[-]{} ,\
& &Q\^2 (1+\^2) = r\_[+]{} r\_[-]{} . $M$ and $Q$ stand for mass and charge parameters, respectively. The surface $r = r_{+}$ is the event horizon. $\beta$ is a dimensionless free parameter which controls the coupling between the dilaton and the Maxwell fields. A change in the sign of $\beta$ is the same as a change in the sign of the dilaton field. Therefore, it is sufficient to discuss only nonnegative values of $\beta$. $\beta = 0$ in GHS solution gives the well known Reissner-Nordström (RN) solution.
It is known that several properties of charged dilaton black holes depend crucially on the coupling parameter $\beta$ $[2-6]$. Recently one of the present authors and Parikh $[7]$ obtained the energy of a static and spherically symmetric charged dilaton black hole for $\beta = 1$. They found that, similar to the case of the Schwarzschild black hole and unlike the RN black hole, the entire energy is confined to the interior of the black hole. It is of interest to investigate the energy associated with charged dilaton black holes for arbitrary value of $\beta$ to see what the energy distribution is for $\beta < 1$ as well as $\beta >1$ and whether or not the energy is confined to the black hole interior for any other value of $\beta$.
The well known energy-momentum pseudotensor of Einstein is $[8]$ \_i\^[ k]{} = H\^[ kl]{}\_[i, l]{} , where H\_i\^[ kl]{} = \_[,m]{} .
Latin indices run from $0$ to $3$. $x^0$ is the time coordinate. The energy and momentum components are P\_i = \_[,]{} dx\^1 dx\^2 dx\^3 , where the Greek index $\alpha$ takes values from $1$ to $3$. $P_0$ and $P_{\alpha}$ stand, respectively, for the energy (say $E$) and momentum components.
It is known that the energy-momentum pseudotensors, for obtaining the energy and momentum associated with asymptotically flat spacetimes, give the correct result if calculations are carried out in quasi-cartesian coordinates ( those coordinates in which the metric $g_{ik}$ approaches the Minkowski metric $\eta_{ik}$ at large distance ) $[8-9]$. Transforming the line element $(2)$ to quasi-cartesian coordinates $t,x,y,z ( x = r \sin\th \cos\ph,
y = r \sin\th \sin\ph, z = r \cos\th ) $ one gets ds\^2 = B dt\^2 - D (dx\^2+dy\^2+dz\^2) - (x dx+y dy +z dz)\^2.
To obtain the energy the required components of $ H_i^{\ kl}$ are H\_0\^[ 01]{} &=& ,\
H\_0\^[ 02]{} &=& ,\
H\_0\^[ 03]{} &=& . By using $(13)$ with $(8)$ in $(11)$, applying the Gauss theorem, and then evaluating the integral over the surface of a sphere of radius $r$, one gets E(r) = M - ( 1 - \^2 ).
Thus one finds that the energy distribution depends on the value of the coupling parameter $\beta$. The energy is confined to its interior [*[only for]{}*]{} $\beta = 1$ and for all other values of $\beta$ the energy is shared by the interior and exterior of the black hole. $\beta=0$ in $(14)$ gives the energy distribution in the RN field (see also ref. $[9]$). $E(r)$ increases with radial distance for $\beta = 0$ (RN spacetime) as well as $\beta<1$, decreases for $\beta>1$, and remains constant for $\beta = 1$. However, the total energy ($r$ approaching infinity in $(14)$) is independent of $\beta$ and is given by the mass parameter of the black hole. The details of the present contribution will be published elsewhere.
[**Acknowledgements**]{}
This work has been partially supported by the Universidad del Pais Vasco under contract UPV 172.310 - EA062/93 (A.C.) and by a Basque Government post-doctoral fellowship (K.S.V.). We thank A. Achúcarro, J. M. Aguirregabiria, and I. Egusquiza for discussions.
[**References**]{}\
$[1]$ D. Garfinkle, G. T. Horowitz and A. Strominger, Phys. Rev. D43\
(1991) 3140; Erratum : Phys. Rev. D45 (1992) 3888.\
$[2]$ J. H. Horne and G. T. Horowitz, Phys. Rev. D46 (1992) 1340.\
$[3]$ K. Shiraishi, Phys. Lett. A166 (1992) 298.\
$[4]$ J. A. Harvey and A. Strominger, Quantum aspects of black holes,\
preprint EFI-92-41, hep-th/9209055 .\
$[5]$ T. Maki and K. Shiraishi, Class. Quant. Grav. 11 (1994) 227.\
$[6]$ C. F. E. Holzhey and F. Wilczek, Nucl. Phys. B380 (1992) 447.\
$[7]$ K. S. Virbhadra and J. C. Parikh, Phys. Lett. B317 (1993) 312.\
$[8]$ C. Møller, Ann. Phys. (NY) 4 (1958) 347.\
$[9]$ K. P. Tod. Proc. Roy. Soc. Lond. A388 (1983) 467;\
K. S. Virbhadra, Phys. Rev. D41 (1990) 1086; Phys. Rev. D42 (1990) 2919;\
F. I. Cooperstock and S. A. Richardson, in Proc. 4th Canadian Conf. on General\
Relativity and Relativistic Astrophysics ( World Scientific, Singapore, 1991 );\
A. Chamorro and K. S. Virbhadra, hep-th/9406148.\
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We study the involution of the real line induced by the outer automorphism of the extended modular group PGL(2,Z). This ‘[modular]{}’ involution is discontinuous at rationals but satisfies a surprising collection of functional equations. It preserves the set of real quadratic irrationals mapping them in a non-obvious way to each other. It commutes with the Galois action on real quadratic irrationals.
More generally, it preserves set-wise the orbits of the modular group, thereby inducing an involution of the moduli space of real rank-two lattices. We give a description of this involution as the boundary action of a certain automorphism of the infinite trivalent tree. It is conjectured that algebraic numbers of degree at least three are mapped to transcendental numbers under this involution.
author:
- 'A. Muhammed Uludağ$^*$, Hakan Ayral[^1]'
bibliography:
- 'references3.bib'
title: 'On the involution of the real line induced by Dyer’s outer automorphism of PGL(2,Z)'
---
*Dedicated to Yılmaz Akyıldız, who shared our enthusiasm about jimm*
Introduction {#sec:introduction}
============
It is (it seems not very well-) known that the group ${\mathrm{PGL}_2({\mathbf Z})}$ has an involutive outer automorphism, which was discovered by Dyer in the late 70’s [@dyer1978automorphism]. It would be very strange if this automorphism had no manifestations in myriad contexts where ${\mathrm{PGL}_2({\mathbf Z})}$ or its subgroups play a major role. Our aim in this paper is to elucidate one of these manifestations.
This paper contains some of the results from our arxiv preprint [@jimmarxiv] which could not be published due to its length and notational issues. For some complementary results, see [@jimmlebesgue].
Let $\widehat{{\mathbf R}}:={\mathbf R}\cup \{\infty\}$. The manifestation in question of ${\mathbf J}$ is a map ${\mathbf J}_{\mathbf R}: \widehat{{\mathbf R}}\rightarrow \widehat{{\mathbf R}}$. Denoting the continued fractions in the usual way $$[n_0,n_1,\dots]=n_0+\cfrac{1}{n_1+\cfrac{1}{{\dots}}},$$ one has, for an irrational number $[n_0,n_1,\dots]$ with $n_0, n_1, \dots \geq 2$, $${\mathbf J}_{\mathbf R}([n_0,n_1,n_2,\dots])=[1_{n_0-1},2,1_{n_1-2},2,1_{n_2-2},\dots]$$ where $1_k$ is the sequence ${1,1,\dots, 1}$ of length $k$. This formula remains valid for $n_0, n_1, \dots \geq 1$, if the emerging $1_{-1}$’s are eliminated in accordance with the rule $[\dots m, 1_{-1},n,\dots]=[\dots m+n-1,\dots]$ and $1_0$ with the rule $[\dots m, 1_{0},n,\dots]=[\dots m,n,\dots]$. See page below for some examples.
It is possible to extend this definition of ${\mathbf J}_{\mathbf R}$ to all of $\widehat{{\mathbf R}}$. If we ignore rationals and the noble numbers (i.e. numbers in the ${\mathrm{PGL}_2({\mathbf Z})}$-orbit of the golden section, $\Phi:=(1+\sqrt{5})/{2}$), then ${\mathbf J}_{\mathbf R}$ is an involution. It is well-defined and continuous at irrationals, but two-valued and discontinuous at rationals (Theorem \[jimmcontini\]).
[**Guide for notation**]{}. In what follows, ${\mathbf J}$ denotes Dyers’ outer automorphism of ${\mathrm{PGL}_2({\mathbf Z})}$, ${\mathbf J}_{\mathcal F}$ denotes the automorphism of the tree $|{\mathcal F}|$ induced by ${\mathbf J}$, ${\mathbf J}_{\partial{\mathcal F}}$ denotes the homeomorphism of the boundary $\partial{\mathcal F}$ induced by ${\mathbf J}_{\mathcal F}$, and ${\mathbf J}_{\mathbf R}$ denotes the involution of ${\mathbf R}$ induced by ${\mathbf J}_{\partial{\mathcal F}}$. However, we reserve the right to drop the subscript and simply write ${\mathbf J}$ when we think that confusion won’t arise.
### Functional equations {#functional-equations .unnumbered}
The involution ${\mathbf J}_{\mathbf R}$ shares the privileged status of the three involutions generating the extended modular group ${\mathrm{PGL}_2({\mathbf Z})}$, $$K:x\rightarrow 1-x, \quad U:x\rightarrow 1/x, \quad V:x\rightarrow -x,$$ as it interacts in a very harmonious way with them. Indeed, removing the rationals and noble numbers from its domain, ${\mathbf J}_{\mathbf R}$ satisfies the following functional equations:
The following set of functional equations are derived from the above ones:
(FE:I) and (FE:III) states that ${\mathbf J}$ is covariant with the operators $U$ and $K$. (FE:IV) is deduced from (FE:II) and (FE:III). The most general form of the equations is (FE:VI) and says that ${\mathbf J}_{\mathbf R}$ conjugates the Möbius map $M\in {\mathrm{PGL}_2({\mathbf Z})}$ to the Möbius map ${\mathbf J}(M)\in {\mathrm{PGL}_2({\mathbf Z})}$. It is deduced from (FE:I-III) by using the involutivity of ${\mathbf J}$.
Before attempting to play with them, beware that the equations are not consistent on ${\mathbf Q}$. On the set of noble numbers ${\mathbf J}$ is 2-to-1 and is not involutive.
${\mathbf J}$ preserves the “[real-multiplication locus]{}", i.e. the set of real quadratic irrationals. It does so in a non-trivial way, though it preserves setwise the ${\mathrm{PGL}_2({\mathbf Z})}$-orbits of real quadratic irrationals. More generally ${\mathbf J}$ preserves setwise the ${\mathrm{PGL}_2({\mathbf Z})}$-orbits on $\widehat{{\mathbf R}}$ thereby inducing an involution of the moduli space of real rank-2 lattices, $\widehat{{\mathbf R}}/{\mathrm{PGL}_2({\mathbf Z})}$. For a description of its fixed points, see Proposition \[fixed\].
One can use (FE:I)-(FE:III) to directly define and study ${\mathbf J}_{\mathbf R}$. However, the nature of ${\mathbf J}_{\mathbf R}$ is best understood by considering it as a homeomorphism of the boundary of the Farey tree, induced by an automorphism of the tree. See the coming sections for details.
Finally, the following consequence of the functional equations is noteworthy: $${\frac{1}{x}+\frac{1}{y}=1\iff \frac{1}{{\mathbf J}(x)}+\frac{1}{{\mathbf J}(y)}=1}.$$ (See page for a complete set of two-variable functional equations.) Hence ${\mathbf J}$ sends harmonic pairs of numbers to harmonic pairs, inducing a duality of Beatty partitions of positive integers. See [@jimmarxiv] for some details.
### Some examples. {#examples .unnumbered}
Here is a list of assorted values of ${\mathbf J}$. Recall that $\Phi$ denotes the golden section. $$\begin{aligned}
\label{jimmphi}
\Phi=[\overline{1}]\implies{\mathbf J}(\Phi)=\infty={\mathbf J}\bigl(-{1}/{\Phi}\bigr),\end{aligned}$$ where by $\overline{v}$ we denote the infinite sequence $v,v,v, \dots$, for any finite sequence $v$. From \[jimmphi\] by using (FE:II) we find $$\begin{aligned}
\label{jimmminusphi}
{\mathbf J}(-\Phi)={\mathbf J}\bigl({1}/{\Phi}\bigr)=0.\end{aligned}$$ Repeated application of (FE:IV) gives $$\begin{aligned}
\label{jimmenplusphi}
{\mathbf J}(n+\Phi)={\mathbf J}([n+1,\overline{1}])={F_{n+1}}/{F_{n}}\end{aligned}$$ where $F_n$ denotes the $n$th Fibonacci number. One has $${\mathbf J}([1_n,2,\overline{1}])=[n+1,\infty]=n+1$$ For the number $\sqrt{2}$ we have something that looks simple $$\sqrt{2}=[1,\overline{2}]\implies{\mathbf J}(\sqrt{2})=[\overline{2}]=1+\sqrt{2}$$ but this is not typical as the next example illustrates: $${\mathbf J}({(3+5\sqrt{2})}/{7})=
{\mathbf J}([1,\overline{2, 3, 1, 1, 2, 1, 1, 1}])=
[2,\overline{2,1,4,5}]
=\frac{-3+2\sqrt{95}}{7}$$ In a similar vein, consider the examples $$\sqrt{11}=[3;\overline{3, 6}]
\implies {\mathbf J}(\sqrt{11})=[\overline{1,1,2,1,2,1,1}]=
\frac{15+\sqrt{901}}{26},$$ $${\mathbf J}(-\sqrt{11})=-1/{\mathbf J}(\sqrt{11})=-\frac{26}{15+\sqrt{901}}=\frac{15-\sqrt{901}}{26}.$$ The last example hints at the following result
The involution ${\mathbf J}$ commutes with the conjugation of real quadratic irrationals; i.e. for every real quadratic irrational $\alpha$ one has $
{\mathbf J}({\alpha}^*)={{\mathbf J}(\alpha)^*}.
$
[*Proof.*]{} Every such $\alpha$ is the fixed point of some $M\in {\mathrm{PSL}_2({\mathbf Z})}$, the Galois conjugate $\alpha^*$ being the other root of the equation $Mx=x$. But then ${\mathbf J}(M\alpha)={\mathbf J}(M){\mathbf J}(\alpha)={\mathbf J}(\alpha)$, i.e. ${\mathbf J}(\alpha)$ is a fixed point of ${\mathbf J}(M)$, the other root being ${\mathbf J}(\alpha)^*$. Finally $M\alpha=\alpha\implies$ $M\alpha^*=\alpha^*\implies$ $ {\mathbf J}(\alpha^*)={\mathbf J}(M\alpha^*)={\mathbf J}(M){\mathbf J}(\alpha^*)$, so ${\mathbf J}(\alpha^*)$ is also a fixed point of ${\mathbf J}(M)$, i.e. it must coincide with ${\mathbf J}(\alpha)^*$. $\Box$
To finish, let us give the ${\mathbf J}$-transform of a non-quadratic algebraic number, $$\begin{aligned}
{\mathbf J}(^3\!\sqrt{2})=
{\mathbf J}([1; 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, 3, 4, 1, \dots])\\
=[2,1,3,1,1,1,4,1,1,4,1_6,3,1_{12},3,1_8,2,3,1,1,2,1_{10},2,2,1,2,3,1,2,1,1,3,\dots ]
\\
=2.784731558662723\dots\end{aligned}$$ and the transforms of two familiar transcendental numbers: $$\begin{aligned}
{\mathbf J}(\pi)=
{\mathbf J}([3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, \dots])=
[1_2, 2, 1_5, 2, 1_{13}, 3, 1_{290}, 5, 3, \dots]\\
=1.7237707925480276079699326494931025145558144289232\dots\end{aligned}$$ $$\begin{aligned}
{\mathbf J}(e)=
{\mathbf J}([2,1,2,1,1,4,1,1,6,1,1,8,\dots])=
[1,3,4,1,1,4,1,1,1,1,\dots, \overline{4,1_{2n}}]\\
=1.3105752928466255215822495496939143349712038085627\dots\end{aligned}$$ We have been unable to relate these numbers to other numbers of mathematics.
We made some numerical experiments on the algebraicity of a few numbers ${\mathbf J}(x)$ where $x$ is an algebraic number of degree $>2$, with a special emphasis on ${\mathbf J}(^3\!\sqrt{2})$. It is very likely that these numbers are transcendental.
Modular group and its automorphism group
========================================
The [*modular group*]{} is the projective group ${\mathrm{PSL}_2({\mathbf Z})}$ of two by two unimodular integral matrices. It acts on the upper half plane by Möbius transformations. It is the free product of its subgroups generated by $S(z)=-1/z$ and $L(z)= (z-1)/z$, respectively of orders 2 and 3. Thus $
{\mathrm{PSL}_2({\mathbf Z})}=\langle S, L \,|\, S^2=L^3=1\rangle.
$ The projective group ${\mathrm{PGL}_2({\mathbf Z})}$ consists of integral matrices of determinant $\pm 1$.
The fact that $Out({\mathrm{PGL}_2({\mathbf Z})})\simeq {\mathbf Z}/2{\mathbf Z}$ has a story with a twist: Hua and Reiner [@hua1952automorphisms] claimed in 1952 that ${\mathrm{PGL}_2({\mathbf Z})}$ has no outer automorphisms. The error was corrected by Dyer [@dyer1978automorphism] in 1978. This automorphism also appears in the work of Djokovic and Miller [@djokovic] from about the same time. Dyer also proved that the automorphism tower of ${\mathrm{PGL}_2({\mathbf Z})}$ stops here. Note that ${\mathrm{PGL}_2({\mathbf Z})}\simeq Aut({\mathrm{PSL}_2({\mathbf Z})})$.
Below is a list of some presentations of ${\mathrm{PGL}_2({\mathbf Z})}$ and ${\mathbf J}$. $$\begin{array}{|l|l|} \hline &\\[-1ex]
\mbox{\bf Presentation of ${\mathrm{PGL}_2({\mathbf Z})}$}\quad &\mbox{\bf The automorphism ${\mathbf J}$}\quad \\[1ex]\hline &\\[-.9ex]
\langle V,U,K\, |\, V^2=U^2=K^2=(VU)^2=(KU)^3=1 \rangle &(V,U,K) \rightarrow (UV,U,K)\\[1ex]\hline &\\[-.9ex]
\langle V,U,L\, |\, V^2=U^2=(LU)^2=(VU)^2=L^3=1 \rangle&(V,U,L) \rightarrow (UV, U, L)\\[1ex]\hline &\\[-.9ex]
\langle S,V,K\, |\, V^2=S^2=K^2=(SV)^2=(KSV)^3=1 \rangle\quad&(S,V,K) \rightarrow (V,S,K)\\[1ex]\hline
\end{array}$$
It is easy to find the ${\mathbf J}$ of an element of ${\mathrm{PGL}_2({\mathbf Z})}$ given as a word in one of the listed presentations. On the other hand, there seems to be no algorithm to compute the ${\mathbf J}$ of an element of ${\mathrm{PGL}_2({\mathbf Z})}$ given in the matrix form, other then actually expressing the matrix in terms of one of the presentations above, finding the ${\mathbf J}$-transform, and then computing the matrix.
The translation $2+x\in {\mathrm{PSL}_2({\mathbf Z})}$ is sent to $(2x+1)/(x+1)\in {\mathrm{PSL}_2({\mathbf Z})}$ under ${\mathbf J}$, i.e. ${\mathbf J}$ may send a parabolic element to a hyperbolic element.
The Farey tree and its boundary
===============================
#### The construction of ${\mathcal F}$.
From ${\mathrm{PSL}_2({\mathbf Z})}$ we construct the [*bipartite Farey tree ${\mathcal F}$*]{} tree on which ${\mathrm{PSL}_2({\mathbf Z})}$ acts, as follows. The set of edges is $E({\mathcal F})={\mathrm{PSL}_2({\mathbf Z})}$. Vertices are the left cosets of the subgroups $\langle S\rangle$ and $\langle L\rangle$. Two distinct vertices $v$ and $v'$ are joined by an edge if and only if $v \cap v'\neq \emptyset$ and in this case the edge between them is the only element in the intersection. Cosets of $\langle S\rangle$ are 2-valent vertices and the cosets of $\langle L\rangle$ are 3-valent. Since distinct cosets are disjoint, ${\mathcal F}$ is a bipartite graph. It is connected and circuit-free since ${\mathrm{PSL}_2({\mathbf Z})}$ is freely generated by $S$ and $L$.
The edges incident to the vertex $\{W,WL,WL^{2}\}$ are $ W, \, WL$ and $ WL^{2} $. These edges inherit a natural cyclic ordering which we fix for all vertices as $(W, WL, WL^2)$. This endows ${\mathcal F}$ with the structure of a ribbon graph. Hence ${\mathcal F}$ is an infinite bipartite tree with a ribbon structure. $M\in {\mathrm{PSL}_2({\mathbf Z})}$ acts on ${\mathcal F}$ from the left by ribbon graph automorphisms by sending the edge labeled $W$ to the edge labeled $MW$. This action is free on $E({\mathcal F})$ but not free on the set of vertices.
#### The boundary of ${\mathcal F}$. {#sec:boundary/of/F}
A [*path*]{} on $\mathcal F$ is a sequence of distinct edges $e_1,e_2,\dots,e_k$ such that $e_i$ and $e_{i+1}$ meet at a vertex, for each $1\leq i< k-1$. Since the edges of ${\mathcal F}$ are labeled by reduced words in the letters $L$ and $S$, a path in ${\mathcal F}$ is a sequence of reduced words $(W_i)$ in $L$ and $S$, such that $W_i^{-1}W_{i+1}\in \{L, L^2, S\}$ for every $i$.
An [*end*]{} of ${\mathcal F}$ is an equivalence class of infinite (but not bi-infinite) paths in ${\mathcal F}$, where eventually coinciding paths are considered as equivalent. The set of ends of ${\mathcal F}$ is denoted $\partial {\mathcal F}$. The action of ${\mathrm{PSL}_2({\mathbf Z})}$ on ${\mathcal F}$ extends to $\partial {\mathcal F}$, the element $M\in {\mathrm{PSL}_2({\mathbf Z})}$ sending the path $(W_i)$ to the path $(MW_i)$.
Given an edge $e$ of ${\mathcal F}$ and an end $b$ of ${\mathcal F}$, there is a unique path in the class $b$ which starts at $e$. Hence for any edge $e$, we may identify the set $\partial {\mathcal F}$ with the set of infinite paths that start at $e$. We denote this latter set by $\partial{\mathcal F}_e$ and endow it with the topology generated by the [*Farey intervals*]{} ${\mathcal O}_{e'}$, i.e. the set of paths through $e'$. The ribbon structure of ${\mathcal F}$ endows $\partial{\mathcal F}_e$ with the structure of a cyclically ordered topological space. Given a second edge $e'$ of ${\mathcal F}$, the spaces $\partial{\mathcal F}_e$ and $\partial{\mathcal F}_{e'}$ are homeomorphic under the order-preserving map which pre-composes with the unique path joining $e$ to $e'$. The action of the modular group on the set $\partial{\mathcal F}$ induces an action of ${\mathrm{PSL}_2({\mathbf Z})}$ by order-preserving homeomorphisms of $\partial{\mathcal F}_e$, for any choice of a base edge $e$.
#### The continued fraction map.
$\partial{\mathcal F}_e$ is homeomorphic to the Cantor set. By exploiting its cyclic order structure, we “smash its holes" as follows. Define a [*rational end*]{} of ${\mathcal F}$ to be an eventually left-turn or eventually right-turn path. Now introduce the equivalence relation $\sim $ on $\partial{\mathcal F}$ as: left- and right- rational paths which bifurcate from the same vertex are equivalent. On $\partial{\mathcal F}_e$ this equivalence sets equal those points which are not separated by a third point.
[]{}\
[ A pair of rational ends.]{}\
On the quotient space $\partial{\mathcal F}_e/\!\sim $ there is the quotient topology induced by the topology on $\partial{\mathcal F}_e$ such that the projection map $$\label{projectionmap}
\partial{\mathcal F}_e\longrightarrow \partial{\mathcal F}_e/\!\sim$$ is continuous. The quotient space $S^1_e:=\partial{\mathcal F}_e/\!\sim $ is a cyclically ordered topological space under the order relation inherited from $\partial{\mathcal F}_e$. The equivalence relation is preserved under the canonical homeomorphisms $\partial{\mathcal F}_e \longrightarrow \partial{\mathcal F}_{e'}$ and is also respected by the ${\mathrm{PSL}_2({\mathbf Z})}$-action. Therefore we have the commutative diagram $$\xymatrix{
\partial{\mathcal F}_e \ar@{->}[r] \ar@{->}[d] & \partial{\mathcal F}_{e'}\ar@{->}[d]\\
S^1_{e} \ar@{->}[r] & S^1_{e'}\\
}$$ where the horizontal arrows are order-preserving homeomorphisms. Moreover, ${\mathrm{PSL}_2({\mathbf Z})}$ acts by homeomorphisms on $S^1_{e}$, for all $e$.
Now, ${\mathcal F}$ is equipped with a distinguished edge, the one marked $I$, the identity element of ${\mathrm{PSL}_2({\mathbf Z})}$. So all spaces $S^1_{e}$ are canonically homeomorphic to $S^1_{I}$.
Any element of $S^1_I$ can be represented by an infinite word in $L$ and $S$. Regrouping occurrences of $LS$ and $L^2S$, any such word $x$ can be written as $$\begin{aligned}
\label{reps}
x&=(LS)^{n_0}(L^2S)^{n_1}(LS)^{n_2}(L^2S)^{n_3}(LS)^{n_4} \cdots \mbox{ or }\\
x&=S(LS)^{n_0}(L^2S)^{n_1}(LS)^{n_2}(L^2S)^{n_3}(LS)^{n_4} \cdots,\end{aligned}$$
where $n_0, n_1 \dots \geq 0$. Since our paths do not have any backtracking we have $n_0\geq0$ and $n_{i}>0$ for $i = 1,2,\cdots$. The pairs of words $$\begin{aligned}
(LS)^{n_0}\cdots (LS)^{n_{k}+1} (L^2S)^{\infty} \mbox{ and } &(LS)^{n_0}\cdots (LS)^{n_{k}}(L^2S) (LS)^{\infty}, & (k\mbox{ even}) \\
(LS)^{n_0}\cdots (L^2S)^{n_{k}+1} (LS)^{\infty} \mbox{ and } &(LS)^{n_0}\cdots (L^2S)^{n_{k}}(LS) (L^2S)^{\infty}, & (k\mbox{ odd}) \end{aligned}$$ correspond to pairs of rational ends and represent the same element of $S_I^1$. For irrational ends this representation is unique.
The ${\mathrm{PSL}_2({\mathbf Z})}$-action on $S^1_I$ is then the pre-composition of the infinite word by the word in $L,S$ representing the element of ${\mathrm{PSL}_2({\mathbf Z})}$. In this picture it is readily seen that this action respects the equivalence relation $\sim $.
Set $T:=LS$, so that $T(x)=1+x$. Note that $$\begin{aligned}
(LS)^n.(L^2S)^m.(LS)^k(x)=(LS)^n.S.[S.(L^2S)^m .S]. S . (LS)^k(x)\\
=(LS)^n.S.[SL^2]^m. S . (LS)^k(x)=(LS)^n.S.[LS]^{-m}. S . (LS)^k(x)\\
=(x+n)\circ (-1/x) \circ (x-m) \circ (-1/x)\circ (x+k) =
n+\cfrac{1}{m+\cfrac{1}{k+x}}\end{aligned}$$ Accordingly, define the [*continued fraction map*]{} $\kappa: S^1_I\rightarrow \hat {\mathbf R} $ by $$\kappa(x)=\left\{ \begin{array}{rl}
[n_0,n_1,n_2,\dots]& \mbox{ if } x=(LS)^{n_0}(L^2S)^{n_1}(LS)^{n_2}(L^2S)^{n_3}(LS)^{n_4}\dots\\
-1/[n_0,n_1,n_2,\dots]& \mbox{ if } x=S(LS)^{n_0}(L^2S)^{n_1}(LS)^{n_2}(L^2S)^{n_3}(LS)^{n_4}\dots
\end{array}\right.$$\
The continued fraction map $\kappa$ is a homeomorphism. \[thm:cfm\]
[*Proof.*]{} To each rational end $[n_0,n_1,n_2,\dots,n_k,\infty]$ we associate the rational number $[n_0,n_1,n_2,\dots,n_k]$. Likewise for the rational ends in the negative sector. This is an order preserving bijection between the set of equivalent pairs of rational ends and ${\mathbf Q}\cup\{\infty\}$. Now observe that an infinite path is no other than a Dedekind cut and conversely every cut determines a unique infinite path, see [@UZD] for details.$\Box$
As a consequence, we see that $\kappa$ conjugates the ${\mathrm{PSL}_2({\mathbf Z})}$-action on $S_I^1$ to its action on $\widehat {\mathbf R}$ by Möbius maps. Furthermore, there is a bijection between $\widehat {{\mathbf Q}}$ and the set of pairs of equivalent rational ends. Any pair of equivalent rational ends determines a unique [*rational horocycle*]{}, a bi-infinite left-turning (or right-turning) path. Hence, there is a bijection between $\widehat {{\mathbf Q}}$ and the set of rational horocycles.
\[correspondences\] Given the base edge $I$ of ${\mathcal F}$,\
(i) There is a natural bijection between the 3-valent vertices of ${\mathcal F}$ and $\widehat {{\mathbf Q}}\setminus\{0,\infty\}$.\
(ii) There is a bijection between the 2-valent vertices and the Farey intervals $[p/q, r/s]$ with $ps-qr=1$.
[*Proof.*]{} (i) On each rational horocycle there lies a unique trivalent vertex which is closest to $I$. If we exclude the two horocycles on which $I$ lies, this gives a bijection between the set of trivalent vertices of ${\mathcal F}$ and the set of horocycles. The map $\kappa$ sends the two horocycles through $I$ to $0$ and $\infty$. Hence, there is a bijection between the set of trivalent vertices of ${\mathcal F}$ and $\widehat {{\mathbf Q}}\setminus \{0, \infty\}$. This bijection sends the trivalent vertex $\{W, WL, WL^2\}$ to $W(1)\in {\mathbf Q}$, where $W$ is a reduced word ending with $S$. (ii) Every 2-valent vertex $\{W, WS\}$ lies exactly on two horocycles, and the set of paths through $\{W, WS\}$ is sent to the interval $[W(0), WS(0)]$ under $\kappa$. $\Box$
#### Periodic paths and the real multiplication set.
Let $\gamma:=(W_1, W_2, \dots, W_n)$ be a finite path in ${\mathcal F}$. Then the [*periodization*]{} of $\gamma$ is the path $\gamma^\omega$ defined as $$\begin{aligned}
W_1, W_2, \dots, W_n, W_nW_1^{-1}W_2, W_nW_1^{-1}W_3, \dots, W_nW_{1}^{-1}W_n, \qquad\qquad\qquad \\
\quad (W_nW_{1}^{-1})^2W_2, \dots, (W_nW_{1}^{-1})^2W_n, (W_nW_{1}^{-1})^3W_2, \dots\end{aligned}$$ In plain words, $\gamma^\omega$ is the path obtained by concatenating an infinite number of copies of a path representing $W_1^{-1}W_n$, starting at the edge $W_1$. If $W_1^{-1}W_n$ is elliptic, then $\gamma^\omega$ is an infinitely backtracking finite path. If not, $\gamma^\omega$ is actually infinite and represents an end of ${\mathcal F}$. We call these [*periodic ends of ${\mathcal F}$*]{}. Thus we have the periodization map $$per:\,\, \{\,finite\, \, hyperbolic \, \, paths\, \} \longrightarrow \{\, ends\, \, of\, \, {\mathcal F}\, \}$$ whose image consists of periodic ends. The map $per$ is not one-to one but its restriction to the set of primitive paths is. The modular group action on $\partial{\mathcal F}$ preserves the set of periodic ends and $per$ is ${\mathrm{PSL}_2({\mathbf Z})}$-equivariant.
Given an edge $e$ of ${\mathcal F}$ and a periodic end $b$ of ${\mathcal F}$, there is a unique path in the class $b$ which starts at $e$. This way the set of periodic ends of ${\mathcal F}$ is identified with the set of eventually periodic paths based at $e$. This set is dense in $\partial{\mathcal F}_e$ and preserved under the canonical homeomorphisms between $\partial{\mathcal F}_e$ and $\partial{\mathcal F}_{e'}$. Every periodic end has a unique ${\mathrm{PSL}_2({\mathbf Z})}$-translate, which is a purely periodic path based at $e$. Finally, the set of periodic ends descends to a well-defined subset of $S^1_e$. The image of this set under $\kappa$ consists of the set of eventually periodic continued fractions, i.e. the set of real quadratic irrationals. These are precisely the fixed points of the ${\mathrm{PSL}_2({\mathbf Z})}$-action on $S^1_e$; a hyperbolic element $M=(LS)^{n_0}(L^2S)^{n_1}\cdots (LS)^{n_{k}}\in {\mathrm{PSL}_2({\mathbf Z})}$ fixing the numbers represented by the infinite words $$\begin{aligned}
(LS)^{n_0}(L^2S)^{n_1}\cdots (LS)^{n_{k}}(LS)^{n_0}(L^2S)^{n_1}\cdots (LS)^{n_{k}}\dots, \mbox{ and}\\
(LS)^{-n_k}\cdots (L^2S)^{-n_1}(LS)^{-n_{k}}(LS)^{-n_k}\cdots (L^2S)^{-n_1}(LS)^{-n_{k}}\dots. \end{aligned}$$
The automorphism group of ${\mathcal F}$
----------------------------------------
Now let us forget about the ribbon structure of ${\mathcal F}$. This gives an abstract graph which we denote by $|{\mathcal F}|$. The automorphism group $Aut(|{\mathcal F}|)$ of $|{\mathcal F}|$ contains $Aut({\mathcal F})$ as a non-normal subgroup. It acts by homeomorphisms on $\partial {\mathcal F}$. However, automorphisms of $|{\mathcal F}|$ do not respect the ribbon structure on ${\mathcal F}$ in general and don’t induce well-defined self-maps of $S_I^1$.
Fix an edge $e$ of ${\mathcal F}$ and denote by $Aut_e(|{\mathcal F}|)$ the group of automorphisms of $|{\mathcal F}|$ that stabilize $e$. For any pair $e$, $e'$ of edges, $Aut_e(|{\mathcal F}|)$ and $Aut_{e'}(|{\mathcal F}|)$ are conjugate subgroups. For $n>0$ let $|{\mathcal F}_n|$ be the finite subtree of $|{\mathcal F}|$ containing vertices of distance $\leq n$ from $e$. Then $|{\mathcal F}_{n}|\subset |{\mathcal F}_{n+1}|$ forms an injective system with respect to inclusion and $Aut_e(|{\mathcal F}_{n+1}|) \rightarrow Aut_e(|{\mathcal F}_{n}|)$ forms a projective system, and $$Aut_e(|{\mathcal F}|)=\lim_{\longleftarrow} Aut_e(|{\mathcal F}_{n}|),$$ showing that $Aut_e(|{\mathcal F}|)$ is a profinite group.
#### A description of tree automorphisms as shuffles.
Let us turn back to the ribbon graph ${\mathcal F}$, with the base edge $I\in{\mathrm{PSL}_2({\mathbf Z})}$. Given any vertex $v$, this base edge permits us to define the subtree (i.e. Farey branch) attached to ${\mathcal F}$ at $v$.
Denote by $V_\bullet$ the set of of degree-3 vertices of ${\mathcal F}$.
The ribbon structure of ${\mathcal F}$ serves as a sort of coordinate system to describe all automorphisms of $|{\mathcal F}|$, as follows. Given a vertex $v=\{W, WL, WL^2\}$ of type $V_\bullet$ of ${\mathcal F}$, the [*shuffle*]{} $\sigma_v$ is the automorphism of $|{\mathcal F}|$ which is defined as: $$\begin{aligned}
\label{twist}
\sigma_v: \mbox{edge labeled } M \longrightarrow
\mbox{edge labeled} \left\{
\begin{array}{ll}
M, & \mbox{ if } M\neq WLX\\
WL^2X,& \mbox{ if } M=WLX\\
WLX,& \mbox{ if } M=WL^2X
\end{array}
\right.\end{aligned}$$ where $M$ and $X$ are reduced words in $S$ and $L$. Thus $\sigma_v$ is the identity away from the Farey branches at $v$, whereas it exchanges the two Farey branches at $v$. Note that $\sigma_v^2=I$, i.e. the shuffle $\sigma_v$ is involutive.
The automorphism $\sigma_v$ is obtained by permuting the two branches attached at $v$, by shuffling these branches one above the other. Beware that $\sigma_v$ is [*not*]{} the automorphism $\theta_v$ of $|{\mathcal F}|$ obtained by rotating in the physical 3-space the branches starting at $WL$ and at $WL^2$ around the vertex $v$. We call this latter automorphism a [*twist*]{}, see below for a precise definition.
We must stress that the definition of $\sigma_v$ requires the ribbon structure of ${\mathcal F}$ as well as a base edge, although $\sigma_v$ is never an automorphism of ${\mathcal F}$.
Evidently, $\sigma_v$ stabilizes $I$, so one has $\sigma_v\in Aut_I(|{\mathcal F}|)$.
Given an arbitrary (finite or infinite) set $\nu$ of vertices in $V_\bullet$, we inductively define the shuffle $\sigma_\nu \in Aut_I(|{\mathcal F}|)$ as follows: First order the elements of $\nu$ with respect to the distance from the base edge $I$, i.e. set $\nu^{(1)}=(v_1^{(1)}, v_2^{(1)},\dots)$, where $v_i^{(1)}\neq v_{i+1}^{(1)}$ and such that $
d(v_i^{(1)},I) \leq d(v_{i+1}^{(1)},I)$ for all $i=1,2,\dots.$ Set $$\sigma^{(i)}_\nu:=\sigma_{v_i^{(i)}} \mbox{ and } v_j^{(i+1)}=\sigma^{(i)}_\nu(v_j^{(i)}) \mbox{ for } j\geq i+1, \quad
i=1,2,\dots$$ Since for any $j>i$, the automorphism $\sigma^{(j)}_\nu$ agrees with $\sigma^{(i)}_\nu$ on ${\mathcal F}_i$, the sequence $\sigma^{(i)}_\nu$ converges in $Aut_I(|{\mathcal F}|)$ and we set $$\sigma_\nu:=\lim_{i\rightarrow \infty} \sigma^{(i)}_\nu$$ for the limit automorphism of $|{\mathcal F}|$. There is some arbitrariness in the initial ordering of $\nu$, concerning its elements of constant distance to the edge $I$, but the limit does not depend on this. The reason is that the shuffles corresponding to those elements commute. $\sigma_\emptyset$ is the identity automorphism by definition.
$Aut_I(|{\mathcal F}|) =\{ \sigma_\nu\, |\, \nu\subseteq V_\bullet({\mathcal F})\}$.
[*Proof.*]{} It suffices to show that the restrictions of shuffles to finite subtrees $|{\mathcal F}_n|$ gives the full group $Aut_I(|{\mathcal F}_n|)$. This is easy. $\Box$
#### The automorphism $\sigma_{V_\bullet}$.
What happens if we shuffle every $\bullet$-vertex of $|{\mathcal F}|$? In other words, what is effect of the automorphism $\sigma_{V_\bullet}$ on $\partial_I F$? If $x\in \partial_I F$ is represented by an infinite word in $L$, $L^2$ and $S$, then according to (\[twist\]), we see that $\sigma_{V_\bullet}$ replaces $L$ by $L^2$ and vice versa. In other words, if $x$ is of the form $$x=(LS)^{n_0}(L^2S)^{n_1}(LS)^{n_2}(L^2S)^{n_3}(LS)^{n_4}\dots,$$ then one has $$\sigma_{V_\bullet}(x)=
(L^2S)^{n_0}(LS)^{n_1}(L^2S)^{n_2}(LS)^{n_3}(L^2S)^{n_4}\dots$$ In terms of the contined fraction representations, we get, for $n_0>0$, $$\sigma_{V_\bullet}([n_0, n_1,n_2,\dots])=[0, n_1,n_2,\dots]$$ and for $n_0=0$ $$\sigma_{V_\bullet}([0, n_1,n_2,\dots])=[n_1,n_2,\dots].$$ A similar formula also holds when $x$ starts with an $S$, and we get $$\sigma_{V_\bullet}(x)=1/x=U(x).$$ Observe (keeping in mind the forthcoming parallelism between ${\mathbf J}$ and $U$) that $U$ is an involution satisfying the equations $$\label{fesforu}
U=U^{-1}, \quad US=SU, \quad
LU=UL^2.$$ Here, $U$, $L$ and $S$ are viewed as operators acting on the boundary $\partial_I{\mathcal F}$. These equations can be re-written in the form of functional equations as below: $$U(Ux)=x, \quad U(-\frac{1}{x})=-\frac{1}{Ux}, \quad U(\frac{1}{1-x})=1-\frac{1}{Ux}.$$ One may derive other functional equations from these, i.e. $UT=ULS=L^2US=L^2SU$ is written as $$U(1+x)=\frac{Ux}{1+Ux}.$$ One may consider $U$ as an “[orientation-reversing]{}" automorphism of the ribbon tree ${\mathcal F}$. In the same vein, $U$ is a homeomorphism of $\partial_I{\mathcal F}$ which reverses its canonical ordering. It is the sole element of $Aut_I(|{\mathcal F}|)$ which respects the equivalence $\sim $ (unlike ${\mathbf J}$) and hence $U$ acts by homeomorphism on $S^1_I$. In fact, taken as a map on $E({\mathcal F})={\mathrm{PSL}_2({\mathbf Z})}$, the involution $U$ is an automorphism of ${\mathrm{PSL}_2({\mathbf Z})}$, the one defined by $$U:\left(\begin{matrix} S\\L\end{matrix}\right) \to
\left(\begin{matrix} S\\L^2\end{matrix} \right)$$ The automorphism group of ${\mathrm{PSL}_2({\mathbf Z})}$ is generated by the inner automorphisms and $U$: this is the group ${\mathrm{PGL}_2({\mathbf Z})}$.
#### Twists.
Let $v\in V_\bullet$ and let $\nu_v$ be the set of all vertices (including $v$) on the Farey branch (with reference to the edge $I$) of ${\mathcal F}$ at $v$. Then the [*twist*]{} of $v$ is the automorphism of ${\mathcal F}$ defined by $$\theta_v:=\sigma_{\nu_v}.$$ In words, $\theta_v$ is the shuffle of every vertex of the Farey branch at $v$. As in the case of shuffles, for any subset $\mu\subset V_\bullet$, one may define the twist $\theta_\mu$ as a limit of convergent sequence of individual twists.
\[twisttoshuffle\] Let $v$ be a vertex in $V_\bullet$ and $v'$, $v''$ its two children (with respect to the ancestor $I$). Put $\mu:=\{v,v',v''\}$. Then $
\sigma_v=\theta_\mu.
$
Hence, shuffles can be expressed in terms of twists and vice versa. This proves:
$Aut_I(|{\mathcal F}|) =\{ \theta_\mu\, |\, \mu\subseteq V_\bullet({\mathcal F})\}$.
\[notatdef\] Let $v^*$ be the trivalent vertex incident to the base edge $I$, i.e. $v^*:=\{I,L,L^2\}$ and set $V_\bullet^*:=V_\bullet\backslash \{v^*\}$. We denote the special automorphism $\theta_{V_\bullet^*}$ by ${\mathbf J}_{\mathcal F}$. In words, this is the automorphism of $|{\mathcal F}|$ obtained by twisting every trivalent vertex except $v^*$.
This ${\mathbf J}_{\mathcal F}$ is the same automorphism obtained by shuffling every other trivalent vertex, such that $v^*$ is not shuffled. More precisely, the following consequence of Lemma \[twisttoshuffle\] holds:
One has $
\theta_{V_\bullet^*}={\mathbf J}_{\mathcal F}=\sigma_J,
$ where $J\subset V_\bullet$ is the set of degree-3 vertices, whose distance to the base edge is an odd number.
Obviously, ${\mathbf J}_{\mathcal F}$ induce an involutive homeomorphism of $\partial_I{\mathcal F}$ but violates its ordering. We denote this homeomorphism by ${\mathbf J}_{\partial{\mathcal F}}$. We must emphasize that ${\mathbf J}_{{\mathcal F}}$ and ${\mathbf J}_{\partial{\mathcal F}}$ are perfectly well-defined mappings on their domain of definition. They don’t exhibit such things as the two-valued behaviour of ${\mathbf J}_{{\mathbf R}}$ at rationals. This two-valued behavior is a consequence of the fact that ${\mathbf J}_{\partial{\mathcal F}}$ do not respect the relation $\sim $.
To see the effect of ${\mathbf J}_{\partial{\mathcal F}}$ on $x\in\partial_I{\mathcal F}$, assume $$x=S^\epsilon (LS)^{n_0}(L^2S)^{n_1}(LS)^{n_2}(L^2S)^{n_3}(LS)^{n_4}\dots, \, (n_0\geq 0, \, n_i>0 \mbox{ if } i>0, \,\epsilon\in\{0, 1\}).$$ We may represent $x$ by a string of 0’s and 1’s (0 for $L$ and 1 for $L^2$): $$\begin{aligned}
x=S^\epsilon
\underbrace{00\dots0}_{n_0}
\underbrace{11\dots1}_{n_1}
\underbrace{00\dots0}_{n_2}
\underbrace{11\dots1}_{n_3}
\dots, \, \end{aligned}$$ Then rationals are represented by the eventually constant strings where for any finite string $a$, the strings $a0111\dots$ and $a1000\dots$ represent the same rational[^2].
Let $\phi:=(01)^\omega$ be the zig-zag path to infinity, and set $\phi^*:=\lnot \phi=(10)^\omega$. Then $${\mathbf J}_{\partial{\mathcal F}}(x)= \begin{cases}
a \xor \phi,& x=a\in \{0,1\}^\omega,\\
S(a \xor \phi*),& x=Sa, \, a\in \{0,1\}^\omega.
\end{cases}$$ where $\xor$ is the operation of term-wise exclusive or (XOR). The involutivity of ${\mathbf J}_{\partial{\mathcal F}}$ then stems from the reversibility of the disjunctive or: $(p\xor q) \xor q=p$.
[**Some Examples.**]{} The string $0(0011)^\omega$ corresponds to the continued fraction $[1,2,2,2,\dots]$, which equals $\sqrt{2}$. One has $$\begin{array}{l|cccccccccccc|c}
\partial{\mathcal F}&&\multicolumn{10}{c}{\{0,1\}^\omega}&&\widehat{{\mathbf R}}\\
\hline
x
&0&1&1&0&0&1&1&0&0&1&1&0\dots &\sqrt{2}\\
\phi
&0&1&0&1&0&1&0&1&0&1&0&1\dots&\Phi\\
{\mathbf J}(x)
&0&0&1&1&0&0&1&1&0&0&1&1\dots&1+\sqrt{2}
\end{array}$$ As for the value of ${\mathbf J}_{\mathbf R}$ at $\infty$, one has $$\begin{array}{l|cccccccccccc|c}
\partial{\mathcal F}&&\multicolumn{10}{c}{\{0,1\}^\omega}&&\widehat{{\mathbf R}}\\
\hline
x
&0&0&0&0&0&0&0&0&0&0&0&0\dots &\infty\\
\phi
&0&1&0&1&0&1&0&1&0&1&0&1\dots&\Phi\\
{\mathbf J}(x)
&0&1&0&1&0&1&0&1&0&1&0&1\dots&\Phi
\end{array}$$ The two values ${\mathbf J}_{\mathbf R}$ assumes at the point $1$ are found as follows: $$\begin{array}{l|cccccccccccc|c}
\partial{\mathcal F}&&\multicolumn{10}{c}{\{0,1\}^\omega}&&\widehat{{\mathbf R}}\\
\hline
x
&0&1&1&1&1&1&1&1&1&1&1&1\dots &1\\
\phi
&0&1&0&1&0&1&0&1&0&1&0&1\dots&\Phi\\
{\mathbf J}(x)
&0&0&1&0&1&0&1&0&1&0&1&0\dots&1+\Phi
\end{array}$$ $$\begin{array}{l|cccccccccccc|c}
\partial{\mathcal F}&&\multicolumn{10}{c}{\{0,1\}^\omega}&&\widehat{{\mathbf R}}\\
\hline
x
&1&0&0&0&0&0&0&0&0&0&0&0\dots &1\\
\phi
&0&1&0&1&0&1&0&1&0&1&0&1\dots&\Phi\\
{\mathbf J}(x)
&1&1&0&1&0&1&0&1&0&1&0&1\dots&1/(1+\Phi)
\end{array}$$ Conversely, one has ${\mathbf J}_{\mathbf R}(1+\Phi)={\mathbf J}_{\mathbf R}(1/(1+\Phi))=1$, illustrating the two-to-oneness of ${\mathbf J}_{\mathbf R}$ on the set of noble numbers.
The noble numbers are the ${\mathrm{PSL}_2({\mathbf Z})}$-translates of the golden section $\Phi$. They correspond to the eventually zig-zag paths in $\partial {\mathcal F}$, and represented by strings terminating with $(01)^\omega$. From the $\xor$-description, it is clear that ${\mathbf J}_{\partial{\mathcal F}}$ sends those strings to rational (i.e. eventually constant) strings and vice versa.
Since the equivalence $\sim $ is not respected by ${\mathbf J}_{\partial{\mathcal F}}$, it does not induce a homeomorphism of $\widehat{{\mathbf R}}$, not even a well-defined map. Nevertheless, if we ignore the pairs of rational ends, then the remaining equivalence classes are singletons and ${\mathbf J}_{\partial{\mathcal F}}$ restricts to a well-defined map $${\mathbf J}_{\mathbf R}:\widehat{{\mathbf R}}\setminus\widehat{{\mathbf Q}}
\to \widehat{{\mathbf R}}$$ If we also ignore the set of “[noble paths]{}", i.e. the set ${\mathrm{PGL}_2({\mathbf Z})}\widehat{{\mathbf Q}}={\mathbf J}^{-1}({\mathbf Q})$, then we obtain an involutive bijection $${\mathbf J}_{\mathbf R}: \widehat{{\mathbf R}}\setminus(\widehat{{\mathbf Q}}\cup {\mathrm{PGL}_2({\mathbf Z})}\widehat{{\mathbf Q}})\to
\widehat{{\mathbf R}}\setminus(\widehat{{\mathbf Q}}\cup {\mathrm{PGL}_2({\mathbf Z})}\widehat{{\mathbf Q}})$$
#### The functional equations.
Since twists and shuffles do not change the distance to the base, we have our first functional equation:
The $|{\mathcal F}|$-automorphisms ${\mathbf J}_{\mathcal F}$ and $U=\sigma_{V_\bullet}$ commute, i.e. ${\mathbf J}_{\mathcal F}U=U{\mathbf J}_{\mathcal F}$. Hence, ${\mathbf J}_{\partial{\mathcal F}}U=U{\mathbf J}_{\partial{\mathcal F}}$ and whenever ${\mathbf J}_{\mathbf R}$ is defined, one has $$\label{jimmu}
{\mathbf J}_{\mathbf R}U= U{\mathbf J}_{\mathbf R}\iff {\mathbf J}_{\mathbf R}\bigl(\frac{1}{x}\bigr)=\frac{1}{{\mathbf J}_{\mathbf R}(x)}$$
In fact, ${\mathbf J}_{\mathcal F}U$ is the automorphism of ${\mathcal F}$ which shuffles every other vertex, starting with the vertex $v^*$. In other words, it shuffles those vertices which are not shuffled by ${\mathbf J}_{\mathcal F}$. We denote this automorphism by ${\mathbf J}_{\mathcal F}^*$. Note that, in terms of the strings, the operation $U$ is nothing but the term-wise negation: $$Ua=\lnot a,$$ and the lemma merely states the fact that $\lnot (\phi \xor a)=\phi \xor \lnot a$. The boundary homeomorphism induced by the automorphism ${\mathbf J}_{\mathcal F}^*$ is thus the map ${\mathbf J}_{\partial {\mathcal F}}^*$ which xors with the string $\phi^*:=(10)^\omega$.
Now, consider the operation $S$. If $x=a\in \{0,1\}^\omega$, then one has ${\mathbf J}_{\partial {\mathcal F}} (Sx)=$ $$\begin{aligned}
= S(a \xor \phi^*)
=S(a\xor \lnot \phi) =S(\lnot(a\xor\phi))=S(U(a\xor\phi)) = SU{\mathbf J}_{\partial {\mathcal F}} (x)=V{\mathbf J}_{\partial {\mathcal F}} (x)\end{aligned}$$ The same equality holds if $x=Sa$, and we get our second functional equation:
The $\partial_I{\mathcal F}$-homeomorphisms ${\mathbf J}_{\partial{\mathcal F}}$ and $S$ satisfy ${\mathbf J}_{\partial{\mathcal F}}S=V{\mathbf J}_{\partial{\mathcal F}}$. Hence, whenever ${\mathbf J}_{\mathbf R}$ is defined, one has $$\label{jimmux}
{\mathbf J}_{\mathbf R}S= V{\mathbf J}_{\mathbf R}\iff {\mathbf J}_{\mathbf R}\bigl(-\frac{1}{x}\bigr)=-{{\mathbf J}_{\mathbf R}(x)}$$
Now we consider the operator $L$. Suppose that $x=Sa$. Then $Lx= LSa=0a$. Hence, noting that $\phi=(01)^\omega=0 (10)^\omega=0\phi^*$ we have $$\begin{aligned}
{\mathbf J}_{\partial {\mathcal F}} (Lx) = 0a \xor \phi=0a\xor 0\phi^*=0(a\xor \phi^*)= LS(a\xor \phi^*)
= L{\mathbf J}_{\partial {\mathcal F}} (x).\end{aligned}$$ Another possibility is that $x=0a$. In other words, $x$ starts with an $L$. Then $Lx$ starts with an $L^2$, i.e. $Lx= 1a$. Hence, $$\begin{aligned}
{\mathbf J}_{\partial {\mathcal F}} (Lx) = 1a \xor \phi=1a\xor 0\phi^*=1(a\xor \phi^*),\\
{\mathbf J}_{\partial {\mathcal F}}(x)=0a \xor \phi=0a\xor 0\phi^*=0(a\xor \phi^*)
\implies L {\mathbf J}_{\partial {\mathcal F}}(x)= 1(a\xor \phi^*).\end{aligned}$$ Finally, if $x=1a$, then $Lx$ starts with an $S$, i.e. $Lx= Sa$. Hence, $$\begin{aligned}
{\mathbf J}_{\partial {\mathcal F}} (Lx) = S(a \xor \phi^*),\\
{\mathbf J}_{\partial {\mathcal F}}(x)=1a \xor \phi=1a\xor 0\phi^*=1(a\xor \phi^*)
\implies L {\mathbf J}_{\partial {\mathcal F}}(x)= S(a\xor \phi^*).\end{aligned}$$ Whence the third functional equation:
The $\partial_I{\mathcal F}$-homeomorphisms ${\mathbf J}_{\partial{\mathcal F}}$ and $L$ satisfy ${\mathbf J}_{\partial{\mathcal F}}L=L{\mathbf J}_{\partial{\mathcal F}}$. Hence, whenever ${\mathbf J}_{\mathbf R}$ is defined, one has $$\label{jimmux}
{\mathbf J}_{\mathbf R}L= L{\mathbf J}_{\mathbf R}\iff {\mathbf J}_{\mathbf R}\bigl(1-\frac{1}{x}\bigr)=1-\frac{1}{{\mathbf J}_{\mathbf R}(x)}$$
Since $U$, $S$ and $L$ generate the group ${\mathrm{PGL}_2({\mathbf Z})}$, these three functional equations forms a complete set, from which the rest can be deduced. For example, $$T=LS\implies {\mathbf J}_{\partial {\mathcal F}}T=LV{\mathbf J}_{\partial {\mathcal F}}\iff
{\mathbf J}_{\mathbf R}(1+x)=1+\frac{1}{{\mathbf J}_{\mathbf R}(x)}$$ This also shows that ${\mathbf J}_{\mathbf R}$ acts as the desired outer automorphism of ${\mathrm{PGL}_2({\mathbf Z})}$.
#### Other forms of the functional equations.
It is possible to derive many alternative forms of the functional equations. Here we record some of them. We leave the task of verifying to the reader. (As usual we drop the subscript ${\mathbf R}$ from ${\mathbf J}_{\mathbf R}$ to increase the readability): $${\mathbf J}\left(1-\frac{1}{x}\right)=1-\frac{1}{{\mathbf J}(x)}, \quad
{\mathbf J}\left(\frac{x}{x+1}\right)+{\mathbf J}\left(\frac{1}{x+1}\right)=1.$$
\[twovariable\]
Diverse facts about ${\mathbf J}$
=================================
${\mathbf J}$ as a limit of piecewise-${\mathrm{PGL}_2({\mathbf Z})}$ functions.
--------------------------------------------------------------------------------
In ${\mathcal F}$, denote by $V_\bullet(n)$ the set of vertices of distance $\leq n$ to the base (excluding $v^*$), and let $V_\bullet^\prime(n)$ be the set of vertices of odd distance $\leq n$ to the base. Then $${\mathbf J}_{\mathcal F}=
\lim_{n\to\infty} \theta_{V_\bullet(n)}
=\lim_{n\to\infty} \sigma_{V_\bullet^\prime(n)}.$$ The maps $\theta_{V_\bullet(n)}$ and $\sigma_{V_\bullet^\prime(n)}$ induce a sort of finitary projective interval exchange maps on $\widehat{{\mathbf R}}$ and thus ${\mathbf J}_{\mathbf R}$ can be written as a limit of such functions. Below we draw the twists $\theta_{V_\bullet(n)}$ for $n=2\dots 5$.
\[jimmiterate\] []{} []{} []{} []{}\
[ The plot of $\theta_{{\mathcal F}_n^*}$ on the interval $[0,1]$, for $n=2,3,4,5$.]{}\
${\mathbf J}$ on rationals {#jimmq}
--------------------------
In virtue of Lemma \[correspondences\], there is a 1-1 correspondence between the set of trivalent vertices and the set ${\mathbf Q}\setminus\{0,1\}$. Since any element of $Aut_I({\mathcal F})$ defines a bijection of $V_\bullet({\mathcal F})$, we see that every automorphism of ${\mathcal F}$ that fixes $I$, defines a unique bijection of ${\mathbf Q}\setminus\{0,1\}$. In particular, this is the case with ${\mathbf J}_{\mathcal F}$. We extend this involution to ${\mathbf Q}\cup\{\infty\}$ by sending $0\leftrightarrow \infty$ and we denote the resulting bijection with ${\mathbf J}_{\mathbf Q}$. This involution satisfies all the functional equations satisfied by ${\mathbf J}_{\mathbf R}$. One has ${\mathbf J}_{\mathbf Q}(1)=1$, and for $x>0$, its values can be computed by using the functional equations ${\mathbf J}_{\mathbf Q}(1+x)=1+1/{\mathbf J}_{\mathbf Q}(x)$ and ${\mathbf J}_{\mathbf Q}(1/x)=1/{\mathbf J}_{\mathbf Q}(x)$. It tends to ${\mathbf J}_{\mathbf R}$ at irrational points and in fact ${\mathbf J}_{\mathbf R}$ can be defined as this limit. However, it must be emphasized that ${\mathbf J}_{\mathbf Q}(x)$ is not the restriction of ${\mathbf J}_{\mathbf R}$ to ${\mathbf Q}$; this latter function is by definition two-valued at rationals[^3].
We show in [@lebesguesym] that ${\mathbf J}_{\mathbf Q}$ “commutes" with Lebesgue measure in a certain sense.
The involution ${\mathbf J}_{\mathbf Q}$ conjugates the multiplication (denoted by $\odot$) on ${\mathbf Q}$ to an operation with 1 as its identity, and such that the inverse of $q$ is $1/q$. The addition (denoted by $\oplus$) is conjugated to an operation with $\infty$ as its neutral element and such that the additive inverse of $q$ is $-1/q$, i.e. $\ominus q=-1/q$.
Fibonacci sequence
------------------
The Fibonacci sequence is defined by the recurrence $
F_0=0, \quad F_1=1,\mbox{ and } F_{n}=F_{n-1}+F_{n-2} \mbox{ for } n\in {\mathbf Z}.
$ One has then $F_{-n}=(-1)^{n+1}F_n$ and $$\widetilde{T}(x)=1+\frac{1}{x}\implies
\widetilde{T}^n=\frac{F_{n+1}x +F_n}{F_nx+F_{n-1}} \quad (n\in {\mathbf Z}).$$ The following lemma is an easy consequence of the functional equations.
\[easyconsequence\] Let $x$ be an irrational number. Then\
(i) $${\mathbf J}(1+x)=1+\frac{1}{{\mathbf J}(x)} \iff {\mathbf J}(Tx)=\widetilde{T}{\mathbf J}(x).$$ (ii) $${\mathbf J}(n+x)=\widetilde{T}^n{\mathbf J}(x)=\frac{F_{n+1}{\mathbf J}(x) +F_n}{F_n {\mathbf J}(x)+F_{n-1}} \quad (n\in {\mathbf Z}).$$ (iii) $${\mathbf J}([1,x])={\mathbf J}(1+\frac{1}{x})=1+{\mathbf J}(x) \implies {\mathbf J}([1_n,x])=n+x \quad (n\in {\mathbf Z}).$$ (iv) $${\mathbf J}([n,x])={\mathbf J}(n+1/x)=\frac{F_{n+1}+F_n{\mathbf J}(x) }{F_n +F_{n-1}{\mathbf J}(x)} \quad (n\in {\mathbf Z}).$$
Continuity
----------
Since the involution ${\mathbf J}_{\partial{\mathcal F}}: \partial{\mathcal F}\to \partial{\mathcal F}$ is a homemorphism and since the subspaces $${\mathbf R}\setminus {\mathbf Q}\simeq \partial{\mathcal F}\setminus \{\mbox{rational paths}\}$$ are also homeomorphic, the function ${\mathbf J}_{\mathbf R}$ is continuous at irrational points:
\[jimmcontini\] The function ${\mathbf J}_{\mathbf R}$ on ${\mathbf R}\setminus {\mathbf Q}$ is continuous.
Moreover, recall that there is a canonical ordering on $\partial{\mathcal F}$ inducing on $\widehat{{\mathbf R}}$ its canonical ordering compatible with its topology. So the notion of lower and upper limits exists on $\partial{\mathcal F}$ and coincides with lower and upper limits on $\widehat{{\mathbf R}}$. This shows that the two values that ${\mathbf J}_{\mathbf R}$ assumes on rational arguments are nothing but the limits $${\mathbf J}(q)^-:=\lim_{x\to q^-} {\mathbf J}(x) \mbox{ and }
{\mathbf J}(q)^+:=\lim_{x\to q^+} {\mathbf J}(x)$$ By choosing one of these values coherently, one can make ${\mathbf J}$ an everywhere upper (or lower) continuous function. Note however that it will (partially) cease to satisfy the functional equations at rational arguments. One has, for irrational $r$ and rational $q$, $$\lim_{q\rightarrow r} {\mathbf J}_{\mathbf Q}(q)=\lim_{q\rightarrow r} {\mathbf J}_{\mathbf R}(q)= {\mathbf J}_{\mathbf R}(r),$$ no matter how we choose the values of ${\mathbf J}_{\mathbf R}(q)$. In fact ${\mathbf J}$ is almost everywhere differentiable with derivative vanishing almost everywhere, see [@jimmarxiv].
Action on the quadratic irrationals
-----------------------------------
Since ${\mathbf J}_{\mathcal F}$ sends eventually periodic paths to eventually periodic paths, ${\mathbf J}_{\mathbf R}$ preserves the real-multiplication set. This is the content of our next result:
\[orbits\] ${\mathbf J}$ defines an involution of the set of real quadratic irrationals, and it respects the ${\mathrm{PGL}_2({\mathbf Z})}$-orbits.
Beyond this theorem, we failed to detect any further arithmetic structure or formula relating the quadratic number to its ${\mathbf J}$-transform. All we can do is to give some sporadic examples, which amounts to exhibit some special real quadratic numbers with known continued fraction expansions. We shall do this below. Before that, however, note that the equation below is solvable for every $n\in {\mathbf Z}$: $${\mathbf J}x=n+ x\implies x={\mathbf J}{\mathbf J}x=\widetilde{T}^n{\mathbf J}x\implies x=\widetilde{T}^n( x+n)$$ $$\frac{F_{n+1}( x+n)+F_n }{F_n( x+n) +F_{n-1}}= x=\frac{F_{n+1} x+F_n+nF_{n+1} }{F_n x^n +F_{n-1}+nF_n}$$ $$\implies F_n x^2+(F_{n-1}+nF_n-F_{n+1}) x -(F_n+nF_{n+1})$$ More generally, the equation ${\mathbf J}x=M x$ is solvable for $M\in{\mathrm{PGL}_2({\mathbf Z})}$. Indeed one has $${\mathbf J}x=M x\implies
x={\mathbf J}{\mathbf J}x=({\mathbf J}M)({\mathbf J}x)=
({\mathbf J}M) M x,$$ and the solutions are the fixed points of $({\mathbf J}M) M$. These points are precisely the words represented by the infinite path $$\begin{aligned}
\label{fixed}
x=({\mathbf J}M) M({\mathbf J}M) M({\mathbf J}M) M({\mathbf J}M) M\dots\end{aligned}$$ where we assume that both $M$ and ${\mathbf J}M$ are expressed as words in $U$ and $T$. Note that these words are precisely the points whose conjugacy classes remain stable under ${\mathbf J}$. $${\mathbf J}x=M({\mathbf J}M) M({\mathbf J}M) M({\mathbf J}M) M({\mathbf J}M) M\dots$$ Hence the ${\mathrm{PGL}_2({\mathbf Z})}$-orbits of the points in (\[fixed\]) remain stable under ${\mathbf J}$. Since the set of these orbits is the moduli space of real lattices, we deduce:
The fixed points of the involution ${\mathbf J}$ on the moduli space of real lattices are precisely the points (\[fixed\]) .
For example, if ${\mathbf J}M=\widetilde{T}^{k+1}$ then $M=T^{k+1}$, and the point (\[fixed\]) is nothing but the point $[\overline{1_k,k+2}]$ mentioned in the introduction.
Examples
--------
#### Example.
In [@manfred] it is shown that $$x=[0;{\overline{1_{n-1},a}}]=\frac{a}{2}\left(\sqrt{1+4\frac{aF_{n-1}+F_{n-2}}{a^2F_n}}-1\right)$$ Hence $
{\mathbf J}(x) = [0;n,\overline{1_{a-2},n+1}],
$ and so ${\mathbf J}(x) = 1/(n+y)$, where $$y=[0;\overline{1_{a-2},n+1}]=\frac{n+1}{2}\left(\sqrt{1+4\frac{(n+1)F_{a-2}+F_{a-3}}{(n+1)^2F_{a-1}}}-1\right)$$
#### Example.
(From Einsiedler & Ward [@ward], Pg.90, ex. 3.1.1) This result of McMullen from [@mcmullen] illustrates how ${\mathbf J}$ behaves on one real quadratic number field. ${\mathbf Q}(\sqrt{5})$ contains infinitely many elements with a uniform bound on their partial quotients, since $
[\overline{1_{k+1},4,5,1_k,3}]\in {\mathbf Q}(\sqrt{5}), \quad \forall k=1,2,\dots.
$ Routine calculations shows that the transforms $
{\mathbf J}([\overline{1_{k+1},4,5,1_k,3}])=[k+2, \overline{2,2,3,k+2,1,k+3}]
$ lies in different quadratic number fields for different $k$’s. Hence, not only ${\mathbf J}$ does not preserve the property of “[belonging to a certain quadratic number field]{}", it sends elements from one quadratic number field to different quadratic number fields. The following result provides even more examples of this nature:
(McMullen [@mcmullen]) For any $s>0$, the periodic continued fractions $$x_m=[\overline{(1,s)^m,1,s+1,s-1, (1,s)^m,1,s+1,s+3)}]$$ lie in ${\mathbf Q}(\sqrt{s^2+4s})$ for any $m\geq 0$.
(Fishman and Miller [@fishman]) One has the following $$\Phi^k=
\begin{cases}
[\overline{F_{k+1}+F_{k-1}}],& \mbox{\it if k is even},\\
[F_{k+1}+F_{k-1}-1, \overline{1,F_{k+1}+F_{k-1}-2}]& \mbox{\it if k is odd}.
\end{cases}$$
The continued fraction of the kth power of the golden section is $${\mathbf J}(\Phi^k)=
\begin{cases}
[1_{F_{k+1}+F_{k-1}-1},\overline{2, 1_{F_{k+1}+F_{k-1}-2}}],& \mbox{\it if k is even},\\
[1_{F_{k+1}+F_{k-1}-2},\overline{3,1_{F_{k+1}+F_{k-1}-4}}]& \mbox{\it if k is odd}.
\end{cases}$$
#### Example.
Suppose $\alpha=p+\sqrt{q}$ be a quadratic irrational with $p,q\in {\mathbf Q}$. Then $\alpha^*=1/\alpha$ provided $$|\alpha|^2=p^2-q=1 \implies q=p^2-1.$$ Suppose $\alpha$ is of this form, i.e. $\alpha=p+\sqrt{p^2-1}$ and suppose ${\mathbf J}(\alpha)=x+\sqrt{y}$. Then since $$x-\sqrt{y}={\mathbf J}(\alpha)^*={\mathbf J}(\alpha^*)={\mathbf J}(1/\alpha)=1/{\mathbf J}(\alpha)=1/(x+\sqrt{y})
\implies x^2-y=1,$$ and we conclude that ${\mathbf J}(\alpha)$ is again of the form $x+\sqrt{x^2-1}$.
On the other hand, suppose ${\mathbf J}(\sqrt{q})=x+\sqrt{y}$. Then $${\mathbf J}(\sqrt{q})^*={\mathbf J}(\sqrt{q}^*)={\mathbf J}(-\sqrt{q})=-1/{\mathbf J}(\sqrt{q})\implies
|{\mathbf J}(\sqrt{q}^*)|=x^2-y=-1$$ Hence, ${\mathbf J}(\sqrt{q})$ is of the form $x+\sqrt{1+x^2}$. Conversely, if $\alpha$ is of the form $p+\sqrt{p^2+1}$, then ${\mathbf J}(\alpha)$ is a quadratic surd.
[**Acknowledgements.**]{} This research is funded by a Galatasaray University research grant and the TÜBİTAK grants 115F412 and 113R017.
[^1]: [Galatasaray University, Department of Mathematics,]{} [Ç[i]{}rağan Cad. No. 36, 34357 Beşiktaş]{} [İstanbul, Turkey]{}
[^2]: The two representations of the number $0\in {\mathbf R}$ are $1^\omega$ and $-0^\omega$ and the two representations of $\infty\in \widehat{\mathbf R}$ are $0^\omega$ and $-1^\omega$.
[^3]: It might have been convenient to declare the values of ${\mathbf J}_{\mathbf R}$ at rational arguments to be given by ${\mathbf J}_{\mathbf Q}(x)$, so that ${\mathbf J}_{\mathbf R}$ would be a well-defined function everywhere (save 0 and $\infty$). However, we have chosen to not to follow this idea, for the sake of uniformity in definitions.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this letter, we consider the problem of detecting a high dimensional signal based on compressed measurements with physical layer secrecy guarantees. We assume that the network operates in the presence of an eavesdropper who intends to discover the state of the nature being monitored by the system. We design measurement matrices which maximize the detection performance of the network while guaranteeing a certain level of secrecy. We solve the measurement matrix design problem under three different scenarios: $a)$ signal is known, $b)$ signal lies in a low dimensional subspace, and $c)$ signal is sparse. It is shown that the security performance of the system can be improved by using optimized measurement matrices along with artificial noise injection based techniques.'
author:
- 'Bhavya Kailkhura, , Sijia Liu, , Thakshila Wimalajeewa, , Pramod K. Varshney, [^1] [^2]'
bibliography:
- 'Conf.bib'
- 'Book.bib'
- 'Journal.bib'
title: '**Measurement Matrix Design for Compressive Detection with Secrecy Guarantees**'
---
Compressive detection, physical layer secrecy, distributed processing, eavesdropper
Introduction
============
The theory of compressive sensing (CS) mostly deals with reconstructing a sparse signal based on a small number of measurements obtained via low dimensional projections [@donoho; @candes; @cssurvey]. The use of CS based measurement schemes in solving inference problems, such as, detection, classification and estimation has also attracted a considerable attention in the recent literature [@dave; @haupt; @thak; @durate]. The random measurement scheme used in [@dave; @haupt; @thak; @durate] provides universality for a wide variety of signal classes, but it fails to exploit the signal structure that may be known *a priori*. To improve performance, optimization of the measurement scheme can be performed by exploiting the signal structure. Measurement matrix design in the context of sparse signal recovery from compressed measurements has been considered in the past [@elad; @abol; @gang; @endra; @rodrigues]. In [@Zahedi], the authors considered the measurement matrix deign problem in the context of sparse signal detection based on compressed measurements. In all of these works [@Zahedi; @elad; @abol; @gang; @endra; @rodrigues], measurement matrices were constrained to the class of tight frames in order to avoid coloring the noise covariance matrix. In [@subspace], the measurement matrix was designed for the detection of a known sparse signal from compressive measurements. In the same context, heuristic or algorithmic approaches were proposed by [@bai] for measurement matrix design.
In this letter, our goal is to design measurement matrices for compressive detection so that a desired performance level is achieved under physical layer secrecy constraints. To that end, we employ the collaborative compressive detection (CCD) framework proposed in our previous paper [@bhavyaphy]. In a CCD framework, the fusion center (FC) receives compressed observation vectors from the nodes and makes the global decision about the presence of the signal vector. The transmissions by the nodes, however, may be observed by an eavesdropper. The secrecy of a detection system against eavesdropping attacks is of utmost importance in many applications [@physecdet]. Recently, a few attempts have been made to address the problem of eavesdropping threats on distributed detection networks [@bkeve]. Security issues with CS based measurement schemes have been considered in [@cssec1; @cssec2; @cssec3], where performance limits of secrecy of CS based measurement schemes were analyzed (under different assumptions). In this work, we investigate the problem from a design perspective and consider the problem of measurement matrix design with secrecy guarantees in an optimization framework. We show that the performance of the CCD framework can be significantly improved by using optimized measurement matrices (which exploit the underlying signal structure) along with artificial noise injection based techniques. More specifically, we design optimal measurement matrices which maximize the detection performance of the network while guaranteeing a certain level of secrecy considering three different scenarios: $1)$ signal of interest $s$ is known, $2)$ $s$ lies in low dimensional subspace, and $3)$ $s$ is sparse.
Observation Model {#sec2}
=================
![System Model[]{data-label="model"}](model_BK){height="0.2\textheight" width="35.00000%"}
Collaborative Compressive Detection
-----------------------------------
Consider two hypotheses $H_{0}$ (signal is absent) and $H_{1}$ (signal is present). Also, consider a parallel network (see Figure \[model\]), comprised of a fusion center (FC) and a set of $N$ nodes, which faces the task of determining which of the two hypotheses is true. At the $i$th node, the observed signal, $u_i$, can be modeled as
$$\begin{aligned}
&& H_0\; :\quad u_i=v_i\\
&& H_1\; :\quad u_i=s+v_i\end{aligned}$$
for $i =1,\cdots,N$, where $u_i$ is the $P \times 1$ observation vector, $s$ is the signal vector to be detected, $v_i$ is the additive Gaussian noise with $v_i \sim \mathcal{N} (0,\sigma^2 I_P)$ where $I_P$ is the $P\times P$ identity matrix. Observations at the nodes are assumed to be conditionally independent and identically distributed. Each node carries out local compression (low dimensional projection) and sends a $M$-length compressed version $y_i$ of its $P$-length observation $u_i$ to the FC. The collection of $M$-length $(< P)$ sampled observations is given by, $y_i=\phi u_i$, where $\phi$ is an $M \times P$ measurement matrix, which is assumed to be the same for all the nodes, and $y_i$ is the $M\times 1$ compressed observation vector (local summary statistic). Under the two hypotheses, the compressed measurements are
$$\begin{aligned}
&& H_0\; :\quad y_i=\phi v_i\\
&& H_1\; :\quad y_i=\phi s+ \phi v_i\end{aligned}$$
for $i = 1,\cdots, N$. The FC makes the global decision about the phenomenon based on the received compressed measurement vectors, $\mathbf{y}=[y_1,\cdots,y_N]$. We also assume that there is an eavesdropper present in the network who intends to discover the state of the nature being monitored by the system.
Collaborative Compressive Detection in the Presence of an Eavesdropper
----------------------------------------------------------------------
To keep the data regarding the presence of the phenomenon secret from the eavesdropper, in our previous work [@bhavyaphy], we used cooperating trustworthy nodes that assist the FC by injecting noise in the signal sent to the eavesdroppers to improve the security performance of the system. It was assumed that $B$ out of $N$ nodes (or $\alpha=B/N$ fraction of the nodes) tamper their data $y_i$ and send $\tilde{y_i}$ as follows:
Under $H_0$: $$\tilde{y_{i}} = \left\{ \begin{array}{rll}
\phi(v_{i}+D_{i}) & \mbox{with probability}\ P_{1}^0 \\
\phi(v_{i}-D_{i}) & \mbox{with probability}\ P_{2}^0 \\
\phi v_{i} & \mbox{with probability}\ (1-P_1^0-P_2^0)\\
\end{array}\right.$$
Under $H_1$: $$\tilde{y_{i}} = \left\{ \begin{array}{rll}
\phi(s+v_{i}+D_{i}) & \mbox{with probability}\ P_{1}^1 \\
\phi(s+v_{i}-D_{i}) & \mbox{with probability}\ P_{2}^1 \\
\phi(s+v_{i}) & \mbox{with probability}\ (1-P_1^1-P_2^1)\\
\end{array}\right.$$
where $D_{i}=\gamma s$ is a $P \times 1$ vector with constant values. The parameter $\gamma>0$ represents the noise strength, which is zero for non noise injecting nodes. We assume that the observation model and noise injection parameters are known to both the FC and the eavesdropper. The only information unavailable at the eavesdropper is the identity of the noise injecting nodes. Thus, the eavesdropper considers each node $i$ to be injecting noise with a certain probability $\alpha$. The FC can distinguish between $y_i$ and $\tilde{y_i}$. Note that, the values of $(P_1^0, P_2^0)$ and $(P_1^1, P_2^1)$ are system dependent and cannot be optimized in many scenarios which limits the secrecy performance of the system. Thus, in this letter, we design optimal measurement matrix $\phi$ which can be used along with artificial noise injection based techniques to improve the security performance of the system.
Problem Formulation
===================
We use the deflection coefficient as the detection performance metric in lieu of the probability of error of the system. Deflection coefficient reflects the output signal to noise ratio and is widely used in optimizing the performance of detection systems. The deflection coefficient at the $i$th node is defined as $$\begin{aligned}
D(y_i)&=& (\mu_1^i-\mu_0^i)^T (\Sigma_0^i)^{-1} (\mu_1^i-\mu_0^i) \end{aligned}$$ where $\mu_j^i$ and $\Sigma_j^i$ are the mean and the covariance matrix of $y_i$ under the hypothesis $H_j$, respectively. Using these notations, the deflection coefficient at the FC can be written as $D(FC)=B D(\tilde{y_i})+(N-B)D(y_i).$ Dividing both sides of the above equation by $N$, we get $D_{FC}=\alpha D(\tilde{y_i})+(1-\alpha)D(y_i)$ where $D_{FC}={D(FC)}/{N}$ and will be used as the performance metric. Similarly, the deflection coefficient at the eavesdropper can be written as $D_{EV}={D(EV)}/{N}=D(\hat{y_i}).$ Notice that both $D_{FC}$ and $D_{EV}$ are functions of the measurement matrix $\phi$ and noise injection parameters $(\alpha,\gamma)$ which are under the control of the FC. This motivates us to design the optimal measurement matrix for fixed noise injection parameters $(\alpha,\gamma)$ under a physical layer secrecy constraint. The problem can be formally stated as: $$\label{opt}
\begin{aligned}
& \underset{\phi}{\text{maximize}}
& & \alpha D(\tilde{y_i})+(1-\alpha)D(y_i) \\
& \text{subject to}
& & D(\hat{y_i})\leq \tau \\
\end{aligned}$$ where $\tau\geq 0$, is referred to as the physical layer secrecy constraint which reflects the security performance of the system. In our previous work [@bhavyaphy], we have derived the expressions for $D_{FC}$ and $D_{EV}$ (see Proposition $1$ and Proposition $2$ in [@bhavyaphy]). Using those expressions reduces to:
$$\label{above}
\begin{aligned}
& \underset{\phi}{\text{maximize}}
& & \frac{\alpha(1-P_b \gamma)^2}{\gamma^2 P_t+\frac{\sigma^2}{\|\hat{P} s\|_2^2}}+(1-\alpha)\frac{\|\hat{P} s\|_2^2}{\sigma^2} \\
& \text{subject to}
& & \frac{(1-\alpha P_b \gamma)^2}{\gamma^2 P_t^E+\frac{\sigma^2}{\|\hat{P} s\|_2^2}}\leq \tau \\
\end{aligned}$$
where $\hat{P}=\phi^T(\phi \phi^T)^{-1}\phi$, $P_b=(P_1^0-P_2^0)+(P_2^1-P_1^1)$\
$P_t=P_1^0+P_2^0-(P_1^0-P_2^0)^2$ and\
$P_t^E=\alpha(P_1^0+P_2^0-\alpha(P_1^0-P_2^0)^2)$ matrix. Next, we solve under various assumptions on the signal structure (e.g., known, low dimensional or sparse).
Optimal Measurement Matrix Design with Physical Layer Secrecy Guarantees
========================================================================
First, we explore some properties of the deflection coefficient at the FC, $D_{FC}$, and at the eavesdropper, $D_{EV}$, which will be used to simplify the measurement matrix design problem.
Deflection coefficient both at the FC and the Eve is a monotonically increasing function of $D_H=\frac{\|\hat{P} s\|_2^2}{\sigma^2}$.
The proof follows from the fact that both $\frac{dD_{FC}}{dD_H}>0$ and $\frac{dD_{EV}}{dD_H}>0$.
The above observation leads to the following equivalent optimal measurement matrix design problem for compressive detection:
$$\label{optimization}
\begin{aligned}
& \underset{\phi}{\text{maximize}}
& & \delta=\|\hat{P} s\|_2^2 \\
& \text{subject to}
& & \|\hat{P} s\|_2^2\leq \frac{\sigma^2}{\frac{(1-\alpha P_b\gamma)^2}{\tau}-\gamma^2P_t^E} \\
\end{aligned}$$
for any arbitrary signal $s$. Note that, for the random measurement matrix $\delta_r=\|\hat{P} s\|_2^2=\frac{M}{N}\|s\|_2^2$ [@dave]. The factor $M/N$ can be seen as the performance loss due to compression as random measurement matrix fails to exploit the signal structure that may be known *a priori*. To improve performance, we consider the optimization of the measurement matrix by exploiting the signal structure while guaranteeing a certain level of secrecy. We show that any arbitrary secrecy constraint can be guaranteed by properly choosing the measurement matrix.
Known Signal Detection
----------------------
First, we consider the case where $s$ is known.
When $s$ is known, the optimal value of the objective function of , is given by $\delta^*=\min\left(\| s\|_2^2,\frac{\sigma^2}{\frac{(1-\alpha P_b \gamma)^2}{\tau}-\gamma^2 P_t^E}\right)$.
The proof follows from the fact that $\hat{P}$ is an orthogonal projection operator, thus, $\|\hat{P}s\|_2^2\leq\|s\|_2^2$.
Let us denote the singular value decomposition of $\phi=U[\pi_M,0]V^T$ where $U$ is an $M\times M$ orthonormal matrix, $[\pi_M,0]$ is an $M\times N$ diagonal matrix and $V$ is an $N\times N$ orthonormal matrix. Now, the optimal $\phi$ which achieves $\delta^*$ is characterized in the following lemma.
When $s$ is known, the optimal $\phi$ which achieves $\delta^*$ in is given by $\phi^*=U[\pi_M,0](V^*R)^{T}$ where $U$ and diagonal $\pi_M>0$ are totally arbitrary,
$$R=\left[ \begin{array}{ccc}
\cos \theta& \mathbf{0} & \sin \theta\\
\mathbf{0} & \mathbf{I} &\mathbf{0}\\
-\sin \theta &\mathbf{0} &\cos \theta \end{array} \right],$$
$\theta$ is the parameter which controls the level of secrecy such that $\theta=0$ if $\| s\|_2^2\leq\frac{\sigma^2}{\frac{(1-\alpha P_b \gamma)^2}{\tau}-\gamma^2 P_t^E}$, and, $\theta=\cos^{-1}\sqrt{\frac{\sigma^2/\| s\|_2^2}{\frac{(1-\alpha P_b \gamma)^2}{\tau}-\gamma^2 P_t^E}}$, otherwise. $V^*=[v_1^*,\cdots,v_N^*]$ is any orthonormal matrix satisfying $v_i^* \perp s, \forall i > M$.
To prove the lemma, notice that
$$\|\hat{P} s\|_2=s^TV\left[ \begin{array}{ccc}
I_M & 0 \\
0 & 0 \end{array} \right]V^Ts=\sum_{i=1}^{M} \tilde{s}_i^2\leq \| s\|_2^2$$
where $\tilde{s}=V^T s$. The upper bound or equality in the above equation can be achieved if and only if $\tilde{s}_i=0,\;\forall i>M$. The corresponding optimal measurement matrix for this case is characterized by $\phi^{*}=U[\pi_M,0]V^T$ where the orthonormal $U$ and diagonal $\pi_M>0$ are totally arbitrary, while $V=[v_1,\cdots,v_N]$, as seen above, has to be an orthonormal matrix satisfying $v_i \perp s, \forall i > M$. Now, the matrix $(VR)$ is a orthonormal matrix for any orthonormal $V$ and observe that the optimal $V^*$ as given in the lemma is also orthonormal. Thus, for optimal $\phi^*$, we have $$\|\hat{P} s\|_2=s^TV^*R\left[ \begin{array}{ccc}
I_M & 0 \\
0 & 0 \end{array} \right](V^*R)^Ts=\cos^2\theta\|s\|_2^2.$$ Next, using the definition of $\theta$, the results in the lemma can be derived.
If we define $\mathrm{Proj}_{u}(w)=\frac{u^Tw}{u^Tu}u$ and $W=[w_1,\cdots,w_N]$ with $w_1=s$ and $w_k$ as any linearly independent set of vectors, one possible solution for $V^*$ in a closed form is: $V^*=[v_1,\cdots,v_N]$, where $v_k=\frac{u_k}{\|u_k\|_2}$ and $u_k=w_k-\sum_{j=1}^{k-1}\mathrm{Proj}_{u_j}(w_k).$ Note that, without physical layer secrecy constraint (or when $\theta=0$) the optimal value of the objective function is $\|s\|_2^2$. Thus, there is no performance loss due to compression. With physical layer secrecy constraint, $\theta$ serves as a tuning parameter to guarantee a certain level of secrecy. This approach provides the optimal measurement matrix with a secrecy guarantee for a known $s$. However, in certain practical scenarios we do not have an exact knowledge of $s$. Next, we consider the cases where $s$ is not completely known.
Low Dimensional Signal Detection
--------------------------------
In this subsection, we consider the case where $s$ is not completely known but is known to lie in a low dimensional subspace and design $\phi$ so that the detection performance at the FC is maximized while ensuring a certain level of secrecy at the eavesdropper. We assume that $s$ resides in a $K$-dimensional subspace where $K<N$. That is to say, $s$ can be expressed as $s=D\beta$ where $D$ is an $N\times K$ matrix, whose columns are orthonormal, and $\beta$ is the $K\times1$ signal vector. Without loss of generality, we assume that $\|\beta\|_2^2=1$. Next, we look at the following two cases: $1)$ $D$ can be designed, $2)$ $D$ is fixed and known. For both the cases, we assume that $\beta$ is deterministic but unknown and find $\phi$ which maximizes the worst case detection performance. Formally, for the case where $D$ is a design parameter, the problem can be stated as
$$\label{opt2}
\begin{aligned}
& \underset{\phi_{M\times N}}{\max}\;\;\underset{D_{N\times K}}{\max}\;\;\underset{\beta_{K\times 1}}{\min}
& & \delta=\|\hat{P} D\beta\|_2^2 \\
& \text{subject to}
& & \|\beta\|_2^2= 1,\;\|\hat{P} D\beta\|_2^2\leq \Delta \\
\end{aligned}$$
where $\Delta=\frac{\sigma^2}{\frac{(1-\alpha P_b\gamma)^2}{\tau}-\gamma^2P_t^E}$.
We state the Courant-Fischer theorem which will be used to solve the above optimization problem.
[(Courant-Fischer[@horn])]{} Let $A$ be a symmetric matrix with eigenvalues $\lambda_1\geq\cdots\geq\lambda_N$ and $S$ denote the any $j$-dimensional linear subspace of $\mathbb{C}^N$. Then, $$\underset{S:\;\text{dim}(S)=j}{\max}\;\;\underset{x\in S}{\min}\;\;\dfrac{x^TAx}{x^Tx}=\lambda_j.$$
When $s$ lies in a low dimensional signal subspace, the optimal value of the objective function of is given by $\delta^*=\min\left(\| \beta\|_2^2,\Delta\right)$ if $K\leq M$, and $\delta^*=0$, otherwise.
Using Courant-Fischer theorem, we can show that the problem without a physical layer secrecy constraint is equivalent to $\underset{\phi_{M\times N}}{\max} \lambda_k(\hat{P})$. Now, the proof follows by observing that $\hat{P}$ is the orthogonal projection operator and its eigenvalues are given by $\lambda_i=1$ for $i=1$ to $M$ and $\lambda_i=0$ for $i=M+1$ to $N$.
Next, we assume that $K\leq M$ and characterize the optimal measurement matrix $\phi^*$ and the optimal subspace $D^*$.
\[lem4\] When $s=D\beta$ and $K\leq M$, the optimal $(\phi^*,D^*)$ which achieves $\delta^*$ should satisfy the following condition: for any arbitrary $\phi=U[\pi_M,0]V^T$ where $V=[v_1,\cdots,v_N]$, the optimal $D^*=\cos\theta D$ with $D=[v_1,\cdots,v_K]$.
Note that for optimal $(\phi^*,D^*)$, we have
$$\begin{aligned}
\|\hat{P} s\|_2&=& \beta^T (D^*)^TV^*\left[ \begin{array}{ccc}
I_M & 0 \\
0 & 0 \end{array} \right]((D^*)^TV^*)^T\beta\\
&=& \beta^T \left[ \begin{array}{ccc}
\cos\theta I_K & 0 \end{array} \right]
\left[ \begin{array}{ccc}
I_M & 0 \\
0 & 0 \end{array} \right]
\left[ \begin{array}{ccc}
\cos\theta I_K \\
0 \end{array} \right]\beta\\
&=&(\cos\theta)^2\sum\limits_{i=1}^{\min(K,M)}(\beta_i)^2\end{aligned}$$
Observe $\underset{\beta}{\min}\sum\limits_{i=1}^{\min(K,M)}(\beta_i)^2=\| \beta\|_2^2$ if $\min(K,M)=K$ and $0$, otherwise. Using the definition of $\theta$, $\delta^*$ can be achieved.
The above lemma can be interpreted as follows: for any fixed $\phi$, one can choose $D$ accordingly, so that the upper bound $\delta^*$ can be achieved. Next, we look at the case where $D$ is fixed and we only optimize measurement matrix $\phi$. Observe that, $$\underset{\phi}{\max}\;\;\underset{\beta}{\min}
\|\hat{P} D\beta\|_2^2\leq \underset{\phi}{\max}\;\;\underset{D}{\max}\;\;\underset{\beta}{\min}
\|\hat{P} D\beta\|_2^2=\| \beta\|_2^2.$$ For a fixed $D$, the optimal value $\delta^*$ of the problem serves as an upper bound. To simplify the problem, we introduce an $(N\times N)$ matrix $P$ to guarantee secrecy in the system. In other words, $y_i=\phi P u_i$ where $P$ is determined to guarantee physical layer secrecy. Next, we find $\phi$ for which this upper bound is achievable for a fixed $D$ and $P$ to secrecy.
For the low dimensional signal case $y_i=\phi P s$ with $s=D\beta$ where $D=[d_1,\cdots,d_K]$ is orthonormal, the optimal measurement matrix $(\phi^*,P^*)$, is given by $P^*=\cos\theta I_{N\times N}$ and $\phi^*=U[\pi_M,0](V^*)^T$ where the orthonormal $U$ and diagonal $\pi_M>0$ are totally arbitrary, while $V^*=[v_1,\cdots,v_N]$ is such that $v_i=d_i$ for $i=1$ to $K$ and $v_i$ for $i=K+1$ to $N$ are such that $V$ forms an orthonormal basis.
The proof is similar to Lemma \[lem4\], thus, omitted.
For both the cases, where $D$ can be designed and where $D$ is fixed and known, without secrecy constraint the optimal value of the objective function is $\|s\|_2^2$. Thus, there is no performance loss due to compression. With secrecy constraint, $\theta$ serves as a tuning parameter to guarantee a certain level of secrecy.
Sparse Signal Detection
-----------------------
In this section, we assume that $s$ is $K$-sparse in the standard canonical basis and $\|s\|_2^2=1$. Also, the exact number of the nonzero entries in $s$, their locations, and their values are assumed to be unknown. We design $\phi$ which maximizes the worst case detection performance by employing a lexicographic optimization approach[^3]. Formally, the problem is
$$\label{opt3}
\begin{aligned}
& \underset{\phi_{M\times N}}{\max}\;\;\underset{s}{\min}
& & \|\hat{P} s\|_2^2 \\
& \text{subject to}
& & \|s\|_2^2= 1,\;\|s\|_0= K, \\
& & &\|\hat{P} s\|_2^2\leq \Delta,\; \phi \in \mathcal{A}_{K-1}\\
\end{aligned}$$
where $\mathcal{A}_{K-1}$ is the set of solutions to the above optimization problem for sparsity level $K-1$ and $\Delta$ is defined in .
There is no performance loss while solving the problem if we restrict our solution space to be matrices on the Stiefel manifold $S_t(M,N)$, where $$S_t(M,N):=\{\phi\in\mathbb{R}^{M\times N}:\phi\phi^T=I\}.$$
The proof follows from the observation that $\pi_M=I_M$ for frames $\phi=U[\pi_M,0]V^T$ in Stiefel manifold and the value of $\|\hat{P}s\|_2^2$ is independent of $\pi_M$ and $U$.
Next, we limit our focus on Stiefel manifolds and establish an upper bound on the value of the objective function in for different sparsity levels. Later we find measurement matrices which can achieve this upper bound.
\[lem7\] For the sparsity level $K=1$, the optimal value of the objective function of is $\min\left(\frac{M}{N},\Delta\right)$. For the sparsity level $K\geq 2$, an upper bound on the value of the objective function is given by $\min\left(\frac{M}{N}(1-\mu),\Delta\right)$, where $\mu=\sqrt{\frac{N-M}{M(N-1)}}$.
The proof is similar to Theorem $1$ and Theorem $3$ as given in [@Zahedi], thus, omitted.
The optimal measurement matrix $(\phi^*,P^*)$, for the $K$ sparse signal case $y_i=\phi P s$ is given by:
- For the sparsity level $K=1$, $\phi^*$ is a uniform tight frame with norm values equal to $\sqrt{M/N}$ and $P^*=\cos\theta I_{N\times N}$,
- For the sparsity level $K\geq2$, $\phi^*$ is an equiangular tight frame with norm values equal to $\sqrt{M/N}$ and $P^*=\cos\theta I_{N\times N}$.
Proof follows from the definition of uniform (or equiangular) tight frames [@Casazza] and observation that the upper bounds in the Lemma \[lem7\] can be reached only by these frames.
Note that, without physical layer secrecy constraint (i.e., $\theta=0$), our results reduce to the ones in [@Zahedi]. With physical layer secrecy constraint, similar to previous cases, $\theta$ serves as a tuning parameter to guarantee an arbitrary level of secrecy. Also, it is shown that a real equiangular tight frame can exist only if $N \leq M(M+1)/2$, and a complex equiangular tight frame requires $N \leq M^2$ [@etf]. When $M$ and $N$ do not satisfy this condition, the bound in Lemma \[lem7\] can not be achieved and one can employ a heuristic or algorithmic approach [@bai].
Discussion and Future Work
==========================
We considered the problem of designing measurement matrices for high dimensional signal detection based on compressed measurements with physical layer secrecy guarantees. It was shown that the optimal design depends on the nature of the signal to be detected. Further, security performance of the system can be improved by using optimized measurement matrices along with artificial noise injection based techniques. In the future, we plan to come up with efficient algorithms to improve the worst-case detection probability for the cases where equiangular tight frames do not exist.
[^1]: This work was supported in part by the National Science Foundation (NSF) under Grant No. 1307775.
[^2]: Authors are with Department of EECS, Syracuse University, Syracuse, NY 13244. (email: [email protected]; [email protected]; [email protected]; [email protected])
[^3]: We first find a set of solutions that are optimal for a $k_1$-sparse signal. Then, within this set, we find a subset of solutions that are also optimal for $(k_1+1)$-sparse signals. This approach is known as a lexicographic optimization.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The Weil representation of the symplectic group associated to a finite abelian group of odd order is shown to have a multiplicity-free decomposition. When the abelian group is $p$-primary of type $\lambda$, the irreducible representations occurring in the Weil representation are parametrized by a partially ordered set which is independent of $p$. As $p$ varies, the dimension of the irreducible representation corresponding to each parameter is shown to be a polynomial in $p$ which is calculated explicitly. The commuting algebra of the Weil representation has a basis indexed by another [partially ordered set]{} which is independent of $p$. The expansions of the projection operators onto the irreducible invariant subspaces in terms of this basis are calculated. The coefficients are again polynomials in $p$. These results remain valid in the more general setting of finitely generated torsion modules over a Dedekind domain.'
address: 'The Institute of Mathematical Sciences, Chennai.'
author:
- Kunal Dutta
- Amritanshu Prasad
title: |
Combinatorics of finite abelian groups\
and Weil representations
---
Introduction
============
Overview {#sec:overview}
--------
[^1]Heisenberg groups were introduced by Weyl [@Weyl50 Chapter 4] in his mathematical formulation of quantum kinematics. Best known among them are the Lie groups whose Lie algebras are spanned by position and momentum operators which satisfy Heisenberg’s commutation relations. Weyl also considered Heisenberg groups which are finite modulo centre, such as the Pauli group (generated by the Pauli matrices), which he used to characterize the kinematics of electron spin.
A fundamental property of Heisenberg groups, predicted by Weyl and proved by Stone [@Stone30] and von Neumann [@vN31] for real Heisenberg groups is known as the Stone-von Neumann theorem. Mackey [@Mackey49] extended this theorem to locally compact Heisenberg groups (see Section \[sec:basic-definitions\] for the case that is pertinent to this paper, and [@SvN] for a more detailed and general exposition). By considering Heisenberg groups associated to finite fields, local fields and adèles, Weil [@MR0165033] demonstrated the importance of Heisenberg groups in number theory.
Weil exploited the Stone-von Neumann-Mackey theorem to construct a projective representation of a group of automorphisms of the Heisenberg group, now commonly known as the Weil representation. Along with parabolic induction and the technique of Deligne and Lusztig [@MR0393266] using $l$-adic cohomology, the Weil representation is one of the most important techniques for constructing representations of reductive groups over finite fields (see Gérardin [@Gerardin77] and Srinivasan [@MR528020]) or local fields (see Gérardin [@MR0396859] and Moeglin-Vigneras-Waldspurger [@MR1041060]).
Tanaka [@MR0219635; @MR0229737] showed how the Weil representation can be used to construct all the irreducible representations of $\mathrm{SL}_2({\mathbf Z}/p^k{\mathbf Z})$ for odd $p$ by looking at Weil representations associated to the abelian groups ${\mathbf Z}/p^k{\mathbf Z}\oplus {\mathbf Z}/p^l{\mathbf Z}$ for $l\leq k$. However, most of the literature on Weil representations associated to finite abelian groups has focused on vector spaces over finite fields and on constructing representations of classical groups over finite fields.
The representation theory of groups over finite principal ideal local rings was initiated by Kloosterman [@MR0021032], who studied $\mathrm{SL}_2({\mathbf Z}/p^k{\mathbf Z})$. In contrast to general linear groups over finite fields, whose character theory was worked out by Green [@MR0072878], the representation theory for general linear groups over these rings is quite hard. It has been shown (Aubert-Onn-Prasad-Stasinski [@MR2607551] and Singla [@Poojareps]) that this problem is intricately related to the problem of understanding the representations of automorphism groups of finitely generated torsion modules over discrete valuation rings. However, explicit constructions have been available either for a very small class of representations (Hill [@MR1334228; @MR1311772]) or for a very small class of groups (Onn [@MR2456275], Stasinski [@MR2588859], Singla[@Poojareps]).
This article concerns the decomposition of the Weil representation of the full symplectic group associated to a finite abelian group of odd order (and more generally, a finite module of odd order over a Dedekind domain) into irreducible representations. When the module in question is elementary (e.g., $({\mathbf Z}/p{\mathbf Z})^n$ for some odd prime $p$), it is well-known that the Weil representation, which may be realized on the space of functions on the abelian group, breaks up into two irreducible subspaces consisting of even and odd functions. Besides this, only the case where all the invariant factors are equal (e.g., $({\mathbf Z}/p^k{\mathbf Z})^n$) has been understood completely (see Prasad [@MR1478492 Theorem 2] for the case where $k$ is even, Cliff-McNeilly-Szechtman [@MR1783635] for the general case). A small part of the decomposition has been explained in the general case (Cliff-McNeilly-Szechtman [@MR1971043]).
In this paper, we describe all the invariant subspaces for the Weil representation for all finite modules of odd order over a Dedekind domain. To be specific, it is shown that the Weil representation has a multiplicity-free decomposition (Theorem \[theorem:multiplicity-free\]). When the underlying finitely generated torsion module is primary of type $\lambda$ as in (\[eq:11\]) or (\[eq:13\]), the irreducible components are parametrized by elements of a partially ordered set which depends only on $\lambda$, and not on the underlying ring. As the local ring varies, for a fixed element of this [partially ordered set]{}, the dimension of the corresponding representation is shown to be a polynomial in the order of its residue field whose coefficients do not depend on the ring (Theorem \[theorem:main\]). These polynomials are computed explicitly (Theorem \[theorem:dim\]). The centralizer algebra of the Weil representation also has a combinatorial basis indexed by a partially ordered set which depends only on $\lambda$ and not on the underlying ring. The projection operators onto the irreducible invariant subspaces, when expressed in terms of this basis are also shown to have coefficients which are polynomials in the order of the residue field whose coefficients also do not depend on the ring (Theorem \[theorem:alpha-qualitative\]), and these polynomials are computed explicitly (Theorems \[theorem:alpha-supp\] and \[theorem:alpha-exact\]). Thus the decomposition of the Weil representation into irreducible invariant subspaces is, despite its apparent complexity, combinatorial in nature.
The results in this paper could serve as a starting point from which more subtle constructions involving the Weil representation (such as Howe duality) which have worked so well in the case of classical groups over finite fields can be extended to groups of automorphisms of finitely generated torsion modules over a discrete valuation ring.
It is worth noting that every Heisenberg group that is finite modulo centre is isomorphic to one of the groups considered here (for the precise statement, see Prasad-Shapiro-Vemuri [@Prasad2010], particularly, Section 3 and Corollary 5.7). For example, the seemingly different Heisenberg groups used by Tanaka [@MR0219635] to construct representations in the principal series and cuspidal series of finite $\mathrm{SL}_2$ are isomorphic. The difference lies in the realization of the special linear group as a group of automorphisms. The decomposition of any Weil representation associated to a finite abelian group will therefore always be a refinement of one of the decompositions described in this paper.
In order to concentrate on the important ideas without being distracted by technicalities, the main body of this paper uses the setting of finite abelian groups. Section \[sec:finite-modules\] explains how to carry over the results to finitely generated torsion modules over discrete valuation rings and even more generally, finite modules over Dedekind domains.
As for prerequisites, we use the fairly simple combinatorial theory of orbits in finite abelian groups developed by us in [@2010arXiv1005.5222D] (the relevant part is recalled in Section \[sec:orbits\]), well-known basic facts about Heisenberg groups and Weil representations which are recalled in Section \[sec:basic-definitions\] (of which simple proofs can be found in [@gdft]), and the standard combinatorial theory of [partially ordered set]{}s, as set out in Chapter 3 of Stanley’s book [@MR1442260].
Basic definitions {#sec:basic-definitions}
-----------------
Let $A$ be a finite abelian group of odd order. Let $\hat A$ denote the Pontryagin dual of $A$. This is the group of all homomorphisms $A\to U(1)$, where $U(1)$ denotes the group of unit complex numbers. Let $K=A\times \hat A$. For each $k=(x,\chi)\in K$, the unitary operator on $L^2(A)$ defined by $$W_k f(u)=\chi(u-x/2)f(u-x) \text{ for all } f\in L^2(A),\; u\in A$$ is called a Weyl operator. These operators satisfy $$W_kW_l=c(k,l)W_{k+l} \text{ for all } k,l\in K,$$ where, if $k=(x,\chi)$ and $l=(y,\lambda)$, then $$c(k,l)=\chi(y/2)\lambda(x/2)^{-1}.$$ The subgroup $$H=\{c W_k|c\in U(1),k\in K\}$$ of the group of unitary operators on $L^2(A)$ is called the Heisenberg group associated to $A$. This group is known to physicists as a generalized Pauli group or a Weyl-Heisenberg group. As defined here, it comes with a unitary representation on $L^2(A)$, called the Schrödinger representation. Mackey’s generalization of the Stone-von Neumann theorem applies:
\[theorem:SvN\] The Schrödinger representation of $H$ is irreducible. Let $\rho:H\to U({\mathcal H})$ be an irreducible unitary representation of $H$ such that $\rho(c W_0)=c\mathrm{Id}_{{\mathcal H}}$ for every $c\in U(1)$. Then there exists, up to scaling, a unique isometry $W:L^2(A)\to {\mathcal H}$ such that $$WW_k=\rho(W_k)W \text{ for all } k\in K.$$
If $g$ is an automorphism of $K$ such that $$\label{eq:2}
c(g k,g l)=c(k,l) \text{ for all } k,l\in K$$ then $\rho_g:H\to U(L^2(A))$ defined by $$\rho_g(c W_k)=c W_{g(k)} \text{ for all } c\in U(1),\; k\in K$$ is an irreducible unitary representation of $H$ on $L^2(A)$ such that $\rho(c W_0)=c\mathrm{Id}_{L^2(A)}$. By Theorem \[theorem:SvN\], there exists a unitary operator $W_g$ on $L^2(A)$ such that $W_g W_k=W_{g(k)}W_g$ for all $k\in K$. In other words, $$\label{eq:1}
W_g W_kW_g^*=W_{g(k)} \text{ for all } k\in K.$$ If $g_1$ and $g_2$ are two such automorphisms, both $W_{g_1g_2}$ and $W_{g_1}W_{g_2}$ intertwine the Schrödinger representation with $\rho_{g_1g_2}$, and hence must differ by a unitary scalar: $$W_{g_1}W_{g_2}=c(g_1,g_2)W_{g_1g_2} \text{ for some } c(g_1,g_2)\in U(1).$$ Let ${\mathrm{Sp}}(K)$ be the group of all automorphisms $g$ of $K$ which satisfy (\[eq:2\]). We have shown that $g\mapsto W_g$ is a projective representation of ${\mathrm{Sp}}(K)$ on $L^2(A)$. This representation is known as the Weil representation.
The operators $W_g$, for $g\in {\mathrm{Sp}}(K)$ can be normalized in such a way that $c(g_1,g_2)=1$ for all $g_1,g_2$ (see Remark \[remark:ordinary-rep\]). Thus the Weil representation can be taken to be an ordinary representation of ${\mathrm{Sp}}(K)$.
The overlap of notation between the Weyl operators and the Weil representation is suggested by (\[eq:1\]), which implies that they can be combined to construct a representation of $H\rtimes {\mathrm{Sp}}(K)$. The operators in this representation are precisely the unitary operators which normalize $H$. The resulting group is sometimes known as a Clifford group or a Jacobi group. It plays a prominent role in the stabilizer formalism for quantum error-correcting codes (see Chapter X of Nielsen and Chuang [@Nielsen-Chuang]).
Formulation of the problem {#sec:problem}
--------------------------
We investigate the decomposition $$\label{eq:12}
L^2(A)=\bigoplus_{\pi\in \widehat{{\mathrm{Sp}}(K)}} m_\pi {\mathcal H}_\pi$$ into irreducible representations. Here $\widehat{{\mathrm{Sp}}(K)}$ denotes the set of equivalence classes of irreducible unitary representations of ${\mathrm{Sp}}(K)$ and, for each $\pi:{\mathrm{Sp}}(K)\to U({\mathcal H}_\pi)$ in $\widehat{{\mathrm{Sp}}(K)}$, $m_\pi$ denotes the multiplicity of $\pi$ in the Weil representation. Although the Weil representation is defined only up to multiplication by a scalar representation, the multiplicities and dimensions of the irreducible representations occurring in the decomposition are invariant under such twists (see Remark \[remark:projective-equiv\]). As explained in Section \[sec:overview\], the outcome of this paper is an understanding of this decomposition.
Product decompositions {#sec:products}
======================
We shall recall and apply a well-known observation on Weil representations associated to a product of abelian groups (see [@Gerardin77 Corollary 2.5]).
Projective equivalence
----------------------
Since Weil representations are defined only up to scalar factors, we use a definition of equivalence of representations that is weaker than unitary equivalence:
\[Projective equivalence\] \[defn:proj-equiv\] Let $G$ be a group and $\rho_i:G\to U({\mathcal H}_i)$ for $i=1,2$ be two unitary representations of $G$. We say that $\rho_1$ and $\rho_2$ are projectively equivalent if there exists a homomorphism $\chi:G\to U(1)$ such that $\rho_2$ is unitarily equivalent to $\rho_1\otimes \chi$.
\[remark:projective-equiv\] If, as a representation of $G$, $${\mathcal H}_i = \bigoplus_{\pi \in \hat G} m^{(i)}_\pi {\mathcal H}_\pi$$ is the decomposition of ${\mathcal H}_i$ into irreducibles for representations as in Definition \[defn:proj-equiv\], then $m^{(2)}_{\pi\otimes\chi}=m^{(1)}_\pi$, so there is a bijection between the sets of irreducible representations of $G$ that appear in ${\mathcal H}_1$ and ${\mathcal H}_2$ which preserves multiplicities and dimensions.
Tensor product decomposition
----------------------------
If $A$ admits a product decomposition $A=A'\times A^{\prime\prime}$, then $$\label{eq:9}
L^2(A)=L^2(A')\otimes L^2(A^{\prime\prime}).$$ Let $K'=A'\times \widehat{A'}$, $K^{\prime\prime}=A^{\prime\prime}\times \widehat{A^{\prime\prime}}$. Thus $K=K'\times K''$. Let $S'$ and $S^{\prime\prime}$ be subgroups of ${\mathrm{Sp}}(K')$ and ${\mathrm{Sp}}(K^{\prime\prime})$ respectively. Then $S=S'\times S''$ is a subgroup of ${\mathrm{Sp}}(K)$.
\[theorem:product-Weil\] The Weil representation of $S$ on $L^2(A)$ is projectively equivalent to the tensor product of the Weil representation of $S'$ on $L^2(A')$ and the Weil representation of $S''$ on $L^2(A'')$.
By (\[eq:1\]), the Weil representations of $S'$ and $S''$ satisfy $$W_{g'}W_{k'}W_{g'}^*=W_{g'(k')}\text{ and } W_{g''}W_{k''}W_{g''}^*=W_{g''(k'')}$$ for all $g'\in S'$, $g''\in S''$, $k'\in K'$ and $k''\in K''$, whence $$(W_{g'}\otimes W_{g''})(W_{k'}\otimes W_{k''})(W_{g'}\otimes W_{g''})^*=W_{g'(k')}\otimes W_{g''(k'')}.$$ Since $W_{k'}\otimes W_{k''}$ coincides with $W_{(k',k'')}$ under the isomorphism (\[eq:9\]), $W_{g'}\otimes W_{g''}$ satisfies the defining identity (\[eq:1\]) for the Weil representation of $S$ on $L^2(A)$.
Primary decomposition {#sec:prim-decomp}
---------------------
Every finite abelian group has primary decomposition $$A = \prod_{p \text{ prime}} A_p,$$ where $A_p$ is the subgroup of elements of $A$ annihilated by some power of $p$. Writing $K_p$ for $A_p\times \widehat{A_p}$, $$K=\prod_p K_p\quad \text{ and } \quad {\mathrm{Sp}}(K)=\prod_p {\mathrm{Sp}}(K_p).$$ Theorem \[theorem:product-Weil\], when applied to the primary decomposition gives
\[cor:primary-decomposition-Weil\] The Weil representation of ${\mathrm{Sp}}(K)$ on $L^2(A)$ is projectively equivalent to the tensor product over those primes $p$ for which $A_p\neq 0$ of the Weil representations of ${\mathrm{Sp}}(K_p)$ on $L^2(A_p)$.
In view of Corollary \[cor:primary-decomposition-Weil\], it suffices to consider the case where $A$ is a finite abelian $p$-group for some odd prime $p$.
Multiplicities and orbits {#sec:mult-orbits}
=========================
We now recall the relation between the decomposition of the Weil representation and orbits in $K$ [@gdft].
An orthonormal basis
--------------------
\[lemma:onb\] The set $\{W_k|k\in K\}$ of Weyl operators is an orthonormal basis of $\operatorname{End}_{\mathbf C}L^2(A)$.
For each $k\in K$ and $T\in \operatorname{End}_{\mathbf C}L^2(A)$, let $$\tau(k)T=W_k T W_k^*.$$ Then $k\mapsto \tau(k)$ is a unitary representation of $K$ on $\operatorname{End}_{\mathbf C}L^2(A)$. If $k=(x,\chi)$ and $l=(y,\lambda)$ are two elements of $K$, then $$\begin{aligned}
\tau(k)W_l&=&W_k W_l W_k^*\\
&=&W_k W_l (W_l W_k)^* W_l\\
&=&c(k,l) W_{k+l} c(l,k)^{-1} W_{l+k}^* W_l\\
&=&\chi(y)\lambda(x)^{-1}W_l.
\end{aligned}$$ Thus the $W_l$’s are eigenvectors for the action of $K$ with distinct eigencharacters. Therefore they form an orthonormal set of operators. Since $|K|=|A|^2=\dim\operatorname{End}_{\mathbf C}L^2(A)$, this orthonormal set is a basis.
Endomorphisms
-------------
By Lemma \[lemma:onb\], every $T\in \operatorname{End}_{\mathbf C}L^2(A)$ has a unique expansion $$\label{eq:3}
T=\sum_{k\in K} T_k W_k, \text{ with each } T_k\in {\mathbf C}.$$
\[theorem:orbit-mult\] For every subgroup $S$ of ${\mathrm{Sp}}(K)$, $$\operatorname{End}_S L^2(A)=\{T\in \operatorname{End}_{\mathbf C}L^2(A)|T_k=T_{g(k)} \text{ for all } g\in S,\; k\in K\}.$$
Note that $T\in \operatorname{End}_S L^2(A)$ if and only if $W_g T W_g^*=T$ for all $g\in S$. Expanding $T$ as in (\[eq:3\]) and using the defining identity (\[eq:1\]) for $W_g$ gives the theorem.
Now suppose that as a representation of $S$, $L^2(A)$ has the decomposition $$L^2(A)=\bigoplus_{\pi\in \hat S}m_{\pi,S}{\mathcal H}_\pi.$$ Then, together with Schur’s lemma, Theorem \[theorem:orbit-mult\] implies
\[cor:orbit-mult\] If $S\backslash K$ denotes the set of $S$-orbits in $K$, $$\sum_{\pi\in \hat S} m_{\pi,S}^2 =|S\backslash K|.$$
Orbits and characteristic subgroups {#sec:counting-orbits}
===================================
We first recall the theory of orbits (under the full automorphism group) and characteristic subgroups in a finite abelian group from [@2010arXiv1005.5222D]. We then see how it applies to ${\mathrm{Sp}}(K)$-orbits in $K$.
Orbits {#sec:orbits}
------
Every finite abelian $p$-group is isomorphic to $$\label{eq:11}
A={\mathbf Z}/p^{\lambda_1}{\mathbf Z}\times \cdots \times {\mathbf Z}/p^{\lambda_l}{\mathbf Z}$$ for a unique sequence $\lambda=(\lambda_1\geq\cdots\geq\lambda_l)$ of positive integers (in other words, a partition). Henceforth, we assume that $A$ is of the above form. For each partition $\lambda$, let $$P_\lambda = \big\{(v,k)|k\in \{\lambda_1,\ldots,\lambda_l\},\; 0\leq v<k\big\}.$$ Say that $(v,k)\geq (v',k')$ if and only if $v'\geq v$ and $k'-v'\leq k-v$. This relation is a partial order on $P_\lambda$. For $x\in {\mathbf Z}/p^k{\mathbf Z}$, let $$v(x)=\max\{0\leq v\leq k|x\in p^v{\mathbf Z}/p^k{\mathbf Z}\}.$$
For $a=(a_1,\ldots,a_l)\in A$, let $I(a)$ be the order ideal in $P_\lambda$ generated by $(v(a_i),\lambda_i)$ with $a_i\neq 0$ in ${\mathbf Z}/p^{\lambda_i}{\mathbf Z}$.
$$\begin{array}{ccc}
\begin{xy}
(0,0)*{\circ};
(5,5)*{\circ}**@{-};
(0,10)*{\circ}**@{-};
(5,15)*{\circ}**@{-};
(0,20)*{\circ}**@{-};
(5,25)*{\circ}**@{-};
(0,30)*{\circ}**@{-};
(5,35)*{\circ}**@{-};
(0,40)*{\circ}**@{-};
(5,5)*{\circ};
(20,20)*{\circ}**@{-};
(5,35)*{\circ}**@{-};
\end{xy}
&\quad\quad\quad &
\begin{xy}
(0,0)*{\bullet};
(5,5)*{\bullet}**@{-};
(0,10)*{\bullet}**@{-};
(5,15)*{\bullet}**@{-};
(0,20)*{\circ}**@{-};
(5,25)*{\circ}**@{-};
(0,30)*{\circ}**@{-};
(5,35)*{\circ}**@{-};
(0,40)*{\circ}**@{-};
(5,5)*{\circ};
(20,20)*{\circ}**@{-};
(5,35)*{\circ}**@{-};
\end{xy}\\
\text{The poset } P_{(5,4,4,1)} & & \text{The order ideal } I(p^4,p^2,p^3,0)
\end{array}$$
\[eg:Plambda\] When $\lambda=(5,4,4,1)$, and $a=(p^4,p^2,p^3,0)$ the Hasse diagram of $P_\lambda$ and the ideal $I(a)$ (represented by black dots) are shown in Figure \[fig:1\]. The elements of $P_\lambda$ are arranged in such a way that $k$ is constant along verticals and decreases from left to right.
[@2010arXiv1005.5222D Theorem 4.1] \[theorem:degeneration\] For $a,b\in A$, $b$ is the image of $a$ under an endomorphism of $A$ if and only if $I(b)\subset I(a)$.
Given $x=(v,k)\in P_\lambda$, let $e(x)$ denote the element in $A$ all of whose entries are zero except for the left-most entry with $\lambda_i=k$, which is $p^v$. For an order ideal $I$ in $P_\lambda$ denote by $\max I$ the set of maximal elements in $I$ and let $$a(I)=\sum_{x\in \max I} e(x).$$ Let $G$ denote the group of all automorphisms of $A$.
[@2010arXiv1005.5222D Theorem 5.4] \[theorem:orbits\] The map $I\mapsto a(I)$ gives rise to a bijection from the set of order ideals in $P_\lambda$ to the set of $G$-orbits in $A$.
The elements $a(I)$, as $I$ varies over the order ideals in $P_\lambda$, can be taken as representatives of the orbits. The inverse of the function of Theorem \[theorem:orbits\] is given by $a\mapsto I(a)$.
Characteristic subgroups {#section:char-sub}
------------------------
For an order ideal $I\subset P_\lambda$ $$A_I=\{a\in A|I(a)\subset I\}$$ is a characteristic subgroup of $A$ of order $p^{[I]}$, where $[I]$ denotes the number of elements in $I$, counted with multiplicity (the multiplicity of $(v,k)$ is the number of times that $k$ occurs in the partition $\lambda$, see [@2010arXiv1005.5222D Theorem 7.3]). Every characteristic subgroup of $A$ is of the form $A_I$ for some order ideal $I\subset P_\lambda$. In fact, $I\mapsto A_I$ is an isomorphism of the lattice of order ideals in $P_\lambda$ onto the lattice of characteristic subgroups of $A$. Thus, the lattice of characteristic subgroups of $A$ is a finite distributive lattice [@MR1442260 Section 3.4]. If $B$ is any group isomorphic to $A$, and $\phi:A\to B$ is an isomorphism, then since $A_I$ is characteristic, the image $B_I=\phi(A_I)$ does not depend on the choice of $\phi$. Consequently, it makes sense to talk of the subgroup $\hat A_I$ of $\hat A$, which is the image of $A_I$ under any isomorphism $A\to \hat A$.
$$\xymatrix{
\begin{xy}
(0,0)*{\bullet};
(5,5)*{\bullet}**@{-};
(0,10)*{\bullet}**@{-};
(5,15)*{\bullet}**@{-};
(0,20)*{\circ}**@{-};
(5,25)*{\circ}**@{-};
(0,30)*{\circ}**@{-};
(5,35)*{\circ}**@{-};
(0,40)*{\circ}**@{-};
(5,5)*{\circ};
(20,20)*{\circ}**@{-};
(5,35)*{\circ}**@{-};
\end{xy}
&\ar[r]^{\text{flip}}&&
\begin{xy}
(0,0)*{\circ};
(5,5)*{\circ}**@{-};
(0,10)*{\circ}**@{-};
(5,15)*{\circ}**@{-};
(0,20)*{\circ}**@{-};
(5,25)*{\bullet}**@{-};
(0,30)*{\bullet}**@{-};
(5,35)*{\bullet}**@{-};
(0,40)*{\bullet}**@{-};
(5,5)*{\circ};
(20,20)*{\circ}**@{-};
(5,35)*{\circ}**@{-};
\end{xy}\\
I \text{ (black dots)} &&& I^\perp \text{ (white dots)}
}$$
For each order ideal $I\subset P_\lambda$, its annihilator $$A_I^\perp:=\{\chi\in \hat A: \chi(a)=1 \text{ for all } a\in A_I\}$$ is a characteristic subgroup of $\hat A$. Therefore, there exists an order ideal $I^\perp\subset P_\lambda$ such that $A_I^\perp=\hat A_{I^\perp}$. Clearly, $I\mapsto I^\perp$ is an order reversing involution of the set of order ideals in $P_\lambda$. The Hasse diagram of $P_\lambda$ has a horizontal axis of symmetry. $I^\perp$ can be visualized as the complement of the reflection of $I$ about this axis of $I$ (see Figure \[fig:2\]).
Symplectic orbits {#sec:symplectic-orbits}
-----------------
\[theorem:symplectic-orbits\] The map $I\mapsto (a(I),0)$ (here $0$ denotes the identity element of $\hat A$) gives rise to a bijection from the set of order ideals in $P_\lambda$ to the set of ${\mathrm{Sp}}(K)$-orbits in $K$.
We first show that each ${\mathrm{Sp}}(K)$-orbit in $K$ intersects $A\times\{0\}$. Let $e_1,\ldots,e_l$ denote the generators of $A$, so $e_i$ is the element whose $i$th coordinate is $1$ and all other coordinates are $0$. Each element $a\in A$ has an expansion $$\label{eq:4}
a=a_1e_1+\cdots+a_le_l \text{ with } 0\leq a_i< p^{\lambda_i} \text{ for each } i\in \{1,\ldots,l\}$$ Let $\epsilon_j$ denote the unique element of $\hat A$ for which $$\epsilon_j(e_k)=e^{2\pi i \delta_{j k} p^{-\lambda_j}}.$$ Then each element $\alpha\in \hat A$ has an expansion $$\label{eq:5}
\alpha=\alpha_1\epsilon_1+\cdots+\alpha_l\epsilon_l \text{ with } 0\leq \alpha_i< p^{\lambda_i} \text{ for each } i\in \{1,\ldots,l\}.$$ Let $k=(a,\alpha)\in K$, with $a$ and $\alpha$ as in (\[eq:4\]) and (\[eq:5\]) respectively. The automorphism of $K$ which takes $e_i\mapsto \epsilon_i$ and $\epsilon_i\mapsto -e_i$ while preserving all other generators $e_j$ and $\epsilon_j$ with $j\neq i$, lies in ${\mathrm{Sp}}(K)$. In terms of coordinates, it has the effect of interchanging $a_i$ and $\alpha_i$ up to sign. Using this automorphism, we may arrange that $v(a_i)\leq v(\alpha_i)$ for each $i$. Therefore, there exists an integer $b_i$ such that $b_i a_i\equiv \alpha_i\mod p^{\lambda_i}$. Let $B_i:A\to \hat A$ be the homomorphism which takes $e_i$ to $b_i\epsilon_i$ and all other generators $e_j$ with $j\neq i$ to $0$. Then the automorphism of $K$ which takes $(a,\alpha)$ to $(a,\alpha-B_i(a))$ also lies in ${\mathrm{Sp}}(K)$. This has the effect of changing $\alpha_i$ to $0$. Repeating this process for each $i$ allows us to reduce $(a,\alpha)$ to $(a,0)$ as claimed.
Now, for every automorphism $g$ of $A$, the automorphism $(a,\alpha)\mapsto (g(a),\hat g^{-1}(\alpha))$ lies in ${\mathrm{Sp}}(K)$ (here $\hat g$ is the automorphism of $\hat A$ defined by $\hat g(\chi)(a)=\chi(g(a))$ for $a\in A$ and $\chi \in \hat A$). Such automorphisms can be used to reduce $(a,0)$ further to an element of the form $(a(I),0)$ for some order ideal $I\subset P_\lambda$. Since, for distinct $I$’s, these elements are in distinct $\operatorname{Aut}(K)$-orbits, they must also be in distinct ${\mathrm{Sp}}(K)$-orbits.
Multiplicity one {#sec:mult-one}
================
Relation to commutativity {#sec:relat-comm}
-------------------------
Suppose that the decomposition of the Weil representation onto irreducible representations is given by $$\label{eq:6}
L^2(A)=\bigoplus_{\pi \in \widehat{{\mathrm{Sp}}(K)}} m_\pi {\mathcal H}_\pi.$$ A standard application of Schur’s lemma to (\[eq:6\]) implies that $m_\pi\leq 1$ if and only if the ring $\operatorname{End}_{{\mathrm{Sp}}(K)}L^2(A)$ of endomorphisms of the Weil representations is commutative. For each order ideal $I\subset P_\lambda$ let $O_I$ denote the ${\mathrm{Sp}}(K)$-orbit of $(a(I),0)$ in $K$ and let $$T_I=\sum_{k\in O_I} W_k.$$ By Theorems \[theorem:orbit-mult\] and \[theorem:symplectic-orbits\], the set of all $T_I$ as $I$ varies over the order ideals $I\subset P_\lambda$ is a basis of $\operatorname{End}_{{\mathrm{Sp}}(K)}L^2(A)$.
Let $K_I=A_I\times \hat A_I$ ($\hat A_I$ as in Section \[section:char-sub\]) and define $$\label{eq:8}
\Delta_I=\sum_{k\in K_I} W_k.$$ The $\Delta_I$’s are obtained from the $T_I$’s by the unipotent upper-triangular transformation $$\Delta_I=\sum_{J\subset I} T_J.$$ Hence, they also form a basis of $\operatorname{End}_{{\mathrm{Sp}}(K)}L^2(A)$.
Therefore, in order to show that $m_\pi\leq 1$ for each $\pi\in \widehat{{\mathrm{Sp}}(K)}$, it suffices to show that for any two order ideals $I,J\subset P_\lambda$, $\Delta_I$ and $\Delta_J$ commute, which will follow from the calculation in Section \[sec:calculation\].
Calculation of the product {#sec:calculation}
--------------------------
\[lemma:product\] For any two order ideals $I,J\subset P_\lambda$, $$\Delta_I\Delta_J=|K_{I\cap J}|\Delta_{(I\cap J)^\perp \cap (I\cup J)}.$$
The coefficient of $W_k$ in $\Delta_I\Delta_J$ is $$\label{eq:7}
\sum_{x\in K_I,y\in K_J, x+y=k} c(x,y).$$ From the definition of $I(a)$, it is easy to see that $I(a+b)\subset I(a)\cup I(b)$. Therefore, $A_I+A_J\subset A_{I\cup J}$ and hence $K_I+K_J\subset K_{I\cup J}$. It follows that the sum (\[eq:7\]) is $0$ unless $k\in K_{I\cup J}$. Suppose $x_0\in K_I$ and $y_0\in K_J$ are such that $x_0+y_0=k$. Then the sum (\[eq:7\]) becomes $$\begin{aligned}
\sum_{l\in K_I\cap K_J} c(x_0+l,y_0-l) & = & c(x_0,y_0)\sum_{l\in K_I\cap K_J} c(l,y_0)c(l,x_0)\\
& = & c(x_0,y_0) \sum_{l\in K_I\cap K_J} c(l,k)\\
& = &
\begin{cases}
c(x_0,y_0)|K_I\cap K_J| &\text{ if } k\in (K_I\cap K_J)^\perp,\\
0 &\text{ otherwise}.
\end{cases}
\end{aligned}$$ Observe that $K_I\cap K_J=K_{I\cap J}$. It remains to show that, for every $k\in K_{I\cup J}$, there exist $x_0\in K_I$ and $y_0\in K_J$ such that $k=x_0+y_0$ and $c(x_0,y_0)=1$. Since $\Delta_I\Delta_J$ is constant on ${\mathrm{Sp}}(K)$-orbits in $K$, we may use Theorem \[theorem:symplectic-orbits\] to assume that $k=(a(I'),0)$ for some order ideal $I'\subset I\cup J$. We have $\max I'\subset I\cup J$. Let $I'_1$ be the order ideal generated by $(\max I')\cap I$, and $I'_2$ be the order ideal generated by $(\max I')-I$. Then $a(I')=a(I'_1)+a(I'_2)$. Clearly $(a(I'_1),0)$ and $(a(I'_2),0)$ have the properties required of $x_0$ and $y_0$.
Multiplicity one {#multiplicity-one}
----------------
We have proved
\[theorem:multiplicity-free\] In the decomposition (\[eq:6\]) of the Weil representation of ${\mathrm{Sp}}(K)$, $m_\pi\leq 1$ for every isomorphism class $\pi$ of irreducible representations of ${\mathrm{Sp}}(K)$.
Every ${\mathrm{Sp}}(K)$-invariant subspace is completely determined by the subset of $\widehat{{\mathrm{Sp}}(K)}$ consisting of representations that occur in it. Therefore
\[cor:Boolean-lattice\] The set of ${\mathrm{Sp}}(K)$-invariant subspaces of $L^2(A)$, partially ordered by inclusion, forms a finite Boolean lattice.
Elementary invariant subspaces {#sec:construction-invariant-subs}
==============================
In this section, we construct some elementary invariant subspaces for the Weil representation of ${\mathrm{Sp}}(K)$ on $L^2(A)$. In Section \[sec:poset-invar-sub\], we will use these subspaces and the results of Section \[sec:components\] to construct enough invariant subspaces to carve out all the irreducible subspaces.
Small order ideals
------------------
\[Small order ideal\] \[defn:small-ideal\] An order ideal $I\subset P_\lambda$ is said to be small if $I\subset I^\perp$, with $I^\perp$ as in Section \[section:char-sub\].
For example, the order ideal $I$ in Figure \[fig:2\] is small.
Interpreting some $\Delta_I$’s
------------------------------
\[lemma:Delta-I\] For each order ideal $I\subset P_\lambda$, let $\Delta_I$ be as in (\[eq:8\]).
1. \[item:2\]$\Delta_{P_\lambda}f(u)=|A|f(-u)$ for all $f\in L^2(A)$ and $u\in A$.
2. \[item:1\]For every small order ideal $I\subset P_\lambda$, $|A_I|^{-2}\Delta_I$ is the orthogonal projection onto the subspace of $L^2(A)$ consisting of functions supported on $A_{I^\perp}$ and invariant under translations in $A_I$.
For any order ideal $I\subset P_\lambda$, we have $$\Delta_I f(u) = \sum_{x\in A_I} \sum_{\chi \in \hat A_I} \chi(u-x/2) f(u-x).$$ The inner sum is $f(u-x)$ times the sum of values of a character of $\hat A_I$, which vanishes if this character is non-trivial, namely if $u-x/2\notin A_{I^\perp}$, and is $|A_I|$ otherwise. Therefore, $$\begin{aligned}
\nonumber
\Delta_If(u) & = & |A_I|\sum_{x\in A_I\cap (2u+A_{I^\perp})} f(u-x)\\
\label{eq:17}
& = & |A_I|\sum_{x\in (u+A_I)\cap (-u+A_{I^\perp})} f(x).
\end{aligned}$$ Taking $I=P_\lambda$ in (\[eq:17\]) gives \[item:2\].
Now suppose that $I\subset I^\perp$. If $u\notin A_{I^\perp}$ then $(u+A_I)\cap (-u+A_{I^\perp})=\emptyset$, so that $\Delta_If(u)=0$. If $u\in A_{I^\perp}$, then the sum (\[eq:17\]) is over $u+A_I$, so $|A_I|^{-2}\Delta_I$ is the averaging over $A_I$-cosets from which \[item:1\] follows.
Even and odd functions {#sec:even-odd-functions}
----------------------
\[theorem:even-odd\] The subspaces of $L^2(A)$ consisting of even and odd functions are invariant under ${\mathrm{Sp}}(K)$.
By \[item:2\], $$\label{eq:22}
[(\mathrm{id}_{L^2(A)}\pm |A|^{-1}\Delta_{P_\lambda})/2] f(u)=(f(u)\pm f(-u))/2.$$ These operators are the orthogonal projections onto the subspaces of even and odd functions in $L^2(A)$. Since these operators commute with ${\mathrm{Sp}}(K)$ (by Theorem \[theorem:orbit-mult\]), their images are ${\mathrm{Sp}}(K)$-invariant subspaces of $L^2(A)$.
\[The Weil representation is an ordinary representation\] \[remark:ordinary-rep\] The subspaces of even and odd functions on $A$ have dimensions $(|A|+1)/2$ and $(|A|-1)/2$ respectively. For each $g\in {\mathrm{Sp}}(K)$, let $W_g^+$ and $W_g^-$ denote the restrictions of $W_g$ to these spaces. Taking the determinants of the identities $$W_{g_1}^\pm W_{g_2}^\pm=c(g_1,g_2)W_{g_1 g_2}^\pm$$ gives the identities $$\begin{gathered}
\det W_{g_1}^+\det W_{g_2}^+=c(g_1,g_2)^{(|A|+1)/2}\det W_{g_1g_2}^+\\
\det W_{g_1}^-\det W_{g_2}^-=c(g_1,g_2)^{(|A|-1)/2}\det W_{g_1g_2}^-.
\end{gathered}$$ Dividing the first equation by the second and rearranging gives: $$c(g_1,g_2)=\frac{\alpha(g_1)\alpha(g_2)}{\alpha(g_1g_2)}$$ for all $g_1,g_2\in {\mathrm{Sp}}(K)$, when $\alpha:G\to U(1)$ is defined by $$\alpha(g)=\det(W_g^+)/\det(W_g^-) \text{ for all } g\in {\mathrm{Sp}}(K).$$ Therefore, if each $W_g$ is replaced by $\alpha(g)^{-1}W_g$, then $g\mapsto W_g$ is a representation of ${\mathrm{Sp}}(K)$ on $L^2(A)$. This argument seems to be well known. It has appeared before in Adler-Ramanan [@MR1621185 Appendix I], and again in Cliff-McNeilly-Szechtman [@MR1783635].
Invariant spaces corresponding to small order ideals {#sec:invar-spac-corr}
----------------------------------------------------
Since $\Delta_I$ commutes with ${\mathrm{Sp}}(K)$, its image is an ${\mathrm{Sp}}(K)$-invariant subspace of $L^2(A)$. An immediate consequence of \[item:1\] is the following theorem:
\[theorem:characteristic-invariant\] For each small order ideal $I\subset P_\lambda$, the subspace of $L^2(A)$ consisting of functions supported on $A_{I^\perp}$ which are invariant under translations in $A_I$ is an ${\mathrm{Sp}}(K)$-invariant subspace of $L^2(A)$.
\[Alternative description\] For $f\in L^2(A)$ recall that its Fourier transform is the function on $\hat A$ defined by $$\hat f(\chi)=\sum_{a\in A} f(a)\overline{\chi(a)} \text{ for each } \chi\in \hat A.$$ For any subgroup $B$ of $A$, the Fourier transforms of functions invariant under translations in $B$ are the functions supported on the annihilator subgroup $B^\perp$ of $A$ (consisting of characters which vanish on $B$), and Fourier transforms of functions supported on $B$ are the functions which are invariant under $B^\perp$. Therefore, the functions supported on $A_{I^\perp}$ which are invariant under $A_I$ are precisely the functions supported on $A_{I^\perp}$ whose Fourier transforms are supported on $\hat A_{I^\perp}$. They are also the functions invariant under translations in $A_I$ whose Fourier transforms are invariant under translations in $\hat A_I$. \[remark:alternative\]
Identify $L^2(A_{I^\perp}/A_I)$ with the space of functions in $L^2(A)$ which are supported on $A_{I^\perp}$ and invariant under translations in $A_I$. Let $K(I)=A_{I^\perp}/A_I\times \widehat{A_{I\perp}/A_I}$. $K(I)$ can be identified with $(A_I{^\perp}\times \hat A_{I^\perp})/(A_I\times \hat A_I)$. Thus $K(I)$ is a quotient of one characteristic subgroup of $K$ by another. Therefore the action ${\mathrm{Sp}}(K)$ on $K$ descends to an action on $K(I)$ giving rise to a homomorphism ${\mathrm{Sp}}(K)\to {\mathrm{Sp}}(K(I))$. The defining condition (\[eq:1\]) for the Weil representation ensures:
\[theorem:invar-I\] For every small order ideal $I\subset P_\lambda$, the Weil representation of ${\mathrm{Sp}}(K)$ on $L^2(A_{I^\perp}/A_I)$ is projectively equivalent to the representation obtained by composing the Weil representation of ${\mathrm{Sp}}(K(I))$ on $L^2(A_{I^\perp}/A_I)$ with the homomorphism ${\mathrm{Sp}}(K)\to {\mathrm{Sp}}(K(I))$.
Component decomposition {#sec:components}
=======================
Connected components of a [partially ordered set]{}
---------------------------------------------------
A [partially ordered set]{} is said to be connected if its Hasse diagram is a connected graph. A connected component of a [partially ordered set]{} is a maximal connected induced subposet. Every [partially ordered set]{} can be written as the disjoint union of its connected components in the sense of [@MR1442260 Section 3.2]. Denote the set of connected components of a poset $P$ by $\pi_0(P)$.
Connected components of $J-I$ {#sec:conn-comp-j-i}
-----------------------------
Suppose that $I\subset J$ are two order ideals in $P_\lambda$. Each connected component $C\in \pi_0(J-I)$ determines a segment (namely, a contiguous set of integers) $S_C$ in $\{1,\ldots,l\}$: $$S_C=\{1\leq k\leq l|(v,k)\in C\text{ for some } v\}.$$ The $S_C$’s are pairwise disjoint, but their union may be strictly smaller than $\{1,\ldots,l\}$. Write $S_0$ for the complement of $\coprod_{C\in \pi_0(I^\perp-I)}S_C$ in $\{1,\ldots,l\}$. It will be convenient to write $\tilde \pi_0(I^\perp-I)=\pi_0(I^\perp-I)\coprod \{0\}$. Define partitions $\lambda(C)=(\lambda_k|k\in S_C)$ for each $C\in \tilde \pi_0(I^\perp-I)$. Then $P_{\lambda(C)}$ is the induced subposet of $P_\lambda$ consisting of those pairs $(v,k)\in P_\lambda$ for which $k\in S_C$. Let $I(C)$ and $J(C)$ be the ideals in $P_{\lambda(C)}$ obtained by intersecting $I$ and $J$ respectively with $P_{\lambda(C)}$.
For example, if $\lambda=(5,4,4,1)$ and $I$ is the order ideal in the diagram on the left in Figure \[fig:2\] and $J=I^\perp$, then $I^\perp-I$ is depicted in the diagram on the left in Figure \[fig:3\]. As the diagram on the right shows, the induced subposet $I^\perp-I$ has two connected components, $C_1$ and $C_2$, with $\lambda(C_1)=(5)$ and $\lambda(C_2)=(1)$. Moreover, $\lambda(0)=(4,4)$.
$$\begin{array}{ccc}
\begin{xy}
(0,0)*{\circ};
(5,5)*{\circ}**@{-};
(0,10)*{\circ}**@{-};
(5,15)*{\circ}**@{-};
(0,20)*{\bullet}**@{-};
(5,25)*{\circ}**@{-};
(0,30)*{\circ}**@{-};
(5,35)*{\circ}**@{-};
(0,40)*{\circ}**@{-};
(5,5)*{\circ};
(20,20)*{\bullet}**@{-};
(5,35)*{\circ}**@{-};
\end{xy}
&\quad\quad\quad
&
\begin{xy}
(0,0)*{};
(0,20)*{\bullet}; (0,40)*{};
(20,20)*{\bullet};
\end{xy}\\
I^\perp-I\text{ inside } P_{5,4,4,1} & & I^\perp-I\text{ by itself}
\end{array}$$
\[lemma:ideals-components\] Let $I\subset J$ be two order ideals in $P_\lambda$. For each $C\in \pi_0(J-I)$ let $L(C)$ be an order ideal in $C$. Let $$L=I\coprod \Big(\coprod_{C\in \pi_0(J-I)} L(C)\Big).$$ Then $L$ is an order ideal in $P_\lambda$.
Since $\coprod_{C\in \pi_0(J-I)} L(C)$ is an order ideal in $J-I$, its union with $I$ is an order ideal in $P_\lambda$.
\[cor:ideal-components\] If $I\subset J$ are two order ideals in $P_\lambda$ and $C$ and $D$ are distinct components of $J-I$, then the intersection with $P_{\lambda(C)}$ of the order ideal in $P_\lambda$ generated by $J(D)$ is contained in $I(C)$.
By Lemma \[lemma:ideals-components\], $J(D)\cup I$ is an order ideal in $P_\lambda$. Therefore, it contains the order ideal in $P_\lambda$ generated by $J(D)$. If $C$ and $D$ are distinct connected components of $J-I$, then $(J(D)\cup I)\cap P_\lambda(C)=I(C)$. Therefore the intersection with $P_{\lambda(C)}$ of the order ideal in $P_\lambda$ generated by $J(D)$ is contained in $I(C)$.
Decomposition of endomorphisms {#sec:deco-endo}
------------------------------
Suppose that $A$ has the form (\[eq:11\]). Then define $A_C$ to be the subgroup $$A_C=\{(a_1,\ldots,a_l)|a_k=0 \text{ if } k\notin S_C\}.$$ Thus $A_C$ is a finite abelian $p$-group of type $\lambda(C)$. We have a decomposition $$\label{eq:18}
A=\prod_{C\in \tilde \pi_0(J-I)} A_C.$$
Denote the characteristic subgroups of $A_C$ corresponding to $I(C)$ and $J(C)$ (which are order ideals in $P_{\lambda(C)}$) by $A_{I,C}$ and $A_{J,C}$ respectively. The decomposition (\[eq:18\]) induces a decomposition $$\label{eq:19}
A_J/A_I=\prod_{C\in \pi_0(J-I)} A_{J,C}/A_{I,C}.$$ There is no contribution from $A_0$ since $A_{I,0}=A_{J,0}$.
With respect to the decomposition (\[eq:18\]), every endomorphism of $A$ can be written as a square matrix $\{\phi_{CD}\}$, where $\phi_{CD}:A_D\to A_C$ is a homomorphism.
\[lemma:sum-endo\] Let $I\subset J$ be order ideals in $P_\lambda$. Then every endomorphism $\phi$ of $A$ induces an endomorphism $$\bar \phi: A_J/A_I\to A_J/A_I$$ such that $\bar\phi(A_{J,C}/A_{I,C})\subset A_{J,C}/A_{I,C}$ for each $C\in \pi_0(J-I)$, and $$\bar \phi= \bigoplus_{C\in \pi_0(J-I)} \overline{\phi_{CC}},$$ where $\overline{\phi_{CC}}$ is the endomorphism of $A_{J,C}/A_{I,C}$ induced by $\phi_{CC}$.
By Theorem \[theorem:degeneration\] and Lemma \[cor:ideal-components\], if $C\neq D$ then $\phi_{CD}(A_{J,D})\subset A_{I,C}$. Therefore, $\bar \phi$ remains unchanged if $\phi_{CD}$ is replaced by $0$ for all $C\neq D$. This amounts to replacing $\phi$ by $\oplus_C \phi_{CC}$ and the lemma follows.
Tensor product decomposition of invariant subspaces
---------------------------------------------------
Let $I$ be a small order ideal. We shall use the notation of Section \[sec:deco-endo\] with $J=I^\perp$. For each $C\in \pi_0(I^\perp-I)$ let $K_C=A_C\times \hat A_C$, and let ${\mathrm{Sp}}(K_C)$ be the corresponding symplectic group. Just as (by Theorem \[theorem:characteristic-invariant\]) $L^2(A_{I^\perp}/A_I)$ is an invariant subspace for the Weil representation of ${\mathrm{Sp}}(K)$ on $L^2(A)$, $L^2(A_{I^\perp,C}/A_{I,C})$ is an invariant subspace for the Weil representation of ${\mathrm{Sp}}(K_C)$ on $L^2(A_C)$.
Now, if $g\in {\mathrm{Sp}}(K)$, we may write $g=
\left(\begin{smallmatrix}
g_{11}& g_{12}\\g_{21}&g_{22}
\end{smallmatrix}\right)
$ with respect to the decomposition $K=A\times \hat A$. For convenience, we identify $\hat A$ with $A$ using $e_i\mapsto \epsilon_i$ for $i=1,\ldots,l$, where $e_i$ and $\epsilon_i$ are as in Section \[sec:symplectic-orbits\]. Hence, we may think of each $g_{ij}$ as an endomorphism of $A$. By Lemma \[lemma:sum-endo\], the resulting endomorphism $\bar g_{ij}$ of $A_{I^\perp}/A_I$ preserves $A_{I^\perp,C}/A_{I,C}$ for each $C$. It follows that the image of ${\mathrm{Sp}}(K)$ in ${\mathrm{Sp}}(K(I))$ (see Section \[sec:invar-spac-corr\]) is the product of the images of the ${\mathrm{Sp}}(K_C)$’s in the ${\mathrm{Sp}}(K_C(I\cap C))$’s as $C$ ranges over $\pi_0(I^\perp-I)$.
Thus, by Theorems \[theorem:product-Weil\] and \[theorem:invar-I\],
\[cor:component-decomposition\] The Weil representation of ${\mathrm{Sp}}(K)$ on $L^2(A_{I^\perp}/A_I)$ is projectively equivalent to the tensor product of the Weil representations of ${\mathrm{Sp}}(K_C)$ on $L^2(A_{I^\perp,C}/A_{I,C})$ as $C$ ranges over $\pi_0(I^\perp-I)$.
Poset of invariant subspaces {#sec:poset-invar-sub}
============================
The invariant subspaces
-----------------------
Let $$Q_\lambda=\big\{(I,\phi)|I\subset P_\lambda\text{ a small order ideal},\;\phi:\pi_0(I^\perp-I)\to {\mathbf Z}/2{\mathbf Z}\text{ any function}\big\}.$$ For each $(I,\phi)\in Q_\lambda$ use the decomposition of Corollary \[cor:component-decomposition\] to define $L^2(A)_{I,\phi}$ as the subspace of $L^2(A_{I^\perp}/A_I)$ given by $$L^2(A)_{I,\phi}=\bigotimes_{C\in \pi_0(I^\perp-I)} L^2(A_{I^\perp,C}/A_{I,C})_{\phi(C)}$$ where $L^2(A_{I^\perp,C}/A_{I,C})_{\phi(C)}$ denotes the space of even or odd functions on $A_{I^\perp,C}/A_{I,C}$ when $\phi(C)$ is $0$ or $1$ respectively. In other words, $L^2(A)_{I,\phi}$ consists of functions on $A_{I^\perp}/A_I$ which, under the decomposition $$\label{eq:20}
A_{I^\perp}/A_I=\prod_{C\in \pi_0(I^\perp-I)} A_{I^\perp,C}/A_{I,C}$$ are even in the components where $\phi(C)=0$ and odd in the components where $\phi(C)=1$. By Theorems \[theorem:even-odd\] and \[theorem:characteristic-invariant\], and by Corollary \[cor:component-decomposition\], $L^2(A)_{I,\phi}$ is an ${\mathrm{Sp}}(K)$-invariant subspace of $L^2(A)$ for each $(I,\phi)\in Q_\lambda$.
The partial order {#sec:partial-order}
-----------------
Clearly,
\[lemma:poset\] For $(I,\phi)$ and $(I',\phi')$ in $Q_\lambda$, $L^2(A)_{I',\phi'}\subset L^2(A)_{I,\phi}$ if and only if the following conditions are satisfied:
1. \[item:3\] $I\subset I'$.
2. \[item:4\] For each $P\in \pi_0(I^\perp-I)$, $$\phi(P)=\sum_{P'\in \pi_0(I^{\prime\perp}-I'),P'\subset P}\phi'(P').$$
Thus the conditions \[item:3\] and \[item:4\] define a partial order on $Q_\lambda$ (which is obviously independent of $p$). Recall that the multiplicity $m(x)$ of an element $x=(v,k)\in P_\lambda$ is the number of times $k$ occurs in the partition $\lambda$. For any subset $S\subset P_\lambda$, let $[S]$ denote the number of elements of $S$, counted with multiplicity: $$[S]=\sum_{x\in S} m(x).$$
\[lemma:dimension\] For each $(I,\phi)\in Q_\lambda$, $$\dim L^2(A)_{I,\phi}=\prod_{C\in \pi_0(I^\perp-I)}\frac{p^{[C]}+(-1)^{\phi(C)}}2.$$
Irreducible subspaces {#sec:parametrization}
=====================
A bijection between $J(P_\lambda)$ and $Q_\lambda$ {#sec:biject-betw-p_lambda}
--------------------------------------------------
Let $J(P_\lambda)$ denote the lattice of order ideals in $P_\lambda$.
\[lemma:cardinalities\] For each partition $\lambda$, $|J(P_\lambda)|=|Q_\lambda|$.
We construct an explicit bijection $Q_\lambda\to J(P_\lambda)$. To $(I,\phi)\in Q_\lambda$, associate the ideal (see Lemma \[lemma:ideals-components\]) $$\Theta(I,\phi)=I\bigcup\Bigg(\coprod_{C\in \pi_0(I^\perp-I)} I_{\phi(C)}\Bigg)$$ where $$I_{\phi(C)}=
\begin{cases}
I\cap C & \text{ if } \phi(C)=0\\
I^\perp\cap C & \text{ if } \phi(C)=1.
\end{cases}$$ In the other direction, given an ideal $J\subset P_\lambda$, $I=J\cap J^\perp$ is a small order ideal. We have $I^\perp=J\cup J^\perp$. For each $C\in \pi_0(I^\perp-I)$ define $$\phi_J(C)=
\begin{cases}
0 & \text{ if } I\cap C=J\cap C,\\
1 & \text{ if } I\cap C=J^\perp\cap C.
\end{cases}$$ Define $\Psi:Q_\lambda\to J(P_\lambda)$ by $\Psi(J)=(J\cap J^\perp,\phi_J)$ where $\phi_J$. It is easy to verify that $\Phi$ and $\Psi$ are mutual inverses.
Existence lemma
---------------
\[lemma:existence\] For every $(I,\phi)\in Q_\lambda$, there exists $f\in L^2(A)_{I,\phi}$ such that $f\notin L^2(A)_{I',\phi'}$ for any $(I',\phi')<(I,\phi)$.
Take as $f$ the unique element in $L^2(A)_{I,\phi}$ whose value at $a(I^\perp)+A_I$ (using the notation of Section \[sec:orbits\]) is $1$, and which vanishes on all elements of $A_{I^\perp}/A_I$ not obtained from $a(I^\perp)+A_I$ by changing the signs of some of its components under the decomposition (\[eq:20\]).
The irreducible invariant subspaces {#sec:irred-invar-subsp}
-----------------------------------
The two lemmas above are enough to give us the main theorem:
\[theorem:main\] For each $(I,\phi)\in Q_\lambda$, there is a unique irreducible subspace for the Weil representation of ${\mathrm{Sp}}(K)$ on $L^2(A)$ which is contained in $L^2(A)_{I,\phi}$ but not $L^2(A)_{I',\phi'}$ for any $(I',\phi')<(I,\phi)$. As $p$ varies, the dimension of this representation is a polynomial in $p$ of degree $[I^\perp-I]$ with leading coefficient $2^{-|\pi_0(I^\perp-I)|}$ and all coefficients in ${\mathbf Z}_{(2)}$.
By Corollary \[cor:Boolean-lattice\], the ${\mathrm{Sp}}(K)$-invariant subspaces of $L^2(A)$ form a Boolean lattice $\Lambda$. Let $R$ denote the set of minimal non-trivial ${\mathrm{Sp}}(K)$-invariant subspaces of $L^2(A)$. These are the atoms of $\Lambda$. By Corollary \[cor:orbit-mult\] and Theorem \[theorem:symplectic-orbits\] the cardinality of $R$ is the same as that of $J(P_\lambda)$. Each invariant subspace is determined by the atoms which are contained in it. The map $(I,\phi)\mapsto L^2(A)_{I,\phi}$ is an order-preserving map $Q_\lambda\to \Lambda$. Let $R_{I,\phi}$ be the set of atoms which occur in $L^2(A)_{I,\phi}$ but not in $L^2(A)_{I',\phi'}$ for any $(I',\phi')<(I,\phi)$. The subsets $R_{I,\phi}$ are $|Q_\lambda|$ pairwise disjoint subsets of $R$, and by Lemma \[lemma:existence\], each of them is non-empty. Therefore, by Lemma \[lemma:cardinalities\], each of them must be singleton and these subspaces exhaust $R$. It follows that there is a unique irreducible representation of ${\mathrm{Sp}}(K)$ that occurs in $L^2(A)_{I,\phi}$ but not in $L^2(A)_{I',\phi'}$ for any $(I',\phi')<(I,\phi)$. Let $V_{I,\phi}$ denote this irreducible subspace.
By Lemma \[lemma:dimension\], $$\sum_{(I',\phi')\leq (I,\phi)}\dim V_{I',\phi'} = \prod_{P\in \pi_0(I^\perp-I)} \frac{p^{[C]}+(-1)^{\phi(C)}}2.$$ By the Möbius inversion formula [@MR1442260 Section 3.7], $$\label{eq:10}
\dim V_{I,\phi}=\sum_{(I',\phi')\leq (I,\phi)} \mu((I,\phi),(I',\phi'))\prod_{C\in \pi_0(I^{\prime\perp}-I')} \frac{p^{[C]}+(-1)^{\phi(C)}}2,$$ where $\mu$ is a the Möbius function of $Q_\lambda$. Since $\mu((I,\phi),(I,\phi))=1$ and the Möbius function is integer-valued, the right hand side of (\[eq:10\]) is indeed a polynomial in $p$ with leading coefficient $2^{-|\pi_0(I^\perp-I)|}$. Clearly, the other coefficients do not have denominators other than powers of $2$.
A combinatorial lemma {#sec:combinatorial-lemma}
---------------------
\[lemma:combinatorial\] Let $P$ be a poset and $J(P)$ be its lattice of order ideals. Let $m:P\to \mathbf N$ be any function (called the multiplicity function). For each $S\subset P$ let $[S]=\sum_{x\in S} m(x)$, the elements of $S$ counted with multiplicity, and $\max S$ denote the set of maximal elements of $S$. If $\alpha:J(P)\to {\mathbf C}[t]$ is a function such that $$\sum_{J\subset I}\alpha(J)=t^{[I]} \text{ for every order ideal } I\subset P,$$ then, $$\label{eq:27}
\alpha(I)=t^{[I]}\prod_{x\in \max I}(1-t^{-m(x)}).$$
By the Möbius inversion formula for a finite distributive lattice [@MR1442260 Example 3.9.6], $$\begin{aligned}
\nonumber
\alpha(I)&=&\sum_{I-\max I\subset J\subset I}(-1)^{|I-J|}t^{[J]}\\
\label{eq:14}
&=&t^{[I]}\sum_{S\subset \max I}(-1)^{|\max I -S|}t^{-[\max I-S]}.
\end{aligned}$$ Each term in the expansion of the product $$\prod_{x\in \max I}(1-t^{-m(x)})$$ is obtained choosing a subset $S\subset \max I$ and taking $$\prod_{x\notin S} (-t^{-m(x)})=(-1)^{|\max I-S|}t^{-[\max I-S]}.$$ Therefore, the expression (\[eq:14\]) for $\alpha(I)$ reduces to (\[eq:27\]) as claimed.
Explicit formula for the dimension {#sec:formula-dimension}
----------------------------------
Recall (from Section \[sec:irred-invar-subsp\]) that for each $(I,\phi)\in Q_\lambda$, $V_{I,\phi}$ denotes the unique irreducible ${\mathrm{Sp}}(K)$-invariant subspace of $L^2(A)$ which lies in $L^2(A)_{I,\phi}$ but not in any proper subspace of the form $L^2(A)_{I',\phi'}$. We shall obtain a nice expression for $\dim V_{I,\phi}$ by applying Lemma \[lemma:combinatorial\] to the induced subposet of $P_\lambda$ given by $$P_\lambda^+=\{(v,k)\in P_\lambda:v< (k-1)/2\}.$$ For each small order ideal $I\subset P_\lambda$, let $I^+=I^\perp\cap P_\lambda^+$. Then $I\mapsto I^+$ is an order reversing isomorphism from the [partially ordered set]{} of small order ideals in $P_\lambda$ to the [partially ordered set]{} $J(P_\lambda^+)$ of all order ideals in $P_\lambda^+$.
Let $$\label{eq:21}
V_I=\bigoplus_{\phi:\pi_0(I^\perp-I)\to {\mathbf Z}/2{\mathbf Z}} V_{I,\phi}.$$ Denote by $V_I^0$ and $V_I^1$ the subspaces of even or odd functions in $V_I$ respectively.
\[theorem:connected-dim\] If $I\subset P_\lambda$ is a small order ideal, then for $\epsilon\in \{0,1\}$, $$\dim V_I^\epsilon=
\begin{cases}
(p^{[I^\perp-I]}+(-1)^\epsilon)/2 & \text{ if } I^+= \emptyset,\\
p^{[I^\perp-I]}\prod_{x\in \max I^+}(1-p^{-2m(x)})/2 & \text{ otherwise.}
\end{cases}$$
Suppose $I\subset P_\lambda$ is a small order ideal. Then, $$\label{eq:16}
L^2(A_{I^\perp}/A_I)=\bigoplus_{J\supset I,\;J\text{ small}} V_J=\bigoplus_{J^+\subset I^+}V_J.$$ Define $\alpha:J(P_\lambda^+)\to {\mathbf C}$ by $\alpha(J^+)=\dim V_J$. Comparing dimensions $$\label{eq:15}
\sum_{J^+\subset I^+} \alpha(J^+)=p^{[I^\perp-I]}.$$ Let $E=\{(v,k)\in I^\perp-I|v=(k-1)/2\}$, the set of points in $I^\perp-I$ which lie on its axis of symmetry. Then $[I^\perp-I]=[E]+2[I^+]$. Therefore (\[eq:15\]) becomes $$\sum_{J^+\subset I^+} \alpha(J^+)=p^{[E]}p^{2[I^+]}.$$ Taking $P=P_\lambda^+$ and setting $t=p^2$ in Lemma \[lemma:combinatorial\] gives $$\begin{aligned}
\dim V_I & = & p^{[E]+2[I^+]}\prod_{x\in \max I^+} (1-p^{-2m(x)})\\
& = & p^{[I^\perp-I]}\prod_{x\in \max I^\perp}(1-p^{-2m(x)}).
\end{aligned}$$ In order to obtain Lemma \[theorem:connected-dim\], it remains to find the dimensions of the spaces of even and odd functions in $V_I$. If $I^+=\emptyset$ then $E=I^\perp-I$. In this case, $V_{I,\phi}$ is just the set of even or odd functions in $L^2(A_{I^\perp}/A_I)$ and has dimension as claimed.
Otherwise, we proceed by induction on $I^+$. Thus assume that Lemma \[theorem:connected-dim\] holds for small order ideals $I'\supsetneq I$. The space of even functions in $L^2(A_{I^\perp}/A_I)$ has dimension one more than the space of odd functions. Breaking up the spaces in (\[eq:16\]) into even and odd functions, we see this difference is accounted for by the summand corresponding to $J^+=\emptyset$, as discussed above. By the induction hypothesis, the dimensions of even and odd parts of the summands corresponding to $\emptyset\subsetneq J^+\subsetneq I^+$ are equal. Therefore, the even and odd parts of $V_I$ must have the same dimension.
\[theorem:dim\] If $I\subset P_\lambda$ is a small order ideal, then $$\dim V_{I,\phi}=\prod_{C\in \pi_0(I^\perp-I)} \dim V_{I(C),\phi(C)},$$ where, since $I(C)^\perp-I(C)$ is connected, $\dim V_{I(C),\phi(C)}$ is given by Lemma \[theorem:connected-dim\].
Examples {#sec:examples}
--------
We begin with the case $A=({\mathbf Z}/p^k{\mathbf Z})^l$, corresponding to $\lambda=(k,\ldots,k)$ (repeated $l$ times). $P_\lambda$ is then a linear order, with $k$ points. $Q_\lambda$ has two linear components, consisting of the even and odd parts. An informative way to display the decomposition of $L^2(A)$ is as the Hasse diagram of $Q_\lambda$, but with the dimension of the corresponding irreducible invariant subspace in place of each vertex. In this case we get $$\xymatrix{
\frac{p^{lk}(1-p^{-2})}2 \ar@{-}[d] & \frac{p^{lk}(1-p^{-2})}2 \ar@{-}[d]\\
\frac{p^{l(k-2)}(1-p^{-2})}2 \ar@{-}& \frac{p^{l(k-2)}(1-p^{-2})}2 \ar@{-}\\
\vdots & \vdots\\
\frac{p^{(k-2(\lfloor k/2\rfloor-1))}(1-p^{-2})}2\ar@{-}[d] & \frac{p^{(k-2(\lfloor k/2\rfloor-1))}(1-p^{-2})}2\ar@{-}[d]\\
\frac{p^{(k-2\lfloor k/2 \rfloor)}+1}2 & \frac{p^{(k-2\lfloor k/2 \rfloor)}-1}2
}$$ The entry at the bottom right is zero when $k$ is even and should be omitted. This is consistent with the previously known results of Prasad [@MR1478492] and Cliff-McNeilly-Szechtman [@MR1783635]. The picture for $\lambda=(2,1)$ is same as that for $\lambda=(3)$. Perhaps the simplest non-trivial example is $\lambda=(3,1)$ (it is the smallest example where $J(P_\lambda)$ is not a chain). We get $$\begin{array}{cc}
\xymatrix{
& \frac{p^4-p^2}2 \ar@{-}[dl] \ar@{-}[d]\\
\frac{(p+1)^2}4 & \frac{(p-1)^2}4
}&
\xymatrix{
\frac{p^4-p^2}2 \ar@{-}[d] \ar@{-}[dr]&\\
\frac{p^2-1}4 & \frac{p^2-1}4
}
\end{array}$$ For $\lambda=(3,2,1)$, we get $$\begin{array}{cc}
\xymatrix{
& \frac{p^6-p^4}2\ar@{-}[d]\\
& \frac{p^4-p^2}2\ar@{-}[dl]\ar@{-}[d]\\
\frac{(p+1)^2}4&\frac{(p-1)^2}4
}
&
\xymatrix{
\frac{p^6-p^4}2\ar@{-}[d]&\\
\frac{p^4-p^2}2\ar@{-}[d]\ar@{-}[dr]&\\
\frac{p^2-1}4&\frac{p^2-1}4
}
\end{array}$$ For $\lambda=(4,2)$, we have $$\begin{array}{cc}
\xymatrix{
& \frac{p^6-p^4}2\ar@{-}[d]\\
& \frac{p^4-2p^2+1}2\ar@{-}[dl]\ar@{-}[d]\\
\frac{p^2-1}2\ar@{-}[dr]&\frac{p^2-1}2\ar@{-}[d]\\
& 1
}
&
\xymatrix{
\frac{p^6-p^4}2\ar@{-}[d]&\\
\frac{p^4-2p^2+1}2\ar@{-}[d]\ar@{-}[dr]&\\
\frac{p^2-1}2&\frac{p^2-1}2\\
&
}
\end{array}$$ For $\lambda=(4,3,2,1)$, we have [$$\begin{array}{cc}
\xymatrix{
& \frac{p^{10}-p^8}2\ar@{-}[d] &\\
& \frac{p^8-p^6}2\ar@{-}[d] &\\
& \frac{p^6-2p^4+p^2}2\ar@{-}[d]\ar@{-}[dl]\ar@{-}[dr] &\\
\frac{p^4-p^2}2\ar@{-}[d]\ar@{-}[dr]&\frac{(p^3-p)(p+1)}4\ar@{-}[dl] &\frac{(p^3-p)(p-1)}4\ar@{-}[dl]\\
\frac{(p+1)^2}4 & \frac{(p-1)^2}4&
}
&
\xymatrix{
& \frac{p^{10}-p^8}2\ar@{-}[d] &\\
& \frac{p^8-p^6}2\ar@{-}[d] &\\
& \frac{p^6-2p^4+p^2}2\ar@{-}[d]\ar@{-}[dl]\ar@{-}[dr] &\\
\frac{(p^3-p)(p+1)}4\ar@{-}[dr]&\frac{(p^3-p)(p-1)}4 \ar@{-}[dr]&\frac{p^4-p^2}2\ar@{-}[dl]\ar@{-}[d]\\
&\frac{p^2-1}4 & \frac{p^2-1}4
}
\end{array}$$ ]{}
Projections onto the irreducible subspaces {#sec:proj-onto-irred}
------------------------------------------
For each $(I,\phi)\in Q_\lambda$, let $E_{I,\phi}$ denote the projection operator onto $V_{I,\phi}$. Recall from Lemma \[lemma:onb\] that the set of Weyl operators $$\{W_k:k\in K\}$$ is an orthonormal basis of $\operatorname{End}_{\mathbf C}L^2(A)$. Therefore, we may write $$E_{I,\phi}=\sum_{k\in K} e_k(I,\phi)W_k$$ for some scalars $e_k(I,\phi)$. The goal of this section is to show that this expansion is completely combinatorial. More precisely, by Theorem \[theorem:symplectic-orbits\], each ${\mathrm{Sp}}(K)$-orbit in $K$ corresponds to an order ideal in $P_\lambda$. We shall show that if $k$ lies in the ${\mathrm{Sp}}(K)$-orbit corresponding to the order ideal $J$, then $e_k(I,\phi)$ is a polynomial in $p$ whose coefficients depend only on the combinatorial data $I$, $\phi$, and $J$.
In Section \[sec:relat-comm\] we saw that $$\{\Delta_L:L\in J(P_\lambda)\}$$ is a basis of $\operatorname{End}_{{\mathrm{Sp}}(K)}L^2(A)$. Therefore, we may write $$E_{I,\phi}=\sum_{L\subset P_\lambda} \alpha_L(I,\phi) \Delta_L,$$ for some constants $\alpha_L(I,\phi)$. If $k$ lies in the orbit corresponding to $J$ then $$e_k(I,\phi)=\sum_{L\supset J}\alpha_L(I,\phi).$$ Therefore, it suffices to show that the $\alpha_L(I,\phi)$ are polynomials in $p$ whose coefficients are determined by the combinatorial data $L$, $I$, and $\phi$ (Theorem \[theorem:alpha-qualitative\]). In fact, Theorems \[theorem:alpha-supp\] and \[theorem:alpha-exact\] compute $\alpha_L(I,\phi)$ explicitly.
To begin with, consider the case where $I^\perp-I$ is connected. If $E_I$ is the projection operator onto $V_I$ (defined by (\[eq:21\])), then by \[item:1\], $$|A|^{-1}p^{[I^\perp-I]}\Delta_I = \sum_{J^+\subset I^+} E_J.$$ Using Möbius inversion for a finite distributive lattice as in Section \[sec:combinatorial-lemma\], $$|A|E_I=\sum_{I^+-\max I^+\subset J^+\subset I^+} (-1)^{|I^+-J^+|}p^{[J^\perp-J]}\Delta_J.$$ Since $V_{I,\phi}$ consists of even or odd functions in $V_I$ (depending on whether $\phi(I^\perp-I)$ is $0$ or $1$), by (\[eq:22\]), $E_{I,\phi}$ is given by $$E_{I,\phi}=E_I(\mathrm{id}_{L^2(A)}+(-1)^{\phi(I^\perp-I)} |A|^{-1}\Delta_{P_\lambda})/2.$$ By Lemma \[lemma:product\], $$(|A|^{-1}p^{[J^\perp-J]}\Delta_J)(|A|^{-1}\Delta_{P_\lambda})=|A|^{-1}\Delta_{J^\perp}.$$ Therefore when $I^\perp-I$ is connected $$\label{eq:23}
2|A|E_{I,\phi}=\sum_{I^+-\max I^+\subset J^+\subset I^+}(-1)^{|I^+-J^+|}(p^{[J^\perp-J]}\Delta_J+(-1)^{\phi(I^\perp-I)}\Delta_{J^\perp}).$$
Now take $I\subset P_\lambda$ to be any small order ideal. The decomposition (\[eq:18\]) gives $$L^2(A)=\bigotimes_{C\in \tilde\pi_0(I^\perp-I)} L^2(A_C)$$ and $$V_{I,\phi}=\Big(\bigotimes_{C\in \pi_0(I^\perp-I)} V_{I(C),\phi(C)}\Big)\otimes L^2(A_{I^\perp(0)}/A_{I(0)}),$$ the last factor being one dimensional (since $I(0)=I^\perp(0)$). So we have $$\label{eq:25}
E_{I,\phi}=\Big(\bigotimes_{C\in \pi_0(I^\perp-I)} E_{I(C),\phi(C)}\Big)\otimes \Delta_{I(0)},$$ where, since $I(C)^\perp-I(C)$ is connected, $E_{I(C),\phi(C)}$ is determined by (\[eq:23\]). A typical term in the expansion (\[eq:25\]) will be of the form $$\label{eq:26}
\Big(\bigotimes_{C\in \pi_0(I^\perp-I)}\Delta_{L(C)}\Big)\otimes \Delta_{I(0)},$$ where, for each $C\in \pi_0(I^\perp-I)$, $I(C)\subset L(C)\subset I^\perp(C)$ with either $L(C)$ or $L(C)^\perp$ is a small order ideal in $P_{\lambda(C)}$. But this is just $\Delta_L$, where $$\label{eq:24}
L=I\bigcup \Big(\coprod_{C\in \pi_0(I^\perp-I)}L(C)\Big),$$ is an order ideal in $P_\lambda$ by Lemma \[lemma:ideals-components\]. We have the qualitative result
\[theorem:alpha-qualitative\] For each $(I,\phi)\in Q_\lambda$, $2^{|\pi_0(I^\perp-I)|}|A|\alpha_L(I,\phi)$ is a polynomial in $p$ whose coefficients are integers which depend only on the combinatorial data $I$, $\phi$ and $L$.
Let $I_L=L\cap L^\perp$. Examining (\[eq:23\]) more carefully gives
\[theorem:alpha-supp\] The coefficient $\alpha_L(I,\phi)$ is non-zero if and only if the following conditions hold:
1. \[item:5\] For each $C\in \pi_0(I^\perp-I)$, either $L(C)$ or $L(C)^\perp$ is a small order ideal in $P_{\lambda(C)}$.
2. \[item:6\] $I^+-\max I^+\subset I_L^+\subset I^+$.
For $\alpha_L(I,\phi)$ to be non-zero, it is necessary that $L$ be of the form (\[eq:24\]) for some order ideals $L(C)$ of $P_{\lambda(C)}$ which occur in the right hand side of (\[eq:23\]). Furthermore, since each order ideal in $P_{\lambda(C)}$ appears at most once in the right hand side of (\[eq:23\]), so each order ideal in $P_\lambda$ appears only once in the expansion (\[eq:25\]). In particular, no cancellation is possible, and for all such ideals $\alpha_L(I,\phi)\neq 0$.
Now $L(C)$ appears on the right hand side of (\[eq:23\]) if and only if \[item:5\] holds, and $I(C)^+-\max I(C)^+\subset I_L(C)^+\subset I(C)^+$. Since $\max I^+=\coprod_C \max I(C)^+$, this amounts to the condition \[item:6\].
If these conditions do hold, then for each $C'\in \pi_0(I_L^\perp-I_L)$ there exists $C\in \pi_0(I^\perp-I)$ such that $C'\subset C$. Furthermore, $\phi_L(C')$ depends only on $C$, so we may denote its value by $\phi_L(C)$. For $I=I(C)$, the right hand side of (\[eq:23\]) can be written as $$\sum_{L(C)}(-1)^{|I(C)^+-L(C)^+|+\phi(C)\phi_L(C)}p^{[I_L(C)^\perp-L(C)]},$$ the sum being over an appropriate set of order ideals $L(C)\subset P_{\lambda(C)}$. Write $\langle \phi_1,\phi_2\rangle$ for $\sum_{C\in \pi_0(I^\perp-I)} \phi_1(C)\phi_2(C)$ for any functions $\phi_i:\pi_0(I^\perp-I)\to {\mathbf Z}/2{\mathbf Z}$. The additive nature of the exponents in the above expression allows us to get an exact expression of $\alpha_L(I,\phi)$:
\[theorem:alpha-exact\] If an order ideal $L\subset P_\lambda$ satisfies the conditions of Theorem \[theorem:alpha-supp\], then $$2^{|\pi_0(I^\perp-I)|}|A|\alpha_L(I,\phi)=(-1)^{|I^+-I_L^+|+\langle\phi,\phi_L\rangle}p^{[I_L^\perp-L]}.$$
Finite modules over a Dedekind domain {#sec:finite-modules}
=====================================
Let $F$ be a non-Archimedean local field with ring of integers $R$. Let $P$ denote the maximal ideal of $R$. Assume that the residue field $R/P$ is of odd order $q$. Fix a continuous character $\psi:F\to U(1)$ whose restriction to $R$ is trivial, but whose restriction to $P^{-1}R$ is not (see, for example, Tate’s thesis [@MR0217026]). Then if $\psi_x(y)=\psi(xy)$, the map $x\mapsto \psi_x$ is an isomorphism of $F$ into $\hat F$. Under this isomorphism, $R$ has image $R^\perp=\widehat{F/R}$. More generally, $P^{-n}$ has image $(P^n)^\perp=\widehat{F/P^n}$ for every integer $n$ (recall that for positive $n$, $P^{-n}$ is the set of elements $x\in F$ such that $x P^n\in R$). Thus, it gives rise to an isomorphism $P^{-n}/R\to \widehat{R/P^n}$ for each positive integer $n$. Since $P^{-n}/R$ inherits the structure of an $R$-module, this isomorphism also allows us to think of $\widehat{R/P^n}$ as an $R$-module. Now suppose $A$ is a finitely generated torsion module over $R$. Then $$\label{eq:13}
A=R/P^{\lambda_1}\times\cdots\times R/P^{\lambda_l}$$ for a unique partition $\lambda$. By the discussion above, $\hat A$ is also an $R$-module (non-canonically isomorphic to $A$). Let $K=A\times \hat A$, and ${\mathrm{Sp}}(K)$ be as in Section \[theorem:SvN\]. Define ${\mathrm{Sp}}_R(K)$ to be the subgroup of ${\mathrm{Sp}}(K)$ consisting of $R$-module automorphisms.
The Weil representation of ${\mathrm{Sp}}_R(K)$ is simply the restriction of the Weil representation of ${\mathrm{Sp}}(K)$ on $L^2(A)$ to ${\mathrm{Sp}}_R(K)$. All the theorems and proofs in this article concerning finite abelian $p$-groups generalize to the Weil representation of ${\mathrm{Sp}}_R(K)$ on $L^2(A)$, so long as $p$ is replaced by $q$ in the formulas. Since every finitely generated torsion module over a Dedekind domain is a product of its primary components, and module automorphisms respect the primary decomposition, the reduction in Section \[sec:prim-decomp\] works for finite modules of odd order over Dedekind domains.
Singla [@Poojareps; @Poojaclassical] has proved that the representation theory of $G(R/P^2)$, where $G$ is a classical group, depends on $R$ only through $q$, the order of the residue field. More precisely, if $R$ and $R'$ are two discrete valuation rings and an isomorphism between their residue fields is fixed (for example, take $R={\mathbf Z}_p$, the ring of $p$-adic integers, and $R'=({\mathbf Z}/p{\mathbf Z})[[t]]$, the ring of Laurent series with coefficients in ${\mathbf Z}/p{\mathbf Z}$), then there is a canonical bijection between the irreducible representations of $G(R/P^2)$ and $G(R'/P^2)$ which preserves dimensions. There is also a canonical bijection between their conjugacy classes which preserves sizes. All existing evidence points towards the existence of a similar correspondence automorphism groups of modules of type $\lambda$ (see, for example [@MR2456275 Conjecture 1.2]). The results in this paper also point in the same direction: for each partition $\lambda$, there is a canonical correspondence between the invariant subspaces of the Weil representations associated to the finitely generated torsion $R$-module of type $\lambda$ and the finitely generated torsion $R'$-module of type $\lambda$ which preserves dimensions.
Acknowledgment {#acknowledgment .unnumbered}
--------------
We thank Uri Onn for some helpful comments on a preliminary version of this paper.
[10]{}
, [*Moduli of abelian varieties*]{}, vol. 1644 of Lecture Notes in Mathematics, Springer-Verlag, 1996.
, [*On cuspidal representations of general linear groups over discrete valuation rings*]{}, Israel J. Math., 175 (2010), pp. 391–420.
, [*Weil representations of symplectic groups over rings*]{}, J. London Math. Soc. (2), 62 (2000), pp. 423–436.
height 2pt depth -1.6pt width 23pt, [*Clifford and [M]{}ackey theory for [W]{}eil representations of symplectic groups*]{}, J. Algebra, 262 (2003), pp. 348–379.
, [*Representations of reductive groups over finite fields*]{}, Ann. of Math. (2), 103 (1976), pp. 103–161.
, [*Degenerations and orbits in finite abelian groups*]{}. (preprint) arXiv:1005.5222v1, 2010.
, [*Construction de séries discrètes [$p$]{}-adiques*]{}, vol. 462 of Lecture Notes in Mathematics, Springer-Verlag, 1975.
height 2pt depth -1.6pt width 23pt, [*Weil representations associated to finite fields*]{}, J. Algebra, 46 (1977), pp. 54–101.
, [*The characters of the finite general linear groups*]{}, Trans. Amer. Math. Soc., 80 (1955), pp. 402–447.
, [*Regular elements and regular characters of [${\rm GL}\sb
n({\mathfrak O})$]{}*]{}, J. Algebra, 174 (1995), pp. 610–635.
height 2pt depth -1.6pt width 23pt, [*Semisimple and cuspidal characters of [${\rm GL}\sb n({\mathfrak O})$]{}*]{}, Comm. Algebra, 23 (1995), pp. 7–25.
, [*The behaviour of general theta functions under the modular group and the characters of binary modular congruence groups*]{}, parts I and II, Ann. of Math. (2), 47 (1946), pp. 317–447.
, [*A theorem of [S]{}tone and von [N]{}eumann*]{}, Duke Mathematical Journal, 16 (1949), pp. 313–326.
, [ *Correspondances de [H]{}owe sur un corps [$p$]{}-adique*]{}, vol. 1291 of Lecture Notes in Mathematics, Springer-Verlag, 1987.
, [*Quantum Computation and Quantum Information*]{}, Cambridge University Press, 2000.
, [*Representations of automorphism groups of finite [$\mathfrak
o$]{}-modules of rank two*]{}, Adv. Math., 219 (2008), pp. 2058–2085.
, [*On character values and decomposition of the [W]{}eil representation associated to a finite abelian group*]{}, J. Analysis, 17 (2009), pp. 73–85.
height 2pt depth -1.6pt width 23pt, [*An easy proof of the [S]{}tone-von [N]{}eumann-[M]{}ackey [T]{}heorem*]{}, Expo. Math., (2010). in press.
, [*Locally compact abelian groups with symplectic self-duality*]{}, Adv. Math., 225 (2010), pp. 2429–2454.
, [*A brief survey on the theta correspondence*]{}, in Number theory ([T]{}iruchirapalli, 1996), vol. 210 of Contemp. Math., Amer. Math. Soc., Providence, RI, 1998, pp. 171–193.
, [*On representations of general linear groups over principal ideal local rings of length two*]{}, J. Algebra, 324 (2010), pp. 2543–2563.
height 2pt depth -1.6pt width 23pt, [*On representations of classical groups over principal ideal local rings of length two*]{}, (personal communication).
, [*Weil representations of finite classical groups*]{}, Invent. Math., 51 (1979), pp. 143–153.
, [*Enumerative combinatorics. [V]{}ol. 1*]{}, vol. 49 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1997. Corrected reprint of the 1986 original.
, [*The smooth representations of [${\rm GL}_2(\mathfrak
o)$]{}*]{}, Comm. Algebra, 37 (2009), pp. 4416–4430.
, [*Linear transformations in [H]{}ilbert space. [III]{}. [O]{}perational methods and group theory*]{}, Proc. Natl. Acad. Sci. USA, 16 (1930), pp. 172–175.
, [*Construction and classification of irreducible representations of special linear group of the second order over a finite field*]{}, Osaka J. Math., 4 (1967), pp. 65–84.
height 2pt depth -1.6pt width 23pt, [*Irreducible representations of the binary modular congruence groups [${\rm
mod}\,p^{\lambda }$]{}*]{}, J. Math. Kyoto Univ., 7 (1967), pp. 123–132.
, [*Fourier analysis in number fields, and [H]{}ecke’s zeta-functions*]{}, in Algebraic [N]{}umber [T]{}heory ([P]{}roc. [I]{}nstructional [C]{}onf., [B]{}righton, 1965), Thompson, 1967, pp. 305–347.
, [*Die [E]{}indeutigkeit der [S]{}chrödingerschen [O]{}peratoren*]{}, Mathematische Annalen, 104 (1931), pp. 570–578.
, [*Sur certains groupes d’opérateurs unitaires*]{}, Acta Mathematica, 111 (1964), pp. 143–211.
, [*The theory of groups and quantum mechanics*]{}, Dover Publications, Inc., 1950.
[^1]: 2010 *Mathematics Subject Classification.* Primary 11F27, 05E10. Secondary 81R05.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
Yutaka [Fujita]{}[^1]\
[*Department of Earth and Space Science, Graduate School of Science, Osaka University,*]{}\
[*Machikaneyama-cho, Toyonaka, Osaka, 560-0043*]{}\
Tomoka [Tosaki]{}, and Akika [Nakamichi]{}\
[*Gunma Astronomical Observatory, 6860-86, Nakayama, Takayama, Agatsuma, Gunma, 377-0702*]{}\
and\
Nario [Kuno]{}\
[*Nobeyama Radio Observatory, Minamimaki, Minamisaku, Nagano, 384-1305*]{}
title: 'CO (J=1-0) Observation of the cD Galaxy of AWM7: Constraints on the Evaporation of Molecular Gas[^2]'
---
Introduction {#sec:intro}
============
The centers of galaxy clusters are usually dominated by very massive ($\sim 10^{13}\;\MO$) galaxies. These galaxies are often called D or cD galaxies. The observations of cold gas ($\ltsim 10^5$ K) give us the clues of the formation and evolution of the cD galaxies.
Cold gas in cD galaxies has been investigated from the viewpoint of cooling flows. The cooling time of intracluster medium (ICM) exceeds the Hubble time ($\sim 10^{10}$ yr) in the most region of clusters (Sarazin 1986). However, around cD galaxies, the density of ICM increases and the cooling time decreases to $t_{\rm cool}\ltsim 10^9$ yr. In the absence of heating, the gas is inferred to be cooling at a rate of $\dot{M}_{\rm
CF}\sim 100\MO\rm\; yr^{-1}$ (Fabian 1994). We will refer to $\dot{M}_{\rm CF}$ as a mass deposition rate from now on. Thus, the total mass accumulated around the cD galaxies would result in $\sim
10^{12}\MO$ if the cooling occurred steadily at the rate over the Hubble time. Although many observers have tried to detect the cooled gas mainly in massive cooling flow clusters $\dot{M}_{\rm CF}> 100\;\MO\rm\;
yr^{-1}$, most of them could not detect such a large amount of cold gas. Using recombination line luminosities, Heckman et al. (1989) estimate that the total mass of $10^4$ K ionized hydrogen is less than $10^8
\MO$. Observations of the atomic hydrogen 21 cm line limit the total mass of optically thin H[I]{} to less than $\sim 10^{9}
\MO$ (Burns et al. 1981; Valentijn, Giovanelli 1982; McNamara et al. 1990). CO observations limit the mass in clouds similar to Galactic molecular clouds to less than $10^{9-10} \MO$ (McNamara, Jaffe 1994; O’Dea et al. 1994; Braine, Dupraz 1994). Among the cD galaxies observed so far ($\sim 30$), only one exception is NGC 1275, the cD galaxy in the Perseus cluster. The molecular gas of $\sim 10^{9-10}\:\MO$ has been detected (e.g. Lazareff et al. 1989; Mirabel et al. 1989; Inoue et al. 1996), although it is smaller than the prediction of the cooling flow model. These observations may imply that most of the cooled gas becomes something other than molecular gas such as low mass stars, or the [*actual*]{} mass deposition rate, $\dot{M}_{\rm CF}$, may be reduced by some heating sources.
Before we move to investigate these possibilities, we should consider another scenario, that is, cooling flows actually exist but the molecular gas deposited by the cooling flows is continuously evaporated by the ambient hot ICM. The evaporation time of a molecular cloud is given by $$\label{eq:evap_1}
t_{\rm evap}\propto n_{\rm c} R_{\rm c}^2 T_{\rm ICM}^{-5/2}
\:,$$ where $n_{\rm c}$ and $R_{\rm c}$ are the density and radius of a molecular cloud, respectively, and $T_{\rm ICM}$ is the temperature of hot ICM (Cowie, McKee 1977). In relation (\[eq:evap\_1\]), the saturation of heat flux is ignored although the following result does not change significantly even in the saturated case (see §4). White et al. (1997) investigate the data of Einstein Observatory and find the relations $T_{\rm ICM}\propto \dot{M}_{\rm CF}^{0.30}$ and $r_{\rm
cool}\propto \dot{M}_{\rm CF}^{0.25}$, where $r_{\rm cool}$ is the cooling radius. Thus, we obtain the relation: $$t_{\rm evap}\propto n_{\rm c} R_{\rm c}^2
\dot{M}_{\rm CF}^{-0.75}$$ On the other hand, if the molecular gas is accumulated by cooling flows and the age of cooling flows is much larger than $t_{\rm evap}$, the mass of molecular gas per unit volume is given by $$m_{\rm mol}\propto (\dot{M}_{\rm CF}/r_{\rm cool}^3) t_{\rm evap}
\propto n_{\rm c} R_{\rm c}^2
\dot{M}_{\rm CF}^{-0.5}$$ Therefore, if $n_{\rm c}$ and $R_{\rm c}$ do not depend on $\dot{M}_{\rm CF}$ too much, clusters with small $\dot{M}_{\rm CF}$ should have large $m_{\rm mol}$. In these clusters, we could find molecular gas.
Since CO has been searched mainly in massive cooling flow clusters ($\dot{M}_{\rm CF}> 100\;\MO\rm\; yr^{-1}$), the observation of clusters with small $\dot{M}_{\rm CF}$ is important because of the above reason. Moreover, we note here another importance of searching CO in small $\dot{M}_{\rm CF}$ clusters. Even if the [*actual*]{} mass deposition rate $\dot{M}_{\rm CF}$ is reduced by some heating sources, molecular gas may be brought to cD galaxies. For example, the capture of gas-rich galaxies is another possible supply route of molecular gas into cD galaxies. In this case, the detection of the molecular gas would be easier for clusters with small $\dot{M}_{\rm CF}$. This is because the X-ray emissions of these clusters are weaker than those of clusters with large $\dot{M}_{\rm CF}$, which means that the heating should be less effective in these clusters and that the cold gas would be less affected by the heating.
We search CO in the cD galaxy NGC1129 at the center of a poor cluster AWM7 with relatively small mass deposition rate ($\dot{M}_{\rm
CF}=41\MO\rm\; yr^{-1}$; Peres et al. 1998). Note that AWM7 is one of the most closely studied clusters in X-ray. Ezawa et al. (1997) and Xu et al. (1997) find the metal abundance excess of the ICM at the center of the cluster. This suggests that the ICM is not well mixed and the cluster has not experienced violent cluster mergers at least recently. Thus, the molecular gas in NGC 1129, if exist, would keep intact. Moreover, the abundance excess, especially in the cD galaxy (Xu et al. 1997), implies that there has been star formation activity at the cluster center. The excess iron mass in the central region ($r<27$ kpc) is $8\times 10^8\;\MO$. Assuming the 1 $\MO$ iron is ejected into ICM per 100 $\MO$ of stars formed, the observation shows that $\sim
10^{11}\;\MO$ of stars have been formed in the region. On the other hand, the present star formation rate of NGC 1129 is $0.04\;\MO\rm\;
yr^{-1}$ within 1.57 kpc form the center (McNamara, O’Connell 1989). If the distribution of stars in the galaxy is $\propto r^{-2}$, the star formation rate for $r<27$ kpc is $0.7\;\MO\;\rm yr^{-1}$. Thus, the present star formation rate in the region is smaller than that the average through the Hubble time ($\sim 10\;\MO\rm\; yr^{-1}$), and the star formation in the past must be larger than that at present. If the ‘starburst’ occurred recently, molecular gas used for it would be left until present. In this paper, we assume $H_0=50\;\rm km\;
s^{-1}\;Mpc^{-1}$ throughout.
Observations {#sec:obs}
============
The $^{12}$CO ($J$ = 1-0) line was observed toward the center of NGC 1129 ($\alpha = 02^{\rm h}51^{\rm m}13^{\rm
s}\hspace{-5pt}.\hspace{2pt}3$, $\delta_{1950} = +41^{\circ}22^{\rm
m}32^{\rm s}$) with the 45-m telescope at Nobeyama Radio Observatory in 1999 March and May. The half-power beam width (HPBW) was $15''$, which corresponds to 7.5 kpc at the distance of NGC 1129 ($z = 0.017325$). The aperture and main beam efficiencies were $\eta_{\rm A}$ = 0.40 and $\eta_{\rm MB}$ = 0.48, respectively.
We used two SIS receivers that can observe two orthogonal linear polarizations simultaneously. Martin-Puplett type SSB filters were used for image sideband rejection. The system noise temperature (SSB) including the atmospheric effect and the antenna ohmic loss was 400-600 K. As receiver backends 2048-channel wide-band acousto-optical spectrometers (AOS) were used. The frequency resolution and channel spacing are 250 kHz and 125 kHz, respectively. Total bandwidth is 250 MHz. Calibration of the line intensity was made by the chopper-wheel method, yielding the antenna temperature ($T^{*}_{\rm A}$) corrected for both atmospheric and antenna ohmic losses. We used the main beam brightness temperature ($T_{\rm mb} \equiv T^{*}_{\rm A}$/$\eta_{\rm
MB}$) in this paper. The telescope pointing was checked and corrected every hour by observing the 43GHz SiO maser emission in a late type star S-Per or W-And. The absolute pointing accuracy was better than $5''$ (peak value) throughout the observations.
Results {#sec:res}
=======
In figure 1, we present the CO(1-0) spectrum which has been binned by $20\;\rm km\; s^{-1}$ and has had baseline removed. The spectrum shows no significant CO(1-0) features either in emission or in absorption. The $3\sigma$ upper limit to the flux integral is given by $$I_{\rm CO}= \frac{3\sigma_{\rm ch}\Delta V}
{\sqrt{\Delta V/\Delta V_{\rm ch}}}
\rm\; K\; km\; s^{-1},$$ where $\sigma_c$ is the channel-to-channel rms noise, $\Delta V_{\rm
ch}$ is the smoothed velocity channel spacing, and $\Delta V$ is the width of line. From figure 1, we obtain $\sigma_{\rm ch}= 0.006$ K.
We assume that the column density of molecular hydrogen is $$N_{\rm H_2} = 2.8\times 10^{20} I_{\rm CO}\;\rm cm^{-2}$$ (O’Dea et al. 1994). The total mass of molecular hydrogen is given by $$\label{eq:mol}
M_{\rm mol} = \frac{\pi r^2}{4\ln 2}N_{\rm H_2}m_{\rm H_2}\;,$$ where $r$ is the beam size at the distance of the source and $m_{\rm
H_2}$ is the mass of the hydrogen molecule (O’Dea et al. 1994). We assume that $\Delta V = 300\;\rm km\; s^{-1}$ for a rectangular line feature. This is the same as McNamara and Jaffe (1994) and nearly corresponds to the internal velocity dispersion of NGC 1129 (McElroy 1995). From equation (\[eq:mol\]), we obtain $M_{\rm mol}<4\times
10^{8}\;\MO$ within 7.5 kpc from the center. This is one of the most sensitive limits for cD galaxies.
Discussion
==========
Although $\dot{M}_{\rm CF}$ of AWM7 is relatively small, the non-detection of molecular gas conflicts with a cooling flow model if the cooled gas becomes molecular gas and if we ignore the effect of the evaporation. Peres et al. (1998) estimate that the mass deposition rate and cooling radius of AWM7 are $\dot{M}_{\rm CF}=41\MO\;\rm yr^{-1}$ and $r_{\rm cool}=103$ kpc, respectively. The analysis based on a cooling flow model shows that the mass deposited within $r$ is $\dot{M}(<r)=\dot{M}_{\rm CF}(r/r_{\rm cool})$ (Fabian 1994). Thus, the mass deposition rate within the beam of Nobeyama 45m is $\dot{m}=3.8
\MO\rm\; yr^{-1}$, considering the projection effect. Thus, molecular gas of $\sim 10^{10}\MO$ would be detected if the cooling flow occurred steadily at the rate over the Hubble time.
As mentioned in §1, when the molecular gas deposited by a cooling flow is continuously evaporated by the ambient hot ICM, the detection of molecular gas would be relatively easy in clusters with small $\dot{M}_{\rm CF}$. Although we cannot detect CO, it constrains the evaporation rate of molecular gas and the heat conduction rate of the ICM. Using the results, we could investigate whether the non-detection of CO is consistent with the evaporation model.
The accumulation time of molecular gas is $$\label{eq:tacc}
t_{\rm acc}=M_{\rm
mol}/\dot{m}\ltsim 1\times 10^8\rm\; yr \;.$$ Although the properties of the molecular clouds deposited by cooling flows are not well-known, we could calculate the evaporation time of the clouds as follows. In disk galaxies, molecular gas is considered to be produced through disk instabilities (e.g. Larson 1987). In elliptical galaxies like cDs, we expect that the mechanism is ineffective. Instead, we expect that the molecular gas is produced through the thermal instability of ICM. One possible seed of the instability is supernova remnants (Fujita et al. 1996, 1997). Thus, we assume that a supernova remnant is the seed of a molecular cloud and that only thermal evaporation affects the cloud after the formation, although these may oversimplify the evolution of molecular clouds (refer to Loewenstein and Fabian \[1990\] for more realistic discussion about the issue). Note that the results in the following can be applied to other formation mechanisms of cloud if the resultant mass is nearly the same. If we can ignore the fragmentation and coalescence of molecular clouds, the mass of a molecular cloud is equal to that of a supernova remnant. Since the radius of a supernova remnant is given by $$\label{eq:snr}
R_{\rm s}\sim
50\;{\rm pc}\left(\frac{P_{\rm ICM}}{4\times
10^5\:\rm cm^{-3}\;K}\right)^{-1/3}
\left(\frac{E_{\rm SN}}{10^{51}\rm\;ergs}\right)^{1/3}$$ (Fujita et al. 1997), the mass of a molecular cloud is $$\begin{aligned}
M_{\rm c}&=&\frac{4}{3}\pi R_{\rm s}^3 m_{\rm H} n_{\rm ICM}\\
\nonumber
&=&130\;\MO
\left(\frac{n_{\rm ICM}}{10^{-2}\;\rm cm^{-2}}\right)\\
& &\times
\left(\frac{P_{\rm ICM}}{4\times
10^5\:\rm cm^{-3}\;K}\right)^{-1}
\left(\frac{E_{\rm SN}}{10^{51}\rm\;ergs}\right)
\label{eq:mass}\;,\end{aligned}$$ where $P_{\rm ICM}$ and $n_{\rm ICM}$ are the pressure and the density of ICM, respectively, $E_{\rm SN}$ is the energy released by a supernova, and $m_{\rm H}$ is the hydrogen mass.
If molecular clouds are in pressure equilibrium with the ambient ICM, the density of the molecular gas is $$\label{eq:nc}
n_{\rm c} = 4\times 10^4\;{\rm cm^{-3}}
\left(\frac{P_{\rm ICM}}{4\times 10^5\:\rm cm^{-3}\;K}\right)
\left(\frac{T_{\rm c}}{10\rm\; K}\right)^{-1}
\;,$$ where $T_{\rm c}$ is the temperature of the molecular gas. Since $n_{\rm
ICM}m_{\rm H}R_{\rm s}^3=n_{\rm c}m_{\rm H_2}R_{\rm c}^3$, the radius of a molecular cloud is $$\begin{aligned}
R_{\rm c}&\sim& 0.25\;{\rm pc}
\left(\frac{R_{\rm s}}{50\;\rm pc}\right)
\left(\frac{T_{\rm c}}{10\;\rm K}\right)^{1/3}
\nonumber\\
& & \times
\left(\frac{T_{\rm ICM}}{4\times 10^7\;\rm K}\right)^{-1/3}
\label{eq:Rc} \;\end{aligned}$$ We assume that heat is supplied to molecular clouds from the isothermal X-ray gas component prevailing even in the central region of clusters (e.g. Ikebe et al. 1999). Moreover, we assume that the isothermality of the component is retained by adiabatic heating or magnetic loops connected to the overall thermal reservoir of the cluster (e.g. Norman, Meiksin 1996) or the large heat conduction rate of the gas constituting the component. The evaporation time of a molecular cloud embedded in ICM is given by $$\begin{aligned}
\label{eq:evap}
t_{\rm evap} &=& 1\times 10^{5}\;{\rm yr}\;
\frac{1}{f}
\left(\frac{n_{\rm c}}{4\times 10^4\rm\; cm^{-3}}\right)\nonumber\\
& & \times
\left(\frac{R_{\rm c}}{0.25\rm\; pc}\right)^2
\left(\frac{T_{\rm ICM}}{4\times 10^7\rm\;K}\right)^{-5/2} \\
&=& 1\times 10^{5}\;{\rm yr}\;
\frac{1}{f}
\left(\frac{n_{\rm c}}{4\times 10^4\rm\; cm^{-3}}\right)^{1/3}\nonumber\\
& & \times
\left(\frac{M_{\rm c}}{130\; \MO}\right)^{2/3}
\left(\frac{T_{\rm ICM}}{4\times 10^7\rm\;K}\right)^{-5/2}
\label{eq:evap_mc}\end{aligned}$$ (Cowie, McKee 1977). Equation (\[eq:evap\_mc\]) shows that if $M_{\rm
c}$ is larger than that given in equation (\[eq:mass\]), $t_{\rm
evap}$ should be larger. The parameter $f$ is the ratio of the heat conduction rate to that of Spitzer (1962) and $0\leq f \leq 1$ (Cowie and McKee \[1977\] assume $f=1$). When $f<1$, the heat conduction rate and the mean free path of an electron are considered to be regulated by plasma instabilities around the cloud (Pistinner et al. 1996; Hattori, Umetsu 1999). If the mean free path of an electron is comparable or even greater than the radius of a cloud, the thermal evaporation is saturated. Defining the saturation parameter, $$\begin{aligned}
\label{eq:sat}
\sigma_0&=&2700\;f\left(\frac{T_{\rm ICM}}{4\times 10^7\;\rm K}\right)^2
\nonumber \\
& & \times
\left(\frac{n_{\rm ICM}}{10^{-2}\rm\; cm^{-3}}\right)^{-1}
\left(\frac{R_{\rm c}}{0.25\rm\; pc}\right)^{-1}
\;,\end{aligned}$$ the saturation occurs when $\sigma_0\gtsim 1$ (Cowie and McKee 1977). In the saturated case, the evaporation time is given by $$\begin{aligned}
\label{eq:evap_sat}
t_{\rm evap} &=& 8\times 10^{6}\;{\rm yr}\;
\left(\frac{n_{\rm c}}{4\times 10^4\rm\; cm^{-3}}\right)\nonumber\\
& & \times
\left(\frac{n_{\rm ICM}}{10^{-2}\rm\; cm^{-3}}\right)^{-1}
\left(\frac{R_{\rm c}}{0.25\rm\; pc}\right)
\nonumber\\
& & \times
\left(\frac{T_{\rm ICM}}{4\times 10^7\rm\;K}\right)^{-1/2}
\left(\frac{\sigma_0}{2700}\right)^{-3/8} \\
&=& 8\times 10^{6}\;{\rm yr}\;
\left(\frac{n_{\rm c}}{4\times 10^4\rm\; cm^{-3}}\right)^{2/3}\nonumber\\
& & \times
\left(\frac{n_{\rm ICM}}{10^{-2}\rm\; cm^{-3}}\right)^{-1}
\left(\frac{M_{\rm c}}{130\; \MO}\right)^{1/3}
\nonumber\\
& & \times
\left(\frac{T_{\rm ICM}}{4\times 10^7\rm\;K}\right)^{-1/2}
\left(\frac{\sigma_0}{2700}\right)^{-3/8}\label{eq:evap_mcs}\end{aligned}$$ (Cowie, McKee 1977).
We will adopt $T_{\rm c}=10\rm\; K$ and $E_{\rm SN}=10^{51}\;\rm ergs$; the temperature of a molecular cloud is the typical one in the Galaxy (Scoville, Sanders 1987). If we adopt the observed values $n_{\rm
ICM}=1.82\times 10^{-2}\rm\; cm^{-3}$ and $T_{\rm ICM}=3.63$ keV (Mohr et al. 1999; Ezawa et al. 1997), equation (\[eq:snr\]) and (\[eq:Rc\]) yield $R_{\rm c}=0.20$ pc and equation (\[eq:sat\]) shows that the saturation occurs when $f\gtsim 5\times 10^{-4}$. If the age of a cooling flow ($t_0\sim 10^{10}$ yr; Fabian 1994) is much larger than the evaporation time of a molecular cloud, the evaporation of molecular gas should be balanced with the accumulation. In this case, the evaporation time should be equal to the accumulation time. From equations (\[eq:tacc\]), (\[eq:evap\]), (\[eq:sat\]), and (\[eq:evap\_sat\]), this requires $f\gtsim 10^{-3}$. However, recent observations of ASCA show that ICM is inhomogeneous in temperature at least in some clusters (e.g. Ikebe et al. 1999). In order to explain this inhomogeneity by the cooling flow model, $f$ must be less than $10^{-5}$ at least around cooler X-ray gas (Pistinner et al. 1996; Hattori, Umetsu 1999) if the cooler gas component is not isolated by something like a magnetic field. Therefore, as long as $f<10^{-5}$, the evaporation cannot account for the non-detection of CO and most of the cooled gas may become something other than molecular gas such as dust (Fabian et al. 1994; Voit, Donahue 1995; Edge et al. 1999) or low mass stars (Sarazin, O’Connell 1983), or there may be something wrong in the cooling flow model, that is, the actual mass deposition rate is much less than the one estimated by X-ray observations.
So far, we have not considered the molecular gas brought by gas-rich galaxies merged into cD galaxies including NGC1129. Finally, we examine the evaporation of this kind of molecular gas. The mass of a molecular cloud brought through the capture would be larger in comparison with the case of cooling flows ($\sim 100\;\MO$; equation \[\[eq:mass\]\]). If molecular clouds in captured galaxies are similar to those in our Galaxy, the masses are typically $M_{\rm c}\gtsim 10^5\;\MO$ (Binney, Tremaine 1987). Thus, if we adopt equation (\[eq:nc\]) and the normalizations therein, $R_{\rm c}\gtsim 2.5$ pc. From equations (\[eq:sat\]) and (\[eq:evap\_mc\]), we expect $t_{\rm evap}\sim
10^{12}$ yr for $f=10^{-5}$. Moreover, even if $f=10^{-3}$, we obtain $t_{\rm evap}\sim 10^{10}$ yr. Thus, the evaporation can be ignored. This means that if molecular gas is brought into a cD galaxy through galaxy captures and $f\ltsim 10^{-3}$, the gas should be left there. (It is to be noted that when $T_{\rm ICM}\sim 10$ keV, equation \[\[eq:evap\_mc\]\] shows that the condition $t_{\rm evap}\gtsim 10^{10}$ requires $f\ltsim 10^{-4}$.) Hence, the non-detection of molecular gas strongly constrains the amount of molecular gas brought into the cD galaxy through galaxy captures. Using a theoretical model based on a hierarchical clustering scenario, Fujita et al. (1999) predict the amount of the molecular gas and compare it with the observations.
Conclusions
===========
We have searched for CO emission from the cD galaxy NGC 1129 in AWM7. We have obtained the upper limit of molecular hydrogen mass ($4\times
10^8\;\MO$). This is one of the most sensitive limits for cD galaxies. We predict the total mass of molecular gas left in the cD galaxy on the assumption that while the gas deposited by a cooling flow once becomes molecular gas, the molecular gas is continuously evaporated by the ambient hot gas. We find that the upper limit of molecular hydrogen mass shows $f\gtsim 10^{-3}$, where $f$ is the ratio of the heat conduction rate to that of Spitzer (1962). However, this is inconsistent with recent X-ray observations showing $f<10^{-5}$. Thus, most of the cooled gas predicted by a cooling flow model does not seem to become molecular gas in the cD galaxy. Therefore, if $f<<1$ as is suggested by the X-ray observations, the ultimate fate of most of the cooled gas may be something other than molecular gas such as dust or low mass stars. Alternatively, the actual mass deposition rate may be much less than the one predicted by a cooling flow model.
We find that molecular clouds brought to a cD galaxy by the gas-rich galaxies captured by the cD should not be evaporated when $f\ltsim
10^{-3}-10^{-4}$. This implies that if we obtain the upper limit of the mass of molecular gas in a cD galaxy, we could constrain the supply of molecular gas brought into through the galaxy captures.
We thank an anonymous referee for invaluable advice and suggestions. This work was supported in part by the JSPS Research Fellowship for Young Scientists.
References {#references .unnumbered}
==========
Binney J., Tremaine S. 1987, Galactic Dinamics (Princeton; New Jersey)
Braine J., Dupraz C. 1994, A&A 283, 407
Burns J.O., White R.A., Haynes M.P. 1981, AJ 86, 1120
Cowie L.L., McKee C.F. 1977, ApJ 211, 135
Edge A.C., Ivison R.J., Smail I., Blain A.W.,Kneib J.-P. 1999, MNRAS 306, 599
Ezawa H., Fukazawa Y., Makishima K., Ohashi T., Takahara F., Xu H., Yamasaki N.Y. 1997, ApJL, 490, 33
Fabian A.C. 1994, ARA&A 32, 77
Fabian A.C., Johnstone R.M., Daines S.J. 1994, MNRAS 271, 737
Fujita Y., Fukumoto J., Okoshi, K. 1996, ApJ 470, 762
Fujita Y., Fukumoto J., Okoshi, K. 1997, ApJ 488, 585
Fujita Y., Nagashima M., Gouda N. 1999, PASJ submitted
Heckman T.M., Baum S.A., van Breugel W.J.M., McCarthy P.J. 1989, ApJ 338, 48
Hattori M, Umetsu K. 1999, ApJ in press
Ikebe Y., Makishima K., Fukazawa Y., Tamura T., Xu H., Ohashi T., Matsushita K. 1999, ApJ 525, 58
Inoue M.Y., Kamono S., Kawabe R., Inoue M., Hasegawa T., Tanaka M.1996, AJ 1111, 1852
Larson R.B. 1987, in Starbursts and Galaxy Formation, ed. Trinh Xuan Thuan, T. Montmerle, and J. Tran Thanh Van (Editions Frontieres; France)
Lazareff B., Castets A., Kim D.W., Jura M. 1989, ApJL 336 13
Loewenstein M., Fabian A.C. 1990, MNRAS, 242, 120
Mirabel I.F., Sanders D.B., Kazes I. 1989, ApJL 340, 9
McElroy D.B. 1995, ApJS 100, 105
McNamara B.R., Bregman J.N., O’Connell R.W. 1990, ApJ 360, 20
McNamara B.R., Jaffe W. 1994, A&A 281, 673
McNamara B.R., O’Connell R.W. 1989, AJ 98, 2018
Norman C., Meiksin A. 1996, ApJ 468, 97
Mohr J.J., Mathiesen B., Evrard A.E. 1999, ApJ 517, 627
O’Dea C.P., Baum S.T., Maloney P.R., Tacconi L.J., Sparks W.B.1994, ApJ 422, 467
Pistinner S., Levinson A., Eichler D. 1996, ApJ 467, 162
Peres C.B., Fabian A.C., Edge A.C., Allen S.W., Johnstone R.M., White D.A. 1998, MNRAS 298, 416
Sarazin C.L. 1986, Phys. Mod. Rev. 58, 1.
Sarazin C.L., O’Connell R.W. 1983, ApJ 268, 552
Scoville N.Z., Sanders D.B. 1987, Interstellar Processes, p21 (Tokyo:Dordrecht)
Spitzer L. 1962, Physics of Fully Ionized Gases (New York: Interscience)
Valentjin E.A., Giovanelli R. 1982, A&A, 114, 208
Voit G.M., Donahue M. 1995, ApJ, 452, 164
White D.A., Jones C., Forman W. MNRAS 292, 419
Xu H., Ezawa H., Fukazawa Y., Kikuchi K., Makishima K., Ohashi T., Tamura T. 1997, PASJ 49, 9
[^1]: JSPS Research Fellow
[^2]: OU-TAP 107
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Ambient neutrons may cause significant background for underground experiments. Therefore, it is necessary to investigate their flux and energy spectrum in order to devise a proper shielding. Here, two sets of altogether ten moderated $^3$He neutron counters are used for a detailed study of the ambient neutron background in tunnel IV of the Felsenkeller facility, underground below 45m of rock in Dresden/Germany. One of the moderators is lined with lead and thus sensitive to neutrons of energies higher than 10MeV. For each $^3$He counter moderator assembly, the energy-dependent neutron sensitivity was calculated with the FLUKA code. The count rates of the ten detectors were then fitted with the MAXED and GRAVEL packages. As a result, both the neutron energy spectrum from 10$^{-9}$ to 300MeV and the flux integrated over the same energy range were determined experimentally.The data show that at a given depth, both the flux and the spectrum vary significantly depending on local conditions. Energy-integrated fluxes of $(0.61 \pm 0.05)$, $(1.96 \pm 0.15)$, and $(4.6 \pm 0.4) \times 10^{-4}$cm$^{-2}$s$^{-1}$, respectively, are measured for three sites within Felsenkeller tunnel IV which have similar muon flux but different shielding wall configurations. The integrated neutron flux data and the obtained spectra for the three sites are matched reasonably well by FLUKA Monte Carlo calculations that are based on the known muon flux and composition of the measurement room walls.'
author:
- 'M. Grieger'
- 'T. Hensel'
- 'J. Agramunt'
- 'D. Bemmerer'
- 'D. Degering'
- 'I. Dillmann'
- 'L.M. Fraile'
- 'D. Jordan'
- 'U. Köster'
- 'M. Marta'
- 'S.E. Müller'
- 'T. Szücs'
- 'J.L. Taín'
- 'K. Zuber'
title: Neutron flux and spectrum in the Dresden Felsenkeller underground facility studied by moderated $^3$He counters
---
Introduction {#sec:Introduction}
============
Experiments striving for the lowest possible background in radiation detectors must be placed underground, in order to efficiently attenuate the direct and indirect effects of cosmic-ray-induced muons [@Heusser95-ARNPS; @Formaggio04-ARNPS]. Underground experiments have proven particularly useful in the study of solar neutrinos [@Davis03-RMP; @Gavrin18-SNC] and of neutrino flavor oscillations [@McDonald16-RMP; @Kajita16-NobelLecture].
In underground laboratories [@Bettini07-arxiv], it is important to precisely know the neutron background: In some cases, because neutrons are part of the solar neutrino-induced signal [@SNO02-PRL], in other cases because they present a possible background for neutrinoless double-beta decay experiments [@Agostini18-PRL; @Legend17-AIPCP; @Zuber01-PLB] and dark matter searches [@Xenon18-PRL], even though they cannot explain [@Klinger15-PRL] a claimed dark matter detection [@Bernabei13-EPJC]. In underground nuclear astrophysics, the experimental study of the neutron source reactions for the astrophysical s-process [@Broggini18-PPNP] requires ultralow ambient neutron background.
Despite the crucial importance of knowing, and then suppressing, the neutron background, there is only a limited number of well-documented studies where both the flux and the energy spectrum of ambient neutrons were determined experimentally for underground laboratories [@Belli89-NCA; @Arneodo99-NCA; @Jordan13-APP; @Jordan13-APP_Corr; @Zhang14-PRD; @Sonay18-PRC]. Other work, for example, concentrates on thermal neutrons [@Best16-NIMA; @Debicki18-NIMA], shows only one [@Niese07-JRNC] to three energy bins [@Rindi88-NIMA] or a limited energy range [@Du18-NIMA], or does not present a deconvoluted neutron energy spectrum [@DaSilva95-NIMA]. It was reported that the flux of muon-induced neutrons seems to be underpredicted by GEANT4 Monte Carlo simulations with the standard physics list [@Du18-APP], whereas FLUKA simulations seemed to fare better [@Kneissl19-APP].
In addition, there are major discrepancies between studies that simply measure the neutron background at one given energy and site. An example is the thermal neutron flux in the Gran Sasso underground laboratory. Reported values vary by up to a factor of 6 between the four individual studies [@Rindi88-NIMA; @Belli89-NCA; @Best16-NIMA; @Debicki18-NIMA], from $(0.32 \pm 0.09 \pm 0.04)\times10^{-6}$cm$^{-2}$s$^{-1}$ [@Best16-NIMA] up to $(2.05 \pm 0.06)\times10^{-6}$cm$^{-2}$s$^{-1}$ [@Rindi88-NIMA]. In a different deep-underground laboratory, Canfranc/Spain, the reported neutron flux [@Jordan13-APP] was recently revised upward by a factor of 4 [@Jordan13-APP_Corr], further underlining the need for new and well-documented experimental efforts.
The aim of the present study is to make a first step to address this unsatisfactory situation with a precisely documented measurement of the ambient neutron flux and energy spectrum in an underground laboratory, matched with a Monte Carlo simulation. The experiment was carried out at Felsenkeller in Dresden/Germany, which is shielded by 45m of hornblende monzonite rock [@Ludwig19-APP; @Szucs19-EPJA].
In underground laboratories, there are two principal sources of ambient neutrons: first, neutrons that are produced in or near the experimental setup by cosmic-ray muons, here called ($\mu,n$) neutrons. Second, neutrons produced by ($\alpha,n$) reactions in the rock, with the $\alpha$ particles supplied by the $^{238}$U and $^{232}$Th decay chains[^1] — these are called ($\alpha,n$) neutrons here. Neutrons due to the spontaneous fission of $^{238}$U have similar characteristics as the ($\alpha,n$) neutrons but usually several orders of magnitude lower flux and are treated together with the ($\alpha,n$) neutrons here. Both ($\mu,n$) and ($\alpha,n$) neutrons show a maximum in the spectral flux near 1MeV neutron energy, as well as a similar overall spectral shape from thermal energies up to 10MeV. At even higher energies, $>$10MeV, the ($\alpha,n$) spectrum quickly drops, whereas the ($\mu,n$) spectrum extends to hundreds of MeV.
At a shallow depth such as the one of this study, it is expected [@Mei06-PRD; @Kneissl19-APP] that ($\mu,n$) dominate over ($\alpha,n$) neutrons. It is noted that the opposite is true for deep-underground laboratories, where instead ($\alpha,n$) neutrons dominate.
The present study was initially motivated by a project to install an ion accelerator and a second low background activity-counting facility [@Bemmerer18-SNC] in tunnels VIII and IX (Fig. \[fig:Map\]) of Felsenkeller [@Ludwig19-APP; @Szucs19-EPJA]. At the time of measurement, the tunnels hosting this new laboratory were not accessible, so the neighboring tunnel IV was studied instead, which presents very similar spatial characteristics.
This work serves to complete the characterization of all background components in Felsenkeller. The measured and simulated muon flux [@Ludwig19-APP] and the measured count rate in large high-purity germanium detectors [@Szucs19-EPJA] were already reported elsewhere.
The present work is organized as follows. The underground site is described in Sec. \[sec:Site\]. FLUKA-based predictions for the neutron spectra and fluxes in three differently shielded rooms in this site are developed in Sec. \[sec:PredictedFlux\]. A detector setup to address these expected fluxes is then developed, and its energy-dependent neutron sensitivity is calculated using FLUKA (Sec. \[sec:Sensitivities\]). The experiment is described in \[sec:Experiment\]. Sec. \[sec:Fit\] shows the obtained neutron fluxes and energy spectra. The data are compared with the literature in Sec. \[sec:Discussion\], and a summary and conclusions are offered in Sec. \[sec:Conclusion\].
Underground site studied {#sec:Site}
========================
The shallow-underground site Felsenkeller is located in the Plauenscher Grund area along the Weißeritz river, in the southwestern corner of Dresden, Germany. Until the 18th century, there was a quarry, then one of the largest breweries of Germany was built there. The brewery closed in 1991, but nine storage tunnels constructed in 1856–1859 remain, as well as a number of overground buildings. The tunnels have horizontal access from the brewery courtyard and are interconnected in a comblike structure (Fig. \[fig:Map\]). They are protected from cosmic rays by an overburden of 45m of hornblende monzonite rock [@Paelchen08-Book]. Rock samples taken inside the tunnels show specific activities of $130 \pm 30$ and $170 \pm 30$Bq/kg of $^{238}$U and $^{232}$Th, respectively [@Grieger16-Master]. Based on the measured vertical muon flux [@Ludwig19-APP], the effective rock overburden is 140m water equivalent (m.w.e.). Above the tunnels is a meadow planted with fruit trees.
The data for the present work were taken in tunnel IV, at the following three different sites inside a facility established in 1982 [@Helbig84-Isotopenpraxis], and enlarged in 1995 [@Niese96-Apradiso]:
1. Messkammer 1 (hereafter called MK1) is shielded by 70cm serpentinite rock and 2cm pre-1945 steel [@Helbig84-Isotopenpraxis] with an estimated total areal density of 160g/cm$^2$ [@Niese98-JRNC]. The serpentinite rock contains just 1.3 and 0.34Bq/kg of $^{238}$U and $^{232}$Th, respectively.
2. The shielding of Messkammer 2 (hereafter called MK2) consists of 1cm steel, 27cm iron pellets , 3.5cm steel, 3cm lead, and 1.2cm steel, from the outside to the inside, in total 210g/cm$^2$ [@Niese96-Apradiso; @Koehler09-Apradiso]. One part of the westernmost wall of MK2 was built with 1cm steel and 20cm lead instead, in total 235g/cm$^2$ for this part of the wall.
3. The workshop (hereafter called WS) is shielded from the surrounding hornblende monzonite rock by a 24cm thick brick wall.
![\[fig:Map\] Map of the tunnels in the Felsenkeller underground site, Dresden/Germany. The arrow indicates geographic northern direction. For this work, the neutron flux was measured in tunnel IV at the three distinct locations WS, MK2, and inside of MK1. The external access to tunnels I–IX is from the west, where a courtyard and the overground buildings of the former brewery (not shown) are located.](Figure1_v1.pdf){width="\columnwidth"}
All three sites are supplied by fresh, climatized air brought in by a ventilation duct from the outside. As a result, the radon concentration in the laboratory air is limited to 100–300Bq/m$^3$.
Predicted neutron flux in Felsenkeller based on a FLUKA simulation {#sec:PredictedFlux}
==================================================================
In order to guide the design of the experiment and the later fit of the experimental data, a FLUKA-based Monte Carlo prediction of the neutron flux and energy spectrum was developed.
General considerations
----------------------
Of the two components dominating the neutron flux in an underground laboratory, the flux of the first component, ($\mu,n$) neutrons, depends on the muon flux, hence the depth. The flux of the second component, ($\alpha,n$) neutrons (and spontaneous fission neutrons), does not depend on depth but on local conditions at the site studied.
The ($\mu,n$) neutrons may originate again from two different processes: either from the capture of stopped negative muons in the rock, or as part of a muon-induced hadronic shower. The relative importance of these two processes depends on muon energy and thus depth [@Mei06-PRD]. In a shallow laboratory such as the one studied here, stopped muon capture is expected to dominate, whereas the opposite is true deep underground [@Mei06-PRD].
The ($\alpha,n$) neutrons are created by $\alpha$ capture on light elements inside the rock, proportional to the specific activities of the natural $^{238}$U and $^{232}$Th decay chains [@Wulandari04-APP]. At great depth, where ($\mu,n$) neutrons are suppressed, ($\alpha,n$) neutrons dominate.
Setup of the FLUKA simulation
-----------------------------
For the ($\mu,n$) neutrons, a simulation was performed using FLUKA [@Ferrari05-FLUKA; @Boehlen14-NDS] version 2011.2x.6. An important physical input, namely the flux and angular distribution of cosmic-ray-induced muons at the sites studied, was recently measured [@Ludwig19-APP], and these data are used here. Using the depth obtained from the measured [@Ludwig19-APP] muon flux, an average muon energy of 17GeV and a parametrized muon energy spectrum are adopted from the literature [@Mei06-PRD]. The rock composition is known from the geological literature [@Paelchen08-Book], which was confirmed by the measured density of 2.75g/cm$^3$ of a rock sample taken.
The wall compositions of the three sites MK1, MK2, and WS are known from construction records and confirmed by visual inspection. The water content of the rock is not known and may vary by season. For the present prediction, the hornblende monzonite rock and the MK1 serpentinite shielding were assumed to contain 3% and 0.7% (by mass) of water, respectively.
In addition to the walls described above, there are also materials inside the rooms. For the sake of simplicity, only those materials inside the three rooms with nuclear charge $Z>20$ were included in the model. In particular, the high-purity germanium detectors in MK1 and MK2 are surrounded by lead castles. This lead may form a significant target for ($\mu,n$) neutron production and is included in the FLUKA model.
A special consideration is necessary for the case of MK2. There, the iron and lead walls provide very little moderation. As a result, actually the detectors used here (see below, Sec. \[sec:Experiment\]) affect the energy spectrum, but to first approximation not the energy-integrated flux. Therefore, for the case of the MK2 prediction, already the moderating effects of the detectors used here are included.
In the FLUKA simulation, muons are started randomly on a virtual hemisphere that is placed inside the rock surrounding the tunnels, randomly distributed from 0.0 to 1.5m deep inside the rock. This value is sufficiently low that the experimental muon flux and angular distribution measured inside the tunnels [@Ludwig19-APP] can still be used. Test runs with 0.00–0.75m and 0–3m deep distributions instead did not significantly alter the neutron yields, showing that the presently assumed starting depth is robust.
For the ($\alpha,n$) neutrons, the known rock composition [@Paelchen08-Book] and the measured specific activities for the $^{232}$Th and $^{238}$U decay chains [@Grieger16-Master], both assumed to be in secular equilibrium, are used. Then, the $\alpha$-induced neutron emission rate and spectrum are calculated with the NeuCBOT [@Westerdale17-NIMA] code and converted to an observed neutron flux using FLUKA.
Predicted neutron spectra
-------------------------
For each of the three sites studied, the predicted spectrum is given by the sum of the ($\mu,n$) and ($\alpha,n$) contributions. The calculated spectra are given in the 296+100 FLUKA energy bins and smoothed to avoid unphysical effects in the later deconvolution (Fig. \[fig:SimulatedSpectra\]). The ($\mu,n$) flux dominates at all energies. For comparison, the ($\alpha,n$) spectrum is shown separately for the case of MK1 (Fig. \[fig:SimulatedSpectra\], brown curve). The energy integrals of the predicted fluxes are 5.8(4), 1.85(13), and 0.51(4) $\times$ 10$^{-4} \,\rm cm^{-2} s^{-1}$ for MK2, WS, and MK1, respectively.
The FLUKA-simulated integrated neutron fluxes scale as $$11.2\,(12) : 3.6\,(4) : 1 \qquad \rm (MK2 : WS : MK1)$$ whereas the measured integrated muon fluxes [@Ludwig19-APP] in these three sites scale as $$1.28\,(14) : 1.49\,(15) : 1 \qquad \rm (MK2 : WS : MK1).$$
The strong differences found in the predicted energy-integrated neutron fluxes can therefore not be explained by the much lower differences in muon flux. The ($\alpha,n$) flux contributes at most 2% to the total flux, again not enough to explain this effect.
Instead, it seems that the high-density shielding of MK2 presents a much more efficient target for ($\mu,n$) neutron production than the rock wall of MK1.
In order to verify this hypothesis, a separate FLUKA simulation was carried out, again using 17GeV muons, and the literature [@Mei06-PRD] energy spectrum. Muons were incident from a spherical shell of 5m radius onto a sphere with 0.1m radius that consists of the material under study, which was given, in turn, by water, serpentinite, reinforced concrete, steel (74% Fe, 18% Cr, 8% Ni), and lead. The neutrons were then detected at 1m distance from the smaller sphere.
The results of this special FLUKA simulation are given in Table \[Table:mu\_n\]. The neutron yield in the major parts of the MK2 walls, steel and lead, is 7 and 15 times higher, respectively, than in serpentinite rock, the material of the MK1 walls, confirming the hypothesis.
At the highest neutron energies under study here, , there is no contribution by ($\alpha,n$) neutrons, and the predicted flux is entirely given by ($\mu,n$) neutrons. Of the FLUKA-predicted neutrons in the energy range, half originate in hadronic showers, and the other half from negative muon capture. At lower neutron energies, , only 20% of the predicted ($\mu,n$) neutrons are from showers, and 80% from capture.
For ($\mu,n$) neutrons from hadronic showers, there may in principle be more than one neutron detected within the time window of the data acquisition system. The FLUKA simulation shows that these effects contribute less than 2% to the neutron count rate.
Detection system design and calculated neutron sensitivity {#sec:Sensitivities}
==========================================================
This section is devoted to the calculation of the neutron sensitivities of the detectors used. First, some general concepts to study isotropic neutron fluxes are recalled for reference (Sec. \[subsec:General\]). Then, a physical motivation for the choice of the detector setup is given (Sec. \[subsec:DetectorSetup\]).
The detector sensitivities are then calculated for isotropic neutron fluxes for all detectors (Sec. \[subsec:FLUKA\]). Finally, possible angular effects are studied (Sec. \[subsec:Anisotropy\]).
General considerations {#subsec:General}
----------------------
It is expected that the present neutron flux is dominated by negative muon capture [@Mei06-PRD] and therefore approximately isotropic (Sec. \[sec:PredictedFlux\]). Therefore, some concepts developed previously for studying the isotropic neutron flux in nuclear reactors [@Fermi46-Pile] may be applied here. For the present purposes, the neutron flux density $\Phi$ is taken as $$\Phi = n_n v \label{eq:Fluxdef}$$ where $n_n$ is the number density of neutrons, and $v$ their average velocity. $\Phi$ is referred to as the “neutron flux,” measured in units cm$^{-2}$s$^{-1}$. Each detector is characterized by its neutron sensitivity $S$, given by $$\label{eq:Sensitivity}
R = S \Phi$$ where $R$ is the count rate, in units s$^{-1}$. The sensitivity is measured in units cm$^2$. For the case of a tube counter, where the counter length $l$ is much larger than the diameter $d$, the sensitivity and side surface area $A = \pi ld$ of the tube are connected by [@Khokonov10-PAN] $$\label{eq:Four}
S = \frac{R}{\Phi} = \frac{A \varepsilon j_+}{\Phi} = A \varepsilon \frac{j_+}{\Phi} = \frac{A \varepsilon}{4}$$ where $\varepsilon$ is the efficiency for the counter to detect a neutron that passes its side-surface area, and $$\begin{aligned}
\label{eq:Current}
j_+ & = & \frac{1}{4 \pi} \int\limits_0^{2\pi}d\phi \int\limits_0^{\infty}dr \int\limits_0^{\pi/2}d\theta \frac{\Phi}{\lambda_s} e^{-r/\lambda_s} \frac{|\cos \theta|}{r^2} r^2 \sin\theta \nonumber \\
& = & \frac{\Phi}{4}\end{aligned}$$ is the directed current which passes through a unit surface area [@Soodak50-Book], with $\lambda_s$ the mean free path for neutrons in the medium surrounding the detector. For a typical $^3$He counter with 1” diameter and a few atmospheres gas pressure, $\varepsilon \sim$ 1 for thermal neutrons. In the case of a nonmonochromatric flux, Eq.(\[eq:Sensitivity\]) becomes $$\label{eq:SpectralSensitivity}
R = \int\limits_{0}^{\infty} S(E) \Phi(E) dE$$ with $\Phi(E)$ the neutron flux per unit energy interval [@Lamarsh18-Book].
When combining Eqs. (\[eq:Sensitivity\]) and (\[eq:Four\]), the neutron flux $\Phi$ is then given by $$\label{eq:FluxAnalysis}
\Phi = \frac{R}{S} = \frac{4R}{\varepsilon A}.$$
These relations are needed when sensitivity data are compared to experiments with a neutron beam [@Beyer12-JINST; @Beyer18-EPJA e.g.], e.g., where typically the side surface area times efficiency $A\varepsilon$ are measured.
Detector setup adopted {#subsec:DetectorSetup}
----------------------
In order to detect neutrons of higher energies than thermal and epithermal, in addition to a bare $^3$He counter, it is necessary to use an array of $^3$He detectors surrounded with polyethylene moderators. As a rule of thumb, it is expected that the energy of the highest neutron sensitivity shifts toward higher energy with increasing amount of moderator, while the overall sensitivity decreases somewhat. As a result, by using several different moderator sizes, the neutron energy spectrum may be determined.
Above 10MeV neutron energy, further increases of the moderator size are impractical, because the decrease in overall sensitivity becomes severe. Instead, a lead-lined polyethylene moderator is needed. The lead acts as a neutron multiplier by way of of ($n,xn$) reactions (with $x \in \{2,3,4,5,...\}$), once the neutron energy exceeds the respective reaction threshold of $E_{{\rm threshold},x}$ = 7.4, 14.2, 22.3, 29.1MeV, respectively, for $x$ = 2–5 secondary neutrons.
In order to enable a physically meaningful fit of the count rates by a neutron energy spectrum, the detector assemblies have been designed so that the structures in the predicted neutron flux (Fig.\[fig:SimulatedSpectra\], Sec.\[sec:PredictedFlux\]) are matched with roughly similar structures in the energy-dependent neutron sensitivities. The adopted array of detector assemblies includes moderators ranging in size from 4.5 to 27.0cm (Table \[Table:Detectors\]). In particular,
- The bare detector A0 addresses the thermal peak expected for all three locations (Fig. \[fig:SimulatedSpectra\]).
- Assembly A3 matches the downscattered ($\mu,n$) peak expected in all sites at 0.3MeV.
- The difference of assemblies B8 and B9 addresses the ($\mu,n$) peak expected around 100MeV (Fig. \[fig:SimulatedSpectra\]).
- Assemblies A2 and B7 with their very wide sensitivity pattern match the flat spectrum from 10$^{-6}$ to 10$^{-1}$MeV expected at all three sites.
-------------- ---------- --------- ------- ------------------
Detector Remarks
Height Width
A0 Unmoderated
A1 4.5 4.5 70.0
A2 7.0 7.0 70.0
A3 12.0 12.0 70.0
A4 18.0 18.0 70.0
A5 22.5 22.5 70.0
A6 27.0 27.0 70.0
B7 7.0 7.0 40.5
B8 22.5 22.5 40.5
B9 21.0 21.0 40.5 0.5cm lead liner
-------------- ---------- --------- ------- ------------------
: \[Table:Detectors\] Dimensions of the polyethylene moderators used together with the $^3$He counters in campaigns A (A0–A6) and B (B7–B9), respectively. See text for details.
Evaluation of sensitivity calculation {#subsec:FLUKA}
-------------------------------------
As a first step for the sensitivity calculation, the geometry of each $^3$He tube and its moderator were modeled with FLUKA [@Ferrari05-FLUKA; @Boehlen14-NDS], 2011.2x.6. Subsequently, the energy-dependent neutron sensitivity $S(E)$ was calculated for an isotropic neutron flux, in 396 energy bins covering the range from $5 \times 10^{-11}$ to 600MeV. Each assembly was simulated separately in vacuum, and neutrons were assumed to come in isotropically from the surface of a spherical shell of 50cm radius. The results of the simulation are shown in Fig. \[fig:Sensitivity\].
The maximum sensitivity for the bare $^3$He detector from the FLUKA simulation was $S(7.5\times10^{-9}\,{\rm MeV})$ = 105cm$^2$. Since the detector was used with 10bar working pressure, this number cannot be directly compared with the data sheet value of $S$ = 144cm$^2$ for the 20bar case. Using analytical expressions given in Ref.[@Khokonov10-PAN], is found in the single-velocity approximation, matching the FLUKA result within 5%.
When comparing the calculated sensitivity curves $S(E)$ for assemblies A0–A6, it is observed that the energy of the peak sensitivity, which is near thermal energies for the bare detector A0, shifts step by step to higher energies. This behavior continues up to assemblies A5 and A6, with a peak sensitivity at $E\sim5$MeV.
In order to also address energies $E>$ 10MeV, the lead-lined assembly B9 is used. Over a wide energy range 10$^{-6}$MeV $< E <$ 10MeV, its sensitivity curve is similar to assembly B8. For $E>$ 10MeV, the sensitivity of assembly B9 rises due to its 0.5cm lead liner.
In order to study the mutual effect of neighboring detector assemblies on each other, for each case the sensitivity calculation was repeated adding the two neighboring detector assemblies as passive materials in the simulation. For assemblies A0 and A1, this causes the peak sensitivity to shift slightly to higher energies. For the other assemblies the sensitivity is reduced by up to 5%.
Study of angular distribution effects {#subsec:Anisotropy}
-------------------------------------
The calculated sensitivities (Fig.\[fig:Sensitivity\], Sec.\[subsec:FLUKA\]) were obtained for isotropic flux, but the $^3$He tubes included in the assemblies have a cylindric geometry, leading to a possible sensitivity to a nonisotropic angular distribution of the incident neutron flux.
The largest such anisotropy is expected for the case of MK1. There, instead of the laboratory walls, the lead castles of the high-purity germanium (HPGe) detectors form the principal ($\mu,n$) target, meaning that there will be more neutrons hitting the assemblies from the sides than from above or below.
Therefore, the sensitivity calculation was repeated for MK1 with the actual simulated neutron angular distribution, as obtained by FLUKA, fed into the FLUKA simulation of the sensitivity. It was found that for assemblies A4–A6, which are most sensitive to the relevant neutron energy range for unmoderated ($\mu,n$) neutrons, the sensitivity increased by 3.6–3.8%. For all other detectors at MK1 and for all assemblies at the other two sites WS and MK2, the relative effect was below $\leq$2%.
In order to take this effect into account, 4% additional systematic uncertainty is added to the error budget.
Experiment {#sec:Experiment}
==========
In the experiment, two sets of $^3$He-filled ionization chambers were used to determine both the total neutron flux and the energy spectrum. These counters are based on the $^3$He(n,p)$^3$H nuclear reaction, which has a $Q$ value of 764keV. The $^3$He(n,p)$^3$H cross section is ($5333 \pm 7$)barn for thermal neutrons and follows the 1/$v$ law (where $v$ is the neutron velocity) over a wide range of energies [@Mughabghab06-Book].
In order to check the performance of the detectors, runs with two different $^{252}$Cf spontaneous fission neutron sources with known activities were recorded prior to each measurement campaign. The $^{252}$Cf runs also serve as a reference for the shape of the neutron-induced pulse height spectrum.
Experimental campaign A {#subsec:CampaignA}
-----------------------
For campaign A, the detectors and moderators of the previous Canfranc neutron flux measurement [@Jordan13-APP; @Jordan13-APP_Corr] are used again here, but with 10bar instead of previously 20bar working pressure. An additional, bare $^3$He counter monitors the thermal neutron flux.
The campaign A detectors are seven $^3$He proportional counter tubes of type LND-252248[^2]. The tubes have an active length and diameter of 60.0 and 2.44cm, respectively, and are filled with 10bar working gas (97% $^3$He, 3% CO$_2$). The tubes were on loan from the BELEN / BRIKEN experiment [@Tarifeno17-JINST] which uses a large array of moderated $^3$He tubes to study $\beta$-delayed neutron emission [@Caballero18-PRC]. Here, the detector-moderator assemblies are called A0–A6, respectively (Table \[Table:Detectors\]).
Each tube was connected to one input channel of one of two 16-fold Mesytec MPR16-HV preamplifiers and supplied with 1450V high voltage. For each preamplifier, only the first group of four channels was used, in order to minimize the length of the cables and limit the noise. The differential output of each preamplifier channel used was sent to a Mesytec STM16+ shaping amplifier and then to a Struck SIS3302 VME 16-bit, 100MS/s sampling digitizer. In order to determine the dead time of the system, a 10Hz pulse generator signal was fed into each preamplifier channel. The digitizer self-triggered separately for each channel and passed the timestamped signal to the GASIFIC data acquisition system [@Agramunt16-NIMA], which then saved it to disk for offline analysis.
This experimental setup, including the data acquisition chain, was designed to be as similar as possible to the one used previously in Canfranc/Spain [@Jordan13-APP; @Jordan13-APP_Corr]. The only differences were, first, the $^3$He gas pressure of 10bar here instead of 20bar in Canfranc, and, second, the unmoderated counter A0 added in the present experiment.
{width="100.00000%"}
Each of the three sites MK2, WS, and MK1 was studied in turn. Counting times of 28.7, 12.5 and 8.6days were performed in the periods 15.12.2014–16.01.2015, 16.01.–03.02.2015, and 03.02.–12.02.2015, respectively. The measured dead time was always $\leq$0.3%.
At each site, the individual $^3$He tube-detector assemblies were placed parallel to each other, with typically 9cm of open space between them (Fig.\[fig:Locations\]). This placement was due to electronic noise considerations, which demand short cable lengths. The star-shaped configuration previously adopted in Canfranc [@Jordan13-APP; @Jordan13-APP_Corr] was not possible here due to space constraints. For the same reason, in the case of MK1, the four smallest assemblies A0–A4 had to be placed in the labyrinth area near the entrance of MK1, so that they were surrounded by the serpentinite only on five out of six sides.
The two measurement bunkers MK1 and MK2 contain three and six lead-shielded HPGe detectors, respectively, that are used for radioactivity measurements. MK2 has a heavy sliding door, and the present data acquisition was stopped typically 150minutes per day for maintenance and sample changes in the HPGe detectors. In MK1 and WS, the data acquisition was running continuously.
Prior to starting the ambient neutron measurement at each of the three sites, data were recorded with a $^{252}$Cf spontaneous fission neutron source placed centrally on top of each detector-moderator assembly, in turn. The neutron emission rate of the $^{252}$Cf source used for campaign A had been determined by Physikalisch-Technische Bundesanstalt(PTB), Braunschweig, Germany. Extrapolated to the time of the current measurement, it was $(6800 \pm 110)$s$^{-1}$.
Experimental campaign B {#subsec:CampaignB}
-----------------------
Campaign B concentrated on the neutron flux above 10MeV neutron energy. Three $^3$He counter tubes of type LND-252189[^3], on loan from Institut Laue-Langevin, Grenoble, France, were used. The tubes have an active length of 30.5cm, an outer diameter of 2.54cm, and are filled with 10bar working gas (97% $^3$He, 3% CO$_2$). The three detector-moderator assemblies are called B7–B9, respectively (Table \[Table:Detectors\]).
Assembly B7 was designed to resemble detector A2, in order to facilitate the connection between campaigns A and B. Detector assemblies B8 and B9 have similar polyethylene moderator sizes, but B9 is additionally fitted with a 0.5cm thick lead liner included at 5cm depth in the polyethylene matrix.
The tubes were operated at 1900V. For each one, the signal was amplified by a Mesytec MRS-2000 preamplifier and an Ortec 671 spectroscopy amplifier, then passed to a histogramming Ortec EtherNIM 919E analog-to-digital converter and multichannel buffer unit. The dead time, as obtained by the Gedcke-Hale algorithm [@Jenkins81-Book] implemented in the 919E unit, was $\leq$0.2%. The pulse height spectrum was saved every 30minutes on hard disk for later analysis.
As in campaign A, also in campaign B, the assemblies were placed subsequently in the three sites MK2, MK1, and WS (Figure \[fig:Locations\]). Data were taken for periods of 10.6, 25.3, and 14.0days in the periods 26.08.–09.09.2016, 09.09.–13.10.2016, and 13.10.–27.10.2016, respectively. The first three days of the MK2 campaign were excluded from the analysis due to excessive electronics noise. In all three sites, data taking was never interrupted, even during the daily maintenance and sample change periods in MK2.
For campaign B, the $^{252}$Cf benchmark measurements were performed at the HZDR Rossendorf surface site, using a $^{252}$Cf source with a neutron emission rate of , corrected for the time of the measurement.
{width="50.00000%"}{width="50.00000%"}
Generation of the pulse height spectra {#subsec:SpectrumGeneration}
--------------------------------------
The shape of the pulse height spectrum in a $^3$He detector is determined by the detection process, which starts with the $^3$He(n,p)$^3$H nuclear reaction. Then, the main peak at 764keV is due to the coincident detection of the full energies of both reaction products $p$ and $^3$H in the gas proportional counter. In addition, there are two steps at 191 and 573keV due to the so-called wall effect, when either the proton or the triton is not stopped inside the sensitive gas volume, but escapes into the detector walls.
These three features are clearly visible in the $^{252}$Cf and MK2 spectra (Fig. \[fig:Spectrum\], first and second rows). In the WS spectra, only the 573 and 764keV features are discernible (Fig. \[fig:Spectrum\], third row). For the MK1 case (Fig. \[fig:Spectrum\], last row), the 764keV peak is always clearly visible but the other features usually not. The lower signal-to-noise ratios for WS and MK1 are due to their lower neutron fluxes.
Given that the $^3$He counters are operated in the proportional regime, the spectra are calibrated linearly in deposited energy $E$ by using the above mentioned features in the $^{252}$Cf runs (Fig. \[fig:Spectrum\]).
The pulser used in the underground measurements of campaign A produced a peak slightly above 1000keV in the spectrum. For the generation of the pulse height spectra, events with data in all seven channels were assumed to be caused by the pulser and gated out. The remaining pulser feature apparent in Fig.\[fig:Spectrum\] at is thus due to nondetection of the pulser signal in another channel and serves to determine the dead time of the DAQ system.
For the analysis, a region of interest ranging typically from 180 to 820keV is adopted (region II in Fig. \[fig:Spectrum\]), encompassing all three neutron-induced features. It is noted that the detectors are mainly sensitive to thermal neutrons ($\sigma$ = 5333barn [@Mughabghab06-Book]). Due to the $1/v$ law, the cross section is 1500 times lower for $E_n$ = 56keV neutrons, which would register just outside the 820keV upper bound of region II. Any significant neutron contribution to the pulse height spectrum outside of region II would thus require a very unlikely neutron energy spectral shape. Instead, the remaining continuum at high energy is assumed to be due to intrinsic detector background [@Amsbaugh07-NIMA].
Determination of the neutron count rates {#subsec:ExpCountingrates}
----------------------------------------
In order to determine the neutron count rates, the observed energy-calibrated pulse height spectrum $D(E)$ in regions I–III (Fig, \[fig:Spectrum\]) is modeled as the sum of the following components [@Reginatto13-AIPCP]: first, a neutron response $N(E)$ determined in the run with the $^{252}$Cf source with negligible background in region II, scaled to match the measured spectrum, and second, a background term $B(E)$ that is given by $$D(E) = c_0 N(E) + B(E) \label{eq:Reginatto}.$$
At low pulse heights (region I in Fig. \[fig:Spectrum\]), the spectrum contains effects of electronic noise which grow exponentially toward the lowest energies and a more slowly background due to residual $\gamma$ rays. These two components are described by the sum of two exponential functions: $$c_1 \, \exp(-c_2 E) + c_3 \, \exp(-c_4 E).$$ At high pulse heights (region III in Fig. \[fig:Spectrum\]), the intrinsic $\alpha$-activity originating in the housing of the $^3$He gas proportional counter dominates, which is parametrized as $$c_5 \, (1 + c_6 E)$$ with $c_6 \ll$ 1 so that the total background is given by [@Reginatto13-AIPCP] $$B(E) = c_1 \, \exp(-c_2 E) + c_3 \, \exp(-c_4 E) + c_5 \, (1 + c_6 E).$$
For each pulse height spectrum, the parameters $c_i$ ($i \in\{\rm 0,1,...,6\}$) were then fit with the WinBUGS1.4 [@Winbugs03-Software] Bayesian Markov Chain Monte Carlo algorithm, using the following spectra $D_{j}(E)$ ($j \in\{\rm I, II, III\}$), depending on the region: $$\begin{aligned}
D_\text{I}(E) &= & c_1 e^{-c_2 E} + &c_3 e^{-c_4 E} + c_5 (1+c_6 E) \\
D_\text{II}(E) &= c_0 N(E)\, + &c_1 e^{-c_2 E} + &c_3 e^{-c_4 E} + c_5 (1+c_6 E) \\
D_\text{III}(E) &= & &c_3 e^{-c_4 E} + c_5 \, (1+c_6 E).\end{aligned}$$
In this way, the rapidly varying part $c_1 e^{-c_2 E}$ of the electronic noise is fitted mainly in region I, and the slowly varying part $c_3 e^{-c_4 E}$ mainly in regions I and III. The $\alpha$-induced background is fitted mainly in region III. The sought after neutron count rate is fitted in region II and encoded in parameter $c_0$. For practical reasons, the fit was performed on the discrete energy bins instead of the energy.
The resulting fitted background $B(E)$ and modeled total response $B(E) + c_0 N(E)$ are also shown in Fig. \[fig:Spectrum\]. For detector A3 (Fig. \[fig:Spectrum\], left column), the signal-to-noise ratio in region II is 16.0, 2.8, and 1.3, respectively, for MK2, WS, and MK1. For the worst case, i.e., detector B8, the signal-to-noise ratio is 5.4, 0.6, and 0.3, respectively, in the same three sites.
Initial interpretation of the count rate data {#sec:Initial}
---------------------------------------------
![\[fig:Countingrates\] Neutron count rates obtained in the present work for Felsenkeller MK2 (red diamonds), WS (red downward triangles), and MK1 (red upwards triangles). For detectors A1–A6, also a comparison with previous rates obtained in Canfranc (blue diamonds, [@Jordan13-APP; @Jordan13-APP_Corr]) is shown.](Figure6_v6.pdf){width="\columnwidth"}
The resulting neutron count rates $R_k(x)$ with $x$ the site and $k$ the number of the assembly are shown in Fig. \[fig:Countingrates\]. The relative uncertainty, as given by the count statistics and the uncertainty on the fitted background, is 2%–6% for MK2, 1%–5% for WS, and 3%–9% (13% for the case of B8) for MK1.
The count rate data already permit some first observations, prior to further analysis. First, assemblies A1–A6 follow the same general pattern in all three sites studied, and also in the previous Canfranc measurement [@Jordan13-APP; @Jordan13-APP_Corr] $$R_k({\rm MK2}) > R_k({\rm WS}) > R_k({\rm MK1}) > R_k({\rm Canfranc}) \nonumber$$ for $k \in \{1,...,6\}$. Second, the differences between sites MK2 and MK1 are similar to the differences between the lower of the two, MK1, and Canfranc. Third, the thermal neutrons in the unmoderated detector A0 break this pattern and show a higher thermal flux in the unshielded WS than in the heavily shielded MK2.
For MK2, WS, and MK1, the pattern observed for A1–A6 is again evident for B7–B9. The pairs of similar assemblies A2–B7 and A5–B8 show again a similar pattern when MK1 and MK2 are compared. The general patterns of the neutron count rates are therefore consistent across campaigns A and B.
Comparison of $^{252}$Cf source data and FLUKA simulation {#subsec:Cf}
---------------------------------------------------------
The FLUKA predictions from the FLUKA simulation (Sec. \[sec:Sensitivities\]) were compared with the data from the $^{252}$Cf source runs. Since the FLUKA code has been extensively validated for its description of neutron interactions [@Agosteo12-NIMA], e.g., the $^{252}$Cf runs serve to verify the correct implementation of detector and moderator geometry in the present simulation.
Here, the $^{252}$Cf source was modeled centrally on top of the detector-moderator assembly in FLUKA. The Mannhart $^{252}$Cf spectrum [@IAEA-Tecdoc-410] with a peak at 0.75MeV was adopted to describe the neutron emission from the $^{252}$Cf source. Other radiations emanating from the source were neglected, because the $^3$He counter is not very sensitive to $\gamma$ rays. Any detected $\gamma$ rays would form a background that is exponentially decreasing with energy and are subtracted from the experimental signal (see above, Sec. \[subsec:ExpCountingrates\]).
Assemblies A3 and B7 show the highest overall sensitivities to the $^{252}$Cf neutrons (Fig.\[fig:Sensitivity\]). For these two cases, the simulated count rates are (3$\pm$2)% and (5$\pm$15)% lower, respectively, than the relevant experimental rate. The uncertainty is in both cases dominated by the calibration of the $^{252}$Cf source used: 2% for the source used for A3 (Sec.\[subsec:CampaignA\]) and 15% for the source used for B7 (Sec.\[subsec:CampaignB\]).
For assemblies A2, A4–A6, and B8–B9, the detector-to-detector ratios of sensitivities are well reproduced, but the simulated rate is up to 10% below the measured rate. In those cases, due to the mismatch between the $^{252}$Cf spectrum and the spectral sensitivity the observed count rate is influenced by features such as laboratory walls, detector stands, or other detectors, which are imperfectly modeled in the simulation. For assemblies A0–A1, thermal or epithermal neutrons dominate over the emitted 0.75MeV neutrons; thus, the $^{252}$Cf data cannot be used.
As a result of these tests, 5% is adopted as systematic uncertainty for the sensitivity calculated by FLUKA.
Determination of the experimental neutron flux in Felsenkeller {#sec:Fit}
==============================================================
![\[fig:All\_spectra\] Unfolded spectra using the MAXED code, for MK2(red), WS(blue), and MK1(green). The $1\sigma$ error bars are shown as shaded areas; see text for details. ](Figure8_v20.pdf){width="\columnwidth"}
The measured count rates (Sec. \[subsec:ExpCountingrates\]) and calculated sensitivities (Sec. \[sec:Sensitivities\]) were used to determine the neutron energy spectrum and flux. The results are shown in this section.
General approach for the fit {#subsec:FitApproach}
----------------------------
As a first step, for each detector $i$, the integral in Eq. (\[eq:SpectralSensitivity\]) is approximated by a sum, using the calculated spectral sensitivities (Fig. \[fig:Sensitivity\], Sec. \[sec:Sensitivities\]) in the 396 energy bins given by FLUKA, $$\label{eq:SpectralSensitivitySum}
R_i^\text{exp} = \sum\limits_{j=1}^{396} S_i(E_j) \Phi(E_j) .$$ Here, $R_i^\text{exp}$ with $i\in\{0,...,9\}$ is the experimental count rate in assembly $i$, $S_i(E_j)$ the calculated sensitivity for assembly $i$ and central energy $E_j$ of energy bin $j$ for , and $\Phi(E_j)$ the sought after neutron flux in the same energy bin. This can also be expressed as a matrix equation
$$\begin{pmatrix}
R_0^\text{exp} \\
\cdots{} \\
R_9^\text{exp} \\
\end{pmatrix}
=
\begin{pmatrix}
S_0(E_1) & \cdots & S_0(E_{396}) \\
\cdots & \cdots & \cdots \\
S_9(E_1) & \cdots & S_9(E_{396}) \\
\end{pmatrix}
\times
\begin{pmatrix}
\Phi(E_1) \\
\cdots{} \\
\Phi(E_{396}) \\
\end{pmatrix}.$$
For the fit to determine $\Phi(E_j)$, this linear inverse problem must be solved. To this end, subsequently two different codes called MAXED [@Reginatto02-NIMA] and GRAVEL [@Matzke94-Gravel], respectively, were used. Both codes are included in the Nuclear Energy Agency’s UMG 3.3 package [@NEA04-UMG]. [*Nota bene*]{} these codes do not give exact mathematical solutions to problem (\[eq:SpectralSensitivitySum\]), because there are only ten measured count rates $R_i^\text{exp}$, one for each detector assembly.
Therefore, the solution space is limited by using physically motivated [*a priori*]{} spectra as starting points of the fit. Here, for each of the three sites studied, the respective predicted spectrum $\Phi^\text{prior}(E)$ (Sec. \[sec:PredictedFlux\]) was taken as a starting point. Both codes then derive a new spectrum based on the starting point and on the measured data.
The first code used here, MAXED, performs the fit by maximizing the entropy function [@Reginatto02-NIMA]: $$\begin{aligned}
S = - \sum\limits_{j=1}^{396} \left( \Phi(E_j)\, \ln \frac{\Phi(E_j)}{\Phi^\text{prior}(E_j)} + \Phi^\text{prior}(E_j) - \Phi(E_j) \right).\end{aligned}$$ In order to remain close to a physically reasonable scenario, the solution is constrained by a limit called $\Omega$ on the $\chi^2$ parameter, $$\begin{aligned}
\Omega \stackrel{!}{\geq} \chi^2 = \sum\limits_{i=0}^{9} \left( \frac{R_i^\text{calc}-R_i^\text{exp}}{\Delta R_i^\text{exp}} \right)^2 \label{eq:ChiSquare}\end{aligned}$$ where $R_i^\text{calc}-R_i^\text{exp}$ is the difference between the calculated and observed count rates for detector $i$ and $\Delta R_i^\text{exp}$ is the experimental uncertainty of $R_i^\text{exp}$. $\Omega$ is usually set equal to the number of detectors. For the present purposes, solutions with $\Omega=10$–16 were used. In the fit process, it is assumed that the errors of $R_i^\text{exp}$ are normally distributed with zero mean and variance $(\Delta R_i^\text{exp})^2$.
The second code, GRAVEL, uses a slightly modified version of the SAND-II code [@McElroy67-Book] and works iteratively. Based on the calculated sensitivities $S_i(E_j)$ and the spectrum $\Phi^\text{prior}(E)$, the expected neutron rates $R_i^\text{calc}$ are calculated and compared to the measured rates $R_i^\text{exp}$ to provide a correction factor $f_i$ for each detector $i$.
For each of the 396 energy bins $j$, the correction factor $f_i$ is then weighted by the detector sensitivity $S_i(E_j)$ for this detector and energy bin and is applied. The resulting spectrum is then used as the starting spectrum for the next iteration. The iteration process stops once the requested value $\Omega$ was reached.
Extracted neutron fluxes and their uncertainties
------------------------------------------------
![\[fig:TestSpectra\] The three extreme-case test spectra called A, B, and C used for the error analysis. See text for details.](Figure7_v11b.pdf){width="\columnwidth"}
The measured neutron fluxes as extracted with MAXED are shown in Fig. \[fig:All\_spectra\]. For both MAXED and GRAVEL, the integral flux is listed in Table \[Table:Flux\].
The error bands in the energy spectra (shaded regions in Fig. \[fig:All\_spectra\]) were obtained by using the IQU software that is contained within the UMG 3.3 package. IQU considers variations of the measured data, quantified by their quoted uncertainty, and performs a sensitivity analysis and uncertainty propagation [@NEA04-UMG].
For all three locations, the relative errors determined by IQU are 10%–15% for energies below 10MeV, and 3–4 times higher above 10 MeV. This increase in uncertainty is due to the declining overall neutron sensitivity (Fig. \[fig:Sensitivity\]) and a resulting decline in detected neutron events, and also due to the fact that the sensitivity of the lead-lined moderator keeps increasing toward higher energies and, thus, cannot be normalized.
Since the IQU error determination is only available for MAXED and depends on the initial unfolding spectrum, an even more conservative approach was used to calculate the uncertainties of the integrated flux values $\Phi_\text{exp}$ shown in Table \[Table:Flux\]. In order to exclude a possible bias due to the characteristics of the FLUKA-derived predicted spectra (Fig. \[fig:SimulatedSpectra\]), three hand-designed extreme test spectra were used: an ($\alpha,n$)-dominated (Test A), a thermal-dominated (Test B), and a flat spectrum , all three shown in Fig. \[fig:TestSpectra\]. These three spectra should not be viewed as real physics cases but as extreme bounds encompassing all plausible physical solutions.
Using the spectra Test A, B, and C, the unfolding was repeated with both MAXED and GRAVEL, resulting in integrated test fluxes $\Phi_i$ ($i \in \{\text{A,B,C}\}$). The error $\Delta\Phi_\text{exp}$ was then taken to be $\Delta\Phi_\text{exp} = \sqrt{\sum_{i}(\Phi_i-\Phi_\text{exp})^2/3}$. This procedure results in 7%–9% uncertainty for the integrated flux. If one were to use only the IQU errors instead, a much lower uncertainty of typically 1% would be found, close to the combined statistical uncertainty of the count rates. Thus, the above described and adopted approach with the test spectra is conservative.
When adding the 5% uncertainty adopted from the $^{252}$Cf test (Sec. \[subsec:Cf\]) and 4% uncertainty from angular effects (Sec. \[subsec:Anisotropy\]), the final flux uncertainty is 9%–11% (Table \[Table:Flux\]).
------------------ ------------- ---------------------- ------------ -- ------------- ------------------ -----------
Location
inside
Felsenkeller ($\mu,n$) Total GRAVEL MAXED
MK2 5.8(4) 0.032(8) 5.8(4) 4.6(4) 4.6(4)
WS 1.65(13) 0.20(5) 1.85(13) 1.96(15) $2.00\,(16)$
MK1 0.50(4) $Out: 0.013\,(3)$ 0.51(4) 0.61(5) $0.63\,(6)$
$In: 0.00007\,(2)$
------------------ ------------- ---------------------- ------------ -- ------------- ------------------ -----------
[|l|l|D[.]{}[.]{}[-1]{}|]{} Location & Reference &\
& &\
Ground level & [@Wiegel02-NIMA_Surface] &\
YangYang, 2000m.w.e. & [@Park13-Apradiso] & 0.67(2)\
Canfranc, 2400m.w.e. (revised) & [@Jordan13-APP; @Jordan13-APP_Corr] & 0.138(14)\
Gran Sasso, 3800m.w.e. & [@Belli89-NCA] & 0.038(2)\
Felsenkeller MK2, 140m.w.e. & Present & 4.6(4)\
Felsenkeller WS, 140m.w.e. & Present & 1.96(15)\
Felsenkeller MK1, 140m.w.e. & Present & 0.61(5)\
Discussion {#sec:Discussion}
==========
Experimental neutron flux at Felsenkeller
-----------------------------------------
The count rates of the individual detectors Fig.\[fig:Countingrates\]) show a general pattern that is similar for all three sites studied and also for the previous measurement in Canfranc, Spain [@Jordan13-APP; @Jordan13-APP_Corr]. There is a significant overall difference, by roughly a factor of 7, between the count rates observed in sites MK1 and MK2. The relative differences are largest for the assemblies moderated by 7–18cm polyethylene, which are mainly sensitive in the neutron energy range from 10$^{-6}$ to 1MeV, and smallest for the unmoderated detector.
This overall trend between the three sites MK2, WS, and MK1 had already been observed previously in a study with two moderated $^3$He counters, both of them with an energetic response similar to the present assembly A3 [@Niese07-JRNC].
The neutron fluxes and energy spectra resulting from the deconvolution algorithm (Fig. \[fig:SimExp\]) are again quite different between the three particular sites studied in Felsenkeller, again with the lowest flux for MK1, the highest for MK2, and WS in between.
![\[fig:SimExp\] Comparison of the experimental (full lines) and predicted (dashed lines) neutron spectra in the three Felsenkeller sites MK2 (red), WS (blue), and MK1 (green). See text for details. ](Figure9_v21.pdf){width="\columnwidth"}
Comparison of data and simulation {#subsec:Comparison_data_simulation}
---------------------------------
The energy-integrated neutron fluxes predicted by FLUKA follow the trend of the data (Table \[Table:Flux\]), but they show relative differences of +(26$\pm$10)%, –(6$\pm$10)%, and –(16$\pm$10)%, for MK2, WS, and MK1, respectively.
When comparing predicted and experimental energy spectra (Fig. \[fig:SimExp\]), it is seen that in all cases, the simulation overpredicts the intermediate to fast energy range . In contrast, the thermal neutron flux ($E$ $\sim$ 2.5$\times$10$^{-8}$MeV) is always underpredicted. These two effects may be due to existing low-density materials such as plexiglass housings for detectors, wooden tables and shelves, and the liquid nitrogen in the dewars of the HPGedetectors that were all neglected in the FLUKA simulation. These materials may moderate higher energy neutrons to lower energies, explaining the disequilibrium between the two above mentioned energy ranges. In addition, thermal neutrons are more likely to be absorbed by structural materials, which may explain some of the overprediction of the observed energy-integrated flux in MK2 and WS.
When considering MK1, where also the energy-integrated flux is somewhat underpredicted, it is noted that for this case of generally very low ($\mu,n$) production, even limited quantities of neglected materials may enhance neutron production. In addition, due to space constraints, the smaller detectors A0–A3 had to be placed in a part of MK1 surrounded only on five out of six sides by serpentinite, with the sixth side showing WS shielding conditions.
For the case of WS, the data show a pronounced thermal peak that is not present in the simulation. This may in principle be caused by moderation in the humid rock walls [@Wulandari04-APP], where the present FLUKA simulation assumed 3% water content (by mass). Since the true humidity changes with weather conditions, i.e. air temperature, humidity and precipitation of the preceding days, it is not easy to model it without daily [*in situ*]{} measurements. Another possibility, at least in principle, are thermal neutrons leaking in through the doors of the laboratory.
In order to assess the effects of humidity, FLUKA simulations with varying water content were performed for all three sites ranging from 3% (adopted value) up to 12% (extreme case). For 12% water content the integrated predicted fluxes change by $-13$% (WS), not at all (inside MK1), and $-7$% (MK2).
In the energy region from 10 to 300MeV, due to the limited statistics and the low observed flux, the flux data are only 2–3$\,\sigma$ above zero (Table \[Table:Flux\]). The prediction is consistent with these limited-precision data.
When considering the matching between simulation and underground neutron flux data, it is important to note that it was recently reported that GEANT4 simulations underpredicted the flux of ($\mu,n$) neutrons [@Du18-APP] in a laboratory with just 13m.w.e. rock overburden. More recent work by the same group also included FLUKA simulations and showed a good match of simulation and data [@Kneissl19-APP]. Earlier work in deep-underground settings reported a good match between simulation and data both for GEANT4[@Zhang14-PRD] and FLUKA[@Empl14-JCAP]. The present data suggest a reasonable description of the ($\mu,n$) flux by FLUKA at 140m.w.e.
Neutron flux at various underground sites
-----------------------------------------
The energy-dependent neutron flux from the present work is compared to previous measured neutron spectra in other sites, using a logarithmic presentation (Fig. \[fig:ExperimentalSpectrum\]). For the same sites, the integrated fluxes are shown in Table \[Table:FluxAllLabs\].
For the Earth’s surface, data by the PTB NEMUS group are used [@Wiegel02-NIMA_Surface]. There, the 100MeV peak is more than 3 orders of magnitude stronger than at Felsenkeller. Since the Felsenkeller muon flux is only 40 times lower than at the Earth’s surface [@Olah16-NPA6; @Ludwig17-Master; @Ludwig19-APP], the remainder of the difference is presumably due to neutrons produced in the atmosphere, which are completely absorbed at Felsenkeller depth.
Now, the present data are briefly compared to previous neutron spectrum measurements in deep-underground laboratories. The first example, the YangYang laboratory in South Korea, is located below 700m of rock (2000m.w.e.). There, the neutron flux was studied using a Bonner sphere spectrometer modeled on the PTB NEMUS system [@Wiegel02-NIMA], including several modified spheres with neutron multipliers [@Park13-Apradiso]. The YangYang spectrum is much flatter than the present one, and the downscattered neutron peak at 0.3MeV is not evident at YangYang. It is interesting that the reported total flux at YangYang is similar to the present MK1 result, even though the depth is much greater and the reported $^{238}$U and $^{232}$Th contents are rather low, 6–23Bq/kg [@Lee06-PLB].
The spectrum in the Canfranc underground laboratory, Spain (2400m.w.e.), is obtained by multiplying the original data [@Jordan13-APP] by a factor of 4 [@Jordan13-APP_Corr]. The Canfranc spectrum has a similar structure as MK2, but a lower overall flux. Neither a bare nor a lead-lined $^3$He counter was used at Canfranc. As a consequence, the flux obtained at thermal energies and above 10MeV are affected by large uncertainties, as discussed in Ref.[@Jordan13-APP].
The Canfranc flux, after revision [@Jordan13-APP; @Jordan13-APP_Corr], is somewhat higher than expected for its depth of 2400m.w.e. This may in principle be due to a very high $^{238}$U/$^{232}$Th content in the rock or due to a different rock chemical composition leading to a more efficient ($\alpha,n$) process. Still, it seems advisable to remeasure the spectral neutron flux at this site.
For Gran Sasso (3800m.w.e.), Belli [*et al.*]{} [@Belli89-NCA] spectrum is shown (Fig. \[fig:ExperimentalSpectrum\]). The Gran Sasso flux, similar to YangYang, shows much less structure than Canfranc or Felsenkeller, in that case possibly due to the limited number of energy bins used. In the future, it would be interesting to obtain better data, in particular near 0.3MeV.
Summary and conclusions {#sec:Conclusion}
=======================
Using two sets of altogether nine moderated $^3$He counters and one unmoderated $^3$He counter, the neutron flux and spectrum were investigated in three sites in tunnel IV of the Felsenkeller underground facility, Dresden, Germany. The resulting energy-integrated fluxes were ($0.61 \pm 0.05$), ($1.96 \pm 0.15$), and ($4.6 \pm 0.4) \times 10^{-4}$cm$^2$s$^{-1}$, for sites MK1, WS, and MK2, respectively.
The data are matched reasonably well by a detailed FLUKA simulation taking into account the known muon flux and angular distribution and the known specific radioactivity of the rock.
In view of the crucial importance of a proper understanding of the underground neutron background, it seems advisable to reinvestigate the underground neutron flux and energy spectrum at several other sites, including deep-underground laboratories.
The present data were instrumental in the planning for the new Felsenkeller underground ion accelerator laboratory, located in tunnelsVIII and IX of the same underground site studied here [@Bemmerer18-SNC]. In particular, the shielding for the new laboratory was designed to resemble the lowest neutron flux site found here, MK1. Neutron background data from the new facility will be reported in due course.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
The authors are indebted to Alfredo Ferrari (CERN) for helpful discussions regarding the FLUKA code. — Financial support by DFG, the Helmholtz Association (NAVI and ), the Spanish Ministerio de Economía y Competitividad (Grants No. FPA2014-52823-C2, No. FPA2017-83946-C2, No. RTI2018-098868-B-I00, and No. SEV-2014-0398 / program Severo Ochoa), and the COST Association (ChETEC CA16117) is gratefully acknowledged.
[10]{}
G. [Heusser]{}, Annu. Rev. Nucl. Part. Sci. [**45**]{}, 543 (1995).
J. A. Formaggio and C. Martoff, Annu. Rev. Nucl. Part. Sci. [**54**]{}, 361 (2004).
R. [Davis]{}, Rev. Mod. Phys. [**75**]{}, 985 (2003).
V. N. Gavrin, , in [*[Proceedings of the 5$^{\rm th}$ International Solar Neutrino Conference]{}*]{}, edited by M. Meyer and K. Zuber [(World Scientific, Singapore, 2019)]{}.
A. B. [McDonald]{}, Rev. Mod. Phys. [**88**]{}, 030502 (2016).
T. [Kajita]{}, Rev. Mod. Phys. [**88**]{}, 030501 (2016).
A. Bettini, J. Phys. Conf. Ser. [**120**]{}, 082001 (2008).
Q. R. [Ahmad]{} [*et al.*]{}, Phys. Rev. Lett. [**89**]{}, 011301 (2002).
M. [Agostini]{} [*et al.*]{}, Phys. Rev. Lett. [**120**]{}, 132503 (2018).
N. Abgrall [*et al.*]{}, , 020027 (2017).
K. [Zuber]{}, Phys. Lett. B [**519**]{}, 1 (2001).
E. [Aprile]{} [*et al.*]{}, Phys. Rev. Lett. [**121**]{}, 111302 (2018).
J. [Klinger]{} and V. A. [Kudryavtsev]{}, Phys. Rev. Lett. [**114**]{}, 151301 (2015).
R. [Bernabei]{} [*et al.*]{}, Eur. Phys. J. C [**73**]{}, 2648 (2013).
C. Broggini, D. Bemmerer, A. Caciolli, and D. Trezzi, Prog. Part. Nucl. Phys. [**98**]{}, 55 (2018).
P. [Belli]{} [*et al.*]{}, Nuovo Cimento A [**101**]{}, 959 (1989).
F. [Arneodo]{} [*et al.*]{}, Nuovo Cimento A [**112**]{}, 819 (1999).
D. [Jordan]{} [*et al.*]{}, Astropart. Phys. [**42**]{}, 1 (2013).
D. [Jordan]{} [*et al.*]{}, , 102372 (2020).
C. [Zhang]{} and D. M. [Mei]{}, Phys. Rev. D [**90**]{}, 122003 (2014).
A. [Sonay]{} [*et al.*]{}, Phys. Rev. C [**98**]{}, 024602 (2018).
A. [Best]{} [*et al.*]{}, Nucl. Instrum. Methods Res., Sect. A [**812**]{}, 1 (2016).
Z. [Debicki]{} [*et al.*]{}, Nucl. Instrum. Methods Res., Sect. A [**910**]{}, 133 (2018).
S. Niese, J. Radioanal. Nucl. Chem. [**272**]{}, 173 (2007).
A. [Rindi]{}, F. [Celani]{}, M. [Lindozzi]{}, and S. [Miozzi]{}, Nucl. Instrum. Methods Res., Sect. A [**272**]{}, 871 (1988).
Q. [Du]{} [*et al.*]{}, Nucl. Instrum. Methods Res., Sect. A [**889**]{}, 105 (2018).
A. [da Silva]{} [*et al.*]{}, Nucl. Instrum. Methods Res., Sect. A [**354**]{}, 553 (1995).
Q. [Du]{} [*et al.*]{}, Astropart. Phys. [**102**]{}, 12 (2018).
R. [Knei[ß]{}l]{} [*et al.*]{}, Astropart. Phys. [**111**]{}, 87 (2019).
F. [Ludwig]{} [*et al.*]{}, Astropart. Phys. [**112**]{}, 24 (2019).
T. [Sz[ü]{}cs]{} [*et al.*]{}, Eur. Phys. J. A [**55**]{}, 174 (2019).
D. [Mei]{} and A. [Hime]{}, Phys. Rev. D [**73**]{}, 053004 (2006).
D. Bemmerer [*et al.*]{}, , in [*[Solar Neutrinos: Proceedings of the 5$^{\rm th}$ International Solar Neutrino Conference]{}*]{}, edited by M. Meyer and K. Zuber ([World Scientific, Singapore, 2019)]{}, pp. 249–263.
W. P[ä]{}lchen and H. Walter, editors, ([Schweizerbart’sche Verlagsbuchhandlung]{}, Stuttgart, 2008).
M. Grieger, , Master’s thesis, Technische Universit[ä]{}t Dresden, 2016.
W. Helbig, S. Niese, and D. Birnstein, Isotopenpraxis [**20**]{}, 60 (1984).
S. Niese, Appl. Radiat. Isot. [**47**]{}, 1109 (1996).
S. Niese, M. Köhler, and B. Gleisberg, J. Radioanal. Nucl. Chem. [**233**]{}, 167 (1998).
M. Köhler [*et al.*]{}, Appl. Radiat. Isot. [**67**]{}, 736 (2009).
H. [Wulandari]{}, J. [Jochum]{}, W. [Rau]{}, and F. [von Feilitzsch]{}, Astropart. Phys. [**22**]{}, 313 (2004).
A. Ferrari, P. Sala, A. Fass[ò]{}, and J. Ranft, , , 2005.
T. T. [B[ö]{}hlen]{} [*et al.*]{}, Nucl. Data Sheets [**120**]{}, 211 (2014).
S. [Westerdale]{} and P. D. [Meyers]{}, Nucl. Instrum. Methods Res., Sect. A [**875**]{}, 57 (2017).
E. Fermi, ([U.S. Atomic Energy Commission]{}, [Oak Ridge, TN.]{}, 1946).
A. K. Khokonov, Y. V. Savoiskii, and A. V. Kamarzaev, Phys. At. Nucl. [**73**]{}, 1482 (2010).
H. Soodak and E. C. Campbell, ([John Wiley & Sons]{}, [New York]{}, 1950).
J. R. Lamarsh and A. J. Baratta, , [4th]{} ed. (Pearson, [Hoboken, NJ]{}, 2018).
R. [Beyer]{} [*et al.*]{}, J. Instrum. [**7**]{}, C02020 (2012).
R. [Beyer]{} [*et al.*]{}, Eur. Phys. J. A [**54**]{}, 58 (2018).
S. F. Mughabghab, ([Elsevier Science]{}, Amsterdam, 2006).
A. Tarife[ñ]{}o-Saldivia [*et al.*]{}, J. Instrum. [**12**]{}, P04006 (2017).
R. [Caballero-Folch]{} [*et al.*]{}, Phys. Rev. C [**98**]{}, 034310 (2018).
J. [Agramunt]{} [*et al.*]{}, Nucl. Instrum. Methods Res., Sect. A [**807**]{}, 69 (2016).
R. Jenkins, R. W. Gould, and D. Gedcke, (Marcel Dekker, New York and Basel, 1981), Chap. 4.8.5, pp. 266–271.
J. F. [Amsbaugh]{} [*et al.*]{}, Nucl. Instrum. Methods Res., Sect. A [**579**]{}, 1054 (2007).
M. [Reginatto]{}, A. [Kasper]{}, H. [Schuhmacher]{}, B. [Wiegel]{}, and A. [Zimbal]{}, AIP Conf. Proc. [**1553**]{}, 77 (2013).
D. J. Spiegelhalter, A. Thomas, N. Best, and D. Lunn, , <https://www.mrc-bsu.cam.ac.uk/software/bugs/the-bugs-project-winbugs/>, 2003.
S. [Agosteo]{} [*et al.*]{}, Nucl. Instrum. Methods Res., Sect. A [**694**]{}, 55 (2012).
W. Mannhart, , in [*[Properties of Neutron Sources. Proceedings of an Advisory Group Meeting on Properties of Neutron Sources organized by the International Atomic Energy Agency and held in Leningrad, USSR, 9-13 June 1986]{}*]{}, [IAEA-Tecdoc-410]{} (International Atomic Energy Agency, Vienna, 1987).
M. [Reginatto]{}, P. [Goldhagen]{}, and S. [Neumann]{}, Nucl. Instrum. Methods Res., Sect. A [**476**]{}, 242 (2002).
M. Matzke, Unfolding of the pulse height spectra: the HEPRO program system (Wirtschaftsverlag NW, Verlag für Neue Wissenscahft, Bremerhaven, 1994).
, , <https://www.oecd-nea.org/tools/abstract/detail/nea-1665>, 2004, .
W. N. McElroy, S. Berg, T. B. Crockett, and R. G. Hawkins, ([Air Force Weapons Laboratory]{}, Kirtland Air Force Base, New Mexico, 1967).
B. [Wiegel]{}, A. V. [Alevra]{}, M. [Matzke]{}, U. J. [Schrewe]{}, and J. [Wittstock]{}, Nucl. Instrum. Methods Res., Sect. A [**476**]{}, 52 (2002).
H. Park, J. Kim, Y. Hwang, and K.-O. Choi, Appl. Radiat. Isot. [**81**]{}, 302 (2013).
A. [Empl]{}, E. V. [Hungerford]{}, R. [Jasim]{}, and P. [Mosteiro]{}, J. Cosmol. Astropart. Phys. **2014**, 064 (2014).
L. [Ol[á]{}h]{} [*et al.*]{}, J. Phys. Conf. Ser. [**665**]{}, 012032 (2016).
F. Ludwig, , Master’s thesis, Technische Universit[ä]{}t Dresden, 2017.
B. [Wiegel]{} and A. V. [Alevra]{}, Nucl. Instrum. Methods Res., Sect. A [**476**]{}, 36 (2002).
H. S. [Lee]{} [*et al.*]{}, Phys. Lett. B [**633**]{}, 201 (2006).
[^1]: A weak contribution by the natural $^{235}$U decay chain is neglected here.
[^2]: LND Inc., Oceanside, New York, USA.
[^3]: Again LND Inc., Oceanside, New York, USA.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In earlier work, Lomeli and Meiss [@lomeli09gff] used a generalization of the symplectic approach to study volume preserving generating differential forms. In particular, for the ${\mathbb{R}}^3$ case, the first to differ from the symplectic case, they derived thirty-six one-forms that generate exact volume preserving maps. In [@xue2014gf], Xue and Zanna studied these differential forms in connection with the numerical solution of divergence-free differential equations: can such forms be used to devise new volume preserving integrators or to further understand existing ones? As a partial answer to this question, Xue and Zanna showed how six of the generating volume form were naturally associated to consistent, first order, volume preserving numerical integrators. In this paper, we investigate and classify the remaining cases. The main result is the reduction of the thirty-six cases to five essentially different cases, up to variable relabeling and adjunction. We classify these five cases, identifying two novel classes and associating the other three to volume preserving vector fields under a Hamiltonian or Lagrangian representation. We demonstrate how these generating form lead to consistent volume preserving schemes for volume preserving vector fields in ${\mathbb{R}}^3$.'
author:
- Olivier Verdier
- Huiyan Xue
- Antonella Zanna
bibliography:
- 'volumepreserving.bib'
title: 'A classification of volume preserving generating forms in $\RR^3$'
---
Introduction and background {#laha}
===========================
The scope of this paper is the study volume preserving generating forms, with the ultimate goal of exploiting these differential forms to obtain consistent numerical methods that preserve volume for arbitrary volume preserving vector fields. This task is particularly hard: there exist *no-go* theorems [@chartier2007preserving; @iserles07] stating that it is not possible to construct volume preserving methods for generic $n$-dimensional volume preserving vector fields within the class of B-series methods, a class that includes classical integrators like Taylor-expansion based methods, Runge–Kutta methods and multistep methods. On the other hand, volume preserving methods can be constructed using the technique of *splitting* [@McLachlan2002]. Splitting methods correspond to P-series (“P” for partitioned systems), a generalization of B-series. Several splitting techniques can be adopted. The earliest and best known splitting consists in decomposing the vector field in 2D Hamiltonian sub-systems [@kang95vpa], which are then solved by a symplectic method. More recently, research has focussed on explicit splitting methods for classes of vector fields, like polynomial or trigonometric, which are wide enough to include most interesting cases, but not as large as the space of all possible vector fields [@quispel03evp; @mclachlan04egi; @mclachlan09evp; @Xue:2012mf; @zanna2014tensor].
Differently from the symplectic case, the generating form approach to generate volume preserving numerical methods is not well understood. Earlier work by [@shangXXgf1; @Shang1994] extends the Hamiltonian technique of [@F1986; @FW1989], using linear maps in the product space, to volume preserving forms, thus obtaining an equivalent of the Hamilton-Jacobi differential equation [@Shang1994]. To obtain a first and second order scheme, Shang had to impose simplifying conditions, requiring the transformation matrix to be a special case of Hadamard matrix. However, the numerical integrators by this approach are quite complicated, as they are defined via implicit maps, although the approach is valid for arbitrary vector fields. Another generating-functions related approach is due to [@GRW199526]: through a special combinations of explicit and implicit maps, Quispel shows that the resulting method is volume preserving. This is a “correction method”: starting from an arbitrary numerical integrator, one has to consider an extra term (the corrrection) for volume preservation. The above mentioned two approaches do not use differential forms directly, rather, they use the equivalent condition on the determinant of the Jacobian of the map. For this reason, they use the terminology of “generating functions” rather than “generating forms”.
More recently, L[ó]{}meli and Meiss [@lomeli09gff] have studied the problem of volume preserving maps using differential forms and generalization of the symplectic approach. They discussed in detail the ${\mathbb{R}}^3$ case, the first to differ from the symplectic case, and described how the generic volume preserving maps can be described by thirty-six one-forms. That paper paves the background for our investigations. In particular, we are interested in understanding how these differential forms are associated to numerical methods (if any) and whether some of these forms can lead to new techniques to obtain volume preserving maps. These questions were partially addressed in [@xue2014gf], where six of the thirty-six differential one forms were identified and associated to splitting methods. The scope of this paper is to discuss and classify the remaining cases.
The main result of the paper is the reduction of the thirty-six cases to five essentially different cases, using equivalence relations (global variable renaming and numerical adjoints). Thereafter, these five cases are classified and three of them associated to known techniques, based on Hamiltonian and/or Lagrangian formalism. In other words, the generating forms are associated to symplectic splitting methods like Symplectic Euler (SE) for Hamiltonian systems, or Discrete Lagrangian (DL) methods for appropriate Lagrangian functions, or a combination of both. We further identify two special classes, $S_1$ and $S_2$, that, to our knowledge, do not have a straightforward and direct mechanical interpretation. It is these two special classes that are of particular interest in the search of new volume preserving algorithms. We give explicit formulas of generating one-forms for the class $S_1$ and $S_2$ corresponding to volume preserving first order methods in the specific case of linear vector fields. A general approach is still unknown and will be the subject of future investigation.
Background and notation {#sec:background}
-----------------------
We consider a differentiable manifold $\mathcal{M}$. Let $\omega$ be a non degenerate $k$-differential form on $\mathcal{M}$, that is, a $k$-linear map, completely skew-symmetric with respect to its arguments. For each $p \in \mathcal{M}$, $\omega(p): (T_p\mathcal{M})^{\times k} \to {\mathbb{R}}$, namely the differential form takes as argument $k$ tangent vectors and returns a number. The coefficients of the differential form might depend on $p\in \mathcal{M}$. Let $d \omega$ be the $k+1$ form obtained with the usual rules of external derivation. Recall that $\omega$ is *closed* if $d\omega = 0$ and that $\omega$ is an exact differential, or simply *exact*, if $\omega = d\nu$, where $\nu$ is a $k-1$ form, called a *primitive*. By application of Stokes’ theorem, $\int_S d \omega = \int_{\partial S} \omega$, valid on any oriented manifold with oriented boundary $\partial S$, to the differential form $d\omega$, it follows that $d^2\omega=0$. Therefore, any exact form, $\omega = d \nu$, is closed, i.e. $d\omega = d^2\nu = 0$. The reverse statement is not true in general, but it holds on contractible manifolds, as explained from the following lemma.
\[th:poincare\] A closed form ($d\omega=0$) is locally exact ($\omega=d\nu$), that is, there is a neighborhood $U$ about each point on which $\omega=d\nu$. The statement is globally true on contractible manifolds.
In the sequel, we focus on volume forms and their primitives. We will assume, otherwise stated, that the differential forms are *non-degenerate*. This means that the coefficients of the form are never simultaneously zero.
A volume form $\Omega$ on a manifold $\mathcal{M}$ is preserved by a $C^{1}$-map $\mathbf{f}: \mathcal{M}\mapsto \mathcal{M}$ if $$\mathbf{f}^{*}\Omega=\Omega,
\label{eq:vol_pres}$$ where $\mathbf{f}^{*}$ denotes the pull-back of $\mathbf{f}$. The map $\mathbf{f}$ is said to be *canonical* or *volume preserving*.
In what follows, we let $\mathcal{M}={\mathbb{R}}^n$. Let $\nu$ be any primitive form of $\Omega$, i.e. $d\nu=\Omega$. Then, condition becomes $\mathbf{f}^{*}d\nu-d\nu=0$ and implies $d(\mathbf{f}^{*}\nu-\nu)=0$, hence $\mathbf{f}^{*}\nu-\nu$ is the exact differential of a $n-2$ form, as a consequence of Lemma \[th:poincare\]. This motivates Definitions \[def:nu\_exact\] and \[df\] below, see [@lomeli09gff].
\[def:nu\_exact\] Let $\nu$ be a primitive of the volume form $\Omega$ and $\mathbf{f}: {\mathbb{R}}^n\mapsto {\mathbb{R}}^n$ an exact volume preserving diffeomorphism such that $$\mathbf{f}^{*}{\nu}-\nu=d\lambda
\label{eq:nu_exact},$$ for a $n-2$ form $\lambda$. The differential form $\lambda$ is called a *generating form with respect to $\nu$.*
Primitives of forms are not uniquely determined: by choosing $\tilde \nu$ another primitive of $\Omega$, the volume preservation condition can be written as $\mathbf{f}^{*}d\nu-d\tilde\nu=0$. A procedure similar to the one just described above leads to:
\[df\] Let $\nu, \tilde \nu$ be two primitives of a volume form $\Omega$, i.e. $d\nu=d\tilde{\nu}=\Omega$ and $\mathbf{f}: {\mathbb{R}}^n\mapsto {\mathbb{R}}^n$ an exact volume preserving diffeomorphism such that $$\mathbf{f}^{*}\tilde{\nu}-\nu=d\lambda\label{eq:0806031},$$ for a $n-2$ form $\lambda$. The $n-2$ differential form $\lambda$ is called a *generating form with respect to $(\nu, \tilde \nu)$.*
We will consider the choice of canonical coordinates $x_1, \ldots, x_n$ in ${\mathbb{R}}^n$ and denote by $\mathbf{x} = (x_1, \ldots, x_n)^T$ the original (old) variables. Given a volume preserving map $\mathbf{f}$, we will denote the transformed (new) variables by uppercase letters, i.e. $\mathbf{X} = (X_1, \ldots, X_n)^T = \mathbf{f}(\mathbf{x})$.
Volume preservation in ${\mathbb{R}}^2$ is equivalent to preservation of area and volume forms are the same as symplectic forms. This case is well understood. The case $n=3$ is the first case for which volume forms and symplectic forms are different.
Generating forms in ${\mathbb{R}}^3$ {#volsec}
====================================
In [@lomeli09gff], Lomeli and Meiss studied in detail exact volume preserving mappings and generating forms in ${\mathbb{R}}^3$. Starting from and using canonical coordinates, they showed that for each choice of primitives $(\nu, \tilde{\nu})$ there are four different generating one-forms, $$\lambda= \phi dx_l + \Phi d X_m, \qquad \phi \in \{A,B\}, \quad \Phi \in \{C,D\}\qquad l, m \in \{1,2,3\},
\label{eq:oneform}$$ which they identified using four generating functions $A, B, C$ and $D$.
As $\nu$ and $\tilde{\nu}$ can be chosen in three different ways ($x_3dx_1\wedge dx_2$, $x_2dx_3\wedge dx_1$ and $x_1dx_2\wedge dx_3$), this approach gives a total of thirty-six generating one-forms. Four of them, corresponding to $\nu= \tilde \nu = x_3 dx_1 \wedge d x_2$, are shown in Table \[tab:123123\]. Each cell in the table is described by: a generating one-form $\lambda$; two *determining conditions*, determining a lowercase and an uppercase variable; a *compatibility condition* and two *twist conditions* to guarantee that the three equations are solvable and they give rise to a well-defined volume preserving map. Altogether, one there are nine such tables, obtained by even permutations of the $x_1,x_2,x_3$ and the $X_1,X_2,X_3$ variables.
---------------------------- ------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------
$X_3 dX_1 \wedge dX_2$ $-$
$x_3 dx_1 \wedge dx_2$ $A dx_1$ $B dx_2$
$C dX_1$ $\begin{array}{ccc} $\begin{array}{ccc}
\lambda = A(x_1, x_2, X_1) dx_1 \\ \lambda = B(x_1, x_2, X_1) dx_2 \\
\qquad \quad \mbox{}+ C(x_1, X_1, X_2) d X_1\\[5pt] \qquad \quad \mbox{}+ C(x_2, X_1, X_2) d X_1\\[5pt]
x_3=\partial_{x_2}A \\ x_3=-\partial_{x_1}B \\
\partial_{X_1} A = \partial_{x_1} C\\ \partial_{X_1} B = \partial_{x_2} C\\
X_3 = -\partial_{X_2} C\\ X_3 = -\partial_{X_2} C\\
\frac{\partial X_1}{\partial x_3} \not=0, \quad \frac{\partial x_1}{\partial X_3} \not=0 \frac{\partial X_1}{\partial x_3} \not=0, \quad \frac{\partial x_2}{\partial X_3} \not=0
\end{array} $ \end{array} $
$D d X_2$ $\begin{array}{ccc} $\begin{array}{ccc}
\lambda = A(x_1, x_2, X_2) dx_1 \\ \lambda = B(x_1, x_2, X_2) dx_2 \\
\qquad\quad \mbox{}+ D(x_1, X_1, X_2) d X_2\\[5pt] \qquad \quad \mbox{}+ D(x_2, X_1, X_2) d X_2\\[5pt]
x_3=\partial_{x_2}A \\ x_3=-\partial_{x_1}B \\
\partial_{X_2} A = \partial_{x_1} D\\ \partial_{X_2} B = \partial_{x_2} D\\
X_3 = \partial_{X_1} D\\ X_3 = \partial_{X_1} D\\
\frac{\partial X_2}{\partial x_3} \not=0, \quad \frac{\partial x_1}{\partial X_3} \not=0 \frac{\partial X_2}{\partial x_3} \not=0, \quad \frac{\partial x_2}{\partial X_3} \not=0
\end{array} $ \end{array} $
---------------------------- ------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------
: The four basic types of generating 1-forms $\lambda$ for $\nu = \tilde \nu = x_3 dx_1 \wedge d x_2$, adapted from [@lomeli09gff]. All the other tables are obtained by applying cyclic even permutations to the variables $(x_1,x_2,x_3)$ in $\nu$ and $(X_1,X_2,X_3)$ in $\mathbf{f}^*\tilde{\nu}$.[]{data-label="tab:123123"}
In [@xue2014gf], Xue and Zanna studied these generating forms with the goal of associating them to volume preserving vector fields and numerical volume preserving integrators. They succeeded in identifying six cases and associating them to splitting methods in using two potential functions: each of the functions gave rise to a two-dimensional Hamiltonian approximated by a Symplectic Euler (SE) method. The six cases are not fundamentally different. First of all, having chosen one case, two of the other cases correspond to a global variable renaming (say, $(x_1,x_2,x_3) \to (x_2,x_3,x_1)$ and $(X_1,X_2,X_3) \to (X_2,X_3,X_1)$ simultaneously). Relabeling the variables in a numerical method does not give a new numerical method. Secondly, the remaining three cases were obtained by exchanging lower cases and reversing time, in other words, they corresponded to the *adjoint* numerical methods of the previous three cases. As it is known how to obtain the adjoint of a given method [@hairer87sod], these cases are not interesting per se either. All this indicates that there is redundancy in the thirty-six cases. To classify and understand which one forms are related to known methods and which forms can lead to genuinely new approaches, we need to establish equivalence classes, so that our search can be restricted to a single differential form for each class.
Problem statement
=================
We consider the ordinary differential equation $$\dot{\mathbf{x}}=\mathbf{a}(\mathbf{x}), ~~\mathbf{x}(0)=\mathbf{x}_{0},
\label{eq:vf}$$ where $\mathbf{x} \in \mathcal{M}$ and $\mathbf{a} :\mathcal{M} \to
T_\mathbf{x} \mathcal{M}$, $\mathbf{a}(\mathbf{x}) = [a_1(\mathbf{x}),
\ldots, a_n(\mathbf{x})]^T$, is a smooth vector field. We denote by $\mathbf{a}^t$ the flow of and by $\omega$ a $k$-form.
Derivatives of differential forms along the flow are called *Lie derivative* and defined as $$L_{\mathbf{a}} \omega = \frac{d}{dt} (\mathbf{a}^t)^* \omega \Big|_{t=0}
\label{eq:lie_deriv}$$ Let $\Omega$ be a volume form on $\mathcal{M}$. We say that the vector field $\mathbf{a}$ is *volume preserving* if $$L_{\mathbf{a}} \Omega =0.
\label{eq:volpres_a}$$ The flow $\mathbf{a}^t$ is volume preserving if its vector field $\mathbf{a}$ is volume preserving. By , this implies that $$(\mathbf{a}^t)^* \Omega = \Omega.$$ Let $i_\mathbf{a}$ a contraction, that is, for any $k$-form $\omega$, $i_\mathbf{a} \omega$ is the $k-1$ form $\omega (\mathbf{a}, \mbox{} \cdot \mbox{})$ obtained by inserting $\mathbf{a}$ in the first slot. By Cartan’s formula for Lie derivatives, $$L_{\mathbf{a}} \Omega = d (i_\mathbf{a} \Omega) + i_{\mathbf{a}} d \Omega,
\label{eq:Cartan}$$ it follows that a vector field $\mathbf{a}$ is volume preserving if $d (i_\mathbf{a} \Omega) =0$, that is, $ i_\mathbf{a} \Omega $ is closed (as $d \Omega =0$, being $\Omega$ a $n$-form).
Let $\lambda$ be a $n-2$ form and $\Omega$ a volume form on $\mathcal{M}$. A vector field $\mathbf{a}$ on $\mathcal{M}$ is *exact volume preserving* with respect to the potential form $\lambda$ if $$i_\mathbf{a} \Omega = d \lambda.
\label{eq:exactvp}$$
In particular, it follows from Poincar[é]{}’s lemma \[th:poincare\] that, when $\mathcal{M}={\mathbb{R}}^n$, globally defined volume preserving vector fields are also exact.
Let us consider the case $n=3$ in more detail. For any vector $\mathbf{v}_i \in {\mathbb{R}}^3$, $i=1,2,3$, we have $\Omega(\mathbf{v}_1, \mathbf{v}_2,\mathbf{v}_3)= dx_1 \wedge dx_2 \wedge dx_3 (\mathbf{v}_1, \mathbf{v}_2,\mathbf{v}_3)= \det[\mathbf{v}_1, \mathbf{v}_2,\mathbf{v}_3]$. By direct computation, we see that $\Omega (\mathbf{a}, \mathbf{v}, \mathbf{w}) = [a_1 dx_2 \wedge dx_3 + a_2 dx_3 \wedge d x_1 + a_3 d x_1 \wedge d x_2] (\mathbf{v}, \mathbf{w})$ for any $\mathbf{v}, \mathbf{w}$. We deduce that $$i_\mathbf{a}\Omega = a_1 dx_2 \wedge dx_3 + a_2 dx_3 \wedge d x_1 + a_3 d x_1 \wedge d x_2.$$ There are three natural choices of the one-form $\lambda$, i.e. $\lambda_i = F^{i}(x_1,x_2,x_3) d x_i$, $i=1,2,3$, where the $F^i$s are arbitrary function. Then, $$d \lambda_i = \sum_{j=1}^3 \partial_{x_j} F^i d x_j \wedge d x_i, \qquad i=1,2,3,$$ and, by , we deduce that a volume preserving vector field, exact with respect to the form $\lambda_i$, must have the following form: $$\begin{aligned}
\label{eq:vf1}
F^1 d x_1 &: \qquad
\begin{array}{ccc}
a_1 &=& 0,\\
a_2 &=& \partial_{x_3} F^{1},\\
a_3 &=& -\partial_{x_2} F^1,
\end{array} \\
\label{eq:vf2}
F^2 d x_2 &: \qquad
\begin{array}{ccc}
a_1 &=& -\partial_{x_3} F^2,\\
a_2 &=& 0,\\
a_3 &=& \partial_{x_1} F^2,
\end{array}\\
\label{eq:vf3}
F^3 d x_3 &:\qquad
\begin{array}{ccc}
a_1 &=& \partial_{x_2} F^3,\\
a_2 &=& -\partial_{x_1} F^3,\\
a_3 &=& 0.\end{array}\end{aligned}$$ A generic three-dimensional volume preserving vector field will be a combination of - above. Note that only two of them are linearly independent, the same yields for the choices of $\lambda_i$: for instance, as long as $F^1, F^2$ depend on all variables, $d \lambda_1$ and $d \lambda_2$ generate all the $dx_i \wedge dx_j$, $i, j=1,2,3$ although $\lambda_1$ contains only $d x_1$ and $\lambda_2$ only $dx_2$.
This is a consequence of a well known result for volume preserving flows in ${\mathbb{R}}^n$: any $n$-dimensional volume preserving differential equation is described by $n-1$ independent potential functions (see for instance [@McLachlan2002; @kang95vpa; @FW1989; @xue2014gf] and discussion therein). One of the earliest normalization of the $n-1$ independent potential functions is due to Weyl [@weyl40tmo].
It is natural to draw a connection between the generating forms and the volume preserving vector field $\vf$. In particular, we address the following questions:
- How do the generating functions $\phi, \Phi$ relate to the vector field $\vf$ and the $F^{i}$s, if there is any relation?
- Can the generating forms be naturally associated to volume preserving numerical methods for whenever the vector field $\mathbf{a}$ in is volume preserving?
An important property of numerical methods is *consistency*. Numerical methods introduce a discrete time step $h$. A fundamental property required to the method is that, in the limit $h\to 0$, $\frac{\mathbf{X}-\mathbf{x}}{h} = \vf$, that is, the method solves the given differential equation.
We now define what is the main goal of this paper: to find suitable potential functions $\pot$ and $\Pot$ for a divergence free vector field $\vf$ which also are a solution of the *consistency problem*, below rephrased in the formalism of this paper for convenience.
The main goal of this section is to show that, subject to consistency, there are only five different classes of generating volume forms in ${\mathbb{R}}^3$.
Defining equations and compatibility conditions
-----------------------------------------------
Consider two arbitrary smooth functions $$\pot,\Pot \colon {\mathbb{R}}^3 \to {\mathbb{R}},$$ and an arbitrary sign, which will be denoted by $\sig$ in the equations.
Define the corresponding mapping $\volpot\colon (\yy1,\yy2,\yy3) \mapsto(\YY1,\YY2,\YY3)$ implicitly by the following equations.
\[eq:volpot\] $$\begin{aligned}
\partial_{2} \pot(\YY3, \yy2, \yy3) &=\yy1 \label{eq:volpot1} \\
\partial_{3} \Pot(\YY3,\YY2,\yy3) &\sig \partial_{1} \pot(\YY3,\yy2, \yy3) = 0 \label{eq:volpot2}\\
\YY1 &= \partial_{2} \Pot(\YY3, \YY2, \yy3). \label{eq:volpot3}\end{aligned}$$
The mapping $\volpot$ is well defined as soon as $\pd[\Pot]{32} \neq 0$ and $\pd[\pot]{21} \neq 0$.
We define the of a permutation $\perm\colon\set{1,2,3}\to\set{1,2,3}$ on an element $\mathbf{x}\in{\mathbb{R}}^3$ by $$\begin{aligned}
(x \act \perm)_i \coloneqq x_{\perm(i)}
.\end{aligned}$$ Consider two permutations $\perm$ and $\Perm$: $$ \perm,\Perm \colon \set{1,2,3} \to \set{1,2,3} .$$ For any map $\volpota \colon {\mathbb{R}}^3 \to {\mathbb{R}}^3$, we define the map $\permact{\perm,\Perm}{\volpota}$ by $$\begin{aligned}
\paren[\big]{\permact{\perm,\Perm}{\volpota}}(x) \coloneqq \volpota(x \act \perm) \act \Perm\inv
.\end{aligned}$$
The consistency problem
-----------------------
\[def:conssol\] We say that a pair of maps $\pot, \Pot \colon \mathfrak{X}_0({\mathbb{R}}^3) \to \mathcal{C}^{\infty}({\mathbb{R}},{\mathbb{R}}^3)$, where $\mathfrak{X}_0({\mathbb{R}}^3)$ denotes the space of divergence-free vector fields on ${\mathbb{R}}^3$, is a for the pair of permutations $(\perm,\Perm)$ if the map $h \mapsto \permact{\perm,\Perm}{\volpot[\pot(h\vf),\Pot(h\vf)]}$ is *consistent* with $\vf$.
This means that $$\begin{aligned}
\lim_{h\to 0} \frac{1}{h}\paren[\Big]{\paren[\big]{\permact{\perm,\Perm}{\volpot[\pot(h\vf),\Pot(h\vf)]}}(\mathbf{x}) - \mathbf{x}} = \vf
.\end{aligned}$$
Note that for any fixed vector field $\vf$, the element $\pot(\vf)$ is a potential, i.e., $\pot(\vf)$ is itself a function from ${\mathbb{R}}^3$ to ${\mathbb{R}}$. Note also that a method obtained from will automatically have order one.
A classification and description of the five generating volume forms in ${\mathbb{R}}^3$
========================================================================================
Having proved that there are only five classes of generating volume forms in ${\mathbb{R}}^3$ (up to relabeling and adjunction) for a given vector field $\vf$ in ${\mathbb{R}}^3$, we proceed with a classification.
Class SE+SE {#sec:SE}
-----------
We commence with case SE+SE. Its generating differential form was already discussed and extended to the $n$-dimensional case in [@lomeli09gff]. This case was also discussed in detail in [@xue2014gf], where it was associated to numerical integrators consisting of combinations of Symplectic Euler methods.
With the notation of this paper, we have $\Sigma(+)=1, \Sigma(\circ)=2,\Sigma(-)=3$ and $\sigma(-)=1,\sigma(\circ)=2,\sigma(+)=3$, see Figure \[fig:triangle\]. From $\tau=\Sigma^{-1}\circ\sigma$, we see that $\mathrm{sign}(\tau)=-1$, therefore (\[eq:-o+general\]) becomes $$\begin{aligned}
x_3&=\partial_{x_2}\phi(x_1,x_2,X_3),\\
\partial_{X_3}\phi(x_1,x_2,X_3)&=\partial_{x_1}\Phi (x_1,X_2,X_3),\\
X_1&=\partial_{X_2}\Phi (x_1,X_2,X_3),
\label{eq:case5}
\end{aligned}$$ with twist conditions $$\frac{\partial X_1}{\partial x_1}\neq 0,~~\frac{\partial x_3}{\partial X_3}\neq 0,$$ and generating form $$\lambda=\phi (x_1,x_2,X_3) dx_1+ \Phi (x_1,X_2,X_3) dX_3.$$ As $\lambda$ is combination of differentials $dx_1$ and $d X_3$, it is natural to consider vector fields generated by and . Setting $\Phi =x_1X_2+\Delta t F^{3}(x_1, X_2,X_3)$ and $\phi=x_2X_3+\Delta tF^{1}(x_1,x_2,X_3)$, leads to the first order volume preserving scheme $$\begin{aligned}
X_1&=x_1+\Delta t \partial_{X_2}F^{3}(x_1,X_2,X_3),\\
X_2&=x_2-\Delta t \partial_{x_1}F^{3}(x_1,X_2,X_3)+\Delta t\partial_{X_3}F^{1}(x_1,x_2,X_3),\\
X_3&=x_3-\Delta t \partial_{x_2}F^{1}(x_1,x_2,X_3),\end{aligned}$$ which is equivalent to two steps of the Symplectic Euler (SE) method to solve each of the two 2D Hamiltonian systems and .
Class DL+SE {#t2}
-----------
We have $\Sigma(+)=1, \Sigma(-)=2,\Sigma(\circ)=3$, and, as for all cases under consideration in this paper, $\sigma(-)=1,\sigma(\circ)=2,\sigma(+)=3$. We have $\mathrm{sign}(\tau)=1$ and (\[eq:-o+general\]) becomes $$\begin{aligned}
x_3&=\partial_{x_2}\phi(x_1,x_2,X_2),\\
\partial_{X_2}\phi(x_1,x_2,X_2)&=\partial_{x_1}\Phi (x_1,X_2,X_3),\\
X_1&=-\partial_{X_3}\Phi (x_1,X_2,X_3),
\label{eq:case4}
\end{aligned}$$ with twist conditions $$\frac{\partial x_1}{\partial X_1}\neq 0,~~\frac{\partial X_2}{\partial x_3}\neq 0,$$ and generating form $$\lambda=\phi(x_1,x_2,X_2) dx_1+ \Phi (x_1,X_2,X_3) dX_2.$$ As $\lambda$ is combination of differentials $dx_1$ and $d X_2$, it is natural to consider vector fields generated by and .
Note that $\phi$ is a function of both $x_2$ and $X_2$ and that $x_3$ is determined by $x_2, X_2$. This points to an interpretation of the Hamiltonian system defined by $H(x_2,x_3)=F^1(x_1, x_2, x_3) $, where $x_2 \equiv q$, $x_3\equiv p$, and $x_1$ is treated as a constant, $$\begin{aligned}
\dot{x}_1&=0,\\
\dot{x}_2&=\partial_{x_3}F^{1}(x_1,x_2,x_3),\\
\dot{x}_3&=-\partial_{x_2}F^{1}(x_1,x_2,x_3).
\end{aligned}$$ by a *Lagrangian* formulation, with Lagrangian function $$L^1(x_1, x_2, \dot x_2) = x_3 \dot x_2 - H = x_3 \dot x_2 - F^1(x_1, x_2,x_3)$$ [@goldstein01cm]. Consider a *discrete Lagrangian* $ L^1_d= \Delta t L^1(x_1, x_2, (X_2-x_2)/\Delta t)$ [@marsden01]. With the choice $$\phi(x_1, x_2, X_2) =L_d^1 (x_1, x_2, X_2),$$ we see that the first equation of is satisfied, and, moreover, an intermediate variable for $x_3$ is obtained, $$\tilde x_3 = - \partial_{X_2} \phi (x_1, x_2, X_2).$$
For the $\Phi$ function, choose $$\Phi(x_1, X_2, X_3) = -x_1X_3 + \Delta t F^2(x_1, X_2, X_3).$$ The third equation of gives $X_1 = x_1 - \Delta t \partial_{X_3} F^2(x_1,X_2,X_3)$ and $ \Phi_{x_1} = - X_3 + \Delta t \partial_{x_1}F^2(x_1,X_2,X_3)$.
Altogether, we obtain $$\begin{aligned}
x_3 &= \partial_{x_2} L_d^1 (x_1, x_2, X_2) \\
\tilde x_3 &= - \partial_{X_2} L_d^1 (x_1, x_2, X_2) \qquad (= - \partial_{X_2} \phi) \\
X_3 &= \tilde x_3 + \Delta t \partial_{x_1}F^2(x_1,X_2,X_3)\qquad ( \hbox{compat.\ cond.\ } \partial_{X_2} \phi = \partial_{x_1} \Phi) \\
X_1 &= x_1 - \Delta t \partial_{X_3} F^2(x_1,X_2,X_3)\end{aligned}$$ which is a combination of a Discrete Lagrangian method (DL) for and a Symplectic Euler (SE) for . As long as the discrete Lagrangian function $L_d^1$ is a consistent approximation to the continuous one, the composed method has at least order one.
Class DL+DL {#type4}
-----------
We have $\Sigma(-) = 2$, $\Sigma(\circ) = 1$, $\Sigma(+) = 3$ and $\mathrm{sign}(\tau) = -1$ and (\[eq:-o+general\]) becomes $$\begin{aligned}
x_3 &= \partial_{x_2} \phi(x_1, x_2, X_2), \\
\partial_{X_2} \phi(x_1, x_2, X_2) &= \partial_{x_1} \Phi(x_1, X_2, X_1), \\
X_3 &= \partial_{X_1} \Phi(x_1, X_2, X_1),
\label{eq:case3}\end{aligned}$$ with twist conditions $$\frac{\partial x_1}{\partial X_3} \not=0, \quad \frac{\partial X_2}{\partial x_3} \not=0,$$ and generating form $$\lambda=\phi(x_1,x_2,X_2) dx_1+\Phi (x_1,X_1,X_2) dX_2.$$ The generating form indicates that one should look for vector fields of the form and .
Similarly to the procedure described above, we choose $\phi = L_d^1$, generating the intermediate approximation $(x_1, X_2,\tilde x_3)$ (recall that $x_1$ is kept constant). Also the system is interpreted as a Lagrangian system, with Lagrangian function $L^2(x_1, \dot x_1, X_2)= \tilde x_3 \dot x_1 - F^2(x_1,X_2, \tilde x_3)$. Now, $X_2$ is kept constant. We set $$\Phi(x_1, X_1, X_2) = -L_d^2 (x_1,X_1,X_2),$$ where $L_d^2 (x_1,X_1,X_2)= \Delta t L^2(x_1, (X_1-x_1)/\Delta t, X_2)$ is a discrete Lagrangian approximation to $L^2$.
Altogether, we obtain $$\begin{aligned}
x_3 &= \partial_{x_2} L_d^1 (x_1, x_2, X_2), \\
\tilde x_3 &= - \partial_{X_2} L_d^1 (x_1, x_2, X_2) \qquad (= - \partial_{X_2} \phi) \\
\tilde x_3 &= \partial_{x_1} L_d^2 (x_1, X_1, X_2) \qquad (\hbox{compat.\ cond.\ } \partial_{X_2} \phi = \partial_{x_1} \Phi)\\
X_3 &= -\partial_{X_1} L_d^2(x_1,X_1,X_2),\end{aligned}$$ which is a combination of a Discrete Lagrangian methods (DL) for and for . As long as the discrete Lagrangian functions $L_d^1, L_d^2$ are consistent approximations to the continuous ones, the composed method has at least order one.
Special classes: $S_1$ and $S_2$ {#type5}
--------------------------------
While all the cases discussed above can be interpreted as splitting in two two-dimensional Hamiltonian systems, either solved by a symplectic method or turned into Lagrangian systems solved by a discrete Lagrangian method, there is no obvious mechanical interpretation for cases $S_1$ and $S_2$. We are not aware of any numerical method for ordinary differential equations that is naturally related to these two generating forms in the same way as all the other cases discussed in this paper. In this respect, cases $S_1$ and case $S_2$ are novel cases.
Cases $S_1$ and $S_2$ are both associated to generating forms of type $$\lambda = \phi d x_1 + \Phi d X_1,$$ which would suggest the choice of two vector fields of the form , clearly a degenerate and not particularly interesting vector field. Are there non-degenerate vector fields for which such generating form gives consistent, non trivial maps? The answer is yes: it is possible to give explicit expressions for $\phi, \Phi$ so that the generating form $\lambda = \phi d x_1 + \Phi d X_1$ of cases $S_1$ and $S_2$ gives consistent, first order, volume preserving numerical methods at least in the case when the vector field $\mathbf{a}$ in is linear: for *linear vector fields*, $\phi, \Phi$ can be taken to be quadratic functions; thus the determining conditions in are linear in the unknown variables.
Hereafter, we restrict our attention to linear divergence-free vector fields $$\begin{aligned}
\label{linear}
\dot{x}_1&=a_1(x_1,x_2,x_3)=a_{11}x_1+a_{12}x_2+a_{13}x_3,\\
\dot{x}_2&=a_2(x_1,x_2,x_3)=a_{21}x_1+a_{22}x_2+a_{23}x_3, \qquad a_{11}+a_{22}+a_{33}=0. \\
\dot{x}_3&=a_3(x_1,x_2,x_3)=a_{31}x_1+a_{32}x_2+a_{33}x_3,\end{aligned}$$
### Class $S_1$
We have that $\Sigma(-)=1, \Sigma(\circ)=2,\Sigma(+)=3$, and $\tau$ is an even permutation. The generating one-form is $$\lambda=\phi(x_1, x_2, X_1) dx_1+ \Phi (x_1, X_1, X_2) d X_1,$$ where $\phi, \Phi $ satisfy $$\begin{aligned}
x_3&=\partial_{x_2}\phi(x_1, x_2, X_1),\\
\partial_{X_1} \phi(x_1, x_2, X_1) &= \partial_{x_1} \Phi (x_1,X_1,X_2),\\
X_3& = -\partial_{X_2} \Phi (x_1,X_1,X_2).
\label{eq:S1}\end{aligned}$$ In addition to assuming linearity of the vector field, we also assume $a_{13}\neq0$ (a twist condition, implying that $x_3$ can be determined from $X_1$, given $x_1, x_2$).
Two possible choices of $(\phi, \Phi)$ yielding first order numerical methods of linear vector fields are given below.
\[th:S1\] Consider the class $S_1$ generating one-form $\lambda = \phi dx_1 + \Phi dX_1$. Let $$\begin{aligned}
\phi(x_1,X_1,x_2)&=\frac{X_1-x_1-\Delta ta_{11}x_1-\Delta t ^2a_{21}a_{12}x_1\frac{k_2}{k_1}}{\Delta t a_{13}+\Delta t^2a_{23}a_{12}\frac{k_2}{k_1}}x_2-\frac{a_{12}}{k_1a_{13}+\Delta ta_{23}a_{12}k_2}\frac{x_2^2}{2},\\
\Phi (x_1,X_1,X_2)&=-\frac{X_1-x_1-\Delta ta_{11}x_1}{\Delta ta_{13}}(1+\Delta ta_{33})X_2+X_2^2\big(- \frac{\Delta ta_{32}}{2}+\frac{a_{12}}{2a_{13}}(1+\Delta ta_{33})\big)\\
&-\Delta t a_{31}X_1X_2-\frac{2X_1x_1-x_1^2(1+\Delta ta_{11})\Delta ta_{23}k_2+\Delta t^2a_{13}a_{21}k_2x_1^2}{2k_3},
\label{eq:S1_quispel}\end{aligned}$$ where $k_1=1+\Delta t^2a_{11}a_{33}-\Delta ta_{22}$, $k_2=1+\Delta t^2\frac{a_{11}a_{33}}{1-\Delta ta_{22}}$, $k_3=\Delta ta_{13}(\Delta ta_{13}+\frac{k_2}{k_1}\Delta ta_{12}a_{23})$, and $$\begin{aligned}
\phi(x_1,X_1,x_2)&=\frac{X_1-(1+\Delta ta_{11})x_1}{\Delta a_{13}}x_2-\frac{1}{2}\frac{a_{12}}{a_{13}}x_2^2,\\
\Phi (x_1,X_1,X_2)&=-\frac{1}{l_1}\big((1+\Delta ta_{33})(\frac{X_1-(1+\Delta ta_{11})x_1}{\Delta ta_{33}}X_2-\frac{1}{2}\frac{a_{12}}{a_{13}}X_2^2)+\Delta ta_{31}x_1X_2\\
&+\frac{1}{2}\Delta ta_{32}X_2^2+\Delta t\frac{a_{12}}{a_{13}}(a_{21}x_1X_1+\frac{1}{2}a_{22}X_2^2)\big)\\
&+\Delta ta_{21}x_1^2/2+\Delta ta_{23}\frac{2X_1x_1-x_1^2(1+\Delta ta_{11})}{2\Delta ta_{13}},
\label{eq:S1_AZ}\end{aligned}$$ where $l_1=1-\Delta t\frac{a_{12}}{a_{13}}a_{23}$.
Both choices and yield first-order volume preserving integrators for the vector field (\[linear\]), provided that $a_{13}\not=0$.
The proof of the result can be found in Appendix \[app:S1\].
### Class $S_2$
This case corresponds to $\Sigma(-)=1, \Sigma(\circ)=3,\Sigma(+)=2$, with $\tau$ an odd permutation. The generating one-form is $$\lambda=\phi(x_1, x_2, X_1) dx_1+ \Phi (x_1, X_1, X_3) d X_1,$$ where $\phi, \Phi $ satisfy $$\begin{aligned}
x_3&=\partial_{x_2}\phi(x_1, x_2, X_1),\\
\partial_{X_1} \phi(x_1, x_2, X_1) &= \partial_{x_1} \Phi (x_1,X_1,X_3),\\
X_2& = \partial_{X_3} \Phi (x_1,X_1,X_3).
\label{eq:S2}\end{aligned}$$ As for the $S_1$ case, we also assume $a_{12}\neq0$. Below we give the explicit expression of a choice $(\phi, \Phi)$ yielding first order numerical methods of linear vector fields.
\[th:S2\] Consider the class $S_2$ generating one-form $\lambda = \phi dx_1 + \Phi dX_1$. Let$$\begin{aligned}
\label{eq:S2_quispel}
\phi(x_1,X_1,x_2)&=\frac{\big(m_1(X_1-x_1-\Delta ta_{11}x_1)-\Delta t^2a_{31}a_{13}m_2 x_1\big)x_2}{\Delta ta_{13}} - \frac{ m_1a_{12}x_2^2}{2a_{13}}-\frac{\Delta ta_{32}m_2 x_2^2}2 \\
\Phi(x_1,X_1,X_3)&= \frac{(1+\Delta ta_{22})\big(X_1-(1+\Delta ta_{11})x_1\big)X_3}{\Delta ta_{12}}-\frac{a_{13}(1+\Delta ta_{22})X_3^2}{2a_{12}}\\
& \quad \mbox{} +\Delta ta_{21}X_1X_3+\frac{\Delta ta_{23}}2 X_3^2+ \frac{m_1(2X_1x_1-\big(1+\Delta ta_{11})x_1^2\big)}{2\Delta t^2 a_{12}a_{13}},\end{aligned}$$ where $m_1=1-\Delta t a_{33}+\Delta t^2a_{11}a_{22}$ and $m_2=1+\Delta t^2a_{11}a_{22}/(1-\Delta ta_{33})$. The choice yields first-order volume preserving integrators for the vector field (\[linear\]), provided that $a_{12}\not=0$.
The proof of this result is similar to that for . For completeness, it can be found in Appendix \[app:S2\].
Conclusions and remarks
=======================
In this paper, we have studied the thirty-six generating one-forms for volume preserving mappings in ${\mathbb{R}}^3$. This is the first $n$-dimensional case for which there is a difference between area preservation, well understood using the tools of symplectic geometry and symplectic forms, and volume preservation.
By imposing equivalence relations (equivalence under relabeling and equivalence under adjunction), we have shown that all cases can be generated by five classes of differential one forms.
We have classified these five cases in terms of known numerical methods that preserve volume and denoted them as SE+SE (already identified by Xue and Zanna in [@xue2014gf]), DL+SE, DL+DL, $S_1$ and $S_2$. Except for the special cases $S_1, S_2$, the classes can be naturally associated to the splitting of the vector field into two $2D$ Hamiltonian systems or into two Lagrangian system, solved by a symplectic method (symplectic Euler, SE) or a discrete Lagrangian approach (DL), or both.
Classes $S_1$ and $S_2$, both defined by a generating one-form of the type $\lambda = \phi dx_1 + \Phi d X_1$, are, to the best of our knowledge, novel cases. Whereas the other classes admit a natural mechanical interpretation (either Hamiltonian or Lagrangian mechanics), it is not clear whether there exists a natural mechanical interpretation for the classes $S_1$ and $S_2$. The corresponding generating forms can be used to generate well defined volume preserving maps, however, for general vector fields, these maps are hightly implicit and do not seem to lead to explicit or efficient numerical methods. For completeness, we have shown possible choices of functions $\phi, \Phi$, that yield consistent methods for linear vector fields. These two cases need a deeper understanding and will be the subject of future research.
Acknowledgement {#acknowledgement .unnumbered}
===============
The work has been supported by NFR grant no. 191178/V30, under the project Geometric Numeric Integration in Applications, by the SpadeACE Project and by the J.C. Kempe memorial fund (grant no. SMK-1238).
Appendix A {#app:S1}
==========
The two choices - correspond to two different techniques to determine $\phi, \Phi$.
The first one, , is based on the *correction method* by Quispel [@GRW199526]: we determine three maps $f_1, f_2, f_3$ that give a volume preserving transformation. Thereafter we invert $f_1$ and use integration, differentiation and some other algebraic manipulations to obtain suitable $\phi, \Phi$.
The second choice, , is based on the following idea: choose a consistent method for $X_1$, depending on $x_3$, say for instance Forward Euler. Because of the linearity of the vector field, this always determines $x_3 = \phi_{x_2}(x_1,X_1,x_2)$, hence $\phi$, up to a function depending only on $x_1,X_1$. Next, we think of $x_3$ as a function of $x_2$. Now, use a consistent map to obtain $X_3$, as function of $X_1, x_1, X_2$ and $x_3(x_1, X_1,s)$, where the occurrences of $x_2$ are replaced by $X_2$). Upon the replacement $x_2 \to X_2$ in $x_3$, the map is not necessarily consistent any longer, and some adjustments must be made, also to ensure consistency for $x_2$.
*\[Prop. \[th:S1\]\]* We commence with the case. Consider the implicit map $$\begin{aligned}
\label{ff}
X_1&=f_1(x_1,X_2,x_3),\\
x_2&=f_2(x_1,X_2,x_3),\\
X_3&=f_3(X_1,X_2,x_3),\end{aligned}$$ and Quispel’s correction method [@GRW199526], to obtain $$\begin{aligned}
\label{t5vp}
X_1&=x_1+\Delta t a_1(x_1,X_2,x_3),\\
X_2&=x_2+\Delta t a_2(x_1,X_2,x_3)- f_{correct}(x_1,X_2,x_3),\\
X_3&=x_3+\Delta t a_3(X_1,X_2,x_3),\end{aligned}$$ where $f_{correct}$ is determined to obtain a volume preserving scheme, $$\begin{aligned}
f_{correct}(x_1,X_2,x_3)&=\int^{X_2}_{const}\Delta t\frac{\partial a_3}{\partial x_3}(x_1+\Delta t a_1(x_1,X_2,x_3),X_2,x_3)-\Delta t\frac{\partial a_3}{\partial x_3}(x_1,X_2,x_3)\\
&+\Delta t^2\frac{\partial a_1}{\partial x_1}(x_1,X_2,x_3)\frac{\partial a_3}{\partial x_3}(x_1+\Delta t a_1(x_1,X_2,x_3),X_2,x_3)dX_2\\
&=\Delta t^2a_{11}a_{33}(X_2-const).\end{aligned}$$ The integration constant should satisfy $const=\Delta t a_2(x_1,const,x_3)$[^1], that is, $$const=\frac{\Delta t a_{21}x_1+\Delta ta_{23}x_3}{1-\Delta ta_{22}}.$$ First of all, we calculate $X_2$ from the second equation of (\[t5vp\]) which has the form, $$\label{x2}
X_2=\frac{x_2+(\Delta ta_{21}x_1+\Delta ta_{23}x_3)k_2}{k_1},$$ where $k_1=1+\Delta t^2a_{11}a_{33}-\Delta ta_{22}$ and $k_2=1+\Delta t^2\frac{a_{11}a_{33}}{1-\Delta ta_{22}}$. Substituting the above equation into the first equation of (\[t5vp\]), we can find $x_3$ in terms of variables $x_1, x_2, X_1$, denoted by $\tilde{x}_3(x_1,X_1,x_2)$. Substituting $\tilde{x}_3$ into the first equation of (\[eq:S1\]) and integrating both sides, we obtain $\phi$ $$\phi=\frac{X_1-x_1-\Delta ta_{11}x_1-\Delta t ^2a_{21}a_{12}x_1\frac{k_2}{k_1}}{\Delta t a_{13}+\Delta t^2a_{23}a_{12}\frac{k_2}{k_1}}x_2-\frac{a_{12}}{k_1a_{13}+\Delta ta_{23}a_{12}k_2}\frac{x_2^2}{2}+\tilde{A}(x_1,X_1).$$ From the first equation of (\[t5vp\]), we see that $x_3$ depends on the variables $x_1, X_1,X_2$. We then can solve $x_3$ in terms of $x_1, X_1,X_2$ from that equation since we assume $a_{13}\neq0$, and denote by $\hat{x}_3(x_1,x_2,X_1)$. We substitute $\hat{x}_3$ into the third equation of (\[t5vp\]). From the third equations of (\[eq:S1\]) and from (\[t5vp\]), we know that $$X_3(x_1,X_1,X_2)=\hat{x}_3(x_1,X_1,X_2)+\Delta t \mathbf{a}_3(X_1,X_2,\hat{x}_3(x_1,X_1,X_2))=-\partial_{X_2}\Phi .$$ From the above equation, we can integrate both sides with respect to $X_2$ and obtain $$\begin{aligned}
\Phi &=-\frac{X_1-x_1-\Delta ta_{11}x_1}{\Delta ta_{13}}(1+\Delta ta_{33})X_2+X_2^2(-\frac{\Delta ta_{32}}{2}+\frac{a_{12}}{2a_{13}}(1+\Delta ta_{33}))\\
&-\Delta t a_{31}X_1X_2+\tilde{C}(x_1,X_1).\end{aligned}$$ Without loss of generality, we set $\tilde{A}=0$. Using the second condition of (\[eq:S1\]) , we obtain $$\tilde{C}_{x_1}=\frac{x_2}{\Delta a_{13}+\Delta ta_{12}a_{23}\frac{k_2}{k_1}}-\frac{(1+\Delta ta_{11})(1+\Delta ta_{33})}{ha_{13}}X_2,$$ From the first equation in (\[t5vp\]), we have $$X_2=\frac{X_1-x_1-\Delta ta_{11}x_1-\Delta ta_{13}x_3}{\Delta ta_{12}},$$ By noticing the relation in (\[x2\]), we obtain $$\tilde{C}_{x_1}=-\frac{X_1-x_1(1+\Delta ta_{11})\Delta ta_{23}k_2+\Delta t^2a_{13}a_{21}k_2x_1}{k_3},$$ where $k_3=\Delta ta_{13}(\Delta ta_{13}+\frac{k_2}{k_1}\Delta ta_{12}a_{23})$. From it is not difficult to see that (\[eq:S1\_quispel\]) gives a first order volume preserving method.
Next, we consider the case. Using the forward Euler method to solve the first equation of (\[linear\]), that is $$X_1=(1+\Delta ta_{11})x_1+\Delta ta_{12}x_2+\Delta ta_{13}x_3.$$ Assuming that $a_{13}\neq0$ and solving for $x_3$, we have $$\label{x3tilde}
x_3=\frac{X_1-(1+\Delta ta_{11})x_1-\Delta ta_{12}x_2}{\Delta ta_{13}}.$$ Integrating both sides, we obtain $$\phi(x_1,X_1,x_2)=\frac{X_1-(1+\Delta ta_{11})x_1}{\Delta a_{13}}x_2-\frac{1}{2}\frac{a_{12}}{a_{13}}x_2^2+\tilde{A}(x_1,X_1),$$ where $\tilde{A}(x_1, X_1)$ is a function to be determined.
Using Euler method to solve the third equation of (\[linear\]), where $x_3$ in (\[x3tilde\]) is replaced by $x_3(x_1,X_1,X_2)$ and using $X_1=x_1+\Delta t a_1$ and $X_2=x_2+\Delta t a_2$, we obtain $$\begin{aligned}
X_3&=(1+\Delta ta_{33})\frac{X_1-(1+\Delta ta_{11})x_1-\Delta ta_{12}X_2}{\Delta ta_{13}}+\Delta ta_{31}x_1+a_{32}X_2\\
&=(1+\Delta ta_{33})x_3+\Delta ta_{31}x_1+\Delta ta_{32}X_2-\Delta t\frac{a_{12}}{a_{13}}a_2+O(\Delta t^2).\end{aligned}$$ There is a problem with the term $a_2$ (as in (\[linear\])) in the above equation for consideration of consistency for $X_3$. So, we take $a_2=a_{21}x_1+a_{22}X_2+a_{23}X_3$ and substitute back into the above equation. Then, we obtain $$(1-\Delta t\frac{a_{12}}{a_{13}}a_{23})X_3=(1+\Delta ta_{33})x_3+\Delta ta_{31}x_1+\Delta ta_{32}X_2-\Delta t \frac{a_{12}}{a_{13}}(a_{21}x_1+a_{22}X_2)$$ Solving $X_3$ from the above equation, substituting it back into the third equation of (\[eq:S1\]) and integrating both sides, we have $$\begin{aligned}
\Phi &=-\frac{1}{l_1}\big((1+\Delta ta_{33})(\frac{X_1-(1+\Delta ta_{11})x_1}{\Delta ta_{33}}X_2-\frac{1}{2}\frac{a_{12}}{a_{13}}X_2^2)+\Delta ta_{31}x_1X_2\\
&+\frac{1}{2}\Delta ta_{32}X_2^2+\Delta t\frac{a_{12}}{a_{13}}(a_{21}x_1X_1+\frac{1}{2}a_{22}X_2^2)\big)+\tilde{C}(x_1,X_1),\end{aligned}$$ where $l_1=1-\Delta t\frac{a_{12}}{a_{13}}a_{23}$ and $\tilde C$ is to be determined. We have two functions $\tilde A, \tilde C$ to be determined and one equation (compatibility condition). The two functions are not independent, hence we set $\tilde{A}=0$. Using the compatibility condition $\partial_{X_1} \phi= \partial_{x_1} \Phi $, we obtain $$\frac{x_2}{\Delta ta_{13}}=\frac{1}{l_1}(1+\Delta ta_{33})(1+\Delta ta_{11})X_2+\tilde{C}_{x_1}.$$ From the divergence-free condition $a_{22}=-(a_{11}+a_{33})$ we have $$X_2=x_2(1+\Delta ta_{22}+O(\Delta t^2))l_1+\tilde{C}_{x_1}.$$ Hence $X_2$ will be consistent provided that $$\tilde{C}=\Delta t a_{21}\frac{x_1^2}{2}+\Delta t a_{23}\frac{2X_1x_1-x_1^2(1+\Delta ta_{11})}{2\Delta ta_{13}}.$$ From the proof, we see that (\[eq:S1\_AZ\]) gives a first order volume preserving method.
Appendix B {#app:S2}
==========
*\[Prop. \[th:S2\]\]* The method generated by and is constructed in the same way as . The implicit map reads $$\begin{aligned}
\label{ffs2}
X_1&=f_1(x_1,x_2,X_3),\\
X_2&=f_2(X_1,X_2,x_3),\\
x_3&=f_3(x_1,x_2,X_3),\end{aligned}$$ and the corrections method is $$\begin{aligned}
\label{t5vps2}
X_1&=x_1+\Delta t a_1(x_1,x_2,X_3),\\
X_2&=x_2+\Delta t a_2(X_1,x_2,X_3),\\
X_3&=x_3+\Delta t a_3(x_1,x_2,X_3)- f_{correct}(x_1,x_2,X_3),\\\end{aligned}$$ where $$\begin{aligned}
f_{correct}(x_1,x_2,X_3)&=\int^{X_3}_{const}\Delta t\frac{\partial a_2}{\partial x_2}(x_1+\Delta t a_1(x_1,x_2,X_3),x_2,X_3)-\Delta t\frac{\partial a_2}{\partial x_2}(x_1,x_2,X_3)\\
&+\Delta t^2\frac{\partial a_1}{\partial x_1}(x_1,x_2,X_3)\frac{\partial a_2}{\partial x_2}(x_1+\Delta t a_1(x_1,x_2,X_3),x_2,X_3)dX_3\\
&=\Delta t^2a_{11}a_{22}(X_3-const).\end{aligned}$$ The integration constant should satisfy $const=\Delta t a_3(x_1,x_2,const1)$, that is, $$const1=\frac{\Delta t a_{31}x_1+\Delta ta_{32}x_2}{1-\Delta ta_{33}}.$$ The rest of the proof is similar to that of Prop. \[th:S1\].
[^1]: Due to consideration of consistency for $X_2$, see more details in [@GRW199526].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We have investigated the vortex dynamics in superconducting thin film devices with non-uniform patterns of artificial pinning centers (APCs). The magneto-transport properties of a conformal crystal and a randomly diluted APC pattern are compared with that of a triangular reference lattice. We have found that in both cases the magneto-resistance below the first matching field of the triangular reference lattice is significantly reduced. For the conformal crystal, the magneto-resistance is below the noise floor indicating highly effective vortex pinning over a wide magnetic field range. Further, we have discovered that for asymmetric patterns the R vs. H curves are mostly symmetric. This implies that the enhanced vortex pinning is due to the commensurability with a stripe in the non-uniform APC pattern and not due to a rearrangement and compression of the whole vortex lattice. (submitted on 04/29/13 to APL)'
author:
- 'S. Guénon'
- 'Y. J. Rosen'
- 'Ali C. Basaran'
- 'Ivan K. Schuller'
title: Highly Effective Superconducting Vortex Pinning in Conformal Crystals
---
Vortex dynamics in superconducting thin films with artificial pinning centers (APCs) have been extensively studied in recent years. A central problem in this field is how to improve the critical current density of a superconducting device by choosing a suitable APC distribution. If the APCs are arranged in a hexagonal or a rectangular lattice, the critical current is increased for certain magnetic matching fields due to the commensurability with the Abrikosov vortex lattice [@Baert1995; @Harada1996; @Martin1997; @Castellanos1997; @Moshchalkov1998; @Morgan1998; @Villegas2003]. In order to increase the field range for vortex pinning, different APC distributions like quasiperiodic [@Kemmler2006; @Misko2010] or random [@Rosen2010; @Kemmler2009] lattices have been investigated. Recent theoretical papers are focusing on non-uniform APC distributions like hyperbolic-tessellation arrays [@Misko2012] and conformal crystals [@Ray2012]. A conformal crystal can be obtained by conformally mapping a semiannular section of a regular hexagonal lattice on a rectangle (see [@Ray2012; @Rothen1993; @Rothen1996]). In this transformation the angles are preserved, but a vertex density gradient along one side of the rectangle is introduced. Hence, a conformal crystal is a non-uniform pattern, in which on a small scale the vertices are arranged triangularly. On a large scale however, the vertex density changes continuously.\
A recent theoretical study [@Ray2012] has shown that the pinning in conformal crystals is significant stronger over a much wider field range than that found for other APCs with an equivalent number of pinning sites. These promising results have not yet been confirmed experimentally.\
In our experimental study we have investigated the vortex dynamics in superconducting thin film devices with non-uniform APC patterns. We compare the magneto-transport properties of conformal crystals and corresponding randomly diluted APC patterns with the properties of a triangular reference lattice. Additionally, we investigate whether an electric current applied to a superconducting micro bridge with a non-uniform APC pattern produces enhanced pinning due to vortex distribution rearrangement and compression. Because the current pushes the vortices against the edge of the sample, an asymmetric APC pattern with respect to the center line of the bridge could produce an asymmetric R vs. H curve when compared to the triangular array. In agreement with theoretical calculations, we find enhanced pinning by non-uniform APC patterns. However, contrary to the expectations the R vs. H curves are essentially symmetric in all cases indicating that the enhanced vortex pinning is not the result of a large scale redistribution of vortices. It appears rather that the vortices are effectively pinned locally in a stripe, in which the vortex lattice and the non-uniform APC pattern is commensurate.\
![(color online) Artificial pinning center patterns in the center of the micro bridge devices. A: symmetric randomly diluted, B and E: triangular, C: symmetric conformal crystal, D: asymmetric randomly diluted, F: asymmetric conformal crystal. Inset: Geometry of the sample under investigation. The micro bridge devices A-C and D-F are in series.[]{data-label="fig:pattern"}](fig_pattern)
Standard e-beam writing was used to prepare magnetic pinning sites [@Velez2008]. First, the APC dot patterns were written into PMMA resist on a SiO$_2$ coated Si substrate. Then, a 40 nm thick Cobalt film was deposited via e-beam evaporation and approximately 120 nm diameter Cobalt dots were obtained in a lift-off step. Micro bridges of a 100 nm thick, RF-sputtered Nb film were prepared using optical lithography and lift-off. A superconducting transition temperature of 8.6 K was measured.\
Fig. \[fig:pattern\] shows the centers of the APC patterns and the geometry of the sample. To avoid misalignment of the APC pattern with respect to the superconducting micro bridges, they were all written in a single scan with an approximate write field of $400 \times 400\,\mu\textnormal{m}^2$. The APC patterns are $100\,\mu$m long and $40\,\mu$m wide, to assure that the whole area between the voltage taps is filled with APCs. Device A and C are symmetric (the pinning site density increases to both edges of the micro bridge), while D and F are asymmetric (the pinning site density increases to the lower edge of the micro bridge). The triangular lattices B and E have a lattice constant of approximately $0.6\,\mu$m. The conformal crystals (C and F) were obtained by mapping the triangular lattice according to [@Ray2012], but, instead of choosing a semiannular region, a partial annular section was chosen with an opening angle of $\pi/2$. Hence, in this study the conformal crystal is deformed less than in [@Ray2012]. For the randomly diluted lattices (A and D) the vertices of the triangular lattice were randomly removed, so that the pinning site density is the same as that of the corresponding conformal crystals. All patterns were compressed by 18% in the direction perpendicular to the micro bridge edges.\
Magneto-resistance was measured in a liquid Helium cryostat with a superconducting magnet and a variable temperature insert. The sample was cooled with evaporated Helium gas and a temperature stability better than $1\,$mK was achieved using a heater in the sample mount. There was a temperature drift of a few mK on a time scale of a few minutes due to a change in the evaporation rate and therefore the time slot for acquiring an R vs. H was about one minute.\
In this study, it was very important to eliminate the different thermo-electric off-sets in the wiring. Therefore, we used for the symmetric patterns the DC polarity reversed mode of the Keithley transport electronics [@Daire2005]. In this mode the current source changes polarity in a low frequency square wave pattern and the voltage is read out synchronously. The signal is averaged by subtracting the opposite polarity voltages. For the asymmetric pattern we used an alternative technique to differentiate between the two current directions. We acquired four R vs. H (DC) curves by changing the current source polarity for consecutive curves. In this technique the average resistivity is given by $$\overline{R}(H)=\left[R_1(H)-R_2(-H)+R_3(H)-R_4(-H)\right]/4$$ The minus sign before the magnetic field in the second and fourth R(H) is necessary, because a change in the current direction changes the direction of the Lorentz-force.\
![(color online) Resistance vs. magnetic field of the micro bridge devices with symmetric APC patterns at different set temperatures (logarithmic scale). The ratio $T/T_c$ was estimated for the triangular reference lattice. Insets: Resistance vs. magnetic fields for small fields (linear scale).[]{data-label="fig:symmetric"}](fig_symmetric)
In order to investigate the flux flow resistance of non-uniform APC patterns we have measured the magneto-resistance at different temperatures near the superconducting transition. In all electrical transport measurements presented, we applied a relatively small $10\,\mu$A current to avoid Joule heating. With the series arrangement, we were able to consecutively measure the R vs. H curves of the three devices at each temperature by changing the voltage contacts. We used the micro bridge devices with a triangular APC pattern as a reference. The ratio $T/T_c$ was determined from the magneto-resistance vs. temperature dependence of the triangular reference pattern.\
Fig. \[fig:symmetric\] shows R vs. H curves of the devices with symmetric patterns at three different temperatures. The triangular reference lattices shows two matching minima at approximately 86 Oe and 172 Oe due to the relatively small diameter to separation ratio of the magnetic pinning dots [@Hoffmann2000]. The resistance of the device with a randomly diluted APC is considerably smaller than that of the triangular lattice below the first matching field. Above the first matching field, R vs. H is almost the same except in vicinity of the second matching field where the triangular lattice is smaller. The flux flow resistance of the device with the conformal crystal is smaller than that of the randomly diluted or the triangular APC pattern for all temperatures and magnetic fields except in vicinity of the matching fields (fig. \[fig:symmetric\] a for $T/T_c=0.994$). For field values smaller than the first matching field, the flux flow resistance is below the noise floor of the measurement. This indicates strong pinning which immobilizes the vortices. We emphasize that this behavior is reproducible and robust for all temperatures in the superconducting state (see fig. \[fig:symmetric\]).\
![(color online) Averaged resistance vs. magnetic field of the micro bridge devices with asymmetric APC patterns at different set temperatures (logarithmic scale). The ratio $T/T_c$ was estimated for the triangular reference lattice. Insets: Averaged resistance vs. magnetic fields for small fields (linear scale).[]{data-label="fig:asymmetric"}](fig_asymmetric)
The R vs. H of the asymmetric devices are very similar to the symmetric ones as shown in fig. \[fig:asymmetric\]. For resistances above approximately $30\,\textnormal{m}\Omega$, the R vs. H of the asymmetric conformal crystal and the randomly diluted pattern are symmetric. For small resistance values the R vs. H curves are slightly asymmetric. For instance, consider the asymmetric conformal crystal at $\textnormal{T}/\textnormal{T}_\textnormal{c}=0.994$ (see fig \[fig:asymmetric\] a). The difference between the magneto-resistance at field values of -30Oe and 30Oe is approximately 6.7m$\Omega$. But in this set of graphs the R vs. H of the triangular reference lattice is asymmetric, too. Between the magneto-resistance at the first matching field the difference is approximately 15m$\Omega$. Hence, the R vs. H asymmetry of the symmetric reference pattern is more pronounced. This indicates that the R vs. H asymmetry of the asymmetric pattern is smaller than the systematic error.\
In conclusion, we discovered that below the first matching field of the triangular reference lattice, the R vs. H of the randomly diluted and the conformal crystal is reduced. For the conformal crystal, the pinning effectively reduces the resistance below the noise floor over a wide magnetic field range.\
In order to investigate whether the enhanced pinning is due to a compression and rearrangement of the vortex lattice, we have measured the magneto-transport of an asymmetric conformal crystal. For a fixed current direction, the vortices are pushed towards the side with either a high or low APC density when the field direction is changed. If the vortex distribution is rearranged and compressed by the Lorentz force when it is pushed against the Bean-Livingston barrier at the micro bridge edge then the commensurability of the vortices with the APC distribution would depend on the field direction. This would result in an asymmetric R vs. H curve. However, our samples do not show a pronounced asymmetry when compared to the triangular lattice. We therefore suggest an alternative explanation:\
For every magnetic field smaller than the first matching field there exists a stripe in the non-uniform APC distribution, in which the APC density is more or less equal to the vortex density. The commensurability of the vortex lattice with the pinning sites in this stripe, which is parallel to the micro bridge edge, increases the vortex pinning and hinders or suppresses the movement of the whole vortex lattice. Although the conformal crystal and the randomly diluted lattice have the same APC density, the commensurability with the vortex lattice is considerably higher in the first one. Therefore, the pinning in the conformal crystal is more effective. This study might be helpful to enhance the critical current density in tape conductors, to reduce the 1/f-noise in SQUIDs [@Woerdenweber2000] or to improve the performance of microwave resonators [@Bothner2011] in magnetic fields.
We thank D. Ray, C.J. Olson Reichhardt, B. Janko and C. Reichhardt for sharing their preprint and useful conversations. This work was supported by the Office of Basic Energy Science, U.S. Department of Energy, under Grant No. DE FG03-87ER-45332.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Free energy is arguably the most important function(al) for understanding of molecular systems. A number of rigorous and approximate methods, and numerous scoring functions for free energy calculation/estimation have been developed over a few decades. However, the continuity of an macrostate (or path) in configurational space has not been well articulated. In this perspective, we discuss the relevance of configurational space continuity in development of more efficient and reliable next generation free energy methodologies.'
author:
- Shiyang Long
- Pu Tian
title: Configurational space continuity and free energy calculations
---
Introduction
============
Despite the fact that we live in a three dimensional world, our intuitive comfortable zone is, unfortunately, limited to one dimension. We may figure out objects and properties in two and three dimensions quite well with some training and aid of visualization tools, which becomes difficult beyond three dimensions. In carrying out computational analysis of physical systems, continuity is a highly important property. There are two aspects of continuity, namely that of a given function itself and that of the domain of a given function, and we focus on the later in this perspective. In one dimension, continuity of the domain of a function is trivial as a line segment is always continuous as long as it is represented by a continuous numerical range $[x_1, x_2]$ (Fig. \[fig:dofcorr\]a). For a function defined on a two dimensional surface or multidimensional hyper-surface, continuity of domain of function is not a big issue when variables on different dimensions are independent, a two dimensional example was given with $x$ in the range $[x_1, x_2]$ and $y$ in the range $[y_1,y_2]$ (Fig. \[fig:dofcorr\]b, the grey rectangular shadow indicates specified domain of a given function $f(x,y)$). However, for a functional defined on correlated variables, specifying range of respective degrees of freedom (DOFs) is no longer sufficient for defining domain of function as illustrated by Fig. \[fig:dofcorr\]c, with grey-shadowed domain of function being a different region from the rectangle specified by ranges of $x$ and $y$, and understanding correlations between (among) involved variables (e.g. $x$ and $y$ here) becomes a necessity. More importantly, continuity of domain of function may start to be a concern as illustrated by the quasi-continuous scenario in Fig. \[fig:dofcorr\]d and discrete regions in Fig. \[fig:dofcorr\]e. With increasing number of dimensions for a hyper-surface, complexity of topology for effective domain of a functional may rapidly become intractable. It is noted that for a multi-dimensional hyper-surface, fixing values of selected DOFs facing the similar problem. For example, one may simply imagine there is a third dimension in Fig. \[fig:dofcorr\]d and Fig. \[fig:dofcorr\]e with corresponding variable fixed. This is an important aspect of the curse of dimensionality for free energy calculation, and we attempt to bring it to the attention of the community.
Free energy for a macrostate $M$ specified by $m$ out of $n$ ($1 \le m \le n$) degrees of freedom (DOFs) of a molecular system is a functional: $$F_{M} = F([r_1], [r_2], \cdots, [r_m])\label {eq:FeTot}$$ here we use square brackets to indicate that each of selected DOF may have a fixed value or vary within given range(s). It is noted that when all DOFs of a molecular system are fixed, we have a well-defined microstate and only energy matters in such cases since entropy for a given microstate is zero. While we may symbolically specify a macrostate with value or range of selected DOF, it is important to note that thermally accessible range of any given DOF is determined by values of all molecular DOFs that interact with it. With no further information, we may safely write that each DOF interacts with all other DOFs as the following: $$\begin{aligned}
[r_i] = R_i(r_1, r_2, \cdots, r_{i-1}, r_{i+1}, \cdots, r_n), (i = 1, 2, \cdots, m)
$$ When one takes it for granted that a macrostate specified by fixed values and ranges of selected DOFs is continuous in configurational space, it is implied that all possible combinations of values within specified range of DOFs are practically thermally accessible, which is not necessarily true. Therefore, it is immediately clear that to ensure understanding continuity of an given region specified by ranges or fixed values of one or more DOFs in such a hyper-surface is a very challenging task, which implicates understanding of the joint distribution $P(r_1, r_2, \cdots, r_n)$ (i.e. the local free energy landscape(FEL)) in given patch(es) of the configurational space. The joint distribution(s) are part of goals rather than start point of our research on any molecular systems. To calculate free energy difference in a straight forward way, we first need to clearly define both end macrostates and therefore are in a dilemma of needing part of what we are looking for (correlations of the underlying variables within each of two end macrostates, which is essentially the excess (non-ideal gas) entropy). When both end macrostates are relatively easy to sample by brute force molecular dynamics (MD) or Monte Carlo (MC) simulations, various forms of principal component analysis (PCA) of trajectories can be effectively utilized to generate information on both free energy difference and corresponding domain in configurational space[@Mu2005; @Zhou2006; @Zhuravlev2010]. Unfortunately, many large scale conformational change of proteins and protein complexes occur on milli-second or longer time scales, which is expected to remain difficult to achieve on a routine basis in the near future. Therefore, more involved free energy methods are necessary.
Historically, free energy calculation has been mainly carried out for two relatively well defined end macrostates with representative structural states for both end macrostates being available from high resolution experimental methods (e.g. X-ray diffraction)[@Monzon2013; @Hrabe2015]. Nonetheless, since no information of relevant DOF correlations are available, we are limited to specifying end macrostates with ranges or fixed values of selected DOFs, and complications may occur. Apparently, thermally accessible domain in configurational space for free energy of a complex molecular system may potentially be much more complex than what illustrated in Fig. \[fig:dofcorr\]. However, for convenience of illustration, we still utilize these simple scenarios in two dimensions as examples. A well behaved continuous macrostate is shown in Fig. \[fig:dofcorr\]c, where specifying range of two dimensions is sufficient to define an easy-to-sample physical macrostate of corresponding system. In the quasi-continuous scenario in Fig. \[fig:dofcorr\]d, one or more well-behaved substates in the specified subspace may be of practical relevance for a complex molecular system, and both situations present difficulty for free energy calculation. The discrete scenario in Fig. \[fig:dofcorr\]e seems to contradict our widely utilized ergodic assumption, it is actually of great relevance for biomolecular systems. One typical type of examples are ligands that bind to a target protein in multiple independent and mutually non-transformable modes[@Mobley2009; @Chodera2011]. More specifically, such ligands have to disassociate from its target protein before adopt an alternative pose of binding. We discuss specific issues of present mainstream free energy methods for the quasi-continuous and discrete scenarios in more details below.
Macrostate (or path) continuity for theoretically rigorous methodologies
========================================================================
Thermodynamic integration (TI)[@Kirkwood1935; @Darve2001] is a well-established methodology for calculation of free energy difference between two end macrostates of a given molecular system. Two fundamental principles underly this type of methods. The first is that change of free energy between two macrostates may be expressed as integration of the ensemble averaged energy over a path connecting them. The second is that free energy is a state function and the resulting integral is in principle not dependent upon specific path one choose, therefore one may choose unphysical integration paths based on operational convenience. In reality, despite this freedom of choosing integration path, one has to face the fact that the path is on a high dimensional hyper space, assure continuity and sufficient smoothness for reducing integration error, is fundamentally difficult (see Fig. \[fig:pathcont\]). Alchemical transformation is widely utilized in thermodynamic integration due to its operational convenience, one importance issue is the well-understood “end point catastrophe”, the singularity is caused essentially by insertion of a new correlated dimension in the original hyper-surface. This issue has been successfully addressed by utilization of soft core potentials[@Steinbrecher2011]. Another major challenge of TI is potential existence of slower DOFs in addition to selected integration path, which may cause convergence difficulty for the calculation if insufficient sampling of such DOFs changes statistically significant configurational subspace of either or both end states. One accompanying problem is that insufficient sampling of such slower DOFs (with dynamics significantly slower than that of integration path) may also influence continuity of both end macrostates in unpredictable way. TI formulations, by utilizing one specific single integration path, require that both end macrostates are continuously-defined region on the relevant high dimensional free energy landscapes. Apparently, when one of the end macrostate specified by fixed values or ranges of various DOFs are quasi-continuous or discrete as shown in Fig. \[fig:dofcorr\]de, TI is problematic and may fail in different ways: i) all quasi-continuous ( or discrete ) regions are functionally important but only one is sampled, resulting in underestimation of statistical weight; ii) the sampled region is different from functionally relevant region but fall in the same “box” as specified by the range of relevant DOFs, both underestimation and overestimation of statistical weight for this end macrostate is possible. In the case that one or both end macrostates are comprising quasi-continuous or discrete substates, the difficulty of ensuring path continuity increases, especially near such end macrostates. In histogram based methods, such as various forms of umbrella sampling\cite{} or weighted histogram analysis\cite{}, a reaction coordinate is selected to connect the start and final macrostates, both of which are specified by ranges or values of selected molecular DOFs. Therefore, as in the case of TI, it is necessary to be careful that proper physical states are sampled in trajectories to ensure correct calculation of free energy differences.
In yet another path-based rigorous free energy calculation method, namely non-equilibrium work (NEW) based analysis[@Jarzynski1997; @Hummer2001; @Goette2008]. Situations change slightly. Since in this method, numerous paths are generated to realize non-equilibrium transitions between the two end macrostates, the absolute continuity is no-longer required for the final macrostate, as sampling of discrete regions in the final macrostate is realizable, at least in theory, when sufficient number of non-equilibrium paths are sampled. However, when trajectories are utilized to sampling the start macrostate, an quasi-continuous or discrete scenario present apparent challenge for sampling. The fundamental difference between integration path in TI and transition paths in NEW is that the former may be any unphysical ones that are convenient for calculation, while the later are physical paths since actual work done along which are exponentially averaged to estimate change of free energy as shown below[@Jarzynski1997]: $$\overline{exp(-\beta W)} = exp(-\beta \Delta F)$$ When the final macrostate has quasi-continuous or discrete substates, relative dynamical accessibility of various substates determines difficulty of convergence for calculated free energy difference. When dynamical accessibility of multiple substates are proportional to their respective statistical weights, convergence may be achieved with relative ease. On the contrary, when heavier substates is more difficult to access dynamically, convergence of NEW methods will deteriorate and significantly more transition processes between two end macrostates need to be recorded.
Free energy perturbation method (FEP)[@Zwanzig1954; @Zwanzig1955; @Bash1987] was originally proposed as an end point method with no need of designing reaction coordinate (or integration pathway). Reliable calculation of free energy difference by FEP requires sufficient overlapping of statistically significant region of configurational space between two end macrostates. When one or both macrostates exhibit quasi-continuous or discrete domain in configurational space, overlapping between two end macrostates is apparently more difficult to achieve when compared with situations that both end macrostates have well-behaved continuity as illustrated in Fig. \[fig:dofcorr\]c. When one or both end macrostate(s) exist as discrete or quasi-continuous regions in FEL with *a priori* unknown correlation of relevant DOFs, multistage FEP[@MFEP1999], which is designed to alleviate insufficient overlapping of statistically significant configurational space of end macrostates via utilization of an effective order parameter, face similar challenges as TI and histogram based methodologies do.
Hypothetical scanning molecular dynamics (Monte Carlo) (HSMD/MC)[@Meirovitch1999; @White2004; @Cheluvaraja2005; @Cheluvaraja2008; @Meirovitch2010] is a recent rigorous end point free energy method based on reconstruction of transition probability to specific configurations generated from regular MD or MC sampling. As reconstruction process is limited to very small configurational space volume in practice, the caveat of including irrelevant region is not a big issue in this case. The potential problem would be from the sufficiency of sampling provided by the starting molecular configurations, which may neglect some discrete or quasi-discrete substates that are of relevance in our interested molecular processes, or alternatively include some discrete or quasi-continuous substates that are essentially irrelevant. Both ways lead to unreliable results.
These rigorous methods usually are utilized to calculate free energy change between end macrostates that are quite well defined structurally (i.e. with structures of both end macrostates available), with the goal of providing atomistic explanations for experimental observations. In these situations, while there is possibility of a functionally relevant state comprising a limited number of quasi-continuous or discrete regions in configurational space, functional robustness and evolution pressure excludes possibility of existence of a large number of statistically significant quasi-continuous or discrete regions. Nonetheless, when starting with structures of both end macrostates, no correlation information is available and care has to be taken for possible configurational discontinuity. Historically, while continuity of macrostate has not been articulated, a number of dimensionality reduction methods have been developed to find the genuine reaction path between interested macrostates\cite{}. As a matter of fact, these methodologies may be utilized to detect potential continuity problem of any give macrostate, albeit with high computational costs. For example, a *posterio* principal component analysis (PCA) of configurational space visited by sampling trajectories in the vicinity of experimental structures, while unable to reveal possible unvisited statistically significant quasi-continuous or discrete regions, should at least disclose the topology and continuity of the visited configurational subspace. One may also using metadynamics approach\cite{} to probe the free energy landscape of the interested molecular systems in the vicinity of both end macrostates to provide guidance for selecting proper rigorous free energy methodology. The most challenging part of metadynamics is to select proper collective variables (CVs), which should be the most critical slow DOFs in molecular systems. A number of strategies\cite{} are available to facilitate definition of CVs. Nonetheless, defining proper CVs is a system specific task and remains to be a major challenge to be tackled.
Macrostate continuity and major approximate methodologies
=========================================================
Due to the prohibitive computational cost, the above mentioned rigorous methods are not practical for high throughput predictive calculation of free energies as in the case of protein folding, design, docking and virtual screening. A number of computationally more economical methods[@LIE1998; @MMPBSA-JMC; @Miller2012] and numerous scoring functions[@Grinter2014; @Liu2015] have been developed over last few decades. We discuss the relevance of configurational space continuity for them below.
In linear interaction energy (LIE) model[@LIE1998], change of free energy upon the binding of a ligand ($L$) to its target protein ($P$) is estimated by simulating the free ligand in solution and the protein-ligand complex ($PL$) with the following equation: $$\Delta G = \beta(\langle E^{L-S}_{ele}\rangle_{PL} - \langle E^{L-S}_{ele}\rangle_L) + \alpha(\langle E^{L-S}_{vdW}\rangle_{PL} - \langle E^{L-S}_{vdW}\rangle_L)$$ with $E^{L-S}_{ele}$ and $E^{L-S}_vdW$ being the electrostatic and van der Waals interaction energies between the ligand and it environment ($S$, including protein and solution), angle brackets indicating ensemble average, and $\alpha$ and $\beta$ are two empirical parameters. For a rather rigid protein-ligand binding, where both free and bound state are likely to be well behaved and continuous in configurational space as illustrated in Fig. \[fig:dofcorr\]c, the approximation would be good given proper parameters $\alpha$ and $\beta$. However, when significant flexibility exist for ligand and/or its target, either free and bound state may have more than one quasi-continuous or discrete substates as illustrated in Fig. \[fig:dofcorr\]d or Fig. \[fig:dofcorr\]e, one set of parameter is likely to fail on some of such substates and reliability of the approximation starts to deteriorate.
The linear response approximation (LRA)[@Sham2000] is quite similar to LIE. The practical difference between them is that non-electrostatic contribution for the former is evaluated with protein dipoles langevin dipoles (PDLD) method (or its semi-microscopic version, the PDLD/s method)[@Singh2009] while for the later is approximated by averaging van der Waals interactions. Nonetheless, both treatments assume that end macrostates are well-defined free energy wells as illustrated by Fig. \[fig:dofcorr\]c. The restraint-release method[@Singh2009], which decomposes conformational change into three contributions, energy difference between two representative (central) structures of the start and final conformation, entropic contribution from local motion around the central structure in start conformation, and entropic contribution from local motion around the central structure in the final conformation, implies the same assumption. One may imagine that it would be highly nontrivial, if ever possible, to pick a “representative (central)” structure for quasi-continuous or discrete scenarios.
In MM/P(G)BSA[@MMPBSA-JMC], free energy of protein-ligand (in a general sense) binding is expressed as: $$\begin{aligned}
\Delta G &= G(PL) - G(P) - G(L) \\
G &= \langle E_{int} + E_{ele} + E_{vdW} + G_{solv} + G_{np} - TS_{MM}\rangle \label{eq:gpart}\end{aligned}$$ the three first terms in equation (\[eq:gpart\]) are the molecular mechanical ($MM$) internal, electrostatics and van der Waals energies, $G_{solv}$ and $G_{np}$ are polar and non-polar solvation free energies respectively. $S_{MM}$ is the configurational entropy estimated with normal mode\cite{}or quasiharmonic analysis\cite{}. Again, the validity of this equation implies that both end macrostates are well-behaved as illustrated in Fig. 1c. Apparently, both normal mode and quasiharmonic analysis are insufficient methods for entropy estimation when a macrostate have quasi-continuous or discrete substates, estimation of solvation terms become significantly more challenging for these scenarios as well.
Flexibility and rugged FEL is essentially a ubiquitous property and widely acknowledged challenge in computational analysis of protein molecular systems[@Panjkovich2012; @Hrabe2015]. Entropic effects of ligands are also demonstrated to be important[@Villa2000]. Therefore, the implicit assumption by these approximate methods that concerned end macrostates have well-behaved continuity in configurational space is not necessarily true, especially for important and/or interesting molecular systems that we have very limited understanding and therefore have strong desire to predict their behavior. In high throughput predictive studies, we usually seek to locate statistically significant configurational subspace by some predesigned procedures. In contrast to application of rigorous methods in comparing macrostates which are already established to be statistically significant, scanning of configurational space faces much greater challenge of continuity. Since we know very little on time scales and correlations of relevant moelcular DOFs, we may well specify “a” macrostate that is physically comprising many quasi-continuous or even discrete regions in configurational space, estimation of collective statistical weight of which would be highly unreliable when utilizing approximate methods that require well-defined continuous end macrostates. In a recent analysis on the utility of various minimum potential terms (minimum protein self energy, minimum protein-solvent interaction energy and their sum) as approximate free energy proxy, correlation of these terms with population based free energy differences deteriorate dramatically when macrostates were defined by projection onto fast torsional DOFs, which essentially correspond to defining many discrete fragments in configurational space as a collective macrostate. Methods that are based on the assumption of well defined continuous macrostates in configurational space will lose their prediction power in such situation.
As discussed above, the effort of free energy calculation has been predominantly focusing on, understandably, the difference between two end macrostates or paths connecting them when using path-based methods. Nonetheless, configurational space continuity within each end macrostate is just as important in terms of reliability of calculation/estimation. When one takes configurational space continuity of “a” macrostate specified by ranges or values of selected DOFs for granted, unexpected errors caused by quasi-continous or discrete configurational substates may severely reduce reliability of calculation.
The most widely utilized free energy methods in high throughput applications (e.g. virtual screening) are various forms of scoring functions\cite{}, parameters for which are derived based upon presently available structures and/or affinity data. Regardless of specific type, scoring functions usually evaluate, compare and search for the best “pose” of molecular interactions. Let’s assume that a given scoring function is sufficiently accurate, the precondition of finding the best “pose” is to have it in a limited list of “poses” to be evaluated. Preparing such a list is an extremely challenging global sampling problem in the configurational space. The key issue here is that we have no idea which specific region(s) of configurational space is represented by a given “pose”, which is essentially a point (or a microscopic patch) in a gigantic multidimensional configurational hyper-surface. Consequently, hierarchical structure of FEL for complex molecular systems (e.g. a protein-ligand interaction system) may not be effectively utilized to accelerate the searching process. Therefore, the capability to measure the statistical weight of an arbitrarily given configurational subspace ( regardless of its actual physical continuity ) is highly desired, a schematic representation of hierarchical configurational space partition is presented in Fig. \[fig:???\]. Achieving this goal makes guaranteed sampling of the global configurational space possible through proper partitions into subspaces. ????
Calculation of free energy difference by direct configurational space discretization
====================================================================================
We recently developed an end point free energy method based on direct discretization of configurational space into explicit conformers[@Wang2016]. More specifically, one first need to define a set of Explicit Conformers with Invariable Statistical Weight Distribution (ECISWD) across the whole configurational space for the interested molecular system, and free energy difference between two end macrostates may be obtained simply by counting thermally accessible such conformers as shown in the following equation: $$\Delta F^{AB} = -k_BTln\frac{N^A_{conf}}{N^B_{conf}}$$ with $k_B$ being the Boltzmann constant, $T$ being the temperature, and $N^{A(B)}_{conf}$ being the number of ECISWD in the two end macrostates $A$ and $B$ respectively. By combining sequential Monte Carlo (SMC) and importance sampling[@Zhang2003; @Zhang2006], counting of conformers may be achieved highly efficiently. In this methodology, continuity of configurational space is not an implied assumption for end macrostates to be compared. However, in importance sampling, each replica apparently may only represent one specific discrete region when many exist. Therefore, while physical continuity is not implied, increase of the number of discrete or quasi-discrete regions in one specified “macrostate” might present challenges and requires larger number of replica for sampling. This is not necessarily the case as when indeed many discrete regions were included in an artificial macrostate, sampling only part of them may provide us with sufficient accuracy already. More investigations are therefore necessary in this regard. While not limited by configurational space continuity assumption, configurational space discretization based free energy method inherently does not ensure continuity for the identified configurational subspace that is the most statistically significant, and one or multiple conformations may be included. Again, metadynamics approach may be utilized to probe the details of FEL within “a” statistically significant region, which may physically correspond to one or a number of quasi-continuous or discrete regions. Development of more efficient methods for identifying possible configurational space (quasi-)discontinuity in “a” specified “macrostate” is highly desired for more effective utility of all present free energy methodologies.
Conclusions
===========
Free energy calculation/estimation is of long-standing interest in computational chemistry, molecular biophysics and biology. There have been numerous excellent original articles, reviews and books on this subject and we apologize for not being able to citing them all here. Readers are encouraged to read more extensively for details of each type of methods. In this perspective, we limited our discussions to challenges of configurational space continuity facing the free energy calculation community. While simple low dimensional schematics were utilized for explanation, real high dimensional molecular systems present similar but conceivably much more complex scenarios.
We illustrate that either explicit consideration for detecting configurational space continuity problem or development of new methods that do not require configurational space continuity of end macrostates ( and/or pathway continuity ) need to be addressed in theoretical and computational advancement of free energy analysis.
This research was supported by National Natural Science Foundation of China under grant number 31270758, and by the Research fund for the doctoral program of higher education under grant number 20120061110019.
@ifundefined
[35]{}
Mu, Y.; Nguyen, P. H.; Stock, G. *Proteins: Structure, Function, and Bioinformatics* **2005**, *58*, 45–52 Zhou, X.; Chou, J.; Wong, S. *BMC Bioinformatics* **2006**, *7*, 40 Zhuravlev, P. I.; Wu, S.; Potoyan, D. a.; Rubinstein, M.; Papoian, G. a. *Methods (San Diego, Calif.)* **2010**, *52*, 115–21 Monzon, A. M.; Juritz, E.; Fornasari, M. S.; Parisi, G. *Bioinformatics* **2013**, *29*, 2512–2514 Hrabe, T.; Li, Z.; Sedova, M.; Rotkiewicz, P.; Jaroszewski, L.; Godzik, A. *Nucleic acids research* **2015**, *44*, D423–428 Mobley, D. L.; Dill, K. A. *Structure* **2009**, *17*, 489–498 Chodera, J. D.; Mobley, D. L.; Shirts, M. R.; Dixon, R. W.; Branson, K.; Pande, V. S. *Current opinion in structural biology* **2011**, *21*, 150–60 Kirkwood, J. G. *Journal of Chemical Physics* **1935**, *3*, 300–313 Darve, E.; Pohorille, A. *The Journal of Chemical Physics* **2001**, *115*, 9169 Steinbrecher, T.; Joung, I.; Case, D. A. *Journal of Computational Chemistry* **2011**, *32*, 3253–3263 Jarzynski, C. *Physical Review Letters* **1997**, *78*, 2690–2693 Hummer, G.; Szabo, a. *Proceedings of the National Academy of Sciences of the United States of America* **2001**, *98*, 3658–61 Goette, M.; Grubmüller, H. *Journal of Computational Chemistry* **2009**, *30*, 447–456 Zwanzig, R. W. *The Journal of Chemical Physics* **1954**, *22*, 1420 Zwanzig, R. W. *The Journal of Chemical Physics* **1955**, *23*, 1915 Bash, P. a.; Singh, U. C.; Langridge, R.; Kollman, P. a. *Science (New York, N.Y.)* **1987**, *236*, 564–8 Lu, N.; Kofke, D. A. *The Journal of Chemical Physics* **1999**, *111* Meirovitch, H. *J. Chem. Phys.* **1999**, *111*, 7215–7224 White, R. P.; Meirovitch, H. *Proceedings of the National Academy of Sciences of the United States of America* **2004**, *101*, 9235–9240 Cheluvaraja, S.; Meirovitch, H. *Journal of Chemical Physics* **2005**, *122* Cheluvaraja, S.; Mihailescu, M.; Meirovitch, H. *Journal of Physical Chemistry B* **2008**, *112*, 9512–9522 Meirovitch, H. *Journal of molecular recognition : JMR* **2010**, *23*, 153–72 Hansson, T.; Marelius, J.; Åqvist, J. *Journal of Computer-Aided Molecular Design* **1998**, *12*, 27–35 Kuhn, B.; Kollman, P. A. *Journal of Medicinal Chemistry* **2000**, *43*, 3786–3791 Miller, B. R.; McGee, T. D.; Swails, J. M.; Homeyer, N.; Gohlke, H.; Roitberg, A. E. *Journal of Chemical Theory and Computation* **2012**, *8*, 3314–3321 Grinter, S. Z.; Zou, X. *Molecules* **2014**, *19*, 10150–10176 Liu, J.; Wang, R. *Journal of Chemical Information and Modeling* **2015**, *55*, 475–482 Sham, Y. Y.; Chu, Z. T.; Tao, H.; Warshel, A. *Proteins* **2000**, *39*, 393–407 Singh, N.; Warshel, A. *Journal of Physical Chemistry B* **2009**, *113*, 7372–7382 Panjkovich, A.; Daura, X. *BMC bioinformatics* **2012**, *13*, 273 Villa, J.; Strajbl, M.; Glennon, T. M.; Sham, Y. Y.; Chu, Z. T.; Warshel, A. *Proceedings of the National Academy of Sciences of the United States of America* **2000**, *97*, 11899–904 Wang, K.; Long, S.; Tian, P. *Scientific Reports* **2016**, *6*, 22217 Zhang, J.; Chen, R.; Tang, C.; Liang, J. *The Journal of Chemical Physics* **2003**, *118*, 6102 Zhang, J.; Liu, J. S. *PLoS computational biology* **2006**, *2*, e168
![Schematic representation of a searching process for the most statistically significant macrostate(s). The rectangle represents the full configurational space of a given molecular system, large dark dots represent the lowest free energy state(s), the small grey dots represent other poses to be examined, it is important to note that size of these dots and the rectangle are not to the proportion. As a matter of fact, the configurational space, the size of which grows exponentially with number of DOFs and is huge for typical biomolecular systems, was intentionally given an extremely small size for the convenience of representation. a) “pose” strategy where many poses were examined and the one with lowest free energy (as indicated by the selected scoring function) is deemed as the answer, the precondition of finding the right answer is that the answer is already in the list of poses to be evaluated. b) A hierarchical strategy to locate for the statistically most significant configurational subspace. On each hierarchy, the most significant subspace is given a darker background that other subspaces, there are four hierarchies in this particular case. c) A configurational space with two statistically most significant states were searched with a hierarchical strategy.[]{data-label="fig:pathcont"}](./csc-FEALL-cut.pdf){width="6in"}
| {
"pile_set_name": "ArXiv"
} |
[**Larbi Alili$^{(1)}$**]{}
\
A class of Volterra transforms, preserving the Wiener measure, with kernels of Goursat type is considered. Such kernels satisfy a self-reproduction property. We provide some results on the inverses of the associated Gramian matrices which lead to a new self-reproduction property. A connection to the classical reproduction property is given. Results are then applied to the study of a class of singular linear stochastic differential equations together with the corresponding decompositions of filtrations. The studied equations are viewed as non-canonical decompositions of some generalized bridges.
[**Keywords:**]{} Brownian motion; Canonical decomposition; Enlargement of filtrations; Goursat kernels; Gramian matrices; Self-reproducing kernels; Stochastic differential equations; Volterra transform.
[**AMS 2000 subject classification:**]{} 26C05; 60J65.
\[section\] \[section\] \[section\] \[section\]
\[section\] \[section\] \[section\]
Introduction and preliminaries
==============================
Gaussian enlargement of filtrations has been extensively studied between the late 70’s and the early 90’s, see [@deheuvels], [@jeulin80], [@jeulinyor85], [@jeulin88] and the references therein. Results stemming from the Gaussian nature of the underlying generalized Gaussian bridges are of interest not only in probability, also in financial mathematics, since they have appeared in an insider trading model developed in [@jurgen98a] and [@karatzas96]. Transforms of Volterra type allow to construct interesting families of Gaussian processes. Volterra-transforms are classified, both from the theory and applications points of view, according to whether their kernels are square-integrable or not. Those with square-integrable kernels play a crucial role in the study of equivalent Gaussian measures, stochastic linear differential equations and the linear Kalman-Bucy filter, see [@hida93] and [@kallianpur80]. To our knowledge, comparably, less interest was given to Volterra transforms with non-square-integrable kernels. Such transforms naturally appear, for instance, in non-canonical representations of some Gaussian processes. They also appear if one forces such transforms to preserve the Wiener measure. The most known examples have corresponding kernels of Goursat type. A few nontrivial ones originate from P. Lévy, see [@Levy-56], [@Levy-57], and serve as a standard reference for showing the importance of the canonical decomposition of semi-martingales. Such constructions have been enriched by people from the Japanese school, see [@hhm], [@Hida-60] and [@Hitsuda-68].
Let us now fix the mathematical setting and summarize results of this paper. We take $B:=(B_t, t\geq 0)$ to be a standard Brownian motion, defined on a complete probability space $(\Omega,
{\mathcal F}, \mathbb{P}_0)$. Denote by $\{\mathcal{F}_t^B, t\geq
0\}$ the filtration it generates. Let $f = (f_1, \cdots, f_n)^*
\in L_{loc}^2(\real_+)=\{ h; \int_0^t h^2(s) \, ds<\infty, \hbox{
for all } t\in [0, \infty)\}$, where $*$ stands for the transpose operator and $n$ is a natural number. Although some of our results extend readily to the cases when $n=\infty$, to simplify the study, we only consider the cases where $n$ is finite. We assume that, for any fixed $t>0$, the covariance matrix $m_t$, of the Gaussian random variable $\int_0^t f^*(s) \, d B_s$, is invertible, i.e., the Gramian matrix $m_t=\int_0^t f(s) \cdot
f^*(s) \, ds$ has an inverse $\alpha_t$. We emphasize that, under the aforementioned condition, it is not difficult to see that $\alpha_t\rightarrow \alpha_{\infty}$, as $t\rightarrow \infty$, where $\alpha_{\infty}$ is a finite matrix. Furthermore, for any $i$, $(\alpha_{\infty})_{ij}=0$ for all $j$, if and only if $\|f_i\|:=(\int_0^{\infty}f_i^2(s) \, ds)^{1/2}=\infty$. With $\phi(t) = \alpha_t \cdot f(t)$ for $t>0$, we shall establish in Theorem \[Theorem29\] that $(\alpha_{t}, t>0)$ is given in terms of $\phi$ by $\alpha_t= \int_t^{\infty} \phi(u) \cdot \phi^*(u) \,
du + \alpha_{\infty}$, for any $t > 0$. This relation has its own right of importance in this work and may have interesting applications to other fields where Gramian matrices together with their inverses are of prime importance, see for instance [@Berlinet-Thomas] and the references therein. In particular, we also refer to [@Andrews-Askey-Roy] for applications to the theory of special functions and to [@Aronszajn-50] and [@Atteia-92] for applications to reproducing kernel-Hilbert spaces and spline functions.
We define the Volterra transform $\Sigma$, associated to a Volterra kernel $k$, on the set of continuous semi-martingales $X$ such that $$\label{condition}
\lim_{\varepsilon \to
0} \int_{\varepsilon}^t \int_0^v k(u,v) \, dX_u \, dv < \infty,
\qquad 0<t<\infty\quad \mbox{\rm{a.s.}},$$ by $$\label{Volterra}
\Sigma(X)_{t} = X_{t} - \int_0^{t} \! \int_0^u k(u,v) \, dX_v \,
du, \qquad 0<t<\infty.$$ Following [@hhm], the kernel $k(t,s) = \phi^*(t)\cdot f(s)$, for $0< s \leq t<+\infty$ is a self-reproducing Volterra kernel. That is equivalent to saying that $\Sigma$, when applied to the Brownian motion $B$, satisfies the following two conditions:
- $\Sigma(B)$ is a standard Brownian motion;
- For any fixed $t\geq 0$, ${\mathcal F}_t^{\Sigma(B)}$ is independent of $\int_0^t f(u) \, dB_u $.
Existence of $\Sigma(B)$ may be justified by using a generalized Hardy inequality discovered in [@hhm], see Remark \[Hardy\] given below. We call $k$ and $\Sigma$, respectively, a [*Goursat-Volterra kernel*]{} and [*transform*]{}, with reproducing basis $f$. The dimension of $Span\{f\}$ is called the order of the Goursat-Volterra kernel $k$. This terminology is formally fixed in Definition \[Definition-g-v-k\].
Next, we bring our focus on conditions [**(i)**]{} and [**(ii)**]{} and think of them in terms of enlargement of filtrations and stochastic differential equations. Condition [**(ii)**]{} says that the orthogonal decomposition $$\label{eqn:C68} {\mathcal F}_t^B = {\mathcal F}_t^{\Sigma(B)}
\otimes \sigma \left( \int_0^t f(u) \, dB_u \right)$$ holds true, for any $t \geq 0$. Here, by $\mathcal{F} \otimes \,
\mathcal{G}$ we mean $\mathcal{F} \vee \mathcal{G}$ with independence between $\mathcal{F}$ and $\mathcal{G}$. We shall show that, for Goursat-Volterra transforms, equation (\[eqn:C68\]) can in fact be rewritten as $$\label{eqn:C70} {\mathcal F}_t^B = {\mathcal F}_t^{\Sigma(B)}
\otimes \sigma \left( Y - \int_t^{\infty} \phi(u) \,
d\Sigma(B)_u\right)$$ valid for any $t \geq 0$, where $Y=(Y_1, \cdots, Y_N)^*$ is a Gaussian random vector which is independent of ${\mathcal
F}_{\infty}^{\Sigma(B)}$ with covariance matrix $E[Y \cdot Y^*] =
\alpha_{\infty} = \lim_{t \to \infty} \alpha_t$ in case $\alpha_{\infty} \not\equiv 0$, and $Y \equiv 0$ otherwise. We allow here $Y$ to have some null or constant components. Going back to condition [**(i)**]{}, we observe that the determination of all continuous semi-martingales which satisfy it amounts to solving equation $$\label{ssde}
X_t = W_t + \int_0^{t} \! \int_0^s \phi^*(s)\cdot f(u) \, dX_u \,
ds, \, \quad X_0 = 0, \qquad t>0,$$ considered on a possibly enlarged probability space, where $W$ is a standard Brownian motion. Note that we only assume $$\label{condition-2}
\lim_{\varepsilon \to 0} \int_{\varepsilon}^t \! \int_0^v
\phi^*(v) \cdot f(u) \, dX_u \, dv < \infty, \qquad 0<t<\infty
\quad \mbox{\rm{a.s.}},$$ and the latter is not absolutely convergent. Because of the singularity at time $0$, we call (\[ssde\]) a singular linear stochastic differential equation. If we take $W=\Sigma(B)$ then, by construction, the original Brownian motion $B$ is one solution. A second one coincides with the associated $f$-generalized bridge on the interval of its finite life-time, introduced in [@alili00]. It follows that the Goursat-Volterra transform $\Sigma$, when defined as above, is not invertible in the sense that (\[ssde\]) has many solutions. This is not a surprising fact. Indeed, $k$ being a self-reproducing kernel implies that it is not square-integrable, as seen in [@fwy99]. Next, Theorem \[thm25\] deals with the investigation of all continuous semi-martingale solutions to (\[ssde\]). In particular, we show that a necessary and sufficient condition for the existence of a strong solution that is Brownian and ${\mathcal
F}_{\infty}^B$-measurable is $\alpha_{\infty}\equiv 0$. In that case ${\mathcal F}_{\infty}^{\Sigma(B)} = {\mathcal
F}_{\infty}^B$. When $\alpha_{\infty}\not\equiv 0$, Theorem \[thm25\] concludes that there exists still a strong solution which is a Brownian motion, in an enlarged space, that involves an independent centered Gaussian vector $Y$ with covariance matrix $\alpha_{\infty}$. Another natural question is a characterization of all continuous semi-martingales that satisfy both conditions [**(i)**]{} and [**(ii)**]{}. This is partially solved in Theorem \[thm-harmonic\] for the case $\alpha_{\infty}\equiv 0$ and the analysis exhibits some connections to certain space-time harmonic functions. The latter are functions $h \in C^{1,2}\left( \real_+
\times \real^n, \real_+\right)$ such that $ h(\cdot, \int_0^{.}
f^*(s) \, dB_s)$ is a continuous $(\mathbb{P}_0,{\mathcal
F})$-martingale with expectation $1$, where $\mathbb{P}_0$ stands for the Wiener measure.
The main results of this paper extend a part of the first chapter of [@yor92] and some results found in [@jeulin88]. Our work offers explicit examples of conditionings and conditioned stochastic differential equations introduced and studied in [@Baudoin]. Furthermore, singular equations of type (\[ssde\]) and the progressive enlargement of filtration given in Corollary \[decomposition\] can easily be applied to insider trading models elaborated in [@jurgen98a], [@Baudoin] and .
**Goursat-Volterra kernels and transforms** {#sec35}
============================================
To a Brownian motion $B$ we associate the centered Gaussian process $\Sigma(B)$ defined by (\[Volterra\]), which we assume is well-defined, where $k$ is a continuous Volterra kernel. That is to say that $k:\mathbb{R}_+^2\rightarrow \mathbb{R}$ satisfies $$k(u,v)=0,\qquad 0< u \le v <\infty$$ and is continuous on $\{(u,v) \in (0,+\infty)\times (0,+\infty): u
> v \}$. We know from [@fwy99] that $\Sigma$ preserves the Wiener measure, or $\Sigma(B)$ is a Brownian motion, if and only if $k$ satisfies the self-reproducing property $$\label{eqn:K10} k(t,s) = \int_0^s k(t,u) k(s,u) \, du, \qquad 0< s
\leq t<\infty.$$ For a connection with reproducing kernels, in the usual sense, we refer to end of this section. Observe that (\[Volterra\]), when applied to $B$, can be viewed as the semi-martingale decomposition of $\Sigma(B)$ with respect to the filtration $(\mathcal{F}^{B}_t,
t\geq 0)$. Now, as a consequence of the Doob-Meyer decomposition of $\Sigma(B)$ in its own filtration, we must have the strict inclusion $${\mathcal F}_t^{\Sigma(B)}
\subsetneqq {\mathcal F}_t^{B},\qquad 0<t<\infty.$$ It is shown in [@jeulin88] that the missing information, called the reproducing Gaussian space, is given in the orthogonal decomposition $${\mathcal F}_t^{B} = {\mathcal F}_t^{\Sigma(B)} \otimes
\sigma(\Gamma_t^{(k)}),$$ where $$\Gamma_t^{(k)} = \left\{ \int_0^t f(u) \, dB_u; f\in L^2\left((0,t]
\right),\: f(s)=\int_0^s k(s,u)f(u) \, du\quad \hbox{ a.e.}
\right\}$$ for any $t>0$. Given a kernel $k$, it is not an easy task to determine a basis of $\Gamma_t^{(k)}$ for each fixed $t>0$, because this amounts to solving explicitly the integral equation $$f(t)
= \int_0^s k(t,u) f(u) \, du,\qquad 0<t<\infty.$$ It is easier to fix the family of spaces $(\Gamma_t^{(k)}, t>0)$ and work out the corresponding Volterra kernel. This procedure, in fact, corresponds to desintegrating the Wiener measure over the interval $[0,t]$, for any fixed $t>0$, along $\Gamma_t^{(k)}$. Recall that a Goursat kernel is a kernel of the form $$k(t,s) =\phi^*(t) \cdot f(s),\qquad 0< s\leq t<\infty,$$ where $\phi=(\phi_1, \cdots,
\phi_n)^*$ and $f=(f_1, \cdots, f_n)^*$ are two vectors of functions defined on $(0,\infty)$ and $n \in \mathbb{N}$. For such kernels it is natural to introduce the following definition.
\[Definition-g-v-k\] A Goursat-Volterra transform $\Sigma$ of order $(n_t, t>0)$ is a Volterra transform preserving the Wiener measure such that, for any Brownian motion $B$ and $t
> 0$, ${\mathcal F}_t^{\Sigma(B)}$ is independent of $\int_0^t
f(u) \, dB_u $ for some vector $f\equiv(f_1, \cdots, f_{n_t})^*$ of $n_t$ linearly independent $L^2_{loc}(\real_+)$ functions. The associated kernel is called a Goursat-Volterra kernel. The objects $f$, $Span\{f\}$ and $Span\{\int_0^{\cdot} f(s) \, dB_s\}$ are called reproducing basis, space and Gaussian space, respectively.
Because for each fixed $t>0$, $m_t$ is positive definite, it can be seen that $t\rightarrow n_t$ is nondecreasing. However, in our setting, we always take the order to be constant and finite. The simplest known example of Goursat-Volterra kernels is $k_1(t,s)=t^{-1}$ and this gives $$\Sigma(B)_{.}=B_.-\int_0^{\cdot} \frac{B_u}{u} \, du.$$ That corresponds to setting $n=1$ and taking $f_1\equiv 1$. It is observed in [@jeulin88] that $\Sigma$ when iterated takes a remarkably simple form. That is with $\Sigma^{(0)}=Id$, $\Sigma^{(1)}=\Sigma$ and $\Sigma^{(m)} = \Sigma^{(m-1)} \circ \Sigma$, for $m \geq 2$, where $\circ$ stands for the composition operation, we have $$\Sigma^{(n)}(B)_{\cdot}=\int_0^{.}L_n(\log{\frac{\cdot}{s}}) \,
dB_s,$$ where $(L_n, n\in\mathbb{N})$ is the sequence of Laguerre polynomials. As a generalization of the above kernel, we quote the following result from [@hhm].
\[hhm-thm\] Let $f$ be a vector of $n$ functions of $L^2_{loc}(\real_+)$ such that for any $t>0$ the Gramian matrix $m_{t}=\int_0^{t} f(s)
\cdot f^*(s) \, ds$ has an inverse denoted by $\alpha_t$. Then, with $\phi(\cdot)=\alpha_{\cdot} \cdot f(\cdot)$, the kernel $k$, defined by $k(t,s)=0$ if $s>t$ and $k(t,s)=\phi^*(t)\cdot f(s)$ otherwise, is a Goursat-Volterra kernel of order $n$.
For a proof of this result, we refer to [@hhm]. Some arguments of the proof are sketched in Remark \[Hardy\] given below. In the remainder of this paper, unless otherwise specified, we work under the setting of Theorem \[hhm-thm\]. The objective of the next result is to obtain an expression of $\alpha_{\cdot}$ in terms of $\phi(\cdot)$. As a straightforward application, we shall show that it allows to obtain a new self-reproducing property satisfied by the kernel $k$. To our knowledge the following result is not known.
\[Theorem29\] $\alpha_t$ converges to a finite matrix $\alpha_{\infty}$ as $t\rightarrow \infty$. Moreover, we have $$\label{expression-alpha}
\alpha_t= \int_t^{\infty}\phi(u)\cdot \phi^*(u) \, du +
\alpha_{\infty}, \qquad 0<t<\infty.$$ Consequently, the self-reproduction property $$k(t,s) = \int_t^{\infty}
k(u,t) k(u,s) \, du + f^*(t)\cdot \alpha_{\infty}\cdot f(s), \quad
0<s\leq t<\infty,$$ holds true.
Fix $t>0$. Observe that the matrices $\alpha_t$ and $m_t$ are symmetric positive definite with absolutely continuous entries. Next, the identity $\alpha_t\cdot m_t=Id_n=m_t\cdot \alpha_t$, when differentiated, yields $\alpha'_t\cdot m_t=-\alpha_t\cdot
m'_t$. It follows that $$\phi(t)\cdot f^*(t)=\alpha_t\cdot f(t)\cdot f^*(t) =\alpha_t\cdot
m'_t =-\alpha'_t\cdot m_t.$$ Consequently, we have $\alpha'_t=-\phi(t)\cdot f^*(t)\cdot
\alpha_t=-\phi(t)\cdot\phi^*(t)$. For any $1\leq j\leq n$, $(\alpha'_t)_{j,j}=-\phi_j^2(t)$ is negative. Hence, $(\alpha_t)_{j,j}$ is decreasing. Because $(\alpha_t)_{j,j}>0$ we get that $ \int_r^{\infty}\phi_{j}^{2}(s) \, ds <\infty$, $r>0$. Since, for $t\geq r$, we can write $\alpha_t =
\alpha_r-\int_r^t\phi(s) \cdot \phi^*(s) \, ds$, by letting $t
\rightarrow +\infty$, we find $\lim_{t\rightarrow \infty}\alpha_t=
\alpha_r-\int_r^{\infty}\phi(s)\cdot \phi^*(s) \,
ds=\alpha_{\infty}$. Thus, $\alpha_{\infty}$ is a matrix with finite entries. The last statement follows from $k(t,s) = f^*(t)
\cdot \alpha_t \cdot f(s)$ where we use the expression of $\alpha_t$ given in (\[expression-alpha\]).
Self-reproducing kernels, in particular Goursat-Volterra kernels, are different from but related to kernel systems and reproducing kernel Hilbert spaces. Our next objective is to outline this connection. For, let us start by fixing a time interval $[0,t]$, for some $t>0$. Let the vector $q_t(u):=(q_{m,t}(u), 0<u\leq t;
1\leq m\leq n)$ be formed by the orthonormal sequence associated to $f_1, f_2,\cdots, f_n$ over the interval $[0,t]$. This system is uniquely characterized by $$\int_0^t q_{m, t}(r)q_{k, t}(r) \, dr=\delta_{m,k}, \qquad 1\leq
m,k\leq n,$$ with the requirement that for each integer $1\leq m\leq n$, $q_{m,t}$ is a linear combination of $f_1$, ..., $f_{m}$ with a positive leading coefficient associated to $f_{m}$. We refer to Lemma 6.3.1, p. 294, in [@Andrews-Askey-Roy] for an expression of the latter in terms of a determinant. The classical kernel system is then given by the symmetric kernel $$\kappa_{t}(u,v)=q_{t}(u) \cdot q_{t}^{*}(v), \qquad 0<u,v\leq t.$$ This is a reproducing kernel in the sense that $$\kappa_{t}(u,v)=\int_0^{t} \kappa_{t}(u,r)\kappa_{t}(v,
r) \, dr,\qquad 0< u, v\leq t.$$ For $1\leq i,j\leq n$, $(\alpha_t)_{i,j}$ is seen to be the coefficient of $f_i(u)f_j(v)$ in the expansion of $\kappa_{t}$. To be more precise, $(\alpha_t)_{i,j}= (b_t \cdot b^*_t )_{i,j}$ where $b$ is an upper diagonal matrix whose entry $(b_t)_{i,k}$ is the coefficient of $f_i(u)$ in $q_{k,t}(u)$ for $i\leq k$. We clearly have $\phi_i^2(t)=-2 (b_t' \cdot b^*_t)_{i,i}$ for all $i$ and it would be interesting to express the matrix $b_t$ in terms of $\phi(t)$. Now, we are ready to state the following result which proof is omitted.
For each fixed $t>0$, the kernel system associated to $f$, over the time interval $[0,t]$, is given by $\kappa_{t}(u,v)=\int_t^{\infty} k(r,u)k(r,v) \, dr +f^*(u) \cdot
\alpha_{\infty} \cdot f(v) $ for $0< u,v \leq t$. In particular, we have $k(t,s)=\kappa_{t}(t,s)$ for all $0 < s \leq t<\infty$.
As in the proof of Theorem \[Theorem29\], the first part of the result follows from the well-known relationship $\kappa_{t}(u,v)=f^*(u) \cdot \alpha_t \cdot f(v)$ for any $0 < u, v \leq t$. The second part follows by taking the limit and using continuity.
To see an example where $\alpha_{\infty}\not\equiv 0$, let us discuss the case $n = 2$. Assume that $f_1$ and $f_2$ are two functions in $L^2_{loc}(\mathbb{R}_+)$. We distinguish four cases and three different forms for $\alpha_{\infty}$. The first corresponds to $\alpha_{\infty}\equiv 0$ when $\|f_1\|=\|f_2\|=+\infty$. The second corresponds to case when $
\|f_1\|$ and $\|f_2\|$ are finite which implies that $\alpha_{\infty}$ is positive-definite. Observe that the off-diagonal entries are zero only when $\int_0^{\infty}
f_1(s)f_2(s)ds=0$. The latter integral is zero if, for instance, we take $f_1=\varphi-\psi$ and $f_2=\varphi+\psi$, where $\|\varphi\|=\|\psi\|<\infty$. In the third case, all the entries of $\alpha_{\infty}$ are zero but $(\alpha_{\infty})_{1,1}=1/\|f_1\|^2$ if $\|f_1\|<+\infty$ and $\|f_2\|=+\infty$. The remaining case is similar by symmetry.
We shall now discuss examples of kernels of order $n$, $n\in \mathbb{N}$, which reproducing spaces are Müntz spaces. We refer to [@alili-wu-02] for proofs of results given below. Take $f_i(s)=s^{\lambda_i}$, $i=1, 2, \cdots $, where $\Lambda=\{\lambda_1,\lambda_2,\cdots\}$ is a sequence of reals such that $\lambda_i\neq \lambda_j$ for $i\neq j$ and $\lambda_i>-1/2$. For a fixed $n<\infty$, the kernel $k_n$ defined by $k_n(t,s)=0$ if $s>t$ and $$\label{eqn:M-K33-2}k_n(t,s)=t^{-1}\sum_{j = 1}^n a_{j,n}
(s/t)^{\lambda_j},\quad a_{j,n} = \frac{\prod_{i = 1}^n (\lambda_i
+ \lambda_j + 1)}{\prod_{i=1, i\neq j}^{n}(\lambda_i -
\lambda_j)}, \quad j=1,..., n,$$ if $0 < s \leq t$, is a Goursat-Volterra kernel of order $n$. Its reproducing Gaussian space, at time $t>0$, is $Span\{ \int_0^t s^i
dB_s; i=1,2,\cdots,n \}$. Going back to the Gramian matrix $(m_t, t\geq 0)$, observe that it has the entries $$\left( m_t \right)_{i, j}=({\lambda_i+\lambda_j
+1})^{-1}{t^{\lambda_i+\lambda_j +1}}, \qquad i, j=1,\cdots, n.$$ Thus if $t=1$ then $m_1$ is a Cauchy matrix. When $\lambda_i=c i$, for some constant $c\neq 0$, and $n=\infty$, $m_1$ is the well-known Hilbert matrix. Note that because $||f_i||= +\infty$, $i=1,\cdots, n $, we have $\alpha_{\infty}\equiv 0$. So we have $\phi_{i}(t)=a_{i,n}t^{-\lambda_i-1}$, $i=1,2,\cdots, n$. Furthermore, the entries of $\alpha_t$ are given by $$(\alpha_t)_{i,j}=
a_{i,n}a_{j,n}(\lambda_i+\lambda_j+1)^{-1}t^{-\lambda_i-\lambda_j-1},
\qquad i,j=1,\cdots n,$$ which follows from the expression of the kernels when compared with Theorem \[Theorem29\]. Note that $\alpha_t$, for $t\neq 1$, can easily be constructed from $\alpha_1$ which is known and can be found in [@Schechter]. Finally, we mention that some results are obtained about infinite order kernels in the Müntz case, see [@alili-wu-02] and [@hm-2004].
\[Hardy\] Observe that we can write $$\Sigma(B)_t= \int_0^{\infty} (I-K^{*}_{f})1_{[0,t]}(u) \, dB_u,
\qquad 0<t<\infty$$ where $K^{*}_{f}$ is the adjoint of the bounded integral operator $K_{f}$ defined on $L^2_{loc}(\mathbb{R}_+)$ by $$K_{f}\alpha(t)=\int_0^t k(t,r) \alpha(r) \, dr, \qquad \alpha\in
L^2_{loc}(\mathbb{R}_+).$$ That $I-K_{f}$ is a partial isometry, with initial subspace $L^2_{loc}(\mathbb{R}_+)\ominus \hbox{Span}\{f\}$ and final subspace $L^2_{loc}(\mathbb{R}_+)$, follows from the generalized Hardy inequality $$\|K_{g}\alpha \|\leq 2\|\alpha\|, \qquad \alpha\in
L^2_{loc}(\mathbb{R}_+).$$ Consequently, the operator $I-K^{*}_f$, when defined on $L^{2}_{loc}(\mathbb{R}_+)$, is isometric which implies the statement of the Theorem \[hhm-thm\]. For the above results, we refer to [@hhm]. We also refer to the comments of Section 3 therein because here we are working with $L^2_{loc}(\mathbb{R}_+)$ instead of $L^2_{loc}([0,1])$.
Many authors work under the condition $$\label{integrability}
\int_0^t \left( \int_0^u k^2(u,v) \, dv \right)^{1/2} du < \infty$$ for all $t> 0$, which is sufficient for $\Sigma(B)$, where $B$ is a standard Brownian motion, to be well-defined, see for instance [@fwy99]. However, condition (\[integrability\]) is too strong for $\Sigma(B)$ to be well-defined. To see that let us fix $b\in L^2_{loc}(\mathbb{R}_+)$. The associated Goursat-Volterra kernel of order $1$ is then found to be $$k(t,v)=b(t)b(v) \left/\int_0^{t}b^2(r) \, dr \right..$$ It satisfies (\[integrability\]) if and only if $\int_0^{t}
|b(s)|/(\int_0^s b^2(r) \, dr)^{1/2} \, ds <\infty$ for all $t<\infty$. For example, the kernel associated to $b(t)=t^{-1}e^{-1/t}$ fails to satisfy (\[integrability\]).
[**On some singular linear stochastic differential equations**]{} {#sec34}
=================================================================
Consider the singular linear stochastic equation (\[ssde\]). Our interest lies in the set of all its continuous semi-martingale solutions which may be defined on a possibly enlarged space. For a particular solution $X$, we recall that (\[ssde\]) is well-defined in the sense that (\[condition-2\]) holds. If we set $W=\Sigma(B)$, where $B$ is a Brownian motion, then the set includes at least two solutions which one shall now briefly describe. First, $B$ is a solution. Second, there is a solution which is defined on $\mathbb{R}_+$ and coincides with the $f$-generalized bridge over its life time. The latter process, denoted by $\left(B_u^{y}, u\leq t_1\right)$, for some $t_1>0$ and a column vector of reals $y$, is defined by $$B^{y}_u=B_u-\psi^*(u)\cdot \int_0^{t_1}f(s) \, dB_s+\psi^*(u)\cdot
y,\qquad 0<u<t_1,$$ where $\psi$ is the unique solution to the linear system $$\int_0^u f(s) \, ds=\psi(u)\cdot \int_0^{t_1}f(s)\cdot f^*(s) \,
ds=\psi(u)\cdot m_{t_1}, \qquad 0<u<t_1.$$ Thus $\psi(u) = \alpha_{t_1} \cdot \int_0^{u}f(s) \, ds$ which implies that $\int_0^{t_1} f(s) \, dB^{y}_s = y$, since $\alpha_{t_1}$ is the inverse of $m_{t_1}$. This is the reason why the above process is called an $f$-generalized bridge over $[0,t_1]$ with endpoint $y$. Now, we have $\Sigma(B^{y}) =
\Sigma(B)$ which is true because $\Sigma$ is linear and $\Sigma(\int_0^{\cdot} f(r) \, dr) \equiv 0$ since $f(t)=\int_0^{t} k(t,v)f(v) \, dv$ for all $0 < t < \infty$. This shows that $B^{y}$ is also a solution to (\[ssde\]) which, in fact, is a non-canonical decomposition. For further results on these processes, such as their canonical decomposition in their own filtrations, we refer to [@alili00]. Now, we consider equation (\[ssde\]) where the driving Brownian motion $W$ is taken to be arbitrary.
\[thm25\] $1)$ $X$ solves equation (\[ssde\]) if and only if there exists a random vector $Y = (Y_1, \cdots, Y_n)^*$ such that $$\label{form}
X = X^0 + \int_0^{\cdot} f^*(u) \, du \cdot Y$$ where $$X^0 = W -
\int_0^{\cdot} \! \int_u^{\infty} \phi^*(v) \cdot f(u) \, dW_v \,
du.$$ In terms of $X$, $Y$ is given by $\displaystyle Y= \lim_{t \to
\infty}
\alpha_t\cdot \int_0^t f(u) \, dX_u$.\
$2)$ $X^0$ is a Brownian motion if and only if $\alpha_{\infty} \equiv 0$. In case $\alpha_{\infty} \not\equiv 0$, a process $X$ solving equation (\[ssde\]) is a Brownian motion if and only if $Y$ is centered Gaussian with covariance matrix $\alpha_{\infty}$ and is independent ${\mathcal F}_{\infty}^{X^0}$.
1\) We proceed by checking first that $X_t^0$ is a particular solution to (\[ssde\]). Using the stochastic Fubini theorem, found for instance in [@protter92], we perform the decompositions $$\begin{aligned}
& & X_t^0 - \int_0^t \! \int_0^u k(u,v) \, dX_v^0 \, du \\
& = & W_t - \int_0^t \! \int_u^{\infty} k(v,u) \, dW_v \, du -
\int_0^t \!
\int_0^u k(u,v) \left( dW_v - \int_v^{\infty} k(\rho,v) \, dW_{\rho} \, dv \right) du \\
& = & W_t - \int_0^t \! \int_u^{\infty} k(v,u) \, dW_v \, du - \int_0^t \! \int_0^u k(u,v) \, dW_v \, du \\
& & \qquad + \int_0^t \! \int_0^u \! \int_0^{\rho} k(u,v)
k(\rho,v) \, dv \, dW_{\rho} \, du + \int_0^t \! \int_u^{\infty}
\! \int_0^u k(u,v) k(\rho,v) \, dv \, dW_{\rho} \, du.\end{aligned}$$ Since $k$ is self-reproducing, the last four terms in the last equation cancel showing that $X_t^0$ solves (\[ssde\]). Next, if $X$ is a solution then by setting $X = X^0 + Z$ we see that $Z$ has to satisfy $$dZ_r = \int_0^r k(r,v) \, dZ_v \, dr, \qquad 0<r<\infty.$$ Multiplying both sides by $f(r)$ and integrating with respect to $r$, along $[0,t]$, yields $$\begin{aligned}
\nonumber
\int_0^t f(v) \, dZ_v &=&\int_0^t f(v) \phi^*(v)\cdot
\int_0^v f(r) \, dZ_r \, dv \\
\nonumber &=& \int_0^t m_v \cdot \phi(v) \phi^*(v)\cdot
\int_0^v f(r) \, dZ_r \, dv \\
\nonumber &=&-\int_0^t m_v \cdot \frac{d}{dv} \, \alpha_v\cdot
\int_0^v f(r) \, dZ_r \, dv\end{aligned}$$ where we used the expression of $\alpha'$ given in the proof of Theorem \[Theorem29\] to obtain the last equality. Because $\alpha$ is the inverse of $m$, the latter relation can be written as $\frac{d}{dt}
\, \alpha_t\cdot \int_0^t f(s) \, dZ_s =0$. This, when integrated, yields $ \alpha_t\cdot \int_0^t f(s) \, dZ_s = Y$ for some random vector $Y$. Hence $\int_0^t f(r) \, dZ_r = m_t\cdot Y
$ which implies that $Z_t=Y^*\cdot \int_0^{t} f(s) \, ds$. This completes the proof of the first part of the first assertion. For the second part, by using Theorem \[Theorem29\] we obtain $$\begin{aligned}
\phi(t) \, dW_t
& = & \phi(t) \, dX_t - \phi(t) \phi^*(t)\cdot \int_0^t f(u) \, dX_u \, dt \\
& = & \alpha_t\cdot d \left(\int_0^t f(u) \, dX_u\right) - \phi(t) \phi^*(t)\cdot \int_0^t f(u) \, dX_u \, dt \\
& = & d \left( \alpha_t\cdot \int_0^t f(u) \, dX_u \right).\end{aligned}$$ Integrating on both sides over $[s,t]$ we obtain $$\int_s^t \phi(u) \, dW_u = \alpha_t\cdot \int_0^t f(u) \, dX_u -
\alpha_s\cdot \int_0^s f(u) \, dX_u.$$ Next, observe that as $t \to \infty$ the left hand side converges almost surely. So the right hand side converges as well to some limit which we denote by $\tilde{Y}$. To be more precise, setting $$\tilde{Y}= \lim_{t \to \infty}\alpha_t\cdot \int_0^t f(u) \, dX_u,$$ we have shown that $$\label{eqn:K42}
\int_t^{\infty} \phi(u) \, dW_u = \tilde{Y} - \alpha_t\cdot
\int_0^t f(u) \, dX_u,\qquad 0< t\leq \infty.$$ Consequently, we have $$\begin{aligned}
\int_0^t \int_u^{\infty } f^*(u)\cdot \phi(v)
\, dW_v \, du &-& {\tilde{Y}}^*\cdot \int_0^t f(u) \, du\\
& = & \int_0^t f^*(u)\cdot \alpha(u) \cdot \int_0^u f(v) \, dX_v \, du \\
& = & \int_0^t \int_0^u \phi^*(u)\cdot f(v) \, dX_v \, du.\end{aligned}$$ Thus, we have $$\begin{aligned}
\int_0^t \! \int_u^{\infty} k(v,u) \, dW_v \, du - \tilde{Y}^* \cdot
\int_0^t f(u) \, du &=& - \int_0^t \! \int_0^u k(u,v) \, dX_v \,
du \\
&=& W_t - X_t.\end{aligned}$$ Comparing with previous calculations yields $Y = \tilde{Y}$, $\mathbb{P}_0$-almost surely.\
2) Theorem \[Theorem29\] implies that $$\label{eqn:K43} E[X_s^0 X_t^0] = s\wedge t -
\int_0^{s\wedge t} \int_0^t f^*(r)\cdot \alpha_{\infty}\cdot f(v)
\, dv \, dr.$$ This clearly shows that $X^0$ is a Brownian motion if and only if $\alpha_{\infty} \equiv 0$. Next, if $X$ is as prescribed then by virtue of (\[eqn:K43\]), and the fact that $\alpha_{\infty}$ is the covariance matrix of $Y$, we have $$\begin{aligned}
E[X_s X_t] & = & s\wedge t - \int_0^{s} \int_0^t f^*(u)\cdot
\alpha_{\infty}\cdot
f(v) \, dv \, du \\
&+& \int_0^s \int_0^t E \left[ (Y^*\cdot f(u)) (Y^*\cdot f(v))
\right] \, dv \, du \\
&=& s\wedge t.\end{aligned}$$ Because $X$ is a continuous Gaussian process we conclude that it is a Brownian motion. Conversely, if $X$ is a Brownian solution to (\[ssde\]) then it has to be of the form (\[form\]). By virtue of the orthogonal properties of Goursat-Volterra transform, we see that $Y$ is independent of ${\mathcal F}_t^{\Sigma(X)} =
{\mathcal F}_t^W$ for any fixed $t
> 0$. Next, by letting $t$ go to $\infty$, we get that $Y$ is independent of ${\mathcal F}_{\infty}^{X^0} \subseteq {\mathcal
F}_{\infty}^W$. Thus, $Y$ is Gaussian vector, with covariance matrix $\alpha_{\infty}$, which is independent of ${\mathcal
F}_{\infty}^{X^0}$ as required.
Thanks to the importance of the symmetric matrix $\alpha_{\infty}$, for instance in Theorem \[Theorem29\], it is natural to look for a description of its structure. The following result, which is hidden in the proof of Theorem \[thm25\], gives a necessary and sufficient condition for a column or a row to be zero.
\[description\] For $1 \le i \le n$, $(\alpha_{\infty})_{i,j} = (\alpha_{\infty})_{j,i} =
0$ for all $j$, if and only if $\| f_i \| = \infty$.
For a fixed $t>0$, $\alpha_t$ is the covariance matrix of $\alpha_t \cdot \int_0^t f(s) \, dB_s$. Furthermore, due to Theorem \[thm25\], we conclude that $\alpha_t \cdot \int_0^t
f(s) \, dB_s$ converges to a Gaussian vector $Y$, possibly with some null components, such that $E(Y \cdot Y^*)=\alpha_{\infty}$. Thus, $Y_i\equiv 0$ for some $i$ if and only if $(\alpha_{\infty})_{i,i}=0$ and if and only if $\|f_i\|=\infty$. Now, $(\alpha_{\infty})_{i,i}=0$ if and only if $(\alpha_{\infty})_{i,j} = 0$ for all $j$. In order to see that, we let $t\rightarrow \infty$ and use continuity in the well-known inequality $|(\alpha_{t})_{i,j}|^2 \leq (\alpha_{t})_{i,i}
(\alpha_{t})_{j,j}$ valid for symmetric positive definite matrices.
Now, we take a look at the orthogonal decompositions of filtrations which arise from Goursat-Volterra transforms and provide their interpretation.
\[decomposition\] The orthogonal decomposition given by (\[eqn:C70\]) holds true. Furthermore, the progressive decomposition $${\mathcal F}_t^{B} = {\mathcal F}_t^{\Sigma(B)} \otimes \sigma
\left( Y- \int_t^{\infty} \phi(u) \, d\Sigma(B)_u \right), \qquad
0<t<\infty$$ holds true, where $Y\equiv 0$ if $\alpha_{\infty}\equiv 0$ and $Y$ is a Gaussian vector independent of ${\mathcal
F}_{\infty}^{\Sigma(B)}$ with covariance matrix $\alpha_{\infty}$ otherwise. Thus, we have ${\mathcal F}_{\infty}^B = {\mathcal
F}_{\infty}^{\Sigma(B)}$ in case $\alpha_{\infty}\equiv 0$ and ${\mathcal F}_{\infty}^B = {\mathcal
F}_{\infty}^{\Sigma(B)}\vee\sigma\{ Y\}$ otherwise.
For a fixed $t> 0$, Theorem \[thm25\] implies that $$B_t = \Sigma(B)_t - \int_0^t \! \int_u^{\infty} k(v,u) \,
d\Sigma(B)_v \, du + Y^*\cdot \int_0^t f(u) \, du$$ where $Y$ is a Gaussian vector with covariance $\alpha_{\infty}$ which is independent of ${\mathcal F}_{\infty}^{\Sigma(B)} $. Hence, we have $$\int_0^t f(u) \, dB_u = m_t\cdot \left( Y - \int_t^{\infty}
\phi(u) \, d\Sigma(B)_u \right)$$ which gives $$\sigma \left\{\int_0^t f(u)
\, dB_u \right\} =\sigma \left\{ Y - \int_t^{\infty} \phi(u) \,
d\Sigma(B)_u \right\}.$$ This implies the first assertion while the last one follows by letting $t$ tend to $+\infty$.
Recall that $\mathcal{F}_0^{B}$ and $\mathcal{F}_0^{\Sigma(B)}$ are trivial. So by letting $t$ converge to $0$, in Corollary \[decomposition\], we see that $\phi^* \in L^2([\varepsilon, \infty)^n)$ for all $\varepsilon
> 0$ but $\phi_i \not\in L^2((0,+\infty))$, for $i=1, \cdots n$. This fact can also be shown by a combination of Theorem \[Theorem29\] and the inequality $$(\alpha_{t})_{i,i} \geq {1}/{(m_t)_{i,i}}={1}/{\|f_i\|^2}$$ which follows from the orthogonal diagonalization of $m_t$ and may be found in Exercise 8, p. 274, in [@D.Harville].
It is clear that if the choice of the vector $f$ allows the use of integration by parts for the integrand in the right hand side of (\[ssde\]) then we obtain a stochastic differential equation which does not involve a stochastic integral. For instance, that is the case for the examples given by P. Lévy, found in [@Levy-56] and [@Levy-57]. These go back to around the middle of last century when stochastic integration was not yet world-widely developed.
**Connections to some positive martingales**
============================================
Let $(k(t,s), t\geq s>0)$ be a Goursat-Volterra kernel of order $n$, where $n$ is a natural number. Assume that $f$ is a reproducing basis for $k$, or for the associated Volterra transform $\Sigma$, and let us keep the notations used in the Introduction. Consider the singular stochastic differential equation (\[ssde\]) associated to $k$ and driven by a given standard Brownian motion $W$. Our aim here is to describe the set $$\begin{aligned}
\Upsilon^{(k)}& = &\{\mathbb{P} \hbox{ is the probability law of a
continuous semi-martingale } X
\\
& & \hbox{on }\; (\mathcal{C}([0,\infty), \mathbb{R}), {\mathcal
F}_{\infty}^*) \; \hbox{solving}\; (\ref{ssde})\;
\hbox{s.t.}\; \Sigma{(X)} \hbox{ is a Brownian motion} \\
& & \hbox{and}\; \mathcal{F}_t^{\Sigma(X)}\; \hbox{is independent
of} \int_0^{t} f(s) \, dX_s, \hbox{ for all } 0<t<\infty \}.\end{aligned}$$ We read from Corollary \[description\] that $\alpha_{\infty}\equiv 0$ if and only $||f_i||=\infty$ for all $i$. Now, we are ready to state the following unified characterization of the set $\Upsilon^{(k)}$.
\[thm-harmonic\] If $\alpha_{\infty}\equiv 0$ then the following assertions are equivalent
- $\mathbb{P} \in \Upsilon^{(k)}$.
- $\mathbb{P}$ is the law of ${\displaystyle B +
Y^*\cdot \int_0^{.} f(s) ds}$, where $B$ is a standard Brownian motion and $Y$ is a vector of random variables which is independent of ${\mathcal F}_{\infty}^B$.
- There exists a positive function $h\in
C^{1,2}\left( \real_+ \times \real^n, \real_+\right)$ such that $
h(., \int_0^{.} f^*(s) \, dB_s)$ is a continuous $(\mathbb{P}_0,{\mathcal F})$-martingale with expectation $1$, and $\mathbb{P} = \mathbb{P}_0^h$ with $$\mathbb{P}_0^h \left.
\right|_{{\mathcal F}_t} = h \left( t, \int_0^t f^*(s) \, dB_s
\right)
\cdot \mathbb{P}_0 \left. \right|_{{\mathcal F}_t}, \qquad 0<t<\infty,$$ where $\mathbb{P}_0$ stands for the Wiener measure,
We split the proof into several steps where we show that $(1)\Longleftrightarrow (2)$ and $(2)\Longleftrightarrow
(3)$. Let us show that $(1) \Longrightarrow (2)$. Let $ \mathbb{P}
\in \Upsilon^{(k)}$. Theorem \[thm25\] implies that there exist a vector $Y$ such that $\mathbb{P}$ is the law of $X_t^0 +
Y^*\cdot \int_0^t f(u) \, du$. That combined with the assumption $\alpha_{\infty}\equiv 0$ leads to the fact that $X^0$ is a Brownian motion. Hence, it suffices to show that $Y$ is independent of $X^0$. From (\[eqn:K42\]) we see that $Y=
\int_t^{\infty} \phi(u) \, dB_u + \alpha_t\cdot \int_0^t f(u) \,
dX_u$, the vector $\int_0^t f(u) dX_u$ is independent of $B$ and, consequently, it is also independent of $X^0$. Thus, whenever $Z\in L^2({\mathcal F}_{\infty}^{X^0})$, for any fixed $t\geq 0$, we have $$\begin{aligned}
E \left[ E \left[ Z \left| {\mathcal F}_t^{X^0} \right. \right]
\phi \left( Y - \int_t^{\infty} \phi(u)
dB_u \right) \right]=E \left[ Z \right] E \left[ \phi \left( Y- \int_t^{\infty}
\phi(u) \, dB_u\right) \right]\end{aligned}$$ for any bounded function $\phi: \real^n \to \real$. By letting $t
\rightarrow \infty$ we conclude that $E \left[ Z \cdot \phi(Y)
\right] = E[Z] E \left[ \phi(Y) \right]$ which implies the required independence. We shall now show that $ (2)
\Longrightarrow (1)$. To this end, let $k$ be a Goursat-Volterra kernel. Denote by $f$ a reproducing basis associated to $k$ and put $X_t= B_t + Y^*\cdot \int_0^{t} f(s) \, ds$ for $t>0$. For a fixed $t>0$, because $\int_0^{t} f(u) \, dB_u \in
\Gamma_{t}^{(k)}$, we can write $$\begin{aligned}
X_t - \int_0^{t} \! \int_0^u k(u,v) \, dX_v \, du &=& B_t -
\int_0^{t} \! \int_0^u k(u,v) \, dB_v \, du = \Sigma(B)_t\end{aligned}$$ which, of course, a Brownian motion. Furthermore, using once more the above argument we can easily see that $ \int_0^t f(u) \, dX_u$ is independent of $\mathcal{F}_t^{\Sigma(B)}$. Next, we deal with $(2) \Longrightarrow (3)$. Denote by $\nu(dy)$ the distribution of $Y$. For any measurable functional $\phi$, we then have that $$\begin{aligned}
& & E \left[ \phi \left( B_s + Y^*\cdot \int_0^s f(u) \, du:
s \le t \right) \right] \\
& = & \int_{\real^n} E \left[ \phi \left( B_s + y^*\cdot \int_0^s
f(u) \, du: s \le t \right) \right] \nu(dy) \\
& = & \int_{\real^n} E \left[ \exp \left(\int_0^t y^*\cdot f(u) \,
dB_u - \frac12 \int_0^t \left( y^*\cdot f(u) \right)^2 \, du
\right) \phi(B_s: s \le t) \right] \nu(dy)\end{aligned}$$ where the last equality is obtained by Girsanov theorem. The required space-time harmonic function is thus given on $\mathbb{R}^+\times \mathbb{R}^n$ by $$h(t, x) = \int_{\real^N} \exp \left( y^* \cdot x - \frac12
\int_0^t \left( y^*\cdot f(s) \right)^2 \, ds \right) \nu(dy).$$ It remains to show that $(3) \Longrightarrow (2)$. For fixed $0<u\leq t<+\infty$, set $\psi(u,t) = \alpha_t\cdot \int_0^u f(s)
\, ds$. Let us write the obvious decomposition $$\begin{aligned}
B_u &=& \left( B_u - \psi^*(u,t)\cdot \int_0^t f(s) \, dB_s
\right) + \psi^*(u,t)\cdot \int_0^t f(s) \, dB_s\end{aligned}$$ and denote by $H_u^t$ the first term of its right hand side. We observe that the process $(H_u^t, u<t)$ has has then the same law under $\mathbb{P}_0$ as under $\mathbb{P}_0^h$. Next, to simplify notations, write $$\hat{H}_u^t= \psi^*(u,t)\cdot \int_0^t f(s) \, dB_s
= Y^*_t\cdot \int_0^u f(s) \, ds,$$ where we set $Y^*_t=\alpha_t \cdot\int_0^t f(r)dB_r$. For any $ 0
\le s \le u \le t$, we have $E[H_s^t H_u^t] = s - \psi^*(s,t)\cdot
\int_0^u f(v) \, dv$ and $ \psi^*(s,t)\cdot \int_0^u f(v) \, dv =
\int_0^u f^*(v) \, dv \cdot \alpha_t \cdot \int_0^s f(r) \,
d\rightarrow 0$ as $t\rightarrow \infty$ because $\alpha_{\infty}\equiv 0$. We conclude that the convergence in distribution $H_{\cdot}^t\rightarrow B^{(h)}_{\cdot}$ holds, where $B^{(h)}$ is a $\mathbb{P}_0^h$-Brownian motion. That implies the convergence of $\hat{H}_{.}^t$ as well to a finite limit. But that can happen if and only if $Y^*_t$ converges to a finite limit which we denote by $Y^*$. Finally, from the above arguments we see that $Y^*$ is independent of $\mathcal{F}_{\infty}^{B^{(h)}}$ which ends the proof.
Unfortunately, for the case $\alpha_{\infty} \not\equiv 0$, the second statement in the above theorem is too strong. For example, $X^0$ satisfies the assertion (1) but it is easily seen that it does not satisfy (2). The implications $(2) \Longrightarrow (1)$ and $(2) \Longrightarrow (3)$ still work in this case. We also can replace $B$ by $X^0$ in statement (2) and prove that $(1)
\Longleftrightarrow (2)$ still holds true. However, $(2)
\Longrightarrow (3)$ fails.
[**Acknowledgment:**]{} This work was partly supported by the Austrian Science Foundation (FWF) under grant Wittgenstein-Prize Z36-MAT. We would like to thank W. Schachermayer and his team for their warm reception. The second author is greatly indebted to the National Science Council Taiwan for the research grant NSC 96-2115-M-009-005-MY2. We thank Th. Jeulin, R. Mansuy and M. Yor for fruitful discussions. Finally, we are grateful for anonymous referees for several reports on previous versions of this paper which lead to its improvement.
[99]{}
Alili, L. Canonical decomposition of certain generalized Brownian bridges. [*Electron. Comm. Probab.*]{}, [**7**]{}, electronic, 2002, pp. 27-36.
Alili, L., Wu, C-T. Müntz linear transforms of Brownian motions. 2007, [*In preparation*]{}.
Andrews, G., Askey, R., Roy, R. [*Special functions.*]{} Encyclopedia of Mathematics and its Applications, [ **71**]{}, 1999, Cambridge University Press.
Aronszajn, N. Theory of reproducing kernels. [*Transactions of the AMS.*]{}, [**68**]{}, 1950, pp. 307-404.
Atteia, M. Hilbertian kernels and spline functions. [*Studies in Computational Mathematics*]{}, [**4**]{}, C. Brezinski and L. Wuytack eds, North-Holland. Amendinger, J. Initial enlargement of filtrations and additional information in financial markets. Ph. D. thesis, 1999, Technische Universität Berlin.
Baudoin, F. Conditioned stochastic differential equations: Theory, Examples and Application to finance. [*Stoch. Proc. and their Appl.*]{}, [**100**]{}, 2002, pp. 109-145.
Berlinet, A., Thomas-Agnan, C. [*Reproducing kernel Hilbert spaces in probability and statistics.*]{} 2004, Kluwer Academic Publishers.
Borwein, B., Erdélyi, T. [*Polynomials and polynomial inequalities.*]{} Graduate Texts in Mathematics, [**161**]{}, 1995, Springer.
Deheuvels, P. Invariance of Wiener processes and of Brownian bridges by integral transforms and applications. [*Stoch. Proc. and their Appl.*]{}, [**13**]{}, no. 3, 1982, pp. 311-318.
Föllmer, H., Wu, C-T, Yor, M. On weak Brownian motions of arbitrary order. [*Ann. Inst. H. Poincaré Probab. Statist.*]{}, [**36**]{}, no. 4, 2000, pp. 447-487.
Harville, D. [*Matrix Algebra from a Statistician’s Perspective*]{}, 1997, Springer.
Hibino, Y. Construction of noncanonical representations of Gaussian process. [*J. of the Fac. of Lib. Arts*]{}, Saga University, no. 28, 1996, pp.1-7. English transl. available at http://www.ms.saga-u.ac.jp/ hibino/paper.htm.
Hibino, Y. A topic on noncanonical representations of Gaussian processes [*Stochastic analysis: classical and quantum*]{}, World Sci. Publ., Hackensack, 2005, pp. 31-34.
Hibino, Y., Hitsuda, M., Muraoka, H. Construction of noncanonical representations of a Brownian motion. [*Hiroshima Math. J.*]{}, [**27**]{}, no. 3, 1997, pp. 439-448.
Hibino, Y., Muraoka, H. Volterra representations of Gaussian processes with an infinite-dimensional orthogonal complement. Quantum Probability and Infinite Dimensional Analysis. From Foundations to Applications, Eds. M. Schürmann and U. Franz, QP-PQ: Quantum Probability and White Noise Analysis, [**18**]{}, World Scientiffic, 2004, pp. 293–302.
Hida, T. Canonical representations of Gaussian processes and their applications. [*Mem. Coll. Sci. Univ. Kyoto Ser. A. Math.*]{}, [**33**]{}, 1960, pp. 109-155.
Hida, T., Hitsuda, M. [*Gaussian processes*]{}. Translated from the 1976 Japanese original by the authors. Translations of Mathematical Monographs, [**120** ]{}, 1993, American Mathematical Society.
Hitsuda, M. Representations of Gaussian processes equivalent to Wiener process. [*Osaka J. Math.*]{}, [**5**]{}, 1968, pp. 299-312.
Jeulin, Th. Semi-martingales et grossissement d’une filtration. [*Lecture Notes in Mathematics*]{}, [**833**]{}, 1980, Springer.
Jeulin, Th., Yor, M. [*Grossissements de filtrations: exemples et applications*]{}. [*Lecture Notes in Mathematics*]{}, [ **1118**]{}, 1985, Springer.
Jeulin, Th., Yor, M. Filtration des ponts browniens et équations differentielles linéaires, in: J. Azéma, P. A. Meyer and M. Yor, eds., Séminaire de Probabilités Vol. XXIV, 1988/89, [*Lecture Notes in Mathematics*]{}, [ **1426**]{}, 1990, Springer.
Jeulin, Th., Yor, M. Moyennes mobiles et semimartingales. [*Sém. de Prob.*]{}, XXVII, 1993, pp. 53-77.
Kallianpur, G. [*Stochastic filtering theory*]{}. Applications of Mathematics, [ bf 13]{}, 1980, Springer.
Karatzas, I., Pikovsky, I. Anticipative portfolio optimization. [*Adv. in Appl. Probab.* ]{}, [**28**]{}, no. 4, 1996, pp. 1095-1122.
Lévy, P. Sur une classe de courbes de l’espace de Hilbert et sur une équation intégrale non linéaire. [*Ann. Sci. Ecole Norm. Sup.*]{}, [**73**]{}, 1956, pp. 121-156.
Lévy, P. Fonctions aléatoires à corrélation linéaires. [*Illinoi J. Math.*]{}, [**1**]{}, 1957, pp. 217-258.
Protter, Ph. [*Stochastic integration and differential equations*]{}. Second edition. Applications of Mathematics, [**21**]{}. Stochastic Modelling and Applied Probability, 2004, Springer.
Schechter, S. On the inversion of certain matrices [*Math. Comp.*]{}, [**13**]{}, 1959, pp. 73-77.
Yor, M. [*Some aspects of Brownian motion, Part I: some special functionals*]{}. Lectures in Mathematics ETH Zürich, 1992, Birkhäuser.
[$^{(1)}$ Department of Statistics, University of Warwick, CV4 7AL, Coventry, UK. [email protected]\
$^{(2)}$ Department of Applied Mathematics, National Chiao Tung University, No. 1001, Ta-Hsueh Road, 300 Hsinchu, Taiwan. [email protected]]{}
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Arjun Bagchi,'
- 'Mirah Gary,'
- 'and Zodinmawia.'
- '\'
title: 'The nuts and bolts of the BMS Bootstrap.'
---
Introduction
============
The modern way of understanding relativistic quantum field theories (QFTs) is through renormalization group flows away from conformal field theories (CFTs). In the parameter space of all QFTs, CFTs arise as fixed points with enhanced scale and conformal symmetry. The very ambitions programme of understanding all QFTs thus is intimately related to the classification of all consistent CFTs. Conformal bootstrap [@Ferrara:1973yt; @Polyakov:1974gs] has emerged as the leading tool in this endeavour.
Any conformal field theory is determined by what has now come to be known as “CFT data", viz. the spectrum of primary operators in the theory, the structure constants that are the constants of the three-point functions of primary operators not fixed by conformal invariance and the central charge of the theory (in case of 2d CFTs). But any random set of data does not constitute a consistent theory. The theory has to obey associativity of the operator algebra. Conformal bootstrap uses conformal symmetry and the consistency of the operator product expansion (OPE) to constrain possible CFTs.
The use of the conformal bootstrap programme was initially limited to two dimensional conformal field theories [@Belavin:1984vu]. Here one has the additional power of infinite dimensional symmetries of the two copies of the underlying Virasoro algebra.
\[Vir\] && \[Ł\_n, Ł\_m\]= (n-m) Ł\_[n+m]{} + \_[n+m, 0]{} (n\^3 - n)\
&& \[[|]{}\_n, [|]{}\_m\]= (n-m) [|]{}\_[n+m]{} + \_[n+m, 0]{} (n\^3 - n)\
&& \[Ł\_n, [|]{}\_m\] =0
The bootstrap equation in 2d CFTs help us solve some CFTs explicitly. The analytical handle that the Virasoro symmetry provides helps put in powerful constraints on the mathematical consistency of theories in 2d. For values of the central charges between 0 and 1, there is a discrete number of CFTs with finite number of primary fields and these are called the minimal models. The bootstrap equations leads to a complete solution of the 2d minimal series.
Following the seminal work of Rattazzi, Rychkov, Tonni and Vichi in 2008 [@Rattazzi:2008pe], who build on earlier work by Dolan and Osborn [@Dolan:2000ut; @Dolan:2003hv], there has been a great flurry of activity in applying conformal bootstrap techniques to spacetime dimensions higher than two. The method of conformal bootstrap has emerged as a very effective tool in calculating things like the critical exponents of Ising model or the $O(N)$ model in 3 dimensions. We refer the reader to the excellent reviews [@Rychkov:2016iqz; @Simmons-Duffin:2016gjk] for a more detailed account of the excitement in this emerging field. See also [@Poland:2016chs] for a very well written overview.
Our present goal is to generalise the ideas and methods of the conformal bootstrap programme to theories with symmetries other than conformal invariance. In this present work, which is a continuation and elaboration of an earlier shorter piece of work [@Bagchi:2016geg], we will concentrate on 2d field theories which are invariant under the following symmetry algebra:
\[gca2d\] && \[L\_n, L\_m\]= (n-m) L\_[n+m]{} + c\_L \_[n+m, 0]{} (n\^3 - n)\
&& \[L\_n, M\_m\]= (n-m) M\_[n+m]{} + c\_M \_[n+m, 0]{} (n\^3 - n)\
&& \[M\_n, M\_m\] =0
This algebra arises as a contraction of two copies of the Virasoro algebra [(\[Vir\])]{} and is called the 2d Galilean Conformal Algebra (GCA) [@Bagchi:2009my; @Bagchi:2009pe]. The algebra also arises as asymptotic symmetries of 3d flat spacetimes and is called the 3d Bondi-Metzner-Sachs (BMS) algebra [@Bondi:1962px; @Sachs:1962zza; @Barnich:2006av]. This isomorphism was first noticed in [@Bagchi:2010zz] and goes under the name of the BMS/GCA correspondence.
We will find that we will be able to construct, in a spirit very similar to that of CFTs, a BMS version of an OPE and then by considering four point functions, we will define the notion of BMS blocks and a BMS crossing equation. This will then lead us to the BMS bootstrap equation. In the limit of large central charge, we will find [*[closed form]{}*]{} expressions for these BMS blocks that form the basis for the solution of the bootstrap equation. We will then go on to recover all our answers as contractions of appropriate quantities in a 2d relativistic CFT. This forms a comprehensive check of our results obtained in the intrinsic method, some of which were first reported in [@Bagchi:2016geg].
To the best of our knowledge, this constitutes the first successful attempt at the construction and concrete steps towards the solution of a bootstrap equation in a theory that is not a relativistic conformal field theory.
Our motivations for addressing field theories with the symmetry algebra [(\[gca2d\])]{} are manifold. This algebra has recently surfaced in various contexts, $viz.$ as symmetries of putative dual field theories to 3d flat space, as conformal symmetries in non-relativistic systems and also as the residual symmetry algebra on the worldsheet of the tensionless closed bosonic string [@Bagchi:2013bga; @Bagchi:2015nca]. Below we address the first two of these applications.
Holography for flat spacetimes
------------------------------
The notion of asymptotic symmetries is a very important concept in the study of gravitational theories, and especially in the context of holographic theories. For a fixed set of boundary conditions, the Asymptotic Symmetry Group (ASG) is the group of allowed diffeomorphisms modded out by the trivial ones (trivial diffeomorphisms are ones that lead to vanishing canonical charges). In a quantum theory of gravity, the states of the theory form representations of the ASG. The ASG also dictates the symmetries of the putative holographically dual field theory.
Infinite dimensional ASGs turn out to be very effective in understanding aspects of the dual field theory. The most studied example of this is the ASG of AdS$_3$, which turns out to be two copies of the infinite dimensional Virasoro algebra. This leads to the conclusion that the dual field theory is a 2d CFT. The canonical analysis by Brown and Henneaux [@Brown:1986nw] can actually be looked upon as the birth of the AdS/CFT correspondence [@Maldacena:1997re].
Interestingly, infinite dimensional ASGs have been known to exist in the context of Minkowski spacetimes long before the discovery of Brown and Henneaux. Bondi, van der Burg, Metzner [@Bondi:1962px] and independently Sachs [@Sachs:1962zza] studied the asymptotic structure of Minkowski spacetime in 4 dimensions at its null boundary and found to their surprise that the symmetries were not dictated by the Poincare group, but an infinite dimensional group which included over and above the Poincare generators, translations of the null direction that depended on the angles of the sphere at infinity. These were called supertranslations and in spite of many efforts to do away with them, it was found that the algebra could not be truncated to just the Poincare algebra. The asymptotic symmetry algebra takes the form
\[bms4\] $$\begin{aligned}
& [ L_n, L_m ] = (n-m) L_{m+n}, \quad [{\bar{L}}_n, {\bar{L}}_m] = (n-m) {\bar{L}}_{n+m} \\
& [L_m, M_{r,s}] = \left( \frac{m+1}{2} - r \right) M_{m+r, s} \ , \quad [{\bar{L}}_m, M_{r,s}] = \left( \frac{m+1}{2} - s\right) M_{r, m+s} \\
& [M_{r,s}, M_{t,u}] = 0\end{aligned}$$
Here $n,m$ range from $-1$ to $+1$ while the other variables can take all integral values. The generators $M_{r,s}$ are the super-translation generators, the translations that depend on the angles of the sphere at infinity.
Later, inspired by possible links to holography, Barnich and Troessaert [@Barnich:2010eb] proposed an extension of the ASG of 4d Minkowski space to include what they called super-rotations. Superrotations are group of all the conformal generators of the sphere at infinity and this extension is essentially the same as the extension of the 2d conformal algebra to include all the generators of the Virasoro algebra from the globally well-defined ones $L_{0, \pm1}$. In the above algebra, this means that $n, m$ now take all integral values. This extended ASG of 4d flatspace is now what is commonly known as the Bondi-Metzner-Sachs (BMS) group. Recently, following Strominger and collaborators [@Strominger:2013jfa], a beautiful story has emerged linking BMS symmetries to soft theorems [@Weinberg:1965nx] and memory effects [@Zeldovich; @Strominger:2014pwa; @Pasterski:2015tva]. We refer the reader to [@Strominger:2017zoo] for a detailed discussion of these aspects.
In the present paper, we are interested in the ASG of 3d Minkowski spacetimes. At null infinity, the ASG is given by the BMS$_3$ algebra [@Barnich:2006av], which, as we have mentioned above, takes the form [(\[gca2d\])]{}. For Einstein gravity, the central terms are $c_L=0, \ c_M = \frac{1}{4G}$. When one considers modifications to Einstein gravity with a gravitational Chern-Simons term, a theory that goes under the name of Topological Massive Gravity, the ASG remains the same but central terms change and $c_L$ and $c_M$ are now both non-zero. Putative duals to theories with 3d gravity with asymptotically flat boundary conditions would thus be given by [(\[gca2d\])]{} with two non-zero central terms. A review of some progress in flat holography in general and in 3d in particular can be found in [@Bagchi:2016bcd; @Riegler:2016hah]. An incomplete list of interesting directions that have been explored in this context are [@Bagchi:2012xr] – [@Bagchi:2015wna].
Our principle goal in this paper is the following. We would like to attempt to constrain 2d field theories with BMS$_3$ symmetry and hence chart out a parameter space for all possible putatively dual theories to asymptotically Minkowskian spacetimes in 3d.
Non-relativistic Conformal Symmetries
-------------------------------------
We live in a world where the everyday things are governed principally by non-relativistic physics. Galilean invariant theories thus are a very good approximation for many real life applications. Thus it is vitally important to understand Galilean field theories. In analogy with relativistic QFTs, it is thus interesting to answer whether all Galilean QFTs can be understood as renormalization group flows away from fixed points governed by the analogue of conformal symmetry. Galilean Conformal Field Theories (GCFT), i.e. field theories with GCA as their symmetry algebra, arise as contractions from relativistic CFTs [@Bagchi:2009my]. It is thus very natural to expect that these non-relativistic fixed points in the parameter space of all Galilean QFTs will be governed by the GCA.
Through the intriguing link of the BMS/GCA correspondence [@Bagchi:2010zz], our programme thus is very useful when we consider applications to non-relativistic QFTs in 2d. This analysis, carried out to its conclusion, would thus help classify all 2d GCFTs and hence lead to an understanding of all 2d Galilean QFTs.
It is interesting here to comment on possible extensions to higher dimensions. It has been claimed in [@Bagchi:2009my] that the GCA is infinite dimensional in all spacetime dimensions. This follows from the observation that the finite contracted algebra can be written in a suggestive form and given an infinite lift in any dimensions. The rather astounding claim is that the non-relativistic limit of a CFT leads to a theory which has an infinite dimensional symmetry. The infinite-dimensional GCA in any arbitrary spacetime dimensions is given by
\[GCA\] && \[L\_n, L\_m\] = (n-m) L\_[n+m]{}, =0,\
&& \[L\_n, M\_m\^i\] = (n-m) M\_[n+m]{}\^i.
Interestingly, it has been shown that field theories like Maxwell’s theory and Yang-Mills theory, which are classically conformally invariant in $D=4$, have non-relativistic versions that exhibit this infinite dimensional symmetry in the Galilean regime [@Bagchi:2014ysa; @Bagchi:2015qcw] [[^1]]{}. Some recent investigations reveal that this classical symmetry enhancement is rather generic and happens in many cases where there are relativistic conformal symmetries to begin with. If there are field theories which also exhibit this infinite dimensional symmetry quantum mechanically, then these systems would be extremely interesting. They could be looked upon as closed sub-sectors in relativistic CFTs that perhaps have the promise to becoming integrable.
In the context of the bootstrap in these higher dimensional theories, it is very possible that our methods here would generalise in a rather simple way to any dimensions. The additional power of infinite symmetries would help in the restriction of the higher dimensional theories.
Outline of the paper
--------------------
The rest of the paper is organised as follows.
In Sec. 2, we give a short summary of the conformal bootstrap programme, specifically focussing on 2d CFTs. This forms a basis for the analysis we will perform for the 2d field theories with BMS symmetry.
In Sec. 3, we look at the 2d field theories with BMS$_3$ symmetries in an intrinsic way. This means that we formulate the analogues of the conformal bootstrap analysis by relying solely on the symmetry structure of the field theory. Some of the results in this section have been reported earlier in [@Bagchi:2016geg]. Here we provide a detailed analysis of those results as well as some more new results which were promised but not presented in [@Bagchi:2016geg].
In Sec. 4, we first discuss the two different limits, viz. the non-relativistic and the ultra-relativistic, of the two copies of the Virasoro algebra to BMS$_3$. We then concentrate on the non-relativistic limit and recover many of the results of Sec. 3 in terms of this limit of the relativistic CFT answers. This serves as a comprehensive check of our results and also stresses the importance of the existence of this limit.
In Sec. 5, we look at a specific subsector of the BMS$_3$ algebra, where the symmetry algebra has previously been shown to reduce to the Virasoro sub-algebra [@Bagchi:2009pe]. We observe that with the specific restrictions on the operator weights and central charges, the bootstrap analysis is consistent with this earlier claim.
We conclude in Sec. 6 with a summary of the paper, some discussions and a list of future directions.
The Conformal bootstrap
=======================
In this section, we revisit some aspects of the conformal bootstrap, which we will specifically need for our analysis in the BMS bootstrap. We will confine ourselves to 2d CFTs, which are governed by two copies of the Virasoro algebra [(\[Vir\])]{}. More details can be found in the original BPZ paper [@Belavin:1984vu] or in some standard CFT text books [@DiFrancesco:1997nk; @Blumenhagen:2009zz].
We will be work exclusively on the plane and hence the form of the generators of the 2d Virasoro algebra will be given by Ł\_n = z\^[n-1]{} \_z, [|]{}\_n = |[z]{}\^[n-1]{} \_[|[z]{}]{} We define a unique vacuum state in the theory $|0{\rangle}$. One defines a state-operator correspondence in the 2d CFT: (0, 0) |0 = | The states in a CFT are labelled by their weights under ${\mathcal{L}}_0$ and ${\bar{\mathcal{L}}}_0$: Ł\_0 |h, [[|h]{}]{} = h |h, [[|h]{}]{}, [|]{}\_0 |h, [[|h]{}]{} = [[|h]{}]{}|h, [[|h]{}]{} One defines a notion of primary fields as the ones which are annihilated by all positively labelled generators: \[hwV\] Ł\_n |h, [[|h]{}]{}\_p = [|]{}\_n |h, [[|h]{}]{}\_p =0 The representations of the Virasoro algebra, called Verma modules, are built by acting on primary fields by raising operators ${\mathcal{L}}_{-n}, {\bar{\mathcal{L}}}_{-n}$. A general state in a CFT is given by: \_[p]{}\^[{,}]{}(z, [[|z]{}]{}) = (Ł\_[-1]{})\^[k\_1]{}...(Ł\_[-l]{})\^[k\_l]{}([|]{}\_[-1]{})\^[[|k]{}\_1]{}...([|]{}\_[-j]{})\^[[|k]{}\_j]{}\_p (z,[[|z]{}]{})(Ł\_[|]{}\_\_p)(z,[[|z]{}]{}).
Operator product expansion
--------------------------
The two and three point functions of primary states are fixed up to constants by invariance under the global part of the algebra ${\mathcal{L}}_{0, \pm1}, {\bar{\mathcal{L}}}_{0, \pm1}$. The two-point function is given by: \_1(z\_1,|[z]{}\_1)\_2(z\_2,|[z]{}\_2) = , The three point function of primary fields is given by: \_1(z\_1,|[z]{}\_1)\_2(z\_2,|[z]{}\_2)\_3(z\_3,|[z]{}\_3) = where $h_{ijk}=-(h_i+h_j-h_k)$. The operator product expansion (OPE) of two primary operators is given by \_[1]{}(z,[[|z]{}]{})\_[2]{}(0,0)=\_[p,{,}]{}\_[12]{}\^[p{,}]{}z\^[h\_[p]{}-h\_1-h\_2+K]{}[[|z]{}]{}\^[[[|h]{}]{}\_[p]{}-[[|h]{}]{}\_1-[[|h]{}]{}\_2+|[K]{}]{}\_[p]{}\^[{,}]{}(0,0), \[opeCFT\] where $K=\sum_{i}ik_{i},$ $\bar{K}=\sum_{j}j\bar{k}_{j}$. Using the OPE to find the three-point function it can be seen that $\frak{C}_{12}^{p\{0,0\}}\equiv \frak{C}_{12}^{p}=\frak{C}_{p12}$. The coefficient $\frak{C}_{12}^{p\{\vec{k},\vec{\bar{k}}\}}$ decouple as \_[12]{}\^[p{,}]{}=\_[p12]{}\_[12]{}\^[p{}]{}|\_[12]{}\^[p{}]{}. The coefficients $\mathcal{B}$ can be obtained by demanding that both sides of the OPE transform the same way under the action of ${\mathcal{L}}_m$ and $\bar{{\mathcal{L}}}_n$. These coefficients for level one and level two are shown in Table .
$\mathcal{B}_{12}^{p\{1\}}=\frac{1}{2}$ $\bar{\mathcal{B}}_{12}^{p\{\bar{1}\}}=\frac{1}{2}$
--------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$\mathcal{B}_{12}^{p\{1,1\}}=\frac{c-12h-4h_p+ch_p+8h_p^2}{4(c-10h_p+2ch_p+16h_p^2)}$ $\bar{\mathcal{B}}_{12}^{p\{\bar{1},\bar{1}\}}=\frac{\bar{c}-12{{\bar h}}-4{{\bar h}}_p+\bar{c}{{\bar h}}_p+8{{\bar h}}_p^2}{4(\bar{c}-10{{\bar h}}_p+2c{{\bar h}}_p+16{{\bar h}}_p^2)}$
$\mathcal{B}_{12}^{p\{2\}}=\frac{2h-h_p+4hh_p+h_p^2}{c-10h_p+2ch_p+16h_p^2}$ $\bar{\mathcal{B}}_{12}^{p\{\bar{2}\}}=\frac{2{{\bar h}}-{{\bar h}}_p+4{{\bar h}}{{\bar h}}_p+{{\bar h}}_p^2}{\bar{c}-10{{\bar h}}_p+2\bar{c}{{\bar h}}_p+16{{\bar h}}_p^2}$
: Coefficients of OPE at level 1 and level 2.[]{data-label="OPE_CFT"}
Conformal blocks and crossing symmetry
--------------------------------------
Invariance under global conformal symmetry is not enough to fix the four-point functions of primary fields. Global invariance can help fix the form of these correlators up to a function of the conformally invariant cross ratios given below. The four-point function has the form \_[i=1]{}\^[4]{}\_[i]{}(z\_[i]{},[[|z]{}]{}\_[i]{})&=&\_[1i<j4]{}z\_[ij]{}\^[\_[k=1]{}\^[4]{}h\_[ijk]{}/3]{}|[z]{}\_[ij]{}\^[\_[k=1]{}\^[4]{}[[|h]{}]{}\_[ijk]{}/3]{}\_[CFT]{}(z,|[z]{}), where $\mathcal{F}_{CFT}(z,\bar{z})$ is an arbitrary coefficients of the cross-ratios $z$ and $\bar{z}$ z=,[[|z]{}]{}=. We can always do a global conformal transformation such that {(z\_i,[[|z]{}]{}\_i)}{(,),(1,1),(z,[[|z]{}]{}),(0,0)}. So we define \_[z\_1,[[|z]{}]{}\_1]{}z\_1\^[2h\_1]{}[[|z]{}]{}\_1\^[2|[h]{}\_1]{}\_1(z\_1,[[|z]{}]{}\_1)\_2(1,1)\_3(z,[[|z]{}]{})\_4(0,0)=\_[34]{}\^[21]{}(z,[[|z]{}]{}), where \_[34]{}\^[21]{}(z,[[|z]{}]{})=h\_1,[[|h]{}]{}\_1|\_2(1,1)\_3(z,[[|z]{}]{})|h\_4,[[|h]{}]{}\_4. Using the OPE on $\phi_3$ and $\phi_4$ inside the correlator, the function $ \mathcal{G}_{34}^{21}(z,{{\bar z}})$ can be written in terms of three-point functions of primaries and their descendants. Specifically, we have \_[34]{}\^[21]{}(z,[[|z]{}]{})=\_p \_[34]{}\^p \_[12]{}\^p \_[34]{}\^[12]{}(p|z,[[|z]{}]{}). The blocks $\mathcal{A}_{34}^{21}(p|z,{{\bar z}})$ factorizes into a holomorphic and an anti-holomorphic parts \_[34]{}\^[21]{}(p|z,[[|z]{}]{})=\_[34]{}\^[12]{}(p|z)|\_[34]{}\^[21]{}(p|[[|z]{}]{}), where \_[34]{}\^[21]{}(p|z,[[|z]{}]{})=z\^[h\_p-h\_3-h\_4]{}\_[{k}]{}\_[34]{}\^[p{k}]{}z\^K.
Inside the correlator we can move the operators around which does not matter except for fermions which would introduce a sign. So apart from $ G_{34}^{21}(z,{{\bar z}})$ we may also define \_[z\_1,[[|z]{}]{}\_1]{}z\_1\^[2h\_1]{}[[|z]{}]{}\_1\^[2|[h]{}\_1]{}\_1(z\_1,[[|z]{}]{}\_1)\_4(1,1)\_3(z,[[|z]{}]{})\_2(0,0) = \_[32]{}\^[41]{}(z,[[|z]{}]{})=h\_1,[[|h]{}]{}\_1|\_4(1,1)\_3(z,[[|z]{}]{})|h\_2,[[|h]{}]{}\_2. It can be seen from their definition that \[confcross\] \_[34]{}\^[21]{}(z,[[|z]{}]{})=\_[32]{}\^[41]{}(1-z,1-[[|z]{}]{}). If we expand both sides in term of the conformal blocks we have the bootstrap equation \_p \_[34]{}\^p \_[21]{}\^p \_[34]{}\^[21]{}(p|z,[[|z]{}]{})=\_q \_[41]{}\^q \_[32]{}\^q \_[32]{}\^[41]{}(p|1-z,1-[[|z]{}]{}).
Global Conformal blocks
-----------------------
The large central charge limit of the Virasoro algebra simplifies a lot of the analysis in 2d CFTs and has been recently pursued actively starting from [@Hartman:2013mia]. There are particular simplifications for the conformal blocks. The global conformal block is the large central charge limit of the Virasoro block [@Fitzpatrick:2014vua; @Fitzpatrick:2015zha]. This is given by \_[34]{}\^[21]{}(p|z,[[|z]{}]{})&=&z\^[h\_p-h\_3-h\_4]{}\_[k]{}\_[34]{}\^[p{k}]{}z\^k&&[[|z]{}]{}\^[[[|h]{}]{}\_p-[[|h]{}]{}\_3-[[|h]{}]{}\_4]{}\_[|[k]{}]{}|\_[34]{}\^[p{|[k]{}}]{}[[|z]{}]{}\^[|[k]{}]{}. The closed form expression of this can be obtained by using the constraint that both sides of the OPE transform the same way under the quadratic Casimirs [@Dolan:2000ut; @Dolan:2003hv] =Ł\_0\^2-(Ł\_1Ł\_[-1]{}+Ł\_[-1]{}Ł\_1),|=|[Ł]{}\_0\^2-(|[Ł]{}\_1|[Ł]{}\_[-1]{}+|[Ł]{}\_[-1]{}|[Ł]{}\_1), of the global subgroup generated by $\{{\mathcal{L}}_{0,\pm 1},\bar{{\mathcal{L}}}_{0,\pm 1}\}$. For simplicity, take all the external operators to be identical and to be a scalar with dimension $\Delta_\phi$. We may write the global block as \_[\_]{}(p|z,[[|z]{}]{})=z\^[-\_]{} |[z]{}\^[-\_]{}\_[h\_p,|[h\_p]{}]{}(z,|[z]{}). The constraint that both sides of the OPE transform the same way under the two quadratic Casimirs gives two differential equations for $\mathcal{K}_{h_p,\bar{h_p}}(z,\bar{z})$ &&\_[h\_p,|[h]{}\_p]{}(z,|[z]{})=h\_p(h\_p+1)\_[h\_p,|[h]{}\_p]{}(z,|[z]{}), &&\_[h\_p,|[h]{}\_p]{}(z,|[z]{})=|[h]{}\_p(|[h]{}\_p+1)\_[h\_p,|[h]{}\_p]{}(z,|[z]{}). \[diff\_eqn\_cft\] Assuming $\mathcal{K}$ holomorphically factorizes $${\ensuremath{\mathcal{K}}}_{h_p,{\ensuremath{\bar{h}}}_p}(z,{\ensuremath{\bar{z}}}) = {\ensuremath{\mathcal{K}}}_{h_p}(z)\bar{{\ensuremath{\mathcal{K}}}}_{{\ensuremath{\bar{h}}}_p}({\ensuremath{\bar{z}}}),$$ the solution are given in terms of gauss hypergeometric function $${\ensuremath{\mathcal{K}}}_{h_p}(z) = \alpha z^{-h_p} {\ensuremath{\ _2F_1}}\left(-h_p,-h_p;-2h_p; z\right) + \gamma z^{h_p+1} {\ensuremath{\ _2F_1}}\left(h_p+1,h_p+1;2h_p+2;z\right)$$ and similarly for the anti-holomorphic sector. We expand around $z=0$ and match to the boundary conditions $\beta_{\phi\phi}^{p,0}=1$, $\beta_{\phi\phi}^{p,1}=\frac{1}{2}$, $${\ensuremath{\mathcal{K}}}_{h_p}(z) = \alpha z^{-h_p}\left(1 - \frac{h_p}{2}z + {\ensuremath{\mathcal{O}}}(z^2)\right) + \gamma z^{h_p+1}\left(1+\frac{h_p+1}{2}z + {\ensuremath{\mathcal{O}}}(z^2)\right).$$ The final result for the global block is [@Dolan:2000ut; @Dolan:2003hv] $${\ensuremath{\mathfrak{g}}}_{\Delta_\phi}(p|z,{\ensuremath{\bar{z}}}) = z^{h_p-2h_\phi}{\ensuremath{\bar{z}}}^{{\ensuremath{\bar{h}}}_p-2{\ensuremath{\bar{h}}}_\phi}{\ensuremath{\ _2F_1}}\left(h_p,h_p;2h_p;z\right){\ensuremath{\ _2F_1}}\left({\ensuremath{\bar{h}}}_p,{\ensuremath{\bar{h}}}_p;2{\ensuremath{\bar{h}}}_p;{\ensuremath{\bar{z}}}\right).$$ More generally, $${\ensuremath{\mathfrak{g}}}^{21}_{34}(p|z,{\ensuremath{\bar{z}}}) = z^{h_p-h_1-h_2}{\ensuremath{\bar{z}}}^{{\ensuremath{\bar{h}}}_p-{\ensuremath{\bar{h}}}_1-{\ensuremath{\bar{h}}}_2}{\ensuremath{\ _2F_1}}(h_p+h_{12},h_p+h_{34};2h_p; z){\ensuremath{\ _2F_1}}({\ensuremath{\bar{h}}}_p+{\ensuremath{\bar{h}}}_{12},{\ensuremath{\bar{h}}}_p+{\ensuremath{\bar{h}}}_{34};2{\ensuremath{\bar{h}}}_p;{\ensuremath{\bar{z}}}),
\label{CFF_global_blocks}$$ where $h_{ij}=h_i-h_j,\,{{\bar h}}_{ij}={{\bar h}}_i-{{\bar h}}_j$.
Bootstrapping BMS symmetries: intrinsic analysis
================================================
In this section, we will construct the bootstrap programme for field theories with BMS symmetries through an intrinsic method. This just means that we will be inspired by the methods of 2d CFTs that we outlined in the previous section, but there will be many crucial differences, as the symmetry algebra [(\[gca2d\])]{} is fundamentally different from two copies of the Virasoro algebra [(\[Vir\])]{}. In the subsequent section we will provide a limiting analysis where we consider the contraction of [(\[Vir\])]{} to [(\[gca2d\])]{} and we will recover some of the answers of this section through the limit. Some of the central results of this section have already appeared in [@Bagchi:2016geg]. In this paper, and particularly in this section, we provide a much more detailed exposition of the basic analysis presented earlier. There are a number of new mathematical details and results that are presented here.
Highest weight representations
------------------------------
We consider 2d field theories that are invariant under the BMS$_3$ algebra. We will call the directions of the field theory $(u, v)$. We will be interested in representation of the algebra [(\[gca2d\])]{} given by \[planevec\] L\_n = - u\^[n+1]{} \_u - (n+1) u\^n v \_v, M\_n = u\^[n+1]{} \_v This will be called the “plane” representation of the BMS$_3$ algebra.
The states of the BMS invariant 2d field theory are by their weights under $L_0$. Since $M_0$ and $L_0$ commute, the states get an additional label under $M_0$ as well. L\_0 |, = |, , M\_0 |, = |, Like in usual 2d CFTs, we will build the representation theory by first defining BMS primary operators. We do this by demanding that the spectrum (defined with respect to ${\Delta}$) be bounded from below. Then the BMS primary operators $|{\Delta}, \xi {\rangle}_p$ are the ones for which \[hwB\] L\_n |, \_p = M\_n |, \_p= 0 n >0. We will assume a state-operator correspondence in the case of BMS field theories as well. While this is not strictly necessary for our analysis, it would be good to have the freedom to talk about operators and states interchangeably. The BMS modules, very much like the Verma modules in the case of the Virasoro algebra, are built by acting creation operators on the BMS primary states.
Operator product expansion
--------------------------
The main objects of physical interest in field theory are the correlation functions. If we know all the correlation functions, we may say that we have completely solved the theory. In finding the form of these functions, symmetries play an important role. It is interesting to know which part of the correlation functions is fixed by symmeties alone and what other parts depend on the dynamics of the theory. In particular, for BMS-invariant theories, the co-ordinate dependence of the two-point and three-point functions are completely fixed simply by invariance under the global subgroup of the BMS group i.e., co-ordinate transformation generated by $L_{0,\pm1},M_{0,\pm1}$. The two-point function is given by [@Bagchi:2009ca; @Bagchi:2009pe] \_1(u\_1,v\_1)\_2(u\_2,v\_2) = \_[12]{} u\_[12]{}\^[-]{} e\^\_[\_1\_2]{}\_[\_1\_2]{} The normalisation of the 2-point function has been fixed to $\delta_{12}$. The three-point function is given by \_1(u\_1,v\_1)\_2(u\_2,v\_2)\_3(u\_3,v\_3) = C\_[123]{} u\_[12]{}\^[\_[123]{}]{} u\_[23]{}\^[\_[231]{}]{} u\_[13]{}\^[\_[312]{}]{}e\^[-\_[123]{}]{}e\^[-\_[231]{}]{} e\^[-\_[312]{}]{}. \[eqn:3-pt\] Here ${\Delta}_{ijk} = - ({\Delta}_i + {\Delta}_j - {\Delta}_k)$ and $\xi_{ijk}$ is defined similarly. $C_{123}$ is an arbitrary parameter called the structure constant. It is not fixed by symmetry but depends on the dynamics (or the details) of the field theory under consideration. So, if these constants are given to us, we can completely determine the three-point function by symmetry consideration alone.
We can also consider higher correlation functions and see how much of their form are fixed by symmetry alone and what other dynamical inputs are needed to fixed the rest. Now, all information about the correlation functions are contained in the operator product algebra, which gives the operator product expansion (OPE) of two primary fields as summation over the primaries and towers of their descendants. So, in order to know how the correlation functions are constrained by symmetries, it is enough to study constraints on the OPE. Indeed, considering these symmetries, we make the following ansatz for the OPE of two primary fields with weights $({\Delta}_1,\xi_1)$ and $({\Delta}_2,\xi_2)$ &&\_[1]{}(u\_1,v\_1)\_[2]{}(u\_2,v\_2)=&& \_[p,{,}]{} u\_[12]{}\^[-\_[1]{}-\_[2]{}+\_[p]{}]{}e\^[(\_[1]{}+\_[2]{}-\_[p]{})]{}(\_[=0]{}\^[K+Q]{} C\_[12]{}\^[p{,},]{}u\_[12]{}\^[K+Q-]{}v\_[12]{}\^)\_[p]{}\^[{,}]{}(u\_2,v\_2).&& \[ope1\] Our notation is that for vectors $\vec{k}=(k_1,k_2,...k_r)$ and $\vec{q}=(q_1,q_2,...q_s)$, descendant fields $\phi_{p}^{\{\vec{k},\vec{q}\}}(u_2,v_2)$ are given by \_[p]{}\^[{,}]{}(u,v)&=&((L\_[-1]{})\^[k\_1]{}...(L\_[-l]{})\^[k\_l]{}(M\_[-1]{})\^[q\_1]{}...(M\_[-j]{})\^[q\_j]{}\_p)(u,v)&&(L\_M\_\_p)(u,v), where $K=\sum_{l}lk_{l},\,Q=\sum_{j}jq_{j}$. So, $\phi_{p}^{\{\vec{k},\vec{q}\}}(u,v)$ is a descendant field at level $K+Q$. For ease of calculation we can take the point $(u_2,v_2)$ in to be the origin, giving us \_[1]{}(u,v)\_[2]{}(0,0)&=&\_[p,{,}]{} u\^[-\_[1]{}-\_[2]{}+\_[p]{}]{}e\^[(\_[1]{}+\_[2]{}-\_[p]{})]{}(\_[=0]{}\^[K+Q]{} C\_[12]{}\^[p{,},]{}u\^[K+Q-]{}v\^)\_[p]{}\^[{,}]{}(0,0)&& [LHS]{} \[ope\] Here the form of the factor $u^{-{\Delta}_{1}-{\Delta}_{2}+{\Delta}_{p}}\,e^{(\xi_{1}+\xi_{2}-\xi_{p})\frac{v}{u}}$ is fixed by the requirement that the OPE gives the correct two-point function and the factor $\sum_{{\alpha}=0}^{K+Q} C_{12}^{p\{\vec{k},\vec{q}\},{\alpha}}u^{K+Q-{\alpha}}v^{{\alpha}}$ is to ensure that both sides of the OPE transform the same way under the action of $L_{0}$. To verify this second requirement, let us act both sides of on the the vacuum $|0,0{\rangle}$ and then see the action of $L_{0}$ on the resulting state. On the LHS we have, L\_[0]{}\_[1]{}(u,v)\_[2]{}(0,0)|0,0 & = & (\[L\_[0]{},\_[1]{}(u,v)\]+\_[1]{}(u,v)L\_[0]{})\_[2]{}(0,0)|0,0 & = & (u\_[u]{}+v\_[v]{}+\_[1]{}+\_[2]{})\_[1]{}(u,v)\_[2]{}(0,0)|0,0. So, if the OPE is correct, the RHS of equation must also transform as above L\_[0]{}([[RHS]{}]{})=(u\_[u]{}+v\_[v]{}+\_[1]{}+\_[2]{})[[RHS]{}]{} \[L0RHS\] If we use the commutator $L_{0}L_{\vec{k}}M_{\vec{q}}=L_{\vec{k}}M_{\vec{q}}L_{0}+(K+Q)L_{\vec{k}}M_{\vec{q}}$ on the LHS of the above equation we have &&L\_[0]{}([RHS]{})=&&\_[p,{,}]{} u\^[-\_[1]{}-\_[2]{}+\_[p]{}]{}e\^[(\_[1]{}+\_[2]{}-\_[p]{})]{}(\_[=0]{}\^[K+Q]{}C\_[12]{}\^[p{,},]{}u\^[K+Q-]{}v\^)(\_[p]{}+K+Q)L\_M\_|\_[p]{},\_[p]{}. && It can also be easily checked that &&(u\_[u]{}+v\_[v]{}+\_[1]{}+\_[2]{})[[RHS]{}]{}=&&\_[p,{,}]{} u\^[-\_[1]{}-\_[2]{}+\_[p]{}]{}e\^[(\_[1]{}+\_[2]{}-\_[p]{})]{}(\_[=0]{}\^[K+Q]{}C\_[12]{}\^[p{,},]{}u\^[K+Q-]{}v\^)(\_[p]{}+K+Q)L\_M\_|\_[p]{},\_[p]{}. && Thus, equation is satisfied, which means that both sides of OPE transform the same way under the action of $L_{0}$. Furthermore, using the OPE inside the three-point functions and comparing the coefficients with it can be seen that C\_[12]{}\^[p{0,0},0]{}C\_[12]{}\^[p]{}=C\_[p12]{}. Therefore, we will rewrite $C_{12}^{p\{\vec{k},\vec{q}\},{\alpha}}$ as C\_[12]{}\^[p{,},]{}=C\_[12]{}\^[p]{}\_[12]{}\^[p{,},]{}, where, by convention, \_[12]{}\^[p{0,0},0]{}=1.
The coefficients ${\beta}_{12}^{p\{\vec{k},\vec{q}\},{\alpha}}$ can be calculated by demanding that both sides of transform the same way under the other generators $L_m$ and $M_n$. Thus, the form of the OPE is completely constrained by symmetries to depend only on external inputs, such as the structure constants, the spectrum of primary operators appearing in the OPE, and the central charge. In other words, if these dynamical inputs are given to us, we can use symmetries to calculate all the correlation functions in a BMS-invariant field theory. These dynamical inputs can be used to classify and completely specify a given BMS-invariant field theory. However, any random sets of these dynamical inputs need not constitute a consistent field theory; they must satisfy a constrain equation given by the BMS bootstrap equation, which arises as a condition for the associativity of the operator product algebra.
Recursion relations
-------------------
Now let us try to find recursion relations for evaluating the coefficients ${\beta}_{12}^{p\{\vec{k},\vec{q}\},{\alpha}}$. For the sake of simplicity we will consider the case ${\Delta}_{1}={\Delta}_{2}={\Delta},\,\xi_{1}=\xi_{2}=\xi$. Applying both sides of equation to the vacuum we have \_[1]{}(u,v)|,=\_[p]{}u\^[-2+\_[p]{}]{}e\^[(2-\_[p]{})]{}\_[N]{}C\_[12]{}\^[p]{}u\^[N-]{}v\^|N,\_p, where the state |N,\_p&=&\_[ ]{}\_[12]{}\^[p{,},]{}L\_M\_|\_p,\_p, is a descendant state at level $N$ in the BMS module, L\_[0]{}|N,\_p=(\_[p]{}+N)|N,\_p. We now act with the generators $L_{n>0}$ on both sides of sides of equation and demand that they should transform in the same way. On the LHS, we have &&L\_[n]{}\_[1]{}(u,v)|, = \[L\_[n]{},\_[1]{}(u,v)\]|,& &= \[u\^[n+1]{}\_[u]{}+(n+1)u\^[n]{}v\_[v]{}+(n+1)(u\^[n]{}-nu\^[n-1]{}v)\]\_[1]{}(u,v)|,. Substituting the RHS of in the above equation, we have & & \_[p]{}C\_[p]{}\^[12]{}u\^[-2+\_[p]{}]{}e\^[(2-\_[p]{})]{}L\_[n]{}\_[N,]{}u\^[N-]{}v\^|N,i\_p & = & \_[p]{}C\_[p]{}\^[12]{}u\^[-2+\_[p]{}]{}e\^[(2-\_[p]{})]{} & & \_[N,]{}u\^[N-+n]{}v\^(N+n -+n +\_p)|N,\_p & & +u\^[N--1]{}v\^[+1]{}(n -n\^2 -n \_p)|N,\_p. If we equate the coefficients of $u^{-2{\Delta}+{\Delta}_{p}}\,e^{(2\xi-\xi_{p})\frac{v}{u}}u^{K+n-{\alpha}}v^{{\alpha}}$ on both sides, we get the recursion relation
L\_[n]{}|N+n,\_p & = & (N+n -+n +\_p)|N,\_p & & +(n -n\^2 -n \_p)|N,-1\_p. \[recurL\] Similarly, demanding that both sides of the OPE transform the same way under $M_0$ and $M_{n>0}$ we get two more recursion relation M\_[0]{}|N,\_p = \_[p]{} |N,,\_p-(+1)|N,+1\_p, \[recurM0\] M\_[n]{}|N+n,\_p = ((n-1) +\_p)|N,\_p-(+1)|N,+1\_p. \[recurM\] These three recursion relations can be used to find all the coefficients ${\beta}_{12}^{p\{\vec{k},\vec{q}\},{\alpha}}$. We have shown this calculation for level 1 and level 2 in the next section.
Finding the coefficients
------------------------
At level zero we have |N=0,=0\_p=\_[12]{}\^[p{0,0},0]{}|\_[p]{},\_[p]{}=|\_[p]{},\_[p]{}.
### Level 1 {#level-1 .unnumbered}
The states in level 1 are given by |1,\_p=\_[12]{}\^[p{1,0},]{}L\_[-1]{}|\_[p]{},\_[p]{}+\_[12]{}\^[p{0,1},]{}M\_[-1]{}|\_[p]{},\_[p]{},=0,1. First let us note that M\_0|1,\_p &=& \_p \_[12]{}\^[p{1,0},]{} L\_[-1]{}|\_[p]{},\_[p]{} + (\_[12]{}\^[p{1,0},]{} + \_p \_[12]{}\^[p{0,1},]{})M\_[-1]{}|\_[p]{},\_[p]{},\
M\_1|1,\_p &=& 2\_p \_[12]{}\^[p{1,0},]{}|\_[p]{},\_[p]{},\
L\_1|1,\_p &=& 2(\_p \_[12]{}\^[p{1,0},]{}+ \_p \_[12]{}\^[p{0,1},]{})|\_[p]{},\_[p]{}. &&M\_[0]{}L\_[-1]{}|\_[p]{},\_[p]{}=\_p L\_[-1]{}|\_[p]{},\_[p]{} + M\_[-1]{}|\_[p]{},\_[p]{},&&M\_[0]{}M\_[-1]{}|\_[p]{},\_[p]{}=\_pM\_[-1]{}|\_[p]{},\_[p]{},M\_[1]{}L\_[-1]{}|\_[p]{},\_[p]{}=2\_p|\_[p]{},\_[p]{},&& M\_[1]{}M\_[-1]{}|\_[p]{},\_[p]{}=0,L\_[1]{}L\_[-1]{}|\_[p]{},\_[p]{}=2\_p|\_[p]{},\_[p]{},&& L\_[1]{}M\_[-1]{}|\_[p]{},\_[p]{}=2\_p|\_[p]{},\_[p]{}.
${\beta}_{12}^{p\{1,0\},0}=\frac{1}{2}$ ${\beta}_{12}^{p\{0,1\},0}=0$
----------------------------------------- ------------------------------------------
${\beta}_{12}^{p\{1,0\},1}=0$ ${\beta}_{12}^{p\{0,1\},1}=-\frac{1}{2}$
: Coefficients of OPE at level 1.[]{data-label="level1"}
Then using the recursion relation , we have M\_[0]{}|1,1\_p = \_[p]{}|1,1\_p \_[12]{}\^[p{1,0},1]{}M\_[-1]{}|\_[p]{},\_[p]{} = 0, M\_[0]{}|1,0\_p = \_[p]{}|1,0\_p-|1,1\_p (\_[12]{}\^[p{1,0},0]{}+\_[12]{}\^[p{0,1},1]{}) M\_[-1]{}|\_[p]{},\_[p]{} = 0, giving us \_[12]{}\^[p{1,0},1]{}=0,\_[12]{}\^[p{1,0},0]{}=-\_[12]{}\^[p{0,1},1]{}. Now, using the recursion relation with $N=0,n=1,{\alpha}=0$, we have M\_[1]{}|1,0\_p = \_[p]{}|\_[p]{},\_[p]{}\_p(2\_[12]{}\^[p{1,0},0]{} -1)|\_[p]{},\_[p]{}=0, giving us the coefficients \_[12]{}\^[p{1,0},0]{}=,\_[12]{}\^[p{0,1},1]{}=-. With $N=0,n=1,{\alpha}=0$, gives the recursion relation L\_[1]{}|1,0\_p = \_[p]{}|\_[p]{},\_[p]{}2\_[12]{}\^[p{0,1},0]{}\_[p]{}|\_[p]{},\_[p]{} = 0, giving us \_[12]{}\^[p{0,1},0]{}=0. The various coefficients are collected above in Table [(\[level1\])]{}. We can see that these match with the coefficients in (A.7) of [@Bagchi:2009pe].
### Level 2 {#level-2 .unnumbered}
The details of the relevant calculations at level 2 are presented in Appendix A. We collect all these coefficients in Table .
[|c|c|c|]{} ${\beta}_{12}^{p\{2,0\},0}=\frac{1}{8}$ & ${\beta}_{12}^{p\{(0,1),0\},0}=\frac{4\xi+\xi_{p}}{8(3c_M+2\xi_{p})}$ & ${\beta}_{12}^{p\{1,1\},0}=-\frac{12 \xi -6 c_M-\xi _p}{16 \xi _p(3 c_M+2 \xi _p)}$\
\
\
${\beta}_{12}^{p\{2,0\},1}=0$ & ${\beta}_{12}^{p\{(0,1),0\},1}=0$ & ${\beta}_{12}^{p\{1,1\},1}=-\frac{1}{4}$\
& ${\beta}_{12}^{p\{0,(0,1)\},1}=-\frac{4\xi+\xi_{p}}{4(3c_M+2\xi_{p})}$\
${\beta}_{12}^{p\{2,0\},2}=0$ & ${\beta}_{12}^{p\{(0,1),0\},2}=0$ & ${\beta}_{12}^{p\{1,1\},2}=0$\
& ${\beta}_{12}^{p\{0,(0,1)\},2}=\frac{1}{8}$\
BMS blocks, crossing symmetry and bootstrap
-------------------------------------------
We have seen that BMS-invariant theories are completely specified by the structure constants, the spectrum of primary fields, and the central charge. However any given sets of these inputs need not always constitute a consistent theory; they have to satisfy an infinite set of equations analogous to the conformal case which we will call the BMS bootstrap equation. This equation comes from self consistency of the OPE, namely that it has to be associative when applying inside the correlator. More precisely, if we use the OPE inside the correlator, the resulting correlator should not depend on which two neighbouring primary operators we applied the OPE. We will study this requirement by considering the four-point function, which for a BMS-invariant theory has the structure \_[i=1]{}\^[4]{}\_[i]{}(u\_[i]{},v\_[i]{})&=&\_[1i<j4]{}u\_[ij]{}\^[\_[k=1]{}\^[4]{}\_[ijk]{}/3]{}e\^[-\_[k=1]{}\^[4]{}\_[ijk]{}/3]{} F\_[BMS]{}(u,v)&& P({\_i,\_i,u\_[ij]{},v\_[ij]{}})F\_[BMS]{}(u,v) \[eqn:4pt\] where the BMS analogues of the cross ratio $u$ and $v$ given by u=,=+-- are invariant under the global coordinate transformation generated by $L_{0,\pm1},M_{0,\pm1}$. We can conveniently do a global coordinate transformation such that {(u\_i,v\_i) } {(,0), (1,0), (u,v), (0,0)}, \[eqn:gt\] where $i=1,...,4$. Correspondingly, we define \_[u\_[1]{},v\_[1]{}0]{}u\_1\^[2\_[1]{}]{}(-)\_[1]{}(u\_[1]{},v\_[1]{})\_[2]{}(1,0)\_[3]{}(u,v)\_[4]{}(0,0)G\_[34]{}\^[21]{}(u,v), which in terms of the in and out states is given by G\_[34]{}\^[21]{}(u,v)=\_[1]{},\_[1]{}|\_[2]{}(1,0)\_[3]{}(u,v)|\_[4]{},\_[4]{}. It can be easily seen that f(u,v)F\_[BMS]{}(u,v)=G\_[34]{}\^[21]{}(u,v), where f(u,v)&=&(1-u)\^[(\_[231]{}+\_[234]{})]{}u\^[(\_[341]{}+\_[342]{})]{} e\^[(\_[231]{}+\_[234]{})]{}e\^[-(\_[341]{}+\_[342]{})]{} So, the four-point function can be expressed in terms of $G_{34}^{21}(u,v)$ as \_[i=1]{}\^[4]{}\_[i]{}(u\_[i]{},v\_[i]{}) = P({\_i,\_i,u\_[ij]{},v\_[ij]{}}) f(u,v)\^[-1]{}G\_[34]{}\^[21]{}(u,v). \[four\_point\] Now, we may also define G\_[32]{}\^[41]{}(u,v)=\_[1]{},\_[1]{}|\_[4]{}(1,0)\_[3]{}(u,v)|\_[2]{},\_[2]{}, and it can be easily seen that these functions $G_{ij}^{kl}(u,v)$ are related by crossing symmetry G\_[34]{}\^[21]{}(u,v)=G\_[32]{}\^[41]{}(1-u,-v). \[crosssym\] It is important to emphasise here that the crossing equation that we have obtained above is [*not the same*]{} as the usual conformal crossing equation [(\[confcross\])]{}.
If we use the OPE between the fields $\phi_3$ and $\phi_4$ in $G_{34}^{21}(u,v)$ we can see that the function can be expressed in terms of the three-point functions of primary fields and their descendants. More precisely, using the OPE, it can be decomposed as G\_[34]{}\^[21]{}(u,v)=\_[p]{}C\_[34]{}\^[p]{}C\_[12]{}\^[p]{}A\_[34]{}\^[21]{}(p|u,v), where the four-point conformal block $A_{34}^{21}(p|u,v)$ is the sum of all contributions coming from the primary field $\phi_p$ and its descendants and is given by[[^2]]{} A\_[34]{}\^[21]{}(p|u,v) &=& (C\_[12]{}\^[p]{})\^[-1]{}u\^[-\_[3]{}-\_[4]{}+\_[p]{}]{}e\^[(\_[3]{}+\_[4]{}-\_[p]{})]{}\_[N]{}u\^[N-]{}v\^\_[1]{},\_[1]{}|\_[2]{}(1,0)|N,\_p &=& u\^[-\_[3]{}-\_[4]{}+\_[p]{}]{}e\^[(\_[3]{}+\_[4]{}-\_[p]{})]{} && \_[{,}]{}(\_[=0]{}\^[K+Q]{}\_[34]{}\^[p{,},]{}u\^[K+Q-]{}v\^) && As we have already seen, the coefficients ${\beta}_{34}^{p\{\vec{k},\vec{q}\},{\alpha}}$ can be calculated recursively using BMS symmetry. Thus, the closed form expression of the BMS blocks are completely determined by symmetry and the only dynamical inputs needed to find the four point functions are the structure constants and the spectrum of primary operators appearing in the OPE.
For the function $G_{32}^{41}(u,v)$ we may use the OPE on $\phi_2$ and $\phi_3$ giving us the expansion G\_[32]{}\^[41]{}(u,v)=\_p C\^p\_[23]{}C\^[p]{}\_[14]{} A\_[32]{}\^[41]{}(p|u,v), where the blocks $A_{32}^{41}(u,v)$ are given by A\_[32]{}\^[41]{}(p|u,v)&=& u\^[-\_[3]{}-\_[2]{}+\_[p]{}]{}e\^[(\_[3]{}+\_[2]{}-\_[p]{})]{} && \_[{,}]{}(\_[=0]{}\^[K+Q]{}\_[32]{}\^[p{,},]{}u\^[K+Q-]{}v\^).&& Now, must be satisfied, even if we expand both sides using OPE in terms of the BMS blocks, giving us the BMS bootstrap equation \[eqn:bootstrap\] This is one of the main initial results of our analysis.
Knowing the BMS blocks, the above equation put a constrain on the structure constants and weights of primary operators in a consistent field theory with BMS symmetry. We can try to solve the bootstrap equation to find all such possible consistent field theories. The only problem is that we do not have a closed form expression of the blocks even though they are fixed by symmetry alone. However, we can find the leading term in a $\frac{1}{c_{L,M}}$ expansion of the blocks. Using this expansion on both sides of , the equation has to be satisfied order by order. The leading order give us the constraint \_[p]{}C\_[34]{}\^[p]{}C\_[12]{}\^[p]{}g\_[34]{}\^[21]{}(p|u,v)=\_[q]{}C\_[32]{}\^[q]{}C\_[41]{}\^[q]{}g\_[32]{}\^[41]{}(q|1-u,-v), where $g_{ij}^{kl}(p|u,v)$ are the large central charge limit of the blocks $A_{ij}^{kl}(p|u,v)$ g\_[ij]{}\^[kl]{}(p|u,v)=\_[c\_[L,M]{}]{} A\_[ij]{}\^[kl]{}(p|u,v). We will find $g_{ij}^{kl}(p|u,v)$ in the next section.
Differential equations for global blocks from quadratic Casimirs
----------------------------------------------------------------
For even dimensional CFTs with $d\geq 4$, the closed form expression of the four point conformal blocks was obtained for scalar operators by Dolan and Osborn in [@Dolan:2000ut; @Dolan:2003hv]. For $2d$ CFTs, their method gives the global conformal blocks, which is the large central charge limit of the full Virasoro conformal blocks, as we have mentioned in the previous section. In this section we will employ this method to obtain the global blocks for BMS algebra, assuming that such a limit will act in a similar manner.
If we take the asymptotic limit $c_L,c_M\rightarrow\infty$ in the OPE , , the leading terms are given by the descendant fields generated by $L_{-1}$ and $M_{-1}$. This can be explicitly seen by looking at the coefficients $\beta$ in the limit $c_L, c_M \to \infty$. For levels 1 and 2, this can be verified by the results obtained in previous sections and outlined in Table [(\[level1\])]{} and Table [(\[level2\])]{}. More precisely, we have && \_[3]{}(u,v)\_[4]{}(0,0)|0,0 = && \_[p,{k,q}]{}u\^[-\_[1]{}-\_[2]{}+\_[p]{}]{}e\^[(\_[3]{}+\_[4]{}-\_[p]{})]{}C\_[34]{}\^[p]{}(\_[=0]{}\^[N=k+q]{}\_[34]{}\^[p{k,q},]{}u\^[k+q-]{}v\^)(L\_[-1]{})\^[k]{}(M\_[-1]{})\^[q]{}|\_p,\_p,&&+(,)+ …, So the function $G_{34}^{21}(u,v)$ has an expansion of the form \_[1]{},\_[1]{}|\_[2]{}(1,0)\_[3]{}(u,v)|\_[4]{},\_[4]{} = \_[p]{}C\_[12]{}\^[p]{}C\_[34]{}\^[p]{}g\_[34]{}\^[21]{}(p|u,v) + (,)+..., where the global block $g_{34}^{21}(p|u,v)$, which is the large central charge limit of $G_{34}^{21}(u,v)$, is given by g\_[34]{}\^[21]{}(p|u,v) &=& u\^[-\_[3]{}-\_[4]{}+\_[p]{}]{}e\^[(\_[3]{}+\_[4]{}-\_[p]{})]{} && \_[{k,q}]{}(\_[=0]{}\^[N=k+q]{}\_[34]{}\^[p{k,q},]{}u\^[N-]{}v\^).\[eqn:cb3\] && More generally, we have &&\_[3]{}(u\_3,v\_3)\_[4]{}(u\_4,v\_4)|0,0 = && \_[p,{k,q}]{}u\_[34]{}\^[-\_[1]{}-\_[2]{}+\_[p]{}]{}e\^[(\_[3]{}+\_[4]{}-\_[p]{})]{}C\_[34]{}\^[p]{}(\_[=0]{}\^[N=k+q]{}\_[34]{}\^[p{k,q},]{}u\^[k+q-]{}v\^)(L\_[-1]{})\^[k]{}(M\_[-1]{})\^[q]{}\_4(u\_4,v\_4)|0,0,&&+ (,)+..., \[eqn:ope3\] with the four-point function given by the $1/c_{L,M}$ expansion \_[i=1]{}\^4\_i(u\_i,v\_i) = \_[p]{}C\_[12]{}\^[p]{}C\_[34]{}\^[p]{}\_[34]{}\^[21]{}(p|u,v) + (,)+..., \[4\_pt\_expand\] where \_[34]{}\^[21]{}(p|u,v)=P({\_i,\_i,u\_[ij]{},v\_[ij]{}})f(u,v)\^[-1]{}g\_[34]{}\^[21]{}(p|u,v). \[eqn:rel\_blocks\]
It is possible to find the blocks $g_{34}^{21}(p|u,v)$ by demanding that both sides of the OPE transform the same way under the action of the quadratic Casimirs belonging to the global algebra generated by $\{L_{-1},L_0,L_1,M_{-1},M_0,M_1\}$. These Casimirs are given by \_1 &=& M\_0\^2-M\_[-1]{}M\_1\
\_2 &=& 2L\_0M\_0-(L\_[-1]{}M\_1+L\_1M\_[-1]{}+M\_1L\_[-1]{}+M\_[-1]{}L\_1). It can be seen that the states $(L_{-1})^{k}(M_{-1})^{q}\phi_4(u_4,v_4)|0,0{\rangle}$ are eigenstates of $\mathcal{C}_1$ and $\mathcal{C}_2$ since the Casimirs commute with $L_{-1}$, $M_{-1}$, \_[1,2]{}(L\_[-1]{})\^[k]{}(M\_[-1]{})\^[q]{}\_4(u\_4,v\_4)|0,0 && \_[1,2]{}\^p (L\_[-1]{})\^[k]{}(M\_[-1]{})\^[q]{}\_4(u\_4,v\_4)|0,0, && where the eigenvalues are given by \_1\^p =\_[p]{}\^[2]{}, \_2\^p = (2\_[p]{}\_[p]{}-2\_[p]{}). Consequently, we have
&&\_[1,2]{} \_[3]{}(u\_3,v\_3)\_[4]{}(u\_4,v\_4)|0,0 = &&\_[p,{k,q}]{}\_[1,2]{}\^pu\_[34]{}\^[-\_[1]{}-\_[2]{}+\_[p]{}]{}e\^[(\_[3]{}+\_[4]{}-\_[p]{})]{}C\_[34]{}\^[p]{}(\_[=0]{}\^[N=k+q]{}\_[34]{}\^[p{k,q},]{}u\^[k+q-]{}v\^)(L\_[-1]{})\^[k]{}(M\_[-1]{})\^[q]{}\_4(u\_4,v\_4)|0,0&&+ (,)+.... After taking the inner product on both sides with ${\langle}\phi_1(u_1,v_1)\phi_2(u_2,v_2)|$, we have \_1(u\_1,v\_1)\_2(u\_2,v\_2)\_[1,2]{}\_3(u\_3,v\_3)\_4(u\_4,v\_4) = \_[p]{} \_[1,2]{}\^p C\_[12]{}\^[p]{}C\_[34]{}\^[p]{}\_[34]{}\^[21]{}(p|u,v) + (,)+.... On the LHS of the above equation, $\mathcal{C}_{1,2}$ act as differential operators $\mathcal{D}_{1,2}$. More precisely, these differential operators are given by && \_1\_3(y\_[3]{})\_4(y\_[4]{})|0 &&= (M\_[0]{}\^[2]{}-M\_[-1]{}M\_[1]{})\_3(y\_[3]{})\_4(y\_[4]{})|0 && = \[(-u\_[3]{}\_[v\_[3]{}]{}+\_3-u\_[4]{}\_[v\_[4]{}]{} + \_4)(-u\_[3]{}\_[v\_[3]{}]{}+ \_3 - u\_[4]{}\_[v\_[4]{}]{} +\_4)&& -(-\_[v\_[3]{}]{}-\_[v\_[4]{}]{})(-u\_[3]{}\^[2]{}\_[v\_[3]{}]{}+2\_3 u\_[3]{}-u\_[4]{}\^[2]{}\_[v\_[4]{}]{}+2\_4 u\_[4]{})\]\_3(y\_[3]{})\_4(y\_[4]{})|0 && = \[2\_3 (u\_[3]{}-u\_4)\_[v\_[4]{}]{}-2\_4 (u\_3-u\_[4]{})\_[v\_[3]{}]{}+(\_3+\_4)\^[2]{}-(u\_3-u\_4)\^2\_[v\_[3]{}]{}\_[v\_[4]{}]{}\](\_3(y\_[3]{})\_4(y\_[4]{}))|0 &&\_1(\_3(y\_[3]{})\_4(y\_[4]{}))|0, &&\_2\_3(y\_[3]{})\_4(y\_[4]{})|0 &&= \[2L\_0M\_0-(L\_[-1]{}M\_1+L\_1M\_[-1]{}+M\_1L\_[-1]{}+M\_[-1]{}L\_1)\]\_3(y\_[3]{})\_4(y\_[4]{})|0 &&= \[2(\_3+\_4-1) (\_3+\_4)+(-2 u\_3 \_3+2 u\_4 \_3) \_[u\_4]{}+(2 u\_3 \_4-2 u\_4 \_4) \_[u\_3]{}&&+(-2 u\_3 \_4+2 u\_4 \_4+2 v\_3 \_4-2 v\_4 \_4) \_[v\_3]{} + (2 u\_3 \_3-2 u\_4 \_3-2 v\_3 \_3+2 v\_4 \_3) \_[v\_4]{}&&+(u\_3\^2-2 u\_3 u\_4+u\_4\^2) \_[v\_4]{}\_[u\_3]{} + (u\_3\^2-2 u\_3 u\_4+u\_4\^2) \_[v\_3]{}\_[u\_4]{}&& + (2 u\_3 v\_3-2 u\_4 v\_3-2 u\_3 v\_4+2 u\_4 v\_4) \_[v\_3]{}\_[v\_4]{}\]\_3(y\_[3]{})\_4(y\_[4]{})|0&&\_2(\_3(y\_[3]{})\_4(y\_[4]{}))|0. Pulling the differential operator outside the four-point function, we have \_[1,2]{}\_1(u\_1,v\_1)\_2(u\_2,v\_2)\_3(u\_3,v\_3)\_4(u\_4,v\_4) = \_[p]{} \_[1,2]{}\^p C\_[12]{}\^[p]{}C\_[34]{}\^[p]{}\_[34]{}\^[21]{}(p|u,v) + (,)+.... We then expand the LHS using . This gives &&\_[1,2]{}\_[p]{}C\_[12]{}\^[p]{}C\_[34]{}\^[p]{}\_[34]{}\^[21]{}(p|u,v) + (,)+... = \_[p]{} \_[1,2]{}\^p C\_[12]{}\^[p]{}C\_[34]{}\^[p]{}\_[34]{}\^[21]{}(p|u,v) + (,)+.... && This equation has to be satisfied order by order. The leading order give us a differential equation for $\tl{g}_{34}^{21}(p|u,v)$ \_[1,2]{}\_[p]{}C\_[12]{}\^[p]{}C\_[34]{}\^[p]{}\_[34]{}\^[21]{}(p|u,v)=\_[p]{} \_[1,2]{}\^p C\_[12]{}\^[p]{}C\_[34]{}\^[p]{}\_[34]{}\^[21]{}(p|u,v) We can decouple this to get differential equations for each block $\tl{g}_{34}^{21}(p|u,v)$, \_[\_[1,2]{}]{}\_[34]{}\^[21]{}(p|u,v) = \_[1,2]{}\^p\_[34]{}\^[21]{}(p|u,v). Using we have \_[1,2]{}(P({\_i,\_i,u\_[ij]{},v\_[ij]{}})f(u,v)\^[-1]{}g\_[34]{}\^[21]{}(p|u,v)) = \_[1,2]{}\^pP({\_i,\_i,u\_[ij]{},v\_[ij]{}})f(u,v)\^[-1]{}g\_[34]{}\^[21]{}(p|u,v). Let us first look at the differential equation associated with $\mathcal{C}_1$, which is given by &&\_1(\_[1i<j4]{}e\^[-(\_[k=1]{}\^[4]{}\_[k]{}-\_[i]{}-\_[j]{})]{}f(u,v)\^[-1]{}g\_[34]{}\^[21]{}(p|u,v))&& = \_[p]{}\^[2]{}\_[1i<j4]{} e\^[-(\_[k=1]{}\^[4]{}\_[k]{}-\_[i]{}-\_[j]{})]{}f(u,v)\^[-1]{}g\_[34]{}\^[21]{}(p|u,v). For simplicity let us consider the case where ${\Delta}_{i=1,2,3,4}={\Delta},\,\xi_{i=1,2,3,4}=\xi$. Then the above equation reduces to &&\_1(\_[1i<j4]{}e\^(1-u)\^u\^e\^e\^[-]{}g\_[,]{}(p|u,v))&&=\_[p]{}\^[2]{}\_[1i<j4]{} e\^(1-u)\^u\^e\^e\^[-]{}g\_[,]{}(p|u,v)), where we have used the notation $g_{{\Delta},\xi}(p|u,v)$ for the blocks $g_{34}^{21}(p|u,v)$ in this special case and $\mathcal{D}_1$ is also taken with $\xi_{i=1,2,3,4}=\xi$. If we combine the functions of $u$ and $v$ into \_[,]{}(p|u,v)=(1-u)\^u\^e\^e\^[-]{}g\_[,]{}(p|u,v), then we have \_1(\_[1i<j4]{}e\^\_[,]{}(p|u,v)) = \_[p]{}\^[2]{}\_[1i<j4]{} e\^\_[,]{}(p|u,v), which explicitly is given by &&\_[,]{}(p|u,v)=0, where $u_{ij}=u_i-u_j$. Under the global conformal transformation , the above equation reduces to \_[,]{}(p|u,v)=0. In terms of the global blocks $g_{{\Delta},\xi}(p|u,v)$, the above equation is given by g\_[,]{}(p|u,v)=0. The differential equation gets simpler if we introduce a function k(p|u,v)=u\^[2]{}e\^[-]{}g\_[,]{}(p|u,v). Plugging this back into the above equation, we have the simplified version: h(p|u,v)=0. \[diffeqn1\] Now let us look at the differential equation associated with $\mathcal{C}_2$. For simplicity we again only consider the case where ${\Delta}_{i=1,2,3,4}={\Delta},\,\xi_{i=1,2,3,4}=\xi$. We have, \_2(\_[1i<j4]{}u\_[ij]{}\^[-]{}e\^\_[\_[p]{},\_[p]{}]{}(u,v))) = (2\_p\_p-2\_p)\_[1i<j4]{}u\_[ij]{}\^[-]{}e\^\_[,]{}(p|u,v). Under the global conformal transformation , the above differential equation reduces to &&\[(2 (-6+8 -2 u\^3 -2 u (-6+8 +v )+u\^2 (-6+10 +v ))&&-9 (-1+u)\^2(-1+\_p) \_p) - 12 (-2+u) (-1+u)\^2 u \_u &&+ 3 (-1+u)\^2 (u\^2 (6+4 ) +8 v -8 u (+v ))\_v + 18 (-1+u)\^3 u\^2 \_v\_u && + 9 (-1+u)\^2 u (-2+3 u) v\_v\^2 \]\_[,]{}(p|u,v)=0. In terms of the global blocks $g_{{\Delta},\xi}(p|u,v)=(1-u)^{-\frac{2{\Delta}}{3}}u^{-\frac{2{\Delta}}{3}}e^{-\frac{2\xi v}{3(1-u)}}e^{\frac{2\xi v}{3u}}\hat{g}_{{\Delta},\xi}(p|u,v)$, we have &&\[2(2 (-1+2 -2 u +v )-(-1+\_p) \_p)+2(u\^2 (1+2 )+2 v -2 u (+2 v ))\_v &&+u (-2+3 u) v \_v\^2-4 (-1+u) u \_u + 2 (-1+u) u\^2 \_v\_u\]g\_[,]{}(p|u,v)=0. In terms of the function $k(p|u,v)$, the differential equation again gets simpler k(p|u,v) =(\_p - 1) \_p k(p|u,v). \[diffeqn2\]
Solution of the BMS Global block
--------------------------------
In this subsection, we find the explicit solution for the differential equation for the global BMS block. The general solutions of the above differential equation [(\[diffeqn1\])]{} are given by k\_1(p|u,v)=A\_[\_[p]{},\_[p]{}]{}(u)e\^[v]{},k\_2(p|u,v)=B\_[\_[p]{},\_[p]{}]{}(u)e\^[-v]{}. \[sol:diff1\] Substituting in the second differential equation we have &&A\_[\_p,\_p]{}(u)(-2(1+)+(2+) u-2 (-1+u) \_p)&&+2 (1-u)\^[3/2]{} u=0, &&B\_[\_p,\_p]{}(u)(2-2 +(-2+) u+2 (-1+u) \_p)&&+2 (1-u)\^[3/2]{} u=0. The solutions of the above differential equations are given by A\_[\_p,\_p]{}(u)=K\_A, B\_[\_p,\_p]{}(u)=K\_B, where $K_A$ and $K_B$ are constant of integration. So the most general solution of $h(p|u,v)$ is given by k(p|u,v)=K\_Ae\^[v]{}+K\_Be\^[-v]{}. \[full\_sol\] Note again that the blocks are defined only for $u^2+v^2<1$. So we don’t have to consider the case $u>1$, where the above equation becomes oscillatory.
Now, we need boundary conditions to find the constant of integration. Looking at , we can see that $k(p|u,v)$ is given by k(p|u,v) &=& u\^[\_[p]{}]{}e\^[-\_p]{} && \_[{k,q}]{}(\_[=0]{}\^[N=k+q]{}\_[34]{}\^[p{k,q},]{}u\^[N-]{}v\^).&& Let us show a few of the terms in the summation. We know that ${\beta}_{12}^{p\{0,0\},0}=1$, ${\beta}_{12}^{p\{1,0\},0}=\frac{1}{2}$, ${\beta}_{12}^{p\{0,1\},0}=-\frac{1}{2}$ and =\_p,=\_p. So, we have k(p|u,v) = u\^[\_[p]{}]{}e\^[-\_p]{}(1+u-v+...). \[cb:expand\] Expanding our solution and comparing with the above equation we can find the values $K_A$ and $K_B$. For $|u|<1$, the expansion of our solution is given by k(p|u,v)&=& K\_Au\^[\_p]{}e\^[-]{}2\^[2-2 \_p]{}u\^[-2 \_p]{}(2\^[-2+2 \_p]{} u\^22\^[-1+2 \_p]{} \_p uv+2\^[-1+2 \_p]{} \_p\^2v\^2) &&+ K\_Bu\^[\_p]{}e\^[-]{}2\^[2-2 \_p]{}(1-\_p v+\_p u+...). Comparing this with , we have K\_A=0,K\_B=2\^[2 \_p-2]{}. So, we have g\_[,]{}(p|u,v)= 2\^[2 \_p-2]{} (1-u)\^[-1/2]{} &&u\^[\_p-2]{} (1+)\^[2-2\_p]{}, && |u|<1. \[BMS\_global\_blocks\] This is the explicit form of the global BMS blocks in the limit of large central charges and is one of the main results of our initial analysis.
BMS Bootstrap: the limiting analysis
====================================
In this section, we will consider the limit of the 2d conformal algebra that leads to the BMS$_3$ algebra, or equivalently, the 2d GCA. There are two distinct limits that do this; one can be looked upon as a non-relativistic limit and the other as an ultra-relativistic limit. We shall first discuss them and then focus on the non-relativistic limit. We shall then proceed to reproduce some of the answers obtained in the previous section in the light of this limit.
Two contractions of 2d conformal algebra
----------------------------------------
There are two distinct contractions of two copies of the relativistic Virasoro algebra that lead to the 2d GCA. At an algebraic level these are given by \[nr\] L\_n = Ł\_n + [|]{}\_n, M\_n = (Ł\_n - [|]{}\_n ) and \[ur\] L\_n = Ł\_n + [|]{}\_[-n]{}, M\_n = (Ł\_n + [|]{}\_[-n]{} ) To see why these contractions are so named, it is instructive to look at the generators of the Virasoro algebra on the cylinder and follow the contraction. The conformal generators on the cylinder are Ł\_n = e\^[in]{} \_, [|]{}\_n = e\^[in[|]{}]{} \_[[|]{}]{}, , [|]{}= . Now in the non-relativistic limit, the co-ordinates on the cylinder scale as $({\sigma}, {\tau}) \to ({\epsilon}{\sigma},{\tau})$. It is now clear that if you take [(\[nr\])]{}, in this limit this combination gives well-behaved vector fields \[nrvec\] L\_n = e\^[in]{}(\_+in\_), M\_n = e\^[in]{} \_. These close to form the algebra [(\[gca2d\])]{}. The ultra-relativistic limit is when we scale the co-ordinates on the cylinder as $({\sigma}, {\tau}) \to ({\sigma},{\epsilon}{\tau})$. It is easy to check that using [(\[ur\])]{}, one now gets well defined vector fields \[urvec\] L\_n = e\^[in]{}(\_+in\_), M\_n = e\^[in]{} \_. It is gratifying to see that [(\[nrvec\])]{} and [(\[urvec\])]{} are related by a swap of ${\sigma}\leftrightarrow {\tau}$, as one would expect. In a combined notation, we can write \[combvec\] L\_n = e\^[inU]{}(\_U+in\_V), M\_n = e\^[inU]{} \_V where U is the un-contracted direction and V is the contracted direction in the field theory. We are interested in the “plane” representations. The mapping between the two representations is given by \[pcmap\] u = e\^[iU]{}, v=iV e\^[iU]{}. This connects us to the notation of the previous section. In particular [(\[combvec\])]{} goes to [(\[planevec\])]{} under the map [(\[pcmap\])]{}.
#### Highest weight representations and limits:
Throughout the previous section on the intrinsic analysis of the construction of the bootstrap equation and the solution of the BMS block in the limit of large central charges, we have worked in the highest weight representation, as was done in the case of the relativistic CFT analysis mentioned previously. We will now attempt to understand some aspects of this through the limit. We seem to have two distinct limits [(\[nr\])]{}, [(\[ur\])]{} and hence two distinct ways of achieving this.
This is, however, not true. Let us consider the ultra-relativistic limit [(\[ur\])]{}. We clearly see that there is a mixture of positive and negative modes of the Virasoro algebra in this limit. The Virasoro highest weight condition [(\[hwV\])]{} thus does not reduce to the BMS highest weight condition [(\[hwB\])]{}. The Virasoro highest weight condition leads to a distinct class of representations called induced representations [@Barnich:2014kra; @Barnich:2015uva; @Campoleoni:2016vsh].
The non-relativistic limit [(\[nr\])]{} is conducive to reproducing our earlier results, as in this case there is no mixing of positive and negative modes and the Virasoro highest weight representations do go over the the BMS highest weight representations. We will thus be focusing on the non-relativistic limit in an attempt to reconstruct the answers previously obtained from the intrinsic analysis.
Reproducing coefficients of OPE
-------------------------------
We will start by attempting to recover the coefficients of the BMS OPE that we obtained in the previous section through the non-relativistic limit.
For simplicity take $h_{1}=h_{2}=h,\,\,{{\bar h}}_{1}={{\bar h}}_{2}={{\bar h}}$. Acting on the vacuum, the RHS is given by \_[p,{,}]{}\_[p12]{} \_[12]{}\^[p{}]{}|\_[12]{}\^[p{}]{}z\^[h\_[p]{}-2h+K]{}[[|z]{}]{}\^[[[|h]{}]{}\_[p]{}-2[[|h]{}]{}+|[K]{}]{}Ł\_[|]{}\_|h\_[p]{},[[|h]{}]{}\_[p]{}. \[opeCFT1\] In terms of the space time co-ordinates $z=t+x$, ${{\bar z}}=t-x$. We will consider the non-relativistic contraction tt,xx. \[contraction\] The reason for doing so, and not considering the ultra-relativistic contraction $(t\to{\epsilon}t, x\to x)$, has already been stated above. We note again that in our notation in the previous section $v$ is the co-ordinate which is contracted. In this case is given by & & \_[p,{,}]{}\_[p12]{} \_[12]{}\^[p{}]{}|\_[12]{}\^[p{}]{}(t+x)\^[h\_[p]{}-2h+K]{}(t-x)\^[[[|h]{}]{}\_[p]{}-2[[|h]{}]{}+|[K]{}]{}Ł\_[|]{}\_|h\_[p]{},[[|h]{}]{}\_[p]{} & = & \_[p,{,}]{}\_[p12]{} \_[12]{}\^[p{}]{}|\_[12]{}\^[p{}]{}t\^[h\_[p]{}-2h+K+[[|h]{}]{}\_[p]{}-2[[|h]{}]{}+|[K]{}]{}(1+)\^[h\_[p]{}-2h+K]{}(1-)\^[[[|h]{}]{}\_[p]{}-2[[|h]{}]{}+|[K]{}]{}Ł\_[|]{}\_|h\_[p]{},[[|h]{}]{}\_[p]{} & = & \_[p,{,}]{}\_[p12]{} \_[12]{}\^[p{}]{}|\_[12]{}\^[p{}]{}t\^[h\_[p]{}+[[|h]{}]{}\_[p]{}-2(h+[[|h]{}]{})+K+|[K]{}]{}(((1+)\^[h\_[p]{}-2h]{}(1-)\^[[[|h]{}]{}\_[p]{}-2[[|h]{}]{}]{})) & & (1+)\^[K]{}(1-)\^[|[K]{}]{}Ł\_[|]{}\_|h\_[p]{},[[|h]{}]{}\_[p]{} & = & \_[p,{,}]{}\_[p12]{} \_[12]{}\^[p{}]{}|\_[12]{}\^[p{}]{}t\^[h\_[p]{}+[[|h]{}]{}\_[p]{}-2(h+[[|h]{}]{})+K+|[K]{}]{}((h\_[p]{}-[[|h]{}]{}\_[p]{}-2(h-[[|h]{}]{}))(+(\^[2]{}))) & & (1+)\^[K]{}(1-)\^[|[K]{}]{}Ł\_[|]{}\_|h\_[p]{},[[|h]{}]{}\_[p]{}. \[opecontrac\] Taking the non-relativistic limit ${\epsilon}\rightarrow0$, we have =\_[0]{}(h+[[|h]{}]{}),=\_[0]{}([[|h]{}]{}-h),Ł\_[n]{}=(L\_[n]{}-M\_[n]{}),[|]{}\_[n]{}=(L\_[n]{}+M\_[n]{}). Putting these in equation , we get & & \_[p,{,}]{}\_[p12]{} \_[12]{}\^[p{}]{}|\_[12]{}\^[p{}]{}t\^[-2\_[p]{}]{}t\^[K+|[K]{}]{}(2-\_[p]{}+(\^[2]{})) & & (1+)\^[K]{}(1-)\^[|[K]{}]{}(L-M)\_(L+M)\_|\_[p]{},\_[p]{}. \[ope\_nrlimit\]
Level 1 {#level-1-1 .unnumbered}
-------
If we look at only the level one states in without the common factor $\frak{C}_{p12} \ t^{{\Delta}-2{\Delta}_{p}}\exp\left(2\xi-\xi_{p}\right)$, we have & & t(\_[12]{}\^[p{1}]{}(1+)(L\_[-1]{}-M\_[-1]{})+\_[12]{}\^[p{|[1]{}}]{}(1-)(L\_[-1]{}+M\_[-1]{}))|\_[p]{},\_[p]{} & = & ((\_[12]{}\^[p{1}]{}+\_[12]{}\^[p{|[1]{}}]{})L\_[-1]{}-(\_[12]{}\^[p{1}]{}+\_[12]{}\^[p{|[1]{}}]{})M\_[-1]{}+(\^[2]{}))|\_[p]{},\_[p]{}. Using the known coefficients of CFT $\mathcal{B}_{12}^{p\{1\}}=\mathcal{B}_{12}^{p\{\bar{1}\}}=\frac{1}{2}$, we can see that \_[12]{}\^[p{1,0},0]{}=,\_[12]{}\^[p{0,1},1]{}=-,\_[12]{}\^[p{1,0},1]{}=\_[12]{}\^[p{0,1},0]{}=0. These match with the coefficients in Table .
Level 2 {#level-2-1 .unnumbered}
-------
The details of the level 2 calculations are presented in Appendix B. We see that the coefficients again match up with the answers previously obtained in the intrinsic method.
Differential equation for blocks from the limiting case
-------------------------------------------------------
Having demonstrated that the coefficients of the BMS OPE can be recovered from a limit of the OPE for the Virasoro algebra, we now go on to demonstrate that some other key features of our intrinsic analysis can also be reproduced in this limit. In this subsection, we concentrate on deriving the differential equations for the global BMS blocks from the corresponding equations for the global CFT blocks.
Under the contraction and the definition in , the differential equations in transforms to &&\_[\_p,\_p]{}(t,x)=0, &&\_[\_p,\_p]{}(t,x)=0. Taking the limit ${\epsilon}\rightarrow 0$, the only remaining part in both differential equations is \_[\_p,\_p]{}(t,x)=0. We can also subtract one differential equation from the other and then take the limit which gives rise to the differential equation \_[\_p,\_p]{}(t,x) =(\_p - 1) \_p \_[\_p,\_p]{}(t,x). These are same as and .
The limit of the Virasoro global block
--------------------------------------
In this sub-section we will analyse the non-relativistic limit of the global CFT blocks [$\mathfrak{g}$]{}\^[21]{}\_[34]{}(p|z,[$\bar{z}$]{}) = z\^[h\_p-h\_1-h\_2]{}[$\bar{z}$]{}\^[[$\bar{h}$]{}\_p-[$\bar{h}$]{}\_1-[$\bar{h}$]{}\_2]{}[$\ _2F_1$]{}(h\_p+h\_[12]{},h\_p+h\_[34]{};2h\_p; z)[$\ _2F_1$]{}([$\bar{h}$]{}\_p+[$\bar{h}$]{}\_[12]{},[$\bar{h}$]{}\_p+[$\bar{h}$]{}\_[34]{};2[$\bar{h}$]{}\_p;[$\bar{z}$]{}), and check if it matches with the global BMS blocks calculated from the intrinsic analysis. In order to take the non-relativistic limit of the global block, we make use of the integral representation of the hypergeometric function $${\ensuremath{\ _2F_1}}(a,b;c;z) = \frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_0^1 dw w^{b-1}(1-w)^{c-b-1}(1-zw)^{-a}$$ which is valid for $|z|<1$ with $\arg(z)<\pi$. The beta function also has an integral representation =\_0\^1 y\^[b-1]{}(1-y)\^[c-b-1]{} dy. So we have [$\ _2F_1$]{}(a,b;c;z)=.
We consider first the anti-holomorphic sector and make the substitutions ${\ensuremath{\bar{h}}}=\frac{\Delta}{2}+\frac{\xi}{2\epsilon}$ and ${\ensuremath{\bar{z}}}=t-\epsilon x$ &&[$\ _2F_1$]{}(+,+;\_p+;t-x)=&&&& . Arranging the integrand in $\bar{I}$ as an exponential of a power series in $\epsilon$, we have &&|[I]{}(\_p,\_p,\_[12]{},\_[34]{},\_[12]{},\_[34]{})&&=\_0\^1 ((+-1)w + (+-1)(1-w). &&.- (+)(1-(t-x)w))dw &&\_0\^1 dw |[f]{}(w)e\^[|[S]{}(w)+Ø()]{}, where |[f]{}(w)&=& w\^[-1]{}(1-w)\^[-1]{}(1-tw)\^[-]{}e\^,|[S]{}(w)&=&((\_p+\_[34]{})w + (\_p - \_[34]{})(1-w) - (\_p + \_[12]{})(1 - t w)). The critical points of the function $\bar{S}(w)$ are given by \_w |[S]{}(w)=0, and are located at w\_=. In particular, the critical point $w_-$ lies on the real axis in the domain of integration. Then, in the limit $\epsilon\rightarrow0$ limit, we can use the saddle point approximation to calculate the integral &&|[I]{}(\_p,\_p,\_[12]{},\_[34]{},\_[12]{},\_[34]{}) e\^[|[S]{}(w\_-)]{}(|[f]{}(w\_-)+Ø())=&&.&& Now, let us do the saddle point analysis for the beta function |[B]{}(\_p,\_p,\_[34]{},\_[34]{}) &=&\_0\^1 y\^[-1]{}(1-y)\^[-1]{}e\^[((\_p+\_[34]{})y +(\_p+\_[34]{})(1-y))]{} dy&&\_0\^1 |[p]{}(y)e\^[|[q]{}(y)]{} dy, where |[p]{}(y)=y\^[-1]{}(1-y)\^[-1]{},|[q]{}(y)=((\_p+\_[34]{})y +(\_p+\_[34]{})(1-y)). The critical point for $\bar{q}(y)$ is at $y_{-}=\frac{\xi _{34}+\xi _p}{2 \xi _p}$. Then we have |[B]{}(\_p,\_p,\_[34]{},\_[34]{}) && e\^[|[q]{}(w\_-)]{} |[p]{}(y\_-)&=& . All the arguments above also follows through for the holomorphic sector where we make the substitution $h=\frac{\Delta}{2}-\frac{\xi}{2\epsilon}$ and $z=t+\epsilon x$ &&[$\ _2F_1$]{}(-,-;\_p-;t+x)&& = , with f(w)&=&w\^[-1]{}(1-w)\^[-1]{}(1-tw)\^[-]{}e\^,S(w)&=& (-(\_p+\_[34]{})w -(\_p - \_[34]{})(1-w) + (\_p + \_[12]{})(1 - t w)),p(y)&=&y\^[-1]{}(1-y)\^[-1]{},q(y)&=& (-(\_p+\_[34]{})y -(\_p+\_[34]{})(1-y)). It can be easily seen that for the integrand in $I(\Delta_p,\xi_p,\Delta_{12},\Delta_{34},\xi_{12},\xi_{34})$, the dominant saddle occurs at the same value $w_-$. Then the saddle point approximation give us &&I(\_p,\_p,\_[12]{},\_[34]{},\_[12]{},\_[34]{})=&&.&& Similarly for the integrand in $B(\Delta_p,\Delta_{34},\xi_{34})$, the saddle point is same as in the anti-holomorphic sector i.e., $y_{-}$. Doing the saddle point approximation, we have B(\_p,\_[34]{},\_[34]{})=. Now, combining the holomorphic and the anti-holomorphic pieces, we have && =&& i&&( )\^[-1]{}. We are interested in the case where all the external operators are identical i.e., $\Delta_{ij}=0,\,\xi_{ij}=0$. Putting the value of $w_{-}$ and $y_{-}$ and taking $\Delta_{ij}=0,\,\xi_{ij}=0$ in the above equation, we have = 2\^[2\_p-2]{} (1-t)\^[-1/2]{} (1+)\^[2-2\_p]{}. Combining this with the factor z\^[h\_p-2h]{}|[z]{}\^[[[|h]{}]{}\_p-2[[|h]{}]{}]{} t\^[\_p-2]{}e\^[-\_p +2]{}, we finally have the global BMS blocks g\_[,]{}(p|t,x)= 2\^[2 \_p-2]{} (1-t)\^[-1/2]{} &&t\^[\_p-2]{} (1+)\^[2-2\_p]{}.&& This matches exactly with global BMS blocks obtained using intrinsic analysis and gives us a very non-trivial and comprehensive check of our intrinsic analysis.
The Chiral Limit
================
In this section, we will explore what is called the Chiral limit of the BMS$_3$ algebra. When one is looking at the BMS$_3$ algebra [(\[gca2d\])]{} with $c_M=0$ and furthermore, restricting to a sector where $\xi=0$, i.e. a sector where all the $M_0$ eigenvalues of the states considered are vanishing, through an analysis of null vectors in the algebra [@Bagchi:2009pe], it can be shown there is a truncation of the BMS$_3$ down to its Virasoro sub-algebra.
A Holographic Interlude: Flatspace Chiral Gravity
-------------------------------------------------
In the context of holography, the phenomenon of truncation of the symmetry algebra in the field theory has been used to construct what is called Flatspace Chiral Gravity [@Bagchi:2012yk]. Topologically Massive Gravity (TMG) is a theory of gravity in 3 dimensions, which, in addition to the usual Einstein Hilbert term, has a gravitational Chern-Simons term. \[acttmg\] S\_[TMG]{} = S\_[EH]{} + S\_[GCS]{} = d\^3x . Here $\Lambda$ is the cosmological constant. When one analyses the asymptotic structure of TMG with asymptotically Minkowskian boundary conditions ($\Lambda=0$), the ASG turns out to be the BMS$_3$ algebra, now with both central charges turned on. They take values: c\_L\^[tmg]{} = , c\_M\^[tmg]{} = . Now if we look at a limit where $G\to \infty, \ \mu\to 0$ with $\mu G = \frac{1}{96k}$ held fixed, the central charges take the value: c\_L = 24 k, c\_M=0. In this limit, the gravitational C-S term in the TMG action [(\[acttmg\])]{} becomes important and the Einstein-Hilbert term is scaled away. This theory is called Chern-Simons Gravity. With asymptotically Minkowskian boundary conditions, this theory has an asymptotic algebra which is just a single copy of a Virasoro algebra with central charge $c=24k$. This theory has been named Flatspace Chiral Gravity (FCG). It can be checked through a gravitational analysis that for FCG, all the $M_n$ charges vanish identically [@Bagchi:2012yk].
Chiral limit and the BMS bootstrap
----------------------------------
In this subsection, we wish to see how our earlier analysis of the BMS bootstrap ties in with the chiral limit described above. To this end, we attempt to construct the coefficients of the BMS OPE where we have vanishing $\xi$ weights and $c_M=0$.
For this special limit, $c_M=0,\,\xi=0,\,\xi_p=0$, the recursion relations given by equation , , reduce to L\_[n]{}|N+n,\_p & = & (N+n -+n +\_p)|N,\_p \[recurVL\] M\_[n]{}|N+n,\_p=-(+1)|N,+1\_p \[recurVM\] M\_[0]{}|N,\_p = -(+1)|N,+1\_p. \[recurVM0\] Using the above equations, let us try to find the coefficients of the OPE upto level two descendants. For level zero we trivially have |N=0,=0=\_[12]{}\^[p{0,0},0]{}|\_[p]{},0=|\_[p]{},0.
#### Level 1:
States at level 1 are given by |1,\_p=\_[12]{}\^[p{1,0},]{}L\_[-1]{}|\_[p]{},0+\_[12]{}\^[p{0,1},]{}M\_[-1]{}|\_[p]{},0,=0,1. Let us note that for $\xi_p=0$ M\_0|1,\_p &=& \_[12]{}\^[p{1,0},]{} M\_[-1]{}|\_[p]{},\_[p]{},\
M\_1|1,\_p &=& 0,\
L\_1|1,\_p &=& 2\_p \_[12]{}\^[p{1,0},]{}|\_[p]{},\_[p]{}. Using we have && L\_[1]{}|1,0\_p = \_[p]{}|0,0\_p 2\_[p]{}\_[12]{}\^[p{1,0},0]{}|\_[p]{},0 = \_[p]{}|\_[p]{},0 && \_[12]{}\^[p{1,0},0]{}=. Using the recursion relation , we have M\_[0]{}|1,1\_p = 0 \_[12]{}\^[p{1,0},1]{}M\_[-1]{}|\_[p]{},0 = 0 \_[12]{}\^[p{1,0},1]{}=0. We see that these are exactly the coefficients that we had for a single Virasoro algebra in the first level as was seen in the table [(\[OPE\_CFT\])]{} in Sec 2. [[^3]]{}
#### Level 2:
We can continue this analysis to any arbitrary level. At level 2 we have |2,,\_[p]{},0 &=& \_[12]{}\^[p{1,1},]{}L\_[-1]{}M\_[-1]{}|\_[p]{},0+\_[12]{}\^[p{2,0},]{}L\_[-1]{}L\_[-1]{}|\_[p]{},0+\_[12]{}\^[p{(0,1),0},]{}L\_[-2]{}|\_[p]{},0 &&+\_[12]{}\^[p{0,2},]{}M\_[-1]{}M\_[-1]{}|\_[p]{},0+\_[12]{}\^[p{0,(0,1)},]{}M\_[-2]{}|\_[p]{},0,=0,1,2. We will focus on determining ${\beta}_{12}^{p\{2,0\},{\alpha}}$ and ${\beta}_{12}^{p\{(0,1),0\},{\alpha}}$. The detailed analysis is presented in an appendix. Here we mention the answers: \_[12]{}\^[p{2,0},0]{}= \_[12]{}\^[p{(0,1),0},0]{}=. These again are the answers we would have expected from a single Virasoro algebra with central charge $c_L$. This provides a cross-check of the chiral truncation of the BMS$_3$ algebra.
Concluding Remarks
==================
A summary of our results
------------------------
In this paper, we have built on our initial analysis of the BMS bootstrap in [@Bagchi:2016geg]. We have provided a lot of detailed calculations that was missing in [@Bagchi:2016geg] and also presented a significant amount of new material, the most significant of which is a comprehensive check of all the main results of the intrinsic analysis by a systematic limiting procedure.
We have first shown, that one could look at the highest weight representations of the BMS$_3$ algebra and, in a manner similar to closely following the conformal bootstrap approach, set up the bootstrap equations in BMS invariant 2d field theories. For this, the central idea was the construction of the BMS operator product expansion, which then allowed us to make statements about the correlation functions of the theory. We made an ansatz for the BMS OPE and showed that this was indeed a consistent choice to make. The OPE could be, e.g., used to check the form of the two and three point functions that were earlier determined from symmetry.
We then went on to construct recursion relations between primary states under the action of the various modes of the BMS algebra. These recursion relations then allowed us to fix the undetermined coefficients in the OPE completely. We showed the results up to the second level and in principle, this is an analysis that could be continued to any arbitrary order. Using the OPE, we then considered BMS four-point functions and constructed a notion of crossing symmetry for these field theories, which turned out to be different from the usual conformal crossing equations.
After this, we constructed the BMS blocks and using crossing symmetry, formulated the BMS bootstrap equation. These equations, when solved, will lead to all possible BMS-invariant field theories in two dimensions. The equations are obviously very difficult to solve. As an important step towards the solution of these equations, we looked at the large central charge limit of the BMS blocks. In this limit, the BMS blocks reduced to what we called the global blocks which were the ones containing descendants of only $L_{-1}$ and $M_{-1}$. We then used the Casimirs of BMS to construct two second order differential equations for the global blocks, which we could solve explicitly. As emphasised above, this is a rather important step in the programme of classifying all BMS invariant field theories with the help of the BMS bootstrap.
Here there is point that we should emphasise. We said that we are interested in BMS invariant field theories as they form putative duals of Minkowski spacetimes. When we consider Einstein gravity in the bulk, we have already state in the beginning that $c_L=0$. Does this mean that our analysis for global blocks would not be valid for Einstein gravity? The answer, interesting, is that it would be. For this, let us look back at Table [(\[level2\])]{}. We see that the coefficients that correspond to the “higher" descendants ($L_{-n} M_{-m} |{\Delta}, \xi{\rangle}_p$ for $n, m \ge 2$) of the BMS primaries are actually suppressed by $c_M$ alone. This can be checked for higher levels as well. So the global block actually requires only $c_M \to \infty$. Hence, this limit works for the theories putatively dual to Einstein gravity.
We went on to recover some of our answers, especially the coefficients of the BMS OPE through the non-relativistic limit of the Virasoro algebra. The fact that the answers obtained in the intrinsic and in the limiting method matched was a check of the correctness of both methods. At the end, we looked at a special case where the central charge $c_M=0$ and also all $\xi=0$. This is a limit where the BMS$_3$ algebra is known to reduce to a single copy of the Virasoro algebra. We found that the coefficients of the BMS OPE also reduce to that of a regular chiral CFT in this case.
Future directions
-----------------
There are several directions that are being currently pursued and others we hope to work on in the near future. Below we present a list of these.
: It is natural to try and generalise our analysis to symmetry algebras with supersymmetry. There exist supersymmetric versions of BMS$_3$ or equivalently, the GCA$_2$. In particular, it is of interest to consider what we call the homogeneous and inhomogeneous Super Galilean Conformal Algebras (SGCAs) [@Bagchi:2016yyf]. The homogeneous SGCA is given by \[sgcah\] && \[L\_n, L\_m\] = (n-m) L\_[n+m]{} + (n\^3 -n) \_[n+m,0]{}\
&& \[L\_n, M\_m\] = (n-m) M\_[n+m]{} + (n\^3 -n) \_[n+m,0]{}\
&& \[L\_n, Q\^\_r\] = ( - r) Q\^\_[n+r]{}, {Q\^\_r, Q\^\_s } = \^ . In the above, we have only written the non-zero commutation relations. The above algebra can be obtained by a contraction of the 2D $\mathcal{N}=(1,1)$ superconformal algebra where the fermionic generators are scaled in a similar fashion. This algebra (stripped of the ${\alpha}, {\beta}$ indices) has also been obtained as the asymptotic symmetries of 3D $\mathcal{N}=1$ supergravity [@Barnich:2014cwa; @Barnich:2015sca]. Another version of the Super GCA is the inhomogeneous one: \[sgcai\] && \[L\_n, L\_m\] = (n-m) L\_[n+m]{} + (n\^3 -n) \_[n+m,0]{},\
&& \[L\_n, M\_m\] = (n-m) M\_[n+m]{} + (n\^3 -n) \_[n+m,0]{},\
&& \[L\_n, G\_r\] = ( -r) G\_[n+r]{}, \[L\_n, H\_r\] = ( -r) H\_[n+r]{}, \[M\_n, G\_r\] = ( -r) H\_[n+r]{},\
&& { G\_r, G\_s } = 2 L\_[r+s]{} + (r\^2 - ) \_[r+s,0]{}, { G\_r, H\_s } = 2 M\_[r+s]{} + (r\^2 - ) \_[r+s,0]{}. Here again the zero commutators are suppressed. This inhomogeneous SGCA can be obtained from the 2D $\mathcal{N}=(1,1)$ superconformal algebra by a different contraction, a contraction which the fermionic generators are scaled in very different ways [@Bagchi:2016yyf]. In the context of supergravity, this leads to a an exotic twisted SUGRA theory [@Lodato:2016alv].
We are at present attempting to construct the bootstrap programme for both these algebras. One of the crucial steps, as in the bosonic case, is the construction of the OPE for the supersymmetric algebra. It is expected that the analysis would generalise to the supersymmetric case in a natural way.
: Liouville theory is a very important example of a 2d CFT that admits a semi-classical limit. This semi-classical limit also connects Liouville theory with AdS$_3$ gravity and $SL(2,R)$ Chern-Simons theory [@Achucarro:1987vz; @Witten:1988hc; @Coussaert:1995zp]. The three point function for general momenta in Liouville theory has been computed and goes under the name of the DOZZ formula [@Dorn:1994xn; @Zamolodchikov:1995aa]. Closed form expression for the structure constants are thus known and it has been explicitly verified that the theory satisfies the conformal bootstrap equation [@Zamolodchikov:1995aa; @Teschner:2001rv]. A lot of the progress on Virasoro blocks in general 2d CFTs hinges on the success of computations in Liouville theory. In the recent emergence of techniques of 2d CFTs for large central charge following [@Hartman:2013mia], Liouville theory has played a central role.
In [@Barnich:2012rz], a contracted version of Liouville theory has been proposed by taking a systematic limit of the parent theory. The Poisson algebra of the conserved charges of this theory turns out to be the BMS algebra [(\[gca2d\])]{}. There are actually two versions of this limiting theory, one with a vanishing $c_M$ and one with $c_M$ non-zero. It is expected that the semi-classical version of the theory with $c_M \neq 0$ would be important for understanding the dual of Einstein gravity in 3d flat spacetimes.
We wish to understand the structure of this theory in detail so that we can compute the equivalent of the DOZZ formula for the three point functions and hence find an explicit example where the BMS bootstrap equations are satisfied. For our explorations of flat holography, this is a vital step. We wish to address questions about BMS blocks and also generalise the recent large $c$ CFT techniques to the BMS case. The BMS Liouville theory would provide us valuable insight into these problems.
: A very natural direction of generalisation of our analysis is to explore a higher dimensional version of our bootstrap analysis. We have made some remarks about this in the introduction. Let us briefly elaborate on some aspects and some possible difficulties. First thing to mention here is that the BMS algebra and the GCA are not isomorphic in higher dimensions, as can be readily seen by looking at equations [(\[bms4\])]{} and [(\[GCA\])]{}. So higher dimensional generalisations of the bootstrap would be different for the two cases.
The structure of BMS algebras in different dimensions is very different. This should be obvious by looking at the 4d case [(\[bms4\])]{} and the 3d case [(\[gca2d\])]{}, which we have addressed in this paper. The systematics of the bootstrap procedure, which depends crucially on the structure of the algebra, would thus be very different and our methods in this paper would not generalise in any natural way for the cases of field theories in 3 and higher dimensions with BMS symmetry. There are indications [@Bagchi:2016bcd] that 3d field theories with BMS$_4$ symmetry actually reduce to 2d CFTs[^4]. So, it is possible that the usual 2d Virasoro bootstrap would be applicable for these field theories. We don’t have any concrete suggestions for field theories with BMS symmetries in even higher dimensions ($D>4$).
The case for GCFTs and the Galilean Conformal Bootstrap in higher dimension is much more encouraging from the point of view of our present analysis. The structure of Galilean conformal symmetry remains very similar as we go up in dimensions. This is evident from [(\[gca2d\])]{} and [(\[GCA\])]{}[[^5]]{}. It is hence expected that constructions similar to what we have attempted in this paper would also work for higher dimensional GCFTs. There would be interesting departures as well. There is no central charge in the $[L_n, M_m^i]$ commutator and hence the semi-classical limit should work differently.
The very interesting thing in this analysis would be the fact that unlike conformal bootstrap in dimensions higher 2, the Galilean conformal bootstrap would benefit from the infinite dimensional symmetry algebra in all dimensions. This would means we would have much more analytical control over our analysis, as compared to the bootstrap programme in the higher dimensional CFT, which has been driven primary with numerical methods. We should be in a very good position to classify non-relativistic quantum field theories by the virtue of this analysis.
: Recently, it has been shown that one can combine Polykov’s original idea about the bootstrap exploiting manifest crossing symmetry with the Mellin representations of CFT amplitudes to get a much better analytical handle on the conformal bootstrap programme for higher dimensions [@Gopakumar:2016wkt; @Gopakumar:2016cpb]. It is very tempting to attempt a similar algorithm for the BMS bootstrap. Here we would need modifications of the Mellin space amplitudes, for which the systematic limit from CFT should be very useful.
There are indeed many other interesting directions to pursue over and above the ones just mentioned. To conclude, we believe the BMS programme that we have initiated in [@Bagchi:2016geg] and elaborated on in this paper is a programme which would be very useful in many diverse fields.
Acknowledgements {#acknowledgements .unnumbered}
----------------
This paper is an invited article for the [*[Classical and Quantum Gravity]{}*]{} Focus issue on “BMS Asymptotic Symmetries" edited by Geoffrey Compere. AB thanks Geoffrey for the invitation and for the initiative for editing this special issue.
It is a pleasure to thank Ivano Lodato, Wout Merbis, Hernan Gonzalez, Glenn Barnich, P. Raman and N. V. Suryanarayana for discussions. AB also wishes to thank the Simons Center for Geometry and Physics for warm hospitality and support during the time of this project. AB is partially supported by an INSPIRE grant from DST, India and by a Max-Planck mobility grant. MG is supported by the FWF project P27396-N27 and the OeAD project IN 03/2015. Z is supported by the India-Israel joint research project UGC/PHY/2014236 and SERB National Post Doctoral Fellowship PDF/2016/002166.
Appendices {#appendices .unnumbered}
==========
Level 2 coefficients: Detailed calculations in intrinsic method
===============================================================
In this appendix, we provide further detailed calculations of the analysis outline in Sec 3.4 for finding the coefficients of the OPE for the BMS$_3$ algebra. Below are the details for the level 2 calculations.
At level 2 we have |2,\_p & = & \_[12]{}\^[p{1,1},]{}L\_[-1]{}M\_[-1]{}|\_[p]{},\_[p]{}+\_[12]{}\^[p{2,0},]{}L\_[-1]{}L\_[-1]{}|\_[p]{},\_[p]{}+\_[12]{}\^[p{(0,1),0},]{}L\_[-2]{}|\_[p]{},\_[p]{} & & +\_[12]{}\^[p{0,2},]{}M\_[-1]{}M\_[-1]{}|\_[p]{},\_[p]{}+\_[12]{}\^[p{0,(0,1)},]{}M\_[-2]{}|\_[p]{},\_[p]{},=0,1,2. Let us note again that &&M\_0L\_[-1]{}M\_[-1]{}|\_[p]{},\_[p]{}=\_p L\_[-1]{}M\_[-1]{}|\_[p]{},\_[p]{} + M\_[-1]{}M\_[-1]{}|\_[p]{},\_[p]{},&&M\_0L\_[-1]{}L\_[-1]{}|\_[p]{},\_[p]{}= 2 L\_[-1]{}M\_[-1]{}|\_[p]{},\_[p]{} + \_p L\_[-1]{}L\_[-1]{}|\_[p]{},\_[p]{},&&M\_0L\_[-2]{}|\_[p]{},\_[p]{}=\_p L\_[-2]{}|\_[p]{},\_[p]{} + 2M\_[-2]{}|\_[p]{},\_[p]{},&&M\_0M\_[-1]{}M\_[-1]{}|\_[p]{},\_[p]{}=\_p M\_[-1]{}M\_[-1]{}|\_[p]{},\_[p]{},M\_0M\_[-2]{}|\_[p]{},\_[p]{}=\_p M\_[-2]{}|\_[p]{},\_[p]{}&& &&M\_1L\_[-1]{}M\_[-1]{}|\_[p]{},\_[p]{}=2\_p M\_[-1]{}|\_[p]{},\_[p]{},M\_1L\_[-1]{}L\_[-1]{}|\_[p]{},\_[p]{}=4\_p L\_[-1]{}|\_[p]{},\_[p]{}+2 M\_[-1]{}|\_[p]{},\_[p]{},&&M\_1L\_[-2]{}|\_[p]{},\_[p]{}=3 M\_[-1]{}|\_[p]{},\_[p]{},M\_1M\_[-1]{}M\_[-1]{}|\_[p]{},\_[p]{}=0,M\_1M\_[-2]{}|\_[p]{},\_[p]{}=0&& &&M\_2L\_[-1]{}M\_[-1]{}|\_[p]{},\_[p]{}=0,M\_2L\_[-1]{}L\_[-1]{}|\_[p]{},\_[p]{}=6\_p|\_[p]{},\_[p]{} ,&&M\_2L\_[-2]{}|\_[p]{},\_[p]{}=(4\_p+c\_M/2) |\_[p]{},\_[p]{},M\_2M\_[-1]{}M\_[-1]{}|\_[p]{},\_[p]{}=0,M\_2M\_[-2]{}|\_[p]{},\_[p]{}=0&& &&L\_1L\_[-1]{}M\_[-1]{}|\_[p]{},\_[p]{}=2\_p L\_[-1]{}|\_[p]{},\_[p]{}+2(\_p+1)M\_[-1]{}|\_[p]{},\_[p]{},&&L\_1L\_[-1]{}L\_[-1]{}|\_[p]{},\_[p]{}= 2(2\_p+1)L\_[-1]{}|\_[p]{},\_[p]{},L\_1L\_[-2]{}|\_[p]{},\_[p]{}=3L\_[-1]{}|\_[p]{},\_[p]{},&&L\_1M\_[-1]{}M\_[-1]{}|\_[p]{},\_[p]{}=4\_p M\_[-1]{} |\_[p]{},\_[p]{},L\_1M\_[-2]{}|\_[p]{},\_[p]{}=3 M\_[-1]{}|\_[p]{},\_[p]{}&& &&L\_2L\_[-1]{}M\_[-1]{}|\_[p]{},\_[p]{}=6\_p|\_[p]{},\_[p]{},L\_2L\_[-1]{}L\_[-1]{}|\_[p]{},\_[p]{}=6\_p |\_[p]{},\_[p]{} ,&&L\_2L\_[-2]{}|\_[p]{},\_[p]{}=(4\_p + 6 c\_L) |\_[p]{},\_[p]{},L\_2M\_[-1]{}M\_[-1]{}|\_[p]{},\_[p]{}=0, && L\_2M\_[-2]{}|\_[p]{},\_[p]{}=(4\_p + c\_M/2) |\_[p]{},\_[p]{} M\_0|2,\_p & = & (\_p\_[12]{}\^[p{1,1},]{}+2\_[12]{}\^[p{2,0},]{})L\_[-1]{}M\_[-1]{}|\_[p]{},\_[p]{}+\_p\_[12]{}\^[p{2,0},]{}L\_[-1]{}L\_[-1]{}|\_[p]{},\_[p]{}&&+\_p\_[12]{}\^[p{(0,1),0},]{}L\_[-2]{}|\_[p]{},\_[p]{}+(\_[12]{}\^[p{1,1},]{}+\_p\_[12]{}\^[p{0,2},]{})M\_[-1]{}M\_[-1]{}|\_[p]{},\_[p]{}&&+(2\_[12]{}\^[p{(0,1),0},]{}+\_p\_[12]{}\^[p{0,(0,1)},]{})M\_[-2]{}|\_[p]{},\_[p]{},\
M\_1|2,\_p &=& (2\_p\_[12]{}\^[p{1,1},]{}+2\_[12]{}\^[p{2,0},]{}+3\_[12]{}\^[p{(0,1),0},]{})M\_[-1]{}|\_[p]{},\_[p]{}&&+4\_p \_[12]{}\^[p{2,0},]{} L\_[-1]{}|\_[p]{},\_[p]{},\
M\_2 |2,\_p &=& (6\_p \_[12]{}\^[p{2,0},]{} +(4\_p+)\_[12]{}\^[p{(0,1),0},]{})|\_[p]{},\_[p]{},\
L\_1|2,\_p &=& (2(\_p+1)\_[12]{}\^[p{1,1},]{}+4\_p\_[12]{}\^[p{0,2},]{}+3\_[12]{}\^[p{0,(0,1)},]{})M\_[-1]{}|\_[p]{},\_[p]{}&&+(2\_p \_[12]{}\^[p{1,1},]{}+2(2\_p+1)\_[12]{}\^[p{2,0},]{}+3\_[12]{}\^[p{(0,1),0},]{}) L\_[-1]{}|\_[p]{},\_[p]{},\
L\_2 |2,\_p &=& (6\_p \_[12]{}\^[p{1,1},]{}+6\_p \_[12]{}\^[p{2,0},]{}+(4\_p+) \_[12]{}\^[p{(0,1),0},]{} . &&+. (4\_p+)\_[12]{}\^[p{0,(0,1)},]{})|\_[p]{},\_[p]{}. Using the recursion relation on the state $|2,2{\rangle}_p$, we get &&M\_[0]{}|2,2\_p = \_[p]{}|2,2\_p&&(2\_[12]{}\^[p{2,0},2]{}L\_[-1]{}M\_[-1]{}+\_[12]{}\^[p{1,1},2]{}M\_[-1]{}M\_[-1]{}+2\_[12]{}\^[p{(0,1),0},2]{}M\_[-2]{})|\_[p]{},\_[p]{} = 0,&& \_[12]{}\^[p{1,1},2]{}=\_[12]{}\^[p{2,0},2]{}=\_[12]{}\^[p{(0,1),0},2]{}=0. Using the above result and with $N=0,n=2,{\alpha}=2$, we have L\_[2]{}|2,2\_p = 0 (4\_p+)\_[12]{}\^[p{0,(0,1)},2]{}|\_[p]{},\_[p]{} = 0 \_[12]{}\^[p{0,(0,1)},2]{}=0. Using with $N=1,n=1,{\alpha}=2$, we have && L\_[1]{}|2,2\_p = -\_[p]{}|1,1\_p&& 4\_p\_[12]{}\^[p{0,2},2]{}M\_[-1]{}|\_[p]{},\_[p]{} = M\_[-1]{}|\_[p]{},\_[p]{}&& \_[12]{}\^[p{0,2},2]{}=. Next, we use on the state $|2,1{\rangle}_p$, giving us && M\_[0]{}|2,1\_p = \_[p]{}|2,1\_p-2|2,2\_p&& (2\_[12]{}\^[p{2,0},1]{}L\_[-1]{}M\_[-1]{} +\_[12]{}\^[p{1,1},1]{}M\_[-1]{}M\_[-1]{}+2\_[12]{}\^[p{(0,1),0},1]{}M\_[-2]{})|\_[p]{},\_[p]{} && = -2\_[12]{}\^[p{0,2},2]{}M\_[-1]{}M\_[-1]{}|\_[p]{},\_[p]{} && \_[12]{}\^[p{1,1},1]{}=-2\_[12]{}\^[p{0,2},2]{}=-,\_[12]{}\^[p{2,0},1]{}=0,\_[12]{}\^[p{(0,1),0},1]{}=0. Now, we use with $N=0,n=2,{\alpha}=1$ && L\_[2]{}|2,1\_p = (-2-2\_[p]{})|0,0\_p&& (-6\_[p]{}+\_[12]{}\^[p{0,(0,1)},1]{}(4\_[p]{}+))|\_[p]{},\_[p]{} = (-2-2\_[p]{})|\_[p]{},\_[p]{}&& \_[12]{}\^[p{0,(0,1)},1]{}=-=-. Using the recursion relation with $N=1,n=1,{\alpha}=1$, we have &&L\_[1]{}|2,1\_p = (2+\_[p]{})|1,1\_p-\_[p]{}|1,0\_p&&(-L\_[-1]{}+(- + 4\_p\_[12]{}\^[p{0,2},1]{}+3\_[12]{}\^[p{0,(0,1)},1]{})M\_[-1]{})|\_[p]{},\_[p]{} &&= (-M\_[-1]{}-L\_[-1]{})|\_[p]{},\_[p]{}&& \_[12]{}\^[p{0,2},1]{}=--\_[12]{}\^[p{0,(0,1)},1]{}=. Now, we use on the state $|2,0{\rangle}_p $, giving us && M\_[0]{}|2,0\_p = \_[p]{}|2,0\_p-|2,1\_p&& (\_[12]{}\^[p{2,0},0]{}2L\_[-1]{}M\_[-1]{} +\_[12]{}\^[p{1,1},0]{}M\_[-1]{}M\_[-1]{} + 2\_[12]{}\^[p{(0,1),0},0]{}M\_[-2]{})|\_[p]{},\_[p]{} & &= -(\_[12]{}\^[p{1,1},1]{}L\_[-1]{}M\_[-1]{}+\_[12]{}\^[p{0,2},1]{}M\_[-1]{}M\_[-1]{}+\_[12]{}\^[p{0,(0,1)},1]{}M\_[-2]{})|\_[p]{},\_[p]{}&&\_[12]{}\^[p{1,1},0]{}=-\_[12]{}\^[p{0,2},1]{}=-,\_[12]{}\^[p{2,0},0]{}=-=,&&\_[12]{}\^[p{(0,1),0},0]{}=-=. Next, we use with $n=2,N=0,{\alpha}=0$ &&L\_[2]{}|2,0\_p=(+\_[p]{})|0,0\_p&& (6\_[12]{}\^[p{1,1},0]{}\_[p]{}+6\_[12]{}\^[p{2,0},0]{}\_[p]{}+\_[12]{}\^[p{(0,1),0},0]{}(4\_[p]{}+)+\_[12]{}\^[p{0,(0,1)},0]{}(4\_[p]{}+))|\_[p]{},\_[p]{}&&=(+\_[p]{})|\_[p]{},\_[p]{} && \_[12]{}\^[p{0,(0,1)},0]{} = &&=.&& Furthermore with $n=1,N=1,{\alpha}=0$ give us the recursion relation
&& L\_[1]{}|2,0\_p = (1+\_[p]{})|1,0\_p&& (2(\_p+1)\_[12]{}\^[p{1,1},0]{}+4\_p\_[12]{}\^[p{0,2},0]{}+3\_[12]{}\^[p{0,(0,1)},0]{})M\_[-1]{}|\_[p]{},\_[p]{}&&+(2\_p \_[12]{}\^[p{1,1},0]{}+2(2\_p+1)\_[12]{}\^[p{2,0},0]{}+3\_[12]{}\^[p{(0,1),0},0]{}) L\_[-1]{}|\_[p]{},\_[p]{}&&=L\_[-1]{}|\_[p]{},\_[p]{} Equating coefficient of $M_{-1}|{\Delta}_{p},\xi_{p}{\rangle}$ in the above equation, we get \_[12]{}\^[p{0,2},0]{} & = & -(2\_[12]{}\^[p{1,1},0]{}(\_[p]{}+1)+3\_[12]{}\^[p{0,(0,1)},0]{})&=&(-36 c\_M\^2 (1+\_p)+24 c\_M (3 +\_p (3 -2 \_p)+(1-3 ) \_p).&&. +\_p (-60 +\_p (96 -4 \_p)+5 \_p-48 \_p+18 c\_L (4 +\_p))). The coefficients are collected in Table 2 in Sec 3.4.
It is clear from the analysis above that, given the recursion relations, we can solve for the $\beta$s for any level. Computational power required obviously increases substantially as we attempt to go higher, but there is no theoretical difficulty in obtaining these coefficients.
Level 2 coefficients: Detailed calculations in limiting method
==============================================================
Using the relation c=(c\_L-),|[c]{}=(c\_L+),h=(-),[[|h]{}]{}=(+), \[hw\] the coefficients of the level two descendant fields given in Table are \_[12]{}\^[p{1,1}]{} & = & =++\^2+(\^[3]{}),\
\_[12]{}\^[p{2}]{} & = & =+ +(\^[2]{}),\
|\_[12]{}\^[p{|[1]{},|[1]{}}]{} & = & =-+\^[2]{}+(\^[3]{}),\
|\_[12]{}\^[p{|[2]{}}]{} & = & =-+(\^[2]{}), where &=&=\_[12]{}\^[p{0,2},1]{},\
&=&(-36 c\_M\^2 (1+\_p)+24 c\_M (3 +\_p (3 -2 \_p)+(1-3 ) \_p)&&+\_p (-60 +\_p (96 -4 \_p)+5 \_p-48 \_p+18 c\_L (4 +\_p)))&=& 2\_[12]{}\^[p{0,2},0]{},\
&=&=-,\
&=&&=&-\_[12]{}\^[p{0,(0,1)},0]{}.
Now collecting all the level two states in the expansion , modulo the common factor $\frak{C}_{p12}t^{{\Delta}-2{\Delta}_{p}}\exp\left(2\xi-\xi_{p})\right)$, we get & &(\_[12]{}\^[p{2}]{}t\^[2]{}(1+)\^[2]{}Ł\_[-2]{}+\_[12]{}\^[p{1,1}]{}t\^[2]{}(1+)\^[2]{}Ł\_[-1]{}Ł\_[-1]{}. && . +\_[12]{}\^[p{1}]{}|\_[12]{}\^[p{|[1]{}}]{}t\^[2]{}(1+)(1-)Ł\_[-1]{}\_[-1]{} . & & . +|\_[12]{}\^[p{|[2]{}}]{}t\^[2]{}(1-)\^[2]{}\_[-2]{}+|\_[12]{}\^[p{|[1]{},|[1]{}}]{}t\^[2]{}(1-)\^[2]{}\_[-1]{}\_[-1]{})|\_[p]{},\_[p]{} & = & (\_[12]{}\^[p{2}]{}t\^[2]{}(1+)\^[2]{}(L\_[-2]{}-M\_[-2]{})+\_[12]{}\^[p{1,1}]{}t\^[2]{}(1+)\^[2]{}(L\_[-1]{}-M\_[-1]{})(L\_[-1]{}-M\_[-1]{}) . & & . +\_[12]{}\^[p{1}]{}|\_[12]{}\^[p{|[1]{}}]{}t\^[2]{}(1+)(1-)(L\_[-1]{}-M\_[-1]{})(L\_[-1]{}+M\_[-1]{}) . & & . +|\_[12]{}\^[p{|[2]{}}]{}t\^[2]{}(1-)\^[2]{}(L\_[-2]{}+M\_[-2]{})+|\_[12]{}\^[p{|[1]{},|[1]{}}]{}t\^[2]{}(1-)\^[2]{}(L\_[-1]{} . &&. +M\_[-1]{})(L\_[-1]{}+M\_[-1]{}) )|\_[p]{},\_[p]{}. \[level2expnasion\] Then the non-relativistic limit of the coefficient of $x^2$ in the above state is given by |2,2\_p &=&\_[0]{}((\^[2]{}L\_[-2]{}-M\_[-2]{})+(\^[2]{}L\_[-1]{}L\_[-1]{}-2L\_[-1]{}M\_[-1]{}+M\_[-1]{}M\_[-1]{}) . & & . -(\^[2]{}L\_[-1]{}L\_[-1]{}-M\_[-1]{}M\_[-1]{}) -(\^[2]{}L\_[-2]{}+M\_[-2]{}) . & &. +(\^[2]{}L\_[-1]{}L\_[-1]{}+2L\_[-1]{}M\_[-1]{}+M\_[-1]{}M\_[-1]{}))|\_p,\_p &=&M\_[-1]{}M\_[-1]{}|\_p,\_p=\_[12]{}\^[p{0,2},2]{}M\_[-1]{}M\_[-1]{}|\_p,\_p. Similarly, the coefficients of $xy$ in gives the state |2,1\_p &=& \_[0]{}(\_[12]{}\^[p{2}]{}(L\_[-2]{}-M\_[-2]{})+(L\_[-1]{}L\_[-1]{}-2L\_[-1]{}M\_[-1]{}+M\_[-1]{}M\_[-1]{}). &&. -|\_[12]{}\^[p{|[2]{}}]{}(L\_[-2]{}+M\_[-2]{})-(L\_[-1]{}L\_[-1]{}+2L\_[-1]{}M\_[-1]{}+M\_[-1]{}M\_[-1]{}))|\_[p]{},\_[p]{}&=&(-L\_[-1]{}M\_[-1]{}-2M\_[-2]{}+M\_[-1]{}M\_[-1]{})|\_[p]{},\_[p]{}&=&(\_[12]{}\^[p{1,1},1]{}L\_[-1]{}M\_[-1]{}+\_[12]{}\^[p{0,(0,1)},1]{}M\_[-2]{}+\_[12]{}\^[p{0,2},1]{}M\_[-1]{}M\_[-1]{})|\_[p]{},\_[p]{}. Lastly, the non-relativistic limit of the coefficients of $t^2$ gives us the state |2,0\_p &=& \_[0]{}((L\_[-2]{}-M\_[-2]{})+(L\_[-1]{}L\_[-1]{}-L\_[-1]{}M\_[-1]{}+M\_[-1]{}M\_[-1]{}) . &&. +(L\_[-1]{}L\_[-1]{}-M\_[-1]{}M\_[-1]{})+(L\_[-2]{}+M\_[-2]{}) . && . +(L\_[-1]{}L\_[-1]{}+L\_[-1]{}M\_[-1]{}+M\_[-1]{}M\_[-1]{}))|\_[p]{},\_[p]{} &=&(L\_[-2]{}-M\_[-2]{}+L\_[-1]{}L\_[-1]{}-L\_[-1]{}M\_[-1]{}+M\_[-1]{}M\_[-1]{})|\_[p]{},\_[p]{}&=&(\_[12]{}\^[p{(0,1),0},0]{} L\_[-2]{}+\_[12]{}\^[p{0,(0,1)},0]{} M\_[-2]{}+\_[12]{}\^[p{2,0},0]{}L\_[-1]{}L\_[-1]{}+\_[12]{}\^[p{1,1},0]{}L\_[-1]{}M\_[-1]{} . &&. +\_[12]{}\^[p{0,2},0]{}M\_[-1]{}M\_[-1]{})|\_[p]{},\_[p]{}. All of these states match with our calculations in the previous section.
Level 2 analysis of coefficients in the Chiral limit
====================================================
|2,,\_[p]{},0 &=& \_[12]{}\^[p{1,1},]{}L\_[-1]{}M\_[-1]{}|\_[p]{},0+\_[12]{}\^[p{2,0},]{}L\_[-1]{}L\_[-1]{}|\_[p]{},0+\_[12]{}\^[p{(0,1),0},]{}L\_[-2]{}|\_[p]{},0 &&+\_[12]{}\^[p{0,2},]{}M\_[-1]{}M\_[-1]{}|\_[p]{},0+\_[12]{}\^[p{0,(0,1)},]{}M\_[-2]{}|\_[p]{},0,=0,1,2. Note that for $\xi_p=0,\,c_M=0$. M\_0|2,\_p & = & 2\_[12]{}\^[p{2,0},]{}L\_[-1]{}M\_[-1]{}|\_[p]{},0 + \_[12]{}\^[p{1,1},]{}M\_[-1]{}M\_[-1]{}|\_[p]{},0&&+2\_[12]{}\^[p{(0,1),0},]{}M\_[-2]{}|\_[p]{},0,\
M\_1|2,\_p &=& (2\_[12]{}\^[p{2,0},]{}+3\_[12]{}\^[p{(0,1),0},]{})M\_[-1]{}|\_[p]{},0\
M\_2 |2,\_p &=& 0,\
L\_1|2,\_p &=& (2(\_p+1)\_[12]{}\^[p{1,1},]{}+3\_[12]{}\^[p{0,(0,1)},]{})M\_[-1]{}|\_[p]{},0&&+(2(2\_p+1)\_[12]{}\^[p{2,0},]{}+3\_[12]{}\^[p{(0,1),0},]{}) L\_[-1]{}|\_[p]{},0,\
L\_2 |2,\_p &=& (6\_p \_[12]{}\^[p{2,0},]{}+(4\_p+ ) \_[12]{}\^[p{(0,1),0},]{})|\_[p]{},0. Using the recursion relation , we have && M\_[0]{}|2,2\_p = 0 && 2\_[12]{}\^[p{2,0},]{}L\_[-1]{}M\_[-1]{}|\_[p]{},0 + \_[12]{}\^[p{1,1},]{}M\_[-1]{}M\_[-1]{}|\_[p]{},0 = 0 && \_[12]{}\^[p{1,1},2]{}=\_[12]{}\^[p{2,0},2]{}=\_[12]{}\^[p{(0,1),0},2]{}=0. Now using we have && L\_[1]{}|2,2\_p=0 \_[12]{}\^[p{0,(0,1)},2]{} = 0. We also have && M\_[0]{}|2,1\_p = -2|2,2\_p && (2\_[12]{}\^[p{2,0},1]{}L\_[-1]{}M\_[-1]{} + \_[12]{}\^[p{1,1},1]{}M\_[-1]{}M\_[-1]{}+2\_[12]{}\^[p{(0,1),0},]{}M\_[-2]{})|\_[p]{},0&& = -2\_[12]{}\^[p{0,2},2]{}M\_[-1]{}M\_[-1]{}|\_[p]{},,&& \_[12]{}\^[p{1,1},1]{}=-2\_[12]{}\^[p{0,2},2]{},\_[12]{}\^[p{2,0},1]{}=0,\_[12]{}\^[p{(0,1),0},1]{}=0, && L\_[1]{}|2,1\_p = (4+\_[p]{})|1,1\_p && 2(1+\_[p]{})\_[12]{}\^[p{1,1},1]{}+3\_[12]{}\^[p{0,(0,1)},1]{} = (4+\_[p]{}) . Furthermore && M\_[0]{}|2,0\_p = -|2,1\_p && (\_[12]{}\^[p{1,1},0]{}M\_[-1]{}M\_[-1]{}+\_[12]{}\^[p{2,0},0]{}2L\_[-1]{}M\_[-1]{}+2\_[12]{}\^[p{(0,1),0},0]{}M\_[-2]{})|\_[p]{},0 &&= -(\_[12]{}\^[p{1,1},1]{}L\_[-1]{}M\_[-1]{}+\_[12]{}\^[p{0,2},1]{}M\_[-1]{}M\_[-1]{}+\_[12]{}\^[p{0,(0,1)},1]{}M\_[-2]{})|\_[p]{},0 &&\_[12]{}\^[p{1,1},0]{}=-\_[12]{}\^[p{0,2},1]{},2\_[12]{}\^[p{2,0},0]{}=-\_[12]{}\^[p{1,1},1]{},2\_[12]{}\^[p{(0,1),0},0]{}=-\_[12]{}\^[p{0,(0,1)},1]{}.&& Using we have && L\_[1]{}|2,0\_p = (1+\_[p]{})|1,0\_p &&((2(\_p+1)\_[12]{}\^[p{1,1},0]{}+3\_[12]{}\^[p{0,(0,1)},0]{})M\_[-1]{} + (2(2\_p+1)\_[12]{}\^[p{2,0},0]{}.. &&.. +3\_[12]{}\^[p{(0,1),0},0]{}) L\_[-1]{}))|\_[p]{},0= L\_[-1]{}|\_[p]{},0 which gives 2(1+2\_[p]{})\_[12]{}\^[p{2,0},0]{}+3\_[12]{}\^[p{(0,1),0},0]{}=(1+\_[p]{}), \[vir1\] 2(1+\_[p]{})\_[12]{}\^[p{1,1},0]{}+3\_[12]{}\^[p{0,(0,1)},0]{}=0. \[vir2\] We also have && L\_[2]{}|2,0\_p = (+\_[p]{})|0,0\_p && (6\_[p]{}\_[12]{}\^[p{2,0},0]{}+\_[12]{}\^[p{(0,1),0},0]{}(4\_[p]{}+c\_L))|\_[p]{}, 0 = (+\_[p]{})|\_[p]{},0&&6\_[p]{}\_[12]{}\^[p{2,0},0]{}+\_[12]{}\^[p{(0,1),0},0]{}(4\_[p]{}+c\_L)=(+\_[p]{}). \[vir3\] Solving and we have \_[12]{}\^[p{2,0},0]{}= \_[12]{}\^[p{(0,1),0},0]{}=. We also have \_[12]{}\^[p{1,1},1]{}=-\_[12]{}\^[p{2,0},0]{}=-, \_[12]{}\^[p{0,(0,1)},1]{}=-\_[12]{}\^[p{(0,1),0},0]{}=-, \_[12]{}\^[p{0,2},2]{}=-\_[12]{}\^[p{1,1},1]{}=, and the coefficients ${\beta}_{12}^{p\{0,2\},1},\,{\beta}_{12}^{p\{1,1\},0},\,{\beta}_{12}^{p\{0,2\},0},\,{\beta}_{12}^{p\{0,(0,1)\},0}$ are undetermined. All the coefficients other than ${\beta}_{12}^{p\{2,0\},0}$ and ${\beta}_{12}^{p\{(0,1),0\},0}$ are coefficients arising from null states and their descendants. In the chiral truncation of the BMS algebra, all of these should just be ignored. It is true that in an ideal situation, we should have found that either these were zero or undetermined in the limit. We don’t understand this aspect of our results completely.
[999]{}
S. Ferrara, A. F. Grillo and R. Gatto, “Tensor representations of conformal algebra and conformally covariant operator product expansion,” Annals Phys. [**76**]{}, 161 (1973). doi:10.1016/0003-4916(73)90446-6 A. M. Polyakov, “Nonhamiltonian approach to conformal quantum field theory,” Zh. Eksp. Teor. Fiz. [**66**]{}, 23 (1974). A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, “Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory,” Nucl. Phys. B [**241**]{}, 333 (1984). doi:10.1016/0550-3213(84)90052-X R. Rattazzi, V. S. Rychkov, E. Tonni and A. Vichi, “Bounding scalar operator dimensions in 4D CFT,” JHEP [**0812**]{}, 031 (2008) doi:10.1088/1126-6708/2008/12/031 \[arXiv:0807.0004 \[hep-th\]\]. F. A. Dolan and H. Osborn, “Conformal four point functions and the operator product expansion,” Nucl. Phys. B [**599**]{}, 459 (2001) doi:10.1016/S0550-3213(01)00013-X \[hep-th/0011040\]. F. A. Dolan and H. Osborn, “Conformal partial waves and the operator product expansion,” Nucl. Phys. B [**678**]{}, 491 (2004) doi:10.1016/j.nuclphysb.2003.11.016 \[hep-th/0011040\].
S. Rychkov, “EPFL Lectures on Conformal Field Theory in D$>=$ 3 Dimensions,” doi:10.1007/978-3-319-43626-5 arXiv:1601.05000 \[hep-th\]. D. Simmons-Duffin, “TASI Lectures on the Conformal Bootstrap,” arXiv:1602.07982 \[hep-th\]. D. Poland and D. Simmons-Duffin, “The conformal bootstrap,” Nature Phys. [**12**]{}, no. 6, 535 (2016). doi:10.1038/nphys3761 A. Bagchi, M. Gary and Zodinmawia, “The BMS Bootstrap,” arXiv:1612.01730 \[hep-th\]. A. Bagchi and R. Gopakumar, “Galilean Conformal Algebras and AdS/CFT,” JHEP [**0907**]{}, 037 (2009) \[arXiv:0902.1385 \[hep-th\]\].
A. Bagchi, R. Gopakumar, I. Mandal and A. Miwa, “GCA in 2d,” JHEP [**1008**]{}, 004 (2010) \[arXiv:0912.1090 \[hep-th\]\]. H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, “Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems,” Proc. Roy. Soc. Lond. A [**269**]{}, 21 (1962). R. Sachs, “Asymptotic symmetries in gravitational theory,” Phys. Rev. [**128**]{}, 2851 (1962). G. Barnich and G. Compere, “Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions,” Class. Quant. Grav. [**24**]{}, F15 (2007) \[gr-qc/0610130\].
A. Bagchi, “The BMS/GCA correspondence,” Phys. Rev. Lett. [**105**]{}, 171601 (2010) \[arXiv:1006.3354 \[hep-th\]\].
A. Bagchi, “Tensionless Strings and Galilean Conformal Algebra,” JHEP [**1305**]{}, 141 (2013) \[arXiv:1303.0291 \[hep-th\]\]. A. Bagchi, S. Chakrabortty and P. Parekh, “Tensionless Strings from Worldsheet Symmetries,” JHEP [**1601**]{}, 158 (2016) doi:10.1007/JHEP01(2016)158 \[arXiv:1507.04361 \[hep-th\]\].
J. D. Brown and M. Henneaux, “Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity,” Commun. Math. Phys. [**104**]{}, 207 (1986). doi:10.1007/BF01211590 J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Int. J. Theor. Phys. [**38**]{}, 1113 (1999) \[Adv. Theor. Math. Phys. [**2**]{}, 231 (1998)\] doi:10.1023/A:1026654312961 \[hep-th/9711200\].
G. Barnich and C. Troessaert, “Aspects of the BMS/CFT correspondence,” JHEP [**1005**]{}, 062 (2010) doi:10.1007/JHEP05(2010)062 \[arXiv:1001.1541 \[hep-th\]\]. A. Strominger, “On BMS Invariance of Gravitational Scattering,” JHEP [**1407**]{}, 152 (2014) doi:10.1007/JHEP07(2014)152 \[arXiv:1312.2229 \[hep-th\]\].
S. Weinberg, “Infrared photons and gravitons,” Phys. Rev. [**140**]{}, B516 (1965). doi:10.1103/PhysRev.140.B516 T. He, V. Lysov, P. Mitra and A. Strominger, “BMS supertranslations and Weinberg’s soft graviton theorem,” JHEP [**1505**]{}, 151 (2015) doi:10.1007/JHEP05(2015)151 \[arXiv:1401.7026 \[hep-th\]\]. D. Kapec, V. Lysov, S. Pasterski and A. Strominger, “Semiclassical Virasoro symmetry of the quantum gravity $ \mathcal{S}$-matrix,” JHEP [**1408**]{}, 058 (2014) doi:10.1007/JHEP08(2014)058 \[arXiv:1406.3312 \[hep-th\]\].
B. Zeldovich and A. G. Polnarev, Ya. Sov.Astron.Lett.,18,17 (1974).
A. Strominger and A. Zhiboedov, “Gravitational Memory, BMS Supertranslations and Soft Theorems,” JHEP [**1601**]{}, 086 (2016) doi:10.1007/JHEP01(2016)086 \[arXiv:1411.5745 \[hep-th\]\].
S. Pasterski, A. Strominger and A. Zhiboedov, “New Gravitational Memories,” arXiv:1502.06120 \[hep-th\]. A. Strominger, “Lectures on the Infrared Structure of Gravity and Gauge Theory,” arXiv:1703.05448 \[hep-th\].
A. Bagchi, R. Basu, A. Kakkar and A. Mehra, “Flat Holography: Aspects of the dual field theory,” JHEP [**1612**]{}, 147 (2016) doi:10.1007/JHEP12(2016)147 \[arXiv:1609.06203 \[hep-th\]\]. M. Riegler, “How General Is Holography?,” arXiv:1609.02733 \[hep-th\]. T. He, P. Mitra, A. P. Porfyriadis and A. Strominger, “New Symmetries of Massless QED,” JHEP [**1410**]{}, 112 (2014) doi:10.1007/JHEP10(2014)112 \[arXiv:1407.3789 \[hep-th\]\].
M. Le Bellac and J.-M. Lévy-Leblond, “Galilean Electromagnetism,” Nuovo Cimento [**14B**]{} (1973), 217
A. Bagchi, R. Basu and A. Mehra, “Galilean Conformal Electrodynamics,” JHEP [**1411**]{}, 061 (2014) doi:10.1007/JHEP11(2014)061 \[arXiv:1408.0810 \[hep-th\]\].
A. Bagchi, R. Basu, A. Kakkar and A. Mehra, “Galilean Yang-Mills Theory,” JHEP [**1604**]{}, 051 (2016) doi:10.1007/JHEP04(2016)051 \[arXiv:1512.08375 \[hep-th\]\]. G. Festuccia, D. Hansen, J. Hartong and N. A. Obers, “Symmetries and Couplings of Non-Relativistic Electrodynamics,” arXiv:1607.01753 \[hep-th\].
P. Di Francesco, P. Mathieu and D. Senechal, “Conformal Field Theory,” doi:10.1007/978-1-4612-2256-9 R. Blumenhagen and E. Plauschinn, “Introduction to conformal field theory : with applications to String theory,” Lect. Notes Phys. [**779**]{}, 1 (2009). doi:10.1007/978-3-642-00450-6 T. Hartman, “Entanglement Entropy at Large Central Charge,” arXiv:1303.6955 \[hep-th\]. A. L. Fitzpatrick, J. Kaplan and M. T. Walters, “Universality of Long-Distance AdS Physics from the CFT Bootstrap,” JHEP [**1408**]{}, 145 (2014) doi:10.1007/JHEP08(2014)145 \[arXiv:1403.6829 \[hep-th\]\]. A. L. Fitzpatrick, J. Kaplan and M. T. Walters, “Virasoro Conformal Blocks and Thermality from Classical Background Fields,” JHEP [**1511**]{}, 200 (2015) doi:10.1007/JHEP11(2015)200 \[arXiv:1501.05315 \[hep-th\]\].
A. Bagchi and I. Mandal, “On Representations and Correlation Functions of Galilean Conformal Algebras,” Phys. Lett. B [**675**]{}, 393 (2009) doi:10.1016/j.physletb.2009.04.030 \[arXiv:0903.4524 \[hep-th\]\].
A. Bagchi, S. Detournay and D. Grumiller, “Flat-Space Chiral Gravity,” Phys. Rev. Lett. [**109**]{}, 151301 (2012) \[arXiv:1208.1658 \[hep-th\]\].
A. Bagchi, S. Detournay, R. Fareghbal and J. Simon, “Holography of 3d Flat Cosmological Horizons,” Phys. Rev. Lett. [**110**]{}, 141302 (2013) \[arXiv:1208.4372 \[hep-th\]\].
G. Barnich, “Entropy of three-dimensional asymptotically flat cosmological solutions,” JHEP [**1210**]{}, 095 (2012) \[arXiv:1208.4371 \[hep-th\]\].
G. Barnich, A. Gomberoff and H. A. Gonzalez, “Three-dimensional Bondi-Metzner-Sachs invariant two-dimensional field theories as the flat limit of Liouville theory,” Phys. Rev. D [**87**]{}, no. 12, 124032 (2013) doi:10.1103/PhysRevD.87.124032 \[arXiv:1210.0731 \[hep-th\]\]. A. Bagchi, S. Detournay, D. Grumiller and J. Simon, “Cosmic Evolution from Phase Transition of Three-Dimensional Flat Space,” Phys. Rev. Lett. [**111**]{}, no. 18, 181301 (2013) doi:10.1103/PhysRevLett.111.181301 \[arXiv:1305.2919 \[hep-th\]\].
A. Bagchi, R. Basu, D. Grumiller and M. Riegler, “Entanglement entropy in Galilean conformal field theories and flat holography,” Phys. Rev. Lett. [**114**]{}, no. 11, 111602 (2015) doi:10.1103/PhysRevLett.114.111602 \[arXiv:1410.4089 \[hep-th\]\].
A. Bagchi, D. Grumiller and W. Merbis, “Stress tensor correlators in three-dimensional gravity,” Phys. Rev. D [**93**]{}, no. 6, 061502 (2016) doi:10.1103/PhysRevD.93.061502 \[arXiv:1507.05620 \[hep-th\]\].
G. Barnich and B. Oblak, “Notes on the BMS group in three dimensions: I. Induced representations,” JHEP [**1406**]{}, 129 (2014) doi:10.1007/JHEP06(2014)129 \[arXiv:1403.5803 \[hep-th\]\].
G. Barnich and B. Oblak, “Notes on the BMS group in three dimensions: II. Coadjoint representation,” JHEP [**1503**]{}, 033 (2015) doi:10.1007/JHEP03(2015)033 \[arXiv:1502.00010 \[hep-th\]\]. A. Campoleoni, H. A. Gonzalez, B. Oblak and M. Riegler, “BMS Modules in Three Dimensions,” Int. J. Mod. Phys. A [**31**]{}, no. 12, 1650068 (2016) doi:10.1142/S0217751X16500688 \[arXiv:1603.03812 \[hep-th\]\].
A. Bagchi, S. Chakrabortty and P. Parekh, “Tensionless Superstrings: View from the Worldsheet,” JHEP [**1610**]{}, 113 (2016) doi:10.1007/JHEP10(2016)113 \[arXiv:1606.09628 \[hep-th\]\]. G. Barnich, L. Donnay, J. Matulich and R. Troncoso, “Asymptotic symmetries and dynamics of three-dimensional flat supergravity,” JHEP [**1408**]{}, 071 (2014) doi:10.1007/JHEP08(2014)071 \[arXiv:1407.4275 \[hep-th\]\]. G. Barnich, L. Donnay, J. Matulich and R. Troncoso, “Super-BMS$_{3}$ invariant boundary theory from three-dimensional flat supergravity,” JHEP [**1701**]{}, 029 (2017) doi:10.1007/JHEP01(2017)029 \[arXiv:1510.08824 \[hep-th\]\].
I. Lodato and W. Merbis, “Super-BMS$_{3}$ algebras from $ \mathcal{N}=2 $ flat supergravities,” JHEP [**1611**]{}, 150 (2016) doi:10.1007/JHEP11(2016)150 \[arXiv:1610.07506 \[hep-th\]\].
A. Achucarro and P. K. Townsend, “A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories,” Phys. Lett. B [**180**]{}, 89 (1986). doi:10.1016/0370-2693(86)90140-1 E. Witten, “(2+1)-Dimensional Gravity as an Exactly Soluble System,” Nucl. Phys. B [**311**]{}, 46 (1988). doi:10.1016/0550-3213(88)90143-5
O. Coussaert, M. Henneaux and P. van Driel, “The Asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant,” Class. Quant. Grav. [**12**]{}, 2961 (1995) doi:10.1088/0264-9381/12/12/012 \[gr-qc/9506019\]. H. Dorn and H. J. Otto, “Two and three point functions in Liouville theory,” Nucl. Phys. B [**429**]{}, 375 (1994) doi:10.1016/0550-3213(94)00352-1 \[hep-th/9403141\]. A. B. Zamolodchikov and A. B. Zamolodchikov, “Structure constants and conformal bootstrap in Liouville field theory,” Nucl. Phys. B [**477**]{}, 577 (1996) doi:10.1016/0550-3213(96)00351-3 \[hep-th/9506136\]. J. Teschner, “Liouville theory revisited,” Class. Quant. Grav. [**18**]{}, R153 (2001) doi:10.1088/0264-9381/18/23/201 \[hep-th/0104158\].
D. Kapec, P. Mitra, A. M. Raclariu and A. Strominger, “A 2D Stress Tensor for 4D Gravity,” arXiv:1609.00282 \[hep-th\]. C. Cheung, A. de la Fuente and R. Sundrum, “4D scattering amplitudes and asymptotic symmetries from 2D CFT,” JHEP [**1701**]{}, 112 (2017) doi:10.1007/JHEP01(2017)112 \[arXiv:1609.00732 \[hep-th\]\]. R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, “Conformal Bootstrap in Mellin Space,” Phys. Rev. Lett. [**118**]{}, no. 8, 081601 (2017) doi:10.1103/PhysRevLett.118.081601 \[arXiv:1609.00572 \[hep-th\]\]. R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, “A Mellin space approach to the conformal bootstrap,” arXiv:1611.08407 \[hep-th\].
[^1]: The reader is referred to [@Festuccia:2016caf] for a slightly different take on infinite symmetries in non-relativistic electrodynamics. Here the authors claim to have a bigger infinity of symmetries that include the GCA and in all dimensions, not only $D=4$.
[^2]: It should be noted that we can only apply the OPE between neighbouring primary fields, so it is understood that the point $(u,v)$ lies between the origin and a circle of radius 1.
[^3]: ${\beta}_{12}^{p\{0,1\},0}$ remains undetermined in this limit and this is to be expected. We also find that one can determine ${\beta}_{12}^{p\{0,1\},1} = -1/2$. But this is a coefficient coming out of a null state $M_{-1}|{\Delta}_{p},0{\rangle}$ and has to be neglected.
[^4]: See also [@Kapec:2016jld; @Cheung:2016iub].
[^5]: There could have been additional complications from an infinite lift of the rotation generators in higher dimensions. But, interestingly, in the field theories that admit the GCA in higher dimensions like Galilean Electrodynamics and Galilean Yang-Mills theories, the symmetry algebra of relevance in the one where rotations don’t get any lift.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
In this investigation we study extreme vortex states defined as incompressible velocity fields with prescribed enstrophy $\E_0$ which maximize the instantaneous rate of growth of enstrophy $d\E/dt$. We provide [an analytic]{} characterization of these extreme vortex states in the limit of vanishing enstrophy $\E_0$ and, in particular, show that the Taylor-Green vortex is in fact a local maximizer of $d\E / dt$ [in this limit]{}. For finite values of enstrophy, the extreme vortex states are computed numerically by solving a constrained variational optimization problem using a suitable gradient method. In combination with a continuation approach, this allows us to construct an entire family of maximizing vortex states parameterized by their enstrophy. We also confirm the findings of the seminal study by [@ld08] that these extreme vortex states saturate (up to a numerical prefactor) the fundamental bound $d\E / dt < C \, \E^3$, for some constant $C >
0$. The time evolution corresponding to these extreme vortex states leads to a larger growth of enstrophy than the growth achieved by any of the commonly used initial conditions with the same enstrophy $\E_0$. However, based on several different diagnostics, there is no evidence of any tendency towards singularity formation in finite time. Finally, we discuss possible physical reasons why the initially large growth of enstrophy is not sustained for longer times.
author:
- |
Diego Ayala$^{1,2}$ and Bartosz Protas$^{2,}$[^1]\
\
$^1$ Department of Mathematics, University of Michigan,\
Ann Arbor, MI 48109, USA\
\
$^2$ Department of Mathematics and Statistics, McMaster University\
Hamilton, Ontario, L8S 4K1, Canada
title: Extreme Vortex States and the Growth of Enstrophy in 3D Incompressible Flows
---
Keywords: Navier-Stokes equations; Extreme Behaviour; Variational methods; Vortex Flows
Introduction {#sec:intro}
============
The objective of this investigation is to study three-dimensional (3D) flows of viscous incompressible fluids which are constructed to exhibit extreme growth of enstrophy. It is motivated by the question whether the solutions to the 3D incompressible Navier-Stokes system on unbounded or periodic domains corresponding to smooth initial data may develop a singularity in finite time [@d09]. By formation of a “singularity” we mean the situation when some norms of the solution corresponding to smooth initial data have become unbounded after a finite time. This so-called “blow-up problem” is one of the key open questions in mathematical fluid mechanics and, in fact, its importance for mathematics in general has been recognized by the Clay Mathematics Institute as one of its “millennium problems” [@f00]. Questions concerning global-in-time existence of smooth solutions remain open also for a number of other flow models including the 3D Euler equations [@gbk08] and some of the “active scalar” equations [@k10].
While the blow-up problem is fundamentally a question in mathematical analysis, a lot of computational studies have been carried out since the mid-’90s in order to shed light on the hydrodynamic mechanisms which might lead to singularity formation in finite time. Given that such flows evolving near the edge of regularity involve formation of very small flow structures, these computations typically require the use of state-of-the-art computational resources available at a given time. The computational studies focused on the possibility of finite-time blow-up in the 3D Navier-Stokes and/or Euler system include [@bmonmu83; @ps90; @b91; @k93; @p01; @bk08; @oc08; @o08; @ghdg08; @gbk08; @h09; @opc12; @bb12; @opmc14], all of which considered problems defined on domains periodic in all three dimensions. Recent investigations by [@dggkpv13; @k13; @gdgkpv14; @k13b] focused on the time evolution of vorticity moments and compared it with the predictions derived from analysis based on rigorous bounds. We also mention the studies by [@mbf08] and [@sc09], along with the references found therein, in which various complexified forms of the Euler equation were investigated. The idea of this approach is that, since the solutions to complexified equations have singularities in the complex plane, singularity formation in the real-valued problem is manifested by the collapse of the complex-plane singularities onto the real axis.
Overall, the outcome of these investigations is rather inconclusive: while for the Navier-Stokes flows most of recent computations do not offer support for finite-time blow-up, the evidence appears split in the case of the Euler system. In particular, the recent studies by [@bb12] and [@opc12] hinted at the possibility of singularity formation in finite time. In this connection we also mention the recent investigations by [@lh14a; @lh14b] in which blow-up was observed in axisymmetric Euler flows in a bounded (tubular) domain.
A common feature of all of the aforementioned investigations was that the initial data for the Navier-Stokes or Euler system was chosen in an ad-hoc manner, based on some heuristic arguments. On the other hand, in the present study we pursue a fundamentally different approach, proposed originally by [@ld08] and employed also by [@ap11a; @ap13a; @ap13b] for a range of related problems, in which the initial data leading to the most singular behaviour is sought systematically via solution of a suitable variational optimization problem. We carefully analyze the time evolution induced by the extreme vortex states first identified by [@ld08] and compare it to the time evolution corresponding to a number of other candidate initial conditions considered in the literature [@bmonmu83; @k93; @p01; @cb05; @opc12]. We demonstrate that the Taylor-Green vortex, studied in the context of the blow-up problem by [@tg37; @bmonmu83; @b91; @cb05], is in fact a particular member of the family of extreme vortex states maximizing the instantaneous rate of enstrophy production in the limit of vanishing enstrophy. In addition, based on these findings, we identify the set of initial data, parameterized by its energy and enstrophy, for which one can a priori guarantee global-in-time existence of smooth solutions. This result therefore offers a physically appealing interpretation of an “abstract” mathematical theorem concerning global existence of classical solutions corresponding to “small” initial data [@lady69]. We also emphasize that, in order to establish a direct link with the results of the mathematical analysis discussed below, in our investigation we follow a rather different strategy than in most of the studies referenced above. While these earlier studies relied on data from a relatively small number of simulations performed at a high (at the given time) resolution, in the present investigation we explore a broad range of cases, each of which is however computed at a more moderate resolution (or, equivalently, Reynolds number). With such an approach to the use of available computational resources, we are able to reveal trends resulting from the variation of parameters which otherwise would be hard to detect. Systematic computations conducted in this way thus allow us to probe the sharpness of the mathematical analysis relevant to the problem.
The question of regularity of solution to the Navier-Stokes system is [usually addressed using “energy” methods which rely on finding upper bounds (with respect to time) on certain quantities of interest, typically taken as suitable Sobolev norms of the solution. A key intermediate step is obtaining bounds on the rate of growth of the quantity of interest, a problem which can be studied with ODE methods.]{} While [for the Navier-Stokes system]{} different norms of the velocity gradient or vorticity can be used to study the regularity of solutions, the use of enstrophy $\E$ (see equation ) is privileged by the well-known result of @ft89, where it was established that if the uniform bound $$\label{eq:RegCrit_FoiasTemam}
\mathop{\sup}_{0 \leq t \leq T} \E(\u(t)) < \infty$$ holds, then the regularity of the solution $\u(t)$ is guaranteed up to time $T$ (to be precise, the solution remains in a suitable Gevrey class). From the computational point of view, the enstrophy $\E(t) := \E(\u(t))$ is thus a convenient indicator of the regularity of solutions, because in the light of , singularity formation must manifest itself by the enstrophy becoming infinite.
While characterization of the maximum possible finite-time growth of enstrophy in the 3D Navier-Stokes flows is the ultimate objective of this research program, analogous questions can also be posed in the context of more tractable problems involving the one-dimensional (1D) Burgers equation and the two-dimensional (2D) Navier-Stokes equation. Although global-in-time existence of the classical (smooth) solutions is well known for both these problems [@kl04], questions concerning the sharpness of the corresponding estimates for the instantaneous and finite-time growth of various quantities are relevant, because these estimates are obtained using essentially the same methods as [employed to derive]{} their 3D counterparts. Since in 2D flows on unbounded or periodic domains the enstrophy may not increase ($d\E/dt \leq 0$), the relevant quantity in this case is the palinstrophy $\P({\mathbf{u}}) := \frac{1}{2}\int_\Omega |
\bnabla{\bm{\omega}}({\mathbf{x}},t) |^2 \,d{\mathbf{x}}$, where ${\bm{\omega}}:={\bnabla\times}{\mathbf{u}}$ is the vorticity (which reduces to a pseudo-scalar in 2D). Different questions concerning sharpness of estimates addressed in our research program are summarized together with the results obtained to date in Table \[tab:estimates\]. We remark that the best finite-time estimate for the 1D Burgers equation was found [*not*]{} to be sharp using the initial data obtained from both the instantaneous and the finite-time variational optimization problems [@ap11a]. On the other hand, in 2D the bounds on both the *instantaneous* and *finite-time* growth of palinstrophy were found to be sharp and, somewhat surprisingly, both estimates were realized by the same family of incompressible vector fields parameterized by energy $\K$ and palinstrophy $\P$, obtained as the solution of an [*instantaneous*]{} optimization problem [@ap13a]. It is worth mentioning that while the estimate for the instantaneous rate of growth of palinstrophy $d\P/dt \leq C\K^{1/2}\P^{3/2}/\nu$ (see Table \[tab:estimates\]) was found to be sharp with respect to variations in palinstrophy, the estimate is in fact not sharp with respect to the prefactor $C_{{\mathbf{u}},\nu} = \K^{1/2}/\nu$ [@ados16], with the correct prefactor being of the form $\widetilde{C}_{{\mathbf{u}},\nu} =
\sqrt{\log\left(\K^{1/2}/\nu\right)}$. We add that what distinguishes the 2D problem, in regard to both the instantaneous and finite-time bounds, is that the RHS of these bounds are expressed in terms of two quantities, namely, energy $\K$ and enstrophy $\E$, in contrast to the enstrophy alone appearing in the 1D and 3D estimates. As a result, the 2D instantaneous optimization problem had to be solved subject to [ *two*]{} constraints.
In the present investigation we advance the research program summarized in Table \[tab:estimates\] by assessing to what extent the finite-time growth of enstrophy predicted by the analytic estimates and can be actually realized by flow evolution starting from different initial conditions, including the extreme vortex states found by [@ld08] to saturate the instantaneous estimate . The key finding is that, at least for the range of modest enstrophy values we considered, the growth of enstrophy corresponding to this initial data, which has the form of two colliding axisymmetric vortex rings, is rapidly depleted and there is no indication of singularity formation in finite time. Thus, should finite-time singularity be possible in the Navier-Stokes system, it is unlikely to result from initial conditions instantaneously maximizing the rate of growth of enstrophy. We also provide a comprehensive characterization of the extreme vortex states which realize estimate together with the resulting flow evolutions.
The structure of the paper is as follows: in the next section we present analytic estimates [on]{} the instantaneous and finite-time growth of enstrophy in 3D flows. In §\[sec:3D\_InstOpt\] we formulate the variational optimization problems which will be solved to find the vortex states with the largest rate of enstrophy production and in §\[sec:3D\_InstOpt\_E0to0\] we provide an asymptotic representation for these optimal states in the limit of vanishing enstrophy. In §\[sec:3D\_InstOpt\_E\] we present numerically computed extreme vortex states corresponding to intermediate and large enstrophy values, while in §\[sec:timeEvolution\] we analyze the temporal evolution corresponding to different initial data in order to compare it with the predictions of estimates and . Our findings are discussed in §\[sec:discuss\], whereas conclusions and outlook are deferred to §\[sec:final\].
[l|c|c]{} &
[Estimate]{}\
&
[Realizability ]{}
\
& &\
& &\
& $\frac{d\P(t)}{dt} \le -\nu\frac{\P^2}{\E} + \frac{C_1}{\nu} \E\,\P$\
$\frac{d\P(t)}{dt} \le \frac{C_2}{\nu} \K^{1/2}\P^{3/2}$ &\
2D Navier-Stokes\
finite-time
& $\max_{t>0} \P(t) \le \P_0 + \frac{C_1}{2\nu^2}\E_0^2$\
$\max_{t>0} \P(t) \le \left(\P_0^{1/2} + \frac{C_2}{4\nu^2}\K_0^{1/2}\E_0\right)^2$ &\
& &\
3D Navier-Stokes\
finite-time
& $\E(t) \le \frac{\E(0)}{\sqrt{1 - 4 \frac{C \E(0)^2}{\nu^3} t}}$\
$\frac{1}{\E(0)} - \frac{1}{\E(t)} \leq \frac{27}{(2\pi\nu)^4}\left[\K(0) - \K(t) \right]$ &\
Bounds on the Growth of Enstrophy in 3D Navier-Stokes Flows {#sec:Bounds3DNS}
===========================================================
We consider the incompressible Navier-Stokes system defined on the 3D unit cube $\Omega = [0,1]^3$ with periodic boundary conditions
\[eq:NSE3D\] $$\begin{aligned}
{2}
\partial_t{\mathbf{u}}+ {\mathbf{u}}\cdot\bnabla{\mathbf{u}}+ \bnabla p - \nu{\Delta}{\mathbf{u}}& = 0 & &\qquad\mbox{in} \,\,\Omega\times(0,T), \\
\bnabla\cdot{\mathbf{u}}& = 0 & & \qquad\mbox{in} \,\,\Omega\times[0,T), \\
{\mathbf{u}}({\mathbf{x}},0) & = {\mathbf{u}}_0({\mathbf{x}}), & &\end{aligned}$$
where the vector ${\mathbf{u}}= [u_1, u_2, u_3]$ is the velocity field, $p$ is the pressure and $\nu>0$ is the coefficient of kinematic viscosity (hereafter we will set $\nu=0.01$ which is the same value as used in the seminal study by [@ld08]). The velocity gradient $\bnabla{\mathbf{u}}$ is the tensor with components $[\bnabla{\mathbf{u}}]_{ij} =
\partial_j u_i$, $i,j=1,2,3$. The fluid density $\rho$ is assumed to be constant and equal to unity ($\rho=1$). The relevant properties of solutions to system can be studied using energy methods, with the energy $\K({\mathbf{u}})$ and its rate of growth given by $$\begin{aligned}
\K({\mathbf{u}}) & := & \frac{1}{2}\int_\Omega |{\mathbf{u}}({\mathbf{x}},t)|^2 \,d{\mathbf{x}}, \label{eq:EnerDef_3D}\\
\frac{d\K({\mathbf{u}})}{dt} & = & -\nu\int_\Omega |\nabla{\mathbf{u}}|^2 \, d{\mathbf{x}}, \label{eq:dK/dt_3D}\end{aligned}$$ where “$:=$” means “equal to by definition”. The enstrophy $\E({\mathbf{u}})$ and its rate of growth are given by $$\begin{aligned}
\E({\mathbf{u}}) & := & \frac{1}{2}\int_\Omega | {\bnabla\times}{\mathbf{u}}({\mathbf{x}},t) |^2 \,d{\mathbf{x}}, \label{eq:EnsDef_3D}\\
\frac{d\E({\mathbf{u}})}{dt} & = & -\nu\int_\Omega |{\Delta}{\mathbf{u}}|^2\,d{\mathbf{x}}+
\int_{\Omega} {\mathbf{u}}\cdot\nabla{\mathbf{u}}\cdot{\Delta}{\mathbf{u}}\, d{\mathbf{x}}=: \R({\mathbf{u}}). \label{eq:dEdt}\end{aligned}$$ For incompressible flows with periodic boundary conditions we also have the following identity [@dg95] $$\int_{\Omega} |{\bnabla\times}{\mathbf{u}}|^2\,d{\mathbf{x}}= \int_{\Omega} |\nabla{\mathbf{u}}|^2\,d{\mathbf{x}}.
\label{eq:duL2}$$ Hence, combining –, the energy and enstrophy satisfy the system of ordinary differential equations
$$\begin{aligned}
\frac{d\K({\mathbf{u}})}{dt} & = -2\nu\E({\mathbf{u}}), \label{eq:dKdt_system}\\
\frac{d\E({\mathbf{u}})}{dt} & = \R({\mathbf{u}}). \label{eq:dEdt_system}\end{aligned}$$
A standard approach at this point is to try to upper-bound $d\E / dt$ and using standard techniques of functional analysis it is possible to obtain the following well-known estimate in terms of $\K$ and $\E$ [@d09] $$\label{eq:dEdt_estimate_KE}
\frac{d\E}{dt} \leq -\nu \frac{\E^2}{\K} + \frac{c}{\nu^3} \E^3$$ for $c$ an absolute constant. A related estimate expressed entirely in terms of the enstrophy $\E$ is given by $$\frac{d\E}{dt} \leq \frac{27}{8\,\pi^4\,\nu^3} \E^3.
\label{eq:dEdt_estimate_E}$$ By simply integrating the differential inequality in with respect to time we obtain the finite-time bound $$\E(t) \leq \frac{\E(0)}{\sqrt{1 - \frac{27}{4\,\pi^4\,\nu^3}\,\E(0)^2\, t}}
\label{eq:Et_estimate_E0}$$ which clearly becomes infinite at time $t_0 = 4\,\pi^4\,\nu^3 /
[27\,\E(0)^2]$. Thus, based on estimate , it is not possible to establish the boundedness of the enstrophy $\E(t)$ globally in time and hence the regularity of solutions. Therefore, the question about the finite-time singularity formation can be recast in terms of whether or not estimate can be saturated. By this we mean the existence of initial data with enstrophy $\E_0 := \E(0)> 0$ such that the resulting time evolution realizes the largest growth of enstrophy $\E(t)$ allowed by the right-hand side (RHS) of estimate . A systematic search for such most singular initial data using variational optimization methods is the key theme of this study. Although different notions of sharpness of an estimate can be defined, e.g., sharpness with respect to constants or exponents in the case of estimates in the form of power laws, the precise notion of sharpness considered in this study is the following
\[def:NotionSharpness\] Given a parameter $p\in\mathbb{R}$ and maps $f,g:\mathbb{R}\to\mathbb{R}$, the estimate $$f(p) \leq g(p)$$ is declared sharp in the limit $p \to p_0\in\mathbb{R}$ if and only if $$\lim_{p \to p_0} \frac{f(p)}{g(p)} \sim \beta, \quad \beta \in \RR.$$
From this definition, the sharpness of estimates in the form $g(p) =
C\, p^{\alpha}$ for some $C \in \RR_+$ and $\alpha \in \RR$ can be addressed in the limit $p \rightarrow \infty$ by studying the adequacy of the exponent $\alpha$.
The question of sharpness of estimate was addressed in the seminal study by [@ld08], see also [@l06], who constructed a family of divergence-free velocity fields saturating this estimate. More precisely, these vector fields were parameterized by their enstrophy and for sufficiently large values of $\E$ the corresponding rate of growth $d\E/dt$ was found to be proportional to $\E^3$. Therefore, in agreement with definition \[def:NotionSharpness\], estimate was declared sharp up to a numerical prefactor. However, the sharpness of the instantaneous estimate alone does not allow us to conclude about the possibility of singularity formation, because for this situation to occur, a sufficiently large enstrophy growth rate would need to be sustained over a [*finite*]{} time window $[0,t_0)$. In fact, assuming the instantaneous rate of growth of enstrophy in the form $d\E / dt =
C \, \E^{\alpha}$ for some $C>0$, any exponent $\alpha > 2$ will produce blow-up of $\E(t)$ in finite time if the rate of growth is sustained. The fact that there is no blow-up for $\alpha \le
2$ follows from Grönwall’s lemma and the fact that one factor of $\E$ in can be bounded in terms of the initial energy using as follows $$\int_0^t \E(s)\, ds = \frac{1}{2\nu} \left[ \K(0) - \K(t)\right] \leq \frac{1}{2\nu} \K(0).
\label{eq:Kt}$$ This relation also leads to an alternative form of the estimate for the finite-time growth of enstrophy, namely $$\begin{aligned}
\frac{d\E}{dt} & \leq \frac{27}{8\pi^4\nu^3}\E^3\quad\Longrightarrow \nonumber \\
\int_{\E(0)}^{\E(t)}\E^{-2}\,d\E & \leq \frac{27}{8\pi^4\nu^3}\int_0^t\E(s)\,ds \quad\Longrightarrow \nonumber \\
\frac{1}{\E(0)} - \frac{1}{\E(t)} & \leq \frac{27}{(2\pi\nu)^4}\left[\K(0) - \K(t) \right] \label{eq:Evs_t_fixE}\end{aligned}$$ which is more convenient than from the computational point of view and will be used in the present study. We note, however, that since the RHS of this inequality cannot be expressed entirely in terms of properties of the initial data, this is [*not*]{} in fact an a priori estimate. Estimate also allows us to obtain a condition on the size of the initial data, given in terms of its energy $\K(0)$ and enstrophy $\E(0)$, which guarantees that smooth solutions will exist globally in time, namely, $$\label{eq:Cond_for_globalReg}
\mathop{\max}_{t \geq 0} \E(t) \leq \frac{\E(0)}{1 - \frac{27}{(2\pi\nu)^4}\K(0)\E(0)}$$ from which it follows that $$\label{eq:K0E0}
\K(0)\E(0) < \frac{(2\pi\nu)^4}{27}.$$ Thus, flows with energy and enstrophy satisfying inequality are guaranteed to be smooth for all time, in agreement with [the]{} regularity results [available under the assumption of]{} small initial data [[@lady69]]{}.
Instantaneously Optimal Growth of Enstrophy {#sec:3D_InstOpt}
===========================================
Sharpness of instantaneous estimate , in the sense of definition \[def:NotionSharpness\], can be probed by constructing a family of “extreme vortex states” ${\widetilde{\mathbf{u}}_{\E_0}}$ which, for each $\E_0 > 0$, have prescribed enstrophy $\E({\widetilde{\mathbf{u}}_{\E_0}}) =
\E_0$ and produce the largest possible rate of growth of enstrophy $\R({\widetilde{\mathbf{u}}_{\E_0}})$. Given the form of , the fields ${\widetilde{\mathbf{u}}_{\E_0}}$ can be expected to exhibit (at least piecewise) smooth dependence on $\E_0$ and we will refer to the mapping $\E_0
\longmapsto {\widetilde{\mathbf{u}}_{\E_0}}$ as a “[maximizing]{} branch”. Thus, information about the sharpness of estimate can be deduced by analyzing the relation $\E_0$ versus $\R({\widetilde{\mathbf{u}}_{\E_0}})$ obtained for a possibly broad range of enstrophy values. A [maximizing]{} branch is constructed by finding, for different values of $\E_0$, the extreme vortex states ${\widetilde{\mathbf{u}}_{\E_0}}$ as solutions of a variational optimization problem defined below.
Hereafter, $H^2(\Omega)$ will denote the Sobolev space of functions with square-integrable second derivatives endowed with the inner product [@af05] $$\forall\,\mathbf{z}_1, \mathbf{z}_2 \in H^2(\Omega) \qquad
\Big\langle \mathbf{z}_1, \mathbf{z}_2 \Big\rangle_{H^2(\Omega)}
= \int_{\Omega} \mathbf{z}_1 \cdot \mathbf{z}_2
+ \ell_1^2 \,\bnabla \mathbf{z}_1 \colon \bnabla \mathbf{z}_2
+ \ell_2^4 \,\Delta \mathbf{z}_1 \cdot \Delta \mathbf{z}_2 \, d{\mathbf{x}}, \label{eq:ipH2}$$ where $\ell_1,\ell_2\in \RR_+$ are parameters with the meaning of length scales (the reasons for introducing these parameters in the definition of the inner product will become clear below). The inner product in the space $L_2(\Omega)$ is obtained from by setting $\ell_1 = \ell_2 = 0$. The notation $H^2_0(\Omega)$ will refer to the Sobolev space $H^2(\Omega)$ of functions with zero mean. For every fixed value $\E_0$ of enstrophy we will look for a divergence-free vector field ${\widetilde{\mathbf{u}}_{\E_0}}$ maximizing the objective function $\R \; : \; H^2_0(\Omega) \rightarrow \RR$ defined in . We thus have the following
\[pb:maxdEdt\_E\] Given $\E_0\in\mathbb{R}_+$ and the objective functional $\R$ from equation , find $$\begin{aligned}
{\widetilde{\mathbf{u}}_{\E_0}}& = & \mathop{\arg\max}_{{\mathbf{u}}\in{\mathcal{S}_{\E_0}}} \, \R({\mathbf{u}}) \\
{\mathcal{S}_{\E_0}} & = & \left\{{\mathbf{u}}\in H_0^2(\Omega)\,\colon\,\nabla\cdot{\mathbf{u}}= 0, \; \E({\mathbf{u}}) = \E_0 \right\}\end{aligned}$$
which will be solved for enstrophy $\E_0$ spanning a broad range of values. This approach was originally proposed and investigated by [@ld08]. In the present study we extend and generalize these results by first showing how other fields considered in the context of the blow-up problem for both the Euler and Navier-Stokes system, namely the Taylor-Green vortex, also arise from variational problem \[pb:maxdEdt\_E\]. We then thoroughly analyze the time evolution corresponding to our extreme vortex states and compare it with the predictions of the finite-time estimates and . As discussed at the end of this section, some important aspects of our approach to solving problem \[pb:maxdEdt\_E\] are also quite different from the method adopted by [@ld08].
The smoothness requirement in the statement of problem \[pb:maxdEdt\_E\] (${\mathbf{u}}\in H_0^2(\Omega)$) follows from the definition of the objective functional $\R$ in equation , where both the viscous term $\nu\int_\Omega
|{\Delta}{\mathbf{u}}|^2\,d{\mathbf{x}}$ and the cubic term $\int_{\Omega}
{\mathbf{u}}\cdot\bnabla{\mathbf{u}}\cdot{\Delta}{\mathbf{u}}\, d{\mathbf{x}}$ contain derivatives of order up to two. The constraint manifold ${\mathcal{S}_{\E_0}}$ can be interpreted as an intersection of the manifold (a subspace) ${\mathcal{S}_{0}}
\in H_0^2(\Omega)$ of divergence-free fields and the manifold $\mathcal{S}'_{\E_0} \in H_0^2(\Omega)$ of fields with prescribed enstrophy $\E_0$. The structure of these constraint manifolds is reflected in the definition of the corresponding projections $\mathbb{P}_{\mathcal{S}}:H_0^2\to\mathcal{S}$ (without a subscript, $\mathcal{S}$ refers to a generic manifold) which is given for each of the two constraints as follows:
- ([*div-free*]{})-constraint: the projection of a field ${\mathbf{u}}$ onto the subspace of solenoidal fields $\mathcal{S}_0$ is performed using the Helmholtz decomposition; accordingly, every zero-mean vector field ${\mathbf{u}}\in H_0^2(\Omega)$ can be decomposed uniquely as $${\mathbf{u}}= \bnabla\phi + {\bnabla\times}{\mathbf{A}},$$ where $\phi$ and ${\mathbf{A}}$ are scalar and vector potentials, respectively; it follows from the identity $\bnabla\cdot({\bnabla\times}{\mathbf{A}})\equiv0$, valid for any sufficiently smooth vector field ${\mathbf{A}}$, that the projection $\mathbb{P}_{\mathcal{S}_0}({\mathbf{u}})$ is given simply by ${\bnabla\times}{\mathbf{A}}$ and is therefore calculated as $$\mathbb{P}_{\mathcal{S}_0}({\mathbf{u}}) = {\mathbf{u}}- \bnabla\left[{\Delta}^{-1}(\bnabla\cdot{\mathbf{u}})\right],
\label{eq:Phodge}$$ where ${\Delta}^{-1}$ is the inverse Laplacian associated with the periodic boundary conditions; the operator $\mathbb{P}_{\mathcal{S}_0}$ is also known as the Leray-Helmholtz projector.
- $(\E_0)$-constraint: the projection onto the manifold $\mathcal{S}'_{\E_0}$ is calculated by the normalization $$\label{eq:FixE0_3D}
\mathbb{P}_{\mathcal{S}'_{\E_0}}({\mathbf{u}}) = \sqrt{\frac{\E_0}{\E\left({\mathbf{u}}\right)}}\,{\mathbf{u}}.$$
Thus, composing with , the projection onto the manifold ${\mathcal{S}_{\E_0}}$ defined in problem \[pb:maxdEdt\_E\] is constructed as $$\mathbb{P}_{\mathcal{S}_{\E_0}}({\mathbf{u}}) = \mathbb{P}_{\mathcal{S}'_{\E_0}}\Big( \mathbb{P}_{\mathcal{S}_0} ({\mathbf{u}})\Big).
\label{eq:P}$$ This approach, which was already successfully employed by [@ap11a; @ap13a], allows one to enforce the enstrophy constraint essentially with the machine precision.
For a given value of $\E_0$, the maximizer ${\widetilde{\mathbf{u}}_{\E_0}}$ can be found as ${\widetilde{\mathbf{u}}_{\E_0}}= \lim_{n\rightarrow \infty} {\mathbf{u}}_{\E_0}^{(n)}$ using the following iterative procedure representing a discretization of a gradient flow projected on $\mathcal{S}_{\E_0}$ $$\begin{aligned}
{\mathbf{u}}_{\E_0}^{(n+1)} & = \mathbb{P}_{\mathcal{S}_{\E_0}}\left(\;{\mathbf{u}}^{(n)}_{\E_0} + \tau_n \nabla\R\left({\mathbf{u}}^{(n)}_{\E_0}\right)\;\right), \\
{\mathbf{u}}_{\E_0}^{(1)} & = {\mathbf{u}}^0,
\end{aligned}
\label{eq:desc}$$ where ${\mathbf{u}}^{(n)}_{\E_0}$ is an approximation of the maximizer obtained at the $n$-th iteration, ${\mathbf{u}}^0$ is the initial guess and $\tau_n$ is the length of the step in the direction of the gradient. It is ensured that the maximizers ${\widetilde{\mathbf{u}}_{\E_0}}$ obtained for different values of $\E_0$ lie on the same [maximizing]{} branch by using the continuation approach, where the maximizer ${\widetilde{\mathbf{u}}_{\E_0}}$ is [employed]{} as the initial guess ${\mathbf{u}}^0$ to compute $\widetilde{\mathbf{u}}_{\E_0+\Delta\E}$ at the next enstrophy level for some sufficiently small $\Delta\E > 0$. As will be demonstrated in §\[sec:3D\_InstOpt\_E0to0\], in the limit $\E_0 \rightarrow 0$ optimization problem \[pb:maxdEdt\_E\] admits a discrete family of closed-form solutions and each of these vortex states is the limiting (initial) member $\widetilde{\mathbf{u}}_{0}$ of the corresponding [maximizing]{} branch. As such, these limiting extreme vortex states are used as the initial guesses ${\mathbf{u}}^0$ for the calculation of $\widetilde{\mathbf{u}}_{\Delta\E}$, i.e., they serve as “seeds” for the calculation of an entire [maximizing]{} branch (as discussed in §\[sec:discuss\], while there exist alternatives to the continuation approach, this technique [in fact results]{} in the fastest convergence of iterations and also ensures that all computed extreme vortex states lie on a single branch). The procedure outlined above is summarized as Algorithm \[alg:optimAlg\], [whereas all]{} details are presented below.
set $\E_0 = 0$ set ${\widetilde{\mathbf{u}}_{\E_0}}= \widetilde{\mathbf{u}}_{0}$ ${\mathbf{u}}_{\E_0}^{(0)} = {\widetilde{\mathbf{u}}_{\E_0}}$ $\E_0 = \E_0 + \Delta \E$ $n = 0$ compute $\R_0 = \R\left({\mathbf{u}}_{\E_0}^{(0)}\right)$
compute the $L_2$ gradient $\nabla^{L_2}\R\left({\mathbf{u}}_{\E_0}^{(n)}\right)$, see equation
compute the Sobolev gradient $\nabla\R\left({\mathbf{u}}_{\E_0}^{(n)}\right)$, see equation
compute the step size $\tau_n$, see equation
set ${\mathbf{u}}_{\E_0}^{(n+1)} = \mathbb{P}_{\mathcal{S}_{\E_0}}\left(\;{\mathbf{u}}_{\E_0}^{(n)} + \tau_n \nabla\R\left({\mathbf{u}}_{\E_0}^{(n)}\right)\;\right)$, see equations –
set $\R_1 = \R\left({\mathbf{u}}_{\E_0}^{(n+1)}\right)$
compute the `relative error` $ = (\R_1 - \R_0)/\R_0$
set $\R_0 = \R_1$
set $n=n+1$
A key step of Algorithm \[alg:optimAlg\] is the evaluation of the gradient $\nabla\R({\mathbf{u}})$ of the objective functional $\R({\mathbf{u}})$, cf. , representing its (infinite-dimensional) sensitivity to perturbations of the velocity field ${\mathbf{u}}$, and it is essential that the gradient be characterized by the required regularity, namely, $\nabla\R({\mathbf{u}}) \in H^2(\Omega)$. This is, in fact, guaranteed by the Riesz representation theorem [@l69] applicable because the Gâteaux differential $\R'({\mathbf{u}};\cdot) : H_0^2(\Omega) \rightarrow \RR$, defined as $\R'({\mathbf{u}};{\mathbf{u}}') := \lim_{\epsilon \rightarrow 0}
\epsilon^{-1}\left[\R({\mathbf{u}}+\epsilon {\mathbf{u}}') - \R({\mathbf{u}})\right]$ for some perturbation ${\mathbf{u}}' \in H_0^2(\Omega)$, is a bounded linear functional on $H_0^2(\Omega)$. The Gâteaux differential can be computed directly to give $$\R'({\mathbf{u}};{\mathbf{u}}') = \int_{\Omega}\left[{\mathbf{u}}'\cdot\bnabla{\mathbf{u}}\cdot{\Delta}{\mathbf{u}}+
{\mathbf{u}}\cdot\bnabla{\mathbf{u}}'\cdot{\Delta}{\mathbf{u}}+
{\mathbf{u}}\cdot\bnabla{\mathbf{u}}\cdot{\Delta}{\mathbf{u}}' \right]\,d{\mathbf{x}}-2\nu\int_{\Omega}{\Delta}^2{\mathbf{u}}\cdot{\mathbf{u}}'\,d{\mathbf{x}}\label{eq:dR}$$ from which, by the Riesz representation theorem, we obtain $$\R'({\mathbf{u}};{\mathbf{u}}')
= \Big\langle \nabla\R({\mathbf{u}}), {\mathbf{u}}' \Big\rangle_{H^2(\Omega)}
= \Big\langle \nabla^{L_2}\R({\mathbf{u}}), {\mathbf{u}}' \Big\rangle_{L_2(\Omega)}
\label{eq:riesz}$$ with the Riesz representers $\nabla\R({\mathbf{u}})$ and $\nabla^{L_2}\R({\mathbf{u}})$ being the gradients computed with respect to the $H^2$ and $L_2$ topology, respectively, and the inner products defined in . We remark that, while the $H^2$ gradient is used exclusively in the actual computations, cf. , the $L_2$ gradient is computed first as an intermediate step. Identifying the Gâteaux differential with the $L_2$ inner product and performing integration by parts yields $$\nabla^{L_2}\R({\mathbf{u}}) = {\Delta}\left( {\mathbf{u}}\cdot\bnabla{\mathbf{u}}\right) + (\bnabla{\mathbf{u}})^T{\Delta}{\mathbf{u}}-
{\mathbf{u}}\cdot\bnabla({\Delta}{\mathbf{u}}) - 2\nu{\Delta}^2{\mathbf{u}}.
\label{eq:gradRL2}$$ Similarly, identifying the Gâteaux differential with the $H^2$ inner product , integrating by parts and using , we obtain the required $H^2$ gradient $\nabla\R$ as a solution of the following elliptic boundary-value problem $$\begin{aligned}
&\left[ {\operatorname{Id}}\, - \,\ell_1^2 \,\Delta + \,\ell_2^4 \,\Delta^2 \right] \nabla\R
= \nabla^{L_2} \R \qquad \text{in} \ \Omega, \\
& \text{Periodic Boundary Conditions}.
\end{aligned}
\label{eq:gradRH2}$$ The gradient fields $\nabla^{L_2}\R({\mathbf{u}})$ and $\nabla\R({\mathbf{u}})$ can be interpreted as infinite-dimensional sensitivities of the objective function $\R({\mathbf{u}})$, cf. , with respect to perturbations of the field ${\mathbf{u}}$. While these two gradients may point towards the same local maximizer, they represent distinct “directions”, since they are defined with respect to different topologies ($L_2$ vs. $H^2$). As shown by @pbh04, extraction of gradients in spaces of smoother functions such as $H^2(\Omega)$ can be interpreted as low-pass filtering of the $L_2$ gradients with parameters $\ell_1$ and $\ell_2$ acting as cut-off length-scales and the choice of their numerical values will be discussed in §\[sec:3D\_InstOpt\_E\].
The step size $\tau_n$ in algorithm is computed as $$\label{eq:tau_n}
\tau_n = \mathop{{\operatorname{argmax}}}_{\tau>0} \left\{ \R\left[\mathbb{P}_{\mathcal{S}_{\E_0}}
\left( \;{\mathbf{u}}^{(n)} + \tau\,\nabla\R({\mathbf{u}}^{(n)}) \;\right)\right] \right\}$$ which is done using a suitable derivative-free line-search algorithm [@r06]. Equation can be interpreted as a modification of a standard line search method where the optimization is performed following an arc (a geodesic) lying on the constraint manifold $\mathcal{S}_{\E_0}$, rather than a straight line. This approach was already successfully employed to solve similar problems in @ap11a [@ap13a].
It ought to be emphasized here that the approach presented above in which the projections – and gradients – are obtained based on the infinite-dimensional (continuous) formulation to be discretized only at the final stage is fundamentally different from the method employed in the original study by [@ld08] in which the optimization problem was solved in a fully discrete setting (the two approaches are referred to as “optimize-then-discretize” and “discretize-then-optimize”, respectively, cf. [@g03]). A practical advantage of the continuous (“optimize-then-discretize”) formulation used in the present work is that the expressions representing the sensitivity of the objective functional $\R$, i.e. the gradients $\nabla^{L_2}\R$ and $\nabla\R$, are independent of the specific discretization approach chosen to evaluate them. This should be contrasted with the discrete (“discretize-then-optimize”) formulation, where a change of the discretization method would require rederivation of the gradient expressions. In addition, the continuous formulation allows us to strictly enforce the regularity of maximizers required in problem \[pb:maxdEdt\_E\]. Finally and perhaps most importantly, the continuous formulation of the maximization problem makes it possible to obtain elegant closed-form solutions of the problem in the limit $\E_0 \rightarrow 0$, which is done in § \[sec:3D\_InstOpt\_E0to0\] below. These analytical solutions will then be used in §\[sec:3D\_InstOpt\_E\] to guide the computation of maximizing branches by numerically solving problem \[pb:maxdEdt\_E\] for a broad range of $\E_0$, as outlined in Algorithm \[alg:optimAlg\].
Extreme Vortex States in the Limit $\E_0 \to 0$ {#sec:3D_InstOpt_E0to0}
===============================================
It is possible to find analytic solutions to problem \[pb:maxdEdt\_E\] in the limit $\E_0 \to 0$ using perturbation methods. To simplify the notation, in this section we will drop the subscript $\E_0$ when referring to the optimal field. The Euler-Lagrange system representing the first-order optimality conditions in optimization problem \[pb:maxdEdt\_E\] is given by [@l69]
\[eq:KKT\_E\] $$\begin{aligned}
{\mathcal{B}}({\widetilde{\mathbf{u}}},{\widetilde{\mathbf{u}}}) - 2\nu{\Delta}^2{\widetilde{\mathbf{u}}}- \lambda{\Delta}{\widetilde{\mathbf{u}}}- \bnabla q & = 0 \qquad\mbox{in}\,\,\Omega , \label{eq:KKT_E_gradR}\\
\nabla\cdot{\widetilde{\mathbf{u}}}& = 0 \qquad\mbox{in}\,\,\Omega , \label{eq:KKT_E_divConstr}\\
\E({\widetilde{\mathbf{u}}}) - \E_0 & = 0, \label{eq:KKT_E_E0Constr}\end{aligned}$$
where $\lambda\in\mathbb{R}$ and $q:\Omega\to\mathbb{R}$ are the Lagrange multipliers associated with the constraints defining the manifold ${\mathcal{S}_{\E_0}}$, and ${\mathcal{B}}({\mathbf{u}},{\mathbf{v}})$, given by $${\mathcal{B}}({\mathbf{u}},{\mathbf{v}}) := {\Delta}\left( {\mathbf{u}}\cdot\bnabla{\mathbf{v}}\right) + (\bnabla{\mathbf{u}})^T{\Delta}{\mathbf{v}}-
{\mathbf{u}}\cdot\bnabla({\Delta}{\mathbf{v}}),$$ is the bilinear form from equation . Using the formal series expansions with $\alpha > 0$
\[eq:series3D\] $$\begin{aligned}
{\widetilde{\mathbf{u}}}& = {\mathbf{u}}_0 + \E_0^{\alpha}{\mathbf{u}}_1 + \E_0^{2\alpha}{\mathbf{u}}_2 + \ldots, \\
\lambda & = \lambda_0 + \E_0^{\alpha}\lambda_1 + \E_0^{2\alpha}\lambda_2 + \ldots, \\
q & = q_0 + \E_0^{\alpha}q_1 + \E_0^{2\alpha}q_2 + \ldots\end{aligned}$$
in and collecting terms proportional to different powers of $\E_0^{\alpha}$, it follows from that, at every order $m=1,2,\dots$ in $\E_0^{\alpha}$, we have $$\E_0^{m\alpha}: \qquad\sum_{j=0}^m {\mathcal{B}}({\mathbf{u}}_j,{\mathbf{u}}_{m-j}) - 2\nu{\Delta}^2{\mathbf{u}}_m -
\sum_{j=0}^m\lambda_j{\Delta}{\mathbf{u}}_{m-j} - \nabla q_m = 0 \quad\mbox{in}\,\,\Omega.$$ Similarly, equation leads to $$\label{eq:Incompressible_Uk}
\nabla\cdot{\mathbf{u}}_m = 0 \quad\mbox{in}\,\,\Omega$$ at every order $m$ in $\E_0^{\alpha}$. It then follows from equation that $$\begin{aligned}
\E({\mathbf{u}}) & = & \E({\mathbf{u}}_0) -\big\langle{\mathbf{u}}_0,{\Delta}{\mathbf{u}}_1\big\rangle_{L_2}\E_0^{\alpha} +
\left[ \E({\mathbf{u}}_1) - \big\langle{\mathbf{u}}_0,{\Delta}{\mathbf{u}}_2\big\rangle_{L_2}\right]\E_0^{2\alpha} + \ldots \\
& = & \E_0,\end{aligned}$$ which, for $\alpha \neq 0$, forces $\E({\mathbf{u}}_0) = 0$. Hence, ${\mathbf{u}}_0
\equiv 0$, $\alpha = 1/2$ and $\E({\mathbf{u}}_1) = 1$. The systems at orders $\E_0^{1/2}$ and $\E_0^1$ are given by:
\[eq:maxdEdt\_Asympt\_1\] $$\begin{aligned}
\E^{1/2}_0:\quad\qquad\qquad\qquad\qquad\qquad 2\nu{\Delta}^2{\mathbf{u}}_1 + \lambda_0{\Delta}{\mathbf{u}}_1 +
\nabla q_1 & = 0 \quad\mbox{in}\,\,\Omega , \label{eq:maxdEdt_Asympt_1_PDE}\\
\nabla\cdot{\mathbf{u}}_1 & = 0 \quad\mbox{in}\,\,\Omega , \label{eq:maxdEdt_Asympt_1_Div0Constr} \\
\E({\mathbf{u}}_1) & = 1, \label{eq:maxdEdt_Asympt_1_E0Constr} \end{aligned}$$
\[eq:maxdEdt\_Asympt\_2\] $$\begin{aligned}
\E_0:\qquad 2\nu{\Delta}^2{\mathbf{u}}_2 + \lambda_0{\Delta}{\mathbf{u}}_2 + \nabla q_2 - {\mathcal{B}}({\mathbf{u}}_1,{\mathbf{u}}_1) +
\lambda_1{\Delta}{\mathbf{u}}_1 & = 0 \quad\mbox{in}\,\,\Omega, \\
\nabla\cdot{\mathbf{u}}_2 & = 0 \quad\mbox{in}\,\,\Omega , \\
\langle {\Delta}{\mathbf{u}}_1, {\mathbf{u}}_2 \rangle_{L_2} & = 0, \end{aligned}$$
where the fact that ${\mathcal{B}}({\mathbf{u}}_0,{\mathbf{u}}_j) = 0$ for all $j$ has been used. While continuing this process to larger values of $m$ may lead to some interesting insights, for the purpose of this investigation it is sufficient to truncate expansions at the order $\O(\E_0)$. The corresponding approximation of the objective functional then becomes $$\label{eq:R03D}
\R({\widetilde{\mathbf{u}}}) = - \nu\E_0\int_{\Omega} \left| {\Delta}{\mathbf{u}}_1 \right|^2 \, d{\mathbf{x}}+ \O(\E_0^{3/2}).$$ It is worth noting that, in the light of relation , the maximum rate of growth of enstrophy in the limit of small $\E_0$ is in fact negative, meaning that, for sufficiently small $\E_0$, the enstrophy itself is a decreasing function for all times. This observation is consistent with the small-data regularity result discussed in Introduction.
As regards problem defining the triplet $\{{\mathbf{u}}_1, q_1,\lambda_0\}$, taking the divergence of equation and using the condition $\nabla\cdot{\mathbf{u}}_1=0$ leads to the Laplace equation ${\Delta}q_1 =
0$ in $\Omega$. Since for zero-mean functions defined on $\Omega$, ${\operatorname{Ker}}({\Delta}) = \{ 0 \}$, it follows that $q_1
\equiv 0$ and equation is reduced to the eigenvalue problem $$\label{eq:maxdEdt_smallE0_eig}
2\nu{\Delta}{\mathbf{u}}_1 + \lambda_0{\mathbf{u}}_1 = 0,$$ with ${\mathbf{u}}_1$ satisfying the incompressibility condition . Direct calculation using equation and condition leads to an asymptotic expression for the objective functional in the limit of small enstrophy $$\label{eq:R0_lambda0_3D}
\R({\widetilde{\mathbf{u}}}) \approx - \lambda_0\E_0.$$ Solutions to the eigenvalue problem in equation can be found using the Fourier expansion of ${\mathbf{u}}_1$ given as (with hats denoting Fourier coefficients) $${\mathbf{u}}_1({\mathbf{x}}) = \sum_{{\mathbf{k}}\in{\mathcal{W}}}\widehat{{\mathbf{u}}}_1({\mathbf{k}}){\textrm{e}^{2\pi i{\mathbf{k}}\cdot{\mathbf{x}}}},$$ where ${\mathcal{W}}\subseteq\mathbb{Z}^3$ is a set of wavevectors ${\mathbf{k}}$ for which $\widehat{{\mathbf{u}}}_1({\mathbf{k}}) \neq 0$. The eigenvalue problem then becomes $$\begin{aligned}
\left[-2\nu(2\pi)^2|{\mathbf{k}}|^2 + \lambda_0\right]\widehat{{\mathbf{u}}}_1({\mathbf{k}}) & = & 0 \qquad\forall\,{\mathbf{k}}\in{\mathcal{W}}, \\
\widehat{{\mathbf{u}}}_1({\mathbf{k}})\cdot{\mathbf{k}}& = & 0 \qquad\forall\,{\mathbf{k}}\in{\mathcal{W}},\end{aligned}$$ with solutions obtained by choosing, for any $k\in\mathbb{Z}\setminus\{0\}$, a set of wavevectors with the following structure $$\label{eq:defW}
{\mathcal{W}}_k = \left\{{\mathbf{k}}\in\mathbb{Z}^3\colon|{\mathbf{k}}|^2 = k \right\}$$ and $\widehat{{\mathbf{u}}}_1({\mathbf{k}})$ with an appropriate form satisfying the incompressibility condition $\widehat{{\mathbf{u}}}_1\cdot{\mathbf{k}}= 0$. For the solutions to equation constructed in such manner it then follows that $\lambda_0 = 2\nu(2\pi)^2|{\mathbf{k}}|^2$ and the optimal asymptotic value of $\R$ is given by $$\label{eq:R0_kvec_3D}
\R({\widetilde{\mathbf{u}}}) \approx - 8\pi^2\nu|{\mathbf{k}}|^2\E_0.$$
Since the fields ${\mathbf{u}}_1$ are real-valued, their Fourier modes must satisfy $\widehat{{\mathbf{u}}}_1(-{\mathbf{k}}) =
\overline{\widehat{{\mathbf{u}}}_1({\mathbf{k}})}$, where $\overline{z}$ denotes the complex conjugate (C.C.) of $z\in\mathbb{C}$. Depending on the choice of ${\mathcal{W}}_k$, a number of different solutions of can be constructed and below we focus on the following three most relevant cases characterized by the largest values of $\R({\widetilde{\mathbf{u}}})$:
1. ${\mathcal{W}}_1 = \{ {\mathbf{k}}_1, {\mathbf{k}}_2, {\mathbf{k}}_3, -{\mathbf{k}}_1, -{\mathbf{k}}_2,
-{\mathbf{k}}_3 \}$, where ${\mathbf{k}}_i = \mathbf{e}_i$, $i=1,2,3$, is the $i^{\textrm{th}}$ unit vector of the canonical basis of $\mathbb{R}^3$; the most general solution can then be constructed as $$\label{eq:uvec_3D_k1}
{\mathbf{u}}_1({\mathbf{x}}) = \mathbf{A}{\textrm{e}^{2\pi i{\mathbf{k}}_1\cdot{\mathbf{x}}}} +
\mathbf{B}{\textrm{e}^{2\pi i{\mathbf{k}}_2\cdot{\mathbf{x}}}} +
\mathbf{C}{\textrm{e}^{2\pi i{\mathbf{k}}_3\cdot{\mathbf{x}}}} + \textrm{C.C.}$$ with the complex-valued constant vectors $\mathbf{A} = [0,A_2,A_3]$, $\mathbf{B} = [B_1,0,B_3]$ and $\mathbf{C} = [C_1,C_2,0]$ suitably chosen so that $\E({\mathbf{u}}_1) = 1$; hereafter we will use the values $A_2 =
A_3 = \ldots = C_2 = 1/(48\pi^2)$; it follows that $|{\mathbf{k}}|^2 =
1$ $\forall\,\,{\mathbf{k}}\in{\mathcal{W}}_1$, and the optimal asymptotic value of $\R$ obtained from equation is given by $$\label{eq:R0_kvec_3D_k1}
\R({\widetilde{\mathbf{u}}}) \approx - 8\pi^2\nu\E_0,$$ \[c1\]
2. ${\mathcal{W}}_2 = {\mathcal{W}}\cup (-{\mathcal{W}})$, where $-{\mathcal{W}}$ denotes the set whose elements are the negatives of the elements of set ${\mathcal{W}}$, for ${\mathcal{W}}= \{
{\mathbf{k}}_1 + {\mathbf{k}}_2, {\mathbf{k}}_1 - {\mathbf{k}}_2, {\mathbf{k}}_1 + {\mathbf{k}}_3, {\mathbf{k}}_1 -
{\mathbf{k}}_3, {\mathbf{k}}_2 + {\mathbf{k}}_3, {\mathbf{k}}_2 - {\mathbf{k}}_3 \}$; the most general solution can be then constructed as $$\begin{aligned}
{\mathbf{u}}_1({\mathbf{x}}) & = & \mathbf{A}{\textrm{e}^{2\pi i[1,1,0]\cdot{\mathbf{x}}}} +
\mathbf{B}{\textrm{e}^{2\pi i[1,-1,0]\cdot{\mathbf{x}}}} +
\mathbf{C}{\textrm{e}^{2\pi i[1,0,1]\cdot{\mathbf{x}}}} + \nonumber \\
& & \mathbf{D}{\textrm{e}^{2\pi i[1,0,-1]\cdot{\mathbf{x}}}} +
\mathbf{E}{\textrm{e}^{2\pi i[0,1,1]\cdot{\mathbf{x}}}} +
\mathbf{F}{\textrm{e}^{2\pi i[0,1,-1]\cdot{\mathbf{x}}}} + \textrm{C.C.}
\label{eq:uvec_3D_k2}\end{aligned}$$ with the constants $\mathbf{A},\mathbf{B},\ldots,\mathbf{F}\in\mathbb{C}^3$ suitably chosen so that $\mathbf{A}\cdot[1,1,0] = 0$, $\mathbf{B}\cdot[1,-1,0]
= 0 ,\ldots,\mathbf{F}\cdot[0,1,-1] = 0$, which ensures that incompressibility condition is satisfied, and that $\E({\mathbf{u}}_1) = 1$; in this case, $|{\mathbf{k}}|^2 =
2$, $\forall\,{\mathbf{k}}\in{\mathcal{W}}_2$, and the optimal asymptotic value of $\R$ is $$\label{eq:R0_kvec_3D_k2}
\R({\widetilde{\mathbf{u}}}) \approx - 16\pi^2\nu\E_0,$$ \[c2\]
3. ${\mathcal{W}}_3 = {\mathcal{W}}\cup (-{\mathcal{W}})$ for ${\mathcal{W}}= \{
{\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{k}}_3,-{\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{k}}_3,{\mathbf{k}}_1-{\mathbf{k}}_2+{\mathbf{k}}_3,{\mathbf{k}}_1+{\mathbf{k}}_2-{\mathbf{k}}_3
\}$; the most general solution can then be constructed as $$\begin{aligned}
{\mathbf{u}}_1({\mathbf{x}}) & = & \mathbf{A}{\textrm{e}^{2\pi i[1,1,1]\cdot{\mathbf{x}}}} +
\mathbf{B}{\textrm{e}^{2\pi i[-1,1,1]\cdot{\mathbf{x}}}} + \nonumber \\
& & \mathbf{C}{\textrm{e}^{2\pi i[1,-1,1]\cdot{\mathbf{x}}}} +
\mathbf{D}{\textrm{e}^{2\pi i[1,1,-1]\cdot{\mathbf{x}}}} + \textrm{C.C.}
\label{eq:uvec_3D_k3}\end{aligned}$$ with the constants $\mathbf{A},\mathbf{B},\mathbf{C},\mathbf{D}\in\mathbb{C}^3$ suitably chosen so that $\mathbf{A}\cdot[1,1,1] = 0$, $\mathbf{B}\cdot[-1,1,1]
= 0$, $\mathbf{C}\cdot[1,-1,1] = 0$ and $\mathbf{D}\cdot[1,1,-1] = 0$, which ensures that incompressibility condition is satisfied, and that $\E({\mathbf{u}}_1) = 1$; in this case, $|{\mathbf{k}}|^2 =3$, $\forall\,{\mathbf{k}}\in{\mathcal{W}}_3$, and the optimal asymptotic value of $\R$ is $$\label{eq:R0_kvec_3D_k3}
\R({\widetilde{\mathbf{u}}}) \approx - 24\pi^2\nu\E_0.$$ \[c3\]
The three constructions of the extremal field ${\mathbf{u}}_1$ given in , and are all defined up to arbitrary shifts in all three directions, reflections with respect to different planes and rotations by angles which are multiples of $\pi / 2$ about the different axes. As a result of this nonuniqueness, there is some freedom in choosing the constants $\mathbf{A},\ldots,\mathbf{F}$. Given that the optimal asymptotic value of $\R$ depends exclusively on the wavevector magnitude $|{\mathbf{k}}|$, cf. , any combination of constants $\mathbf{A},\ldots,\mathbf{F}$ will produce the [*same*]{} optimal rate of growth of enstrophy. Thus, to fix attention, in our analysis we will set $\mathbf{A}=\mathbf{B}=\mathbf{C}$ in case (i), $\mathbf{A}=\mathbf{B}=\ldots=\mathbf{F}$ in case (\[c2\]) and $\mathbf{A}=\ldots=\mathbf{D}$ in case (\[c3\]). With these choices, the contribution from each component of the field ${\mathbf{u}}_1$ to the total enstrophy is the same. The maximum (i.e., least negative) value of $\R$ can be thus obtained by choosing the smallest possible $|{\mathbf{k}}|^2$. This maximum is achieved in case (\[c1\]) with the wavevectors ${\mathbf{k}}_1 = [1,0,0]$, ${\mathbf{k}}_2 = [0,1,0]$, ${\mathbf{k}}_3 =
[0,0,1]$, and $-{\mathbf{k}}_1$, $-{\mathbf{k}}_2$ and $-{\mathbf{k}}_3$, for which $|{\mathbf{k}}|^2 = 1$. Because of this maximization property, this is the field we will focus on in our analysis in §\[sec:3D\_InstOpt\_E\] and §\[sec:timeEvolution\].
The three fields constructed in , and are visualized in figure \[fig:maxdEdt\_vortexCells\]. This analysis is performed using the level sets $\Gamma_{s}(F)\subset\Omega$ defined as $$\label{eq:levelSets}
\Gamma_{s}(F) := \{{\mathbf{x}}\in\Omega : F({\mathbf{x}}) = s \},$$ for a suitable function $F:\Omega\to\mathbb{R}$. In figures \[fig:maxdEdt\_vortexCells\](a–c) we choose $F({\mathbf{x}}) =
|{\bnabla\times}{\mathbf{u}}_1|({\mathbf{x}})$ with $s = 0.95||{\bnabla\times}{\mathbf{u}}_1||_{L_\infty}$. To complement this information, in figures \[fig:maxdEdt\_vortexCells\](d–f) we also plot the isosurfaces and cross-sectional distributions of the $x_1$ component of the field ${\mathbf{u}}_1$.
The fields shown in figure \[fig:maxdEdt\_vortexCells\] reveal interesting patterns involving well-defined “vortex cells”. More specifically, we see that in case (\[c1\]), given by equation and shown in figures \[fig:maxdEdt\_vortexCells\](a,d), the vortex cells are staggered with respect to the orientation of the cubic domain $\Omega$ in all three planes, whereas in case (\[c3\]), given by equation and shown in figures \[fig:maxdEdt\_vortexCells\](c,f), the vortex cells are aligned with the domain $\Omega$ in all three planes. On the other hand, in case (\[c2\]), given by equation and shown in figures \[fig:maxdEdt\_vortexCells\](b,e), the vortex cells are staggered in one plane and aligned in another with the arrangement in the third plane resulting from the arrangement in the first two. These geometric properties are also reflected in the $x_1$-component of the field ${\mathbf{u}}_1$ which is independent of $x_1$ in cases (\[c1\]) and (\[c2\]), but exhibits, respectively, a staggered and aligned arrangement of the cells in the $y-z$ plane in these two cases. In case (\[c3\]) the cells exhibit an aligned arrangement in all three planes. The geometric properties of the extreme vortex states obtained in the limit $\E_0 \rightarrow 0$ are summarized in Table \[tab:E0\]. We remark that an analogous structure of the optimal fields, featuring aligned and staggered arrangements of vortex cells in the limiting case, was also discovered by [@ap13a] in their study of the maximum palinstrophy growth in 2D. While due to a smaller spatial dimension only two optimal solutions were found in that study, the one characterized by the staggered arrangement also lead to a larger (less negative) rate of palinstrophy production.
[l|c|c|c|c]{}
[Case]{}\
&
[Formula for the velocity field]{}
&
[Arrangement of cells in $y-z$ plane]{}
&
[Dependence of $x_1$ component of ${\mathbf{u}}_1$ on $x_1$]{}
&
[Remarks]{}
\
(\[c1\])\
&
&
staggered
&
uniform
&
staggered ABC flow
\
(\[c2\])\
&
&
aligned
&
uniform
&
aligned ABC flow
\
(\[c3\])\
&
&
aligned
&
cell-like
&
Taylor-Green vortex
\
It is also worth mentioning that the initial data for two well-known flows, namely, the Arnold-Beltrami-Childress (ABC) flow [@mb02] and the Taylor-Green flow [@tg37], are in fact particular instances of the optimal field ${\mathbf{u}}_1$ corresponding to, respectively, cases (\[c1\]) and (\[c3\]). Following the notation of @df86, general ABC flows are characterized by the following velocity field $$\label{eq:ABC_flow}
\begin{array}{r@{\,\,}c@{\,\,}l}
u_1(x_1,x_2,x_3) & = & A'\sin(2\pi x_3) + C'\cos(2\pi x_2), \\
u_2(x_1,x_2,x_3) & = & B'\sin(2\pi x_1) + A'\cos(2\pi x_3), \\
u_3(x_1,x_2,x_3) & = & C'\sin(2\pi x_2) + B'\cos(2\pi x_1),
\end{array}$$ where $A'$, $B'$ and $C'$ real constants. The vector field in equation can be obtained from equation by choosing $\mathbf{A} = (B'/2)[0,-i,1]$, $\mathbf{B} = (C'/2)[1,0,-i]$ and $\mathbf{C} = (A'/2)[-i, 1,0]$. By analogy, we will refer to the state described by as the “aligned ABC flow”. Likewise, the well-known Taylor-Green vortex can be obtained as a particular instance of the field ${\mathbf{u}}_1$ from equation again using a suitable choice of the constants $\mathbf{A},\mathbf{B},\mathbf{C},\mathbf{D}$. Traditionally, the velocity field ${\mathbf{u}}= [u_1,u_2,u_3]$ characterizing the Taylor-Green vortex is defined as [@bmonmu83] $$\label{eq:TaylorGreen_vortex}
\begin{array}{r@{\,\,}c@{\,\,}l}
u_1(x_1,x_2,x_3) & = & \gamma_1\sin(2\pi x_1)\cos(2\pi x_2)\cos(2\pi x_3), \\
u_2(x_1,x_2,x_3) & = & \gamma_2\cos(2\pi x_1)\sin(2\pi x_2)\cos(2\pi x_3), \\
u_3(x_1,x_2,x_3) & = & \gamma_3\cos(2\pi x_1)\cos(2\pi x_2)\sin(2\pi x_3), \\
0 & = & \gamma_1+\gamma_2+\gamma_3,
\end{array}$$ for $\gamma_1,\gamma_2,\gamma_3\in\mathbb{R}$. For given values of $\gamma_1$,$ \gamma_2$ and $\gamma_3$ in , the corresponding constants $\mathbf{A},\mathbf{B},\mathbf{C},\mathbf{D}$ in can be found by separating them into their real and imaginary parts denoted, respectively, ${\mathbf{A}_{\textrm{Re}}},{\mathbf{B}_{\textrm{Re}}},{\mathbf{C}_{\textrm{Re}}},{\mathbf{D}_{\textrm{Re}}}$ and ${\mathbf{A}_{\textrm{Im}}},{\mathbf{B}_{\textrm{Im}}},{\mathbf{C}_{\textrm{Im}}},{\mathbf{D}_{\textrm{Im}}}$. Then, after choosing $${\mathbf{A}_{\textrm{Re}}}={\mathbf{B}_{\textrm{Re}}}={\mathbf{C}_{\textrm{Re}}}={\mathbf{D}_{\textrm{Re}}}=\mathbf{0} = [0,0,0],$$ the imaginary parts can be determined by solving the following system of linear equations $$2\left[
\begin{array}{cccc}
I_3 & -I_3 & -I_3 & -I_3 \\
-I_3 & I_3 & -I_3 & -I_3 \\
-I_3 & -I_3 & I_3 & -I_3 \\
-I_3 & -I_3 & -I_3 & I_3
\end{array}
\right]
\left[
\begin{array}{c}
{\mathbf{A}_{\textrm{Im}}} \\
{\mathbf{B}_{\textrm{Im}}} \\
{\mathbf{C}_{\textrm{Im}}} \\
{\mathbf{D}_{\textrm{Im}}}
\end{array}
\right] =
\left[
\begin{array}{c}
\mathbf{0} \\
\gamma_1\mathbf{e}_1 \\
\gamma_2\mathbf{e}_2 \\
\gamma_3\mathbf{e}_3
\end{array}
\right],$$ where $I_3$ is the $3 \times 3$ identity matrix. The values of ${\mathbf{A}_{\textrm{Im}}},\ldots,{\mathbf{D}_{\textrm{Im}}}$ are thus given by $${\mathbf{A}_{\textrm{Im}}} = -\frac{1}{8}\left[ \begin{array}{c}\gamma_1 \\ \gamma_2 \\ \gamma_3 \end{array} \right],\,
{\mathbf{B}_{\textrm{Im}}} = -\frac{1}{8}\left[ \begin{array}{c}-\gamma_1 \\ \gamma_2 \\ \gamma_3 \end{array} \right],\,
{\mathbf{C}_{\textrm{Im}}} = -\frac{1}{8}\left[ \begin{array}{c}\gamma_1 \\ -\gamma_2 \\ \gamma_3 \end{array} \right],\,
{\mathbf{D}_{\textrm{Im}}} = -\frac{1}{8}\left[ \begin{array}{c}\gamma_1 \\ \gamma_2 \\ -\gamma_3 \end{array} \right].$$ A typical choice of the parameters used in the numerical studies performed by [@bmonmu83] and [@b91] is $\gamma_1 = - \gamma_2
= 1$ and $\gamma_3 = 0$.
We remark that the Taylor-Green vortex has been employed as the initial data in a number of studies aimed at triggering singular behaviour in both the Euler and Navier-Stokes systems [@tg37; @bmonmu83; @b91; @bb12]. It is therefore interesting to note that it arises in the limit $\E_0 \rightarrow 0$ as one of the extreme vortex states in the variational formulation considered in the present study. It should be emphasized, however, that out of the three optimal states identified above (see Table \[tab:E0\]), the Taylor-Green vortex is characterized by the smallest (i.e., the most negative) instantaneous rate of enstrophy production $d\E/dt$. On the other hand, we are not aware of any prior studies involving ABC flows in the context of extreme behaviour and potential singularity formation. The time evolution corresponding to these states and some other initial data will be analyzed in detail in §\[sec:timeEvolution\].
Extreme Vortex States with Finite $\E_0$ {#sec:3D_InstOpt_E}
========================================
In this section we analyze the optimal vortex states ${\widetilde{\mathbf{u}}_{\E_0}}$ obtained for finite values of the enstrophy in which we extend the results obtained in the seminal study by [@ld08]. As was also the case in the analogous study in 2D [@ap13a], there is a distinct branch of extreme states ${\widetilde{\mathbf{u}}_{\E_0}}$ parameterized by the enstrophy $\E_0$ and corresponding to each of the three limiting states discussed in §\[sec:3D\_InstOpt\_E0to0\] (cf. figure \[fig:maxdEdt\_vortexCells\] and Table \[tab:E0\]). Each of these branches is computed using the continuation approach outlined in Algorithm \[alg:optimAlg\]. As a key element of the gradient-based maximization technique , the gradient expressions – are approximated pseudo-spectrally using standard dealiasing of the nonlinear terms and with resolutions varying from $128^3$ in the low-enstrophy cases to $512^3$ in the high-enstrophy cases, which necessitated a massively parallel implementation using the Message Passing Interface (MPI). As regards the computation of the Sobolev $H^2$ gradients, cf. , we set $\ell_1 = 0$, whereas the second parameter $\ell_2$ was adjusted during the optimization iterations and was chosen so that $\ell_2 \in [\ell_{\min},
\ell_{\max}]$, where $\ell_{\min}$ is the length scale associated with the spatial resolution $N$ used for computations and $\ell_{\max}$ is the characteristic length scale of the domain $\Omega$, that is, $\ell_{\min} \sim \O( 1/N) $ and $\ell_{\max} \sim \O(1)$. We remark that, given the equivalence of the inner products corresponding to different values of $\ell_1$ and $\ell_2$ (as long as $\ell_2 \neq 0$), these choices do not affect the maximizers found, but only how rapidly they are approached by iterations . For further details concerning the computational approach we refer the reader to the dissertation by [@a14]. As was the case in the analogous 2D problem studied by @ap13a, the largest instantaneous growth of enstrophy is produced by the states with vortex cells staggered in all planes, cf. case (\[c1\]) in Table \[tab:E0\]. Therefore, in our analysis we will focus exclusively on this branch of extreme vortex states which has been computed for $\E_0 \in [10^{-3},2\times10^2]$.
The optimal instantaneous rate of growth of enstrophy $\R_{\E_0} =
\R({\widetilde{\mathbf{u}}_{\E_0}})$ and the energy of the optimal states $\K({\widetilde{\mathbf{u}}_{\E_0}})$ are shown as functions of $\E_0$ for small $\E_0$ in figures \[fig:RvsE0\_FixE\_small\](a) and \[fig:RvsE0\_FixE\_small\](b), respectively. As indicated by the asymptotic form of $\R$ in and the Poincaré limit [$\K_0=\E_0/(2\pi)^2$]{}, both of which are marked in these figures, the behaviour of $\R_{\E_0}$ and $\K({\widetilde{\mathbf{u}}_{\E_0}})$ as $\E_0 \rightarrow 0$ is correctly captured by the numerically computed optimal states. In particular, we note that $\R_{\E_0}$ is negative for $0 \le
\E_0 \lessapprox 7$ and exhibits the same trend as predicted in for $\E_0 \rightarrow 0$. For larger values of $\E_0$ the rate of growth of enstrophy becomes positive. Likewise, the asymptotic behaviour of the energy of the optimal fields does not come as a surprise since, as discussed in §\[sec:3D\_InstOpt\_E0to0\], in the limit $\E_0 \to 0$ the maximizers of $\R$ are eigenfunctions of the Laplacian, which also happen to saturate Poincaré’s inequality.
The structure of the optimal vortex states ${\widetilde{\mathbf{u}}_{\E_0}}$ is analyzed next. They are visualized using in which the vortex cores are identified as regions $\Sigma := \{\Gamma_{s}(Q): s
\ge 0 \}$ for $Q$ defined as [@davidson:turbulence] $$\label{eq:Q_3D}
Q({\mathbf{x}}) := \, \frac{1}{2}\left[ {\operatorname{tr}}(\bm{\Omega}\bm{\Omega}^T) - {\operatorname{tr}}(\mathbf{S}\mathbf{S}^T) \right],$$ where $\mathbf{S}$ and $\bm{\Omega}$ are the symmetric and anti-symmetric parts of the velocity gradient tensor $\nabla{\mathbf{u}}$, that is, $[\mathbf{S}]_{ij} = \frac{1}{2}(\partial_j u_i + \partial_i
u_j)$ and $[\bm{\Omega}]_{ij} = \frac{1}{2}(\partial_j u_i -
\partial_i u_j )$, $i,j=1,2,3$. The quantity $Q$ can be interpreted as the local balance between the strain rate and the vorticity magnitude. The isosurfaces $\Gamma_{0}(Q - 0.5||Q||_{L_\infty})$ representing the optimal states ${\widetilde{\mathbf{u}}_{\E_0}}$ with selected values of $\E_0$ are shown in figures \[fig:RvsE0\_FixE\_small\](c)-(e). For the smallest values of $\E_0$, the optimal fields exhibit a cellular structure already observed in figure \[fig:maxdEdt\_vortexCells\](a). For increasing values of $\E_0$ this cellular structure transforms into a vortex ring, as seen in figure \[fig:RvsE0\_FixE\_small\](e). The component of vorticity normal to the plane $P_x = \{{\mathbf{x}}\in\Omega : \mathbf{n}\cdot({\mathbf{x}}-{\mathbf{x}}_0) = 0
\}$ for $\mathbf{n} = [1,0,0]$ and ${\mathbf{x}}_0 = [1/2,1/2,1/2]$ is shown in figures \[fig:RvsE0\_FixE\_small\](f)-(h), where the transition from cellular structures to a localized vortex structure as enstrophy increases is evident.
The results corresponding to large values of $\E_0$ are shown in figure \[fig:RvsE0\_FixE\_large\] with the maximum rate of growth of enstrophy $\R_{\E_0}$ plotted as a function of $\E_0$ in figure \[fig:RvsE0\_FixE\_large\](a). We observe that, as $\E_0$ increases, this relation approaches a power law of the form $\R_{\E_0} = C_1' \,\E_0^{\alpha_1}$. In order to determine the prefactor $C_1'$ and the exponent $\alpha_1$ we perform a local least-squares fit of the power law to the actual relation $\R_{\E_0}$ versus $\E_0$ for increasing values of $\E_0$ starting with $\E_{0} = 20$ (this particular choice the starting value is justified below). Then, the exponent $\alpha_1$ is computed as the average of the exponents obtained from the local fits with their standard deviation providing the error bars, so that we obtain $$\label{eq:RvsE0_powerLaw}
\R_{\E_0} = C'_1\E_0^{\,\alpha_1}, \qquad C'_1 = 3.72 \times 10^{-3} , \ \alpha_1 = 2.97 \pm 0.02$$ (the same approach is also used to determine the exponents in other power-law relations detected in this study). We note that the exponent $\alpha_1$ obtained in is in fact very close to 3 which is the exponent in estimate . For the value of the viscosity coefficient used in the computations ($\nu=0.01$), the constant factor $C_1 = 27/(8\pi^4\nu^3)$ in estimate has the value $C_1 \approx 3.465 \times 10^4$ which is approximately seven orders of magnitude larger than $C'_1$ given in . To shed more light at the source of this discrepancy, the objective functional $\R$ from equation can be separated into a negative-definite viscous part $\R_{\nu}$ and a cubic part $\R_{\textrm{cub}}$ defined as
$$\begin{aligned}
\R_{\nu}({\mathbf{u}}) & := -\nu\int_\Omega |{\Delta}{\mathbf{u}}|^2\,d{\mathbf{x}}, \\
\R_{\textrm{cub}}({\mathbf{u}}) & := \int_{\Omega} {\mathbf{u}}\cdot\nabla{\mathbf{u}}\cdot{\Delta}{\mathbf{u}}\, d{\mathbf{x}}, \label{eq:Rcub}\end{aligned}$$
so that $\R({\mathbf{u}}) = \R_{\nu}({\mathbf{u}}) + \R_{\textrm{cub}}({\mathbf{u}})$. The values of $\R_{\textrm{cub}}({\widetilde{\mathbf{u}}_{\E_0}})$ are also plotted in figure \[fig:RvsE0\_FixE\_large\](a) and it is observed that this quantity exhibits a power-law behaviour of the form $$\label{eq:RcubvsE0_powerLaw}
\R_{\textrm{cub}}({\widetilde{\mathbf{u}}_{\E_0}}) = C''_1\E_0^{\,\alpha_2}, \qquad C''_1 = 1.38\times 10^{-2},\ \alpha_2 = 2.99 \pm 0.05.$$ While the value of $C''_1$ is slightly larger than the value of $C'_1$ in , it is still some six orders of magnitude smaller than the constant factor $C_1 = 27/(8\pi^4\nu^3)$ from estimate . These differences notwithstanding, we may conclude that estimate is sharp in the sense of definition \[def:NotionSharpness\]. The power laws from equations and are consistent with the results first presented by @l06 [@ld08], where the authors reported a power-law with exponent $\alpha_{LD} = 2.99$ and a constant of proportionality $C_{LD} = 8.97\times 10^{-4}$. The energy of the optimal fields $\K({\widetilde{\mathbf{u}}_{\E_0}})$ for large values of $\E_0$ is shown in figure \[fig:RvsE0\_FixE\_large\](b) in which we observe that the energy stops to increase at about $\E_0 \approx 20$. This transition justifies using $\E_0 = 20$ as the lower bound on the range of $\E_0$ where the power laws are determined via least-square fits.
Figures \[fig:RvsE0\_FixE\_large\](c)-(e) show the isosurfaces $\Gamma_{0}(Q - 0.5||Q||_{L_\infty})$ representing the optimal fields ${\widetilde{\mathbf{u}}_{\E_0}}$ for selected large values of $\E_0$. The formation of these localized vortex structures featuring two rings as $\E_0$ increases is evident in these figures. The formation process of localized vortex structures is also visible in figures \[fig:RvsE0\_FixE\_large\](f)–(h), where the component of vorticity normal to the plane $P_{xz} = \{{\mathbf{x}}\in\Omega : \mathbf{n}\cdot({\mathbf{x}}- {\mathbf{x}}_0) = 0 \}$ for $\mathbf{n} = [1,0,-1]$ and ${\mathbf{x}}_0 =
[1/2,1/2,1/2]$ is shown (we note that the planes used in figures \[fig:RvsE0\_FixE\_small\](c)–(e) and \[fig:RvsE0\_FixE\_large\](c)–(e) have different orientations).
\
\
\
\
Next we examine the variation of different diagnostics applied to the extreme states ${\widetilde{\mathbf{u}}_{\E_0}}$ as enstrophy $\E_0$ increases. The maximum velocity $||{\widetilde{\mathbf{u}}_{\E_0}}||_{L_\infty}$ and maximum vorticity $||{\bnabla\times}{\widetilde{\mathbf{u}}_{\E_0}}||_{L_\infty}$ of the optimal fields are shown, respectively, in figures \[fig:ScalingLaws\_fixE\](a) and \[fig:ScalingLaws\_fixE\](b) as functions of $\E_0$. For each quantity, two distinct power laws are observed in the forms
$$\begin{aligned}
{4}
||{\widetilde{\mathbf{u}}_{\E_0}}||_{L_\infty} & \sim C_1\E^{\alpha_1}_0,& \quad
C_1 &= 0.263,& \alpha_1 & = 0.5 \pm 0.023,& \quad &
\mbox{as }\, \E_0\to 0, \\
||{\widetilde{\mathbf{u}}_{\E_0}}||_{L_\infty} & \sim C_2\E^{\alpha_2}_0,& \quad
C_2 &= 6.3\times 10^{-2},& \quad \alpha_2 &= 1.04 \pm 0.13,& \qquad &
\mbox{as }\, \E_0\to \infty, \label{eq:uLinf}\end{aligned}$$
and
$$\begin{aligned}
{4}
||{\bnabla\times}{\widetilde{\mathbf{u}}_{\E_0}}||_{L_\infty} & \sim C_1\E_0^{\alpha_1},& \quad
C_1 &= 2.09,& \alpha_1 & = 0.54 \pm 0.03,& \quad &
\mbox{as }\, \E_0\to 0, \\
||{\bnabla\times}{\widetilde{\mathbf{u}}_{\E_0}}||_{L_\infty} & \sim C_2\E_0^{\alpha_2},& \quad
C_2 &= 6.03\times 10^{-2},& \quad \alpha_2 &= 1.99 \pm 0.17,& \quad &
\mbox{as }\, \E_0\to \infty.
\label{eq:omegaLinf}\end{aligned}$$
In order to quantify the variation of the relative size of the vortex structures, we will introduce two characteristic length scales. The first one is based on the energy and enstrophy, and was defined by [@dg95] as $$\label{eq:Lambda_def}
\Lambda := \frac{1}{2\pi}\left[ \frac{\K({\widetilde{\mathbf{u}}_{\E_0}})}{\E({\widetilde{\mathbf{u}}_{\E_0}})} \right]^{1/2}.$$ It is therefore equivalent to the Taylor microscale $\lambda^2
= 15\int_\Omega |{\mathbf{u}}|^2 d{\mathbf{x}}/ \int_\Omega |{\bm{\omega}}|^2 d{\mathbf{x}}$ used in turbulence research [@davidson:turbulence]. Another length scale, better suited to the ring-like vortex structures shown in figures \[fig:RvsE0\_FixE\_large\](c)-(e), is the average radius $R_{\Pi}$ of one of the vortex rings calculated as $$\label{eq:VortexRadius_def}
R_{\Pi} := \frac{ \int_\Omega r({\mathbf{x}})\chi_{\Pi}({\mathbf{x}}) \,d{\mathbf{x}}}{ \int_\Omega \chi_{\Pi}({\mathbf{x}})d{\mathbf{x}}},
\ \text{where} \ \
r({\mathbf{x}}) = |{\mathbf{x}}- \overline{{\mathbf{x}}}|,\quad \overline{{\mathbf{x}}} = \frac{\int_{\Omega} {\mathbf{x}}\chi_{\Pi}({\mathbf{x}}) d{\mathbf{x}}}{\int_{\Omega} \chi_{\Pi}({\mathbf{x}})d{\mathbf{x}}},$$ and $\chi_{\Pi}$ is the characteristic function of the set $$\begin{aligned}
\Pi & = & \{ \Gamma_s( Q ) : s > 0.9|| Q ||_{L_\infty}\} \cap \\
& & \{ {\mathbf{x}}\in\Omega : {\mathbf{n}}\cdot({\mathbf{x}}-{\mathbf{x}}_0) > 0, \ {\mathbf{n}}= [1,1,1], \ {\mathbf{x}}_0 = [1/2,1/2,1/2] \}.\end{aligned}$$ In the above definition of the set $\Pi$, the intersection of the two regions is necessary to restrict the set $R_{\Pi}$ to only one of the two ring structures visible in figures \[fig:RvsE0\_FixE\_large\](c)–(e). The quantity $\overline{{\mathbf{x}}}$ can be therefore interpreted as the geometric centre of one of the vortex rings. The dependence of $\Lambda$ and $R_\Pi$ on $\E_0$ is shown in figures \[fig:ScalingLaws\_fixE\](c,d) in which the following power laws can be observed
$$\begin{aligned}
{4}
\Lambda &\sim \O(1)\quad\mbox{and}& \quad R_\Pi &\sim \O(1)&&& \quad &
\mbox{as } \,\E_0\to 0, \\
\Lambda &\sim C_1\E_0^{\alpha_1},& \quad
C_1 &= 10.96,& \alpha_1 &= -0.886 \pm 0.105,& \qquad &
\mbox{as } \, \E_0\to \infty, \label{eq:Lambda_powerLaw_largeE0} \\
R_\Pi &\sim C_2\E_0^{\alpha_2},& \quad
C_2 &= 2.692,& \quad \alpha_2 &= -1.01 \pm 0.16,& \quad &
\mbox{as } \, \E_0\to \infty.
\label{eq:Radius_powerLaw_largeE0}\end{aligned}$$
By comparing the error bars in the key power laws and with the error bars in power-law relations , , and , we observe that there is less uncertainty in the first case, indicating that the quantities $||{\widetilde{\mathbf{u}}_{\E_0}}||_{L_\infty}$, $||{\bnabla\times}{\widetilde{\mathbf{u}}_{\E_0}}||_{L_\infty}$, $\Lambda$ and $R_\Pi$ tend to be more sensitive to approximation errors than $\R_{\E_0}({\widetilde{\mathbf{u}}_{\E_0}})$. Non-negligible error bars may also indicate that, due to modest enstrophy values attained in our computations, the ultimate asymptotic regime corresponding to $\E_0 \rightarrow
\infty$ has not been reached in some power laws.
A useful aspect of employing the average ring radius $R_{\Pi}$ as the characteristic length scale is that its observed scaling with respect to $\E_0$ can be used as an approximate indicator of the resolution $1/N$ required to numerically solve problem \[pb:maxdEdt\_E\] for large values of enstrophy. From the scaling in relation , it is evident that a two-fold increase in the value of $\E_0$ will be accompanied by a similar reduction in $R_\Pi$, thus requiring an eight-fold increase in the resolution (a two-fold increase in each dimension). This is one of the reasons why computation of extreme vortex states ${\widetilde{\mathbf{u}}_{\E_0}}$ for large enstrophy values is a very challenging computational task. In particular, this relation puts a limit on the largest value of $\E_0$ for which problem \[pb:maxdEdt\_E\] can be in principle solved computationally at the present moment: a value of $\E_0 =
2000$, a mere order of magnitude above the largest value of $\E_0$ reported here, would require a resolution of $8192^3$ used by some of the largest Navier-Stokes simulations to date.
To summarize, as the enstrophy increases from $\E_0 \approx 0$ to $\E_0 = \O(10^2)$, the optimal vortex states change their structure from cellular to ring-like. While with the exception of $\R({\widetilde{\mathbf{u}}_{\E_0}})$ and $\K({\widetilde{\mathbf{u}}_{\E_0}})$, all of the diagnostic quantities behave in a monotonous manner, the corresponding power laws change at about $\E_0
\approx 20$, which approximately marks the transition from the cellular to the ring-like structure (cf. figure \[fig:RvsE0\_FixE\_small\](e) vs. \[fig:RvsE0\_FixE\_large\](c)). This is also the value of the enstrophy beyond which the energy $\K({\widetilde{\mathbf{u}}_{\E_0}})$ starts to decrease (figure \[fig:RvsE0\_FixE\_large\](b)). This transition also coincides with a change of the symmetry properties of the extreme vortex states ${\widetilde{\mathbf{u}}_{\E_0}}$ — while in the limit $\E_0 \rightarrow 0$ these fields feature reflection and discrete rotation symmetries (cf. § \[sec:3D\_InstOpt\_E0to0\]), for $20 \lessapprox \E_0 \rightarrow
\infty$ the optimal states are characterized by axial symmetry. The asymptotic (as $\E_0 \rightarrow \infty$) extreme vortex states on locally maximizing branches corresponding to the aligned ABC flow and the Taylor-Green vortex (cf. Table \[tab:E0\]) are similar to the fields shown in figures \[fig:RvsE0\_FixE\_large\](c)–(h), except for a different orientation of their symmetry axes with respect to the periodic domain $\Omega$ (these results are not shown here for brevity). The different power laws found here are compared to the corresponding results obtained in 2D in §\[sec:discuss\]. It is also worth mentioning that, as shown by [@ad16], all power laws discussed in this section, cf. , , , and , can be deduced rigorously using arguments based on dimensional analysis under the assumption of axisymmetry for the optimal fields ${\widetilde{\mathbf{u}}_{\E_0}}$.
\
Finally, the findings of this section allow us to shed some light on the “small data” result which provides the conditions on the size of the initial data ${\mathbf{u}}_0$, given in terms of its energy $\K(0)$ and enstrophy $\E(0)$, in the Navier-Stokes system guaranteeing that smooth solutions exist globally in time. The power-law fits and allow us to sharpen condition be replacing the constant on the RHS with either $2\nu / C'_1$ or $2\nu / C''_1$, so that we obtain $$\label{eq:K0E02}
\K(0)\E(0) < \left\{ \frac{2\nu}{C'_1} \ \ \text{or} \ \ \frac{2\nu}{C''_1} \right\}.$$ The region of the “phase space” $\{\K,\E\}$ described by condition is shown in white in figure \[fig:K0E0\]. The gray region represents the values of $\K(0)$ and $\E(0)$ for which long-time existence of smooth solutions cannot be a priori guaranteed (the two shades of gray correspond to the two constants which can be used in ). Solid circles represent the different extreme states found in this section, whereas the thin curves mark the time-dependent trajectories which will be analyzed in §\[sec:timeEvolution\]. We conclude from figure \[fig:K0E0\] that the change of the properties of the optimal states ${\widetilde{\mathbf{u}}_{\E_0}}$ discussed above occurs in fact for the values of enstrophy ($\E(0) \approx 20$) for which the states ${\widetilde{\mathbf{u}}_{\E_0}}$ are on the boundary of the region of guaranteed long-time regularity.
Time Evolution of Extreme Vortex States {#sec:timeEvolution}
=======================================
The goal of this section is to analyze the time evolution, governed by the Navier-Stokes system , with extreme vortex states identified in §\[sec:3D\_InstOpt\_E\] used as the initial data ${\mathbf{u}}_0$. In particular, we are interested in the finite-time growth of enstrophy $\E(t)$ and how it relates to estimates , and . We will compare these results with the growth of enstrophy obtained using other types of initial data which have also been studied in the context of the blow-up problem for both the Euler and Navier-Stokes systems, namely, the Taylor-Green vortex [@tg37; @bmonmu83; @b91; @bb12], the Kida-Pelz flow [@bp94; @p01; @ghdg08], colliding Lamb-Chaplygin dipoles [@opc12] and perturbed antiparallel vortex tubes [@k93; @k13]. Precise characterization of these different initial conditions is provided in Table \[tab:InitialConditions\] and, for the sake of completeness, the last three states are also visualized in figure \[fig:SummaryIC\]. We comment that, with the exception of the Taylor-Green vortex which was shown in §\[sec:3D\_InstOpt\_E0to0\] to be a local maximizer of problem \[pb:maxdEdt\_E\] in the limit $\E_0 \rightarrow 0$, all these initial conditions were postulated based on rather ad-hoc physical arguments. We also add that, in order to ensure a fair comparison, the different initial conditions listed in Table \[tab:InitialConditions\] are rescaled to have the same enstrophy $\E_0$, which is different from the enstrophy values used in the original studies where these initial conditions were investigated [@opc12; @k13; @dggkpv13; @opmc14]. As regards our choices of the initial enstrophy $\E_0$, to illustrate different possible behaviours, we will consider initial data located in the two distinct regions of the phase space $\{\K,\E\}$ shown in figure \[fig:K0E0\], corresponding to values of $\K_0$ and $\E_0$ for which global regularity may or may not be a priori guaranteed according to estimates –.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$${\mathbf{u}}_0({\mathbf{x}}) = [u, v, w]$$
------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------
Instantaneous
optimizer ${\widetilde{\mathbf{u}}_{\E_0}}$ $${\mathbf{u}}_0 = \mathop{{\operatorname{argmax}}}_{{\mathbf{u}}\in\mathcal{S}_{\E_0}} \R({\mathbf{u}})$$ See problem
Taylor-Green
vortex $$\begin{aligned} $\bm{\gamma}=(1,-1,0)$ in
u(x,y,z) & = & A\sin(2\pi x)\cos(2\pi y)\cos(2\pi z) \\
v(x,y,z) & = & -A\cos(2\pi x)\sin(2\pi y)\cos(2\pi z) \\
w(x,y,z) & = & 0
\end{aligned}$$
equation ,
$A$ chosen so that $\E({\mathbf{u}}_0) = \E_0$.
Kida-Pelz
flow $$\begin{aligned} Taken from
u(x,y,z) & = & A\sin(2\pi x)[ \cos(6\pi y)\cos(2\pi z) - \\
& & \cos(2\pi y)\cos(6\pi z) ] \\
v(x,y,z) & = & A\sin(2\pi y)[ \cos(6\pi z)\cos(2\pi x) - \\
& & \cos(2\pi x_3)\cos(6\pi x_1) ] \\
w(x,y,z) & = & A\sin(2\pi z)[ \cos(6\pi x)\cos(2\pi y) - \\
& & \cos(2\pi x)\cos(6\pi y) ] \\
\end{aligned}$$
@bp94,
$A$ chosen so that $\E({\mathbf{u}}_0) = \E_0$.
Lamb-Chaplygin
dipoles $$-{\Delta}{\mathbf{u}}_0 = {\bnabla\times}{\bm{\omega}}_0, \quad {\bm{\omega}}_0 = [\, 0, \, \omega(x,z), \,\omega(x,y)\,]$$ $$\omega(x(r,\theta),y(r,\theta)) = \left\{ Taken from
\begin{array}{c@{\,\,}r}
-2U\kappa \frac{J_1(\kappa r)}{J_0(\kappa a)}\sin(\theta) & (r \leq a) \\
0 & (r > a)
\end{array}
\right.$$
@opc12.
$a = 0.15$, $\kappa a = z_1$,
the first zero of $J_1$
$U = \sqrt{\frac{\E_0}{2\pi z^2_1}}$
Perturbed
anti-parallel
vortex tubes $$-{\Delta}{\mathbf{u}}_0 = {\bnabla\times}{\bm{\omega}}_0, \quad {\bm{\omega}}_0 = \omega(x,y)\frac{\bm{\sigma'}}{|\bm{\sigma}'|}(s)$$ $$\omega(x(r,\theta),y(r,\theta)) = \frac{A}{(r/a)^{4} + 1}$$ $$\bm{\sigma}(s) = [2a, 2b/\cosh(s^2/c^2)-b, s]$$ Taken from
@k13.
$a= 0.05$, $b=a/2$,
$c=a$, $s$ is the arc-length parameter.
$A$ chosen so that $\E({\mathbf{u}}_0) = \E_0$.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Characterization of the different initial data used in time evolution studies in §\[sec:timeEvolution\].[]{data-label="tab:InitialConditions"}
System is solved numerically with an approach combining a pseudo-spectral approximation of spatial derivatives with a third-order semi-implicit Runge-Kutta method [@NumRenaissance] used to discretize the problem in time. In the evaluation of the nonlinear term dealiasing was used based on the $2/3$ rule together with the Gaussian filtering proposed by [@hl07]. Massively parallel implementation based on MPI and using the [ fftw]{} routines [@fftw] to perform Fourier transforms allowed us to use resolutions varying from $256^3$ to $1024^3$ in the low-enstrophy and high-enstrophy cases, respectively. A number of different diagnostics were checked to ensure that all flows discussed below are well resolved. We refer the reader to the dissertation by [@a14] for additional details and a validation of this approach.
The time-dependent results will be shown with respect to a normalized time defined as $\tau := U_c t/\ell_c$ with $U_c := \|{\widetilde{\mathbf{u}}_{\E_0}}\|_{L_2}$ and $\ell_c = \Lambda$ (cf. equation ) playing the roles of the characteristic velocity and length scale. We begin by showing the time evolution of the enstrophy $\E(\tau)$ corresponding to the five different initial conditions listed in Table \[tab:InitialConditions\] with $\E_0 = 10$ and $\E_0 = 100$ in figures \[fig:Fvs\_t\_fixE\](a) and \[fig:Fvs\_t\_fixE\](b), respectively (because of the faster time-scale, the time axis in the latter figure is scaled logarithmically). We see that the maximizers ${\widetilde{\mathbf{u}}_{\E_0}}$ of problem \[pb:maxdEdt\_E\] are the only initial data which triggers growth of enstrophy for these values of the initial enstrophy and, as expected, this growth is larger for $\E_0 = 100$ than for $\E_0 = 10$. The other initial condition which exhibits some tendency for growth when $\E_0 = 100$ is the Taylor-Green vortex. In all cases the enstrophy eventually decays to zero for large times.
Next we examine whether the flow evolutions starting from the instantaneous maximizers ${\widetilde{\mathbf{u}}_{\E_0}}$ as the initial data saturate the finite-time estimate . We do this by defining functions
\[eq:fg\] $$\begin{aligned}
f(\tau) & := \frac{1}{\E(0)} - \frac{1}{\E(\tau)}\qquad\mbox{and} \label{eq:fga} \\
g(\tau) & := \frac{C}{2\nu}\left[ \K(0) - \K(\tau) \right] \label{eq:fgb}\end{aligned}$$
representing, respectively, the left- and right-hand side of the estimate and then plotting them with respect to the normalized time $\tau$, which is done in figures \[fig:fg\](a) and \[fig:fg\](b) for $\E_0 = 10$ and $\E_0 = 100$, respectively. The constant $C>0$ in the definition of $g(\tau)$ is numerically computed from the power-law fit in . It follows from estimate that $f(\tau) \leq g(\tau)$ pointwise in time. The hypothetical extreme event of a finite-time blow-up can be represented graphically by an intersection of the graph of $f(\tau)$ with the horizontal line $y = 1/\E_0$, which is also shown in figures \[fig:fg\](a)–(b). The behaviour of $g(\tau)$, representing the upper bound in estimate , is quite distinct in figures \[fig:fg\](a) and \[fig:fg\](b) reflecting the fact that the initial data ${\widetilde{\mathbf{u}}_{\E_0}}$ in the two cases comes from different regions of the phase diagram in figure \[fig:K0E0\]. In figure \[fig:fg\](a), corresponding to $\E_0 = 10$, the upper bound $g(\tau)$ never reaches $1/\E_0$, in agreement with the fact that the finite-time blow-up is a priori ruled out in this case. On the other hand, in figure \[fig:fg\](b), corresponding to $\E_0 = 100$, the upper bound $g(\tau)$ does intersect $1/\E_0$ implying that, in principle, finite-time blow-up might be possible in this case. The sharpness of estimate can be assessed by analyzing how closely the behaviour of $f(\tau)$ matches that of $g(\tau)$. In both figures \[fig:fg\](a) and \[fig:fg\](b) we observe that for a short period of time $f(\tau)$ exhibits a very similar growth to the upper bound $g(\tau)$, but then this growth slows down and $f(\tau)$ eventually starts to decrease short of ever approaching the limit $1/\E_0$.
We further characterize the time evolution by showing the maximum enstrophy increase $\delta\E := \mathop{\max}_{t \geq 0} \, \{ \E(t) -
\E(0) \}$ and the time when the maximum is achieved $T_{\max} :=
\mathop{\arg\max}_{t \geq 0}\, \E(t)$ as functions of $\E_0$ in figures \[fig:Emax\_vsE0\_fixE\](a) and \[fig:Emax\_vsE0\_fixE\](b), respectively. In both cases approximate power laws in the form $$\delta\E \sim \E^{\alpha_1}_0, \quad \alpha_1 = 0.95 \pm 0.06 \qquad\mbox{and}\qquad
T_{\max} \sim \E^{\alpha_2}_0, \quad \alpha_2 = -2.03 \pm 0.02$$ are detected in the limit $\E_0 \rightarrow \infty$ (as regards the second result, we remark that $T_{\max}$ is not equivalent to the time until which the enstrophy grows at the sustained rate proportional to $\E_0^3$, cf. figure \[fig:fg\]). To complete presentation of the results, the dependence of the quantities $$\mathop{\max}_{t \geq 0} \, \left\{\frac{1}{\E_0} - \frac{1}{\E(t)}\right\} \qquad\mbox{and}\qquad [\K(0) - \K(T_{\max})]$$ on the initial enstrophy $\E_0$ is shown in figures \[fig:Emax\_vsE0\_fixE\](c) and \[fig:Emax\_vsE0\_fixE\](d), respectively. It is observed that both quantities approximately exhibit a power-law behaviour of the form $\E^{-1}_0$. Discussion of these results in the context of the estimates recalled in §\[sec:intro\] is presented in the next section.
\
Discussion {#sec:discuss}
==========
In this section we provide some comments about the results reported in §§\[sec:3D\_InstOpt\_E0to0\], \[sec:3D\_InstOpt\_E\] and \[sec:timeEvolution\]. First, we need to mention that our gradient-based approach to the solution of optimization problem \[pb:maxdEdt\_E\] can only yield local maximizers and, due to nonconvexity of the problem, it is not possible to guarantee a priori that the maximizers found are global. To test for the possible presence of branches other than the ones found using the continuation approach described in §\[sec:3D\_InstOpt\], cf. Algorithm \[alg:optimAlg\], we tried to find new maximizers by initializing the gradient iterations with different initial guesses ${\mathbf{u}}^0$. They were constructed as solenoidal vector fields with prescribed regularity and random structure, which was achieved by defining the Fourier coefficients of ${\mathbf{u}}^0$ as $\widehat{{\mathbf{u}}}^0({\mathbf{k}}) = F(|{\mathbf{k}}|)e^{i\phi({\mathbf{k}})}$ with the amplitude $F(|{\mathbf{k}}|) \sim 1/|{\mathbf{k}}|^2$ and the phases $\phi({\mathbf{k}})$ chosen as random numbers uniformly distributed in $[0,2\pi]$. However, in all such tests conducted for $\E_0 = O(1)$ the gradient optimization algorithm would always converge to maximizers ${\widetilde{\mathbf{u}}_{\E_0}}$ belonging to one of the branches discussed in §\[sec:3D\_InstOpt\_E\] (modulo possible translations in the physical domain). While far from settling this issue definitely, these observations lend some credence to the conjecture that the branch identified in §\[sec:3D\_InstOpt\_E\] corresponds in fact to the global maximizers. These states appear identical to the maximizers found by [@ld08] and our search has also yielded two additional branches of locally maximizing fields, although we did not capture the lower branch reported by [@ld08]. However, since that branch does not appear connected to any state in the limit $\E_0 \rightarrow 0$, we speculate that it might be an artifact of the “discretize-then-optimize” formulation used by [@ld08], in contrast to the “optimize-then-discretize” approach employed in our study which provides a more direct control over the analytic properties of the maximizers. We add that the structure of the maximizing branches found here is in fact quite similar to what was discovered by [@ap13a] in an analogous problem in 2D. Since the 2D problem is more tractable from the computational point of view, in that case we were able to undertake a much more thorough search for other maximizers which did not however yield any solutions not associated with the main branches.
The results reported in §\[sec:3D\_InstOpt\_E\] and §\[sec:timeEvolution\] clearly exhibit two distinct behaviours, depending on whether or not global-in-time regularity can be guaranteed a priori based on estimates –. These differences are manifested, for example, in the power laws evident in figures \[fig:ScalingLaws\_fixE\] and \[fig:Emax\_vsE0\_fixE\], as well as in the different behaviours of the RHS of estimate with respect to time in figures \[fig:fg\](a) and \[fig:fg\](b). However, for the initial data for which global-in-time regularity cannot be ensured a priori there is no evidence of sustained growth of enstrophy strong enough to signal formation of singularity in finite time. Indeed, in figure \[fig:Emax\_vsE0\_fixE\](c) one sees that the quantity $\mathop{\max}_{t \geq 0} \, \left\{1/\E_0 - 1/\E(t)\right\}$ behaves as $C_1 / \E_0$, where $C_1 < 1$, when $\E_0$ increases, revealing no tendency to approach $1/\E_0$ which is a necessary precursor of a singular behaviour (cf. discussion in §\[sec:timeEvolution\]). To further illustrate how the rate of growth of enstrophy achieved initially by the maximizers ${\widetilde{\mathbf{u}}_{\E_0}}$ is depleted in time, in figure \[fig:dEdtE\] we show the flow evolution corresponding to ${\widetilde{\mathbf{u}}_{\E_0}}$ with $\E_0 = 100$ as a trajectory in the coordinates $\{\E, d\E/dt\}$. From the discussion in Introduction we know that in order for the singularity to occur in finite time, the enstrophy must grow at least at a sustained rate $d\E / dt \sim
\E^\alpha$ for some $\alpha > 2$. In other words, a “blow-up trajectory” will be realized only if the trajectory of the flow, expressed in $\{\E,d\E/dt \}$ coordinates, is contained in the region $\mathcal{M} = \{ (\E,d\E/dt) \; : \; C_1\E^2 < d\E/dt \leq C_2\E^3
\}$, for some positive constants $C_1$ and $C_2$. For the flow corresponding to the instantaneous optimizers ${\widetilde{\mathbf{u}}_{\E_0}}$, the initial direction of a trajectory in $\{\E,d\E/dt \}$ coordinates is determined by the vector ${\mathbf{v}}= \left[1,
\left.\tfrac{d\R}{d\E}\right|_{\E_0} \right]$ and, for initial conditions ${\mathbf{u}}_0$ satisfying $\R({\mathbf{u}}_0) = C\E^3({\mathbf{u}}_0)$, it follows that $$\left.\frac{d\R}{d\E}\right|_{\E_0} = 3C\E_0^2.$$ Since the optimal rate of growth is sustained only over a short interval of time, the trajectory of the flow in the $\{\E,d\E/dt\}$ coordinates approaches the region $\mathcal{M}$ only tangentially following the direction of the lower bound $C_1\E^2$, and remains outside $\mathcal{M}$ for all subsequent times. This behaviour is clearly seen in the inset of figure \[fig:dEdtE\].
An interpretation of this behaviour can be proposed based on equation from which it is clear that the evolution of the flow energy is closely related to the growth of enstrophy. In particular, if the initial energy $\K(0)$ is not sufficiently large, then its depletion due to the initial growth of enstrophy may render the flow incapable of sustaining this growth over a longer period of time. This is in fact what seems to be happening in the present problem as evidenced by the data shown in figure \[fig:K0E0\]. We remark that, for a prescribed enstrophy $\E(0)$, the flow energy cannot be increased arbitrarily as it is upper-bounded by Poincaré’s inequality [$\K(0) \le (2\pi)^2 \E(0)$]{}. This behaviour can also be understood in terms of the geometry of the extreme vortex states ${\widetilde{\mathbf{u}}_{\E_0}}$. Figure \[fig:ring\] shows a magnification of the pair of vortex rings corresponding to the optimal field ${\widetilde{\mathbf{u}}_{\E_0}}$ with $\E_0=100$. It is observed that the vorticity field ${\bnabla\times}{\widetilde{\mathbf{u}}_{\E_0}}$ inside the vortex core has an azimuthal component only which exhibits no variation in the azimuthal direction. Thus, in the limit $\E_0
\rightarrow \infty$ the vortex ring shrinks with respect to the domain $\Omega$ (cf. figure \[fig:ScalingLaws\_fixE\](d)) and the field ${\widetilde{\mathbf{u}}_{\E_0}}$ ultimately becomes axisymmetric (i.e., in this limit boundary effects vanish). At the same time, it is known that the 3D Navier-Stokes problem on an unbounded domain and with axisymmetric initial data is globally well posed [@k03], a results which is a consequence of the celebrated theorem due to @ckn83.
{width="90.00000%"}
. \[fig:dEdtE\]
![Vortex lines inside the region with the strongest vorticity in the extreme vortex state ${\widetilde{\mathbf{u}}_{\E_0}}$ with $\E_0 = 100$. The colour coding of the vortex lines is for identification purposes only.[]{data-label="fig:ring"}](Figs2/dnsNS3D_K00_E47_Instant_plane_vorLns){width="60.00000%"}
We close this section by comparing the different power laws characterizing the maximizers ${\widetilde{\mathbf{u}}_{\E_0}}$ and the corresponding flow evolutions with the results obtained in analogous studies of extreme behaviour in 1D and 2D (see also Table \[tab:estimates\]). First, we note that the finite-time growth of enstrophy $\delta\E$ in 3D, cf. figure \[fig:Emax\_vsE0\_fixE\](a), exhibits the same dependence on the enstrophy $\E_0$ of the instantaneously optimal initial data as in 1D, i.e., is directly proportional to $\E_0$ in both cases [@ap11a]. This is also analogous to the maximum growth of palinstrophy $\P$ in 2D which was found by [@ap13a] to scale with the palinstrophy $\P_0$ of the initial data, when the instantaneously optimal initial condition was computed subject to [ *one*]{} constraint only (on $\P_0$). When the instantaneously optimal initial data was determined subject to [*two*]{} constraints, on $\K_0$ and $\P_0$, then the maximum finite-time growth of palinstrophy was found to scale with $\P_0^{3/2}$ [@ap13b]. On the other hand, the time $T_{\max}$ when the maximum enstrophy is attained, cf. figure \[fig:Emax\_vsE0\_fixE\](b), scales as $\E_0^{-2}$, which should be contrasted with the scalings $\E_0^{-1/2}$ and $\P_0^{-1/2}$ found in the 1D and 2D cases, respectively. This implies that the time interval during which the extremal growth of enstrophy is sustained in 3D is shorter than the corresponding intervals in 1D and 2D.
Conclusions and Outlook {#sec:final}
=======================
By constructing the initial data to exhibit the most extreme behaviour allowed for by the mathematically rigorous estimates, this study offers a fundamentally different perspective on the problem of searching for potentially singular solutions from most earlier investigations. Indeed, while the corresponding flow evolutions did not reveal any evidence for finite-time singularity formation, the initial data obtained by maximizing $d\E/dt$ produced a significantly larger growth of enstrophy in finite time than any other candidate initial conditions (cf. Table \[tab:InitialConditions\] and figure \[fig:Fvs\_t\_fixE\]). Admittedly, this observation is limited to the initial data with $\E_0 \le 100$, which corresponds to Reynolds numbers $Re = \sqrt{\E_0\, \Lambda} / \nu \lessapprox 450$ lower than the Reynolds numbers achieved in other studies concerned with the extreme behaviour in the Navier-Stokes flows [@opc12; @k13; @dggkpv13; @opmc14]. Given that the definitions of the Reynolds numbers applicable to the various flow configurations considered in these studies were not equivalent, it is rather difficult to make a precise comparison in terms of specific numerical values, but it is clear that the largest Reynolds numbers attained in these investigations were at least an order of magnitude higher than used in the present study; for Euler flows such a comparison is obviously not possible at all. However, from the mathematical point of view, based on estimates –, there is no clear indication that a very large initial enstrophy $\E_0$ (or, equivalently, a high Reynolds number) should be a necessary condition for singularity formation in finite time. In fact, blow-up cannot be a priori ruled out as soon as condition is violated, which happens for all initial data lying on the gray region of the phase space in figure \[fig:K0E0\]. We remark that additional results were obtained (not reported in this paper) by studying the time evolution corresponding to the optimal initial data ${\widetilde{\mathbf{u}}_{\E_0}}$, but using smaller values of the viscosity coefficient $\nu=10^{-3},
10^{-4}$, thereby artificially increasing the Reynolds number at the price of making the initial data suboptimal. Although these attempts did increase the amplification of enstrophy as compared to what was observed in figures \[fig:fg\] and \[fig:Emax\_vsE0\_fixE\], no signature of finite-time singularity formation could be detected either.
Our study confirmed the earlier findings of [@ld08] about the sharpness of the instantaneous estimate . We also demonstrated that the finite-time estimate is saturated by the flow evolution corresponding to the optimal initial data ${\widetilde{\mathbf{u}}_{\E_0}}$, but only for short times, cf. figure \[fig:fg\], which are not long enough to trigger a singular behaviour.
In §\[sec:discuss\] we speculated that a relatively small initial energy $\K(0)$, cf. figure \[fig:RvsE0\_FixE\_large\](b), might be the property of the extreme vortex states ${\widetilde{\mathbf{u}}_{\E_0}}$ preventing the resulting flow evolutions from sustaining a significant growth of enstrophy over long times. On the other hand, in Introduction we showed that estimate need not be saturated for blow-up to occur in finite time and, in fact, sustained growth at the rate $d\E/dt = C \, \E^\alpha$ with any $\alpha > 2$ will also produce singularity in finite time. Thus, another strategy to construct initial data which could lead to a more sustained growth of enstrophy in finite time might be to increase its kinetic energy by allowing for a smaller instantaneous rate of growth (i.e., with an exponent $2 < \alpha \le 3$ instead of $\alpha = 3$). This can be achieved by prescribing an additional constraint in the formulation of the variational optimization problem, resulting in
\[pb:maxdEdt\_KE\] $$\begin{aligned}
{\widetilde{\mathbf{u}}_{\K_0,\E_0}}& = & \mathop{\arg\max}_{{\mathbf{u}}\in{\mathcal{S}_{\K_0,\E_0}}} \, \R({\mathbf{u}}) \\
{\mathcal{S}_{\K_0,\E_0}} & = & \left\{{\mathbf{u}}\in H_0^2(\Omega)\,\colon\,\nabla\cdot{\mathbf{u}}= 0, \; \K({\mathbf{u}}) = \K_0, \; \E({\mathbf{u}}) = \E_0 \right\}.\end{aligned}$$
It differs from problem \[pb:maxdEdt\_E\] in that the maximizers are sought at the intersection of the original constraint manifold ${\mathcal{S}_{\E_0}}$ and the manifold defined by the condition $\K({\mathbf{u}})
= \K_0$, where $\K_0 \le (2\pi)^2 \E_0$ is the prescribed energy. While computation of such maximizers is more complicated, robust techniques for the solution of optimization problems of this type have been developed and were successfully used in the 2D setting by [@ap13a]. Preliminary results obtained in the present setting by solving problem \[pb:maxdEdt\_KE\] for $\K_0 =
1$ are indicated in figure \[fig:K0E0\], where we see that the flow evolutions starting from ${\widetilde{\mathbf{u}}_{\K_0,\E_0}}$ do not in fact produce a significant growth of enstrophy either. An alternative, and arguably more flexible approach, is to formulate this problem in terms of multiobjective optimization [@k99] in which the objective function $\R({\mathbf{u}})$ in problem \[pb:maxdEdt\_E\] would be replaced with $$\R_{\eta}({\mathbf{u}}) := \eta\, \R({\mathbf{u}}) + (1-\eta)\, \K({\mathbf{u}}),
\label{eq:Rm}$$ where $\eta \in [0,1]$. Solution of such a multiobjective optimization problem has the form of a “Pareto front” parameterized by $\eta$. Clearly, the limits $\eta \rightarrow 1$ and $\eta \rightarrow 0$ correspond, respectively, to the extreme vortex states already found in §\[sec:3D\_InstOpt\_E0to0\] and §\[sec:3D\_InstOpt\_E\], and to the Poincaré limit. Another interesting possibility is to replace the energy $\K({\mathbf{u}})$ with the helicity [$\H({\mathbf{u}}) :=
\int_{\Omega} {\mathbf{u}}\cdot(\bnabla\times{\mathbf{u}})\,d\Omega$]{} in the multiobjective formulation , as this might allow one to obtain extreme vortex states with a more complicated topology (i.e., a certain degree of “knottedness”). We note that all the extreme vortex states found in the present study were “unknotted”, i.e., were characterized by $\H({\widetilde{\mathbf{u}}_{\E_0}}) = 0$, as the vortex rings were in all cases disjoint (cf. figure \[fig:ring\]).
Finally, another promising possibility to find initial data producing a larger growth of enstrophy is to solve a [*finite-time*]{} optimization problem of the type already studied by [@ap11a] in the context of the 1D Burgers equation, namely
\[pb:maxdE\] $$\tilde{{\mathbf{u}}}_{0;\E_0,T} = \mathop{\arg\max}_{{\mathbf{u}}_0\in{\mathcal{S}_{\E_0}}} \, \E(T),$$
where $T>0$ is the length of the time interval of interest and ${\mathbf{u}}_0$ the initial data for the Navier-Stokes system . In contrast to problems \[pb:maxdEdt\_E\] and \[pb:maxdEdt\_KE\], solution of problem \[pb:maxdE\] is more complicated as it involves flow evolution. It represents therefore a formidable computational task for the 3D Navier-Stokes system. However, it does appear within reach given the currently available computational resources and will be studied in the near future.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are indebted to Charles Doering for many enlightening discussions concerning the research problems studied in this work. The authors are grateful to Nicholas Kevlahan for making his parallel Navier-Stokes solver available, which was used to obtain the results reported in §\[sec:timeEvolution\]. Anonymous referees provided many insightful comments which helped us improve this work. This research was funded through an Early Researcher Award (ERA) and an NSERC Discovery Grant, whereas the computational time was made available by SHARCNET under its Dedicated Resource Allocation Program. Diego Ayala was funded in part by NSF Award DMS-1515161 at the University of Michigan and by the Institute for Pure and Applied Mathematics at UCLA.
[54]{} natexlab\#1[\#1]{}
2005 [*[Sobolev]{} Spaces*]{}. Elsevier.
2014 [Extreme Vortex States and Singularity Formation in Incompressible Flows]{}. PhD thesis, McMaster University, available at .
2016 Extreme enstrophy growth in axisymmetric flows. In preparation.
2016 An estimate for the growth of palinstrophy in [2D]{} incompressible flows. In preparation.
2011 On maximum enstrophy growth in a hydrodynamic system. [*Physica D*]{} [**240**]{}, 1553–1563.
2014 Maximum palinstrophy growth in [2D]{} incompressible flows. [*Journal of Fluid Mechanics*]{} [ **742**]{}, 340–367.
2014 Vortices, maximum growth and the problem of finite-time singularity formation. [*Fluid Dynamics Research*]{} [**46**]{} (3), 031404.
2009 [*Numerical Renaissance*]{}. Renaissance Press.
1994 Direct numerical simulation of transition to turbulence from a high-symmetry initial condition. [*Physics of Fluids*]{} [**6**]{}, 2757–2784.
1991 Direct simulation of three-dimensional turbulence in the [Taylor-Green]{} vortex. [*Fluid Dynamics Research*]{} [**8**]{}, 1–8.
1983 Small-scale structure of the [Taylor-Green]{} vortex. [ *Journal of Fluid Mechanics*]{} [**130**]{}, 411–452.
2012 Interplay between the [Beale-Kato-Majda]{} theorem and the analyticity-strip method to investigate numerically the incompressible [Euler]{} singularity problem. [*Phys. Rev. E*]{} [**86**]{}, 066302.
2008 3d euler about a 2d symmetry plane. [*Physica D*]{} [**237**]{}, 1912–1920.
1983 [Partial regularity of suitable weak solutions of the Navier-Stokes equations]{}. [*Communications on Pure and Applied Mathematics*]{} [**35**]{}, 771–831.
2005 Evolution of complex singularities in [Kida-Pelz]{} and [Taylor-Green]{} inviscid flows. [*Fluid Dynamics Research*]{} [**36**]{}, 239–248.
2004 [*Turbulence. An introduction for scientists and engineers*]{}. Oxford University Press.
2009 The [3D Navier-Stokes]{} problem. [*Annual Review of Fluid Mechanics*]{} pp. 109–128.
1995 [*Applied Analysis of the [Navier-Stokes]{} Equations*]{}. Cambridge University Press.
1986 Chaotic streamlines in the [ABC]{} flows. [*Journal of Fluid Mechanics*]{} [**167**]{}, 353–391.
2013 Vorticity moments in four numerical simulations of the [3D Navier-Stokes]{} equations. [*Journal of Fluid Mechanics*]{} [**732**]{}, 316–331.
2000 Existence and smoothness of the [Navier-Stokes]{} equation. available at , [Clay Millennium Prize Problem Description]{}.
1989 Gevrey class regularity for the solutions of the [Navier–Stokes]{} equations. [*Journal of Functional Analysis*]{} [ **87**]{}, 359–369.
2003 [*[FFTW]{} User’s Manual*]{}. Massachusetts Institute of Technology.
2014 [Regimes of nonlinear depletion and regularity in the 3D Navier-Stokes equations]{}. [*Nonlinearity*]{} [**27**]{} (1–19).
2008 The three–dimensional [Euler]{} equations: singular or non–singular? [*Nonlinearity*]{} [**21**]{}, 123–129.
2008 Numerical simulations of possible finite-time singularities in the incompressible [Euler]{} equations: comparison of numerical methods. [*Physica D*]{} [ **237**]{}, 1932–1936.
2003 [*Perspectives in Flow Control and Optimization*]{}. SIAM.
2009 Blow-up or no blow-up? a unified computational and analytic approach to [3D]{} incompressible [Euler]{} and [Navier–Stokes]{} equations. [*Acta Numerica*]{} pp. 277–346.
2007 Computing nearly singular solutions using pseudo-spectral methods. [*Journal of Computational Physics*]{} [**226**]{}, 379–397.
1993 Evidence for a singularity of the three-dimensional, incompressible [Euler]{} equations. [*Phys. Fluids A*]{} [**5**]{}, 1725–1746.
2013 Bounds for [Euler]{} from vorticity moments and line divergence. [*Journal of Fluid Mechanics*]{} [**729**]{}, R2.
2013 Swirling, turbulent vortex rings formed from a chain reaction of reconnection events. [*Physics of Fluids*]{} [**25**]{}, 065101.
2003 [Remarks for the axisymmetric Navier-Stokes equations]{}. [ *Journal of Differential Equations*]{} [**187**]{}, 226–239.
2010 Regularity and blow up for active scalars. [*Math. Model. Nat. Phenom.*]{} [**5**]{}, 225–255.
2004 [*Initial-Boundary Value Problems and the [Navier-Stokes]{} Equations*]{}, [*Classics in Applied Mathematics*]{}, vol. 47. SIAM.
1969 [*The mathematical Theory of Viscous Incompressible Flow*]{}. Gordon and Breach.
2006 Bounds on the enstrophy growth rate for solutions of the [3D Navier-Stokes]{} equations. PhD thesis, University of Michigan.
2008 Limits on enstrophy growth for solutions of the three-dimensional [Navier–Stokes]{} equations. [*Indiana University Mathematics Journal*]{} [**57**]{}, 2693–2727.
1969 [*Optimization by Vector Space Methods*]{}. John Wiley and Sons.
2014 [Potentially Singular Solutions of the 3D Axisymmetric Euler Equations]{}. [*Proceedings of the National Academy of Sciences*]{} (in press).
2014 [Toward the Finite-Time Blowup of the 3D Incompressible Euler Equations: a Numerical Investigation]{}. [*SIAM: Multiscale Modeling and Simulation*]{} (in press).
2002 [*Vorticity and Incompressible Flow*]{}. Cambridge University Press.
2008 Complex-space singularities of [2D Euler]{} flow in lagrangian coordinates. [*Physica D*]{} [**237**]{}, 1951–1955.
1999 [*Nonlinear Multiobjective Optimization*]{}. Springer.
2008 A miscellany of basic issues on incompressible fluid equations. [*Nonlinearity*]{} [**21**]{}, 255–271.
2008 [Numerical study of the Eulerian–Lagrangian analysis of the [Navier-Stokes]{} turbulence]{}. [*Phys. Fluids*]{} [**20**]{}, 1–11.
2014 A minimal flow unit for the study of turbulence with passive scalars. [ *Journal of Turbulence*]{} [**15**]{}, 731–751.
2012 Vortex events in [Euler]{} and [Navier-Stokes]{} simulations with smooth initial conditions. [ *Journal of Fluid Mechanics*]{} [**690**]{}, 288–320.
2001 Symmetry and the hydrodynamic blow-up problem. [ *Journal of Fluid Mechanics*]{} [**444**]{}, 299–320.
2004 A comprehensive framework for the regularization of adjoint analysis in multiscale [PDE]{} systems. [ *Journal of Computational Physics*]{} [**195**]{}, 49–89.
1990 Collapsing solutions to the [3D Euler]{} equations. [*Phys. Fluids A*]{} [**2**]{}, 220–241.
2006 [*Nonlinear Optimization*]{}. Princeton University Press.
2009 Calculation of complex singular solutions to the [3D]{} incompressible [Euler]{} equations. [*Physica D*]{} [**238**]{}, 2368–2379.
1937 Mechanism of the production of small eddies from large ones. [*Proceedings of the Royal Society of London A*]{} [**158**]{}, 499–521.
[^1]: Email address for correspondence: [email protected]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We discuss the properties of quantum state reactivity as a measure for quantum correlation. This information geometry–based definition is a generalization of the two qu$b$it construction of Schumacher to multipartite quantum states. It requires a generalization of information distance to information areas as well as to higher–dimensional volumes. The reactivity is defined in the usual chemistry way as a ratio of surface area to volume. The reactivity is an average over all detector settings. We show that this measure posses the key features required for a measure of quantum correlation. We show that it is invariant under local unitary transformations, non–increasing under local operations and classical communication, and monotonic. Its maximum bound can’t be obtained using only classical correlation. Furthermore, reactivity is an analytic function of measurement probabilities and easily extendable to higher multipartite states.'
author:
- |
Shahabeddin Mostafanazhad Aslmarand$^{1}$, Warner A. Miller$^{1}$,\
Tahereh Razaei$^{1}$, Paul M. Alsing$^{2}$ & Verinder S. Rana$^{3}$
bibliography:
- 'qig4-2019.bib'
title: |
Properties of Quantum Reactivity\
for a Multipartite State
---
Information Geometry in Quantum Mechanics {#sec:intro}
=========================================
The central role that information plays is in physical laws are well established and captured by two pithy phrases, “Information is Physical," and “It–from–Bit.”[@Wheeler:1990; @Landauer:1991; @Zurek:1989] Within this framework, Schumacher introduced a triangle inequality that is based on measurements of a singlet state.[@Schumacher:1991] His approach was an innovative application of quantum information geometry that highlighted quantum entanglement between two qu$b$its. Schumacher’s original construction for the singlet state was based on the quantum information distance measure of Rokhlin and Rajski. This distance formula was first used in quantum mechanics by Zurek and Bennett et al.[@Rokhlin:1967; @Rajski:1961; @Zurek:1989; @Bennett:1998]
Quantum entanglement is the key resource for quantum information processing.[@Preskill:2012] This has been accompanied by a wealth of studies of entanglement and entanglement measures. Here we report on the properties of a recent information geometry measure of entanglement arising from an extension of Schumacher’s construction from bipartite to multipartite states[@Miller:2018; @Aslmarand:2019] In particular, we introduced a geometric-based measure of [*reactivity*]{} that is a ratio of surface area to volume.[@QIG:1990; @Miller:2019]
We show that the quantum state reactivity satisfies the properties required for such a measure of quantum correlation. We discuss these properties in Sec. \[sec:properties\]. However, we will first provide a brief outline of Schumacher’s information distance–based geometry in Sec. \[sec:Schumacher\] in order to motivate our generalization to multipartite quantum states. In Sec. \[sec:Schumacher\], we also extend this distance to information area and volumes. These higher–dimensional volumes can be used to define the [*reactivity*]{} measure for quantum correlation. The reactivity is directly related to the definition used in chemistry as the ratio of surface area divided by volume. This is easily generalizable to higher dimensional multipartite quantum states. In Sec. \[fini\] we discuss and summarize our results.
Definition of Mean Reactivity to Generalized Multipartite Quantum States {#sec:Schumacher}
========================================================================
Schumacher showed that geometries created for entangled quantum states are not ordinarily embeddable in Euclidian space.[@Schumacher:1991] He utilized an information metric based on Shannon entropy as: $$\label{Eq:D}
{\mathcal D}_{AB} = H_{A|B} + H_{B|A} = 2H_{AB}-H_{A}-H_{B},$$ where $H_{A|B} $ is conditional entropy of $A$ given $B$.[@Rokhlin:1967; @Rajski:1961; @Zurek:1989; @Bennett:1998]
This measure of distance satisfies all three properties of a metric,
1. It is constructed so as to be symmetric, ${\mathcal D}_{AB} = {\mathcal D}_{BA}$
2. It obeys the triangle inequality, ${\mathcal D}_{AB}\geq {\mathcal D}_{AC}+ {\mathcal D}_{CB}$.
3. It is non-negative, ${\mathcal D}_{AB}\geq 0$, and equal to 0 when $A$“=”$B$.
In order to connect this distance to a measure of quantum correlation, Schumacher created a trapezoidal structure for a bipartite system, by assigning two detector for each of the two observers measuring each qubit as illustrated in Fig. \[fig:Schumacher\].
![The trapezoidal information geometry of an ensemble of singlet states, $|S\rangle$, that was first introduced by Schumacher.[@Schumacher:1991] Here Alice employs two detectors $A_1$ and $A_2$ measuring one of the two entangled photons, and Bob also has two distinct detectors $B_1$ and $B_2$. Alice and Bob record a “1” if their detectors triggers, otherwise they measure a “0.” From their string of measurements they can define an information distance ${\mathcal D}_{A_iB_j}$ between each of the four pairs of detectors. Schumacher found that for some choice of detector angles, namely $\{\alpha_1,\beta_1,\alpha_2,\beta_2\} =\{0,\pi/8,\pi/4,3\pi/8 \}$, that the trapezoidal geometry could not be embedded into the Euclidean plane. The triangle equality would be violated for this entangled state. ](TIsinglet "fig:"){width="3"} \[fig:Schumacher\]
In this geometry, and for a range of detector settings, the direct distance between the two detectors $A$ and $D$ is larger than the sum of the three indirect distances. In other words, Schumacher showed that the inequality can be violated for maximally entangled states. $$\label{eq:D}
{\mathcal D}_{A_1B_2} \leq{\mathcal D}_{A_1B_1}+{\mathcal D}_{A_2B_1}+{\mathcal D}_{A_2B_2}.$$ This is equivalent to the non-embeddibility of the trapezoid into the Euclidean plane. In this simple gedanken experiment, he showed that it’s possible to capture the none-classicality of the quantum correlation of a quantum system by looking at the Shannon-based information geometry applied directly to the space of measurements obtained by the four detectors of Alice and Bob.
Following Schumacher’s approach, we generalized this approach to a multipartite system containing an arbitrarily large number of qu$b$its.[@Miller:2018; @Aslmarand:2019; @Miller:2019] In so doing, we defined a reactivity $\mathcal{R}$ for a quantum network, and for a bipartite quantum state it is expressed in terms of the information distance, $$\label{eq:R1}
{\mathcal R}:=\frac{1}{\overline{{\mathcal D}_{AB}}}.$$ Here, the average is taken over all detector settings, $$\label{eq:Dbar}
\overline{D_{AB}} := \frac{1}{4\pi^2} \int_0^{2\pi} \int_0^{2\pi} D_{AB}(\alpha,\beta) d\alpha d\beta.$$ In Fig. \[fig:cdr\] we compare the reactivity ($\mathcal R$) to the other commonly–used measures, concurrence ($\mathcal C$)[@Wootters:1998; @Rungta:2003] and discord ($\mathcal D$)[@Zurek:2001; @Rulli:2011] for a bipartite Werner state, $$|W\rangle=\lambda |S\rangle\langle S | + \frac{1}{4} \left(1-\lambda\right)I,$$ where $\lambda\in [0,1]$ is the entanglement, and $|S\rangle=(|00\rangle+|11\rangle)/\sqrt{2}$ is a singlet state.
![Comparison of concurrence and discord with our definition of reactivity for a bipartite Werner state. Concurrence provides a measure for entanglement in that it is zero for separable states; however, it may be difficult to implement for higher–dimensional multipartite states. Both global quantum discord and reactivity are measures for quantum correlation and not entanglement. Global quantum discord will always be an upper bound for reactivity; however, it may increase under LOCC in some cases. Reactivity is non-increasing under LOCC.[@Vedral:2017] ](Werner2BW "fig:"){width="5"} \[fig:cdr\]
In addition to information distance in Eq. \[eq:D\], we can analogously assign an information area of a tripartite quantum state[@QIG:1990], where $$\label{Eq:A}
{\mathcal A}_{ABC} := H_{A|BC}H_{B|CA}+H_{B|CA}H_{C|AB}+H_{C|AB}H_{A|BC}.$$ This can be extended analogously to higher-dimensional simplexes, e.g. the information volume for a 4–qubit quantum state[@QIG:1990; @Miller:2018; @Aslmarand:2019] such that $$\label{Eq:V}
\begin{array}{ll}
{\mathcal V}_{ABCD}& := H_{A|BCD}H_{B|CDA}H_{C|DAB} +H_{B|CDA}H_{C|DAB}H_{D|ABC}\\
&\ \ +H_{C|DAB}H_{D|ABC}H_{A|CDB}+H_{D|ABC}H_{A|BCD}H_{B|CDA}.
\end{array}$$ Such higher–dimensional volumes enable us to define the reactivity, $$\label{Eq:C}
{\mathcal R} :=
\left(
\frac{ \overline{ {}^{(d\!-\!2)}\!Area } }{ \overline{ {}^{(d\!-\!1)}\!V\!olume. } }
\right)
,$$ for higher number of qu$d$its. Here for qu$d$it state the volume is $(d\!-\!1)$-dimensional volume and the area is its $(d\!-\!2)$–dimensional boundary. We showed that for Werner state that reactivity increases as quantum correlation increases.[@Aslmarand:2019] In this paper, we suggest that this reactivity satisfies the requisite properties for a measure of quantum correlation.
Mean Reactivity as Candidate Measure of Quantum Correlation {#sec:properties}
===========================================================
We show in this section that this measure of mean reactivity ${\mathcal R}$ satisfies the four established properties for a measure of quantum correlation.[@Bennett:1996]. Our proposed geometrical measure of correlation satisfies the following properties:
1. Reactivity is invariant under unitary transformations.
2. Reactivity is non-increasing under LOCC’s.
3. Reactivity is a monotonic function in quantum correlation, and the maximum bound on this curvature can’t be obtained using only classical correlation.
We address each of these in order in the next four subsections, Sec. \[sec:property1\]-Sec. \[sec:property3\]. Its very definition based on the ratio of analytic functions shows that it is extendable to higher–dimensional multipartite quantum states.
Invariance under Local Unitary Operators {#sec:property1}
----------------------------------------
We know that the reactivity defined in Eq. \[eq:R1\] depends on the information distance which has the following properties $$\begin{split}
D_{AB} &= H_{A|B} + H_{B|A}\\
&= Tr\left(M_A \otimes M_B \rho\right)\, \log\left[ \frac {Tr(M_A \otimes M_B\, \rho) ^2}{Tr\left(M_A Tr_B (\rho)\right) \, Tr\left(M_B Tr_A(\rho)\right)}\right]
\end{split}$$ Which $M_A$ and $M_B$ are the projecton operators for Alice and Bob, now if we apply a unitary transformation to initial state $$\rho \longrightarrow U_A \otimes U_B \, \rho \, {U_A}^T \otimes {U_B}^T$$ then $$\begin{split}
D_{AB} &= Tr\left(M_A \otimes M_B \left(U_A \otimes U_B \, \rho \, {U_A}^T \otimes {U_B}^T \right)\right) \\
&\log{\left[ \frac {Tr\left(M_A \otimes M_B \left(U_A \otimes U_B \, \rho \, {U_A}^T \otimes {U_B}^T \right)\right) ^2}{Tr\left(M_A U_A Tr_B (\rho){U_A}^T\right) \,Tr\left(M_B {U_B} Tr_A(\rho){U_B}^T \right)}\right]}\\
& = Tr\left( \left({U_A}^T \otimes {U_B}^T M_{A} \otimes M_B U_A \otimes U_B \right) \, \rho \right)\\
&\log\left[ \frac { Tr\left( \left({U_A}^T \otimes {U_B}^T M_{A} \otimes M_B U_A \otimes U_B \right) \, \rho \right)^2}{Tr\left({U_A}^TM_A U_A Tr_B (\rho)\right) \, Tr\left({U_B}^T M_B {U_B} Tr_A(\rho)\right)}\right]\\
\end{split}$$ Any arbitrary $2\times 2$ unitary quantum gate can be written as a phase shift multiplied by a rotation. This rotation can be expressed as a composition of a rotation and a rotation about the z-axis, $$\begin{split}
&M_A = \sum \lambda_a |a \rangle \langle a |\\
&{U_A}^TM_A U_A = \sum \lambda_a R(z)^T R(\alpha)^T |a \rangle \langle a | R(z) R(\alpha)
\end{split}$$ In other words, we rotate the detector of Alice by some fixed angle about some axis for all measurements. Likewise, we rotate the detector of Bob independently by some other fixed angle and other rotation axis. This operation will not change $\kappa$ since we took an average of the information distance over all possible configurations of Alice and Bob in EQ. \[eq:Dbar\]. Then unitary LOCC’s will not change our measure of correlation.
Reactivity is Non-Increasing under LOCC’s {#sec:property2}
-----------------------------------------
It is accepted that any definition of a measure for quantum correlations that it should not increase under any LOCC. We argue that for our reactivity measure in Eq. \[eq:R1\] for a bipartite state that, on average, $H_{A|B}$ will increase under any LOCC. In particular, we wish to prove that for a given state and any two LOCC operators $L_A$ and $L_B$, $$|\psi' \rangle = M_{AB} |\psi \rangle = L_A \otimes L_B |\psi \rangle,$$ that both the entropy and distance increase, $$\begin{split}
&H_{A|B}(\psi') \geq H_{A|B}(\psi) \\
&D_{AB}(\psi') \geq D_{AB}(\psi);
\end{split}$$ respectively. Consider the local operations acting on the density matrix $$M \rho M^\dag = M |\psi \rangle \langle \psi | M^\dag,$$ then necessarily $$\rho'= \sum_{i,j,k,l} Tr \left(M_{\!{}_{A_i\!B_j}}\, \rho\, M^\dag_{\!{}_{A_k\!B_l}}\right) \left[M_{\!{}_{A_i\!B_j}}\, \rho\, M^\dag_{\!{}_{A_k\!B_l}}\right]$$ Using the convexity of Shannon entropy we have $$H_{A|B}(\rho)\geq \sum_{i,j,k,l} Tr \left(M_{\!{}_{A_i\!B_j}}\, \rho\, M^\dag_{\!{}_{A_k\!B_l}}\right) H_{AB} \left(M_{\!{}_{A_i\!B_j}}\, \rho\, M^\dag_{\!{}_{A_k\!B_l}}\right)$$ Then $\mathcal{R}(\psi')\leq \mathcal{R}(\psi)$. Although all the proofs in this section are written for two qu$b$it systems they are easily generalized to a larger number of qu$b$its. Since in in higher dimensions the reactivity for $n$ observers, $\{C_i\}_{i,1,2,\ldots,n}$ will still satisfy $${\mathcal R} \propto \frac{1}{\overline H_{C_1 C_2 \ldots C_n}}.$$
Reactivity is Monotonic, its Maximum Bound Requires Quantum Correlations. {#sec:property3}
-------------------------------------------------------------------------
It is well established that quantum correlation is a resource for quantum processing.[@Preskill:2012] The correlation in entangled quantum states is stronger than any classical correlation. Unlike classical correlation, quantum correlation is non-vanishing in more than one basis. In particular, a bipartite state with large quantum correlation may yield the same information distance ${\mathcal D}_{AB}$ as classically correlated quantum state for a given measurement operator $M_{AB}$. This is not acceptable for a measure of quantum correlation. However, our definition of reactivity in Eqs. \[eq:R1\]&\[Eq:C\] averages over all measurements and the information distance will then be larger for the classically–correlated quantum state. Therefore, the reactivity of the quantum correlated system will be larger since its inversely proportional to the information distance. For example, even though the two states, $$|\psi \rangle = |HH \rangle\ \ \hbox{(no quantum correlation)}$$ and $$|\psi' \rangle = \frac{|HH \rangle + |VV \rangle}{\sqrt{2}}\ \ \hbox{(maximal quantum correlation)}$$ have the same distance $D_{AB}=D'_{AB}$ for Alice and Bob given the horizontal–horizontal ($HH$) measurement basis measurement; nevertheless, when we take an average over all configurations of Alice and Bob we showed that $$\overline{D_{AB}}(\psi) \geq \overline{D_{AB}}(\psi')$$ then maximum bound created by maximally entangled state can’t be created by any classical correlation.
Summary of the Properties of Reactivity {#fini}
=======================================
We showed in this paper that the geometrically-defined reactivity over the space of measurements satisfies the major properties required of a measure of quantum correlation. The reactivity measure is scalable in the sense that it can be generalized to higher number of qu$b$its. As a measure of correlation it has the advantage of being interpretation free unlike quantum discord for multipartite states. Its expression is a relatively straightforward analytic function probabilities, and it does not require any global minimization procedure or matrix inversion. In other words, it appears to us to be relatively easy to calculate in comparison to other measures of correlation. Nevertheless, its computational complexity is driven by the need to compute joint entropies over the observers measurement outcomes. For a qudit state this would require a $d$–fold summation that scales exponentially. However, we may be able to extract an accurate measure of the correlation for a multipartite system with high fidelity by using only a fraction of the possible measurements. An analysis of the fidelity of this and other entanglement measures under partial–measures, and its impact on reducing the computational complexity is beyond the scope of this paper. Nevertheless, we are animated by Quantum Sanov’s Theorem that shows that the fidelity of distinguishing two quantum density matrices pure $\rho_2$ from $\rho_1$ improves exponentially with the number of measurements, $N$,[@Vedral:1997] $$\label{eq:Sanov}
\left(
\begin{array}{c}
Fidelity\ of\ \rho_1 \rightarrow \rho_2\\
with\ N\ measurements
\end{array}
\right)
= 1-e^{-N S\left(\rho_1||\rho_2\right)}.$$ Here, $S(\rho_1||\rho_2) = Tr(\rho_1 \log \rho_1 - \rho_1 \log \rho_2)$ is a relative entropy. We hope that this theorem’s “information thermodynamic" structure can be extended to a larger classes of quantum states.
Acknowledgment {#acknowledgment .unnumbered}
==============
In here we thank David Meyer and Alexander Meill from UCSD for probing questions on our previous paper that motivated us to expound on the four properties of the mean reactivity. PMA and WAM would like thank support from the Air Force Office of Scientific Research (AFOSR). WAM research was supported under AFOSR/AOARD grant \#FA2386-17-1-4070. One of us (WAM) would also like to acknowledge the support from the Griffiss Institute and the Air Force Research Laboratory at Rome Labs under the Visiting Faculty Research Program. Any opinions, findings, conclusions or recommendations expressed in this mate- rial are those of the author(s) and do not necessarily reflect the views of AFRL.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We show that two-photon transport is strongly correlated in one-dimensional waveguide coupled to a two-level system. The exact S-matrix is constructed using a generalized Bethe-Ansatz technique. We show that the scattering eigenstates of this system include a two-photon bound state that passes through the two-level system as a composite single particle. Also, the two-level system can induce effective attractive or repulsive interactions in space for photons. This general procedure can be applied to the Anderson model as well.'
author:
- 'Jung-Tsung Shen'
- Shanhui Fan
title: 'Strongly Correlated Two-Photon Transport in One-Dimensional Waveguide Coupled to A Two-Level System'
---
Creating strong photon-photon interaction at few-photon level is of great interest for quantum information sciences. In atomic gases, such interaction can be accomplished either with systems exhibiting electromagnetically induced transparency (EIT) [@Harris:1998; @Imamoglu:1997], or by reaching the strong-coupling regime of a two-level atom in a high-Q cavity [@Birnbaum:2005]. However, in an on-chip, solid-state environment, which is crucial for practical applications, there have been significant challenges in implementing these concepts. For example, it is difficult to create the long-lifetime dark state, which is required for EIT effects, in most practical solid-state environments [@Turukhin:2001]. While the strong-coupling regime has been reached by placing a quantum dot in a high-Q photonic crystal microcavity [@Reithmaier:2004; @Yoshie:2004a], doing so requires very accurate tuning of both the electronic and optical resonances to ensure simultaneous spectral and spatial overlaps [@Badolato:2005].
In this Letter we propose and analyze in details an alternative scheme to create strong photon-photon interaction. Our approach exploits a unique one-dimensional feature for photon states in many nano-photonic structures. In a photonic crystal with a complete photonic band gap, for example, a line-defect waveguide forms a true one-dimensional continuum for photons, since there is no other states within the gap. Here we show that by coupling a two-level system to such continuum, strong photon-photon interactions can be created (Fig. \[Fi:Geometry\]). (Below we refer to the two-level system as the “atom”). In this system, the strong interaction arises from the fact that in a one-dimensional system, the re-emitted and scattered waves from the atom inevitably interfere with the incident waves. Moreover, since the atom, intuitively speaking, can at most absorb only one-photon at a time, the transport properties of multi-photon are strongly correlated.
Compared with previous solid-state approaches, our scheme does not require the presence of long-lifetime dark state. Neither does this scheme necessitate detailed spectral tuning or spatial control of the two-level system, since it operates in the weak-coupling regime, and thus the one-dimensional continuum can be broadband. Moreover, the Hamiltonian of the system actually describes an exact photonic analogue of the Kondo effect, which is important for processing electronic quantum bits [@Zumbuhl:2004]. Our approach may therefore open a new avenue towards practical photon-based quantum information processing on-chip.
The system in Fig. \[Fi:Geometry\] is modeled by the Hamiltonian [@Shen:2005; @Shen:2005a]: $$\begin{aligned}
\label{E:hamiltonian}
H&=\int dx \left\{-i v_g c_{R}^{\dagger}(x)\frac{\partial}{\partial
x} c_{R}(x)
+ i v_g c_{L}^{\dagger}(x)\frac{\partial}{\partial x} c_{L}(x)\right. \notag\\
&\quad \left.+ \bar{V} \delta(x) \left(c^{\dagger}_{R}(x) \sigma_{-} +
c_{R}(x) \sigma_{+}
+ c^{\dagger}_{L}(x) \sigma_{-} + c_{L}(x) \sigma_{+}\right)\right\} \notag\\
&\quad +E_{e} a^{\dagger}_{e} a_{e}+ E_{g} a^{\dagger}_{g} a_{g}\end{aligned}$$ where $v_g$ is the group velocity of the photons, and $c_{R}^{\dagger}(x)$($c_{L}^{\dagger}(x)$) is a bosonic operator creating a right-going(left-going) photon at $x$. $\bar{V}$ is the coupling constant, $a^{\dagger}_{g}$($a^{\dagger}_{e}$) is the creation operator of the ground (excited) state of the atom, $\sigma_{+}=a^{\dagger}_{e}
a_{g}$($\sigma_{-}=a^{\dagger}_{g} a_{e}$) is the atomic raising (lowering) ladder operator satisfying $\sigma_{+}|n,-\rangle =|n,+\rangle$ and $\sigma_{+}|n,+\rangle =0$, where $|n, \pm\rangle$ describes the state of the system with $n$ photons and the atom in the excited ($+$) or ground ($-$) state. $E_{e}-E_{g}
(\equiv\Omega)$ is the transition energy. This Hamiltonian describes the situation where the propagating photons can run in both directions, and is referred to as “two-mode” model.
By employing the following transformation $c^{\dagger}_e (x)\equiv \frac{1}{\sqrt{2}}(c^{\dagger}_{R}(x)+ c^{\dagger}_{L}(-x))$, $c^{\dagger}_o (x)\equiv
\frac{1}{\sqrt{2}}(c^{\dagger}_{R}(x)- c^{\dagger}_{L}(-x))$, the original Hamiltonian is transformed into two decoupled “one-mode” Hamiltonians, *i.e.,* $H=H_e + H_o$, where $$\begin{aligned}
H_e &= \int dx (-i) v_g c_{e}^{\dagger}(x)\frac{\partial}{\partial x}
c_{e}(x) + \int dx V \delta(x)\left(c^{\dagger}_{e}(x)
\sigma_{-} + c_{e}(x) \sigma_{+}\right)+E_{e} a^{\dagger}_{e} a_{e}+
E_{g} a^{\dagger}_{g}
a_{g}\notag\\
H_o &= \int dx (-i) v_g c_{o}^{\dagger}(x)\frac{\partial}{\partial x}
c_{o}(x).\end{aligned}$$$H_o$ is an interaction-free one-mode Hamiltonian, while $H_e$ describes a non-trivial one-mode interacting model with coupling strength $V\equiv\sqrt{2} \bar{V}$. $H_e$ is identical in form to the s-d model [@Anderson:1961; @Wiegmann:1983], which describes the S-wave scattering of electrons off a magnetic impurity in three dimensions. Here, however, instead of a fermionic operators describing electrons, we have bosonic operators describing photons.
The one-photon eigenstate for $H_e$ takes the form $|k\rangle \equiv \int dx [e^{i k x} \left(\theta(-x)+ t_k \theta(x)\right)c^{\dagger}(x) + e_k \sigma_{+}] |0, -\rangle$ [@Shen:2005; @Shen:2005a], where $$t_k \equiv\frac{k - \Omega - i \Gamma/2}{k -\Omega + i \Gamma/2}, \quad \Gamma\equiv V^2$$ is the transmission amplitude of magnitude 1, and $e_k = \frac{\sqrt{\Gamma}}{k-\Omega+i \Gamma/2}$ is the excitation amplitude. The single photon experiences resonance when its energy $k$ is close to the transition energy $\Omega$ of the atom. For notational simplicity, $v_g$ and $\hbar$ are set to 1, and the subscript “$e$” in $c^{\dagger}_e$ is dropped hereafter.
In this Letter we focus on the transport properties of the interacting Hamiltonian $H_e$ with two incident photons. For this Hamiltonian, as well as the Anderson model and the interacting resonance level model in condensed matter physics, attempts to diagonalize using the Bethe Ansatz have been published [@Wiegmann:1983; @Rupasov:1984a; @Mehta:2006]. As we will emphasize below, however, a complete and correct description of the transport properties requires a careful re-examination of these solutions. In particular, the Bethe Ansatz solution constructed following the procedures in Ref. \[\] is in fact not complete for this purpose. Rather, to construct the scattering matrix, one needs *one* additional two-photon bound state. These can all be derived by a systematic approach detailed below.
We first describe the general features of the scattering problem. Before and after the scattering, the photons are away from the atom, and thus the two-photon Hilbert spaces of the “*in*” (before scattering) and “*out*” (after scattering) states [@Greiner:1996] are the same space of *free* photons and consists of *all* symmetric functions of the coordinates of the photons, $x_1$ and $x_2$. This Hilbert space is spanned by a complete basis $\{|S_{k,p}\rangle: k\leq p\}$ defined as $$\langle x_1, x_2|S_{k, p}\rangle \equiv \frac{1}{2\pi}\frac{1}{\sqrt{2}}\left(e^{i k x_1} e^{i p x_2} +e^{i k x_2} e^{i p x_1}\right)= \frac{\sqrt{2}}{2\pi} e^{i E x_c}\cos\left(\Delta x\right),$$where $E=k+p$ is the total energy of the photon pair, $x_c \equiv 1/2(x_1 + x_2)$, $x\equiv x_1 - x_2$, and $\Delta \equiv (k - E/2) = 1/2 (k-p) \leq 0$. Alternatively, the *same* Hilbert space can instead be spanned by another basis $\{|A_{k,p}\rangle: k\leq p\}$ defined as $$\langle x_1, x_2|A_{k,p}\rangle \equiv \frac{1}{2\pi}\frac{1}{\sqrt{2}}\,\mbox{sgn}(x)\left(e^{i k x_1} e^{i p x_2} -e^{i k x_2} e^{i p x_1}\right)
= \frac{\sqrt{2}i}{2\pi}\,\mbox{sgn}(x) \, e^{i E x_c}\sin\left(\Delta x\right)$$ where $\mbox{sgn}(x)\equiv \theta(x)-\theta(-x)$ is the sign function. We emphasize that, while both $\{|S_{k,p}\rangle: k\leq p\}$ and $\{|A_{k,p}\rangle: k\leq p\}$ are complete [@Schulz:1982], arbitrary linear combination $\{ a_{k, p} |S_{k,p}\rangle + b_{k, p} |A_{k,p}\rangle: k\leq p\}$ may not be.
The transport properties of two photons, in the presence of the atom, are described by the S-matrix ($\mathbf{S}$) that maps between the Hilbert space of the in and out states: $|\mbox{out}\rangle = \mathbf{S} |\mbox{in}\rangle$. The matrix element of the S-matrix, for example, $\langle S_{k, p}|\mathbf{S}|S_{k', p'}\rangle$ is the transition amplitude of the process [@Greiner:1996].
The S-matrix of the two-photon case, as will be derived below, can be diagonalized as $$\label{E:SMatrix}
\mathbf{S}\equiv\sum_{k< p} t_k t_p |W_{k,p}\rangle \langle W_{k, p}| + \sum_{E} t_E |B_E\rangle\langle B_E|,$$ with $$\begin{aligned}
|W_{k,p}\rangle &\equiv \frac{1}{\sqrt{(k-p)^2 +\Gamma^2}}\left[(k-p)|S_{k,p}\rangle + i \Gamma |A_{k,p}\rangle\right],\notag\\
\langle x_1, x_2 |B_E\rangle &\equiv \frac{\sqrt{\Gamma}}{\sqrt{4 \pi}} e^{i E x_c -\Gamma |x|/2}, \quad t_E \equiv \frac{E-2\Omega-2 i \Gamma}{E-2\Omega+ 2 i \Gamma}\label{E:Def}.\end{aligned}$$
Now we prove Eqs. (\[E:SMatrix\])-(\[E:Def\]) by first showing that $|W_{k,p}\rangle$ and $|B_{E}\rangle$ are eigenstates of the scattering matrix. A two-photon eigenstate for $H_e$ has the general form: $$|\Phi\rangle \equiv\left(\int dx_1 dx_2 \, g(x_1, x_2) c^{\dagger}(x_1) c^{\dagger}(x_2) + \int dx \, e(x) c^{\dagger}(x) \sigma_{+}\right)|0, -\rangle,$$ where $e(x)$ is the probability amplitude of the atom in the excited state. Due to the boson statistics, the wavefunction satisfies $g(x_1, x_2) = +g(x_2, x_1)$. ($g(x_1, x_2)$ is continuous on the line $x_1 = x_2$ for bosons.)
From $H_e |\Phi\rangle = E |\Phi\rangle$, we obtain the equations of motion: $$\begin{aligned}
\left(-i \frac{\partial}{\partial x_1} -i \frac{\partial}{\partial x_2} - E\right) g(x_1, x_2) &+ \frac{V}{2} \left(e(x_1) \delta(x_2) + e(x_2) \delta(x_1)\right) =0, \notag\\
\left(-i \frac{\partial}{\partial x} - E + \Omega\right) e(x) &+ V \left(g(0, x) + g(x,0)\right) = 0, \end{aligned}$$ where $g(0, x) = g(x, 0) \equiv 1/2 \times(g(0^-, x) + g(0^+, x))$. The functions $g(x_1, x_2)$ and $e(x)$ are piecewise continuous. The interactions occur on the coordinate axes: $x_1=0$, and $x_2=0$. Applying the equations of motions gives the following boundary conditions on the boundary of quadrants II ($x_1< 0 < x_2$) and III ($x_1, x_2 < 0$): $$\begin{aligned}
\label{E:1}
-i \left(g(x_1, 0^+) - g(x_1, 0^-)\right) &+\frac{V}{2} e(x_1)=0,\notag\\
\left(-i\frac{\partial}{\partial x_1}-(E-\Omega)\right) e(x_1) &+ V(g(x_1, 0^+) + g(x_1, 0^-))=0,\end{aligned}$$and on the boundary of quadrants II ($x_1< 0 < x_2$) and I ($0 < x_1, x_2$): $$\begin{aligned}
\label{E:2}
-i \left(g(0^+, x_2) - g(0^-, x_2)\right) &+\frac{V}{2} e(x_2)=0,\notag\\
\left(-i\frac{\partial}{\partial x_2}-(E-\Omega)\right) e(x_2) &+ V(g(0^+, x_2) + g(0^-, x_2))=0.\end{aligned}$$These boundary conditions must be supplemented by a further condition$$\label{E:3}
e(0^-) = e(0^+),$$which ensures the self-consistency.
By boson symmetry we only need to consider the half space $x_1 \leq x_2$. In this half space, suppose $g(x_1, x_2) =B_3 e^{i k x_1 + i p x_2} + A_3 e^{i p x_1 + i k x_2}$ for $x_1 < x_2 <0$, using Eqs. (\[E:1\]-\[E:3\]), we obtain $g(x_1, x_2) = t_k t_p (B_3 e^{i k x_1 + i p x_2} + A_3 e^{i p x_1 + i k x_2})$ for $0< x_1 <x_2$, provided $B_3/A_3 = (k - p - i \Gamma)/(k - p + i \Gamma)$ as required from the continuity condition of $e(x)$. Therefore, in the *full* quadrant III, the in-state, $|W_{k,p}\rangle$ as defined by $$\begin{aligned}
\langle x_1, x_2|W_{k,p}\rangle &=\left(A_3 e^{i k x_1 + i p x_2} + B_3 e^{i p x_1 + i k x_2}\right) \theta(x_1 - x_2)+ \left(B_3 e^{i k x_1 + i p x_2} + A_3 e^{i p x_1 + i k x_2}\right) \theta(x_2 - x_1)\notag\\
%&\propto (k-p) \left[e^{i k x_1 + i p x_2}+e^{i p x_1 + i k x_2}\right] + i \Gamma \mbox{sgn}(x)\left[e^{i k x_1 + i p x_2}-e^{i p x_1 + i k x_2}\right]\notag\\
&\propto (k-p)\langle x_1, x_2 |S_{k, p}\rangle + i \Gamma \langle x_1, x_2|A_{k, p}\rangle,\end{aligned}$$is an eigenstate of the S-matrix with eigenvalue $t_k t_p$. This construction and the form of the solution is in essence the Bethe Ansatz method [@Wiegmann:1983; @Mehta:2006].
The set $\{|W_{k,p}\rangle: k < p\}$ however does not form a complete set of basis of the free two-photon Hilbert space. Instead, there exists *one* additional eigenstate of S-matrix, $|B_{E}\rangle$, defined by Eq. (\[E:Def\]). To see that $|B_{E}\rangle$ is an eigenstate of the S-matrix, suppose $g(x_1, x_2) = e^{i E x_c} e^{- \Gamma |x|/2}$ in quadrant III, again using Eqs. (\[E:1\]-\[E:3\]), we obtain $g(x_1, x_2) = t_E e^{i E x_c} e^{- \Gamma |x|/2}$ in quadrant I. Such bound state is important when calculating the ground-state energy in the Anderson model [@Kawakami:1981]. We show here that it is also crucial to the scattering and transport properties.
The set of eigenstates $\{|W_{k,p}, |B_{E}\rangle\}$ forms a complete and orthonormal basis that spans the free two-photon Hilbert space. The orthonormality check is straightforward: $\langle W_{k', p'}|W_{k,p}\rangle = \delta(k-k')\delta(p-p') = \delta(\Delta-\Delta')\delta(E-E')$, $\langle B_E|B_E\rangle = \delta(E-E')$, and $\langle W_{k, p}|B_{E}\rangle =0$. The completeness can be proven by checking that $$\mathbf{W}\equiv\sum_{k< p} |W_{k, p}\rangle\langle W_{k, p}| + \sum_{E}|B_E\rangle\langle B_E|,$$ is indeed an identity operator. This, together with the eigenvalues $t_k t_p$ and $t_E$, prove Eq. (\[E:SMatrix\]).
We note that the two-photon bound state described by $|B_{E}\rangle$, of which the spatial extent is $1/\Gamma$, behaves as an effective single composite particle with an energy $k+p$, and remains integral when passing through the atom. The two-level system therefore provides the capability of manipulating composite particles of photons [@Jacobson:1995] without destroying them. This capability is important in quantum cryptography [@Ekert:1991] and quantum lithography [@Boto:2000].
For an arbitrary in-state of $|\mbox{in}\rangle = |S_{k_1, p_1}\rangle$, the momenta distribution of the out-state $\langle S_{k_2, p_2} |\mbox{out}\rangle$ is $$\label{E:SMatrixElement}
\langle S_{k_2, p_2}|\mathbf{S}|S_{k_1, p_1}\rangle = t_{k_1}t_{p_1}\delta(\Delta_1 -\Delta_2)\delta(E_1 - E_2) + t_{k_1} t_{p_1}\delta(\Delta_1 +\Delta_2)\delta(E_1 - E_2)+ B\delta(E_1 -E_2)$$ where the first two terms are the direct and exchange terms of each individual incident momentum; the third term with $$\begin{aligned}
B &=
\frac{16 i \Gamma^2}{\pi}\frac{E_1-2\Omega + i\Gamma}{\left[4\Delta_1^2 -(E_1 - 2\Omega + i\Gamma)^2\right] \left[4\Delta_2^2 -(E_1 - 2\Omega + i\Gamma)^2\right]}.\end{aligned}$$ represents the background fluorescence due to the scattering. When $\Delta_1 \neq \Delta_2$, $|B(E_1, \Delta_1, \Delta_2)|^2$ is the probability density for the outgoing photon pair in $(E_1, \Delta_2)$ state, when the incoming photon pair is in $(E_1, \Delta_1)$ state.
The emergence of the background fluorescence is completely different from the well-known resonance fluorescence phenomenon where a strong laser beam is scattering off an ensemble of two-level systems [@Scully:1997]. In the current two-photon case, the background fluorescence results from the fact that the momentum of each photon is not conserved. Consequently the interactions with the two-level system redistribute the momenta of the photons over a continuous range, under the total energy and momentum conservation constraint. Furthermore, the locations of the poles in $B$, at $k_{1,2}=p_{1,2}=\Omega -i\Gamma/2$, correspond approximately to either one of the photons having an energy at $\Omega$. Thus, the background fluorescence arises as one photon inelastically scatters off a composite transient object formed by the atom absorbing the other photon.
Fig. \[Fi:Background\_3D\] plots normalized $|B(E, \Delta_1, \Delta_2)|^2$ for various photon pair energy $E$. $|B(E, \Delta_1, \Delta_2)|^2$ is an even function of $E-2\Omega$. When $|E-2\Omega|\leq \Gamma$, there is a single peak centered at $\Delta_1=\Delta_2=0$. The height of the peak reaches maximum at $E=2\Omega$ (Fig. \[Fi:Background\_3D\](a)), and gradually decreases as $|E-2\Omega|$ increases. When $|E-2\Omega| = \Gamma$, the top of the peak becomes flat (Fig. \[Fi:Background\_3D\](b)). When $|E-2\Omega| > \Gamma$, there are four peaks centered at $(\pm\sqrt{(E-2\Omega)^2-\Gamma^2}/2, \pm\sqrt{(E-2\Omega)^2-\Gamma^2}/2)$, respectively (Fig. \[Fi:Background\_3D\](c) and (d)). For any $E$ and $\Delta_1$, the locations of the peaks for $|B(E, \Delta_1, \Delta_2)|^2$ are independent of $\Delta_1$. In contrast, the $\delta$-functions in the S-matrix (Eq. (\[E:SMatrixElement\])) are located on the $\Delta_1=\Delta_2$ line.
The emergence of the background fluorescence also manifests as an effective spatial interaction between the photons. For an in-state $|\mbox{in}\rangle = |S_{E_1, \Delta_1}\rangle$, the out-state is $$\langle x_c, x |\mbox{out}\rangle=
%t_{k_1} t_{p_1} \langle x_c, x|S_{E_1, \Delta_1}\rangle+\int_{-\infty}^{0} B \langle x_c, x|S_{E_1, \Delta_2}\rangle d\Delta_2=
%e^{i E_1 x_c} \sqrt{2}/2\pi\times\left(t_{k_1} t_{p_1} \cos\left(\Delta_1 x\right)+\int_{-\infty}^{0} B \cos\left(\Delta_2 x\right) d\Delta_2\right)\equiv e^{i E_1 x_c} \langle x|\phi\rangle$,
e^{i E_1 x_c}\frac{\sqrt{2}}{2\pi}\left(t_{k_1} t_{p_1} \cos\left(\Delta_1 x\right)-\frac{4\Gamma^2}{4\Delta_1^2 -(E_1-2\Omega + i\Gamma)^2} e^{i (E_1-2\Omega) |x|/2 -\Gamma |x|/2}\right)$$ which takes the form $e^{i E_1 x_c} \langle x|\phi\rangle$, where $\langle x |\phi\rangle$ is the wavefunction in the relative coordinate $x$. The deviation of the out-state wavefunctions from that of interaction-free case is large when $\Delta_1 \simeq \pm (E_1/2 -\Omega)$, i.e., when at least one of the incident photons is close to resonance. Fig. \[Fi:Bunching\](a) plots the normalized deviation of $|\langle x=0|\phi\rangle|^2$ from the interaction-free case as a function of $E_1$ and $\Delta_1$. A positive (negative) deviation implies that the two photons bunch (anti-bunch) after scattering. The hyperbola $4\Delta_1^2 -(E_1-2\Omega)^2 = \Gamma^2$ indicate where the deviation is zero, thereby separate the bunching and anti-bunching regions. The deviation reaches maximum at $E_1 -2\Omega=\Delta_1=0$, when both incident photons are on resonance with the atom. The wavefunction for this case is shown in Fig. \[Fi:Bunching\](b), which exhibits the exponentially decaying feature in $x$. The two photons form a bound state after scattering, with half-width in space about $1/\Gamma$. When $E_1 -2\Omega$ is kept at zero, the height of the peak at $x=0$ decreases with increasing $|\Delta_1|$ (Fig. \[Fi:Bunching\](c)(d)). Fig. \[Fi:Bunching\](d) shows the case for $\Delta_1=-\sqrt{3}\Gamma/2$, where the peak at $x=0$ is completely depleted. Both bunching and anti-bunching behavior occur at other non-resonant $E_1$ and $\Delta_1$ as well but is generally weaker. The resonance thus can induce either an effective repulsion or attraction between two photons.
This work is supported by a Packard Fellowship.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We are going to study the limiting spectral measure of fixed dimensional Hermitian block-matrices with large dimensional Wigner blocks. We are going also to identify the limiting spectral measure when the Hermitian block-structure is Circulant. Using the limiting spectral measure of a Hermitian Circulant block-matrix we will show that the spectral measure of a Wigner matrix with $k-$weakly dependent entries need not to be the semicircle law in the limit.'
author:
- Tamer Oraby
title: 'The spectral laws of Hermitian block-matrices with large random blocks'
---
Preliminaries and main results {#sec2}
==============================
Let ${\mathcal{M}}_n(\mathbb{C})$ be the space of all $n \times n$ matrices with complex-valued entries. Define the normalized trace of a matrix ${\mathbf A}=\left(A_{ij}\right)_{i,j=1}^{n} \in {\mathcal{M}}_n(\mathbb{C})$ to be $ {{\rm tr}}_n({\mathbf A}):=\frac1n \sum_{i=1}^n A_{ii}$.
The spectral measure of a Hermitian $n\times n$ matrix ${\mathbf A}$ is the probability measure $\mu_{{\mathbf A}}$ given by $$\mu_{{\mathbf A}}=\frac{1}{n}\sum_{j=1}^n \delta_{{\lambda}_j}$$ where ${\lambda}_1\leq{\lambda}_2\leq\cdots\leq{\lambda}_n$ are the eigenvalues of ${\mathbf A}$ and $\delta_x$ is the point mass at $x$.
The weak limit of the spectral measures $\mu_{{\mathbf A}_n}$ of a sequence of matrices $\{{\mathbf A}_n\}$ is called the limiting spectral measure. We will denote the weak convergence of a probability measure $\mu_n$ to $\mu$ by $$\mu_n \xrightarrow{{\mathcal{D}}} \mu \mbox{ as $n\to \infty$}.$$
A finite symmetric block-structure ${\mathbb{B}}_k(a,b,c,\dots)$ (or shortly ${\mathbb{B}}_k$) over a finite alphabet $\mathcal{K}=\{a,b,c,\dots\}$ is a $k\times k$ symmetric matrix whose entries are elements in $\mathcal{K}$.
If ${\mathbb{B}}_k$ is a $k\times k$ symmetric block-structure and ${\mathbf A},{\mathbf B},{\mathbf C},\dots$ are $n\times n$ Hermitian matrices, then ${\mathbb{B}}_k({\mathbf A},{\mathbf B},{\mathbf C},\dots)$ is an $nk\times nk$ Hermitian matrix. One of the interesting block structures is the $k\times k$ symmetric Circulant over $\{a_1,a_2,\ldots,a_k\}$ that is defined as $$\label{circulant}
{\mathbb{C}}_k(a_1,a_2,\ldots,a_k)= \frac{1}{\sqrt{k}}\left[
{\begin{array}{*{20}c}
a_1 & a_2 & a_3 & {\dots} & a_k \\
a_k & a_1 & a_2 & {\dots} & a_{k-1} \\
a_{k-1} & a_k & a_1& {\dots} &a_{k-2} \\
{\vdots} & {\vdots} & {\vdots} & {\ddots} &{\vdots} \\
a_2 & a_3& a_4 & {\dots} & a_1 \\
\end{array} } \right]$$ where $a_j=a_{k-j+2}$ for $j=2,3,\ldots,k$.
A random matrix ${\mathbf A}$ is a matrix whose entries are random variables. If ${\mathbb{B}}_k$ is a block-structure and ${\mathbf A},{\mathbf B},{\mathbf C},\dots$ are random matrices, then ${\mathbb{B}}_k({\mathbf A},{\mathbf B},{\mathbf C},\dots)$ is a random block-matrix.
\[wigner\] We call an $n\times n$ Hermitian random matrix ${\mathbf A}=\frac{1}{\sqrt{n}}(X_{ij})_{i,j=1}^n$ a Wigner matrix if $\{X_{ij}; 1 \leq i < j\}$ is a family of independent and identically distributed complex random variables such that $E(X_{12})=0$ and $E(|X_{12}|^2)=\sigma^2$. In addition, $\{X_{ii};
i \geq 1\}$ is a family of independent and identically distributed real random variables that is independent of the upper-diagonal entries. We will denote all such Wigner matrices of order $n$ by $\mathbf{Wigner}(n,\sigma^2)$.
If $\{{\mathbf A}_n\} $ is a sequence of $\mathbf{Wigner}(n,\sigma^2)$ matrices, then by Wigner’s Theorem (*cf.* [@Bai99]), $$\mu_{{\mathbf A}_n}\xrightarrow{{\mathcal{D}}} \gamma_{0,\sigma^2} \: \mbox{ as
$n\to \infty$} \qquad a.s.$$ where $\gamma_{\alpha,\sigma^2}$ is the semicircle law centered at $\alpha$ and of variance $\sigma^2$ which is given as $$\gamma_{\alpha,\sigma^2}(dx)=\frac{1}{2\pi \sigma^2} \sqrt{4 \sigma^2 - (x-\alpha)^2} \:\:
\mathbf{1}_{[\alpha-2\sigma, \alpha+2\sigma]}(x) dx.$$ Now we are ready to state the main result of this paper.
\[T1\] Consider a family of independent Wigner matrices $\left (
\{{\mathbf A}_n(i)\} ;i=1,\ldots,h \right )$ for which $E(|A_{12}(i)|^4)<
\infty$ and $E(A_{11}^2(i))<\infty$ for every $i$. For a fixed $k\times k$ symmetric block-structure ${\mathbb{B}}_k$, define $${\mathbf X}_{n,k}:={\mathbb{B}}_k({\mathbf A}_n(1), {\mathbf A}_n(2),\dots, {\mathbf A}_n(h)).$$ Then there exists a non-random symmetric probability measure $\mu_{{\mathbb{B}}_k}$ which depends only the block- structure ${\mathbb{B}}_k$ such that $$\mu_{{\mathbf X}_{n,k}}\xrightarrow{{\mathcal{D}}} \mu_{{\mathbb{B}}_k} \quad\mbox{as $n\to
\infty$} \qquad a.s.$$
The proof of Theorem \[T1\] relies on free probability theory and will be given in Section \[Section: Proof of T1\].
Consider the symmetric Circulant block-matrix ${\mathbb{C}}_k$ defined in . If ${\mathbf A}_n(1),{\mathbf A}_n(2),\ldots,$ ${\mathbf A}_n(\lfloor \frac{k}{2} \rfloor+1)$ are independent $\mathbf{Wigner}(n,1)$ for every $n$, then Theorem \[T1\] insures the existence of a non-random probability measure $\nu_k$ such that $$\mu_{{\mathbb{C}}_{k}({\mathbf A}_n(1),{\mathbf A}_n(2),\dots,{\mathbf A}_n(\lfloor \frac{k}{2}
\rfloor+1))}\xrightarrow{{\mathcal{D}}}
\nu_k \mbox{ as $n\to \infty$} \qquad a.s.$$ However, Theorem \[T1\] doesn’t specify $\nu_k$ but we will identify it in the following proposition.
\[limitcirculant\] If ${\mathbf A}_n(1),{\mathbf A}_n(2),\ldots,{\mathbf A}_n(\lfloor \frac{k}{2} \rfloor+1)$ are independent $\mathbf{Wigner}(n,1)$ for every $n$, then $$\mu_{{\mathbb{C}}_{k}({\mathbf A}_n(1),{\mathbf A}_n(2),\dots,{\mathbf A}_n(\lfloor \frac{k}{2}
\rfloor+1))}\xrightarrow{{\mathcal{D}}}
\nu_k \mbox{ as $n\to \infty$} \qquad a.s.$$ where $$\nu_k=\left\{
\begin{array}{ll}
\frac{k-1}{k}\;
\gamma_{0,\frac{k-1}{k}}+\frac{1}{k}\;\gamma_{0,\frac{2k-1}{k}}, &
\hbox{if $k$ is odd;} \\ \\
\frac{k-2}{k}\;
\gamma_{0,\frac{k-2}{k}}+\frac{2}{k}\;\gamma_{0,\frac{2k-2}{k}}, &
\hbox{if $k$ is even.}
\end{array}
\right.$$
Since ${\mathbf A}_n(j)={\mathbf A}_n(k-j+2)$ for $j=2,3,\ldots,k$; then [@davis Theorem 3.2.2.] implies that ${\mathbb{C}}_{k}({\mathbf A}_n(1),{\mathbf A}_n(2),\dots,{\mathbf A}_n(\lfloor \frac{k}{2}
\rfloor+1))$ has the same eigenvalues as $\{{\mathbf B}_n(j); \;j=1\dots,k
\}$ where $$\label{oddcir}
{\mathbf B}_n(j):=\frac{1}{\sqrt{k}}[{\mathbf A}_n(1)+2 \sum_{\ell=2}^{(k+1)/2}
\cos(\frac{2\pi (\ell-1) (j-1)}{k}){\mathbf A}_n(\ell)]$$ if $k$ is odd and $$\label{evencir}
{\mathbf B}_n(j):=\frac{1}{\sqrt{k}}[{\mathbf A}_n(1)+2 \sum_{\ell=2}^{k/2}
\cos(\frac{2\pi (\ell-1) (j-1)}{k}){\mathbf A}_n(\ell)+\cos((j-1)\pi)
{\mathbf A}_n(\frac{k}{2}+1)]$$ if $k$ is even. Hence, $$\mu_{{\mathbb{C}}_{k}({\mathbf A}_n(1),{\mathbf A}_n(2),\dots,{\mathbf A}_n(\lfloor \frac{k}{2}
\rfloor+1))}=\frac1k \sum_{j=1}^k \mu_{{\mathbf B}_n(j)}.$$
Using the well known trigonometric sum $\sum_{\ell=0}^N \cos (\ell
x)=\frac12 (\frac{\sin ((N+\frac12)x)}{\sin \frac{x}{2}}+1)$, one can easily show that $$\label{trigono} \sum_{\ell=0}^N \cos^2 (\ell x)=\frac12 (N+\frac32+\frac{\sin
((2N+1)x)}{\sin x}).$$
Consider the case when $k$ is odd. In Equation , for $j \neq 1$, ${\mathbf B}_n(j)$ is a $\mathbf{Wigner}(n,\frac{k-1}{k})$ where the variance of the off-diagonal entries of ${\mathbf B}_n(j)$ is given by $\frac{1}{k}[1+4 \sum_{\ell=2}^{(k+1)/2} \cos^2(\frac{2\pi (\ell-1)
(j-1)}{k})]$ which turns out to be $\frac{k-1}{k}$ by Equation . For $j=1$, ${\mathbf B}_n(1)$ is simply a $\mathbf{Wigner}(n,\frac{2k-1}{k})$. Hence, Wigner’s theorem for ${\mathbf B}_n(1)$ and the rest $k-1$ Wigner matrices ${\mathbf B}_n(j);j=2,\dots,k$ finishes the proof of the odd case.
The case when $k$ is even follows from a similar argument by showing that for $j=1,\frac{k}{2}+1$; ${\mathbf B}_n(j)$ is a $\mathbf{Wigner}(n,\frac{2k-2}{k})$ and for $j\neq 1,\frac{k}{2}+1$; ${\mathbf B}_n(j)$ is a $\mathbf{Wigner}(n,\frac{k-2}{k})$.
In [@Bai99 p.626], Bai raised the question of whether Wigner’s theorem is still holding true when the independence condition in the Wigner matrix is weakened. Schenker and Schulz-Baldes [@sculz] provided an affirmative answer under some dependency assumptions in which the number of correlated entries doesn’t grow too fast and the number of dependent rows is finite. After the first draft of the underlying paper was completed, we learnt that Anderson and Zeitouni [@AZ-2006] showed that it doesn’t hold in general and they gave an example in which the limiting spectral distribution is the free multiplicative convolution of the semicircle law and shifted arcsine law. In the rest of this section, we are going to use the following corollary of Proposition \[limitcirculant\] to give another example.
Let ${\mathbb{W}}(a_{11},a_{12},\ldots,a_{nn})$ be the Wigner symmetric block-structure, *i.e.*, $${\mathbb{W}}(a_{11},a_{12},\ldots,a_{nn})=
\left[ {\begin{array}{*{20}c}
a_{11} & a_{12} & {\dots} & a_{1n} \\
a_{12} & a_{22} & {\dots} & a_{2n} \\
{\vdots} & {\vdots} & {\ddots} &{\vdots} \\
a_{1n} & a_{2n}& {\dots} & a_{nn} \\
\end{array} } \right].$$ Consider the family of $k \times k$ random matrices $\{{\mathbf A}_{ij}:i,j \geq 1 \}$ such that ${\mathbf A}_{ij}={\mathbf A}_{ji}$ and ${\mathbf A}_{ij}={\mathbb{C}}_k(a_{ij},b_{ij},c_{ij}, \ldots)$ where $\{a_{ij},b_{ij},c_{ij}, \ldots:i,j \geq 1 \}$ are independent and identically distributed random variables with variance one. Then ${\mathbb{K}}_{n,k}:={\mathbb{W}}({\mathbf A}_{11},{\mathbf A}_{12},\ldots,{\mathbf A}_{nn})$ is an $kn \times
kn$ symmetric matrix.
\[weak\] Fix $k \in \mathbb{N}$. The limiting spectral measure of ${\mathbb{K}}_{n,k}$ is given by $$\mu_{{\mathbb{K}}_{n,k}}\xrightarrow{{\mathcal{D}}}
\nu_k \mbox{ as $n\to \infty$} \qquad a.s.$$
In order to prove this corollary we need the following definitions. Let ${\mathbf A}$ and ${\mathbf B}$ be $n\times m$ and $k\times \ell$ matrices, respectively. By $\otimes$ we mean here the Kronecker product for which ${\mathbf A}\otimes {\mathbf B}=(A_{ij}{\mathbf B})_{i=1,\ldots,n;j=1,\ldots,m}$ is an $nk \times m \ell$ matrix. The $(p,q)$-commutation matrix ${\mathbf P}_{p,q}$ is a $pq \times pq$ matrix defined as $${\mathbf P}_{p,q}=\sum_{i=1}^p \sum_{j=1}^q {\mathbf E}_{ij} \otimes {\mathbf E}_{ij}^T$$ where ${\mathbf E}_{ij}$ is the $p \times q$ matrix whose entries are zero’s except the $(i,j)-$entry is 1. It is known that ${\mathbf P}_{p,q}^{-1}={\mathbf P}_{p,q}^T={\mathbf P}_{q,p}$ and ${\mathbf P}_{n,k}({\mathbf A}\otimes
{\mathbf B}){\mathbf P}_{\ell,m}={\mathbf B}\otimes {\mathbf A}$ (*cf.* [@searle]).
Since ${\mathbb{K}}_{n,k}= \sum_{i,j=1}^n \widetilde{{\mathbf E}}_{ij} \otimes
{\mathbf A}_{ij}$ where $\widetilde{{\mathbf E}}_{ij}$ is the $n \times n$ matrix whose entries are zero’s except the $(i,j)-$entry is 1. Hence $$\begin{array}{l c l}
{\mathbf P}_{k,n}{\mathbb{K}}_{n,k}{\mathbf P}_{n,k}&=&\sum_{i,j=1}^n {\mathbf A}_{ij} \otimes
\widetilde{{\mathbf E}}_{ij} \\ &=& \sum_{i,j=1}^n
{\mathbb{C}}_k(a_{ij},b_{ij},c_{ij}, \ldots) \otimes \widetilde{{\mathbf E}}_{ij} \\
&=& \sum_{i,j=1}^n
{\mathbb{C}}_k(a_{ij}\widetilde{{\mathbf E}}_{ij},b_{ij}\widetilde{{\mathbf E}}_{ij},c_{ij}\widetilde{{\mathbf E}}_{ij},
\ldots) \\ &=&
{\mathbb{C}}_k(\sum_{i,j=1}^n a_{ij}\widetilde{{\mathbf E}}_{ij}, \sum_{i,j=1}^n b_{ij}\widetilde{{\mathbf E}}_{ij},\sum_{i,j=1}^n c_{ij}\widetilde{{\mathbf E}}_{ij}, \ldots) \\
&=&{\mathbb{C}}_k({\mathbf A}_n,{\mathbf B}_n,{\mathbf C}_n, \ldots)
\end{array}$$ where ${\mathbf A}_n=(a_{ij})_{i,j=1}^{ n}$, ${\mathbf B}_n=(b_{ij})_{i,j=1}^{ n}$, ${\mathbf C}_n=(c_{ij})_{i,j=1}^{ n}$, $\ldots$ are independent $\mathbf{Wigner}(n,1)$ matrices. Therefore, ${\mathbb{K}}_{n,k}$ and ${\mathbb{C}}_k({\mathbf A}_n,{\mathbf B}_n,{\mathbf C}_n, \ldots)$ are similar to each other and so have the same eigenvalues. Thus the result follows.
Now, we define the distance on $\mathbb{N}^2$ by $d \left(
(i,j),(i',j')\right)=max \{|i-i'|,|j-j'|\} $ and for $S,T \subset
\mathbb{N}^2$; $d \left( S,T \right) =min \{d \left(
(i,j),(i',j')\right): (i,j)\in S, \, (i',j') \in T \}$. We say the random field $\{X_{ij}:(i,j)\in \mathbb{N}^2_{\leq}\}$ is $(k-1)$-dependent if the $\sigma$-fields $\mathcal{F}_S=\sigma(\{X_{ij}:(i,j)\in S\})$ and $\mathcal{F}_T=\sigma(\{X_{ij}:(i,j)\in T\})$ are independent for all $S,T \subset \mathbb{N}^2_{\leq}$ such that $d\left( S,T
\right)>k-1$.
The matrix ${\mathbb{K}}_{n,k}={\mathbb{W}}({\mathbf A}_{11},{\mathbf A}_{12},\ldots,{\mathbf A}_{nn})$, defined in Corollary \[weak\], is an $kn \times kn$ matrix with $(k-1)$-dependent entries, up to symmetry. That is, if we write ${\mathbb{K}}_{n,k}=(X_{ij})_{i,j=1}^{nk}$, then $\{X_{ij}:(i,j)\in
\mathbb{N}^2_{\leq}\}$ is a $(k-1)$-dependent random field. However, the limiting spectral measure of ${\mathbb{K}}_{n,k}$ is not the semicircle law but rather a mixture of two semicircle laws due to Corollary \[weak\]. Our example violates the conditions imposed on the Wigner matrix by Schenker and Schulz-Baldes in [@sculz] in both the number of correlated entries and the number of dependent rows grow as $O(n^2)$ and not $o(n^2)$.
Unfortunately, $\{X_{ij}:(i,j)\in \mathbb{N}^2_{\leq}\}$, in our example, is not strictly stationary as the distributions remain the same only when shifts are made by multiple of $k$.
Proofs {#sec4}
======
In order to prove Theorem \[T1\] we need to introduce some definitions from free probability theory.
A noncommutative probability space $(\mathcal{A},\tau)$ is a pair of a unital algebra $\mathcal{A}$ with a unit element ${\mathbb I}$ and a linear functional $\tau$, called the state, for which $\tau({\mathbb I})=1$. We call an element ${\mathbf{a}}\in \mathcal{A}$ a noncommutative random variable and call $\tau({\mathbf{a}}^n)$ its $n^{th}$ moment. We say that $\mathcal{A}$ is a \*-algebra if the involution \* is defined on $\mathcal{A}$. In addition, we assume that $\tau({\mathbf{a}}^*)=\overline{\tau({\mathbf{a}})}$ and $\tau({\mathbf{a}}^* {\mathbf{a}})\geq 0$. Henceforth, we will consider only \*-algebras. We say that ${\mathbf{a}}\in \mathcal{A} $ is selfadjoint if ${\mathbf{a}}^*={\mathbf{a}}$.
Fix a noncommutative probability space $(\mathcal{A},\tau)$. For each selfadjoint ${\mathbf{a}}\in \mathcal{A}$ there exists a probability measure $\mu_{\mathbf{a}}$ on $\mathbb{R}$ such that $$\tau({\mathbf{a}}^n)=\int_{\mathbb{R}} x^n \mu_{\mathbf{a}}(dx)$$ for all $n\geq 1$, see [@Meyer p.2]. The probability measure $\mu_{\mathbf{a}}$ is unique if $|\tau({\mathbf{a}}^n)| \leq M^n$ for some $M>0$ and for all $n \geq 1$.
A family of subalgebras $(\mathcal{A}_j; j\in J)$ of $\mathcal{A}$, which contain ${\mathbb I}$, is said to be free with respect to $\tau$ if for every $k\geq 1$ and $j_1 \neq j_2 \neq \ldots \neq j_k \in J \subset
\mathbb{N}$ $$\tau({\mathbf{a}}_1 {\mathbf{a}}_2\cdots {\mathbf{a}}_k)=0$$ for all ${\mathbf{a}}_i \in
\mathcal{A}_{j_i}$ whenever $\tau({\mathbf{a}}_i)=0$ for every $1\leq i \leq
k$.
Random variables in a noncommutative probability space $(\mathcal{A},\tau)$ are said to be free if the subalgebras generated by them and ${\mathbb I}$ are free.
We say that a family of sequences of random matrices $(\{{\mathbf A}_n(l)\};l=1,\ldots,m)$ is asymptotically free (*cf.* [@Hiai-Petz]) if for every noncommutative polynomial $p$ in $m$ variables $${{\rm tr}}_n \left( p({\mathbf A}_n(1),\ldots,{\mathbf A}_n(m))\right)
\xrightarrow{n\to \infty} \tau \left( p({\mathbf{a}}_1,\ldots,{\mathbf{a}}_m)\right)
\qquad a.s.$$ where $({\mathbf{a}}_1,\ldots,{\mathbf{a}}_m)$ is a family of free noncommutative random variables in some noncommutative probability space $(\mathcal{A},\tau)$.
\[folk\] If $(\{{\mathbf A}_n(l)\};l=1,\dots,m)$ is a family of independent $\mathbf{Wigner}(n,1)$ matrices for which $E(|A_{12}(l)|^4)< \infty$ and $E(A_{11}^2(l))<\infty$, then $(\{{\mathbf A}_n(l)\};l=1,\dots,m)$ is asymptotically free.
Proof of Theorem \[folk\]
-------------------------
In [@capitaine-2005], Capitaine and Donati-Martin showed the asymptotic freeness for independent Wigner matrices when the distribution of the entries is symmetric and satisfies Poincaré inequality. Recently, Guionnet [@Guionnet2006] gave a proof where she assumes that all the moments of the entries exist. Szarek [@Szarek] showed us a proof for symmetric and non-symmetric matrices with uniformly bounded entries. Szarek’s proof, in brief, is based on concentration inequalities and some tools of operator theory. In this paper, we are going to give a combinatorial proof for the case of Hermitian Wigner matrices with finite variance and fourth moment of the entries.
The Schatten $p$-norm of a matrix ${\mathbf A}$ is defined as $\|{\mathbf A}\|_p:=({{\rm tr}}_n|{\mathbf A}|^p)^{\frac1p}$ whenever $1\leq p < \infty$, where $|{\mathbf A}|=({\mathbf A}^T {\mathbf A})^{\frac12}$. The operator norm is defined as $\|{\mathbf A}\|:=\max_{1\leq i \leq n} |{\lambda}_i|$ where ${\lambda}_i$; $i=1,2,\ldots,n$ are the eigenvalues of ${\mathbf A}$. The following three inequalities hold true;
1. Domination inequality [@Hiai-Petz p.154] $$\label{domiantion}
|{{\rm tr}}_n({\mathbf A})|\leq \|{\mathbf A}\|_1 \leq \|{\mathbf A}\|_p \leq \|{\mathbf A}\|$$
2. Hölder’s inequality [@Hiai-Petz p.154] $$\label{hol}
\|{\mathbf A}{\mathbf B}\|_r \leq \|{\mathbf A}\|_p \|{\mathbf B}\|_q$$ whenever $\frac1r=\frac1p+\frac1q$ for $p,q>1$ and $r\geq 1$.
3. Generalized Hölder’s inequality $$\label{genhol}
\|{\mathbf A}{(1)} {\mathbf A}{(2)} \cdots {\mathbf A}{(m)}\|_1 \leq \|{\mathbf A}{(1)}\|_{p_1}
\|{\mathbf A}{(2)}\|_{p_2} \cdots \|{\mathbf A}{(m)}\|_{p_m}$$ where ${\mathbf A}{(1)}, {\mathbf A}{(2)}, \ldots, {\mathbf A}{(m)}$ are $n\times n$ matrices and $\sum_{i=1}^m \frac{1}{p_i}=1$. This inequality follows from (\[hol\]) by induction.
Let ${\mathbf A}=\frac{1}{\sqrt{n}}(X_{i,j})_{i,j=1}^n$ be a $\mathbf{Wigner}(n,1)$ matrix. We define $\widetilde{{\mathbf A}}=\frac{1}{\sqrt{n}}(\widetilde{X}_{i,j})_{i,j=1}^n$ to be the matrix whose off-diagonal entries are those of ${\mathbf A}$ truncated by $c/\sqrt{n}$ and standardized. We will also assume that the diagonal entries of $\widetilde{{\mathbf A}}$ are zero’s. In other words, $$\widetilde{X}_{i,j}=\left\{
\begin{array}{ll}
\frac{1}{\sigma(c)} \left [ X_{i,j}\mathbf{1}_{(|X_{i,j}|\leq
c)}-E(X_{i,j}\mathbf{1}_{(|X_{i,j}|\leq
c)}) \right], & \hbox{for $i<j$;} \\
0, & \hbox{for $i=j$.}
\end{array}
\right.$$ where $\mathbf{1}_{(|X_{i,j}|\leq c)}$ is equal to one if $|X_{i,j}|\leq c$ and zero otherwise; and $$\sigma^2(c)=E \left[ X_{i,j}\mathbf{1}_{(|X_{i,j}|\leq
c)}-E(X_{i,j}\mathbf{1}_{(|X_{i,j}|\leq c)}) \right]^2.$$ Note that $\sigma^2(c)\to 1$ as $c\to \infty$ and ${{\rm Var}}(X_{1,2}{(j)}\mathbf{1}_{(|X_{1,2}(j)|> c)}) \leq
1-\sigma^2(c)$.
The proof of Theorem \[folk\] resembles the proof of Wigner’s theorem given in [@Bai99]. We will split it into a number of lemmas.
\[prop21\] If $\left (\{{\mathbf A}_n(l)\};l=1,\ldots m \right)$ is a family of independent sequences of $\mathbf{Wigner}(n,1)$ matrices for which $E(|A_{12}(l)|^4)< \infty$ and $E(A_{11}^2(l))<\infty$ for every $l$, then $$\label{eq1} \lim_{n\to \infty}
|{{\rm tr}}_n \left({\mathbf A}_n{(1)} {\mathbf A}_n{(2)} \cdots {\mathbf A}_n{(m)}\right)-{{\rm tr}}_n
(\widetilde{{\mathbf A}}_n{(1)} \widetilde{{\mathbf A}}_n{(2)} \cdots
\widetilde{{\mathbf A}}_n{(m)})|=0 \qquad a.s.$$
First, $${\mathbf A}_n{(1)} {\mathbf A}_n{(2)} \cdots
{\mathbf A}_n{(m)}-\widetilde{{\mathbf A}}_n{(1)} \widetilde{{\mathbf A}}_n{(2)} \cdots
\widetilde{{\mathbf A}}_n{(m)}=\sum_{j=1}^m \prod_{k=1}^{j-1}
\widetilde{{\mathbf A}}_n{(k)} ({\mathbf A}_n{(j)}-\widetilde{{\mathbf A}}_n{(j)})
\prod_{l=j+1}^{m} {\mathbf A}_n{(l)}$$ with the convention that $\prod_{k=1}^{0} \widetilde{{\mathbf A}}_n{(k)}=\prod_{l=m+1}^{m}
{\mathbf A}_n{(l)}={\mathbf I}$. But,
$$\begin{array}{l l} |{{\rm tr}}_n \left( \prod_{k=1}^{j-1} \widetilde{{\mathbf A}}_n{(k)}
\;({\mathbf A}_n{(j)}-\widetilde{{\mathbf A}}_n{(j)})\; \prod_{l=j+1}^{m} {\mathbf A}_n{(l)}
\right)| &= \\ |{{\rm tr}}_n \left( \prod_{l=j+1}^{m} {\mathbf A}_n{(l)}\;
\prod_{k=1}^{j-1} \widetilde{{\mathbf A}}_n{(k)}\;
({\mathbf A}_n{(j)}-\widetilde{{\mathbf A}}_n{(j)})
\right)| &\leq \\
\| \prod_{l=j+1}^{m} {\mathbf A}_n{(l)} \; \prod_{k=1}^{j-1}
\widetilde{{\mathbf A}}_n{(k)}\|_2 \; \|
{\mathbf A}_n{(j)}-\widetilde{{\mathbf A}}_n{(j)}\|_2
&\leq \\ \prod_{l=j+1}^{m} \| {\mathbf A}_n{(l)} \|_{2(m-1)}\; \prod_{k=1}^{j-1}
\| \widetilde{{\mathbf A}}_n{(k)}\|_{2(m-1)}\; \|
{\mathbf A}_n{(j)}-\widetilde{{\mathbf A}}_n{(j)}\|_2
\end{array}$$
for all $1 \leq j \leq m$ with the convention that $\prod_{k=1}^{0}
\| \widetilde{{\mathbf A}}_n{(k)}\|_p = \prod_{l=m+1}^{m} \| {\mathbf A}_n{(l)}
\|_p=1$. The last two inequalities are due to the generalized Hölder’s inequality .
It is enough to show that $$\lim_{n \to \infty} \|
{\mathbf A}_n{(j)}-\widetilde{{\mathbf A}}_n{(j)}\|_2 = 0 \qquad a.s.$$ for all $j$’s, since $\lim_{n\to \infty}\| {\mathbf A}_n{(l)} \|_{2(m-1)}$ and $
\lim_{n\to \infty}\| \widetilde{{\mathbf A}}_n{(k)}\|_{2(m-1)} $ are finite almost surely (*cf.* [@Bai99 Theorem 2.12]) for every $l$ and $k$ due to the domination inequality and that $E(|A_{12}(l)|^4)< \infty$ and $E(A_{11}^2(l))<\infty$ for every $l$. Let $\widehat{{\mathbf A}}_n{(j)}:={\mathbf A}_n{(j)}-\sigma(c)
\widetilde{{\mathbf A}}_n{(j)}$ or $\widehat{X}_{r,s}{(j)}:=X_{r,s}{(j)}-\sigma(c)
\widetilde{X}_{r,s}{(j)}$ for every $r$ and $s$. Thus, $$\|
{\mathbf A}_n{(j)}-\widetilde{{\mathbf A}}_n{(j)}\|_2 \leq \|
\widehat{{\mathbf A}}_n{(j)}\|_2+|1-\sigma(c)| \;
\|\widetilde{{\mathbf A}}_n{(j)}\|_2$$
By definition, $$\| \widehat{{\mathbf A}}_n{(j)}\|_2^2= \frac{1}{n^2} \sum_{r=1}^{n} \sum_{s=1}^{n}
|\widehat{X}_{r,s}{(j)}|^2=\frac{1}{n^2} \sum_{r=1}^{n}
|\widehat{X}_{r,r}{(j)}|^2+\frac{1}{n^2} \sum_{r\neq s}
|\widehat{X}_{r,s}{(j)}|^2$$ Note that $$\widehat{X}_{r,s}{(j)}=\left\{
\begin{array}{ll}
X_{r,s}{(j)}\mathbf{1}_{(|X_{r,s}(j)|> c)}
-E(X_{r,s}{(j)}\mathbf{1}_{(|X_{r,s}(j)|> c)}), & \hbox{for $r<s$;} \\
X_{r,r}(j), & \hbox{for $r=s$.}
\end{array}
\right.$$ Since $E(X_{1,1}^2(j))<\infty$ then $\lim_{n \to \infty}
\frac{1}{n^2} \sum_{r=1}^{n} X_{r,r}^2{(j)}=0$ almost surely due to the Strong Law of Large Numbers (*SLLN*). Once more the *SLLN* implies that $$\lim_{n \to \infty} \frac{1}{n^2} \sum_{r\neq s}
| \widehat{X}_{r,s}{(j)}|^2
={{\rm Var}}(X_{1,2}{(j)}\mathbf{1}_{(|X_{1,2}(j)|> c)}) \quad a.s.$$ Hence, $\lim_{n\to \infty}\|
\widehat{{\mathbf A}}_n{(j)}\|_2^2={{\rm Var}}(X_{1,2}{(j)}\mathbf{1}_{(|X_{1,2}(j)|>
c)})$ almost surely. It is also evident that $\lim_{n\to
\infty}\|\widetilde{{\mathbf A}}_n{(j)}\|_2=1$ almost surely. Therefore, for arbitrary small $\epsilon<0$ and sufficiently large $c$, $$\limsup_{n\to \infty} \|
{\mathbf A}_n{(j)}-\widetilde{{\mathbf A}}_n{(j)}\|_2 \leq
1-\sigma^2(c)+|1-\sigma(c)| <\epsilon$$ which completes the proof.
Henceforth we will assume that for all $l$ the entries $|X_{i,j}{(l)}|\leq c$ for every $i<j$ and $X_{i,i}{(l)}=0$.
\[rate\] If $\left (\{{\mathbf A}_n(l)\};l=1,\ldots m \right)$ is a family of independent sequences of $\mathbf{Wigner}(n,1)$ matrices whose entries are bounded, then $$\label{dy} \lim_{n \to \infty} E \left({{\rm tr}}_n \left({\mathbf A}_n(1)
{\mathbf A}_n(2)\cdots {\mathbf A}_n(m)\right)\right )=\tau \left({\mathbf{a}}_{1} {\mathbf{a}}_{2}
\cdots {\mathbf{a}}_{m}\right)$$ where ${\mathbf{a}}_{i}$’s are some free noncommutative random variables in $(\mathcal{A},\tau)$ such that ${\mathbf{a}}_{i}$ has the semicircle law $\gamma_{0,1}$ for all $i$.
We say that a partition $\pi=\{ B_1,\ldots,B_p \}$ of a set of integers is non-crossing if $a<b<c<d$ is impossible for $a,c\in B_i$ and $b,d \in B_j$ when $i \neq j$. We denote the family of all non-crossing partitions of $\{1,\ldots,k\}$ by $\text{NC}(k)$. Also let $\text{NC}_2(k)$ be the family of all non-crossing pair partitions which is empty unless $k$ is even. The Catalan number $$C_k=\frac{1}{k+1}
\left( \begin{matrix}
2k \\
k
\end{matrix} \right)$$ is equal to the size of $\text{NC}(k)$ and also the size of $\text{NC}_2(2k)$.
If $\left ({\mathbf{a}}_l;l=1,\ldots m \right)$ is a family of free semicircular random variables which have mean zero and variance one, then (*cf.* [@Ryan98a Equation (8)])
$$\label{multimoment} \tau \left({\mathbf{a}}_{i_1} {\mathbf{a}}_{i_2} \cdots
{\mathbf{a}}_{i_k}\right)= \left\{
\begin{array}{ll}
\sum_{\pi \in \text{NC}_2(k)} \prod_{\{p,q\}\in
\pi} \mathbf{1}_{i_p=i_q}, & \hbox{if $k$ is even;} \\
0, & \hbox{otherwise.}
\end{array}
\right.$$
for any $i_1,\dots,i_k \in \{1,\dots,m\}$.
\[C30\] If $\left (\{{\mathbf A}_n(l)\};l=1,\ldots m \right)$ is a family of independent sequences of $\mathbf{Wigner}(n,1)$ matrices whose entries are bounded and have zero diagonal entries, then $$\label{eq12}
\sum_{n=1}^\infty {{\rm Var}}\left( {{\rm tr}}_n \left(\prod_{i=1}^m
{\mathbf A}_n(i) \right)\right)<\infty$$
It is enough to show that
$${{\rm Var}}\left( {{\rm tr}}_n\left(\prod_{i=1}^m {\mathbf A}_n(i)
\right)\right)=O(n^{-2})$$
We will denote the number of distinct integers among $(i_1,\ldots,i_m)$ by $\langle\langle i_1,\ldots,i_m
\rangle\rangle$.
$${{\rm Var}}\left( {{\rm tr}}_n\left(\prod_{i=1}^m {\mathbf A}_n(i) \right)\right)=E
\left( {{\rm tr}}_n\left(\prod_{i=1}^m {\mathbf A}_n(i) \right)\right)^2-\left[ E
\left( {{\rm tr}}_n\left(\prod_{i=1}^m {\mathbf A}_n(i) \right) \right)
\right]^2=$$
$$\frac{1}{n^{m+2}}\sum_{{\mathbb I}(m,n),{\mathbb J}(m,n)} [ E\left( \prod_{r=1}^m
X_{i_r,i_{r+1}}(r) \prod_{s=1}^m X_{j_s,j_{s+1}}(s) \right)$$ $$\qquad\qquad\qquad\qquad\qquad -E\left( \prod_{r=1}^m
X_{i_r,i_{r+1}}(r)\right) E\left( \prod_{s=1}^m X_{j_s,j_{s+1}}(s)
\right)]$$ where ${\mathbb I}(m,n)=\{(i_1,\ldots,i_m):1 \leq i_1,\ldots,i_m
\leq n \}$ and ${\mathbb J}(m,n)=\{(j_1,\ldots,j_m):1 \leq j_1,\ldots,j_m
\leq n \}$ with the convention that $i_{m+1}=i_1$ and $j_{m+1}=j_1$. The term under summation is zero unless:
1. Each one of the unordered pairs $\left( \{i_1,i_2\},\ldots,\{i_m,i_1\} ,
\{j_1,j_2\},\ldots,\{j_m,j_1\} \right)$ appears at least twice.
2. At least one of the unordered pairs $\left( \{i_1,i_2\},\ldots,\{i_m,i_1\} \right)$ is identical to one of the unordered pairs $\left( \{j_1,j_2\},\ldots,\{j_m,j_1\} \right)$.
The first condition implies that $\langle\langle
i_1,\ldots,i_m,j_1,\ldots,j_m \rangle\rangle \leq m+2$. Adding the second condition forces at least two more integers to be replications which implies that $\langle\langle
i_1,\ldots,i_m,j_1,\ldots,j_m \rangle\rangle \leq m$. Since $|X_{i,j}(l)|\leq c$ then
$${{\rm Var}}\left( {{\rm tr}}_n\left(\prod_{i=1}^m {\mathbf A}_n(i) \right)\right) \leq
\frac{C}{n^2}.$$
Lemma \[C30\] implies that the limit in is holding in the almost sure sense due to Borel-Cantelli lemma. In other words, if $\left (\{\widetilde{{\mathbf A}}_n(l)\};l=1,\ldots m \right)$ is a family of independent sequences of $\mathbf{Wigner}(n,1)$ matrices whose entries are bounded, then $$\label{dy1} \lim_{n \to \infty} {{\rm tr}}_n \left(\widetilde{{\mathbf A}}_n(1)
\widetilde{{\mathbf A}}_n(2)\cdots \widetilde{{\mathbf A}}_n(m)\right)=\tau
\left({\mathbf{a}}_{1} {\mathbf{a}}_{2} \cdots {\mathbf{a}}_{m}\right) \qquad a.s.$$ where ${\mathbf{a}}_{i}$’s are some free noncommutative random variables in $(\mathcal{A},\tau)$ such that ${\mathbf{a}}_{i}$ has the semicircle law $\gamma_{0,1}$ for all $i$.
Now, let $\left (\{{\mathbf A}_n(l)\};l=1,\ldots m \right)$ be a family of independent sequences of $\mathbf{Wigner}(n,1)$ matrices for which $E(|A_{12}(l)|^4)< \infty$ and $E(A_{11}^2(l))<\infty$ for every $l$. Then by Lemma \[prop21\] and Equation $$\label{dy2} \lim_{n \to \infty} {{\rm tr}}_n \left({\mathbf A}_n(1)
{\mathbf A}_n(2)\cdots {\mathbf A}_n(m)\right)=\tau \left({\mathbf{a}}_{1} {\mathbf{a}}_{2} \cdots
{\mathbf{a}}_{m}\right) \qquad a.s.$$
Finally, since any noncommutative polynomial $p$ can be written as a linear combination of noncommutative monomials, then $$\lim_{n \to
\infty} {{\rm tr}}_n \left( p \left( {\mathbf A}_n(1), \ldots, {\mathbf A}_n(m) \right)
\right) =\tau \left( p \left({\mathbf{a}}_{1},\ldots,{\mathbf{a}}_{m} \right) \right)
\qquad a.s.$$
Proof of Theorem \[T1\] {#Section: Proof of
T1}
-----------------------
$\newline$
Fix $k \geq 1$ and a symmetric block-structure ${\mathbb{B}}_k$. Let $\left (
\{{\mathbf A}_n(i)\} ;i=1,\ldots,h \right )$ be a family of independent Wigner matrices such that $E(|A_{12}(i)|^4)< \infty$ and $E(A_{11}^2(i))<\infty$ for every $i$.
Let’s introduce the noncommutative probability space $(\mathcal{A}
\bigotimes {\mathcal{M}}_k({\mathbb{C}}),\tau \bigotimes {{\rm tr}}_k)$, where $\bigotimes$ stands for the tensor product. A typical element in $\mathcal{A}
\bigotimes {\mathcal{M}}_k({\mathbb{C}})$ is a $k \times k$ matrix whose entries are noncommutative random variables in $\mathcal{A}$. For example, ${\mathbb{B}}_k({\mathbf{a}}_1,\ldots,{\mathbf{a}}_h)\in \mathcal{A} \bigotimes {\mathcal{M}}_k({\mathbb{C}})$ for any ${\mathbf{a}}_1,\ldots,{\mathbf{a}}_h \in \mathcal{A}$. The state $\tau
\bigotimes {{\rm tr}}_k$ is defined by $\tau \bigotimes {{\rm tr}}_k
({\mathbf A})=\frac{1}{k} \sum_{i=1}^k \tau(A_{ii})$ for any ${\mathbf A}\in
\mathcal{A} \bigotimes {\mathcal{M}}_k({\mathbb{C}})$.
The proof of Theorem \[T1\] is based on the method of moments. First, we are going to show that for every $s\in \mathbb{N}$, the limit of ${{\rm tr}}_{nk}
\left({\mathbb{B}}_k\left({\mathbf A}_n(1),\ldots,{\mathbf A}_n(h)\right)^s\right) $ exists as $n \to \infty$, almost surely.
Fix $s \geq 1$. We can see that the trace for the $s$-power of ${\mathbf X}_{n,k}:={\mathbb{B}}_k \left({\mathbf A}_n(1),\ldots,{\mathbf A}_n(h)\right)$ is the trace of some noncommutative polynomial in the matrices ${\mathbf A}_n(1),\ldots,{\mathbf A}_n(h)$. In other words, $${{\rm tr}}_{nk} \left(
{\mathbf X}_{n,k}^s \right)=\frac1k \sum_{i=1}^k {{\rm tr}}_n \left( p_i\left(
{\mathbf A}_n(1),\ldots,{\mathbf A}_n(h) \right) \right)$$ for some noncommutative polynomial $p_i$ and $1 \leq i \leq k$. Theorem \[folk\] implies that for each $i$ $${{\rm tr}}_n \left( p_i\left( {\mathbf A}_n(1),\ldots,{\mathbf A}_n(h) \right) \right)
\to \tau \left( p_i\left( {\mathbf{a}}_1,\ldots,{\mathbf{a}}_h \right) \right) \mbox{
as $n\to \infty$} \qquad a.s.$$ where $\left ({\mathbf{a}}_l;l=1,\ldots m
\right)$ is a family of free semicircular random variables. Therefore $${{\rm tr}}_{nk} \left({\mathbb{B}}_k\left({\mathbf A}_n(1),\ldots,{\mathbf A}_n(h)\right)^s\right)
\to \frac1k \, \tau \left( \sum_{i=1}^k p_i\left(
{\mathbf{a}}_1,\ldots,{\mathbf{a}}_h \right) \right) \mbox{ as $n\to \infty$} \qquad
a.s.$$ Thus, $$\label{Bconv}
{{\rm tr}}_{nk} \left( {\mathbf X}_{n,k}^s \right) \to \tau \bigotimes {{\rm tr}}_k
\left({\mathbb{B}}_k\left({\mathbf{a}}_1,\ldots,{\mathbf{a}}_h\right)^s\right) \mbox{ as $n\to
\infty$} \qquad a.s.$$ Note that if $s$ is an odd integer then $\tau \bigotimes {{\rm tr}}_k
\left({\mathbb{B}}_k\left({\mathbf{a}}_1,\ldots,{\mathbf{a}}_h\right)^s\right)$ is zero by Equation .
To complete the proof, it would be enough to show that there exist $M> 0$ and $C >0$ such that $ \tau \bigotimes {{\rm tr}}_k \left(
{\mathbb{B}}_k\left({\mathbf{a}}_1,\ldots,{\mathbf{a}}_h\right)^{2s} \right) \leq C \, M^{2s}$ for all $s \geq 1$. However, for a fixed $s \geq 1$ $$\label{eqnbound1}
\tau \bigotimes {{\rm tr}}_k \left( {\mathbb{B}}_k\left({\mathbf{a}}_1,\ldots,{\mathbf{a}}_h\right)^{2s} \right)
= \sum_{{\mathbb J}(2s,k)} \tau(B_{j_1 j_2} B_{j_2 j_3} \cdots B_{j_{2s}
j_1})$$ where $B_{ij}\in \{{\mathbf{a}}_1,\ldots,{\mathbf{a}}_h\}$ and ${\mathbb J}(m,k):=\{(j_1,\ldots,j_m):1 \leq j_1,\ldots,j_m \leq k \}$. But again by Equation , $$\label{eqnbound2}
\sum_{{\mathbb J}(2s,k)} \tau(B_{j_1 j_2} B_{j_2 j_3} \cdots B_{j_{2s}
j_1}) \leq k^{2s}C_s=C (2k)^{2s}$$ for some constant $C>0$ where $C_s$ is the Catalan number.
Hence, there exists a non-random symmetric probability measure $\mu_{{\mathbb{B}}_k}$ with a compact support in $\mathbb{R}$ that has the moments $\tau \bigotimes {{\rm tr}}_k
\left({\mathbb{B}}_k\left({\mathbf{a}}_1,\ldots,{\mathbf{a}}_h\right)^s\right)$, for every $s\geq 1$, such that $$\label{Fconv}
\mu_{{\mathbf X}_{n,k} } \xrightarrow{{\mathcal{D}}} \,\, \mu_{{\mathbb{B}}_k} \mbox{ as
$n\to \infty$} \qquad a.s.$$
Concluding remarks
==================
1. We have shown in Proposition \[limitcirculant\] that the limiting spectral measure of Hermitian Circulant block-matrices with Wigner blocks is a mixture of two semicircle laws. See Figure \[F1\].
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Histograms of the eigenvalues of $100$ ${\mathbb{C}}_k$ block -matrices with Wigner blocks of dimension $n=200$ for $k=4$ and $5$. The solid curves are for the exact probability density functions provided in Proposition \[limitcirculant\]. \[F1\]](C4c.eps "fig:"){height="4.5cm"} ![Histograms of the eigenvalues of $100$ ${\mathbb{C}}_k$ block -matrices with Wigner blocks of dimension $n=200$ for $k=4$ and $5$. The solid curves are for the exact probability density functions provided in Proposition \[limitcirculant\]. \[F1\]](C5c.eps "fig:"){height="4.5cm"}
$k=4$ $k=5$
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
We can also read from the simulation, see Figure \[F2\], of the $3
\times 3$ Toeplitz block-matrix $${\mathbb{T}}_3({\mathbf A},{\mathbf B},{\mathbf C})= \left[
{\begin{array}{*{20}c}
{\mathbf A}& {\mathbf B}& {\mathbf C}\\
{\mathbf B}& {\mathbf A}& {\mathbf B}\\
{\mathbf C}& {\mathbf B}& {\mathbf A}\\
\end{array} } \right]$$ that the limiting spectral measure is a mixture of two distributions. It is evident that one of them is the semicircle law $\gamma_{0,2}$.
![Histograms of the eigenvalues of $100$ ${\mathbb{T}}_3$ block -matrices with Wigner blocks of dimension $n=200$. \[F2\]](T3.eps){height="4.5cm"}
2. If we change the blocks of the Circulant block-matrix in Proposition \[limitcirculant\] into random symmetric circulant matrices then from the proof of the proposition and the limit in [@Bose-Mitra Remark 2], the limiting spectral measure will be a mixture of two normal distributions.
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank my advisor professor Bryc for his suggestions and his continuous and sincere help and support. I am also grateful to professor Szarek for references pertinent to Theorem \[folk\] and its proof.
[10]{}
G. Anderson and O. Zeitouni. A law of large numbers for finite-range dependent random matrices. , 2006.
Z. Bai. Methodologies in spectral analysis of large-dimensional random matrices, a review. , 9(3):611–677, 1999. With comments by G. J. Rodgers and Jack W. Silverstein; and a rejoinder by the author.
Arup Bose and Joydip Mitra. Limiting spectral distribution of a special circulant. , 60:111–120, 2002.
M. Capitaine and C. Donati-Martin. Strong asymptotic freeness for wigner and wishart matrices.
P. Davis. . Pure and applied mathematics. New York : Wiley, 1979.
A. Guionnet. . Saint-Flour. July 2006.
H. Henderson and S. Searle. The vec-permutation matrix, the vec operator and [K]{}ronecker products: a review. , 9(4):271–288, 1980/81.
F. Hiai and D. Petz. , volume 77 of [*Mathematical Surveys and Monographs*]{}. American Mathematical Society, Providence, RI, 2000.
P. Meyer. , volume 1538 of [ *Lecture Notes in Mathematics*]{}. Springer-Verlag, 1995.
. Ryan. On the limit distributions of random matrices with independent or free entries. , 193(3):595–626, 1998.
J. Schenker and H. Schulz-Baldes. Semicircle law and freeness for random matrices with symmetries or correlations. , 12(4):531–542, 2005.
S. Szarek. Private communiactions. 2005.
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} |
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abstract: 'Given a sequence of distinct positive integers $v_1,v_2,\ldots$ and any positive integer $n$, the discriminator $D_v(n)$ is defined as the smallest positive integer $m$ such $v_1,\ldots,v_n$ are pairwise incongruent modulo $m$. We consider the discriminator for the sequence $u_1,u_2,\ldots$, where $u_j$ equals the absolute value of $((-3)^j-5)/4$, that is $u_j=(3^j-5(-1)^j)/4$. We prove a 2012 conjecture of Sabin Salajan characterizing the discriminator of the sequence $u_1,u_2,\ldots$.'
author:
- Pieter Moree and Ana Zumalacárregui
title: 'Salajan’s conjecture on discriminating terms in an exponential sequence'
---
Introduction
============
Given a sequence of distinct positive integers $v_1,v_2,\ldots$ and any positive integer $n$, the discriminator $D_v(n)$ is defined as the smallest positive integer $m$ such $v_1,\ldots,v_n$ are pairwise incongruent modulo $m$. Browkin and Cao [@BC] relate it to cancellations algorithms similar to the sieve of Eratosthenes.\
The main problem is to give an easy description or characterization of $D_v(n)$ (in many cases such a characterization does not seem to exist). Arnold, Benkoski and McCabe [@ABM] might have been the first to consider this type of problem (they introduced also the name). They considered the case where $v_j=j^2$. Subsequently various authors, see e.g. [@BSW; @PP; @Sun; @Zieve], studied the discriminator for polynomial sequences.\
It is a natural problem to study the discriminator for non-polynomial sequences. Very little work has been done in this direction. E.g. there are some conjectures due to Sun [@Sun] in case $v_j=j!$, $v_j={\binom{2j}{j}}$ and $v_j=a^j$.\
In this paper we study the discriminator for a closely related sequence $u_1,u_2,\ldots$ with $u_j=|(-3)^j-5|/4=(3^j-5(-1)^j)/4$. This sequence satisfies the binary recurrence $u_n=2 u_{n-1} + 3 u_{n-2}$, for every $n \geq 3$. with starting values $u_1=2$ and $u_2=1$. The first few terms are $$2, 1, 8, 19, 62, 181, 548, 1639, 4922,\ldots$$ Note that for $j\ge 2$ we have $u_{j+1}>u_j$ and that all $u_j$ are distinct. It is almost immediate that the terms are of alternating parity. Since all $u_j$ are distinct the number $$D_S(n)=\min\{m\ge 1:u_1,\ldots,u_n{\rm~are~pairwise~distinct~modulo~}m\}$$ is well-defined. Note that $D_S(n)\ge n$. In Table 1 we give the values of $D_S(n)$ for $1\le n\le 32768$ (with the powers of $5$ underlined).\
**TABLE 1**
range value range value
----------- ------------------- --------------- ---------------------
$1$ $1$ $129-256$ $256$
$2$ $2$ $257-512$ $512$
$3-4$ $4$ $513-1024$ $1024$
$5-8$ $8$ $1025-2048$ $2048$
$9-16$ $16$ $2049-2500$ $\underline{3125}$
$17-20$ $\underline{25}$ $2501-4096$ $4096$
$21-32$ $32$ $4097-8192$ $8192$
$33-64$ $64$ $8193-12500$ $\underline{15625}$
$65-100$ $\underline{125}$ $12501-16384$ $16384$
$101-128$ $128$ $16385-32768$ $32768$
Based on this table Sabin Salajan, who at the time was an intern with the first author, proposed a conjecture that we will prove in this paper to be true. The first author had asked Sabin to find second order linear recurrences for which the discriminator values have a nice structure. After an extensive search Sabin came up with the sequence $u_1,u_2,\ldots$. For convenience we call this sequence the [*Salajan sequence*]{} $S$ and its associated discriminator $D_S$ the [*Salajan discriminator*]{}. If $m=D_S(n)$ for some $n\ge 1$, then we say that $m$ is a [*Salajan value*]{}, otherwise it is a [*Salajan non-value*]{}.
\[main\] Let $n\ge 1$. Put $e=\lceil{\log_2(n)}\rceil$ and $f=\lceil{\log_{5}({5n/4)}}\rceil$. Then $$D_S(n)=\min\{2^e,5^f\}.$$
If the interval $[n,5n/4)$ contains a power of $2$, say $2^a$, then we have $D_S(n)=2^a$.
Note that $2^e$ is the smallest power of $2$ which is $\ge n$ and that $5^f$ is the smallest power of $5$ which is $\ge 5n/4$.\
From Table 1 one sees that not all powers of $5$ are Salajan values. Let ${\cal F}$ be the set of integers $b\ge 1$ such that the interval $[4\cdot5^{b-1},5^b]$ does not contain a power of $2$. Then it is not difficult to show that the image of $D_S$ is given by $\{2^a:a\ge 0\}\cup \{5^b:b\in {\cal F}\}$. Using Weyl’s criterion one can easily establish (see Section \[Izabela\]) the following proposition.
\[iza\] As $x$ tends to infinity we have $\#\{b\in {\cal F}:b\le x\}\sim \beta x$, with $\beta = 3-\log 5/\log 2 = 0.678\ldots$.
Strategy of the proof of Theorem \[main\]
=========================================
For the benefit of the reader we describe the strategy of the (somewhat lengthy) proof of Theorem \[main\].\
We first show that if $2^e\ge n$ and $5^f\ge 5n/4$, then $D_S(n)\le \min\{2^e,5^f\}$. This gives us the absolutely crucial upper bound $D_S(n)<2n$.\
Next we study the periodicity of the sequence modulo $d$ and determine its period $\rho(d)$. The idea is to use the information so obtained to show that many $d$ are Salajan non-values. In case $3\nmid d$ the sequence turns out to be purely periodic with even period that can be given precisely. This is enough for our purposes as we can show that $3|D_S(n)$ does not occur.
Now we restrict to the $d$ with $3\nmid d$. Using that $D_S(n)<2n$ one easily sees that if $\rho(d)\le d/2$, then $d$ is a Salajan non-value. The basic property (\[per2\]) of the period together with the evenness of the period now excludes composite values of $d$. Thus we have $d=p^m$, with $p$ a prime.
In order for $\rho(p^m)>p^m/2$ to hold we find that we must have ord$_9(p)=(p-1)/2$, that is $9$ must have maximal possible order modulo $p$. Moreover, $9$ must have maximal possible order modulo $p^m$, that is ord$_9(p)=\varphi(p^m)/2$. (A square cannot have a multiplicative order larger than $\varphi(p^m)/2$ modulo $p^m$.) This is about as far as the study of the periodicity will get us. To get further we will use a more refined tool, the [*incongruence index*]{}. Given an integer $m$, this is the maximum $k$ such that $u_1,\ldots,u_k$ are pairwise distinct modulo $m$. We write $\iota(m)=k$. For $3\nmid m$, $\iota(m)\le \rho(m)$. Using that $D_S(n)<2n$ one notes that if $\iota(d)\le d/2$, then $d$ is a Salajan non-value.
For the primes $p>3$ we show by a lifting argument that if $\iota(p)<\rho(p)$, then $p^2,p^3,\ldots$ are Salajan non-values. Likewise, we prove that if $\iota(p)\le p/2$, then $p,p^2,p^3,\ldots$ are Salajan non-values. We then show that except for $p=5$, all primes with ord$_9(p)=(p-1)/2$ satisfy $\iota(p)<\rho(p)$. At this point we are left with the primes $p>5$ satisfying ord$_9(p)=(p-1)/2$ as only possible Salajan values. Then using classical exponential sums techniques, and some combinatorial arguments, we infer that $\iota(p)<4p^{3/4}$. Using this bound, after some computational work, we then conclude that $\iota(p)\le p/2$ for every $p>5$.
Thus we are left with $D_S(n)=2^a$ for some $a$ or $D_S(n)=5^b$ for some $b$. By Lemma \[prop2n\] and Lemma \[prop5n\] it now follows that $2^a\ge n$ and $5^b\ge 5n/4$. This then completes the proof.
Preparations for the proof {#preparations}
==========================
We will show that $2^e$ with $2^e\ge n$ and $5^f$ with $5^f\ge 5n/4$ are admissible discriminators. That is, we will show that the sequence $u_1,\dots, u_n$ lie in distinct residue classes modulo $2^e$ and in distinct residue classes modulo $5^f$.
Let $p$ be a prime. If $p^a|n$ and $p^{a+1}\nmid n$, then we put $\nu_p(n)=a$. The following result is well-known, for a proof see, e.g., Beyl [@Beyl].
\[Beyl\] Let $p$ be a prime, $r\ne -1$ an integer satisfying $r\equiv 1({\rm mod~}p)$ and $n$ a natural number. Then $$\nu_p(r^n-1) =
\begin{cases}
\nu_2(n)+\nu_2(r^2-1)-1 & \text{if $p=2$ and $n$ is even};\\
\nu_p(n)+\nu_p(r-1) & \text{otherwise}.
\end{cases}$$
\[hakbijl\] Let $f\ge 2$ and $p$ be an odd prime. If $g$ is a primitive root modulo $p$, then $g$ is a primitive root modulo $p^f$ if and only if $g^{p-1}\not\equiv 1({\rm mod~}p^2)$.
Corollary \[hakbijl\] is a classical result from elementary number theory. For an alternative proof see, e.g., Apostol [@Apostel Theorem 10.6].
\[alpha\] Let $n\ge 1$ be an integer, $p$ a prime and put $e_p=\lfloor{\log_p(n-1)}\rfloor$. Let $r\equiv 1({\rm mod~}p)$ be an integer $\ne -1$. Put $r_p=\nu_p(r-1)$. If $p=2$, we assume in addition that $r$ is a square. The integers $r,\ldots,r^{n}$ are pairwise distinct modulo $p^{e_p+r_p+1}$.
[*Proof*]{}. Write $m=p^{e_p+r_p+1}$. Let $1\le i<j\le n$ and suppose that $r^{i}\equiv r^{j}({\rm mod~}m)$, thus $r^{j-i}\equiv 1({\rm mod~}m)$ and hence $\nu_p(r^{j-i}-1)\ge e_p+r_p+1$. Note that $\nu_p(k)\le e_p$ for $1\le k\le n-1$. Thus $\nu_p(j-i)\le e_p$ and, by Lemma \[Beyl\], we deduce that $\nu_p(r^{j-i}-1)\le e_p+r_p$. Contradiction.[$\Box$]{}
\[9and81\]$~$\
The integers $9,\ldots,9^{n}$ are pairwise distinct modulo $2^{e_2+4}$.\
The integers $81,\ldots,81^n$ are pairwise distinct modulo $5^{e_5+2}$.
\[prop2n\] Let $n \geq 2$ be an integer with $n\le 2^m$. Then, we have that $u_1,\dots,u_n$ are pairwise distinct modulo $2^m$.
[*Proof*]{}. For $n=2$ the result is obvious. So assume that $n\ge 3$. Since the terms of the sequence alternate between even and odd, it suffices to compare the remainders $({\rm mod~}2^m)$ of the terms having an index with the same parity. Thus assume that we have $$u_{2j+\alpha} \equiv u_{2k+\alpha}({\rm mod~}2^m){\rm ~with~}1\le 2j+\alpha<2k+\alpha \leq n,~\alpha\in \{1,2\}.$$ It follows from this that $9^{k-j}\equiv 1 ({\rm mod~}2^{m+2})$. We have $\nu_2(9^{k-j}-1)=\nu_2(k-j)+3$ by Lemma \[Beyl\]. Further, $2k-2j \leq n-1<2^m$, so $\nu_2(k-j) \leq m-2$ (here we used that $n\ge 3$). Therefore $\nu_2(9^{k-j}-1)=\nu_2(k-j)+3 \leq (m-2) +3 = m+1$, which implies that $9^{k-j}-1$ cannot be divisible by $2^{m+2}$. Contradiction. [$\Box$]{}\
[Remark]{}. The incongruence of $u_i$ and $u_j$ (mod $2^m$) with $i$ and $j$ of the same parity and $1\le i<j\le n$ is equivalent with $9,9^2,\ldots,9^{\lfloor (n-1)/2\rfloor}$ being pairwise incongruent mod $2^m$. Using this observation and Corollary \[9and81\] we obtain an alternative proof of Lemma \[prop2n\].\
On noting that trivially $D_S(n)\ge n$ and that for $n\ge 2$ the interval $[n,2n-1]$ always contains some power of $2$, we obtain the following corollary to Lemma \[prop2n\].
\[bertie\] We have $n\leq D_S(n)\le 2n-1$.
\[prop5n\] The integers $u_1,\dots,u_n$ are pairwise distinct modulo $5^{m}$ iff $$5^m\ge 5n/4.$$
[*Proof*]{}. If $5^m<5n/4$, then $1+4\cdot 5^{m-1}\le n$. By Lemma \[Beyl\] we have $$81^{5^{m-1}}\equiv 1({\rm mod~}5^m),$$ which ensures that $u_1\equiv u_{1+4\cdot 5^{m-1}}({\rm mod~}5^m)$. Next let us assume that $5^m\ge 5n/4$. This ensures that $m\ge 1$. The remainders of the sequence modulo $5$ are $2,1,4,3,2,1,\dots$ and so the sequence has period $4$ modulo $5$. Thus we may assume that $m\ge 2$. It suffices to show that $u_{j_1}\not\equiv u_{k_1}({\rm mod~}5^{m})$ with $1\le j_1< k_1\le n$ in the same congruence class modulo $4$. We will argue by contradiction. Thus we assume that $$u_{4j+\alpha} \equiv u_{4k+\alpha}({\rm mod~}5^{m}){\rm ~with~}1\le 4j+\alpha<4k+\alpha \leq n,~\alpha\in \{1,2,3,4\}.$$ From this it follows that $81^{k-j}\equiv 1 ({\rm mod~}5^{m})$, where $k-j\le (n-\alpha)/4<n/4\le 5^{m-1}$ by hypothesis and hence $\nu_5(k-j)\le m-2$. On invoking Lemma \[Beyl\] we now infer that $\nu_5(81^{k-j}-1)=\nu_5(k-j) + 1 \leq m-2+1=m-1$. Contradiction. [$\Box$]{}\
[Remark]{}. The incongruence of $u_i$ and $u_j$ (mod $5^m$) with $i$ and $j$ in the same residue class modulo $4$ and $1\le i<j\le n$ is equivalent with $81,81^2,\ldots,81^{\lfloor (n-1)/4\rfloor}$ being pairwise incongruent mod $5^m$. Using this observation and Corollary \[9and81\] we obtain an alternative proof of Lemma \[prop5n\].\
In order to determine whether a given $m$ discriminates $u_1,\ldots,u_n$ modulo $m$, we can separately consider whether $u_i\not\equiv u_j({\rm mod~}m)$ with $1\le i<j\le n$ of the same parity (case 1) and with distinct parity (case 2). The first case is easy and covered by Lemma \[oneventje\], the second case is trivial in case $m$ is a power of $2$ or $5$, but in general much harder than the first case.
\[oneventje\] Suppose that $3\nmid m$ and $1\le \alpha\le n$. We have $u_i\not\equiv u_j({\rm mod~}m)$ for every pair $(i,j)$ satisfying $\alpha\le i< j\le n$ with $i\equiv j({\rm mod~}2)$ iff ord$_9(4m)>(n-\alpha)/2$.
[*Proof*]{}. We have $u_i\not\equiv u_{i+2k}({\rm mod~}m)$ iff $9^k\not\equiv 1({\rm mod~}4m)$. Thus $u_i\not\equiv u_j({\rm mod~}m)$ for every pair $(i,j)$ with $\alpha\le i<j\le n$ and $i\equiv j({\rm mod~}2)$ iff $9^k\not\equiv 1({\rm mod~}4m)$ for $1\le k\le (n-\alpha)/2$. [$\Box$]{}\
[*Alternative proof of Lemma*]{} \[prop2n\]. If $i$ and $j$ are of different parity, then $u_i\not\equiv u_j({\rm mod~}2)$. Hence we may assume that $i$ and $j$ are of the same parity. On invoking Lemma \[oneventje\] we then obtain that $u_1,\ldots,u_{n}$ are distinct modulo $2^m$ iff ord$_9(2^{m+2})>(n-1)/2$. By Lemma \[Beyl\] we have ord$_9(2^{m+2})=2^{m-1}$, concluding the proof. [$\Box$]{}\
[*Alternative proof of Lemma*]{} \[prop5n\]. The remainders of the sequence modulo $5$ are $2,1,4,3,2,1,\dots$ and so terms $u_i$ and $u_j$ with $i$ and $j$ of different parity are incongruent. Now by Lemma \[oneventje\] the integers $u_1,\dots,u_n$ are pairwise distinct modulo $5^{f}$ iff ord$_9(4\cdot 5^f)>(n-1)/2$. Since $3$ is a primitive root modulo $5$ and $3^4\not\equiv 1({\rm mod~}5^2)$, we have by Corollary \[hakbijl\] that $3$ is a primitive root modulo $5^f$ and hence ord$_3(5^f)=4\cdot 5^{f-1}=\varphi(5^f)$, with $\varphi$ Euler’s totient function. On making use of the trivial observation that, for integers $m$ coprime to $3$, $$\label{lcmorder}
2{\rm ord}_9(4m)={\rm lcm}(2,{\rm ord}_3(4m)),$$ we infer that ord$_9(4\cdot 5^f)={\rm ord}_9(5^f)={\rm ord}_3(5^f)/2=2\cdot 5^{f-1}$. The proof is now finished by noting that the condition ord$_9(4\cdot 5^f)>(n-1)/2$ is equivalent to $5^f\ge 5n/4$. [$\Box$]{}
Periodicity and discriminators
==============================
Generalities
------------
We say that a sequence of integers $\{v_j\}_{j=1}^{\infty}$ is [*(eventually) periodic*]{} modulo $d$ if there exist integers $n_0\ge 1$ and $k\ge 1$ such that $$\label{per1}
v_n\equiv v_{n+k}({\rm mod~}d)$$ for every $n\ge n_0$. The minimal choice for $n_0$ is called the [*pre-period*]{}. The smallest $k\ge 1$ for which (\[per1\]) holds for every $n\ge n_0$ is said to be the [*period*]{} and denoted by $\rho_v(d)$. In case we can take $n_0=1$ we say that the sequence is [*purely periodic*]{} modulo $d$.
Let $\{v_j\}_{j=1}^{\infty}$ be a second order linear recurrence with the two starting values and the coefficients of the defining equation being integers. Note that, for a given $d$, there must be a pair $(a,b)$ such hat $a\equiv v_n$ and $b\equiv v_{n+1}$ modulo $d$ for infinitely many $n$. Since a pair of consecutive terms determines uniquely all subsequent ones, it follows that the sequence is periodic modulo $d$. If we consider $n$-tuples instead of pairs modulo $d$, we see that an $n$th order linear recurrence with the $n$ starting values and the coefficients of the defining equation being integers, is always periodic modulo $d$.\
If a sequence $v$ is periodic modulo $d_1$ and modulo $d_2$ and $(d_1,d_2)=1$, then we obviously have $$\label{per2}
\rho_v(d_1d_2)={\rm lcm}(\rho_v(d_1),\rho_v(d_2)).$$ If the sequence is purely periodic modulo $d_1$ and modulo $d_2$ and $(d_1,d_2)=1$, then it is also purely periodic modulo $d_1d_2$. Another trivial property of $\rho_v$ is that if the sequence $v$ is periodic modulo $d_2$, then for every divisor $d_1$ of $d_2$ we have $$\label{per3}
\rho_v(d_1)|\rho_v(d_2).$$ The following result links the period with the discriminator. Its moral is that if $\rho_v(d)$ is small enough, we cannot expect $d$ to occur as $D_v$-value, i.e. $d$ does not belong to the image of $D_v$.
\[gee\] Assume that $D_v(n)\le g(n)$ for every $n\ge 1$ with $g$ non-decreasing. Assume that the sequence $v$ is purely periodic modulo $d$ with period $\rho_v(d)$. If $g(\rho_v(d))<d$, then $d$ is a $D_v$-non-value.
[*Proof*]{}. Since $v_1\equiv v_{1+\rho_v(d)}({\rm mod~}d)$ we must have $\rho_v(d)\ge n$. Suppose that $d$ is a $D_v$-value, that is for some $n$ we have $D_v(n)=d$. Then $d=D_v(n)\le g(n)\le g(\rho_v(d))$. Contradiction. [$\Box$]{}
Periodicity of the Salajan sequence {#peri}
-----------------------------------
The purpose of this section is to establish Theorem \[generalperiod\], which gives an explicit formula for the period $\rho(d)$ and the pre-period for the Salajan sequence. Since it is easy to show that $3\nmid D_S(n)$, it would be actually enough to study those integers $d$ with $3\nmid d$ (in which case the Salajan sequence is purely periodic modulo $d$). However, for completeness we discuss the periodicity of the Salajan sequence for [*every*]{} d.
\[generalperiod\] Suppose that $d>1$. Write $d=3^{\alpha}\cdot \delta$ with $(\delta,3)=1$. The period of the Salajan sequence modulo $d$, $\rho(d)$, exists and satisfies $\rho(d)=2{\rm ord}_9(4\delta)$. The pre-period equals $\max(1,\alpha)$.
\[9\] The Salajan sequence is purely periodic iff $9\nmid d$.
Write $d=3^{\alpha}\cdot \delta$ with $(\delta,3)=1$. The Salajan sequence is purely periodic iff $9\nmid d$. Furthermore, if $9\nmid d$, then $\rho(d)\,|\,2{\rm ord}_9(\delta)$.
[*Proof*]{}. Since $u=2,{\overline{1,8}}({\rm mod~}9)$ the condition $9\nmid d$ is necessary for the Salajan sequence to be purely periodic modulo $d$.
We will now show that it is also sufficient. Let us first consider the case where $\alpha=0$. We note that $u_n\equiv u_{n+2k} ({\rm mod~}d)$ iff $3^n\equiv 3^{n+2k}({\rm mod~}4\delta)$. It follows that $\rho(d)\,|\,2{\rm ord}_9(4\delta)\,|\,2k$. If $\alpha=1$, then we use (\[per2\]) and the observation that $2=\rho(3)\,|\,2{\rm ord}_9(4\delta)$. [$\Box$]{}\
[Remark]{}. The above proof shows that if $\rho(d)$ is even, then $\rho(d)=2{\rm ord}_9(4\delta)$.
\[even\] Assume that $9\nmid d$ and $d>1$. The Salajan sequence is purely periodic with period $\rho(d)=2{\rm ord}_9(4\delta)$, where $d=3^{\alpha}\cdot \delta$ with $(\delta,3)=1$.
[*Proof*]{}. By the previous remark it suffices to show that $\rho(d)$ is even. If $\alpha=1$, then $2=\rho(3)\,|\,\rho(d)$ (here we use (\[per3\])) and we are done, so we may assume that $\alpha=0$. If $5\,|\,d$, then $4=\rho(5)\,|\,\rho(d)$ and so we may assume that $(5,d)=1$. Suppose that $\rho(d)$ is odd. Then $$\label{per4}
u_{n}\equiv u_{n+\rho(d)}({\rm mod~}d)$$ iff $3^n-5(-1)^n\equiv 3^{n+\rho(d)}+5(-1)^n({\rm mod~}4d)$ iff $5^*(1-3^{\rho(d)})/2\equiv (-3)^{-n}({\rm mod~}2d)$, where $5^*$ is the inverse of 5 modulo $2d$. Now if (\[per4\]) is to hold for every $n\ge 1$, then $(-3)^n$ assumes only one value as $n$ ranges over the positive integers. Since $(-3)^{\phi(2d)}\equiv 1({\rm mod~}2d)$ we must have $(-3)^n\equiv 1({\rm mod~}2d)$ for every $n\ge 1$. This implies that $d=2$ or $d=1$. Since $5^*(1-3^2)/2\not\equiv 1({\rm mod~}4)$ it follows that $d=1$. Contradiction. [$\Box$]{}\
[*Proof of Theorem*]{} \[generalperiod\]. It is an easy observation that modulo $3^{\alpha}$ the Salajan sequence has pre-period $\max(\alpha,1)$ and period two. This in combination with Lemma \[even\] and (\[per2\]) then completes the proof. [$\Box$]{}
Comparison of $\rho(d)$ with $d$
--------------------------------
\[liftje\] Let $p>3$. We have $\rho(p^m)\,|\,\rho(p)p^{m-1}$.
[*Proof*]{}. Since $3^{\rho(p)}\equiv 1({\rm mod~}p)$ we have $3^{\rho(p)p^{m-1}}\equiv 1({\rm mod~}p^m)$ and, provided that $\rho(p)$ is even, this implies that $u_k\equiv u_{k+\rho(p)p^{m-1}}({\rm mod~}p^m)$ for every $k\ge 1$. [$\Box$]{}
\[liftje2\] Either $\rho(p^2)=\rho(p)$ or $\rho(p^2)=p\rho(p)$.
[*Proof*]{}. We have $\rho(p)|\rho(p^2)|p\rho(p)$.
\[previous\] We have $\rho(2^e)=2^e$ and $\rho(3^e)=2$. If $p$ is odd, then $\rho(p^e)|\varphi(p^e)$.
[*Proof*]{}. From Lemma \[Beyl\] and Lemma \[even\] we infer that ord$_9(2^{e+2})=2^{e-1}$ and hence $\rho(2^e)=2^e$. For $n$ large enough modulo $3^e$ the sequence alternates between $-5/4$ and $5/4$ modulo $3^e$. Since these are different residue classes, we have $\rho(3^e)=2$.\
It remains to prove the final claim. If $p=3$ it is clearly true and thus we may assume that $p>3$. Note that $\rho(p^e)=2{\rm ord}_9(4p^e)=2{\rm ord}_9(p^e)$ and that $2{\rm ord}_9(p^e)\,|\,2(\varphi(p^e)/2)=\varphi(p^e)$. [$\Box$]{}
\[previous1\] We have $\rho(d)\le d$.
\[uppierho\] Suppose that $d_1,d_2>1$ and $(d_1,d_2)=1$. Then $$\rho(d_1d_2)\le \rho(d_1)\rho(d_2)/2\le d_1d_2/2.$$
[*Proof*]{}. We have $\rho(d_1d_2)={\rm lcm}(\rho(d_1),\rho(d_2))$. By Lemma \[even\] both $\rho(d_1)$ and $\rho(d_2)$ are even. It thus follows that $\rho(d_1d_2)\le \rho(d_1)\rho(d_2)/2$. The final estimate follows by Corollary \[previous1\]. [$\Box$]{}
Non-values of $D_S(n)$ {#non-values}
======================
Recall that if $m=D_S(n)$ for some $n\ge 1$ we call $m$ a Salajan value and otherwise a Salajan non-value.\
Most of the following proofs rely on the simple fact that for certain sets of integers we have that if $u_1,\dots,u_n$ are in $n$ distinct residue classes modulo $m$, then $m\ge 2n$ contradicting Corollary \[bertie\].
$D_S(n)$ is not a multiple of $3$
---------------------------------
\[notdrie\] We have $3\nmid D_S(n)$.
[*Proof*]{}. We argue by contradiction and so assume that $D_S(n)=3^{\alpha}m$ with $(m,3)=1$ and $\alpha \ge 1$. Since by definition $u_{\alpha}\not\equiv u_{\alpha+2t}({\rm mod~}3^{\alpha}m)$ for $t=1,\ldots,\lfloor{(n-\alpha)/2}\rfloor$ and $u_{\alpha}\equiv u_{\alpha+2t}({\rm mod~}3^{\alpha})$ for every $t\ge 1$, it follows that $u_{i}\not\equiv u_{j}({\rm mod~}m)$ with $\alpha \le i<j\le n$ and $i$ and $j$ of the same parity. By Lemma \[oneventje\] it then follows that ord$_9(4m)>(n-\alpha)/2$. By Lemma \[even\], Corollary \[previous1\] and Corollary \[bertie\] we then find that $n-\alpha+1\le 2{\rm ord}_9(4m)=\rho(m)\le m\le 2n/3^{\alpha}$. This implies that $n\le 3^{\alpha}(\alpha-1)/(3^{\alpha}-1)$. On the other hand, by Corollary \[bertie\] we have $3^{\alpha}m\le 2n$ and hence $n\ge 3^{\alpha}/2$. Combining the upper and the lower bound for $n$ yields $3^{\alpha}\le 2\alpha-1$, which has no solution with $\alpha\ge 1$. [$\Box$]{}\
[Remark]{}. It is not difficult to show directly that if $3\nmid m$, then $2{\rm ord}_9(4m)\le m$ and thus a proof of Lemma \[notdrie\] can be given that is free of periodicity considerations and only involves material from Section \[preparations\].
$D_S(n)$ is a prime-power
-------------------------
Assume $9\nmid d$. By Corollary \[9\] and Corollary \[bertie\] we can take $g(n)=2n-1$ in Lemma \[gee\]. This yields Lemma \[nonnie\]. However, for the convenience of the reader we give a more direct proof.
\[nonnie\] Suppose that $d$ with $9\nmid d$ satisfies $\rho(d)\le d/2$, then $d$ is a Salajan non-value.
*Proof*. Suppose that $d=D_S(n)$ for some integer $n$. By Corollary \[bertie\] we have $d<2n$. By Lemma \[generalperiod\] the condition $9\nmid d$ guarantees that the Salajan sequence is purely periodic modulo $d$. Since $u_1\equiv u_{1+\rho(d)}({\rm mod~d})$ we must have $\rho(d)\ge n$. Now suppose that $d\ge 2\rho(d)$. It then follows that $d\ge 2n$, contradicting $d=D_S(n)<2n$. [$\Box$]{}\
We now have the necessary ingredients to establish the following result. Let $p$ be odd. On noting that in $(\mathbb Z/p^m\mathbb Z)^*$ a square has maximal order $\varphi(p^m)/2$, we see that the following result says that a Salajan value is either a power of two or prime power $p^m$ with $9$ having maximal multiplicative order in $(\mathbb Z/p^m\mathbb Z)^*$.
\[redpp\] A Salajan value $>1$ must be of the form $p^m$, with $p=2$ or $p>3$ and $m\ge 1$. Further, one must have ord$_9(p^m)=\varphi(p^m)/2$ and ord$_9(p)=(p-1)/2$. If $m\ge 2$ we must have $3^{p-1}\not\equiv 1({\rm mod~}p^2)$.
[*Proof*]{}. Suppose that $d>1$ is a Salajan value that is not a prime power. Thus we can write $d=d_1d_2$ with $d_1,d_2>1$, $(d_1,d_2)=1$. By Lemma \[notdrie\] we have $3\nmid d_1d_2$. By Lemma \[uppierho\] we have $\rho(d_1d_2)\le d_1d_2/2$, which by Lemma \[nonnie\] implies that $d=d_1d_2$ is a non-value. Thus $d$ is a prime power $p^m$. By Lemma \[notdrie\] we have $p=2$ or $p>3$. Now let us assume that $p>3$. By Lemma \[previous\] we have either $\rho(p^m)=\varphi(p^m)$ or $\rho(p^m)\le \varphi(p^m)/2$. The latter inequality leads to $\rho(p^m)\le p^m/2$ and hence to $p^m$ being a non-value. Using Theorem \[generalperiod\] we infer that ord$_9(p^m)=\varphi(p^m)/2$. Now if ord$_9(p^m)<(p-1)/2$, this leads to ord$_9(p^m)<\varphi(p^m)$ and hence we must have ord$_9(p^m)=(p-1)/2$. Finally, suppose that $m\ge 2$ and $3^{p-1}\equiv 1({\rm mod~}p^2)$. Then ord$_9(p^m)<\varphi(p^m)/2$. This contradiction shows that if $m\ge 2$ we must have $3^{p-1}\not\equiv 1({\rm mod~}p^2)$. [$\Box$]{}\
The possible Salajan values can be further limited by using some results on a quantity we will baptise as the [*incongruence index*]{}.
$D_S(n)$ is a prime or a small prime power {#primesets}
------------------------------------------
Put ${\cal P}=\{p:p>3,~{\rm ord}_9(p)=(p-1)/2\}$. If a prime $p>3$ is a Salajan value, then by Lemma \[redpp\] we must have $p\in {\cal P}$. If $p\in {\cal P}$, then by Theorem \[generalperiod\] we have $\rho(p)=p-1$. This will be used a few times in the sequel. Let $${\cal P}_j=\{p:p>3,~p\equiv j({\rm mod~}4),~{\rm ord}_3(p)=p-1\},~j\in \{1,3\}$$ and $${\cal P}_2=\{p:p>3,~p\equiv 3({\rm mod~}4),~{\rm ord}_3(p)=(p-1)/2\}.$$ By equation (\[lcmorder\]) we have $2{\rm ord}_9(p)={\rm lcm}(2,{\rm ord}_3(p))$. From this we infer that ${\cal P}={\cal P}_1 \cup {\cal P}_2 \cup {\cal P}_3$. We have $${\cal P}_1=\{5,17,29,53,89,101,113,137,149,173,197,233,257,269,281,293,\ldots\},$$ $${\cal P}_2=\{11,23,47,59,71,83,107,131,167,179,191,227,239,251,263,\ldots\},$$ $${\cal P}_3=\{7,19,31,43,79,127,139,163,199,211,223,283,
\ldots\}.$$ (The reader interested in knowing the natural densities of these sets, under GRH, is referred to the appendix.)\
The aim of this section is to establish the following result, the proof of which makes use of properties of the incongruence index and is given in Section \[5.4.1\].
\[reducetoprime\] Let $d>1$ be an integer coprime to $10$. If $d$ is a Salajan value, then $d\in {\cal P}_1\cup {\cal P}_2$.
### The incongruence index
Let $\{v_j\}_{j=1}^{\infty}$ be a sequence of integers and $m$ an integer. Then the largest number $k$ such that $v_1,\ldots,v_{k}$ are pairwise incongruent modulo $m$, we call the incongruence index, $\iota_v(m)$, of $v$ modulo $m$.
Note that $\iota_v(m)\le m$. In case the sequence $v$ is purely periodic modulo $d$, we have $\iota_v(d)\le \rho_v(d)$. A minor change in the proof of Lemma \[gee\] yields the following result.
\[gee2\] Assume that $D_v(n)\le g(n)$ for every $n\ge 1$ with $g$ non-decreasing. If $d>g(\iota_v(d))$, then $d$ is a $D_v$-non-value.
Likewise a minor variation in the proof of Lemma \[nonnie\] gives the following result, which will be of vital importance in order to discard possible Salajan values. (For the Salajan sequence $u$ we write $\iota(d)$ instead of $\iota_u(d)$.)
\[nonnie2\] If $\iota(d)\le d/2$, then $d$ is a Salajan non-value.
### Lifting from $p^m$ to $p^{m+1}$
\[indelift\] If $p>3$ and $\iota(p^m)<\rho(p^m)$, then $\iota(p^{m+1})<p^{m+1}/2$.
[*Proof*]{}. Either $\rho(p^{m+1})=\rho(p^m)$ or $\rho(p^{m+1})=p\rho(p^m)$. In the first case $$\iota(p^{m+1})\le \rho(p^{m+1})=\rho(p^m)\le p^m<p^{m+1}/2,$$ so we may assume that $\rho(p^{m+1})=p\rho(p^m)$. This implies that $$\label{nonmirimanoff}
3^{\rho(p^m)}\equiv 1+kp^m({\rm mod~}p^{m+1})$$ with $p\nmid k$. From this we infer that $u_{i+j\rho(p^m)}$ assumes $p$ different values modulo $p^{m+1}$ as $j$ runs through $0,1,\ldots,p-1$. Put $j_1=\iota(p^{m})+1$. By assumption there exists $1\le i_1<j_1$ such that $u_{i_1}\equiv u_{j_1}({\rm mod~}p^{m})$. Modulo $p^{m+1}$ we have $$\{u_{i_1+j\rho(p^m)}:0\le j\le p-1\}=\{u_{j_1+j\rho(p^m)}:0\le j\le p-1\}.$$ The cardinality of these sets is $p$. Now let us consider the subsets obtained from the above two sets if we restrict $j$ to be $\le p/2$. Each contains $(p+1)/2$ different elements. It follows that these sets must have an element in common. Say we have $$u_{i_1+k_1\rho(p^m)}\equiv u_{j_1+k_2\rho(p^m)}({\rm mod~}p^{m+1}),~0\le k_1,k_2\le p/2.$$ Since by assumption $i_1\not\equiv j_1({\rm mod~}\rho(p^m))$, we have that $$i_1+k_1 \rho(p^m)\ne j_1+k_2\rho(p^m).$$ The proof is completed on noting that $i_1+k_1 \rho(p^m)$ and $j_1+k_2\rho(p^m)$ are bounded above by $$\iota(p^m)+1+(p-1){\rho(p^m)\over 2}\le (p+1){\rho(p^m)\over 2}\le (p+1){\varphi(p^m)\over 2}=
p^{m-1}{(p^2-1)\over 2}<{p^{m+1}\over 2},$$ where we used that by assumption $\iota(p^m)+1\le \rho(p^m)$ and Lemma \[previous\]. [$\Box$]{}
\[Bb\] Suppose that $l\ge 1$. If $\iota(p^l)\le p^l/2$, then $\iota(p^m)\le p^m/2$ for every $m>l$.
[*Proof*]{}. Note that $p>5$. If $p\not\in {\cal P}$, then $\rho(p^{m})\le p^{m-1}\rho(p)\le p^{m-1}(p-1)/2
\le p^m/2$ and hence $\iota(p^m)\le \rho(p^m)\le p^m/2$, so we may assume that $p\in {\cal P}$. Now we proceed by induction. Suppose that we have established that $\iota(p^k)\le p^k/2$ for $l\le k\le m-1$. By Corollary \[liftje2\] there are two cases to be considered.\
Case 1. $\rho(p^2)=\rho(p)=p-1$.\
In this case $\rho(p^m)\le p^{m-2}\rho(p)=\varphi(p^{m-1})\le p^{m-1}\le p^m/2,$ and hence $\iota(p^{m})\le p^m/2$.\
Case 2. We have $\rho(p^2)=p\rho(p)$ and hence $\rho(p^{m})=p^{m-1}\rho(p)=\varphi(p^{m})$. By assumption we have $\iota(p^{m-1})\le p^{m-1}/2<p^{m-2}(1-1/p)=\rho(p^{m-1})$. By Lemma \[indelift\] it then follows that $\iota(p^m)\le p^m/2$. [$\Box$]{}\
On combining the latter two lemmas with Lemma \[nonnie2\] we arrive at the following more appealing result.
\[18\]$~$\
[1)]{} If $p>3$ and $\iota(p)<\rho(p)$, then $p^2,p^3,\ldots$ are all Salajan non-values.\
[2)]{} If $\iota(p)\le p/2$, then $p,p^2,p^3,\ldots$ are all Salajan non-values.
[*Proof*]{}. 1) If the conditions on $p$ are satisfied, then by Lemma \[indelift\] it follows that $\iota(p^2)\le p^2/2$, which by Lemma \[Bb\] implies that $\iota(p^m)\le p^m/2$ for every $m\ge 2$. By Lemma \[nonnie2\] it then follows that $p^m$ is a non-value.\
2) If $\iota(p)\le p/2$, then $\iota(p^m)\le p^m/2$ for every $m\ge 1$ by Lemma \[Bb\] and by Lemma \[nonnie2\] it then follows that $p^m$ is a non-value. [$\Box$]{}\
We will see in Proposition \[sharpiotap\] that actually $\iota(p)\le p/2$ for $p>5$.
If ord$_9(p)=(p-1)/2$, then $\iota(p)<\rho(p)$ unless $p=5$ {#tip}
-----------------------------------------------------------
Lemma \[nonnie2\] in combination with the following lemma shows that every $p\in {\cal P}_3$ is a Salajan non-value. Recall that if $p\in {\cal P}$, then $\rho(p)=p-1$.
\[halfpforP\_3\] Suppose that $p\in {\cal P}_3$. Then $\iota(p)\le p/2<p-1=\rho(p)$.
[*Proof*]{}. Since by assumption $3$ is a primitive root modulo $p$, we have that $({3\over p})=-1$. It follows that $$1=u_2={({3\over p})+5\over 4}\equiv {3^{(p-1)/2}+5\over 4}=u_{p-1\over 2}({\rm mod~}p).$$ We infer that $\iota(p)\le (p-1)/2$.[$\Box$]{}
On using Lemma \[indelift\] the following result can be used to show that if $p\in {\cal P}_1$ and $m\ge 2$, then $p^m$ is a Salajan non-value.
\[remainder\] If $p>5$ and $p\in {\cal P}_1\cup {\cal P}_2$, then there exists $k\le p-3$ such that $u_{k}\equiv u_{k+1}({\rm mod~}p)$ and hence $\iota(p)<p-1=\rho(p)$.
[*Proof*]{}. Note that $$u_{2m-1}\equiv u_{2m} ({\rm mod~}p){\rm ~iff~}3^{2m}\equiv 15({\rm mod~}p)$$ and $$u_{2m}\equiv u_{2m+1} ({\rm mod~}p){\rm ~iff~}3^{2m}\equiv -5({\rm mod~}p).$$ If $p\in {\cal P}_1$, then $3$ is a primitive root modulo $p$, hence $({3\over p})=-1$ and $({-3\over p})=-1$ as $p\equiv 1({\rm mod~}4)$. If $p\in {\cal P}_2$, then $({3\over p})=1$ and $({-3\over p})=-1$ as $p\equiv 3({\rm mod~}4)$. We see that $({15\over p})=({-3\over p})({-5\over p})=-({-5\over p})$ and hence either $15$ or $-5$ is a square modulo $p$. Since by assumption ord$_9(p)=(p-1)/2$, every square $s\ne 0$ modulo $p$ is of the form $s=3^{2k}$ for some $1\le k\le (p-1)/2$. It follows that either $3^{2k}\equiv -5({\rm mod~}p)$ or $3^{2k}\equiv 15({\rm mod~}p)$ for some $1\le k\le (p-1)/2$. Since $3^{p-1}\equiv 1 ({\rm mod~}p)$ and, modulo $p$, $-5$ and $15$ are not congruent to $1$, it follows that $2k\le p-3$ and so $\iota(p)\le p-3+1=p-2$. [$\Box$]{}\
[Remark.]{} We have $({15\over p})=({-5\over p})$ in case $p\in {\cal P}_3$ and $({-5\over p})=-1$ iff $p\equiv \pm 1({\rm mod~}5)$. We infer that if $p>5$ and $p\in {\cal P}$, then there exists $k\le p-3$ such that $u_{k}\equiv u_{k+1}({\rm mod~}p)$, except when $p\in {\cal P}_3$ and $p\equiv \pm 1({\rm mod~}5)$.\
[Remark.]{} It is not true in general that $\iota(p)<\rho(p)$, there are many counter-examples, e.g., $p=193,307,1093,1181,1871$. It is an open problem whether there are infinitely many prime numbers $p$ such that $\iota(p)=\rho(p)$.
### Proof of Proposition \[reducetoprime\] {#5.4.1}
Suppose that $(d,10)=1$. By Lemma \[redpp\] it follows that $d=p^m$ with $p>5$ and $p\in {\cal P}$. It follows from Lemmas \[halfpforP\_3\] and \[remainder\] that $\iota(p)<\rho(p)$ for every $p\in \mathcal P$ with $p>5$, which implies by Lemma \[18\] that $m=1$ and $d=p$.
By Lemma \[nonnie\] and Lemma \[halfpforP\_3\] every prime $p\in {\cal P}_3$ is a Salajan non-value. On recalling that ${\cal P}={\cal P}_1 \cup {\cal P}_2 \cup {\cal P}_3$ the proof is then completed. [$\Box$]{}
$D_S(n)$ is not a ‘big’ prime
-----------------------------
We will now use classical exponential sum techniques to show that, for sufficiently large primes, the condition given in Corollary \[prop2n\] is not satisfied. Therefore, big primes are Salajan non-values.
Let us denote by $\psi$ the additive characters of the group $G$ and $\psi_0$ the trivial character. For any non-empty subset $A\subseteq G$, let us define the quantity $$\label{def_S(A)}
|\widehat{A}|=\max_{\psi\neq \psi_0} \left| \sum_{a\in A} \psi(a)\right|,$$ where the maximum is taken over all non-trivial characters in $G$.
\[B+B\_zero\] Let $G$ be a finite abelian group. For any given non-empty subsets $A,B\subseteq G$, whenever $A\cap (B+B)=\emptyset$ we have $$|B|\le {|\widehat{A}||G|\over |A|+|\widehat{A}|},$$ where $|\widehat{A}|$ is the quantity defined in .
[*Proof*]{}. The number $N$ of pairs $(b,b')\in B\times B$ such that $b+b'\in A$ equals $$\label{characters}
N=\frac{1}{|G|}\sum_{\psi}\sum_A\sum_{B\times B} \psi(b+b' -a) = \frac{|B|^2|A|}{|G|} + R$$ where, by the orthogonality of the characters, $$\begin{aligned}
|R|& = \left| \frac{1}{|G|}\sum_{\psi\ne \psi_0}\sum_A\sum_{B\times B}
\psi(b+b' -a) \right|
\le \frac{1}{|G|}\sum_{\psi\ne \psi_0}\left|\sum_A
\psi(a)\right|\left|\sum_B\psi(b)\right|^2 \\
&\le \frac{|\widehat{A}|}{|G|}\sum_{\psi\ne
\psi_0}\left|\sum_B\psi(b)\right|^2.\end{aligned}$$ Note that $$\left| \sum_B \psi(b) \right|^2 = \sum_{b,b'\in B}\psi (b-b'),$$ since as complex numbers $\overline{\psi(b)}=\psi(-b)$, and that by orthogonality of the characters $$\sum_{\psi} \sum_{b,b'\in B} \psi(b-b') = \left\{
\begin{array}{lc} 0&\text{ if } b\neq b',\\
|G|& \text{ if } b=b'.
\end{array}
\right.$$ Thus $$\label{Error}
|R|\le \frac{|\widehat{A}|}{|G|} \sum_{\psi\neq \psi_0} \left| \sum_B
\psi(b) \right|^2 = \frac{|\widehat{A}|}{|G|} \left( |G||B| -
|B|^2\right).$$ Since by assumption $N=0$, it follows from and that $$\frac{|B|^2|A|}{|G|}\le \frac{|\widehat{A}|}{|G|}(|G||B|-|B|^2),$$ which concludes the proof. [$\Box$]{}
We will need the following auxiliary result, which can be found in [@Ana].
\[S(A)\] Let $p$ be a prime and $g$ be a primitive root modulo $p$. The set $$A=\{ (x,y):\, 3g^x-g^{y} \equiv 30 \pmod p\}\subset \mathbb Z_{p-1} \times \mathbb Z_{p-1}$$ has $p-2$ elements and satisfies $|\widehat{A}|<p^{1/2}$.
[Remark]{}. It is easy to see that any subset of an abelian group satisfies that $|A|^{1/2}\le |\widehat{A}|$, so the bound in Lemma \[S(A)\] is essentially best possible.
[Remark]{}. In fact this result is true in a more general context (see for example [@Ana]): let $g$ be a primitive root in a finite field $\mathbb F_q$ and $a$, $b$ and $c$ be non-zero elements in the field. Then, the set $A_g(a,b,c)=\{(x,y): ag^x-bg^y = c\}$ in $\mathbb F_q$ has $q-2$ elements and satisfies $|\widehat{A}_g(a,b,c)|<q^{1/2}$.
\[prime\_lemma\] Let $p>3$ be a prime. Suppose that $u_1,\ldots,u_n$ are pairwise distinct modulo $p$. Then $p> \left\lfloor \frac{n}{4} \right\rfloor^{4/3}$.
[*Proof*]{}. First observe that if two elements have the same parity index, then $u_i\not\equiv u_{i+2k} ({\rm mod~}p)$ iff $
9^k\not\equiv 1({\rm mod~}p),$ thus ord$_9(p)\ge n/2$. (Alternatively one might invoke Lemma \[oneventje\] to obtain this conclusion.) By hypothesis, comparing elements with distinct parity index, it follows that $$\label{power3} 3\cdot 9^{k}-9^{s} \equiv 30 \pmod p,\ 1\leq
k,s\leq \left\lfloor \tfrac{n}{2}\right\rfloor$$ has no solution (otherwise $u_{2k}\equiv u_{2s-1} \pmod p$, with $1\le 2k,
2s-1\le n$).
We will now show that the non existence of solutions to equation implies that $p >\lfloor \frac{n}{4} \rfloor^{4/3}$. Let $g$ be a primitive root modulo $p$ and let $A$ be the set defined in Lemma \[S(A)\]. Let $m$ be the smallest integer such that $g^m\equiv 9\pmod p$ and $$B=\{(mx,my):1\le x,y\le \lfloor n/4\rfloor\} \subset \mathbb Z_{p-1} \times \mathbb Z_{p-1}.$$ Note that, since ord$_9(p)\ge n/2$, it follows that $|B|=\left\lfloor
\frac{n}{4} \right\rfloor^2$ (since $m$ generates a subgroup of order at least $n/2$ modulo $p-1$).
Observe that the non existence of solutions to equation implies that $$\label{power4} 3\cdot g^{mk}-g^{ms} \equiv 30 \pmod p,\ 1\leq
k,s\leq \left\lfloor \tfrac{n}{2}\right\rfloor$$ has no solutions and in particular $A\cap (B+B)=\emptyset$ (since clearly $B+B\subseteq \{(mx,my): 1\le x,y\le \left\lfloor n/2\right\rfloor \}$). It follows from Lemma \[B+B\_zero\] and Lemma \[S(A)\] that $$\label{bijna}
|B|=\left\lfloor \frac{n}{4} \right\rfloor^2 \le
\frac{|\widehat{A}||G|}{|A|+|\widehat{A}|}\le
\frac{p^{1/2}(p-1)^2}{p-2+p^{1/2}}<
p^{3/2},$$ which concludes the proof. [$\Box$]{}
\[Nobigprimes\] If $p > 5$ is a prime number, then $p$ is a Salajan non-value.
*Proof.* First observe that, if $n\ge 2060$ then it follows from Proposition \[prime\_lemma\] that if, for some prime $p\ge n$ the elements $u_1,\ldots,u_n$ are pairwise distinct modulo $p$ then $$p> \left\lfloor \frac{n}{4} \right\rfloor^{4/3} \ge 2n,$$ and by Corollary \[prop2n\] it follows that $p$ is a Salajan non-value. For primes $5\le p \le 2060$, the result follows from the calculations included in Table 1. [$\Box$]{}\
Taking $n=\iota(p)$ in Proposition \[prime\_lemma\] we obtain, after some numerical work, the following estimate. Since $\iota(29)=14$ the bound is sharp.
\[sharpiotap\] Let $p>5$ be a prime. Then $\iota(p)\le \min((p-1)/2,4p^{3/4})$.
[*Proof*]{}. By Proposition \[prime\_lemma\] we infer that $\iota(p)<3+4p^{3/4}$. A tedious analysis using the one but last estimate for $|B|$ in (\[bijna\]) gives the more elegant bound $\iota(p)<4p^{3/4}$. For $p<4111$ one verifies the claimed bound by direct computation. Since $4p^{3/4}<(p-1)/2$ for $p\ge 4111$, we are done. [$\Box$]{}
The proof of Salajan’s conjecture
=================================
In Section \[preparations\], we established that powers of $2$ and powers of $5$ were candidates for Salajan values. Finally, after studying the characteristics of the period and the incongruence index of the Salajan sequence, we discard in Section \[non-values\] any other possible candidates.
[*Proof of Theorem*]{} \[main\]. It follows from Proposition \[reducetoprime\] that if $d>1$ is a Salajan value, then either $(10,d)>1$ or $d\in \mathcal P_1\cup \mathcal P_2$. It follows from Corollary \[Nobigprimes\] that no prime greater than $5$ can be a Salajan value and hence $(10,d)>1$. By Lemma \[nonnie\] it follows that $d$ has to be a prime power. Therefore, since $(10,d)>1$, the discriminator must be a power of $2$ or a power of $5$.
First suppose that $D_S(n)=2^e$. On invoking Lemma \[prop2n\] it then follows that $e=\min\{a:2^a\ge n\}$. Next suppose that $D_S(n)=5^f$. By Lemma \[prop5n\] it then follows that $f=\min\{a:2^a\ge 5n/4\}$. So we have $D_S(n)=2^e$ or $D_S(n)=5^f$. By the definition of the discriminator we now infer that $D_S(n)=\min\{2^e,5^f\}$. [$\Box$]{}
Appendix
========
The natural density of the sets ${\cal P}_i$
--------------------------------------------
Standard methods allow one to determine, assuming the Generalized Riemann Hypothesis, the densities of the sets ${\cal P}_i$ defined in Section \[primesets\]. (For a survey of related material see Moree [@Asurvey].)
Assume GRH. We have $$\#\{p\le x:p\in {\cal P}_i\}=\delta({\cal P}_i){x\over \log x}+O\Big({x\log \log x\over \log^2 x}\Big),$$ with $\delta({\cal P}_1)=\delta({\cal P}_2)=3A/5=0.224373488\ldots$ and $\delta({\cal P}_3)=2A/5=0.149582325\ldots$ and $$A=\prod_p\left(1-{1\over p(p-1)}\right)=0.3739558136\ldots,$$ the Artin constant.
The result also holds for the set ${\cal P}$, where we find $\delta({\cal P})=\delta({\cal P}_1)+\delta({\cal P}_2)+\delta({\cal P}_3)=8A/5=0.598329301\ldots$.
[*Proof*]{}. These three results can be obtained by a variation of the classical result of Hooley [@H] and this yields the estimate with $\delta({\cal P}_i)$ yet to be determined. We note that the sets ${\cal P}_i$ are mutually disjunct. By [@PU Theorem 4] we have $\delta({\cal P}_1)=3A/5$ and $\delta({\cal P}_1\cup {\cal P}_3)=A$. This gives $\delta({\cal P}_3)=2A/5$. By [@PNear Theorem 3] we have $\delta({\cal P})=8A/5$ and hence $\delta({\cal P}_2)=\delta({\cal P})-
\delta({\cal P}_1\cup {\cal P}_3)=3A/5$. [$\Box$]{}\
For the benefit of the reader we give a perhaps more insightful argument why $\delta({\cal P}_2)=3A/5$.\
Assuming GRH we have, cf. Moree [@PNear], $$\delta({\cal P}_2)=\sum_{n=1}^{\infty}{\mu(n)\over [\mathbb Q(\zeta_{2n},3^{1/2n}):\mathbb Q]}
-\sum_{n=1}^{\infty}{\mu(n)\over [\mathbb Q(i,\zeta_{2n},3^{1/2n}):\mathbb Q]},$$ where the first sum gives the density of the primes $p$ such that ord$_3(p)=(p-1)/2$ and the second sum the density of the primes $p$ such that $p\equiv 1({\rm mod~}4)$ and ord$_3(p)=(p-1)/2$. Since for $n$ even, $i\in \mathbb Q(\zeta_{2n})$, we find that $$\delta({\cal P}_2)=\sum_{(n,2)=1}^{\infty}{\mu(n)\over [\mathbb Q(\zeta_{2n},3^{1/2n}):\mathbb Q]}
-\sum_{(n,2)=1}^{\infty}{\mu(n)\over [\mathbb Q(i,\zeta_{2n},3^{1/2n}):\mathbb Q]}.$$ Now suppose that $n$ is odd. If $3|n$, then $\sqrt{-3}\in \mathbb Q(\zeta_{2n})$. Since $\sqrt{3}\in \mathbb Q(\zeta_{2n},3^{1/2n})$, it follows that $\mathbb Q(i,\zeta_{2n},3^{1/2n})=\mathbb Q(\zeta_{2n},3^{1/2n})$. On the other hand, if $(n,3)=1$ one infers that $[\mathbb Q(i,\zeta_{2n},3^{1/2n}):\mathbb Q]=2[\mathbb Q(\zeta_{2n},3^{1/2n}):\mathbb Q]$. This leads to $$\delta({\cal P}_2)={1\over 2}\sum_{(n,6)=1}^{\infty}{\mu(n)\over [\mathbb Q(\zeta_{2n},3^{1/2n}):\mathbb Q]}
={1\over 4}\sum_{(n,6)=1}{\mu(n)\over n\varphi(n)}={3\over 5}A,$$ where we used that $[\mathbb Q(\zeta_{2n},3^{1/2n}):\mathbb Q]=\varphi(2n)2n=2\varphi(n)n$ if $(n,6)=1$ and the identity $$\sum_{(n,6)=1}{\mu(n)\over n\varphi(n)}=\prod_{p>3}\left(1-{1\over p(p-1)}\right)={12\over 5}A.$$
Counting the elements $\le x$ in ${\cal F}$ {#Izabela}
-------------------------------------------
In this section, written jointly with Izabela Petrykiewicz, we will establish Proposition \[iza\] from the introduction.\
Recall that ${\cal F}=\{f~:~[4\cdot 5^{f-1}, 5^f]\textnormal{ contains no power of 2}\}$. Consider ${\cal G}={\mathbb N}\setminus {\cal F}$. We have that $g$ is in ${\cal G}$ iff $4\cdot 5^{g-1}\leq 2^k \leq 5^g$ for some $k\in {\mathbb N}$. Thus we have $g$ is in ${\cal G}$ iff $2\log2 + (g-1)\log 5 \leq k \log 2 \leq g\log 5$, that is iff $2+(g-1) \alpha\leq k \leq g \alpha,$ where $\alpha=\log 5/\log 2$. Since $k$ is an integer, we may replace $g\alpha$ by $[g\alpha]$ and the condition becomes $k\in [[g\alpha]+\{g\alpha\}+2-\alpha,[g\alpha]]$. Note that there can be only an integer in this interval iff $\{g\alpha\}\le \alpha-2$. Note that $\alpha$ is irrational. Now it is a consequence of Weyl’s criterion, see, e.g., [@Bug; @KN], that for a fixed $0<\beta<1$ we have $$\#\{g\le x:\{g\alpha\}\le \beta\}\sim \beta x,~x\rightarrow \infty.$$ On applying this with $\beta=\alpha-2$ the proof of Proposition \[iza\] is easily completed. [$\Box$]{}\
[Acknowledgement]{}. This project was started in the context of an internship of Sabin Salajan at MPIM in 2012 and a visit of the first author to the second author to ICMAT in Madrid. The project was taken up again during a two week visit of Bernadette Faye (Senegal) at MPIM. The first author thanks Bernadette for discussions and her help with some computer experiments. Further he thanks David Brink, Igor Shparlinski and Arne Winterhof for helpful e-mail correspondence and Paul Tegelaar for comments on an earlier version. The idea of the proof of Proposition \[iza\] is due to Izabela Petrykiewicz.
[99]{} T. Apostol, [*Introduction to analytic number theory*]{}, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. L.K. Arnold, S.J. Benkoski and B.J. McCabe, The discriminator (a simple application of Bertrand’s postulate), [*Amer. Math. Monthly*]{} [**92**]{} (1985), 275–-277. R.F. Beyl, Cyclic subgroups of the prime residue group, [*Amer. Math. Monthly*]{} [**84**]{} (1977), 46–-48. P.S. Bremser, P.D. Schumer and L.C. Washington, A note on the incongruence of consecutive integers to a fixed power, [*J. Number Theory*]{} [**35**]{} (1990), 105–-108. J. Browkin and H.-Q. Cao, Modifications of the Eratosthenes sieve, [*Colloq. Math.*]{} [**135**]{} (2014), 127–138. Y. Bugeaud, [*Distribution modulo one and Diophantine approximation*]{}, Cambridge Tracts in Mathematics [**193**]{}, Cambridge University Press, Cambridge, 2012. J. Cilleruelo and A. Zumalacárregui, An additive problem in finite fields with powers of elements of large multiplicative order, [*Rev. Mat. Complut.*]{} [**27**]{} (2014), 501–508. C. Hooley, On Artin’s conjecture, [*J. Reine Angew. Math.*]{} [**225**]{} (1967), 209–220. L. Kuipers and H. Niederreiter, [*Uniform distribution of sequences*]{}, Pure and Applied Mathematics, Wiley-Interscience, New York-London-Sydney, 1974 H.B. Mann, [*Addition theorems: The addition theorems of group theory and number theory*]{}, Interscience Publishers John Wiley and Sons, New York-London-Sydney, 1965. P. Moree, The incongruence of consecutive values of polynomials, [*Finite Fields Appl.*]{} [**2**]{} (1996), 321–-335. P. Moree, Uniform distribution of primes having a prescribed primitive root, [*Acta Arith.*]{} [**89**]{} (1999), 9–21. P. Moree, Artin’s primitive root conjecture – a survey, [*Integers*]{} [**12A**]{} (2012), No. 6, 1305–1416. P. Moree, Near-primitive roots, [*Funct. Approx. Comment. Math.*]{} [**48.1**]{} (2013), 133–145. P. Moree and G. L. Mullen, Dickson polynomial discriminators, [*J. Number Theory*]{} [**59**]{} (1996), 88–105. Zhi-Wei Sun, On functions taking only prime values, [*J. Number Theory*]{} [**133**]{} (2013), 2794–2812. M. Zieve, A note on the discriminator, [*J. Number Theory*]{} [**73**]{} (1998), 122–-138.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Person Re-identification (ReID) is to identify the same person across different cameras. It is a challenging task due to the large variations in person pose, occlusion, background clutter, How to extract powerful features is a fundamental problem in ReID and is still an open problem today. In this paper, we design a Multi-Scale Context-Aware Network (MSCAN) to learn powerful features over full body and body parts, which can well capture the local context knowledge by stacking multi-scale convolutions in each layer. Moreover, instead of using predefined rigid parts, we propose to learn and localize deformable pedestrian parts using Spatial Transformer Networks (STN) with novel spatial constraints. The learned body parts can release some difficulties, pose variations and background clutters, in part-based representation. Finally, we integrate the representation learning processes of full body and body parts into a unified framework for person ReID through multi-class person identification tasks. Extensive evaluations on current challenging large-scale person ReID datasets, including the image-based Market1501, CUHK03 and sequence-based MARS datasets, show that the proposed method achieves the state-of-the-art results.'
author:
- |
Dangwei Li$^{1,2}$, Xiaotang Chen$^{1,2}$, Zhang Zhang$^{1,2}$, Kaiqi Huang$^{1,2,3}$\
$^{1}$CRIPAC$\;\&\;$NLPR, CASIA $^{2}$University of Chinese Academy of Sciences\
$^{3}$CAS Center for Excellence in Brain Science and Intelligence Technology\
[@nlpr.ia.ac.cn]{}
bibliography:
- 'egbib.bib'
title: |
Learning Deep Context-aware Features over Body and Latent Parts\
for Person Re-identification
---
Introduction
============
Person re-identification aims to search for the same person across different cameras with a given probe image. It has attracted much attention in recent years due to its importance in many practical applications, such as video surveillance and content-based image retrieval. Despite of years of efforts, it still has many challenges, such as large variations in person pose, illumination, and background clutter. In addition, similar appearance of clothes among different people and imperfect pedestrian detection results further increase its difficulty in real applications.
![The schematic of typical feature learning framework with deep learning. As shown in black dashed boxes, current approaches focus on the full body or rigid body parts for feature learning. Different from them, we use the spatial transformer networks to learn and localize pedestrian parts and use multi-scale context-aware convolutional networks to extract full-body and body-parts representations for ReID. Best viewed in color. []{data-label="fig:framework_simple"}](figs/PipelineNetwork.pdf){width="0.35\paperwidth"}
Most existing methods for ReID focus on developing a powerful representation to handle the variations of viewpoint, body pose, background clutter, [@xu2014person; @GrayECCV08; @FarenzenaCVPR10; @YangyangECCV14; @Yang2017aaai; @KviatkovskyPAMI13color; @ZhaoruiCVPR13unsupervised; @zhaoruiCVPR14learning; @LiaoshengcaiCVPR15; @MatsukawaCVPR16; @LidangweiArxiv16], or learning an effective distance metric [@KostingerCVPR12; @ProsserBMVC10person; @ZhengweishiPAMI13reid; @LizhenCVPR13; @LiaoshengcaiCVPR15; @ZhangLiCVPR16; @ChendapengCVPR16similarity]. Some of existing methods learn both of them jointly [@LiWeiCVPR14; @YiICPR14DML; @AhmedCVPR15improved; @Shihanlin2016Embedding]. Recently, deep feature learning based methods [@DingPR15deep; @Chengde2016person; @Varior2016Siamese; @VariorECCV16Gated], which learn a global pedestrian feature and use Euclidean metric to measure two samples, have obtained the state-of-the-art results. With the increasing sample size of ReID dataset, the learning of features from multi-class person identification tasks [@XiaotongCVPR16Domain; @ZhengliangECCV16; @XiaotongARXIV16end; @ZhengliangArxiv16; @SchumannArxiv16deep], denoted as ID-discriminative Embedding (IDE) [@ZhengliangArxiv16], has shown great potentials on current large-scale person ReID datasets, such as MARS [@ZhengliangECCV16] and PRW [@ZhengliangArxiv16], where the IDE features are taken from the last hidden layer of Deep Convolutional Neural Networks (DCNN). In this paper, we aim to learn the IDE feature for person ReID using DCNN.
Existing DCNN models for person ReID typically learn a global full-body representation for input person image (Full body in Figure \[fig:framework\_simple\]), or learn a part-based representation for predefined rigid parts (Rigid body parts in Figure \[fig:framework\_simple\]) or learn a feature embedding for both of them. Although these DCNN models have obtained impressive results on existing ReID datasets, there are still two problems. **First**, for feature learning, current popular DCNN models typically stack single-scale convolution and max pooling layers to generate deep networks. With the increase of the number of layers, these DCNN models could easily miss some small scale visual cues, such as sunglasses and shoes. However, these fine-grained attributes are very useful to distinguish the pedestrian pairs with small inter-class variations. Thus these DCNN models are not the best choice for pedestrian feature learning. **Second**, due to the pose variations and imperfect pedestrian detectors, the pedestrian image samples may be misaligned. Sometimes they may have some backgrounds or lack some parts, legs. In these cases, for part-based representation, the predefined rigid grids may fail to capture correct correspondence between two pedestrian images. Thus the rigid predefined grids are far from robust for effective part-based feature learning.
In this paper, we propose to learn the features of full body and body parts jointly. **To solve the first problem**, we propose a Multi-Scale Context-Aware Network (MSCAN). As shown in Figure \[fig:framework\_simple\], for each convolutional layer of the MSCAN, we adopt multiple convolution kernels with different receptive fields to obtain multiple feature maps. Feature maps from different convolution kernels are concatenated as current layer’s output. To decrease the correlations among different convolution kernels, the dilated convolution [@YuKoltun2016] is used rather than general convolution kernels. Through this way, multi-scale context knowledge is obtained at the same layer. Thus the local visual cues for fine-grained discrimination is enhanced. In addition, through embedding contextual features layer-by-layer (convolution operation across layers), MSCAN can obtain more context-aware representation for input image. **To solve the second problem**, instead of using rigid body parts, we propose to localize latent pedestrian parts through Spatial Transform Networks (STN) [@JaderbergNIPS15spatial], which is originally proposed to learn image transformation. To adapt it to the pedestrian part localization task, we propose three new constraints on the learned transformation parameters. With these constraints, more flexible parts can be localized at the informative regions, so as to reduce the distraction of background contents.
Generally, the features of full body and body parts are complementary to each other. The full-body features pay more attention to the global information while the body-part features care more about the local regions. To better utilize these two types of representations, in this paper, features of full body and body parts are concatenated to form the final pedestrian representation. In test stage, the Euclidean metric is adopted to compute the distance between two L2 normalized person representations for person ReID.
The contributions of this paper are summarized as follows: (a) We propose a multi-scale context-aware network to enhance the visual context information for better feature representation of fine-grained visual cues. (b) Instead of using rigid parts, we propose to learn and localize pedestrian parts using spatial transformer networks with novel prior spatial constraints. Experimental results show that fusing the global full-body and local body-part representations greatly improves the performance of person ReID.
Related Work {#relatedwork}
============
Typical person ReID methods focus on two key points: developing a powerful feature for image representation and learning an effective metric to make the same person be close and different persons far away. Recently, deep learning approaches have achieved the state-of-the-art results for person ReID [@XiaotongCVPR16Domain; @ZhengliangECCV16; @VariorECCV16Gated; @zheng2016personreview; @zhang2015bit]. Here we mainly review the related deep learning methods.
Deep learning approaches for person ReID tend to learn person representation and similarity (distance) metric jointly. Given a pair of person images, previous deep learning approaches learn each person’s features followed by a deep matching function from the convolutional features [@LiWeiCVPR14; @AhmedCVPR15improved; @Chen2017aaai; @Chen2017cvprid] or the Fully Connected (FC) features [@YiICPR14DML; @wang2016dari; @Shihanlin2016Embedding]. In addition to the deep metric learning, some work directly learns image representation through pair-wise contrastive loss or triplet ranking loss, and use Euclidean metric for comparison [@DingPR15deep; @Chengde2016person; @Varior2016Siamese; @VariorECCV16Gated].
With the increasing sample size of ReID dataset, the IDE feature which is learned through multi-class person identification tasks, has shown great potentials on current large-scale person ReID datasets. Xiao [@XiaotongCVPR16Domain] propose the domain guided dropout to learn features over multiple datasets simultaneously with identity classification loss. Zheng [@ZhengliangECCV16] learn the IDE feature for the video-based person re-identification. Xiao [@XiaotongARXIV16end] and Zheng [@ZhengliangArxiv16] learn the IDE feature to jointly solve the pedestrian detection and person ReID tasks. Schumann [@SchumannArxiv16deep] learn the IDE feature for domain adaptive person ReID. The similar phenomenon has also been validated on face recognition [@SunyiCVPR14deep1]. As we know, previous DCNN models usually adopt the layer-by-layer single-scale convolution kernels to learn the context information. Some DCNN models [@YiICPR14DML; @Chengde2016person; @Shihanlin2016Embedding] adopt rigid body parts to learn local pedestrian features. Different from them, we improve the classical models in two ways. Firstly, we propose to enhance the context knowledge through multi-scale convolutions at the same layer. The relationship among different context knowledge are learned by embedding feature maps layer-by-layer (convolution or FC operation). Secondly, instead of using rigid parts, we utilize the spatial transformer networks with proposed prior constraints to learn and localize latent human parts.
{width="0.70\paperwidth"}
Proposed Method {#proposedmethod}
===============
The focus of this approach is to learn powerful feature representations to describe pedestrians. The overall framework of the proposed method is shown in Figure \[fig:framework\_all\]. In this section, we introduce our model from four aspects: a multi-scale context-aware network for efficient feature learning (Section \[MSCAN\]), the latent parts learning and localization for better local part-based feature representation (Section \[LatentPartLoc\]), the fusion of global full-body and local body-part features for person ReID (Section \[FeatureFusion\]), and our final objective function in Section \[ObjectiveFunction\].
Multi-scale Context-aware Network {#MSCAN}
---------------------------------
Visual context is an important component to assist visual-related tasks, such as object recognition [@LintsungyiECCV14microsoft] and object detection [@ZhengPAMI12context; @ZengArxiv16crafting]. Typical convolutional neural networks model context information through hierarchical convolution and pooling [@KrizhevskyNIPS12; @HekaimingCVPR16Residual]. For person ReID task, the most important visual cues are visual attribute knowledge, such as clothes color and types. However, they have large variations in scale, shape and position, such as the hat/glasses at small local scale and the cloth color at the larger scale. Directly using bottom-to-up single-scale convolution and pooling may not be effective to handle these complex variations. Especially, with the increase number of layers, the small visual regions, such as hat, will be easily missed in top layers. To better learn these diverse visual cues, we propose the Multi-scale Context-Aware Network.
layer dilation kernel pad \#filters output
------- ---------- ------------ ------- ----------- -------------------------
input - - - - 3$\times$160$\times$64
conv0 1 5$\times$5 2 32 32$\times$160$\times$64
pool0 - 2$\times$2 - - 32$\times$80$\times$32
conv1 1/2/3 3$\times$3 1/2/3 32/32/32 96$\times$80$\times$32
pool1 - 2$\times$2 - - 96$\times$40$\times$16
conv2 1/2/3 3$\times$3 1/2/3 32/32/32 96$\times$40$\times$16
pool2 - 2$\times$2 - - 96$\times$20$\times$8
conv3 1/2/3 3$\times$3 1/2/3 32/32/32 96$\times$20$\times$8
pool3 - 2$\times$2 - - 96$\times$10$\times$4
conv4 1/2/3 3$\times$3 1/2/3 32/32/32 96$\times$10$\times$4
pool4 - 2$\times$2 - - 96$\times$5$\times$2
: Model architecture of MSCAN.[]{data-label="tab:mscan"}
The architecture of the proposed MSCAN is shown in Tabel \[tab:mscan\]. It has an initial convolution layer with kernel size $5\times5$ to capture the low-level visual features. Then we use four multi-scale convolution layers to obtain the complex image context information. In each multi-scale convolution layer, we use a convolution kernel with size $3\times3$. To obtain multi-scale receptive fields, we adopt dilated convolution [@YuKoltun2016] for the convolution filters. We use three different dilation ratios, i.e. 1,2 and 3, to capture different scale context information. The feature maps from different dilation ratios are concatenated along the channel axis to form the final output of the current convolution layer. Thus, the visual context information are enhanced explicitly. To integrate different context information together, the feature maps of current convolution layer are embedded through layer-by-layer convolution or FC operation. As a result, the visual cues at different scales are fused in a latent way. Besides, we adopt Batch Normalization [@Ioffe15batch] and ReLU neural activation units after each convolution layer.
In this paper, we use the dilated convolutions with dilation ratios 1, 2 and 3 instead of the classic convolution filters with kernel size $3\times3$, $5\times5$ and $7\times7$. The main reason is that the classic convolution filters with kernel size $3\times3$, $5\times5$ and $7\times7$ overlap with each other at the same output position and produce redundant information. To make it clearer, we show the dilated convolution kernel (size $3\times3$) with dilation ratio ranging from $1$ to $3$ in Figure \[fig:DilationCov\]. For the same output position which is shown in red circle, the convolution kernel with larger dilation ratio has larger receptive field, while only the center position is overlapped with other convolution kernels. This can reduce the redundant information among filters with different receptive fields.
In summary, as shown in Figure \[fig:framework\_all\], we use MSCAN to learn the multi-scale context representation for full body and body parts. In addition, it is also used for feature learning in spatial transformer networks mentioned below.
![Example of dilated convolution for the same input feature map. The convolutional kernel is $3\times3$ and the dilation ratio from left to right is 1, 2, and 3. The blue boxes are effective positions for convolution at the red circle. Best viewed in color. []{data-label="fig:DilationCov"}](figs/DilationCov.pdf){width="0.35\paperwidth"}
Latent Part Localization {#LatentPartLoc}
------------------------
Pedestrian parts are important in person ReID. Some existing work [@GrayECCV08; @LiaoshengcaiCVPR15; @YiICPR14DML; @Chengde2016person] has explored rigid body parts to develop robust features. However, due to the unsatisfying pedestrian detection algorithms and large pose variations, the method of using rigid body parts for local feature learning is not the optimal solution. As shown in Figure \[fig:framework\_simple\], when using rigid body parts, the top part consists of large amount of background. This motivates us to learn and localize the pedestrian parts automatically.
We integrate STN [@JaderbergNIPS15spatial] as the part localization net in our proposed model. The original STN is proposed to explicitly learn the image transformation parameters, such as translation and scale. It has two main advantages: (1) it is fully differentiable and can be easily integrated into existing deep learning frameworks, (2) it can learn to translate, scale, crop or warp an interesting region without explicit region annotations. These facts make it very suitable for pedestrian parts localization.
STN includes two components, the spatial localization network to learn the transformation parameters, and the grid generator to sample the input image using an image interpolation kernel. More details about STN can be seen in [@JaderbergNIPS15spatial]. In our implementation of STN, the bilinear interpolation kernel is adopted to sample the input image. And four transformation parameters $\theta=[s_x, t_x, s_y, t_y]$ are used, where $s_x$ and $s_y$ are the horizontal and vertical scale transformation parameters, and $t_x$ and $t_y$ are the horizontal and vertical translation parameters. The image height and width are normalized to be in $[-1, 1]$. Only scale and translation parameters are learned because these two types of transformations serve enough to crop the pedestrian parts effectively. The transformation is applied as an inverse warping to generate the output body part regions: $$\begin{gathered}
\begin{pmatrix} x^{in}_{i} \\ y^{in}_{i} \end{pmatrix} =
\begin{bmatrix} s_x & 0 & t_x \\ 0 & s_y & t_y \end{bmatrix}
\begin{pmatrix} x^{out}_{i} \\ y^{out}_{i} \\ 1 \end{pmatrix}\end{gathered}$$
where $x^{in}$ and $y^{in}$ are the input image coordinates, $x^{out}$ and $y^{out}$ are the output part image coordinates, and $i$ indexes the pixels in the output body part image.
In this paper, we expect STN to learn three parts corresponding to the head-shoulder, upper body and lower body. Each part is learned by an independent STN from the original pedestrian image. For the spatial localization network, firstly we use MSCAN to extract the global image feature maps. Then we learn the high-level abstract representation by a 128-dimension FC layer (FC\_[loc]{} in Figure \[fig:framework\_all\]). At last, we learn the transformation parameters $\theta$ with a 4-dimension FC layer based on the FC\_loc. The MSCAN and FC\_[loc]{} are shared among three spatial localization networks. The grid generator can crop the learned pedestrian parts based on the learned transformation parameters. In this paper, the resolution of the cropped part image is $96\times64$.
For the part localization network, it is hard to learn three groups of parameters for part localization. There are three problems. First, the predicted parts from STN can easily fall into the same region, , the center region of a person, and result in redundance. Second, the scale parameters can easily become negative and the pedestrian part will be mirrored vertically or horizontally or both. This is not consistent with general human cognition. Because few person will stand upside down in surveillance scenes. At last, the cropped parts may fall out of the person image, thus the network would be hard to converge. To solve the above problems, we propose three prior constraints on the transformation parameters in the part localization network.
The first constraint is for the positions of predicted parts. We expect the predicted parts to be near the prior center points, so that the learned parts would be complementary to each other. This is termed as the center constraint, which is formalized as follows: $$%L_{cen} = \frac{1}{2}\{(t_x-C_x)^2 + (t_y-C_y)^2\}
L_{cen} = \frac{1}{2}\max\{0, (t_x-C_x)^2 + (t_y-C_y)^2 - \alpha\}$$ where $C_x$ and $C_y$ are prior center points for each part. $\alpha$ is the threshold to control the translation between estimated and prior center points. In our experiments, we set the prior center point ($C_x, C_y$) to $(0, 0.6)$, $(0, 0)$, and $(0, -0.6)$ for each part. The threshold $\alpha$ is set to $0.5$.
The second one is the value range constraint on the predicted scale parameter. We hope the scale to be positive, so that the predicted parts have a reasonable extent. The value range constraint on the scale parameter is formalized as follows: $$L_{pos} = \max\{0, \beta - s_x \} + \max\{0, \beta - s_y \}$$ where $\beta$ is threshold parameter and is set to 0.1 in this paper.
The last one is to make the localization network focus on the inner region of an image. It is formalized as follows: $$\renewcommand\arraystretch{1.5}
\begin{array}{r}
L_{in} = \frac{1}{2}\max\{0, ||s_x \pm t_x||^2 - \gamma \} \\
+ \frac{1}{2}\max\{0, ||s_y \pm t_y||^2 - \gamma \}
\end{array}$$ where $\gamma$ is the boundary parameter. $\gamma$ is set to 1.0 in our paper, which means the cropped parts should be inside the pedestrian image.
Finally the loss for the transformation parameters in the part localization network is described as follows: $$\label{equ:los_loc}
L_{loc} = L_{cen} + \xi_1 L_{pos} + \xi_2 L_{in}$$ where $\xi_1$ and $\xi_2$ are hyperparameters. The hyperparameters $\xi_1$ and $\xi_2$ are both set to 1.0 in our experiments.
Feature Extraction and Fusion {#FeatureFusion}
-----------------------------
The features of full body and body parts are learned by separate networks and then are fused in a unified framework for multi-class person identification tasks. For the body-based representation, we use MSCAN to extract the global feature maps and then learn a 128-dimension feature embedding (denoted as FC\_body in Figure \[fig:framework\_all\]). For the part-based representation, first, for each body part, we use the MSCAN to extract its feature maps and learn a 64-dimension feature embedding (denoted as FC\_part1, FC\_part2, FC\_part3). Then, we learn a 128-dimension feature embedding (denoted as FC\_part) based on features of each body part. The Dropout [@srivastava2014dropout] is adopted after each FC layer to prevent overfitting. At last, the features of global full body and local body parts are concatenated to be a 256-dimension feature as the final person representation.
Objective Function {#ObjectiveFunction}
------------------
In this paper, we adopt the softmax loss as the objective function for multi-class person identification tasks. $$\label{equ:los_cls}
L_{cls} = -\sum_{i=1}^{N}log\frac{\exp(W_{y_i}^{T}x_i+b_{y_i})}{\sum\nolimits_{j=1}^{C}\exp(W_j^Tx_i+b_j)}$$ where $i$ is the index of person images, $x_i$ is the feature of $i$-th sample, $y_i$ is the identity of $i$-th sample, $N$ is the number of person images, $C$ is the number of person identities, $W_j$ is the classifier for $j$-th identity.
For the overall network training, we use the classification and localization loss jointly. The final objective function is as follows. $$\label{equ:los_all}
L = L_{cls} + \lambda L_{loc}$$ where the $\lambda$ is the hyperparameter, which is set to 0.1 in our experiments.
Experiments
===========
In this paragraph, the datasets and evaluation protocols are introduced in Section \[exp:dataset\]. Implementation details are described in Section \[exp:detail\]. Comparisons with state-of-the-art methods are discussed in Section \[exp:stateoftheart\]. The effectiveness of proposed model is analyzed in Section \[exp:effmscan\] and Section \[exp:efflpl\]. Cross-dataset evaluation is described in Section \[exp:discussion\].
Datasets and Protocols {#exp:dataset}
----------------------
**Datasets.** In this paper, we evaluate our proposed method on current largest person ReID datasets, including Market1501 [@ZhengliangICCV15], CUHK03 [@LiWeiCVPR14] and MARS [@ZhengliangECCV16]. We do not directly train our model on small datasets, such as VIPeR [@Gray07VIPeR]. It would be easily overfitting and insufficient to learn such a large capacity network on small datasets from scratch. However, we give some results through fine-tuneing the model from Market1501 to VIPeR and make cross-dataset ReID on VIPeR for generalization validation. Related experimental results are discussed in Section \[exp:discussion\].
Market1501: It contains 1,501 identities which are captured by six manually set cameras. There are 32,368 pedestrian images in total. Each person has 3.6 images on average at each viewpoint. It provides two types of images, including cropped and automatically detected pedestrians by the Deformable Part based Model (DPM) [@FelzenszwalbPAMI10object]. Following [@ZhengliangICCV15], 751 identities are used for training and the rest 750 identities are used for testing.
CUHK03: It contains 1,360 identities which are captured by six surveillance cameras in campus. Each identity is captured by two disjoint cameras. Totally it consists of 13,164 person images and each identity has about 4.8 images at each viewpoint. This dataset provides two types of annotations, including manually annotated bounding boxes, and bounding boxes detected using DPM. We validate our proposed model on both types of data. Following [@LiWeiCVPR14], we use 1,260 person identities for training and the rest 100 identities for testing. Experiments are conducted 20 times and the mean result is reported.
MARS: It is the largest sequence-based person ReID dataset. It contains 1,261 identities with each identity captured by at least two cameras. It consists of 20,478 tracklets and 1,191,003 bounding boxes. Following [@ZhengliangECCV16], we use 625 identities for training and the rest 631 identities for testing.
**Protocols.** Following original evaluation protocols in each dataset, we adopt three evaluation protocols for fair comparison with existing methods. The first one is Cumulated Matching Characteristics (CMC) which is adopted on the CUHK03 and MARS datasets. The second one is Rank-1 identification rate on the Market1501 dataset. The third one is mean Average Precision (mAP) on the Market1501 and MARS datasets. mAP considers both precision and recall rate, which could be complementary to CMC.
Implementation Details {#exp:detail}
----------------------
**Model:** We try to learn the pedestrian representation through multi-class person identification tasks using full body and body parts. To evaluate the effectiveness of full body and body parts independently, we extract two sub-models from the whole network of Figure \[fig:framework\_all\]. The first one only uses the full body to learn the person representation with identity classification loss. The second one only uses the parts to learn the person representation with identity classification and body parts localization loss. For person re-identification, we use the L2 normalized person representation and Euclidean metric to measure the distance between two pedestrian samples.
**Optimization:** Our model is implemented based on Caffe [@JiaMM14caffe]. We use all the available training identities for training and randomly select one sample for each identity for validation. As the dataset can be quite large, in practice we use a stochastic approximation of the objective function. Training data is randomly divided into mini-batches with a batch size of 64. The model performs forward propagation on each mini-batch and computes the loss. Backpropagation is then used to compute the gradients on each mini-batch and the weights are updated with stochastic gradient descent. We start with a base learning rate of $\eta = 0.01 $ and gradually decrease it after each $1\times10^4$ iterations. It should be noted that the learning rate of part localization network is 1% of that in feature learning network. We use a momentum of $\mu = 0.9$ and weight decay $\lambda = 5\times10^{-3}$. For overall network training, we initialize the network using pretrained body-based and part-based model and then follow the same training strategy as described above. We use the model at $5\times10^4$ iterations for testing.
**Data Preprocessing:** For each image, we resize it to $160\times64$, subtract the mean value on each channel (B, G and R), and then normalize it with scale $1.0/256$ for network training. To prevent overfitting, we randomly reflect each image horizontally in the training stage.
Comparison with State-of-the-art Methods {#exp:stateoftheart}
----------------------------------------
**Market1501:** For the Market1501 dataset, several state-of-the-art methods are compared, including Bag of Words (BOW) [@ZhengliangICCV15], Weighted Approximate Rank Component Analysis (WARCA) [@Jose2016scalable], Discriminative Null Space (DNS) [@ZhangLiCVPR16], Spatially Constrained Similarity function on Polynomial feature map (SCSP) [@ChendapengCVPR16similarity], and deep learning based approaches, such as PersonNet [@Wulin2016Personnet], Comparative Attention Network (CAN) [@Liu2016end], Siamese Long Short-Term Memory (S-LSTM) [@Varior2016Siamese], Gated Siamese Convolutional Neural Network (Gate-SCNN) [@VariorECCV16Gated]. The experimental results are shown in Table \[tab:marketresults\].
Compared with existing full body-based convolutional neural networks, such as CAN and Gate-SCNN, the proposed network structure can better capture pedestrian features with multi-class person identification tasks. Our full-body representation improves Rank-1 identification rate by 9.57% on the state-of-the-art results produced by the Gate-CNN in single query. Compared with the full body, our body-part representation increase 0.80%. The main reason is that the pedestrians detected by DPM consists much more background information and the part-based representation can better reduce the influences of background clutter.
The full-body and body-part representations are complementary to each other. The full-body representation cares more about the global information, such as the background and body shape. The body-part representation pays more attention to parts, such as head, upper body and lower body. As shown in Table \[tab:marketresults\], the fusion model of full body and body parts improves Rank-1 identification rate by more than 4.00% compared with the body and parts-based models separately in single query. The mAP improves about 17.98% compared with the best result produced by Gate-CNN.
Query
--------------------------------- ----------- ----------- ----------- -----------
Evaluation metrics R1 mAP R1 mAP
BOW [@ZhengliangICCV15] 34.38 14.1 42.64 19.47
BOW + HS [@ZhengliangICCV15] - - 47.25 21.88
WARCA [@Jose2016scalable] 45.16 - - -
PersonNet [@Wulin2016Personnet] 37.21 26.35 - -
S-LSTM [@Varior2016Siamese] - - 61.6 35.3
SCSP [@Chengde2016person] 51.9 26.35 - -
CAN [@liu2017end] 48.24 24.43 - -
DNS [@ZhangLiCVPR16] 55.43 29.87 71.56 46.03
Gate-SCNN [@VariorECCV16Gated] 65.88 39.55 76.04 48.45
Our-Part 76.25 53.33 84.12 62.90
Our-Body 75.45 52.41 83.43 62.03
Our-Fusion **80.31** **57.53** **86.79** **66.70**
: Experimental results on the Market1501 dataset. - means that no reported results are available.[]{data-label="tab:marketresults"}
**CUHK03:** For the CUHK03 dataset, we compare our method with many existing approaches, including Filter Pair Neural Networks (FPNN) [@LiWeiCVPR14], Improved Deep Learning Architecture (IDLA) [@AhmedCVPR15improved], Cross-view Quadratic Discriminant Analysis (XQDA) [@LiaoshengcaiCVPR15], PSD constrained asymmetric metric learning (denoted as MLAPG) [@LiaoshengcaiICCV15], Sample-Specific SVM (SS) [@ZhangCVPR16sample], Single image and Cross image representation (SI-CI) [@WangfaqiangCVPR16JSC], Embedding Deep Metric (EDM) [@Shihanlin2016Embedding], Domain Guided Dropout (DGD) [@XiaotongCVPR16Domain], DNS, S-LSTM and Gate-SCNN. On this dataset, we conduct experiments on both the detected and the labeled datasets. As presented in previous work [@LiWeiCVPR14], we use the CMC curve in the single shot case to evaluate the performance. The overall results are shown in the Table \[tab:cuhk03detectedresults\] and Table \[tab:cuhk03labeledresults\]. The full CMC curves are shown in supplementary materials.
Compared with metric learning methods, such as the state-of-the-art approach DNS, the proposed fusion model improves the Rank-1 identification rate by 11.66% and 13.29% on the labeled and detected datasets respectively. Compared with the similar multi-class person identification network DGD, the Rank-1 identification rate improves by 1.63% using our fusion model on the labeled dataset. It should be noted that we only use the labeled sets for training, while the DGD is trained on both the labeled and detected datasets. This demonstrates the effectiveness of the proposed model.
Dataset
-------------------------------- ----------- ----------- ----------- -----------
Rank 1 5 10 20
FPNN [@LiWeiCVPR14] 19.89 50.00 64.00 78.50
IDLA [@AhmedCVPR15improved] 44.96 76.01 83.47 93.15
XQDA [@LiaoshengcaiCVPR15] 46.25 78.90 88.55 94.25
MLAPG [@LiaoshengcaiICCV15] 51.15 83.55 92.05 96.90
SS-SVM [@ZhangCVPR16sample] 51.20 80.80 89.60 95.50
SI-CI [@WangfaqiangCVPR16JSC] 52.17 84.30 92.30 95.00
DNS [@ZhangLiCVPR16] 54.70 84.75 94.80 95.20
S-LSTM [@Varior2016Siamese] 57.30 80.10 88.30 -
Gate-SCNN [@VariorECCV16Gated] 61.80 80.90 88.30 -
EDM [@Shihanlin2016Embedding] 52.09 82.87 91.78 97.17
Our-Part 62.74 88.53 93.97 97.21
Our-Body 64.95 89.82 94.58 97.56
Our-Fusion **67.99** **91.04** **95.36** **97.83**
: Experimental results on the CUHK03 detected dataset.[]{data-label="tab:cuhk03detectedresults"}
Dataset
------------------------------------ ----------- ----------- ----------- -----------
Rank 1 5 10 20
FPNN [@LiWeiCVPR14] 20.65 51.50 66.50 80.00
IDLA [@AhmedCVPR15improved] 54.74 86.50 93.88 98.10
XQDA [@LiaoshengcaiCVPR15] 52.20 82.23 92.14 96.25
MLAPG [@LiaoshengcaiICCV15] 57.96 87.09 94.74 98.00
Ensemble [@PaisitkriangkraiCVPR15] 62.10 89.10 94.80 98.10
SS-SVM [@ZhangCVPR16sample] 57.00 85.70 94.30 97.80
DNS [@ZhangLiCVPR16] 62.55 90.05 94.80 98.10
EDM [@Shihanlin2016Embedding] 61.32 88.90 96.44 **99.94**
DGD [@XiaotongCVPR16Domain] 72.58 91.59 95.21 97.72
Our-Part 69.41 92.68 96.68 99.02
Our-Body 71.88 93.66 97.46 99.18
Our-Fusion **74.21** **94.33** **97.54** 99.25
: Experimental results on the CUHK03 labeled dataset.[]{data-label="tab:cuhk03labeledresults"}
**MARS:** This dataset is the largest sequence-based person ReID dataset. On this dataset, we compare the proposed method with several classical methods, including Keep It as Simple and straightforward Metric (KISSME) [@KostingerCVPR12], XQDA [@LiaoshengcaiCVPR15], and CaffeNet [@KrizhevskyNIPS12]. Similar to previous work [@ZhengliangECCV16], both single query and multiple query are evaluated on MARS. The overall experimental results on the MARS are shown in Table \[tab:marsresults\_single\] and Table \[tab:marsresults\_mutiple\]. The full CMC curves are shown in supplementary materials.
Compared with CaffeNet, a similar multi-class person identification network, our body-based model improves the Rank-1 identification rate by 2.93% and mAP by 4.22% using XQDA in single query. It should be noticed that our network does not use any pre-training with additional data. Usually deep learning network can obtain better results when pretrained with on ImageNet classification task. Our fusion model improves Rank-1 identification rate and mAP by 6.47% and by 8.45% in single query. This illustrates the effectiveness of our model.
Query
---------------------------------- ----------- ----------- ----------- -----------
Evaluation metrics 1 5 20 mAP
CNN+Eulidean [@ZhengliangECCV16] 58.70 77.10 86.80 40.40
CNN+KISSME [@ZhengliangECCV16] 65.00 81.10 88.90 45.60
CNN+XQDA [@ZhengliangECCV16] 65.30 82.00 89.00 47.60
Our-Fusion+Eulidean 68.38 84.19 91.52 51.13
Our-Fusion+KISSME 69.24 85.15 92.17 53.00
Our-Part+XQDA 66.62 82.07 90.76 49.74
Our-Body+XQDA 68.23 83.99 92.17 51.82
Our-Fusion+XQDA **71.77** **86.57** **93.08** **56.05**
: Experimental results on the MARS with single query.
\[tab:marsresults\_single\]
Query
----------------------------------- ----------- ----------- ----------- -----------
Evaluation metrics 1 5 20 mAP
CNN+KISSME+MQ [@ZhengliangECCV16] 68.30 82.60 89.40 49.30
Our-Fusion+Euclidean+MQ 78.28 91.97 96.87 61.62
Our-Fusion+KISSME+MQ 80.51 93.18 97.22 63.50
Our-Fusion+XQDA+MQ **83.03** **93.69** **97.63** **66.43**
: Experimental results on the MARS with multiple query.
\[tab:marsresults\_mutiple\]
Effectiveness of MSCAN {#exp:effmscan}
----------------------
To determine the effectiveness of MSCAN, we explore four variants of MSCANs to learn IDE feature based on the whole body image, which is denoted as MSCAN-$k$, $k = \{1,2,3,4\}$. $k$ is the number of dilation ratios. For example, MSCAN-$3$ means for each convolution layer in Conv1-Conv4, there are three convolution kernels with dilation ratio 1, 2, and 3 respectively. With the increase of $k$, the MSCAN captures larger context information at the same convolution layer.
The experimental results based on these four types of MSCANs on the Market1501 dataset are shown in Table \[tab:mscanmarket\]. As we can see, with the increase of the number of dilation ratios, the Rank-1 identification rate and mAP improve stably in single query case. For multiple query case, which means fusing all images belonging to the same query person at the same camera through average pooling in feature space, the Rank-1 identification rate and mAP also improves step by step. However, the Rank-1 identification rate and mAP increase not much when $K$ increase from 3 to 4. We think there is a suitable number of dilation ratios for feature learning. Considering the model complexity and accuracy improvements in Rank-1 identification rate, we adopt the MSCAN-3 as our final MSCAN model in this paper.
Query type
-------------------- ----------- ----------- ----------- -----------
Evaluation metrics Rank-1 mAP Rank-1 mAP
MSCAN-1 65.38 41.85 75.21 51.14
MSCAN-2 72.21 49.19 82.22 59.03
MSCAN-3 75.45 52.41 83.43 62.03
MSCAN-4 **76.25** **53.14** **84.09** **62.95**
: Experimental results of four types of MSCAN using body-based representation for ReID on the Market1501 dataset.
\[tab:mscanmarket\]
Effectiveness of Latent Part Localization {#exp:efflpl}
-----------------------------------------
**Learned parts rigid parts** To compare with popular rigid pedestrian parts, we divide the pedestrian into three overlapped regions as predefined rigid parts. We use the rigid body parts instead of the learned latent body parts for part-based feature learning. Experimental results with rigid and learned body parts are shown in Table \[tab:partmarket\]. Compared with rigid body parts, the learned body parts improve Rank-1 identification rate and mAP by 3.27% and 3.73% in single query, and by 1.70% and by 2.67% in multiple query. This validate the effectiveness of learned person parts.
For better understanding the learned pedestrian parts, we visualize the localized latent parts in Figure \[fig:visualizationloc\] using our fusion model. For these detected person with large background (the first row in Figure \[fig:visualizationloc\]), the proposed model can learn foreground information with complementary latent pedestrian parts. As we can see, the learned parts consist of three main components, including upper body, middle body (combination of upper body and lower body), and lower body. Similar results can be achieved when original detection pedestrians contain less background or occlusion (the second row in Figure \[fig:visualizationloc\]). It is easy to see that, the automatically learned pedestrian parts are not strictly head-shoulder, upper body and lower-body. But it indeed consists of these three parts with large overlap. Compared with rigid parts, the proposed model can automatically localize the appropriate latent parts for feature learning.
![Six samples of original image, rigid predefined parts and learned latent pedestrian parts. Samples in each column are the same person with different backgrounds. Best viewed in color. []{data-label="fig:visualizationloc"}](figs/part_loc.pdf){width="0.35\paperwidth"}
Query type
-------------------- ----------- ----------- ----------- -----------
Evaluation metrics Rank-1 mAP Rank-1 mAP
Rigid parts 72.98 49.60 82.42 60.23
Latent parts **76.25** **53.33** **84.12** **62.90**
: Experimental results of rigid parts and learned parts for ReID on the Market1501 dataset.
\[tab:partmarket\]
**Effectiveness of localization loss** To evaluate the effectiveness of the proposed constraints on the latent part localization network, we conduct additional experiments by adding or deleting proposed $L_{loc}$ in the training stage of body parts network for ReID. Experimental results are shown in Table \[tab:stnconstrictmarket\]. As we can see, with the additional $L_{loc}$, the Rank-1 accuracy increases by 9.03%. We owe the improvements to the effectiveness of the proposed constraints on the part localization network.
Query type
-------------------- ----------- ----------- ----------- -----------
Evaluation metrics Rank-1 mAP Rank-1 mAP
$L_{cls}$ 67.22 45.27 77.55 55.40
$L_{cls}+L_{loc}$ **76.25** **53.33** **84.12** **62.90**
: The influences of $L_{loc}$ on part-based network on the Market1501 dataset.
\[tab:stnconstrictmarket\]
Cross-dataset Evaluation {#exp:discussion}
------------------------
Similar with typical image classification task with CNN, our approach requires large scale of data, not only more identities, but also more instances for each identity. So we do not train the proposed model on each single small person ReID dataset, such as VIPeR. Instead, we conduct cross-dataset evaluation from the pretrained model on the Market1501, CUHK03 and MARS datasets to the VIPeR dataset. The experimental results are shown in Table \[tab:crossviper\]. Compared with other methods, such as Domain Transfer Rank Support Vector Machines [@MaICCV13domain] and DML [@YiICPR14DML], the models trained on large-scale datasets have better generalization ability and have better Rank-1 identification rate.
To take further analysis of the proposed method, we also fine-tune the model from large dataset Market1501 to small dataset VIPeR. Experimental results are shown in Table \[tab:markettransfertoviper\]. Our fusion-based model obtains better Rank-1 identification rate than existing deep models, IDLA [@AhmedCVPR15improved] (34.8%), Gate-SCNN [@VariorECCV16Gated] (37.8%), SI-CI [@WangfaqiangCVPR16JSC] (35.8%), and achieves comparable results with DGD [@XiaotongCVPR16Domain] (38.6%).
Methods Training Set 1 10 20 30
-------------------------- ----------------- ----------- ----------- ----------- -----------
DTRSVM [@MaICCV13domain] i-LIDS 8.26 31.39 44.83 53.88
DTRSVM [@MaICCV13domain] PRID 10.90 28.20 37.69 44.87
DML [@YiICPR14DML] CUHK Campus 16.17 45.82 57.56 64.24
Ours-Fusion CUHK03 detected 17.30 44.58 55.51 61.77
Ours-Fusion CHUK03 labeled 19.44 **49.99** **60.78** **66.74**
Ours-Fusion MRAS 18.46 43.65 52.96 59.34
Ours-Fusion Market1501 **22.21** 47.24 57.13 62.26
: Cross-dataset person ReID on the VIPeR dataset
\[tab:crossviper\]
Method Rank-1 Rank-5 Rank-10 Rank-20
------------ ----------- ----------- ----------- -----------
Our-Part 32.70 57.49 67.62 78.90
Our-Body 33.12 60.23 72.05 82.59
Our-Fusion **38.08** **64.14** **73.52** **82.91**
: Experimental results on VIPeR through fine-tuneing the model from Market1501 to VIPeR.
\[tab:markettransfertoviper\]
Conclusion {#conclusions}
==========
In this work, we have studied the problem of person ReID in three levels: 1) a multi-scale context-aware network to capture the context knowledge for pedestrian feature learning, 2) three novel constraints on STN for effective latent parts localization and body-part feature representation, 3) the fusion of full-body and body-part identity discriminative features for powerful pedestrian representation. We have validated the effectiveness of the proposed method on current large-scale person ReID datasets. Experimental results have demonstrated that the proposed method achieves the state-of-the-art results.
**Acknowledgement** This work is funded by the National Key Research and Development Program of China (2016YFB1001005), the National Natural Science Foundation of China (Grant No. 61673375, Grant No. 61403383 and Grant No. 61473290), and the Projects of Chinese Academy of Science (Grant No. QYZDB-SSW-JSC006, Grant No. 173211KYSB20160008).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The [*concordance*]{} cosmological model has been successfully tested over the last decades. Despite its successes, the fundamental nature of dark matter and dark energy is still unknown. Modifications of the gravitational action have been proposed as an alternative to these dark components. The straightforward modification of gravity is to generalize the action to a function, $f(R)$, of the scalar curvature. Thus one is able to describe the emergence and the evolution of the Large Scale Structure without any additional (unknown) dark component. In the weak field limit of the $f(R)$-gravity, a modified Newtonian gravitational potential arises. This gravitational potential accounts for an extra force, generally called fifth force, that produces a precession of the orbital motion even in the classic mechanical approach. We have shown that the orbits in the modified potential can be written as Keplerian orbits under some conditions on the strength and scale length of this extra force. Nevertheless, we have also shown that this extra term gives rise to the precession of the orbit. Thus, comparing our prediction with the measurements of the precession of some planetary motions, we have found that the strength of the fifth force must be in the range $[2.70-6.70]\times10^{-9}$ with the characteristic scale length to fix to the fiducial values of $\sim 5000$ AU.'
author:
- Ivan De Martino
- Ruth Lazkoz
- Mariafelicia De Laurentis
title: 'Analysis of the Yukawa gravitational potential in $f(R)$ gravity I: semiclassical periastron advance.'
---
Introduction {#uno}
============
Standard cosmology is entirely based on General Relativity. It is capable of explaining both the present period of accelerated expansion and the dynamics of self-gravitating systems resorting to Dark Energy and Dark Matter, respectively. The model has been confirmed by observations carried out over the last decades [@Planck16_13]. The need of having recourse to Dark Matter to explain the dynamics of stellar clusters, galaxies, groups and clusters of galaxies, among others astrophysical objects, has been well known for many decades now. Such systems show a deficit of mass when the photometric and spectroscopic estimates are compared with the dynamical one. Early astronomical candidates proposed to solve this problem of [*missing*]{} mass were MAssive Compact Halo Objects (MACHOs) and ReAlly Massive Baryon Objects (RAMBOs), sub-luminous compact objects (or clusters of objects) like Black Holes and Neutron Stars that could not have been observed due to several selection effects. Since the number of the observed sub-luminous objects was not enough to account for the [*missing matter*]{}, the idea that this matter was hidden in some exotic particles, weakly interacting with ordinary matter, emerged. Many candidates have been proposed such as Weakly Interacting Massive Particle (WIMP), axions, neutralino, Q-balls, gravitinos and Bose-Einstein condensate, among the others [@Bertone2005; @Capolupo2010; @Schive2014; @demartino2017b; @demartino2018; @Lopes2018; @Panotopoulos2018], but there are no experimental evidences of their existence so far [@Feng2010].
An alternative approach is to modify Newton’s law. Such a modification naturally arises in the weak field limit of some modified gravity models [@Moffat2006; @demartino2017; @PhysRept; @manos; @sergei] that attempt to explain the nature of Dark Matter and Dark Energy as an effect of the space-time curvature. These theories predict the existence of massive gravitons that may carry the gravitational interaction over a certain scale depending by the mass of these particles [@Bogdanos:2009tn; @felix; @Bellucci:2008jt; @graviton]. Thus, in their weak field limit, a Yukawa-like modification to Newton’s law emerges. One of those models is $f(R)$-gravity where the Einstein-Hilbert action, which is linear in the Ricci scalar $R$, is replaced with a more general function of the curvature, $f(R)$. In its the weak field limit, the modified Newtonian potential has the following functional form [@Annalen]: $$\label{eq:potyuk}
\Phi(r) = -\frac{G M}{(1+\delta)r}(1+\delta e^{-r/\lambda}),$$ where $M$ is the mass of the point-like source, $r$ the distance of a test particle ($m$) from the source, $G$ is Newton’s constant, $\delta$ is the strength of the Yukawa correction and the $\lambda$ represents the scale over which the Yukawa-force acts. Since $f(R)$-gravity is a fourth-order theory, the Yukawa scale length arises from the extra degrees of freedom (in the general paradigm, a $(2k+2)$-order theory of gravity gives rise to $k$ extra gravitational scales [@Quandt1991]). Both parameters are also related to the $f(R)$-Lagrangian as [@Annalen; @PhysRept]: $$\begin{aligned}
\delta = f'_0 - 1, \qquad \lambda = \sqrt{-\frac{6f''_0}{f'_0}},\end{aligned}$$ where $$f'_0=\frac{df(R)}{dR}\biggr|_{R=R_0}\,, \qquad f''_0=\frac{d^2f(R)}{dR^2}\biggr|_{R=R_0}.$$ Next, considering the field equations and trace of $f(R)$ gravity at the first order approximation in terms of the perturbations of the metric tensor, and choosing a suitable transformation and a gauge condition, one can relate the massive states of the graviton to the $f(R)$-Lagrangian and to the Yukawa-length: $$\begin{aligned}
m_g^2 \propto \frac{-f'_0}{f''_0} =\frac{2}{\lambda^2}.\end{aligned}$$ Therefore, it is customary to identify the Yukawa-length with the Compton wavelength of the massive graviton $\lambda_c = hc/m_g$. Thus, for example, we have $\lambda\sim10^{3}$ km with a mass of gravitons $m_g\sim 10^{-22}$ eV [@Lee2010; @Abbott2017]. Therefore, the effect of a modification of the Newtonian potential must naturally act at galactic and extragalactic scales, where $f(R)$-gravity has been successfully tested [@demartino2014; @demartino2015; @demartino2016]. Nevertheless, smaller effects could be detected at shorter scales [@Talmadge1988] where the strength of the Yukawa-correction has been bounded using the Pioneer anomaly [@Anderson1998; @Anderson2002] and S2 star orbits [@Borka20012; @Borka20013; @Zakharov2016; @Hees2017; @Zakharov2018; @Iorio2005]. Obviously, the most interesting systems to test gravitational theories are binary systems composed by coalescing compact stars, such as neutron stars, white dwarfs and/or black holes [@deLa_deMa2014; @deLa_deMa2015; @LeeS2017], but the study of stable orbits is equally important since it allows us to study possible variations of the gravitational interaction in the weak field limit. In the last decades, the orbital precession has been used to probe General Relativity [@Will2014; @Iorio2009], as well as to place bounds on “anti-gravity” due to the cosmological constant [@islam1983; @Iorio2006; @Sereno2006], on forces proposed as alternatives to dark matter [@Gron1996; @Khriplovich2006] and/or induced from extensions of General Relativity [@Capozziello2001; @Moffat2006; @Sanders2006; @Battat2008; @Nyambuya2010; @Ozer2017; @Liu2018].
In this paper we show, in a semi-classical approach, that the orbital motion under the modified gravitational potential in Eq. can be traced back to a Keplerian orbit with modified eccentricity, but with an orbital precession due to the Yukawa-term. We consider two point-like masses orbiting around each other and we use a Newtonian approach to compute the equation of the orbit, and a perturbative approach to compute the precession of the orbit. Finally, we use the current limits on the orbital precession of the planetary orbits to place a bound on the strength of the Yukawa-term. The paper is organized as follows: in Sec. \[due\] we introduce the equations of motion, in Sec. \[tre\] we compute the equation of the orbits, in Sec. \[quattro\] we compute analytically the precession effect due to the Yukawa potential, and we use current measurements of the orbital precession of Solar System’s planets to bound the parameter $\delta$ in Eq. . We consider, for each planet, a 3$\sigma$ interval around the best fit value of the precession, and we compute the lower and an upper limit on $\delta$ so that the predicted precession relies in the observed interval. In Sec. \[cinque\] we discuss some consequences of our results. Finally, in Sec. \[sei\] we give our conclusions.
Newtonian approach to two body problem in Yukawa potential {#due}
==========================================================
The starting point is the equation of motion of a massive point-like particle, $m$, in the gravitational potential well generated by the particle $M$, and given in Eq. . In polar coordinates $(r,\varphi)$ and with respect to the center of mass, the equations of motion read $$\begin{aligned}
\label{eq:1} & \ddot{r} = -\nabla\Phi(r)\,,\\
\label{eq:2} & \frac{d}{dt}(r^2 \dot{\varphi}) = 0 \,, \end{aligned}$$ and the total energy of the system can be written as [@deLa2011] $$\label{eq:energy}
E_T = \frac{1}{2}\mu(\dot{r}^2 + r^2\dot{\varphi}^2)-\frac{Gm M}{(1+\delta)r}(1+\delta e^{-r/\lambda}),$$ where $\displaystyle{\mu=\frac{mM}{m+M}}$ is the reduced mass, and $\Phi(r)$ is the modified gravitational potential of Eq. . Using the conservation of the angular momentum $L$ expressed in Eq. , it is straightforward to recast the total energy as a function of the radial coordinate: $$\label{eq:energy2}
E_T = \frac{1}{2}\mu\dot{r}^2 + \frac{L^2}{2\mu r^2} -\frac{Gm M}{(1+\delta)}\frac{(1+\delta e^{-r/\lambda})}{r}.$$
Eq. is the only one needed to compute the equation of motion for an unperturbed orbit. Nevertheless, we can learn more about the orbits by defining an effective potential as $$\label{eq:Veff}
V_{\rm eff}(r) = \frac{L^2}{2\mu r^2} - \frac{Gm M}{(1+\delta) r} - Gm M \frac{\delta}{(1+\delta) r} e^{-r/\lambda}.$$ Here, the first term accounts for the repulsive force associated to the angular momentum, the second term represents the gravitational attraction, and the third term can be interpreted as an additional force due to the Yukawa-like term in the gravitational potential acting on the particle. The effective potential demands other considerations: first, one needs $\delta \neq -1$ in order to avoid a singularity in the second and third terms; second, if $\delta$ assumes negative values, the second term stays attractive as far as the condition $ \delta > -1 $ is satisfied, and the last term becomes repulsive; third, the condition $\delta <-1$ makes the second term repulsive, rendering the third term attractive; fourth, if $\delta >0$ then both second and third terms are attractive.
For illustration, in Fig. \[fig1\](a) and (b) we plot the potential and the effective potential as a function of $r/\lambda$ showing their dependence on the strength of the Yukawa term. Notice that the minimum of the effective potential depends on the strength parameter $\delta$ of the Yukawa-term (Fig. \[fig1\](b)). As expected, a negative value of $\delta$ makes the potential well deeper as compared to the Newtonian case ($\delta=0$), while a positive value makes it flatter. This can be understood looking at Eq. , for $-1<\delta<0$ the effective mass $M'= M/(1+\delta)$ becomes larger, while for $\delta>0$ it becomes smaller than the “Newtonian mass" $M$.
{width="8.6cm"} {width="8.6cm"}
Differentiating with respect to the radial coordinate and looking for the minimum, one finds the condition: $$\frac{d V_{\rm eff}(r)}{dr} =0 \Rightarrow \frac{L^2}{\mu r} = \frac{G m M \left(\delta e^{-\frac{r}{\lambda}}+1\right)}{(\delta +1)}+\frac{\delta G m M e^{-\frac{r}{\lambda}}}{(\delta +1) \lambda}r\,.$$ The second derivative and the previous condition on the angular momentum leads to obtain the following expression $$\frac{d^2V_{\rm eff}(r)}{dr^2}= \frac{G m M e^{-\frac{r}{\lambda}}}{(\delta +1)r^3}\biggl[ \delta(-\lambda^{-2}r^2
+ \lambda^{-1}r +1) + e^{\frac{r}{\lambda}} \biggr].$$ A minimum in the effective potential exists if the following condition is satisfied $$\label{eq:condition1}
g(x)\equiv \delta(-x^2 + x +1) + e^x>0\,,$$ where we have defined $x\equiv r/\lambda$. Eq. is satisfied in the following cases: (i) $\delta>-1$ for $x\rightarrow0$, (ii) $\delta>-e$ for $x\rightarrow1$, and (iii) $\forall\,\delta$ in the limit $x\rightarrow\infty$. Let us notice that the first case, meaning $r\ll \lambda$, is the common configuration of an astrophysical system with its dynamics happening at scales much lower than the Compton wavelength of the massive graviton, such as planetary motion around the Sun. On the contrary, the second case ($r\sim\lambda$) represents systems such as S-Stars around the Galactic center, with their dynamics happening at scales of the order of parsecs. Finally, the last case ($r\gg \lambda$) can be associated to the extragalactic and cosmological scales. Since we are interested in studying systems on distance scales much smaller than the Compton scale of a massive graviton, the exponential term in previous equations Taylor expanded as $$\label{eq:approx_exp}
e^{\pm x} \approx 1 \pm x + \frac{x^2}{2} + \mathcal{O}\bigl(x^3\bigr).$$ When replacing Eq. in to Eq. , the first term gives the Newtonian force, the second term induces a shift in the energy of the system, and the third term gives rise to a constant radial acceleration (often called fifth force) that can be written as follows $$a_{\rm corr} = -\frac{a^*\delta}{2(1+\delta)}\frac{r^{*2}}{\lambda^2},$$ where $a^*$ is the Newtonian acceleration of an object at distance $r^*$. As an example, this correction can be applied to the Pioneer anomaly, thus obtaining a strength $|\delta|\leq 1.7\times10^{-4}$ at $\lambda\sim200$ AU [@Anderson1998; @Anderson2002]. It is important to remark that the approximation in Eq. is valid only for dynamics at the scale of planetary systems or stars with orbits having their semi-major axis much smaller than $\lambda$. In contrast, to study the dynamics of systems on larger scales, one cannot use the approximation in Eq. but rather, the equations of motion must be integrated numerically.
Let us analyze the condition for the existence a minimum in the effective potential at both $\mathcal{O}(x^{2})$ and $\mathcal{O}(x^{3})$ orders:
$\mathcal{O}(x^{2})$ order
: at this order of approximation, the effective potential becomes $$V_{eff}(r) = \frac{L^2}{2 \mu r^2} - \frac{G m M}{r}+\frac{\delta G m M}{(\delta +1) \lambda},$$ and we find the minimum at the radius $$\label{eq:rmin_order1}
r_{min} = \frac{L^2}{2\mu G m M},$$ which is the same as the one of Newtonian gravity (as expected), while the effective potential at the minimum is shifted with respect to the Newtonian one $$\begin{aligned}
\label{eq:veffmin_order1}
&&V_{eff, min}= - \frac{1}{2} G m M \left(\frac{G \mu m M}{L^2} -\frac{2 \delta }{(\delta +1)\lambda}\right)\,,\nonumber\\
\end{aligned}$$
$\mathcal{O}(x^{3})$ order
: the effective potential can be recast as $$\begin{aligned}
\label{eq:veff_order2}
&&V_{eff}(r) = \frac{L^2}{2 \mu r^2} - \frac{G m M}{r}+\frac{\delta G m M}{(\delta +1) \lambda} -\frac{\delta G m M r}{2 (\delta +1) \lambda^2}\,.\nonumber\\
\end{aligned}$$ Since we are looking for a strength force in the regime $\delta\ll1$, thus meaning a small deviation from Newtonian dynamic, the shift in $r_{min}$ is absolutely negligible. Thus, replacing Eq. in to Eq. we get $$\begin{aligned}
\label{eq:veffmin_order2}
V_{eff, min}& =& - \frac{G m M }{2} \left(\frac{G \mu m M}{L^2} -\frac{2 \delta }{(\delta+1)\lambda}\right) \nonumber\\
&-&\frac{L^2 \delta }{2 (1+\delta ) \lambda ^2 \mu }\,.
\end{aligned}$$
Therefore, at both $\mathcal{O}(\lambda^{-2})$ and $\mathcal{O}(\lambda^{-3})$ orders, the minimum of the effective potential always exists and it is located at the same radius ($r_{min}$) than in the Newtonian case. Finally, Eqs. and show that the minimum of the effective potential is shifted as qualitatively explained above and shown in Fig. \[fig1\]b.
Equation of the orbits {#tre}
======================
Hereby, we compute the equation of the closed orbit in both $\mathcal{O}(x^{2})$ and $\mathcal{O}(x^{3})$ approximations, and we show that, under some conditions on the eccentricity and the position of latus rectum, the orbit can be recast into the usual Keplerian form, where the correction due to the Yukawa-term getting hidden into the orbital parameters. We work in the regime $r\ll\lambda$ in order to replace the exponential term in Eq. with Eq. .
Approximation at $\mathcal{O}(x^{2})$-order {#treA}
-------------------------------------------
To compute the equation of the orbit we rewrite the radial component of the velocity as $$\dot{r}=-\dfrac{L}{\mu}\dfrac{d}{d\varphi}\dfrac{1}{r}\,,$$ then, at second order in the approximation of the Yukawa-term, the total energy of the system can be recast as $$\label{eq:energy3}
E_T = \frac{L^2}{2\mu}\left(\dfrac{d}{d\varphi}\dfrac{1}{r}\right)^2 + \frac{L^2}{2\mu r^2} -\frac{Gm M}{r} + \frac{Gm M \delta}{(1+\delta)\lambda}.$$ From the previous equation we can obtain the following differential equation $$\label{eq:energy4}
u'^2+u^2 - 2\beta_0 u = \beta_1,$$ where $u\equiv1/r$, $u'=du/d\varphi$ and $$\label{eq:betas01}
\gamma = G m M; \qquad \beta_0=\frac{\mu\gamma}{L^2}; \qquad \beta_1= \frac{2\mu E_T}{L^2}- \frac{2\mu\gamma}{L^2\lambda}\frac{\delta}{1+\delta}.$$ Differentiating Eq. , we get $$\label{eq:energy5}
u'\biggr(u''+u- \beta_0\biggr) = 0.$$ As we are looking for a Keplerian solution, we make the following [*ansatz*]{}: $$\label{eq:orbit0}
u\equiv\frac{1}{r}=\frac{1}{l}(1+\epsilon\cos\varphi),$$ where $l$ is the [*latus rectum*]{} and $\epsilon$ is the eccentricity. Therefore, inserting the Eq. in Eq. , we obtain the following condition for the [*latus rectum*]{}: $$\label{eq:orbit1}
l=\frac{1}{\beta_0}.$$ Then, we substitute Eqs. into Eq. thus obtaining the following expression for the eccentricity: $$\label{eq:orbit2}
\epsilon^2=1 + l^2 \beta_1,$$ that in terms of energy of the system is $$\label{eq:orbit3}
\epsilon^2 = 1 - \frac{2 L^2}{\mu\gamma} \frac{\delta}{(1+\delta)\lambda} + \frac{2 E_T L^2 \mu}{\mu^2\gamma^2},$$ which for $\delta=0$ gets reduced to the Newtonian value: $$\label{eq:orbit3_1}
\epsilon^2 = 1 + \frac{2 E_T L^2 \mu}{\mu^2\gamma^2}.$$ This shift is clearly not testable with observations given that we measure the orbital parameters, whereas the total energy is a theory dependent parameter. Nevertheless, looking at Eq. , it is straightforward to understand that, if the total energy and angular momentum are fixed then they correspond to an orbital motion with an eccentricity that would vary depending on the strength of the Yukawa correction, as shown in Figure \[fig2\].
![Illustration of the effect of the modified gravitational potential on the orbital parameters. Panel (a) shows the orbits for different values of $\delta$. The angular momentum and the total energy are set to those values that give rise to an elliptical orbit with eccentricity $\epsilon=0.5$ in Newtonian mechanics ($\delta=0$) showing that such an orbital solution would show a difference in the eccentricity when the Yukawa term is taken in to account. Panel (b) shows the relative difference with the Newtonian mechanics along the orbit.[]{data-label="fig2"}](fig2a.eps "fig:"){width="8.6cm"} ![Illustration of the effect of the modified gravitational potential on the orbital parameters. Panel (a) shows the orbits for different values of $\delta$. The angular momentum and the total energy are set to those values that give rise to an elliptical orbit with eccentricity $\epsilon=0.5$ in Newtonian mechanics ($\delta=0$) showing that such an orbital solution would show a difference in the eccentricity when the Yukawa term is taken in to account. Panel (b) shows the relative difference with the Newtonian mechanics along the orbit.[]{data-label="fig2"}](fig2b.eps "fig:"){width="8.6cm"}
Approximation at $\mathcal{O}(x^{3})$ order {#treB}
-------------------------------------------
Approximating the Yukawa-term at third order, the differential equation becomes $$\label{eq:energy6}
u'^2+u^2- 2\beta_0u - \beta_2\frac{1}{u} = \beta_1,$$ where $\beta_0$ and $\beta_1$ are given in Eq. , and $$\label{eq:beta2}
\beta_2=\frac{\mu\gamma\delta}{2 L^2\lambda(1+\delta)}.$$ By taking the derivative of Eq. we obtain $$\label{eq:energy7}
u'\left(u''+u +\frac{\beta_2}{u^2}-\beta_0\right)=0\,.$$
Let us introduce Eq. into Eq. and evaluate the expression at $\varphi=[0; \pi]$, which respectively correspond to the minimum and maximum distance between the two masses. Thus, we obtain two conditions: $$\begin{aligned}
& \label{eq:orbit4} (1 - l \beta_0) \epsilon^2 + 2 (1 - l \beta_0) \epsilon - l \beta_0 + l^3 \beta_2 +1 =0\,, \\
& \label{eq:orbit5} (1 - l \beta_0) \epsilon^2 - 2 (1 - l \beta_0) \epsilon - l \beta_0 + l^3 \beta_2 +1 =0\,.\end{aligned}$$ Subtracting Eqs. and we obtain the latus rectum which turns out to have the same expression as in Eq. . Finally, introducing Eq. in Eq. and evaluating it, once again, at $\varphi=[0; \pi]$ we obtain the following condition for the eccentricity $$\epsilon^2= 1+ l^2\beta_1-4 \beta_2.$$ Let us note that the previous expression reduces to Eq. when $\beta_2=0$, and thus to the Newtonian value when $\delta=0$. The previous equation can be straightforwardly recast in terms of energy of the system as $$\label{eq:orbit6}
\epsilon^2 = 1 + \frac{2 E_T L^2 \mu}{\mu^2\gamma^2} - \frac{2 L^2}{\mu\gamma} \frac{\delta}{(1+\delta)\lambda} - \frac{2\mu\gamma\delta}{L^2\lambda(1+\delta)}.$$ The $\mathcal{O}(x^{3})$ order the shift is larger than at $\mathcal{O}(x^{2})$ order, and the difference due to the order of approximation is not negligible (see Fig. \[fig3\]).
![The plots and panels replicate the ones in Fig. \[fig2\] for the $\mathcal{O}(\lambda^{-3})$ approximation order.[]{data-label="fig3"}](fig3a.eps "fig:"){width="8.6cm"} ![The plots and panels replicate the ones in Fig. \[fig2\] for the $\mathcal{O}(\lambda^{-3})$ approximation order.[]{data-label="fig3"}](fig3b.eps "fig:"){width="8.6cm"}
Precession in Yukawa potential {#quattro}
==============================
To compute analytically the periastron advance due to the Yukawa-like term in the gravitational potential, we study small perturbations to the circular orbit. Thus, let us recast the total energy as $$\label{eq:prec1}
u'^2+u^2+ \frac{g(u)}{L^2} = \frac{2\mu E_T}{L^2} - \frac{2\mu\gamma}{L^2\lambda}\frac{\delta}{1+\delta},$$ where $ g(u)$ account for the gravitational interaction. Let us impose a close orbit defined by a minimum and a maximum distance from the center: $r_-|_{\varphi=0} =a(1-\epsilon) $ and $r_+|_{\varphi=\pi} =a(1+\epsilon)$, respectively. Here $a$ is the semi-major axis of the orbit. Thus, those correspond to $u_0=1/r_-$ and $u_1=1/r_+$. Being $u'|_{u=u_0}=u'|_{u=u_1}=0$, the Eq. gives rise to the following two conditions $$\begin{aligned}
& u_0^2+ \frac{g(u_0)}{L^2} = \frac{2\mu E_T}{L^2} - \frac{2\mu\gamma}{L^2\lambda}\frac{\delta}{1+\delta}\,,\\
& u_1^2+ \frac{g(u_1)}{L^2} = \frac{2\mu E_T}{L^2} - \frac{2\mu\gamma}{L^2\lambda}\frac{\delta}{1+\delta}\,,\end{aligned}$$ from which one obtains $$\begin{aligned}
& L^2 = \frac{g(u_0) - g(u_1)}{u_1^2 - u_0^2}\,,\\
& E_T = \frac{u_1^2g(u_0) - u_0^2g(u_1)}{2\mu(u_1^2 - u_0^2)} + \frac{\mu\gamma\delta}{\mu(1+\delta)\lambda}\,.\end{aligned}$$ Then, the differential equation becomes $$\label{eq:prec2}
u' = \sqrt{G(u_0,u_1,u)}\,,$$ where $G(u_0,u_1,u)$ is
$$\begin{aligned}
G(u_0,u_1,u)=\frac{g(u_0)(u_1^2-u^2) + g(u_1)(u^2-u_0^2)-(b^2 - u_0^2)g(u)}{g(u_0) - g(u_1)}\, .\end{aligned}$$
We can find the amount of angle required to pass from $r_-$ to $r_+$ by integrating equation : $$\label{eq:prec_angle}
\varphi(r_+)-\varphi(r_-) = \int_{u_0}^{u_1} G(u_0,u_1,u)^{-1/2}du\,.$$ Hence the particle will move from $r_-$ to $r_+$ and back every time $\varphi\rightarrow\varphi+2\pi$, thus $r(\varphi)$ is periodic with period $2\pi$. Therefore, the precession for each revolution is $$\omega = 2|\varphi(r_+)-\varphi(r_-)| - 2\pi.$$
In the case of approximating the exponential term at $\mathcal{O}(x^{2})$ order, the function $g(u)$ only depends by a Newtonian term: $$\label{eq:prec3}
g(u) = -2\mu\gamma u = 2 \mu \Phi_N(1/u),$$ where $\Phi_N(1/u)$ is the classical Newtonian potential. Thus, the precession does not exist as expected for the Newtonian potential.
Nevertheless, when approximating the exponential term at $\mathcal{O}(\lambda^{-3})$ order we have $$\label{eq:prec4}
g(u) = 2 \mu \Phi_N(1/u) - \frac{\mu\gamma\delta}{\lambda(1+\delta)} \frac{1}{u}.$$ In order to solve the integral in Eq. we perform a change of variables $$u_1 = u_0+\eta; \qquad u= u_0+\eta\upsilon\,,$$ with $0<\upsilon<1$. Then, the Eq. can be recast as $$\label{eq:prec7}
\Delta \varphi \equiv \varphi(r_+)-\varphi(r_-) = \eta \int_0^1 g(u_0,u_0+\eta,\lambda, \delta, u_0+\eta\upsilon)d\upsilon\,,$$ where $$g(u_0,u_0+\eta,\lambda, \delta, u_0+\eta\upsilon) = \frac{1}{\sqrt{G(u_0,u_0+\eta,\lambda, \delta, u_0+\eta\upsilon)}}.$$
Finally, defining the auxiliary variable $\xi\equiv(1+\delta)\lambda^2$, we find
$$\begin{aligned}
\label{eq:precYuk}
\Delta \varphi &= \pi\sqrt{1+\frac{2 \delta }{-3 \delta +2 u_0^2 \xi }} \biggl(1 -\frac{2 u_0 \delta \xi}{3 \delta ^2-8 u_0^2 \delta \xi +4 u_0^4 \xi^2} \eta
+ \frac{ \delta \left(-3 \delta ^3+16 u_0^2 \delta ^2 \xi -124 u_0^4 \delta \xi^2+144 u_0^6 \xi ^3\right)}{16 \left(3 u_0 \delta ^2-8 u_0^3 \delta \xi +4 u_0^5 \xi^2\right)^2}\eta ^2\biggr)\end{aligned}$$
To bound the strength $\delta$, we use the motion of the Solar system’s planets. Specifically, we use Mercury, Venus, Earth, Mars, Jupiter and Saturn for which the orbital precession has been measured [@Nyambuya2010]. We identify the allowed region of $\delta$ for which the predicted precession does not contradict the data. In Figure \[fig4\] we show the allowed zone of parameter space for each planet (light blue shades), and we also show that for $-1<\delta<0$ the precession in ongoing in the opposite direction with respect the observed one, while $\delta>0$ give rise to a precession in the right direction confirming the results found for $R^n$ gravity using the S-stars orbiting around the Galactic Center [@Borka20012; @Borka20013]. This results was rather expected since the effect produced by the modification of the gravitational potential must be greater or lower than the Newtonian one that is zero. The scale length has been fixed to the confidence value $\lambda=5000$ AU [@Zakharov2016], and a lower and upper limit on $\delta$ is inferred, and reported in Table \[tab:1\]. The tightest interval on $\delta$ is obtained with Saturn that is located at the highest distance from the Sun. This restricts $\delta$ to vary in the range from $2.70\times10^{-9}$ to $6.70\times10^{-9}$. With these values of the strength we have also predicted the precession for Uranus, Neptune and Pluto[^1], and we found a precession up to three order of magnitude larger than the one predicted by General Relativity meaning that the strength must be even smaller than $fews \times 10^{-9}$ to match the general relativistic constraints. All results are summarized in Table \[tab:2\].
{width="8.9cm"} {width="8.9cm"}\
{width="8.9cm"} {width="8.9cm"}\
{width="8.9cm"} {width="8.9cm"}\
[l c c c c c c c]{}\
&\
&\
**Planet** & & & $i$ & $\epsilon$ & $\dot{\omega}_{obs}$ & $\dot{\omega}_{GR}$ & $[\delta_{min}; \delta_{max}]$\
& ([AU]{}) & ([yrs]{}) & (degrees) & &\
\
[**Mercury**]{} & ${0.39}$ & ${0.24}$ & ${7.0}$ & ${0.206}$ & $43.1000\pm0.5000$ & $43.5$ & $[1.02;\, 1.09]\times10^{-2}$\
[**Venus**]{} & ${0.72}$ & ${0.62}$ & ${3.4}$ & ${0.007}$ & $8.0000\pm5.0000$ & $\,\,\,8.62$ & $[-0.76;\, 2.51]\times10^{-3}$\
[**Earth**]{} & ${1.00}$ & ${1.00}$ & ${0.0}$ & ${0.017}$ & $5.0000\pm1.0000$ & $\,\,\,3.87$ & $[1.45;\, 5.79]\times10^{-4}$\
[**Mars**]{} & ${1.52}$ & ${1.88}$ & ${1.9}$ & ${0.093}$ & $1.3624\pm0.0005$ & $\,\,\,1.36$ & $[5.90;\, 5.92]\times10^{-5}$\
[**Jupiter**]{} & ${5.20}$ & ${11.86}$ & ${1.3}$ & ${0.048}$ & $0.0700\pm0.0040$ & $\,\,\,0.0628$ & $[0.92;\, 1.30]\times10^{-7}$\
[**Saturn**]{} & ${9.54}$ & ${29.46}$ & ${2.5}$ & ${0.056}$ & $0.0140\pm0.0020$ & $\,\,\,0.0138$ & $[2.70;\, 6.70]\times10^{-9}$\
\
\
[l c c c c c c]{}\
&\
&\
**Planet** & & & $i$ & $\epsilon$ & $\dot{\omega}_{GR}$ & $[\dot{\omega}_{min}; \dot{\omega}_{max}]|$\
& ([AU]{}) & ([yrs]{}) & (degrees) &\
\
[**Uranus**]{} & ${19.2}$ & ${84.10}$ & ${0.8}$ & ${0.046}$ & $\,\,\,0.0024$ & $[0.05;\,0.12]$\
[**Neptune**]{} & ${30.1}$ & ${164.80}$ & ${1.8}$ & ${0.009}$ & $\,\,\,0.00078$ & $[0.18;\,0.45]$\
[**Pluto**]{} & ${39.4}$ & ${247.70}$ & ${17.2}$ & ${0.250}$ & $\,\,\,0.00042$ & $[0.11;\,0.30]$\
\
\
Implications for $f(R)$ gravity {#cinque}
===============================
To make compatible $f(R)$ models with local gravity constraints, these theories usually require a “screening mechanism”. When considering theories with a non-minimally coupled scalar field, one has to impose strong conditions on the effective mass of the scalar field that must depend on the space-time curvature or, alternatively, on the matter density distribution of the environment [@Khoury2004; @Khoury2009]. Thus, the scalar field can have a short range at Solar System scale escaping the experimental constraints, and have a long range at the cosmological scale, where it can propagate freely affecting the cosmological dynamics, and driving the accelerated expansion (see for details [@defelice2010]). With the same aim, similar mechanisms have been proposed for other models, such as the symmetron and the braneworld [@Dvali2000; @Nicolis2009; @Hinterbichler2010]. [ Nevertheless, these mechanisms are introduced [*ad hoc*]{} and particularized for each theory. In $f(R)$ gravity, the need of introducing a screening mechanism arises when, instead of working with higher order field equations, one performs a conformal transformation from the Jordan to the Einstein frame, where the field equations are of second order but a scalar field, related to the $f'(R)$ term, appears. Although it is simpler to work with second order field equations, and the two frames are mathematically equivalent, one should remember that the physical equivalence is not guaranteed in general [@Magnano; @Faraoni; @darkmetric]. Thus, one could prefer to work with high order field equations, staying in the Jordan frame, and handling the extra degrees of freedom as free parameters to be constrained by the data. In such a case, the scale dependence of these parameters plays the role of the screening mechanism. The screening mechanism is traced by the density of the self gravitating systems [@chameleon].]{}
Relatively, the results in Table \[tab:1\] can be straightforwardly interpreted as the fact that the Yukawa correction term to the Newtonian gravitational potential is screened at planetary scales. Indeed, the departure from Newtonian gravity is of the order of $10^{-9}$ in $\delta$. Finally, the values of the strength and the scale of the Yukawa potential highly degenerate at such small scales. To illustrate this degeneracy we have computed the lower and upper limit on $\delta$ varying $\lambda$ from 100 AU to $10^{4}$ AU. The results are shown in Fig. \[fig5\], where we have highlighted the parts of the parameter space that are (and are not) allowed. We show that a change of one order of magnitude in the scale length is reflected in change up to two order of magnitude in $\delta$. The plot is particularized for Saturn.
![Degeneracy between the strength and the scale length of the Yukawa gravitational potential. The plot is particularized for the case of Saturn.[]{data-label="fig5"}](fig5.eps){width="8.6cm"}
Conclusions and Remarks {#sei}
=======================
Measurements of the orbital precession of Solar System bodies can be used to compare observations with theoretical predictions arising from alternative theories of gravity. Specifically, $f(R)$ gravity models that, in their weak field limit, show a Yukawa-like correction to the Newtonian gravitational potential can be used to compute the orbital precession with a classical mechanics approach. We have computed an analytical expression for the orbital precession and compared its prediction with the values for the Solar System’s planets. We found that, fixing the characteristic scale length to $\lambda=5000$ AU [@Zakharov2016], the strength must rely in the range $[2.70; 6.70]\times10^{-9}$. Nevertheless, we must point out the presence of a degeneracy between the strength and the scale of the Yukawa potential. We find the direction of the orbital precession changing with the sign of the strength, confirming previous results [@Borka20012; @Borka20013]. If the change of the direction of the orbital precession can be used as an effective way to discriminate between General Relativity and its alternative, should be studied in a full relativistic approach where the motion happens along the geodesics [@pII].
Acknowledgements {#acknowledgements .unnumbered}
================
I.D.M acknowledge financial supports from University of the Basque Country UPV/EHU under the program “Convocatoria de contratación para la especialización de personal investigador doctor en la UPV/EHU 2015”, from the Spanish Ministerio de Economía y Competitividad through the research project FIS2017-85076-P (MINECO/AEI/FEDER, UE), and from the Basque Government through the research project IT-956-16. M.D.L. is supported by the ERC Synergy Grant “BlackHoleCam” – Imaging the Event Horizon of Black Holes (Grant No. 610058). M.D.L. acknowledge INFN Sez. di Napoli (Iniziative Specifiche QGSKY and TEONGRAV). This article is based upon work from COST Action CA1511 Cosmology and Astrophysics Network for Theoretical Advances and Training Actions (CANTATA), supported by COST (European Cooperation in Science and Technology).
[99]{}
Planck Collaboration, [Astron. Astrophys.]{} 594, A13 (2016).
G. Bertone, D. Hooper, J. Silk, Phys. Rept., 405, 279 (2005)
A. Capolupo, Advances in High Energy Physics, 2016, 8089142 (2010)
Hsi-Yu Schive, T. Chiueh, T. Broadhurst, Nature Physics, 10, 7, 496-499 (2014)
I. De Martino, T. Broadhurst, S.-H. H. Tye, T. Chiueh, Hsi-Yu Schive, R. Lazkoz, , 119, 221103 (2017)
I. De Martino, T. Broadhurst, S.-H. H. Tye, T. Chiueh, Hsi-Yu Schive, R. Lazkoz, Galaxies, 6, 10 (2018)
I. Lopes, G. Panotopoulos (2018) arXiv:1801.05031
G. Panotopoulos, I. Lopes, (2018) arXiv:1801.03387
J.L. Feng, Ann. Rev. Astron. Astrophys. 48, 495, (2010).
J. W. Moffat, J. Cosmol. Astropart. Phys. 0603, 004 (2006).
I. de Martino, M. De Laraurentis, [Phys. Lett. B]{} [770]{}, 440 (2017)
S. Capozziello, M. De Laurentis, Phys. Rept. 509, 167 (2011).
Y.-F. Cai, S. Capozziello, M. De Laurentis, E. N. Saridakis, Rept. Prog. Phys. [**79**]{}, 106901 (2016).
S. Nojiri and S. D. Odintsov, Phys. Rept. [**505**]{}, 59 (2011)
C. Bogdanos, S. Capozziello, M. De Laurentis and S. Nesseris, Astropart. Phys. [**34**]{} 236 (2010).
M. De Laurentis, O. Porth, L. Bovard, B. Ahmedov and A. Abdujabbarov, Phys. Rev. D [**94**]{} no.12, 124038 (2016).
S. Bellucci, S. Capozziello, M. De Laurentis and V. Faraoni, Phys. Rev. D [**79**]{}, 104004 (2009)
S. Capozziello, M. De Laurentis, S.Nojiri, S.D. Odintsov,Phys. Rev. D [**95**]{} 083524 (2017)
S. Capozziello, M. De Laurentis, Annalen der Physik 524, 545 (2012).
I. Quandt, H. J. Schmidt, Astron. Nachr., 312, 97 (1991).
K. Lee, F. A. Jenet, R. H. Price, N. Wex, M. Kramer 722, 1589-1597 (2010)
B. P. Abbott et al. (LIGO Scientific and Virgo Collaboration) 118, 221101 (2017)
S. Capozziello, E. De Filippis, V.Salzano, [Mon. Not. R. Astron. Soc.]{}[**394**]{}, 947 (2009)
I. de Martino, M. De Laurentis, F. Atrio-Barandela, S. Capozziello, [Mon. Not. R. Astron. Soc.]{} 442, 2, 921-928 (2014)
I. de Martino, M. De Laurentis, S. Capozziello, Universe 1, 123 (2015)
I. de Martino, 93, 124043 (2016)
C. Talmadge, J.-P. Berthias, R. W. Hellings and E. M. Standish, 61, 1159 (1988).
J. D. Anderson, P. A. Laing, E. L. Lau, A. S. Liu, M. M. Nieto, and S. G. Turyshev, 81, 2858 (1998).
J. D. Anderson, P. A. Laing, E. L. Lau, A. S. Liu, M. M. Nieto, and S. G. Turyshev, 65, 082004 (2002).
D. Borka, P. Jovanović, V. Borka Jovanović and A. F. Zakharov, 85 124004 (2012)
D. Borka, P. Jovanović, V. Borka Jovanović, A. F. Zakharov, [J. Cosmol. Astropart. Phys.]{} 11 050 (2013).
A.F. Zakharov and P. Jovanović and D. Borka and V. Borka Jovanović, [J. Cosmol. Astropart. Phys.]{} 05,045, (2016)
A. Hees, T. Do, A. M. Ghez, G. D. Martinez, S. Naoz, E. E. Becklin, A. Boehle, S. Chappell, D. Chu, A. Dehghanfar, et al. 118, 211101, (2017)
A. F. Zakharov, P. Jovanović, D. Borka, V. Borka Jovanović, (2018) arXiv:1801.04679
L. Iorio, [Class. Quant. Grav.]{}, 22, 5271 (2005); L. Iorio, High Energy Physics, 2012, 73, (2012); L. Iorio, Phys. Lett. A, 298, 315 (2002);L. Iorio, Planetary and Space Science, 55, 1290 (2007).
M. De Laurentis, I. de Martino, [Mon. Not. R. Astron. Soc.]{} 431, 1, 741 (2013)
M. De Laurentis, I. de Martino, Int. J. Goem. Math. Meth., 1250040D (2015)
S. Lee, (2017) arXiv:1711.09038.
C. Will, Living Reviews in Relativity, 17 (2014)
L. Iorio, Astron. J. 137, 3 (2009)
J. N. Islam, Phys. Lett. A 97, 239 (1983).
L. Iorio, Int. J. Mod. Phys. D 15, 473 (2006).
M. Sereno and P. Jetzer, 73, 063004 (2006).
Ø. Grøn and H. H. Soleng, 456, 445 (1996).
I. B. Khriplovich and E. V. Pitjeva, Int. J. Mod. Phys. D 15 , 615 (2006).
S. Capozziello, S. De Martino, S. De Siena, and F. Illuminati, Mod. Phys. Lett. A 16, 693 (2001).
R. H. Sanders, [Mon. Not. R. Astron. Soc.]{} 370 , 1519 (2006).
J. B. R. Battat, C. W. Stubbs, J. F. Chandler, 78, 022003, (2008)
G. G. Nyambuya, [Mon. Not. R. Astron. Soc.]{} 403, 1381 (2010)
H. Özer, Ö. Delice, arXiv:1708.05900 (2017)
Tan Liu, Xing Zhang, Wen Zhao [Phys. Lett. B]{}B 777 286-293 (2018)
M. De Laurentis, The Open Astronomy Journal, 4, 108-150 (2011)
J. Khoury, and A. Weltman, 93, 171104, (2004).
J. Khoury, and M. Wyman, 80, 064023, (2009).
A. de Felice, and S. Tsujikawa, Living Reviews in Relativity, 13, 3, (2010).
G. Dvali, G. Gabadadze, M. Porrati, [Phys. Lett. B]{} 485, 208-214, (2000).
A. Nicolis, R. Rattazzi, E. Trincherini, 79, 064036, (2009).
K. Hinterbichler, J. Khoury , , 104, 231301, (2010).
M. De Laurentis, I. De Martino, R. Lazkoz, arXiv:1801.08136 (2017).
G. Magnano, L. Sokolowski, Phys. Rev. D, [**50**]{} 5039, (1994) V. Faraoni, E. Gunzig, Int.J.Theor.Phys., 38:217?225 (1999) S. Capozziello, M. De Laurentis, M. Francaviglia, S. Mercadante, Foundations of Physics [**39**]{}, 1161 (2009). S. Capozziello, S. Tsujikawa , Phys. Rev. D, [**77**]{},107501 (2008).
[^1]: Although the latter is not a planet, its large distance from the Sun and its small mass makes the object very useful to show the impact of the modified gravitational potential.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We construct examples of countable linear groups $\Gamma < {{\operatorname{SL}}}_n({{\mathbf{R}}})$ with no non-trivial normal abelian subgroup that admit a faithful, sharply $2$-transitive action on a set. The stabilizer of a point in the this action does not contain an involution.'
author:
- 'Yair Glasner and Dennis D. Gulko'
bibliography:
- 'yair.bib'
title: |
Non-split linear sharply 2-transitive groups,\
after Rips-Segev-Tent
---
Introduction
============
A sharply $2$-transitive group is, by definition, a permutation group $\Gamma \curvearrowright X$ which acts transitively and freely on ordered pairs of distinct points. Such a group is called [*[split]{}*]{} if it admits a non-trivial normal abelian subgroup. The following question, that has attracted the attention of algebraists for many years, was recently answered negatively, by Rips Segev and Tent in [@RST:s2t]
\[q:main\] Is every sharply $2$-transitive group split?
In Theorem \[thm:s2t\] we show that the answer remains negative even in the setting of countable linear groups. This contrasts nicely with the prior results of [@GlGu; @GMS:s2t] that show that the answer to the same question is positive for linear groups when the permutational characteristic of $\Gamma$ is not $2$; or in other words under the additional assumption that involutions in $\Gamma$ fix a point.
Splitting implies a tame, algebraic, structure theory. In particular with every split sharply $2$-transitive group $\Gamma$ one can associate a near field $N$, which is by definition a division ring that is distributive only from the right[^1]. In this case $\Gamma$ indeed [*[splits]{}*]{} as a semi-direct product $\Gamma= N^* \ltimes N$ of the multiplicative and additive groups of the near field. Moreover the given sharply $2$-transitive action is the unique faithful primitive permutation representation of this group and it is isomorphic to the natural action $N^* \ltimes N \curvearrowright N$ by affine transformation $x^{(a,b)} = x\cdot a+b$.
When $\Gamma \curvearrowright X$ is sharply $2$-transitive there exists an element flipping any two points in $X$, whose square must be trivial. This gives rise to a large set of involutions ${{\operatorname{Inv}}}(\Gamma) \subset \Gamma$. Such a pair of points determines the involution and since $\Gamma$ is transitive on pairs of points all involutions are conjugate. Any nontrivial element, and in particular any involution, can have either $0$ or $1$ fixed points. If an involution does fix a point then the map ${{\operatorname{Inv}}}(\Gamma) {\rightarrow}X$ taking an involution to its fixed point is a $\Gamma$ invariant bijection. Consequently the $\Gamma$ action on ${{\operatorname{Inv}}}(\Gamma)$ by conjugation is $2$-transitive and in particular, the order of the product of two different involutions is independent of the choice of the specific involutions. This gives rise to the following definition:
Let $\Gamma$ be a sharply $2$-transitive permutation group on $X$. If the stabilizer of a point contains an involution let $p = {{\operatorname{Ord}}}(\sigma \tau)$ be the order of the product of two distinct involutions. We define the [*[permutational characteristic of $\Gamma$]{}*]{} to be $${{{\it {p}}\operatorname{-char}}}(\Gamma)=\left\{ \begin{array}{ c l }
2\hspace{8pt} & {\text{Involutions do not fix a point}}\\
p \hspace{8pt} & p < \infty \\
0\hspace{8pt} & p =\infty
\end{array}\right.$$
It is not difficult to verify that ${{{\it {p}}\operatorname{-char}}}(\Gamma)$ is either $0$ or prime, and that it coincides with the characteristic of the near field whenever $\Gamma$ splits. We refer the readers to [@Kerby:mst], [@GlGu] for more details.
In two papers [@Z1:finite_s2t; @Z2:finite_s2t] from 1936 H. Zassenhaus completed a full classification of finite sharply $2$-transitive groups. He started by showing that every such group splits, and then gave a complete classification of finite near fields. Contrary to the situation for skew fields, the latter classification involves non-trivial examples of finite near fields. In [@T1:cont_s2t; @T2:cont_s2t] Tits proved that every locally compact connected sharply $2$-transitive group splits. The near fields here are just ${{\mathbf{R}}},{{\mathbf{C}}}$ or $\mathbf{H}$. In [@Turk:s2t_char3] Türkelli proved that every sharply $2$-transitive group $\Gamma$ with ${{{\it {p}}\operatorname{-char}}}(\Gamma)=3$ splits. In [@GlGu; @GMS:s2t] it was shown that every linear sharply $2$-transitive group $\Gamma < {{\operatorname{GL}}}_n(k)$ with ${{{\it {p}}\operatorname{-char}}}(\Gamma) \ne 2$ splits. This was shown by us under the additional assumption that $\operatorname{char}(k) \ne 2$, an assumption that was deemed unnecessary by [@GMS:s2t] who also relaxed the linearity assumption.
Let $\Delta \curvearrowright Y$, $\Gamma \curvearrowright X$ be two actions of groups on sets. An [*[embedding]{}*]{} of such actions $(\phi,\iota): (\Delta \curvearrowright Y) {\hookrightarrow}(\Gamma \curvearrowright X)$ is a monomorphism $\phi: \Delta {\rightarrow}\Gamma$ and a $\phi$-equivariant injective map $\iota: Y {\rightarrow}X$.
Such an embedding is obtained, in particular, whenever $\Delta < \Gamma$ is a subgroup and there is a $\Delta$-orbit in $Y$ that is permutation isomorphic to the given action of $\Delta$ on $X$. At the group theoretic level if the two actions are identified with coset actions $Y = A {\backslash}\Delta, X = B {\backslash}\Gamma$ then we are just looking at an embedding $\phi: \Delta < \Gamma$ such that $A = \Delta \cap B$.
Recently the first examples of non-split sharply $2$-transitive groups were given by Rips-Segev-Tent in [@RST:s2t]. Their examples are very general in the sense that they show that every transitive group action on a set $G \curvearrowright X$ with the properties that the stabilizer of a point does not contain an involution and the stabilizer of every pair of distinct points is trivial can be embedded into a non-split sharply $2$-transitive action of permutational characteristic $2$. Clearly these two conditions are also necessary for such an embedding.
Our main theorem comes to show that such non-split sharply $2$-transitive groups of permutational characteristic $2$ can be constructed even within the realm of linear groups. Our main theorem is:
\[thm:s2t\] (Linear s-2-t groups) Let $H < {{\operatorname{SL}}}_n({{\mathbf{R}}})$ be a countable group that contains neither involutions nor nontrivial scalar matrices. Assume that $H \curvearrowright X$ is a transitive permutation action with the property that the stabilizer of every pair of distinct points is trivial. Then there exists a larger countable group $H < H_1 < {{\operatorname{SL}}}_n({{\mathbf{R}}})$ which admits a sharply $2$-transitive non-split permutation representation $H_1 \curvearrowright X_1$ of permutational characteristic $2$ and an embedding: $$(H \curvearrowright X) {\hookrightarrow}(H_1 \curvearrowright X_1).$$
In some examples the resulting group $H_1$ admits uncountably many non-isomorphic faithful primitive permutation representations $H_1 \stackrel{\rho_{\alpha}}{\curvearrowright} X_{\alpha}$.
The last statement of the theorem should be contrasted with the well known fact that a split sharply $2$-transitive group admits a unique (up to isomorphism of permutations representations) faithful primitive permutation representation. Thus the nonsplit examples of [@RST:s2t] are the first natural candidates for sharply $2$-transitive groups that admit multiple faithful primitive actions. In order to actually construct such examples linearity comes in handy. We appeal to the results of [@GG:AOS] which ensure that any Zariski dense countable subgroup of ${{\operatorname{SL}}}_n({{\mathbf{R}}})$ with trivial center admits uncountably many non-isomorphic faithful primitive actions. But linearity is not essential here. Using the methods of [@RST:s2t], it is possible to construct groups that admit many faithful non-isomorphic sharply $2$-transitive actions! This will be shown in subsequent paper.
Reductions
==========
Given an involution $t \in {{\operatorname{SL}}}_n({{\mathbf{R}}})$ let ${{\mathbf{R}}}^n = W^+(t) \oplus W^-(t)$ denote the decomposition of ${{\mathbf{R}}}^n$ into its $\pm 1$ eigenspaces and ${\pi}_t = \frac{1 + t}{2} :{{\mathbf{R}}}^n {\rightarrow}W^{+}(t)$ the projection on $W^{+}(t)$ along $W^{-}(t)$. When possible, we will omit $t$ from the notation writing $W^{\pm}$ instead of $W^{\pm}(t)$. Our main technical theorem is the following, linear version of [@RST:s2t Theorem 1.1]:
\[thm:tec\] Let $G < {{\operatorname{SL}}}_n({{\mathbf{R}}})$ be a countable group, $t \in G$ an involution and $A < G$ a malnormal subgroup containing no involutions. Assume further that all the involutions in $G$ are conjugate (in $G$), that $\dim(W^{+}(t)) \ge \dim(W^{-}(t))+2$, and whenever $W^{+}(t)$ is contained in an eigenspace of ${\pi}_t \circ g$ for some $g \in G$ then either $g =1$ or $g =t$.
Then for any element $v \in G \setminus A$ there exists a countable extension $G \le G_1 < {{\operatorname{SL}}}_n({{\mathbf{R}}})$ with a malnormal subgroup $A_1 < G_1$ containing no involutions, such that $A_1 \cap G = A$ and an element $f \in A_1$ such that $A_1 t f = A_1 v$. Moreover all the involutions in $G_1$ are conjugate and the only elements of $G_1$ such that $W^{+}(t)$ is contained in an eigenspace of ${\pi}_t \circ g$ are $g =1$ and $g =t$.
The significant difference between this theorem and [@RST:s2t Theorem 1.1] is the linearity requirement $G,G_1 < {{\operatorname{SL}}}_n({{\mathbf{R}}})$. As we plan to use this theorem infinitely many times within an induction argument, it is particularly important for us that $G_1$ is realized as a linear group within the same ambient matrix group as the original group $G$. A few “auxiliary requirements” become necessary in order to prove the theorem in the linear setting: That $G$ be countable, with conjugate involutions and that if $g \not \in \{1,t\}$ then $W^+(t)$ is not contained in an eigenspace of ${\pi}_t \circ g$.
In the proof we follow the strategy of [@RST:s2t]. For the convenience of the reader we quote here three of their propositions. The first proposition reduces the main theorem to two special cases.
\[prop:hyp\] It is enough to prove Theorem \[thm:tec\] under the additional assumption that $v,v^{-1} \not \in AtA$ and either $v^{-1} \not \in AvA$ or $v$ is an involution.
See [@RST:s2t Section 2], the exact same reduction works for us here.
The next two theorems treat these two cases respectively:
\[thm:RST\_free\] ([@RST:s2t Theorem 3.1]) Let $G$ be a group, $A < G$ a malnormal subgroup containing no involutions, $t,v \in G$ two elements with $t$ an involution and $v,v^{-1} \not \in AtA$ and $v^{-1} \not \in AvA$. If $A$ already contains an element $f$ such that $Atf=Av$ then take $G_1 =G, A_1 =A, f=f$. Otherwise set $$G_1 := G * \langle f \rangle, \qquad A_1 = \langle A, f, tfv^{-1} \rangle.$$ Where $\langle f \rangle = {{\mathbf{Z}}}$.
Then $A_1$ is malnormal in $G_1$, $A_1tf=A_1v$ and $G \cap A_1 = A$.
\[thm:RST\_HNN\] [@RST:s2t Theorem 4.1] Let $G$ be a group, $A < G$ a malnormal subgroup containing no involutions, $t,v \in G$ two involutions such that $v,v^{-1} \not \in AtA$. If $A$ already contains an element $f$ such that $Atf=Av$ then take $G_1 =G, A_1 =A, f=f$. Otherwise set $$G_1= \langle G,f \ | \ f^{-1}tf=v \rangle, \qquad A_1 := \langle A, f \rangle.$$
Then $A_1$ is malnormal in $G_1$, $A_1tf = A_1 v$ and $G \cap A_1 = A$.
The contribution of the current paper is in showing that in the setting of the above two theorems; when the group $G < {{\operatorname{SL}}}_n({{\mathbf{R}}})$ is countable linear, and subject to all of the auxiliary conditions described in Theorem \[thm:tec\], then $G_1$ is still linear. In fact $G_1$ is still contained in the same ambient matrix group ${{\operatorname{SL}}}_n({{\mathbf{R}}})$. This is achieved in Theorems \[thm:free\_prod\] and \[thm:HNN\] appearing in Section \[sec:linearity\]. In turn this observation is enough to carry out the full Rips-Segev-Tent construction inside the ambient matrix group.
Some projective dynamics
========================
All our notation is taken from [@BG:Dense_Free Section 3] and we refer the readers to that paper for more details.
We use extensively the action $SL_n({{\mathbf{R}}}) \curvearrowright {{\mathbb{P}}}:= {{\mathbf{R}}}^n/R^{*}$ on the projective space. If $0 \ne v \in {{\mathbf{R}}}^n, \langle 0 \rangle \ne W < {{\mathbf{R}}}^n$ are a nontrivial vector and subspace in ${{\mathbf{R}}}^n$ we denote the corresponding projective point, and subspace by $[v] \in {{\mathbb{P}}}, [W] < {{\mathbb{P}}}$. Fix a norm ${\left\Vert\cdot\right\Vert}$ on ${{\mathbf{R}}}^n$, this gives rise in to a norm on the exterior product ${{\mathbf{R}}}^n \wedge {{\mathbf{R}}}^n$ which is used in turn to endow the projective space ${{\mathbb{P}}}= {{\mathbf{R}}}^{n}/R^{*}$ with the metric $$d([v],[w]) = \frac{{\left\Vertv \wedge w\right\Vert}}{{\left\Vertv\right\Vert} \cdot {\left\Vertw\right\Vert}}$$ We will denote the $\epsilon$ neighborhood of a set in this metric by $(\Omega)_{\epsilon} = \{x \in {{\mathbb{P}}}\ | \ d(x,\Omega) < \epsilon\}$.
Any nontrivial matrix $0 \ne M \in M_n({{\mathbf{R}}})$ gives rise to a partially defined map $[M]: {{\mathbb{P}}}\setminus [\ker(M)] {\rightarrow}{{\mathbb{P}}}$. A projective point $[v] \in {{\mathbb{P}}}$ is moved by $M$ if and only if $v$ is not an eigenvector of $M$. Indeed $[M]$ is not defined on $[v]$ if and only if $v$ is an eigenvector with eigenvalue $0$ and $[v]$ is a fixed point of $[M]$ if and only if $v$ is an eigenvector with a nonzero eigenvalue. For example our auxiliary condition in Theorem \[thm:tec\] requires that whenever $g \not \in \{1,t\}$ then the matrix ${\pi}_t \circ g = \frac{g + tg}{2}$ should move at least one projective point in $[W^{+}(t)]$. If $M \in {{\operatorname{GL}}}_n({{\mathbf{R}}})$ then the map $[M]$ is defined on the whole projective space and is in fact a bilipschitz homeomorphism with respect to the metric defined above.
\[lem:general\_pos\] Let $B \in {{\operatorname{PGL}}}_r({{\mathbf{R}}})$. If $v_0,v_1,\ldots, v_r < {{\mathbf{R}}}^r$ are $r+1$ vectors in general position that are all eigenvectors of $B$. then $B$ is a scalar matrix.
By definition, $r+1$ vectors in $R^n$ are in [*[general position]{}*]{} if any $i$ of them span an $i$-dimensional subspace as long as $i \le r$. By counting considerations there must be at least one eigenspace $V< {{\mathbf{R}}}^r$ of dimension $l$ containing at least $l+1$ of the eigenvectors $\{v_0,v_1,\ldots, v_r\}$. Since the vectors are in general position then $l=r$, so $B$ has an eigenvalue with $r$-linearly independent eigenvectors and is hence a scalar.
Linearity proofs {#sec:linearity}
================
\[prop:free\_prod\] Let $n \ge 3$ and $G,H < {{\operatorname{SL}}}_n({{\mathbf{R}}})$ be two countable groups that contain no nontrivial scalar matrices. Let ${\pi}: {{\mathbf{R}}}^n {\rightarrow}W$ be a linear projection with $\dim(\Im({\pi})) - \dim(\ker({\pi})) \ge 2$. For $f \in {{\operatorname{SL}}}_n({{\mathbf{R}}})$ let $\Phi_f:G*H {\rightarrow}{{\operatorname{SL}}}_n({{\mathbf{R}}})$ be defined by $\Phi_f(g)=g, \forall g \in G$ and $\phi_f(h) = fhf^{-1}, \ \forall h \in H$.
Then for a Baire generic choice of $f \in {{\operatorname{SL}}}_n({{\mathbf{R}}})$ the map $\Phi_f$ is injective and its image contains no nontrivial scalar matrices. Moreover whenever $\go \not \in G$ then $W$ is not contained in an eigenspace of ${\pi}\circ \Phi_f(\go)$.
Let $\go \in G*H$ and write it as a reduced word $\go = h_1 g_1 \cdot...\cdot h_k g_k,$ with $g_i, \in G, h_i \in H$, all of them nontrivial except possibly $g_k$ and $h_1$. We may (and shall) also assume that $\go \not \in G$ since in this case there is nothing to prove. Set $${{\mathcal{U}}}(\go) ={\left\{f\in{{\operatorname{SL}}}_n({{\mathbf{R}}}): {\pi}\Phi_f(\go) {\text{ is scalar on }} W\right\}}, \qquad \qquad {{\mathcal{U}}}= \bigcap_{\go \in G*H \setminus G} {{\mathcal{U}}}(\go).$$ By Lemma \[lem:general\_pos\] ${{\mathcal{U}}}(\go)$ is Zariski open in ${{\operatorname{SL}}}_n$. Indeed if $\dim(W^{+})=r$ and $\{w_0,w_1,\ldots, w_r \}$ are $r+1$ vectors in general position in $W^{+}$, the complement is characterized by the equations $\{{\pi}\Phi_f (\go) w_i \wedge w_i = 0\}_{0 \le i \le r}$. Since ${{\operatorname{SL}}}_n({{\mathbf{R}}})$ is connected, if we show that ${{\mathcal{U}}}(\go)$ is nonempty it will follow from the Baire category theorem that ${{\mathcal{U}}}$ is nonempty and the theorem will be proved.
Choose a basis $\{v_1,v_2,\ldots, v_n\}$ for ${{\mathbf{R}}}^n$ a vector $v \in W$ and a number $L >> 0$. Let $a^{+} = [v_1]$, $a^{-} = [v_2]$, $x = [v]$, $H^{+} = [{{\operatorname{Span}}}\{v_2,v_3,\ldots, v_n\}]$ and $H^{-} = [{{\operatorname{Span}}}\{v_1,v_3,\ldots, v_n\}]$ three projective points and two projective hyperplanes corresponding to these vectors. Define $f = f(L) \in {{\operatorname{SL}}}_n({{\mathbf{Z}}})$ by the requirements: $$f v_1 = L v_1, \qquad f v_2 = \frac{1}{L} v_2, \qquad f v_i = v_i , \ \ \ \forall 3 \le i \le n.$$ The dynamics of $f$ on ${{\mathbb{P}}}$ is proximal in the sense that $\lim_{L {\rightarrow}\infty} f(L)^{\pm 1}(x) = a^{\pm}, \forall x \not \in H^{\pm}$ respectively. Note that $${\pi}\Phi_f(\go)(x) = \pi f h_1 f^{-1} g_1 f h_2 f^{-1} \ldots f h_k f^{-1} g_k (x).$$ Consider the sequence of points $f^{-1} g_k(x)$, $f h_k f^{-1} g_k(x)$, up to $f h_1 f^{-1} g_1 f \ldots f h_k f^{-1} g_k (x)$. Our strategy is to choose all of the above data in such a way that these points alternate, coming arbitrarily close to $a^{-}$ and $a^{+}$ respectively when $L {\rightarrow}\infty$. If we insist also that $x \ne {\pi}(a^{+})$ (or that $x \ne {\pi}g_1 (a^{+})$ when $h_1 = 1$) it will follow that when $L$ is large enough ${\pi}\Phi_f(\go)$ does not fix the projective point $x \in W$ and the claim will follow. To carry this out we require the following:
1. $g_k(x) \not \in H^{-}$ (in particular $x \not \in H^{-}$ if it so happens that $g_k=1$)
2. $g_i (a^{+}) \not \in H^{-}, {\text{ for }} 1 \le i < k$
3. $h_i (a^{-}) \not \in H^{+}, {\text{ for }} 1 < i \le k$
4. ${\pi}(a^{+}) \ne x,$ if $h_1 \ne 1$ and ${\pi}g_1 (a^{+}) \ne x$ otherwise
Since $\{g_1,g_2,\ldots, g_{k-1}\}$ are by assumption non scalar matrices we can choose $v_1$ which is not an eigenvector of any of them. $v_2$ is chosen to be linearly independent of $v_1$ and not an eigenvector of any of the matrices $\{h_2,\ldots,h_k\}$. Now choose the rest of the basis $\{v_3,\ldots,v_n\}$ in such a way that $H^{\pm}$ stay away from the finitely many points as required in the second and third conditions. In addition we choose these hyperplanes in such a way that $g_k([W^{+}]) \not \subset H^{+}$ in order to make the first condition possible. Now choose $v \in W$ so as to actually satisfy the first and fourth conditions. The latter is always possible by our assumption that $\dim(W) > \dim(\ker({\pi}))+1 = \dim {\pi}^{-1}({{\operatorname{Span}}}\{v_1\}) = \dim {\pi}^{-1}({{\operatorname{Span}}}\{g_1 v_1\})$.
It is clear by all these choices, arguing by induction on $m$, that $$\lim_{L {\rightarrow}\infty} {\pi}\Phi_f(L)(\go)(x) =
\left \{
\begin{array}{ll}
{\pi}(a^{+}) \ne x, \ & {\text{if }} h_1 \ne 1 \\
{\pi}g_1 (a^{+}), \ & {\text{if }} h_1 = 1
\end{array} \right \}
\ne x$$ Which concludes the proof.
Similar linearity statements for free products are well known and established for example in [@shalen:free]. However since we didn’t find the exact statement we needed we included the complete proof.
The previous theorem gives rise to our linear version of Theorem \[thm:RST\_free\]
(Free product) \[thm:free\_prod\] Under the assumptions of Theorem \[thm:tec\] set $G_1 = G*{{\mathbf{Z}}}$ where the group ${{\mathbf{Z}}}= \langle f \rangle$. Then there exists an element of infinite order $\ell \in {{\operatorname{SL}}}_n({{\mathbf{C}}})$ such that the natural map $\Phi_{\ell}: G_1 {\rightarrow}{{\operatorname{SL}}}_n({{\mathbf{R}}})$ defined by $g \mapsto g, \forall g \in G$ and $f \mapsto \ell$ is an isomorphism onto its image $\langle G, \ell \rangle < {{\operatorname{SL}}}_n({{\mathbf{R}}})$. Moreover all the involutions in $G_1$ are conjugate, and the only elements of $G_1$ such that $W^{+}(t)$ is contained in an eigenspace of ${\pi}_t \circ \Phi_{\ell}(g)$ are $g =1$ and $g =t$.
(Of Theorem \[thm:free\_prod\]) Let $H$ be any infinite cyclic group in ${{\operatorname{SL}}}_n({{\mathbf{R}}})$ and $G < {{\operatorname{SL}}}_n({{\mathbf{R}}})$ the given group. Applying Proposition \[prop:free\_prod\] to these groups $G,H$ yields everything we need leaving only the verification of the fact that all involutions in $G*H$ are conjugate. Let $\sigma \in G*{{\mathbf{Z}}}$ be any involution. Being an element of finite order it must stabilize a vertex in the Bass-Serre tree. Since ${{\mathbf{Z}}}$ contains no involutions $\sigma$ is conjugate into $G$ and by our assumption all involutions in $G$ are already conjugate.
And here is our linear version of Theorem \[thm:RST\_HNN\]
(HNN extension) \[thm:HNN\] Under the assumptions of Theorem \[thm:tec\] let $t \ne v \in G$ be an involution and set $G_1= \langle G,f \ | \ f^{-1}tf=v \rangle$ and $A_1 := \langle A, f \rangle.$ There exists an element of infinite order $\ell \in {{\operatorname{SL}}}_n({{\mathbf{C}}})$ such that $\ell^{-1} t \ell = v$ and such that the natural map $\Phi_{\ell}: G_1 {\rightarrow}{{\operatorname{SL}}}_n({{\mathbf{R}}})$ defined by $g \mapsto g, \forall g \in G$ and $f \mapsto \ell$ is an isomorphism onto its image. Moreover all the involutions in $G_1$ are conjugate and the only elements of $G_1$ such that $W^{+}(t)$ is contained in an eigenspace of ${\pi}_t \circ g$ are $g =1$ and $g =t$.
(Of Theorem \[thm:HNN\]). Assume that we are in the setting of that theorem and $t,v \in G$ are the two given involutions. Conjugating $G$, if necessary, we may assume that $t = {{\operatorname{Diag}}}{\left(1,...,1,-1,...,-1\right)}$ is a diagonal matrix with $r$ ones and $n-r$ minus-ones along the diagonal. If $e_1,...,e_n$ denote the vectors of the standard basis then $W^{+}:=W^{+}(t) = \langle e_1, e_2, \ldots,e_r \rangle,$ and $W^{-}:=W^{-}(t) =\langle e_{r+1},e_{r+2}, \ldots, e_n \rangle$ are, respectively, the $\pm 1$ eigenspaces of $t$. By our assumption $r > n-r$ which immediately implies also that $r\ge 2$.
Recall that $G_1 = \langle G,f \ | \ f^{-1}tf=v\rangle$, let $G_2 = \langle G,k \ | \ k^{-1}tk=t\rangle$. It follows from a standard argument involving the universal properties of these two HNN extensions that they are isomorphic. Indeed by assumption all the involutions in $G$ are already conjugate within $G$ so there is an element $h \in G$ such that $h^{-1}th = v$. Let $F: G_1 {\rightarrow}G_2$ be the homomorphism defined by the requirement that $F(g) = g, \forall g \in G, F(f) = k h$ and let $I: G_2 {\rightarrow}G_1$ be the homomorphism defined by the requirement $I(g)=g, \forall g \in G, I(k) = f h^{-1}$. Since $I \circ F$ fixes pointwise both $G$ and $f$ it must be the identity of $G_1$ and similarly $F \circ I$ is the identity of $G_2$. We will hence identify these two groups and in particular we will identify $G_1$ as the HNN extension $G_1 = \langle G,k \ | \ k^{-1}tk=t \rangle$ where $k = fh^{-1}$.
Let $Z = {{\operatorname{SL}}}(W^{+}) \times {{\operatorname{SL}}}(W^{-}) < C_{{{\operatorname{SL}}}_n({{\mathbf{R}}})}(t)$. This group is isomorphic to ${{\operatorname{SL}}}_{r}({{\mathbf{R}}}) \times {{\operatorname{SL}}}_{n-r}({{\mathbf{R}}})$ and in particular it is a connected closed subgroup of ${{\operatorname{SL}}}_n({{\mathbf{R}}})$. For any $u \in Z$ we obtain a homomorphism. $\Phi_u : G_1 {\rightarrow}{{\operatorname{SL}}}_n({{\mathbf{R}}})$ given by $g \mapsto g, \forall g \in G$ and $k \mapsto u$.
Our goal is to find some $u \in Z$ such that for every $w \in G_1, \ w \not \in \{1,t\}$ the element $\Phi_u(w)$ does not fix $[W^{+}]$ pointwise. This will show that $\Phi_u: G_1 {\rightarrow}\langle G, u \rangle$ is an isomorphism and at the same time it will establish the auxiliary requirement that the only elements of $G_1$ to fix $[W^{+}]$ pointwise be $1$ and $t$.
Let $w \in G_1 \setminus \{1,t\}$. Let $\{v_0,v_1,\ldots, v_r\}$ be any $r+1$ vectors in general position within $W^{+}$. By Lemma \[lem:general\_pos\] we see that $$\begin{aligned}
{{\mathcal{U}}}_{w} & := & \{u \in Z \ | \ W^{+} {\text{ is not contained in an eigenspace of }} {\pi}_t \circ \Phi_u(w) \} \\
& = & Z \setminus \left(\bigcap_{i=0}^r \{u \in Z \ | \ ({\pi}_t \circ g (v_i)) \wedge v_i = 0 \} \right)\end{aligned}$$ is the complement of a closed subvariety of $Z$. If we can show that ${{\mathcal{U}}}_{w}$ is nonempty it will follow from the Baire category theorem that ${{\mathcal{U}}}= \bigcap_{w \in G_1 \setminus \{1,t\}} {{\mathcal{U}}}_{w}$ is a dense $G_{\delta}$ subset of $Z$. In particular we would have found some $u \in Z$ satisfying all of our requirements.
From here on we will fix $w \in G_1 \setminus \{1,t\}$ and write this element in a reduced canonical form as $w=g_1 k^{\delta_1} g_2 \ldots k^{\delta_m}g_{m+1}$ where $g_i \in G, ,\delta_i \in \{\pm 1\}$. That the word is reduced means that it is subject to the additional restrictions that $g_i \ne 1$, for $i \in \{2,\ldots,m\}$, and that $k^{\delta_{i}} g_{i+1} k^{\delta_{i+1}}$ is neither of the form $k^{-1}t k$ nor of the form $k t k^{-1}$ for any $i \in \{1 \ldots m\}$. Thus $$\Phi_u(w) = g_1 u^{\delta_1} g_2 \ldots u^{\delta_m}g_{m+1}.$$ Let $S = \{g_1,g_1^{-1}, g_2,g_2^{-1}, \ldots, g_{m+1}^{-1}\}$ and set $S_0 = S \setminus \{1,t\}$.
By our assumption the set $$\Omega := \{v \in W^{+} \ | \ v {\text{ is not an eigenvector of }} {\pi}_t \circ g {\text{ for any }} g \in S_0 \}$$ is the complement of the union of finitely many proper linear subspaces of $W^{+}$. In particular $\Omega$ is a dense open subset of the vector space $W^{+}$. Let $v_1 \in \Omega$ be any vector and $v_2 \in \Omega$ a linearly independent vector that also avoids the finitely many one dimensional subspaces $\{ {{\operatorname{Span}}}\{{\pi}_t \circ g (v_1) \} \ | \ g \in S_0 \}$. Since $S_0$ is symmetric, these choices ensure that ${\pi}_t \circ g(v_i)$ and $v_j$ are linearly independent for every $i,j \in \{1,2\}$. Thus we may complete this pair of vectors to a basis $\{v_1,v_2, \ldots, v_r\}$ for $W^{+}$ in such a way that the two codimension one spaces ${{\operatorname{Span}}}\{v_1,v_3,v_4,\ldots, v_r\}$ and ${{\operatorname{Span}}}\{v_2,v_3,\ldots, v_r\}$ of $W^{+}$ do not contain any of the points $\{{\pi}_t \circ g (v_i) \ | \ g \in S_0, i \in \{1,2\}\}$. Finally let $\{v_{r+1}, v_{r+2},\ldots, v_n\}$ be a basis for $W^{-}$. So that $\{v_1,v_2,\ldots, v_n\}$ becomes a basis for the whole space which is compatible with the direct sum decomposition $W^{+} \oplus W^{-}$. Let $a^{+} = [v_1], a^{-} = [v_2]$, $H^{+} = [{{\operatorname{Span}}}\{v_2,v_3,\ldots, v_n\}]$ and $H^{-} = [{{\operatorname{Span}}}\{v_1,v_3,\ldots, v_n\}]$ be two projective points and two projective hyperplanes corresponding to these vectors. Our choices above imply that $$\label{eqn:prop_vi}
g(a^{\pm}) \not \in H^{-} \cup H^{+}, \qquad \forall g \in S_0.$$
Pick a number $L >> 0$ and define $u = u(L) \in Z$ by the requirements: $$u v_1 = L v_1, \qquad u v_2 = \frac{1}{L} v_2, \qquad u v_i = v_i , \ \ \ \forall 3 \le i \le n.$$ The dynamics of the element $u$ on the projective plane ${{\mathbb{P}}}$ is very proximal in the sense that $$\label{eqn:prox}
\lim_{L {\rightarrow}\infty} u(L)^{\pm 1}(x) = a^{\pm}, \qquad \forall x \not \in H^{\pm} {\text{ respectively}}.$$ We denote by $g(L) = \Phi_{u(L)}(w)$ in order to stress the dependence of this element on $L$.
If $m = 0$ then $g = g_1$ and as long as $g \not \in \{1,t\}$ any element $x \in \Omega$ will satisfy ${\pi}_t \circ g (x) \ne x$ as required. So let us assume that $m \ge 1$ and choose $x \in [W^{+}]$ to be any projective point such that $g_{m+1} (x) \not \in H^{+} \cup H^{-}$ and $g_1 (a^{\pm}) \not \in [{\pi}_t^{-1}(x)]$. Such a choice is possible, regardless of the value of $g_1$ in view of our assumption that $\dim W^{+} > \dim W^{-}$.
Now argue by induction on $m$ that $\lim_{L {\rightarrow}\infty} g_1^{-1} g(L) (x) = a^{\delta_1}$. If $m = 1$, $g_1^{-1}g(L) = u^{\delta_1} g_2$. Since by our choice of $x$, $g_2 \cdot x \not \in H^{+} \cup H^{-}$ the desired property follows directly from Equation (\[eqn:prox\]). Now applying the induction hypothesis to the word $\tilde{w} = g_2 k_2^{\delta_2} g_3 \ldots k_m^{\delta_m} g_{m+1}$ and $\tilde{g}(L) = \Phi_{u(L)}(\tilde{w})$ we obtain $$\lim_{L {\rightarrow}\infty} g_1^{-1} g(L) (x) = \lim u(L)^{\delta_1} g_2 g_2^{-1} \tilde{g}(L) (x)
= \lim_{L {\rightarrow}\infty} u(L)^{\delta_1} g_2 a^{\delta_2} = a^{\delta_1}$$ The equality before last, uses the induction hypothesis. The last equality is justified as follows. If $g_2 \ne t$ then by Equation (\[eqn:prop\_vi\]) $g_2 a^{\delta_2} \not \in H^{+} \cup H^{-}$, while if $g_2 = t$ then by the fact that the word $w$ is reduced we know that $\delta_1 = \delta_2$ and hence $a^{\delta_2} \not \in H^{\delta_1}$. In both cases the claim follows directly from Equation (\[eqn:prox\]). This completes the proof of the induction. By the Baire theoretic argument we have thus found $u \in Z$ such that $\Phi_u: G_1 {\rightarrow}{{\operatorname{SL}}}_n({{\mathbf{R}}})$ becomes an isomorphism and such that the induction hypothesis, according to which $W^{+}$ is not contained in an eigenspace of $\Phi_u(w)$ for any $w \in G_1 \setminus \{1,k\}$, is also satisfied.
It remains only to show that all the involutions in $G_1$ are conjugate. We argue on the Bass-Serre tree corresponding to the HNN extension $G_1 = \langle G, f \ | \ f^{-1}tf = v \rangle$. If $\sigma \in G_1$ is an involution then, being an element of finite order, it must fix a vertex in the Bass-Serre tree. Since the action of $G_1$ is transitive on the vertices all vertex stabilizers are conjugate to $G$. But by the induction hypothesis all involutions in $G$ itself are conjugate and the theorem is proved.
Conclusion
==========
(of Theorem \[thm:s2t\]) Let $H \curvearrowright X$ be the given action, fix any basepoint $x \in X$ and let $A = H_x$ be its stabilizer. The condition that the action on (ordered) pairs of distinct points is free is equivalent to the condition that $A$ be malnormal in H. We fix any involution $t \in SL_n({{\mathbf{R}}})$ and denote by $W^{\pm}$ its $\pm 1$ eigenspaces and by ${\pi}= {{\mathbf{R}}}^n {\rightarrow}W^+$ the projection on $W^{+}$ along $W^{-}$. We impose also the restriction $\dim(W^{+}) \ge \dim(W^{-})+2$. Applying Proposition \[prop:free\_prod\] to the groups $H$ and $\langle t \rangle$ and then replacing if necessary the given group $H$ by its conjugate obtained in that theorem we may assume that $G := \langle H, t \rangle \cong H * {{\mathbf{Z}}}/2{{\mathbf{Z}}}$ and that furthermore: (i) $A$ is malnormal in $G$, (ii) all involutions in $G$ are conjugate to $t$ and (iii) if $W^{+}$ is contained in an eigenspace of some $g \in G$ then $G\in \{1,t\}$. Indeed (i),(ii) are guaranteed by the properties of free products and (iii) follows from Proposition \[prop:free\_prod\].
The conditions (i),(ii),(iii) are exactly these needed in order to apply Theorem \[thm:tec\]. Let us enumerate the elements of the group $G$ as follows $G = \{v_1,v_2, \ldots\}$. Applying this theorem inductively to obtain a sequence of extensions $(A,G) < (A_1,G_1) < (A_2,G_2) < \ldots$, such that $G_i \cap A_{i+1} = A_i$. These come together with elements $f_i \in A_i$ such that $A_i t f_i = A_i v_i$. Let $G_{\omega} = \cup_{i=1}^{\infty} G_i$ and $A_{\omega} = \cup_{i=1}^{\infty} A_i$. These groups satisfy the properties (i),(ii),(iii). In addition $G \cap A_{\omega} = A$ and for every $v \in G$ there exists some $f \in A$ such that $Atf=Av$.
Applying the procedure of the previous paragraph to $(A_{\omega},G_{\omega})$ and continuing inductively we obtain a sequence $(A_{\omega}, G_{\omega}) < (A_{2\omega},G_{2\omega}), < \ldots$ and the union $H_1 = G_{\omega \cdot \omega} = \cup_{i=1}^{\infty} G_{i \omega}$ and $B_1 = A_{\omega \cdot \omega} = \cup_{i =1}^{\infty}A_{i\omega}$ satisfy the following conditions:
- $B_1$ is malnormal in $H_1$
- For every $v \in H_1$ there exits an element $f \in B_1$ such that $B_1 t f = B_1 v$
- $H \cap B_1 = A$
The first two properties are respectively equivalent to the action $H_1 \curvearrowright B_1{\backslash}H_1$ being free and transitive on ordered pairs of distinct points. The last property implies that the action of $H$ on $B_1 \cdot H$ is isomorphic to the original given action $H \curvearrowright A {\backslash}H$.
We know that ${{{\it {p}}\operatorname{-char}}}(H_1)=2$ because the stabilizer of a point is $B_1$ and this group contains no involutions by construction. If $M \lhd H_1$ is a normal Abelian subgroup then $M \cap G_{\alpha}$ is a normal abelian subgroup of $G_{\alpha}$ for every ordinal $\alpha$ and by construction these groups clearly have non nontrivial normal abelian subgroups. Thus $M$ itself must be trivial. This proves that $H_1$ does not split. A similar argument shows that if $M,N \lhd H_1$ are two commuting normal subgroups then one of them must be trivial.
Finally we appeal to the main theorem of [@GG:AOS] (see also [@GG:Primitive]) to deduce that $H_1$ admits uncountably many non-isomorphic primitive permutation representations. We have to verify that $H_1$ is not of Affine or diagonal type. And this immediately follows from the fact that $H_1$ contains neither normal abelian subgroups, nor pairs of commuting normal subgroups respectively.
[Yair Glasner.]{} Department of Mathematics. Ben-Gurion University of the Negev. P.O.B. 653, Be’er Sheva 84105, Israel. [[email protected]]{}
[Dennis D. Gulko.]{} Department of Mathematics. Ben-Gurion University of the Negev. P.O.B. 653, Be’er Sheva 84105, Israel. [[email protected]]{}
[^1]: Following [@RST:s2t] all the actions in this paper will be right actions. This choice is also responsible for the choice of right near fields here.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we explore propagation of energy flux in the future Poincaré patch of de Sitter spacetime. We present two results. First, we compute the flux integral of energy using the symplectic current density of the covariant phase space approach on hypersurfaces of constant radial physical distance. Using this computation we show that in the tt-projection, the integrand in the energy flux expression on the cosmological horizon is same as that on the future null infinity. This suggests that propagation of energy flux in de Sitter spacetime is sharp. Second, we relate our energy flux expression in tt-projection to a previously obtained expression using the Isaacson stress-tensor approach.'
---
****
Sk Jahanur Hoque$^1$ and Amitabh Virmani$^{1,2,3, }$[^1]
$^1$Chennai Mathematical Institute, H1 SIPCOT IT Park,\
Kelambakkam, Tamil Nadu, India 603103\
$^2$Institute of Physics, Sachivalaya Marg,\
Bhubaneswar, Odisha, India 751005\
$^3$Homi Bhabha National Institute, Training School Complex,\
Anushakti Nagar, Mumbai 400085, India\
\
$ \, $\
Introduction
============
The era of gravitational wave astronomy has begun [@Abbott:2016blz; @TheLIGOScientific:2017qsa; @GBM:2017lvd; @Coulter]. It is now all the more important that our theoretical understanding be at par with the impressive experimental developments that have gone into the discovery of gravitational waves. There are several theoretical aspects that are potentially important in relation to generation and propagation of gravitational waves but have not been fully explored. One such aspect is the effect of the positive cosmological constant on the propagation of gravitational waves.
The discovery of the accelerated expansion of the universe from distant supernovae and cosmic microwave background surveys have shown that around 68% of the energy density of the universe is dark energy. While at a fundamental level dark energy is poorly understood, the positive cosmological constant is the simplest explanation of it. From the theoretical point of view, positive cosmological constant posses numerous challenges in relation to study of gravitational waves. In a recent series of papers Ashtekar, Bonga, and Kesavan [@ABKI; @ABKII; @ABKIIIPRL; @ABKIII] have systematically initiated the study of gravitational waves focusing on the numerous effects that the presence of a positive cosmological constant brings. Subsequently, several authors have contributed to the development of the subject [@DHI; @DHII; @Bishop:2015kay; @Bonga; @JA]. The primary aim of this work is to expand on some of these studies, in particular on some aspects of [@DHI; @DHII], and to clarify their relation to [@ABKII; @ABKIII].
In comparison to Minkowski spacetime there are several effects that the positive cosmological constant brings on the propagation of linearised gravitational field. For a detailed discussion of these points, we refer the reader to [@ABKII; @ABKIII]; here we wish to focus on two points especially. First, while wavelengths of linear waves remain constant in flat space, they increase in de Sitter spacetime as the universe undergoes de Sitter expansion. So much so that in the asymptotic region of interest, the wavelengths diverge. Naively, this seems to invalidate the geometrical optics approximation commonly used in the gravitational waves literature. Secondly, due to the curvature of the background spacetime, the linear gravitational field satisfies a *massive* wave equation, i.e., propagation of waves in de Sitter spacetime is not on the light cone. Due to backscattering from the background curvature, in general, there is a tail term.
Partial understanding of these effects is already available. Our study expands on that knowledge. Firstly, although in the asymptotic region of interest, wavelengths diverge, reference [@DHII] made precise how the geometrical optics approximation is still useful. They arrived at an effective stress tensor for gravitational waves following the original work of Isaacson [@Isaacson:1968zza; @Isaacson:1967zz]. An aim of this paper is to re-obtain appropriate version of those expressions from the covariant phase space approach, thus clarifying their relation to [@ABKII; @ABKIII]. The second aim of the paper is to make precise the notion of the “sharp" propagation of energy flux in de Sitter spacetime, i.e., to understand in what sense the tail term mentioned above does not matter for radiated energy flux.
The rest of the paper is organized as follows. We start with a brief review of linearised gravity on de Sitter spacetime in section \[sec:linear\] and write various identities involving derivatives of the radiative field that we need in later sections. In section \[sec:current\_and\_energy\] we compute the symplectic current density for linearised gravity on de Sitter spacetime and write a general expression for the energy flux through a hypersurface $\Sigma$. Since symplectic current density is conserved, it allows us to compute energy flux through any hypersurface.
In section \[sec:tt\] we use the general expression obtained in section \[sec:current\_and\_energy\] to compute the flux integrals on hypersurfaces of constant radial physical distance. These hypersurfaces allow us to interpolate between the cosmological horizon and the future null infinity. We show that in the tt-projection, the integrand in the energy flux expression on the cosmological horizon is same as that on the future null infinity. This suggests that the propagation of energy flux in de Sitter spacetime is sharp. We also relate our energy flux expression to the previously obtained expression of reference [@DHII]. This section constitutes the main results of our work.
We close with a discussion in section \[sec:disc\].
![The full square is the Penrose diagram of global de Sitter spacetime, with each point representing a 2-sphere. In this paper we exclusively work in the future Poincaré patch of de Sitter spacetime — the upper triangular region (red triangle) of this diagram. Blue lines denote hypersurfaces of constant radial physical distance. These hypersurfaces are generated by the time-translational (dilatation) Killing vector. On these hypersurfaces, $\tau$ is a Killing parameter that runs from $-\infty$ to $\infty$. The dotted lines are lines of constant retarded time. Green line is the worldline of the radiating source. []{data-label="PoincarePatch"}](GlobalConformalChart.pdf){width="70.00000%"}
Linearised gravity and various identities involving radiative field {#sec:linear}
===================================================================
Linearised gravity on de Sitter spacetime
-----------------------------------------
We are interested in linearised gravity over de Sitter background. We exclusively work in the future Poincaré patch of de Sitter spacetime. The background de Sitter metric in the Poincaré patch is ds\^2 = |g\_ dx\^dx\^= a\^[2]{} ( - d\^2 + dx\^2), \[background\] a=-(H)\^[-1]{}, H=, where $\Lambda$ is the positive cosmological constant. Linearised perturbations over the background are written as g\_ = |g\_ + \_. Coordinates $x_i$, with $i = 1,2,3$, ranges from $(-\infty, \infty)$, whereas coordinate $
\eta$ takes values in the range $(-\infty, 0)$, with $\eta = 0$ at the future null infinity $\mathcal{I}^+$. The future infinity $\mathcal{I}^+$ is a spacelike surface, see figure \[PoincarePatch\].
For the background metric the Christoffel symbol is \^\_[c]{} = - ( \^0\_\^\_+ \^0\_c\^\_+ \^\_0 \_[c]{}). Using this useful expressions for the d’Alembertian and for various other derivative operators can be written, see e.g. [@deVega:1998ia; @DHI]. In terms of the trace reversed combination $\hat{\gamma}_{\alpha\beta}:=\gamma_{\alpha\beta}-\frac{1}{2}\bar{g}
_{\alpha\beta} \ (\bar{g}^{\mu\nu}\gamma_{\mu\nu})$, the linearised Einstein equations take the form, $$\frac{1}{2}\left[ - \overline{\Box} \hat{\gamma}_{\mu\nu} + \left\{
\overline{\nabla}_{\mu}B_{\nu} + \overline{\nabla}_{\nu}B_{\mu} -
\bar{g}_{\mu\nu}(\overline{\nabla}^{\alpha}B_{\alpha})\right\}\right] +
\frac{\Lambda}{3}\left[\hat{\gamma}_{\mu\nu} -
\hat{\gamma}\bar{g}_{\mu\nu}\right] ~ = ~ 8\pi G T_{\mu\nu} \label{LinEqn}
$$ where $B_{\mu} := \overline{\nabla}_{\alpha}\hat{\gamma}^{\alpha}_{~\mu}$ and $\overline{\nabla}_{\alpha}$ is the metric compatible covariant derivative with respect to the background metric $\bar g_{\a \b}$.
As is well known in the literature [@deVega:1998ia; @DHI; @ABKIII], these equations written in terms of a rescaled variables leads to a great deal of simplification. We define, \_:=a\^[-2]{}\_, and using the gauge condition [@deVega:1998ia], \^\_ + (2 \_[0]{} + \_\^0 \_\^[ ]{} ) = 0, \[ChiGauge\] equation becomes, - 16 G T\_ = \_ + \_0\_ - (\_\^0\_\^0\_\^[ ]{} + \_\^0\_[0]{} + \_\^0\_[0]{} ) \[LinEqnChi\], where $\Box$ is simply the d’Alembertian with respect to Minkowski metric in cartesian coordinates, $\Box = - \partial_\eta^2 + \partial_i^2$.
In terms of variables $
\hat{\chi} :=\chi_{00}+\chi_{~i}^{i}, ~\chi_{0i},~ \chi_{ij}
$ equation (\[LinEqnChi\]) decomposes into three decoupled equations () &=& - , \[decoupled1\]\
()&=&- , \[decoupled2\]\
\_[ij]{} + \_0 \_[ij]{} &=& - 16G T\_[ij]{}, \[decoupled3\] where $\hat{T}:=T_{00}+T_{~i}^{i}$.
Under a linearised diffeomorphism $\xi^{\mu}$, $\chi_{\mu\nu}$ transforms as, \_ = (\_\_ + \_\_ - \_\^\_) - \_\_0, where \_ := a\^[-2]{} \_ = \_\^. \[ChiGaugeTrans\] A small calculation then shows that the gauge condition (\[ChiGauge\]) is preserved under transformations generated by vector fields $\xi^{\mu}$ — the residual gauge transformations — satisfying, $$\Box\underline{\xi}_{\mu} +
\frac{2}{\eta}\partial_0\underline{\xi}_{\mu} -
\frac{2}{\eta^2}\delta_{\mu}^0\underline{\xi}_0= 0.
\label{ResidualChiGauge}$$ Under these residual gauge transformations equation is also invariant.
We can exhaust the residual gauge freedom as follows. We note that $\delta{\hat{\chi}}$ satisfies, $$\begin{aligned}
\Box~{\big(\delta{\hat{\chi}}\big)}&=& 4~\bigg[\Box~\bigg(\partial_0 \underline{\xi}_0-
\frac{\underline{\xi}_0}{\eta}\bigg)\bigg] \\
&=&-\frac{2}{\eta}\partial_0~
\big(\delta{\hat{\chi}}\big)+\frac{2}{\eta^{2}}~\big(\delta{\hat{\chi}}\big)
\eea
where in going from the first step to the second step we have used \eqref{ResidualChiGauge}. This form of the equation implies that,
\be
\Box~\bigg(\frac{\delta{\hat{\chi}}}{\eta}\bigg)=0,
\ee
i.e., $\delta{\hat{\chi}}$ satisfies the wave equation \eqref{decoupled1} outside the source.
Therefore, using an appropriate residual gauge transformation we can set $
\hat{\chi}=0$
outside the source. Similarly $\chi_{0i}$ can be set to zero outside the source \cite{deVega:1998ia}.
Gauge condition (\ref{ChiGauge}) then implies
$\partial^0\chi_{00} = 0$. Choosing $\chi_{00}$ to be zero at some
initial $\eta = $ constant hypersurface we can take $\chi_{00} = 0$ everywhere. Doing so,
gauge condition (\ref{ChiGauge}) becomes
\be
\partial^{i}\chi_{ij} = \chi^{i}_{~i} = 0. \label{TTconditions}
\ee
\subsection{TT-gauge vs tt-projection}
With conditions \eqref{TTconditions} imposed there are no further gauge transformations allowed.
Thus, transverse and
traceless (TT) solutions are fully gauge fixed. Therefore, away from the source it suffices to focus on equation (\ref{decoupled3}).
In general, solutions of this inhomogeneous equation do not satisfy the TT conditions. However, any
spatial rank-2 symmetric tensor can be decomposed into its irreducible components as,
\begin{equation}
\chi_{ij}=\frac{1}{3}\delta_{ij}\delta^{kl}\chi_{kl}+(\partial_{i}\partial_{j}-
\frac{1}{3}\delta_{ij}
\nabla^{2})B+\partial_{i}B_{j}^{\mathrm{T}}+\partial_{j}B_{i}^{\mathrm{T}}+\chi_{ij}^{\mathrm{TT}},
\end{equation}
where $\chi_{ij}^{\mathrm{TT}} $ refers to the transverse-traceless part of the field $\chi_{ij}$, i.e., it satisfies,
\be
\partial^{i}
\chi_{ij}^{\mathrm{TT}}=\delta^{ij} \chi_{ij}^{\mathrm{TT}}=0.
\ee
The vector $B_{i}^{\mathrm{T}}$ is transverse,
$\partial^i B_i^{\mathrm{T}} = 0.$ In this decomposition only $
\chi_{ij}^{\mathrm{TT}}$ is
the gauge invariant piece. Hence, $\chi_{ij}^{\mathrm{TT}}$ is best regarded as the physical
component of the field $\chi_{ij}$.
Given a tensor $\chi_{ij}$, in general it is highly non-trivial to extract $\chi_{ij}^{\mathrm{TT}}$; see \cite{Bonga} for an explicit example.
In the context of gravitational waves, another conceptually distinct notion of transverse-traceless tensors is often used in the literature. This notion is operationally simpler but inequivalent to the above notion. Here one `extracts' the `transverse-traceless' part of a rank-two tensor simply by defining an algebraic projection operator,
\begin{align}
P_i^{~j} &= \delta_i^{~j} - \hat{x}_i\hat{x}^{j}, &
\Lambda_{ij}^{~~kl} &= \frac{1}{2}(P_i^{~k}P_j^{~l} +
P_i^{~l}P_j^{~k} - P_{ij}P^{kl}),
\end{align}
where $\hat x^i = x^i/r$ with $r= \sqrt{x^i x_i}$.
In order to distinguish it from the the above notion, we use the notation $\chi_{ij}^{\mathrm{tt}}$,
\be
\chi_{ij}^{\mathrm{tt}} := \Lambda_{ij}^{~~kl}\chi_{kl}
\label{ttProjection}
\ee
For a detailed discussion of the differences between these two notions see \cite{Ashtekar:2017ydh, Ashtekar:2017wgq}. For asymptotically flat
space-times the two notions match only at null infinity $\mathcal{I}^+$ \cite{Ashtekar:2017wgq, Bonga}. The tt-projection is well tailored to the $1/r$ expansion commonly used for asymptotically flat spacetimes.
The global structure of de Sitter spacetime is very different from Minkowski spacetime. Expansion in powers of $1/r$ is not a useful tool to analyse asymptotically de Sitter spacetimes. In particular, the radial tt-projection is not a valid operation to extract the transverse-traceless part of a rank-2 tensor on the full $\mathcal{I}^+$. The TT-tensor is the correct notion of transverse traceless tensors. However, if one restricts oneself to large radial distances away from the source, one may expect that the tt-projection also gives useful answers. In fact, it appears to work better than expected. In
the context of the power radiated by a spatially compact circular binary system analysed in
\cite{Bonga,JA} the difference does not seem to matter.
The tt-projection being algebraic allows us to do various non-trivial computations which seem difficult to perform otherwise. In particular, this simplicity allows us to gain a physical understanding of the propagation of gravitational waves in de Sitter spacetime. In this paper we mostly restrict ourselves to tt-projection, with the understanding that our results need to be generalised to TT gauge. A detailed study of this we leave for future research.
\subsection{Derivatives of radiative field}
\label{Identity}
In order to compute energy flux through different slices, we need various
derivatives of radiative field $\chi_{ij}$. In this subsection we establish those identities.
The expression for radiative $\chi_{ij}$ we use was obtained in references \cite{ABKIII, DHI},
\be
\chi_{ij} (\eta, r) = 4 G\frac{\eta}{r (\eta - r)} \int d^3 x' T_{ij}(\eta - r, x') + 4 G \int_{-\infty}^{\eta - r} d\eta' \frac{1}{\eta'{}^2}\int d^3 x' T_{ij}(\eta', x'),
\ee
where $T_{ij}$ is the source energy-momentum tensor. In arriving at this expression, Green's functions for the differential operator in \eqref{decoupled3} is used together with the approximation
\be
\eta - | \vec x - \vec x'| \approx \eta - | \vec x| = \eta - r,
\ee
in order to pull the factor of $\frac{1}{r(\eta - r)}$ out from the integral.
The integral of the stress tensor can be expressed in terms of the mass and pressure quadrupole moments $Q_{ij}$ and $\overline{Q}
_{ij}$ at the retarded time $\eta_{\mathrm{ret}}:=\eta - r$ \cite{ABKIII, DHI},
\be
\int d^3 x' T_{ij}(\eta - r, x') = \frac{1}{2 a(\eta_{\mathrm{ret}})} \left( \ddot{Q}_{ij} +
2 H \dot Q_{ij} + 2 H \dot{\overline{Q}}_{ij} + 2 H^2 {\overline{Q}}
_{ij} \right) (\eta_{\mathrm{ret}}), \label{moments}
\ee
where dots denote Lie derivatives with respect to time-translation (dilatation) Killing vector
\be
T^{\mu} \partial_\mu = - H ( \eta \partial_\eta + r \partial_r). \label{time_translation}
\ee
The mass and pressure quadrupole moments $Q_{ij}$ and $\overline{Q}
_{ij}$ are defined as an integrals over the source at some fixed time $\eta$,
\bea
Q_{ij} (\eta)= \int \ a^3 (\eta) T_{00} (\eta, x) x_i x_j d^3 x,
\eea
\bea
\overline{Q}_{ij} (\eta)= \int \ a^3 (\eta) \delta^{kl}T_{kl}(\eta, x) x_i x_j d^3 x.
\eea
Using these expressions, we get the identities
\be
\partial_\eta \chi_{ij} (\eta, x) = 4G\frac{\eta}{(\eta - r)r} \partial_\eta
\left[ \int d^3 x' T_{ij}(\eta - r, x') \right] =: \frac{2G\eta}{(\eta - r)r}
R_{ij} (\eta_{\mathrm{ret}}).
\ee
where
\be
R_{ij}(\eta_{\mathrm{ret}}) ~ = ~\bigg[\dddot{Q}_{ij} + 3H\ddot{Q}_{ij} +
2H^2\dot{Q}_{ij} + H\ddot{\overline{Q}}_{ij} + 3H^2\dot{\overline{Q}}_{ij} +
2H^3\overline{Q}_{ij}\bigg](\eta_{\mathrm{ret}}) .
\ee
Similarly,
\bea
\partial_r \chi_{ij} &=& - \partial_\eta \chi_{ij} - \frac{4}{r^2}\int d^3 x' T_{ij}
(\eta - r, x') \label{useful_identity_DH}
\\
&=&-\frac{2G\eta}{r (\eta - r)} R_{ij}(\eta_{\mathrm{ret}})+ 2H G \ \frac{(\eta -r)}{r^{2}}\left(
\ddot{Q}_{ij} + 2 H \dot Q_{ij} + 2 H \dot{\overline{Q}}_{ij} + 2 H^2
\overline{Q}
_{ij} \right)(\eta_{\mathrm{ret}}).
\eea
As a result
\bea
(T \cdot \partial) \chi_{ij} &=& - H (\eta \partial_\eta + r \partial_r ) \chi_{ij} \\
&=&-\frac{2G H \eta}{r} R_{ij} (\eta_{\mathrm{ret}}) -2GH^{2} \bigg(\frac{\eta - r }{r}\bigg)\left(
\ddot{Q}_{ij} + 2 H \dot Q_{ij} + 2 H \dot{\overline{Q}}_{ij} + 2 H^2
\overline{Q}_{ij} \right) (\eta_{\mathrm{ret}}). \label{T_dot_partial_chi}
\eea
For later convenience we also define
\be
A_{ij}=\ddot{Q}_{ij}+2H\dot{Q}_{ij}+H\dot{\overline{Q}}_{ij}+2H^{2}
\overline{Q}_{ij}. \label{A_def}
\ee
This quantity is interesting as it satisfies the relations
\be
R_{ij}=\dot{A}_{ij}+HA_{ij}= (T \cdot \partial) A_{ij}-HA_{ij},
\label{R_A_old}
\ee
which we will need later.
On the future cosmological horizon of the source defined by
\be
\cH^+: \qquad \eta + r =0,
\ee
equation \eqref{moments} simplifies to,
\be
\left[\int d^3 x' T_{ij}(\eta - r, x')\right]\Bigg{|}_{\cH^+} = (H r)\left( \ddot{Q}_{ij} + 2 H \dot Q_{ij} + 2 H \dot{\overline{Q}}_{ij} + 2 H^2 \overline{Q}_{ij} \right) (\eta_{\mathrm{ret}}),
\ee
and equation \eqref{T_dot_partial_chi} simplifies to,
\be
(T \cdot \partial) \chi_{ij} \Big{|}_{\cH^+} = 2G H R_{ij} (\eta_{\mathrm{ret}}) + 4G H^2 \left( \ddot{Q}_{ij} + 2 H \dot Q_{ij} + 2 H \dot{\overline{Q}}_{ij} + 2 H^2 \overline{Q}_{ij} \right) (\eta_{\mathrm{ret}}).
\ee
\section{Symplectic current density and energy flux}
\label{sec:current_and_energy}
We are interested in computing energy flux through any Cauchy surface and more generally through other surfaces. Perhaps the most convenient way to do this is via the covariant phase space approach. For linearised gravity, the covariant phase space can be taken to be simply the space of solutions $\gamma_{ab}$ of the linearised Einstein's equations together with appropriate gauge conditions \cite{ABKII}. A standard procedure \cite{ABR, LeeWald} then gives a symplectic structure.
When restricted to cosmological slices, the symplectic structure was computed and used in \cite{ABKII, ABKIII}. In this work we are interested in other slices. In our discussion below we focus on the symplectic current density and its integrals, rather than on the careful construction of the phase space itself. The phase space construction is somewhat subtle \cite{ABKII} due to certain divergences as the future null infinity $\mathcal{I}^+$ is approached. Some of our intermediate expressions below are formally divergent as the future null infinity is approached, however, our final answers are all finite and have a well defined limit at $\mathcal{I}^+$.
We start with an expression of symplectic current of linearised Einstein gravity with a cosmological constant, which we can evaluate on different slices. A convenient form is \cite{Hollands:2005wt},
\be
\omega^\a =\frac{1}{32\pi G} \ P^{ \a \b \c \d \e \f }
\left( \delta_1 g_{ \b \c } \overline{\nabla}_{ \d } \delta_2 g_{\e \f } - \delta_2 g_{\b \c } \overline{\nabla}_{\d} \delta_1 g_{\e \f }\right), \label{omega_exp_1}
\ee
where
\be
P^{\a \b \c \d \e \f} = \bar g^{\a \e} \bar g^{\f \b} \bar g^{\c \d} - \frac{1}{2} \bar g^{\a \d} \bar g^{\b \e} \bar g^{\f \c} - \frac{1}{2} \bar g^{\a \b} \bar g^{\c \d} \bar g^{\e \f} - \frac{1}{2} \bar g^{\b \c} \bar g^{\a \e} \bar g^{\f \d} + \frac{1}{2} \bar g^{\b \c} \bar g^{\a \d} \bar g^{\e \f}.
\ee
We use the notation
\bea
\delta_1 g_{\a \b} &=& \gamma_{\a \b}, \\
\delta_2 g_{\a \b} &=& \widetilde \gamma_{\a \b},
\eea
where $\gamma_{\a \b}$ and $\widetilde \gamma_{\a \b}$ are fully gauge fixed physical solutions of the (homogeneous) linearised Einstein equations. We take them to satisfy
Lorentz and radiation gauge,
\be
\overline{\nabla}^\a {\gamma}_{\a\b} = 0, \qquad \gamma_{0\a} = 0, \qquad \bar{g}^{\a\b} \gamma_{\a\b} = 0. \label{gauge_gamma}
\ee
These gauge conditions are the same as \eqref{TTconditions}.
Since $\gamma_{\a\b}$ and $\widetilde \gamma_{\a\b}$ are both traceless, the last three terms in $P^{\a \b \c \d \e \f}$ do not contribute to the symplectic current $\omega^\a$. We effectively have
\be
P^{\a \b \c \d \e \f} = \bar g^{\a \e} \bar g^{\f \b} \bar g^{\c \d} - \frac{1}{2} \bar g^{\a \d} \bar g^{\b \e} \bar g^{\f \c}. \label{P_simple}
\ee
Expanding out the covariant derivatives in \eqref{omega_exp_1} in terms of the Christoffel symbols we get a simplified expression,
\bea
\omega^\a
&=& \frac{1}{32\pi G} \ P^{\a \b \c \d \e \f} \gamma_{\b \c} \left(\partial_\d \widetilde \gamma_{\e \f} - \overline{\Gamma}^\m_{\d \e} \widetilde \gamma_{\m \f} - \overline{\Gamma}^\m_{\d \f} \widetilde \gamma_{\e \m} \right) - (1 \leftrightarrow 2),
\eea
with $P^{\a \b \c \d \e \f}$ given in \eqref{P_simple}.
\subsubsection*{Time component}
Using the simplified expressions above, the time component of the symplectic current is
\be
\omega^\eta = \frac{1}{64\pi G} (H^2 \eta^2) \left(\gamma^{\b \c} \partial_\eta \widetilde \gamma_{\b \c} -\widetilde \gamma^{\b \c} \partial_\eta \gamma_{\b \c} \right).
\ee
We note that due to the gauge conditions \eqref{gauge_gamma}, $\gamma_{\a \b}$ has only spatial components. In terms of the rescaled field $\gamma_{ij} = a^2 \chi_{ij}$,
we have
\be \label{SympCur1}
\omega^\eta = \frac{1}{64 \pi G} (H^2 \eta^2) \left(\chi^{ij} \partial_\eta \widetilde \chi_{ij} -\widetilde \chi_{ij} \partial_\eta \chi_{ij}\right).
\ee
This expression matches with the corresponding expression in reference \cite{ABKII}. In such expressions TT superscript on $\chi_{ij}$ is implicit.
\subsubsection*{Space components}
A similar calculation gives
\be \label{SympCur2}
\omega^i = \frac{1}{32\pi G}a^{-2} \delta^{ij} \left\{ \chi^{lm} \partial_m \widetilde \chi_{jl} - \frac{1}{2} \chi^{lm} \partial_j \widetilde \chi_{lm} - \widetilde \chi^{lm} \partial_m \chi_{jl} + \frac{1}{2} \widetilde \chi^{lm} \partial_j \chi_{lm} \right\}.
\ee
\subsubsection*{Energy flux}
From general results on the covariant phase space approach \cite{ABR, LeeWald, ABKII},
it follows that the energy flux (Hamiltonian for time-translation symmetry $T$) is given as
\be
E_T (\gamma) = -\int_{\Sigma} \omega^\alpha (\gamma, \pounds_T \gamma) \left(n_{\alpha} \sqrt{h_\Sigma}d^3 \x\right), \label{ET}
\ee
where $h_\Sigma$ is the determinant of the induced metric on the slice $\Sigma$ with coordinates $\xi^i$ and $n^\alpha$ is the future directed normal vector to the slice $\Sigma$. In this expression we have evaluated the symplectic current density with $\widetilde \gamma_{\a\b} = \pounds_T \gamma_{\a\b}$. For use in equations \eqref{SympCur1} and \eqref{SympCur2}, we need to evaluate $\widetilde \chi_{ij} = a^{-2} \widetilde \gamma_{ij} = a^{-2} \pounds_T \gamma_{ij}$. This quantity is computed to be
\bea
\widetilde \chi_{ij}
= a^{-2} \ \pounds_T (a^2 \chi_{ij})
= (T \cdot \partial) \chi_{ij}.
\eea
As a result we have the following components of the current $j^\a := \omega^\alpha (\gamma, \pounds_T \gamma)$ for computing the energy flux,
\bea
j^\eta &=& \frac{1}{64\pi G} \ (H^2 \eta^2) \left(\chi^{ij} \partial_\eta \left[ (T \cdot \partial) \chi_{ij} \right] - (T \cdot \partial) \chi_{ij} \partial_\eta \chi^{ij}\right), \label{EFI} \\
j^i &=& \frac{1}{32\pi G} \ (H^2 \eta^2) \delta^{ik} \left\{ \chi^{lm} \partial_m \left[ (T \cdot \partial) \chi_{kl} \right] - \frac{1}{2} \chi^{lm} \partial_k \left[ (T \cdot \partial) \chi_{lm} \right] \right. \nn \\
& &\qquad \qquad \qquad \qquad \left. - \left[ (T \cdot \partial) \chi^{lm} \right] \partial_m \chi_{kl} + \frac{1}{2} \left[ (T \cdot \partial) \chi^{lm} \right] \partial_k \chi_{lm} \right\}.\label{EFII}
\eea
Since $j^\a$ is conserved, we can use it to compute flux across any hypersurface. In this paper we will restrict ourselves to hypersurfaces generated by the time-translation Killing vector $T$. Near the future null infinity $\mathcal{I}^+$ these hypersurfaces are spacelike. Inside the cosmological horizon $\mathcal{H}^+$ these hypersurfaces are timelike. See figure~\ref{PoincarePatch}.
\section{Energy flux in tt-projection}
\label{sec:tt}
In this section we compute the energy flux across hypersurfaces generated by the time-translation Killing vector $T$. We exclusively work with tt-projection.
We start by observing some useful properties of the tt-projection,
\begin{eqnarray}
\partial_{\eta}(\chi_{ij}^{\mathrm{tt}}(\eta, r)) & = & (\partial_{\eta}\chi_{ij}(\eta, r))^{\mathrm{tt}}, \label{tt_commute1}
\\
\partial_{r}(\chi_{ij}^{\mathrm{tt}}(\eta, r)) &=& (\partial_{r}\chi_{ij}(\eta, r))^{\mathrm{tt}}, \label{tt_commute2}
\eea
i.e., tt-projection commutes with $\partial_\eta$ and $\partial_r$.
Moreover,
\be
\partial_m(\chi_{ij}^{\mathrm{tt}}(\eta, r)) =
(\partial_{m}\Lambda_{ij}^{~~kl})\chi_{kl}(\eta, r) +
\hat{x}_m(\partial_r\chi_{ij}(\eta, r))^{\mathrm{tt}} ,
\ee
as a result we have,
\bea
\partial^j(\chi_{ij}^{\mathrm{tt}}(\eta, r)) & = &
\hat{x}^j\Lambda_{ij}^{~~kl}\partial_r\chi_{ij}(\eta, r) +
(\partial^{j}\Lambda_{ij}^{~~kl})\chi_{kl}(\eta, r) \nn \\ &=& \mathcal{O}(r^{-1}),
\label{Transversality}
\eea
where we used,
\bea
\partial_{m}\Lambda_{ij}^{~~kl} & = & -
\frac{1}{r}\left[\hat{x}_i\Lambda_{mj}^{~~~kl} +
\hat{x}_j\Lambda_{mi}^{~~~kl} + \hat{x}^k\Lambda_{ijm}^{~~~~l} +
\hat{x}^l\Lambda_{ijm}^{~~~~k}\right] = \mathcal{O}(r^{-1}) \ . \label{ProjectorDerivative} \end{aligned}$$ The traceless-ness of $\chi_{ij}^{\mathrm{tt}}$ is manifest, but $\chi_{ij}^{\mathrm{tt}}$ satisfies the spatial transversality condition to $\mathcal{O}(r^{-1})$ only, cf. .
Energy flux across hypersurfaces of constant radial physical distance {#energy_flux}
---------------------------------------------------------------------
Hypersurfaces of constant radial physical distance can be defined as, \_: := a () r = - = . These hypersurfaces are generated by the time-translation Killing vector $T$, cf. , T = T\^\_= - H (\_+ r \_r).Let $\tau$ be the Killing parameter along the integral curves of the Killing vector $T$ satisfying, $$\begin{aligned}
\frac{d\eta}{d\tau}&=-H\eta,&
\frac{d x^i}{d\tau}&=-H x^i, & \label{KillingParameter}\end{aligned}$$ then, coordinates on $\Sigma_\rho$ can be taken to be $\tau, \theta,$ and $\phi$. The induced metric on $\Sigma_\rho$ is, h\_[ab]{} = (H\^2\^2 - 1, \^2, \^2 \^2). This metric is of Lorentzian signature for $H\rho < 1$ (inside the cosmological horizon), is degenerate for $H\rho = 1$ (the cosmological horizon), and is of Euclidean signature for $H\rho > 1$ (outside the cosmological horizon); see figure \[PoincarePatch\]. In this subsection we work with the timelike and spacelike cases; the case of the null cosmological horizon is considered in the next subsection (section \[cosmological\_horizon\]).
The volume element for the non-null cases is = \^2 , and the unit normal is n\_ = a | H\^2\^2-1|\^[-1/2]{}(H, x\_i/r). Here $\epsilon =
+1$ for time-like hypersurfaces $H\rho < 1$, and $-1$ for space-like hypersurfaces $H\rho > 1$. Therefore, the infinitesimal volume element vector field is [@Poisson] d \_= n\_ d\^3 = a\^[3]{} r\^2 (H, ) d d d . The hypersurface integral for energy flux is then written as, \[flux\_physical\_radius\] E\_T = -\_[\_]{}d \_ j\^ & = & -\_[-]{}\^[+]{}d\_[S\^2]{}d r\^[2]{} a\^[3]{} (H j\^+ ), where $\tau$ is the Killing parameter defined in . Using , this expression can be rewritten as, E\_T =- \_[\_ ]{} a\^[4]{} ( j\^+ j\^r ) d\^3 x. \[omega\_rho\] In this expression, both terms diverge as $\eta
\rightarrow 0$, or as $\rho \to \infty$. It is easily seen from and that $j^{\eta}$ term diverges as $\eta^{-2}
$ while $j^r$ term diverges as $\eta^{-1}$. This situation is similar to $E_T$ evaluated on constant $\eta$ slices in [@ABKII]. We will see below that, as in [@ABKII], the divergent pieces turn out to be total derivative.
### $j^{\eta}$ contribution {#jeta-contribution .unnumbered}
Let us first look at the $j^{\eta}$ part of integral , we call it $E_T^{(1)}$, E\_T\^[(1)]{} &=&-\_[-]{}\^[+]{}d\_[S\^2]{}d r\^[2]{} a\^[3]{} (H j\^)\
&=& - \_[-]{}\^[+]{} d \_[S\^2]{} d r\^[2]{} a\^[3]{} H a\^[-2]{}\
&=& H\^2 \^[3]{} \_[-]{}\^[+]{} d \_[S\^2]{} d\
&=&-H\^2 \^[3]{} {\_[-]{}\^[+]{} d \_[S\^2]{} d - \_[-]{}\^[+]{} d d },\
\[omega\_eta\_final\] where we have done the following manipulations. In the first step we have substituted . In the second step we have used the property that $T \cdot \partial = \frac{d}{d\tau}$ and $
\partial_{\eta}[\frac{d}{d\tau}]=\frac{d}{d\tau}[\partial_{\eta}]-H
\partial_{\eta}$. In the third step we have done integrations by part with respect to the Killing parameter $\tau$ and have made use of equation . This integration by parts is valid because $\frac{d}
{d\tau}$ is tangential to $\Sigma_\rho$.
The second term in expression is a total derivative. This integral is zero for the following reasons. On timelike $\rho$ = constant hypersurfaces, $
\tau=+\infty$ corresponds to future timelike infinity $i^+ $ and $\tau=-\infty$ corresponds to past timelike infinity $i^- $. Assuming that the source is static at the boundary points [@ABKIII; @DHI], i.e., $ \frac{dQ_{ij}}{d\tau}\big|_{\tau=\pm\infty}=0$ and $\frac{d\overline{Q}
_{ij}}{d\tau}\big|_{\tau= \pm\infty}=0$, $\chi_{ij}$ vanishes at $i^+$ and $i^-$. Hence the end point contributions in the integral vanish for timelike hypersurfaces.
On spacelike $\rho$ = constant hypersurfaces, $\tau=+\infty$ corresponds to future timelike infinity $i^+ $ and $\tau=-\infty$ corresponds to spatial infinity $i^0$; see figure \[PoincarePatch\]. $\chi_{ij}$ vanishes at $i^0$ due to no incoming radiation boundary conditions at $\eta = - \infty$. Hence the end point contributions in the integral also vanish for spacelike hypersurfaces.
### $j^{i}$ contributions {#ji-contributions .unnumbered}
Let us now look at the $j^{i}$ part of integral . We call this piece $E_T^{(2)}$. Upon substituting we get four terms. We separate the contributions of these terms based on their derivative structures. Two of these terms are, $E_T^{(2, I)}$, E\_T\^[(2, I)]{}&=&-\_[-]{}\^[+]{} d\_[S\^2]{} d r\^[2]{} a\^[3]{} a\^[-2]{} { \^[lm]{} \_[m]{}(T ) \_[kl]{}- (T ) \^[lm]{} \_[m]{}\_[kl]{} }\
&=& - \_[-]{}\^[+]{} d\_[S\^2]{} d [x\^k]{} { \^[lm]{} \_[m]{}(T ) \_[kl]{}- (T ) \^[lm]{} \_[m]{}\_[kl]{} }\
&=&- d\^[3]{}x { \^[lm]{} \_[m]{}(T ) \_[kl]{}- (T ) \^[lm]{} \_[m]{}\_[kl]{} }. Since we are working with tt-projection, we have for the integrand, $$\begin{aligned}
&{x^k}\left\{ \chi^{lm}_{\mathrm{tt}}~ \partial_m \left[ (T \cdot \partial)
\chi_{kl}^{\mathrm{tt}} \right] - \left[ (T \cdot \partial) \chi^{lm}_{\mathrm{tt}} \right] \partial_m \chi_{kl}
^{\mathrm{tt}} \right\}
\\
&= {x^k}\left\{ \chi^{lm}_{\mathrm{tt}}~ \partial_m \left[ (T \cdot
\partial) (\Lambda^{ij}_{kl}~\chi_{ij}) \right]
-\left[ (T \cdot \partial) \chi^{lm}_{\mathrm{tt}} \ \partial_{m}(\Lambda^{ij}_{kl} \chi_{ij}) \right] \right\} \nn \\
&=r \ \hat{x}^k \left\{ \chi^{lm}_{\mathrm{tt}}\bigg[(\partial_{m}\Lambda^{ij}_{kl})
(T \cdot \partial) \chi_{ij}
+\Lambda^{ij}_{kl}~\partial_{m}(T \cdot \partial)\chi_{ij}\bigg]
-(T \cdot \partial) \chi_{lm}^{\mathrm{tt}} \bigg[ \Lambda^{ij}_{kl} \ (\partial_{m}\chi_{ij})
+\chi_{ij}(\partial_{m} \Lambda^{ij}_{kl})
\bigg] \right\} \nn \\
&= r \ \chi^{lm}_{\mathrm{tt}} \left\{-\frac{1}{r}\Lambda_{lm}^{ij} (T \cdot \partial)
\chi_{ij}\right\}- r \ (T \cdot \partial) \chi_{lm}^{\mathrm{tt}} \left\{ -\frac{1}{r}\Lambda^{ij}_{lm} \
\chi_{ij}\right\} \nn \\
&=- \chi^{lm}_{\mathrm{tt}} (T \cdot \partial)\chi_{lm}^{\mathrm{tt}}+\chi^{lm}_{\mathrm{tt}} (T
\cdot \partial)\chi_{lm}^{\mathrm{tt}} \nn\\
&= 0,
\end{aligned}$$ i.e., these two terms cancel each other.
The remaining terms $E_T^{(2,II)}$ in the $j^i$ integral are, E\_T\^[(2,II)]{} &= & \_[-]{}\^[+]{} d \_[S\^2]{} d r\^[2]{} a\^[3]{} a\^[-2]{} \
&=& - H\^2 \_[-]{}\^[+]{} d \_[S\^2]{} d\
&=& H\^2 { \_[-]{}\^[+]{} d \_[S\^2]{} d \_[r]{} \^[ij]{} - \_[-]{}\^[+]{} d \_[S\^2]{} d },\
\[omega\_i\_final\] where in arriving at these expressions we have done manipulations similar to ones done above. In the first step we have used the property that $T \cdot
\partial = \frac{d}{d\tau}$ and $\partial_{r}[\frac{d}{d\tau}]=\frac{d}
{d\tau}[\partial_{r}]-H\partial_{r}$. In the second step we have done integrations by part with respect to the Killing parameter $\tau$ and have made use of equation .
The second term in expression is a total derivative. On $\rho$ = constant surfaces, $\tau=+\infty$ corresponds to the boundary point $i^+$. For $\rho$ = constant timelike (spacelike) surfaces, $\tau=-\infty$ corresponds to $i^-(i^0)$, see figure \[PoincarePatch\]. At all these points the field $\chi_{ij}$ vanishes. Hence contributions from the total derivative term are zero in .
### Adding the two contributions {#adding-the-two-contributions .unnumbered}
The non zero contributions from $j^{\eta}$ and $j^{i}$ to the flux integral are E\_T = \^[ik]{}\^[jl]{}. \[Flux\_Phys\] At this stage we can use various identities from section \[Identity\] to get, E\_T &=& \_[S\^2]{} d\_[-]{}\^ d \^[ik]{}\^[jl]{}\
& & , \[energy\_main\_sec\] where we recall that $A_{ij}$ is defined in A\_[ij]{}=\_[ij]{}+2H\_[ij]{}+H\_[ij]{}+2H\^[2]{} \_[ij]{}, and it satisfies identities R\_[ij]{}=\_[ij]{}+HA\_[ij]{}=A\_[ij]{}-HA\_[ij]{}. \[R\_A\] In arriving at expression we have used the fact that $\partial_{r}$ and $\partial_{\eta}$ commute with the tt-projection, cf. equations –. We also note that the operation of tt-projection commutes with the dot operation.
Interestingly, all the other terms except the $RR$ term in expression combine into a total derivative. Substituting $R_{ij}$ in terms of $A_{ij}$ in the other terms in we get, \^[ik]{}\^[jl]{} =\^[ik]{}\^[jl]{}, which is a total derivative on $\Sigma_\rho$. Like in the previous subsection, contributions from this total derivatives terms vanish. This is so because $A_{ij}$ vanishes due to the staticity assumption of the source at the boundary points. Hence, these terms do not contribute to the energy flux. Note that, formally several of these total derivative terms do not have a good limit as $\rho \rightarrow
\infty$, reflecting the fact that the hypersurface integral of the symplectic current density itself does not have a good limit on $\mathcal{I}^{+}.$ However, the divergent terms turn out to be total derivatives, as in [@ABKII]. A final expression is therefore, E\_T = \_[S\^2]{} d\_[-]{}\^ d \^[ik]{}\^[jl]{}. \[energy\_flux\_final\]
Flux integral on cosmological horizon {#cosmological_horizon}
-------------------------------------
The analog of the above computation can also be done on the cosmological horizon. The cosmological horizon is a null surface at, \^+ : + r = 0. The fact that it is a null surface brings about some non-trivial changes to the computation of subsection \[energy\_flux\], which we highlight below. On the cosmological horizon $\sqrt{h} =
H^{-2} \sin\theta,
$ and we fix the normalisation of the normal vector as, n\_ = - |H|\^[-1]{}(1, x\_i/r) , so that $n^{\mu} = T^{\mu}$ at $\cH^+$. The flux integral is therefore,
E\_[T]{}=-\_[\^+]{} d\_ j\^ &=&- \^[+]{}\_[-]{} \_[S\^[2]{}]{} (j\^+). \[energy\_main\_H\]
### $j^{\eta}$ contribution {#jeta-contribution-1 .unnumbered}
The $j^{\eta}$ terms in integral are E\_T\^[(1)]{}&=&\^[+]{}\_[-]{} d \_[S\^2]{} d (\^[ij]{} \_- (T ) \_[ij]{} \_\^[ij]{}). Following the step similar to the previous subsection, this contribution becomes, E\_T\^[(1)]{}&=& \^\_[-]{} d \_[S\^2]{} d r \^[ik]{} \^[jl]{}.
### $j^{i}$ contributions {#ji-contributions-1 .unnumbered}
The $j^{i}$ part of integral again has two types of terms. The terms with the derivative structure of the form \[tt\_zero\] -\^[+]{}\_[-]{} d\_[S\^2]{} d x\^k { \^[lm]{} \_m - \_m \_[kl]{} } . cancel with each other like in the previous subsection. The remaining terms in the integral become, E\_T\^[(2)]{} &=&-\_[-]{}\^[+]{} d\_[S\^2]{} d { \^[lm]{} \_r - \_r \_[lm]{} }\
&=& d\_[S\^2]{} d \_r \_[lm]{}\
&=& d\_[S\^2]{} d \^[ik]{}\^[jl]{}.
### Adding the two contributions {#adding-the-two-contributions-1 .unnumbered}
The energy flux across $\mathcal{H}^+$ is, E\_T = \_[-]{}\^ d\_[S\^2]{} d \^[ik]{}\^[jl]{}. \[temp\] Now using the identities from section \[Identity\] and substituting $H\rho=1$ on the cosmological horizon, the energy flux expression becomes, $$\begin{aligned}
E_T = \frac{G}{8\pi}\int_{S^2} d\Omega \int_{-\infty}^{\infty} d\tau \bigg[ R_{ij}^{\mathrm{tt}}R_{kl}^{\mathrm{tt}}
+4H A_{ij}^{\mathrm{tt}}R_{kl}^{\mathrm{tt}}+4 H^{2}
A_{ij}^{\mathrm{tt}}A_{kl}^{\mathrm{tt}}\bigg] \delta^{ik}\delta^{jl}.
\end{aligned}$$ Again terms other than the $RR$ term combine into a total derivative. Using , we note that, 4H \_[-]{}\^ d\^[ik]{}\^[jl]{} = 2 H \_[-]{}\^ d= 0. Hence, the energy flux across $\mathcal{H}^+$ is simply, E\_T = \_[-]{}\^ d\_[S\^2]{} d R\_[ij]{}\^ R\_[kl]{}\^ \^[ik]{}\^[jl]{}. \[EF\_CH\]
Sharp propagation of energy
---------------------------
The integrands in integrals and are exactly the same. In particular, the integrand is independent of $\rho$. Hence the power radiated P = = \_[S\^2]{} d R\_[ij]{}\^ R\_[kl]{}\^ \^[ik]{}\^[jl]{} \[power\] is independent of $\rho$.
The power is a function of retarded time alone. Along the outgoing null rays, retarded time is constant, see figure \[PoincarePatch\]. Specifically, the power can be computed at a cross-section of the cosmological horizon or at a cross-section of the future null infinity. As long as the cross-sections are on the same retarded time the two expressions are identical. This is the sense in which propagation of energy flux is sharp in de Sitter spacetime. See also [@DHII; @Bonga] for related comments.
Comparison with the stress-tensor approach of [@DHII]
------------------------------------------------------
Reference [@DHII] also obtained an expression for the energy flux across hypersurfaces of constant radial physical distance. It uses the Isaacson stress-tensor approach. To compare our energy flux expression to theirs, we first use = (T ) = - H (\_+ r \_r), and then expand out the resulting expression to get, E\_T = - d\_[S\^2]{} d H\^[2]{}\^[2]{}r { \_ \_[ij]{}\_\_[kl]{}+\_[r]{}\_[ij]{}\_[r]{}\_[kl]{} - \_[r]{}\_[ij]{}\_\_[kl]{} } \^[ik]{}\^[jl]{}. \[energy\_flux\_DH\] This expression matches with that of [@DHII] (equations (52) and (53)), modulo the ‘averaging’. The averaging is part of the Isaacson stress-tensor approach.
Our analysis differs from [@DHII] in another technical aspect. In reference [@DHII], to obtain energy flux in the form of equation from , the approximation \_ \_[ij]{} -\_[r]{}\_[ij]{} \[approximation\] was used. From the computation of section \[energy\_flux\], we note that this approximation is not needed. The terms it ignores combine into a total derivative.
Discussion {#sec:disc}
==========
We have explored propagation of energy flux in the future Poincaré patch of de Sitter spacetime. We computed energy flux integral on hypersurfaces of constant radial physical distance. We showed that in the tt-projection, the integrand in the energy flux expression on the cosmological horizon is same as that on the other hypersurfaces of constant physical radial distance. This strongly suggests that the energy flux propagates sharply in de Sitter spacetime. We also related our flux expression to a previously obtained expression of [@DHII], where a Isaacson stress-tensor approach was used.
Our work can be extended in several directions. Perhaps the most pressing extension is to generalise our computations in TT-gauge and clarify their relation to [@ABKII; @ABKIII]. To systematically study this problem, it will be useful to carefully define the covariant phase space using hypersurfaces of constant radial physical distance. Such an approach offers advantages over [@ABKII; @ABKIII], as in this foliation, slices near future null infinity do not intersect source’s worldvolume. Hence the covariant phase space based on homogeneous solutions of Einstein’s equations is better defined. It can perhaps also be useful to compute the electric and magnetic parts of the Weyl tensors adapted to $\rho =$ constant slicing and write the flux expression in terms of these tensors. We hope to return to some of these problems in our future work.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We thank Ghanashyam Date and Alok Laddha for discussions. We are grateful to Ghanashyam Date for carefully reading a version of the manuscript, and for his detailed comments. This research is supported in part by the DST Max-Planck partner group project “Quantum Black Holes” between CMI, Chennai and AEI, Golm.
Addendum: Energy flux in TT gauge {#app:TT}
=================================
In this appendix we evaluate expression in TT-gauge. Although we are not able to match our final answer to that of [@ABKIII], the computations involved are sufficiently interesting to include this discussion as an appendix. This appendix is not included in the journal version of the paper. For ease of reference we write the energy flux expression again, E\_T = -\_[\_]{}d \_ j\^ & = & -\_[-]{}\^[+]{}d\_[S\^2]{}d r\^[2]{} a\^[3]{} (H j\^+ ), \[flux\_physical\_radius\_TT\] where recall that $\tau$ is the Killing parameter defined in .
### $j^{\eta}$ contribution {#jeta-contribution-2 .unnumbered}
Computation of the $j^{\eta}$ part of integral is identical to the corresponding computation presented in section \[energy\_flux\]. A final answer is E\_T\^[(1)]{} &=&- H\^2 \^[3]{} \_[-]{}\^[+]{} d \_[S\^2]{} d \^[ik]{} \^[jl]{}.
### $j^{i}$ contributions {#ji-contributions-2 .unnumbered}
Let us first look at the $j^{i}$ part of integral . We call this piece $E_T^{(2)}$. Upon substituting we get four terms. We separate the contributions of these terms based on their derivative structures. Two of these terms are, $E_T^{(2, I)}$, E\_T\^[(2, I)]{}&=& \_[-]{}\^[+]{} d\_[S\^2]{} d r\^[2]{} a\^[3]{} a\^[-2]{} { \^[lm]{} \_[m]{}(T ) \_[kl]{}- (T ) \^[lm]{} \_[m]{}\_[kl]{} }\
&=& \_[-]{}\^[+]{} d\_[S\^2]{} d [x\^k]{} { \^[lm]{} \_[m]{}(T ) \_[kl]{}- (T ) \^[lm]{} \_[m]{}\_[kl]{} }\
&=& d\^[3]{}x { \^[lm]{} \_[m]{}(T ) \_[kl]{}- (T ) \^[lm]{} \_[m]{}\_[kl]{} }. Upon an integration by parts we get, E\_T\^[(2, I)]{}&=&\
&=&0, i.e., these two terms exactly cancel each other in TT gauge since $\partial^{m}
\chi_{lm}^{\mathrm{TT}}=0$.
For the remaining terms in the $j^i$ integral, computation is identical to the corresponding computation presented in section \[energy\_flux\]. A final answer is E\_T\^[(2,II)]{} &=& H\^2 \_[-]{}\^[+]{} d \_[S\^2]{} d \_[r]{} \_[ij]{}\^ \^[ik]{} \^[jl]{}.
### Adding the two contributions {#adding-the-two-contributions-2 .unnumbered}
A final expression for the energy flux in TT gauge is, E\_T &=& H\^[2]{}{d\_[S\^2]{} d (r\_\_[kl]{}\^+\_[r]{}\_[kl]{}\^)} \^[ik]{}\^[jl]{}\
&=& H\^[2]{}{d\_[S\^2]{} d\^ \^}\^[ik]{}\^[jl]{}, \[TT\_energy\]\
where we have used the fact that $\partial_\eta$ and $r\partial_r$ commute with the TT operation.
Although we do not have a clear interpretation of , neither a detailed understanding of its relation of [@ABKIII], we make the following (possibly interesting/useful) observation. Under the integral sign, we can first evaluate the expressions at $\rho$ = constant surface and then take its TT part[^2]. Thought of it in this way, it appears appropriate to pull out factors of $\rho$ from bracketed expressions in . Then, we can express energy flux as, E\_T= d\_[S\^2]{} d\^[ik]{}\^[jl]{}, as the remaining three terms in energy expressions can be written as a total derivative, (1+H)\^[2]{} (A\_[ij]{}\^A\_[kl]{} \^) \^[ik]{}\^[jl]{}.
[99]{}
B. P. Abbott [*et al.*]{}, “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett. [**116**]{}, no. 6, 061102 (2016) doi:10.1103/PhysRevLett.116.061102 \[arXiv:1602.03837 \[gr-qc\]\]. B. P. Abbott [*et al.*]{}, “GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral,” Phys. Rev. Lett. [**119**]{}, no. 16, 161101 (2017) doi:10.1103/PhysRevLett.119.161101 \[arXiv:1710.05832 \[gr-qc\]\]. B. P. Abbott [*et al.*]{}, “Multi-messenger Observations of a Binary Neutron Star Merger,” Astrophys. J. [**848**]{}, no. 2, L12 (2017) doi:10.3847/2041-8213/aa91c9 \[arXiv:1710.05833 \[astro-ph.HE\]\].
D. A. Coulter [*et al.*]{}, “Swope Supernova Survey 2017a (SSS17a), the Optical Counterpart to a Gravitational Wave Source,” Science doi:10.1126/science.aap9811 \[arXiv:1710.05452 \[astro-ph.HE\]\].
A. Ashtekar, B. Bonga and A. Kesavan, “Asymptotics with a positive cosmological constant: I. Basic framework,” Class. Quant. Grav. [**32**]{}, no. 2, 025004 (2015) doi:10.1088/0264-9381/32/2/025004 \[arXiv:1409.3816 \[gr-qc\]\]. A. Ashtekar, B. Bonga and A. Kesavan, “Asymptotics with a positive cosmological constant. II. Linear fields on de Sitter spacetime,” Phys. Rev. D [**92**]{}, no. 4, 044011 (2015) doi:10.1103/PhysRevD.92.044011 \[arXiv:1506.06152 \[gr-qc\]\].
A. Ashtekar, B. Bonga and A. Kesavan, “Gravitational waves from isolated systems: Surprising consequences of a positive cosmological constant,” Phys. Rev. Lett. [**116**]{}, no. 5, 051101 (2016) doi:10.1103/PhysRevLett.116.051101 \[arXiv:1510.04990 \[gr-qc\]\].
A. Ashtekar, B. Bonga and A. Kesavan, “Asymptotics with a positive cosmological constant: III. The quadrupole formula,” Phys. Rev. D [**92**]{}, no. 10, 104032 (2015) doi:10.1103/PhysRevD.92.104032 \[arXiv:1510.05593 \[gr-qc\]\].
G. Date and S. J. Hoque, “Gravitational waves from compact sources in a de Sitter background,” Phys. Rev. D [**94**]{}, no. 6, 064039 (2016) doi:10.1103/PhysRevD.94.064039 \[arXiv:1510.07856 \[gr-qc\]\].
G. Date and S. J. Hoque, “Cosmological Horizon and the Quadrupole Formula in de Sitter Background,” Phys. Rev. D [**96**]{}, no. 4, 044026 (2017) doi:10.1103/PhysRevD.96.044026 \[arXiv:1612.09511 \[gr-qc\]\]. N. T. Bishop, “Gravitational waves in a de Sitter universe,” Phys. Rev. D [**93**]{}, no. 4, 044025 (2016) doi:10.1103/PhysRevD.93.044025 \[arXiv:1512.05663 \[gr-qc\]\].
B. Bonga, J. S. Hazboun, “Power radiated by a binary system in a de Sitter Universe,” \[arXiv:1708.05621\]. S. J. Hoque and A. Aggarwal, “Quadrupolar power radiation by a binary system in de Sitter Background,” \[arXiv:1710.01357\]
R. A. Isaacson, “Gravitational Radiation in the Limit of High Frequency. II. Nonlinear Terms and the Ef fective Stress Tensor,” Phys. Rev. [**166**]{}, 1272 (1968). doi:10.1103/PhysRev.166.1272 R. A. Isaacson, “Gravitational Radiation in the Limit of High Frequency. I. The Linear Approximation and Geometrical Optics,” Phys. Rev. [**166**]{}, 1263 (1967). doi:10.1103/PhysRev.166.1263 H. J. de Vega, J. Ramirez and N. G. Sanchez, “Generation of gravitational waves by generic sources in de Sitter space-time,” Phys. Rev. D [**60**]{}, 044007 (1999) doi:10.1103/PhysRevD.60.044007 \[astro-ph/9812465\].
A. Ashtekar and B. Bonga, “On a basic conceptual confusion in gravitational radiation theory,” Class. Quant. Grav. [**34**]{}, no. 20, 20LT01 (2017) doi:10.1088/1361-6382/aa88e2 \[arXiv:1707.07729 \[gr-qc\]\]. A. Ashtekar and B. Bonga, “On the ambiguity in the notion of transverse traceless modes of gravitational waves,” Gen. Rel. Grav. [**49**]{}, no. 9, 122 (2017) doi:10.1007/s10714-017-2290-z \[arXiv:1707.09914 \[gr-qc\]\].
A. Ashtekar, L. Bombelli, and O. Reula, “Covariant phase space of asymptotically flat gravitational fields,” (M. Francaviglia and D. Holm (eds.)). Mechanics, analysis and geometry: 200 years after Lagrange - 1991. North-Holland. In: Amsterdam.
J. Lee and R. M. Wald, “Local symmetries and constraints,” J. Math. Phys. [**31**]{}, 725 (1990). doi:10.1063/1.528801
S. Hollands, A. Ishibashi and D. Marolf, “Comparison between various notions of conserved charges in asymptotically AdS-spacetimes,” Class. Quant. Grav. [**22**]{}, 2881 (2005) doi:10.1088/0264-9381/22/14/004 \[hep-th/0503045\]. E. Poisson, *A Relativist’s Toolkit – The Mathematics of Black-Hole Mechanics*, Cambridge University Press, 2004.
[^1]: Currently on lien from Institute of Physics, Sachivalaya Marg, Bhubaneswar, Odisha, India 751005.
[^2]: Although the TT conditions are tailored to $\eta =$ constant slices.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study the divergence form second-order elliptic equations with mixed Dirichlet-conormal boundary conditions. The unique $W^{1,p}$ solvability is obtained with $p$ being in the optimal range $(4/3,4)$. The leading coefficients are assumed to have small mean oscillations and the boundary of domain is Reifenberg flat. We also assume that the two boundary conditions are separated by some Reifenberg flat set of co-dimension $2$ on the boundary.'
address:
- |
School of Mathematics\
Korea Institute for Advanced Study\
85 Hoegiro\
Dongdaemun-gu\
Seoul 02455\
Republic of Korea
- ' Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA'
- 'Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA'
author:
- Jongkeun Choi
- Hongjie Dong
- Zongyuan Li
title: 'optimal regularity for a Dirichlet-conormal problem in Reifenberg flat domain'
---
[^1]
Introduction
============
In this paper, we discuss the mixed boundary value problem for second-order elliptic operators: $$\label{180126@eq1}
\begin{cases}
Lu =f+ D_i f_i & \text{in }\, \Omega,\\
Bu = f_i n_i & \text{on }\, {\mathcal{N}},\\
u =0 & \text{on }\, {\mathcal{D}},
\end{cases}$$ where $\Omega$ is a domain (not necessarily bounded) in ${\mathbb{R}}^d$, $d\ge 2$ with the boundary divided into two non-intersecting portions ${\mathcal{D}}$ and ${\mathcal{N}}$. The differential operator $L$ is in divergence form acting on real valued functions $u$ as follows: $$Lu=D_i(a_{ij}(x)D_j u+b_i(x) u)+\hat{b}_i(x) D_i u+c(x)u.$$ Here, all the coefficients are assumed to be bounded measurable, and the leading coefficients $a_{ij}$ are symmetric and uniformly elliptic. We denote by $Bu=(a_{ij}D_j u+b_i u)n_i$ the conormal derivative of $u$ on ${\mathcal{N}}$ associated with the operator $L$. Dirichlet and conormal boundary conditions are prescribed on the portions ${\mathcal{D}}$ and ${\mathcal{N}}$ respectively, which are separated by their relative boundary $\Gamma\subset{\partial}\Omega$. Both the equation and the boundary conditions are understood in the weak sense. For precise definition, see Definition \[def-weak-sln\].
As is well known, solutions to purely Dirichlet/conormal boundary value problems are smooth when coefficients, data, and boundaries of domains are smooth. However, for mixed boundary value problems, such a regularity result does not hold near the interface $\Gamma$, and the regularity of solutions depends also on that of $\Gamma$ and the way two boundary conditions meet (e.g., the meeting angle and certain compatibility conditions). For instance, the best possible regularity of derivatives of solutions to is $$Du \in L_p \quad \text{for }\, p<4$$ when the two boundary portions meet tangentially (the angle between ${\mathcal{D}}$ and ${\mathcal{N}}$ is $\pi$); see Example \[eg-classical\] for a classical counterexample. In this paper, we investigate minimal regularity assumptions of $a_{ij}$, $\partial \Omega$, and $\Gamma$, which guarantee the above optimal regularity as well as the solvability of the mixed problem .
Regularity theory for mixed problems has been studied for a long time. For the case when the two boundary portions ${\mathcal{D}}$ and ${\mathcal{N}}$ meet tangentially, we refer the reader to Shamir [@Sh] and Savaré [@Sav]. In [@Sh], the author proved $W^{1,4-\varepsilon}$ regularity for non-divergence form elliptic equations with smooth coefficients in half space. He also obtained $W^{s,p}$ regularity on a smooth bounded domain with the indices $p>4$ and $s<1/2+2/p$. At one end, the optimal $C^{1/2-{\varepsilon}}$-Hölder regularity can be obtained by passing $p\nearrow\infty$, which improved a general Hölder regularity result of De Giorgi’s type by Stamppachia in [@Sta]. It is also worth mentioning that in [@Sav], the author proved optimal regularity in Besov space $B^{3/2}_{2,\infty}$ for the divergence form elliptic equations with Lipschitz coefficients on a $C^{1,1}$ domain.
For the case when ${\mathcal{D}}$ and ${\mathcal{N}}$ do not meet tangentially, we refer the reader to I. Mitrea-M. Mitrea [@MR2309180], where the authors studied the mixed problem with $L u=\Delta u$. They proved the $W^{1,p}$ solvability with $$\label{190201@eq1}
\frac{3}{2+\varepsilon}<p<\frac{3}{1-\varepsilon} \quad \text{for some }\, \varepsilon=\varepsilon(\Omega, {\mathcal{D}}, {\mathcal{N}})\in (0,1)$$ on the so-called creased domains in ${\mathbb{R}}^d$, $d\ge 3$, which means that ${\mathcal{D}}$ and ${\mathcal{N}}$ are separated by a Lipschitz interface and the angle between ${\mathcal{D}}$ and ${\mathcal{N}}$ is less than $\pi$. This class of domains was introduced by Brown in [@B] to answer a question raised by Kenig in [@K] regarding the non-tangential maximal function estimate $${\lVert(\nabla u)^{*}\rVert}_{L_2({\partial}\Omega)}\le C\big({\lVert\nabla_{\rm{tan}} u\rVert}_{L_2({\mathcal{D}})}+{\lVert{\partial}u/ {\partial}n\rVert}_{L_2({\mathcal{N}})}\big)$$ of harmonic functions. As mentioned in [@K], the above regularity result can be false when $\Omega$ is smooth so that ${\mathcal{D}}$ and ${\mathcal{N}}$ meet tangentially, whereas it holds for purely Dirichlet/Neumann problem. For further work in this direction, see [@MR2503013; @OB] and the references therein.
In this paper, we work on the so-called “Reifenberg flat” domain, which is, roughly speaking, at every small scale the boundary is close to certain hyperplane. A Reifenberg flat domain is much more general than a Lipschitz domain with small Lipschitz constant: locally it is not given by a graph, and typically it contains fractal structures. The Reifenberg flat domain was introduced by Reifenberg in [@R] when he worked on the Plateau problem. Since then, there has been a lot of work on Reifenberg flat domains regarding minimal surfaces, harmonic measures, regularity of free boundaries, and divergence form elliptic/parabolic equations. An important fact for studying divergence form equation in such domains is that any small Reifenberg flat domain is a $W^{1,p}$-extension domain for every $p\in[1,\infty]$. Hence we have all the Sobolev inequalities up to the first order. For this result and the history of studying Reifenberg flat domains, one may refer to [@AMS].
Notice that although on Reifenberg flat domain, neither the outer normal nor the trace operator of $W^{1,p}$ is defined, the weak formulation in Definition \[def-weak-sln\] still makes sense due to the fact that no boundary integral term appears when $\Omega$ is smooth enough so that the outer normal and the trace operator are well defined.
We prove the solvability in Sobolev spaces $W^{1,p}$ and the $L_p$-estimates with $p$ being in the optimal range $4/3<p<4$ for the mixed problem with BMO coefficients on Reifenberg flat domains. The two boundary portions ${\mathcal{D}}$ and ${\mathcal{N}}$ are assumed to meet [*[almost]{}*]{} tangentially, which means ${\mathcal{D}}$ and ${\mathcal{N}}$ are separated by some Reifenberg flat set of co-dimension $2$ on the boundary. We note that our result holds for both bounded and unbounded domains. For the bounded domain case, we can further relax the assumptions on the source term. As mentioned before, since Lipschitz domains with small Lipschitz constant are Reifenberg flat, our results can be applied also on creased domains. Therefore, we see that in the restriction , the best possible range of $\varepsilon$ is $0<\varepsilon<1/4$ for creased domains with small Lipschitz constant.
This paper is a continuation of [@DK11; @DK12], in which elliptic systems on Reifenberg flat domains with rough coefficients and purely Dirichlet/conormal boundary conditions were studied. See also the series [@BW1; @BW2] regarding second-order equations on bounded domains. Our proof is mainly based on a perturbation argument suggested in [@Ca] by Caffarelli and Peral, by studying the level sets of maximal functions. The key step in our proof is to carefully design an approximation function near $\Gamma$, which combines the cut-off and reflection techniques in [@DK11; @DK12]. Compared to the purely Dirichlet or purely conormal problems, the approximation function in our problem is less regular, which is only $W^{1,4-{\varepsilon}}$, not Lipschitz. This situation is similar to [@arXiv:1806.02635], where dedicated decay rates of the level sets are required.
The paper is organized as follows. In Section 2, we introduce the basic notation, definitions, and assumptions. Our main results are given in Theorem \[thm-well-posedness\] for both bounded and unbounded domains and in Theorem \[thm-bounded-domain\] for bounded domains. In Section 3, we prove two useful tools for our problem: the local Sobolev-Poincaré inequality and the reverse Hölder inequality. Then in Section 4, we study a model problem, which is the $W^{1,4-{\varepsilon}}$ regularity of harmonic functions on the upper half space with mixed boundary conditions. With all these preparation, the proof of the main theorem including the approximation via cut-off and reflection, and the level set argument is presented in Section 5. In Section 6, we relax the regularity assumptions on the source term for the bounded domain case, mainly by solving a divergence form equation.
Notation and Main Results
=========================
Let $d$ be the space dimension. We write a typical point $x\in {\mathbb{R}}^d$ as $x=(x',x'')$, where $$x'=(x_1,x_2)\in {\mathbb{R}}^2, \quad x''=(x_3,\cdots,x_d)\in {\mathbb{R}}^{d-2}.$$ In the same spirit, for a domain $\Omega\subset {\mathbb{R}}^d$ and $p,q\ge 1$, we define the anisotropic space $L_{p,x'}L_{q,x''}(\Omega)$ as the set of all measurable functions $u$ on $\Omega$ having a finite norm $$\|u\|_{L_{p,x'}L_{q,x''}(\Omega)}=\Bigg(\int_{{\mathbb{R}}^2}\Bigg(\int_{{\mathbb{R}}^{d-2}}{\lvertu\rvert}^q {\mathbb{I}}_{\Omega}\,dx''\Bigg)^{p/q}\,dx'\Bigg)^{1/p},$$ where ${\mathbb{I}}$ is the usual indicator function. We abbreviate $L_{p, x'}L_{p,x''}(\Omega)=L_p(\Omega)$. We will also use the notation $$\label{181224@eq1}
{\mathbb{R}}^d_{+}={\{x=(x_1,\cdots,x_d)\in{\mathbb{R}}^d : x_1>0\}},\quad B_R^{+}={\mathbb{R}}^d_+\cap B_R(0),$$ $$B'_{R}=\{x'\in {\mathbb{R}}^2:|x'|<R\}, \quad (B'_R)^+={\mathbb{R}}^2_+\cap B'_R,$$ and $\Omega_R(x)=\Omega\cap B_R(x)$ for all $x\in {\mathbb{R}}^d$ and $R>0$.
Now we formulate our mixed boundary value problem. We consider domain $\Omega \subset {\mathbb{R}}^d$ with boundary divided into two non-intersecting portions, and $\Gamma$ being the boundary of ${\mathcal{D}}$ relative to ${\partial}\Omega$: $${\partial}\Omega={\mathcal{D}}\cup{\mathcal{N}}, \quad {\mathcal{D}}\cap {\mathcal{N}}=\emptyset,\quad \Gamma={\partial}_{{\partial}\Omega}{\mathcal{D}}.$$ We need the following notation for Sobolev spaces with boundary conditions prescribed on the whole or part of the boundary. For $1\le p\le \infty$, we denote by $W^{1,p}(\Omega)$ the usual Sobolev space and by $W^{1,p}_0(\Omega)$ the completion of $C^\infty_0(\Omega)$ in $W^{1,p}(\Omega)$, where $C^\infty_0(\Omega)$ is the set of all smooth, compactly supported functions in $\Omega$. Similarly, we let $W^{1,p}_{\mathcal{D}}(\Omega)$ be the completion of $C^\infty_{\mathcal{D}}(\Omega)$ in $W^{1,p}(\Omega)$, where $C_{\mathcal{D}}^\infty(\Omega)$ is the set of all smooth functions on $\overline{\Omega}$ which vanish in a neighborhood of ${\mathcal{D}}$.
Let $L$ be a second-order elliptic operator in divergence form $$L u=D_i(a_{ij}(x)D_j u+b_i(x) u)+\hat{b}_i(x) D_i u+c(x)u,$$ where the coefficients ${\boldsymbol{A}}=(a_{ij})_{i,j=1}^d$, ${\boldsymbol{b}}=(b_1,\ldots,b_d)$, $\hat{{\boldsymbol{b}}}=(\hat{b}_1,\ldots,\hat{b}_d)$, and $c$ are bounded measurable functions defined on $\overline{\Omega}$: for some positive constants $\Lambda$ and $K$, we have $$|{\boldsymbol{A}}|\le \Lambda^{-1}, \quad |{\boldsymbol{b}}|+|\hat{{\boldsymbol{b}}}|+|c|\le K.$$ Note that the summation convention is adopted throughout this paper. The leading coefficients ${\boldsymbol{A}}=(a_{ij})$ are also assumed to be symmetric, satisfy the uniformly ellipticity condition: $$\sum_{i,j=1}^d a_{ij}(x)\xi_j\xi_i\ge \Lambda |\xi|^2, \quad \forall \xi\in {\mathbb{R}}^d, \quad \forall x\in \overline{\Omega}.$$ We denote by $$Bu=({\boldsymbol{A}} Du+{\boldsymbol{b}} u)\cdot n=(a_{ij}D_j u+b_i u)\,n_i$$ the conormal derivative operator on the boundary of $\Omega$ associated with the operator $L$, where $n=(n_1,\ldots,n_d)$ is the outward unit normal to $\partial \Omega$. We will see that in the weak formulation, this boundary condition is still well defined even when the outer unit normal is not defined point-wise. Now we give the formal definition of weak solutions. Let $p\in (1,\infty)$.
\[def-weak-sln\] For $f, f_i\in L_p(\Omega)$, $i\in \{1,\ldots,d\}$, we say that $u\in W^{1,p}_{\mathcal{D}}(\Omega)$ is a weak solution to the mixed boundary value problem $$\label{180417@eq2}
\begin{cases}
Lu =f+ D_i f_i & \text{in }\, \Omega,\\
Bu =f_i n_i & \text{on }\, {\mathcal{N}},\\
u =0 & \text{on }\, {\mathcal{D}},
\end{cases}$$ if $$\int_\Omega (- a_{ij}D_j u - b_i u) D_i \phi + (\hat{b}_i D_i u + cu)\phi\,dx=\int_\Omega f \phi\,dx-\int_\Omega f_i D_i\phi\,dx$$ holds for any $\phi\in W^{1,p/(p-1)}_{{\mathcal{D}}}(\Omega)$.
In this paper, we will work on the so-called Reifenberg flat domains, which is defined below in $(i)$. In $(ii)$, we assume that locally the two types of boundary conditions are almost separated: the relative boundary $\Gamma$ is also Reifenberg flat.
\[ass-RF\] There exists a positive constant $R_1$ such that the following hold.
1. For any $x_0\in \partial \Omega$ and $R\in (0,R_1]$, there is a coordinate system depending on $x_0$ and $R$ such that in this new coordinate system (called the coordinate system associated with $(x_0, R)$), we have $$\{y:x_{01}+\gamma R<y_1\}\cap B_R(x_0)\subset \Omega_R(x_0)\subset \{y:x_{01}-\gamma R<y_1\}\cap B_R(x_0).$$
2. Let $\Gamma$ be the boundary (relative to $\partial \Omega$) of ${\mathcal{D}}$. If $x_0\in \Gamma$ and $R\in (0, R_1]$, we can further require that the coordinate system defined in $(i)$ satisfy $$\Gamma \cap B_R(x_0)\subset \{y: |y'-x'|< \gamma R\}\cap B_R(x_0),$$ $$\big(\partial \Omega \cap B_R(x_0)\cap \{y: y_2>x_{02}+\gamma R\}\big)\subset {\mathcal{D}},$$ $$\big(\partial \Omega \cap B_R(x_0)\cap \{y: y_2<x_{02}-\gamma R\}\big)\subset {\mathcal{N}}.$$
In this paper, we always assume that ${\mathcal{D}},{\mathcal{N}}\neq \emptyset$, since otherwise the boundary condition becomes purely conormal or Dirichlet. Corresponding results have been included in [@DK11; @DK12]. See also [@BW1; @BW2].
We consider the equations with small “BMO” leading coefficients with a small parameter $\theta\in (0,1)$ to be specified later.
\[ass-smallBMO\] There exists $R_2\in (0,1]$ such that for any $x\in \overline{\Omega}$ and $r\in (0, R_2]$, we have $$\dashint_{\Omega_r(x)}|a_{ij}(y)-(a_{ij})_{\Omega_r(x)}|\,dy < \theta.$$
In the following, we denote $R_0:= \min{\{R_1,R_2\}}$.
Now we can present our main result. First, in $\Omega$ (bounded or unbounded) we consider the existence and uniqueness of $W^{1,p}_{{\mathcal{D}}}$ weak solution to the following equation: $$\label{eqn-main-large-lambda}
\begin{cases}
Lu-\lambda u= f+ D_i f_i & \text{in }\, \Omega,\\
Bu =f_i n_i & \text{on }\, {\mathcal{N}},\\
u=0 &\text{on }\, {\mathcal{D}}.
\end{cases}$$ Compared to , here we introduce the $-\lambda u$ term to create the required decay at infinity for the unbounded domain case. For simplicity, we will use the following notation with $\lambda > 0$: $$U:={\lvertDu\rvert}+\sqrt{\lambda}{\lvertu\rvert},\quad F:=\sum_{i=1}^d{\lvertf_i\rvert}+\frac{1}{\sqrt{\lambda}}{\lvertf\rvert}.$$
\[thm-well-posedness\] For any $p\in (4/3,4)$, we can find positive constants $$(\gamma_0,\theta_0)=(\gamma_0,\theta_0)(d,p,\Lambda), \quad \lambda_0=\lambda_0(d,p,\Lambda, R_0, K),$$ such that the following holds. If Assumptions \[ass-RF\] $(\gamma_0)$ and \[ass-smallBMO\] $(\theta_0)$ are satisfied, and $\lambda > \lambda_0$, then for any $(f_i)_{i=1}^d\in (L_p(\Omega))^d$, $f\in L_p(\Omega)$ there exists a unique weak solution $u\in W^{1,p}_{\mathcal{D}}(\Omega)$ to satisfying $$\label{eqn-general}
{\lVertU\rVert}_{L_p(\Omega)} \leq N {\lVertF\rVert}_{L_p(\Omega)},$$ where $N=N(d,p,\Lambda)$ is a constant.
When $\Omega$ is bounded, we have better results: instead of taking large $\lambda$, we can assume the usual sign condition $L1\le 0$, which is understood in the weak sense: $$\int_{\Omega} (-b_i D_i \phi+c \phi)\,dx\le 0$$ for any $\phi\in W^{1,p/(p-1)}_{{\mathcal{D}}}(\Omega)$ satisfying $\phi\ge 0$. Also, the integrability of the non-divergence form source term $f$ can be generalized to $L_{p_{*}}$, where $$\label{eqn-def-p*}
p_{*}=\begin{cases}
pd/(p+d) &\text{when }\, p>d/(d-1),\\
1+{\varepsilon}&\text{when }\, p\leq d/(d-1)
\end{cases}$$ for any ${\varepsilon}>0$.
\[thm-bounded-domain\] Let $\Omega$ be a bounded domain in ${\mathbb{R}}^d$. For any $p\in(4/3,4)$, we can find positive constants $\gamma_0, \theta_0$ depending on $(d,p,\Lambda)$, such that the following holds. If Assumptions \[ass-RF\] $(\gamma_0)$ and \[ass-smallBMO\] $(\theta_0)$ are satisfied, and $L1\leq0$ in the weak sense, then for any $(f_i)_{i=1}^d\in (L_p(\Omega))^d$, $f\in L_{p_*}(\Omega)$ there exists a unique weak solution $u\in W^{1,p}_{\mathcal{D}}(\Omega)$ to satisfying $$\label{eqn-a-priori-no-dependence}
{\lVertu\rVert}_{W^{1,p}(\Omega)} \leq N\Bigg(\sum_{i=1}^d{\lVertf_i\rVert}_{L_p(\Omega)} + {\lVertf\rVert}_{L_{p_{*}}(\Omega)}\Bigg),$$ where $N$ is a constant independent of $u,f_i$ and $f$.
In the above theorems, we always assume that $$\label{190109@eq1}
p\in (4/3,4), \quad \text{${\boldsymbol{A}}=(a_{ij})_{i,j=1}^d$ is symmetric}.$$ Indeed, by the Lax-Milgram Lemma and the reverse Hölder’s inequality, when $p$ is close to $2$, the symmetry of ${\boldsymbol{A}}$ is not needed. Otherwise, by the following two examples, we see that the restrictions in are optimal for the solvability of mixed boundary value problems. Precisely, based on a duality argument, Example \[eg-classical\] shows the restriction $p\in (4/3,4)$ is optimal, and Example \[190109@ex1\] shows the symmetry of ${\boldsymbol{A}}$ is required for the solvability in $W^{1,p}(\Omega)$ when $p$ is away from $2$. Here, for the reader’s convenience, we temporarily set $${\mathbb{R}}^2_+=\{x=(x_1,x_2)\in {\mathbb{R}}^2:x_2>0\},$$ which is different from that in . Note that the examples below are applicable to higher dimensional cases by a trivial extension.
\[eg-classical\] In ${\mathbb{R}}^2_+$, let $u(x_1,x_2)={\rm Im}(x_1+ix_2)^{1/2}$. One can simply check that $$\Delta u =0 \text{ in } {\mathbb{R}}^2_{+}, \quad u=0 \text{ on } {\partial}{\mathbb{R}}^2_{+}\cap{\{x_1>0\}}, \quad \frac{{\partial}u}{{\partial}x_2}=0 \text{ on } {\partial}{\mathbb{R}}^2_{+}\cap{\{x_1<0\}}.$$ Since $Du$ is of order $r^{-1/2}$, one could also check that near the origin $Du\in L_p$ for any $p\in [1,4)$, but $Du\notin L_4$.
\[190109@ex1\] In ${\mathbb{R}}^2_+$, let $u(x_1,x_2)={\rm Im}(x_1+ix_2)^s$ with $s \in (0,1/2)$. We have $$D_i(a_{ij}D_j u)=0 \text{ on } {\mathbb{R}}^2_{+},\quad u=0 \text{ on } {\partial}{\mathbb{R}}^2_{+}\cap{\{x_1>0\}},\quad a_{ij}D_j u n_i=0 \text{ on }{\partial}{\mathbb{R}}^2_{+}\cap{\{x_1<0\}},$$ where $$(a_{ij})_{i,j=1}^{2} = \begin{bmatrix}
1 & \cot(\pi s)\\-\cot(\pi s) & 1
\end{bmatrix}.$$ Since $Du$ is of order $r^{s-1}$, near the origin we only have $Du\in L_p$ only if $p<\frac{2}{1-s}$. Note that $\frac{2}{1-s}<4$ and $\frac{2}{1-s}\searrow 2$ as $s\searrow 0$.
Local Poincaré Inequality and Reverse Hölder’s Inequality
=========================================================
In this section, we introduce two useful tools for our problem. The first one is the local Sobolev-Poincaré inequality. Notice that a Reifenberg flat domain intersecting with a ball might no longer be Reifenberg flat. We cannot simply localize to obtain the required local version, although Sobolev inequalities of $W^{1,p}$ hold for the Reifenberg flat domain since it is an extension domain.
\[180514@thm1\] Let $\gamma\in [0,1/48]$ and $\Omega\subset {\mathbb{R}}^d$ be a Reifenberg flat domain satisfying Assumption \[ass-RF\] $(\gamma)$ $(i)$. Let $x_0\in \partial \Omega$ and $R\in (0, R_1/4]$. Then, for any $p\in (1,d)$ and $u\in W^{1,p}(\Omega_{2R}(x_0))$, we have $$\|u-(u)_{\Omega_R(x_0)}\|_{L_{dp/(d-p)}(\Omega_R(x_0))}\le N\|Du\|_{L_p(\Omega_{2R}(x_0))},$$ where $N=N(d,p)$.
See [@CDK18 Theorem 3.5].
\[180514@cor1\] Let $\gamma\in [0,1/48]$ and $\Omega\subset {\mathbb{R}}^d$ be a Reifenberg flat domain satisfying Assumption \[ass-RF\] $(\gamma)$ $(i)$. Let $x_0\in \overline{\Omega}$, $R\in (0, R_1/4]$, and ${\mathcal{D}}\subset \partial \Omega$ with ${\mathcal{D}}\cap B_R(x_0)\neq \emptyset$. If there exist $z_0\in {\mathcal{D}}\cap B_R(x_0)$ and $\alpha\in (0,1)$ such that $$\label{eqn-mainly-Dirichlet}
B_{\alpha R}(z_0)\subset B_R(x_0), \quad \big(\partial \Omega\cap B_{\alpha R}(z_0)\big) \subset \big({\mathcal{D}}\cap B_R(x_0)\big),$$ then the following hold.
1. For any $p\in (1,d)$ and $u\in W^{1,p}_{{\mathcal{D}}}(\Omega)$, we have $$\|u\|_{L_{dp/(d-p)}(\Omega_R(x_0))}\le N\|Du\|_{L_p(\Omega_{2R}(x_0))},$$ where $N=N(d,p, \alpha)$.
2. For any $u\in W^{1,2}_{{\mathcal{D}}}(\Omega)$, we have $$\|u\|_{L_2(\Omega_R(x_0))}\le NR\|Du\|_{L_2(\Omega_{2R}(x_0))},$$ where $N=N(d,\alpha)$.
The assertion $(b)$ is a simple consequence of the assertion $(a)$. Indeed, by taking $p\in \big(\frac{2d}{d+2}, 2\big)$, and using Hölder’s inequality and the assertion $(a)$, we have $$\begin{aligned}
\|u\|_{L_2(\Omega_R(x_0))}&\le NR^{d/2-d/p+1}\|u\|_{L_{dp/(d-p)}(\Omega_R(x_0))}\\
&\le NR^{d/2-d/p+1}\|Du\|_{L_p(\Omega_{2R}(x_0))}\le N R\|Du\|_{L_2(\Omega_{2R}(x_0))}.
\end{aligned}$$ To prove the assertion $(a)$, we extend $u$ by zero on $B_{\alpha R}(z_0)\setminus \Omega$ so that $u\in W^{1,p}(B_{\alpha R}(z_0))$. Since $|B_{\alpha R}(z_0)\setminus \Omega|\ge N(d)(\alpha R)^d$, by the boundary Poincaré inequality, we have $$\label{180514@A1}
\|u\|_{L_{dp/(d-p)}(\Omega_{\alpha R}(z_0))}\le N(d,q)\|Du\|_{L_p(\Omega_{\alpha R}(z_0))}.$$ Notice from the triangle inequality and Hölder’s inequality that $$\begin{aligned}
&\|u\|_{L_{dp/(d-p)}(\Omega_R(x_0))}\\
&\le \|u-(u)_{\Omega_R(x_0)}\|_{L_{dp/(d-p)}(\Omega_R(x_0))} +\|(u)_{\Omega_R(x_0)}-(u)_{\Omega_{\alpha R}(z_0)}\|_{L_{dp/(d-p)}(\Omega_R(x_0))}\\
&\quad +\|(u)_{\Omega_{\alpha R}(z_0)}\|_{L_{dp/(d-p)}(\Omega_R(x_0))}\\
&\le N\alpha^{1-d/p}\big(\|u-(u)_{\Omega_R(x_0)}\|_{L_{dp/(d-p)}(\Omega_R(x_0))} + \|u\|_{L_{dp/(d-p)}(\Omega_{\alpha R}(z_0))}\big).
\end{aligned}$$ This combined with Theorem \[180514@thm1\] and gives the desired estimate.
In the rest of the section, we shall prove the reverse Hölder’s inequality for the following mixed boundary value problem without lower order terms $$\label{eqn-main-no-lower}
\begin{cases}
D_i(a_{ij}D_j u)-\lambda u= f+ D_i f_i & \text{in }\, \Omega,\\
a_{ij}D_j u n_i = f_i n_i & \text{on }\, {\mathcal{N}},\\
u=0 & \text{on }\, {\mathcal{D}}.
\end{cases}$$ Here, we do not impose any regularity assumption (including the symmetry condition) on the coefficients $a_{ij}$. Recall the notation that for $\lambda>0$, $$U:={\lvertDu\rvert}+\sqrt{\lambda}{\lvertu\rvert},\quad F:=\sum_{i=1}^d{\lvertf_i\rvert}+\frac{1}{\sqrt{\lambda}}{\lvertf\rvert}.$$
\[180514@lem1\] Let $\gamma\in (0,1/48]$, $\frac{2d}{d+2}<p<2$, and $\Omega\subset {\mathbb{R}}^d$ be a Reifenberg flat domain satisfying Assumption \[ass-RF\] $(\gamma)$. Suppose that $u\in W^{1,2}_{\mathcal{D}}(\Omega)$ satisfies with $f_i,f\in L_2(\Omega)$. Let $x_0\in \overline{\Omega}$ and $R\in (0, R_1]$, satisfying either $$B_{R/16}(x_0)\subset \Omega \quad \text{or}\quad x_0\in \partial \Omega.$$ Then, when $\lambda>0$, we have $$\int_{\Omega_{R/32}(x_0)}U^2\,dx\le NR^{d(1-2/p)}\Bigg(\int_{\Omega_{R}(x_0)}U^p\,dx\Bigg)^{2/p}+N\int_{\Omega_{R}(x_0)}F^2\,dx.$$ When $\lambda=0$ and $f\equiv0$, we have $$\int_{\Omega_{R/32}(x_0)}|Du|^2\,dx\le NR^{d(1-2/p)}\Bigg(\int_{\Omega_{R}(x_0)}|Du|^p\,dx\Bigg)^{2/p}+N\int_{\Omega_{R}(x_0)}|f_i|^2\,dx.$$ In the above, the constant $N$ depends only on $d$, $p$, and $\Lambda$.
Here we only prove for the case $\lambda>0$. When $\lambda=0$, the proof still works if we replace $U$ by $|Du|$ and $F$ by $\sum_i |f_i|$. Also, we prove only the case $x_0\in \partial \Omega$ because the proof for the interior case is similar to the one in case (ii) for purely conormal boundary conditions. Without loss of generality, we assume that $x_0=0$. Let us fix $R\in (0, R_1]$. We consider the following two cases: $$B_{R/16}\cap \Gamma \neq \emptyset, \quad B_{R/16}\cap \Gamma=\emptyset.$$
1. $B_{R/16}\cap \Gamma\neq \emptyset$. We take $y_0\in \Gamma$ such that $\operatorname{dist}(0, \Gamma)=|y_0|$, and observe that $$\label{180515@eq1a}
B_{R/16}\subset B_{R/8}(y_0)\subset B_{R/2}(y_0)\subset B_{R}.$$ Since $u\in W^{1,2}_{{\mathcal{D}}}(\Omega)$, as a test function to , we can use $\eta^2 u\in W^{1,2}_{{\mathcal{D}}}(\Omega)$, where $\eta$ is a smooth function on ${\mathbb{R}}^d$ satisfying $$0\le \eta\le 1, \quad \eta\equiv 1 \,\text{ on }\, B_{R/8}(y_0), \quad \operatorname{supp} \eta \subset B_{R/4}(y_0), \quad |\nabla \eta|\le NR^{-1}.$$ Now, using Hölder’s inequality and Young’s inequality, we have $$\label{180515@eq1}
\int_{\Omega_{R/4}(y_0)} \eta^2 U^2\,dx\le \frac{N}{R^2}\int_{\Omega_{R/4}(y_0)} |u|^2\,dx+N\int_{\Omega_{R/4}(y_0)}F^2\,dx,$$ where $N=N(d,\Lambda)$. We fix a coordinate system associated with $(y_0, R/4, \Gamma)$ satisfying the properties in Assumption \[ass-RF\] $(\gamma)$ $(ii)$. Since we have $$\big(\partial \Omega \cap B_{R/4}(y_0)\cap \{y:y_2>\gamma R/4\}\big)\subset {\mathcal{D}},$$ there exists $z_0\in {\mathcal{D}}$ satisfying $$B_{R/16}(z_0)\subset B_{R/4}(y_0), \quad
\big(\partial \Omega \cap B_{R/16}(z_0) \big)\subset \big({\mathcal{D}}\cap B_{R/4}(y_0)\big).$$ Note that because $\frac{2d}{d+2}<p<2$, we have $\frac{dp}{d-p}>2$. Then by Hölder’s inequality and Corollary \[180514@cor1\] $(a)$, we see that $$\begin{aligned}
\nonumber
\frac{1}{R^2}\int_{\Omega_{R/4}(y_0)}|u|^2\,dx&\le NR^{d(1-2/p)}\Bigg(\int_{\Omega_{R/4}(y_0)} |u|^{dp/(d-p)}\,dx\Bigg)^{2(d-p)/dp}\\
\label{180515@eq1b}
&\le N R^{d(1-2/p)}\Bigg(\int_{\Omega_{R/2}(y_0)}|Du|^p\,dx\Bigg)^{2/p},\end{aligned}$$ where $N=N(d,p)$. Combining this inequality and , and using , we obtain the desired estimate.
2. $B_{R/16}\cap \Gamma=\emptyset$. Then ${\partial}\Omega\cap B_{R/16}$ is contained in either ${\mathcal{D}}$ or ${\mathcal{N}}$. When it is in ${\mathcal{D}}$, the proof for the previous case still works if we simply choose any $y_0\in{\partial}\Omega\cap B_{R/16}$. When it is contained in ${\mathcal{N}}$, as a test function to , we can use $\zeta^2(u-c)\in W^{1,2}_{{\mathcal{D}}}(\Omega)$, where $c=(u)_{\Omega_{R/16}}$ and $\zeta$ is a smooth function on ${\mathbb{R}}^d$ satisfying $$0\le \zeta\le 1, \quad \zeta\equiv 1 \,\text{ on }\, B_{R/32}, \quad \operatorname{supp} \zeta \subset B_{R/16}, \quad |\nabla \zeta|\le NR^{-1}.$$ By testing with $\zeta^2 (u-c)$, we have $$\int_{\Omega_{R/16}} (\zeta U)^2\,dx\le \frac{N}{R^2}\int_{\Omega_{R/16}} \big|u-(u)_{\Omega_{R/16}}\big|^2\,dx+\frac{N}{R^d} \Bigg(\int_{\Omega_{R/16}} \sqrt{\lambda}|u|\,dx\Bigg)^2+N\int_{\Omega_{R/16}} F^2\,dx,$$ where $N=N(d,\Lambda)$. Similar to , we get from Theorem \[180514@thm1\] that $$\frac{1}{R^2}\int_{\Omega_{R/16}}\big|u-(u)_{\Omega_{R/16}}\big|^2\,dx \le NR^{d(1-2/p)}\Bigg(\int_{\Omega_{R/8}}|Du|^p\,dx\Bigg)^{2/p},$$ where $N=N(d,p)$. By Hölder’s inequality, we also have $$\frac{1}{R^d}\Bigg(\int_{\Omega_{R/16}} \sqrt{\lambda}|u|\,dx\Bigg)^2\le NR^{d(1-2/p)}\Bigg(\int_{\Omega_{R/16}} \big(\sqrt{\lambda}|u|\big)^p\,dx\Bigg)^{2/p}.$$ Combining these together, we obtain the desired estimate.
The lemma is proved.
Based on Lemma \[180514@lem1\] and Gehring’s lemma, we get the following reverse Hölder’s inequality.
\[lem-reverse-holder\] Let $\gamma\in (0, 1/48]$, $p>2$, and $\Omega\subset {\mathbb{R}}^d$ be a Reifenberg flat domain satisfying Assumption \[ass-RF\] $(\gamma)$. Suppose that $u\in W^{1,2}_{\mathcal{D}}(\Omega)$ satisfies with $f_i, f\in L_p(\Omega)\cap L_2(\Omega)$. Then there exist constants $p_0\in (2,p)$ and $N>0$, depending only on $d$, $p$, and $\Lambda$, such that for any $x_0\in {\mathbb{R}}^d$ and $R\in (0, R_1]$, the following hold. When $\lambda>0$, we have $$\big(\overline{U}^{p_0}\big)^{1/p_0}_{B_{R/2}(x_0)}\le N \big(\overline{U}^{2}\big)^{1/2}_{B_{R}(x_0)}+N\big(\overline{F}^{p_0}\big)^{1/p_0}_{B_{R}(x_0)}.$$ When $\lambda=0$ and $f\equiv 0$, we have $$\big(|D\overline{u}|^{p_0}\big)^{1/p_0}_{B_{R/2}(x_0)}\le N \big(|D\overline{u}|^{2}\big)^{1/2}_{B_{R}(x_0)}+N\big(|\overline{f_i}|^{p_0}\big)^{1/p_0}_{B_{R}(x_0)},$$ where $\overline{U}$, $\overline{F}$, $D\overline{u}$, and $\overline{f_i}$ are the extensions of $U$, $F$, $Du$, and $f_i$ to ${\mathbb{R}}^d$ so that they are zero on ${\mathbb{R}}^d\setminus \Omega$.
Again, we only prove for the case $\lambda>0$. Let us fix a constant $p_1\in \big(\frac{2d}{d+2},2\big)$, and set $$\Phi=\overline{U}^{p_1}, \quad \Psi=\overline{F}^{p_1}.$$ Then by Lemma \[180514@lem1\], we have $$\label{180515@eq2}
\int_{B_{R/112}(x_0)} \Phi^{2/p_1}\,dx\le NR^{d(1-2/p_1)}\Bigg(\int_{B_R(x_0)} \Phi\,dx\Bigg)^{2/p_1}+N \int_{B_R(x_0)} \Psi^{2/p_1}\,dx$$ for any $x_0\in {\mathbb{R}}^d$ and $R\in (0, R_1]$, where $N=N(d,\Lambda, p)=N(d,\Lambda)$. Indeed, if $B_{R/56}(x_0)\subset \Omega$, then follows from Lemma \[180514@lem1\]. In the case when $B_{R/56}(x_0)\cap \partial \Omega\neq \emptyset$, there exists $y_0\in \partial \Omega$ such that $|x_0-y_0|=\operatorname{dist}(x_0, \partial \Omega)$ and $$B_{R/112}(x_0)\subset B_{3R/112}(y_0) \subset B_{6R/7}(y_0)\subset B_{R}(x_0).$$ Using this together with Lemma \[180514@lem1\], we get . If $B_{R/56}(x_0)\subset {\mathbb{R}}^d\setminus \Omega$, by the definition of $\overline{U}$, holds.
By and a covering argument, we have $$\dashint_{B_{R/2}(x_0)} \Phi^{2/p_1}\,dx\le N\Bigg(\dashint_{B_R(x_0)} \Phi\,dx\Bigg)^{2/p_1}+N \dashint_{B_R(x_0)} \Psi^{2/p_1}\,dx$$ for any $x_0\in {\mathbb{R}}^d$ and $R\in (0, R_1]$, where $N=N(d,\Lambda)$. Therefore, by Gehring’s lemma (see, for instance, [@G Ch. V]), we get the desired estimate. The lemma is proved.
Harmonic functions in half space with mixed boundary condition
==============================================================
In this section, we prove a regularity result for harmonic functions with mixed Dirichlet-Neumann boundary conditions on half space. We denote $$B_R=B_R(0),\quad \Gamma_R^{+}:= B_R\cap {\{x_1=0,x_2>0\}},\quad \Gamma_R^{-}:= B_R\cap {\{x_1=0,x_2<0\}}.$$
\[thm-harmonic-mixed-halfspace\] Suppose $u \in W^{1,2}_{\Gamma^{+}_R}(B_R^{+})$ is a weak solution to $$\begin{cases}
\Delta u -\lambda u =0 & \text{in }B^{+}_R,\\
\frac{{\partial}u}{{\partial}x_1}=0 & \text{on } \Gamma_R^{-},\\
u=0 &\text{on } \Gamma_R^{+},
\end{cases}$$ where $\lambda >0$. Then for any $p\in[2,4)$, we have $u\in W^{1,p}(B^{+}_{R/4})$ with $$(U^p)_{B^{+}_{R/4}}^{1/p} \leq N(d,p) (U^2)^{1/2}_{B_R^{+}}.$$ In the case when $\lambda=0$, the same estimate holds with $|Du|$ in place of $U$.
In Theorem \[thm-harmonic-mixed-halfspace\], the boundary condition is only prescribed on the flat part of the boundary. Hence the meaning of “weak solution” is slightly different. In the theorem and throughout the paper, a $W^{1,2}_{\Gamma_R^{+}}(B^{+}_R)$ weak solution means: for any $\phi\in W^{1,2}(B^{+}_R)$ satisfying $\phi=0$ on ${\partial}B^{+}_R\setminus \Gamma_R^{-}$, $$\int_{B_R^{+}} (\nabla u \cdot \nabla \phi + \lambda u\phi)\,dx = 0.$$ It is clear that, as a test function, one can use $\eta u$, where $\eta\in C^\infty_c(B_R)$.
For the proof of Theorem \[thm-harmonic-mixed-halfspace\], we will use the following two dimensional regularity result.
\[lem-laplacian-2d\] In the half ball $B_R^{+}\subset {\mathbb{R}}^2$, consider $u\in W^{1,2}_{\Gamma_R^+}(B_R^{+})$ which solves $$\begin{cases}
\Delta u = f &\text{in }B^{+}_R,\\
\frac{{\partial}u}{{\partial}x_1}=0 &\text{on } \Gamma_R^{-},\\
u=0 &\text{on } \Gamma_R^{+},\\
\end{cases}$$ where $f\in L_2(B_R^{+})$. Then for any $p\in[2,4)$, we have $u \in W^{1,p}(B^{+}_{R/2})$ with $$\label{2destimate}
({\lvertDu\rvert}^p)_{B^{+}_{R/2}}^{1/p} \leq N(p) \big(({\lvertDu\rvert}^2)^{1/2}_{B_R^{+}} + R({\lvertf\rvert}^2)^{1/2}_{B_R^{+}}\big).$$
From the proof below, it is clear that in Lemma \[lem-laplacian-2d\], $R/2$ can be replaced with any $r\in (0,R)$. In this case, the constant $N$ also depends on $r$ and $R$.
By a scaling argument, we may assume $R=1$. We consider the following change of variables: $(y_1,y_2)\in B_{1}\cap {\{y_1>0,y_2>0\}}\mapsto (x_1,x_2)\in B^{+}_1$ : $$x_1=2y_1y_2,\quad x_2=y_2^2-y_1^2,$$ or in complex variables: $$x_2+ix_1=(y_2+iy_1)^2.$$ Write $\widetilde{u}(y_1,y_2)=u(x_1,x_2)$ and $\widetilde{f}(y_1,y_2)=f(x_1,x_2)$. Then we can rewrite the equation as $$\begin{cases}
\Delta_y \widetilde{u}=4{\lverty\rvert}^2\widetilde{f} & \text{in } B_1^{++},\\
\frac{{\partial}\widetilde{u}}{{\partial}y_2}=0 & \text{on } B_{1}\cap {\{y_1>0, y_2=0\}},\\
\widetilde{u}=0 &\text{on } B_{1}\cap {\{y_1=0,y_2>0\}},
\end{cases}$$ where $B_1^{++}:=B_1\cap \{y_1>0,y_2>0\}$. Next, we take an even extension of $\widetilde{u}$ and $\widetilde{f}$ with respect to $y_2$-variable. Still denote the extended functions on $B_{1}^{+}$ by $\widetilde{u}$ and $\widetilde{f}$. Then the following equation is satisfied: $$\begin{cases}
\Delta_y \widetilde{u}=4{\lverty\rvert}^2\widetilde{f} & \text{in } B_{1}^{+},\\
\widetilde{u}=0 & \text{on } B_{1} \cap \{y_1=0\}.
\end{cases}$$ Note that $$|D_x u|\le \frac{N}{|y|}|D_y \widetilde{u}|
,\quad dx=4{\lverty\rvert}^2\,dy.$$ By the Sobolev embedding theorem, the local $W^2_2$ estimate for elliptic equations, and the boundary Poincaré inequality, we obtain $$\begin{aligned}
{\lVertD_y \widetilde{u}\rVert}_{L_q(B^{+}_{\sqrt 2/2})}
\le {\lVert\widetilde{u}\rVert}_{W^{2,2}(B_{\sqrt 2/2}^{+})}
&\leq N \big({\lVert\widetilde{u}\rVert}_{L_2(B^{+}_{1})} + {\lVert4{\lverty\rvert}^2 \widetilde{f}\rVert}_{L_2(B^{+}_{1})}\big)\\
&\leq N \big({\lVertD_y \widetilde{u}\rVert}_{L_2(B^{+}_{1})} + {\lVert4{\lverty\rvert}^2 \widetilde{f}\rVert}_{L_2(B^{+}_{1})}\big)\\
&\leq N \big({\lVertD_y \widetilde{u}\rVert}_{L_2(B^{++}_{1})} + {\lVert4{\lverty\rvert}^2 \widetilde{f}\rVert}_{L_2(B^{++}_{1})}\big),\end{aligned}$$ where $N=N(p)>0$ and $q=q(p)$ is a constant with $$\label{181225@A1}
q>\frac{2p}{4-p}\ge p.$$ Here we also used the fact that $\widetilde{u}$ and $\widetilde{f}$ are both even functions in $y_2$. Translating back to $x$-variables, we obtain $$\begin{aligned}
{\lVert{\lvertx\rvert}^{\frac{q-2}{2q}}D_xu\rVert}_{L_q(B^{+}_{1/2})}
&\leq N \big({\lVertD_x u\rVert}_{L_2(B^{+}_{1})} + {\lVert{\lvertx\rvert}^{1/2}f\rVert}_{L_2(B^{+}_{1})}\big)\nonumber\\
&\leq N \big({\lVertD_x u\rVert}_{L_2(B^{+}_1)} + {\lVertf\rVert}_{L_2(B^{+}_{1})}\big).\label{2dmidstep}\end{aligned}$$ By Hölder’s inequality and , we get $$\begin{aligned}
{\lVertD_x u\rVert}_{L_p(B^{+}_{1/2})}
&\leq {\lVert{\lvertx\rvert}^{\frac{q-2}{2q}}D_x u\rVert}_{L_q(B^{+}_{1/2})} {\lVert{\lvertx\rvert}^{-\frac{q-2}{2q}}\rVert}_{L_{qp/(q-p)}(B_{1/2}^{+})}\\
&\leq N{\lVert{\lvertx\rvert}^{\frac{q-2}{2q}}D_{x}u\rVert}_{L_q(B^{+}_{1/2})}.\end{aligned}$$ Combining this with , we obtain $${\lVertD_x u\rVert}_{L_p(B^{+}_{1/2})} \leq N\big({\lVertD_x u\rVert}_{L_2(B^{+}_{1})}+{\lVertf\rVert}_{L_2(B^{+}_{1})}\big),$$ which is exactly . The lemma is proved.
We are now ready to present the proof of Theorem \[thm-harmonic-mixed-halfspace\].
We first prove the theorem for $\lambda=0$. By a scaling argument and Lemma \[lem-laplacian-2d\], we may assume $R=1$ and $d\ge 3$. Noting that we can differentiate both the equation and the boundary condition in $x''$-direction, the following Caccioppoli type inequality holds: $$\|D_x (D^k_{x''}u)\|_{L_2(B_s^+)}\le \frac{N(d,k)}{|t-s|^{k}}\|Du\|_{L_2(B_t^+)}$$ for $0<s<t\le 1$ and $k\in \{0,1,2,\ldots\}$. Thus by anisotropic Sobolev embedding, we can increase the integrability in $x''$-variables so that $$D_{x'}u\in L_{2,x'}L_{p,x''}(B^+_{r}),
\quad D_{x''}u,\,D_{x''}^2u\in L_{p}(B^+_{r})\quad \forall r<1,$$ with the estimate $$\label{181207@eq1}
\|D_{x'}u\|_{L_{2,x'}L_{p,x''}(B^+_{r})}
+\|D_{x''}u\|_{L_{p}(B^+_{r})}
+\|D_{x''}^2u\|_{L_{p}(B^+_{r})}
\le N(d,p,r)\|Du\|_{L_2(B^+_1)}.$$ It remains to estimate $D_{x'}u$. From , for almost every ${\lvertx''\rvert}<1/2$, we have $$D^2_{x''}u(\cdot,x'')\in L_2\big((B'_{2/3})^+\big).$$ Now we rewrite the equation as a $2$-dimension problem in $x'$-variables: $$\begin{cases}
\Delta_{x'}u(\cdot,x'')= -\Delta_{x''}u(\cdot,x'') & \text{in }(B'_{2/3})^{+},\\
\frac{{\partial}u}{{\partial}x_1}=0 & \text{on } B'_{2/3}\cap {\{x_1=0,x_2<0\}},\\
u=0 &\text{on } B'_{2/3}\cap {\{x_1=0,x_2>0\}}.
\end{cases}$$ We apply a properly rescaled version of Lemma \[lem-laplacian-2d\] to see that for almost every $|x''|<1/2$, $$\|D_{x'}u(\cdot,x'')\|_{L_p((B'_{1/2})^+)} \le N(p)\big(\|D_{x'}u(\cdot,x'')\|_{L_2((B'_{2/3})^+)}
+\|\Delta_{x''}u(\cdot,x'')\|_{L_2((B'_{2/3})^+)}\big).$$ Taking $L_p$ norm in $\{x''\in {\mathbb{R}}^{d-2}:|x''|<1/2\}$ for both sides, and using the Minkowskii inequality and with $r=\sqrt 3/2$, we obtain $D_{x'}u\in L_p(B_{1/2}^+)$ and $$\|D_{x'}u\|_{L_p(B_{1/2}^+)}\le N(d,p)\|Du\|_{L_2(B_1^+)}.$$ This gives the desired estimate for $\lambda=0$.
For a general $\lambda>0$, we use an idea by S. Agmon. We define $$v(x,\tau)=u(x)\cos(\sqrt{\lambda}\tau+\pi/4),$$ and observe that $v$ satisfies $$\label{181122@eq1}
\begin{cases}
\Delta_{(x,\tau)} v = 0 & \text{in } \hat{B}^{+}_1,\\
\frac{{\partial}v}{{\partial}x_1}=0 & \text{on } \hat{\Gamma}_1^{-},\\
v=0 & \text{on } \hat{\Gamma}_1^{+}.
\end{cases}$$ where $$\hat{B}_1=\{(x,\tau)\in {\mathbb{R}}^{d+1}: |(x,\tau)|<1\}, \quad \hat{B}_1^+=\hat{B}_1\cap \{x_1>0\},$$ $$\hat{\Gamma}_1^+=\hat{B}_1\cap \{x_1=0, x_2>0\}, \quad \hat{\Gamma}^-_1=\hat{B}_1\cap \{x_1=0, x_2<0\}.$$ By applying the result for $\lambda=0$ to , we have $$\label{181122@eq1a}
(|D_{(x,\tau)}v|^p)^{1/p}_{\hat{B}_{1/2}^+}\le N(|D_{(x,\tau)}v|^2)^{1/2}_{\hat{B}_{1}^+},$$ where $N=N(d,p)$. Note that the function $\Phi$ given by $$\Phi(\lambda)=\int_0^{1/4} \big|\cos(\sqrt{\lambda} \tau +\pi/4)\big|^p\,d\tau$$ has a positive lower bound depending only on $p$. Thus by using and the fact that $$\big|Du(x)\cos(\sqrt{\lambda} \tau+\pi/4)\big|\le |D_{(x,\tau)}v(x,\tau)|\le U(x),$$ we have $$\begin{aligned}
\int_{B_{1/4}^+} |Du|^p\,dx&\le N\int_0^{1/4} \int_{B_{1/4}^+} |Du|^p \big|\cos(\sqrt{\lambda} \tau +\pi/4)\big|^p \,dx\,d\tau \\
&\le N\int_{\hat{B}^+_{1/2}} |D_{(x,\tau)}v|^p \,dx \, d\tau\le N \Bigg(\int_{B_1^+}U^2\,dx\Bigg)^{p/2},\end{aligned}$$ where $N=N(d,p)$. Similarly, from the fact that $$\big|\sqrt{\lambda}u(x)\sin(\sqrt{\lambda} \tau+\pi/4)\big|\le |D_{(x,\tau)}v(x,\tau)|\le U(x),$$ we obtain $$\int_{B_{1/4}^+} \big|\sqrt{\lambda}u\big|^p \,dx\le N\Bigg(\int_{B_1^+}U^2\,dx\Bigg)^{p/2}.$$ Combining these together we get the desired estimate. The theorem is proved.
Regularity of $W^{1,2}_{{\mathcal{D}}}$ weak solutions
======================================================
The crucial step in proving unique $W^{1,p}$ solvability is the following improved regularity result. As in Theorem \[thm-well-posedness\], we consider a domain $\Omega\subset {\mathbb{R}}^d$ which can be bounded or unbounded, together with nonempty boundary portions ${\mathcal{D}},{\mathcal{N}}$. Again, recall the notation that for $\lambda>0$, $$U:={\lvertDu\rvert}+\sqrt{\lambda}{\lvertu\rvert},\quad F:=\sum_{i=1}^d{\lvertf_i\rvert}+\frac{1}{\sqrt{\lambda}}{\lvertf\rvert}.$$
\[prop-regularity\] For any $p\in (2,4)$, we can find positive constants $\gamma_0,\theta_0$ depending on $(d,p,\Lambda)$, such that if Assumptions \[ass-RF\] $(\gamma_0)$ and \[ass-smallBMO\] $(\theta_0)$ are satisfied, the following holds. For any $W^{1,2}_{{\mathcal{D}}}(\Omega)$ weak solution $u$ to with $\lambda>0$ and $f_i, f\in L_p(\Omega)\cap L_2(\Omega)$, we have $u \in W^{1,p}_{{\mathcal{D}}}(\Omega)$ and $$\label{est-no-lower-order}
{\lVertU\rVert}_{L_p(\Omega)} \leq N (R_0^{d(1/p-1/2)}{\lVertU\rVert}_{L_2(\Omega)} + {\lVertF\rVert}_{L_p(\Omega)}).$$ Furthermore, if we also have $f\equiv 0$, then we can take $\lambda=0$, and the following estimate holds: $$\label{eqn-no-nondiv}
{\lVertDu\rVert}_{L_p(\Omega)} \leq N(R_0^{d(1/p-1/2)}{\lVertDu\rVert}_{L_2(\Omega)} + {\lVertf_i\rVert}_{L_p(\Omega)}).$$ In the above, the constant $N$ only depends on $d$, $p$, and $\Lambda$.
Based on Proposition \[prop-regularity\], we obtain the following a priori estimate for the equations with lower order terms and large $\lambda$, which will be useful for the unique solvability in Theorem \[thm-well-posedness\].
\[cor-no-u-rhs\] Let $p\in (2,4)$ and $\gamma_0,\theta_0$ be the constants from Proposition \[prop-regularity\]. Under Assumptions \[ass-RF\] $(\gamma_0)$ and \[ass-smallBMO\] $(\theta_0)$, there exists a positive constant $\lambda_1$ depending on $(d, p,\Lambda, R_0, K)$ such that if $u$ is a $W^{1,p}_{{\mathcal{D}}}$ weak solution to the equation with $f_i,f\in L_p(\Omega)$ and $\lambda>\lambda_1$, then we have $$\label{181222@eq1}
\|U\|_{L_p(\Omega)}\le N\|F\|_{L_p(\Omega)},$$ where $N=N(d,p,\Lambda)$.
The rest of this section is devoted to the proofs of Proposition \[prop-regularity\] and Corollary \[cor-no-u-rhs\].
Decomposition of $Du$ {#subsection-decom}
---------------------
We will use an interpolation argument to prove Proposition \[prop-regularity\]. The key step is the following decomposition (approximation).
\[prop-decom\] Suppose that $u\in W^{1,2}_{{\mathcal{D}}}(\Omega)$ satisfies with $\lambda>0$ and $f_i,f\in L_p(\Omega)\cap L_2(\Omega)$, where $p>2$. Then under Assumptions \[ass-RF\] $(\gamma)$ and \[ass-smallBMO\] $(\theta)$ with $\gamma<1/(32\sqrt{d+3})$ and $\theta\in (0,1)$, for any $x_0 \in \overline{\Omega}$ and $R<R_0$, there exist positive functions $W, V \in L_2(\Omega_{R/32}(x_0))$ such that $$U \leq W+V \quad \text{in } \Omega_{R/32}(x_0).$$ Moreover, we have for any $q < 4$, $$\begin{aligned}
(W^2)^{1/2}_{\Omega_{R/32}(x_0)} &\leq N\big( (\theta^{\frac{1}{2\mu'}}+ \gamma^{\frac{1}{2\mu'}})(U^2)^{1/2}_{\Omega_R(x_0)} + (F^{2\mu})^{\frac{1}{2\mu}}_{\Omega_R(x_0)}\big),\label{eqn-est-W}\\
(V^q)^{1/q}_{\Omega_{R/32}(x_0)} &\leq N\big( (U^2)^{1/2}_{\Omega_R(x_0)} + (F^{2\mu})^{\frac{1}{2\mu}}_{\Omega_R(x_0)}\big).\label{eqn-est-V}\end{aligned}$$ Here $\mu$ is a constant satisfying $2\mu=p_0$, where $p_0=p_0(d,p,\Lambda)>2$ comes from Lemma \[lem-reverse-holder\], and $\mu'$ satisfies $1/\mu+1/\mu'=1$. The constant $N$ only depends on $d$, $p$, $q$, and $\Lambda$.
The rest of Section \[subsection-decom\] will be devoted to the proof of this proposition.
According to the relative position of $x_0$ to ${\mathcal{D}}, {\mathcal{N}}$, we will discuss the following 3 cases.
[*Case 1*]{}: ${\operatorname{dist}}(x_0,{\partial}\Omega)\geq R/32$.
In this case, our decomposition is only concerned with the interior of $\Omega$. We can do the usual “freezing coefficient” approximation. The existence of such $W,V$ can be found in [@DK11 Lemma 8.3 (i)] or [@DK12 Lemma 5.1 (i)].
In the next two cases, we also need to approximate the Reifenberg flat boundary by hyperplane and deal with corresponding boundary conditions.
[*Case 2*]{}: ${\operatorname{dist}}(x_0,{\partial}\Omega)<R/32$, ${\operatorname{dist}}(x_0,\Gamma)\geq R/24$.
In this case, either we have $B_{R/24}(x_0)\cap{\mathcal{N}}=\emptyset$ or $B_{R/24}(x_0)\cap{\mathcal{D}}=\emptyset$. Correspondingly, we only deal with purely Dirichlet or purely conormal boundary condition. The functions $W$ and $V$ are constructed in $\Omega_{R/24}(x_0)$. Then for the estimates, we need to shrink the radius to $R/32$. Such construction and estimates can be found in [@DK11 Lemma 8.3 (ii)] and [@DK12 Lemma 5.1 (ii)]. Briefly, we approximate the neighborhood of $\Omega_{R/24}(x_0)$ by a half ball thanks to the small Reifenberg flat assumption. Then we apply a cutoff technique for the Dirichlet case, or a reflection technique for the conormal case. All these two techniques will be introduced in Case 3 below.
In both Cases 1 and 2, actually we can take $q=\infty$ in .
[*Case 3*]{}: ${\operatorname{dist}}(x_0,{\partial}\Omega)<R/32$, ${\operatorname{dist}}(x_0,\Gamma) < R/24$.
In this case, we deal with the “mixed” boundary condition. Take $y_0\in \Gamma$ with ${\operatorname{dist}}(y_0, x_0)<R/24$. Consider the coordinate system associated with $(y_0,R/4)$ as in Assumption \[ass-RF\] $(\gamma)$. For simplicity, we shift the origin in $x'=(x_1,x_2)-$hyperplane, such that $${\partial}\Omega\cap B_{R/4}(y_0)\subset{\{-\gamma R/2<x_1<0\}},$$ $$\label{eqn-moving-coordinates}
\Gamma \cap B_{R/4}(y_0)\subset{\{-\gamma R/2<x_2<0\}}.$$
In the following, we will omit the center when it is $y_0$. For example, $$\Omega_{R/4}:= \Omega\cap B_{R/4}(y_0),\quad \Omega_{R/4}^{+}:=\Omega_{R/4}\cap{\mathbb{R}}^d_{+},\quad \Omega_{R/4}^{-}:=\Omega_{R/4}\cap{\mathbb{R}}^d_{-},$$ where ${\mathbb{R}}^d_-=\{x\in {\mathbb{R}}^d:x_1<0\}$. Note that this is slightly different from the usual convention that we omit the center when it is the coordinate origin. The following inclusion relation will be useful: $$\label{eq11.37}
\Omega_{R/32}(x_0)\subset\Omega_{R/8}\subset\Omega_{R/2}\subset\Omega_R(x_0).$$
Now we start to construct the decomposition. First, we introduce a cut-off function $\chi\in C^\infty({\mathbb{R}}^d)$ with $D\chi$ supported in a “L-shaped” domain, satisfying $$\begin{cases}
\chi =0, \quad \text{on }{\{x_2>-\gamma R\}}\cap {\{x_1<\gamma R\}},\\
\chi =1, \quad \text{on }{\{x_2<-2\gamma R\}}\cup {\{x_1>2\gamma R\}},\\
0\leq \chi \leq 1, |D\chi|\leq \frac{2}{\gamma R}.
\end{cases}$$
The following two lemmas should be read as parts of the proof of Proposition \[prop-decom\]. The first one is an important estimate of a typical term in our proof. Both the inequality itself and the decomposition technique in the proof will be used later.
\[lem-usingpoincare\] We have $$({\lvertD\chi u\rvert}^2)^{1/2}_{\Omega_{R/4}} \leq N \gamma^{1/(2\mu')}\big((U^2)^{1/2}_{\Omega_R(x_0)} + (F^{2\mu})^{1/(2\mu)}_{\Omega_R(x_0)}\big),$$ where $N=N(d,p,\Lambda)$.
From the construction of $\chi$, we have $${\lVertD\chi u\rVert}_{L_2(\Omega_{R/4})}\leq \frac{2}{\gamma R}{\lVert{\mathbb{I}}_{{\operatorname{supp}}{\{D\chi\}}}u\rVert}_{L_2(\Omega_{R/4})}.$$ Now we decompose the set ${\operatorname{supp}}{\{D\chi\}}\cap \Omega_{R/4}$ to obtain the required smallness. Consider the following grid points on ${\partial}{\mathbb{R}}^d_{+}$: $${\mathcal{D}}_{grid} := {\{z\in {\mathbb{R}}^d : z=(0,k\gamma R) \text{ for }k=(k_2,\ldots,k_d) \in \mathbb{Z}^{d-1},k_2\ge -1\}}\cap \Omega_{R/4}.$$ Clearly $\bigcup_{z\in{\mathcal{D}}_{grid}}\Omega_{\sqrt {d+3}\gamma R}(z)$ covers ${\operatorname{supp}}{\{D\chi\}}\cap \Omega_{R/4},$ and because $\gamma<1/(32\sqrt{d+3})$ $$\bigcup_{z\in{\mathcal{D}}_{grid}}\Omega_{\sqrt {d+3}\gamma R}(z)\subset \Omega_{R/3},$$ with each point covered by at most $N(d)$ of such neighborhoods. Due to , we know that in each $\Omega_{\sqrt{d+3}\gamma R}(z)$, is satisfied with $$(z_0,\alpha)=(z+(c,(-1+\sqrt{d+3})\gamma R/2,0,\cdots,0), 1/4),$$ where $c\in(-\gamma R/2,0)$ is chosen carefully to guarantee $z_0\in {\partial}\Omega$. Hence we can apply the Poincaré inequality stated in Corollary \[180514@cor1\] and Hölder’s inequality to obtain $$\begin{aligned}
{\lVert{\mathbb{I}}_{{\operatorname{supp}}{\{D\chi\}}} u\rVert}_{L_2(\Omega_{R/4})}^2
&\leq N\sum_{z\in {\mathcal{D}}_{grid}}{\lVertu\rVert}_{L_2(\Omega_{\sqrt {d+3}\gamma R}(z))}^2\nonumber \\
&\leq N(\gamma R)^2\sum_{z\in {\mathcal{D}}_{grid}}{\lVertDu\rVert}_{L_2(\Omega_{2\sqrt {d+3}\gamma R}(z))}^2\nonumber\\
&\leq N(\gamma R)^2{\lVert{\mathbb{I}}_{|x_1|<2\sqrt {d+3}\gamma R}Du\rVert}_{L_2(\Omega_{R/3})}^2\label{eqn-overlap}\\
&\leq N(\gamma R)^2\cdot(\gamma R^d)^{1/\mu'}{\lVertDu\rVert}_{L_{2\mu}(\Omega_{R/3})}^2.
\label{eqn-holder-area-suppchi-smallness}\end{aligned}$$ To obtain , we used Hölder’s inequality. Now we rewrite this using the notation of average and use a properly rescaled version of Lemma \[lem-reverse-holder\] as well as to obtain $$\begin{aligned}
({\lvertD\chi u\rvert}^2)^{1/2}_{\Omega_{R/4}} &\leq N\gamma^{1/(2\mu')}({\lvertDu\rvert}^{2\mu})^{1/(2\mu)}_{\Omega_{R/3}}\\
&\leq N \gamma^{1/(2\mu')}\big((U^2)^{1/2}_{\Omega_R(x_0)} + (F^{2\mu})^{1/(2\mu)}_{\Omega_R(x_0)}\big).\end{aligned}$$ The lemma is proved.
The second lemma shows how we “freeze” the boundary to be a hyperplane using a cut-off technique together with a reflection.
\[lem-RF-as-pert-flat\] The function $\chi u \in W^{1,2}(\Omega_{R/4}^{+})$ satisfies the following equation in the weak sense $$\label{eqn-reflected}
\begin{cases}
D_i(a_{ij}D_j(\chi u)) - \lambda \chi u= D_ig_i^{(1)} + D_ig_i^{(2)} + g_i^{(3)}D_i\chi + g_i^{(4)}D_i\widetilde{\chi} + g^{(5)} & \text{in } \Omega_{R/4}^{+},\\
a_{ij}D_j(\chi u)n_i = g_i^{(1)}n_i +g_i^{(2)}n_i & \text{on } \Gamma^{-},\\
\chi u=0 &\text{on } \Gamma^{+},
\end{cases}$$ where $$\begin{aligned}
g_i^{(1)}&=a_{ij}uD_j\chi + f_i\chi ,\quad g_i^{(2)}=(-{\varepsilon}_{i}{\varepsilon}_{j}\widetilde{a_{ij}}\widetilde{\chi}D_j\widetilde{u} + {\varepsilon}_{i}\widetilde{\chi}\widetilde{f_i}){\mathbb{I}}_{(-x_1,x_2,x'')\in\Omega_{R/4}^{-}},\\
g_i^{(3)}&=a_{ij}D_ju-f_i,\quad g_i^{(4)}=({\varepsilon}_i{\varepsilon}_j \widetilde{a_{ij}}D_j\widetilde{u}-{\varepsilon}_i\widetilde{f_i}){\mathbb{I}}_{(-x_1,x_2,x'')\in\Omega_{R/4}^{-}},\\
g^{(5)}&= \chi f + \widetilde{\chi}\widetilde{f}{\mathbb{I}}_{(-x_1,x_2,x'')\in\Omega_{R/4}^{-}}+ \lambda\widetilde{\chi}\widetilde{u}{\mathbb{I}}_{(-x_1,x_2,x'')\in\Omega_{R/4}^{-}},\\
\Gamma^+&=\partial \Omega_{R/4}^+\cap \{x_1=0,x_2>0\}, \quad \Gamma^-=\partial \Omega_{R/4}^+\cap \{x_1=0,x_2<0\}.\end{aligned}$$ Here we denote $\widetilde{f}(x_1,x_2,x''):=f(-x_1,x_2,x'')$, and similarly for $\widetilde{a_{ij}}, \widetilde{\chi}$, and $\widetilde{u}$. We also use the following notation $${\varepsilon}_{i}:=\begin{cases}
-1& \text{if }\,\, i=1,\\
1&\text{if }\,\, i\neq 1.
\end{cases}$$
Take any test function $\psi \in W^{1,2}_{{\partial}\Omega_{R/4}^{+}\setminus\Gamma^{-}}(\Omega_{R/4}^{+})$. We extend $\psi$ to ${\mathbb{R}}^d_+$ by setting $\psi\equiv 0$ on ${\mathbb{R}}^d_+\setminus \Omega_{R/4}^+$, and then, we again extend evenly to ${\mathbb{R}}^d$. Denote this extended function by ${\mathcal{E}}\psi$, and note that $\chi{\mathcal{E}}\psi\in W^{1,2}_{{\mathcal{D}}}(\Omega)$. Testing with $\chi{\mathcal{E}}\psi$ and rearranging terms will give us .
We continue the proof of Proposition \[prop-decom\]. Solve the following equation $$\label{eqn-w}
\begin{cases}
\begin{aligned}
D_{i}(\overline{a_{ij}}D_j\hat{w}) - \lambda \hat{w}&=
D_i((\overline{a_{ij}}-a_{ij})D_j(\chi u)) + D_ig_i^{(1)} + D_i g_i^{(2)}\\
&\quad + g_i^{(3)}D_i\chi + g_i^{(4)}D_i\widetilde{\chi} +g^{(5)}
\end{aligned}
&\text{in }\Omega_{R/4}^{+},\\
\overline{a_{ij}}D_j\hat{w}\cdot n_i = (\overline{a_{ij}}-a_{ij})D_j (\chi u) n_i + g_i^{(1)}n_i +g_i^{(2)}n_i &\text{on }\Gamma^{-},\\
\hat{w}=0 & \text{on }{\partial}\Omega_{R/4}^{+}\setminus\Gamma^{-},
\end{cases}$$ for $\hat{w}\in W^{1,2}_{{\partial}\Omega_{R/4}^{+}\setminus\Gamma^{-}}(\Omega_{R/4}^{+})$, where $\overline{a_{ij}}=(a_{ij})_{\Omega_{R/4}}$ are constants. Due to the Lax-Milgram lemma, such $\hat{w}$ exists. For simplicity, we denote $$\hat{W}:={\lvertD\hat{w}\rvert} + \sqrt{\lambda}{\lvert\hat{w}\rvert}.$$ Testing by $\hat{w}$, and using the ellipticity and Hölder’s inequality, we have $$\begin{aligned}
&{\lVert\hat{W}\rVert}^2_{L_2(\Omega^{+}_{R/4})} \leq {\lVert(\overline{a_{ij}}-a_{ij})D_j(\chi u)\rVert}_{L_2(\Omega_{R/4}^{+})}{\lVertD\hat{w}\rVert}_{L_2(\Omega^{+}_{R/4})}\label{eqn-hat-w-1}\\
&\quad+ \big\|g_i^{(1)}\big\|_{L_2(\Omega_{R/4}^{+})}{\lVertD\hat{w}\rVert}_{L_2(\Omega^{+}_{R/4})} + {\bigl\lVertg_i^{(2)}\bigr\rVert}_{L_2(\Omega_{R/4}^{+})}{\lVertD\hat{w}\rVert}_{L_2(\Omega^{+}_{r/4})}\label{eqn-hat-w-2}\\
&\quad+ {\bigl\lVert{\mathbb{I}}_{{\operatorname{supp}}{\{D\chi\}}}g_i^{(3)}\bigr\rVert}_{L_2(\Omega_{R/4}^{+})}{\lVertD_i\chi\hat{w}\rVert}_{L_2(\Omega_{R/4}^{+})}
+ {\bigl\lVertg_i^{(4)}\bigr\rVert}_{L_2(\Omega_{R/4}^{+})}{\bigl\lVertD_i\widetilde{\chi}\hat{w}{\mathbb{I}}_{(-x_1,x_2,x'')\in\Omega_{R/4}^{-}}\bigr\rVert}_{L_2(\Omega_{R/4}^{+})}\label{eqn-hat-w-3}\\
&\quad+ {\left\lVert\lambda^{-1/2}{g^{(5)}}\right\rVert}_{L_2(\Omega_{R/4}^{+})}{\lVert\lambda^{1/2}\hat{w}\rVert}_{L_2(\Omega_{R/4}^{+})}\label{eqn-hat-w-4}.\end{aligned}$$ For the term in , we use Assumption \[ass-smallBMO\], Hölder’s inequality, Lemma \[lem-reverse-holder\], and Lemma \[lem-usingpoincare\]. Noting that ${\lvert\Omega_{R/4}^{+}\rvert}$, ${\lvert\Omega_{R/4}\rvert}$ and ${\lvert\Omega_{R}(x_0)\rvert}$ are all comparable to $R^d$, we have: $$\label{eqn-est-aij}
\begin{split}
\big({\lvert(\overline{a_{ij}}-a_{ij})D_j(\chi u)\rvert}^2\big)^{1/2}_{\Omega_{R/4}^{+}}
&\leq N\big({\lvert\overline{a_{ij}}-a_{ij}\rvert}^{2\mu'}\big)^{1/(2\mu')}_{\Omega_{R/4}} \big({\lvertDu\rvert}^{2\mu}\big)^{1/(2\mu)}_{\Omega_{R/4}}
+N\big({\lvertD\chi u\rvert}^2\big)^{1/2}_{\Omega_{R/4}^{+}}\\
&\leq N\big({\lvert\overline{a_{ij}}-a_{ij}\rvert}\big)^{1/(2\mu')}_{\Omega_{R/4}}
\big({\lvertDu\rvert}^{2\mu}\big)^{1/(2\mu)}_{\Omega_{R/4}}+N\big({\lvertD\chi u\rvert}^2\big)^{1/2}_{\Omega_{R/4}^{+}}\\
&\leq N \big(\theta^{1/(2\mu')}+\gamma^{1/(2\mu')}\big)\big((U^2)^{1/2}_{\Omega_R(x_0)} + (F^{2\mu})^{1/(2\mu)}_{\Omega_R(x_0)}\big),
\end{split}$$ where $N=N(d,p,\Lambda)$ is a constant.
For the terms in , we first estimate $g_i^{(1)}$. This is simply due to Lemma \[lem-usingpoincare\] and Hölder’s inequality: $$\label{eqn-est-g}
\begin{split}
\big({\bigl\lvertg_i^{(1)}\bigr\rvert}^2\big)^{1/2}_{\Omega_{R/4}^{+}}
&\leq N({\lvertuD\chi\rvert}^2)^{1/2}_{\Omega_{R/4}} + N({\lvertf_i\rvert}^2)^{1/2}_{\Omega_{R/4}}\\
&\leq N \gamma^{1/(2\mu')}(U^2)^{1/2}_{\Omega_R(x_0)} + N(F^{2\mu})^{\frac{1}{2\mu}}_{\Omega_R(x_0)}.
\end{split}$$ Now we estimate $g_i^{(2)}$ as follows: $$\label{eqn-est-G}
\begin{split}
\big({\bigl\lvertg_i^{(2)}\bigr\rvert}^2\big)^{1/2}_{\Omega_{R/4}^{+}}
&\le \big({\bigl\lvert\widetilde{a_{ij}}\widetilde{\chi}D_j\widetilde{u}
{\mathbb{I}}_{(-x_1,x_2,x'')\in \Omega_{R/4}^-}\bigr\rvert}^2\big)^{1/2}_{\Omega_{R/4}^{+}} +
\big({\bigl\lvert\widetilde{\chi}\widetilde{f_i}
{\mathbb{I}}_{(-x_1,x_2,x'')\in \Omega_{R/4}^-}\bigr\rvert}^2\big)^{1/2}_{\Omega_{R/4}^{+}}\\
&\leq N\big({\bigl\lvert{\mathbb{I}}_{\Omega_{R/4}^{-}} Du\bigr\rvert}^2\big)^{1/2}_{\Omega_{R/4}} + N({\lvertf_i\rvert}^2)^{1/2}_{\Omega_{R/4}}\\
&\leq N\gamma^{1/(2\mu')}({\lvertDu\rvert}^{2\mu})^{1/(2\mu)}_{\Omega_{R/4}} + N({\lvertf_i\rvert}^{2\mu})^{1/(2\mu)}_{\Omega_{R/4}}\\
&\leq N\gamma^{1/(2\mu')}\big((U^2)^{1/2}_{\Omega_R(x_0)} + (F^{2\mu})^{1/(2\mu)}_{\Omega_R(x_0)}\big) + N({\lvertf_i\rvert}^{2\mu})^{1/(2\mu)}_{\Omega_{R/4}},
\end{split}$$ where in the last line, we used Lemma \[lem-reverse-holder\].
For the terms in , we first use the same decomposition technique together with Poincaré’s inequality as in the proof of Lemma \[lem-usingpoincare\] (until the step ) to obtain: $${\lVertD_i\chi\hat{w}\rVert}_{L_2(\Omega_{R/4}^{+})}\leq N {\lVertD\hat{w}\rVert}_{L_2(\Omega_{R/4}^{+})}.$$ To avoid the problem of increased integrating domain, here we need to modify the decomposition to be $$\bigcup_{z\in{\mathcal{D}}_{grid}}\big(\Omega_{\sqrt {d+3}\gamma R}(z)\cap\Omega_{R/4}^{+}\big),$$ and the same proof still applies. Again, with the help of Hölder’s inequality and Lemma \[lem-reverse-holder\], we can estimate $g_i^{(3)}$ as follows: $$\begin{aligned}
{\bigl\lVert{\mathbb{I}}_{{\operatorname{supp}}{\{D\chi\}}}g_i^{(3)}\bigr\rVert}_{L_2(\Omega_{R/4}^{+})}
&\leq N{\bigl\lVert{\mathbb{I}}_{{\operatorname{supp}}{\{D\chi\}}}Du\bigr\rVert}_{L_2(\Omega_{R/4}^{+})} + N {\bigl\lVert{\mathbb{I}}_{{\operatorname{supp}}{\{D\chi\}}}f_i\bigr\rVert}_{L_2(\Omega_{R/4}^{+})}\\
&\leq N\gamma^{1/(2\mu')}({\lVertU\rVert}_{L_2(\Omega_R(x_0))} + R^{d/(2\mu')}{\lVertF\rVert}_{L_{2\mu}(\Omega_R(x_0))}).\end{aligned}$$ Hence, $$\label{eqn-est-h}
\begin{split}
{\bigl\lVert{\mathbb{I}}_{{\operatorname{supp}}{\{D\chi\}}}&g_i^{(3)}\bigr\rVert}_{L_2(\Omega_{R/4}^{+})}{\lVertD_i\chi\hat{w}\rVert}_{L_2(\Omega_{R/4}^{+})} \\
&\leq N\gamma^{1/(2\mu')}({\lVertU\rVert}_{L_2(\Omega_R(x_0))} + R^{d/(2\mu')}{\lVertF\rVert}_{L_{2\mu}(\Omega_R(x_0))})
{\lVertD\hat{w}\rVert}_{L_2(\Omega_{R/4}^{+})}.
\end{split}$$ Using similar techniques as in , we can deduce that $$\label{eqn-est-H}
\begin{split}
{\bigl\lVertg_i^{(4)}\bigr\rVert}_{L_2(\Omega_{R/4}^{+})}&{\bigl\lVertD_i\widetilde{\chi}\hat{w}{\mathbb{I}}_{(-x_1,x_2,x'')\in\Omega_{R/4}^{-}}\bigr\rVert}_{L_2(\Omega_{R/4}^{+})}\\
&\leq N\gamma^{1/(2\mu')}({\lVertU\rVert}_{L_2(\Omega_R(x_0))} + R^{d/(2\mu')}{\lVertF\rVert}_{L_{2\mu}(\Omega_R(x_0))})
{\lVertD\hat{w}\rVert}_{L_2(\Omega_{R/4}^{+})}.
\end{split}$$ We are left to estimate the one last term in : $$\begin{aligned}
\label{eqn-est-non-div}
&{\left\lVert\lambda^{-1/2}{g^{(5)}}\right\rVert}_{L_2(\Omega_{R/4}^{+})}\nonumber\\
&\leq {\left\lVert\lambda^{-1/2}\chi f\right\rVert}_{L_2(\Omega_{R/4}^{+})} + {\left\lVert\lambda^{-1/2}{\widetilde{f}}
\widetilde{\chi}{\mathbb{I}}_{(-x_1,x_2,x'')\in\Omega_{R/4}^{-}}\right\rVert}_{L_2(\Omega_{R/4}^{+})} +{\bigl\lVert\lambda^{1/2}\widetilde{\chi}\widetilde{u}{\mathbb{I}}_{(-x_1,x_2,x'')
\in\Omega_{R/4}^{-}}\bigr\rVert}_{L_2(\Omega_{R/4}^{+})}\nonumber\\
&\leq 2{\lVertF\rVert}_{L_2(\Omega_{R/4})} + N\gamma^{1/(2\mu')}({\lVertU\rVert}_{L_2(\Omega_R(x_0))} + {\lVertF\rVert}_{L_2(\Omega_R(x_0))}),\end{aligned}$$ where for the last term, we applied similar techniques as we did to estimate $g_i^{(2)}$.
Substituting - back, we obtain $$\label{eqn-est-w-hat-final}
(\hat{W}^2)^{1/2}_{\Omega_{R/4}^{+}}\leq N\big( (\theta^{\frac{1}{2\mu'}}+ \gamma^{\frac{1}{2\mu'}})(U^2)^{1/2}_{\Omega_R(x_0)} + (F^{2\mu})^{\frac{1}{2\mu}}_{\Omega_R(x_0)}\big).$$ Now we define $$W:=\begin{cases}
\hat{W} + {\lvertD((1-\chi)u)\rvert} + \sqrt{\lambda}{\lvert(1-\chi)u\rvert} &\text{in } \Omega_{R/32}(x_0)\cap {\mathbb{R}}^d_{+},\\
{\lvertDu\rvert} + \sqrt{\lambda}{\lvertu\rvert} &\text{in } \Omega_{R/32}(x_0)\cap {\mathbb{R}}^d_{-}.
\end{cases}$$ Using , Hölder’s inequality, Lemma \[lem-reverse-holder\], and Lemma \[lem-usingpoincare\], we can obtain .
To construct $V$, we set $$v:=\chi u-\hat{w}.$$ Clearly $v\in W^{1,2}_{\Gamma^{+}}(\Omega_{R/4}^{+})$. Simple computation using Lemma \[lem-RF-as-pert-flat\] and shows that $v$ satisfies $$\begin{cases}
D_{i}(\overline{a_{ij}}D_jv) - \lambda v= 0 &\text{in }\Omega_{R/4}^{+},\\
\overline{a_{ij}}D_jv \cdot n_i = 0 &\text{on }\Gamma^{-},\\
v=0 &\text{on }\Gamma^{+}.
\end{cases}$$ Now we define $$V:=\begin{cases}
{\lvertDv\rvert} + \sqrt{\lambda}{\lvertv\rvert}& \text{in } \Omega_{R/4}^{+},\\
0 & \text{in } \Omega_{R/4}^{-}.
\end{cases}$$ Then we have $$U\le W+V \quad \text{in } \Omega_{R/32}(x_0)$$ from the fact that $$u=v+\hat{w}+(1-\chi)u\quad \text{in } \Omega_{R/32}(x_0)\cap {\mathbb{R}}^d_+, \quad W=U \quad \text{in }\Omega_{R/32}(x_0)\cap {\mathbb{R}}^d_-.$$ Using , we can apply a properly rescaled version of Theorem \[thm-harmonic-mixed-halfspace\] with a change of variables to obtain that for any $q\in[2,4)$, $V\in L_q(\Omega_{R/32}(x_0))$ satisfying $$\label{eqn-est-v}
\begin{split}
(V^q)^{1/q}_{\Omega_{R/32}(x_0)}
\leq&
(V^q)^{1/q}_{\Omega_{R/8}^{+}}
\le N (V^2)^{1/2}_{\Omega_{R/4}^+}\\
\leq& N (({\lvertD(\chi u)\rvert}^2)^{1/2}_{\Omega_{R/4}^+} + \sqrt{\lambda}({\lvert\chi u\rvert}^2)^{1/2}_{\Omega_{R/4}^+} + {\lvert\hat{W}^2\rvert}^{1/2}_{\Omega_{R/4}^+})\\
\leq& N (U^2)^{1/2}_{\Omega_R(x_0)} + N\big((\theta^{1/(2\mu')}+\gamma^{1/(2\mu')})(U^2)^{1/2}_{\Omega_R(x_0)} + (F^{2\mu})^{1/(2\mu)}_{\Omega_R(x_0)}\big)\\
\leq& N (U^2)^{1/2}_{\Omega_R(x_0)} + N(F^{2\mu})^{1/(2\mu)}_{\Omega_R(x_0)}.
\end{split}$$ Here we used the estimates for $uD\chi$ and $\hat{W}$ in previous steps. Clearly, from we obtain . This finishes the proof of Proposition \[prop-decom\].
Level Set Argument
------------------
In previous steps, we treat the perturbation problem by decomposing $U$ into two parts, with $L_2$ and $L_q$ estimates respectively. Now we interpolate using a level set argument to obtain the required $L_p$ estimate for Proposition \[prop-regularity\]. Such argument was suggested by Caffarelli in [@Ca] for a “kernel free” approach to $W^{1,p}$ estimate of divergence form second-order elliptic equations. Note that our estimate is not an a priori estimate, i.e., we do not need to assume $Du \in L_p$ in advance.
Define $$\begin{aligned}
&{\mathcal{A}}(s):= {\{x\in\Omega:{\mathcal{M}}_{\Omega}(U^2)^{1/2} >s\}},\\
&{\mathcal{B}}(s):= {\{x\in\Omega: (\gamma^{1/(2\mu')}+\theta^{1/(2\mu')})^{-1}
{\mathcal{M}}_{\Omega}(F^{2\mu})^{1/(2\mu)} + {\mathcal{M}}_{\Omega}(U^2)^{1/2} >s \}},\end{aligned}$$ where $\mu,\mu'\in (1,\infty)$ are the constants from Proposition \[prop-decom\]. Here we denote ${\mathcal{M}}_{\Omega}$ to be the Hardy-Littlewood maximal operator restricted on $\Omega$, i.e., for $f\in L_{1,\rm{loc}}(\Omega)$ and $x\in\Omega$: $${\mathcal{M}}_{\Omega}(f)(x):=\sup_{r>0}\fint_{B_r(x)}|f|
{\mathbb{I}}_{\Omega}.$$ By the Hardy-Littlewood theorem, for any $f\in L_q(\Omega)$ with $q\in [1,\infty)$, we have $$\label{eqn-weak-est}
{\lvert{\{x\in\Omega:{\mathcal{M}}_{\Omega}(f)(x)>s\}}\rvert} \leq N\frac{{\lVertf\rVert}_{L_q(\Omega)}^q}{s^q},$$ where $N=N(d,q)$.
Proposition \[prop-decom\] leads to the following lemma.
\[lem-levelset\] Under the same hypothesis of Proposition \[prop-decom\], for any $q\in[2,4)$, there exists a constant $N$ depending on $(d,p,q,\Lambda)$, such that for all $\kappa>2^{d/2}$ and $s>0$, the following holds: if for some $R<R_0, x_0\in \overline{\Omega}$, $$\label{eqn-stoptime}
{\lvert\Omega_{R/128}(x_0)\cap{\mathcal{A}}(\kappa s)\rvert} \geq N\big(\kappa^{-q} + \kappa^{-2}(\gamma^{1/\mu'}+\theta^{1/\mu'})\big){\lvert\Omega_{R/128}(x_0)\rvert},$$ then $\Omega_{R/128}(x_0)\subset {\mathcal{B}}(s)$.
Without loss of generality, we assume $s=1$. We also extend $U$ and $F$ to be zero outside $\Omega$. We will prove the contrapositive of the above statement.
Suppose there exists a point $z_0$, with $$z_0\in \Omega_{R/128}(x_0), \quad z_0\notin{\mathcal{B}}(1),$$ then by the definition of ${\mathcal{B}}$, we have $$(\gamma^{1/(2\mu')}+\theta^{1/(2\mu')})^{-1}
{\mathcal{M}}_{\Omega}(F^{2\mu})^{1/(2\mu)}(z_0) + {\mathcal{M}}_{\Omega}(U^2)^{1/2}(z_0) \leq 1.$$ In particular, for any $r>0$, we have $$(\gamma^{1/(2\mu')}+\theta^{1/(2\mu')})^{-1}
(F^{2\mu})^{1/(2\mu)}_{B_r(z_0)} + (U^2)^{1/2}_{B_r(z_0)} \leq 1.$$ Using Proposition \[prop-decom\] with $z_0$ in place of $x_0$, we can find $W,V$ defined on $\Omega_{R/32}(z_0)$, such that for any $q\in [2,4)$, $$\label{eqn-WV-bound}
\begin{split}
&U\leq V+W \quad \text{in } \Omega_{R/32}(z_0),\\
&(W^2)^{1/2}_{\Omega_{R/32}(z_0)}\leq N(\gamma^{1/(2\mu')}+\theta^{1/(2\mu')}),\quad (V^q)^{1/q}_{\Omega_{R/32}(z_0)}\leq N.
\end{split}$$ Notice that we have the following inclusion $$\label{eqn-inclusion}
{\Omega_{R/128}(x_0)\subset\Omega_{R/64}(z_0)\subset\Omega_{R/32}(z_0)}.$$ Now for any $y_0\in \Omega_{R/128}(x_0)\cap{\mathcal{A}}(\kappa)$, by the definition of ${\mathcal{A}}$, we can find some $r>0$ such that $$\Bigg(\fint_{B_r(y_0)}U^2\,dx\Bigg)^{1/2}>\kappa.$$ We claim that $r<R/64$. Otherwise noting $y_0\in \Omega_{R/64}(z_0)$, we have $\Omega_r(y_0)\subset\Omega_{2r}(z_0)$. Hence we can deduce that $$\begin{aligned}
\Bigg(\fint_{B_r(y_0)}U^2\,dx\Bigg)^{1/2}\leq 2^{d/2}\Bigg(\fint_{B_{2r}(z_0)}U^2\,dx\Bigg)^{1/2}\leq 2^{d/2} {\mathcal{M}}_{\Omega}(U^2)^{1/2}(z_0)\leq 2^{d/2}< \kappa,\end{aligned}$$ which is a contradiction.
Now, since $r<R/64$, the decomposition $U\leq W+V$ is defined in $\Omega_r(y_0)\subset\Omega_{R/32}(z_0)$. Extending $W$ and $V$ to be zero outside $\Omega$, we have $$\begin{aligned}
\Bigg(\fint_{B_r(y_0)}U^2\,dx\Bigg)^{1/2}
&\leq \Bigg(\fint_{B_r(y_0)}W^2\,dx\Bigg)^{1/2}+\Bigg(\fint_{B_r(y_0)}V^2\,dx\Bigg)^{1/2}\\
&\leq {\mathcal{M}}_{\Omega}(W^2 {\mathbb{I}}_{\Omega_{R/32}(z_0)})^{1/2}(y_0)+ {\mathcal{M}}_{\Omega}(V^2 {\mathbb{I}}_{\Omega_{R/32}(z_0)})^{1/2}(y_0).\end{aligned}$$ Then by , and , we obtain $$\begin{aligned}
{\lvert\Omega_{R/128}(x_0)\cap{\mathcal{A}}(\kappa)\rvert}
\leq& {\lvert\Omega_{R/32}(z_0)\cap{\mathcal{A}}(\kappa)\rvert}\\
\leq&
{\bigl\lvert{\{{\mathcal{M}}_{\Omega}(W^2 {\mathbb{I}}_{\Omega_{R/32}(z_0)})^{1/2}>\kappa/2\}}\bigr\rvert} + {\bigl\lvert{\{{\mathcal{M}}_{\Omega}(V^2 {\mathbb{I}}_{\Omega_{R/32}(z_0)})^{1/2}>\kappa/2\}}\bigr\rvert}\\
\leq& N\frac{{\lVertW\rVert}^2_{L_2(\Omega_{R/32}(z_0))}}{(\kappa/2)^2} + N\frac{{\lVertV\rVert}^q_{L_q(\Omega_{R/32}(z_0))}}{(\kappa/2)^q}\\
\leq& N{\lvert\Omega_{R/32}(z_0)\rvert}\big(\kappa^{-2}(\gamma^{1/\mu'}+\theta^{1/\mu'})+\kappa^{-q}\big)\\
\leq& N \big(\kappa^{-2}(\gamma^{1/\mu'}+\theta^{1/\mu'})+\kappa^{-q}\big){\lvert\Omega_{R/128}(x_0)\rvert}.\end{aligned}$$ Here $N=N(d,p ,q,\Lambda)$ is exactly what we aim to find.
Using a lemma in measure theory called “crawling of the ink spot” which was first introduced by Krylov and Safonov in [@KS; @S], we obtain the following decay estimate from Lemma \[lem-levelset\].
\[cor-decay\] Under the same hypothesis of Proposition \[prop-decom\], for any $q\in [2,4)$, there exists a constant $N$ depending on $(d,p,q,\Lambda)$, such that for any $\kappa>\max{\{2^{d/2},\kappa_0\}}$ and $$\label{eqn-s0}
s>s_0(d,p,q,\Lambda,\kappa,R_0,{\lVertU\rVert}_{L_2(\Omega)}):=\Bigg(\frac{{\lVertU\rVert}_{L_2(\Omega)}^2}{N\kappa^2(\kappa^{-q} + \kappa^{-2}(\gamma^{1/\mu'}+\theta^{1/\mu'}))|B_{R_0/128}|}\Bigg)^{1/2},$$ we have $${\lvert{\mathcal{A}}(\kappa s)\rvert} \leq N\big(\kappa^{-q} + \kappa^{-2}(\gamma^{1/\mu'}+\theta^{1/\mu'})\big){\lvert{\mathcal{B}}(s)\rvert},$$ where $\kappa_0$ is the constant satisfying $$\label{eqn-kappa}
N\big(\kappa_0^{-q} + \kappa_0^{-2}(\gamma^{1/\mu'}+\theta^{1/\mu'})\big) <1/3.$$
Here we only sketch the proof. The key idea is to use a stopping time argument (or the Calderón-Zygmund decomposition as in [@Ca]). Different from Krylov and Safonov’s original version, we cover $\Omega$ by balls instead of dyadic cubes. For any $x_0\in{\mathcal{A}}(\kappa s)$, by , , and , we see that does not hold with $R_0$ in place of $R$. We shrink the “ball” $\Omega_{R/128}(x_0)$ from $R=R_0$ until the first time holds. Due to , and the Lebesgue differentiation theorem, such $R$ exists and $R\in (0,R_0)$. We are left to use the Vitali covering lemma to pick a “almost disjoint” cover.
Proof of Proposition \[prop-regularity\] and Corollary \[cor-no-u-rhs\]
-----------------------------------------------------------------------
Now we are ready to give the proof of Proposition \[prop-regularity\].
Let us fix $p\in (2,4)$, and let $\gamma$, $\theta$, and $\kappa$ be positive constants to be chosen later, such that $$\gamma<1/(32\sqrt{d+3}), \quad \theta<1, \quad \kappa >\max\{2^{d/2},\kappa_0\},$$ where $\kappa_0=\kappa_0(d,p,\Lambda)$ is a constant satisfying with $q=(p+4)/2$. It suffices to prove $$\label{eqn-rewrite-Lp}
\lim_{S\rightarrow \infty} \int_0^Sp{\lvert{\mathcal{A}}(s)\rvert}s^{p-1}\,ds \leq N\big(R_0^{d(1-p/2)}{\lVertU\rVert}_{L_2(\Omega)}^p + {\lVertF\rVert}_{L_p(\Omega)}^p\big)$$ under Assumptions \[ass-RF\] $(\gamma)$ and \[ass-smallBMO\] $(\theta)$. The left-hand side becomes $$\lim_{S\rightarrow \infty} \int_0^{S/\kappa}p\kappa^p{\lvert{\mathcal{A}}(\kappa s)\rvert}s^{p-1}\,ds.$$ We bound the integrand by using Corollary \[cor-decay\] with $q=(p+4)/2$ when $s>s_0$. For $s\leq s_0$, we apply Chebyshev’s inequality. Then we have $$\begin{aligned}
&\int_0^{S/\kappa} {\lvert{\mathcal{A}}(\kappa s)\rvert}\kappa^ps^{p-1}\,ds \\
&\leq N\int_0^{s_0}\frac{{\lVertU\rVert}^2_{L_2(\Omega)}}{(\kappa s)^2}\kappa^ps^{p-1}\,ds + N\big(\kappa^{-q} + \kappa^{-2}(\gamma^{\frac 1 {\mu'}}+\theta^{\frac 1 {\mu'}})\big) \int_0^{S/\kappa}{\lvert{\mathcal{B}}(s)\rvert}\kappa^p s^{p-1}\,ds\\
&\leq N_{0}R_0^{d(1-p/2)}{\lVertU\rVert}_{L_2(\Omega)}^{p}+ N_{1}\big(\kappa^{-q} + \kappa^{-2}(\gamma^{\frac 1 {\mu'}}+\theta^{\frac 1 {\mu'}})\big)\kappa^p \int_0^{S/\kappa}{\lvert{\mathcal{A}}(s/2)\rvert} s^{p-1}\,ds+ N{\lVertF\rVert}_{L_p(\Omega)}^p,\end{aligned}$$ where $N_1=N_1(d,p,\Lambda)$ and $N_0$ depends also on $\kappa$. Here in the last line, we used the following relationship: $${\mathcal{B}}(s)\subset{\mathcal{A}}(s/2)\cup{\{(\gamma^{1/(2\mu')}+\theta^{1/(2\mu')})^{-1}
{\mathcal{M}}_{\Omega}(F^{2\mu})^{1/(2\mu)} >s/2\}}$$ and the Hardy-Littlewood inequality, noting that $2\mu<p$. Now we choose $\kappa$ sufficient large such that $N_{1}\kappa^{p-q}<2^{-p-2}$, and then $\theta$ and $\gamma$ sufficient small such that $N_{1}(\kappa^{p-2}(\gamma^{1/\mu'}+\theta^{1/\mu'}))<2^{-p-2}$. Then we have $$\int_0^Sp{\lvert{\mathcal{A}}(s)\rvert}s^{p-1}\,ds
\leq NR_0^{d(1-p/2)}{\lVertU\rVert}_{L_2(\Omega)}^{p} + N{\lVertF\rVert}_{L_p(\Omega)}^p+ \frac{p}{2} \int_0^{S/(2\kappa)}{\lvert{\mathcal{A}}(s)\rvert}s^{p-1}\,ds,$$ where $N=N(d,p,\Lambda)$. This yields . Hence, we have $u \in W^{1,p}(\Omega)$ satisfying . Note that all the previous proof including the reverse Hölder inequality, the estimate for harmonic functions, the decomposition lemma, and the level set argument, also work when $\lambda=0$ if we substitute $U$ by $|Du|$ and $F$ by $\sum_i|f_i|$. Thus we can also obtain when $\lambda=0$ and $f=0$.
To end this section, we give the proof of Corollary \[cor-no-u-rhs\].
Consider the usual smooth cut-off function $\zeta\in C^\infty_c(B_{{\varepsilon}})$ with $\zeta \in [0,1], {\lvertD\zeta\rvert}\le N/{\varepsilon}$. From , we can obtain the equation for $\zeta u$: $$\begin{cases}
D_i(a_{ij}D_j(u\zeta))-\lambda (u\zeta) = D_i(f_i\zeta + h_i)+ (f\zeta + h) & \text{in }\, \Omega,\\
a_{ij}D_j (u\zeta) n_i =(f_i\zeta + h_i) n_i &\text{on }\, {\mathcal{N}},\\
u\zeta=0 & \text{on }\, {\mathcal{D}},
\end{cases}$$ where $h_i$ and $h$ are given as follows: $$\begin{split}
h_i&:= a_{ij}uD_j\zeta - b_i u\zeta,\\
h&:=a_{ij}D_juD_i\zeta + b_iuD_i\zeta - \hat{b}_iD_iu\zeta-cu\zeta-f_iD_i \zeta.
\end{split}$$ Hence by , for any $\lambda>0$, we have $$\label{190112@eq3}
\begin{aligned}
&{\lVertD(u\zeta)\rVert}_{L_p(\Omega)}+\sqrt{\lambda}{\lVertu\zeta\rVert}_{L_p(\Omega)}\\
&\leq N_0 R_0^{d(1/p-1/2)}\big({\lVertD(u\zeta)\rVert}_{L_2(\Omega)} + \sqrt{\lambda}{\lVertu\zeta\rVert}_{L_2(\Omega)}\big)\\
&\quad+N_0 \|f_i \zeta\|_{L_p(\Omega)}+\frac{N_0}{\sqrt{\lambda}}\big(\|f_i D\zeta\|_{L_p(\Omega)}+\|f\zeta\|_{L_p(\Omega)}\big)\\
& \quad +N_1\big({\lVertuD\zeta\rVert}_{L_p(\Omega)}+{\lVertu\zeta\rVert}_{L_p(\Omega)}\big)\\
&\quad+\frac{N_1}{\sqrt{\lambda}}\big({\lVertDu\cdot D\zeta\rVert}_{L_p(\Omega)}+{\lVertuD\zeta\rVert}_{L_p(\Omega)}+{\lVertDu\zeta\rVert}_{L_p(\Omega)}+\|u\zeta\|_{L_p(\Omega)}\big),
\end{aligned}$$ where $N_0=N_0(d,p, \Lambda)$ and $N_1$ depends also on $K$. Using Hölder’s inequality, we obtain $${\lVertD(u\zeta)\rVert}_{L_2(\Omega)} + \sqrt{\lambda}{\lVertu\zeta\rVert}_{L_2(\Omega)} \leq {\varepsilon}^{d/2-d/p}\big({\lVertD(u\zeta)\rVert}_{L_p(\Omega)} + \sqrt{\lambda}{\lVertu\zeta\rVert}_{L_p(\Omega)}\big).$$ Thus by taking $\varepsilon=\varepsilon(d,p,\Lambda, R_0)>0$ sufficiently small such that $N_0(\varepsilon/R_0)^{d/2-d/p}<1/2$, we can absorb the first two terms on the right-hand side of to the left-hand side. Then by using the standard partition of unity technique and choosing $\lambda$ large enough, we conclude . The corollary is proved.
Solvability and General $p$
===========================
With the regularity result in hand, we are now going to prove Theorem \[thm-well-posedness\] concerning the solvability. Note that in this section, we deal with more general cases $p\in(4/3,4)$. We first state the following $L_2$ well-posedness result, which is a direct consequence of the Lax-Milgram lemma.
\[lem-L2-solvability\] Let $\Omega$ be a domain with ${\partial}\Omega={\mathcal{D}}\cup{\mathcal{N}}$. Consider the equation with $b_i,\hat{b}_i,c\in L_\infty(\Omega)$. Then for any $$f,f_i\in L_2(\Omega),\quad \lambda > \lambda_2:= 4\Big({\lVertb_i\rVert}^2_{L_\infty(\Omega)}/\Lambda
+{\lVert\hat{b}_i\rVert}^2_{L_\infty(\Omega)}/\Lambda+{\lVertc\rVert}_{L_\infty(\Omega)}\Big),$$ there exists a unique $W^{1,2}_{{\mathcal{D}}}(\Omega)$ weak solution $u$ to , satisfying $${\lVertU\rVert}_{L_2(\Omega)}\leq N {\lVertF\rVert}_{L_2(\Omega)},$$ where $N=N(\Lambda)$ is a constant.
We prove by three cases under Assumptions \[ass-RF\] $(\gamma_0)$ and \[ass-smallBMO\] $(\theta_0)$, where $\gamma_0, \theta_0$ are the constants from Proposition \[prop-regularity\]. Assume that $\lambda>\lambda_0$, where $\lambda_0$ is a constant to be chosen below, which satisfies $$\lambda_0\ge\max\{\lambda_1, \lambda_2\}.$$ Here, $\lambda_1$ and $\lambda_2$ are the constants from Corollary \[cor-no-u-rhs\] and Lemma \[lem-L2-solvability\], respectively.
[*Case 1*]{}: $p=2$. This is Lemma \[lem-L2-solvability\].
[*Case 2*]{}: $p\in(2,4)$. Due to the method of continuity and the a priori estimate (see Corollary \[cor-no-u-rhs\]), it suffices to prove the theorem when all the lower order coefficients are zero, i.e., $b_i\equiv \hat{b}_i\equiv c\equiv 0$.
Now we approximate $f,f_i$ by $f^{(n)}, f^{(n)}_i$ strongly in $L_p(\Omega)$, where $f^{(n)}, f^{(n)}_i\in L_2(\Omega)\cap L_p(\Omega)$. Then by Lemma \[lem-L2-solvability\], there exist $W^{1,2}_{{\mathcal{D}}}(\Omega)$ weak solutions $u^{(n)}$ to (without lower order terms) with $f^{(n)}, f_i^{(n)}$ in place of $f, f_i$. Moreover, it follows from Proposition \[prop-regularity\] and Corollary \[cor-no-u-rhs\] that $\{u^{(n)}\}$ is a Cauchy sequence in $W^{1,p}_{{\mathcal{D}}}(\Omega)$. Denote its limit by $u\in W^{1,p}_{{\mathcal{D}}}(\Omega)$. Clearly $u$ is a weak solution, and is satisfied.
[*Case 3*]{}: $p\in(4/3,2)$. We first use a duality argument to prove the a priori estimate, i.e. assuming $u\in W^{1,p}_{{\mathcal{D}}}(\Omega)$ is a solution to with $f,f_i\in L_p(\Omega)$, we are to prove the estimate . For simplicity, we consider the following equivalent norm for the space $L_p(\Omega)\times (L_p(\Omega))^d$ with $\lambda>0$: $${\lVert(f,(f_i)_{i=1}^d)\rVert}_{p,\lambda}:= \lambda^{-1/2}{\lVertf\rVert}_{L_p(\Omega)} + \sum_{i=1}^d {\lVertf_i\rVert}_{L_p(\Omega)},$$ and its dual space $L_{p'}(\Omega)\times (L_{p'}(\Omega))^d$: $${\lVert(f,(f_i)_{i=1}^d)\rVert}_{p',1/\lambda}:= \lambda^{1/2}{\lVertf\rVert}_{L_{p'}(\Omega)} + \sum_{i=1}^d {\lVertf_i\rVert}_{L_{p'}(\Omega)},$$ where $p'\in(2,4)$ satisfying $1/p+1/p'=1$.
By duality, to prove , it suffices to prove $$\sup_{\substack{\varphi_i,\varphi \in C^\infty_c(\Omega)\\
{\lVert(\varphi,(\varphi_i)_{i=1}^d)\rVert}_{p',1/\lambda}=1}}{\left\lvert\int_\Omega (D_i u \varphi_i + \lambda u \varphi )\, dx\right\rvert} \leq N\left({\lVertf_i\rVert}_{L_p(\Omega)}+ \lambda^{-1/2}{\lVertf\rVert}_{L_p(\Omega)}\right).$$ For this, we solve for $v\in W^{1,p'}_{{\mathcal{D}}}(\Omega)$ to the following adjoint problem: $$\label{eqn-dual}
\begin{cases}
D_i(a_{ji}D_j v - \hat{b}_i v) - b_i D_i v + cv-\lambda v=\lambda \varphi - D_i \varphi_i & \text{in }\, \Omega,\\
a_{ji}D_j v n_i - \hat{b}_i v n_i=-\varphi_i \cdot n_i & \text{on }\, {\mathcal{N}},\\
v=0 & \text{on }\, {\mathcal{D}}.
\end{cases}$$ Noting that $u$ is a test function for , and $v$ is a test function for , we have $$\begin{aligned}
{\left\lvert\int_\Omega (D_i u \varphi_i + \lambda u \varphi)\,dx\right\rvert} &= {\left\lvert\int_\Omega (-a_{ji}D_j v D_iu + \hat{b}_iv D_iu - b_iD_ivu+cvu -\lambda vu)\,dx\right\rvert}\\
&={\left\lvert\int_\Omega (-f_iD_iv + fv)\,dx\right\rvert}\\
&\leq \big(\|f_i\|_{L_p(\Omega)}+\lambda^{-1/2}\|f\|_{L_p(\Omega)}\big)\big(\|Dv\|_{L_{p'}(\Omega)}+\lambda^{1/2}\|v\|_{L_{p'}(\Omega)}\big)\nonumber
\\
&= N\big(\|f_i\|_{L_p(\Omega)}+\lambda^{-1/2} \|f\|_{L_p(\Omega)}\big).\end{aligned}$$ Here, we use the following $W^{1,p'}$ estimate for the equation : $${\lVertD_iv\rVert}_{L_{p'}(\Omega)} + \sqrt{\lambda}{\lVertv\rVert}_{L_{p'}(\Omega)} \leq N\big({\lVert\varphi_i\rVert}_{L_{p'}(\Omega)}
+\lambda^{-1/2}{\lVert\lambda\varphi\rVert}_{L_{p'}(\Omega)}\big)=N.$$ This gives us the $W^{1,p}$-a priori estimate, i.e., when $p\in(4/3,4)$.
To see the solvability, we approximate $f,f_i$ by $$f^{(n)},f^{(n)}_i \in C^\infty_c (\subset L_p), \quad f^{(n)}\rightarrow f, \ f^{(n)}_i\rightarrow f_i \quad \text{in } L_p.$$ Let $u^{(n)}$ be the unique $W^{1,2}_{\mathcal{D}}$ weak solution associated with $f^{(n)}$ and $f^{(n)}_i$. Due to the $W^{1,p}$-a priori estimate that we just obtained, and the same argument as in case 2, it is enough to show $u^{(n)}\in W^{1,p}_{\mathcal{D}}(\Omega)$. Due to Hölder’s inequality, this can be further reduced to showing the following: $$\label{eqn-holder-decay}
\sum_k{\lVertu\rVert}_{W^{1,2}(\Omega_{k+1}\setminus\Omega_{k})}\cdot k^{(d-1)(1/p-1/2)}<\infty.$$ Here, we denoted $\Omega_k:=\Omega_k(0)$. For this, we use a classical “hole-filling” technique. Take $\eta\in C^\infty_c(B_k^c)$, $\eta =1$ in $B_{k+1}^c$, ${\lvertD\eta\rvert}\leq 2$. Testing the equation by $u\eta^2$ and rearranging terms, we obtain that there exits some $\lambda_0=\lambda_0(d,p,\Lambda,R_0,{\lVertb_i\rVert}_\infty,{\lVert\hat{b_i}\rVert}_\infty,{\lVertc\rVert}_\infty)$, such that for $\lambda>\lambda_0$, $$\int_{\Omega_{k+1}^c} ({\lvertDu\rvert}^2 + \lambda{\lvertu\rvert}^2)\,dx \leq N \int_{\Omega_{k+1}\setminus\Omega_{k}} {\lvertu\rvert}^2\,dx.$$ Clearly, this leads to $${\lVertDu\rVert}^2_{L_2(\Omega_{k+1}^c)} + \lambda{\lVertu\rVert}^2_{L_2(\Omega_{k+1}^c)} \leq \frac{N}{N+\lambda} ({\lVertDu\rVert}^2_{L_2(\Omega_{k}^c)} + \sqrt{\lambda}{\lVertu\rVert}^2_{L_2(\Omega_{k}^c)}).$$ Hence, ${\lVertu\rVert}_{W^{1,2}(\Omega_{k+1}\setminus\Omega_{k})}$ decays exponentially and in particular, holds. This finishes our proof.
Bounded Domain Case
===================
In this section, we deal with the bounded domain case, i.e., Theorem \[thm-bounded-domain\]. First, we reduce the problem to the case $f=0$ by solving a divergence equation. This reduction has also been used in [@CDK18]. Note that in the following, we use a key fact that a Reifenberg flat domain is also a so-called John domain, which can be found in [@DK18-JDE Remark 3.3].
Let us first recall the definition of John domains.
\[def-John\] A bounded set $\Omega\subset{\mathbb{R}}^d$ is a John domain, if there exist $x_0\in \Omega$ and $\lambda>0$ such that for every $x\in\Omega$ there exists a continuous rectifiable curve $\gamma: [0,1]\mapsto \Omega$, such that $\gamma(0)=x,\gamma(1)=x_0$, and $$\label{eqn-John}
{\operatorname{dist}}(\gamma(t),\Omega^c) \geq \lambda \cdot{\lvert\gamma[0,t]\rvert}$$ for all $t\in[0,1]$, where ${\lvert\gamma[0,t]\rvert}$ represents the arc length.
\[lem-solve-divergence-eqn\] Assume $\Omega$ is a bounded Reifenberg flat domain with ${\partial}\Omega={\mathcal{D}}\cup{\mathcal{N}}, {\mathcal{D}}, {\mathcal{N}}\neq \emptyset$, satisfying Assumption \[ass-RF\] $(\gamma)$, $\gamma<1/2$. Let $p>1$ and $p_*$ be given as in . Then for every $f\in L_{p_{*}}(\Omega)$, there exists $\phi=(\phi_1,\cdots,\phi_d)\in (W^{1,p_*}_{\mathcal{N}}(\Omega))^d$ ($\subset(L_p(\Omega))^d$, by the Sobolev embedding), such that $$\label{eqn-est-div}
D_i \phi_i = f \text{ in } \Omega, \quad {\lVert\phi_i\rVert}_{L_p(\Omega)}\leq N{\lVertf\rVert}_{L_{p_{*}}(\Omega)},$$ where $N=N(d,p,{\operatorname{diam}}(\Omega),R_1)$ is a constant.
Noting that ${\mathcal{D}},{\mathcal{N}}\neq\emptyset$, we can choose a point $x_0\in\Gamma$. Taking the coordinate system in $B_{R_1}(x_0)$ from Assumption \[ass-RF\], we extend $\Omega$ beyond ${\mathcal{D}}$ as follows.
We first take the Whitney decomposition of the open set $$\widetilde{\Omega}_{R_1}:=\Omega_{R_1}(x_0)\cap{\{x_{02}+23/32R_1 < y_2 < x_{02}+25/32R_1\}}$$ as in [@St Chapter IV], i.e., $\widetilde{\Omega}_{R_1}=\cup_k Q_k$, where the disjoint cubes $Q_k$ satisfy $${\operatorname{diam}}(Q_k)\leq {\operatorname{dist}}(\overline{Q_k},(\widetilde{\Omega}_{R_1})^c) \leq 4{\operatorname{diam}}(Q_k).$$ Denote the center of $Q_k$ to be $x_k$. We extend $Q_k$ to $\hat{Q}_k$ in the way that $$\hat{Q}_k - x_k = 8(Q_k - x_k).$$ Let $\hat{\Omega}=\Omega\cup(\cup_k \hat{Q}_k)$. It is easy to see that $${\mathcal{N}}\subset {\partial}\hat{\Omega}, \quad C_1 R_1^d\leq{\lvert\hat{\Omega}\setminus\Omega\rvert}\leq C_2 R_1^d,$$ where $C_1, C_2$ are constants only depending on the space dimension $d$. Next, we check that $\hat{\Omega}$ is still a John domain, i.e., for any $\hat{x} \in \hat{\Omega}$ we construct the path connecting $\hat{x}$ and $x_0$, which satisfies the conditions in Definition \[def-John\].
[*Case 1*]{}: $\hat{x} \in \Omega$. Noting that any Reifenberg flat domain is also a John domain, we take the same path as in Definition \[def-John\]. Noting that for any $x\in\Omega$, ${\operatorname{dist}}(x,\hat{\Omega}^c)\geq {\operatorname{dist}}(x,\Omega^c)$, is satisfied with the same $\lambda$.
[*Case 2*]{}: $\hat{x} \in \hat{\Omega}\setminus\Omega$. We assume that $\hat{x}$ lies in the extended cube $\hat{Q}_k$ with center $x_k$ and ${\operatorname{diam}}(\hat{Q}_k)=8r_k$. Let $x_0$ be the point defined in Definition \[def-John\]. If $x_0\in Q_k$, we can take the straight line path. In this case, is satisfied with the constant $7/(9\sqrt{d})$. Now, if $x_0\notin Q_k$, we first consider the straight line path $$\gamma_1: [0,1/2]\mapsto \hat{Q}_k,\quad \gamma_1(0)=\hat{x},\quad \gamma_1(1/2)=x_k.$$ Since $\hat{x}\notin Q_k$, we have $$\frac{1}{2}r_k\leq{\lvert\gamma_1[0,1/2]\rvert}\leq 4\sqrt{d}r_k.$$ Noting that $x_k\in \Omega$, we consider the re-parametrized path coming from Definition \[def-John\]: $$\gamma_2:[0,1/2]\mapsto\Omega, \quad \gamma_2(0)=x_k,\quad \gamma_2(1/2)=x_0.$$ Take $\gamma =\gamma_1\circ\gamma_2$ be the path connecting $\gamma_1$ and $\gamma_2$. Now, when $t\in[0,1/2]$, again is satisfied with the constant $1/\sqrt{d}$. When $t\in[1/2,1]$, we consider the following two cases: $\gamma(t)\in Q_k$ or $\gamma(t)\notin Q_k$.
If $\gamma(t)\in Q_k$, we have $$\begin{split}
\frac{{\lvert\gamma[0,t]\rvert}}{{\operatorname{dist}}(\gamma(t),\hat{\Omega}^c)}
&\leq \frac{{\lvert\gamma[0,1/2]\rvert}}{{\operatorname{dist}}(\gamma(t),\hat{\Omega}^c)} + \frac{{\lvert\gamma[1/2,t]\rvert}}{{\operatorname{dist}}(\gamma(t),\Omega^c)}\\
&\leq \frac{4\sqrt{d}r_k}{7r_k/2} + \lambda^{-1}= \frac{8\sqrt{d}}{7}+\lambda^{-1}.
\end{split}$$ When $\gamma(t)\notin Q_k$, we have ${\lvert\gamma[1/2,t]\rvert}\geq r_k/2$. Hence, $$\begin{split}
\frac{{\lvert\gamma[0,t]\rvert}}{{\operatorname{dist}}(\gamma(t),\hat{\Omega}^c)}
&=\frac{{\lvert\gamma[0,t]\rvert}}{{\lvert\gamma[1/2,t]\rvert}}\cdot\frac{{\lvert\gamma[1/2,t]\rvert}}{{\operatorname{dist}}(\gamma(t),\hat{\Omega}^c)}\\
&=(1+\frac{{\lvert\gamma[0,1/2]\rvert}}{{\lvert\gamma[1/2,t]\rvert}}) \cdot \frac{{\lvert\gamma[1/2,t]\rvert}}{{\operatorname{dist}}(\gamma(t),\hat{\Omega}^c)}\\
&\leq (1+\frac{4\sqrt{d}r_k}{r_k/2})\cdot \frac{{\lvert\gamma[1/2,t]\rvert}}{{\operatorname{dist}}(\gamma(t),\hat{\Omega}^c)}\\
&\leq (1+8\sqrt{d})\lambda^{-1}.
\end{split}$$ With all above, we have proved that $\hat{\Omega}$ is still a John domain. Now we extend $f$ to $\hat{\Omega}$ as $$\begin{cases}
\hat{f}:= f& \text{in }\Omega,\\
\hat{f}:= -\frac{1}{{\lvert\hat{\Omega}\setminus\Omega\rvert}}\int_\Omega f & \text{in }\hat{\Omega}\setminus\Omega.
\end{cases}$$ Then we have $$\int_{\hat{\Omega}} \hat{f} = 0,\quad {\lVert\hat{f}\rVert}_{L_{p_{*}}(\hat{\Omega})}\leq N(R_1,{\lvert\Omega\rvert}){\lVertf\rVert}_{L_{p_{*}}(\Omega)}.$$ Since $\hat{\Omega}$ is a John domain, we apply the result in [@ADM Theorem 4.1] to find $\phi=(\phi_1,\cdots,\phi_d)\in (W^{1,{p_{*}}}_0(\hat{\Omega}))^d$ satisfying $$D_i\phi_i = \hat{f}\quad \text{in }\hat{\Omega}, \quad {\lVert\phi_i\rVert}_{W^{1,{p_{*}}}(\hat{\Omega})}\leq N({\operatorname{diam}}(\hat{\Omega}),d,p){\lVert\hat{f}\rVert}_{L_{p_{*}}(\hat{\Omega})}.$$
Now by Sobolev inequalities and our construction of $\hat{\Omega},\hat{f}$, we obtain that $\phi\in (W^{1,p_{*}}_{\mathcal{N}}(\Omega))^d$, and $${\lVert\phi_i\rVert}_{L_p(\Omega)}\leq N({\operatorname{diam}}(\Omega),R_1,d,p){\lVert\hat{f}\rVert}_{L_{p_{*}}(\hat{\Omega})}.$$ The lemma is proved
Now we are ready to give the proof of Theorem \[thm-bounded-domain\].
Using Lemma \[lem-solve-divergence-eqn\], for every $f\in L_{p_{*}}(\Omega)$, we can find $$(\phi_i)_{i=1}^d\in (W^{1,p_{*}}_{\mathcal{N}}(\Omega))^d\subset (L_p(\Omega))^d$$ satisfying . Now we consider the following problem: $$\label{eqn-no-div}
\begin{cases}
Lu= D_i (f_i+\phi_i) &\text{in }\, \Omega,\\
Bu =(f_i+\phi_i) \, n_i & \text{on }\, {\mathcal{N}},\\
u=0 & \text{on }\, {\mathcal{D}}.
\end{cases}$$ Since $\phi =0$ on ${\mathcal{N}}$, one can easily check that any solution to is also a solution to . Hence, without loss of generality, we may assume $f=0$.
We aim to use the Fredholm alternative. For this, we first introduce some operators. From Theorem \[thm-well-posedness\], for fixed large enough $\lambda$, we can find a unique weak solution $u\in W^{1,p}_{\mathcal{D}}(\Omega)$ to satisfying , and hence .
We write $R(\lambda,L)$ as this solution operator, i.e., $$R(\lambda,L):(L_p(\Omega))^d\times L_p(\Omega)\mapsto W^{1,p}_{\mathcal{D}}(\Omega),\quad R(\lambda,L)(f_i,f)=u.$$ In particular, for any $L_p$ function $f$, we write $$R_\lambda(f):=R(\lambda,L)(0,f).$$ From , $R_\lambda$ is a bounded linear operator from $L_p(\Omega)$ to $W^{1,p}_{\mathcal{D}}(\Omega)$. Denote $I$ as the compact embedding from $W^{1,p}_{\mathcal{D}}(\Omega)$ to $L_p(\Omega)$. Now, we write $$T:W^{1,p}_{\mathcal{D}}(\Omega)\rightarrow W^{1,p}_{\mathcal{D}}(\Omega),\quad T(u)= R_\lambda\circ I(u).$$ From our construction, $T$ is a compact operator. Noting that we have assumed that $f=0$, applying the operator $R(\lambda,L)$ to both sides of $$(L-\lambda)u +\lambda u= D_i f_i,$$ we can rewrite as $$\label{eqn-fredholm}
(Id + \lambda T)u=R(\lambda,L)(f_i,0).$$ By the Fredholm alternative, has a unique $W^{1,p}_{\mathcal{D}}(\Omega)$ solution satisfying $${\lVertu\rVert}_{W^{1,p}(\Omega)}\leq N{\lVertf_i\rVert}_{L_p(\Omega)},$$ if the following homogeneous equation only has zero solution $$\label{eqn-zero}
\begin{cases}
Lv = 0& \text{in }\Omega,\\
Bv=0& \text{on }{\mathcal{N}},\\
v=0& \text{on }{\mathcal{D}}.
\end{cases}$$ When $v\in W^{1,2}(\Omega)$, this is true due to the weak maximum principle, noting that the proof in [@GT Section 8.1] actually shows that $\sup_{\overline{\Omega}}{\lvertv\rvert}$ has to be achieved at the Dirichlet boundary. Hence the uniqueness of is proved for the case $p\geq 2$.
When $p<2$, we can use Theorem \[thm-well-posedness\] and a bootstrap argument to improve the regularity. Suppose $v\in W^{1,p}(\Omega)$ is a solution to . Take $\lambda$ large enough, noting that $v$ is also a $W^{1,p}$ solution to $$\begin{cases}
(L-\lambda) v= -\lambda v &\text{in }\Omega,\\
Bv=0& \text{on }{\mathcal{N}},\\
v=0& \text{on }{\mathcal{D}}.
\end{cases}$$ By the Sobolev embedding, $-\lambda v\in L_{pd/(d-p)}(\Omega)$. Take $p^{*}=\min{\{pd/(d-p),2\}}$. By the uniqueness of $W^{1,p^{*}}$ solutions in Theorem \[thm-well-posedness\], we obtain $v \in W^{1,p^{*}}$. Repeating this process if needed, in finite steps, we can reach $v\in W^{1,2}$. Hence we can use the weak maximum principle to deduce $v=0$ as before. This finishes the proof of the uniqueness. Hence, by the Fredholm alternative, Theorem \[thm-bounded-domain\] is proved.
[10]{}
Gabriel Acosta, Ricardo G. Durán, and María A. Muschietti. Solutions of the divergence operator on [J]{}ohn domains. , 206(2):373–401, 2006.
Russell Brown. The mixed problem for [L]{}aplace’s equation in a class of [L]{}ipschitz domains. , 19(7-8):1217–1233, 1994.
Russell Brown and Irina Mitrea. The mixed problem for the [L]{}amé system in a class of [L]{}ipschitz domains. , 246(7):2577–2589, 2009.
Sun-Sig Byun and Lihe Wang. Elliptic equations with [BMO]{} coefficients in [R]{}eifenberg domains. , 57(10):1283–1310, 2004.
Sun-Sig Byun and Lihe Wang. The conormal derivative problem for elliptic equations with [BMO]{} coefficients on [R]{}eifenberg flat domains. , 90(1):245–272, 2005.
L. A. Caffarelli and I. Peral. On [$W^{1,p}$]{} estimates for elliptic equations in divergence form. , 51(1):1–21, 1998.
Jongkeun Choi, Hongjie Dong, and Doyoon Kim. Conormal derivative problems for stationary [S]{}tokes system in [S]{}obolev spaces. , 38(5):2349–2374, 2018.
Hongjie Dong and Doyoon Kim. Higher order elliptic and parabolic systems with variably partially [BMO]{} coefficients in regular and irregular domains. , 261(11):3279–3327, 2011.
Hongjie Dong and Doyoon Kim. The conormal derivative problem for higher order elliptic systems with irregular coefficients. In [*Recent advances in harmonic analysis and partial differential equations*]{}, volume 581 of [*Contemp. Math.*]{}, pages 69–97. Amer. Math. Soc., Providence, RI, 2012.
Hongjie Dong and Doyoon Kim. Weighted [$L_q$]{}-estimates for stationary [S]{}tokes system with partially [BMO]{} coefficients. , 264(7):4603–4649, 2018.
Hongjie Dong and Doyoon Kim. ${L}_p$-estimates for time fractional parabolic equations with coefficients measurable in time. , 345:289–345, 2019.
Mariano Giaquinta. , volume 105 of [*Annals of Mathematics Studies*]{}. Princeton University Press, Princeton, NJ, 1983.
David Gilbarg and Neil S. Trudinger. . Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.
Carlos E. Kenig. , volume 83 of [*CBMS Regional Conference Series in Mathematics*]{}. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994.
N. V. Krylov and M. V. Safonov. A property of the solutions of parabolic equations with measurable coefficients. , 44(1):161–175, 239, 1980.
Antoine Lemenant, Emmanouil Milakis, and Laura V. Spinolo. On the extension property of [R]{}eifenberg-flat domains. , 39(1):51–71, 2014.
Irina Mitrea and Marius Mitrea. The [P]{}oisson problem with mixed boundary conditions in [S]{}obolev and [B]{}esov spaces in non-smooth domains. , 359(9):4143–4182, 2007.
Katharine A. Ott and Russell M. Brown. The mixed problem for the [L]{}aplacian in [L]{}ipschitz domains. , 38(4):1333–1364, 2013.
E. R. Reifenberg. Solution of the [P]{}lateau [P]{}roblem for [$m$]{}-dimensional surfaces of varying topological type. , 104:1–92, 1960.
M. V. Safonov. Harnack’s inequality for elliptic equations and [H]{}ölder property of their solutions. , 96:272–287, 312, 1980. Boundary value problems of mathematical physics and related questions in the theory of functions, 12.
Giuseppe Savaré. Regularity and perturbation results for mixed second order elliptic problems. , 22(5-6):869–899, 1997.
Eliahu Shamir. Regularization of mixed second-order elliptic problems. , 6:150–168, 1968.
Guido Stampacchia. Problemi al contorno ellitici, con dati discontinui, dotati di soluzionie hölderiane. , 51:1–37, 1960.
Elias M. Stein. . Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.
[^1]: H. Dong and Z. Li were partially supported by the NSF under agreement DMS-1600593.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
In recent years, bootstrap methods have drawn attention for their ability to approximate the laws of “max statistics” in high-dimensional problems. A leading example of such a statistic is the coordinate-wise maximum of a sample average of $n$ random vectors in ${\mathbb{R}}^p$. Existing results for this statistic show that the bootstrap can work when $n\ll p$, and rates of approximation (in Kolmogorov distance) have been obtained with only logarithmic dependence in $p$. Nevertheless, one of the challenging aspects of this setting is that established rates tend to scale like $n^{-1/6}$ as a function of $n$.
The main purpose of this paper is to demonstrate that improvement in rate is possible when extra model structure is available. Specifically, we show that if the coordinate-wise variances of the observations exhibit decay, then a nearly $n^{-1/2}$ rate can be achieved, *independent of $p$*. Furthermore, a surprising aspect of this dimension-free rate is that it holds even when the decay is *very weak*. Lastly, we provide examples showing how these ideas can be applied to inference problems dealing with functional and multinomial data.
author:
-
-
-
title: 'Bootstrapping Max Statistics in High Dimensions: Near-Parametric Rates Under Weak Variance Decay and Application to Functional and Multinomial Data'
---
,
Introduction
============
One of the current challenges in theoretical statistics is to understand when bootstrap methods work in high-dimensional problems. In this direction, there has been a surge of recent interest in connection with “max statistics” such as $$T=\max_{1\leq j\leq p} S_{n,j},$$ where $S_{n,j}$ is the $j$th coordinate of the sum involving i.i.d. vectors $X_1,\dots,X_n$ in ${\mathbb{R}}^p$.
This type of statistic has been a focal point in the literature for at least two reasons. First, it is an example of a statistic for which bootstrap methods can succeed in high dimensions under mild assumptions, which was established in several pathbreaking works . Second, the statistic $T$ is closely linked to several fundamental topics, such as suprema of empirical processes, nonparametric confidence regions, and multiple testing problems. Likewise, many applications of bootstrap methods for max statistics have ensued at a brisk pace in recent years .
One of the favorable aspects of bootstrap approximation results for the distribution $\mathcal{L}(T)$ is that rates have been established with only logarithmic dependence in $p$. For instance, the results in [@CCK:AOP] imply that under certain conditions, the Kolmogorov distance ${d_{\textup{K}}}$ between $\mathcal{L}(T)$ and its bootstrap counterpart $\mathcal{L}(T^*|X)$ satisfies the bound $$\label{eqn:introrate}
{d_{\textup{K}}}\Big( {\mathcal{L}}(T) \, , {\mathcal{L}}(T^*|X)\Big) \ \leq \frac{c \log(p)^{b}}{n^{1/6}}$$ with high probability, where $c,b>0$ are constants not depending on $n$ or $p$, and $X$ denotes the matrix whose rows are $X_1,\dots,X_n$. (In the following, symbols such as $c$ will be often re-used to designate a positive constant not depending on $n$ or $p$, possibly with a different value at each occurrence.) Additional refinements of this result can be found in the same work, with regard to the choice of metric, or choice of bootstrap method. Also, recent progress in sharpening the exponent $b$ has been made by [@Deng:2017]. However, this mild dependence on $p$ is offset by the $n^{-1/6}$ dependence on $n$, which differs from the $n^{-1/2}$ rate in the multivariate Berry-Esseen theorem when $p\ll n$.
Currently, the general problem of determining the best possible rates for Gaussian and bootstrap approximations is largely open in the high-dimensional setting. In particular, if we let $\tilde T$ denote the counterpart of $T$ that arises from replacing $X_1,\dots,X_n$ with independent Gaussian vectors satisfying $\text{cov}(X_i)=\text{cov}(\tilde X_i)$, then a conjecture of [@CCK:AOP] indicates that a bound of the form $d_{\text{K}}({\mathcal{L}}(T),{\mathcal{L}}(\tilde T))\leq cn^{-1/6}\log(p)^b$ is optimal under certain conditions. A related conjecture in the setting of high-dimensional U-statistics may also be found in [@ChenUstat:2018]. (Further discussion of related work on Gaussian approximation is given in Appendix \[app:Gaussian\].) Nevertheless, the finite-sample performance of bootstrap methods for max statistics is often more encouraging than what might be expected from the $n^{-1/6}$ dependence on $n$ [see, e.g. @ChengJASA:2017; @fan:18; @Belloni:2018]. This suggests that improved rates are possible in at least some situations.
The purpose of this paper is to quantify an instance of such improvement when additional model structure is available. Specifically, we consider the case when the coordinates of $X_1,\dots,X_n$ have decaying variances. If we let $\sigma_j^2={\operatorname{var}}(X_{1,j})$ for each $1\leq j\leq p$, and write $\sigma_{(1)}\geq \dots\geq \sigma_{(p)}$, then this condition may be formalized as $$\label{eqn:varcondintro}
\sigma_{(j)} \leq c \, j^{-\alpha} \ \ \ \ \text{for all} \ \ \ j\in\{1,\dots,p\},$$ where $\alpha>0$ is a parameter not depending on $n$ or $p$. (A complete set of assumptions, including a weaker version of , is given in Section \[sec:prelim\].) This type of condition arises in many contexts, and in Section \[sec:prelim\] we discuss examples related to principal component analysis, count data, and Fourier coefficients of functional data. Furthermore, this condition can be assessed in practice, due to the fact that the parameters $\sigma_1,\dots,\sigma_p$ can be accurately estimated, even in high dimensions (cf. Lemma \[lem:corbasic\]).
Within the setting of decaying variances, our main results show that a nearly parametric rate can be achieved for both Gaussian and bootstrap approximation of ${\mathcal{L}}(T)$. More precisely, this means that for any fixed , the bound ${d_{\textup{K}}}( {\mathcal{L}}(T) , {\mathcal{L}}(\tilde T))\leq cn^{-1/2+\delta}$ holds, and similarly, the event $$\label{eqn:improved}
{d_{\textup{K}}}\Big( {\mathcal{L}}(T) \, , {\mathcal{L}}(T^*|X)\Big) \ \leq \ c\, n^{-1/2+\delta}$$ holds with high probability. Here, it is worth emphasizing a few basic aspects of these bounds. First, they are non-asymptotic and *do not depend on $p$*. Second, the parameter $\alpha$ is allowed to be *arbitrarily small*, and in this sense, the decay condition is very weak. Third, the result for $T^*$ holds when it is constructed using the standard multiplier bootstrap procedure [@CCK:2013].
With regard to the existing literature, it is important to clarify that our near-parametric rate does not conflict with the conjectured optimality of the rate $n^{-1/6}$ for Gaussian approximation. The reason is that the $n^{-1/6}$ rate has been established in settings where the values $\sigma_1,\dots,\sigma_p$ are restricted from becoming too small. A basic version of such a requirement is that $$\label{eqn:lower}
\min_{1\leq j\leq p} \sigma_j \geq c.$$ Hence, the conditions and are complementary. Also, it is interesting to observe that the two conditions “intersect” in the limit $\alpha\to 0^+$, suggesting there is a phase transition in rates at the “boundary” corresponding to $\alpha=0$.
Another important consideration that is related to the conditions and is the use of standardized variables. Namely, it is of special interest to approximate the distribution of the statistic $$T'=\max_{1\leq j\leq p} S_{n,j}/\sigma_j,$$ which is equivalent to approximating ${\mathcal{L}}(T)$ when each $X_{i,j}$ is standardized to have variance 1. Given that standardization eliminates variance decay, it might seem that the rate $n^{-1/2+\delta}$ has no bearing on approximating ${\mathcal{L}}(T')$. However, it is still possible to take advantage of variance decay, by using a basic notion that we refer to as “partial standardization”.
The idea of partial standardization is to slightly modify $T'$ by using a fractional power of each $\sigma_j$. Specifically, if we let $\tau_n\in [0,1]$ be a free parameter, then we can consider the partially standardized statistic $$\label{eqn:mdef}
M=\max_{1\leq j\leq p} S_{n,j}/\sigma_j^{\tau_n},$$ which interpolates between $T$ and $T'$ as $\tau_n$ ranges over $[0,1]$. This statistic has the following significant property: If $X_1,\dots,X_n$ satisfy the variance decay condition , and if $\tau_n$ is chosen to be slightly less than 1, then our main results show that the rate $n^{-1/2+\delta}$ holds for bootstrap approximations of ${\mathcal{L}}(M)$. In fact, this effect occurs even when $\tau_n\to 1$ as $n\to\infty$. Further details can be found in Section \[sec:main\]. Also note that our main results are formulated entirely in terms of $M$, which covers the statistic $T$ as a special case.
In practice, simultaneous confidence intervals derived from approximations to ${\mathcal{L}}(M)$ are just as easy to use as those based on ${\mathcal{L}}(T')$. Although there is a slight difference between the quantiles of $M$ and $T'$ when $\tau_n<1$, the important point is that the quantiles of ${\mathcal{L}}(M)$ may be preferred, since faster rates of bootstrap approximation are available. (See also Figure \[fig:tau\] in Section \[sec:expt\].) In this way, the statistic $M$ offers a simple way to blend the utility of standardized variables with the beneficial effects of variance decay.
#### Outline
The remainder of the paper is organized as follows. In Section \[sec:prelim\], we outline the problem setting, with a complete statement of the theoretical assumptions, as well as some motivating facts and examples. Our main results are given in Section 3, which consist of a Gaussian approximation result for ${\mathcal{L}}(M)$ (Theorem \[THM:G\]), and a corresponding bootstrap approximation result (Theorem \[THM:BOOT\]). To provide a numerical illustration of our results, we discuss a problem in functional data analysis in Section \[sec:expt\], where the variance decay condition naturally arises. Specifically, we show how bootstrap approximations to ${\mathcal{L}}(M)$ can be used to derive simultaneous confidence intervals for the Fourier coefficients of a mean function. A second application to high-dimensional multinomial models is described in Section \[sec:multinomial\], which offers both a theoretical bootstrap approximation result, as well as some numerical results. Lastly, our conclusions are summarized in Section \[sec:conc\]. All proofs are given in the appendices, found in the supplementary material.
#### Notation
The standard basis vectors in ${\mathbb{R}}^p$ are denoted $e_1,\dots,e_p$, and the identity matrix of size $p\times p$ is denoted $\mathbf{I}_p$. For any symmetric matrix $A\in{\mathbb{R}}^{p\times p}$, the ordered eigenvalues are denoted $\lambda(A)=(\lambda_1(A),\dots,\lambda_p(A))$, where $\lambda_{\max}(A)=\lambda_1(A)\geq \cdots \geq \lambda_p(A)=\lambda_{\min}(A)$. The operator norm of a matrix $A$, denoted $\|A\|_{\text{op}}$, is the same as its largest singular value. If $v\in{\mathbb{R}}^p$ is a fixed vector, and $r>0$, we write $\|v\|_r=(\sum_{j=1}^p |v_j|^r)^{1/r}$. In addition, the weak-$\ell_r$ (quasi) norm is given by $\|v\|_{w\ell_r}=\max_{1\leq j\leq p} j^{1/r}|v|_{(j)},$ where are the sorted absolute entries of $v$. Likewise, the notation $v_{(1)}\geq\cdots\geq v_{(p)}$ refers to the sorted entries. In a slight abuse of notation, we write $\|\xi\|_r={\mathbb{E}}[|\xi|^r]^{1/r}$ to refer to the $L^r$ norm of a scalar random variable $\xi$, with $r\geq 1$. The $\psi_1$-Orlicz norm is $\|\xi\|_{\psi_1}=\inf\{t>0\, | \, {\mathbb{E}}[\exp(|\xi|/t)]\leq 2\}$. If $\{a_n\}$ and $\{b_n\}$ are sequences of non-negative real numbers, then the relation $a_n\lesssim b_n$ means that there is a constant $c>0$ not depending on $n$, and an integer $n_0\geq 1$, such that $a_n\leq c b_n$ for all $n\geq n_0$. Also, we write $a_n\asymp b_n$ if $a_n\lesssim b_n$ and $b_n\lesssim a_n$. Lastly, define the abbreviations $a_n\vee b_n=\max\{a_n,b_n\}$ and $a_n\wedge b_n=\min\{a_n,b_n\}$.
Setting and preliminaries {#sec:prelim}
=========================
We consider a sequence of models indexed by $n$, with all parameters depending on $n$, except for those that are stated to be fixed. In particular, the dimension $p=p(n)$ is regarded as a function of $n$, and hence, if a constant does not depend on $n$, then it does not depend on $p$ either.
\[A:model\] \
1. There is a vector $\mu=\mu(n)\in{\mathbb{R}}^p$ and positive semi-definite matrix $\Sigma=\Sigma(n)\in{\mathbb{R}}^{p\times p}$, such that the observations $X_1,\dots,X_n\in{\mathbb{R}}^p$ are generated as $X_i=\mu+\Sigma^{1/2}Z_i$ for each $1\leq i\leq n$, where the random vectors $Z_1,\dots,Z_n\in{\mathbb{R}}^p$ are i.i.d.\
2. The random vector $Z_1$ satisfies ${\mathbb{E}}[Z_1]=0$ and ${\mathbb{E}}[Z_1Z_1{^{\top}}]=\mathbf{I}_p$, as well as $\sup_{\|u\|_2=1}\|Z_1{^{\top}}u\|_{\psi_1}\leq c_0$, for some constant $c_0>0$ that does not depend on $n$.
#### Remarks
Note that no constraints are placed on the ratio $p/n$. Also, the sub-exponential tail condition in part *(ii)* is similar to other tail conditions that have been used in previous works on bootstrap methods for max statistics . To state our next assumption, it is necessary to develop some notation. For any $d\in\{1,\dots,p\}$, let ${\mathcal{J}}(d)$ denote a set of indices corresponding to the $d$ largest values among $\sigma_1,\dots,\sigma_p$, i.e., $\{\sigma_{(1)},\dots,\sigma_{(d)}\}= \{\sigma_j| \ j\in{\mathcal{J}}(d)\}$. In addition, let $R(d)\in{\mathbb{R}}^{d\times d}$ denote the correlation matrix of the random variables $\{X_{1,j}\,|\, j\in{\mathcal{J}}(d)\}$. Lastly, let $a\in(0,{\textstyle}\frac{1}{2})$ be a constant fixed with respect to $n$, and define the integers $\ell_n$ and $k_n$ according to $$\begin{aligned}
\
\ell_n&=\big\lceil (1\vee \log(n)^3)\wedge p\big\rceil\\[0.2cm]
k_n&=\big\lceil \big(\ell_n\vee n^{\frac{1}{\log(n)^{a}}}\big)\wedge p\big\rceil.\label{eqn:kndef}\end{aligned}$$ Note that both $\ell_n$ and $k_n$ grow slower than any fractional power of $n$, and always satisfy $1\leq \ell_n\leq k_n\leq p$.
\[A:cor\] \
1. The parameters $\sigma_1,\dots,\sigma_p$ are positive, and there are positive constants $\alpha$, $c$, and $c_{\circ}\in(0,1)$, not depending on $n$, such that $$\label{eqn:varcond1}
\ \sigma_{(j)} \leq c \, j^{-\alpha} \ \ \ \ \text{for all} \ \ \ j\in\{k_n,\dots,p\},$$ $$\label{eqn:varcond2}
\ \sigma_{(j)} \geq {\textstyle}c_{\circ} \, j^{-\alpha}\ \ \ \text{for all} \ \ \ j\in\{1,\dots,k_n\}.$$
2. There is a constant ${\epsilon}_0\in(0,1)$, not depending on $n$, such that $$\label{eqn:cormax}
\max_{i\neq j}R_{i,j}(\ell_n)\leq 1-{\epsilon}_0.$$ Also, the matrix $R^+(\ell_n)$ with $(i,j)$ entry given by $\max\{R_{i,j}(\ell_n),0\}$ is positive semi-definite, and there is a constant $C>0$ not depending on $n$ such that $$\label{eqn:corbound}
\sum_{1\leq i<j\leq \ell_n}\!\!R_{i,j}^+(\ell_n) \ \leq \ C \ell_n.$$
#### Remarks
Since $\ell_n,k_n\ll n$, it is possible to accurately estimate the parameters $\sigma_{(1)},\dots,\sigma_{(k_n)}$, as well as the matrix $R(\ell_n)$, even when $p$ is large (cf. Lemmas \[lem:cor\] and \[lem:corbasic\]). In this sense, it is possible to empirically assess the conditions above. When considering the size of the decay parameter $\alpha$, note that if $\Sigma$ is viewed as a covariance operator acting on a Hilbert space, then the condition $\alpha>1/2$ essentially corresponds to the case of a trace-class operator — a property that is typically assumed in functional data analysis . From this perspective, the condition $\alpha>0$ is very weak, and allows the trace of $\Sigma$ to diverge as $p\to\infty$.
With regard to the conditions on the correlation matrix $R(\ell_n)$, it is important to keep in mind that they only apply to a small set of variables of size $\mathcal{O}(\log(n)^3)$ — and the dependence among the variables outside of ${\mathcal{J}}(\ell_n)$ is *completely unrestricted*. The interpretation of is that it prevents excessive dependence among the coordinates with the largest variances. Meanwhile, the condition that $R^+(\ell_n)$ is positive semi-definite is more technical in nature, and is only used in order to apply a specialized version of Slepian’s lemma (Lemma \[lem:slepian\]). Nevertheless, this condition always holds in the important case where $R(\ell_n)$ is non-negative. Perturbation arguments may also be used to obtain other examples where some entries of $R(\ell_n)$ are negative.
Examples of correlation matrices
--------------------------------
Some correlation matrices satisfying Assumption (ii) are given below. $$\begin{aligned}
\ \ \ \ \ \ & \text{$\bullet$ \emph{Autoregressive:}} && R_{i,j}=\rho_{0}^{|i-j|} \, , \quad\quad\quad\quad\quad\quad\quad \, \ \text{for any } \rho_0\in(0,1). &&& \\[0.7cm]
\ \ \ \ \ \ &\text{$\bullet$ \emph{Algebraic decay:}} && R_{i,j}=1\{i=j\}+{\textstyle}\frac{1\{i\neq j\}}{4|i-j|^{\gamma}} \, , \, \ \, \, \ \ \ \text{ for any} \ \gamma \geq 2. &&& \\[0.7cm]
\ \ \ \ \ \ &\text{$\bullet$ \emph{Banded:}} && R_{i,j}=\Big(1-{\textstyle}\frac{|i-j|}{c_0}\Big)_+ \ ,\quad\quad\quad \,\ \ \text{ for any } c_0>0.\\[0.7cm]
\ \ \ \ \ \ &\text{$\bullet$ \emph{Multinomial:}} && R_{i,j}=1\{i=j\}-\sqrt{{\textstyle}\frac{\pi_i\pi_j}{(1-\pi_i)(1-\pi_j)}}1\{i\neq j\}\,\\[0.1cm]
& \ && \text{where $(\pi_1,\dots,\pi_p)$ is a probability vector.}\end{aligned}$$ By combining these types of correlation matrices with choices of $(\sigma_1,\dots,\sigma_p)$ that satisfy and , it is straightforward to construct examples of $\Sigma$ that satisfy all aspects of Assumption \[A:cor\].
Examples of variance decay {#sec:decayexamples}
--------------------------
To provide additional context for the decay condition , we describe some general situations where it occurs.
- *Principal component analysis (PCA).* The broad applicability of PCA rests on the fact that many types of data have an underlying covariance matrix with weakly sparse eigenvalues. Roughly speaking, this means that most of the eigenvalues of $\Sigma$ are small in comparison to the top few. Similar to the condition , this situation can be modeled with the decay condition $$\label{eqn:eigdecay}
\lambda_j(\Sigma) \leq c j^{-\gamma},$$ for some parameter $\gamma>0$ [e.g. @Bunea:Xiao:2015]. Whenever this holds, it can be shown that the variance decay condition *must* hold for some associated parameter $\alpha>0$, and this is done in Proposition \[PROP:DECAY\] below. So, in a qualitative sense, this indicates that if a dataset is amenable to PCA, then it is also likely to fall within the scope of our setting.\
Another way to see the relationship between PCA and variance decay is through the measure of “effective rank”, defined as $${\tt{r}}(\Sigma)={\textstyle}\frac{{\operatorname{tr}}(\Sigma)}{\|\Sigma\|_{\text{op}}}.$$ This quantity has played a key role in a substantial amount of recent work on PCA, because it offers a useful way to describe covariance matrices with an “intermediate” degree of complexity, which may be neither very low-dimensional, nor very high-dimensional. We refer to [@Vershynin:2012], [@Lounici:2014], [@Bunea:Xiao:2015], [@Reiss:2016], [@Koltchinskii:Bernoulli:2017; @Koltchinskii:2017], [@Nickl:2017], [@Naumov:2017], and [@Jung:2018], among others. Many of these works have focused on regimes where $$\label{eqn:effrank1}
{\tt{r}}(\Sigma)=o(n),$$ which conforms naturally with variance decay. Indeed, within a basic setup where $n\asymp p$ and $\|\Sigma\|{_{\textup{op}}}\asymp 1$, the condition holds under $\sigma_{(j)}\leq cj^{-\alpha}$ for any $\alpha>0$.\
- *Count data.* Consider a multinomial model based on $p$ cells and $n$ trials, parameterized by a vector of cell proportions $\boldsymbol\pi=(\pi_1,\dots,\pi_p)$. If the $i$th trial is represented as a vector $X_i\in{\mathbb{R}}^p$ in the set of standard basis vectors $\{e_1,\dots,e_p\}$, then the marginal distributions of $X_i$ are binomial with $\sigma_j^2=\pi_j(1-\pi_j)$. In particular, it follows that *all* multinomial models satisfy the variance decay condition , because if we let $\boldsymbol \sigma=(\sigma_1,\dots,\sigma_p)$, then the weak-$\ell_2$ norm of $\boldsymbol \sigma$ must satisfy $\|\boldsymbol\sigma\|_{w\ell_2}\leq \|\boldsymbol\sigma\|_2\leq 1$, which implies $$\sigma_{(j)} \ \leq \ j^{-1/2}$$ for all $j\in\{1,\dots,p\}$. In order to study the consequences of this further, we offer some detailed examples in Section \[sec:multinomial\]. More generally, the variance decay condition also arises for other forms of count data. For instance, in the case of a high-dimensional distribution with sparse Poisson marginals, the relation ${\operatorname{var}}(X_{i,j})={\mathbb{E}}[X_{i,j}]$ shows that sparsity in the mean vector can lead to variance decay.\
- *Fourier coefficients of functional data.* Let $Y_1,\dots,Y_n$ be an i.i.d. sample of functional data, taking values in a separable Hilbert space $\mathcal{H}$. In addition, suppose that the covariance operator $\mathcal{C}=\text{cov}(Y_1)$ is trace-class, which implies an eigenvalue decay condition of the form . Lastly, for each $i\in\{1,\dots,n\}$, let $X_i\in{\mathbb{R}}^p$ denote the first $p$ generalized Fourier coefficients of $Y_i$ with respect to some fixed orthonormal basis $\{\psi_j\}$ for $\mathcal{H}$. That is, $X_i=(\langle Y_i,\psi_1\rangle,\dots,\langle Y_i,\psi_p\rangle).$
Under the above conditions, it can be shown that no matter which basis $\{\psi_j\}$ is chosen, the vectors $X_1,\dots, X_n$ always satisfy the variance decay condition. (This follows from Proposition \[PROP:DECAY\] below.) In Section \[sec:expt\], we explore some consequences of this condition as it relates to simultaneous confidence intervals for the Fourier coefficients of the mean function ${\mathbb{E}}[Y_1]$.
To conclude this section, we state a proposition that was used in the examples above. This basic result shows that decay among the eigenvalues $\lambda_1(\Sigma),\dots,\lambda_p(\Sigma)$ requires at least some decay among $\sigma_1,\dots,\sigma_p$.
\[PROP:DECAY\] Fix two numbers $s\geq 1$, and $r\in(0,s)$. Then, there is a constant $c_{r,s}>0$ depending only on $r$ and $s$, such that for any symmetric matrix $A\in{\mathbb{R}}^{p\times p}$, we have $$\|\textup{diag}(A) \|_{w\ell_s}\leq c_{r,s} \| \lambda(A)\|_{w\ell_r}.$$ In particular, if $A=\Sigma$, and if there is a constant $c_0>0$ such that the inequality $$\lambda_j(\Sigma)\leq c_0\, j^{-1/r}$$ holds for all $1\leq j\leq p$, then the inequality $$\sigma_{(j)}^2 \, \leq \, c_0c_{r,s} \, j^{-1/s}$$ holds for all $1\leq j\leq p$.
The proof is given in Appendix \[app:intro\], and follows essentially from the Schur-Horn majorization theorem, as well as inequalities relating $\|\cdot\|_r$ and $\|\cdot\|_{w\ell_r}$.
Main results {#sec:main}
============
In this section, we present our main results on Gaussian approximation and bootstrap approximation.
Gaussian approximation
----------------------
Let $\tilde S_n\sim N(0,\Sigma)$ and define the Gaussian counterpart of the partially standardized statistic $M$ (\[eqn:mdef\]) according to $$\label{eqn:Mtildedef}
\tilde{M}=\max_{1\leq j\leq p} \tilde{S}_{n,j}/\sigma_j^{\tau_n}.$$ Our first theorem shows that in the presence of variance decay, the distribution $\mathcal{L}(\tilde M)$ can approximate $\mathcal{L}(M)$ at a nearly parametric rate in Kolmogorov distance. Recall that for any random variables $U$ and $V$, this distance is given by ${d_{\textup{K}}}({\mathcal{L}}(U),{\mathcal{L}}(V))=\sup_{t\in{\mathbb{R}}}|{\mathbb{P}}(U\leq t)-{\mathbb{P}}(V\leq t)|$.
\[THM:G\] Fix any number $\delta\in(0,1/2)$, and suppose that Assumptions \[A:model\] and \[A:cor\] hold. In addition, suppose that $\tau_n\in [0,1)$ with $(1-\tau_n)\sqrt{\log(n)}\gtrsim 1$. Then, $$\label{eqn:thmG}
d_{\textup{K}}\big(\mathcal{L}(M) \, , \, \mathcal{L}(\tilde{M})\big) \ \lesssim \ n^{-\frac 12+\delta}.$$
#### Remarks
As a basic observation, note that the result handles the ordinary max statistic $T$ as a special case with $\tau_n=0$. In addition, it is worth emphasizing that the rate does not depend on the dimension $p$, or the variance decay parameter $\alpha$, provided that it is positive. In this sense, the result shows that even a small amount of structure can have a substantial impact on Gaussian approximation (in relation to existing $n^{-1/6}$ rates that hold when $\alpha=0$). Lastly, the reason for imposing the lower bound on $1-\tau_n$ is that if $\tau_n$ quickly approaches 1 as $n\to\infty$, then the variances ${\operatorname{var}}(S_{n,j}/\sigma_j^{\tau_n})$ will also quickly approach 1, thus eliminating the beneficial effect of variance decay.
Multiplier bootstrap approximation {#sec:MB}
----------------------------------
In order to define the multiplier bootstrap counterpart of $\tilde M$, first define the sample covariance matrix $$\label{eqn:sighat}
{\widehat{\Sigma}}_n=\frac{1}{n}\sum_{i=1}^n(X_i-\bar X)(X_i-\bar X){^{\top}},$$ where $\bar X=\frac 1n \sum_{i=1}^n X_i$. Next, let $S_n^{\star}\sim N(0,{\widehat{\Sigma}}_n)$, and define the associated max statistic as $$\label{eqn:Mstardef}
M^{\star}=\max_{1\leq j\leq p}S_{n,j}^{\star}/{\widehat{\sigma}}_j^{\tau_n},$$ where $({\widehat{\sigma}}_1^2,\dots,{\widehat{\sigma}}_p^2)=\text{diag}({\widehat{\Sigma}}_n)$. In the exceptional case when ${\widehat{\sigma}}_j=0$ for some $j$, the expression $S_{n,j}^{\star}/{\widehat{\sigma}}_j$ is understood to be 0. This convention is natural, because the event $S_{n,j}^{\star}=0$ holds with probability 1, conditionally on ${\widehat{\sigma}}_j=0$.
#### Remarks
The above description of $M^{\star}$ differs from some previous works insofar as we have suppressed the role of “multiplier variables”, and have defined $S_n^{\star}$ as a sample from $N(0,{\widehat{\Sigma}}_n)$. From a mathematical standpoint, this is equivalent to the multiplier formulation [@CCK:2013], where $S_n^{\star}= \frac{1}{\sqrt n}\sum_{i=1}^n \xi_i^{\star}(X_i-\bar X)$ and $\xi_1^{\star},\dots,\xi_n^{\star}$ are i.i.d. $N(0,1)$ random variables, generated independently of $X$.
\[THM:BOOT\] Fix any number $\delta\in(0,1/2)$, and suppose the conditions of Theorem \[THM:G\] hold. Then, there is a constant $c>0$ not depending on $n$, such that the event $$d_{\textup{K}}\big({\mathcal{L}}( \tilde M)\, ,\, {\mathcal{L}}(M^{\star}|X)\big) \ \leq c\, n^{-\frac 12+\delta}$$ occurs with probability at least $1-\frac cn$.
#### Remarks
At a high level, the proofs of Theorems \[THM:G\] and \[THM:BOOT\] are based on the following observation: When the variance decay condition holds, there is a relatively small subset of $\{1,\dots,p\}$ that is likely to contain the maximizing index for $M$. In other words, if ${\widehat{\j}}\in\{1,\dots,p\}$ denotes a random index satisfying $M=S_{n,{\widehat{\j}}}/\sigma_{{\widehat{\j}}}^{\tau_n}$, then the “effective range” of ${\widehat{\j}}$ is fairly small. Although this situation is quite intuitive when the decay parameter $\alpha$ is large, what is more surprising is that the effect persists even for small values of $\alpha$.
Once the maximizing index ${\widehat{\j}}$ has been localized to a small set, it becomes possible to use tools that are specialized to the regime where $p\ll n$. For example, Bentkus’ multivariate Berry-Esseen theorem [@Bentkus:2003] (cf. Lemma \[lem:bentkus\]) is helpful in this regard. Another technical aspect of the proofs worth mentioning is that they make essential use of the sharp constants in Rosenthal’s inequality, as established in [@Zinn:1985] (Lemma \[lem:rosenthal\]).
Numerical illustration with functional data {#sec:expt}
===========================================
Due to advances in technology and data collection, functional data have become ubiquitous in the past two decades, and statistical methods for their analysis have received growing interest. General references and surveys may be found in . The purpose of this section is to present an illustration of how the partially standardized statistic $M$ and the bootstrap can be employed to do inference on functional data. More specifically, we consider a one-sample test for a mean function, which proceeds by constructing simultaneous confidence intervals (SCI) for its Fourier coefficients. With regard to our theoretical results, this is a natural problem for illustration, because the Fourier coefficients of functional data typically satisfy the variance decay condition , as explained in the third example of Section \[sec:decayexamples\]. Additional background and recent results on mean testing for functional data may be found in , as well as the references therein.
Tests for the mean function
---------------------------
To set the stage, let ${\mathcal{H}}$ be a separable Hilbert space of functions, and let $Y\in {\mathcal{H}}$ be a random function with mean ${\mathbb{E}}[Y]=\mu$. Given a sample $Y_{1},\ldots,Y_{n}$ of i.i.d. realizations of $Y$, a basic goal is to test $$\label{eqn:test}
H_0: \mu=\mu^{\circ} \text{ \ \ \ \ \ versus \ \ \ \ \ } H_1: \mu\neq \mu^{\circ},$$ where $\mu^{\circ}$ is a fixed function in ${\mathcal{H}}$.
This testing problem can be naturally formulated in terms of SCI, as follows. Let $\{\psi_j\}$ denote any fixed orthonormal basis for ${\mathcal{H}}$. Also, let $\{u_j\}$ and $\{u_{j}^{\circ}\}$ respectively denote the generalized Fourier coefficients of $\mu$ and $\mu^{\circ}$ with respect to $\{\psi_j\}$, so that $$\mu={\textstyle\sum}_{j=1}^{\infty}u_{j}\psi_{j} \text{\ \ \ \ \ and \ \ \ \ \ } \mu^{\circ}={\textstyle\sum}_{j=1}^{\infty}u_{j}^{\circ}\,\psi_{j}.$$ Then, the null hypothesis is equivalent to $u_{j}=u_{j}^{\circ}$ for all $j\geq 1$. To test this condition, one can construct a confidence interval ${{\widehat{\mathcal{I}}}}_{j}$ for each $u_{j}$, and reject the null if $u_{j}^{\circ}\not\in {{\widehat{\mathcal{I}}}}_{j}$ for at least one $j\geq 1$. In practice, due to the infinite dimensionality of ${\mathcal{H}}$, one will choose a sufficiently large integer $p$, and reject the null if for at least one $j\in\{1,\dots,p\}$.
Recently, a similar general strategy was pursued by [@Choi2018], hereafter CR, who developed a test for the problem based on a hyper-rectangular confidence region for $(u_1,\dots,u_p)$ — which is equivalent to constructing SCI. In the CR approach, the basis is taken to be the eigenfunctions $\{\psi_{\mathcal{C},j}\}$ of the covariance operator , and $p$ is chosen as the number of eigenfunctions $\psi_{\mathcal{C},1},\dots,\psi_{\mathcal{C},p}$ required to explain a certain fraction (say 99%) of variance in the data. However, since $\mathcal{C}$ is unknown, the eigenfunctions must be estimated from the available data.
When $p$ is large, estimating the eigenfunctions $\psi_{\mathcal{C},1},\dots,\psi_{\mathcal{C},p}$ is a well-known challenge in functional data analysis. For instance, a large choice of $p$ may be needed to explain 99% of the variance if the sample paths of $Y_1,\dots, Y_n$ are not sufficiently smooth. Another example occurs when $H_1$ holds but $\mu$ and $\mu^{\circ}$ are not well separated, which may require a large choice of $p$ in order to distinguish $(u_1,\dots,u_p)$ and $(u_1^{\circ},\dots,u_p^{\circ})$. In light of these considerations, we will pursue an alternative approach to constructing SCI that does not require estimation of eigenfunctions.
Applying the bootstrap {#sec:applyboot}
----------------------
Let $\{\psi_j\}$ be any pre-specified orthonormal basis for ${\mathcal{H}}$. For instance, when ${\mathcal{H}}=L^2[0,1]$, a natural option is the standard Fourier basis. For a sample $Y_1,\dots,Y_n\in{\mathcal{H}}$ as considered before, define random vectors in ${\mathbb{R}}^p$ according to $$X_i=(\langle Y_i,\psi_{1}\rangle,\ldots, \langle Y_i,\psi_{p}\rangle),$$ and note that ${\mathbb{E}}[X_1]=(u_{1},\ldots,u_{p})$. For simplicity, we retain the previous notations associated with $X_1,\dots,X_n$, so that $S_{n,j}=n^{-1/2}\sum_{i=1}^{n}(X_{i,j}-u_{j})$, and likewise for other quantities. In addition, for any $\tau_n\in[0,1]$, let $$L=\min_{1\leq j\leq p}S_{n,j}/\sigma_{j}^{\tau_{n}} \text{ \ \ \ and \ \ \ }M=\max_{1\leq j\leq p}S_{n,j}/\sigma_{j}^{\tau_{n}}.$$ For a given significance level ${\varrho}\in (0,1)$, the ${\varrho}$-quantiles of $L$ and $M$ are denoted $q_L({\varrho})$ and $q_M({\varrho})$. Thus, the following event occurs with probability at least $1-{\varrho}$, $$\label{eq:sci}
\bigcap_{j=1}^{p}\bigg\{ {\textstyle}\frac{q_L({\varrho}/2)\sigma_{j}^{\tau_{n}}}{\sqrt{n}}\leq\bar{X}_{j}-u_{j}\leq {\textstyle}\frac{q_M(1-{\varrho}/2) \sigma_{j}^{\tau_{n}}}{\sqrt{n}}\bigg\},$$ which leads to theoretical SCI for $(u_1,\dots,u_p)$.
We now apply the bootstrap from Section \[sec:MB\] to estimate $q_L({\varrho}/2)$ and $q_M(1-{\varrho}/2)$. Specifically, we generate $B\geq 1$ independent samples of $M^{\star}$ as in , and then define ${\widehat{q}}_M(1-{\varrho}/2)$ to be the empirical $(1-{\varrho}/2)$-quantile of the $B$ samples (and similarly for ${\widehat{q}}_L({\varrho}/2)$), leading to the bootstrap SCI $$\label{eqn:SCIdef}
{\widehat{\mathcal{I}}}_j=\bigg[\bar X_j - {\textstyle}\frac{{\widehat{q}}_M(1-{\varrho}/2) {\widehat{\sigma}}_{j}^{\tau_{n}}}{\sqrt{n}} \ , \ \bar X_j-{\textstyle}\frac{{\widehat{q}}_L({\varrho}/2) {\widehat{\sigma}}_{j}^{\tau_{n}}}{\sqrt{n}}\bigg]$$ for each $j\in\{1,\dots,p\}$.
It remains to select the value of $\tau_n$, for which we adopt the following simple rule. For each choice of $\tau_n$ in a set of possible candidates, say $\mathcal{T}=\{0,0.1,\ldots,0.9,1\}$, we construct the associated intervals ${\widehat{\mathcal{I}}}_1,\dots,{\widehat{\mathcal{I}}}_p$ as in , and then select the value $\tau_n\in\mathcal{T}$ for which the average width $\frac 1p \sum_{j=1}^p |\,{\widehat{\mathcal{I}}}_j|$ is the smallest, where $|[a,b]|=b-a$.
In Figure \[fig:tau\], we illustrate the influence of $\tau_n$ on the shape of the SCI. There are two main points to notice: (1) The intervals change very gradually as a function of $\tau_n$, which shows that partial standardization is at most a mild adjustment of ordinary standardization. (2) The choice of $\tau_n$ involves a tradeoff, which controls the “allocation of power” among the $p$ intervals. When $\tau_n$ is close to 1, the intervals are wider for the leading coefficients (small $j$), and narrower for the subsequent coefficients (large $j$). However, as $\tau_n$ decreases from 1, the widths of the intervals gradually become more uniform, and the intervals for the leading coefficients become narrower. Hence, if the vectors $(u_1,\dots,u_p)$ and $(u_1^{\circ},\dots,u_p^{\circ})$ differ in the leading coefficients, then choosing a smaller value of $\tau_n$ may lead to a gain in power. One last interesting point to mention is that in the simulations reported below, the selection rule of “minimizing the average width” typically selected values of $\tau_n$ around 0.8, and hence strictly less than 1.
![Illustration of the impact of $\tau_n$ on the shape of simultaneous confidence intervals (SCI). The curves represent upper and lower endpoints of the respective SCI, where the Fourier coefficients are indexed by $j$. Overall, the plot shows that the SCI change very gradually as a function of $\tau_n$, and that there is a trade-off in the widths of the intervals. Namely, as $\tau_n$ decreases, the intervals for the leading coefficients (small $j$) become tighter, while the intervals for the subsequent coefficients (large $j$) become wider. []{data-label="fig:tau"}](tau-plot-2.pdf){width="50.00000%" height="\textheight"}
Simulation settings {#sec:simstudy}
-------------------
To study the numerical performance of the SCI described above, we generated i.i.d. samples from a Gaussian process on $[0,1]$, with population mean function $$\mu_{\omega,\rho,\theta}(t)=(1+\rho)\cdot\left(\exp[-\{g_{\omega}(t)+2\}^{2}]+\exp[-\{g_{\omega}(t)-2\}^{2}]\right)+\theta$$ indexed by parameters $(\omega,\rho,\theta)$, where $g_{\omega}(t):=8h_{\omega}(t)-4$, and $h_{\omega}(t)$ denotes the Beta distribution function with shape parameters $(2+\omega,2)$. This family of functions was considered in @Chen2012. To interpret the parameters, note that $\omega$ determines the shape of the mean function (see Figure \[fig:mean-func-family\]), whereas $\rho$ and $\theta$ are scale and shift parameters. In terms of these parameters, the null hypothesis corresponds to $\mu=\mu^{\circ}:=\mu_{0,0,0}$.
![Left: Mean functions for varying shape parameters $\omega$ with $\rho=\theta=0$. Middle: Mean functions for varying scale parameters $\rho$ with $\omega=\theta=0$. Right: Mean functions with different shift parameters $\theta$ with $\omega=\rho=0$.[]{data-label="fig:mean-func-family"}](mean-func-family.pdf){width="\textwidth" height="\textheight"}
The population covariance function was taken to be the Matérn function $${\mathcal{C}}(s,t)={\textstyle}\frac{(\sqrt{2\nu}|t-s|)^{\nu}}{16\Gamma(\nu)2^{\nu-1}}K_{\nu}(\sqrt{2\nu}|t-s|),$$ which was previously considered in CR, with $K_\nu$ being a modified Bessel function of the second kind. We set $\nu=0.1$, which results in relatively rough sample paths, as illustrated in the left panel of Figure \[fig:raw-data\]. Also, the significant presence of variance decay is shown in the right panel.
When implementing the bootstrap in Section \[sec:applyboot\], we used the first $p=100$ functions from the standard Fourier basis on \[0,1\]. (In principle, an even larger value $p$ could have been selected, but we chose $p=100$ to limit computation time.) For comparison purposes, we also implemented the ‘$R_{zs}$’ version of the method proposed in CR, using the accompanying R package [@fregion] under default settings, which typically utilized estimates of the first $p\approx 50$ eigenfunctions of $\mathcal{C}$.
![Left: A sample of the functional data $Y_1,\dots,Y_n$ in the simulation study. Right: The ordered values $\sigma_{(j)}=\sqrt{{\operatorname{var}}(X_{1,j})}$ are represented by dots, which are approximated by the decay profile $0.15j^{-0.69}$ (solid line).[]{data-label="fig:raw-data"}](raw-data-50.pdf "fig:"){width="45.00000%" height="\textheight"} ![Left: A sample of the functional data $Y_1,\dots,Y_n$ in the simulation study. Right: The ordered values $\sigma_{(j)}=\sqrt{{\operatorname{var}}(X_{1,j})}$ are represented by dots, which are approximated by the decay profile $0.15j^{-0.69}$ (solid line).[]{data-label="fig:raw-data"}](sigma-1.pdf "fig:"){width="45.00000%" height="\textheight"}
#### Results on type I error
The nominal significance level was set to $5\%$ in all simulations. To assess the actual type I error, we carried out 5,000 simulations under the null hypothesis, for both $n=50$ and $n=200$. When $n=50$, the type I error was 6.7% for the bootstrap method, and 1.6% for CR. When $n=200$, the results were 5.7% for the bootstrap method, and 2.6% for CR. So, in these cases, the bootstrap respects the nominal significance level relatively well. In addition, our numerical results support the idea that partial standardization can be beneficial, because in the fully standardized case where $\tau_n=1$, we observed less accurate type I error rates of 7.0% for $n=50$, and 6.4% for $n=200$.
#### Results on power
To consider power, we varied each of the parameters $\omega$, $\rho$ and $\theta$, one at a time, while keeping the other two at their baseline value of zero. In each parameter setting, we carried out 1,000 simulations with sample size $n=50$. The results are summarized in Figure \[fig:power\], showing that the bootstrap achieves relative gains in power — especially with respect to the shape ($\omega$) and scale ($\rho$) parameters. In particular, it seems that using a large number of basis functions can help to catch small differences in these parameters (see also Figure \[fig:mean-func-family\]).
![Empirical power for the partially standardized bootstrap method (solid) and the CR method (dotted) Left: Empirical power for varying shape parameters $\omega$ while $\rho=\theta=0$. Middle: Empirical power for varying scale parameters $\rho$ while $\omega=\theta=0$. Right: Empirical power for varying shift parameters $\theta$ while $\omega=\rho=0$.[]{data-label="fig:power"}](power-analysis.pdf){width="\textwidth" height="\textheight"}
Examples with multinomial data {#sec:multinomial}
==============================
When multinomial models are used in practice, it is not uncommon for the number of cells $p$ to be quite large. Indeed, the challenges of this situation have been a topic of sustained interest, and many inferential questions remain unresolved [e.g. @Hoeffding:1965; @Holst:1972; @Fienberg:1973; @cres:84; @zelt:87; @Paninski2008; @Chafai:2009; @Balakrishnan:2019]. A recent survey is [@Balakrishnan:2018]. As one illustration of how our approach can be applied to such models, this section will look at the task of constructing SCI for the cell proportions. Although this type of problem has been studied from a variety of perspectives over the years [e.g. @Quesenberry:1964; @Goodman:1965; @Fitzpatrick:1987; @Glaz:1995; @Wang:2008; @Chafai:2009], relatively few theoretical results directly address the high-dimensional setting — and in this respect, our example offers some progress. Lastly, it is notable that multinomial data are of a markedly different character than the functional data considered in Section \[sec:expt\], which demonstrates how our approach has a broad scope of potential applications.
Theoretical example {#sec:M1}
-------------------
Recall from Section \[sec:decayexamples\] that we regard the observations in the multinomial model as lying in the set of standard basis vectors $\{e_1,\dots,e_p\}\subset{\mathbb{R}}^p$. In this context, we also write ${\widehat{\pi}}_j=\bar{X}_j$ to indicate that the $j$th coordinate of the sample mean is an estimate of the $j$th cell proportion $\pi_j$. In addition, it is important to clarify that a variance decay condition of the form is automatically satisfied in this model (as explained in Section \[sec:decayexamples\]), and so it is not necessary to include this as a separate assumption. Below, we retain the definition of $k_n$ in .
\[A:mult\] \
(i) The observations $X_1,\dots,X_n\in{\mathbb{R}}^p$ are i.i.d., with ${\mathbb{P}}(X_1=e_j)=\pi_j$ for each $j\in\{1,\dots,p\}$, where $\boldsymbol\pi=(\pi_1,\dots,\pi_p)$ is a probability vector that may vary with $n$.\
(ii) There are constants $\alpha>0$ and ${\epsilon}_0\in(0,1)$, with neither depending on $n$, such that $$\sigma_{(j)} \, \geq \, {\epsilon}_0 \, j^{-\alpha} \ \ \ \text{ for all } \ \ \ j\in \{1,\dots,(k_n+1)\wedge p\}.$$
#### Remarks
A concrete set of examples satisfying the conditions of Assumption \[A:mult\] is given by probability vectors of the form $\pi_{(j)}\propto j^{-\eta}$, with $\eta>1$. Furthermore, the condition $\eta>1$ is mild, since the inequality $\pi_{(j)}\leq j^{-1}$ is satisfied by every probability vector.
#### Applying the bootstrap {#applying-the-bootstrap}
In the high-dimensional setting, the multinomial model differs in an essential way from the model in Section \[sec:prelim\], because there will often be many empty cells (indices) $j\in\{1,\dots,p\}$ for which ${\widehat{\sigma}}_j=0$. For the indices where this occurs, the usual confidence intervals of the form have zero width, and thus cannot be used. More generally, if the number of observations in cell $j$ is small, then it is inherently difficult to construct a good confidence interval around $\pi_j$. Consequently, we will restrict our previous SCI by focusing on a set of cells that contain a sufficient number of observations. For theoretical purposes, such a set may be defined as $$\label{eq:Jn}
{\widehat{\mathcal{J}}}_n=\Big\{j\in \{1,\dots,p\} \ \Big| \ {\widehat{\pi}}_j\geq \sqrt{{\textstyle}\frac{\log(n)}{n}}\Big\}.$$ Accordingly, the max statistic and its bootstrapped version are defined by taking maxima over the indices in ${\widehat{\mathcal{J}}}_n$, and we denote them as $${\mathcal{M}}= \max_{j\in{\widehat{{\mathcal{J}}}}_n} S_{n,j}/\sigma_j^{\tau_n}$$ and $${\mathcal{M}}^{\star} = \max_{j\in{\widehat{{\mathcal{J}}}}_n} S_{n,j}^{\star}/{\widehat{\sigma}}_j^{\tau_n},$$ where we arbitrarily take $\mathcal{M}$ and $\mathcal{M}^{\star}$ to be zero in the exceptional case when ${\widehat{\mathcal{J}}}$ is empty.
Although the presence of the random index set ${\widehat{{\mathcal{J}}}}_n$ complicates the distributions of $\mathcal{M}$ and $\mathcal{M}^{\star}$, it is a virtue of the bootstrap that this source of randomness is automatically accounted for in the resulting inference. In addition, the following result shows that the bootstrap continues to achieve a near-parametric rate of approximation.
\[THM:M\] Fix any $\delta\in(0,1/2)$, and suppose that Assumption \[A:mult\] holds. In addition, suppose that $\tau_n\in[0,1)$ with $(1-\tau_n)\sqrt{\log(n)}\gtrsim 1$. Then, there is a constant $c>0$ not depending on $n$ such that the event $$d_{\textup{K}}\big({\mathcal{L}}({\mathcal{M}}), {\mathcal{L}}({\mathcal{M}}^{\star}|X)\big) \ \leq \ c \, n^{-1/2+\delta},$$ occurs with probability at least $1-\frac cn$.
#### Remarks
The proof of this result shares much of the same structure as the proofs of Theorems \[THM:G\] and \[THM:BOOT\], but there are a few differences. First, the use of the random index set ${\widehat{{\mathcal{J}}}}_n$ in the definition of $\mathcal{M}$ and $\mathcal{M}^{\star}$ entails some extra technical considerations, which are handled with the help of Kiefer’s inequality (Lemma \[lem:Kiefer\]). Second, we develop a lower bound for $\lambda_{\min}(\Sigma(k_n))$, where $\Sigma(k_n)$ is the covariance matrix of the variables indexed by ${\mathcal{J}}(k_n)$ (see Lemma \[lem:lambdamin\]). This bound may be of independent interest for problems involving multinomial distributions, and does not seem to be well known; see also [@Benasseni:2012] for other related eigenvalue bounds.
Numerical example {#sec:M2}
-----------------
We illustrate the bootstrap procedure in the case of the model $\pi_j\propto j^{-1}$, which was considered in a recent numerical study of [@Balakrishnan:2018]. Taking $p=1000$ and we applied the bootstrap method to construct 95% SCI for the proportions $\pi_j$ corresponding to the cells with at least 5 observations. The cutoff value of $5$ is based on a guideline that is commonly recommended in textbooks, e.g., [@Agresti:2003 p.19], [@Rice:2007 p.519]. Lastly, the parameter $\tau_n$ was chosen in the same way as described in Section \[sec:applyboot\].
Based on 5000 Monte Carlo runs, the average coverage probability was found to be 93.7% for $n=500$, and 94.4% for $n=1000$, demonstrating satisfactory performance. Regarding the parameter $\tau_n$, the selection rule typically produced values close to 0.8, for both $n=500$ and $n=1000$. As a point of comparison, it is also interesting to mention the coverage probabilities that occurred when $\tau_n$ was set to 1 (which eliminates all variance decay). In this case, the coverage probabilities became less accurate, with values of $92.7\%$ for $n=500$, and $93.1\%$ for $n=1000$. Hence, this shows that taking advantage of variance decay can enhance coverage probability.
Conclusions {#sec:conc}
===========
The main conclusion to draw from our work is that a modest amount of variance decay in a high-dimensional model can substantially improve rates of bootstrap approximation for max statistics — which helps to reconcile some of the empirical and theoretical results in the literature. In particular, there are three aspects of this type of model structure that are worth emphasizing. First, the variance decay condition is very weak, in the sense that the parameter $\alpha>0$ is allowed to be arbitrarily small. Second, the condition is approximately checkable in practice, since the parameters $\sigma_1,\dots,\sigma_p$ can be accurately estimated when $n\ll p$. Third, this type of structure arises naturally in a variety of contexts.
Beyond our main theoretical focus on rates of bootstrap approximation, we have also shown that the technique of partial standardization leads to favorable numerical results. Specifically, this was illustrated with examples involving both functional and multinomial data, where variance decay is an inherent property that can be leveraged. Finally, we note that these applications are by no means exhaustive, and the adaptation of the proposed approach to other types of data may provide further opportunities for future work.
#### Organization of appendices
In Appendix \[app:intro\] we prove Proposition \[PROP:DECAY\], and in Appendices \[app:thmg\] and \[app:thmboot\] we prove Theorems \[THM:G\] and \[THM:BOOT\] respectively. These proofs rely on numerous technical lemmas, which are stated and proved in Appendix \[app:technical\]. Next, the proof of Theorem \[THM:M\] for the multinomial model is given in Appendix \[app:mult\], and the associated technical lemmas are given in Appendix \[app:lemmamult\]. Lastly, in Appendix \[app:background\] we provide statements of background results, and in Appendix \[app:Gaussian\] we provide a discussion of related work in the Gaussian approximation literature.
#### General remarks and notation
Based on the formulation of Theorems \[THM:G\], \[THM:BOOT\], and \[THM:M\], it is sufficient to show that these results hold for all large values of $n$, and it will simplify some of the proofs to make use of this reduction. For this reason, it is understood going forward that $n$ is sufficiently large for any given expression to make sense. Another convention is that all proofs in Appendices \[app:thmg\], \[app:thmboot\], \[app:technical\], \[app:mult\], and \[app:lemmamult\] will implicitly assume that $p>k_n$ (unless otherwise stated), because once the proofs are given for this case, it will follow that the low-dimensional case where $p\leq k_n$ can be handled as a direct consequence (which is explained on page ).
To fix some notation that will be used throughout the appendices, let $d\in\{1,\dots,p\}$, and define a generalized version of $M$ as $$M_d=\max_{j\in{\mathcal{J}}(d)}S_{n,j}/\sigma_j^{\tau_n}.$$ In particular, the statistic $M$ defined in equation is the same as $M_p$. Similarly, the Gaussian and bootstrap versions of $M_d$ are defined as $$\tilde M_d=\max_{j\in{\mathcal{J}}(d)}\tilde S_{n,j}/\sigma_j^{\tau_n},$$ and $$M_d^{\star} =\max_{j\in{\mathcal{J}}(d)}S_{n,j}^{\star}/{\widehat{\sigma}}_j^{\tau_n}.$$ In addition, define the parameter $$\beta_n=\alpha(1-\tau_n).$$ Lastly, we will often use the fact that if a random variable $\xi$ satisfies the bound $\|\xi\|_{\psi_1}\leq c$ for some constant $c$ not depending on $n$, then there is another constant $C>0$ not depending on $n$, such that $\|\xi\|_r\leq C\, r$ for all $r\geq 1$ [@VershyninHDP Proposition 2.7.1].
Proof of Proposition \[PROP:DECAY\] {#app:intro}
===================================
It is a standard fact that for any $s\geq 1$, the $\ell_s$ norm dominates its $w\ell_s$ counterpart, and so $\|\text{diag}(A)\|_{w\ell_s}\leq \|\text{diag}(A)\|_s$. Next, since $A$ is symmetric, the Schur-Horn Theorem implies that the vector $\text{diag}(A)$ is majorized by $\lambda(A)$ [@Olkin:2011 p.300]. Furthermore, when $s\geq 1$, the function $\|\cdot \|_s$ is Schur-convex on ${\mathbb{R}}^p$, which means that if $u\in{\mathbb{R}}^p$ is majorized by $v\in{\mathbb{R}}^p$, then $\|u\|_s\leq \|v\|_s$ [@Olkin:2011 p.138]. Hence, $$\|\text{diag}(A)\|_{w\ell_s}\leq \| \lambda(A)\|_s.$$ Finally, if $r\in(0,s)$, then for any $v\in{\mathbb{R}}^p$, the inequality $$\|v\|_s \leq \big(\zeta(s/r)\big)^{1/s}\,\|v\|_{w\ell_r}$$ holds, where $\zeta(x):=\sum_{j=1}^{\infty}j^{-x}$ for $x>1$. This bound may be derived as in [@Johnstone:2017 p.257], $$\|v\|_s^s \ = \ \sum_{j=1}^p |v|_{(j)}^s \ \leq \ \sum_{j=1}^p \big(\|v\|_{w\ell_r} j^{-1/r}\big)^s \ \leq \ \zeta(s/r)\cdot \|v\|_{w\ell_r}^s,$$ which completes the proof.
Proof of Theorem \[THM:G\] {#app:thmg}
==========================
Consider the inequality $$d_{\textup{K}}(\mathcal{L}(M_p),\mathcal{L}(\tilde{M}_p)) \, \leq \, {\textsc{I}}_n+{\textsc{II}}_n+{\textsc{III}}_n,$$ where we define $$\begin{aligned}
{\textsc{I}}_n &= d_{\text{K}}\Big(\mathcal{L}(M_p) \, , \, \mathcal{L}(M_{k_n})\Big)\\[0.2cm]
{\textsc{II}}_n &=d_{\text{K}}\Big( \mathcal{L}(M_{k_n})\, , \, \mathcal{L}(\tilde M_{k_n})\Big)\\[0.2cm]
{\textsc{III}}_n &=d_{\text{K}}\Big( \mathcal{L}(\tilde M_{k_n})\, ,\, \mathcal{L}(\tilde M_p)\Big).\label{eqn:IIIdef}\end{aligned}$$ Below, we show that the term ${\textsc{II}}_n$ is at most of order $n^{-\frac{1}{2}+\delta}$ in Proposition \[prop:bentkus\]. Later on, we establish a corresponding result for ${\textsc{I}}_n$ and ${\textsc{III}}_n$ in Proposition \[prop:IandIII\]. Taken together, these results complete the proof of Theorem \[THM:G\].
\[prop:bentkus\] Fix any number $\delta\in(0,1/2)$, and suppose the conditions of Theorem \[THM:G\] hold. Then, $$\label{eqn:IInbound}
{\textsc{II}}_n \, \lesssim \, n^{-\frac 12+\delta}.$$
Let $\Pi_{k_n}\in{\mathbb{R}}^{k_n\times p}$ denote the projection onto the coordinates indexed by ${\mathcal{J}}(k_n)$. This means that if we write ${\mathcal{J}}(k_n)=\{j_1,\dots,j_{k_n}\}$ so that $(\sigma_{j_1},\dots,\sigma_{j_{k_n}})=(\sigma_{(1)},\dots,\sigma_{(k_n)})$, then the $l$th row of $\Pi_{k_n}$ is the standard basis vector $e_{j_l}\in{\mathbb{R}}^p$. Next, define the diagonal matrix $D_{k_n}={\textup{diag}}(\sigma_{(1)},\dots,\sigma_{(k_n)})$. It follows that $$M_{k_n}=\max_{1\leq j\leq k_n} e_j{^{\top}}D_{k_n}^{-\tau_n}\Pi_{k_n} S_n.$$ Define the matrix $\mathfrak{C}{^{\top}}=D_{k_n}^{-\tau_n} \Pi_{k_n}\Sigma^{1/2}$, which is of size $k_n\times p$. Also, let $r$ denote the rank of $\mathfrak{C}$, and note that $r\leq k_n$, since the matrix $\Sigma$ need not be invertible. Next, consider a decomposition $$\mathfrak{C}=QR,$$ where the columns of $Q\in{\mathbb{R}}^{p\times r}$ are an orthonormal basis for the image of $\mathfrak{C}$, and $R\in{\mathbb{R}}^{r\times k_n}$. Hence, if we define the random vector $$\label{eqn:Zrep}
\breve Z={\textstyle}\frac{1}{\sqrt n}\sum_{i=1}^n Q{^{\top}}Z_i,$$ then we have $$D_{k_n}^{-\tau_n}\Pi_{k_n} S_n =R{^{\top}}\breve Z.$$ It is simple to check that for any fixed $t\in{\mathbb{R}}$, there exists a Borel convex set $\mathcal{A}_t\subset{\mathbb{R}}^{r}$ such that ${\mathbb{P}}(M_{k_n}\leq t)={\mathbb{P}}(\breve Z\in\mathcal{A}_t)$. By the same reasoning, we also have ${\mathbb{P}}(\tilde M_{k_n}\leq t)=\gamma_r(\mathcal{A}_t)$, where $\gamma_{r}$ is the standard Gaussian distribution on ${\mathbb{R}}^{r}$. Therefore, the quantity ${\textsc{II}}_n$ satisfies the bound $$\begin{split}
{\textsc{II}}_n
& \ \leq \ \sup_{\mathcal{A}\in\mathscr{A}}\, \Big| {\mathbb{P}}\big(\breve Z\in \mathcal{A} \big)- \gamma_{r}(\mathcal{A})\Big|,
\end{split}$$ where $\mathscr{A}$ denotes the collection of all Borel convex subsets of ${\mathbb{R}}^{r}$.
We now apply Theorem 1.1 of [@Bentkus:2003] (Lemma \[lem:bentkus\]), to handle the supremum above. First observe that the definition of $\breve Z$ in satisfies the conditions of that result, since the terms $Q{^{\top}}Z_1,\dots, Q{^{\top}}Z_n$ are i.i.d. with zero mean and identity covariance matrix. Therefore, $${\textsc{II}}_n \ \lesssim \ r^{1/4}\cdot {\mathbb{E}}\big[ \|Q{^{\top}}Z_1\|_2^3] \cdot n^{-1/2}.$$ It remains to bound the middle factor on the right side. By Lyapunov’s inequality, $$\label{eqn:fourthcalc}
\begin{split}
{\mathbb{E}}\big[\|Q{^{\top}}Z_1\|_2^3\big] &\leq \Big({\mathbb{E}}\Big[\big(Z_1{^{\top}}QQ{^{\top}}Z_1\big)^2\Big]\Big)^{3/4}.
\end{split}$$ Next, if $v_1,\dots,v_r$ denote the orthonormal columns of $Q$, then we have $$QQ{^{\top}}= {\textstyle\sum}_{j=1}^r v_jv_j{^{\top}}.$$ Hence, if we put $\zeta_j=Z_1{^{\top}}v_j$, then $$\begin{split}
{\mathbb{E}}\Big[(Z_1{^{\top}}QQ{^{\top}}Z_1)^2\Big] \ &= \ \Big\|{\textstyle\sum}_{j=1}^r \zeta_j^2\Big\|_2^2\\[0.2cm]
& \ \leq \ \Big({\textstyle\sum}_{j=1}^r \|\zeta_j^2\|_2\Big)^2\\[0.2cm]
& \ \lesssim \ k_n^2,
\end{split}$$ where we have used the fact that $r\leq k_n$ and $\|\zeta_j^2\|_2=\|Z_1{^{\top}}v_j\|_4^2\lesssim 1$, based on Assumption \[A:model\]. Combining the last few steps gives ${\mathbb{E}}[\|Q{^{\top}}Z_1\|_2^3]\lesssim k_n^{6/4}$, and hence $${\textsc{II}}_n \ \lesssim \ k_n^{7/4}n^{-1/2}\ \lesssim \ n^{-\frac 12+\delta},$$ as needed.
\
\[prop:IandIII\] Fix any number $\delta\in(0,1/2)$, and suppose the conditions of Theorem \[THM:G\] hold. Then, $$\label{IandIII}
{\textsc{I}}_n \, \lesssim \, n^{-\frac 12+\delta} \ \ \ \text{ and } \ \ \ {\textsc{III}}_n \, \lesssim \, n^{-\frac 12+\delta}.$$
We only prove the bound for ${\textsc{I}}_n$, since the same argument applies to ${\textsc{III}}_n$. It is simple to check that for any fixed real number $t$, $$\Big|{\mathbb{P}}\Big(\max_{1\leq j\leq p} S_{n,j}/\sigma_j^{\tau_n} \leq t\Big)-{\mathbb{P}}\Big(\max_{j\in{\mathcal{J}}(k_n)} S_{n,j}/\sigma_j^{\tau_n} \leq t\Big)\Big| = \ {\mathbb{P}}\Big(A(t)\cap B(t)\Big),$$ where we define the events $$\label{inclusion}
\small
A(t)=\Big\{\max_{j\in{\mathcal{J}}(k_n)} S_{n,j}/\sigma_j^{\tau_n} \leq t\Big\} \ \ \ \text{ and } \ \ \ \ B(t)=\Big\{\max_{j\in{\mathcal{J}}(k_n)^c} S_{n,j}/\sigma_j^{\tau_n}> t\Big\},$$ and ${\mathcal{J}}(k_n)^c$ denotes the complement of ${\mathcal{J}}(k_n)$ in $\{1,\dots,p\}$. Also, for any pair of real numbers $t_{1,n}$ and $t_{2,n}$ satisfying $t_{1,n}\leq t_{2,n}$, it is straightforward to check that the following inclusion holds for all $t\in{\mathbb{R}}$, $$\label{inclusion}
A(t)\cap B(t) \ \subset \ A(t_{2,n})\cup B(t_{1,n}).$$ Applying a union bound, and then taking the supremum over $t\in{\mathbb{R}}$, we obtain $${\textsc{I}}_n\,\leq \, {\mathbb{P}}(A(t_{2,n}))\,+\, {\mathbb{P}}(B(t_{1,n})).$$
The remainder of the proof consists in selecting $t_{1,n}$ and $t_{2,n}$ so that $t_{1,n}\leq t_{2,n}$ and that the probabilities ${\mathbb{P}}(A(t_{2,n}))$ and ${\mathbb{P}}(B(t_{1,n}))$ are sufficiently small. Below, Lemma \[lem:bounds\] shows that if $t_{1,n}$ and $t_{2,n}$ are chosen as $$\begin{aligned}
t_{1,n}&=c\cdot k_n^{-\beta_n}\cdot \log(n) \label{eqn:t1}\\[0.3cm]
t_{2,n}&=c_{\circ}\cdot \ell_n^{-\beta_n}\cdot\sqrt{\log(\ell_n)},\label{eqn:t2}\end{aligned}$$ for a certain constant $c>0$, and $c_{\circ}$ as in , then ${\mathbb{P}}(A(t_{2,n}))$ and ${\mathbb{P}}(B(t_{1,n}))$ are at most of order $n^{-\frac 12+\delta}$. Furthermore, the inequality $t_{1,n}\leq t_{2,n}$ holds for all large $n$, due to the definitions of $\ell_n$, $k_n$, and $\beta_n$, as well as the condition $(1-\tau_n)\sqrt{\log(n)}\gtrsim 1$.
\[lem:bounds\] Fix any number $\delta\in(0,1/2)$, and suppose the conditions of Theorem \[THM:G\] hold. Then, there are positive constants $c$ and $c_{\circ}$, not depending on $n$, that can be selected in the definitions of $t_{1,n}$ and $t_{2,n}$, so that $$\label{aboundlem}
{\mathbb{P}}(A(t_{2,n})) \, \lesssim \, n^{-\frac 12+\delta},\tag{a}$$ and $$\label{bboundlem}
{\mathbb{P}}(B(t_{1,n})) \, \lesssim \, n^{-1}.\tag{b}$$
#### <span style="font-variant:small-caps;">Proof of Lemma \[lem:bounds\] part</span>
Due to Proposition \[prop:bentkus\] and the fact that ${\mathcal{J}}(\ell_n)\subset {\mathcal{J}}(k_n)$, we have $$\begin{split}
{\mathbb{P}}(A(t_{2,n})) &\ \leq \ {\mathbb{P}}\Big(\max_{j\in{\mathcal{J}}(k_n)} \tilde S_{n,j}/\sigma_j^{\tau_n}\leq t_{2,n}\Big)+{\textsc{II}}_n\\[0.2cm]
& \ \leq \ \ {\mathbb{P}}\Big(\max_{j\in{\mathcal{J}}(\ell_n)} \tilde S_{n,j}/\sigma_j^{\tau_n}\leq t_{2,n}\Big)+ c\, n^{-\frac 12+\delta}.
\end{split}$$ To bound the probability in the last line, we will make use of the Gaussianity of $\tilde S_n$ to apply certain results based on Slepian’s lemma, as contained in Lemmas \[lem:slepian2\] and \[lem:slepian\] below. As a preparatory step, consider some generic random variables $\{Y_j\}$ and positive scalars $\{a_j\}$ indexed by ${\mathcal{J}}(\ell_n)$, as well as a constant $b$ such that $\max_{j\in{\mathcal{J}}(\ell_n)}a_j\leq b$. Then, $$\begin{split}
{\mathbb{P}}\Big(\max_{j\in{\mathcal{J}}(\ell_n)} Y_j\leq t_{2,n}\Big) & \leq \ {\mathbb{P}}\Big(\max_{j\in{\mathcal{J}}(\ell_n)}a_jY_j \leq b\, t_{2,n}\Big),
\end{split}$$ which can be seen by expressing the left side in terms of $\cap_j\{a_jY_j\leq a_j t_{2,n}\}$, and noting that this set is contained in $\cap_j\{a_jY_j\leq b t_{2,n}\}$. Due to Assumption \[A:cor\] with $c_{\circ}\in(0,1)$, we have the inequality $\sigma_j^{\tau_n-1}\leq \ell_n^{\beta_n}/c_{\circ}$ for all $j\in{\mathcal{J}}(\ell_n)$, and so we may apply the previous observation with $a_j=\sigma_j^{\tau_n-1}$, and $b=\ell_n^{\beta_n}/c_{\circ}$. Furthermore, the definition of $t_{2,n}$ gives $b\, t_{2,n}=\sqrt{\log(\ell_n)}$, and so we if we let $Y_j=\tilde S_{n,j}/\sigma_j^{\tau_n}$, it follows that $${\mathbb{P}}\Big(\max_{j\in{\mathcal{J}}(\ell_n)} \tilde S_{n,j}/\sigma_j^{\tau_n}\leq t_{2,n}\Big) \ \leq \ {\mathbb{P}}\Big(\max_{j\in{\mathcal{J}}(\ell_n)} \tilde S_{n,j}/\sigma_j\leq \sqrt{\log(\ell_n)}\Big).$$ The proof is completed by applying the next result (Lemma \[lem:slepian2\]) in conjunction with the conditions of Assumption \[A:cor\]. (Take $m=\ell_n$ in the statement of Lemma \[lem:slepian2\].)
#### Remark
The lemma below may be of independent interest, and so we have stated it in a way that can be understood independently of the context of our main assumptions. Also, the constants $1/2$ and $1/3$ in the exponent of the bound can be improved slightly, but we have left the result in this form for simplicity.
\[lem:slepian2\] For each integer $m\geq 1$, let $\mathsf{R}=\mathsf{R}(m)$ be a correlation matrix in ${\mathbb{R}}^{m\times m}$, and let $\mathsf{R}^+=\mathsf{R}^+(m)$ denote the matrix with $(i,j)$ entry given by $\max\{\mathsf{R}_{i,j},0\}$. Suppose the matrix $\mathsf{R}^+$ is positive semi-definite for all $m$, and that there are constants ${\epsilon}_0\in(0,1)$ and $c>0$, not depending on $m$, such that the inequalities $$\begin{aligned}
\ \max_{i\neq j}\mathsf{R}_{i,j} & \ \leq \, 1-{\epsilon}_0\\[0.2cm]
{\textstyle}\sum_{i\neq j} \mathsf{R}_{i,j}^+ & \ \leq \ c\, m\label{eqn:rpluscondn}\end{aligned}$$ hold for all $m$. Lastly, let $(\zeta_1,\dots,\zeta_m)$ be a Gaussian vector drawn from $N(0,\mathsf{R})$. Then, there is a constant $C>0$, not depending on $m$, such that the inequality $$\label{eqn:slepian2result}
{\mathbb{P}}\Big(\max_{1\leq j\leq m} \zeta_j\leq \sqrt{\log(m)}\Big) \ \leq \ C\exp\big(-{\textstyle}\frac{1}{2}m^{1/3}\big)$$ holds for all $m$.
It is enough to show that the result holds for all large $m$, because if $m\leq m_0$, then the result is clearly true when $C= \exp(\frac{1}{2}m_0^{1/3})$. To begin the argument, we may introduce a Gaussian vector $(\xi_1,\dots,\xi_m)\sim N(0,\mathsf{R}^+)$, since the matrix $\mathsf{R}^+$ is positive semi-definite. In turn, the version of Slepian’s Lemma given in Lemma \[lem:slepian\] leads to $$\label{eqn:splitfactor}
\begin{split}
{\mathbb{P}}\Big(\max_{1\leq j\leq m}\zeta_j \leq \sqrt{\log(m)}\Big)
& \ \leq \ {\mathbb{P}}\Big(\max_{1\leq j\leq \ell_n}\xi_j \leq \sqrt{\log(m)}\Big)\\[0.3cm]
& \ \leq \ K_m\cdot \Phi^{m}\big( \sqrt{\log(m)}\big),
\end{split}$$ where we put $$K_m= \exp\Bigg\{\sum_{1\leq i<j\leq m} \log\Big({\textstyle}\frac{1}{1-\frac{2}{\pi}\arcsin(\mathsf{R}_{i,j}^+)}\Big)\exp\Big(-{\textstyle}\frac{\log(m)}{1+\mathsf{R}_{i,j}^+}\Big)\Bigg\}.$$ Next, we apply the assumption $\max_{i\neq j}\mathsf{R}_{i,j}^+\leq 1-{\epsilon}_0$. Since the functions $x\mapsto \log(1/(1-x))$ and $x\mapsto\frac{2}{\pi}\arcsin(x)$ have bounded derivatives on any closed subinterval of $[0,1)$, it follows that $$\log\Big({\textstyle}\frac{1}{1-\frac{2}{\pi}\arcsin(\mathsf{R}_{i,j}^+)}\Big) \ \leq c\, \mathsf{R}_{i,j}^+,$$ for some constant $c>0$ not depending on $m$. Therefore, by possibly increasing $c$, the condition gives $$\begin{split}
K_m
& \ \leq \ \exp\Big\{ cm \cdot \exp\Big(-{\textstyle}\frac{\log(m)}{1+(1-{\epsilon}_0)}\Big)\Big\}\\[0.3cm]
& \ = \ \exp\Big\{cm^{1-\frac{1}{2-{\epsilon}_0}}\Big\}.
\end{split}$$ To bound the earlier factor involving $\Phi^m(\sqrt{\log(m)})$, let $\eta_0\in(0,1)$ be a small constant to be optimized below, and note that the following inequality holds for all sufficiently large $s>0$, $$\label{numericalineq}
\begin{split}
\Phi\Big(\sqrt{(2-\eta_0{\epsilon}_0)\log(s)}\Big) \ \leq \ 1-{\textstyle}\frac{1}{s},
\end{split}$$ which may be found in [@Massart:2013 p.337]. Taking $s=m^{\kappa_0}$ with $\kappa_0:=\frac{1}{2-\eta_0{\epsilon}_0}$ shows that for all large $m$, $$\begin{split}
\Phi^{m}\big(\sqrt{\log(m)}\big) & \ \leq \ \Big(1-{\textstyle}\frac{1}{m^{\kappa_0}}\Big)^{m}\\[0.3cm]
& \ \leq \ \exp\big(-m^{1-\kappa_0}\big).
\end{split}$$ We now collect the last several steps. If we observe that $\kappa_0<\frac{1}{2-{\epsilon}_0}$, then the following inequalities hold for all large $m$, $$\begin{split}
K_m\cdot \Phi^{m}\big( \sqrt{\log(m)}\big) & \ \leq \ \exp\Big\{cm^{1-\frac{1}{2-{\epsilon}_0}}-m^{1-\kappa_0}\Big\}\\[0.3cm]
& \ \leq \ \exp\Big\{-(1-\eta_0)m^{1-\kappa_0}\Big\}.
\end{split}$$ So, by possibly further decreasing $\eta_0$, we have $(1-\kappa_0)>1/3$, as well as $(1-\eta_0)>1/2$. This leads to the stated result.
\
#### <span style="font-variant:small-caps;">Proof of Lemma \[lem:bounds\] part</span>
Define the random variable $$V=\max_{j\in{\mathcal{J}}(k_n)^c} S_{n,j}/\sigma_j^{\tau_n},$$ and let $$q=\max\big\{{\textstyle}\frac{2}{\beta_n}, \log(n), 3\big\}.$$ Clearly, for any $t>0$, we have the tail bound $$\label{qchebyshev}
{\mathbb{P}}\big( V\geq t)\leq \frac{\|V\|_q^q}{t^q},$$ and furthermore $$\begin{split}
\|V\|_q^q &= {\mathbb{E}}\bigg[\Big|\max_{j\in{\mathcal{J}}(k_n)^c} S_{n,j}/\sigma_j^{\tau_n}\Big|^q\bigg]\\[0.3cm]
&\leq \sum_{j\in{\mathcal{J}}(k_n)^c} \sigma_j^{q(1-\tau_n)}\,{\mathbb{E}}\big[|{\textstyle}\frac{1}{\sigma_j}S_{n,j}|^q\big].
\end{split}$$ By Lemma \[lem:Snjnorm\], we have $\|\frac{1}{\sigma_j}S_{n,j}\|_{q}\leq cq$, and so $$\begin{split}
\|V\|_q^q \ &\leq \ (c q)^q \sum_{j\in{\mathcal{J}}(k_n)^c} \sigma_j^{q(1-\tau_n)} \\[0.3cm]
&\lesssim \ (c q)^q \sum_{j=k_n+1}^p j^{-q \beta_n }\\[0.3cm]
&\leq \ (cq)^q \int_{k_n}^p x^{-q\beta_n} dx\\[0.3cm]
&\leq \ {\textstyle}\frac{(c q)^q }{q \beta_n-1}\ k_n^{-q \beta_n +1},
\end{split}$$ where we recall $\beta_n=\alpha(1-\tau_n)$, and note that $ q\beta_n\geq 2$, which holds by the definition of $q$. Hence, if we put $C_n:={\textstyle}\frac{c}{(q\beta_n -1)^{1/q}} \cdot k_n^{1/q}$, then $$\|V\|_q \leq \ C_n \cdot q\cdot k_n^{-\beta_n}.$$ Furthermore, it is simple to check that $C_n\lesssim 1$, and that the assumption $(1-\tau_n)\sqrt{\log(n)}\gtrsim 1$ implies $q\lesssim \log(n).$ Therefore, from the inequality with $t=e\|V\|_q$, as well as the definition of $q$, we obtain $${\mathbb{P}}\Bigg(V\geq c\cdot \log(n)\cdot k_n^{-\beta_n}\Bigg) \ \leq \ e^{-q} \ \leq \ {\textstyle}\frac 1n,$$ for some constant $c>0$ not depending on $n$, as needed.
\
Proof of Theorem \[THM:BOOT\] {#app:thmboot}
=============================
Consider the inequality $$d_{\textup{K}}(\mathcal{L}(\tilde M_p),\mathcal{L}(M_p^{\star}| X)) \ \leq \ {\textsc{I}}'_n \ + \ {\textsc{II}}'_n(X) \ + \ {\textsc{III}}'_n(X),$$ where we define $$\begin{aligned}
{\textsc{I}}'_n & \ = \ d_{\text{K}}\Big(\mathcal{L}(\tilde M_p) \, , \, \mathcal{L}(\tilde M_{k_n})\Big)\\[0.2cm]
{\textsc{II}}'_n(X) &\ = \ d_{\text{K}}\Big(\mathcal{L}(\tilde M_{k_n}) \, , \, \mathcal{L}( M^{\star}_{k_n}|X\big)\Big)\\[0.2cm]
{\textsc{III}}'_n(X) & \ = \ d_{\text{K}}\Big(\mathcal{L}(M^{\star}_{k_n}| X\big) \, , \, \mathcal{L}( M_p^{\star} |X\big)\Big).\end{aligned}$$ Note that ${\textsc{I}}_n'$ is deterministic, whereas ${\textsc{II}}_n'(X)$ and ${\textsc{III}}_n'(X)$ are random variables depending on $X$. The remainder of the proof consists in showing that each of these terms are at most of order $n^{-\frac 12+\delta}$, with probability at least $1-\frac cn$. The terms ${\textsc{II}}_n'(X)$ and ${\textsc{III}}_n'(X)$ are handled in Sections \[sec:II’\] and \[sec:III’\] respectively. The first term ${\textsc{I}}_n'$ requires no further work, due to Proposition \[prop:IandIII\] (since ${\textsc{I}}_n'$ is equal to ${\textsc{III}}_n$, defined in equation ).
Handling the term ${\textsc{III}}_n'(X)$ {#sec:III'}
----------------------------------------
The proof of Proposition \[prop:IandIII\] can be partially re-used to show that for any fixed realization of $X$, and any real numbers $t_{1,n}'\leq t_{2,n}'$, the following bound holds $${\textsc{III}}_n'(X) \ \leq \ {\mathbb{P}}\big(A'(t_{2,n}')\big| X\big) \ + \ {\mathbb{P}}\big(B'(t_{1,n}')\big| X\big),$$ where we define the following events for any $t\in{\mathbb{R}}$, $$\label{inclusionboot}
\small
A'(t)=\Big\{\max_{j\in{\mathcal{J}}(k_n)} S^{\star}_{n,j}/{\widehat{\sigma}}_j^{\tau_n}\leq t\Big\} \ \ \ \text{ and } \ \ \ \ B'(t)=\Big\{\max_{j\in{\mathcal{J}}(k_n)^c} S^{\star}_{n,j}/{\widehat{\sigma}}_j^{\tau_n}> t\Big\}.$$ Below, Lemma \[lem:Bboundboot\] ensures that $t_{1,n}'$ and $t_{2,n}'$ can be chosen so that the random variables ${\mathbb{P}}(B'(t_{1,n}')\big|X)$ and ${\mathbb{P}}(A'(t_{2,n}')\big|X)$ are at most $c n^{-\frac 12+\delta}$, with probability at least $1-\frac cn$. Also, it is straightforward to check that under Assumption \[A:cor\], the choices of $t_{1,n}'$ and $t_{2,n}'$ given in Lemma \[lem:Bboundboot\] satisfy $t_{1,n}'\leq t_{2,n}'$ when $n$ is sufficiently large.
\[lem:Bboundboot\] Fix any number $\delta\in(0,1/2)$, and suppose the conditions of Theorem \[THM:G\] hold. Then, there are positive constants $c_1$, $c_2$, and $c$, not depending on $n$, for which the following statement is true:\
If $t_{1,n}'$ and $t_{2,n}'$ are chosen as $$\begin{aligned}
t_{1,n}' &= c_1\cdot k_n^{-\beta_n}\cdot\log(n)^{3/2} \ \ \ \ \text{ and }
\label{eqn:t1prime}\\[0.2cm]
t_{2,n}' &=c_2\cdot \ell_n^{-\beta_n}\cdot\sqrt{\log(\ell_n)},
\label{eqn:t2prime}\end{aligned}$$ then the events $$\label{abound}\tag{a}
{\mathbb{P}}(A'(t'_{2,n})\big|X) \leq c\, n^{-\frac 12+\delta}$$ and $$\label{bbound}\tag{b}
{\mathbb{P}}(B'(t'_{1,n})\big| X) \leq n^{-1}$$ each hold with probability at least $1-\frac cn$.
#### <span style="font-variant:small-caps;">Proof of Lemma \[lem:Bboundboot\] part</span>
Using the definition of ${\textsc{II}}_n'(X)$, followed by ${\mathcal{J}}(\ell_n)\subset {\mathcal{J}}(k_n)$, we have $$\begin{split}
{\mathbb{P}}(A'(t'_{2,n})|X)
& \ \leq \ {\mathbb{P}}\Big(\max_{j\in {\mathcal{J}}(\ell_n)} \tilde S_{n,j}/\sigma_j^{\tau_n}\leq t'_{2,n}\Big) +{\textsc{II}}_n'(X).
\end{split}$$ Taking $t_{2,n}'=t_{2,n}$ as in , the proof of Lemma \[lem:bounds\] part shows that the first term is $\mathcal{O}(n^{-1/2})$. With regard to the second term, Proposition \[prop:IIprime\] in the next subsection shows that there is a constant $c>0$ not depending on $n$ such that the event $${\textsc{II}}_n'(X) \leq c\, n^{-\frac 12+\delta}$$ holds with probability at least $1-\frac cn$. This completes the proof.
\
#### <span style="font-variant:small-caps;">Proof of Lemma \[lem:Bboundboot\] part</span>
Define the random variable $$\label{eqn:Vstar}
V^{\star}:=\max_{j\in{\mathcal{J}}(k_n)^c} S_{n,j}^{\star}/{\widehat{\sigma}}_j^{\tau_n},$$ and as in the proof of Lemma \[lem:bounds\], let $q =\max\big\{{\textstyle}\frac{2}{\beta_n},\log(n),3\big\}.$ The idea of the proof is to construct a function $b(\cdot)$ such that the following bound holds for every realization of $X$, $$\Big( {\mathbb{E}}\big[|V^{\star}|^q\big | X\big]\Big)^{1/q} \leq b(X),$$ and then Chebyshev’s inequality gives the following inequality for any number $b_n$ satisfying $b(X)\leq b_n$, $${\mathbb{P}}\Big(V^{\star} \geq e b_n\, \Big | X\Big) \ \leq \ e^{-q} \ \leq \ {\textstyle}\frac 1n.$$ In turn, we will derive an expression for $b_n$ such that the event $\{b(X)\leq b_n\}$ holds with high probability. This will lead to the statement of the lemma, because it will turn out that $t_{1,n}'\asymp b_n$.\
To construct the function $b(\cdot)$, observe that the initial portion of the proof of Lemma \[lem:bounds\] shows that for any realization of $X$, $$\label{eqn:Vstarbound}
{\mathbb{E}}\big[|V^{\star}|^q\big | X\big]\ \leq \sum_{j\in{\mathcal{J}}(k_n)^c} {\widehat{\sigma}}_j^{q(1-\tau_n)}\,{\mathbb{E}}\big[|{\textstyle}\frac{1}{{\widehat{\sigma}}_j}S_{n,j}^{\star}|^q|X\big].$$ Next, Lemma \[lem:Snjnorm\] ensures that for every $j\in\{1,\dots,p\}$, the event $${\mathbb{E}}\big[|{\textstyle}\frac{1}{{\widehat{\sigma}}_j}S_{n,j}^{\star}|^q|X\big] \ \leq (c\,q)^q,$$ holds with probability 1. Consequently, if we let $s=q(1-\tau_n)$ and consider the random variable $${\widehat{\mathfrak{s}}}:=\Bigg(\sum_{j\in{\mathcal{J}}(k_n)^c}{\widehat{\sigma}}_j^{s}\Bigg)^{\frac{1}{s}},$$ as well as $$b(X):=c\cdot q\cdot {\widehat{\mathfrak{s}}}^{(1-\tau_n)},$$ then we obtain the bound $$\Big( {\mathbb{E}}\big[|V^{\star}|^q\big | X\big]\Big)^{1/q} \ \leq b(X),$$ with probability 1. To proceed, Lemma \[lem:mathfrak\] implies $${\mathbb{P}}\bigg(b(X) \geq q\cdot {\textstyle}\frac{(c\sqrt q)^{1-\tau_n}}{(q\beta_n-1)^{1/q}}\cdot k_n^{-\beta_n+1/q}\bigg) \ \leq e^{-q}\leq \frac 1n,$$ for some constant $c>0$ not depending on $n$. By weakening this tail bound slightly, it can be simplified to $${\mathbb{P}}\bigg(b(X) \geq C_n'\cdot q^{3/2} \cdot k_n^{-\beta_n}\bigg) \ \leq {\textstyle}\frac 1n,$$ where $C_n':={\textstyle}\frac{ c \,k_n^{1/q}}{(q\beta_n-1)^{1/q}}$, and we recall $\beta_n=\alpha(1-\tau_n)$. To simplify further, it can be checked that $C_n'\lesssim 1$, and that the assumption $(1-\tau_n)\sqrt{\log(n)}\gtrsim 1$ gives $q\lesssim \log(n)$. It follows that there is a constant $c$ not depending on $n$ such that if $$b_n:=c\cdot \log(n)^{3/2} \cdot k_n^{-\beta_n},$$ then $${\mathbb{P}}(b(X)\geq b_n) \ \leq {\textstyle}\frac 1n,$$ which completes the proof.
Handling the term ${\textsc{II}}_n'(X)$ {#sec:II'}
---------------------------------------
\[prop:IIprime\] Fix any number $\delta\in(0,1/2)$, and suppose the conditions of Theorem \[THM:G\] hold. Then, there is a constant $c>0$ not depending on $n$ such that the event $${\textsc{II}}_n'(X) \leq c\, n^{-\frac 12+\delta}$$ holds with probability at least $1-\frac cn$.
Define the random variable $$\breve M_{k_n}^{\star}:=\max_{j\in {\mathcal{J}}(k_n)} S_{n,j}^{\star}/\sigma_j^{\tau_n},$$ which differs from $M_{k_n}^{\star}$, since $\sigma_j^{\tau_n}$ is used in place of ${\widehat{\sigma}}_j^{\tau_n}$. Consider the triangle inequality $$\label{eqn:II'}
\small
{\textsc{II}}_n'(X) \ \leq \ {d_{\textup{K}}}\Big({\mathcal{L}}(\tilde M_{k_n})\, , {\mathcal{L}}(\breve M_{k_n}^{\star}|X)\Big) \ + \ {d_{\textup{K}}}\Big({\mathcal{L}}(\breve M_{k_n}^{\star}|X) \, , \, {\mathcal{L}}(M_{k_n}^{\star}|X)\Big).$$ The two terms on the right will bounded separately.
To address the first term on the right side of , we will apply Lemma \[lem:hellinger\], for which a substantial amount of notation is needed. Recall the matrix $\mathfrak{C}=\Sigma^{1/2}\Pi_{k_n}{^{\top}}D_{k_n}^{-\tau_n}$ of size $p\times k_n$, where the projection matrix $\Pi_{k_n}\in{\mathbb{R}}^{k_n\times p}$ is defined in the proof of Proposition \[prop:bentkus\]. Note that $\tilde M_{k_n}$ is the coordinate-wise maximum of a Gaussian vector drawn from $N(0,\mathfrak{S})$, with $\mathfrak{S}=\mathfrak{C}{^{\top}}\mathfrak{C}$. To address $\breve{M}_{k_n}^{\star}$, let $$W_n={\textstyle}\frac 1n \sum_{i=1}^n (Z_i-\bar Z)(Z_i-\bar Z){^{\top}}$$ where $\bar Z=\frac 1n \sum_{i=1}^n Z_i$, and observe that $\breve{M}_{k_n}^{\star}$ is the coordinate-wise maximum of Gaussian vector drawn from $N(0,\breve{\mathfrak{S}})$, with $\breve{\mathfrak{S}}=\mathfrak{C}{^{\top}}W_n\mathfrak{C}$. Next, consider the s.v.d., $$\mathfrak{C}=U\Lambda V{^{\top}},$$ where if $r$ denotes the rank of $\mathfrak{C}$, then we may take $U\in{\mathbb{R}}^{p\times r}$ to have orthonormal columns, $\Lambda\in{\mathbb{R}}^{r\times r}$ to be invertible, and $V{^{\top}}\in{\mathbb{R}}^{r\times k_n}$ to have orthonormal rows. In order to apply Lemma \[lem:hellinger\] for a given realization of $\breve{\mathfrak{S}}$, it is necessary that the columns of $\mathfrak{S}$ and $\breve{\mathfrak{S}}$ span the same subspace of ${\mathbb{R}}^{k_n}$. This occurs with probability at least $1-\frac cn$, because the matrix $\breve{\mathfrak{S}}$ is equal to $V\Lambda (U{^{\top}}W_n U)\Lambda V{^{\top}}$, and the matrix $(U{^{\top}}W_n U)$ is invertible with probability at least $1-\frac cn$ (due to Lemma \[lem:white\]). Another ingredient for applying Lemma \[lem:hellinger\] is the following algebraic relation, which is a direct consequence of the definitions just introduced, $$\Big(V{^{\top}}\mathfrak{S} V\Big)^{-1/2}\Big( V{^{\top}}\breve{\mathfrak{S}}V\Big) \Big(V{^{\top}}\mathfrak{S}V\Big)^{-1/2} \ = \ U{^{\top}}W_n U.$$ Building on this relation, Lemma \[lem:hellinger\] shows that if the event $$\label{eqn:tempevent2}
\|U{^{\top}}W_n U-\mathbf{I}_{r}\|_{\text{op}} \ \leq \ {\epsilon},$$ holds for some number ${\epsilon}>0$, then the event $${d_{\textup{K}}}\Big({\mathcal{L}}(\tilde M_{k_n})\, , {\mathcal{L}}(\breve M_{k_n}^{\star}|X)\Big) \leq c\cdot k_n^{1/2} \cdot {\epsilon}$$ also holds, where $c>0$ is a constant not depending on $n$ or ${\epsilon}$. Thus, it remains to specify ${\epsilon}$ in the event . For this purpose, Lemma \[lem:white\] shows that if ${\epsilon}=c \cdot n^{-1/2}\cdot k_n\cdot \log(n)$, then the event holds with probability at least $1-\frac cn$. So, given that $$n^{-1/2}\cdot k_n^{3/2}\cdot \log(n) \ \lesssim \ n^{-\frac 12+\delta},$$ the first term in the bound requires no further consideration.\
To deal with the second term in , we proceed by considering the general inequality $${d_{\textup{K}}}({\mathcal{L}}(\xi),{\mathcal{L}}(\zeta)) \ \leq \ \sup_{t\in{\mathbb{R}}}{\mathbb{P}}\big(|\zeta-t|\leq r\big) \ + \ {\mathbb{P}}(|\xi-\zeta|>r),$$ which holds for any random variables $\xi$ and $\zeta$, and any real number $r>0$ (cf. @CCK:SPA [Lemma 2.1]). Specifically, we will let ${\mathcal{L}}(\breve M_{k_n}^{\star}|X)$ play the role of ${\mathcal{L}}(\xi)$, and let ${\mathcal{L}}(M_{k_n}^{\star}|X)$ play the role of ${\mathcal{L}}(\zeta)$. In other words, we need to establish an anti-concentration inequality for $\mathcal{L}(M_{k_n}^{\star}|X)$, as well as a coupling inequality for $ M_{k_n}^{\star}$ and $\breve M_{k_n}^{\star}$, conditionally on $X$.
To establish the coupling inequality, if we put $$r_n= c \cdot n^{-1/2}\cdot \log(n)^{5/2},$$ for a suitable constant $c$ not depending on $n$, then Lemma \[lem:coupling\] shows that the event $$\label{eqn:maincouple}
{\mathbb{P}}\Big(\big|\breve M_{k_n}^{\star} - M_{k_n}^{\star}\big| > r_n\, \Big| X\Big) \ \leq \ {\textstyle}\frac cn$$ holds with probability at least $1-\frac cn$.
Lastly, the anti-concentration inequality can be derived from Nazarov’s inequality (Lemma \[lem:Nazarov\]), since $M_{k_n}^{\star}$ is obtained from a Gaussian vector, conditionally on $X$. For this purpose, let $${\widehat{\underline{\sigma}}}_{k_n}=\min_{j\in\mathcal{J}(k_n)}{\widehat{\sigma}}_j.$$ In turn, Nazarov’s inequality implies that the event $$\sup_{t\in{\mathbb{R}}}\, {\mathbb{P}}\Big(|M_{k_n}^{\star} -t|\leq r_n \, \Big| X\Big) \ \leq \ c\cdot \frac{r_n}{{\widehat{\underline{\sigma}}}_{k_n}^{1-\tau_n}}\cdot \sqrt{\log(k_n)},$$ holds with probability 1. Meanwhile, Lemma \[lem:cor\] and Assumption \[A:cor\] imply that the event $$\label{eqn:minsigevent}
\frac{1}{{\widehat{\underline{\sigma}}}_{k_n}^{1-\tau_n}} \ \leq c\, k_n^{\beta_n}$$ holds with probability at least $1-\frac cn$. Combining the last few steps, we conclude that the following bound holds with probability at least $1-\frac cn$, $$\small
\begin{split}
\sup_{t\in{\mathbb{R}}}\, {\mathbb{P}}\Big(|M_{k_n}^{\star} -t|\leq r_n \, \Big| X\Big) \ & \leq {\textstyle}c\cdot n^{-1/2}\cdot k_n^{\beta_n}\cdot \log(n)^{5/2}\cdot\sqrt{\log(k_n)}\\[0.2cm]
& \ \leq c\,n^{-\frac 12 +\delta},
\end{split}$$ as needed.
Technical lemmas for Theorems \[THM:G\] and \[THM:BOOT\] {#app:technical}
========================================================
\[lem:sighatrnorm\] Fix any number $\delta\in(0,1/2)$, and suppose the conditions of Theorem \[THM:G\] hold. Also, let $q=\max\{\frac{2}{\beta_n}, \log(n),3\}$. Then, there is a constant $c>0$ not depending on $n$, such that for any $j\in\{1,\dots,p\}$, we have $$\|{\widehat{\sigma}}_{j}\|_q \ \leq \ c \cdot \sigma_j \cdot \sqrt{q}.$$
Define the vector $u:=\frac{1}{\sigma_j}\Sigma^{1/2}e_j\in{\mathbb{R}}^p$, which satisfies $\|u\|_2=1$. Observe that $$\label{eqn:sigsteps}
\begin{split}
{\textstyle}\frac{1}{\sigma_j}\|{\widehat{\sigma}}_{j}\|_q &= \ \Bigg\|\Big({\textstyle}\frac 1n \sum_{i=1}^n (Z_i{^{\top}}u)^2-(\bar Z{^{\top}}u)^2\Big)^{1/2}\Bigg\|_q\\[0.2cm]
&\leq \ \bigg\|\Big({\textstyle}\frac 1n \sum_{i=1}^n ( Z_i{^{\top}}u)^2\Big)^{1/2}\bigg\|_q\\[0.2cm]&= \ \Big\|{\textstyle}\frac 1n \sum_{i=1}^n (Z_i{^{\top}}u)^2\Big\|_{q/2}^{1/2}\ .
\end{split}$$ Since the random variables $(Z_1{^{\top}}u)^2,\dots, (Z_n{^{\top}}u)^2$ are independent and non-negative, part *(i)* of Rosenthal’s inequality in Lemma \[lem:rosenthal\] implies the $L^{q/2}$ norm in the last line satisfies $$\small
\Big\|{\textstyle}\frac 1n \sum_{i=1}^n (Z_i{^{\top}}u)^2\Big\|_{q/2} \ \leq \ c \cdot q \cdot \max\bigg\{\big\|(Z_1{^{\top}}u)^2\big\|_1\, , \, n^{-1+2/q} \big\| (Z_1{^{\top}}u)^2\big\|_{q/2}\bigg\},$$ for an absolute constant $c>0$. For the first term inside the maximum, observe that since $\|u\|_2=1$ and $Z_1$ is isotropic, we have $\|(Z_1{^{\top}}u)^2\|_1=1$. To handle the second term inside the maximum, Assumption \[A:model\] implies $\|(Z_1{^{\top}}u)^2\|_{q/2}\lesssim q^2$. Combining the last few steps, and noticing the square root on the $L^{q/2}$ norm in the last line of , we obtain $$\begin{split}
{\textstyle}\frac{1}{\sigma_j}\|{\widehat{\sigma}}_{j}\|_q & \ \lesssim \sqrt q \cdot \max\Big\{ 1\, , \, n^{-1/2+1/q}q\Big\},
\end{split}$$ and this implies the statement of the lemma.
\[lem:mathfrak\] Fix any number $\delta\in(0,1/2)$, and suppose the conditions of Theorem \[THM:G\] hold. Also, let $q=\max\{\frac{2}{\beta_n},\log(n),3\}$, and $s=q(1-\tau_n)$, and consider the random variables ${\widehat{\mathfrak{s}}}$ and ${\widehat{\mathfrak{t}}}$ defined by $${\widehat{\mathfrak{s}}}=\Bigg(\sum_{j\in{\mathcal{J}}(k_n)^c} {\widehat{\sigma}}_{j}^s\Bigg)^{1/s} \text{ \ \ \ \ and \ \ \ \ } {\widehat{\mathfrak{t}}}=\Bigg(\sum_{j\in{\mathcal{J}}(k_n)} {\widehat{\sigma}}_{j}^s\Bigg)^{1/s}.$$ Then, there is a constant $c>0$ not depending on $n$ such that $${\mathbb{P}}\Bigg({\widehat{\mathfrak{s}}}\geq {\textstyle}\frac{c\sqrt q}{(q\beta_n-1)^{1/s}}\cdot k_n^{-\alpha+1/s}\Bigg) \ \leq \ e^{-q},$$ and $$\label{eqn:boundmathfrakt}
{\mathbb{P}}\Big(\, {\widehat{\mathfrak{t}}}\, \geq {\textstyle}\frac{c \sqrt q}{(q\beta_n-1)^{1/s}}\Big) \ \leq \ e^{-q}.$$
In light of the Chebyshev inequality ${\mathbb{P}}\big({\widehat{\mathfrak{s}}}\geq e\|{\widehat{\mathfrak{s}}}\|_q\big) \leq e^{-q}$, it suffices to bound $\|{\widehat{\mathfrak{s}}}\|_q$ (and similarly for ${\widehat{\mathfrak{t}}}$). We proceed by direct calculation, $$\footnotesize
\begin{split}
\|{\widehat{\mathfrak{s}}}\|_q \ & = \ \Bigg\|\sum_{j\in{\mathcal{J}}(k_n)^c}{\widehat{\sigma}}_{j}^s\Bigg\|_{q/s}^{1/s}\\[0.2cm]
&\leq \ \Bigg(\sum_{j\in{\mathcal{J}}(k_n)^c} \big\|{\widehat{\sigma}}_{j}^s\big\|_{q/s}\Bigg)^{1/s} \ \ \ \ \ \ \ \text{(triangle inequality for $\|\cdot\|_{q/s}$, with $q/s\geq 1$)}\\[0.2cm]
&=\ \ \Bigg(\sum_{j\in{\mathcal{J}}(k_n)^c} \big\|{\widehat{\sigma}}_{j}\big\|_q^s\Bigg)^{1/s}\\[0.2cm]
&\lesssim \ \sqrt{q} \cdot\Bigg( \sum_{j\in{\mathcal{J}}(k_n)^c} \sigma_j^{s}\Bigg)^{1/s} \ \ \ \ \ \ \ \ \ \text{(Lemma~\ref{lem:sighatrnorm})}\\[0.2cm]
&\lesssim \ \sqrt{q} \cdot \Bigg(\int_{k_n}^p x^{-s\alpha}dx\Bigg)^{1/s} \\[0.2cm]
&\lesssim \ \sqrt{q}\cdot \frac{k_n^{-\alpha+1/s}}{(s\alpha-1)^{1/s}},
\end{split}$$ and in the last step we have used the fact that $s\alpha=q\beta_n>1$, which holds since $q$ is defined to satisfy $q\beta_n>1$. The calculation for ${\widehat{\mathfrak{t}}}$ is essentially the same, except that we use $\sum_{j\in{\mathcal{J}}(k_n)}\sigma_j^s\lesssim 1$. \
#### Remark
The following result is a variant of Lemma A.7 in the paper [@Zhilova:2015].
\[lem:hellinger\] Let $A$ and $B$ be positive semi-definite matrices in whose columns span the same subspace of ${\mathbb{R}}^d$. Define two multivariate normal random vectors $\xi\sim N(0, A)$ and $\zeta\sim N(0,B)$. Let $r\leq d$ be the dimension of the subspace spanned by the columns of $A$ and $B$, and let $Q\in{\mathbb{R}}^{d\times r}$ have columns that are an orthonormal basis for this subspace. Define the $r\times r$ positive definite matrices $\tilde A=Q{^{\top}}A Q$ and $\tilde B=Q{^{\top}}B Q$, and let $H$ be any square matrix satisfying $H{^{\top}}H=\tilde A$. Finally, let ${\epsilon}>0$ be a number such that $\|(H^{-1}){^{\top}}\tilde B (H^{-1}) -\mathbf{I}_r\|{_{\textup{op}}}\leq {\epsilon}$. Then, there is an absolute constant $c>0$ such that $$\label{hellenger}
\sup_{t\in{\mathbb{R}}}\bigg|{\mathbb{P}}\Big(\max_{1\leq j\leq d} \xi_j\leq t\Big)-{\mathbb{P}}\Big(\max_{1\leq j\leq d} \zeta_j\leq t\Big)\bigg| \ \leq c\sqrt{r}\, {\epsilon}.$$
We may assume that $\sqrt{r} \,{\epsilon}\leq 1/2$, for otherwise the claim trivially holds with $c=2$. Define the $r$-dimensional random vectors $\tilde \xi=Q{^{\top}}\xi$ and $\tilde \zeta=Q{^{\top}}\zeta$. As a consequence of the assumptions, the random vector $\xi$ lies in the column-span of $Q$ almost surely, which gives $Q\tilde \xi=\xi$ almost surely. It follows that for any $t\in{\mathbb{R}}$, the event $\{\max_{1\leq j\leq d} \xi_j \leq t\}$ can be expressed as $\{\tilde \xi\in\mathcal{A}_t\}$ for some Borel set $\mathcal{A}_t\subset {\mathbb{R}}^r$. Likewise, we also have $\{\max_{1\leq j\leq d} \zeta_j \leq t\} = \{\tilde \zeta\in\mathcal{A}_t\}$. Hence, the left hand side of is upper-bounded by the total variation distance between ${\mathcal{L}}(\tilde \xi)$ and ${\mathcal{L}}(\tilde \zeta)$, and in turn, Pinsker’s inequality implies this is upper-bounded by $c\sqrt{d_{\textup{KL}}({\mathcal{L}}(\tilde \zeta),{\mathcal{L}}(\tilde \xi))}$, where $c>0$ is an absolute constant, and $d_{\textup{KL}}$ denotes the KL divergence. Since the random vectors $\tilde \xi$ and $\tilde \zeta$ are Gaussian, the following exact formula is available if we let $\tilde C=(H{^{\top}})^{-1}\tilde B(H^{-1})-\mathbf{I}_r$, $$\begin{split}
d_{\textup{KL}}({\mathcal{L}}(\tilde \zeta),{\mathcal{L}}(\tilde \xi)) & \ = \ {\textstyle}\frac{1}{2}\Big({\operatorname{tr}}(\tilde C)-\log\det(\tilde C+\mathbf{I}_r)\Big)\\[0.2cm]
& \ = \ {\textstyle}\frac{1}{2}\sum_{j=1}^r \lambda_j(\tilde C)-\log(\lambda_j(\tilde C)+1).
\end{split}$$ Using the basic inequality $|x-\log(x+1)|\leq x^2/(1+x)$ that holds for any $x\in(-1,\infty)$, as well as the condition $|\lambda_j(\tilde C)|\leq {\epsilon}\leq 1/2$, we have $$\begin{split}
d_{\textup{KL}}({\mathcal{L}}(\tilde \zeta),{\mathcal{L}}(\tilde \xi)) & \ \leq \ c \, r\, \|\tilde C\|{_{\textup{op}}}^2\\[0.2cm]
& \ \leq c \, r \, {\epsilon}^2,
\end{split}$$ for some absolute constant $c>0$.
\[lem:Snjnorm\] Suppose the conditions of Theorem \[THM:G\] hold, and let $q=\max\{\frac{2}{\beta_n}, \log(n),3\}$. Then, there is a constant $c>0$ not depending on $n$ such that for any $j\in\{1,\dots,p\}$, we have $$\label{eqn:firstsnjnorm}
\|{\textstyle}\frac{1}{\sigma_j}S_{n,j}\|_q \leq c\, q,$$ and the following event holds with probability 1, $$\Big({\mathbb{E}}\big[|{\textstyle}\frac{1}{{\widehat{\sigma}}_j}S_{n,j}^{\star}|^q|X\big] \Big)^{1/q} \ \leq c\, q.$$
We only prove the first bound, since the second one can be obtained by repeating the same argument, conditionally on $X$. Since $q>2$, Lemma \[lem:rosenthal\] gives $$\label{eqn:lemrosenthalfirstD}
\|{\textstyle}\frac{1}{\sigma_j}S_{n,j}\|_q \ \lesssim \ q\cdot \max\Big\{ \|{\textstyle}\frac{1}{\sigma_j}S_{n,j}\|_2 \, , \, n^{-1/2+1/q} \|{\textstyle}\frac{1}{\sigma_j}(X_{1,j}-\mu_j)\|_q\Big\}.$$ Clearly, $$\|{\textstyle}\frac{1}{\sigma_j}S_{n,j}\|_2^2 \ = \ {\operatorname{var}}({\textstyle}\frac{1}{\sigma_j}S_{n,j}) \ = \ 1.$$ Furthermore, if we define the vector $u:={\textstyle}\frac{1}{\sigma_j}\Sigma^{1/2}e_j$ in ${\mathbb{R}}^p$, which satisfies $\|u\|_2=1$, then $$\begin{split}
\big\|{\textstyle}\frac{1}{\sigma_j}(X_{1,j}-\mu_j)\big\|_q
&=\big\| Z_1{^{\top}}u\big\|_q \ \lesssim \ q \\[0.2cm]
\end{split}$$ where the last step follows from Assumption \[A:model\]. Applying the work above to the bound gives $$\|{\textstyle}\frac{1}{\sigma_j}S_{n,j}\|_q \ \lesssim \ q\cdot \max\Big\{ 1, n^{-1/2+1/q}\cdot q\Big\}.$$ Finally, the stated choice of $q$ implies that the right side in the last display is of order $q$.
\[lem:white\] Let the random vectors $Z_1,\dots,Z_n\in{\mathbb{R}}^p$ be as in Assumption \[A:model\], and let $Q\in {\mathbb{R}}^{p\times r}$ be a fixed matrix having orthonormal columns with $r\leq k_n$. Lastly, let $$\label{eqn:Wndef}
W_n=\frac 1n \sum_{i=1}^n (Z_i-\bar Z)(Z_i-\bar Z){^{\top}},$$ where $\bar Z=\frac 1n \sum_{i=1}^n Z_i$. Then, there is a constant $c>0$ not depending on $n$, such that the event $$\label{eqn:opnormevent}
\big\|Q{^{\top}}W_nQ-\mathbf{I}_{r}\big\|{_{\textup{op}}}\ \leq \ {\textstyle}\frac{c\log(n)k_n}{\sqrt n},$$ holds with probability at least $1-\frac cn$.
Let ${\epsilon}\in (0,1/2)$, and let $\mathcal{N}$ be an ${\epsilon}$-net (with respect to the $\ell_2$-norm) for the unit $\ell_2$-sphere in ${\mathbb{R}}^{r}$. It is well known that $\mathcal{N}$ can be chosen so that $\text{card}(\mathcal{N})\leq (3/{\epsilon})^{r}$, and the inequality $$\big\|Q{^{\top}}W_n Q- \mathbf{I}_{r}\big\|{_{\textup{op}}}\leq {\textstyle}\frac{1}{1-2{\epsilon}}\cdot \displaystyle \max_{u\in\mathcal{N}} \Big|u{^{\top}}\Big(Q{^{\top}}W_n Q-\mathbf{I}_{r}\Big)u\Big|,$$ holds with probability 1 [@Vershynin:2012 Lemmas 5.2 and 5.4]. For a fixed $u\in\mathcal{N}$, put $\xi_{u,i}:=Z_i{^{\top}}Qu$, and consider the simple algebraic relation $$u{^{\top}}\Big(Q{^{\top}}W_n Q-\mathbf{I}_{r}\Big)u = \underbrace{\Big({\textstyle}\frac{1}{n}\sum_{i=1}^n \xi_{i,u}^2-1\Big)}_{=: \Delta(u)}- \underbrace{\Big({\textstyle}\frac{1}{n}\sum_{i=1}^n \xi_{i,u}\Big)^2}_{=:\Delta'(u)}.$$ We will show that both terms on the right side are small with high probability, and then take a union bound over $u\in\mathcal{N}$. The high-probability bounds will be obtained by using Lemma \[lem:rosenthal\] to control $\|\Delta(u)\|_q$ and $\|\Delta'(u)\|_q$ when $q$ is sufficiently large.\
To apply Lemma \[lem:rosenthal\], first observe the following bounds, which are consequences of Assumption \[A:model\], $$\|\xi_{i,u}\|_q \ \lesssim \ q,$$ and $$\begin{split}
\|\xi_{i,u}^2-1\|_q \ \lesssim \ q^2.
\end{split}$$ Therefore, when $q>2$, Lemma \[lem:rosenthal\] gives $$\label{eqn:justbefore}
\begin{split}
\|\Delta(u)\|_q & \ \lesssim \ q \max\Big\{ \|\Delta(u)\|_2 \, , \, {\textstyle}\frac{1}{n}\big({\textstyle\sum}_{i=1}^n \|\xi_{i,u}^2-1\|_q^q\big)^{1/q}\Big\}\\[0.3cm]
& \lesssim \ q \max\Big\{ {\textstyle}\frac{1}{\sqrt n} \, , \, n^{-1+1/q} \cdot q^2\Big\}.
\end{split}$$ Due to Chebyshev’s inequality, $${\mathbb{P}}\Big( |\Delta(u)| \ \geq e \|\Delta(u)\|_q\Big) \ \leq \ e^{-q},$$ and so if we take $q= \max\{C\log(n)k_n,3\}$ for some constant $C>0$ to be tuned below, then gives $\|\Delta(u)\|_q\lesssim q/\sqrt{n}$, and the following inequality holds for all $u\in\mathcal{N}$, $${\mathbb{P}}\Big( |\Delta(u)| \ \geq {\textstyle}\frac{c\log(n) k_n}{\sqrt n}\Big) \ \leq \ \exp\Big\{-C\log(n) k_n\Big\}.$$ The random variable $\Delta'(u)$ can be analyzed with a similar set of steps, which leads to the following inequality for all $u\in\mathcal{N}$, $${\mathbb{P}}\bigg( |\Delta'(u)| \ \geq c\Big({\textstyle}\frac{\log(n) k_n}{\sqrt n}\Big)^2\bigg) \ \leq \ \exp\Big\{-C\log(n) k_n\Big\}.$$ Combining the previous work with a union bound, if we consider the choice ${\epsilon}=\min\{c\log(n)k_n/\sqrt{n},{\textstyle}\frac{1}{4}\}$, then $${\mathbb{P}}\Big(\big\|Q{^{\top}}W_n Q- \mathbf{I}_{r}\big\|{_{\textup{op}}}\geq {\epsilon}\Big) \ \leq \ 2\exp\Big\{ - C\cdot k_n\cdot\log(n) \ + \ r\cdot \log(3/{\epsilon})\Big\}.$$ Finally, choosing $C$ sufficiently large implies the stated result.
#### Remark
For the next results, define the correlation $$\rho_{j,j'}={\textstyle}\frac{\Sigma_{j,j'}}{\sigma_j\sigma_{j'}},$$ and its sample version $${\widehat{\rho}}_{j,j'}={\textstyle}\frac{{\widehat{\Sigma}}_{j,j'}}{{\widehat{\sigma}}_j{\widehat{\sigma}}_{j'}},$$ for any $j,j'\in\{1,\dots,p\}$.
\[lem:cor\] Suppose the conditions of Theorem \[THM:G\] hold. Then, there is a constant $c>0$ not depending on $n$ such that the three events $$\label{eqn:firstcor}
\max_{j\in{\mathcal{J}}(k_n)}\Big| {\textstyle}\frac{{\widehat{\sigma}}_j}{\sigma_j}-1\Big| \leq {\textstyle}\frac{c\log(n)}{\sqrt n},$$ $$\min_{j\in{\mathcal{J}}(k_n)}{\widehat{\sigma}}_j^{1-\tau_n} \ \geq \ \Big(\min_{j\in{\mathcal{J}}(k_n)} \sigma_j^{1-\tau_n}\Big)\cdot \Big(1-{\textstyle}\frac{c\log(n)}{\sqrt n}\Big),$$ and $$\label{eqn:firstcor}
\max_{j,j'\in{\mathcal{J}}(k_n)}\big| {\widehat{\rho}}_{jj'}-\rho_{jj'}\big| \leq {\textstyle}\frac{c\log(n)}{\sqrt n}$$ each hold with probability at least $1-{\textstyle}\frac{c}{n}$.
The result is a direct consequence of Lemma \[lem:corbasic\] below. The details are essentially algebraic manipulations, and so are omitted.
\[lem:corbasic\] Suppose the conditions of Theorem \[THM:G\] hold, and fix any two (possibly equal) indices $j,j'\in\{1,\dots,p\}$. Then, for any number $\kappa\geq 1$, there are positive constants $c$ and $c_1(\kappa)$ not depending on $n$ such that the event $$\label{eqn:corevent}
\Big| {\textstyle}\frac{{\widehat{\Sigma}}_{j,j'}}{\sigma_j\sigma_{j'}}-\rho_{j,j'}\Big| \leq \frac{c_1(\kappa)\log(n)}{\sqrt n}$$ holds with probability at least $1-cn^{-\kappa}$.
#### Remark
The event in the lemma has been formulated to hold with probability at least $1-cn^{-\kappa}$, rather than $1-\frac{c}{n}$, in order to accommodate a union bound for proving Lemma \[lem:cor\].
Consider the $\ell_2$-unit vectors $u=\Sigma^{1/2}e_j/\sigma_j$ and $v=\Sigma^{1/2}e_{j'}/\sigma_{j'}$ in ${\mathbb{R}}^p$. Letting $W_n$ be as defined in , observe that $$\label{eqn:decomp}
\begin{split}
\frac{{\widehat{\Sigma}}_{j,j'}}{\sigma_j\sigma_{j'}} -\rho_{j,j'}
& \ = u{^{\top}}(W_n-\mathbf{I}_p)v.
\end{split}$$ For each $1\leq i\leq n$, define the random variables $\zeta_{i,u}=Z_i{^{\top}}u$ and $\zeta_{i,v}=Z_i{^{\top}}v$. In this notation, the relation becomes $$\frac{{\widehat{\Sigma}}_{j,j'}}{\sigma_j\sigma_{j'}} -\rho_{j,j'} =\underbrace{\Big({\textstyle}\frac 1n \sum_{i=1}^n \zeta_{i,u}\zeta_{i,v}- u{^{\top}}v\Big)}_{=:\Delta(u,v)} \ - \ \underbrace{\Big({\textstyle}\frac 1n \sum_{i=1}^n \zeta_{i,u}\Big)\Big({\textstyle}\frac 1n \sum_{i=1}^n \zeta_{i,v}\Big)}_{=:\Delta'(u,v)}.$$ Note that ${\mathbb{E}}[\zeta_{i,u}\zeta_{i,v}]=u{^{\top}}v$. Also, if we let $q=\max\{\kappa\log(n),3\}$, then $$\|\zeta_{i,u}\zeta_{i,v}-u{^{\top}}v\|_q \ \lesssim \ q^2,$$ which follows from Assumption \[A:model\]. Therefore, Lemma gives the following bound for $q>2$, $$\begin{split}
\|\Delta(u,v)\|_q & \ \lesssim \ q \max\Big\{ \|\Delta(u,v)\|_2 \, , \, {\textstyle}\frac{1}{n}\big(\sum_{i=1}^n \|\zeta_{i,u}\zeta_{i,v}-u{^{\top}}v\|_q^q\big)^{1/q}\Big\}\\[0.3cm]
& \ \lesssim \ q \max\Big\{{\textstyle}\frac{1}{\sqrt n} \, , \, n^{-1+1/q}\cdot q^2\Big\}\\[0.3cm]
& \ \lesssim \ {\textstyle}\frac{\log(n)}{\sqrt n}.
\end{split}$$ Using the Chebyshev inequality $${\mathbb{P}}(|\Delta(u,v)|\geq e \|\Delta(u,v)\|_q) \ \leq e^{-q},$$ we have $${\mathbb{P}}\Big(|\Delta(u,v)|\geq {\textstyle}\frac{c\kappa\log(n)}{\sqrt n}\Big) \ \leq \ {\textstyle}\frac{1}{n^{\kappa}}.$$ Similar reasoning leads to the following tail bound for $\Delta'(u,v)$, $${\mathbb{P}}\bigg(|\Delta'(u,v)|\geq \big({\textstyle}\frac{c\kappa\log(n)}{\sqrt n}\big)^2\bigg) \ \leq \ {\textstyle}\frac{1}{n^{\kappa}},$$ and combining with the previous tail bound gives the stated result.
\[lem:coupling\] Fix any number $\delta\in(0,1/2)$, and suppose the conditions of Theorem \[THM:G\] hold. Then, there is a constant $c>0$ not depending on $n$ such that the event holds with probability at least $1-\frac cn$.
Let $(a_1,\dots,a_{k_n})$ and $(b_1,\dots,b_{k_n})$ be real vectors, and note the basic fact $$\bigg|\max_{1\leq j\leq k_n} a_j -\max_{1\leq j\leq k_n}b_j\bigg| \ \leq \ \max_{1\leq j\leq k_n} |a_j-b_j|.$$ From this, it is simple to derive the inequality $$\big| \breve M_{k_n}^{\star}-M_{k_n}^{\star}\big| \leq \max_{j\in{\mathcal{J}}(k_n)}\Big|\big({\textstyle}\frac{{\widehat{\sigma}}_j}{\sigma_j}\big)^{\tau_n}-1\Big|\cdot \displaystyle \max_{j\in{\mathcal{J}}(k_n)}\Big|S_j^{\star}/{\widehat{\sigma}}_j^{\tau_n}\Big|.$$ To handle the first factor on the right side, it follows from Lemma \[lem:corbasic\] that the event $$\max_{j\in{\mathcal{J}}(k_n)}\Big|\big({\textstyle}\frac{{\widehat{\sigma}}_j}{\sigma_j}\big)^{\tau_n}-1\Big| \ \leq c\cdot n^{-1/2}\cdot \log(n)$$ holds with probability at least $1-\frac cn$. Next, consider the random variable $$\label{eqn:Ustar}
U^{\star}:=\max_{j\in{\mathcal{J}}(k_n)} |S_{n,j}^{\star}/{\widehat{\sigma}}_j^{\tau_n}|.$$ It suffices to show there is possibly larger constant $c>0$, such that the event $$\label{eqn:Ustarbound}
{\mathbb{P}}\Big( U^{\star} \geq c \log(n)^{3/2} \Big| X\Big) \leq {\textstyle}\frac 1n$$ holds with probability at least $1-\frac cn$. Using Chebyshev’s inequality with $q\geq \log(n)$ gives $${\mathbb{P}}\Big(U^{\star} \geq e\, \big({\mathbb{E}}[|U^{\star}|^q|X]\big)^{1/q}\Big| X\Big)\leq e^{-q}.$$ Likewise, if the event $$\label{eqn:UstarLq}
({\mathbb{E}}[|U^{\star}|^q|X])^{1/q} \ \leq c\log(n)^{3/2}$$ holds for some constant $c>0$, then the event also holds. For this purpose, the argument in the proof of Lemma \[lem:Bboundboot\] can be essentially repeated with $q=\max\{\frac{2}{\beta_n},\log(n),3\big\}$ to show that the event holds with probability at least $1-\frac cn$. The main detail to notice when repeating the argument is that $U^{\star}$ involves a maximum over ${\mathcal{J}}(k_n)$, whereas the argument for Lemma \[lem:Bboundboot\] involves a maximum over ${\mathcal{J}}(k_n)^c$. This distinction can be handled by using the bound in Lemma \[lem:mathfrak\].
#### The case when {#lowdimcase}
The previous proofs relied on the condition $p>k_n$ only insofar as this implies $k_n\geq n^{\frac{1}{\log(n)^a}}$ and $\ell_n\geq \log(n)^3$. (These conditions are used in the analyses of ${\textsc{I}}_n$ and ${\textsc{III}}_n$, as well as ${\textsc{I}}_n'$ and ${\textsc{III}}_n'(X)$.) However, if $p\leq k_n$, then the definition of $k_n$ implies that $p=k_n$, which causes the quantities ${\textsc{I}}_n$, ${\textsc{III}}_n$, ${\textsc{I}}_n'(X)$ and ${\textsc{III}}_n'(X)$ to become exactly 0. In this case, the proofs of Theorems \[THM:G\] and \[THM:BOOT\] reduce to bounding ${\textsc{II}}_n$ and ${\textsc{II}}_n'(X)$, and these arguments can be repeated as before.
Proof of Theorem \[THM:M\] {#app:mult}
==========================
#### Notation and remarks
An important piece of notation for this appendix (and the next one) is the integer $d_n$, which we define to be the largest index $d_n\in\{1,\dots,p\}$ such that $\sigma_{(d_n)}^2\geq {\epsilon}_0^2/\sqrt{n}$, with ${\epsilon}_0$ as in Assumption \[A:mult\]. (Such an index must exist under Assumption \[A:mult\].) Also, it is simple to check that $k_n\leq d_n$ holds for all large $n$ under Assumption \[A:mult\]. Furthermore, as in the previous appendices, we will assume $p>k_n$, since the case $p\leq k_n$ can be handled using similar reasoning to that explained in the previous paragraph. Lastly, it will be helpful to note that $\sigma_{(j)}^2=\pi_{(j)}(1-\pi_{(j)})$, since it can be checked that $\pi_i\leq \pi_j$ implies $\sigma_i\leq \sigma_j$.
#### Outline of proof
Since the number $d_n$ will often play the role that $p$ did in previous proofs, we will use a slightly different notation for the analogues of the earlier quantities ${\textsc{I}}_n$, ${\textsc{II}}_n$, ${\textsc{II}}_n'(X)$, and ${\textsc{III}}_n'(X)$. This will also serve as a reminder that new details are involved in the context of the multinomial model. The new quantities are: $$\begin{aligned}
{\textup{\texttt{I}}}_n &= d_{\text{K}}\big(\mathcal{L}(M_{d_n}) \, , \, \mathcal{L}(M_{k_n})\big)\\[0.2cm]
{\textup{\texttt{II}}}_n &=d_{\text{K}}\big( \mathcal{L}(M_{k_n})\, , \, \mathcal{L}(\tilde M_{k_n})\big)\\[0.2cm]
{\textup{\texttt{II}}}'_n(X) &\ = \ d_{\text{K}}\big(\mathcal{L}(\tilde M_{k_n}) \, , \, \mathcal{L}( M^{\star}_{k_n}|X\big)\big)\\[0.2cm]
{\textup{\texttt{III}}}'_n(X) & \ = \ d_{\text{K}}\big(\mathcal{L}(M^{\star}_{k_n}| X\big) \, , \, \mathcal{L}( M_{d_n}^{\star} |X\big)\big).\end{aligned}$$ The overall structure of the proof is based on the simple bound $$\label{eqn:threepartsM}
\begin{split}
d_{\textup{K}}\big({\mathcal{L}}({\mathcal{M}})\,,\,{\mathcal{L}}({\mathcal{M}}^{\star}|X)\big) \ \leq & \ \ d_{\textup{K}}\big({\mathcal{L}}({\mathcal{M}}) \, ,\,{\mathcal{L}}(M_{d_n})\big)\\[0.2cm]
& \ + {\textup{\texttt{I}}}_n +{\textup{\texttt{II}}}_n+{\textup{\texttt{II}}}_n'(X)+{\textup{\texttt{III}}}_n'(X)\\[0.2cm]
& \ + d_{\textup{K}}\big({\mathcal{L}}(M_{d_n}^{\star}|X) \, , \, {\mathcal{L}}({\mathcal{M}}^{\star}|X)\big).
\end{split}$$ Most of the proof will be completed through four separate lemmas, showing that each of the quantities in Roman numerals is at most of order $n^{-1/2+\delta}$. These lemmas are labeled as \[lem:IIM\] (for ${\textup{\texttt{II}}}_n$), \[lem:IM\] (for ${\textup{\texttt{I}}}_n$), \[lem:IIprimeM\] (for ${\textup{\texttt{II}}}'_n(X)$) and \[lem:IIIprimeM\] (for ${\textup{\texttt{III}}}_n'(X)$). Finally, the other two quantities in the first and third lines will be shown to be at most of order $n^{-1/2+\delta}$ in Lemma \[lem:otherM\]. \
\[lem:IIM\]Fix any number $\delta\in(0,1/2)$, and suppose the conditions of Theorem \[THM:M\] hold. Then, $${\textup{\texttt{II}}}_n \ \lesssim \ n^{-1/2+\delta}.$$
The argument is rougly similar to the proof of Proposition \[prop:bentkus\], and we retain some of the notation used there. For the new proof, define the $k_n\times k_n$ matrix $$\label{eqn:sigmakdef}
\Sigma(k_n):=\Pi_{k_n}\Sigma\Pi_{k_n}{^{\top}},$$ where $\Pi_{k_n}\in{\mathbb{R}}^{k_n\times p}$ is the projection onto the coordinates indexed by ${\mathcal{J}}(k_n)$, as explained on page . Let $r_n$ denote the rank of $\Sigma(k_n)$, and let $$\Sigma(k_n) = Q\Lambda_{r_n} Q{^{\top}}$$ be a spectral decomposition of $\Sigma(k_n)$, where $Q\in{\mathbb{R}}^{k_n\times r_n}$ has orthonormal columns, and $\Lambda\in{\mathbb{R}}^{r_n\times r_n}$ is diagonal and invertible. In addition, define $$\mathfrak{C}=\Lambda_{r_n}^{1/2}Q{^{\top}}D_{k_n}^{-\tau_n},$$ as well as the $r_n$-dimensional random vector $$Z_i'=\Lambda_{r_n}^{-1/2}Q{^{\top}}\Pi_{k_n}(X_i-\boldsymbol\pi),$$ and the sample average $$\breve Z_n'= {\textstyle}\frac{1}{\sqrt n}\sum_{i=1}^n Z_i'.$$ Since the random vector $\Pi_{k_n}(X_i-\boldsymbol \pi)$ lies in the column span of $\Sigma(k_n)$ almost surely, it follows that the relation $QQ{^{\top}}\Pi_{k_n}(X_i-\boldsymbol\pi)=\Pi_{k_n}(X_i-\boldsymbol\pi)$ holds almost surely. In turn, this gives $$Q\Lambda_{r_n}^{1/2} Z_i' = \Pi_{k_n}(X_i-\boldsymbol \pi),$$ and hence $$D_{k_n}^{-\tau_n}\Pi_{k_n}S_n = \mathfrak{C}{^{\top}}\breve Z_n'.$$ Consequently, for any $t\in{\mathbb{R}}$, there is a Borel convex set $\mathcal{A}_t\subset{\mathbb{R}}^{r_n}$ such that $${\mathbb{P}}\big(M_{k_n}\leq t) \ = \ {\mathbb{P}}\Big(\max_{j\in{\mathcal{J}}(k_n)}D_{k_n}^{-\tau_n}\Pi_{k_n}S_{n,j} \leq t\Big) \ = \ {\mathbb{P}}\big(\breve Z_n'\in\mathcal{A}_t\big).$$ Similar reasoning also can be applied to $\tilde M_{k_n}$ in order to obtain the expression , where $\gamma_{r_n}$ is the standard Gaussian distribution on ${\mathbb{R}}^{r_n}$. Therefore, we have the bound $${\textup{\texttt{II}}}_n \ \leq \ \sup_{\mathcal{A}\in\mathscr{A}}\Big| {\mathbb{P}}(\breve Z' \in\mathcal{A}) - \gamma_{r_n}(\mathcal{A})\Big|,$$ with $\mathscr{A}$ being the collection of Borel convex subsets of ${\mathbb{R}}^{r_n}$.
Since the vectors $Z_1',\dots,Z_n'$ are i.i.d., with mean 0 and identity covariance matrix, we may apply the Berry-Esseen bound of Bentkus (Lemma \[lem:bentkus\]). The only remaining detail is to bound ${\mathbb{E}}[\|Z_1'\|_2^3]$, and show that it is at most a fixed power of $k_n$. To do this, Lemma \[lem:lambdamin\] implies there is a constant $c>0$ not depending on $n$ such that the following inequalities hold with probability 1, $$\begin{split}
\|Z_1'\|_2^2 & \ \leq \ {\textstyle}\frac{1}{\lambda_{r_n}(\Sigma(k_n))}\big\|\Pi_{k_n}(X_i-\boldsymbol\pi)\big\|_2^2\\[0.2cm]
& \ \leq \ ck_n^c \cdot k_n,
\end{split}$$ where we have used the facts that $\|Q\|{_{\textup{op}}}\leq 1$, and the entries of $\Pi_{k_n}(X_i-\boldsymbol\pi)$ are bounded in magnitude by 1. Thus, we have ${\mathbb{E}}[\|Z_1'\|_2^3]\lesssim k_n^{(3/2)(c+1)}$, which completes the proof.
\[lem:IM\] Fix any number $\delta\in(0,1/2)$, and suppose the conditions of Theorem \[THM:M\] hold. Then, $${\textup{\texttt{I}}}_n \ \lesssim \ n^{-1/2+\delta}.$$
By repeating the proof of Proposition \[prop:IandIII\], it follows that if $\texttt{t}_{1,n}$ and $\texttt{t}_{2,n}$ are any two real numbers with $\texttt{t}_{1,n}\leq \texttt{t}_{2,n}$, then $${\textup{\texttt{I}}}_n \ \leq \ {\mathbb{P}}(\texttt{A}(\texttt{t}_{2,n})) \ + \ {\mathbb{P}}(\texttt{B}(\texttt{t}_{1,n})),$$ where we write $$\texttt{A}(t)=\Big\{\max_{j\in{\mathcal{J}}(k_n)} S_{n,j}/\sigma_j^{\tau_n} \leq t\Big\} \ \ \ \text{ and } \ \ \ \ \texttt{B}(t)=\Big\{\max_{j\in{\mathcal{J}}(d_n)\setminus{\mathcal{J}}(k_n)} S_{n,j}/\sigma_j^{\tau_n}> t\Big\},$$ for any $t\in {\mathbb{R}}$. (Note that $\texttt{B}(t)$ differs from $B(t)$ only insofar as ${\mathcal{J}}(d_n)\setminus{\mathcal{J}}(k_n)$ replaces ${\mathcal{J}}(k_n)^c$.)
To handle the probability ${\mathbb{P}}(\texttt{A}(\texttt{t}_{2,n}))$, we mimic the definition and let $$\texttt{t}_{2,n}={\epsilon}_0\cdot \ell_n^{-\beta_n}\cdot \sqrt{\log(\ell_n)},$$ with ${\epsilon}_0\in(0,1)$ as in Assumption \[A:mult\]. Having made this choice, the proof of Lemma \[lem:bounds\] may be repeated essentially verbatim to show that ${\mathbb{P}}(\texttt{A}(\texttt{t}_{2,n}))\lesssim n^{-1/2+\delta}$. In particular, it is important to note that the correlation matrix $R(\ell_n)$ in the multinomial case satisfies the conditions needed for that argument to work, because $R^+(\ell_n)=\mathbf{I}_{\ell_n}$. Also, this argument relies on Lemma \[lem:IIM\] in the same way that the proof of Lemma \[lem:bounds\] relies on Proposition \[prop:bentkus\].
To handle ${\mathbb{P}}(\texttt{B}(\texttt{t}_{1,n}))$, the proof of Lemma \[lem:bounds\] can be mostly repeated to show that this probability is of order $1/n$. However, there are a few differences. First, we may regard the $\alpha$ from the context of Lemma \[lem:bounds\] as being equal to $1/2$, due to the basic fact that $\sigma_{(j)}\leq j^{-1/2}$ always holds in the multinomial model. Likewise, we define $$\texttt{t}_{1,n}=c\cdot k_n^{-(1-\tau_n)/2}\cdot \log(n)$$ as the analogue of $t_{1,n}$ in . The only other issues to notice are that the set ${\mathcal{J}}(k_n)^c$ is replaced by ${\mathcal{J}}(d_n)\setminus{\mathcal{J}}(k_n)$, and that we must verify the following condition. Namely, there is a constant $c>0$ not depending on $n$ such that the inequality $$\max_{j\in{\mathcal{J}}(d_n)} \big\|{\textstyle}\frac{1}{\sigma_j}S_{n,j}\big\|_q \ \leq c q$$ holds when $q=\max\{\frac{2}{(1/2)(1-\tau_n)},\log(n),3\}$. This will be verified later in Lemma \[lem:SnjnormM\]. Finally, it is simple to check that $\texttt{t}_{1,n} \leq \texttt{t}_{2,n}$ holds for all large $n$. \
\[lem:IIprimeM\] Fix any number $\delta\in(0,1/2)$, and suppose the conditions of Theorem \[THM:M\] hold. Then, there is a constant $c>0$ not depending on $n$ such that the event $${\textup{\texttt{II}}}_n'(X) \ \leq \ c\, n^{-1/2+\delta}$$ holds with probability at least $1-\frac cn$.
Let the random variable $\breve M_{k_n}^{\star}$ be as defined in the proof of Proposition \[prop:IIprime\], and consider the triangle inequality $$\label{eqn:IIprimeM}
\small
{\textup{\texttt{II}}}_n'(X) \ \leq \ {d_{\textup{K}}}\Big({\mathcal{L}}(\tilde M_{k_n})\, , {\mathcal{L}}(\breve M_{k_n}^{\star}|X)\Big) \ + \ {d_{\textup{K}}}\Big({\mathcal{L}}(\breve M_{k_n}^{\star}|X) \, , \, {\mathcal{L}}(M_{k_n}^{\star}|X)\Big).$$ Regarding the second term on the right, the proof of Proposition \[prop:IIprime\] shows that this term can be controlled with a coupling inequality for $\breve{M}_{k_n}^{\star}$ and $M_{k_n}^{\star}$, as well as an anti-concentration inequality for $M_{k_n}^{\star}$. In the multinomial context, both of these inequalities can be established using the same overall approach as before. The main items that need to be updated are to replace ${\mathcal{J}}(k_n)^c$ with ${\mathcal{J}}(d_n)\setminus {\mathcal{J}}(k_n)$, and to control quantities involving by using Lemmas (\[lem:corM\], \[lem:sighatrnormM\]) instead of Lemmas (\[lem:cor\], \[lem:sighatrnorm\]).
Next, to control the first term in the bound , we will use an argument based on Lemma \[lem:hellinger\], which requires quite a few pieces of notation. First, let $\Sigma(k_n)=\Pi_{k_n}\Sigma\Pi_{k_n}{^{\top}}\in{\mathbb{R}}^{k_n\times k_n}$, and let the rank of this matrix be denoted by $r_n$. Next, write the spectral decomposition of $\Sigma(k_n)$ as $$\Sigma(k_n)=Q\Lambda_{r_n}Q{^{\top}},$$ where $Q\in{\mathbb{R}}^{k_n\times r_n}$ has orthonormal columns and $\Lambda_{r_n}\in{\mathbb{R}}^{r_n\times r_n}$ is diagonal. In addition define $$\label{eqn:whitemultdef}
\begin{split}
{\widehat{\Sigma}}(k_n)&=\Pi_{k_n}{\widehat{\Sigma}} \Pi_{k_n}{^{\top}}\\[0.2cm]
W_n&= \Lambda_{r_n}^{-1/2}Q{^{\top}}{\widehat{\Sigma}}(k_n) Q\Lambda_{r_n}^{-1/2}.
\end{split}$$ Since each vector $\Pi_{k_n}(X_i-\bar X)$ lies in the column span of $\Sigma(k_n)$ almost surely, it follows that the relation $QQ{^{\top}}{\widehat{\Sigma}}(k_n)QQ{^{\top}}={\widehat{\Sigma}}(k_n)$ holds almost surely, which is equivalent to $${\widehat{\Sigma}}(k_n)=Q\Lambda_{r_n}^{1/2}W_n\Lambda_{r_n}^{1/2}Q{^{\top}}.$$ With this in mind, let $\mathfrak{C}=\Lambda_{r_n}^{1/2}Q{^{\top}}D_{k_n}^{-\tau_n}$, and also define $$\begin{split}
\mathfrak{S}&=\mathfrak{C}{^{\top}}\mathfrak{C}\\[0.2cm]
\breve{\mathfrak{S}} &= \mathfrak{C}{^{\top}}W_n\mathfrak{C}.
\end{split}$$ It is straightforward to check that $\tilde M_{k_n}$ is the coordinate-wise maximum of a Gaussian vector drawn from $N(0,\mathfrak{S})$, and similarly, $\breve M_{k_n}^{\star}$ is the coordinate-wise maximum of a Gaussian vector drawn from $N(0,\breve{\mathfrak{S}})$.
We will now compare $\tilde M_{k_n}$ and $\breve M_{k_n}^{\star}$ by applying Lemma \[lem:hellinger\]. For this purpose, let $$\label{eqn:Udef}
\mathfrak{C}=ULV{^{\top}}$$ be an s.v.d. for $\mathfrak{C}$, where the matrix $L\in{\mathbb{R}}^{r_n\times r_n}$ is diagonal and invertible, and the matrices $U\in{\mathbb{R}}^{r_n\times r_n}$ and $V\in{\mathbb{R}}^{k_n\times r_n}$ each have orthonormal columns. From these definitions, it follows that $$\Big(V{^{\top}}\mathfrak{S} V\Big)^{-1/2}\Big( V{^{\top}}\breve{\mathfrak{S}}V\Big) \Big(V{^{\top}}\mathfrak{S}V\Big)^{-1/2} \ = \ U{^{\top}}W_n U.$$ Thus, the matrix $(V{^{\top}}\mathfrak{S} V)^{1/2}$ will play the role of $H$ in the statement of Lemma \[lem:hellinger\]. Also, in order to apply that lemma, we need that the columns of $\mathfrak{S}$ and $\breve{\mathfrak{S}}$ span the same subspace of ${\mathbb{R}}^{k_n}$ (with high probability). Noting that $\mathfrak{S}=VL^2 V{^{\top}}$ and $\breve{\mathfrak{S}} =VL (U{^{\top}}W_n U)L V{^{\top}}$, it follows that $\mathfrak{S}$ and $\breve{\mathfrak{S}}$ have the same column span whenever $U{^{\top}}W_n U$ is invertible, and by Lemma \[lem:whitemult\], this holds with probability at least $1-c/n$. Therefore, Lemma \[lem:hellinger\] ensures that if the event $$\label{eqn:tempevent}
\|U{^{\top}}W_n U-\mathbf{I}_{r_n}\|_{\text{op}} \ \leq \ {\epsilon},$$ holds for some number ${\epsilon}>0$, then the event $${d_{\textup{K}}}\Big({\mathcal{L}}(\tilde M_{k_n})\, , {\mathcal{L}}(\breve M_{k_n}^{\star}|X)\Big) \leq c\cdot k_n^{1/2} \cdot {\epsilon}$$ also holds. Finally, Lemma \[lem:whitemult\] shows that if we take ${\epsilon}= c\log(n)k_n^c/\sqrt{n}$, then the event holds with probability at least $1-c/n$, which completes the proof.
\[lem:IIIprimeM\] Fix any number $\delta\in(0,1/2)$, and suppose the conditions of Theorem \[THM:M\] hold. Then, there is a constant $c>0$ not depending on $n$ such that the event $${\textup{\texttt{III}}}_n'(X) \ \leq \ c\, n^{-1/2+\delta}$$ holds with probability at least $1-\frac cn$.
The proof follows the argument outlined in Section \[sec:III’\]. The main details to be updated for the multinomial context arise in controlling the quantities $\{{\widehat{\sigma}}_j | j\in{\mathcal{J}}(d_n)\}$. Specifically, the index set ${\mathcal{J}}(k_n)^c$ must be replaced with ${\mathcal{J}}(d_n)\setminus {\mathcal{J}}(k_n)$, and Lemmas (\[lem:corM\], \[lem:sighatrnormM\]) must be used in place of Lemmas (\[lem:cor\], \[lem:sighatrnorm\]).
\[lem:otherM\] Suppose the conditions of Theorem \[THM:M\] hold. Then, there is a constant $c>0$ not depending on $n$, such that $$\label{eqn:reductionfirst}
d_{\textup{K}}\big({\mathcal{L}}({\mathcal{M}}), {\mathcal{L}}(M_{d_n})\big) \ \leq \ {\textup{\texttt{I}}}_n+{\textstyle}\frac{c}{n},$$ and the event $$\label{eqn:reductionsecond}
d_{\textup{K}}\big({\mathcal{L}}({\mathcal{M}}^{\star}|X), {\mathcal{L}}(M_{d_n}^{\star}|X)\big) \ \leq \ {\textup{\texttt{III}}}_n'(X)$$ occurs with probability at least $1-\frac cn$.
Fix any $t\in{\mathbb{R}}$. By intersecting the event $\{\max_{j\in{\widehat{{\mathcal{J}}}}_n}S_{n,j}/\sigma_j^{\tau_n}\leq t\}$ with the events $\{{\mathcal{J}}(k_n)\subset {\widehat{{\mathcal{J}}}}_n\}$ and $\{{\mathcal{J}}(k_n)\not \subset{\widehat{{\mathcal{J}}}}_n\}$, and noting that the maximum can only become smaller on a subset, we have $$\footnotesize
\begin{split}
{\mathbb{P}}\Big(\max_{j\in{\widehat{{\mathcal{J}}}}_n}S_{n,j}/\sigma_j^{\tau_n}\leq t\Big) \ \leq \
{\mathbb{P}}\Big(\max_{j\in{\mathcal{J}}(k_n)}S_{n,j}/\sigma_j^{\tau_n}\leq t\Big) +{\mathbb{P}}\big({\mathcal{J}}_{k_n}\not\subset{\widehat{{\mathcal{J}}}}_n).
\end{split}$$ Therefore, by subtracting from both sides the probability involving the maximum over ${\mathcal{J}}(d_n)$, we have $$\label{eqn:firstsign}
{\mathbb{P}}\Big(\max_{j\in{\widehat{{\mathcal{J}}}}_n}S_{n,j}/\sigma_j^{\tau_n}\leq t\Big) - {\mathbb{P}}\Big(\max_{j\in{\mathcal{J}}(d_n)}S_{n,j}/\sigma_j^{\tau_n}\leq t\Big) \ \leq \ {\textup{\texttt{I}}}_n+{\mathbb{P}}\big({\mathcal{J}}_{k_n}\not\subset{\widehat{{\mathcal{J}}}}_n).$$ Similarly, by intersecting with the events $\{{\widehat{{\mathcal{J}}}}\subset {\mathcal{J}}(d_n)\}$ and $\{{\widehat{{\mathcal{J}}}}_n\not \subset{\mathcal{J}}(d_n)\}$, we have $${\mathbb{P}}\Big(\max_{j\in{\mathcal{J}}(d_n)}S_{n,j}/\sigma_j^{\tau_n}\leq t\Big)
\ \leq \ {\mathbb{P}}\Big(\max_{j\in{\widehat{{\mathcal{J}}}}_n}S_{n,j}/\sigma_j^{\tau_n}\leq t\Big)+ {\mathbb{P}}({\widehat{{\mathcal{J}}}}_n\not \subset{\mathcal{J}}(d_n)).
$$ Next, subtracting the probability involving the maximum over ${\widehat{{\mathcal{J}}}}_n$ gives $$\label{eqn:secondsign}
{\mathbb{P}}\Big(\max_{j\in{\mathcal{J}}(d_n)}S_{n,j}/\sigma_j^{\tau_n}\leq t\Big)
\ - \ {\mathbb{P}}\Big(\max_{j\in{\widehat{{\mathcal{J}}}}_n}S_{n,j}/\sigma_j^{\tau_n}\leq t\Big) \ \leq \ 0 + {\mathbb{P}}({\widehat{{\mathcal{J}}}}_n\not \subset{\mathcal{J}}(d_n)).
$$ Combining and implies $$d_{\textup{K}}\big({\mathcal{L}}({\mathcal{M}})\, ,\, {\mathcal{L}}(M_{d_n})\big) \ \leq \ {\textsc{I}}_n+{\mathbb{P}}\big({\mathcal{J}}_{k_n}\not\subset{\widehat{{\mathcal{J}}}}_n)+{\mathbb{P}}({\widehat{{\mathcal{J}}}}_n\not \subset{\mathcal{J}}(d_n)),$$ and furthermore, Lemma \[lem:Jhat\] shows that the last two probabilities on the right are at most $c/n$. This proves . The inequality follows by similar reasoning, and is actually easier, because conditioning on $X$ allows us to work under the assumption that the events $\{{\mathcal{J}}(k_n)\subset {\widehat{{\mathcal{J}}}}_n\}$ and $\{{\widehat{{\mathcal{J}}}}_n\subset{\mathcal{J}}(d_n)\}$ hold, since they occur with probability at least $1-\frac cn$.
Technical Lemmas for Theorem \[THM:M\] {#app:lemmamult}
======================================
\[lem:Jhat\] Suppose the conditions of Theorem \[THM:M\] hold. Then, with probability at least $1-\frac cn$, the following two events hold simultaneously, $$\label{eqn:firstJhat}
{\mathcal{J}}(k_n) \ \subset \ {\widehat{\mathcal{J}}}_n,$$ and $$\label{eqn:secondJhat}
{\widehat{\mathcal{J}}}_n \ \subset \ {\mathcal{J}}(d_n).$$
We first address the event . By a union bound, the following inequalities hold for all large $n$, $$\begin{split}
{\mathbb{P}}\Big({\mathcal{J}}(k_n)\not\subset {\widehat{\mathcal{J}}}_n \Big)
& \ \leq \ \sum_{j\in{\mathcal{J}}(k_n)} {\mathbb{P}}\Big({\widehat{\pi}}_j <\sqrt{\log(n)/n}\Big)\\[0.3cm]
& \leq \ k_n\cdot \max_{j\in{\mathcal{J}}(k_n)}{\mathbb{P}}\Big(\sqrt{n}|{\widehat{\pi}}_j-\pi_j| >n^{1/4}\Big)
\end{split}$$ where the last step follows from the fact that if $j\in{\mathcal{J}}(k_n)$, then the crude (but adequate) inequality $\sqrt{n}\pi_j- \sqrt{\log(n)}\geq n^{1/4}$ holds for all large $n$. In turn, Hoeffding’s inequality [@vanderVaart:Wellner:2000 p.460] implies $$k_n\cdot\max_{j\in{\mathcal{J}}(k_n)}{\mathbb{P}}\Big(\sqrt{n}|{\widehat{\pi}}_j-\pi_j| >n^{1/4}\Big) \ \leq \ 2k_n e^{-2\sqrt{n}} \ \lesssim \ {\textstyle}\frac{1}{n},$$ which establishes .
We now turn to . Under Assumption \[A:mult\], it is simple to check that ${\mathcal{J}}(k_n)\subset{\mathcal{J}}(d_n)$ holds for all large $n$. Consequently, in the low-dimensional situation where $k_n=p$, we must have ${\mathcal{J}}(d_n)={\mathcal{J}}(p)$ for all large $n$, and then is clearly true. Hence, we may work in the situation where $k_n<p$. To begin, observe that Assumption \[A:mult\] gives $\pi_{(1)}(1-\pi_{(1)})=\sigma_{(1)}^2\geq {\epsilon}_0^2$, which implies $1-\pi_{(1)}\geq {\epsilon}_0^2$, and hence $1-\pi_j\geq {\epsilon}_0^2$ for all $j\in\{1,\dots,p\}$. Based on this observation, if we consider the set $${\mathcal{J}}_n':=\Big\{j\in\{1,\dots,p\}\, \Big| \, \pi_j\geq 1/\sqrt{n}\Big\},$$ then $j\in{\mathcal{J}}_n'$ implies $\sigma_j^2=\pi_j(1-\pi_j)\geq {\epsilon}_0^2/\sqrt{n}$. Now, recall that ${\mathcal{J}}(d_n)$ is defined so that $j\in{\mathcal{J}}(d_n) \Longleftrightarrow \sigma_j^2\geq {\epsilon}_0^2/\sqrt{n}$. As a result of this definition, we have ${\mathcal{J}}_n'\subset{\mathcal{J}}(d_n)$. Therefore, in order to show that the event $\{{\widehat{{\mathcal{J}}}}_n\subset{\mathcal{J}}(d_n)\}$ holds with probability at least $1-c/n$, it suffices to show that the event $\{{\widehat{{\mathcal{J}}}}_n\subset{\mathcal{J}}_n'\}$ holds with at least the same probability.
To proceed, observe that the following inclusion always holds, $$\begin{split}
{\widehat{\mathcal{J}}}_n
& \ \subset \ \Big\{ j\in\{1,\dots,p\} \ \Big| \ \sqrt{n}\pi_j \ \geq \sqrt{\log(n)} - \max_{1\leq j\leq p}\sqrt{n}({\widehat{\pi}}_j-\pi_j)\Big\}.
\end{split}$$ Therefore, if we can show that the event $$\mathcal{E}:=\Big\{ \sqrt{\log(n)} - \max_{1\leq j\leq p}\sqrt{n}({\widehat{\pi}}_j-\pi_j) \geq 1\Big\}$$ occurs with probability at least $1-\frac cn$, then the event $\{{\widehat{{\mathcal{J}}}}_n\subset {\mathcal{J}}_n'\}$ will also occur with probability at least $1-\frac cn$. Now, consider the union bound $${\mathbb{P}}(\mathcal{E}^c) \ \leq \ \sum_{j=1}^p {\mathbb{P}}\Big(\sqrt{n}({\widehat{\pi}}_j-\pi_j) > \sqrt{\log(n)}-1\Big).$$ We will bound this sum by considering two different sets of indices. For the indices $j\in{\mathcal{J}}(k_n)^c$, the values $\pi_j$ are mostly small. This motivates the use of Kiefer’s inequality (Lemma \[lem:Kiefer\]), which implies there is some $c>0$ not depending on $n$, such that the following bound holds for all large $n$, and $j\in{\mathcal{J}}(k_n)^c$, $$\begin{split}
{\mathbb{P}}\Big(\sqrt{n}({\widehat{\pi}}_j-\pi_j) > \sqrt{\log(n)}-1\Big) & \ \leq \ 2\exp\Big\{-c\log(n)\log({\textstyle}\frac{1}{\pi_j})\Big\}\\[0.2cm]
& \ = \ 2 \pi_j^{c\log(n)}.
\end{split}$$ On the other hand, if $j\in{\mathcal{J}}(k_n)$, then $\pi_j$ is of moderate size, and Hoeffding’s inequality [@vanderVaart:Wellner:2000 p.460] implies the following inequality for all large $n$, $$\begin{split}
{\mathbb{P}}\Big(\sqrt{n}({\widehat{\pi}}_j-\pi_j) > \sqrt{\log(n)}-1\Big) & \ \leq \ 2\exp\big\{-(3/2)\log(n)\big\}\\[0.2cm]
& \ = \ 2 n^{-3/2}.
\end{split}$$ (The constant $3/2$ in the exponent has been chosen for simplicity, and is not of special importance.) Combining the two different types of bounds, and using the fact that any probability vector satisfies $\pi_{(j)}\leq j^{-1}$ for all $j\in\{1,\dots,p\}$, we obtain $$\begin{split}
{\mathbb{P}}(\mathcal{E}^c)
& \ \lesssim \ k_n n^{-3/2} \ + \ \sum_{j=k_n+1}^p \pi_{(j)}^{c\log(n)}\\[0.3cm]
& \ \lesssim \ {\textstyle}\frac{1}{n} \ + \displaystyle\int_{k_n}^p x^{-c\log(n)}dx\\[0.3cm]
& \ \lesssim \ {\textstyle}\frac{1}{n} \ + k_n^{-c\log(n)+1}\\[0.3cm]
& \ \lesssim \ {\textstyle}\frac{1}{n},
\end{split}$$ as needed.
\[lem:sighatrnormM\] Suppose the conditions of Theorem \[THM:M\] hold, and let $q=\max\{{\textstyle}\frac{2}{(1/2)(1-\tau_n)}, \log(n),3\}$. Then, there is a constant $c>0$ not depending on $n$, such that for any $j\in\mathcal{J}(d_n)$, we have $$\|{\widehat{\sigma}}_{j}\|_q \ \leq \ c \cdot \sigma_j \cdot \sqrt{q}.$$
By direct calculation $$\begin{split}
{\textstyle}\frac{1}{\sigma_j}\|{\widehat{\sigma}}_{j}\|_q
&= \ {\textstyle}\frac{1}{\sigma_j} \Bigg\|{\textstyle}\frac 1n \sum_{i=1}^n (X_{i,j}^2-\pi_j)+\big(\pi_j-\pi_j^2\big)+\big(\pi_j^2-\bar{X}_j^2\big)\Bigg\|_{q/2}^{1/2}\\[0.2cm]
&\leq \ {\textstyle}\frac{1}{\sigma_j} \Bigg(\big\|{\textstyle}\frac 1n \sum_{i=1}^n (X_{i,j}^2-\pi_j)\big\|_{q/2}^{1/2}+(\pi_j-\pi_j^2)^{1/2}+\big\|\pi_j^2-\bar{X}_j^2\big\|_{q/2}^{1/2}\Bigg)\\[0.3cm]
&\leq \ {\textstyle}\frac{1}{\sigma_jn^{1/4}}\big\| S_{n,j}\big\|_{q/2}^{1/2} \ + \ 1 \ + \ {\textstyle}\frac{\sqrt{2}}{\sigma_jn^{1/4}}\big\| S_{n,j}\big\|_{q/2}^{1/2}.
\end{split}$$ Since $1/(\sigma_j n^{1/4})\leq 1/{\epsilon}_0$ when $j\in{\mathcal{J}}(d_n)$, it follows from Lemma \[lem:rosenthal\] that the first and third terms are at most of order $\sqrt{q}$.
#### Remark
For the next lemma, put $\boldsymbol \pi_{k_n}:=(\pi_{(1)},\dots,\pi_{(k_n)})$, and recall the definition $\Sigma(k_n)=\Pi_{k_n}\Sigma\Pi_{k_n}{^{\top}}$ from line .
\[lem:lambdamin\] If $r_n$ denotes the rank of $\Sigma(k_n)$, and the conditions of Theorem \[THM:M\] hold, then there is a constant $c>0$ not depending on $n$ such that $$\lambda_{r_n}(\Sigma(k_n)) \ \gtrsim \ k_n^{-c}.$$
We first consider the case $k_n<p$, and then handle the case $k_n=p$ separately at the end of the proof. Under the multinomial model, we have for all $i,j\in\{1,\dots,k_n\}$, $$\Sigma_{i,j}(k_n) = \begin{cases} \ \pi_{(i)}(1-\pi_{(i)}) & \text{ if } i=j, \\ \ -\pi_{(i)}\pi_{(j)} & \text{ if } i\neq j. \end{cases}$$ For each $i\in\{1,\dots,k_n\}$, define the “deleted row sum”, $$\varrho_i:=\sum_{j\neq i}|\Sigma_{i,j}(k_n)|.$$ By the Geršgorin disc theorem [@Horn:Johnson:1990 Sec. 6.1], $$\begin{split}
\lambda_{r_n}(\Sigma(k_n)) & \ \geq \ \lambda_{\min}(\Sigma(k_n))\\[0.2cm]
& \ \geq \ \min_{1\leq i\leq k_n}\Big\{\Sigma_{i,i}(k_n) - \varrho_i\Big\}\\[0.2cm]
& \ = \ \min_{1\leq i\leq k_n} \Big\{(\pi_{(i)} -\pi_{(i)}{\textstyle\sum}_{j=1}^{k_n}\pi_{(j)}\Big\},\\[0.2cm]
& \ = \ \pi_{(k_n)}\Big(1-{\textstyle\sum}_{j=1}^{k_n} \pi_{(j)}\Big).
\end{split}$$ When $k_n<p$, it follows from Assumption \[A:mult\] that $$\begin{split}
1-{\textstyle\sum}_{j=1}^{k_n}\pi_{(j)}&\geq \pi_{(k_n+1)}\\[0.2cm]
&\geq \sigma_{(k+1)}^2\\[0.2cm]
&\geq {\epsilon}_0^2 (k_n+1)^{-2\alpha}\\[0.2cm]
&\gtrsim k_n^{-2\alpha}.
\end{split}$$ Hence, the previous steps show that $\lambda_{r_n}(\Sigma(k_n))$ is at least of order $k_n^{-4\alpha}$.
Finally, consider the case when $k_n=p$. In this case, it is a basic fact that the rank of $\Sigma=\Sigma(k_n)$ satisfies $r_n=p-1$ (where we note that Assumption \[A:mult\] ensures $\pi_{(p)}>0$). Also, it is known from matrix analysis [@Benasseni:2012 Theorem 1] that $$\lambda_{p-1}(\Sigma)\geq \pi_{(p)},$$ and therefore Assumption \[A:mult\] leads to $$\begin{split}
\lambda_{r_n}(\Sigma(k_n))&\geq \pi_{(k_n)}\\[0.2cm]
&\geq\sigma_{(k_n)}^2\\[0.2cm]
&\geq {\epsilon}_0^2 k_n^{-2\alpha},
\end{split}$$ as needed.
\[lem:whitemult\] Let the deterministic matrix $U\in{\mathbb{R}}^{r_n\times r_n}$ and the random matrix $W_n\in{\mathbb{R}}^{r_n\times r_n}$ be as defined in and respectively. Also, suppose that the conditions of Theorem \[THM:M\] hold. Then, there is a constant $c>0$ not depending on $n$ such that the event $$\big\|U{^{\top}}W_n U-\mathbf{I}_{r_n}\big\|_{\textup{op}}\ \leq \ {\textstyle}\frac{c\, k_n^c \log(n)}{\sqrt n}$$ holds with probability at least $1-\frac cn$.
Let the notation from the proof of Lemma \[lem:IIprimeM\] be in force, and observe that $$\begin{split}
\big\|U{^{\top}}W_n U-\mathbf{I}_{r_n}\big\|_{\textup{op}} & \ = \ \Big\|U{^{\top}}\Big(\Lambda_{r_n}^{-1/2}Q{^{\top}}{\widehat{\Sigma}}(k_n) Q\Lambda_{r_n}^{-1/2} - \mathbf{I}_{r_n}\Big)U\Big\|{_{\textup{op}}}\\[0.2cm]
& \ \leq \ \Big\|\Lambda_{r_n}^{-1/2}Q{^{\top}}\Big({\widehat{\Sigma}}(k_n) -\Sigma(k_n)\Big)Q\Lambda_{r_n}^{-1/2}\Big\|{_{\textup{op}}}\\[0.2cm]
& \ \leq \ \frac{1}{\lambda_{r_n}(\Sigma(k_n))} \, \big\|{\widehat{\Sigma}}(k_n)-\Sigma(k_n)\big\|{_{\textup{op}}}.
\end{split}$$ With regard to the first factor in the previous line, Lemma \[lem:lambdamin\] implies $${\textstyle}\frac{1}{\lambda_{r_n}(\Sigma(k_n))} \lesssim k_n^c.$$ To complete the proof, let $\boldsymbol\pi_{k_n}=\Pi_{k_n}\boldsymbol \pi$, and let $u\in{\mathbb{R}}^{k_n}$ be a generic unit vector. Consider the decomposition $$\footnotesize
\begin{split}
|u{^{\top}}\big({\widehat{\Sigma}}(k_n)-\Sigma(k_n)\big)u| & \leq \bigg|\frac1n\sum_{i=1}^n \Big((X_i{^{\top}}\Pi_{k_n}{^{\top}}u)^2-u{^{\top}}\text{diag}(\boldsymbol\pi_{k_n})u\Big)\bigg| + \Big| (\bar X{^{\top}}\Pi_{k_n}{^{\top}}u)^2-(\boldsymbol\pi_{k_n}{^{\top}}u)^2\Big|\\[0.3cm]
& \ =: \ \Delta_n(u)+\Delta_n'(u).
\end{split}$$ In order to control these terms, note that each random variable $(X_i{^{\top}}\Pi_{k_n}{^{\top}}u)^2$ is bounded in magnitude by 1, and has expectation equal to $u{^{\top}}\text{diag}(\boldsymbol \pi_{k_n})u$. In addition, we have $$|\Delta_n'(u)| \ \leq \ 2|\bar X{^{\top}}\Pi_{k_n}{^{\top}}u-\boldsymbol \pi_{k_n}{^{\top}}u|.$$ Based on these observations, the proof of Lemma \[lem:white\] can be essentially repeated to show that there is a constant $c>0$ not depending on $n$ such that the event $$\big\|{\widehat{\Sigma}}(k_n)-\Sigma(k_n)\big\|_{\text{op}} \ \leq \ {\textstyle}\frac{c\log(n) k_n}{\sqrt{n}}$$ holds with probability at least $1-\frac cn$. This completes the proof.\
\[lem:SnjnormM\] Let $q=\max\{{\textstyle}\frac{2}{(1/2)(1-\tau_n)}, \log(n),3\}$, and suppose that the conditions of Theorem \[THM:M\] hold. Then, there is a constant $c>0$ not depending on $n$, such that $$\max_{j\in\mathcal{J}(d_n)}\|{\textstyle}\frac{1}{\sigma_j}S_{n,j}\|_q \leq cq.$$ In addition, the following event holds with probability 1, $$\max_{j\in\mathcal{J}(d_n)} \Big({\mathbb{E}}\big[|{\textstyle}\frac{1}{{\widehat{\sigma}}_j}S_{n,j}^{\star}|^q|X\big] \Big)^{1/q} \ \leq c\, q.$$
The second inequality can be obtained by repeating the proof of Lemma \[lem:Snjnorm\], since $S_n^{\star}$ is still Gaussian under the setup of Assumption \[A:mult\]. To prove the first inequality, note that since $q>2$, Lemma \[lem:rosenthal\] gives $$\label{eqn:lemrosenthalfirst}
\|{\textstyle}\frac{1}{\sigma_j}S_{n,j}\|_q \ \lesssim \ q\cdot \max\Big\{ \|{\textstyle}\frac{1}{\sigma_j}S_{n,j}\|_2 \, , \, n^{-1/2+1/q} \|{\textstyle}\frac{1}{\sigma_j}(X_{1,j}-\pi_j)\|_q\Big\}.$$ Clearly, $$\|{\textstyle}\frac{1}{\sigma_j}S_{n,j}\|_2^2 \ = \ {\operatorname{var}}({\textstyle}\frac{1}{\sigma_j}S_{n,j}) \ = \ 1.$$ For the stated choice of $q$, the second term inside the maximum satisfies $$n^{-1/2+1/q} \|{\textstyle}\frac{1}{\sigma_j}(X_{1,j}-\pi_j)\|_q \ \lesssim \frac{1}{\sqrt n \sigma_j},$$ and also, since $j\in\mathcal{J}(d_n)$, we have $\frac{1}{\sqrt{n}\sigma_j}\lesssim \frac{1}{n^{1/4}}$, which leads to the stated claim.
\[lem:corM\] Suppose the conditions of Theorem \[THM:M\] hold. Then, there is a constant $c>0$ not depending on $n$ such that the events $$\label{eqn:firstcor}
\max_{j\in{\mathcal{J}}(k_n)}\Big| {\textstyle}\frac{{\widehat{\sigma}}_j}{\sigma_j}-1\Big| \leq {\textstyle}\frac{c\cdot k_n^c\cdot\sqrt{\log(n)}}{n^{1/2}},$$ and $$\label{eqn:minvarhattemp}
\min_{j\in{\mathcal{J}}(k_n)}{\widehat{\sigma}}_j^{1-\tau_n} \ \geq \ \Big(\min_{j\in{\mathcal{J}}(k_n)} \sigma_j^{1-\tau_n}\Big)\cdot \Big(1-{\textstyle}\frac{c\cdot k_n^c\cdot \sqrt{\log(n)}}{n^{1/2}}\Big)$$ each hold with probability at least $1-{\textstyle}\frac{c}{n}$.
Note that if occurs, then also occurs, and so we only deal with the former event. Fix any number $\kappa\geq 2$. By a union bound, it suffices to show there are positive constants $c$ and $c_1(\kappa)$ not depending on $n$, such that for any $j\in{\mathcal{J}}(k_n)$, the event $$\label{eqn:corevent}
\Big| {\textstyle}\frac{{\widehat{\sigma}}_j^2}{\sigma_j^2}-1\Big| \ \leq \ \frac{c_1(\kappa)\cdot k_n^c\cdot \sqrt{\log(n)}}{n^{1/2}}$$ holds with probability at least $1-\frac{c}{n^{\kappa}}$. To this end, observe that $$\begin{split}
\Big| {\textstyle}\frac{{\widehat{\sigma}}_j^2}{\sigma_j^2}-1\Big| & \ \leq \ \Big|{\textstyle}\frac{1}{\sigma_j^2n} \sum_{i=1}^n (X_{i,j}^2-\pi_j) \Big| \ + \ {\textstyle}\frac{1}{\sigma_j^2}\big|\pi_j^2-\bar{X}_j^2\big|\\[0.3cm]
& \ \leq \ {\textstyle}\frac{3}{\sigma_j^2 \sqrt{n}}|S_{n,j}|.
\end{split}$$ Due to Assumption \[A:mult\], and the fact that $j\in{\mathcal{J}}(k_n)$, we have $${\textstyle}\frac{1}{\sigma_j^2\sqrt n} \ \lesssim \ {\textstyle}\frac{k_n^{2\alpha}}{\sqrt n}.$$ In addition, Hoeffding’s inequality ensures there is a constant $c_1(\kappa)$ such that the event $$|S_{n,j}| \ \leq \ c_1(\kappa)\sqrt{\log(n)}$$ holds with probability at least $1-\frac{c}{n^{\kappa}}$.
Background results {#app:background}
==================
The following result is a multivariate version of the Berry-Esseen theorem due to [@Bentkus:2003].
\[lem:bentkus\] Let $V_1,\dots,V_n$ be i.i.d. random vectors ${\mathbb{R}}^d$, with zero mean, and identity covariance matrix. Furthermore, let $\gamma_d$ denote the standard Gaussian distribution on ${\mathbb{R}}^d$, and let $\mathscr{A}$ denote the collection of all Borel convex subsets of ${\mathbb{R}}^d$. Then, there is an absolute constant $c>0$ such that $$\sup_{\mathcal{A}\in\mathscr{A}} \Big| {\mathbb{P}}\big({\textstyle}\frac{1}{\sqrt{n}}(V_1+\dots+V_n)\in \mathcal{A}\big) -\gamma_d(\mathcal{A})\Big| \ \leq \ \displaystyle \frac{c\cdot d^{1/4}\cdot {\mathbb{E}}\big[\|V_1\|_2^3\big]}{n^{1/2}}.$$
The following is a version of Nazarov’s inequality [@Nazarov:2003; @Klivans:2008], as formulated in [@CCK:SPA Lemma 4.3].
\[lem:Nazarov\] Let $(\xi_1,\dots,\xi_m)$ be a multivariate normal random vector, and suppose the parameter $\underline{\sigma}^2:=\min_{1\leq j\leq m} {\operatorname{var}}(\xi_j)$ is positive. Then, for any $r>0$, $$\sup_{t\in{\mathbb{R}}}\, {\mathbb{P}}\Big(\Big|\max_{1\leq j\leq m} \xi_j -t\Big| \ \leq r \Big) \ \leq \ \frac{2r}{\underline{\sigma}}\cdot (\sqrt{2\log(m)}+2).$$
The result below is a version of Slepian’s lemma, which is adapted from [@Li:Shao:2002 Theorem 2.2]. (See references therein for earlier versions of this result.)
\[lem:slepian\] Let $m\geq 3$, and let $\mathsf{R}\in{\mathbb{R}}^{m\times m}$ be a correlation matrix with $\max_{i\neq j}\mathsf{R}_{i,j}<1$. Also, let $\mathsf{R}^+$ be the matrix with $(i,j)$ entry given by $\max\{\mathsf{R}_{i,j},0\}$, and suppose $\mathsf{R}^+$ is positive semi-definite. Furthermore, let $\zeta\sim N(0,\mathsf{R})$ and $\xi\sim N(0,\mathsf{R}^+)$. Then, the following inequalities hold for any $t\geq 0$, $${\mathbb{P}}\Big(\max_{1\leq j\leq m} \zeta_j \leq t\Big) \ \leq \ {\mathbb{P}}\Big(\max_{1\leq j\leq m} \xi_j\leq t\Big) \ \leq \ K_m(t) \cdot \Phi^m(t),$$ where $$K_m(t)=\exp\Bigg\{\sum_{1\leq i<j\leq m} \log\Big({\textstyle}\frac{1}{1-{\textstyle}\frac{2}{\pi}\arcsin(\mathsf{R}_{i,j}^+)}\Big)\exp\Big(-{\textstyle}\frac{t^2}{1+\mathsf{R}_{i,j}^+}\Big)\Bigg\}.$$
The following inequalities are due to [@Zinn:1985].
\[lem:rosenthal\] Fix $r\geq 1$ and put $\textup{Log}(r):=\max\{\log(r),1\}$. Let $\xi_1,\dots,\xi_m$ be independent random variables satisfying ${\mathbb{E}}[|\xi_j|^r]<\infty$ for all $1\leq j\leq m$. Then, there is an absolute constant $c>0$ such that the following two statements are true.
- If $\xi_1,\dots,\xi_m$ are non-negative random variables, then $$\big\|{\textstyle\sum}_{j=1}^m \xi_i\big\|_r\leq c\cdot {\textstyle}\frac{r}{\textup{Log}(r)} \cdot \max\bigg\{ \big\|{\textstyle\sum}_{j=1}^m \xi_j\big\|_1 \, , \, \big({\textstyle\sum}_{j=1}^m \big\|\xi_i\|_r^r\big)^{1/r}\bigg\}.$$
- If $r>2$, and the random variables $\xi_1,\dots,\xi_m$ all have mean 0, then $$\big\|{\textstyle\sum}_{j=1}^m \xi_i\big\|_r\leq c\cdot {\textstyle}\frac{r}{\textup{Log}(r)} \cdot \max\bigg\{ \big\|{\textstyle\sum}_{j=1}^m \xi_j\big\|_2 \, , \, \big({\textstyle\sum}_{j=1}^m \big\|\xi_i\|_r^r\big)^{1/r}\bigg\}.$$
#### Remark
The non-negative case is handled in [@Zinn:1985 Theorem 2.5]. With regard to the mean 0 case, the statement above differs slightly from [@Zinn:1985 Theorem 4.1], which requires symmetric random variables, but the remark on page 247 of that paper explains why the variables $\xi_1,\dots,\xi_m$ need not be symmetric as long as they have mean 0.\
The result below is a sharpened version of Hoeffding’s inequality for handling the binomial distribution when the success probability is small [@vanderVaart:Wellner:2000 Corollary A.6.3].
\[lem:Kiefer\] Let $\xi_1,\dots,\xi_m$ be independent Bernoulli random variables with success probability $\pi_0\in(0,1/e)$, and let $\bar\xi=\frac{1}{m}\sum_{i=1}^m\xi_i$. Then, the following inequality holds for any $m\geq 1$ and $t>0$, $${\mathbb{P}}\Big(\sqrt{m}|\bar\xi-\pi_0|\geq t\Big) \ \leq \ 2\exp\Big\{-t^2\big[\log\big({\textstyle}\frac{1}{\pi_0}\big)-1\big]\Big\}.$$
Related work on Gaussian approximation {#app:Gaussian}
======================================
Although our focus is primarily on rates of bootstrap approximation, this topic is closely related to rates of Gaussian approximation in the central limit theorem — for which there is a long line of work in finite-dimensional and infinite-dimensional settings. We refer to the chapter [@Bentkus:2000] for a general survey, as well as Appendix L of the paper [@CCK:2013] for a discussion that is oriented more towards high-dimensional statistics.
To describe how our work fits into this literature, we fix some notation. Let $\mathbb{B}$ denote a Banach space, with $\mathscr{A}$ being a collection of its subsets, and $\varphi:\mathbb{B}\to{\mathbb{R}}$ being a function. For the moment, we will regard the summands of $S_n$ as centered i.i.d. random elements of $\mathbb{B}$, and let $\tilde S_n$ denote a centered Gaussian element of $\mathbb{B}$ with the same covariance as $S_n$. Broadly speaking, the literature on rates of Gaussian approximation is concerned with bounding the quantities $\rho_n(\mathscr{A})=\sup_{A\in\mathscr{A}}\rho_n(A)$, or $\|\Delta_n\|_{\infty}=\sup_{t\in{\mathbb{R}}}|\Delta_n(t)|$, where $$\begin{aligned}
\rho_n(A) & \ = \ \Big|{\mathbb{P}}(S_n\in A) - {\mathbb{P}}(\tilde S_n\in A)\Big|,\label{eqn:rhodef}\\[0.2cm]
\Delta_n(t) & \ = \ \Big|{\mathbb{P}}\big(\varphi(S_n)\leq t\big) - {\mathbb{P}}\big(\varphi(\tilde S_n)\leq t\big)\Big|.\label{eqn:deltadef}
\end{aligned}$$ In particular, note that if $\mathbb{B}={\mathbb{R}}^p$ and $\tilde T=\max_{1\leq j\leq p} \tilde S_{n,j}$, then the distance $d_{\text{K}}(\mathcal{L}(T),\mathcal{L}(\tilde T))$ can be represented as either $\rho_n(\mathscr{A})$ or $\|\Delta_n\|_{\infty}$, by taking $\mathscr{A}$ to be a class of rectangles, or by taking $\varphi$ to be the coordinate-wise maximum function. Typically, the rates at which $\rho_n(\mathscr{A})$ and $\|\Delta_n\|_{\infty}$ decrease with $n$ is dependent on the distribution of $S_n$, the dimension of $\mathbb{B}$, as well as the smoothness of $\mathscr{A}$ and $\varphi$, among other factors. Although the study of $\rho_n(\mathscr{A})$ and $\|\Delta_n\|_{\infty}$ is highly multifaceted, it is a general principle that the smoothness of $\mathscr{A}$ and $\varphi$ tends to be much more influential when $\mathbb{B}$ is infinite-dimensional, as compared to the finite-dimensional case. (A discussion may be found in [@Bentkus:1993].) Indeed, this is worth emphasizing in relation to our work, since the choices of $\mathscr{A}$ and $\varphi$ corresponding to $d_{\text{K}}(\mathcal{L}(T),\mathcal{L}(\tilde T))$ are notable for their lack of smoothness.
To illustrate this point, consider the following two facts that are known to hold when $\mathbb{B}$ is separable and infinite-dimensional [@Bentkus:1993]: (1) If ${\mathbb{E}}[\|X_1\|_{\mathbb{B}}^3]<\infty$, and $\varphi$ satisfies smoothness conditions stronger than having three Fréchet derivatives, then $\|\Delta_n\|_{\infty}$ is of order $\mathcal{O}(n^{-1/2})$. (2) If the previous conditions hold, except that $\varphi$ is only assumed to have one Fréchet derivative, then an example can be constructed so that $\|\Delta_n\|_{\infty}$ is bounded below by a sequence that converges to 0 arbitrarily slowly. (Other examples of lower bounds may be found in the papers [@Bentkus:1986; @Talagrand:1984], among others.) By contrast, in the finite-dimensional case where $\mathbb{B}={\mathbb{R}}^p$ with $p$ held fixed as $n\to\infty$ and ${\mathbb{E}}[\|X_1\|_2^3]<\infty$, it is known that non-smooth choices of $\varphi$ and $\mathscr{A}$ can lead to a $n^{-1/2}$ rate. Namely, it is known that $\|\Delta_n\|_{\infty}$ and $\rho_n(\mathscr{A})$ are of order $\mathcal{O}(n^{-1/2})$ when $\varphi$ is convex, or when $\mathscr{A}$ is the class of Borel convex sets. Beyond the case where $p$ is held fixed, many other works allow $n$ and $p$ to diverge together. When $p$ grows relatively slowly compared to $n$, the leading rate for the choices of $\varphi$ and $\mathscr{A}$ just mentioned is essentially $\mathcal{O}(p^{7/4}/n^{1/2})$ [@Bentkus:2003; @Bentkus:2005]. See also [@Sazonov:1968; @Nagaev:1976; @Senatov:1981; @Portnoy:1986; @Gotze:1991; @Chen:2011; @Zhai:2018] for additional background. Meanwhile, when $p\gg n$, it has recently been established that rates of the form $\mathcal{O}(\log(p)^b/n^{1/6})$ can be achieved if $\varphi$ is the coordinate-wise maximum function, or if $\mathscr{A}$ is a class of “sparsely convex sets”, such as rectangles [@CCK:AOP].
In light of the previous paragraph, it is natural that results in the infinite-dimensional setting have focused predominantly on smooth choices of $\mathscr{A}$ and $\varphi$. Nevertheless, there are some special cases where notable results have been obtained for non-smooth choices of $\varphi$ and $\mathscr{A}$ that correspond to max statistics. One such result is established in the paper [@Asriev:1986], which deals with bounds on $\rho_n(A(\mathbf{r}))$ for rectangular sets such as $A(\mathbf{r}):= \prod_{j=1}^{\infty}(-\infty,r_j]$, with $\mathbf{r}=(r_1,r_2,\dots)$, in the infinite-dimensional Euclidean space ${\mathbb{R}}^{\infty}$. More specifically, if we continue to let $\sigma_j^2={\operatorname{var}}(X_{1,j})$ for $j=1,2,\dots$, and define the “effective dimension” parameter $ {\tt{d}}(\mathbf{r})=\sum_{j=1}^{\infty} {\textstyle}\frac{\sigma_j}{\sigma_j+r_j},$ then the following bound holds under certain conditions on the distribution of $S_n$, $$\label{eqn:asriev}
\rho_n(A(\mathbf{r})) \ \lesssim \ {\textstyle}\frac{\log(n)^{3/2}}{n^{1/2}} \cdot {\tt{d}}(\mathbf{r})\cdot(1+\tt{d}(\mathbf{r})^2).$$ To comment on how the bound relates with our Gaussian approximation result in Theorem 3.1, note that they both involve near-parametric rates, and are governed by the parameters $(\sigma_1,\sigma_2,\dots)$. However, there are also some crucial differences. First, the bound is *non-uniform* with respect to the set $A(\mathbf{r})$, whereas Theorem 3.1 is a uniform result. In particular, this difference becomes apparent by setting all $r_j$ equal to $\sigma_{(n)}$, which implies ${\tt{d}}(\mathbf{r})\geq n/2$, and causes the bound diverge as $n\to\infty$. Second, the bound relies on the assumption that $S_n$ has a *diagonal covariance* matrix, whereas Theorem 3.1 allows for much more general covariance structures.
Some further examples related to max statistics arise in the context of empirical process theory. Let ${\mathbb{G}}_n(f)=\sqrt{n}\sum_{i=1}^n (f(X_i)-{\mathbb{E}}[f(X_i)])$ denote an empirical process that is generated from i.i.d. observations $X_1,\dots,X_n$, and indexed by a class of functions $f\in\mathscr{F}$. Also let the Gaussian counterpart of ${\mathbb{G}}_n$ be denoted by $\tilde{{\mathbb{G}}}$ (i.e. a Brownian bridge) [@vanderVaart:Wellner:2000]. In this setting, the paper [@Norvaivsa:1991] studies the quantity $$\label{eqn:Deltanew}
\Delta_n(t)=\Big|{\mathbb{P}}\big(\|{\mathbb{G}}_n\|_{\mathscr{F}}\leq t\big) - {\mathbb{P}}\big(\|\tilde{\mathbb{G}}\|_{\mathscr{F}}\leq t\big)\Big|,$$ which can be understood in terms of the earlier definition by letting $\varphi({\mathbb{G}}_n)=\|{\mathbb{G}}_n\|_{\mathscr{F}}=\sup_{f\in\mathscr{F}}|{\mathbb{G}}_n(f)|$. Under the assumption that $\mathscr{F}$ is a VC subgraph class of uniformly bounded functions, it is shown in the paper [@Norvaivsa:1991] that the bound $$\label{eqn:empbound1}
\Delta_n(t) \ \lesssim \ {\textstyle}\frac{1}{(1+t)^3}{\textstyle}\frac{\log(n)^2}{n^{1/6}}$$ holds for all $t\geq 0$. In relation to the current work, the condition that the functions in $\mathscr{F}$ are uniformly bounded is an important distinction, because the max statistic $T=\max_{1\leq j\leq p} {\mathbb{G}}_n(f_j)$ arises from the functions $f_j(x)=x_j$, which are unbounded.
More recently, the papers [@CCK:suprema; @CCK:SPA] have derived bounds on the quantity $\|\Delta_n\|_{\infty}$ (as well as coupling probabilities) under weaker assumptions on the class $\mathscr{F}$. For instance, these works allow for classes of functions that are non-Donsker or unbounded, provided that a suitable envelope function is available. Nevertheless, the resulting bounds on $\|\Delta_n\|_{\infty}$ involve restrictions on how quickly the parameter $\inf_{f\in\mathscr{F}}{\operatorname{var}}({\mathbb{G}}_n(f))$ can decrease with $n$ — whereas our results do not involve such restrictions. Also, the rates developed in these works are broadly similar to , and it is unclear if a modification of the techniques would lead to near-parametric rates in our setting. Lastly, a number of earlier results that are related to bounding $\|\Delta_n\|_{\infty}$, such as invariance principles and couplings, can be found in the papers [@Csorgo:1986; @Massart:1986; @Massart:1989; @Paulauskas:Stieve:1990; @Bloznelis:1997 and references therein]. However, these works are tailored to fairly specialized processes (e.g., dealing with càdlàg functions on the unit interval, rectangles in ${\mathbb{R}}^p$ when $p$ is fixed, or processes generated by Uniform\[0,1\] random variables).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The relativistic proton-neutron quasiparticle random phase approximation (PN-RQRPA) is applied in the calculation of $\beta$-decay half-lives of neutron-rich nuclei in the $Z\approx 28$ and $Z\approx 50$ regions. The study is based on the relativistic Hartree-Bogoliubov calculation of nuclear ground-states, using effective Lagrangians with density-dependent meson-nucleon couplings, and also extended by the inclusion of couplings between the isoscalar meson fields and the derivatives of the nucleon fields. This leads to a linear momentum dependence of the scalar and vector nucleon self-energies. The residual QRPA interaction in the particle-hole channel includes the $\pi + \rho$ exchange plus a Landau-Migdal term. The finite-range Gogny interaction is employed in the $T=1$ pairing channel, and the model also includes a proton-neutron particle-particle interaction. The results are compared with available data, and it is shown that an extension of the standard relativistic mean-field framework to include momentum-dependent nucleon self-energies naturally leads to an enhancement of the effective (Landau) nucleon mass, and thus to an improved PN-QRPA description of $\beta^-$-decay rates.'
author:
- 'T. Marketin'
- 'D. Vretenar'
- 'P. Ring'
title: 'Calculation of $\beta$-decay rates in a relativistic model with momentum-dependent self-energies'
---
Weak-interaction processes in exotic nuclei far from stability play an important role in stellar explosive events. In particular, $\beta$-decay rates of very neutron-rich nuclei set the time scale of the $r$-process nucleosynthesis, i.e. the multiple neutron capture process which determines the synthesis of nearly half of the nuclei heavier than Fe. Since the vast majority of nuclides which lie on the path of the $r$-process will not be experimentally accessible in the foreseeable future, it is important to develop microscopic nuclear structure models that can provide accurate predictions of weak-interaction rates of thousands of nuclei with large neutron to proton asymmetry. There are basically two microscopic approaches that can be employed in large-scale calculations of $\beta$-decay rates: the interacting shell model and the quasiparticle random phase approximation (QRPA). While the advantage of using the shell model is the ability to take into account the detailed structure of the $\beta$-strength function [@LM.03], the QRPA approach is based on global effective interactions and provides a systematic description of $\beta$-decay properties of arbitrarily heavy nuclei along the $r$-process path [@Bor.06]. In a recent review of modern QRPA calculations of $\beta$-decay rates for astrophysical applications [@Bor.06], Borzov has emphasized the importance of performing calculations based on self-consistent mean-field models, rather than on empirical mean-field potentials, e.g. the Woods-Saxon potential. In a self-consistent framework both the nuclear ground states, i.e. the masses which determine the possible $r$-process path, and the corresponding $\beta$-decay properties are calculated from the same energy density functional or effective nuclear interaction. This approach ensures the consistency of the nuclear structure input for astrophysical modeling, and allows reliable extrapolations of the nuclear spin-isospin response to regions of very neutron-rich nuclei.
The fully consistent proton-neutron (PN) relativistic QRPA [@VPNR.03; @Paa.04] has recently been employed in the calculation of $\beta$-decay half-lives of neutron-rich nuclei in the N$\approx$50 and N$\approx$82 regions [@NMV.05]. The model is based on the relativistic QRPA [@Paa.03], formulated in the canonical basis of the relativistic Hartree-Bogoliubov (RHB) framework [@VALR.05]. The RHB+RQRPA model is fully self-consistent. For the interaction in the particle-hole channel modern effective Lagrangians with density-dependent meson-nucleon couplings are used, and pairing correlations are described by the pairing part of the finite range Gogny interaction. Both in the particle-hole ($ph$) and particle-particle ($pp$) channels, the same interactions are used in the RHB equations which determine the nuclear ground-state, and in the matrix equations of the RQRPA. This is important because the energy weighted sum rules are only satisfied if the pairing interaction is consistently included both in the static RHB and in the dynamical RQRPA calculations. In both channels the same strength parameters of the interactions are used in the RHB and RQRPA calculations. The formulation of the RHB+RQRPA model in the canonical quasiparticle basis enables the description of weakly-bound neutron-rich nuclei far from stability, because this basis diagonalizes the density matrix and includes both the bound states and the positive-energy single-particle continuum [@Paa.03].
In the corresponding proton-neutron RQRPA [@VPNR.03; @Paa.04] the spin-isospin-dependent interaction terms are generated by the $\pi$- and $\rho$-meson exchange. Although the direct one-pion contribution to the nuclear ground state vanishes at the mean-field level because of parity conservation, it must be included in the calculation of spin-isospin excitations. In addition, the derivative type of the pion-nucleon coupling necessitates the inclusion of the zero-range Landau-Migdal term, which accounts for the contact part of the nucleon-nucleon interaction, with the strength parameter $g^{\prime}$ adjusted to reproduce experimental data on the excitation energies of Gamow-Teller resonances (GTR). The model also includes the $T=0$ proton-neutron pairing interaction: a short-range repulsive Gaussian function combined with a weaker longer-range attractive Gaussian [@Eng.99]. In general the calculated $\beta$-decay half-lives are very sensitive to the strength of the $T=0$ pairing which, in the case of $\beta^-$-decay, enhances the Gamow-Teller strength in the $Q_{\beta^-}$ energy window.
Standard relativistic mean-field models are based on the static approximation, i.e. the nucleon self-energy is real, local and energy-independent. Consequently, these models describe correctly the ground-state properties and the sequence of single-particle levels in finite nuclei, but not the level density around the Fermi surface. The reason is the low effective nucleon mass $m^*$ which, in the relativistic framework, is also related to the Dirac mass $m_D = m + S(\bm{r})$, where $m$ is the bare nucleon mass and $S(\bm{r})$ denotes the scalar nucleon self-energy, and thus constrained by the empirical spin-orbit energy splittings. The difference between the vector and scalar nucleon self-energies determines the spin-orbit potential, whereas their sum defines the effective single-nucleon potential, and is constrained by the nuclear matter binding energy at saturation density. The energy spacings between spin-orbit partner states in finite nuclei, and the nuclear matter binding and saturation, place the following constraints on the values of the Dirac mass and the nucleon effective mass: $0.55 m \le m_D \le 0.6 m$, $0.64
m \le m^* \le 0.67 m$, respectively. These values have been used in most standard relativistic mean-field effective interactions. However, when these interactions are used in the calculation of $\beta^-$ decay rates, the resulting half-lives are usually more than an order of magnitude longer than the empirical values. This is because the low effective nucleon mass implies a low density of states around the Fermi surface, and therefore in a self-consistent relativistic QRPA calculation of $\beta$-decay the transition energies will be low, resulting in long lifetimes. In order to reproduce the empirical half-lives, it is thus necessary to employ relativistic effective interactions with higher values of the nucleon effective mass. We note that in the case of non-relativistic global effective interactions such as, for instance, Skyrme-type interactions, calculation of ground-state properties and excitation energies of quadrupole giant resonances have shown that a realistic choice for the nucleon effective mass is in the interval $m^*/m = 0.8\pm 0.1$ [@Rei.99; @Cha.97].
In Ref. [@NMV.05] we have used the RHB+RQRPA model to calculate $\beta$-decay half-lives of neutron-rich nuclei in the N$\approx$50 and N$\approx$82 regions. Starting from the standard density-dependent effective interaction DD-ME1 [@NVFR.02] ($m_D = 0.58\,m$, $m^*=0.66\,m$), a new effective interaction was adjusted with higher values for the Dirac mass and the nucleon effective mass: $m_D = 0.67\,m$, $m^* = 0.76\, m$. However, a standard RMF interaction with such a high value of the Dirac mass would systematically underestimate the empirical spin-orbit splittings in finite nuclei. To compensate the reduction of the effective spin-orbit potential caused by the increase of the Dirac mass, the DD-ME1 interaction was further extended by including an additional interaction term: the tensor coupling of the $\omega$-meson to the nucleon. The resulting interaction was used in the relativistic Hartree-Bogoliubov calculation of nuclear ground states. With the Gogny D1S interaction in the $T=1$ pairing channel, and also including the $T=0$ particle-particle interaction in the PN-QRPA, it was possible on one hand to reproduce the empirical values of the energy spacings between spin-orbit partner states in spherical nuclei, and on the other hand the calculated $\beta$-decay half-lives were in reasonable agreement with the experimental data for the Fe, Zn, Cd, and Te isotopic chains.
With the model developed in Ref. [@NMV.05] the problems of the low effective mass and long $\beta$-decay half-lives were solved on an ad hoc basis. The effective interaction was adjusted with the particular purpose of increasing the effective nucleon mass, and the resulting problem of the reduction of the effective spin-orbit potential was solved by the inclusion of an additional interaction term. A much better solution is provided by the recently introduced relativistic mean-field model with momentum-dependent nucleon self-energies [@Typ.03; @Typ.05]. In this model the standard effective Lagrangian with density-dependent meson-nucleon coupling vertices is extended by including a particular form of the couplings between the isoscalar meson fields and the derivatives of the nucleon fields. This leads to a linear momentum dependence of the scalar and vector self-energies in the Dirac equation for the in-medium nucleon. Even though the extension of the standard mean-field framework is phenomenological, it is nevertheless based on Dirac-Brueckner calculations of in-medium nucleon self-energies, and consistent with the relativistic optical potential in nuclear matter, extracted from elastic proton-nucleus scattering data. In the extended model it is possible to increase the effective nucleon mass, while keeping a small Dirac mass which is required to reproduce the empirical strength of the effective spin-orbit potential.
In the very recent work of Ref. [@Typ.05], in particular, an improved Lagrangian density of the model with density-dependent and derivative couplings (D$^3$C) has been introduced. The parameters of the coupling functions were adjusted to ground-state properties of eight doubly-magic spherical nuclei, and the results for nuclear matter, neutron matter, and finite nuclei were compared to those obtained with conventional RMF models. It was shown that the new effective interaction improves the description of binding energies, nuclear shapes and spin-orbit splittings of single-particle levels. More important, it was possible to increase the effective nucleon mass ($m^* = 0.71\,m$) and, correspondingly, the density of single-nucleon levels close to the Fermi surface as compared to standard RMF models. At the same time the Dirac mass was kept at the small value $m_D = 0.54m$, which ensures that the model reproduces the empirical spin-orbit splittings. The momentum dependence of the nucleon self-energies provides also a correct description of the empirical Schroedinger-equivalent central optical potential.
In this work we employ the model with density-dependent and derivative couplings (D$^3$C) of Ref. [@Typ.05] in the calculation of $\beta$-decay rates of neutron-rich nuclei in several isotopic chains in the $Z \approx 28$ and $Z \approx 50$ regions. The results of fully consistent RHB plus proton-neutron QRPA will be compared with those obtained with the standard density-dependent RMF interaction DD-ME1 and, in addition, with a new effective interaction based on the D$^3$C model, but with an even higher value of the effective nucleon mass. We will analyze the dependence of the $\beta$-decay half-lives on the choice of the effective particle-hole interaction, and the strength of the $T=0$ pairing interaction.
The functional forms of the density dependence of the $\sigma$, $\omega$ and $\rho$ meson-nucleon couplings are identical for the conventional DD-ME1 effective interaction and the D$^3$C model. The latter includes momentum-dependent isoscalar scalar and vector self-energies, and thus contains two additional coupling functions $\Gamma_{S}$ and $\Gamma_{V}$. In Ref. [@Typ.05] these have been parametrized with the following functional form: $$\Gamma_{i}(x) = \Gamma_{i}(\rho_{ref}) x^{-a_{i}} \quad \textrm{for} \quad i=S, V \; ,$$ where $x = \rho_v / \rho_{ref}$, $\rho_v$ is the vector density, and the reference density $\rho_{ref}$ corresponds to the vector density determined at the saturation point of symmetric nuclear matter. In the parameterization of Ref. [@Typ.05] $a_S = a_V = 1$, and we will retain these values in the following calculation. The parameters $ \Gamma_{S}(\rho_{ref})$ and $\Gamma_{V}(\rho_{ref})$ have been constrained by the requirement that the resulting optical potential in symmetric nuclear matter at saturation density has the value 50 MeV at a nucleon energy of 1 GeV. In total there are 10 adjustable parameters in the D$^3$C model, compared to eight for the standard density-dependent RMF models, e.g. the DD-ME1 parameterization.
The effective nucleon mass of the D$^3$C model is $m^* = 0.71\, m$, compared to $m^* = 0.66 \,m$ for DD-ME1. In addition, starting from D$^3$C, for the purpose of calculating $\beta$-decay rates we have adjusted a new parameterization with $m^* = 0.79\, m$, which is much closer to the effective masses used in non-relativistic Skyrme effective interactions [@Rei.99; @Cha.97]. The new effective interaction which, for simplicity we denote D$^3$C, has been adjusted following the original procedure of Ref. [@Typ.05], with an additional constraint on the effective nucleon mass. Even though we have tried to increase the effective mass as much as possible, $m^* = 0.79\, m$ is the highest value for which a realistic description of nuclear matter and finite nuclei is still possible, and the quality of the calculated nuclear matter equation of state and of ground-state properties of spherical nuclei is comparable to that of the DD-ME1 and D$^3$C interactions. The three interactions are compared in Table \[TabA\], where we include the characteristics of the corresponding nuclear matter equations of state at saturation point: the saturation density $\varrho_{\rm sat}$, the binding energy per particle $a_{V}$, the symmetry energy $a_{4}$, the nuclear matter compression modulus $K_{\infty}$, the Dirac mass $m_{D}$, and the effective (Landau) mass $m^{*}$. In addition, for the two interactions with energy-dependent single-nucleon potentials, we compare the values of $ \Gamma_{S}(\rho_{ref})$ and $\Gamma_{V}(\rho_{ref})$. We notice a pronounced increase of the strength of the scalar field. This is, however, compensated by the corresponding decrease of the strength of the vector coupling, so that the difference $ \Gamma_{V}(\rho_{ref}) - \Gamma_{S}(\rho_{ref})$ is practically the same for D$^3$C and D$^3$C. For both interactions the optical potential at 1 GeV nucleon energy has been constrained to 50 MeV. With the increase of the effective nucleon mass from DD-ME1 to D$^3$C and D$^3$C$^*$, we also note the corresponding decrease of the nuclear matter compression modulus $K_{\infty}$ . This correlation between $K_{\infty}$ and $m^*$ is also well known in non-relativistic Skyrme effective interactions [@Cha.97].
DD-ME1 D$^{3}$C D$^3$C
------------------------------------- -------- ---------- ----------
$\varrho_{\rm sat}$ \[fm${}^{-3}$\] 0.152 0.151 0.152
$a_{V}$ \[MeV\] -16.20 -15.98 -16.30
$a_{4}$ \[MeV\] 33.1 31.9 33.0
$K_{\infty}$ \[MeV\] 244.5 232.5 224.9
$m_{D}/m$ 0.58 0.54 0.57
$m^{*}/m$ 0.66 0.71 0.79
$\Gamma_{S}$ 0.0 -21.632 -146.089
$\Gamma_{V}$ 0.0 302.188 180.889
: \[TabA\] Properties of symmetric nuclear matter at saturation density calculated with the models DD-ME1, D$^{3}$C, and D$^3$C.
In Fig. \[Fig1\] we display the neutron and proton single-particle levels in $^{132}$Sn calculated in the relativistic mean-field model with the DD-ME1, D$^{3}$C, and D$^3$C effective interactions, in comparison with available data for the levels close to the Fermi surface [@Isa.02]. Compared to the DD-ME1 interaction, the enhancement of the effective mass in D$^{3}$C and D$^3$C results in the increase of the density of states around the Fermi surface, and the calculated spectra are in much better agreement with the empirical energy spacings.
![(Color online) Neutron (left panel) and proton (right panel) single-particle levels in $^{132}\textrm{Sn}$ calculated with the DD-ME1 (a), D$^{3}$C (b) and D$^3$C (c) interactions, compared to experimental levels (d) [@Isa.02].[]{data-label="Fig1"}](Fig1.eps)
In the next step the three effective interactions have been tested and compared in RHB plus proton-neutron relativistic QRPA calculations of $\beta$-decay half-lives for the isotopic chains: Fe, Ni, Zn, Cd, Sn and Te. The nuclear ground-states have been calculated in the RHB model with the DD-ME1, D$^{3}$C, and D$^3$C effective interactions in the particle-hole channel, and the pairing part of the Gogny force, $$V^{pp}(1,2)~=~\sum_{i=1,2}e^{-((\mathbf{r}_{1}-\mathbf{r}_{2})/{\mu _{i}})^{2}}\,(W_{i}~+~B_{i}P^{\sigma }-H_{i}P^{\tau }-M_{i}P^{\sigma }P^{\tau })
\label{Gogny}$$ in the particle-particle channel, with the set D1S [@BGG.91] for the parameters $\mu _{i}$, $W_{i}$, $B_{i} $, $H_{i}$ and $M_{i}$ $(i=1,2)$. This force has been very carefully adjusted to pairing properties of finite nuclei all over the periodic table. In particular, the basic advantage of the Gogny force is the finite range, which automatically guarantees a proper cut-off in momentum space. In the following calculations we have also used the Gogny interaction in the $T=1$ $pp$-channel of the PN-RQRPA.
The RHB ground-state solution determines the single-nucleon canonical basis, i.e. the configuration space in which the matrix equations of the relativistic QRPA are expressed (see Refs. [@Paa.03; @Paa.04] for a detailed presentation of the formalism). The particle-hole residual interaction of the PN-RQRPA is derived from the following Lagrangian density: $$\mathcal{L}_{\pi + \rho}^{int} =
- g_\rho \bar{\psi}\gamma^{\mu}\vec{\rho}_\mu \vec{\tau} \psi
- \frac{f_\pi}{m_\pi}\bar{\psi}\gamma_5\gamma^{\mu}\partial_{\mu}
\vec{\pi}\vec{\tau} \psi \; .
\label{lagrres}$$ The coupling between the $\rho$-meson and the nucleon is already contained in the RHB effective Lagrangian, and the same interaction is consistently used in the isovector channel of the QRPA. The direct one-pion contribution to the ground-state RHB solution vanishes because of parity-conservation, but it must be included in the calculation of the Gamow-Teller strength. For the pseudovector pion-nucleon coupling we have used the standard values: $$m_{\pi}=138.0~{\rm MeV}~~~~\;\;\;\;\frac{\;f_{\pi}^{2}}{4\pi}=0.08\;.$$ In addition, the zero-range Landau-Migdal term accounts for the contact part of the isovector channel of the nucleon-nucleon interaction $$V_{\delta \pi} = g' {\left}( \frac{f_{\pi}}{m_{\pi}} {\right})^{2} \vec{\tau}_{1}
\vec{\tau}_{2} \bm{\Sigma_{1}}\cdot \bm{\Sigma_{2}} \delta{\left}( \bm{r}_{1}-\bm{r}_{2} {\right}) \; .$$ For each effective interaction, the strength parameter $g^\prime$ is adjusted to reproduce the excitation energy of the Gamow-Teller resonance in $^{208}$Pb. In the present calculation these values are: $g^\prime = 0.55, 0.54$ and $0.76$, for DD-ME1, D$^{3}$C, and D$^3$C, respectively.
Finally, the proton-neutron QRPA interaction is completely determined by the choice of the $T=0$ pairing interaction [@Eng.99]: $$\label{eq2}
V_{12}
= - V_0 \sum_{j=1}^2 g_j \; {\rm e}^{-r_{12}^2/\mu_j^2} \;
\hat\Pi_{S=1,T=0}
\quad ,
\label{pn-pair}$$ where $\hat\Pi_{S=1,T=0}$ projects onto states with $S=1$ and $T=0$. The ranges $\mu_1$=1.2fm and $\mu_2$=0.7fm of the two Gaussians are the same as for the Gogny interaction Eq. (\[Gogny\]), and the relative strengths $g_1 =1$ and $g_2 = -2$ are adjusted so that the force is repulsive at small distances. The only remaining free parameter is $V_0$, the overall strength.
The half-life of the $\beta^-$-decay of an even-even nucleus in the allowed Gamow-Teller approximation is calculated from the following expression: $$\label{halflife}
\frac{1}{T_{1/2}}=\sum_m{\lambda_{if}^m} = D^{-1}g_A^2\sum_m
\int{dE_e {\left}| \sum_{pn} <1^+_m||\bm{\sigma}\tau_-||0^+>{\right}|^2
\frac{dn_m}{dE_e}}\;,$$ where $D=6163.4\pm 3.8$ s [@BG.00]. ${\left| 0^+ \right\rangle}$ denotes the ground state of the parent nucleus, and ${\left| 1^+_m \right\rangle}$ is a state of the daughter nucleus. The sum runs over all final states with an excitation energy smaller than the $Q_{\beta^-}$ value. In order to account for the universal quenching of the Gamow-Teller strength function, we have used the effective weak axial nucleon coupling constant $g_A=1$, instead of $g_A=1.26$ [@BM.75]. The kinematic factor in Eq. (\[halflife\]) can be written as $$\label{kinematic}
\frac{dn_m}{dE_e}=E_e\sqrt{E_e^2-m_e^2}(\omega -E_e)^2 F(Z,A,E_e)\;,$$ where $\omega$ denotes the energy difference between the initial and the final state. The Fermi function $F(Z,A,E_e)$ corrects the phase-space factor for the nuclear charge and finite nuclear size effects [@KR.65].
![(Color online) $\beta$-decay half-lives of Fe (left panel), Ni (middle panel), and Zn (right panel) nuclei, calculated with the DD-ME1, D$^{3}$C, and D$^3$C effective interactions, compared with the experimental values [@NUDAT]. Open symbols correspond to PN-QRPA values calculated without the inclusion of the $T=0$ pairing interaction. The filled squares are half-lives calculated with the D$^3$C interaction and $T=0$ pairing, with the strength parameter $V_0 = 125$ MeV for Fe, and $V_0 = 300$ MeV for Zn isotopes.[]{data-label="Fig2"}](Fig2.eps)
In Figure \[Fig2\] we display the $\beta^-$-decay half-lives of iron, nickel and zinc isotopes calculated with the DD-ME1, D$^{3}$C, and D$^3$C, and compare them with the experimental values taken from NUDAT database [@NUDAT]. The data for $^{76}$Ni and $^{78}$Ni are from Ref. [@Hos.05]. Open symbols correspond to values calculated without the inclusion of $T=0$ pairing. Since the $\beta^-$-decay rates are generally very sensitive to the proton-neutron pairing, and its strength is usually adjusted separately for each isotopic chain, we will first discuss the results obtained without the $T=0$ pairing interaction. For all three isotopic chains, the shortest half-lives are obtained with the interaction with the highest effective mass, i.e. D$^3$C, even though these are still far from the experimental values. For the Fe nuclei all three interactions give similar results, whereas more pronounced differences are found for the Ni and Zn isotopic chains. In the two latter cases similar results are obtained with DD-ME1 and D$^{3}$C and, in fact, longer half-lifes are predicted by D$^{3}$C, even though it has a higher effective nucleon mass. Much shorter half-lives for the Ni and Zn nuclei are calculated with the D$^3$C effective interaction. The origin of these large differences in the calculated rates can be understood from Table \[TabB\], where we list the transition energies for the strongest transition in the Zn isotopes with $76 \leq A \leq 82$: $\nu 2p_{1/2} \to \pi 2p_{3/2}$. We note that the transition energies for the DD-ME1 and D$^{3}$C interactions are comparable and, in particular, those calculated with DD-ME1 are slightly larger, resulting in faster $\beta^-$-decay rates. Both interaction predict a $\beta$-stable $^{76}\textrm{Zn}$. On the other hand, the transition energies predicted by the interaction D$^3$C are much larger and, correspondingly, the calculated half-lives are at least an order of magnitude shorter.
DD-ME1 D$^{3}$C D$^3$C
-------------------- -------- ---------- --------
$^{76}\textrm{Zn}$ 0.15 -0.05 1.74
$^{78}\textrm{Zn}$ 0.93 0.72 2.65
$^{80}\textrm{Zn}$ 2.01 1.80 3.69
$^{82}\textrm{Zn}$ 2.69 2.51 4.58
: Transition energies (in MeV) for the strongest transition in the Zn isotopes: $\nu 2p_{1/2} \to \pi 2p_{3/2}$.
\[TabB\]
A similar situation is found in the neutron-rich nuclei in the $Z \approx 50$ region. The calculated half-lives of Cd, Sn and Te nuclei are plotted in Fig. \[Fig3\], in comparison with available data [@NUDAT]. The Cd isotopes, in particular, are calculated as $\beta$-stable with the D$^{3}$C interaction, because the predicted transition energies are smaller than the electron rest mass. Much better results are obtained with the modified interaction D$^3$C, which clearly reproduces the isotopic trend of the half-lives of neutron-rich Cd nuclei. DD-ME1 and D$^{3}$C produce almost identical results for Sn and Te nuclei. Shorter half-lives, especially for Sn, are calculated with D$^3$C, but these are still orders of magnitude from the experimental values. It appears that all three interactions reproduce the isotopic trend in the Te chain.
![(Color online) Same as in Fig. \[Fig2\], but for the Cd (left panel), Sn (middle panel), and Te (right panel) isotopic chains. For the D$^3$C effective interaction, in all three isotopic chains the strength of the $T=0$ pairing interaction is $V_0 = 235$ MeV (filled squares).[]{data-label="Fig3"}](Fig3.eps)
We have considered the effect of the $T=0$ pairing interaction on the calculated $\beta$-decay half-lives only for the D$^3$C effective interaction which, with the effective nucleon mass $m^*/m = 0.79$ comparable to those of non-relativistic effective interactions, gives the shortest half-lives. Even without the inclusion of the proton-neutron $pp$ interaction, for the Fe nuclei the calculated half-lives are already close to the experimental values, except for $^{64}$Fe (see Fig. \[Fig2\]). By adjusting the value of the strength parameter of the $T=0$ pairing to $V_0 = 125$ MeV, the PN-QRPA calculation reproduces the $\beta$-decay half-lives of $^{66}$Fe, $^{68}$Fe and $^{70}$Fe (filled squares). In the case of Ni isotopes the $T=0$ interaction in the $pp$-channel is not effective because of the $Z=28$ and $N=40$ closures [@Eng.99; @NMV.05]. The $\pi 1f_{7/2}$ orbit is completely occupied, and the transition $\nu 1f_{5/2} \to \pi 1f_{7/2}$ is thus blocked. The $T=0$ pairing could only have an effect on the $\nu 1g_{9/2} \to \pi 1g_{9/2}$ transition, but the $\pi 1g_{9/2}$ orbital is located high above the Fermi surface. Thus the $T=0$ $pp$ interaction cannot shift the low-energy GT strength and enhance the $\beta$-decay rates. Even using the D$^3$C interaction, the calculated half-lives are an order of magnitude longer than the experimental values.
The principal advantage of the self-consistent approach to the modeling of $\beta$-decay rates is the use of universal (A independent) interactions in the $ph$-channel and, in many cases including the model used in this work, in the $T=1$ $pp$-channel. Unfortunately, this is not possible in the $T=0$ $pp$-channel, for which no information is contained in the ground-state data. The strength of this interaction is adjusted separately for each isotopic chain or, in the best case, a single value of the strength can be used in a certain mass region [@Eng.99; @NMV.05]. It is especially difficult to keep the same strength of the $T=0$ pairing when crossing a closed shell. Thus in going from the Fe to the Zn isotopic chain we had to increase the strength parameter $V_0$ by more than a factor two. The value $V_0 = 300$ MeV has been adjusted to reproduce the half-life of $^{78}$Zn (filled squares in the right panel of Fig. \[Fig2\]) but, even though the calculated values are in qualitative agreement with the data, with the inclusion of the $T=0$ pairing the PN-QRPA results do not reproduce the isotopic dependence of the experimental half-lives. In other words, it was not possible to find a single value of the proton-neutron pairing strength that could reproduce the half-lives of neutron-rich Zn isotopes.
The filled squares in Fig. \[Fig3\] correspond to the half-lives calculated with the D$^3$C effective interaction, the $\pi + \rho$ plus Landau-Migdal interaction in the $ph$-channel, the Gogny interaction Eq. (\[Gogny\]) in the $T=1$ $pp$-channel, and the $T=0$ pairing Eq. (\[pn-pair\]). The strength of the latter: $V_0=235$ MeV, has been adjusted to the half-life of $^{128}$Cd, and this value has been used for the Cd, Sn and Te isotopic chains. The effect of the $T=0$ pairing is especially pronounced for Cd and Te nuclei, and the results are in qualitative agreement with the available data, although the calculation does not reproduce the isotopic trend for the Cd chain, and overestimates the half-lives of Te isotopes. On the other hand, for the proton closed-shell Sn nuclei the $T=0$ pairing interaction is much less effective, and the calculated half-lives of $^{134}$Sn and $^{136}$Sn are two orders of magnitude longer than the experimental values. Better results could be obtained, of course, by adjusting $V_0$ separately for each isotopic chain.
The calculations performed in this work have shown that the extension of the standard relativistic mean-field framework to include momentum-dependent (energy-dependent in stationary systems) nucleon self-energies naturally leads to an enhancement of the effective (Landau) nucleon mass, and thus to an improved PN-QRPA description of $\beta^-$-decay rates. However, even though the momentum-dependent RMF model with density-dependent meson-nucleon couplings, adjusted here to $m^* = 0.79\, m$, predicts half-lives of neutron-rich medium-mass nuclei in qualitative agreement with data, the results are not as good as those obtained in the most advanced non-relativistic self-consistent density-functional plus continuum-QRPA calculations [@Bor.03; @Bor.05; @Bor.06], or with the self-consistent HFB+QRPA model with Skyrme interactions of Ref. [@Eng.99]. Namely, although we have been able to increase the effective mass of the interaction used in the RHB calculations of nuclear ground states to $m^* = 0.79\, m$, a value which is sufficient for the description of giant resonances [@Rei.99; @Cha.97], the detailed description of the low-energy Gamow-Teller strength necessitates an even higher value of $m^*$. In fact, the effective mass of the Skyrme SkO’ interaction used in Ref. [@Eng.99] is $m^* = 0.9\, m$, whereas the continuum-QRPA calculations by Borzov are based on the Fayans phenomenological density functional with the bare nucleon mass, i.e. $m^* = m$ [@Bor.03; @Bor.05; @Bor.06]. However, it would be very difficult to further increase the effective nucleon mass in the framework of the model used in this work, i.e. on the nuclear matter level, without destroying the good agreement with empirical ground-state properties of finite nuclei. On the other hand, this would not even be the correct procedure because the additional enhancement of the effective nucleon mass is due to the coupling of single-nucleon levels to low-energy collective vibrational states, an effect which goes entirely beyond the mean-field approximation and is not included in the present model. In principle, the effect of two- and three-phonon states on the weak-interaction rates could be taken into account by explicitly considering the coupling of single-quasiparticle states to phonons, and the resulting complex configurations would certainly lead to a redistribution of low-energy Gamow-Teller strength. Even though such extended (second) RPA approaches have been routinely used for many years in the calculation of widths of isoscalar and isovector giant resonances, no systematic large-scale calculations of $\beta$-decay rates have been reported so far. We have therefore started to develop a new self-consistent model based on the recently introduced covariant theory of particle-vibration coupling [@LR.06], and this framework will be applied in the calculation of $\beta$-decay half-lives of neutron-rich medium mass nuclei.
In heavier nuclei, or in nuclei with an even higher neutron to proton asymmetry, in addition to allowed Gamow-Teller transitions, first-forbidden transitions must be taken into account in the calculation of $\beta$-decay half-lives. As it has been shown in recent studies by Borzov using the density-functional plus continuum-QRPA framework [@Bor.03; @Bor.05; @Bor.06], the first-forbidden decays have a pronounced effect on the $\beta$-decay characteristics of $r$-process nuclei in the $Z\approx 28$, $N>50$; $Z\geq 50$, $N>82$; and $Z = 60 - 70$, $N \approx 126$ regions. For studies of weak-interaction rates in $r$-process nuclei very far from stability, it will therefore be important to include first-forbidden transitions in the relativistic PN-QRPA model.
This work has been supported in part by the Bundesministerium für Bildung und Forschung - project 06 MT 246, by the Gesellschaft für Schwerionenforschung GSI - project TM-RIN. T. Marketin and D. Vretenar would like to acknowledge the support from the Alexander von Humboldt - Stiftung.
[999]{}
K. Langanke and G. Mart' inez-Pinedo, Rev. Mod. Phys. 75, 819 (2003).
I.N. Borzov, Nucl. Phys. A 777, 645 (2006).
D. Vretenar, N. Paar, T. Nikši' c, and P. Ring, Phys. Rev. Lett. 91, 262502 (2003).
N. Paar, T. Nikši' c, D. Vretenar, and P. Ring Phys. Rev. C 69, 054303 (2004).
T. Nikši' c, T. Marketin, D. Vretenar, N. Paar, and P. Ring, Phys. Rev. C 71, 014308 (2005).
N. Paar, P. Ring, T. Nikši' c, and D. Vretenar, Phys. Rev. C 67, 034312 (2003).
D. Vretenar, A. V. Afanasjev, G. A. Lalazissis, P. Ring, Phys. Rep. 409, 101 (2005).
J. Engel, M. Bender, J. Dobaczewski, W. Nazarewicz, and R. Surman, Phys. Rev. C 60, 014302 (1999).
P.-G. Reinhard, Nucl. Phys. A 649, 305c (1999).
E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R. Schaeffer, Nucl. Phys. A 627, 710 (1997); ibid. 635, 231 (1998).
T. Nikši' c, D. Vretenar, P. Finelli, and P. Ring, Phys. Rev. C 66, 024306 (2002).
S. Typel, T. v. Chossy, and H.H. Wolter, Phys. Rev. C 67, 034002 (2003).
S. Typel, Phys. Rev. C 71, 064301 (2005).
V. I. Isakov, K. I. Erokhina, H. Mach, M. Sanchez-Vega, and B. Fogelberg, Eur. Phy. J. A 14, 29 (2002).
J. F. Berger, M. Girod, and D. Gogny, Comp. Phys. Comm. 63, 365 (1991).
I.N. Borzov and S. Goriely, Phys. Rev. C 62, 035501 (2000).
A. Bohr and B.R. Mottelson, Nuclear Structure (Benjamin, New York, 1975), Voll II.
E.J. Konopinski and M.E. Rose, in $\alpha$-, $\beta$-, and $\gamma$-ray spectroscopy, ed. by K. Siegbahn (North-Holland Pub. Co., Amsterdam, 1965) p. 1327.
NUDAT database, National Nuclear Data Center, http://www.nndc.bnl.gov/nndc/nudat/
P.T. Hosmer [*et al.*]{}, Phys. Rev. Lett. 94, 112501 (2005).
I.N. Borzov, Phys. Rev. C 67, 025802 (2003).
I.N. Borzov, Phys. Rev. C 71, 065801 (2005).
E. Litvinova and P. Ring, Phys. Rev. C 73, 044328 (2006).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
#### Abstract.
We define piecewise-linear and birational analogues of the toggle-involutions on order ideals of posets studied by Striker and Williams and use them to define corresponding analogues of rowmotion and promotion that share many of the properties of combinatorial rowmotion and promotion. Piecewise-linear rowmotion (like birational rowmotion) admits an alternative definition related to Stanley’s transfer map for the order polytope; piecewise-linear promotion relates to Sch[ü]{}tzenberger promotion for semistandard Young tableaux. The three settings for these dynamical systems (combinatorial, piecewise-linear, and birational) are intimately related: the piecewise-linear operations arise as tropicalizations of the birational operations, and the combinatorial operations arise as restrictions of the piecewise-linear operations to the vertex-set of the order polytope. In the case where the poset is of the form $[a] \times [b]$, we exploit a reciprocal symmetry property recently proved by Grinberg and Roby to show that birational rowmotion (and consequently piecewise-linear rowmotion) is of order $a+b$. This yields a new proof of a theorem of Cameron and Fon-der-Flaass. Our proofs make use of the correspondence between rowmotion and promotion orbits discovered by Striker and Williams, which we make more concrete. We also prove some homomesy results, showing that for certain functions $f$, the average value of $f$ over each rowmotion/promotion orbit is independent of the orbit chosen.
address: 'Department of Mathematical Sciences, UMass Lowell, USA'
author:
- 'David Einstein[^1]'
- 'James Propp[^2]'
bibliography:
- 'fpsac.bib'
nocite: '[@*]'
title: 'Piecewise-linear and birational toggling'
---
[Note:]{} This is essentially a synopsis of the longer article-in-progress [@einsteinpropp]. It was prepared for FPSAC 2014, and will appear along with the other FPSAC 2014 extended abstracts in a special issue of the journal Discrete Mathematics and Theoretical Computer Science.
\[theorem\][Lemma]{} \[theorem\][Corollary]{} \[theorem\][Proposition]{} \[theorem\][Fact]{} \[theorem\][Observation]{} \[theorem\][Claim]{}
\[theorem\][Definition]{} \[theorem\][Example]{} \[theorem\][Conjecture]{} \[theorem\][Open Problem]{} \[theorem\][Problem]{} \[theorem\][Question]{}
\[theorem\][Remark]{} \[theorem\][Note]{}
\#1\#2[(-2pt[\#1\#2]{} -2pt)]{}
[Background]{} \[sec:background\]
We assume readers are familiar with the definition of a finite poset $(P, \leq)$, as for instance given in Ch. 3 of [@stanley2011enumerative]. Much of our work involves the case $P = [a] \times [b] =
\{(i,j) \in {\naturals}\times {\naturals}: \ 1 \leq i \leq a, \ 1 \leq j \leq b\}$ with ordering defined by $(i,j) \leq (i',j')$ iff $i \leq i'$ and $j \leq j'$. We put $n=a+b$.
We write $x {\lessdot}y$ (“$x$ is covered by $y$”) or equivalently $y {\gtrdot}x$ (“$y$ covers $x$”) when $x < y$ and no $z \in P$ satisfies $x < z < y$. We say $P$ is [*ranked*]{} if there is a function $r: P \rightarrow \{0,1,2,\dots\}$ so that all minimal elements of $P$ have rank 0 and such that $x {\lessdot}y$ implies $r(x) = r(y)-1$.
An [*rc-embedding*]{} of a poset $P$ is defined by [@prorow] as a map $\pi: P \rightarrow {\integers}\times {\integers}$ such that $x$ covers $y$ iff $\pi(x)-\pi(y)$ is $(1,1)$ or $(-1,1)$. This yields a Hasse diagram for $P$ in which all covering relations are edges of slope $\pm 1$. In the case $P = [a] \times [b]$, we will adopt the rc-embedding $\pi$ that sends $(i,j) \in P$ to $(j-i,i+j-2) \in {\integers}^2$. The [*ranks*]{} (or, in the terminology of [@prorow], [*rows*]{}) are the subsets of $P$ that consist of all $x \in P$ at a given height, or vertical position, relative to the rc-embedding. We define the [*files*]{} (or, in the terminology of [@prorow], [*columns*]{}) as the subsets of $P$ that consist of all $x \in P$ at a given horizontal position relative to the rc-embedding. For example, let $P = [2] \times [2]$, and write $(1,1),(2,1),(1,2),(2,2)$ as $w,x,y,z$ for short, with $w < x < z$ and $w < y < z$. Our rc-embedding has $\pi(w) = (0,0)$, $\pi(x) = (-1,1)$, $\pi(y) = (1,1)$, and $\pi(z) = (0,2)$.
(-1,-.5)(1,2.5) (0,0)(-1,1) (0,0)(1,1) (-1,1)(0,2) (1,1)(0,2) (0,0)[.05]{} (0,2)[.05]{} (-1,1)[.05]{} (1,1)[.05]{} (0,-.2)[$w$]{} (0,2.2)[$z$]{} (-1.2,1)[$x$]{} (1.2,1)[$y$]{}
The ranks are $\{w\}$, $\{x,y\}$, and $\{z\}$, and the files are $\{x\}$, $\{w,z\}$, and $\{y\}$. We number the ranks of $[a] \times [b]$ from 0 (bottom) to $n-1$ (top), and we number the files of $[a] \times [b]$ from 1 (left) to $n$ (right). That is, for $P = [a] \times [b]$, $(i,j) \in P$ belongs to the $(i+j-2)$nd rank of $P$ and the $(j-i+a)$th[^3] file of $P$.
We call $S \subseteq P$ an [*order ideal*]{} (or [*downset*]{}) of $P$ when $x \in S$ and $y \leq x$ imply $y \in S$. We call $S \subseteq P$ a [*filter*]{} (or [*upset*]{}) of $P$ when $x \in S$ and $y \geq x$ imply $y \in S$. We call $S \subseteq P$ an [*antichain*]{} when $x, y \in S$ and $x \neq y$ imply that $x$ and $y$ are [*incomparable*]{} (i.e., neither $x \leq y$ nor $y \leq x$). The sets consisting of the order ideals, filters, and antichains of $P$ are respectively denoted by $J(P)$, ${{\mathcal{F}}}(P)$, and ${{\mathcal{A}}}(P)$.
There are natural bijections $\alpha_1: J(P) \rightarrow {{\mathcal{F}}}(P)$, $\alpha_2: {{\mathcal{F}}}(P) \rightarrow {{\mathcal{A}}}(P)$, and $\alpha_3: {{\mathcal{A}}}(P) \rightarrow J(P)$ given by the following recipes:
for $I \in J(P)$, let $\alpha_1(I)$ be the complement $P \setminus I$;
for $F \in {{\mathcal{F}}}(P)$, let $\alpha_2(F)$ be the set of minimal elements of $F$ (i.e., the set of $x \in F$ such that $y < x$ implies $y \not\in F$); and
for $A \in {{\mathcal{A}}}(P)$, let $\alpha_3(A)$ be the downward saturation of $A$ (i.e., the set of $y \in P$ such that $y \leq x$ for some $x \in A$).
The composition ${\rho}:= \alpha_3 \circ \alpha_2 \circ \alpha_1: J(P) \rightarrow J(P)$ is not the identity map; e.g., it sends the full order ideal $I=P$ to the empty order ideal $I=\eset$. (Note that [@brouwerschrijver] studied the closely related map $F = \alpha_2 \circ \alpha_1 \circ \alpha_3: {{\mathcal{A}}}(P) \rightarrow {{\mathcal{A}}}(P)$.)
[@cameron1995orbits] gave an alternative characterization of ${\rho}$. Given $x \in P$ and $I \in J(P)$, let $\tau_x(I)$ (“$I$ toggled at $x$” in Striker and Williams’ terminology) denote the set $I {\bigtriangleup}\{x\}$ if this set is in $J(P)$ and $I$ otherwise. Equivalently, $\tau_x(I)$ is $I$ unless $y \in I$ for all $y {\lessdot}x$ and $y \not\in I$ for all $y {\gtrdot}x$, in which case $\tau_x(I)$ is $I {\bigtriangleup}\{x\}$. (We will sometimes say that “toggling $x$ turns $I$ into $\tau_x(I)$”.) Clearly $\tau_x$ is an involution. It is easy to show that $\tau_x$ and $\tau_y$ commute unless $x {\lessdot}y$ or $x {\gtrdot}y$. If $x_1,x_2,\dots,x_{|P|}$ is any [*linear extension*]{} of $P$ (that is, a listing of the elements of $P$ such that $x_i < x_j$ in $P$ implies $i<j$ in $\naturals$), then the composition $\tau_{x_1} \circ \tau_{x_2} \circ \cdots \circ \tau_{x_{|P|}}$ coincides with ${\rho}$. In the case where the poset $P$ is [*ranked*]{}, one natural way to linearly extend $P$ is to list the elements in order of increasing rank. Given the right-to-left order of composition of $\tau_{x_1} \circ \tau_{x_2} \circ \cdots \circ \tau_{x_{|P|}}$, this corresponds to toggling the top rank first, then the next-to-top rank, and so on, lastly toggling the bottom rank. When $x$ and $y$ belong to the same rank of $P$, the toggle operations $\tau_x$ and $\tau_y$ commute, so even without using the theorem of Cameron and Fon-der-Flaass, we can see that this composite operation on $J(P)$ is well-defined. [@prorow] use the term “row” as a synonym for “rank”, and they refer to ${\rho}$ as [*rowmotion*]{}.
In the example above, under the action of $\alpha_1$, $\alpha_2$, and $\alpha_3$, the order ideal $\{w,x\} \in J(P)$ gets successively mapped to $\{y,z\} \in {{\mathcal{F}}}(P)$, $\{y\} \in {{\mathcal{A}}}(P)$, and $\{w,y\} \in J(P)$. Under the action of $\tau_z$, $\tau_y$, $\tau_x$, and $\tau_w$, the order ideal $\{w,x\} \in J(P)$ gets successively mapped to $\{w,x\}$, $\{w,x,y\}$, $\{w,y\}$, and $\{w,y\}$ (all in $J(P)$). In both cases we obtain ${\rho}(\{w,x\}) = \{w,y\}$.
Note that if $x$ and $y$ belong to the same file, the toggle operations $\tau_x$ and $\tau_y$ commute, since neither of $x,y$ can cover the other. Thus the composite operation of toggling the elements of $P$ from left to right is well-defined; [@prorow] call this operation [*promotion*]{}, and show that it is conjugate to rowmotion in the toggle group (the group generated by the toggle involutions). We denote this map by ${\pi}$.
[Piecewise-linear toggling]{} \[sec:PL\]
Given a poset $P = \{x_1,\dots,x_p\}$ (with $p=|P|$) and an rc-embedding of $P$, let ${\reals}^P$ denote the set of functions $f : P \rightarrow {\reals}$; we can represent such an $f$ as a [*$P$-array*]{} (or [*array*]{} for short) in which the values of $f(x)$ for all $x \in P$ are arranged on the page according to the rc-embedding of $P$ in the plane. We will sometimes identify ${\reals}^P$ with ${\reals}^{p}$, associating $f \in {\reals}^P$ with $v = (f(x_1),\dots,f(x_p))$, though this depends on the chosen ordering of the elements of $P$. Let $\widehat{P}$ denote the augmented poset obtained from $P$ by adding two extra elements $\widehat{0}$ and $\widehat{1}$ (which we sometimes denote by $x_0$ and $x_{p+1}$) satisfying $\widehat{0} < x < \widehat{1}$ for all $x \in P$. The [*order polytope*]{} ${{\mathcal{O}}}(P) \subset {\reals}^p$ (see [@stanley86]) is the set of vectors $(\widehat{f}(x_1),\dots,\widehat{f}(x_p))$ in ${\reals}^{p}$ arising from functions $\widehat{f} : \widehat{P} \rightarrow {\reals}$ that satisfy $\widehat{f}(\widehat{0}) = 0$ and $\widehat{f}(\widehat{1}) = 1$ and are [*order-preserving*]{} ($x \leq y$ in $P$ implies $\widehat{f}(x) \leq \widehat{f}(y)$ in ${\reals}$). In some cases it is better to work with the augmented vector $(\widehat{f}(x_0),\widehat{f}(x_1),\dots,\widehat{f}(x_p),\widehat{f}(x_{p+1}))$ in ${\reals}^{p+2}$. In either case we have a convex compact polytope.
For example, if $P = [2] \times [2] = \{w,x,y,z\}$, then ${{\mathcal{O}}}(P) = \{ v=(v_1,v_2,v_3,v_4) \in {\reals}^4$ : $0 \leq v_1$, $v_1 \leq v_2$, $v_1 \leq v_3$, $v_2 \leq v_4$, $v_3 \leq v_4$, and $v_4 \leq 1$}; each such $v$ can be depicted as the $P$-array $$\begin{array}{ccc}
& v_4 & \\[3mm]
v_2 & & v_3 \\[3mm]
& v_1 &
\end{array}$$ ${{\mathcal{O}}}(P)$ is the convex hull of the vectors $(0,0,0,0)$, $(0,0,0,1)$, $(0,0,1,1)$, $(0,1,0,1)$, $(0,1,1,1)$, and $(1,1,1,1)$, which are precisely the vectors associated with the filters of $P$. It is shown in [@stanley86] that for any poset $P$, the vertices of ${{\mathcal{O}}}(P)$ correspond to the indicator functions of the filters of $P$.
Given a convex compact polytope ${K}$ in ${\reals}^p$ (we are only concerned with the case ${K}={{\mathcal{O}}}(P)$ here but the definition makes sense more generally), we define the [*piecewise-linear toggle operation*]{} $\tau_i$ ($1 \leq i \leq p$) as the unique map from ${K}$ to itself whose action on the 1-dimensional cross-sections of ${K}$ in the $i$th coordinate direction is the linear map that switches the two endpoints of the cross-section. That is, given $v = (v_1,\dots,v_p) \in {K}$, we define $$\label{eq:phiv}
\tau_i(v) = (v_1,\dots,v_{i-1},L+R-v_i,v_{i+1},\dots,v_p),$$ where the real numbers $L$ and $R$ are respectively the left and right endpoints of the set $\{t \in {\reals}: \ (v_1,\dots,v_{i-1},t,v_{i+1},\dots,v_p)
\in {K}\}$, which is a bounded interval because $K$ is convex and compact.[^4] Since $L+R-(L+R-v_i)=v_i$, each toggle operation is an involution.
Similar involutions were studied by [@kirillov1996groups] in the context of Gelfand-Tsetlin triangles. Indeed, one can view their action in our piecewise-linear toggling framework, where instead of looking at the rectangle posets $[a] \times [b]$ one looks at the triangle posets with elements $\{(i,j): \ 1 \leq i \leq j \leq N\}$ and covering-relations $(i,j-1) {\lessdot}(i,j)$ (for $1 \leq i \leq j \leq N$) and $(i+1,j+1) {\lessdot}(i,j)$ (for $1 \leq i \leq j \leq N-1$). Their “elementary transformations” (Definition 0.1) are our “toggles”.
In the case where ${K}$ is the order polytope of $P$ and a particular element $x \in P$ has been indexed as $x_i$, we write $\tau_i$ as $\tau_x$. The $L$ and $R$ that appear in (\[eq:phiv\]) are given by $$\label{eq:L}
L = \max \{v_j: \ 0 \leq j \leq p+1, \ x_j {\lessdot}x_i\}$$ and $$\label{eq:R}
R = \min \{v_j: \ 0 \leq j \leq p+1, \ x_j {\gtrdot}x_i\} .$$ (One also has $L = \max \{v_j : x_j < x_i\}$ and $R = \min \{v_j : x_j > x_i\}$, but the formulas (\[eq:phiv\])–(\[eq:R\]) turn out to be the right ones to use when extending the operations $\tau_i$ from ${{\mathcal{O}}}(P)$ to all of ${\reals}^p$, as well as the right ones to use when lifting toggling to the birational setting as described in the next section.) It is easy to show that $\tau_x$ and $\tau_y$ commute unless $x {\lessdot}y$ or $x {\gtrdot}y$. These piecewise-linear toggle operations $\tau_x$ are analogous to the combinatorial toggle operations $\tau_x$ (and indeed the former generalize the latter in a sense to be made precise below), so it is natural to define piecewise-linear rowmotion ${\rho_{{{\mathcal{P}}}}}: {{\mathcal{O}}}(P) \rightarrow {{\mathcal{O}}}(P)$ as the composite operation accomplished by toggling from top to bottom (much as ordinary rowmotion ${\rho}: J(P) \rightarrow J(P)$ can be defined as the composite operation obtained by toggling from top to bottom). Likewise we can define piecewise-linear promotion ${\pi_{{{\mathcal{P}}}}}: {{\mathcal{O}}}(P) \rightarrow {{\mathcal{O}}}(P)$ as the composite operation accomplished by toggling from left to right.
Continuing the example $P = [2] \times [2] = \{w,x,y,z\}$ from section \[sec:background\], let $v = (.1,.2,.3,.4) \in {{\mathcal{O}}}(P)$. Under the action of $\tau_z$, $\tau_y$, $\tau_x$, and $\tau_w$, the vector $v$ gets successively mapped to $(.1,.2,.3,.9)$, $(.1,.2,.7,.9)$, $(.1,.8,.7,.9)$, and $(.6,.8,.7,.9) = {\rho_{{{\mathcal{P}}}}}(v)$, while under the action of $\tau_x$, $\tau_w$, $\tau_z$, and $\tau_y$, the vector $v$ gets successively mapped to $(.1,.3,.3,.4)$, $(.2,.3,.3,.4)$, $(.2,.3,.3,.9)$, and $(.2,.3,.8,.9) = {\pi_{{{\mathcal{P}}}}}(v)$.
If $f$ is the indicator function of the filter $P \setminus I$, then ${\rho_{{{\mathcal{P}}}}}(v)$ (resp. ${\pi_{{{\mathcal{P}}}}}(v)$) is the indicator function of the filter $P \setminus {\rho}(I)$ (resp. $P \setminus {\pi}(I)$); in this way ${\rho_{{{\mathcal{P}}}}}$ and ${\pi_{{{\mathcal{P}}}}}$ generalize ${\rho}$ and ${\pi}$.
In the full version of the article ([@einsteinpropp]), we extend ${\rho_{{{\mathcal{P}}}}}$ and ${\pi_{{{\mathcal{P}}}}}$ to all of ${\reals}^p$, not just ${{\mathcal{O}}}(P)$. We also study a variant of these extended operations in which one takes $(\widehat{f}(\widehat{0}), \widehat{f}(\widehat{1})) = (0,0)$ instead of $(0,1)$; although there is no longer an order polytope in the picture, these “homogeneous” actions are easier to understand, and capture most of the behavior of the general inhomogeneous case. One can show that the action of the Sch[ü]{}tzenberger promotion operator (which we denote by ${\pi_{{{\mathcal{S}}}}}$) on the set of semistandard Young tableaux of rectangular shape with $A$ rows and $B$ columns having entries between 1 and $n$ is naturally conjugate to the action of the piecewise-linear promotion operator ${\pi_{{{\mathcal{P}}}}}$ on the rational points in the order polytope of $P = [A] \times [n-A]$ with denominator dividing $B$. (We are grateful to Alex Postnikov and Darij Grinberg for explaining this to us. For the original definition of promotion, see [@schutzenberger1972promotion]; for more modern treatments, see [@stanley2009promotion] and [@vanleeuwen].) For example, take $A=2$, $B=3$, and $n=5$, and consider the semistandard Young tableau $$\begin{array}{ccc}
1 & 2 & 2 \\
3 & 5 & 5
\end{array}$$ We represent the tableau $T$ as a Gelfand-Tsetlin triangle whose $i$th row ($1 \leq i \leq n$) lists, in decreasing order (with 0’s appended or deleted from the end as needed), the number of parts less than or equal to $n-i+1$ in the successive rows of the tableau: $$\begin{array}{ccccccccc}
\overline{3} & & \overline{3} & & \underline{0} & & \underline{0} & & \underline{0} \\
& \overline{3} & & 1 & & \underline{0} & & \underline{0} & \\
& & 3 & & 1 & & \underline{0} & & \\
& & & 3 & & 0 & & & \\
& & & & 1 & & & &
\end{array}$$ This tableau splits into three parts: a triangle of $B$’s (overlined, with top row of length $A$), a triangle of 0’s (underlined, with top row of length $n-A$), and an $(n-A)$-by-$(A)$ rectangle. If we flip this rectangle across the line $x+y=0$, so that the top corner becomes the left corner and vice versa, we get a $P$-array with entries between 0 and $B$: $$\begin{array}{cccc}
& 3 & & \\
1 & & 3 & \\
& 1 & & 1 \\
& & 0 &
\end{array}$$ If we divide each entry by $B$, we get a point $v(T)$ in ${{\mathcal{O}}}(P)$ from which one can recover $T$ by reversing all the above steps. One can show that $v({\pi_{{{\mathcal{S}}}}}(T))={\pi_{{{\mathcal{P}}}}}(v(T))$. Indeed, the file-toggle operations (in which one performs piecewise-linear toggling at all $x \in P$ belonging to the $i$th file of $[n-A] \times [A]$, with $1 \leq i \leq n$; see \[sec:filetoggle\]) can be shown to correspond respectively to the $n$ Bender-Knuth involutions on the Young tableau, whose composition gives ${\pi_{{{\mathcal{S}}}}}$.
The vertices of ${{\mathcal{O}}}(P)$ correspond to the 0,1-valued functions $f$ on $P$ with the property that $x \leq y$ in $P$ implies $f(x) \leq f(y)$ in $\{0,1\}$; these are precisely the indicator functions of filters. Filters are in bijection with order ideals by way of the complementation map, so the vertices of ${{\mathcal{O}}}(P)$ are in bijection with the elements of the lattice $J(P)$. Each toggle operation acts as a permutation on the vertices of ${{\mathcal{O}}}(P)$. Indeed, if we think of each vertex of ${{\mathcal{O}}}(P)$ as determining a cut of the poset $P$ into an upset (filter) ${S_{\rm up}}$ and a complementary downset (order ideal) ${S_{\rm down}}$ (the pre-image of 1 and 0, respectively, under the order-preserving map from $P$ to $\{0,1\}$), then the effect of the toggle operation $\tau_x$ ($x \in P$) is just to move $x$ from ${S_{\rm up}}$ to ${S_{\rm down}}$ (if $x$ is in ${S_{\rm up}}$) or from ${S_{\rm down}}$ to ${S_{\rm up}}$ (if $x$ is in ${S_{\rm down}}$) unless this would violate the property that ${S_{\rm up}}$ must remain an upset and ${S_{\rm down}}$ must remain a downset. In particular, we can see that when our point $v \in {{\mathcal{O}}}(P)$ is a vertex associated with the cut $({S_{\rm up}},{S_{\rm down}})$, the effect of $\tau_x$ on ${S_{\rm down}}$ is just toggling the order ideal ${S_{\rm down}}$ at the element $x \in P$.
[@cameron1995orbits] showed that rowmotion acting on $J([a] \times [b])$ is of order $a+b$. (Subsequently [@prorow] gave a simpler proof, by showing that promotion is of order $a+b$ and that rowmotion is conjugate to promotion.) The same is true of piecewise-linear rowmotion and promotion acting on ${{\mathcal{O}}}([a] \times [b])$:
\[thm:pl-order\] For $P = [a] \times [b]$, the maps ${\rho_{{{\mathcal{P}}}}}$ and ${\pi_{{{\mathcal{P}}}}}$ are of order $a+b$.
It seems plausible that one might be able to deduce the order of ${\rho_{{{\mathcal{P}}}}}$ and ${\pi_{{{\mathcal{P}}}}}$ from the order of ${\rho}$ and ${\pi}$, but we have not been able to find such an argument.[^5] Instead, our proof of Theorem \[thm:pl-order\] detours through the notions of birational promotion and rowmotion.
[Birational toggling]{} \[sec:birational\]
The definition of the piecewise-linear toggling operation via formulas (\[eq:phiv\])–(\[eq:R\]) involves only addition, subtraction, min, and max. Consequently one can define birational transformations on $({\reals}^+)^P$ with formal resemblance to the toggle operations on ${{\mathcal{O}}}(P)$. This transfer makes use of a dictionary in which 0, addition, subtraction, max, and min are respectively replaced by 1, multiplication, division, addition, and parallel addition (defined below), resulting in a subtraction-free rational expression.[^6] Parallel addition can be expressed in terms of the other operations, but taking a symmetrical view of the two forms of addition turns out to be fruitful. Indeed, in setting up the correspondence we have a choice to make: by “series-parallel duality”, one could equally well use a dictionary that switches the roles of addition and parallel addition. We hope the choice that we have made here will prove to be convenient.
For $x,y$ satisfying $x+y \neq 0$, we define the parallel sum of $x$ and $y$ as $x {\mathbin{\|}}y = xy/(x+y)$. In the case where $x$, $y$ and $x+y$ are all nonzero, $xy/(x+y)$ is equal to $1/(\frac1x+\frac1y)$, which clarifies the choice of notation and terminology: if two electrical resistors of resistance $x$ and $y$ are connected in parallel, the compound circuit has an effective resistance of $x {\mathbin{\|}}y$. If $x$ and $y$ are in ${\reals}^+$, then $x+y$ and $x {\mathbin{\|}}y$ are in ${\reals}^+$ as well. Also, ${\mathbin{\|}}$ is commutative and associative, so that a compound parallel sum $x {\mathbin{\|}}y {\mathbin{\|}}z {\mathbin{\|}}\cdots$ is well-defined; it equals the product $x y z \cdots$ divided by the sum of all products that omit exactly one of the variables, and in the case where $x,y,z,\dots$ are all positive, it can also be written as $1/(\frac1x+\frac1y+\frac1z+\cdots)$.
Given a non-empty set $S = \{s_1,s_2,\dots\}$, let ${{\sum}}^+ S$ denote $s_1 + s_2 + \cdots$ and ${{\sum}}^{{\mathbin{\|}}} S$ denote $s_1 {\mathbin{\|}}s_2 {\mathbin{\|}}\cdots$. Then for $v = (v_0,v_1,\dots,v_p,v_{p+1}) \in ({\reals}^+)^{p+2}$ with $v_0 = v_{p+1} = 1$ and for $1 \leq i \leq p$ we define $$\label{eq:phib}
\tau_i(v) = (v_0,v_1,\dots,v_{i-1},LR/v_i,v_{i+1},\dots,v_p,v_{p+1}),$$ with $$\label{eq:Lb}
L = {{\sum}}^{+} \{v_j: \ 0 \leq j \leq p+1, \ x_j {\lessdot}x_i\}$$ and $$\label{eq:Rb}
R = {{\sum}}^{{\mathbin{\|}}} \{v_j: \ 0 \leq j \leq p+1, \ x_j {\gtrdot}x_i\}.$$ We call the maps $\tau_i: ({\reals}^+)^P \rightarrow ({\reals}^+)^P$ given by (\[eq:phib\])–(\[eq:Rb\]) [*birational toggle operations*]{}, as opposed to the piecewise-linear toggle operations treated in the previous section.[^7] As the $0$th and $p+1$st coordinates of $v$ are not affected by any of the toggle operations, we can just omit those coordinates, reducing our toggle operations to actions on $({\reals}^+)^p$. Since $LR/(LR/v_i) = v_i$, each birational toggle operation is an involution on the orthant $({\reals}^+)^{p}$. As in the preceding section, we identify $({\reals}^+)^p$ with $({\reals}^+)^P$. The birational toggle operations are analogous to the piecewise-linear toggle operations (in a sense to be made precise below), so it is natural to define [*birational rowmotion*]{} ${\rho_{{{\mathcal{B}}}}}: ({\reals}^+)^P \rightarrow ({\reals}^+)^P$ as the composite operation accomplished by toggling from top to bottom, and to define [*birational promotion*]{} ${\pi_{{{\mathcal{B}}}}}: ({\reals}^+)^P \rightarrow ({\reals}^+)^P$ as the composite operation accomplished by toggling from left to right.
Continuing our running example $P = [2] \times [2] = \{w,x,y,z\}$, let $v = (1,2,3,4) \in {\reals}^P$, corresponding to the positive function $f$ that maps $w,x,y,z$ to $1,2,3,4$, respectively, with $f(\widehat{0}) = f(\widehat{1}) = 1$. Under the action of $\tau_z$, $\tau_y$, $\tau_x$, and $\tau_w$, the vector $v = (1,2,3,4)$ gets successively mapped to $(1,2,3,\frac{5}{4})$, $(1,2,\frac{5}{12},\frac{5}{4})$, $(1,\frac{5}{8},\frac{5}{12},\frac{5}{4})$, and $(\frac{1}{4},\frac{5}{8},\frac{5}{12},\frac{5}{4}) = {\rho_{{{\mathcal{B}}}}}(v)$. For simplicity, we have defined ${\pi_{{{\mathcal{B}}}}}$ as a map from $({\reals}^+)^P$ to itself. However, ${\pi_{{{\mathcal{B}}}}}$ can be extended to a map from a dense open subset of ${\reals}^P$ to itself, and indeed, from a dense open subset $U$ of ${\complexes}^P$ to itself. All expressions we consider are well-defined on the open orthant $({\reals}^+)^P$, and all the theorems we prove amount to identities that are valid when all variables lie in this orthant; this implies that the identities hold outside of some singular variety in ${\complexes}^P$. Identifying the singular subvariety on which ${\pi_{{{\mathcal{B}}}}}$ (or one of its powers) is undefined seems like an interesting question, but it is one that we leave to others. Alternatively, Tom Roby has pointed out that one can replace ${\reals}^+$ by a ring of rational functions in formal indeterminates indexed by the elements of $P$, thereby avoiding the singularity issue (once one checks that the rational functions in question can be expressed as ratios of polynomials with positive coefficients).
Piecewise-linear rowmotion and promotion can be viewed as tropicalizations of birational rowmotion and promotion. To the extent that facts about birational toggling can be formulated as (complicated but finite) identities in subtraction-free arithmetic, the dictionary alluded to at the start of section \[sec:birational\] allows one to carry the identities to the “max, min, plus” setting.[^8] For instance, when in a later section we prove that ${\rho_{{{\mathcal{B}}}}}^n$ and ${\pi_{{{\mathcal{B}}}}}^n$ act trivially on $({\reals}^+)^P$ (with $P=[a] \times [b]$ and $n=a+b$), it will follow immediately that ${\rho_{{{\mathcal{P}}}}}^n$ and ${\pi_{{{\mathcal{P}}}}}^n$ act trivially on ${\reals}^P$. (Here we gloss over the role that $\widehat{0}$ and $\widehat{1}$ play. Our treatment of birational toggling assumes $\widehat{f}(\widehat{0}) = \widehat{f}(\widehat{1}) = 1$ but our treatment of piecewise-linear toggling assumes $\widehat{f}(\widehat{0}) = 0 \neq 1 = \widehat{f}(\widehat{1})$. The full version of the paper addresses this issue with an appropriate dehomogenization lemma.)
[Birational rowmotion and Stanley’s transfer map]{}
\[sec:transfer\]
Although most of our work with rowmotion treats it as a composition of $|P|$ toggles (from the top to the bottom of $P$), we noted in section \[sec:background\] that ${\rho}$ can also be defined as a composition of three operations $\alpha_1$, $\alpha_2$, $\alpha_3$.[^9] This alternative definition can be lifted to the piecewise-linear and birational settings.
For the piecewise-linear setting, we first recall the definition of the [*chain polytope*]{} ${{\mathcal{C}}}(P)$ of a poset $P$ as defined by [@stanley86]. A [*chain*]{} in a poset $P$ is a totally ordered subset of $P$, and a [*maximal chain*]{} in a poset $P$ is a chain that is not a proper subset of any other chain. If the poset $P$ is ranked, with all maximal elements having the same rank, then the maximal chains in $P$ are precisely those chains that contain an element of every rank. The [*chain polytope*]{} of a poset $P$ is the set of maps from $P$ to $[0,1]$ such that for every chain $C$ in $P$ (or, equivalently, for every maximal chain $C$ in $P$), $$\label{eq:chain}
\sum_{x \in C} f(x) \leq 1.$$ Just as the vertices of the order polytope of $P$ correspond to the indicator functions of the filters of $P$, the vertices of the chain polytope of $P$ correspond[^10] to the indicator functions of the antichains of $P$.
Stanley defines the [*transfer map*]{} $\Phi: {{\mathcal{O}}}(P) \rightarrow {\reals}^P$ via the formula $$\label{eq:transfer}
(\Phi f)(x) = \min \{ f(x)-f(y): y \in \widehat{P}, \ x {\gtrdot}y \}$$ for all $x \in P$ (recall that we have $f(\widehat{0}) = 0$). Stanley proves that $\Phi$ is a bijection between ${{\mathcal{O}}}(P)$ and ${{\mathcal{C}}}(P)$ that carries the vertices of the former to the vertices of the latter. The inverse of $\Phi$ is given by[^11] $$\label{eq:inverse}
(\Psi g)(x) = \max \{ g(y_1)+g(y_2)+\cdots+g(y_k): \widehat{0} {\lessdot}y_1
{\lessdot}y_2 {\lessdot}\cdots {\lessdot}y_k = x \} .$$
Let $\tilde{{{\mathcal{O}}}}(P)$ be the set of order-reversing maps from $P$ to $[0,1]$. We now define bijections $\alpha_1: \tilde{{{\mathcal{O}}}}(P) \rightarrow {{\mathcal{O}}}(P)$, $\alpha_2: {{\mathcal{O}}}(P) \rightarrow {{\mathcal{C}}}(P)$, and $\alpha_3: {{\mathcal{C}}}(P) \rightarrow \tilde{{{\mathcal{O}}}}(P)$ given by the following recipes:
for $f \in \tilde{{{\mathcal{O}}}}(P)$, let $\alpha_1(f)$ be defined by $$(\alpha_1(f))(x) = 1-f(x);$$
for $f \in {{\mathcal{O}}}(P)$, let $\alpha_2(f)$ be defined by $$(\alpha_2 f)(x) =
\min \{ f(x)-f(y): y \in \widehat{P}, \ x {\gtrdot}y \};$$ and
for $f \in {{\mathcal{C}}}(P)$, let $\alpha_3(f)$ be defined by $$\label{eq:longsum}
(\alpha_3 f)(x) = \max \{ f(y_1)+f(y_2)+\cdots+f(y_k): \ x = y_1
{\lessdot}y_2 {\lessdot}\cdots {\lessdot}y_k {\lessdot}\widehat{1} \} .$$
Note that $\alpha_2$ is $\Phi$ and that $\alpha_3$ is $\Psi$ (aka $\Phi^{-1}$) “turned upside down”. It is not hard to check that (\[eq:longsum\]) can be replaced by the recursive definition $$\label{eq:shortsum}
(\alpha_3 f)(x) = f(x) +
\max \{ (\alpha_3 f)(y): \ y \in \widehat{P}, \ y {\gtrdot}x \}$$ which turns out to be the form most suitable for lifting to the birational setting.
\[thm:pl-three\] ${\rho_{{{\mathcal{P}}}}}= \alpha_1 \circ \alpha_3 \circ \alpha_2$.
(Note that $\alpha_1 \circ {\rho_{{{\mathcal{P}}}}}\circ \alpha_1 =
\alpha_3 \circ \alpha_2 \circ \alpha_1$, as in the original definition of ${\rho}$.)
Similarly, in the birational setting put $$\begin{aligned}
(\alpha_1 f)(x) & = & 1/f(x), \label{eq:alph1}\\
(\alpha_2 f)(x) & = &
{{\sum}}^{{\mathbin{\|}}} \{ f(x)/f(y): \ y \in x^- \}, \ {\rm and} \label{eq:alph2}\\
(\alpha_3 f)(x) & = &
f(x) \ {{\sum}}^{+} \{ (\alpha_3 f)(y): \ y \in x^+ \}\label{eq:alph3},\end{aligned}$$ where $x^+$ denotes $\{ y \in \widehat{P}: \ y {\gtrdot}x \}$ and $x^-$ denotes $\{ y \in \widehat{P}: \ x {\gtrdot}y \}$. (Note that definition (\[eq:alph3\]), like definition (\[eq:shortsum\]), is recursive.)
\[thm:birational-three\] ${\rho_{{{\mathcal{B}}}}}= \alpha_1 \circ \alpha_3 \circ \alpha_2$.
Of course the $\alpha$’s in Theorem \[thm:birational-three\] are not the $\alpha$’s in Theorem \[thm:pl-three\] but their birational counterparts.
In the full paper, we derive Theorem \[thm:pl-three\] from Theorem \[thm:birational-three\] by tropicalization and dehomogenization.
[Recombination and Reciprocal Symmetry]{} \[sec:recombine\]
As was noted by [@prorow], there is an intimate relationship between rowmotion and promotion in rc-embedded posets: the two maps have the same orbit structure because they are conjugate as elements of the toggle group. This relationship becomes even clearer in the piecewise-linear and birational settings. Let $P = [2] \times [2]$. Here is the ${\rho_{{{\mathcal{B}}}}}$-orbit of $(1,2,3,4)$: $$\begin{array}{ccccccccc}
( & 1 & , & 2 & , & 3 & , & 4 & )\\[5pt]
( & 1/4 & , & 5/8 & , & 5/12 & , & 5/4 & )\\[5pt]
( & 4/5 & , & 1/3 & , & 1/2 & , & 5/6 & )\\[5pt]
( & 6/5 & , & 12/5 & , & 8/5 & , & 1 & )
\end{array}$$ Here is the ${\pi_{{{\mathcal{B}}}}}$-orbit of $(1,2,5/12,5/4)$: $$\begin{array}{ccccccccc}
( & 1 & , & 2 & , & 5/12 & , & 5/4 & )\\[5pt]
( & 1/4 & , & 5/8 & , & 1/2 & , & 5/6 & )\\[5pt]
( & 4/5 & , & 1/3 & , & 8/5 & , & 1 & )\\[5pt]
( & 6/5 & , & 12/5 & , & 3 & , & 4 & )
\end{array}$$ Note that the same numbers appear as entries in both orbits, with the same multiplicity. More specifically, given $P = [a] \times [b]$, define the [*recombination map*]{} $D$ as the map from the set of $P$-arrays to itself such that for every $P$-array $f$, the $(i,j)$ entry in $D(f)$ is the $(i,j)$ entry in ${\rho_{{{\mathcal{B}}}}}^{i-1}(f)$.
\[thm:recombine\] [(the “recombination lemma”):]{} $D \circ {\pi_{{{\mathcal{B}}}}}= {\rho_{{{\mathcal{B}}}}}\circ D$.
It follows from Theorem \[thm:recombine\] that $D$ is invertible and that ${\pi_{{{\mathcal{B}}}}}$ and ${\rho_{{{\mathcal{B}}}}}$ have the same orbit-structure.
A seemingly much deeper fact is the following consequence of the work of [@grinbergroby] (Theorem 10.6 in particular).
\[thm:recip\] [(reciprocal symmetry):]{} The $(a-i+1,b-j+1)$ entry in ${\rho_{{{\mathcal{B}}}}}^{a+b+1-i-j}(f)$ is the reciprocal of the $(i,j)$ entry in $f$.
Applying this theorem twice yields the conclusion that for $n=(a+b+1-i-j)+(a+b+1-(a-i+1)-(b-j+1))=a+b$, the $(i,j)$ entry in ${\rho_{{{\mathcal{B}}}}}^{n}$ is the reciprocal of the reciprocal of the $(i,j)$ entry in $f$. This implies that ${\rho_{{{\mathcal{B}}}}}^{n}$ is the identity map (and recombination then assures us that that ${\pi_{{{\mathcal{B}}}}}^{n}$ is the identity map as well). The fact that ${\rho}^{n}$ acts trivially on $J([a] \times [b])$ was first proved by [@fon1993orbits].
These facts have implications in the piecewise-linear setting. The recombination property says that the $(i,j)$ entry in $D(f)$ is the $(i,j)$ entry in ${\rho_{{{\mathcal{B}}}}}^{i-1}(f)$, and reciprocal symmetry says that the $(a-i+1,b-j+1)$ entry in ${\rho_{{{\mathcal{P}}}}}^{a+b+1-i-j}(f)$ is 1 minus the $(i,j)$ entry in $f$. We also may conclude that ${\rho_{{{\mathcal{P}}}}}^{n}$ and ${\pi_{{{\mathcal{P}}}}}^{n}$ are the identity map. The last of these conclusions, in combination with our remarks in section \[sec:PL\] linking certian $P$-arrays with semistandard Young tableaux, gives us a new proof of the standard fact that Sch[ü]{}tzenberger promotion on standard tableaux of fixed rectangular shape with entries bounded by $n$ has order $n$.
We stress that recombination is not specific to $[a] \times [b]$, but applies to any rc-embedded poset, even in cases where rowmotion is not of finite order. The recombination lemma is heavily based on Theorem 5.4 in [@prorow] (construction of an equivariant bijection).
[File-toggling and promotion]{} \[sec:filetoggle\]
Here we restrict to $P$ of the form $[a] \times [b]$, with $n=a+b$. The birational toggle operations $\tau_i$, combined in unconstrained fashion, generate a group that is infinite when $a>1$ or $b>1$ (we prove this in detail in the full article for the case $a=b=2$), and its structure is likely to be quite complicated, but some of the subgroups admit homomorphisms to the symmetric group $S_n$, and they can be useful for understanding rowmotion and promotion. One such subgroup, generated by $n-1$ involutions associated with the respective ranks of $P$, was discovered by [@grinbergroby]. Here we study a different subgroup, generated by $n-1$ involutions associated with the respective files of $P$.
Recall that $[a] \times [b]$ can be partitioned into files numbered 1 through $n-1$ from left to right. Given $f: \widehat{P} \rightarrow {\reals}^+$ with $f(\widehat{0}) = f(\widehat{1}) = 1$, let $p_i$ ($1 \leq i \leq n-1$) be the product of the numbers $f(x)$ with $x$ belonging to the $i$th file of $P$, let $p_0 = p_n = 1$, and for $1 \leq i \leq n$ let $q_i = p_i/p_{i-1}$. Call $q_1,\dots,q_n$ the [*quotient sequence*]{} associated with $f$, and denote it by $Q(f)$. This is analogous to the difference sequence introduced in [@propproby]. Note that the product $q_1 \cdots q_n$ telescopes to $p_n/p_0=1$. For $i$ between 1 and $n-1$, let $\tau_i^*$ be the product of the commuting involutions $\tau_x$ for all $x$ belonging to the $i$th file. Lastly, given a sequence of $n$ numbers $w = (w_1,\dots,w_n)$, and given $1 \leq i \leq n-1$, define $\sigma_i (w) = (w_1,\dots,w_{i-1},w_{i+1},w_{i},w_{i+2},\dots,w_n)$; that is, $\sigma_i$ switches the $i$th and $i+1$st entries of $w$.
\[lem:swap\] For all $1 \leq i \leq n-1$, and for all $f$, $$Q(\tau_i^* f) = \sigma_i Q(f).$$ That is, toggling the $i$th file of $f$ swaps the $i$th and $i+1$st entries of the quotient sequence of $f$.
Recalling that ${\pi_{{{\mathcal{B}}}}}$ is the composition $\tau_{n-1}^* \circ \cdots \circ \tau_{1}^*$, we have:
\[cor:shift\] $Q({\pi_{{{\mathcal{B}}}}}f)$ is the leftward cyclic shift of $Q(f)$.
[Homomesy]{} \[sec:homomesy\]
Given a set $X$, an operation $T : X \rightarrow X$ whose $n$th power is the identity map on $X$, and a function $F$ from $X$ to a field ${{\mathcal{K}}}$ of characteristic 0, we say that $F$ is [*homomesic*]{} relative to (or under the action of) $T$, or that the triple $(X,T,F)$ exhibits [*homomesy*]{}, if for all $x \in X$ the average $$\frac{1}{n} \sum_{k=0}^{n-1} F(T^k(x))$$ equals some $c$ independent of $x$. We also say in this situation that the function $F$ (which we will sometimes call a [*functional*]{} on $X$) is $c$-[*mesic*]{} relative to the map $T$. The article by [@propproby] gives examples of combinatorial situations in which homomesy holds. See also [@bloom2013homomesy]. Theorem \[thm:recip\] yields as a corollary that $(({\reals}^+)^P,{\rho_{{{\mathcal{B}}}}},F)$ is 0-mesic, where $F(f) = \log(f(i,j) f(a+1-i,b+1-j))$ (factors cancel in pairs). Applying recombination, we see that the same is true if rowmotion is replaced by promotion. In both cases, tropicalizing yields homomesy for $F(f) = f(i,j) + f(a+1-i,b+1-j)$ under piecewise-linear rowmotion and promotion.
A different sort of homomesy comes from the files of $[a] \times [b]$. Using Corollary \[cor:shift\], one can show that for each $i$ between 1 and $n-1$, if one defines $F_i(f)$ as the logarithm of the product of the values of $f(x)$ as $x$ ranges over the $i$th file of $[a] \times [b]$, then $(({\reals}^+)^P,{\rho_{{{\mathcal{B}}}}},F_i)$ is 0-mesic. This can be carried to the piecewise-linear setting as well. Restricting to the vertices of ${{\mathcal{O}}}(P)$, one obtains the main homomesy theorem of [@propproby].
We can see both forms of homomesy on display in the rowmotion orbit shown at the start of section \[sec:recombine\]. For instance, the middle file of the poset consists of the elements $w$ and $z$, associated with the entries $v_1$ and $v_4$ of each vector $v$. Defining $F(f)$ as $\log f(w)f(z)$, we see that over the orbit the function $F$ takes on the values $\log 4$, $\log 5/16$, $\log 2/3$, and $\log 6/5$, which sum to 0.
\[pl-homomesy\] Given $P = [a] \times [b]$, with $n=a+b$, define functionals $F_{i,j}$ ($1 \leq i \leq a$, $1 \leq j \leq b$) and $F_k$ ($1 \leq k \leq n-1$) by $$F_{i,j}(f) = f(i,j) + f(a+1-i,b+1-j)),$$ $$F_k(f) = \sum_{j-i=k-a} f(i,j).$$ These functionals are all homomesic under the action of ${\rho_{{{\mathcal{P}}}}}$ and ${\pi_{{{\mathcal{P}}}}}$.
\[birational-homomesy\] Given $P = [a] \times [b]$, with $n=a+b$, define functionals $F_{i,j}$ ($1 \leq i \leq a$, $1 \leq j \leq b$) and $F_k$ ($1 \leq k \leq n-1$) by $$F_{i,j}(f) = \log(f(i,j) f(a+1-i,b+1-j)),$$ $$F_k(f) = \log(\prod_{j-i=k-a} f(i,j)).$$ These functionals are all homomesic under the action of ${\rho_{{{\mathcal{B}}}}}$ and ${\pi_{{{\mathcal{B}}}}}$.
The recombination lemma easily implies that a functional $F$ is homomesic under rowmotion if and only if it is homomesic under promotion. Also, any linear combination of homomesic functions is homomesic.
In the full version of the article, a kind of converse of Theorem \[pl-homomesy\] will be proved:
\[pl-converse\] Given $P = [a] \times [b]$, with $p=ab$, let $F$ be some function in the span of the $p$ evaluation functions $f \mapsto f(i,j)$ (with $1 \leq i \leq a$, $1 \leq j \leq b$), such that $F$ is homomesic under the action of ${\rho_{{{\mathcal{P}}}}}$ (or equivalently under the action of ${\pi_{{{\mathcal{P}}}}}$); then $F$ must be a linear combination of the functional $F_{i,j}$ and $F_k$ defined in Theorem \[pl-homomesy\].
Let $V$ be the vector space spanned by the functionals $F_{i,j}$ and $F_k$. It should be noted that the functionals $F_{i,j}$ and $F_k$ have linear dependencies, so although they span $V$, they are not a basis of $V$.
Although we have restricted ourselves to $({\reals}^+)^P$ for simplicity, to the extent that our main results are complicated but finite subtraction-free identities, results like these homomesy theorems, or the fact that rowmotion and promotion are of order $n$, apply throughout the complement of some proper subvariety of ${\complexes}^P$ (though we need to use $\log |z|$ in place of $\log z$). Also note that our birational maps are homogeneous, so projective counterparts of rowmotion and promotion can be defined and are likely to be helpful.
\[sec:ack\] This work was supported by a grant from NSF. The authors are grateful to Arkady Berenstein, Darij Grinberg, Alex Postnikov, Tom Roby, Richard Stanley, and Jessica Striker for helpful conversations, and to the referees for helpful suggestions.
\[sec:biblio\]
[^1]: Email: .
[^2]: Homepage: <http://jamespropp.org>.
[^3]: Note that $j-i+a$ ranges from $1$ to $a+b-1=n-1$; this is slightly different from the indexing in [@propproby].
[^4]: Note that $L$ and $R$ depend on $v_1,\dots,v_{i-1},v_{i+1},\dots,v_p$, though our notation suppresses this dependence.
[^5]: The Coxeter hyperplane arrangement of type $A$ divides the order polytope into simplices, and on each simplex the maps ${\rho_{{{\mathcal{P}}}}}$ and ${\pi_{{{\mathcal{P}}}}}$ are not just piecewise-linear but actually linear (by which we really mean “affine”), and one might hope to base a proof of Theorem \[thm:pl-order\] on this; unfortunately, the images of these simplices under ${\rho_{{{\mathcal{P}}}}}$ and ${\pi_{{{\mathcal{P}}}}}$ are not themselves simplices in this dissection, so the most simple sort of proof one might imagine does not work.
[^6]: The authors are indebted to Arkady Berenstein for pointing out the details of this transfer of structure from the piecewise-linear setting to the birational setting.
[^7]: In principle we should use a different symbol than $\tau_i$, but in practice it should always be clear whether we are referring to piecewise-linear operations or birational operations.
[^8]: We are indebted to Colin McQuillan and Will Sawin for clarifying this point; see [@mathoverflow1].
[^9]: Indeed this was the way in which Brouwer and Schrijver originally defined their operation $F$, in the context of the Boolean lattices $[2] \times [2] \times \cdots \times [2]$.
[^10]: One direction of this claim is easy: since every antichain intersects every chain of $P$ in at most one element of $P$, the indicator function of an antichain must correspond to a point in ${{\mathcal{C}}}(P)$. For the other direction, see Theorem 2.2 of [@stanley86].
[^11]: This is not precisely the definition of $\Psi$ that Stanley gives, but the two definitions are easily seen to be equivalent.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The generalized decoration-iteration transformation is adopted to treat exactly a hybrid model of doubly decorated two-dimensional lattices, which have localized Ising spins at their nodal lattice sites and itinerant electrons delocalized over pairs of decorating sites. Under the assumption of a half filling of each couple of the decorating sites, the investigated model system exhibits a remarkable spontaneous antiferromagnetic long-range order with an obvious quantum reduction of the staggered magnetization. It is shown that the critical temperature of the spontaneously long-range ordered quantum antiferromagnet displays an outstanding non-monotonic dependence on a ratio between the kinetic term and the Ising-type exchange interaction.'
author:
- Jozef Strečka
- Akinori Tanaka
- Lucia Čanová
- Taras Verkholyak
title: 'Exact evidence for the spontaneous antiferromagnetic long-range order in the two-dimensional hybrid model of localized Ising spins and itinerant electrons'
---
\[sec:intro\] Introduction
==========================
Exactly solvable lattice-statistical models are of appreciable scientific interest as they bring a valuable insight into diverse aspects of quantum, cooperative, and critical phenomena.[@baxt82; @matt93; @sach99; @lavi99; @lieb04; @suth04; @diep04; @wu09] It should be mentioned, however, that sophisticated mathematical methods must be usually employed when searching for an exact treatment of even relatively simple interacting many-body systems, while an exact treatment of more realistic or more complex models is often quite unfeasible or is accompanied with a substantial increase of computational difficulties. From this point of view, the exact mapping technique based on generalized algebraic transformations [@fish59; @syoz72; @roja09] belongs to the simplest mathematical methods, which allows to obtain the exact solution of more complicated model from a precise mapping relationship with a simpler exactly solved model. Following Fisher’s ideas,[@fish59] an arbitrary statistical-mechanical system (even of quantum nature) that merely interacts with either two or three outer Ising spins may be in principle replaced by the effective interactions between the outer Ising spins through appropriately chosen decoration-iteration [@syoz51] or star-triangle [@onsa44] mapping transformations. Even although the concept based on the generalized algebraic transformations has been worked out by Fisher more than a half century ago,[@fish59] the algebraic mapping transformations were initially widely used to treat only lattice-statistical models consisting of the Ising spins (see Ref. and references cited therein) before this conceptually simple approach was finally applied to hybrid models composed of the Ising and classical Heisenberg spins,[@gonc82; @hori83; @gonc84; @gonc86; @sant86; @sant95] as well as, the Ising and quantum Heisenberg spins.[@stre02; @jasc02; @stre04; @stre06; @stre08; @yao08; @valv09; @stre09]
Another interesting application of the algebraic mapping transformations has recently been suggested by Pereira *et al*. [@pere08; @pere09] when applying the generalized decoration-iteration transformation to an intriguing diamond-chain model of interacting spin-electron system. In this diamond-chain model, the nodal lattice sites are occupied by the localized Ising spins and the mobile electrons can freely move on a couple of interstitial decorating sites symmetrically placed in between two localized Ising spins. The main aim of this work is to treat exactly an analogous two-dimensional (2D) hybrid model defined on doubly decorated planar lattices, which should provide a deeper insight into how itinerant character of the mobile electrons will influence phase transitions and critical phenomena of this interacting spin-electron system.
The outline of this paper is as follows. In Section \[sec:model\], we first provide a rather detailed description of the model under investigation together with the most crucial steps of the exact mapping method, which enables us to obtain exact closed-form expressions for the critical temperature, order parameter and other relevant thermodynamic quantities. The most interesting results for the ground state, the finite-temperature phase diagram and thermodynamics are presented and detailed discussed in Section \[sec:result\]. Finally, the summary of the most important scientific achievement is mentioned with several concluding remarks in Section \[sec:conc\].
\[sec:model\] Model and method
==============================
Let us consider a hybrid lattice-statistical model of interacting spin-electron system on doubly decorated 2D lattices, which have one localized Ising spin at each nodal lattice site and two delocalized mobile electrons at each couple of decorating sites. The magnetic structure of the model under investigation is schematically depicted in figure \[fig1\] on the particular example of the doubly decorated square lattice.
![A part of the doubly decorated square lattice, which has one localized Ising spin at each nodal lattice site (full circle) and two mobile electrons at each couple of decorating sites (empty circles). The ellipse demarcates the localized Ising spins and itinerant electrons described through the $k$th bond Hamiltonian (\[ham\]).[]{data-label="fig1"}](fig1.eps){width="8cm"}
For further convenience, the total Hamiltonian can be defined as a sum over bond Hamiltonians $\hat{\cal H} = \sum_k \hat{\cal H}_k$, where each bond Hamiltonian $\hat{\cal H}_k$ involves all the interaction terms of two itinerant electrons from the $k$th bond of the doubly decorated 2D lattice $$\begin{aligned}
\hat{\cal H}_k = \! \! \! &-& \! \! \! t \left( c^{\dagger}_{k1, \uparrow} c^{}_{k2, \uparrow}
+ c^{\dagger}_{k1, \downarrow} c^{}_{k2, \downarrow}
+ c^{\dagger}_{k2, \uparrow} c^{}_{k1, \uparrow}
+ c^{\dagger}_{k2, \downarrow} c^{}_{k1, \downarrow} \right) \nonumber \\
\! \! \! &-& \! \! \!
\frac{J}{2} \hat{\sigma}_{k1}^z \left( c^{\dagger}_{k1, \uparrow} c^{}_{k1, \uparrow}
- c^{\dagger}_{k1, \downarrow} c^{}_{k1, \downarrow} \right)
\nonumber \\
\! \! \! &-& \! \! \!
\frac{J}{2} \hat{\sigma}_{k2}^z \left( c^{\dagger}_{k2, \uparrow} c^{}_{k2, \uparrow}
- c^{\dagger}_{k2, \downarrow} c^{}_{k2, \downarrow} \right). \label{ham}\end{aligned}$$ Here, $c^{\dagger}_{k \alpha, \gamma}$ and $c^{}_{k \alpha, \gamma}$ ($\alpha = 1,2$, $\gamma = \uparrow, \downarrow$) denote usual creation and annihilation fermionic operators and $\hat{\sigma}_{k \alpha}^z$ is the standard spin-1/2 operator with the eigenvalues ${\sigma}_{k \alpha}^z = \pm 1/2$. The hopping parameter $t$ takes into account kinetic energy of the mobile electrons and the exchange integral $J$ describes the Ising-type interaction between the itinerant electrons and their nearest Ising neighbors.
A crucial step of our approach lies in the calculation of the partition function. Owing to a validity of the commutation relation between different bond Hamiltonians $[\hat{\cal H}_i, \hat{\cal H}_j] = 0$, the total partition function ${\cal Z}$ can be partially factorized into a product of the bond partition functions ${\cal Z}_k$ $$\begin{aligned}
{\cal Z} = \displaystyle \sum_{\{ \sigma_i \}} \prod_{k=1}^{Nq/2}
\mbox{Tr}_{k} \exp \left(- \beta \hat{\cal H}_k \right)
= \displaystyle \sum_{\{ \sigma_i \}} \prod_{k=1}^{Nq/2} {\cal Z}_k,
\label{pf}\end{aligned}$$ where $\beta = 1/(k_{\rm B} T)$, $k_{\rm B}$ is Boltzmann’s constant, $T$ is the absolute temperature, $N$ is the total number of the Ising spins (i.e. the nodal lattice sites) and $q$ is their coordination number (i.e. the number of nearest-neighbor decorating sites). Next, the symbol $\sum_{\{\sigma_i \}}$ denotes a summation over all possible spin configurations of the localized Ising spins and the symbol $\mbox{Tr}_{k}$ stands for a trace over degrees of freedom of two mobile electrons from the $k$th couple of decorating sites. An explicit form of the bond partition function ${\cal Z}_k$ can be subsequently acquired by a direct diagonalization of the bond Hamiltonian (\[ham\]). The matrix representation of the bond Hamiltonian $\hat{\cal H}_k$ in the orthonormal basis of states $| \psi_i \rangle = \{c^{\dagger}_{k1, \uparrow} c^{\dagger}_{k2, \uparrow}| 0 \rangle$, $c^{\dagger}_{k1, \downarrow} c^{\dagger}_{k2, \downarrow} | 0 \rangle$, $c^{\dagger}_{k1, \uparrow} c^{\dagger}_{k2, \downarrow} | 0 \rangle$, $c^{\dagger}_{k1, \downarrow} c^{\dagger}_{k2, \uparrow} | 0 \rangle$, $c^{\dagger}_{k1, \uparrow} c^{\dagger}_{k1, \downarrow}| 0 \rangle$, $c^{\dagger}_{k2, \uparrow} c^{\dagger}_{k2, \downarrow}| 0 \rangle \}$ ($| 0 \rangle$ labels the empty state) reads
$$\begin{aligned}
\langle \psi_j | \hat{{\cal{H}}}_{k}| \psi_i \rangle = \left(
\begin{array}{cccccc}
-h_{k1} - h_{k2} & 0 & 0 & 0 & 0 & 0 \\
0 & h_{k1} + h_{k2} & 0 & 0 & 0 & 0 \\
0 & 0 & -h_{k1} + h_{k2} & 0 & -t & -t \\
0 & 0 & 0 & h_{k1} - h_{k2} & t & t \\
0 & 0 & -t & t & 0 & 0 \\
0 & 0 & -t & t & 0 & 0 \\
\end{array}
\right),
\label{matrix}\end{aligned}$$
where we have defined two new parameters $h_{k1} = J \sigma_{k1}^z/2$ and $h_{k2} = J \sigma_{k1}^z/2$ that include the Ising-type interaction between the mobile electrons and their nearest-neighbor localized Ising spins. It is noteworthy that the parameters $h_{k1}$ and $h_{k2}$ can alternatively be viewed also as generally non-uniform effective field, which is produced by the localized Ising spins on the itinerant electrons situated at the nearest-neighbor decorating sites. The straightforward diagonalization of the Hamiltonian matrix (\[matrix\]) yields six eigenenergies $$\begin{aligned}
E_{k1,k2} \! \! \! &=& \! \! \! \pm (h_{k1} + h_{k2}), \qquad E_{k3,k4} = 0, \nonumber \\
E_{k5,k6} \! \! \! &=& \! \! \! \pm \sqrt{\left(h_{k1} - h_{k2} \right)^2 + 4 t^2},
\label{eigen}\end{aligned}$$ which can be further used for the relevant calculation of the bond partition function ${\cal Z}_k$. After tracing out the degrees of freedom of the itinerant electrons, the bond partition function ${\cal Z}_k$ merely depends on spin states of two localized Ising spins and, besides, its explicit form immediately implies a possibility of performing the generalized decoration-iteration transformation [@fish59; @syoz72; @roja09] $$\begin{aligned}
\! \! \! \! \! \! \! && \! \! \! \! \! {\cal Z}_k = \sum_{i=1}^6 \exp \left( - \beta E_{ki} \right) =
2 + 2 \cosh \! \! \left[ \frac{\beta J}{2} (\sigma_{k1}^z + \sigma_{k2}^z) \right] + \nonumber \\
\! \! \! \! \! \! \! && \! \! \! \! \!
2 \cosh \! \! \left[ \frac{\beta}{2} \sqrt{J^2 (\sigma_{k1}^z - \sigma_{k2}^z)^2 + (4t)^2} \right]
= A \exp(\beta R \sigma_{k1}^z \sigma_{k2}^z). \nonumber \\
\label{dit}\end{aligned}$$ The physical meaning of the mapping transformation (\[dit\]) is to replace the bond partition function ${\cal Z}_k$ by the equivalent expression, which would contain the effective pair interaction $R$ between the localized Ising spins only. The mapping parameters $A$ and $R$ are unambiguously given by the ’self-consistency’ condition of the algebraic transformation (\[dit\]), which must hold for any combination of spin states of two Ising spins $\sigma_{k1}^z$ and $\sigma_{k2}^z$ involved therein. It can be readily proved that the decoration-iteration transformation (\[dit\]) indeed represents a set of two independent equations, which directly determine the mapping parameters $A$ and $R$ $$\begin{aligned}
A = (V_1 V_2)^{1/2}, \qquad \beta R = 2 \ln (V_1/V_2),
\label{mp} \end{aligned}$$ that are for the sake of brevity expressed in terms of the newly defined functions $V_1$ and $V_2$ $$\begin{aligned}
V_1 \!\!\! &=& \!\!\! 2 + 2 \cosh (\beta J/2) + 2 \cosh (2 \beta t), \nonumber \\
V_2 \!\!\! &=& \!\!\! 4 + 2 \cosh \left[\beta \sqrt{J^2 + (4t)^2}/2 \right].
\label{fun} \end{aligned}$$ Substituting the algebraic transformation (\[dit\]) into the factorized form of the partition function (\[pf\]), which physically corresponds to performing the decoration-iteration mapping at each bond of the doubly decorated 2D lattice, consequently leads to a simple mapping relationship between the partition function ${\cal Z}$ of the interacting spin-electron system on the doubly decorated 2D lattice and, respectively, the partition function ${\cal Z}_{{\rm IM}}$ of the simple spin-1/2 Ising model on the corresponding undecorated lattice with the effective (temperature-dependent) nearest-neighbor interaction $R$ $$\begin{aligned}
{\cal Z} (\beta, J, t) = A^{Nq/2} {\cal Z}_{{\rm IM}} (\beta, R).
\label{mr}\end{aligned}$$ The mapping relation (\[mr\]) essentially completes our exact calculation of the partition function ${\cal Z}$, since the partition function of the nearest-neighbor spin-1/2 Ising model has been precisely calculated for several 2D lattices (for reviews see Refs. ). For brevity, let us therefore merely quote the respective results for the partition functions of the spin-1/2 Ising model on the square lattice [@onsa44] $$\begin{aligned}
\lim_{N \to \infty} \!\!\!\!\! && \!\!\!\!\! \frac{1}{N} \ln {\cal Z}_{{\rm IM}} = \ln 2
+ \frac{1}{4 \pi^2} \int_0^{2 \pi} \!\!\! \int_0^{2 \pi} \!\!\! \ln [\cosh^2 \left(\beta R/2 \right) \nonumber \\
\!\!\! &-& \!\!\! \sinh \left(\beta R/2 \right) \left(\cos \theta + \cos \phi \right)]
{\rm d} \theta {\rm d} \phi
\label{pfsquare}\end{aligned}$$ and the honeycomb lattice [@hout50] $$\begin{aligned}
\!\!\!\! && \!\!\!\! \lim_{N \to \infty} \!\! \frac{1}{N} \ln {\cal Z}_{{\rm IM}} = \ln 2
+ \frac{1}{16 \pi^2} \int_0^{2 \pi} \!\!\! \int_0^{2 \pi} \!\!\! \ln [\{1 +
\cosh^3 \! \left(\beta R/2 \right) \nonumber \\
\!\!\!\! && \!\!\!\! -
\sinh^2 \left(\beta R/2 \right) \left[\cos \theta + \cos \phi + \cos(\theta + \phi) \right]\}/2]
{\rm d} \theta {\rm d} \phi.
\label{pfhoney}\end{aligned}$$ In what follows, the precise mapping relationship (\[mr\]) between the partition functions will be used as a starting point for performing a rather comprehensive analysis of the critical behavior, the order parameter, and basic thermodynamic quantities.
Critical condition
------------------
In order to locate a critical point of the investigated spin-electron system, one may take advantage of the fact that the partition function always becomes non-analytic at a critical point. It can be easily understood from Eqs. (\[mp\])–(\[mr\]) that the mapping parameter $A$ cannot cause a non-analytic behavior of the partition function ${\cal Z}$ and thus, the investigated spin-electron system becomes critical if and only if the corresponding spin-1/2 Ising model with the effective coupling $\beta R$ becomes critical as well. Accordingly, the critical points of the investigated spin-electron system can be straightforwardly obtained from a comparison of the effective temperature-dependent coupling $\beta R$ with the relevant critical point of the corresponding spin-1/2 Ising model on the undecorated lattice. However, the mathematical structure of the mapping parameter $R$ has another important consequences on a critical behavior. More specifically, one may easily prove from Eq. (\[fun\]) a general validity of the inequality $V_1<V_2$, which in compliance with the definition (\[mp\]) implies the antiferromagnetic nature of the effective interaction ($R<0$) in the spin-1/2 Ising model on the corresponding undecorated lattice. Owing to this fact, the interacting spin-electron system is always mapped on the spin-1/2 Ising model with the antiferromagnetic nearest-neighbor interaction, which has a non-zero critical temperature only on the 2D loose-packed lattices such as square [@onsa44] and honeycomb [@hout50; @temp50; @newe50; @husi50; @syoz50] lattices. As a matter of fact, it is well known dictum that the spin-1/2 Ising model with the antiferromagnetic nearest-neighbor interaction on close-packed lattices like triangular [@hout50; @syoz50; @wann50] and kagomé [@syoz51; @kano53] lattices does not exhibit a spontaneous long-range order at any finite temperature and consequently, it does not have any finite-temperature critical point that would correspond to the order-disorder transition.
Bearing all this in mind, we will consider hereafter only the interacting spin-electron system on the doubly decorated 2D loose-packed lattices. It is worthwhile to remark that the critical temperature of the antiferromagnetic spin-1/2 Ising model on the loose-packed lattices is equal to the one of the ferromagnetic model and hence, the critical points for the spin-electron system on the doubly decorated square and honeycomb lattices readily follow from the conditions $\beta_{\rm c} |R| = 2 \ln (1 + \sqrt{2})$[@onsa44] and, respectively, $\beta_{\rm c} |R| = 2 \ln(2 + \sqrt{3})$,[@hout50] where $\beta_{\rm c} = 1/(k_{\rm B} T_{\rm c})$ and $T_{\rm c}$ is the critical temperature.
Staggered magnetization
-----------------------
As a direct consequence of the mapping equivalence with the antiferromagnetic spin-1/2 Ising model on the relevant undecorated loose-packed lattice, one should anticipate predominantly antiferromagnetic character of the interacting spin-electron system on the doubly decorated 2D loose-packed lattice as well. Let us therefore calculate the staggered magnetization as the most common order parameter inherent to the antiferromagnetic spin alignment. Using the exact mapping theorems developed by Barry *et al*.,[@barr88; @khat90; @barr91; @barr95] the spontaneous staggered magnetization of the localized Ising spins can easily be calculated from the exact spin identity $$\begin{aligned}
m_i \equiv \frac{1}{2} \langle \hat{\sigma}_{k1}^z - \hat{\sigma}_{k2}^z \rangle_{t,J}
= \frac{1}{2} \langle \hat{\sigma}_{k1}^z - \hat{\sigma}_{k2}^z \rangle_{R}
\equiv m_{\rm IM} (\beta R),
\label{mi} \end{aligned}$$ where the symbols $\langle \ldots \rangle_{t,J}$ and $\langle \ldots \rangle_{R}$ denote standard canonical ensemble average performed within the interacting spin-electron model on the doubly decorated 2D lattice and, respectively, its equivalent spin-1/2 Ising model on the corresponding undecorated 2D lattice.[@acom] The exact spin identity (\[mi\]) furnishes an accurate proof that the staggered magnetization of the Ising sublattice in the interacting spin-electron model on the doubly decorated 2D lattice directly equals the staggered magnetization of the spin-1/2 Ising model on the corresponding undecorated 2D lattice with the antiferromagnetic nearest-neighbor interaction $R<0$. However, the spontaneous staggered magnetization of the antiferromagnetic spin-1/2 Ising model on the loose-packed 2D lattices precisely coincides with the spontaneous magnetization of the ferromagnetic model, i.e. the quantity, which has exactly been determined for several 2D Ising models (see Ref. and references cited therein). In this regard, the staggered magnetization of the antiferromagnetic spin-1/2 Ising model on the square [@yang52] and honeycomb [@naya54] lattices for instance read $$\begin{aligned}
m_{\rm IM} \! \! \! &=& \! \! \! \frac{1}{2} \left[ 1 - \frac{16 x^4}{(1-x^2)^4} \right]^{1/8}
\! \! \! \! \! \!,
\qquad {\rm (square)} \label{m0} \\
m_{\rm IM} \! \! \! &=& \! \! \!
\frac{1}{2} \left[ 1 - \frac{16 x^3 (1 + x^3)}{(1-x)^3(1-x^2)^3} \right]^{1/8}\! \! \! \! \! \!,
\qquad {\rm (honeycomb)} \nonumber \end{aligned}$$ where $x = \exp(- \beta R/2)$. The above formulas complete our calculation of the staggered magnetization of the localized Ising spins when substituting the exact expression (\[m0\]) with the appropriately chosen effective coupling (\[mp\])–(\[fun\]) into the exact spin identity (\[mi\]).
On the other hand, the staggered magnetization of the itinerant electrons delocalized over pairs of decorating sites can be calculated with the aid of the generalized Callen-Suzuki identity [@call63; @suzu65; @saba81; @saba85; @balc02] $$\begin{aligned}
\langle f (c^{\dagger}_{k \alpha, \gamma}, c^{}_{k \alpha, \gamma}) \rangle_{t,J}
\! = \left \langle
\frac{\mbox{Tr}_k f (c^{\dagger}_{k \alpha, \gamma}, c^{}_{k \alpha, \gamma})
\exp(- \beta \hat{\cal H}_k)}{\mbox{Tr}_k \exp(- \beta \hat{\cal H}_k)}
\right \rangle_{\! \! t,J}\! \! \! \! \! \!, \nonumber \\
\label{csi}\end{aligned}$$ where $\alpha=1,2$, $\gamma = \uparrow,\downarrow$, and $f$ is in principle an arbitrary function of the creation and annihilation fermionic operators from the $k$th bond Hamiltonian (\[ham\]). With the help of the exact identity (\[csi\]), the spontaneous staggered magnetization of the itinerant electrons can be calculated from the expression $$\begin{aligned}
m_e = \frac{1}{2} \langle (\hat{S}_{k1}^z - \hat{S}_{k2}^z) \rangle_{t,J}
\! = \left \langle \frac{1}{4 \beta {\cal Z}_k} \left( \frac{\partial {\cal Z}_k}{\partial h_{k1}} -
\frac{\partial {\cal Z}_k}{\partial h_{k2}} \right)
\right \rangle_{t,J}\! \! \! \! \! \!, \nonumber \\
\label{me}\end{aligned}$$ where $\hat{S}_{k \alpha}^z = (c^{\dagger}_{k \alpha, \uparrow} c^{}_{k \alpha, \uparrow} - c^{\dagger}_{k \alpha, \downarrow} c^{}_{k \alpha, \downarrow})/2$ marks the $z$th component of the spin operator of the mobile electron. After a straightforward calculation based on the differential operator technique $\exp(a \partial/\partial x + b \partial/\partial y) g(x,y) = g(x+a, y+b)$ [@honm79; @kane93] and the exact van der Waerden identity $\exp(c \sigma^z)
= \cosh(c/2) + 2 \sigma^z \sinh(c/2)$, the staggered magnetization of the itinerant electrons can be related to the staggered magnetization of the localized Ising spins through the precise formula $$\begin{aligned}
m_e = m_i \frac{J}{\sqrt{J^2 + (4t)^2}}
\frac{\sinh \left[\frac{\beta}{2} \sqrt{J^2 + (4t)^2} \right]}
{2 + \cosh \left[\frac{\beta}{2} \sqrt{J^2 + (4t)^2} \right]}.
\label{mag}\end{aligned}$$ Since the exact expression for the staggered magnetization of the localized Ising spins is already known from Eqs. (\[mi\])–(\[m0\]), the formula (\[mag\]) provides the relevant exact result for the staggered magnetization of the itinerant electrons hopping between the decorating sites.
Thermodynamics
--------------
Before concluding this section, it is worthy of notice that several basic thermodynamic quantities can also be easily derived from the exact mapping equivalence (\[mr\]) between the partition functions ${\cal Z}$ and ${\cal Z}_{\rm IM}$. For instance, the Helmholtz free energy $F$, the internal energy $U$, the entropy $S$, and the specific heat $C$, can directly be computed from the basic relations of thermodynamics and statistical physics such as $$\begin{aligned}
F = - k_{\rm B} T \ln {\cal Z}, \, \, \, U = - \frac{\partial \ln {\cal Z}}{\partial \beta}, \, \, \,
S = - \frac{\partial F}{\partial T}, \,\, \, C = \frac{\partial U}{\partial T}. \nonumber
\label{tsp}\end{aligned}$$
\[sec:result\] Results and discussion
=====================================
Now, let us proceed to a discussion of the most interesting results obtained for the interacting spin-electron system on the doubly decorated 2D lattices. Before doing this, however, the relations (\[mp\])–(\[fun\]) might serve in evidence that the effective nearest-neighbor interaction $R$ of the spin-1/2 Ising model on the corresponding undecorated lattice is invariant under the transformation $J \to -J$. A change of the ferromagnetic Ising interaction $J>0$ to the antiferromagnetic one $J<0$ actually causes only a rather trivial change of the mutual spin orientation of the itinerant electrons and their nearest Ising neighbors. This observation would suggest that the critical temperature as well as other thermodynamic quantities remain unchanged under the sign change $J \to -J$ and thus, one may further consider the ferromagnetic interaction $J>0$ without loss of the generality.
Ground state
------------
First, let us take a closer look at the ground-state behavior. It is quite evident that the ground state will correspond to the lowest-energy eigenvalue of the bond Hamiltonian (\[ham\]), which can be obtained from the eigenenergies (\[eigen\]) by considering four available configurations of the Ising spins $\sigma_{k1}$ and $\sigma_{k2}$ explicitly involved therein. It turns out that the lowest-energy eigenstate constitutes a peculiar four-sublattice quantum antiferromagnet, which can be characterized through the eigenvector $$\begin{aligned}
|{\rm AF} \rangle = \displaystyle \prod_{k=1}^{Nq/2} \! \! \! \! \! \! && \! \! \! \! \! \!
|\!\! \uparrow, \downarrow \rangle_{\sigma_{k1}, \sigma_{k2}} \Biggl[ \frac{1}{2} \left(1 + \frac{J}{\sqrt{J^2 + (4t)^2}} \right) c^{\dagger}_{k1, \uparrow} c^{\dagger}_{k2, \downarrow}
\nonumber \\
\! \! \! &-& \! \! \! \frac{1}{2} \left(1 - \frac{J}{\sqrt{J^2 + (4t)^2}} \right)
c^{\dagger}_{k1, \downarrow} c^{\dagger}_{k2, \uparrow} \label{gs} \\
\! \! \! &+& \! \! \! \frac{2t}{\sqrt{J^2 + (4t)^2}} \left(c^{\dagger}_{k1, \uparrow}
c^{\dagger}_{k1, \downarrow} + c^{\dagger}_{k2, \uparrow} c^{\dagger}_{k2, \downarrow}
\right) \Biggr] | 0 \rangle,
\nonumber\end{aligned}$$ where the product runs over all bonds of the doubly decorated 2D lattice, the former ket vector determines spin states of the localized Ising spins, and the latter one spin states of mobile electrons. Interestingly, one may also find a much simpler goniometric representation of the lowest-energy eigenstate $|{\rm AF} \rangle$ by introducing the mixing angle $\phi$ through the definition $\tan 2 \phi = 4t/J$, which yields for the particular case with $J>0$ [@note] $$\begin{aligned}
|{\rm AF} \rangle \! \! \! &=& \! \! \! \displaystyle \prod_{k=1}^{Nq/2}
|\!\! \uparrow, \downarrow \rangle_{\sigma_{k1}, \sigma_{k2}}
\Bigl[ \cos^2 \! \phi \, c^{\dagger}_{k1, \uparrow} c^{\dagger}_{k2, \downarrow} -
\sin^2 \! \phi \, c^{\dagger}_{k1, \downarrow} c^{\dagger}_{k2, \uparrow} \nonumber \\
\! \! \! &+& \! \! \! \sin \phi \cos \phi \left( c^{\dagger}_{k1, \uparrow}
c^{\dagger}_{k1, \downarrow} + c^{\dagger}_{k2, \uparrow} c^{\dagger}_{k2, \downarrow} \right)
\Bigr] |0 \rangle.
\label{gsg} \end{aligned}$$ Altogether, the four-sublattice quantum antiferromagnet $|{\rm AF} \rangle$ can be characterized by a perfect Néel order of the Ising spins situated at the nodal sites of some loose-packed 2D lattice, while the mobile electrons delocalized over its decorating sites rest in the entangled state composed of two intrinsic antiferromagnetic states $c^{\dagger}_{k1, \uparrow} c^{\dagger}_{k2, \downarrow} |0 \rangle$, $c^{\dagger}_{k1, \downarrow} c^{\dagger}_{k2, \uparrow} |0 \rangle$, and, two non-magnetic ionic states $c^{\dagger}_{k1, \uparrow} c^{\dagger}_{k1, \downarrow} |0 \rangle$, $c^{\dagger}_{k2, \uparrow} c^{\dagger}_{k2, \downarrow} |0 \rangle$. The quantum entanglement of those four microstates arises from a virtual hopping process of the itinerant electrons, which is diagrammatically illustrated in Fig. \[fig2\]. The respective probability distribution of the four entangled microstates is displayed in Fig. \[fig3\] as a function of a relative strength of the hopping term.
![\[fig3\]The probability distribution for the four entangled microstates as a function of the ratio between the hopping term $t$ and the Ising exchange constant $J$.](fig2.eps){width="8cm"}
![\[fig3\]The probability distribution for the four entangled microstates as a function of the ratio between the hopping term $t$ and the Ising exchange constant $J$.](fig3.eps){width="8cm"}
Let us make a few comments on an origin of the remarkable four-sublattice quantum antiferromagnet emerging in $|{\rm AF} \rangle$. The hopping process of both itinerant electrons energetically favors their antiparallel spin alignment and this antiferromagnetic correlation is subsequently mediated through the Ising-type exchange interaction $J$ also on their two nearest-neighbor Ising spins (hence, the antiferromagnetic character of the effective interaction $R<0$ between the Ising spins). If a relative strength of the hopping term is sufficiently weak, the Ising spins prefer the ferromagnetic alignment with respect to their nearest-neighbor mobile electrons as it can be clearly seen from the occurrence probability of the majority microstate $c^{\dagger}_{k1, \uparrow} c^{\dagger}_{k2, \downarrow} |0 \rangle$ that converges to $p_{\uparrow,\downarrow} \to 1$ in the limit $t/J \to 0$. It is nevertheless worth mentioning that the occurrence probabilities of three minority microstates monotonically increase upon strengthening the hopping term at the expense of the occurrence probability of the majority microstate until they asymptotically reach the same value $p_{\uparrow,\downarrow} = p_{\downarrow,\uparrow} = p_{\uparrow \downarrow, 0} = p_{\downarrow \uparrow, 0} \to 1/4$ in the limit $t/J \to \infty$.
Finite-temperature phase diagram
--------------------------------
Next, let us examine critical phenomena associated with finite-temperature phase transitions of the spontaneously long-range ordered phase $|{\rm AF} \rangle$. It is worthwhile to recall that the critical temperature can easily be obtained by solving numerically the critical conditions derived in the foregoing section. The finite-temperature phase diagram in the form of the critical temperature versus the kinetic term dependence is shown in Fig. \[fig4\] for the interacting spin-electron system on doubly decorated square and honeycomb lattices. It is quite obvious from this figure that the interacting spin-electron system on any loose-packed doubly decorated 2D lattice exhibits the same general trends in the relevant dependences of the critical temperature. The critical temperature initially exhibits a relatively rapid increase from zero temperature with increasing the ratio between the hopping term $t$ and the exchange constant $J$
![The dimensionless critical temperature as a function of the relative strength of the hopping term for the interacting spin-electron system on the doubly decorated square and honeycomb lattices.[]{data-label="fig4"}](fig4.eps){width="8cm"}
until it achieves its maximum value. The critical temperature then gradually decreases upon further increase of a relative strength of the kinetic energy before it again tends to zero temperature in the other particular limit $t/J \to \infty$. In the limit $t/J \to 0$, the observed zero critical temperature can be attributed to a localization of the itinerant electrons at particular decorating sites. In fact, there does not exist any other interaction between the itinerant electrons within the tight-binding model described by the Hamiltonian (\[ham\]) except the effective interaction originating from their virtual hopping process. If the hopping process of the itinerant electrons is ignored, the interacting spin-electron system then effectively splits into independent fragments each of them having one central Ising spin coupled to the $q$ localized electrons from its nearest-neighbor decorating sites. In the other particular limit $t/J \to \infty$, the zero critical temperature results from the same probabilities of the four entangled microstates of $|{\rm AF} \rangle$ as it has been already discussed by the ground-state analysis. Owing to this fact, the decorating sites occupied by the mobile electrons have non-magnetic character, which is compatible with a disappearance of the effective interaction $R=0$ between the localized Ising spins that occurs in the particular limit $t/J \to \infty$.
Order parameter
---------------
Temperature variations of both sublattice staggered magnetizations $m_i$ and $m_e$ are depicted in Fig. \[fig5\] for three different values of a relative strength of the kinetic energy. The most obvious difference between the sublattice staggered magnetizations $m_i$ and $m_e$ pertinent to the Ising spins and itinerant electrons, respectively, lies in the quantum reduction of the latter staggered magnetization. The greater a relative strength of the hopping term is, the greater quantum reduction of the staggered magnetization $m_e$ can be observed in Fig. \[fig5\] in concordance with the relevant ground-state prediction of the lowest-energy eigenstate (\[gs\]), as well as, the zero-temperature limit of the expression (\[mag\]) both yielding $$\begin{aligned}
m_e (T=0) = \frac{1}{2} \frac{J}{\sqrt{J^2 + (4t)^2}}.
\label{maggs}\end{aligned}$$ On the other hand, the sublattice staggered magnetization $m_i$ pertinent to the Ising spins always starts from its maximum possible value that implies a perfect Néel long-range order of the localized Ising spins. For completeness, it is also worthy of notice that both sublattice staggered magnetizations tend to zero in a vicinity of the critical temperature with the critical exponent $\beta = 1/8$ from the standard Ising universality class.
![Thermal dependences of both sublattice staggered magnetizations for the interacting spin-electron system on the doubly decorated square lattice at three different values of a relative strength of the hopping term $t/J = 0.2$, $0.4$, and $1.0$.[]{data-label="fig5"}](fig5.eps){width="8cm"}
Specific heat
-------------
Finally, thermal dependences of the specific heat are plotted in Fig. \[fig6\] for the interacting spin-electron system on the doubly decorated square lattice at three different relative strengths of the kinetic term $t/J = 0.1$, $0.25$, and $0.5$. It is quite evident that the investigated model system exhibits the familiar logarithmic singularity at critical temperature of the order-disorder transition, which provides another independent confirmation of the critical behavior from the standard Ising universality class.
![Temperature variations of the specific heat for the interacting spin-electron system on the doubly decorated square lattice at three different values of a relative strength of the hopping term $t/J = 0.1$, $0.25$, and $0.5$.[]{data-label="fig6"}](fig6.eps){width="8cm"}
In addition to the logarithmic divergence observed in a close vicinity of the critical point, the specific heat also displays a round Schottky-type maximum in the high-temperature tail of the specific heat curve. This maximum is well separated from the logarithmic singularity at relatively weak hopping terms (see solid line for $t/J = 0.1$), while it becomes superimposed on a logarithmic divergence at moderate strengths of the kinetic term (see dotted line for $t/J = 0.25$). Last but not least, the round high-temperature maximum again separates from the logarithmic singularity upon further increase of a relative strength of the hopping term. The round maximum then shifts towards higher temperatures and it becomes the broader, the stronger a relative strength of the kinetic term is (see dashed line for $t/J = 0.5$).
\[sec:conc\]Concluding remarks
==============================
In this article, the hybrid lattice-statistical model of the interacting spin-electron system on doubly decorated 2D lattices has exactly been solved by the use of generalized decoration-iteration transformation under the constraint of a half-filling of each couple of the decorating sites. It has been shown that the ground state of the investigated model system defined on any doubly decorated loose-packed 2D lattice represents an interesting four-sublattice quantum antiferromagnetic phase. It is worth noticing, moreover, that this four-sublattice quantum antiferromagnet exhibits a remarkable combination of the spontaneous long-range order manifested through a non-trivial criticality at non-zero temperatures with obvious macroscopic features of quantum origin such as the quantum reduction of the staggered magnetization pertinent to the itinerant electrons. To the best of our knowledge, the model under investigation thus represents a rather rare example of the exactly solved model with such an intriguing combination of otherwise hardly compatible properties. To compare with, the interacting spin-electron system at a quarter filling (i.e. with one mobile electron per each couple of decorating sites) merely exhibits a classical ferromagnetic or ferrimagnetic spontaneous long-range order depending on whether the ferromagnetic or antiferromagnetic interaction between the localized Ising spins and itinerant electrons is assumed.[@icm] This would indicate that an existence of the four-sublattice quantum antiferromagnetic phase is closely related to a collective motion of electrons even if there does not exist any direct interaction between mobile electrons within our tight-binding model.
Besides the purely academic interest in searching exactly solvable quantum spin models, the considered model system has also been suggested in order to bring insight into a magnetism of hybrid systems consisting of both localized spins as well as mobile electrons. Among the most famous magnetic materials, which obey this specific requirement, one could mention the magnetic metal SrCo$_6$O$_{11}$ [@ishi05; @muku06; @ishi07a; @ishi07b] or the series of polymeric coordination compounds \[Ru$_2$(OOC$t$Bu)$_4$\]$_3$\[M(CN)$_6$\] (M = Fe[@yosh02; @miku06; @vos05], Cr[@vos05; @mill05]). In the latter family of magnetic materials, one and three unpaired electrons of the trivalent metal ions such as Fe$^{3+}$ ($S = 1/2$) and Cr$^{3+}$ ($S = 3/2$) are localized at the corners of a simple square lattice, whereas three unpaired electrons are delocalized over the mixed-valent dimeric unit Ru$_2^{5+}$ residing each bond of the square lattice.[@norm79] In this respect, the magnetic structure of this series of coordination compounds closely resembles the one of the suggested model system even if the electronic structure of the mixed-valent dimeric unit Ru$_2^{5+}$ would surely require more complex Hamiltonian in order to describe the double-exchange mechanism in the mixed-valent Ru$_2^{5+}$ dimeric unit. In this direction will continue our further work.
This work was supported by the Slovak Research and Development Agency under the contract LPP-0107-06. The financial support provided under the grants VEGA 1/2009/05 and VVGS 2/09-10 is also gratefully acknowledged.
[100]{}
R.J. Baxter, [*Exactly solved models in statistical mechanics*]{} (Academic Press, New York, 1982).
D.C. Mattis, [*The Many-Body Problem: An Encyclopedia of Exactly Solved Models in One Dimension*]{} (World Scientific, Singapore, 1993).
S. Sachdev, *Quantum Phase Transitions* (Cambrige University Press, Cambridge, 1999).
D.A. Lavis and G.M. Bell, [*Statistical Mechanics of Lattice Systems*]{} (Springer, Berlin, 1999), Vol.1.
E.H. Lieb, [*Condensed Matter Physics and Exactly Soluble Models*]{}, edited by B. Nachtergaele, J.P. Solovej, and J. Yngvason (Springer, Berlin, 2004).
B. Sutherland, [*Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems*]{} (World Scientific, Singapore, 2004).
H.T. Diep and H. Giacomini, in [*Frustrated Spin Systems*]{}, edited by H.T. Diep (World Scientific, Singapore, 2004).
F.Y. Wu, *Exactly Solved Models: A Journey in Statistical Mechanics* (World Scientific, Singapore, 2009).
M.E. Fisher, Phys. Rev. **113**, 969 (1959).
I. Syozi, in [*Phase Transitions and Critical Phenomena*]{}, edited by C. Domb and M.S. Green (Academic Press, New York, 1972), Vol.1.
O. Rojas, J.S. Valverde, S.M. de Souza, Physica A **388**, 1419 (2009).
I. Syozi, Progr. Theor. Phys. [**6**]{}, 306 (1951).
L. Onsager, Phys. Rev. [**65**]{}, 117 (1944).
L.L. Gonçalves, Physica A [**110**]{}, 339 (1982).
T. Horiguchi and L.L. Gonçalves, Physica A [**120**]{}, 600 (1983).
L.L. Gonçalves and T. Horiguchi, Physica A [**127**]{}, 587 (1984).
L.L. Gonçalves and T. Horiguchi, J. Phys. A: Math. Gen. [**19**]{}, 1449 (1986).
R.J.V. dos Santos, S. Coutinho, and J.R.L. de Almeida, J. Phys. A: Math. Gen. [**19**]{}, 3049 (1986).
R.J.V. dos Santos and S. Coutinho, Int. J. Mod. Phys. B **9**, 3387 (1995).
J. Strečka and M. Jaščur, Phys. Rev. B **66**, 174415 (2002).
J. Strečka and M. Jaščur, Phys. Stat. Solidi B **233**, R12 (2002).
J. Strečka and M. Jaščur, J. Magn. Magn. Mater. **272-276**, 987 (2004).
J. Strečka and M. Jaščur, Acta Phys. Slovaca **56**, 65 (2006).
J. Strečka, L. Čanová, M. Jaščur, and M. Hagiwara, Phys. Rev. B **78**, 024427 (2008).
D.X. Yao, Y.L. Loh, E.W. Carlson, and M. Ma, Phys. Rev. B **78**, 024428 (2008).
J.S. Valverde, O. Rojas, and S.M. de Souza, Phys. Rev. E **79**, 041101 (2009).
J. Strečka, L. Čanová, and K. Minami, Phys. Rev. E **79**, 051103 (2009).
M.S.S. Pereira, F.A.B.F. de Moura, and M.L. Lyra, Phys. Rev. B **77**, 024402 (2008).
M.S.S. Pereira, F.A.B.F. de Moura, and M.L. Lyra, Phys. Rev. B **79**, 054427 (2009).
C. Domb, Adv. Phys. **9**, 149 (1960).
B.M. McCoy and T.T. Wu, *The Two-Dimensional Ising Model* (Harvard University Press, Cambridge, 1973).
R.M.F. Houtappel, Physica [**16**]{}, 425 (1950).
H.N.V. Temperley, Proc. Roy. Soc. A [**203**]{}, 202 (1950).
G.F. Newell, Phys. Rev. [**79**]{}, 876 (1950).
K. Husimi and I. Syozi, Prog. Theor. Phys. [**5**]{}, 177 (1950).
I. Syozi, Prog. Theor. Phys. [**5**]{}, 341 (1950).
G.H. Wannier, Phys. Rev. [**79**]{}, 357 (1950); erratum: Phys. Rev. B **7**, 5017 (1973).
K. Kano and S. Naya, Prog. Theor. Phys. [**10**]{}, 158 (1953).
J.H. Barry, M. Khatun, and T. Tanaka, Phys. Rev. B [**37**]{}, 5193 (1988).
M. Khatun, J.H. Barry, and T. Tanaka, Phys. Rev. B [**42**]{}, 4398 (1990).
J.H. Barry, T. Tanaka, M. Khatun, C.H. Múnera, Phys. Rev. B [**44**]{}, 2595 (1991).
J.H. Barry and M. Khatun, Phys. Rev. B [**51**]{}, 5840 (1995).
Here and in what follows, the staggered magnetization is assumed to be calculated by adding a symmetry-breaking field to the Ising spin sites and taking the limit of zero field.
K.Y. Lin, Chinese J. Phys. **30**, 287 (1992).
C.N. Yang, Phys. Rev. **85**, 808 (1952).
S. Naya, Prog. Theor. Phys. **11**, 53 (1954).
H.B. Callen, Phys. Lett. [**4**]{}, 161 (1963).
M. Suzuki, Phys. Lett. [**19**]{}, 267 (1965).
F.C. SáBarreto, I.P. Fittipaldi, and B. Zeks, Ferroelectrics **39**, 1103 (1981).
F.C. SáBarreto and I.P. Fittipaldi, Physica A **129**, 360 (1985).
T. Balcerzak, J. Magn. Magn. Mater. **246**, 213 (2002).
R. Honmura and T. Kaneyoshi, J. Phys. C: Solid St. Phys. [**12**]{}, 3979 (1979).
T. Kaneyoshi, Acta Phys. Pol. A [**83**]{}, 703 (1993).
The relevant eigenstate for the other particular case $J<0$ can be obtained from the eigenvector (\[gsg\]) under the reversal of all Ising spins $\sigma_{k}^z \to - \sigma_{k}^z$.
J. Strečka, A. Tanaka, M. Jaščur, presented at International Conference on Magnetism, to be held on July 27 – 31, 2009, in Karlsruhe, Germany. Submitted to J. Phys.: Conf. Ser., preprint arxiv: 0908.2880.
S. Ishiwata, D. Wang, T. Saito, and M. Takano, Chem. Mater. **17**, 2789 (2005).
H. Mukuda, Y. Kitaoka, S. Ishiwata, T. Saito, Y. Shimakawa, H. Harima, and M. Takano, J. Phys. Soc. Jpn. **75**, 094715 (2006).
S. Ishiwata, I. Terasaki, F. Ishii, N. Nagaosa, H. Mukuda, Y. Kitaoka, T. Saito, and M. Takano, Phys. Rev. Lett. **98**, 217201 (2007).
S. Ishiwata, M. Lee, Y. Onose, N.P. Ong, M. Takano, and I. Terasaki, J. Magn. Magn. Mater. **310**, 1989 (2007).
D. Yoshioka, M. Mikuriya, and M. Handa, Chem. Lett., 1044 (2002).
M. Mikuriya, D. Yoshioka, and M. Handa, Coord. Chem. Rev. **250**, 2194 (2006).
T.E. Vos and J.S. Miller, Angew. Chem. Int. Ed. **44**, 2416 (2005).
J.S. Miller, T.E. Vos, and W.W. Shum, Adv. Mater. **17**, 2251 (2005).
J.G. Norman, G.E. Renzoni, and D.A. Case, J. Am. Chem. Soc. **101**, 5256 (1979).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present multiloop flow equations in the functional renormalization group (fRG) framework for the four-point vertex and self-energy, formulated for a general fermionic many-body problem. This generalizes the previously introduced vertex flow \[F. B. Kugler and J. von Delft, [Phys. Rev. Lett. **120**, 057403 (2018)](http://link.aps.org/doi/10.1103/PhysRevLett.120.057403)\] and provides the necessary corrections to the self-energy flow in order to complete the derivative of all diagrams involved in the truncated fRG flow. Due to its iterative one-loop structure, the multiloop flow is well suited for numerical algorithms, enabling improvement of many fRG computations. We demonstrate its equivalence to a solution of the (first-order) parquet equations in conjunction with the Schwinger-Dyson equation for the self-energy.'
author:
- 'Fabian B. Kugler'
- Jan von Delft
bibliography:
- 'references.bib'
date: 31 January 2018
nocite: '[@Rubtsov2008; @Brener2008; @Hafermann2009; @Rohringer2013; @Ribic2017; @Ribic2017a]'
title: Multiloop functional renormalization group for general models
---
Introduction
============
Two of the most powerful generic methods in the study of large or open many-body systems at intermediate coupling strength are the parquet formalism [@Bickers2004; @Roulet1969] and the functional renormalization group (fRG) [@Metzner2012; @Kopietz2010]. As is commonly known, these frameworks are intimately related. However, their equivalence has only recently been established via multiloop fRG (mfRG) flow equations, introduced in a case study of the X-ray-edge singularity [@Kugler2017]. In this paper, we consolidate this equivalence and formulate the mfRG flow for the general many-body problem. For this, we generalize the multiloop vertex flow from Ref. , and, to ensure full inclusion of the self-energy, we present two multiloop corrections to the self-energy flow. Altogether, the mfRG flow is shown to fully generate all parquet diagrams for the vertex and self-energy; it is thus equivalent to solving the (first-order) parquet equations in conjunction with the Schwinger-Dyson equation (SDE) for the self-energy. The parquet equations (together with the SDE) provide exact, self-consistent equations for the four-point vertex and self-energy, allowing one to describe one-particle and two-particle correlations [@Bickers2004]. The only input is the totally irreducible (four-point) vertex. Approximating it by the bare interaction yields the first-order parquet equations [@Roulet1969] (or parquet approximation [@Bickers2004]), a solution of which generates the so-called parquet diagrams for the four-point vertex and self-energy. The functional renormalization group provides an infinite hierarchy of exact flow equations for vertex functions, depending on an RG scale parameter $\Lambda$. During the flow, high-energy ($ \gtrsim \Lambda $) modes are successively integrated out, and the full solution is obtained at $\Lambda=0$, such that one is free in the specific way the $\Lambda$ dependence (regulator) is chosen [@Metzner2012; @Kopietz2010]. If one restricts the fRG flow equations to the four-point vertex and self-energy, one is left with the six-point vertex as input. In the typical approximation, the six-point vertex is neglected, implying that all diagrams contributing to the flow are of the parquet type [@Kugler2017; @Kugler2017a]. However, due to this truncation, the flow equations (for both self-energy and four-point vertex) no longer form a total derivative of diagrams w.r.t. the flow parameter $\Lambda$. This limits the predictive power of fRG and yields results that actually depend on the choice of regulator. The mfRG corrections to the fRG flow simulate the effect of six-point vertex contributions on parquet diagrams, by means of an iterative multiloop construction. They complete the derivative of diagrams in the flow equations of both self-energy and four-point vertex, which are otherwise only partially contained. As it achieves a full resummation of all parquet diagrams in a numerically efficient way, the mfRG flow allows for significant improvement of fRG computations and overcomes weaknesses of the formalism experienced hitherto. The paper is organized as follows. In [Sec. \[sec:setup\]]{}, we give the setup with all notations, before we recall the basics of the parquet formalism in [Sec. \[sec:parquet\]]{}. In [Sec. \[sec:mfrg\]]{}, we present the mfRG flow equations for the four-point vertex and self-energy. We show that they fully generate all parquet diagrams to arbitrary order in the interaction and comment on computational and general properties of the flow equations. Finally, we present our conclusions in [Sec. \[sec:conclusion\]]{}.
Setup {#sec:setup}
=====
We consider a general theory of interacting fermions, defined by the action $$\begin{aligned}
S & = - \sum_{x{^{\prime}}, x} \bar{c}_{x{^{\prime}}} \big[ (G^0)^{-1} \big]_{x{^{\prime}}, x} c_{x}
- \tfrac{1}{4}
\!\!\!\! \sum_{x{^{\prime}},x,y{^{\prime}},y} \!\!\!\!
\Gamma^0_{x{^{\prime}},y{^{\prime}};x,y} \bar{c}_{x{^{\prime}}} \bar{c}_{y{^{\prime}}} c_{y} c_{x}
{,}\end{aligned}$$ with a bare propagator $G^0$ and a bare four-point vertex $\Gamma^0$, which is antisymmetric in its first and last two arguments. The index $x$ denotes all quantum numbers of the Grassmann field $c_x$. If we choose, e.g., Matsubara frequency, momentum, and spin, with $x = (i\omega, {\bm{k}}, \sigma) = (k, \sigma)$, and consider a translationally invariant system with interaction $U_{|{\bm{k}}|}$, the bare quantities read
$$\begin{aligned}
G^0_{x{^{\prime}},x}
&
\overset{\textrm{e.g.}}{=}
G^0_{k,\sigma} \delta_{k{^{\prime}},k{^{\phantom\prime}}} \delta_{\sigma{^{\prime}},\sigma{^{\phantom\prime}}}
\\
- \Gamma^0_{x_1{^{\prime}},x_2{^{\prime}};x_1{^{\phantom\prime}},x_2{^{\phantom\prime}}}
&
\overset{\textrm{e.g.}}{=}
(
U_{|{\bm{k}}_1{^{\prime}}-{\bm{k}}_1{^{\phantom\prime}}|} \delta_{\sigma_1{^{\prime}},\sigma_1{^{\phantom\prime}}}
\delta_{\sigma_2{^{\prime}},\sigma_2{^{\phantom\prime}}}
\nonumber \\ & \, \ -
U_{|{\bm{k}}_1{^{\prime}}-{\bm{k}}_2{^{\phantom\prime}}|} \delta_{\sigma_1{^{\prime}},\sigma_2{^{\phantom\prime}}}
\delta_{\sigma_2{^{\prime}},\sigma_1{^{\phantom\prime}}}
) \,
\delta_{k_1{^{\prime}}+k_2{^{\prime}},k_1{^{\phantom\prime}}+k_2{^{\phantom\prime}}}
{.}\label{eq:vertex_U}\end{aligned}$$
Correlation functions of fields, corresponding to time-ordered expectation values of operators, are given by the path integral $$\langle c_{x_1} \cdots \bar{c}_{x_n} \rangle =
\frac{1}{Z}
\int \! \mathcal{D}[\bar{c}] \mathcal{D}[c] \,
c_{x_1} \cdots \bar{c}_{x_n} e^{-S}
{,}$$ where $Z$ ensures normalization, such that $\langle 1 \rangle=1$. Two-point correlation functions are represented by the full propagator $G$. Via Dyson’s equation, $G$ is expressed in terms of the bare propagator $G^0$ and the self-energy $\Sigma$ \[cf. [Fig. \[fig:dyson\]]{}(a)\], according to $$\begin{aligned}
G_{x, x{^{\prime}}} = - \langle c_{x} \bar{c}_{x{^{\prime}}} \rangle
{,}\quad
G = G^0 + G^0 \cdot \Sigma \cdot G
{,}\label{eq:dyson}\end{aligned}$$ using the matrix product $
( A \cdot B )_{x, x{^{\prime}}} = \sum_{y} A_{x, y} B_{y, x{^{\prime}}}
{.}$ In a diagrammatic expansion, the lowest-order contribution to the self-energy is given by the diagram in [Fig. \[fig:dyson\]]{}(b), making use of the bare objects $G^0$, $\Gamma^0$. For later purposes, we define a *self-energy loop* ($L$) as $$\begin{aligned}
L(\Gamma, G)_{x{^{\prime}}, x}
& =
- \sum_{y{^{\prime}}, y} \Gamma_{x{^{\prime}}, y{^{\prime}}; x, y} G_{y, y{^{\prime}}}
{.}\label{eq:sigmaloop}\end{aligned}$$ With this, we can write the first-order contribution from [Fig. \[fig:dyson\]]{}(b) generally and in the above example as
$$\begin{aligned}
\Sigma^{1\textrm{st}}_{x{^{\prime}}, x}
&
\overset{\hphantom{\textrm{e.g.}}}{=}
L(\Gamma^0, G^0)_{x{^{\prime}}, x}
\\
&
\overset{\textrm{e.g.}}{=}
\big(
U_{0} \sum_{\tilde{k},\tilde{\sigma}} G^0_{\tilde{k},\tilde{\sigma}}
-
\sum_{\tilde{k}} U_{|{\bm{k}}-\tilde{{\bm{k}}}|}^{\phantom0} G^0_{\tilde{k},\sigma}
\big)
\delta_{k{^{\prime}},k}
\delta_{\sigma{^{\prime}},\sigma}
{.}\end{aligned}$$
![(a) Dyson’s equation relating the full propagator $G_{x,x{^{\prime}}}$ (black, thick line) to the bare propagator $G^0$ (gray, thin line) and the self-energy $\Sigma$ (circle). (b) First-order diagram for the self-energy using the bare vertex $\Gamma^0$ (solid dot).[]{data-label="fig:dyson"}](pyfeynnb_prop.pdf){width=".48\textwidth"}
Four-point correlation functions can be expressed via the full (one-particle-irreducible) four-point vertex $\Gamma$: $$\begin{aligned}
\langle c_{x_1{^{\phantom\prime}}} c_{x_2{^{\phantom\prime}}} \bar{c}_{x_2{^{\prime}}} \bar{c}_{x_1{^{\prime}}} \rangle
& =
G_{x_1{^{\phantom\prime}}x_1{^{\prime}}} G_{x_2{^{\phantom\prime}}x_2{^{\prime}}} - G_{x_1{^{\phantom\prime}}x_2{^{\prime}}} G_{x_2{^{\phantom\prime}}x_1{^{\prime}}}
\nonumber \\
& \ +
G_{x_1{^{\phantom\prime}}y_1{^{\prime}}} G_{x_2{^{\phantom\prime}}y_2{^{\prime}}}
\Gamma_{y_1{^{\prime}}, y_2{^{\prime}}; y_1{^{\phantom\prime}}, y_2{^{\phantom\prime}}}
G_{y_1{^{\phantom\prime}}x_1{^{\prime}}} G_{y_2{^{\phantom\prime}}x_2{^{\prime}}}
{.}\end{aligned}$$ Note that we omit the superscript compared to the usual notation ($\Gamma^{(4)}$) [@Kugler2017; @Kugler2017a; @Metzner2012; @Kopietz2010] and often refer to the four-point vertex simply as the vertex. Our definition of $\Gamma$ [^1] agrees with that of Ref. and therefore contains a relative minus sign compared to Ref. . The diagrammatic expansion of $\Gamma$ up to second order in the interaction is shown in [Fig. \[fig:vertex\]]{}. In such diagrams, the position of the external legs will always be fixed and labeled in correspondence to the four arguments of a vertex. Let us define *bubble functions* ($B$), distinguished between the three two-particle channels $r \in \{ a, p, t \}$, as
\[eq:bubbles\] $$\begin{aligned}
B_a(\Gamma, \Gamma')_{x_1{^{\prime}}, x_2{^{\prime}}; x_1{^{\phantom\prime}}, x_2{^{\phantom\prime}}}
& =
\sum_{y_1{^{\prime}}, y_1{^{\phantom\prime}}, y_2{^{\prime}}, y_2{^{\phantom\prime}}}
\Gamma_{x_1{^{\prime}}, y_2{^{\prime}}; y_1{^{\phantom\prime}}, x_2{^{\phantom\prime}}}
\nonumber \\
& \ \times
G_{y_1{^{\phantom\prime}}, y_1{^{\prime}}} G_{y_2{^{\phantom\prime}}, y_2{^{\prime}}}
\Gamma'_{y_1{^{\prime}}, x_2{^{\prime}}; x_1{^{\phantom\prime}}, y_2{^{\phantom\prime}}}
\label{eq:abubble} \\
B_p(\Gamma, \Gamma')_{x_1{^{\prime}}, x_2{^{\prime}}; x_1{^{\phantom\prime}}, x_2{^{\phantom\prime}}}
& = \tfrac{1}{2}
\sum_{y_1{^{\prime}}, y_1{^{\phantom\prime}}, y_2{^{\prime}}, y_2{^{\phantom\prime}}}
\Gamma_{x_1{^{\prime}}, x_2{^{\prime}}; y_1, y_2{^{\phantom\prime}}}
\nonumber \\
& \ \times
G_{y_1{^{\phantom\prime}}, y_1{^{\prime}}} G_{y_2{^{\phantom\prime}}, y_2{^{\prime}}}
\Gamma'_{y_1{^{\prime}}, y_2{^{\prime}}; x_1{^{\phantom\prime}}, x_2{^{\phantom\prime}}}
\label{eq:pbubble} \\
B_t(\Gamma, \Gamma')_{x_1{^{\prime}}, x_2{^{\prime}}; x_1{^{\phantom\prime}}, x_2{^{\phantom\prime}}}
& = -
\sum_{y_1{^{\prime}}, y_1{^{\phantom\prime}}, y_2{^{\prime}}, y_2{^{\phantom\prime}}}
\Gamma_{y_1{^{\prime}}, x_2{^{\prime}}; y_1{^{\phantom\prime}}, x_2{^{\phantom\prime}}}
\nonumber \\
& \ \times
G_{y_2{^{\phantom\prime}}, y_1{^{\prime}}} G_{y_1{^{\phantom\prime}}, y_2{^{\prime}}}
\Gamma'_{x_1{^{\prime}}, y_2{^{\prime}}; x_1{^{\phantom\prime}}, y_2{^{\phantom\prime}}}
{.}\label{eq:tbubble}\end{aligned}$$
The translation of [Fig. \[fig:vertex\]]{} is then simply given by $$\textstyle
\Gamma^{\textrm{2nd}}=
\Gamma^0+
\sum_r
B_r(\Gamma^0, \Gamma^0){.}\label{eq:vertex2ndOrder}$$
![Diagrammatic expansion of the four-point vertex $\Gamma$ (square) up to second order in the interaction (i.e., these diagrams define $\Gamma^{\textrm{2nd}}$). The positions of the external (amputated) legs refer to the arguments of $\Gamma_{x_1{^{\prime}}, x_2{^{\prime}}; x_1{^{\protect\phantom\prime}}, x_2{^{\protect\phantom\prime}}}$.[]{data-label="fig:vertex"}](pyfeynnb_vertex.pdf){width=".48\textwidth"}
Following the conventions of Bickers [@Bickers2004], the factor of $1/2$ in [Eq. (\[eq:pbubble\])]{} ([Fig. \[fig:vertex\]]{}) makes sure that, when summing over all internal indices, one does not overcount the effect of the two indistinguishable (parallel) lines. The minus sign in [Eq. (\[eq:tbubble\])]{} ([Fig. \[fig:vertex\]]{}) stems from the fact that the antiparallel bubbles [(\[eq:abubble\])]{} and [(\[eq:tbubble\])]{} are related by exchange of fermionic legs. Indeed, using the antisymmetry of $\Gamma$ and $\Gamma'$ in their arguments (crossing symmetry), we find $$B_a(\Gamma, \Gamma')_{x_1{^{\prime}}, x_2{^{\prime}}; x_1{^{\phantom\prime}}, x_2{^{\phantom\prime}}}
= -
B_t(\Gamma, \Gamma')_{x_2{^{\prime}}, x_1{^{\prime}}; x_1{^{\phantom\prime}}, x_2{^{\phantom\prime}}}
{.}\quad
\label{eq:a_t_sym}$$ The channel label $r \in \{ a, p, t \}$ refers to the fact that the individual diagrams are reducible—i.e., they fall apart into disconnected diagrams—by cutting two *antiparallel* lines, two *parallel* lines, or two *transverse* (antiparallel) lines, respectively. (The term transverse itself refers to a horizontal space-time axis.) In using the terms antiparallel and parallel, we adopt the nomenclature used in the seminal application of the parquet equations to the X-ray-edge singularity by Roulet et al. [@Roulet1969]. Equivalently, a common notation [@Rohringer2017; @Wentzell2016] for the channels $a, p, t$ is $ph, pp, \overline{ph}$, referring to the (longitudinal) particle-hole, the particle-particle, and the transverse (or vertical) particle-hole channel, respectively. One also finds the labels $x, p, d$ in the literature [@Jakobs2010], referring to the so-called exchange, pairing, and direct channel, respectively. In the context of fRG (cf. [Sec. \[sec:mfrg\]]{}), functions such as $G$, $\Sigma$, $\Gamma$ develop a scale ($\Lambda$) dependence (which will be suppressed in the notation). If we write the bubble functions also symbolically as $$\begin{aligned}
B_r(\Gamma, \Gamma')
& =
\big[ \Gamma \circ G \circ G \circ \Gamma' \big]_r
{,}\label{eq:gbubble}\end{aligned}$$ we can immediately define bubbles with differentiated propagators (but undifferentiated vertices) according to $$\begin{aligned}
\dot{B}_r(\Gamma, \Gamma')
& =
\big[ \Gamma \circ
\big( \partial_{\Lambda} ( G \circ G ) \big) \circ \Gamma' \big]_r
{,}\label{eq:gdotbubble}\end{aligned}$$ In the fRG flow equations, we will further need the (so-called) single-scale propagator, defined by ($\mathbbm{1}_{x,y}=\delta_{x,y}$) $$\begin{aligned}
S =
\partial_{\Lambda} G|_{\Sigma=\textrm{const.}}
= ( \mathbbm{1} + G \cdot \Sigma ) \cdot
\big( \partial_{\Lambda} G^0 \big) \cdot
( \Sigma \cdot G + \mathbbm{1} )
{.}\label{eq:singlescale}\end{aligned}$$ Before moving on to the mfRG flow, let us next review the basics of the parquet formalism.
Parquet formalism {#sec:parquet}
=================
The parquet formalism [@Bickers2004; @Roulet1969] provides exact, self-consistent equations for both four-point vertex and self-energy. Focusing on the vertex first, the central parquet equation represents a classification of diagrams distinguished by reducibility in the three two-particle channels: $$\begin{aligned}
\Gamma
& =
R + \sum_r \gamma_r
{,}\quad
I_r = R + \sum_{r' \neq r} \gamma_{r'}
{.}\label{eq:parquet}\end{aligned}$$ Diagrams of $\Gamma$ are either reducible in one of the three channels (i.e., part of $\gamma_r$ for $r \in \{ a, p, t \}$, cf. [Fig. \[fig:vertex\]]{}), or they belong to the class of totally irreducible diagrams $R$ \[cf. [Fig. \[fig:sd\]]{}(a)\]. (The notation again refers to Ref. .) As a diagram cannot simultaneously be reducible in more than one channel [@Roulet1969], one collects diagrams that are not reducible in $r$ lines into the irreducible vertex $I_r$ of that channel. Reducible and irreducible vertices are further related by the self-consistent Bethe-Salpeter equations (BSEs) $$\begin{aligned}
\gamma_r
& =
B_r(I_r, \Gamma)
{,}\label{eq:BetheSalpeter}\end{aligned}$$ the graphical representations of which are given in [Fig. \[fig:bs\]]{}.
![(a) Vertex diagram irreducible in all two-particle channels (i.e., it belongs to $R$) and thus not part of $\Gamma$ in the parquet approximation. (b) Schwinger-Dyson equation, relating the self-energy to the four-point vertex self-consistently.[]{data-label="fig:sd"}](pyfeynnb_sigma_sd.pdf){width=".48\textwidth"}
The BSEs [(\[eq:BetheSalpeter\])]{} are computed with full propagators $G$. Thus, they require knowledge of the self-energy, which itself can be determined by the self-consistent SDE depending on the four-point vertex \[cf. [Fig. \[fig:sd\]]{}(b)\]: $$\begin{aligned}
\Sigma
& =
L(\Gamma^0, G)
+
L\big[B_p(\Gamma^0,\Gamma),G\big]
\nonumber \\
& =
L(\Gamma^0, G)
+
\tfrac{1}{2}
L\big[B_a(\Gamma^0,\Gamma),G\big]
{.}\label{eq:SchwingerDyson}\end{aligned}$$
![Bethe-Salpeter equations in the three two-particle channels, relating the reducible ($\gamma_r$) and irreducible ($I_r$) vertices self-consistently in the parquet formalism.[]{data-label="fig:bs"}](pyfeynnb_all_bs.pdf){width=".455\textwidth"}
The only input required for solving the parquet equations is the totally irreducible vertex $R$. All remaining contributions to the vertex and self-energy are determined self-consistently. The simplest way to solve the parquet equations is to approximate $R$ by the bare vertex $\Gamma^0$. This is called the first-order parquet solution [@Roulet1969], or parquet approximation [@Bickers2004], and corresponds to a summation of the leading logarithmic diagrams in logarithmically divergent perturbation theories. The diagrams generated by the first-order parquet solution are called parquet diagrams. For $\Gamma$, these can be obtained by successively replacing bare vertices by one of the three bubbles from [Eq. (\[eq:bubbles\])]{} (connected by full lines), starting from the bare vertex. For $\Sigma$, the parquet diagrams are obtained by inserting the parquet vertex into the SDE. They can also be characterized by the property that one needs to cut at most one bare line to obtain a *parquet* vertex with possible dressing at the external legs. By this, we mean that, instead of an ingoing or outgoing amputated leg, the external line is of the type $\mathbbm{1} + \Sigma \cdot G$ or $\mathbbm{1} + G \cdot \Sigma$, respectively, using again a parquet self-energy.
Multiloop fRG flow {#sec:mfrg}
==================
The functional renormalization group [@Metzner2012; @Kopietz2010] provides a hierarchy of exact flow equations for vertex functions, depending on an RG parameter $\Lambda$, serving as infrared cutoff in the bare propagator. A typical choice for the $\Lambda$ dependence, in order to flow from the trivially uncorrelated to the full theory, is characterized by the boundary conditions $G_{\Lambda_i}=0$ and $G_{\Lambda_f}=G$, implying $\Gamma_{\Lambda_i}=\Gamma^0$. Restring the flow to $\Sigma$ and $\Gamma$, the six-point vertex remains as input and is neglected in the standard approximation.
{width=".99\textwidth"}
Here, we view fRG as a tool to resum diagrams which does not necessarily rely on the original fRG hierarchy deduced from the flow of the (quantum) effective action. In previous works [@Kugler2017; @Kugler2017a], we have used the X-ray-edge singularity as an example to show that the standard truncation of fRG restricts the flow to parquet diagrams of the vertex, and that the derivatives of those diagrams are only partially contained. Using the same model, we have introduced multiloop fRG flow equations for the vertex which complete the derivative of parquet diagrams in an iterative manner, as organized by the number of loops connecting full vertices, and thus do achieve a full summation of all parquet diagrams [@Kugler2017]. The X-ray-edge singularity facilitates diagrammatic arguments as it allows one to consider only two two-particle channels and to neglect self-energies. Here, we give the details of how the mfRG flow of the vertex is generalized to all three two-particle channels with indistinguishable particles (as already indicated in Ref. ) and formulate the mfRG corrections to the self-energy flow (not discussed in Ref. ). We first pose the mfRG flow equations and motivate them by showing examples of diagrams, which are otherwise only partially contained. Then, we justify the extensions of the truncated fRG flow by arguing that all diagrams are of the appropriate type without any overcounting. Subsequently, we give a recipe for counting the number of diagrams generated by the parquet and mfRG flow equations. This allows one to check that the mfRG flow fully captures all parquet diagrams order for order in the interaction. Finally, we discuss computational and general properties of the flow equations.
Flow equations for the vertex
-----------------------------
The mfRG flow of the vertex proposed in Ref. makes use of the channel classification known from the parquet equations and is organized by the *loop order* $\ell$. We write $$\begin{aligned}
\partial_{\Lambda} \Gamma =
\sum_r \partial_{\Lambda} \gamma_r
{,}\ \partial_{\Lambda} \gamma_r
=
\sum_{\ell \geq 1} \dot{\gamma}_r^{(\ell)}
{,}\ \dot{\gamma}_{\bar{r}}^{(\ell)} = \sum_{r' \neq r} \dot{\gamma}_{r'}^{(\ell)}
{,}\end{aligned}$$ where $\dot{\gamma}_r^{(\ell)}$ contains differentiated diagrams reducible in channel $r$ with $\ell$ loops connecting full vertices and will be constructed iteratively; $\bar{r}$ represents the complementary channels to channel $r$. Using the bubble functions [(\[eq:bubbles\])]{} and the channel decomposition, the multiloop flow for $\Gamma$ is compactly stated as ($\ell \geq 1$)
\[eq:multiloop\_flow\] $$\begin{aligned}
\dot{\gamma}_r^{(1)}
& =
\dot{B}_r(\Gamma, \Gamma)
\label{eq:one-loop_flow} {,}\\
\dot{\gamma}_r^{(2)}
& =
B_r \big( \dot{\gamma}_{\bar{r}}^{(1)}, \Gamma \big)
+
B_r \big( \Gamma, \dot{\gamma}_{\bar{r}}^{(1)} \big)
\label{eq:two-loop_flow} {,}\\
\dot{\gamma}_r^{(\ell+2)}
& =
B_r \big( \dot{\gamma}_{\bar{r}}^{(\ell+1)}, \Gamma \big)
+
\dot{\gamma}_{r,\textrm{C}}^{(\ell+2)}
+
B_r \big( \Gamma, \dot{\gamma}_{\bar{r}}^{(\ell+1)} \big)
\label{eq:higher-loop_flow} {,}\\
\dot{\gamma}_{r,\textrm{C}}^{(\ell+2)}
& =
B_r \big[ \Gamma,
B_r \big( \dot{\gamma}_{\bar{r}}^{(\ell)}, \Gamma \big) \big]
=
B_r \big[ B_r \big( \Gamma, \dot{\gamma}_{\bar{r}}^{(\ell)} \big),
\Gamma \big]
\label{eq:vertex_center_part}\end{aligned}$$
and illustrated in [Fig. \[fig:vertex\_flow\]]{}.
The standard truncated, one-loop flow of $\Gamma$ is simply given by [Eq. (\[eq:one-loop\_flow\])]{} \[[Fig. \[fig:vertex\_flow\]]{}(a)\]. A simplified version of this equation, in which one uses the single-scale propagator $S$ [(\[eq:singlescale\])]{} instead of $\partial_{\Lambda}G$ in the differentiated bubble [(\[eq:gdotbubble\])]{}, corresponds to the result obtained from the exact flow equation upon neglecting the six-point vertex [^2]. The form given here, with $\partial_{\Lambda}G$ instead of $S$ (also known as Katanin substitution [@Metzner2012; @Katanin2004]), already includes corrections to this originating from vertex diagrams containing differentiated self-energy contributions. In the exact flow equation, these contributions are contained in the six-point vertex $\Gamma^{(6)}$ and excluded in $S$; omitting $\Gamma^{(6)}$, they are incorporated again by $\partial_{\Lambda}G = S + G \cdot (\partial_{\Lambda} \Sigma) \cdot G$. Comparing [Eqs. (\[eq:vertex2ndOrder\])]{}, [(\[eq:gbubble\])]{}, [(\[eq:gdotbubble\])]{} with [Eq. (\[eq:one-loop\_flow\])]{} \[or [Fig. \[fig:vertex\]]{} with [Fig. \[fig:vertex\_flow\]]{}(a)\], it is clear that the one-loop flow is correct up to second order, for which only bare vertices are involved. Indeed, all differentiated diagrams of $\Gamma^{2^{\textrm{nd}}}$, which are obtained by summing all copies of diagrams in which one $G^0$ line is replaced by $\partial_{\Lambda}G^0$, are contained in $\sum_r \dot{\gamma}_r^{(1)}$. However, starting at third order, the one-loop flow [(\[eq:one-loop\_flow\])]{} does not fully generate all (parquet) diagrams, since, in the exact flow, the six-point vertex starts contributing. In mfRG, the two-loop flow \[[Eq. (\[eq:two-loop\_flow\])]{}, [Fig. \[fig:vertex\_flow\]]{}(b)\] completes the derivative of third-order diagrams of $\Gamma$ (i.e., it contains all diagrams needed to ensure that $\dot{\gamma}_r^{(1)}+\dot{\gamma}_r^{(2)}$ fully represent $\partial_{\Lambda} \gamma_r^{\textrm{3rd}}$). An example is given in [Fig. \[fig:examples\]]{}(a), which shows a parquet diagram reducible in channel $a$. The differentiated diagram in [Fig. \[fig:examples\]]{}(d), as part of the derivative of [Fig. \[fig:examples\]]{}(a), is not included in the one-loop flow. The reason is that $\dot{\gamma}_{a}^{(1)}$ only contains vertices connected by antiparallel $G^0$-$\partial_{\Lambda}G^0$ lines, and not parallel ones, as would be necessary for this differentiated diagram. It is, however, included in the two-loop correction to the flow, as can be seen by inserting the lowest-order contributions for all vertices into the first summand on the r.h.s. of $\dot{\gamma}_{a}^{(2)}$ (using $\dot{\gamma}_{p}^{(1)}$) in [Fig. \[fig:vertex\_flow\]]{}(b).
![(a-c) Some diagrams that are included in the parquet approximation and only partially contained in one-loop fRG. (d-f) One particular differentiated diagram for each of the diagrams (a-c) \[the (gray, thin) line with a dash stands for $\partial_{\Lambda} G^0$\] that is not part of the standard truncated flow, but included in mfRG.[]{data-label="fig:examples"}](pyfeynnb_examples.pdf){width=".48\textwidth"}
At all higher loop orders ($\ell+2 \geq 3$) \[[Eq. (\[eq:higher-loop\_flow\])]{}, [Fig. \[fig:vertex\_flow\]]{}(c)\], we iterate this scheme and further add the *center part* [(\[eq:vertex\_center\_part\])]{} of the vertex flow. This connects the $\ell$-loop flow from the complementary ($\bar{r}$) channels by $r$ bubbles on both sides, and is needed to complete the derivative of parquet diagrams starting at fourth order. Since $\dot{\gamma}_{r,\textrm{C}}^{(\ell+2)}$ raises the loop order by two, it was still absent in the two-loop flow. The three summands in $\dot{\gamma}_{r}^{(\ell+2)}$, including $\dot{\gamma}_{r,\textrm{C}}^{(\ell+2)}$, exhaust all possibilities to obtain differentiated vertex diagrams in channel $r$ at loop order $\ell+2$ in an iterative one-loop procedure. The mfRG vertex flow up to loop order $\ell$ therefore fully captures all parquet diagrams up to order $n=\ell+1$ in the interaction (cf. [Sec. \[sec:diagr\_count\]]{}).
Flow equation for the self-energy
---------------------------------
The self-energy has an *exact* fRG flow equation, which simply connects the four-point vertex with the single-scale propagator (cf. [Fig. \[fig:sigma\_flow\]]{}). However, if a vertex obtained from the truncated vertex flow is inserted into this standard self-energy flow equation, it generates diagrams that are only partially differentiated. In fact, even after correcting the vertex flow via mfRG to obtain all parquet diagrams of $\Gamma$, $\dot{\Sigma}_{\textrm{std}}$ does not yet form a total derivative. Although $\dot{\Sigma}_{\textrm{std}}$ is in principle exact \[as is the SDE [(\[eq:SchwingerDyson\])]{}\], using the *parquet* vertex in this flow gives a less accurate result than inserting it into the SDE: All diagrams obtained from $\dot{\Sigma}_{\textrm{std}}$ are of the parquet type, but their derivatives are not fully generated by the standard flow equation. This problem can be remedied by adding multiloop corrections to the self-energy flow, which complete the derivative of all involved diagrams. The corrections consist of two additions that build on the center parts [(\[eq:vertex\_center\_part\])]{} of the vertex flow in the $a$ and $p$ channels, $$\dot{\gamma}_{\bar{t},\textrm{C}} =
\sum_{\ell \geq 1}
\big(
\dot{\gamma}_{a,\textrm{C}}^{(\ell)} +
\dot{\gamma}_{p,\textrm{C}}^{(\ell)} \big)
{.}\label{eq:gamma_tbar_c}$$ Using the self-energy loop [(\[eq:sigmaloop\])]{}, the mfRG flow equation for $\Sigma$ is then given by (cf. [Fig. \[fig:sigma\_flow\]]{})
\[eq:sigma\_flow\]
\_ & = \_ + \_[|[t]{}]{} + \_[t]{} [,]{}& \_ & = L(, S) [,]{}\[eq:sigma\_flow\_std\]\
\_[|[t]{}]{} & = L(\_[|[t]{},]{}, G) [,]{}& \_[t]{} & = L ( , G \_[|[t]{}]{} G ) [.]{}\[eq:sigma\_flow\_t\]
Note that self-energy diagrams in $\dot{\Sigma}_{t}$ and $\dot{\Sigma}_{\bar{t}}$ are reducible and irreducible in the $t$ channel, respectively. However, here, this property is not exclusive; $\dot{\Sigma}_{\textrm{std}}$, too, contains diagrams that are reducible and irreducible in the $t$ channel, as is directly seen by inserting the second-order vertex from [Fig. \[fig:vertex\]]{} into the first summand of [Fig. \[fig:sigma\_flow\]]{}.
![Multiloop flow equation for the self-energy, adding two corrections ($\dot{\Sigma}_{\bar{t}}$, $\dot{\Sigma}_{t}$) to the standard fRG flow, $\dot{\Sigma}_{\textrm{std}}$. The (black, thick) line with a dash denotes the single-scale propagator $S$.[]{data-label="fig:sigma_flow"}](pyfeynnb_sigma_flow.pdf){width=".4\textwidth"}
To motivate the addition of $\dot{\Sigma}_{\bar{t}}$ and $\dot{\Sigma}_{t}$, let us consider the first examples where multiloop corrections are needed to complete the derivative of diagrams, which occur at fourth and fifth order, respectively. The diagram in [Fig. \[fig:examples\]]{}(b) is obtained by inserting the $\gamma_a$ diagram from [Fig. \[fig:examples\]]{}(a) (and the symmetry-related $\gamma_t$ diagram) into the SDE \[[Fig. \[fig:sd\]]{}(b)\]. The differentiated diagram in [Fig. \[fig:examples\]]{}(e) is part of the derivative of [Fig. \[fig:examples\]]{}(b), but not contained in the standard flow. In fact, the vertex needed for this diagram to be part of $\dot{\Sigma}_{\textrm{std}}$ \[i.e., the vertex obtained by cutting the differentiated line in [Fig. \[fig:examples\]]{}(e)\] is a so-called envelope vertex, the lowest-order realization of a nonparquet vertex \[cf. [Fig. \[fig:sd\]]{}(b)\] [^3]. The diagram from [Fig. \[fig:examples\]]{}(e) is, however, included in the first correction $\dot{\Sigma}_{\bar{t}}$, as can be seen by inserting the lowest-order contributions of all vertices in the center part of $\dot{\gamma}_{a}^{(3)}$ (using again $\dot{\gamma}_{p}^{(1)}$) in [Fig. \[fig:vertex\_flow\]]{}(c) and connecting the top lines.
{width=".99\textwidth"}
Inserting the self-energy diagram from [Fig. \[fig:examples\]]{}(b) into the full propagator of the first summand in the SDE \[[Fig. \[fig:sd\]]{}(b)\] yields the diagram in [Fig. \[fig:examples\]]{}(c). Similar to the previous discussion, one finds that the differentiated diagram in [Fig. \[fig:examples\]]{}(f), needed for the full derivative of [Fig. \[fig:examples\]]{}(c), is neither contained in $\dot{\Sigma}_{\textrm{std}}$ nor $\dot{\Sigma}_{\bar{t}}$. It is, however, included in the second mfRG correction, $\dot{\Sigma}_{t}$, as one of the lowest-order realizations of the last summand in [Fig. \[fig:sigma\_flow\]]{}. The two extra terms of the mfRG self-energy flow, $\dot{\Sigma}_{\bar{t}}$ and $\dot{\Sigma}_{t}$, incorporate the whole multiloop hierarchy of differentiated vertex diagrams via $\dot{\gamma}_{\bar{t},\textrm{C}}$ \[[Eq. (\[eq:gamma\_tbar\_c\])]{}\]. As is discussed in the following subsections, they suffice to generate all parquet diagrams of $\Sigma$ and, therefore, provide the full dressing of the parquet vertex in return.
Justification
-------------
We will now justify our claim that the mfRG flow fully generates all parquet diagrams for $\Gamma$ and $\Sigma$. We will first show that all differentiated diagrams in mfRG are of the parquet type and that there is no overcounting of diagrams. Concerning the vertex, this has already been done for the two-channel case of the X-ray-edge singularity [@Kugler2017]. The arguments for the general case are in fact completely analogous and repeated here for the sake of completeness. The self-energy is discussed thereafter. The only totally irreducible contribution to the four-point vertex in the mfRG flow is the bare interaction stemming from the initial condition of the vertex, $\Gamma_{\Lambda_i}=\Gamma^0$. All further diagrams on the r.h.s. of the flow equations are obtained by iteratively combining two vertices by one of the three bubbles from [Eq. (\[eq:bubbles\])]{}. Hence, they correspond to differentiated *parquet* diagrams in the respective channel. The fact that there is no overcounting in mfRG, i.e., that each diagram occurs at most once, can be seen employing arguments of diagrammatic reducibility and the unique position of the differentiated line in the diagrams. To be specific, let us consider here the $a$ channel; the arguments for the other channels are completely analogous. First, we note that diagrams in the one-loop term always differ from higher-loop ones. The reason is that, in higher-loop terms, the differentiated line appears in the vertex coming from $\partial_{\Lambda} \gamma_{\bar{a}}$. This can never contain two vertices connected by an $a$ $G$-$\partial_{\Lambda}G$ bubble, since such terms only originate upon differentiating $\gamma_{a}$, the vertex reducible in $a$ lines. Second, diagrams in the left, center, or right part \[first, second, and third summand in [Fig. \[fig:vertex\_flow\]]{}(c), respectively\] of an $\ell$-loop contribution always differ. This is because the vertex $\gamma^{(\ell)}_{\bar{a}}$ is irreducible in $a$ lines. The left part is then reducible in $a$ lines *only after* the differentiated line appeared, the right part *only before*, and the center part is reducible in this channel *before and after* $\partial_{\Lambda}G$. Third, the same parts (say, the left parts) of different-order loop contributions ($\ell \neq \ell'$) are always different. Assume they agreed: As the $a$ bubble induces the first reducibility in this channel, already $\gamma^{(\ell)}_{\bar{a}}$ and $\gamma^{(\ell')}_{\bar{a}}$ would have to agree. For these, only the same parts can agree, as mentioned before. The argument then proceeds iteratively until one compares the one-loop part to a higher-loop ($|\ell - \ell'| + 1$) one. These are, however, distinct according to the first point. Concerning the self-energy, all diagrams of the flow belong to the parquet type, since they are constructed from (differentiated) parquet vertices by closing loops of external legs in an iterative one-loop procedure. By cutting one $G^0$ or the $\partial_{\Lambda}G^0$ line in such a self-energy diagram, one can always obtain a (differentiated) parquet vertex with possibly dressed amputated legs. First, there is no overcounting between $\dot{\Sigma}_{\textrm{std}}$ and $\dot{\Sigma}_{\bar{t}}$ because cutting the differentiated line in $\dot{\Sigma}_{\textrm{std}}$ generates a parquet vertex (with possibly dressed amputated legs coming from the single-scale propagator; cf. [Fig. \[fig:sigma\_flow\]]{}), whereas this is not the case for $\dot{\Sigma}_{\bar{t}}$. To illustrate this statement, we consider in [Fig. \[fig:sigma\_proof1\]]{} a typical case of a $\dot{\Sigma}_{\bar{t}}$ correction, where we take the $a$ part of $\dot{\gamma}_{\bar{t},\textrm{C}}$ \[cf. [Eq. (\[eq:gamma\_tbar\_c\])]{}\] with $\partial_{\Lambda} \gamma_{t}$ in the center. We can insert the BSE $\gamma_{t} = B_t(I_t, \Gamma)$ ([Fig. \[fig:bs\]]{}) and consider simultaneously all scenarios where the differentiated line, originating from $\partial_{\Lambda} \gamma_{t}$, is contained in any of the dashed parts. To be even more specific, we take a specific part of $I_{t}=R+\gamma_a+\gamma_p$, namely $\gamma_{a}=B_a(I_a,\Gamma)$ ([Fig. \[fig:bs\]]{}), and consider the cases where the differentiated line, if contained in $I_{t}$, is contained in the corresponding bubble. If one now cuts any of the dashed lines, as candidates for the differentiated line, one finds that the remaining vertex is *not* of the parquet type, as it is not reducible in any of the two-particle channels. The same irreducibility in three lines, when starting to cut the differentiated line in $\dot{\gamma}_{\bar{t},\textrm{C}}$, occurs in all diagrammatic realizations of $\dot{\Sigma}_{\bar{t}}$. Since the standard flow $\dot{\Sigma}_{\textrm{std}}$ with the *full* instead of the *parquet* vertex is exact, it follows that the $\dot{\Sigma}_{\bar{t}}$ part can be written similarly as $\dot{\Sigma}_{\textrm{std}}$, but using a *nonparquet* (np) vertex \[[Fig. \[fig:sigma\_proof2\]]{}(a)\]. As a consequence, $\dot{\Sigma}_{t}$, obtained by connecting $\dot{\Sigma}_{\bar{t}}$ and $\Gamma$ by a $t$ bubble, can similarly be written with a nonparquet vertex \[[Fig. \[fig:sigma\_proof2\]]{}(b)\]. Thus, there cannot be any overcounting between $\dot{\Sigma}_{\textrm{std}}$ and $\dot{\Sigma}_{t}$, either. Finally, there is likewise no overcounting between $\dot{\Sigma}_{\bar{t}}$ and $\dot{\Sigma}_{t}$: After removing the differentiated line in $\dot{\Sigma}_{\bar{t}}$, the remaining nonparquet vertex $\Gamma_{\textrm{np}}$ is in particular irreducible in the $t$ channel (as was discussed above). However, removing the differentiated line in $\dot{\Sigma}_{t}$ after expressing $\dot{\Sigma}_{\bar{t}}$ via $\Gamma_{\textrm{np}}$ \[cf. [Fig. \[fig:sigma\_proof2\]]{}(b)\], the remaining vertex $\Gamma'_{\textrm{np}}$ is by construction reducible in $t$ lines (although not a parquet vertex).
![Rewriting of the corrections to the self-energy flow: (a) $\dot{\Sigma}_{\bar{t}}$ can be expressed by a nonparquet vertex $\Gamma_{\textrm{np}}$ contracted with the single-scale propagator $S$. (b) $\dot{\Sigma}_{t}$, obtained by connecting $\dot{\Sigma}_{\bar{t}}$ and $\Gamma$ by a $t$ bubble, then involves a bubble connecting a nonparquet and parquet vertex, which yields another nonparquet vertex $\Gamma'_{\textrm{np}}$, contracted with $S$.[]{data-label="fig:sigma_proof2"}](pyfeynnb_sigma_proof2.pdf){width=".48\textwidth"}
In summary, all diagrams of the four-point vertex and self-energy generated by the mfRG flow belong to the parquet class and are included at most once. To show that the mfRG flow generates *all* differentiated parquet diagrams, we will demonstrate next that, at any given order in the interaction, their number is equal to the number of diagrams generated by the mfRG flow.
Counting of diagrams {#sec:diagr_count}
--------------------
In order to count the number of diagrams in all involved functions, we make use of either exact, self-consistent equations or the mfRG flow equations. As a first example, we count the number of diagrams in the full propagator $G$ at order $n$ in the interaction, ${\mathcal{N}}_G(n)$, given the number of diagrams in the self-energy, ${\mathcal{N}}_{\Sigma}(n)$. Concerning the bare propagator and self-energy, we know ${\mathcal{N}}_{G^0}(n) = \delta_{n,0}$ and ${\mathcal{N}}_{\Sigma}(0) = 0$. From Dyson’s equation [(\[eq:dyson\])]{}, we then get $${\mathcal{N}}_G(n) = \delta_{n,0} + \sum_{m=1}^{n} {\mathcal{N}}_{\Sigma}(m) {\mathcal{N}}_G(n-m)
{.}\label{eq:NG}$$ Defining a convolution of sequences, according to $${\mathcal{N}}_1 = {\mathcal{N}}_2 \ast {\mathcal{N}}_3
\ \Leftrightarrow \
{\mathcal{N}}_1(n) = \sum_{m=0}^{n} {\mathcal{N}}_2(m) {\mathcal{N}}_3(n-m)
\ \forall n
{,}$$ we can write [Eq. (\[eq:NG\])]{} in direct analogy to the original equation [(\[eq:dyson\])]{} as $${\mathcal{N}}_G = {\mathcal{N}}_{G^0} + {\mathcal{N}}_{G^0} \ast {\mathcal{N}}_{\Sigma{^{\vphantom{(6)}}}} \ast {\mathcal{N}}_{G{^{\vphantom{(6)}}}}
{.}$$ Similar relations for the self-energy and vertex can be obtained from the SDE [(\[eq:SchwingerDyson\])]{}, the parquet equation [(\[eq:parquet\])]{}, and the BSEs [(\[eq:BetheSalpeter\])]{}. The number of diagrams in the bare vertex is ${\mathcal{N}}_{\Gamma^0}=\delta_{n,1}$ (one can also take any ${\mathcal{N}}_{\Gamma^0} \propto \delta_{n,1}$). From the SDE [(\[eq:SchwingerDyson\])]{}, we get for the self-energy $$\begin{aligned}
{\mathcal{N}}_{\Sigma} = {\mathcal{N}}_{\Gamma^0} \ast {\mathcal{N}}_{G{^{\vphantom{(6)}}}}
+ \tfrac{1}{2}\,
{\mathcal{N}}_{\Gamma^0} \ast {\mathcal{N}}_{G{^{\vphantom{(6)}}}} \ast {\mathcal{N}}_{G{^{\vphantom{(6)}}}} \ast {\mathcal{N}}_{G{^{\vphantom{(6)}}}} \ast {\mathcal{N}}_{\Gamma{^{\vphantom{(6)}}}}
{.}\label{eq:diagr_count_sd}\end{aligned}$$ Note that, when counting diagrams, we can ignore the extra minus signs but must keep track of prefactors of magnitude not equal to unity. These prefactors avoid double counting of the antisymmetric vertex [@Bickers2004] and originate from the way the diagrams are constructed [^4]. Concerning the full vertex, we can use that the symmetry relation between the $a$ and $t$ bubble given in [Eq. (\[eq:a\_t\_sym\])]{} holds for the full reducible vertices $\gamma_{a}$ and $\gamma_{t}$ [@Bickers2004], such that ${\mathcal{N}}_{\gamma_{a}}={\mathcal{N}}_{\gamma_{t}}$. In the parquet approximation $R=\Gamma^0$, and the parquet equation [(\[eq:parquet\])]{} and the BSEs [(\[eq:BetheSalpeter\])]{} yield
$$\begin{aligned}
{\mathcal{N}}_{\Gamma}
& =
{\mathcal{N}}_{R} + 2\, {\mathcal{N}}_{\gamma_a} + {\mathcal{N}}_{\gamma_p}
\label{eq:diagr_count_R}
\\
{\mathcal{N}}_{\gamma_a}
& =
( {\mathcal{N}}_{\Gamma} - {\mathcal{N}}_{\gamma_a} ) \ast {\mathcal{N}}_G \ast {\mathcal{N}}_G \ast {\mathcal{N}}_{\Gamma}
\label{eq:diagr_count_gamma_a} \\
{\mathcal{N}}_{\gamma_p}
& =
\tfrac{1}{2} ( {\mathcal{N}}_{\Gamma} - {\mathcal{N}}_{\gamma_p} ) \ast {\mathcal{N}}_G \ast {\mathcal{N}}_G \ast {\mathcal{N}}_{\Gamma}
{.}\label{eq:diagr_count_gamma_p}\end{aligned}$$
Since ${\mathcal{N}}_{\Gamma^0}(0) = 0$, these equations, just like the original equations, can be solved iteratively. Knowing the number of diagrams in all quantities up to order $n-1$ allows one to calculate them at order $n$. This can also be done numerically. Table \[tab:num\_diagr\] (first two lines) shows the number of parquet diagrams up to order 6. For large interaction order $n$, we find that the number of diagrams in the parquet vertex and self-energy grows exponentially in $n$ \[cf. [Fig. \[fig:diagrcount\]]{}(a)\]. To prove our claim that the mfRG flow generates all parquet diagrams, we must count the number of diagrams, ${\mathcal{N}}_{\dot{\Sigma}}(n)$ and ${\mathcal{N}}_{\dot{\gamma}_r}(n)$, obtained by differentiating the set of all corresponding parquet graphs. Then, we check that these numbers are exactly reproduced by the number of diagrams contained on the r.h.s.of the mfRG flow equations. A diagram of the full propagator at order $n$ has $2n+1$ internal lines, a self-energy diagram $2n-1$, and vertex diagram $2n-2$. According to the product rule, the number of differentiated diagrams is thus
$$\begin{aligned}
{\mathcal{N}}_{\dot{G}}(n) & = {\mathcal{N}}_G(n) (2n+1)
{,}\\
{\mathcal{N}}_{\dot{\Sigma}}(n) & = {\mathcal{N}}_{\Sigma}(n) (2n-1)
\label{eq:diagr_count_sigma_dot}
{,}\\
{\mathcal{N}}_{\dot{\gamma}_r}(n) & = {\mathcal{N}}_{\gamma_r}(n) (2n-2)
{.}\end{aligned}$$
[0.48]{}[@ l c c c c c c ]{}
------------------------------------------------------------------------
$n$ & 1 & 2 & 3 & 4 & 5 & 6\
\
------------------------------------------------------------------------
${\mathcal{N}}_{\Gamma}$ & 1 & 2$\frac{1}{2}$ & 15$\tfrac{1}{4}$ & 108$\tfrac{1}{8}$ & 832$\tfrac{1}{16}$ & 6753$\tfrac{21}{32}$\
${\mathcal{N}}_{\Sigma}$ & 1 & 1$\frac{1}{2}$ & 5$\frac{1}{4}$ & 25$\frac{7}{8}$ & 156$\frac{1}{16}$ & 1073$\frac{3}{32}$\
------------------------------------------------------------------------
${\mathcal{N}}_{\dot{\Gamma}}$ & 0 & 5 & 61 & 648$\frac{3}{4}$ & 6656$\frac{1}{2}$ & 67536$\frac{9}{16}$\
${\mathcal{N}}_{\dot{\Gamma}^{(1\ell)}}$ & 0 & 5 & 45 & 373$\frac{3}{4}$ & 3117$\frac{1}{2}$ & 26519$\frac{1}{16}$\
${\mathcal{N}}_{\dot{\Gamma}^{(2\ell)}}$ & 0 & 0 & 16 & 216 & 2264 & 21972\
${\mathcal{N}}_{\dot{\Gamma}^{(3\ell)}}$ & 0 & 0 & 0 & 59 & 1062 & 13481$\frac{1}{2}$\
${\mathcal{N}}_{\dot{\Gamma}^{(4\ell)}}$ & 0 & 0 & 0 & 0 & 213 & 4792$\frac{1}{2}$\
${\mathcal{N}}_{\dot{\Gamma}^{(5\ell)}}$ & 0 & 0 & 0 & 0 & 0 & 771$\frac{1}{2}$\
------------------------------------------------------------------------
${\mathcal{N}}_{\dot{\Sigma}}$ & 1 & 4$\frac{1}{2}$ & 26$\frac{1}{4}$ & 181$\frac{1}{8}$ & 1404$\frac{9}{16}$ & 11804$\frac{1}{32}$\
${\mathcal{N}}_{\dot{\Sigma}_{\textrm{std}}}$ & 1 & 4$\frac{1}{2}$ & 26$\frac{1}{4}$ & 177$\frac{1}{8}$ & 1311$\frac{9}{16}$ & 10348$\frac{1}{32}$\
${\mathcal{N}}_{\dot{\Sigma}_{\bar{t}}}$ & 0 & 0 & 0 & 4 & 89 & 1349\
${\mathcal{N}}_{\dot{\Sigma}_{t}}$ & 0 & 0 & 0 & 0 & 4 & 107\
\
From the mfRG flow of the vertex \[[Eq. (\[eq:multiloop\_flow\])]{}\], we deduce
$$\begin{aligned}
{\mathcal{N}}_{\dot{\gamma}_{a}^{(1)}}
& = 2\, {\mathcal{N}}_{\Gamma{\vphantom{\dot{G}}}} \ast {\mathcal{N}}_{\dot{G}} \ast {\mathcal{N}}_{G{\vphantom{\dot{G}}}} \ast {\mathcal{N}}_{\Gamma{\vphantom{\dot{G}}}}
{,}\\
{\mathcal{N}}_{\dot{\gamma}_{p}^{(1)}}
& = {\mathcal{N}}_{\Gamma{\vphantom{\dot{G}}}} \ast {\mathcal{N}}_{\dot{G}} \ast {\mathcal{N}}_{G{\vphantom{\dot{G}}}} \ast {\mathcal{N}}_{\Gamma{\vphantom{\dot{G}}}}
{,}\\
{\mathcal{N}}_{\dot{\gamma}_{a}^{(2)}}
& = 2\, ( {\mathcal{N}}_{\dot{\gamma}^{(1)}_a} + {\mathcal{N}}_{\dot{\gamma}^{(1)}_p} )
\ast {\mathcal{N}}_{\Pi} \ast {\mathcal{N}}_{\Gamma}
{,}\\
{\mathcal{N}}_{\dot{\gamma}_{p}^{(2)}}
& = 2\, {\mathcal{N}}_{\dot{\gamma}^{(1)}_a} \ast {\mathcal{N}}_{\Pi} \ast {\mathcal{N}}_{\Gamma}
{,}\end{aligned}$$
where ${\mathcal{N}}_{\Pi} = {\mathcal{N}}_G \ast {\mathcal{N}}_G$ denotes the number of diagrams in a bubble. For $\ell+2 \geq 3$, we have
$$\begin{aligned}
{\mathcal{N}}_{\dot{\gamma}_{a}^{(\ell+2)}}
& = 2\, ( {\mathcal{N}}_{\dot{\gamma}^{(\ell+1)}_a} + {\mathcal{N}}_{\dot{\gamma}^{(\ell+1)}_p} )
\ast {\mathcal{N}}_{\Pi} \ast {\mathcal{N}}_{\Gamma}
\nonumber \\
& \ +
{\mathcal{N}}_{\Gamma} \ast {\mathcal{N}}_{\Pi} \ast
( {\mathcal{N}}_{\dot{\gamma}^{(\ell)}_a} + {\mathcal{N}}_{\dot{\gamma}^{(\ell)}_p} )
\ast {\mathcal{N}}_{\Pi} \ast {\mathcal{N}}_{\Gamma}
{,}\\
{\mathcal{N}}_{\dot{\gamma}_{p}^{(\ell+2)}}
& = 2\, {\mathcal{N}}_{\dot{\gamma}^{(\ell+1)}_a}
\ast {\mathcal{N}}_{\Pi} \ast {\mathcal{N}}_{\Gamma}
\nonumber \\
& \ +
\tfrac{1}{2} \,
{\mathcal{N}}_{\Gamma} \ast {\mathcal{N}}_{\Pi} \ast
{\mathcal{N}}_{\dot{\gamma}^{(\ell)}_a}
\ast {\mathcal{N}}_{\Pi} \ast {\mathcal{N}}_{\Gamma}
{.}\end{aligned}$$
Summing all loop contributions yields $$\textstyle
{\mathcal{N}}_{\dot{\gamma}_a}^{\textrm{mfRG}}
=
\sum_{\ell \geq 1} {\mathcal{N}}_{\dot{\gamma}^{(\ell)}_a}
{,}\quad
{\mathcal{N}}_{\dot{\gamma}_p}^{\textrm{mfRG}}
=
\sum_{\ell \geq 1} {\mathcal{N}}_{\dot{\gamma}^{(\ell)}_p}
{.}$$ For the flow of the self-energy [(\[eq:sigma\_flow\])]{}, we need the center part of the vertex flow in the $a$ and $p$ channel, for which the number of diagrams sums up to $$\begin{aligned}
{\mathcal{N}}_{\dot{\gamma}_{\bar{t},\textrm{C}}}
& =
{\mathcal{N}}_{\Gamma} \ast {\mathcal{N}}_{\Pi} \ast
\Big( \tfrac{3}{2} \,
{\mathcal{N}}_{\dot{\gamma}_a}^{\textrm{mfRG}} + {\mathcal{N}}_{\dot{\gamma}_p}^{\textrm{mfRG}}
\Big)
\ast {\mathcal{N}}_{\Pi} \ast {\mathcal{N}}_{\Gamma}
{.}\end{aligned}$$ The number of diagrams in the single-scale propagator $S$ [(\[eq:singlescale\])]{} can be obtained from two equivalent relations
$$\begin{aligned}
{\mathcal{N}}_{S{\vphantom{\dot{G}}}}
& =
{\mathcal{N}}_{\dot{G}} - {\mathcal{N}}_{G{\vphantom{\dot{G}}}} \ast N_{\dot{\Sigma}} \ast {\mathcal{N}}_{G{\vphantom{\dot{G}}}}
\\
& =
( {\mathcal{N}}_{\mathbbm{1}{\vphantom{\dot{G}}}} + {\mathcal{N}}_{G{\vphantom{\dot{G}}}} \ast {\mathcal{N}}_{\Sigma{\vphantom{\dot{G}}}} ) \ast {\mathcal{N}}_{\dot{G}^0} \ast
( {\mathcal{N}}_{\mathbbm{1}{\vphantom{\dot{G}}}} + {\mathcal{N}}_{\Sigma{\vphantom{\dot{G}}}} \ast {\mathcal{N}}_{G{\vphantom{\dot{G}}}} )
{,}\end{aligned}$$
with ${\mathcal{N}}_{\dot{G}^0}(n) = \delta_{n,0} = {\mathcal{N}}_{\mathbbm{1}}(n)$. From [Eq. (\[eq:sigma\_flow\])]{}, we then get
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\_[\_[|[t]{}]{}]{} & = \_[\_[|[t]{},]{}]{} \_[G]{} [,]{}& \_[\_[t]{}]{} & = \_ \_ \_[\_[|[t]{}]{}]{} [.]{}\[eq:diagr\_count\_sigma\_flow\]
![Logarithmic plots for the number of diagrams at interaction order $n$ for both vertex and self-energy. (a) ${\mathcal{N}}_{\Gamma}$, ${\mathcal{N}}_{\Sigma}$ grow exponentially for large $n$ (inset: the ratio of subsequent elements approaches a constant). (b) The cumulative low-loop vertex flows ($1\ell$ up to $5\ell$) and the self-energy flows $\dot{\Sigma}_{\textrm{std}}$ (labeled std) and $\dot{\Sigma}_{\textrm{std}}+\dot{\Sigma}_{\bar{t}}$ (labeled $\bar{t}$) miss differentiated parquet diagrams. However, the full multiloop flow for vertex and self-energy generates all differentiated parquet diagrams to arbitrary order in the interaction.[]{data-label="fig:diagrcount"}](diagr_count.pdf){width=".48\textwidth"}
Numerically, one can check order for order in the interaction \[cf. Table \[tab:num\_diagr\] and [Fig. \[fig:diagrcount\]]{}(b)\] that, indeed, the mfRG flow generates exactly the same number of diagrams as obtained by differentiating all parquet diagrams, i.e., $$\begin{aligned}
{\mathcal{N}}_{\dot{\gamma}_r}^{\textrm{\vphantom{G}}}(n)
& =
{\mathcal{N}}_{\dot{\gamma}_r}^{\textrm{mfRG}}(n)
{,}\quad
{\mathcal{N}}_{\dot{\Sigma}}^{\textrm{\vphantom{G}}}(n)
=
{\mathcal{N}}_{\dot{\Sigma}}^{\textrm{mfRG}}(n)
\quad
\forall n
{.}\end{aligned}$$ This demonstrates the equivalence between solving the multiloop fRG flow and solving the (first-order) parquet equations for a general model.
Computational aspects {#sec:mfRG_comp}
---------------------
All contributions to the mfRG flow—for the vertex as well as for the self-energy—are of an iterative one-loop structure and hence well suited for numerical algorithms. In fact, by keeping track of the left (L) and right (R) summands in the higher-loop vertex flow [(\[eq:higher-loop\_flow\])]{} $$\dot{\gamma}_{r,\textrm{L}}^{(\ell+2)}
=
B_r \big( \gamma_{\bar{r}}^{(\ell+1)}, \Gamma \big)
{,}\quad
\dot{\gamma}_{r,\textrm{R}}^{(\ell+2)}
=
B_r \big( \Gamma, \gamma_{\bar{r}}^{(\ell+1)} \big)
{,}$$ the center part [(\[eq:vertex\_center\_part\])]{} can be efficiently computed as $$\dot{\gamma}_{r,\textrm{C}}^{(\ell+2)}
=
B_r \big( \Gamma, \gamma_{r,\textrm{L}}^{(\ell+1)} \big)
=
B \big( \gamma_{r,\textrm{R}}^{(\ell+1)}, \Gamma \big)
{.}$$ Consequently, the numerical effort in the multiloop corrections of the vertex flow scales linearly in $\ell$. The self-energy flow [(\[eq:sigma\_flow\])]{} is already stated with one integration only.
The (standard) fRG hierarchy of flow equations constitutes a (first-order) ordinary differential equation. Neglecting the six-point vertex, it can be written as $$\partial_{\Lambda} \Sigma = f_{\Sigma}^{\textrm{std}} (\Lambda, \Sigma, \Gamma)
{,}\quad
\partial_{\Lambda} \Gamma = f_{\Gamma}^{\textrm{std}} (\Lambda, \Sigma, \Gamma)
{,}$$ where, here and henceforth, $f$ denotes the part of the r.h.s. of the flow equation corresponding to its indices. Improving this approximation by adding differentiated self-energy contributions in the vertex flow (as is also done in mfRG), $f_{\Gamma}^{\textrm{std}}$ is replaced by another function $\tilde{f}_{\Gamma}^{\textrm{std}} (\Lambda, \Sigma, \Gamma, \partial_{\Lambda} \Sigma)$, which further depends on the $\Lambda$ derivative of the self-energy. Such a differential equation is still feasible for many algorithms as one can simply compute $\partial_{\Lambda} \Sigma$ first and use it in the calculation of $\partial_{\Lambda} \Gamma$. However, the full mfRG flow for the vertex and self-energy has the form $$\partial_{\Lambda} \Sigma =
f_{\Sigma} (\Lambda, \Sigma, \Gamma, \partial_{\Lambda} \Gamma)
{,}\quad
\partial_{\Lambda} \Gamma =
f_{\Gamma} (\Lambda, \Sigma, \Gamma, \partial_{\Lambda} \Sigma)
{,}$$ in which derivatives occur on all parts of the r.h.s., yielding an algebraic (as opposed to ordinary) differential equation. Techniques to solve algebraic differential equations exist, but a discussion of them exceeds the scope of this paper. Let us merely suggest an approximate solution strategy that reduces the mfRG flow to an ordinary differential equation, has no computational overhead, and deviates from the exact flow starting at sixth order in the interaction, summarized as follows:
\[eq:approx\_mfRG\_sol\] $$\begin{aligned}
\dot{\Sigma}_{\textrm{std}}
& =
f_{\dot{\Sigma}_{\textrm{std}}} (\Lambda, \Sigma, \Gamma)
{,}\label{eq:approx_mfRG_sol_1} \\
\partial_{\Lambda} \Gamma
& \approx
\dot{\Gamma}_{\textrm{approx}}
=
f_{\Gamma}
(\Lambda, \Sigma, \Gamma, \partial_{\Lambda} \Sigma=\dot{\Sigma}_{\textrm{std}} )
{,}\label{eq:approx_mfRG_sol_2} \\
\partial_{\Lambda} \Sigma
& \approx
\dot{\Sigma}_{\textrm{std}} +
f_{\dot{\Sigma}_{\bar{t}}}
(\Lambda, \Sigma, \partial_{\Lambda} \Gamma = \dot{\Gamma}_{\textrm{approx}})
\nonumber \\ & \ +
f_{\dot{\Sigma}_{t}}
(\Lambda, \Sigma, \partial_{\Lambda} \Gamma = \dot{\Gamma}_{\textrm{approx}})
{.}\label{eq:approx_mfRG_sol_3}\end{aligned}$$
According to this scheme, one computes first the standard flow of the self-energy, which deviates from the full $\Sigma$ flow at interaction order $U^4$. Inserting this into the vertex flow yields an approximate vertex derivative, $\dot{\Gamma}_{\textrm{approx}}$, where deviations from the full flow, induced by the approximate form of $\partial_{\Lambda} \Sigma$, start at order $U^6$. The center part of the vertex flow involves at least four vertices, such that deviations, induced by the self-energy, start at order $U^8$. The resulting, approximate $\dot{\gamma}_{\bar{t},\textrm{C}}$ can then be used to complete $\partial_{\Lambda} \Sigma$, adding the terms $\dot{\Sigma}_{\bar{t}}$ and $\dot{\Sigma}_{t}$, such that the self-energy flow is correctly computed up to errors of order $U^8$. Evidently, this scheme can also be iterated \[using [Eqs. (\[eq:approx\_mfRG\_sol\_2\])]{} and [(\[eq:approx\_mfRG\_sol\_3\])]{}\], increasing the accuracy by four orders with each step. We have attached a pseudocode for such a solution strategy of the mfRG flow in [Appendix \[appendix1\]]{}.
[General aspects]{}
-------------------
Since the standard fRG flow for the self-energy and four-point vertex—*including* the six-point vertex—is exact, all mfRG corrections can be understood as fully simulating the effect of the six-point vertex on parquet diagrams of $\Sigma$ and $\Gamma$. For instance, the two-loop corrections to the vertex flow and the Katanin substitution in the improved one-loop flow equation contain all third-order contributions of the six-point vertex [@Kugler2017a; @Katanin2004; @Eberlein2014]. Nevertheless, in the standard fRG hierarchy of flow equations, the parquet graphs comprise $n$-point vertices of arbitrary order ($n$) [@Kugler2017a], such that a non-diagrammatic derivation of mfRG based on this hierarchy appears rather difficult. Conversely, the derivation of the mfRG flow does not rely on the fRG hierarchy or properties of the (quantum) effective action; it can thus be understood independently and without prior knowledge of fRG. The mfRG flow at the two- or higher-loop level is exact up to third order in the interaction and therefore naturally fulfills Ward identities with accuracy $\textit{O}(\Gamma^4)$, compared to $\textit{O}(\Gamma^3)$ in the case of one-loop fRG [@Katanin2004]. Yet, since the parquet self-energy is exact up to *fourth* order but the parquet vertex only up to *third* order, such identities are typically violated starting at fourth order. One can think of schemes to extend mfRG beyond the parquet approximation. However, we find those rather impracticable and only briefly mention them in [Appendix \[appendix2\]]{}. Furthermore, the mfRG flow is applicable for any initial condition of the vertex functions. Whereas the choice $G_{\Lambda_i}=0$ used here leads to a summation of all parquet diagrams, starting the mfRG flow from the local quantities of dynamical mean-field theory (DMFT) [@Georges1996; @Taranto2014] allows one to add nonlocal correlations, similarly to solving the parquet equations in the dynamical vertex approximation (D$\Gamma$A) [@Toschi2007; @Held2008; @Valli2010]. However, contrary to D$\Gamma$A, the mfRG flow is built on the full vertex $\Gamma^{(4)}_{\textrm{DMFT}}$ and does not require the *diagrammatic* decomposition of the *nonperturbative* vertex [^5] $\Gamma^{(4)}_{\textrm{DMFT}} = R + \sum_r \gamma_r$ that leads to diverging results close to a quantum phase transition [@Schaefer2013; @Schaefer2016; @Gunnarsson2017]. Inspecting the one-loop flow equations of the vertex once more, we observe that diagrams on the r.h.s.contain the differentiated propagator *only* in the two-particle lines that induce the reducibility. Propagators which appear in two-particle lines which do not induce the reducibility are not differentiated. Therefore, only those diagrams that are reducible in *all* positions of two-particle lines—the so-called ladder diagrams—are fully included. It follows that the standard truncated, one-loop fRG flow is biased towards *ladder* constructions of the four-point vertex. For a constant interaction $U$ and a transfer energy-momentum $\Omega$, ladder diagrams of a certain channel can easily be summed to $\Gamma^{\textrm{ladder}}_{\Omega} = U ( 1 - U \Pi_{\Omega} )^{-1}$, where $\Pi_{\Omega}$ is the corresponding bubble. Ladder diagrams are therefore particularly prone to divergences with increasing $U$ or increasing values of $\Pi_{\Omega}$ (as can occur upon lowering the cutoff scale $\Lambda$) and can thus be responsible for premature vertex divergences in fRG. Indeed, so far, fRG computations have often suffered from such vertex divergences, and the flow had be stopped at finite RG scale $\Lambda_c$ [@Metzner2012; @Eberlein2014a]. In this context, the two-loop corrections have been found to significantly reduce the critical scale of vertex divergences $\Lambda_c$ [@Eberlein2014; @Rueck2017]. This suggests that it would be worthwhile to study the effect of higher-loop mfRG corrections—we expect that they reduce $\Lambda_c$ even further. Throughout this paper, we have taken a perspective that views fRG as a tool to resum diagrams (say, *physical* diagrams) by integrating a collection of *differentiated* (and thus $\Lambda$-dependent) diagrams. In this regard, the mfRG corrections do not add new *physical* diagrams to the flow, they only add *differentiated* diagrams to complete those derivatives of physical diagrams that are only partially contained by one-loop fRG. In other words, for any physical diagram to which a differentiated diagram of mfRG contributes, there also exists a differentiated diagram in one-loop fRG. The differentiated diagrams of the higher-loop corrections *and* the one-loop flow all contribute the *same* set of physical diagrams—the parquet diagrams. Whereas the *one-loop* flow of the vertex contains differentiated propagators at the two-particle-reducible positions, the *multiloop* flow iteratively adds those parts for which the differentiated line is increasingly nested. Such non-ladder contributions are crucial to suppress vertex divergences originating from the summation of ladder diagrams [@Kugler2017]. Similarly, the *standard* self-energy flow does not form a total derivative any more if one has only the parquet vertex at one’s disposal. All diagrams of the standard flow are of the parquet type, but differentiated lines in heavily nested positions are omitted (cf. [Fig. \[fig:examples\]]{}). The mfRG corrections incorporate all remaining contributions by two additions that build up on the multiloop vertex flow. Altogether, the mfRG flow achieves a full summation of all parquet diagrams of the vertex and self-energy. Consequently, mfRG solutions are no longer dependent on the specific way the $\Lambda$ dependence (regulator) was introduced [@Kugler2017] and thus fully implement the meaning of the original fRG idea.
Conclusion {#sec:conclusion}
==========
We have presented multiloop fRG flow equations for the four-point vertex and self-energy, formulated for the general fermionic many-body problem. The mfRG corrections fully simulate the effect of the six-point vertex on parquet diagrams, completing the derivatives of diagrams that are only partially contained in the standard truncated fRG flow. Whereas one-loop fRG contains differentiated propagators only at the two-particle-reducible positions and the standard self-energy flow does not suffice to form a total derivative when having only the parquet vertex at one’s disposal, the multiloop iteration adds all remaining parts, where the differentiated line appears at increasingly nested positions. We have motivated the multiloop corrections at low orders and ruled out any overcounting of diagrams. Moreover, we have put forward a simple recipe to count diagrams and numerically check that the mfRG flow generates all differentiated parquet diagrams for the vertex and self-energy, order for order in the interaction. Due to its iterative one-loop structure, the mfRG flow is well suited for efficient numerical computations. We have given a simple approximation, which renders the algebraic differential equation accessible to standard solvers for ordinary differential equations and exhibits only minor deviations from the full mfRG flow. Given the general formulation, the benefits of mfRG on physical problems can be exploited in a large number of fRG applications. The full resummation of parquet diagrams via mfRG eliminates the bias of fRG computations towards divergent ladder constructions of the vertex and restores the independence on the choice of regulator. We expect that this will generically enhance the usefulness of the truncated fRG framework and increase the robustness of the physical conclusions drawn from fRG results.
We thank S. Jakobs for pointing out the need for multiloop corrections to the self-energy flow and W. Metzner and A. Toschi for useful discussions. We acknowledge support by the Cluster of Excellence Nanosystems Initiative Munich; F.B.K. acknowledges funding from the research school IMPRS-QST.
Pseudocode implementation {#appendix1}
=========================
**Function** $f(\Lambda, \Psi$):
$S = S(\Lambda,\Psi.\Sigma)$ $G = G(\Lambda,\Psi.\Sigma)$ ${\textrm{d}}\Sigma_{\textrm{std}} = L( \Psi.\Gamma, S) $ ${\textrm{d}}\Psi.\Sigma = {\textrm{d}}\Sigma_{\textrm{std}} $
${\textrm{d}}G = S + G\cdot {\textrm{d}}\Psi.\Sigma \cdot G$ ${\textrm{d}}\gamma_r = \dot{B}_r(\Psi.\Gamma, \Psi.\Gamma, G, {\textrm{d}}G )$
/\* *jump to line 41 for one-loop fRG* \*/
${\textrm{d}}\gamma_r^{\textrm{L}} = B_r\big( \sum_{r'\neq r}{\textrm{d}}\gamma_{r'},
\Psi.\Gamma, G \big)$ ${\textrm{d}}\gamma_r^{\textrm{R}} = B_r\big( \Psi.\Gamma,
\sum_{r'\neq r}{\textrm{d}}\gamma_{r'}, G \big)$ ${\textrm{d}}\gamma_r^{\textrm{T}} = {\textrm{d}}\gamma_r^{\textrm{L}} + {\textrm{d}}\gamma_r^{\textrm{R}}$ ${\textrm{d}}\gamma_r \leftarrow {\textrm{d}}\gamma_r + {\textrm{d}}\gamma_r^{\textrm{T}}$
/\* *jump to line 41 for two-loop fRG* \*/
${\textrm{d}}\gamma_{\bar{t}}^{\textrm{C}} = 0$ ${\textrm{d}}\gamma_r^{\textrm{C}} = B_r( \Psi.\Gamma, {\textrm{d}}\gamma_{r}^{\textrm{L}},
G )$ ${\textrm{d}}\gamma_r^{\textrm{L}} = B_r\big( \sum_{r'\neq r}{\textrm{d}}\gamma_{r'}^{\textrm{T}}, \Psi.\Gamma, G \big)$ ${\textrm{d}}\gamma_r^{\textrm{R}} = B_r\big( \Psi.\Gamma, \sum_{r'\neq r}{\textrm{d}}\gamma_{r'}^{\textrm{T}}, G \big)$ ${\textrm{d}}\gamma_r^{\textrm{T}} =
{\textrm{d}}\gamma_r^{\textrm{L}} + {\textrm{d}}\gamma_r^{\textrm{C}} + {\textrm{d}}\gamma_r^{\textrm{R}}$ ${\textrm{d}}\gamma_r \leftarrow {\textrm{d}}\gamma_r + {\textrm{d}}\gamma_r^{\textrm{T}}$ ${\textrm{d}}\gamma_{\bar{t}}^{\textrm{C}} \leftarrow
{\textrm{d}}\gamma_{\bar{t}}^{\textrm{C}} + {\textrm{d}}\gamma_a^{\textrm{C}}+ {\textrm{d}}\gamma_p^{\textrm{C}}$
**break**
/\* *jump to line 41 for $\ell_f$-loop fRG without corrections to the self-energy flow* \*/
${\textrm{d}}\Sigma_{\bar{t}} =
L( {\textrm{d}}\gamma_{\bar{t}}^{\textrm{C}}, G) $ ${\textrm{d}}\Sigma_{t} =
L( \Psi.\Gamma, G \cdot {\textrm{d}}\Sigma_{\bar{t}} \cdot G)$ ${\textrm{d}}\Psi.\Sigma = {\textrm{d}}\Sigma_{\textrm{std}} + {\textrm{d}}\Sigma_{\bar{t}} + {\textrm{d}}\Sigma_{t}$
**break**
${\textrm{d}}\Psi.\Gamma = \sum_r {\textrm{d}}\gamma_r$ **return** ${\textrm{d}}\Psi$
\[alg\]
In this section, we present a pseudocode for the approximate solution strategy of the mfRG flow explained in [Sec. \[sec:mfRG\_comp\]]{}. Generally, an ordinary differential equation (ODE) is of the form $\partial_{\Lambda} \Psi(\Lambda) = f(\Lambda, \Psi)$, and numerous numerical ODE solvers are available. The only input required for such an ODE solver, apart from stating the initial condition $\Psi(\Lambda_i) = \Psi_i$ and the extremal points $\Lambda_i$, $\Lambda_f$, is an implementation of the function $f(\Lambda, \Psi)$. In the case of mfRG, $\Psi$—describing the state of the physical system at a specified value of the flow parameter $\Lambda$—is a vector that contains the self-energy (say, $\Psi.\Sigma$) and the vertex (say, $\Psi.\Gamma$) for all configurations of quantum numbers (e.g., Matsubara frequency, momenta, and spin). In order to use an ODE solver to compute the mfRG flow, we only need to specify a way to compute $f(\Lambda, \Psi)$. This is provided by [Algorithm \[alg\]]{}, written in pseudocode. [Algorithm \[alg\]]{} makes use of functions outlined in the main text, for which we also include dependencies that have been suppressed earlier. This applies to the single-scale propagator $S$ \[[Eq. (\[eq:singlescale\])]{}\] in line 1, the Dyson equation for $G$ \[[Eq. (\[eq:dyson\])]{}\] in line 2, the differentiated bubble $\dot{B}$ \[[Eq. (\[eq:gdotbubble\])]{}\] in line 8, and the bubble $B$ \[[Eq. (\[eq:bubbles\])]{}\], which is used several times. For a good numerical performance, an efficient implementation of the bubble functions appearing in [Algorithm \[alg\]]{} using vertex symmetries and high-frequency asymptotics is crucial [@Wentzell2016; @SuppKugler2017]. The algorithm has a few external parameters: $\ell_f$ (line 19) denotes the maximal loop order, and $it_f$ (line 5) the number of iterations that improve the accuracy of the flow by four orders of the interaction with each step (cf. [Sec. \[sec:mfRG\_comp\]]{}). These parameters can also be used dynamically via the break conditions of the loops depending on the tolerance $\epsilon$ (lines 30, 37). Note that, typically, one also specifies a tolerance for the numerical ODE solver, say $\epsilon_{\textrm{ODE}}$. If $\epsilon$ is chosen in accordance with $\epsilon_{\textrm{ODE}}$ and the number of loops ($\ell_f$) or iterations ($it_f$) is not fixed a priori, this algorithm yields a solution of the *full* mfRG flow and thus a full summation of *all* parquet diagrams—to the specified numerical accuracy. The straightforward implementation as given by the pseudocode in [Algorithm \[alg\]]{} demonstrates the feasibility of the mfRG flow for almost any fRG application.
Multiloop flow beyond the parquet approximation {#appendix2}
===============================================
The mfRG flow as described so far achieves a full summation of all parquet diagrams of the vertex and self-energy. The first deviations from the exact quantities, i.e., the first nonparquet diagrams, occur at fourth order for the vertex—these are the envelope vertices, such as the one shown in [Fig. \[fig:sd\]]{}(a)—and, as follows by use of the SDE [(\[eq:SchwingerDyson\])]{}, at fifth order for the self-energy. One can in principle add terms to the mfRG flow equations that go beyond the parquet approximation. The flow equation of $\Gamma$ then also needs to generate differentiated diagrams of envelope vertices. This is achieved by adding the differentiated envelope vertices, i.e., all envelope diagrams of $\Gamma$ with one $G$ line replaced by $\partial_{\Lambda} G$ at all possible positions, to the flow equation. Subsequently, one performs the replacement $\Gamma^0 \to \Gamma$ to generate contributions at all interaction orders. (Note that the mfRG corrections of the self-energy flow have to be changed accordingly.) However, such contributions to the vertex flow are—by the very fact that they are of nonparquet type—not of an iterative one-loop structure anymore \[i.e., their evaluation requires the computation of two or more (nested) integrals\] and are thus computationally unfavorable. Another possibility to obtain nonparquet diagrams from mfRG is to keep the flow equations unchanged and modify the initial condition. One can then add scale-independent envelope vertices, i.e., envelope vertices computed in the final theory (at $\Lambda_f$) with some approximation of the self-energy, to the initial condition of the vertex: $\Gamma_{\Lambda_i}^{\vphantom{a}} = \Gamma^0 + \Gamma^{\textrm{envelope}}_{\Lambda_f}$. (Hence, $\Gamma^{\textrm{envelope}}$ must be computed only once.) This yields contributions to the flow that are not actually differentiated diagrams at a given scale $\Lambda$. Nevertheless, the initial vertex $\Gamma_{\Lambda_i}$ constitutes a new totally irreducible building block in the mfRG flow. After completion of the flow, one obtains a summation of all “parquet” diagrams with the totally irreducible vertex $R = \Gamma_{\Lambda_i}$ instead of $R = \Gamma^0$; i.e., one obtains vertex and self-energy at one level beyond the parquet approximation \[cf. [Eq. (\[eq:parquet\])]{}\]. Such results deviate from the exact quantities starting at fifth and sixth order for $\Gamma$ and $\Sigma$, respectively. This scheme of adding nonparquet contributions can also be iterated and used with expressions for $R = \Gamma_{\Lambda_i}$ of even higher order. However, it appears rather tedious and is more in the spirit of an iterative solution of the parquet equations than of an actual fRG flow.
[^1]: Defining the four-point vertex as expansion coefficient of the (quantum) effective action ${\bm{\Gamma}}$, we use $\Gamma_{x{^{\prime}}, y{^{\prime}}; x, y} = \delta^4 {\bm{\Gamma}} / ( \delta \bar{c}_{x{^{\prime}}} \delta \bar{c}_{y{^{\prime}}} \delta c_{x} \delta c_{y})$ at zero fields. Via antisymmetry, we have $\Gamma_{x{^{\prime}}, y{^{\prime}}; x, y} = - \Gamma_{x{^{\prime}}, y{^{\prime}}; y, x}$, and all of the standard relations in our paper agree precisely with those of Ref. .
[^2]: Note that in the flow equation of the vertex in Ref. , Eq. (52), a minus sign in the first line on the r.h.s. is missing.
[^3]: The third-order diagram of $R$ in Fig. 1(b) of Ref. is of nonparquet type only in the X-ray-edge singularity, where reducibility is required in interband two-particle lines. The corresponding diagram with identical lines belongs to the $t$ channel.
[^4]: In the SDE [(\[eq:SchwingerDyson\])]{}, e.g., the self-energy is constructed in a way that involves two parallel lines connected to the same vertex, requiring a factor of $1/2$ to avoid double counting. In the standard flow equation [(\[eq:sigma\_flow\_std\])]{} for $\Sigma$, no such lines exist and hence no extra factor either.
[^5]: Alternatives to D$\Gamma$A which do not require the totally irreducible vertex are the dual fermion [@Rubtsov2008; @Brener2008; @Hafermann2009] and the related 1PI approach [@Rohringer2013]. However, upon transformation to the dual variables, the bare action contains $n$-particle vertices for all $n$. Recent studies [@Ribic2017; @Ribic2017a] show that the corresponding six-point vertex yields sizable contributions for the (physical) self-energy, and it remains unclear how a truncation in the (dual) bare action can be justified.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We propose that the best sites to search for SGRs outside the Local group are galaxies with active massive star formation. Different possibilities to observe SGR activity from these sites are discussed. In particular we searched for giant flares from M82, M83, NGC 253, and NGC 4945 in the BATSE data. We present a list of potential candidates, however, in our opinion no good candidates alike giant SGR flares were found. Still, hyperflares similar to the one of 27 December 2004 can be observed from larger distances. From the BATSE data we select 5 candidates coincident with the galaxies Arp 299 and NGC 3256 which have very high rate of starformation, and propose that they can be examples of hyperflares; however, this result has low statistical significance.'
author:
- |
S.B. Popov$^{1}$ [^1]\
$^1$[*Sternberg Astronomical Institute, Universitetski pr. 13, Moscow 119992, Russia*]{}
date: 'Accepted ...... Received ......; in original form ...... '
title: Soft gamma repeaters and starforming galaxies
---
=-0.5 in
gamma rays: bursts —
Introduction
============
Sources of soft gamma repeaters (SGRs) are one of the most puzzling types of neutron stars. Now at least four of them are known in our Galaxy and in the Large Magellanic Cloud (we refer to @wt2004 for a recent summary of all properties of SGRs). SGRs show three main types of bursts:
- weak bursts, $L\la 10^{41}$ erg s$^{-1}$;
- intermediate bursts, $L\sim10^{41}$–$10^{43}$ erg s$^{-1}$;
- giant flares, $L\la 10^{45}$ erg s$^{-1}$.
Weak bursts are relatively frequent. About several hundreds were detected from 4 sources during $\sim 25$ yrs, i.e. the average rate is about 1 per month per source. However, these bursts appear in groups during periods of activity of a SGR. Duration is very short, $\sim 0.1$ s. Intermediate bursts have typical durations $\sim$ few seconds and are much more rare. These two types of bursts will not be discussed below.
Giant flares (GFs) are very rare. Only four GFs were observed (however, some authors do not include the burst of SGR 1627-41 on June 18 1998 into this list as it is slightly weaker then the others and shows a different pulse structure). These flares are extremely energetic. Typical duration of the initial spike is about one second or smaller. The rate of GFs is very uncertain as only four were detected, usualy it is assumed to be just (1/20 - 1/30) yrs$^{-1}$ per source. The latest GF detected on 27 December 2004 was suggested to be a representative the forth class of bursts – “supergiant flares” or “hyperflares” (HFs). It exceeds previous GFs in the energy release by two orders of magnitude, and below we consider this kind of events separately as they can be observed from larger distances, and potentially can contribute to short-hard GRBs detected by BATSE and other satellites (see @ngpf2005 [@h2005] and references therein).
Being very interesting objects SGRs are very rare, probably due to a short life cycle, $10^4$ yrs, which can be connected with a short duration of a magnetar activity. It would be very important to detect these sources outside the Local group of galaxies to increase the sample. Especially it is interesting to understand the birth rate of SGRs and the fraction of NSs which appear as these sources. Any solid data (even upper limits) on the number of GFs and HFs detected from outside of the Local group can help here a lot.
Here we want to discuss a possibility to observe SGRs outside the Local group of galaxies (for previous discussions of extragalactic SGRs see @d2001 and recent e-prints by @ngpf2005 [@h2005]). Detection of such objects can give an opportunity to estimate a fraction of NSs which appear as SGRs on larger statistics. In this short note [^2] we mainly focus on regions of active starformation.
We will discuss two approaches to find GFs and/or HFs from sources outside the Local group:
- Close-by ($\la$10 Mpc) galaxies with high starformation rate should give the main contribution for detection of GFs.
- Few galaxies with extreme values of starformation rate (so-called “supernova factories”) are the best sights to search for rare HFs.
Giant flares from local galaxies in the BATSE catalogue
=======================================================
As it is discussed by Heckman (1998) inside 10 Mpc about 25% of starformation is due to only four well known galaxies: M82, NGC 253, NGC 4945, and M83 (see Table 1). As probably BATSE could detect GFs similar to the prototype event of 5 March 1979 [@m1979] only from short distances ($\la 3$-$5$ Mpc), the contribution of these four galaxies can be even more important. The main idea which we put forward here is the following: close-by galaxies with high present day star formation rate are the best sites to search for SGRs outside the Local group.
Name Distance, Mpc SN rate per year
---------- --------------- ------------------
M 82 3.4 0.1-0.6
NGC 253 2.5 0.1-0.3
NGC 4945 3.7 0.1-0.5
M 83 3.7 0.1-0.5
: Close galaxies with high starformation rate
Supernova (SN) rates presented in the table are approximate ones as there are no estimates better than a factor of 2-3 precision. [^3] We collected data from different papers. For example, the rate for NGC253 is taken from @e1998 (see also @p2001). In the following we will use the conservative intermediate values: 0.4, 0.2, 0.2, and 0.1 for M82, NGC 253, NGC 4945, and M83 correspondently. In comparison with the Galactic rate of SN, these galaxies have significant enhancement (roughly 12, 6, 6, and 3 times correspondently), so we can expect proportionally higher number of SGRs (and GFs from them). With the galactic rate $\sim $ 4 flares in 25 years, for BATSE (4.75 years equivalent of all-sky coverage) we can expect roughly 8-9 GFs from M82, 4-5 GFs from each of NGC 253 and NGC 4945, and 2 GFs from M83 (it total about 20 GFs from four galaxies during the BATSE life cycle).
It is useful to check if in the BATSE catalogue[^4] there are potential SGR-candidates in these four galaxies. We have to look for short bursts with $T_{90}$ at least less than 2 seconds (the burst from SGR 1627-41 was longer than initial strong spikes from three other SGRs). Another criterium is fluence. Taking into account large distances to host galaxies we do not expect SGR candidates to be bright (expected fluences $\la 10^{-7}$ erg cm$^{-2}$). Then we have to select only relatively soft bursts, as GFs are softer than typical GRBs[^5].
All GRBs for which at least one of the four galaxies appears inside the error box are given in Table 2. For each burst we give its number, coordinates, error box radius, $T_{50}$ and $T_{90}$, maximum fluence (maximum among four channels), and softness (the ratio of the 1st to the 3rd channel). Coordinates and error box radii are given in degrees.
---------- ---------- ---------- ------- ----------- ----------- ------------ ----------
Trigger $\alpha$ $\delta$ Error $T_{50}$, $T_{90}$, Maximum Softness
number box s s fluence
M82
2054 164.33 66.15 17.91 — — — —
2160 150.84 68.81 2.15 11.072 123.136 — —
2660 157.28 70.08 3.02 6.464 16.896 79.451e-08 0.613
2821 124.84 60.59 13.53 0.152 0.392 67.355e-09 0.122
3118 117.57 80.37 34.15 0.136 0.232 18.628e-08 0.059
3915 96.66 65.27 61.91 0.080 0.200 — —
6219 170.66 70.18 9.54 1.856 2.752 1.1506e-07 0.014
6255 148.68 60.79 12.71 — — — —
6488 155.76 76.41 9.94 1.152 2.240 6.3462e-07 0.333
6547 155.18 62.23 13.58 0.029 0.097 1.0350e-07 0.129
7297 140.07 76.39 9.53 0.438 1.141 6.5662e-07 0.050
7552 137.56 65.9 6.15 11.648 63.296 5.8479e-07 1.122
7970 136.87 64.49 8.48 0.157 0.387 6.0876e-08 0.343
M83
1485 202.01 -29.98 9.65 — — 50.284e-08 0.501
1510 198.84 -34.35 7.29 — — — —
2349 203.15 -31.88 6.19 1.696 4.032 21.409e-08 0.426
2384 203.8 -18.21 17.81 0.128 0.192 84.657e-09 0.309
2596 211.51 -27.07 19.74 — — — —
2756 209.39 -23.16 9.65 — — — —
5444 199.44 -31.51 4.94 — — — —
6447 191.44 -36.6 14.77 0.256 1.024 6.0229e-08 0.159
6708 206.73 -29.7 3.02 3.904 12.160 1.0452e-06 0.051
7361 204.17 -28.29 7.28 0.960 1.856 7.8501e-08 0.121
7385 203.02 -27.81 3.59 — — — —
8076 199.39 -29.98 7.39 0.075 0.218 3.1075e-07 0.072
NGC 253
2140 9.08 -24.03 4.34 6.784 19.072 48.766e-08 1.070
2312 14.72 -33.56 8.93 0.112 0.272 45.113e-08 0.082
2325 12.7 -24.78 13.55 16.448 22.528 39.642e-08 0.218
3908 11.56 -27.25 7.96 5.344 13.920 59.805e-08 0.203
6135 9.06 -22.84 8.79 1.856 4.032 1.4354e-06 0.119
6648 10.26 -26.97 2.52 24.512 88.384 1.7639e-06 0.061
6918 4.86 -30.74 13.24 — — — —
7551 13.78 -24.45 3.75 53.248 119.616 6.8409e-07 1.665
NGC 4945
108 201.31 -45.41 13.78 1.280 3.136 14.604e-07 0.259
1167 209.4 -45.74 10.04 7.552 24.576 22.754e-08 0.330
2405 197.74 -44.32 11.28 25.088 64.014 83.646e-08 0.155
2800 200.29 -47.94 15.92 0.320 0.448 35.153e-08 0.072
3895 189.39 -47.72 6.99 0.384 0.768 42.483e-09 0.135
6447 191.44 -36.6 14.77 0.256 1.024 6.0229e-08 0.159
---------- ---------- ---------- ------- ----------- ----------- ------------ ----------
There are many short burst coincident with one of the four selected galaxies. (still, a significant number can be projected onto the region around a particular galaxy by chance). Only one of them – 7970 – is soft. So, only this burst can be considered as a reasonable candidate. However, $T_{90}=0.387$ s seems to be long for a typical GF. The main conclusion of this analysis can be such that we found no good candidates GFs from hypothetical SGRs in M82, M83, NGC 253, and NGC 4945. However, if our assumption about softness of GFs can be soften several candidates can be proposed. This result can be used to put constraints on the number of SGR sources in these galaxies, on the frequency of GFs or/and on the maximum flux during a GF (Popov, Stern in prep.).
[@lrrrrrrr@]{} Trigger &$\alpha$ & $\delta$ & Error & $T_{50}$,& $T_{90}$, & Maximum &\
number & & & box & s & s & fluence\
&&&&&&\
&&&&&&\
Arp 299 &&&&&&\
2265 & 180.2 & 59.57 & 8.32 & 0.256 & 0.456 &10.557e-08\
3118 & 117.57 & 80.37 & 34.15& 0.136 & 0.232 & 18.628e-08\
6547 & 155.18 & 62.23 & 13.58 & 0.029 & 0.097 & 1.0350e-07\
&&&&&&\
NGC 3256 &&&&&&\
2372 & 161.1& -36.01& 8.84& 0.072 & 0.256& 64.178e-09\
2485 & 173.45 & -40.09 & 18.39& 0.128 & 0.176 & 34.750e-09\
\
&&&&&&\
Hyperflares from Virgo and from galaxies with extreme starformation rates
=========================================================================
The situation with HFs (like the 27 Dec 2004 one) is quite different as the BATSE maximum detection distance for such events is larger by about an order of magnitude. Having a limiting distance $\approx 50$ Mpc we can roughly estimate a number of GFs and HFs from this volume. We assume that this number can be scaled from the galactic one using the number of galaxies or the starformation rate (SFR). Inside 50 Mpc we can use several estimates. For example, @d2001 uses the following expression to obtain an estimate of a number of galaxies similar to the Milky Way: $
N_{Gal}=0.0117 \, h_{65}^3 \, R_{Mpc}^3. $ For $R=50$ Mpc we obtain about 1500 galaxies. So, for 4.5 years of observation we can expect nearly 800 GFs and about 200 HFs assuming 3 GFs and 1 HF observed in the Milky Way in 25 years. Similar estimates can be obtained using estimates of @bcw2004 and @gza1995. @bcw2004 provide the following value for SFR density at $z=0.1$: $0.01915\, M_{\odot}/{\rm yr}/{\rm Mpc}^3$. Inside 50 Mpc it gives $\approx 10^4\, M_{\odot}/{\rm yr}/{\rm Mpc}^3$. SFR for the Milky way is estimated to be few solar masses per year. So, the ratio is about few thousands. @gza1995 estimate SFR in star-forming galaxies for $z\la0.045$ as $0.013 \, M_{\odot}/{\rm yr}/{\rm Mpc}^3$. It gives $\approx 6800 \,
M_{\odot}/{\rm yr}/{\rm Mpc}^3$ inside 50 Mpc. All three estimates are in good correspondence. So, having one HF in $\sim$30 years in our Galaxy we can expect few hundreds HFs during the BATSE lifetime potentially detectable by this satellite.
The largest structure up to $R \sim 50$ Mpc is the Virgo cluster of galaxies (see @bts1987 for all details about the cluster). It includes about 1300 galaxies (including 130 spirals). BATSE should be able to detect HFs from Virgo cluster as fairly strong bursts. It is important to estimate an expected number of GFs and HFs from Virgo. However, there are several galaxies with significantly enhanced starformation. So, roughly we can estimate that the SFR in Virgo is about few hundreds time larger than in our galaxy. We can expect up to one hundred HFs during the BATSE lifetime if we assume the rate in the Galaxy about 1 in 30-40 years.
Despite these optimistic predictions no anisotropy in distribution of short GRBs or any correlations with known type of objects were found. Here we want to adress another possibility — observations of HFs by BATSE from particular galaxies outside the Virgo cluster. Of course, this large volume ($R\la 50$ Mpc) cannot be dominated by few starforming galaxies, nevertheless, some peculiar objects can be considered as important targets to search for HFs from extragalactic SGRs.
In the Universe there is a small amount of galaxies with extreme SFR – “supernova factories”. They have core collapse SN rate up to two orders of magnitude higher than in the Milky Way. Up to the limiting distance of detection of a HF by BATSE there are two prominent objects of that kind: Arp 299 [@n2004] and NGC 3256 [@l2004]. We propose to look for HF candidates in the direction of these two galaxies. [^6]
Knowing a SN rate in a given galaxy we can obtain an estimate of a HF rate. In the Galaxy only one HF in 30 years was observed. (By the way it is roughly coincedent with the core collapse SN rate.) The SN rates in Arp 299 and NGC 3256 are about 1 per year. So, we can expect about 1 HF per year from each of them. During the BATSE lifetime few such events can be expected.
The fluence for the 27 Dec event was estimated in @h2005 as 1.36 erg cm$^{-2}$. The distance is about 15 kpc. For 40 Mpc (distance to Arp 299 and NGC 3256) we can expect fluences about $10^{-7}$ erg cm$^{-2}$.
In the BATSE catalogue [^7] we looked for short GRBs with known timing and fluences coincident with Arp 299 and NGC 3256. On the whole (among 2704 GRB coordinates of which we used) error boxes of 12 appeared to be coincident with Arp 299 and 6 with NGC 3256. From these set five short hard bursts with known fluences were selected (Table 3): three from the direction of Arp 299 (for one another very short burst – the trigger number 3915 – fluences are not given) and two from NGC 3256 (however, another burst with the trigger number 6278 can be a possible candidate). All five GRBs look like short spikes in the BATSE data. We propose that these 5 GRBs can be good candidates to be HFs. Of course, some amount of events can be coincident with these galaxies by chance, and such a probability is not low.[^8] Still, we think that our finding is worth discussing. Future and present day missions with better angular resolustion (like [*Swift*]{} and, probably, [*Integral*]{} and [*HETE*]{}) can shed light on the association of short GRBs with galaxies with high SFR.
Conclusions
===========
We discussed the connection between SGRs and starforming galaxies. Our suggestion is that few well know galaxies with large SFR are the best candidate sites to look for SGR flares. For the case of GFs we especially mention such close-by galaxies as M82, M83, NGC 253, NGC 4945. For HFs we point to “supernova factories” Arp 299 and NGC 3256.
We found 5 candidates (3 in the direction of Arp 299 and 2 in the direction of NGC 3256) which can be HFs.
Acknowledgments {#acknowledgments .unnumbered}
===============
I want to thank Drs. B.E. Stern and M.E. Prokhorov for many discussions and suggestions. The work was supported by the RFBR grants 04-02-16720 and 03-02-16068.
[99]{} Binggeli B., Tammann G. A., Sandage, A., 1987, AJ 94, 251 Brinchmann J., Charlot S., White S.D.M. et al., 2004, MNRAS 351, 1151 Duncan R.C., 2001, in Proc. 20th Texas Symposium on relativistic astrophysics, AIP conference proceedings, Vol. 586. Eds. J. Craig Wheeler and H. Martel, p. 495 \[astro-ph/0103235\] Engelbracht C.W., Rieke, M.J., Rieke, G.H., Kelly, D.M., Achtermann, J.M., 1998, ApJ 505, 639 Gallego J., Zamorano J., Aragon-Salamanca A., Rego M., 1995, ApJ 455, L1 Golentskii, S. V., Mazets E.P., Il’inskii, V. N., Guryan, Iu. A., 1979, Soviet Astron. Let. 5, 340 Heckman T., 1998, in “Origins”, ASP Conference Series, Vol. 148, 1998, Eds. Charles E. Woodward, J. Michael Shull, and Harley A. Thronson, Jr., p.127 \[astro-ph/9708263\] Hurley K., Boggs S.E., Smith D.M. et al., 2005, Nature (in press) =astro-ph/0502329 Lipari S.L., Diaz, R.J., Forte, J.C., et al., 2004, MNRAS 354, L1 Mazets E.P., Golentskii, S. V., Il’inskii, V. N., Aptekar, R. L., Guryan, Iu. A., 1979, Nature 282, 587 Nakar E., Gal-Yam A., Piran T., Fox D.B., 2005, astro-ph/0502148 Neff S.G., Ulvestad, J.S., Teng, S.H., 2004, ApJ 611, 186 Pietsch W., Roberts, T.P., Sako, M., et al. 2001, A&A 365, L174 Woods P.M., Thompson C., 2004, astro-ph/0406133
[^1]: E-mail: [email protected]
[^2]: This is just a brief discussion note, it is submitted only to the ArXiv, and should be refered by its astro-ph number. More elaborated results will be presented elsewhere (Popov, Stern in prep.).
[^3]: Note, that distances are also uncertain, but the precision for them is much better.
[^4]: http://cossc.gsfc.nasa.gov/batse/
[^5]: A hard tail in the spectrum of 5 March event starts at $E>430$ keV [@g1979], so it doesn’t influence the third channel with $100<E<300$ keV; see also fig. 14.8 in [@wt2004]
[^6]: @h2005 suggested that flares even stronger then the one of Dec 27 can be expected from younger SGRs. Without any doubts the best place to look for them are these two galaxies (and objects similar to them).
[^7]: http://cossc.gsfc.nasa.gov/batse/
[^8]: For example, three bursts with large error boxes – 3118, 3915 and 6547 – appeared to be coincident both with M82 and Arp 299; and 6447 with M83 and NGC 4945.
| {
"pile_set_name": "ArXiv"
} |
[**Asymmetric and Moving-Frame Approaches to** ]{}
[**the 2D and 3D Boussinesq Equations**]{}[^1]
[Xiaoping Xu]{}
[Institute of Mathematics, Academy of Mathematics & System Sciences]{}
[Chinese Academy of Sciences, Beijing 100190, P.R. China]{} [^2]
[*Dedicated to 2008 Beijing Olympic Games*]{}
[**Abstract**]{}
Introduction
============
Both the atmospheric and oceanic flows are influenced by the rotation of the earth. In fact, the fast rotation and small aspect ratio are two main characteristics of the large scale atmospheric and oceanic flows. The small aspect ratio characteristic leads to the primitive equations, and the fast rotation leads to the quasi-geostropic equations (cf. \[2\], \[6\], \[7\], \[9\]). A main objective in climate dynamics and in geophysical fluid dynamics is to understand and predict the periodic, quasi-periodic, aperiodic, and fully turbulent characteristics of the large scale atmospheric and oceanic flows (e.g., cf. \[4\], \[5\]).
The Boussinesq system for the incompressible fluid follows in ${\mathbb}{R}^2$ is $$u_t+uu_x+vu_y-\nu{\Delta}u=-p_x,\qquad v_t+uv_x+vv_y-\nu{\Delta}v-{\theta}=-p_y,\eqno(1.1)$$ $${\theta}_t+u{\theta}_x+v{\theta}_y-\kappa {\Delta}{\theta}=0,\qquad
u_x+v_y=0,\eqno(1.2)$$ where $(u,v)$ is the velocity vector field, $p$ is the scalar pressure, ${\theta}$ is the scalar temperature, $\nu\geq 0$ is the viscosity and $\kappa\geq 0$ is the thermal diffusivity. The above system is a simple model in atmospheric sciences (e.g., cf. \[8\]). Chae \[1\] proved the global regularity, and Hou and Li \[3\] obtained the well-posedness of the above system.
Aonther slightly simplified version of the system of primitive equations is the three-dimensional stratified rotating Boussinesq system (e.g., cf. \[7\], \[9\]): $$u_t+uu_x+vu_y+wu_z-\frac{1}{R_0}v={\sigma}({\Delta}u-p_x),\eqno(1.3)$$ $$v_t+uv_x+vv_y+wv_z+\frac{1}{R_0}u={\sigma}({\Delta}v-p_y),\eqno(1.4)$$ $$w_t+uw_x+vw_y+ww_z-{\sigma}R T={\sigma}({\Delta}w-p_z),\eqno(1.5)$$ $$T_t+uT_x+vT_y+wT_z={\Delta}T+w,\eqno(1.6)$$ $$u_x+v_y+w_z=0,\eqno(1.7)$$ where $(u,v,w)$ is the velocity vector filed, $T$ is the temperature function, $p$ is the pressure function, ${\sigma}$ is the Prandtle number, $R$ is the thermal Rayleigh number and $R_0$ is the Rossby number. Moreover, the vector $(1/R_0)(-v,u,0)$ represents the Coriolis force and the term $w$ in (1.6) is derived using stratification. So the above equations are the extensions of Navier-Stokes equations by adding the Coriolis force and the stratified temperature equation. Due to the Coriolis force, the two-dimensional system (1.1) and (1.2) is not a special case of the above three-dimensional system. Hsia, Ma and Wang \[4\] studied the bifurcation and periodic solutions of the above system (1.3)-(1.7).
In \[10\], we used the stable range of nonlinear term to solve the equation of nonstationary transonic gas flow. Moreover, we \[11\] solved the three-dimensional Navior-Stokes equations by asymmetric techniques and moving frames. Based on the algebraic characteristics of the above equations, we use in this paper asymmetric ideas and moving frames to solve the above two Boussinesq systems of partial differential equations. New families of explicit exact solutions with multiple parameter functions are obtained. Many of them are the periodic, quasi-periodic, aperiodic solutions that may have practical significance. Using Fourier expansion and some of our solutions, one can obtain discontinuous solutions. The symmetry transformations for these equations are used to simplify our arguments.
For convenience, we always assume that all the involved partial derivatives of related functions always exist and we can change orders of taking partial derivatives. The parameter functions are so chosen that the involved expressions make sense. We also use prime $'$ to denote the derivative of any one-variable function.
Observe that the two-dimensional Boussinesq system (1.1) and (1.2) is invariant under the action of the following symmetry transformation: $${\cal T}(u)=a^{-1}{\epsilon}_1u(a^2(t+b),a{\epsilon}_1(x+{\alpha}),a{\epsilon}_2(y+{\beta}))-{\alpha}',\eqno(1.8)$$ $${\cal T}(v)=a^{-1}{\epsilon}_2v(a^2(t+b),a(x+{\alpha}),a(y+{\beta}))-{\beta}',\eqno(1.9)$$ $${\cal T}(p)=a^{-2}p(a^2(t+b),a{\epsilon}_1(x+{\alpha}),a{\epsilon}_2(y+{\beta}))+{{\alpha}'}'x+{{\beta}'}'y+{\gamma},\eqno(1.10)$$ $${\cal T}({\theta})=a^{-3}{\epsilon}_2{\theta}(a^2(t+b),a{\epsilon}_1(x+{\alpha}),a{\epsilon}_2(y+{\beta})),\eqno(1.11)$$ where $a,b\in{\mathbb}{R}$ with $a\neq 0$, ${\epsilon}_1,{\epsilon}_2\in\{1,-1\}$ and ${\alpha},{\beta},{\gamma}$ are arbitrary functions of $t$. The above transformation transforms a solution of the equation (1.1) and (1.2) into another solution with additional three parameter functions.
Denote $\vec x=(x,y)$. The three-dimensional stratified rotating Boussinesq system is invariant under the following transformations: $${\cal T}_1[(u,v,w)]=((u(t+b,\vec x A,{\epsilon}z),v(t+b,\vec x
A),{\epsilon}z)A,{\epsilon}w),\eqno(1.12)$$ $${\cal T}_1(p)=p(t+b,\vec x A,{\epsilon}z),\qquad
{\cal T}_1(T)=T(t+b,\vec x A,{\epsilon}z);\eqno(1.13)$$ $${\cal T}_2(u)=u(t,x+{\alpha},y+{\beta},z+{\gamma})-{\alpha}',\qquad
{\cal T}_2(v)=v(t,x+{\alpha},y+{\beta},z+{\gamma})-{\beta}',\eqno(1.14)$$ $${\cal T}_2(w)=w(t,x+{\alpha},y+{\beta},z+{\gamma})-{\gamma}',\qquad
{\cal T}_2(T)=T(t,x+{\alpha},y+{\beta},z+{\gamma})-{\gamma},\eqno(1.15)$$ $${\cal
T}_2(p)=p(t,x+{\alpha},y+{\beta},z+{\gamma})+{\sigma}^{-1}({{\alpha}'}'x+{{\beta}'}'y+{{\gamma}'}'z)-R{\gamma}z+\mu;\eqno(1.16)$$ where ${\epsilon}=\pm 1$, $b\in{\mathbb}{R}$, $A\in
O(2,{\mathbb}{R})$, and ${\alpha},{\beta},{\gamma},\mu$ are arbitrary functions of $t$. The above transformations transform a solution of the equation (1.3)-(1.7) into another solution. In particular, applying the transformation ${\cal T}_2$ to any solution in this paper yields another solution with extra four parameter functions.
To simplify problems, we always solve the Boussinesq systems modulo the above corresponding symmetry transformations, which is an idea that geometers and topologists often use.
The paper is organized as follows. In Section 2, we solve the two-dimensional Boussinesq equations (1.1)-(1.2) and obtain four families of explicit exact solutions. In Section 3, we present an approach with $u,v,w,T$ linear in $x,y$ to the equations (1.3)-(1.7), and obtain two families of explicit exact solutions. Assuming $u_z=v_z=w_{zz}=T_{zz}=0$ in Section 4, we find another two families of explicit exact solutions of the equations (1.3)-(1.7). In Section 5, we obtain a family of explicit exact solutions of (1.3)-(1.7) that are independent of $x$. The status can be changed by applying the transformation in (1.12) and (1.13) to them.
Solutions of the 2D Boussinesq Equations
========================================
In this section, we solve the two-dimensional Boussinesq equations (1.1)-(1.2) by an asymmetric method and by an moving frame.
According to the second equation in (1.2), we take the potential form: $$u=\xi_y,\qquad v=-\xi_x\eqno(2.1)$$ for some functions $\xi$ in $t,x,y$. Then the two-dimensional Boussinesq equations become $$\xi_{yt}+\xi_y\xi_{xy}-\xi_x\xi_{yy}-\nu{\Delta}\xi_y=-p_x,\qquad \xi_{xt}+\xi_y\xi_{xx}-\xi_x\xi_{xy}-\nu{\Delta}\xi_x+{\theta}=p_y,\eqno(2.2)$$ $${\theta}_t+\xi_y{\theta}_x-\xi_x{\theta}_y-\kappa {\Delta}{\theta}=0.\eqno(2.3)$$ By our assumption $p_{xy}=p_{yx}$, the compatible condition of the equations in (2.2) is $$({\Delta}\xi)_t+\xi_y({\Delta}\xi)_x-\xi_x({\Delta}\xi)_y-\nu{\Delta}^2\xi+{\theta}_x=0.\eqno(2.4)$$ Now we first solve the system (2.3) and (2.4).
Our asymmetric approach is to assume $${\theta}={\varepsilon}(t,y),\qquad\xi=\phi(t,y)+x\psi(t,y)\eqno(2.5)$$ for some functions ${\varepsilon},\phi$ and $\psi$ in $t,y$. Then (2.3) becomes $${\varepsilon}_t-\psi{\varepsilon}_y-\kappa{\varepsilon}_{yy}=0.\eqno(2.6)$$ Moreover, (2.4) becomes $$\phi_{yyt}+x\psi_{yyt}+(\phi_y+x\psi_y)\psi_{yy}-\psi(\phi_{yyy}+x\psi_{yyy})-\nu(\phi_{yyyy}+x\psi_{yyyy})=0,
\eqno(2.7)$$ equivalently, $$\phi_{yyt}+\phi_y\psi_{yy}-\psi\phi_{yyy}-\nu\phi_{yyyy}=0,
\eqno(2.8)$$ $$\psi_{yyt}+\psi_y\psi_{yy}-\psi\psi_{yyy}-\nu\psi_{yyyy}=0.
\eqno(2.9)$$ The above two equations are equivalent to: $$\phi_{yt}+\phi_y\psi_y-\psi\phi_{yy}-\nu\phi_{yyy}={\alpha}_1,
\eqno(2.10)$$ $$\psi_{yt}+\psi_y^2-\psi\psi_{yy}-\nu\psi_{yyy}={\alpha}_2
\eqno(2.11)$$ for some functions ${\alpha}_1$ and ${\alpha}_2$ of $t$ to be determined.
Let $c$ be a fixed real constant and let ${\gamma}$ be a fixed function of $t$. We define $$\zeta_1(s)=\frac{e^{{\gamma}s}-ce^{-{\gamma}s}}{2},\qquad \eta_1=\frac{e^{{\gamma}s}+ce^{-{\gamma}s}}{2},\eqno(2.12)$$ $$\zeta_0(s)=\sin{\gamma}s,\qquad \eta_0(s)=\cos{\gamma}s.\eqno(2.13)$$ Then $$\eta_r^2(s)+(-1)^r\zeta_r^2(s)=c^r\eqno(2.14)$$ and $${\partial}_s(\zeta_r(s))={\gamma}\eta_r(s),\qquad
{\partial}_s(\eta_r(s))=-(-1)^r{\gamma}\zeta_r(s),\eqno(2.15)$$ where we treat $0^0=1$ when $c=r=0$. First we assume $$\psi={\beta}_1y+{\beta}_2\zeta_r(y)\eqno(2.16)$$ for some functions ${\beta}_1$ and ${\beta}_2$ of $t$, where $r=0,1$. Then (2.11) becomes $$\begin{aligned}
\hspace{2cm}& &{\beta}_1'+c^r{\beta}_2^2{\gamma}^2+{\beta}_1^2+[({\beta}_2{\gamma})'+(-1)^r\nu{\beta}_2{\gamma}^3+2{\beta}_1{\beta}_2{\gamma}]\eta_r(y)
\\ & &+(-1)^r{\beta}_2{\gamma}({\beta}_1{\gamma}-{\gamma}')y\zeta_r(y)={\alpha}_2,\hspace{6.2cm}
(2.17)\end{aligned}$$ which is implied by the following equations: $${\beta}_1'+c^r{\beta}_2^2{\gamma}^2+{\beta}_1^2={\alpha}_2,\qquad{\beta}_1{\gamma}-{\gamma}'=0,\eqno(2.18)$$ $$({\beta}_2{\gamma})'+(-1)^r\nu{\beta}_2{\gamma}^3+2{\beta}_1{\beta}_2{\gamma}=0.\eqno(2.19)$$ For convenience, we assume ${\gamma}=\sqrt{{\alpha}'}$ for some function ${\alpha}$ of $t$. Thus we have $${\beta}_1=\frac{{\gamma}'}{{\gamma}}=\frac{{{\alpha}'}'}{2{\alpha}'},\qquad
{\beta}_2=\frac{b_1e^{-(-1)^r\nu{\alpha}}}{\sqrt{({\alpha}')^3}},\qquad
b_1\in{\mathbb}{R}.\eqno(2.20)$$ To solve (2.10), we assume $$\phi={\beta}_3\eta_r(y)\eqno(2.21)$$ for some function ${\beta}_3$, modulo the transformation in (1.8)-(1.11). Now (2.10) becomes $$[(-1)^r(({\beta}_3{\gamma})'+{\beta}_1{\beta}_3{\gamma})+\nu{\beta}_3{\gamma}^3]\zeta_r(y)=-{\alpha}_1, \eqno(2.22)$$ which is implied by $$(-1)^r(({\beta}_3{\gamma})'+{\beta}_1{\beta}_3{\gamma})+\nu{\beta}_3{\gamma}^3=0.\eqno(2.23)$$ Thus $${\beta}_3=\frac{b_2e^{-(-1)^r\nu{\alpha}}}{{\alpha}'},\eqno(2.24)$$ where $b_2$ is a real constant.
In order to solve (2.6), we assume $${\varepsilon}=be^{{\gamma}_1\eta_r(y)},\eqno(2.25)$$ where $b$ is a real constant and ${\gamma}_1$ is a function of $t$. Then (2.6) is implied by $${\gamma}_1'\eta_r(y)+(-1)^r{\beta}_2{\gamma}{\gamma}_1\zeta_r^2(y)+\kappa{\gamma}^2{\gamma}_1((-1)^r\eta_r(y)-{\gamma}_1\zeta_r^2(y))=0,
\eqno(2.26)$$ which is implied by $${\gamma}_1'+(-1)^r\kappa{\gamma}^2{\gamma}_1=0,\qquad(-1)^r{\beta}_2-\kappa{\gamma}{\gamma}_1=0.\eqno(2.27)$$ Then the first equation implies $${\gamma}_1=b_3e^{-(-1)^r\kappa{\alpha}}\eqno(2.28)$$ for some constant $b_3$. By the second equations in (2.20) and (2.27), we have: $$(-1)^r\frac{b_1e^{-(-1)^r\nu{\alpha}}}{\sqrt{({\alpha}')^3}}=b_3\kappa\sqrt{{\alpha}'}e^{-(-1)^r\kappa{\alpha}}.\eqno(2.29)$$ For convenience, we take $$b_1=(-1)^rb_0^2\kappa b_3,\qquad b_0\in{\mathbb}{R}.\eqno(2.30)$$ Then (2.29) is implied by $${\alpha}'e^{(-1)^r(\nu-\kappa){\alpha}/2}=b_0.\eqno(2.31)$$ If $\nu=\kappa$, then we have ${\alpha}=b_0t+c_0$. Modulo the transformation in (1.8)-(1.11), we take $b_0=1$ and $c_0=0$, that is, ${\alpha}=t$. When $\nu\neq
\kappa$, we similarly take $b_0=1$ and $${\alpha}=\frac{2(-1)^r}{\nu-\kappa}\ln[(-1)^r(\nu-\kappa)t/2+c_0],\qquad c_0\in{\mathbb}{R}.\eqno(2.32)$$
Suppose $\nu=\kappa$. Then ${\gamma}=1$ and $$\phi=b_2e^{-(-1)^r\nu t}\eta_r(y),\qquad\psi=(-1)^rb_3\nu e^{-(-1)^r\nu
t}\zeta_r(y).\eqno(2.33)$$ Moreover, $${\theta}=b\exp(b_3e^{-(-1)^r\nu t}\eta_r(y)),\eqno(2.34)$$ $$\xi=b_2e^{-(-1)^r\nu t}\eta_r(y)+(-1)^rb_3\nu e^{-(-1)^r\nu
t}x\zeta_r(y)\eqno(2.35)$$ by (2.5). According to (2.1), $$u=\xi_y=(-1)^r[-b_2e^{-(-1)^r\nu t}\zeta_r(y)+b_3\nu e^{-(-1)^r\nu
t}x\eta_r(y)],\eqno(2.36)$$ $$v=-\xi_x=-(-1)^rb_3\nu e^{-(-1)^r\nu
t}\zeta_r(y).\eqno(2.37)$$ Note $$u_t+uu_x+vu_y-\nu{\Delta}u=
b_3^2\nu^2c^r e^{-(-1)^r2\nu t}x,\eqno(2.38)$$ $$v_t+uv_x+vv_y-\nu{\Delta}v-{\theta}=vv_y-b\exp(b_3e^{-(-1)^r\nu t}\eta_r(y)).\eqno(2.39)$$ By (1.1), we have $$p=
b\int\exp(b_3e^{-(-1)^r\nu t}\eta_r(y))dy-\frac{1}{2}b_3^2\nu^2
e^{-(-1)^r2\nu t}(c^rx^2+\zeta_r^2(y))\eqno(2.40)$$ modulo the transformation in (1.8)-(1.11).
Consider the case $\nu\neq \kappa$. Then $${\gamma}=\sqrt{{\alpha}'}=\frac{1}{\sqrt{(-1)^r(\nu-\kappa)t/2+c_0}}\eqno(2.41)$$ by (2.32). Moreover, $$\phi=b_2[(-1)^r(\nu-\kappa)t/2+c_0]^{2\nu/(\kappa-\nu)+1}\eta_r(y)\eqno(2.42)$$ by (2.21) and (2.24). Furthermore, $$\psi=\frac{(-1)^r(\kappa-\nu)y}{4[(-1)^r(\nu-\kappa)t/2+c_0]}+
(-1)^rb_3\kappa
[(-1)^r(\nu-\kappa)t/2+c_0]^{2\nu/(\kappa-\nu)+3/2}\zeta_r(y)\eqno(2.43)$$ by (2.16), (2.20) and (2.30). According to (2.25), (2.28) and (2.32), $${\theta}=be^{b_3[(-1)^r(\nu-\kappa)t/2+c_0]^{2\kappa/(\kappa-\nu)}\eta_r(y)}.\eqno(2.44)$$ Similarly, we have $$\begin{aligned}
\hspace{1cm}u_t+uu_x+vu_y-\nu{\Delta}u&=&b_3^2c^r\kappa^2
[(-1)^r(\nu-\kappa)t/2+c_0]^{4\nu/(\kappa-\nu)+2}x
\\ & &+\frac{3(\nu-\kappa)^2x}{16[(-1)^r(\nu-\kappa)t/2+c_0]^2},
\hspace{3.8cm}(2.45)\end{aligned}$$ $$\begin{aligned}
& &v_t+uv_x+vv_y-\nu{\Delta}-{\theta}=-\psi_t+\psi\psi_y+\nu\psi_{yy}-{\theta}\\
&=&-be^{b_3[(-1)^r(\nu-\kappa)t/2+c_0]^{2\kappa/(\kappa-\nu)}\eta_r(y)}
+\frac{3}{4}b_3\kappa(\kappa-\nu)
[(-1)^r(\nu-\kappa)t/2+c_0]^{2\nu/(\kappa-\nu)+1/2}\zeta_r(y)
\\ & &+\frac{3(\nu-\kappa)^2y}{16[(-1)^r(\nu-\kappa)t/2+c_0]^2}+
\frac{b_3^2}{2}\kappa^2
[(-1)^r(\nu-\kappa)t/2+c_0]^{4\nu/(\kappa-\nu)+3}{\partial}_y\zeta_r^2(y).\hspace{0.6cm}(2.46)\end{aligned}$$ According (1.1), we have $$\begin{aligned}
p&=&b\int
e^{b_3[(-1)^r(\nu-\kappa)t/2+c_0]^{2\kappa/(\kappa-\nu)}\eta_r(y)}dy
-\frac{b_3^2}{2}c^r\kappa^2
[(-1)^r(\nu-\kappa)t/2+c_0]^{4\nu/(\kappa-\nu)+2}x^2
\\ & &-\frac{3(\nu-\kappa)^2(x^2+y^2)}{32[(-1)^r(\nu-\kappa)t/2+c_0]^2}
-\frac{b_3^2}{2}\kappa^2
[(-1)^r(\nu-\kappa)t/2+c_0]^{4\nu/(\kappa-\nu)+3}\zeta_r^2(y)
\\
& &+\frac{3}{4}(-1)^rb_3\kappa(\kappa-\nu)
[(-1)^r(\nu-\kappa)t/2+c_0]^{2\nu/(\kappa-\nu)+1}\eta_r(y)\hspace{3.6cm}(2.47)\end{aligned}$$ modulo the transformation in (1.8)-(1.11).
[**Theorem 2.1**]{}. [*Let $b,b_2,b_3,c,c_0\in{\mathbb}{R}$ and let $r=0,1$. If $\nu=\kappa$, we have the solution (2.34), (2.36), (2.37) and (2.40) of the two-dimensional Boussinesq equations (1.1)-(1.2), where $\zeta_r(y)$ and $\eta_r(y)$ are defined in (2.12)-(2.13) with ${\gamma}=1$. When $\nu\neq\kappa$, we have the following solutions of the two-dimensional Boussinesq equations (1.1)-(1.2): $$\begin{aligned}
u&=&\frac{(-1)^r(\kappa-\nu)x}{4[(-1)^r(\nu-\kappa)t/2+c_0]}+
(-1)^rb_3\kappa
[(-1)^r(\nu-\kappa)t/2+c_0]^{2\nu/(\kappa-\nu)+1}x\eta_r(y)\\ &&
-(-1)^rb_2[(-1)^r(\nu-\kappa)t/2+c_0]^{2\nu/(\kappa-\nu)+1/2}\zeta_r(y),
\hspace{5cm}(2.48)\end{aligned}$$ $$v=\frac{(-1)^r(\nu-\kappa)y}{4[(-1)^r(\nu-\kappa)t/2+c_0]}-
(-1)^rb_3\kappa
[(-1)^r(\nu-\kappa)t/2+c_0]^{2\nu/(\kappa-\nu)+3/2}\zeta_r(y),\eqno(2.49)$$ ${\theta}$ is given in (2.44) and $p$ is given in (2.47), where $\zeta_r(y)$ and $\eta_r(y)$ are defined in (2.12)-(2.13) with ${\gamma}=[(-1)^r(\nu-\kappa)t/2+c_0]^{-1/2}$.*]{}
Observe that $$\psi=6\nu y^{-1}\eqno(2.50)$$ is another solution of (2.11). In order to solve (2.10), we assume $$\phi=\sum_{i=1}^\infty{\gamma}_iy^i\eqno(2.51)$$ modulo the transformation in (1.8)-(1.11), where ${\gamma}_i$ are functions of $t$ to be determined. Now (2.10) becomes $$-6\nu{\gamma}_1y^{-2}-18\nu{\gamma}_2y^{-1}+\sum_{i=1}^\infty[i{\gamma}_i'-\nu(i+2)(i+3)(i+4){\gamma}_{i+2}]
y^{i-1}={\alpha}_1,\eqno(2.52)$$ equivalently, $${\gamma}_1={\gamma}_2=0,\qquad {\alpha}_1=-60\nu{\gamma}_3,\eqno(2.53)$$ $$i{\gamma}_i'-\nu(i+2)(i+3)(i+4){\gamma}_{i+2}=0,\qquad
i> 1.\eqno(2.54)$$ Thus $${\gamma}_{2i+2}=\frac{2i{\gamma}_{2i}'}{\nu(2i+2)(2i+3)(2i+4)}=0,\qquad
i\geq 1,\eqno(2.55)$$ $${\gamma}_{2i+3}=\frac{(2i+1){\gamma}_{2i+1}'}{\nu(2i+3)(2i+4)(2i+5)}=\frac{360{\gamma}_3^{(i)}}{\nu^i(2i+2)(2i+5)!},\qquad
i\geq 1.\eqno(2.56)$$ Hence $$\phi=360\sum_{i=0}^\infty
\frac{{\alpha}^{(i)}y^{2i+3}}{\nu^i(2i+3)(2i+5)!},\eqno(2.57)$$ where ${\alpha}$ is an arbitrary function of $t$ such that the series converges, say, a polynomial in $t$.
To solve (2.6), we also assume $${\varepsilon}=\sum_{i=0}^\infty{\beta}_i y^i,\eqno(2.58)$$ where ${\beta}_i$ are functions of $t$. Then (2.6) becomes $$6\nu{\beta}_1y^{-1}+\sum_{i=0}^\infty[{\beta}_i'+(i+2)(6\nu-(i+1)\kappa){\beta}_{i+2}]y^i=0,\eqno(2.58)$$ that is, ${\beta}_1=0$ and $${\beta}_i'-(i+2)(6\nu+(i+1)\kappa){\beta}_{i+2}=0,\qquad i\geq 0.\eqno(2.59)$$ Hence $${\theta}={\beta}+\sum_{i=1}^\infty\frac{{\beta}^{(i)}y^{2i}}{2^ii!\prod_{r=1}^i(6\nu+(2r-1)\kappa)},\eqno(2.60)$$ where ${\beta}$ is an arbitrary function of $t$ such that the series converges, say, a polynomial in $t$. In this case, $$u_t+uu_x+vu_y-\nu{\Delta}u=-60\nu{\alpha},\eqno(2.61)$$ $$v_t+uv_x+vv_y-\nu{\Delta}-{\theta}=-36\nu^2
y^{-3}-{\beta}-\sum_{i=1}^\infty\frac{{\beta}^{(i)}y^{2i}}{2^ii!\prod_{r=1}^i(6\nu+(2r-1)\kappa)}.
\eqno(2.62)$$ According (1.1), we have $$p=60\nu{\alpha}x-18\nu^2 y^{-2}+{\beta}y+
\sum_{i=1}^\infty\frac{{\beta}^{(i)}y^{2i+1}}{2^ii!(2i+1)\prod_{r=1}^i(6\nu+(2r-1)\kappa)}\eqno(2.63)$$ modulo the transformation in (1.8)-(1.11).
[**Theorem 2.2**]{}. [*We have the following solutions of the two-dimensional Boussinesq equations (1.1)-(1.2): $$u=360\sum_{i=0}^\infty
\frac{{\alpha}^{(i)}y^{2i+2}}{\nu^i(2i+5)!}-6\nu xy^{-2},\qquad v=-6\nu
y^{-1},\eqno(2.64)$$ ${\theta}$ is given in (2.60) and $p$ is given in (2.63), where ${\alpha}$ and ${\beta}$ are arbitrary functions of $t$ such that the related series converge, say, polynomials in $t$.*]{}
Let ${\gamma}$ be a function of $t$. Denote the moving frame $${\tilde}\varpi=x\cos{\gamma}+y\sin{\gamma},\qquad \hat\varpi=y\cos{\gamma}-x\sin{\gamma}.\eqno(2.65)$$ Then $${\partial}_t({\tilde}\varpi)={\gamma}'\hat\varpi,\qquad
{\partial}_t(\hat\varpi)=-{\gamma}'{\tilde}\varpi.\eqno(2.66)$$ Moreover, $${\partial}_{{\tilde}\varpi}=\cos{\gamma}\:{\partial}_x+\sin{\gamma}\:{\partial}_y,\qquad
{\partial}_{\hat\varpi}=-\sin{\gamma}\:{\partial}_x+\cos{\gamma}\:{\partial}_y.\eqno(2.67)$$ In particular, $${\Delta}={\partial}_x^2+{\partial}_y^2={\partial}_{{\tilde}\varpi}^2+{\partial}_{\hat\varpi}^2.\eqno(2.68)$$
We assume $$\xi=\phi(t,{\tilde}\varpi)-\frac{{\gamma}'}{2}(x^2+y^2)
,\qquad{\theta}=\psi(t,{\tilde}\varpi),\eqno(2.69)$$ where $\phi$ and $\psi$ are functions in $t,{\tilde}\varpi$. Then (2.3) becomes $$\psi_t-\kappa\psi_{{\tilde}\varpi{\tilde}\varpi}=0\eqno(2.70)$$ and (2.4) becomes $$-2{{\gamma}'}'+\phi_{t{\tilde}\varpi{\tilde}\varpi}
-\nu\phi_{{\tilde}\varpi{\tilde}\varpi{\tilde}\varpi{\tilde}\varpi}+\psi_{{\tilde}\varpi}\cos{\gamma}=0.\eqno(2.71)$$ Modulo the transformation in (1.8)-(1.11), the above equation is equivalent to $$-2{{\gamma}'}'{\tilde}\varpi+\phi_{t{\tilde}\varpi}
-\nu\phi_{{\tilde}\varpi{\tilde}\varpi{\tilde}\varpi}+\psi\cos{\gamma}=0.\eqno(2.72)$$
Assume $\nu=\kappa$. We take the following solution of (2.70): $$\psi=\sum_{i=1}^m a_id_ie^{a_i^2\kappa t\cos 2b_i+a_i{\tilde}\varpi\cos
b_i}\sin(a_i^2\kappa t\sin 2b_i+a_i{\tilde}\varpi\sin
b_i+b_i+c_i)\eqno(2.73)$$ where $a_i,b_i,c_i,d_i$ are real numbers. Moreover, (2.72) is equivalent to solving the following equation: $$\begin{aligned}
\hspace{1.5cm}& &2\nu{\gamma}'-{{\gamma}'}'{\tilde}\varpi^2+\phi_t
-\nu\phi_{{\tilde}\varpi{\tilde}\varpi}+[\sum_{i=1}^md_ie^{a_i^2\kappa t\cos
2b_i+a_i{\tilde}\varpi\cos b_i}\\ & &\times\sin(a_i^2\kappa t\sin
2b_i+a_i{\tilde}\varpi\sin
b_i+c_i)]\cos{\gamma}=0\hspace{4.9cm}(2.74)\end{aligned}$$ by (2.1). Thus we have the following solution of (2.74): $$\begin{aligned}
\phi&=&-[\sum_{i=1}^md_ie^{a_i^2\kappa
t\cos 2b_i+a_i{\tilde}\varpi\cos b_i}\sin(a_i^2\kappa t\sin
2b_i+a_i{\tilde}\varpi\sin b_i+c_i)]\int \cos{\gamma}\:dt\\ &
&+{\gamma}'{\tilde}\varpi^2+\sum_{s=1}^n\hat d_se^{\hat a_s^2\kappa t\cos
2\hat b_s+\hat a_s{\tilde}\varpi\cos \hat b_s}\sin(\hat a_s^2\kappa t\sin
2\hat b_s+\hat a_s{\tilde}\varpi\sin \hat b_s+\hat
c_s),\hspace{1.6cm}(2.75)\end{aligned}$$ where $\hat a_s,\hat
b_s,\hat c_s,\hat d_s$ are real numbers.
Suppose $\nu\neq \kappa$. To make (2.72) solvable, we choose the following solution of (2.70): $$\psi=\sum_{i=1}^m a_id_ie^{a_i^2\kappa t+a_i{\tilde}\varpi}.\eqno(2.76)$$ Now (2.72) is equivalent to solving the following equation: $$\nu{\gamma}'-{{\gamma}'}'{\tilde}\varpi^2+\phi_t
-\nu\phi_{{\tilde}\varpi{\tilde}\varpi}+\sum_{i=1}^md_ie^{a_i^2\kappa
t+a_i{\tilde}\varpi}\cos{\gamma}=0\eqno(2.77)$$ by (2.1). We obtain the following solution of (2.77): $$\begin{aligned}
\hspace{1cm}\phi&=&{\gamma}'{\tilde}\varpi^2+\sum_{s=1}^n\hat
d_se^{\hat a_s^2\kappa t\cos 2\hat b_s+\hat a_s{\tilde}\varpi\cos \hat
b_s}\sin(\hat a_s^2\kappa t\sin 2\hat b_s+\hat a_s{\tilde}\varpi\sin \hat
b_s+\hat c_s)\\
& &-\sum_{i=1}^md_ie^{a_i^2\nu t+a_i{\tilde}\varpi}\int
e^{a_i^2(\kappa-\nu)t}\cos{\gamma}\:dt.\hspace{6.2cm}(2.78)\end{aligned}$$
Note $$u=\phi_\varpi\sin{\gamma}-{\gamma}'y,\qquad v={\gamma}'x
-\phi_\varpi\cos{\gamma}.\eqno(2.79)$$ By (2.72), $$\begin{aligned}
\hspace{1cm}& & u_t+uu_x+vu_y-\nu{\Delta}u\\&=&(\phi_{\varpi
t}-\nu\phi_{\varpi\varpi\varpi})\sin{\gamma}+2{\gamma}'\phi_\varpi\cos{\gamma}-{\gamma}'^2x-{{\gamma}'}'y
\\
&=&(2{{\gamma}'}'{\tilde}\varpi-\psi\cos{\gamma})\sin{\gamma}+2{\gamma}'\phi_\varpi\cos{\gamma}-{\gamma}'^2x-{{\gamma}'}'y,
\\ &=&{{\gamma}'}'(x\sin 2{\gamma}-y\cos 2{\gamma})
+(2{\gamma}'\phi_\varpi-\psi\sin{\gamma}) \cos{\gamma}-{\gamma}'^2x, \hspace{3cm}(2.80)\end{aligned}$$ $$\begin{aligned}
\hspace{1cm}& &v_t+uv_x+vv_y-\nu{\Delta}v-{\theta}\\ &=&(\nu\phi_{\varpi\varpi\varpi}-\phi_{\varpi
t})\cos{\gamma}+2{\gamma}'\phi_\varpi\sin{\gamma}-{\gamma}'^2y+{{\gamma}'}'x -\psi\\
&=&(\psi\cos{\gamma}-2{{\gamma}'}'{\tilde}\varpi)\cos{\gamma}+2{\gamma}'\phi_\varpi\sin{\gamma}-{\gamma}'^2y+{{\gamma}'}'x
-\psi\\ &=&-{{\gamma}'}'(x\cos 2{\gamma}+y\sin
2{\gamma})+(2{\gamma}'\phi_\varpi-\psi\sin{\gamma})\sin{\gamma}-{\gamma}'^2y.\hspace{2.8cm}(2.81)\end{aligned}$$ According to (1.1), $$p=\frac{{{\gamma}'}^2-{{\gamma}'}'\sin 2{\gamma}}{2}x^2+\frac{{{\gamma}'}^2+{{\gamma}'}'\sin
2{\gamma}}{2}y^2+{{\gamma}'}'xy\cos2{\gamma}+\int\psi d{\tilde}\varpi\:\sin{\gamma}-2{\gamma}'\phi
\eqno(2.82)$$ modulo the transformation in (1.8)-(1.11).
[**Theorem 2.3**]{}.
*Let ${\gamma}$ be any function of $t$ and denote ${\tilde}\varpi=x\cos{\gamma}+y\sin{\gamma}$. Take $$\{a_i,b_i,c_i,d_i,\hat a_s,\hat
b_s,\hat c_s,\hat d_s\mid
i=1,...,m;s=1,...,n\}\subset{\mathbb}{R}.\eqno(2.83)$$ If $\nu=\kappa$, we have the following solutions of the two-dimensional Boussinesq equations (1.1)-(1.2): $$\begin{aligned}
u=-{\gamma}' y+\sin{\gamma}\{2{\gamma}'{\tilde}\varpi+\sum_{s=1}^n\hat a_s\hat d_se^{\hat a_s^2\kappa
t\cos 2\hat b_s+\hat a_s{\tilde}\varpi\cos \hat b_s}\sin(\hat a_s^2\kappa
t\sin 2\hat b_s+\hat a_s{\tilde}\varpi\sin \hat b_s+\hat b_s+\hat c_s)\\
-[\sum_{i=1}^m a_id_ie^{a_i^2\kappa t\cos 2b_i+a_i{\tilde}\varpi\cos
b_i}\sin(a_i^2\kappa t\sin 2b_i+b_i+a_i{\tilde}\varpi\sin b_i+c_i)]\int
\cos{\gamma}\:dt\},\hspace{0.7cm}(2.84)\end{aligned}$$ $$\begin{aligned}
v={\gamma}'x-\cos{\gamma}\{2{\gamma}'{\tilde}\varpi+\sum_{s=1}^n\hat a_s\hat d_se^{\hat a_s^2\kappa
t\cos 2\hat b_s+\hat a_s{\tilde}\varpi\cos \hat b_s}\sin(\hat a_s^2\kappa
t\sin 2\hat b_s+\hat a_s{\tilde}\varpi\sin \hat b_s+\hat b_s+\hat
c_s)\\-[\sum_{i=1}^m a_id_ie^{a_i^2\kappa t\cos
2b_i+a_i{\tilde}\varpi\cos b_i}\sin(a_i^2\kappa t\sin
2b_i+a_i{\tilde}\varpi\sin b_i+b_i+c_i)]\int
\cos{\gamma}\:dt\},\hspace{1cm}(2.85)\end{aligned}$$ ${\theta}=\psi$ in (2.73), and $$\begin{aligned}
p&=&(\sin{\gamma}+2{\gamma}'\int\cos{\gamma})[\sum_{i=1}^md_ie^{a_i^2\kappa t\cos
2b_i+a_i{\tilde}\varpi\cos b_i}\sin(a_i^2\kappa t\sin
2b_i+a_i{\tilde}\varpi\sin b_i+c_i)]\\ & &+\frac{{{\gamma}'}^2-{{\gamma}'}'\sin
2{\gamma}}{2}x^2+\frac{{{\gamma}'}^2+{{\gamma}'}'\sin
2{\gamma}}{2}y^2+{{\gamma}'}'xy\cos2{\gamma}-2{\gamma}'^2{\tilde}\varpi^2\\ &
&-2{\gamma}'\sum_{s=1}^n\hat d_se^{\hat a_s^2\kappa t\cos 2\hat b_s+\hat
a_s{\tilde}\varpi\cos \hat b_s}\sin(\hat a_s^2\kappa t\sin 2\hat b_s+\hat
a_s{\tilde}\varpi\sin \hat b_s+\hat
c_s).\hspace{2.4cm}(2.86)\end{aligned}$$*
When $\nu\neq\kappa$, we have the following solutions of the two-dimensional Boussinesq equations (1.1)-(1.2): $$\begin{aligned}
\hspace{1cm}u&=&\{\sum_{s=1}^n\hat a_s\hat
d_se^{\hat a_s^2\kappa t\cos 2\hat b_s+\hat a_s{\tilde}\varpi\cos \hat
b_s}\sin(\hat a_s^2\kappa t\sin 2\hat b_s+\hat a_s{\tilde}\varpi\sin \hat
b_s+\hat b_s+\hat c_s)\\
& &+2{\gamma}'{\tilde}\varpi-\sum_{i=1}^ma_id_ie^{a_i^2\nu t+a_i{\tilde}\varpi}\int
e^{a_i^2(\kappa-\nu)t}\cos{\gamma}\:dt
\}\sin{\gamma}-{\gamma}'y,\hspace{2.3cm}(2.87)\end{aligned}$$ $$\begin{aligned}
\hspace{1cm}v&=&-\{\sum_{s=1}^n\hat a_s\hat
d_se^{\hat a_s^2\kappa t\cos 2\hat b_s+\hat a_s{\tilde}\varpi\cos \hat
b_s}\sin(\hat a_s^2\kappa t\sin 2\hat b_s+\hat a_s{\tilde}\varpi\sin \hat
b_s+\hat b_s+\hat c_s)\\
& &+2{\gamma}'{\tilde}\varpi-\sum_{i=1}^m a_id_ie^{a_i^2\nu
t+a_i{\tilde}\varpi}\int e^{a_i^2(\kappa-\nu)t}\cos{\gamma}\:dt
\}\cos{\gamma}+{\gamma}'x,\hspace{2.3cm}(2.88)\end{aligned}$$ ${\theta}=\psi$ in (2.76), and $$\begin{aligned}
p&=&\frac{{{\gamma}'}^2-{{\gamma}'}'\sin
2{\gamma}}{2}x^2+\frac{{{\gamma}'}^2+{{\gamma}'}'\sin
2{\gamma}}{2}y^2+{{\gamma}'}'xy\cos2{\gamma}-2{\gamma}'^2{\tilde}\varpi^2
\\&&-2{\gamma}'\sum_{s=1}^n\hat
d_se^{\hat a_s^2\kappa t\cos 2\hat b_s+\hat a_s{\tilde}\varpi\cos \hat
b_s}\sin(\hat a_s^2\kappa t\sin 2\hat b_s+\hat a_s{\tilde}\varpi\sin \hat
b_s+\hat c_s)\\ & &+\sum_{i=1}^m d_ie^{a_i^2\nu
t+a_i{\tilde}\varpi}(2{\gamma}'+\sin{\gamma})\int
e^{a_i^2(\kappa-\nu)t}\cos{\gamma}\:dt).\hspace{5cm}(2.89)\end{aligned}$$
[**Remark 2.4**]{}. By Fourier expansion, we can use the above solution to obtain the one depending on two piecewise continuous functions of ${\tilde}\varpi$.
Asymmetric Approach I to the 3D Equations
=========================================
Starting from this section, we use asymmetric approaches developed in \[11\] to solve the stratified rotating Boussinesq equations (1.3)-(1.7).
For convenience of computation, we denote $$\Phi_1=u_t+uu_x+vu_y+wu_z-\frac{1}{R_0}v-{\sigma}(u_{xx}+u_{yy}+u_{zz}),\eqno(3.1)$$ $$\Phi_2=v_t+uv_x+vv_y+wv_z+\frac{1}{R_0}u-{\sigma}(v_{xx}+v_{yy}+v_{zz}),\eqno(3.2)$$ $$\Phi_3=w_t+uw_x+vw_y+ww_z-{\sigma}R T-{\sigma}(w_{xx}+w_{yy}+w_{zz}).\eqno(3.3)$$ Then the equations (1.3)-(1.5) become $$\Phi_1+{\sigma}p_x=0,\qquad
\Phi_2+{\sigma}p_y=0,\qquad
\Phi_3+{\sigma}p_z=0.
\eqno(3.4)$$ Our strategy is to solve the following compatibility conditions: $${\partial}_y(\Phi_1)={\partial}_x(\Phi_2),\qquad
{\partial}_z(\Phi_1)={\partial}_x(\Phi_3),\qquad{\partial}_z(\Phi_2)={\partial}_y(\Phi_3).
\eqno(3.5)$$
First we assume $$u=\phi_z(t,z) x+{\varsigma}(t,z) y+\mu(t,z),\qquad v=\tau(t,z)
x+\psi_z(t,z) y+{\varepsilon}(t,z),\eqno(3.6)$$ $$w=-\phi(t,z)-\psi(t,z),\qquad T={\vartheta}(t,z)+z,\eqno(3.7)$$ where $\phi,{\vartheta},{\varsigma},\mu,\tau,$ and ${\varepsilon}$ are functions of $t,z$ to be determined. Then $$\begin{aligned}
\Phi_1&=&\phi_{tz}x+{\varsigma}_t y+\mu_t+
\phi_z(\phi_z x+{\varsigma}y+\mu)+({\varsigma}-1/R_0)(\tau x+\psi_zy+{\varepsilon})\\ &
&-(\phi+\psi)(\phi_{zz}x+{\varsigma}_z y+\mu_z)
-{\sigma}(\phi_{zzz}x+{\varsigma}_{zz} y+\mu_{zz})\\
&=&[\phi_{tz}+\phi_z^2+\tau({\varsigma}-1/R_0)-\phi_{zz}(\phi+\psi)-{\sigma}\phi_{zzz}]x\\
& &+[{\varsigma}_t+{\varsigma}\phi_z+\psi_z({\varsigma}-1/R_0)-{\varsigma}_z(\phi+\psi)-{\sigma}{\varsigma}_{zz}]y\\
& &+\mu_t+ \mu\phi_z+({\varsigma}-1/R_0){\varepsilon}-\mu_z(\phi+\psi)-{\sigma}\mu_{zz},
\hspace{5.3cm}(3.8)\end{aligned}$$ $$\begin{aligned}
\Phi_2&=&\tau_tx+\psi_{tz}y+{\varepsilon}_t+\psi_z(\tau x+\psi_zy+{\varepsilon})+
(\tau+1/R_0)(\phi_zx+{\varsigma}y+\mu)\\
& &-(\phi+\psi)(\tau_zx+\psi_{zz}y+{\varepsilon}_z)
-{\sigma}(\tau_{zz}x+\psi_{zzz}y+{\varepsilon}_{zz})\\
&=&[\psi_{tz}+\psi_z^2+{\varsigma}(\tau+1/R_0)-(\phi+\psi)\psi_{zz}-{\sigma}\psi_{zzz}]y\\
& &+[\tau_t+\tau\psi_z+(\tau+1/R_0)\phi_z-(\phi+\psi)\tau_z-{\sigma}\tau_{zz}]x\\ & &+{\varepsilon}_t+
{\varepsilon}\psi_z+(\tau+1/R_0)\mu-(\phi+\psi){\varepsilon}_z-{\sigma}{\varepsilon}_{zz},
\hspace{5.3cm}(3.9)\end{aligned}$$ $$\Phi_3=-\phi_t-\psi_t+(\phi+\psi)(\phi_z+\psi_z)-{\sigma}R({\vartheta}+z)+{\sigma}(\phi_{zz}+\psi_{zz}).\eqno(3.10)$$ Thus (3.5) is equivalent to the following system of partial differential equations: $$\phi_{tz}+\phi_z^2+\tau({\varsigma}-1/R_0)-\phi_{zz}(\phi+\psi)-{\sigma}\phi_{zzz}={\alpha}_1,\eqno(3.11)$$ $${\varsigma}_t+{\varsigma}\phi_z+\psi_z({\varsigma}-1/R_0)-{\varsigma}_z(\phi+\psi)-{\sigma}{\varsigma}_{zz}={\alpha},\eqno(3.12)$$ $$\mu_t+
\mu\phi_z+({\varsigma}-1/R_0){\varepsilon}-\mu_z(\phi+\psi)-{\sigma}\mu_{zz}={\alpha}_2,\eqno(3.13)$$ $$\psi_{tz}+\psi_z^2+{\varsigma}(\tau+1/R_0)-(\phi+\psi)\psi_{zz}-{\sigma}\psi_{zzz}={\beta}_1,\eqno(3.14)$$ $$\tau_t+\tau\psi_z+(\tau+1/R_0)\phi_z-(\phi+\psi)\tau_z-{\sigma}\tau_{zz}={\alpha},\eqno(3.15)$$ $${\varepsilon}_t+{\varepsilon}\psi_z+(\tau+1/R_0)\mu-(\phi+\psi){\varepsilon}_z-{\sigma}{\varepsilon}_{zz}={\beta}_2\eqno(3.16)$$ for some ${\alpha},{\alpha}_1,{\alpha}_2,{\beta}_1,{\beta}_2$ are functions of $t$.
Let $0\neq b$ and $c$ be fixed real constants. Recall the notions in (2.12) and (2.13) with ${\gamma}=b$. We assume $$\phi=b^{-1}{\gamma}_1\zeta_r(z),\qquad
\psi=b^{-1}({\gamma}_2\zeta_r(z)+{\gamma}_3\eta_r(z)),\eqno(3.17)$$ $${\varsigma}={\gamma}_4({\gamma}_2\eta_r(z)-(-1)^r{\gamma}_3\zeta_r(z)),\qquad\tau={\gamma}_5{\gamma}_1\eta_r(z),\qquad{\gamma}_4{\gamma}_5=1,\eqno(3.18)$$ where ${\gamma}_i$ are functions of $t$ to be determined. Moreover, (3.11) becomes $$({\gamma}_1'+(-1)^rb^2{\sigma}{\gamma}_1-{\gamma}_1{\gamma}_5/R_0)\eta_r(z)+({\gamma}_1+{\gamma}_2){\gamma}_1c^r
={\alpha}_1,\eqno(3.19)$$ which is implied by $${\alpha}_1=({\gamma}_1+{\gamma}_2){\gamma}_1c^r,\eqno(3.20)$$ $${\gamma}_1'+(-1)^rb^2{\sigma}{\gamma}_1-{\gamma}_1{\gamma}_5/R_0=0.\eqno(3.21)$$ On the other hand, (3.15) becomes $$[({\gamma}_1{\gamma}_5)'+ {\gamma}_1/R_0+(-1)^rb^2{\sigma}{\gamma}_1{\gamma}_5]\eta_r+
{\gamma}_1{\gamma}_5({\gamma}_1+{\gamma}_2)c^r={\alpha},\eqno(3.22)$$ which gives $${\alpha}={\gamma}_1{\gamma}_5({\gamma}_1+{\gamma}_2)c^r,\eqno(3.23)$$ $$({\gamma}_1{\gamma}_5)'+(-1)^rb^2{\sigma}{\gamma}_1{\gamma}_5+ {\gamma}_1/R_0=0.\eqno(3.24)$$ Solving (3.21) and (3.24) for ${\gamma}_1$ and ${\gamma}_1{\gamma}_5$, we get $${\gamma}_1=b_1e^{-(-1)^rb^2{\sigma}t}\sin\frac{t}{R_0},\qquad{\gamma}_1{\gamma}_5=
b_1e^{-(-1)^rb^2{\sigma}t}\cos\frac{t}{R_0},\eqno(3.25)$$ where $b_1$ is a real constant. In particular, we take $${\gamma}_5=\cot\frac{t}{R_0}.\eqno(3.26)$$
Observe that (3.12) becomes $$\begin{aligned}
\hspace{1cm}& &[({\gamma}_2{\gamma}_4)'+(-1)^rb^2{\sigma}{\gamma}_2{\gamma}_4-{\gamma}_2/R_0]\eta_r(z)
+{\gamma}_4({\gamma}_1{\gamma}_2+{\gamma}_2^2+(-1)^r{\gamma}_3^2)c^r\\
&&-(-1)^r[({\gamma}_3{\gamma}_4)'+(-1)^rb^2{\sigma}{\gamma}_2{\gamma}_4-{\gamma}_3/R_0]\zeta_r(z)={\alpha}\hspace{4.3cm}(3.27)\end{aligned}$$ and (3.14) becomes $$\begin{aligned}
\hspace{1cm}& &[{\gamma}_2'+(-1)^rb^2{\sigma}{\gamma}_2+{\gamma}_2{\gamma}_4/R_0]\eta_r(z)
+({\gamma}_1{\gamma}_2+{\gamma}_2^2+(-1)^r{\gamma}_3^2)c^r\\
&&-(-1)^r[{\gamma}_3'+(-1)^rb^2{\sigma}{\gamma}_3+{\gamma}_3{\gamma}_4/R_0]\zeta_r(z)={\beta}_1,
\hspace{4.8cm}(3.28)\end{aligned}$$ equivalently, $${\alpha}={\gamma}_4({\gamma}_1{\gamma}_2+{\gamma}_2^2+(-1)^r{\gamma}_3^2)c^r,\eqno(3.29)$$ $${\beta}_1=({\gamma}_1{\gamma}_2+{\gamma}_2^2+(-1)^r{\gamma}_3^2)c^r,\eqno(3.30)$$ $$({\gamma}_2{\gamma}_4)'+(-1)^rb^2{\sigma}{\gamma}_2{\gamma}_4-{\gamma}_2/R_0=0,\eqno(3.31)$$ $${\gamma}_2'+(-1)^rb^2{\sigma}{\gamma}_2+{\gamma}_2{\gamma}_4/R_0=0,\eqno(3.32)$$ $$({\gamma}_3{\gamma}_4)'+(-1)^rb^2{\sigma}{\gamma}_2{\gamma}_4-{\gamma}_3/R_0=0,\eqno(3.33)$$ $${\gamma}_3'+(-1)^rb^2{\sigma}{\gamma}_3+{\gamma}_3{\gamma}_4/R_0=0.\eqno(3.34)$$ Solving (3.31)-(3.34) under the assumption ${\gamma}_4{\gamma}_5=1$, we obtain $${\gamma}_2{\gamma}_4=b_2e^{-(-1)^rb^2{\sigma}t}\sin\frac{t}{R_0},\qquad{\gamma}_2=
b_2e^{-(-1)^rb^2{\sigma}t}\cos\frac{t}{R_0},\eqno(3.35)$$ $${\gamma}_3{\gamma}_4=b_3e^{-(-1)^rb^2{\sigma}t}\sin\frac{t}{R_0},\qquad{\gamma}_3=
b_3e^{-(-1)^rb^2{\sigma}t}\cos\frac{t}{R_0}.\eqno(3.36)$$ In particular, we have: $${\gamma}_4=\tan\frac{t}{R_0}.\eqno(3.37)$$
According to (3.23) and (3.29), $${\gamma}_1{\gamma}_5({\gamma}_1+{\gamma}_2)c^r={\gamma}_4({\gamma}_1{\gamma}_2+{\gamma}_2^2+(-1)^r{\gamma}_3^2)c^r,
\eqno(3.38)$$ equivalently $$-2b_1b_2\cos\frac{2t}{R_0}+(b_2^2-b_1^2+(-1)^rb_3^2)\sin\frac{2t}{R_0}
=0.\eqno(3.39)$$ Thus $$b_1b_2=0,\qquad b_2^2-b_1^2+(-1)^rb_3^2=0.\eqno(3.40)$$ So $$r=0,\qquad b_2=0,\qquad b_1=b_3\eqno(3.41)$$ or $$r=1,\qquad b_1=0,\qquad b_2=b_3.\eqno(3.42)$$
Assume $r=0$ and $b_1\neq 0$. Then $$\phi=b^{-1}b_1e^{-b^2{\sigma}t}\sin bz\:\sin\frac{t}{R_0},\qquad \psi=
b^{-1}b_1e^{-b^2{\sigma}t}\cos bz\:\cos\frac{t}{R_0},\eqno(3.43)$$ $${\varsigma}=-b_1e^{-b^2{\sigma}t}\sin bz\:\sin\frac{t}{R_0},\qquad\tau=b_1e^{-b^2{\sigma}t}\cos bz\:\cos\frac{t}{R_0}.\eqno(3.44)$$ Moreover, we take $\mu={\varepsilon}={\vartheta}=0$. Then $$\Phi_1={\gamma}_1^2(x+{\gamma}_5y)=b_1^2e^{-2b^2{\sigma}t}\sin\frac{t}{R_0}\left(x\sin\frac{t}{R_0}+y\cos\frac{t}{R_0}\right)\eqno(3.45)$$ by (3.8), (3.11)-(3.12), (3.20) and (3.23). Similarly $$\Phi_2=b_1^2e^{-2b^2{\sigma}t}\cos\frac{t}{R_0}\left(x\sin\frac{t}{R_0}+y\cos\frac{t}{R_0}\right).\eqno(3.46)$$
According to (3.10) $$\Phi_3=\left[b^{-1}R_0^{-1}b_1e^{-b^2{\sigma}t}-b^{-1}b_1^2e^{-2b^2{\sigma}t}\cos
\left(bz-\frac{t}{R_0}\right)\right]\sin\left(bz-\frac{t}{R_0}\right)-R{\sigma}z.\eqno(3.47)$$ By (3.4), we have $$\begin{aligned}
\hspace{1cm}p&=&\frac{Rz^2}{2}+\frac{b_1e^{-b^2{\sigma}t}}{b^2{\sigma}R_0}\cos
\left(bz-\frac{t}{R_0}\right)-\frac{b_1^2e^{-2b^2{\sigma}t}}{2{\sigma}b^2}\cos ^2\left(bz-\frac{t}{R_0}\right)\\
& &-\frac{b_1^2e^{-2b^2{\sigma}t}}{2{\sigma}}\left(y^2\cos^2\frac{t}{R_0}+x^2\sin^2\frac{t}{R_0}+xy\sin\frac{2t}{R_0}
\right)\hspace{3.7cm}(3.48)\end{aligned}$$ modulo the transformation in (1.14)-(1.16).
Suppose $r=1$ and $b_2\neq 0$. Then $$\phi=\tau=\mu={\varepsilon}={\vartheta}=0,\;\;\psi=b^{-1}b_2e^{bz+b^2{\sigma}t}\cos\frac{t}{R_0},
\qquad{\varsigma}=b_2e^{bz+b^2{\sigma}t}\sin\frac{t}{R_0}.\eqno(3.49)$$ Moreover, $$\Phi_1=\Phi_2=0,\;\;\Phi_3=b^{-1}b_2R_0^{-1}e^{bz+b^2{\sigma}t}\sin\frac{t}{R_0}+b^{-1}b_2^2e^{2(bz+b^2{\sigma}t)}\cos^2\frac{t}{R_0}-R{\sigma}z.\eqno(3.50)$$ According to (3.4), $$p=\frac{Rz^2}{2}-\frac{b_2e^{bz+b^2{\sigma}t}}{b^2{\sigma}R_0}\sin\frac{t}{R_0}-\frac{b_2^2e^{2(bz+b^2{\sigma}t)}}{2b^2{\sigma}}\cos^2\frac{t}{R_0}\eqno(3.51)$$ modulo the transformation (1.14)-(1.16).
[**Theorem 3.1**]{}. [*Let $b,b_1,b_2\in{\mathbb}{R}$ with $b\neq 0$. We have the following solutions of the three-dimensional stratified rotating Boussinesq equations (1.3)-(1.7): (1) $$u=b_1e^{-b^2{\sigma}t}(x\cos bz-y\sin bz)\sin\frac{t}{R_0},\qquad v=
b_1e^{-b^2{\sigma}t}(x\cos bz-y\sin bz)\cos\frac{t}{R_0},\eqno(3.52)$$ $$w=-b^{-1}b_1e^{-b^2{\sigma}t}\cos\left(bz-\frac{t}{R_0}\right),\qquad
T=z\eqno(3.53)$$ and $p$ is given in (3.48); (2) $$u=b_2e^{bz+b^2{\sigma}t}y\sin\frac{t}{R_0},\qquad v=b_2e^{bz+b^2{\sigma}t}y\cos\frac{t}{R_0},\eqno(3.54)$$ $$w=-b^{-1}b_2e^{bz+b^2{\sigma}t}\cos\frac{t}{R_0}\qquad T=z\eqno(3.55)$$ and $p$ is given in (3.51).*]{}
Next we assume $\phi={\varsigma}=\psi=\tau=0$. Then $$\mu_t-\frac{1}{R_0}{\varepsilon}-{\sigma}\mu_{zz}={\alpha}_2,\;\;
{\varepsilon}_t+\frac{1}{R_0}\nu-{\sigma}{\varepsilon}_{zz}={\beta}_2,\;\;{\vartheta}_t-{\vartheta}_{zz}=0.\eqno(3.56)$$ Solving them, we get:
[**Theorem 3.2**]{}. [*Let $a_i,b_i,c_i,d_i,\hat a_r,\hat b_r,\hat
c_r,\hat d_r,{\tilde}a_s,{\tilde}b_s,{\tilde}c_s,{\tilde}d_s$ be real numbers. We have the following solutions of the three-dimensional stratified rotating Boussinesq equations (1.3)-(1.7): $$\begin{aligned}
u&=&\cos\frac{t}{R_0}\;\sum_{i=1}^md_ie^{a_i^2{\sigma}t\cos
2b_i+ a_iz\cos b_i}\sin(a_i^2{\sigma}t\sin 2b_i+a_iz\sin b_i+c_i)\\
& &+\sin\frac{t}{R_0}\;\sum_{r=1}^n \hat d_re^{\hat a_r^2{\sigma}t\cos
2\hat b_r+a_rz\cos\hat b_r}\sin(\hat a_r^2{\sigma}t\sin 2\hat b_r+\hat
a_rz\sin \hat b_r+\hat c_r),\hspace{1.7cm}(3.57)\end{aligned}$$ $$\begin{aligned}
v&=&-\sin\frac{t}{R_0}\;\sum_{i=1}^md_ie^{a_i^2{\sigma}t\cos
2b_i+ a_iz\cos b_i}\sin(a_i^2{\sigma}t\sin 2b_i+a_iz\sin b_i+c_i)\\
& &+\cos\frac{t}{R_0}\;\sum_{r=1}^n \hat d_re^{\hat a_r^2{\sigma}t\cos
2\hat b_r+a_rz\cos\hat b_r}\sin(\hat a_r^2{\sigma}t\sin 2\hat b_r+\hat
a_rz\sin \hat b_r+\hat c_r),\hspace{1.6cm}(3.58)\end{aligned}$$ $$w=0,\;\;T=z+\sum_{s=1}^k{\tilde}a_s{\tilde}d_s e^{{\tilde}a_s^2 t\cos
2{\tilde}b_s+ {\tilde}a_sz\cos {\tilde}b_s}\sin({\tilde}a_s^2 t\sin 2{\tilde}b_s+{\tilde}a_sz\sin {\tilde}b_s+{\tilde}b_s+{\tilde}c_s),\eqno(3.59)$$ $$p=\frac{R z^2}{2}+R\sum_{s=1}^{m_3}{\tilde}d_s e^{{\tilde}a_s^2 t\cos
2{\tilde}b_s+ {\tilde}a_sz\cos {\tilde}b_s}\sin({\tilde}a_s^2 t\sin 2{\tilde}b_s+{\tilde}a_sz\sin {\tilde}b_s+{\tilde}c_s).\eqno(3.60)$$* ]{}
[**Remark 3.3**]{}. By Fourier expansion, we can use the above solution to obtain the one depending on three arbitrary piecewise continuous functions of $z$.
Asymmetric Approach II to the 3D Equations
==========================================
In this section, we solve the stratified rotating Boussinesq equations (1.4)-(1.7) under the assumption $$u_z=v_z=w_{zz}=T_{zz}=0.\eqno(4.1)$$
Let ${\gamma}$ be a function of $t$ and we use the moving frame ${\tilde}\varpi$ in (2.65). Assume $$u=f(t,{\tilde}\varpi)\sin{\gamma}-{\gamma}'y,\qquad
v=-f(t,{\tilde}\varpi)\cos{\gamma}+{\gamma}'x,\eqno(4.2)$$ According to (4.3), we assume $$w=\phi(t,\varpi),\qquad
T=\psi(t,\varpi)+z,\eqno(4.3)$$ for some functions $f,\;\phi$ and $\psi$ in $t$ and ${\tilde}\varpi$. Using (2.66)-(2.68), we get $$\Phi_1=-({\gamma}'^2+{\gamma}'/R_0)x-{{\gamma}'}'y+f_t\sin{\gamma}+(2{\gamma}'+1/R_0)f\cos{\gamma}-{\sigma}f_{{\tilde}\varpi{\tilde}\varpi}\sin{\gamma},\eqno(4.4)$$ $$\Phi_2=-({\gamma}'^2+{\gamma}'/R_0)y+{{\gamma}'}'x-f_t\cos{\gamma}+(2{\gamma}'+1/R_0)f\sin{\gamma}+{\sigma}f_{{\tilde}\varpi{\tilde}\varpi}\cos{\gamma},\eqno(4.5)$$ $$\Phi_3=\phi_t-{\sigma}\phi_{{\tilde}\varpi{\tilde}\varpi}-{\sigma}R(\psi+z).\eqno(4.6)$$ By (3.5), we have $$-2{{\gamma}'}'+f_{{\tilde}\varpi t}-{\sigma}f_{{\tilde}\varpi{\tilde}\varpi{\tilde}\varpi}=0,\eqno(4.7)$$ $$\phi_t-{\sigma}\phi_{{\tilde}\varpi{\tilde}\varpi}-{\sigma}R\psi=0.\eqno(4.8)$$ Moreover, (1.6) becomes $$\psi_t-\psi_{{\tilde}\varpi{\tilde}\varpi}=0.\eqno(4.9)$$
Solving (4.7), we have: $$f=2{\gamma}'{\tilde}\varpi+\sum_{i=1}^m a_id_ie^{a_i^2\kappa t\cos 2b_i+a_i{\tilde}\varpi\cos
b_i}\sin(a_i^2\kappa t\sin 2b_i+a_i{\tilde}\varpi\sin
b_i+b_i+c_i),\eqno(4.10)$$ where $a_i,b_i,c_i,d_i$ are arbitrary real numbers. Moreover, (4.8) and (4.9) yield $$\phi=\sum_{r=1}^n \hat d_re^{\hat a_r^2 t\cos 2\hat b_r+\hat a_r{\tilde}\varpi\cos
\hat b_r}\sin(\hat a_r^2 t\sin 2\hat b_i+\hat a_r{\tilde}\varpi\sin \hat
b_r+\hat c_r)+{\sigma}Rt\psi,\eqno(4.11)$$ $$\psi=\sum_{s=1}^k{\tilde}d_se^{{\tilde}a_s^2t\cos 2{\tilde}b_s+{\tilde}a_s{\tilde}\varpi\cos {\tilde}b_s}\sin({\tilde}a_s^2 t\sin 2{\tilde}b_s+{\tilde}a_s{\tilde}\varpi\sin {\tilde}b_s+{\tilde}c_s)\eqno(4.12)$$ if ${\sigma}=1$, and $$\begin{aligned}
\phi&=&\sum_{r=1}^n \hat d_re^{\hat a_r^2{\sigma}t\cos 2\hat b_r+\hat a_r{\tilde}\varpi\cos
\hat b_r}\sin(\hat a_r^2{\sigma}t\sin 2\hat b_i+\hat a_r{\tilde}\varpi\sin
\hat b_r+\hat c_r)\\ & &+\frac{{\sigma}R}{1-{\sigma}}\sum_{s=1}^k{\tilde}d_se^{{\tilde}a_s^2t\cos 2{\tilde}b_s+{\tilde}a_s{\tilde}\varpi\cos {\tilde}b_s}\sin({\tilde}a_s^2 t\sin 2{\tilde}b_s+{\tilde}a_s{\tilde}\varpi\sin {\tilde}b_s+{\tilde}c_s),\hspace{2.2cm}(4.13)\end{aligned}$$ $$\psi=\sum_{s=1}^k{\tilde}a_s^2{\tilde}d_se^{{\tilde}a_s^2t\cos 2{\tilde}b_s+{\tilde}a_s{\tilde}\varpi\cos
{\tilde}b_s}\sin({\tilde}a_s^2 t\sin 2{\tilde}b_s+{\tilde}a_s{\tilde}\varpi\sin {\tilde}b_s+2{\tilde}b_s+{\tilde}c_s)\eqno(4.14)$$ when ${\sigma}\neq 1$, where $\hat
a_r,\hat b_r,\hat c_r,\hat d_r,{\tilde}a_s,{\tilde}b_s,{\tilde}c_s,{\tilde}d_s$ are arbitrary real numbers.
Now $$\Phi_1=({{\gamma}'}'\sin2{\gamma}-{\gamma}'^2-{\gamma}'/R_0)x-{{\gamma}'}'y\cos2{\gamma}+(2{\gamma}'+1/R_0)f\cos{\gamma},\eqno(4.15)$$ $$\Phi_2=-({{\gamma}'}'\sin2{\gamma}+{\gamma}'^2+{\gamma}'/R_0)y-{{\gamma}'}'x\cos2{\gamma}+(2{\gamma}'+1/R_0)f\sin{\gamma}\eqno(4.16)$$ and $\Phi_3=-{\sigma}Rz$. According (3.4), we have $$\begin{aligned}
p&=&
-\frac{2{\gamma}'+1/R_0}{{\sigma}}[{\gamma}'{\tilde}\varpi^2+\sum_{i=1}^m
d_ie^{a_i^2\kappa t\cos 2b_i+a_i{\tilde}\varpi\cos b_i}\sin(a_i^2\kappa
t\sin 2b_i+a_i{\tilde}\varpi\sin b_i+c_i)]\\
&&+\frac{R}{2}z^2+\frac{({\gamma}'^2+{\gamma}'/R_0)(x^2+y^2)+{{\gamma}'}'(y^2-x^2)\sin2{\gamma}}
{2{\sigma}}+\frac{{{\gamma}'}'}{{\sigma}}xy\cos2{\gamma}\hspace{1.8cm}(4.17)\end{aligned}$$ modulo the transformation in (1.14)-(1.16).
[**Theorem 4.1**]{}. [*Let $a_i,b_i,c_i,d_i,\hat a_r,\hat b_r,\hat
c_r,\hat d_r,{\tilde}a_s,{\tilde}b_s,{\tilde}c_s,{\tilde}d_s$ be real numbers and let ${\gamma}$ be any function of $t$. Denote ${\tilde}\varpi=x\cos{\gamma}+y\sin{\gamma}$. We have the following solutions of the three-dimensional stratified rotating Boussinesq equations (1.3)-(1.7): $$\begin{aligned}
u&=&[\sum_{i=1}^m a_id_ie^{a_i^2\kappa
t\cos 2b_i+a_i{\tilde}\varpi\cos b_i}\sin(a_i^2\kappa t\sin
2b_i+a_i{\tilde}\varpi\sin b_i+b_i+c_i)\\ & &+2{\gamma}'{\tilde}\varpi]\sin{\gamma}-{\gamma}'
y,\hspace{10.1cm}(4.18)\end{aligned}$$ $$\begin{aligned}
v&=&[-\sum_{i=1}^m a_id_ie^{a_i^2\kappa
t\cos 2b_i+a_i{\tilde}\varpi\cos b_i}\sin(a_i^2\kappa t\sin
2b_i+a_i{\tilde}\varpi\sin b_i+b_i+c_i)\\ & &+2{\gamma}'{\tilde}\varpi]\cos{\gamma}+{\gamma}'
x,\hspace{10.1cm}(4.19)\end{aligned}$$ $p$ is given in (4.17); $$\begin{aligned}
w&=&\sum_{r=1}^n \hat d_re^{\hat a_r^2 t\cos 2\hat b_r+\hat a_r{\tilde}\varpi\cos
\hat b_r}\sin(\hat a_r^2 t\sin 2\hat b_i+\hat a_r{\tilde}\varpi\sin \hat
b_r+\hat c_r)\\ & &+{\sigma}Rt\sum_{s=1}^k{\tilde}d_se^{{\tilde}a_s^2t\cos 2{\tilde}b_s+{\tilde}a_s{\tilde}\varpi\cos {\tilde}b_s}\sin({\tilde}a_s^2 t\sin 2{\tilde}b_s+{\tilde}a_s{\tilde}\varpi\sin {\tilde}b_s+{\tilde}c_s),\hspace{2.5cm}(4.20)\end{aligned}$$ $$T=z+\sum_{s=1}^k{\tilde}d_se^{{\tilde}a_s^2t\cos 2{\tilde}b_s+{\tilde}a_s{\tilde}\varpi\cos {\tilde}b_s}\sin({\tilde}a_s^2 t\sin 2{\tilde}b_s+{\tilde}a_s{\tilde}\varpi\sin {\tilde}b_s+{\tilde}c_s)\eqno(4.21)$$ if ${\sigma}=1$, and $$\begin{aligned}
w&=&\sum_{r=1}^n \hat d_re^{\hat a_r^2{\sigma}t\cos 2\hat b_r+\hat a_r{\tilde}\varpi\cos
\hat b_r}\sin(\hat a_r^2{\sigma}t\sin 2\hat b_i+\hat a_r{\tilde}\varpi\sin
\hat b_r+\hat c_r)\\ & &+\frac{{\sigma}R}{1-{\sigma}}\sum_{s=1}^k{\tilde}d_se^{{\tilde}a_s^2t\cos 2{\tilde}b_s+{\tilde}a_s{\tilde}\varpi\cos {\tilde}b_s}\sin({\tilde}a_s^2 t\sin 2{\tilde}b_s+{\tilde}a_s{\tilde}\varpi\sin {\tilde}b_s+{\tilde}c_s),\hspace{2.2cm}(4.22)\end{aligned}$$ $$T=z+\sum_{s=1}^k{\tilde}a_s^2{\tilde}d_se^{{\tilde}a_s^2t\cos 2{\tilde}b_s+{\tilde}a_s{\tilde}\varpi\cos
{\tilde}b_s}\sin({\tilde}a_s^2 t\sin 2{\tilde}b_s+{\tilde}a_s{\tilde}\varpi\sin {\tilde}b_s+2{\tilde}b_s+{\tilde}c_s)\eqno(4.23)$$ when ${\sigma}\neq 1$.* ]{}
[**Remark 4.2**]{}. By Fourier expansion, we can use the above solution to obtain the one depending on three arbitrary piecewise continuous functions of ${\tilde}\varpi$.
Next we let ${\alpha}$ be any fixed function of $t$ and set $$\varpi={\alpha}(x^2+y^2).\eqno(4.24)$$ We assume $$u=y\phi(t,\varpi)-\frac{{\alpha}'}{2{\alpha}}x,\qquad
v=-x\phi(t,\varpi)-\frac{{\alpha}'}{2{\alpha}}y,\eqno(4.25)$$ $$w=\psi(t,\varpi)+\frac{{\alpha}'}{{\alpha}} z,\qquad T={\vartheta}(t,\varpi)+z\eqno(4.26)$$ where $\phi,\psi$ and ${\vartheta}$ are functions in $t,\varpi$. Note $$\Phi_1=-\frac{{{\alpha}'}^2+2{\alpha}{{\alpha}'}'}{4{\alpha}^2}x
+\frac{{\alpha}'}{2R_0{\alpha}}y+y\phi_t+\left(\frac{x}{R_0}
-\frac{{\alpha}'}{{\alpha}}y \right)\phi-x\phi^2-4{\sigma}{\alpha}y(\varpi\phi)_{\varpi \varpi},\eqno(4.27)$$ $$\Phi_2=-\frac{{{\alpha}'}^2+2{\alpha}{{\alpha}'}'}{4{\alpha}^2}y
-\frac{{\alpha}'}{2R_0{\alpha}}x-x\phi_t+\left(\frac{y}{R_0}
+\frac{{\alpha}'}{{\alpha}}x \right)\phi-y\phi^2+4{\sigma}{\alpha}x(\varpi\phi)_{\varpi \varpi}.\eqno(4.28)$$ According to the first equation in (3.5), $$\left[\varpi\left(\phi_t-\frac{{\alpha}'}{{\alpha}}\phi-4{\sigma}{\alpha}(\varpi\phi)_{\varpi
\varpi}\right)\right]_\varpi+\frac{{\alpha}'}{2R_0{\alpha}}=0,\eqno(4.29)$$ equivalently, $$(\varpi\phi)_t-\frac{{\alpha}'}{{\alpha}}\varpi\phi-4{\sigma}{\alpha}\varpi(\varpi\phi)_{\varpi
\varpi}+\frac{{\alpha}'\varpi}{2R_0{\alpha}}={\alpha}{\beta}'\eqno(4.30)$$ for some function ${\beta}$ of $t$. Write $$\hat\phi=\frac{\varpi\phi}{{\alpha}}+\frac{\varpi}{2R_0{\alpha}}-{\beta}.\eqno(4.31)$$ Then (4.30) becomes $$\hat\phi_t-4{\sigma}{\alpha}\varpi\hat\phi_{\varpi \varpi}=0.\eqno(4.32)$$ Suppose $$\hat\phi=\sum_{i=1}^\infty{\gamma}_i\varpi^i,\eqno(4.33)$$ where ${\gamma}_i$ are functions of $t$ to be determined. Equation (4.32) yields $$({\gamma}_i)_t=4i(i+1){\sigma}{\alpha}{\gamma}_{i+1}.\eqno(4.34)$$ Hence $${\gamma}_{i+1}=\frac{({\alpha}^{-1}{\partial}_t)^i({\gamma})}{i!(i+1)!(4{\sigma})^i}\eqno(4.35)$$ for some function ${\gamma}$ of $t$. Thus $$\hat\phi=
\sum_{i=0}^\infty\frac{({\alpha}^{-1}{\partial}_t)^i({\gamma})\varpi^{i+1}}{i!(i+1)!(4{\sigma})^i}.
\eqno(4.36)$$ By (4.31), we get $$\phi=\frac{{\alpha}{\beta}}{\varpi}-\frac{1}{2R_0}+
{\alpha}\sum_{i=0}^\infty\frac{({\alpha}^{-1}{\partial}_t)^i({\gamma})\varpi^i}
{i!(i+1)!(4{\sigma})^i}.\eqno(4.37)$$
Note $$\Phi_3=\psi_t+\frac{{\alpha}'}{{\alpha}}\psi-4{\sigma}(\varpi\psi_{\varpi})_{\varpi}-{\sigma}R({\vartheta}+z).\eqno(4.38)$$ By the last two equations in (3.5), $$\psi_t+\frac{{\alpha}'}{{\alpha}}\psi-4{\sigma}(\varpi\psi_{\varpi})_{\varpi}-{\sigma}R{\vartheta}=0\eqno(4.39)$$ modulo the transformation in (1.14)-(1.16). On the other hand, (1.6) becomes $${\vartheta}_t-4(\varpi{\vartheta}_{\varpi})_{\varpi}=0.\eqno(4.40)$$ Hence $${\vartheta}=\sum_{i=0}^\infty\frac{{\theta}_1^{(i)}\varpi^{i+1}}{4^i((i+1)!)^2}\eqno(4.41)$$ modulo the transformation in (1.14)-(1.16), where ${\theta}_1$ is an arbitrary function of $t$. Substituting (4.41) into (4.39), we obtain $$\psi=
{\alpha}^{-1}{\theta}_2\varpi+ {\alpha}^{-1}\sum_{i=1}^\infty\frac{{\theta}_2^{(i)}+
R\sum_{r=0}^{i-1}{\sigma}^{i-r}({\alpha}{\theta}_1^{(i-s-1)})^{(s)}}{(4{\sigma})^i((i+1)!)^2}
\varpi^{i+1},\eqno(4.42)$$ where ${\theta}_2$ is another arbitrary function of $t$.
Now $$\Phi_1=-\frac{{{\alpha}'}^2+2{\alpha}{{\alpha}'}'}{4{\alpha}^2}x
+\frac{{\alpha}{\beta}' y}{\varpi}+\frac{x}{R_0}\phi-x\phi^2,\eqno(4.43)$$ $$\Phi_2=-\frac{{{\alpha}'}^2+2{\alpha}{{\alpha}'}'}{4{\alpha}^2}y
-\frac{{\alpha}{\beta}' x}{\varpi}+\frac{y}{R_0}\phi-y\phi^2\eqno(4.44)$$ by (4.27) and (4.28), and $$\Phi_3=({\alpha}^{-1}{\alpha}'-{\sigma}R)z\eqno(4.45)$$ by (4.38). According to (3.4), we obtain $$\begin{aligned}
p&=&\left(\frac{{{\alpha}'}^2+2{\alpha}{{\alpha}'}'}{4{\sigma}{\alpha}^2}+\frac{3}{8{\sigma}R_0^2}\right)(x^2+y^2)
+\frac{{\beta}'}{{\sigma}}\arctan\frac{y}{x} +\frac{(R_0{\alpha}{\gamma}-1){\beta}}{{\sigma}R_0}\ln{\alpha}(x^2+y^2)\\ &
&-\frac{{\sigma}^{-1}{\beta}^2}{2(x^2+y^2)}+\frac{{\sigma}R-{\alpha}^{-1}{\alpha}'R}{2{\sigma}}z^2 -\frac{1}{{\sigma}R_0}\sum_{i=0}^\infty\frac{({\alpha}^{-1}{\partial}_t)^i({\gamma}){\alpha}^{i+1}(x^2+y^2)^{i+1}}
{((i+1)!)^2(4{\sigma})^i}\\ & &+\frac{{\alpha}}{2{\sigma}}\sum_{i,j=0}^\infty
\frac{({\alpha}^{-1}{\partial}_t)^i({\gamma})({\alpha}^{-1}{\partial}_t)^j({\gamma})({\alpha}(x^2+y^2))^{i+j+1}}
{i!j!(i+1)!(j+1)!(i+j+1)(4{\sigma})^{i+j}}\\ &
&+\frac{{\alpha}{\beta}}{2{\sigma}}\sum_{i=1}^\infty\frac{({\alpha}^{-1}{\partial}_t)^i({\gamma})({\alpha}(x^2+y^2))^i}
{i!(i+1)!i(4{\sigma})^i}\hspace{7.6cm}(4.46)\end{aligned}$$ modulo the transformation in (1.14)-(1.16). By (4.25), (4.26), (4.37), (4.41) and (4.42), we have:
[**Theorem 4.3**]{} [*Let ${\alpha},{\beta},{\gamma},{\theta}_1,{\theta}_2$ be any function of $t$ such that the following involved series converge. We have the following solutions of the three-dimensional stratified rotating Boussinesq equations (1.3)-(1.7): $$u=\frac{{\beta}y}{x^2+y^2}-\frac{y}{2R_0}-\frac{{\alpha}'}{2{\alpha}}x+
{\alpha}y\sum_{i=0}^\infty\frac{({\alpha}^{-1}{\partial}_t)^i({\gamma}){\alpha}^i(x^2+y^2)^i}
{i!(i+1)!(4{\sigma})^i},\eqno(4.47)$$ $$v=\frac{x}{2R_0}-\frac{{\alpha}'}{2{\alpha}}y-\frac{{\beta}x}{x^2+y^2}+
{\alpha}x\sum_{i=0}^\infty\frac{({\alpha}^{-1}{\partial}_t)^i({\gamma}){\alpha}^i(x^2+y^2)^i}
{i!(i+1)!(4{\sigma})^i},\eqno(4.48)$$ $$w={\theta}_2(x^2+y^2)+\frac{{\alpha}'}{{\alpha}} z+\frac{1}{{\alpha}}\sum_{i=1}^\infty\frac{{\theta}_2^{(i)}+
R\sum_{r=0}^{i-1}{\sigma}^{i-r}({\alpha}{\theta}_1^{(i-s-1)})^{(s)}}{(4{\sigma})^i((i+1)!)^2}
{\alpha}^{i+1}(x^2+y^2)^{i+1},\eqno(4.49)$$ $$T=z+\sum_{i=0}^\infty\frac{{\theta}_1^{(i)}{\alpha}^{i+1}(x^2+y^2)^{i+1}}{4^i((i+1)!)^2}
\eqno(4.50)$$ and $p$ is given in (4.46).*]{}
Asymmetric Approach III to the 3D Equations
===========================================
In this section, we solve (1.3)-(1.7) with $v_x=w_x=T_x=0$.
Let $c$ be a real constant. Set $$\varpi=y\cos c+z\sin c.\eqno(5.1)$$ Suppose $$u=f(t,\varpi),\qquad
v=\phi(t,\varpi)\sin c,\eqno(5.2)$$ $$w=-\phi(t,\varpi)\cos c,\qquad
T=\psi(t,\varpi)+z,\eqno(5.3)$$ where $f,\;\phi$ and $\psi$ are functions in $t$ and $\varpi$. Then $$\Phi_1=f_t-{\sigma}f_{\varpi\varpi}-\frac{\sin c}{R_0}\phi,\eqno(5.4)$$ $$\Phi_2=(\phi_t-{\sigma}\phi_{\varpi\varpi})\sin c+\frac{1}{R_0}f,\eqno(5.5)$$ $$\Phi_3=({\sigma}\phi_{\varpi\varpi}-\phi_t)\cos c-{\sigma}R(\psi+z).\eqno(5.6)$$ By (3.5), $$f_{\varpi t}-{\sigma}f_{\varpi\varpi\varpi}-\frac{\sin
c}{R_0}\phi_{\varpi}=0,\eqno(5.7)$$ $$(\phi_t-{\sigma}\phi_{\varpi\varpi})_{\varpi}+\frac{\sin c}{R_0}f_{\varpi}+{\sigma}R\psi_{\varpi}\cos c=0.\eqno(5.8)$$ Modulo (1.14)-(1.16), we have $$f_t-{\sigma}f_{\varpi\varpi}-\frac{\sin c}{R_0}\phi=0,\eqno(5.9)$$ $$\phi_t-{\sigma}\phi_{\varpi\varpi}+\frac{\sin c}{R_0}f+{\sigma}R\psi\cos c=0.\eqno(5.10)$$
Denote $$\left(\begin{array}{c}\hat
f\\\hat\phi\end{array}\right)=\left(\begin{array}{cc}\cos\frac{t\sin
c}{R_0}&-\sin\frac{t\sin c}{R_0}\\ \sin\frac{t\sin
c}{R_0}&\cos\frac{t\sin
c}{R_0}\end{array}\right)\left(\begin{array}{c}
f\\\phi\end{array}\right).\eqno(5.11)$$ Then (5.9) and (5.10) become $$\hat f_t-{\sigma}\hat f_{\varpi\varpi}-{\sigma}R\psi\cos c\;\sin\frac{t\sin
c}{R_0}=0,\eqno(5.12)$$ $$\hat\phi_t-{\sigma}\hat\phi_{\varpi\varpi}+{\sigma}R\psi\cos c\;\cos\frac{t\sin
c}{R_0}=0.\eqno(5.13)$$ On the other hand, (1.6) becomes $$\psi_t-\psi_{\varpi\varpi}=0.\eqno(5.14)$$
Assume ${\sigma}=1$. We have the following solution: $$\psi=\sum_{i=1}^m a_id_ie^{a_i^2t\cos 2b_i+a_i\varpi\cos
b_i}\sin(a_i^2t\sin 2b_i+a_i\varpi\sin b_i+b_i+c_i),\eqno(5.15)$$ $$\begin{aligned}
\hat f&=&- RR_0\cot c\;\cos\frac{t\sin
c}{R_0}\;\sum_{i=1}^m a_id_ie^{a_i^2 t\cos 2b_i+a_i\varpi\cos
b_i}\sin(a_i^2t\sin 2b_i+a_i\varpi\sin b_i+b_i+c_i)\\ &
&+\sum_{r=1}^n \hat a_r\hat d_re^{\hat a_r^2 t\cos 2\hat b_r+\hat
a_r\varpi\cos \hat b_r}\sin(\hat a_r^2t\sin 2\hat b_i+\hat
a_r\varpi\sin \hat b_r+\hat b_r+\hat
c_r),\hspace{2.1cm}(5.16)\end{aligned}$$ $$\begin{aligned}
\hat \phi&=&- RR_0\cot c\;\sin\frac{t\sin
c}{R_0}\;\sum_{i=1}^m a_id_ie^{a_i^2t\cos 2b_i+a_i\varpi\cos
b_i}\sin(a_i^2t\sin 2b_i+a_i\varpi\sin b_i+b_i+c_i)\\ &
&+\sum_{s=1}^k{\tilde}a_s{\tilde}d_se^{{\tilde}a_s^2t\cos 2{\tilde}b_s+{\tilde}a_s\varpi\cos {\tilde}b_s}\sin({\tilde}a_s^2 t\sin 2{\tilde}b_s+{\tilde}a_s\varpi\sin {\tilde}b_s+{\tilde}b_s+{\tilde}c_s),\hspace{2.1cm}(5.17)\end{aligned}$$ where $a_i,b_i,c_i,\hat
a_r,\hat b_r,\hat c_r,\hat d_r, {\tilde}a_s,{\tilde}b_s,{\tilde}c_s,{\tilde}d_s$ are arbitrary real numbers. According to (5.11), $$\begin{aligned}
f=-RR_0\cot c\;\cos\frac{2t\sin
c}{R_0}\;\sum_{i=1}^m a_id_ie^{a_i^2t\cos 2b_i+a_i\varpi\cos
b_i}\sin(a_i^2t\sin 2b_i+a_i\varpi\sin b_i+b_i+c_i)
\\ +\cos\frac{t\sin c}{R_0}\;\sum_{r=1}^n \hat a_r\hat d_re^{\hat a_r^2 t\cos 2\hat b_r+\hat
a_r\varpi\cos \hat b_r}\sin(\hat a_r^2t\sin 2\hat b_i+\hat
a_r\varpi\sin \hat b_r+\hat b_r+\hat c_r)\hspace{2.6cm} \\
+\sin\frac{t\sin c}{R_0}\;\sum_{s=1}^k{\tilde}a_s{\tilde}d_se^{{\tilde}a_s^2t\cos 2{\tilde}b_s+{\tilde}a_s\varpi\cos {\tilde}b_s}\sin({\tilde}a_s^2 t\sin
2{\tilde}b_s+{\tilde}a_s\varpi\sin {\tilde}b_s+{\tilde}b_s+{\tilde}c_s),\hspace{1.5cm}(5.18)\end{aligned}$$ $$\begin{aligned}
& &\phi=-\sin\frac{t\sin c}{R_0}\;\sum_{r=1}^n \hat a_r\hat d_re^{\hat a_r^2 t\cos 2\hat b_r+\hat
a_r\varpi\cos \hat b_r}\sin(\hat a_r^2t\sin 2\hat b_i+\hat
a_r\varpi\sin \hat b_r+\hat b_r+\hat c_r)\\ & &+\cos\frac{t\sin
c}{R_0}\;\sum_{s=1}^k{\tilde}a_s{\tilde}d_se^{{\tilde}a_s^2t\cos 2{\tilde}b_s+{\tilde}a_s\varpi\cos {\tilde}b_s}\sin({\tilde}a_s^2 t\sin 2{\tilde}b_s+{\tilde}a_s\varpi\sin {\tilde}b_s+{\tilde}b_s+{\tilde}c_s).\hspace{0.8cm}(5.19)\end{aligned}$$
Suppose ${\sigma}\neq 1$. We take the following solution of (5.11)-(5.14): $$\psi=\sum_{i=1}^m a_id_ie^{a_i^2t+a_i\varpi},\eqno(5.20)$$ $$\begin{aligned}
\hat f&=&{\sigma}R\sum_{i=1}^m
a_id_ie^{a_i^2t+a_i\varpi}\frac{\cos
c\:\left[a_i^2(1-{\sigma})\sin\frac{t\sin c}{R_0}-R_0^{-1}\sin
c\:\cos\frac{t\sin c}{R_0}\right]}{a_i^4(1-{\sigma})^2+R_0^{-2}\sin^2c}
\\ & &+\sum_{r=1}^n \hat a_r\hat d_re^{\hat a_r^2{\sigma}t\cos
2\hat b_r+\hat a_r\varpi\cos \hat b_r}\sin(\hat a_r^2{\sigma}t\sin
2\hat b_i+\hat a_r\varpi\sin \hat b_r+\hat b_r+\hat
c_r),\hspace{1.5cm}(5.21)\end{aligned}$$ $$\begin{aligned}
\hat \phi&=&{\sigma}R\sum_{i=1}^m
a_id_ie^{a_i^2t+a_i\varpi}\frac{\cos
c\:\left[a_i^2({\sigma}-1)\cos\frac{t\sin c}{R_0}-R_0^{-1}\sin
c\:\sin\frac{t\sin c}{R_0}\right]}{a_i^4(1-{\sigma})^2+R_0^{-2}\sin^2c}
\\ &
&+\sum_{s=1}^k{\tilde}a_s{\tilde}d_se^{{\tilde}a_s^2{\sigma}t\cos 2{\tilde}b_s+{\tilde}a_s\varpi\cos {\tilde}b_s}\sin({\tilde}a_s^2{\sigma}t\sin 2{\tilde}b_s+{\tilde}a_s\varpi\sin {\tilde}b_s+{\tilde}b_s+{\tilde}c_s),\hspace{1.4cm}(5.22)\end{aligned}$$ where $a_i,b_i,c_i,\hat
a_r,\hat b_r,\hat c_r,\hat d_r, {\tilde}a_s,{\tilde}b_s,{\tilde}c_s,{\tilde}d_s$ are arbitrary real numbers. According to (5.11), $$\begin{aligned}
f&=&\cos\frac{t\sin c}{R_0}\;\sum_{r=1}^n \hat a_r\hat d_re^{\hat a_r^2{\sigma}t\cos 2\hat b_r+\hat
a_r\varpi\cos \hat b_r}\sin(\hat a_r^2{\sigma}t\sin 2\hat b_i+\hat
a_r\varpi\sin \hat b_r+\hat b_r+\hat c_r) \\&& +\sin\frac{t\sin
c}{R_0}\;\sum_{s=1}^k{\tilde}a_s{\tilde}d_se^{{\tilde}a_s^2{\sigma}t\cos 2{\tilde}b_s+{\tilde}a_s\varpi\cos {\tilde}b_s}\sin({\tilde}a_s^2{\sigma}t\sin 2{\tilde}b_s+{\tilde}a_s\varpi\sin {\tilde}b_s+{\tilde}b_s+{\tilde}c_s)\\
& &-{\sigma}R\sum_{i=1}^m\frac{a_id_ie^{a_i^2t+a_i\varpi}\sin
2c}{2R_0(a_i^4(1-{\sigma})^2+R_0^{-2}\sin^2c)}
,\hspace{6.6cm}(5.23)\end{aligned}$$ $$\begin{aligned}
\phi&=&-\sin\frac{t\sin c}{R_0}\;\sum_{r=1}^n \hat a_r\hat d_re^{\hat a_r^2{\sigma}t\cos 2\hat b_r+\hat
a_r\varpi\cos \hat b_r}\sin(\hat a_r^2{\sigma}t\sin 2\hat b_i+\hat
a_r\varpi\sin \hat b_r+\hat b_r+\hat c_r)\\ & &+\cos\frac{t\sin
c}{R_0}\;\sum_{s=1}^k{\tilde}a_s{\tilde}d_se^{{\tilde}a_s^2{\sigma}t\cos 2{\tilde}b_s+{\tilde}a_s\varpi\cos {\tilde}b_s}\sin({\tilde}a_s^2{\sigma}t\sin 2{\tilde}b_s+{\tilde}a_s\varpi\sin {\tilde}b_s+{\tilde}b_s+{\tilde}c_s)\\
& &-{\sigma}R\sum_{i=1}^m\frac{a_i^3d_i({\sigma}-1)e^{a_i^2t+a_i\varpi}\cos
c}{a_i^4(1-{\sigma})^2+R_0^{-2}\sin^2c}
.\hspace{7.5cm}(5.24)\end{aligned}$$
By (5.4)-(5.6), (5.9) and (5.10), $\Phi_1=0$, $$\Phi_2=\left(\frac{\cos c}{R_0}f-{\sigma}R\psi\sin c\right)\cos
c,\eqno(5.25)$$ $$\Phi_3=\left(\frac{\cos c}{R_0}f-{\sigma}R\psi\sin c\right)\sin
c-{\sigma}Rz.\eqno(5.26)$$ According to (3.4), $$\begin{aligned}
p&=&\frac{R\cos^2 c}{\sin c}\cos\frac{2t\sin
c}{R_0}\;\sum_{i=1}^m d_ie^{a_i^2t\cos 2b_i+a_i\varpi\cos
b_i}\sin(a_i^2t\sin 2b_i+a_i\varpi\sin b_i+c_i)
\\ & &-\frac{\cos c}{ R_0}\cos\frac{t\sin c}{R_0}\;\sum_{r=1}^n \hat d_re^{\hat a_r^2 t\cos 2\hat b_r+\hat
a_r\varpi\cos \hat b_r}\sin(\hat a_r^2t\sin 2\hat b_i+\hat
a_r\varpi\sin \hat b_r+\hat c_r) \\&& -\frac{\cos c}{
R_0}\sin\frac{t\sin c}{R_0}\;\sum_{s=1}^k{\tilde}d_se^{{\tilde}a_s^2t\cos
2{\tilde}b_s+{\tilde}a_s\varpi\cos {\tilde}b_s}\sin({\tilde}a_s^2 t\sin 2{\tilde}b_s+{\tilde}a_s\varpi\sin {\tilde}b_s+{\tilde}c_s)\\ & &+R\sin c\:\sum_{i=1}^m
d_ie^{a_i^2t\cos 2b_i+a_i\varpi\cos b_i}\sin(a_i^2t\sin
2b_i+a_i\varpi\sin
b_i+c_i)+\frac{R}{2}z^2\hspace{1.2cm}(5.27)\end{aligned}$$ modulo if ${\sigma}=1$, and $$\begin{aligned}
p&=&-\frac{\cos c}{{\sigma}R_0}\cos\frac{t\sin c}{R_0}\;\sum_{r=1}^n \hat d_re^{\hat a_r^2{\sigma}t\cos 2\hat b_r+\hat
a_r\varpi\cos \hat b_r}\sin(\hat a_r^2{\sigma}t\sin 2\hat b_i+\hat
a_r\varpi\sin \hat b_r+\hat c_r) \\&& -\frac{\cos c}{{\sigma}R_0}\sin\frac{t\sin c}{R_0}\;\sum_{s=1}^k{\tilde}d_se^{{\tilde}a_s^2{\sigma}t\cos 2{\tilde}b_s+{\tilde}a_s\varpi\cos {\tilde}b_s}\sin({\tilde}a_s^2{\sigma}t\sin
2{\tilde}b_s+{\tilde}a_s\varpi\sin {\tilde}b_s+{\tilde}c_s)\\
& &+ \sum_{i=1}^m\frac{d_iRe^{a_i^2t+a_i\varpi}\sin 2c\;\cos
c}{2R_0^2(a_i^4(1-{\sigma})^2+R_0^{-2}\sin^2c)}+R\sin c\;\sum_{i=1}^m
d_ie^{a_i^2t+a_i\varpi}+\frac{R}{2}z^2
,\hspace{1.7cm}(5.28)\end{aligned}$$ modulo the transformation in (1.14)-(1.16).
In summary, we have:
[**Theorem 5.1**]{}. [*Let $a_i,b_i,c_i,\hat a_r,\hat b_r,\hat
c_r,\hat d_r, {\tilde}a_s,{\tilde}b_s,{\tilde}c_s,{\tilde}d_s,c$ be arbitrary real numbers. Denote $\varpi=y\cos x+z\sin c$. We have the following solutions of the three-dimensional stratified rotating Boussinesq equations (1.3)-(1.7): $$u=f,\qquad
v=\phi\sin c,\qquad w=-\phi\cos c,\qquad T=\psi+z,\eqno(5.29)$$ where (1) $f$ is given in (5.18), $\phi$ is given in (5.19), $\psi$ is given in (5.15) and $p$ is given in (5.27) if ${\sigma}=1$; (2) $f$ is given in (5.23), $\phi$ is given in (5.24), $\psi$ is given in (5.20) and $p$ is given in (5.28) when ${\sigma}\neq 1$.*]{}
[**Remark 5.2**]{}. By Fourier expansion, we can use the above solution to obtain the one depending on three arbitrary piecewise continuous functions of $\varpi$. Applying the transformation ${\cal
T}_1$ in (1.12)-(1.13) to the above solution, we get a solution involving all the variables $t,x,y,z$.
[10]{}
D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity, [*Adv. Math.*]{} [**203**]{} (2006), 497-513.
M. Gill and S. Childress, [*Topics in Geophysical Fluid Dynamics, Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics*]{}, Springer-verlag, New York, 1987.
T. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations, [*Discrete Contin. Dyn. Syst.*]{} [**12**]{} (2005), 1-12.
C. Hsia, T. Ma and S. Wang, Stratified rotating Boussinesq equations in geophysical fluid dynamics: dynamic bifurcation and periodic solutions, [*J. Math. Phys.*]{} [**48**]{} (2007), no. 6, 06560.
E. N. Lorenz, Deterministic nonperiodic flow, [*J. Atmos. Sci.*]{} [**20**]{} (1963), 130-141.
J. Lions, R. Teman and S. Wang, New formulations of the primitive equations of the atmosphere and applications, [*Nonlinearity*]{} [**5**]{} (1992), 237-288.
J. Lions, R. Teman and S. Wang, On the equations of large-scale ocean, [*Nonlinearity*]{} [**5**]{} (1992), 1007-1053.
A. Majda, [*Introduction to PDEs and Waves for the Atmosphere and Ocean*]{}, Courant Lecture Note in Mathematics, Vol. 9, AMS and CIMS, 2003.
J. Pedlosky, [*Geophsical Fluid Dynamics*]{}, 2rd Edition, Springer-verlag, New York, 1987.
X. Xu, Stable-Range approach to the equation of nonstationary transonic gas flows, [*Quart. Appl. Math.*]{} [**65**]{} (2007), 529-547.
X. Xu, Asymmetric and moving-frame approaches to Navier-Stokes equations, [*Quart. Appl. Math.*]{}, in press, arXiv:0706.1861.
[^1]: 2000 Mathematical Subject Classification. Primary 35C05, 35Q55; Secondary 37K10.
[^2]: Research supported by China NSF 10431040
| {
"pile_set_name": "ArXiv"
} |
=1
Introduction
============
The origin of Hom-structures may be found in the physics literature around 1990, concerning $q$-deformations of algebras of vector fields, especially Witt and Virasoro algebras, see for instance [@AizawaSato; @ChaiIsKuLuk; @CurtrZachos1; @Liu1]. Hartwig, Larsson and Silvestrov studied this kind of algebras in [@HLS; @LS1] and called them Hom-Lie algebras because they involve a homomorphism in the defining identity. More precisely, a Hom-Lie algebra is a linear space $L$ endowed with two linear maps $[-]\colon L\otimes L\rightarrow L$ and $\alpha\colon L\rightarrow L$ such that $[-]$ is skew-symmetric and $\alpha$ is an algebra endomorphism with respect to the bracket satisfying the so-called Hom-Jacobi identity $$\begin{gathered}
[\alpha (x),[y,z]]+[\alpha (y),[z,x]]+[\alpha (z),[x,y]]=0,\qquad \forall
\, x,y,z\in L.\end{gathered}$$ Since any associative algebra becomes a Lie algebra by taking the commutator $[a, b]=ab-ba$, it was natural to look for a Hom-analogue of this property. This was accomplished in [@ms1], where the concept of Hom-associative algebra was introduced, as being a linear space $A$ endowed with a multiplication $\mu \colon A\otimes A\rightarrow A$, $\mu (a\otimes b)=ab$, and a linear map $\alpha \colon A\rightarrow A$ satisfying the so-called Hom-associativity condition $$\begin{gathered}
\alpha (a)(bc)=(ab)\alpha (c), \qquad \forall\, a, b, c\in A.\end{gathered}$$ If $A$ is Hom-associative then $(A, [a, b]=ab-ba, \alpha)$ becomes a Hom-Lie algebra, denoted by $L(A)$. Notice that Hom-Lie algebras, in this paper, were considered without the assumption of multiplicativity of $\alpha$.
In subsequent literature (see for instance [@yau2]) were studied subclasses of these classes of algebras where the linear maps $\alpha $ involved in the definition of a Hom-Lie algebra or Hom-associative algebra are required to be multiplicative, that is $\alpha ([x,y])=[\alpha(x),\alpha(y)]$ for all , respectively $\alpha (ab)=\alpha (a)\alpha (b)$ for all $a, b\in A$, and these subclasses were called multiplicative Hom-Lie algebras, respectively multiplicative Hom-associative algebras. Since we will always assume multiplicativity of the maps $\alpha $ and to simplify terminology, we will call Hom-Lie or Hom-associative algebras what was called above multiplicative Hom-Lie or Hom-associative algebras.
The Hom-analogues of coalgebras, bialgebras and Hopf algebras have been introduced in . The original definition of a Hom-bialgebra involved two linear maps, one twisting the associativity condition and the other one the coassociativity condition. Later, two directions of study on Hom-bialgebras were developed, one in which the two maps coincide (these are still called Hom-bialgebras) and another one, started in [@stef], where the two maps are assumed to be inverse to each other (these are called monoidal Hom-bialgebras).
In the last years, many concepts and properties from classical algebraic theories have been extended to the framework of Hom-structures, see for instance [@AC1; @AC2; @said; @stef; @chenwangzhang; @hassanzadeh; @LB; @mp1; @ms3; @ms4; @sheng; @yau1; @yau2].
The main tool for constructing examples of Hom-type algebras is the so-called “twisting principle” introduced by D. Yau for Hom-associative algebras and extended afterwards to other types of Hom-algebras. For instance, if $A$ is an associative algebra and $\alpha \colon A\rightarrow A$ is an algebra map, then $A$ with the new multiplication defined by $a\ast b=\alpha (a)\alpha (b)$ is a Hom-associative algebra, called the *Yau twist* of $A$.
A categorical interpretation of Hom-associative algebras has been given by Caenepeel and Goyvaerts in [@stef]. First, to any monoidal category $\mathcal{C}$ they associate a new monoidal category $\widetilde{\mathcal{H}}(\mathcal{C})$, called a Hom-category, whose objects are pairs consisting of an object of $\mathcal{C}$ and an automorphism of this object ($\widetilde{\mathcal{H}}(\mathcal{C})$ has nontrivial associativity constraint even if the one of $\mathcal{C}$ is trivial). By taking $\mathcal{C}$ to be ${_{\Bbbk }\mathcal{M}}$, the category of linear spaces over a base field $\Bbbk $, it turns out that an algebra in the (symmetric) monoidal category $\widetilde{\mathcal{H}}({_{\Bbbk }\mathcal{M}})$ is the same thing as a Hom-associative algebra $(A,\mu ,\alpha )$ with bijective $\alpha $. The bialgebras in $\widetilde{\mathcal{H}}({_{\Bbbk }\mathcal{M}})$ are the monoidal Hom-bialgebras we mentioned before.
In [@giacomo], the first author extended the construction of the Hom-category $\widetilde{\mathcal{H}}(\mathcal{C})$ to include the action of a given group $\mathcal{G}$. Namely, given a monoidal category $\mathcal{C}$, a group $\mathcal{G}$, two elements $c,d\in Z(\mathcal{G})$ and $\nu $ an automorphism of the unit object of $\mathcal{C}$, the group Hom-category $\mathcal{H}^{{c,d,\nu}}(\mathcal{G},\mathcal{C})$ has as objects pairs $(A,f_{A})$, where $A$ is an object in $\mathcal{C}$ and $f_{A}\colon \mathcal{G} \rightarrow \operatorname{Aut}_{\mathcal{C}}(A)$ is a group homomorphism. The associativity constraint of $\mathcal{H}^{{c,d,\nu}}(\mathcal{G},\mathcal{C})$ is naturally defined by means of $c$, $d$, $\nu $ (see Claim \[Cl:monoidal\] and Theorem \[Th:Monoidal\]) and it is, in general, non trivial. A braided structure is also defined on $\mathcal{H}^{{c,d,\nu }}(\mathcal{G},\mathcal{C})$ (see Claim \[cl:braid\] and Theorem \[Theo:braidcat\]) turning it into a braided category which is symmetric whenever $\mathcal{C}$ is. When $\mathcal{G}=\mathbb{Z}$, $c=d=1_{\mathbb{Z}}$ and $\nu
=\operatorname{id}_{\mathbf{1}}$ one gets the category $\mathcal{H}(\mathcal{C})$ from [@stef], while for $c=1_{\mathbb{Z}}$, $d=-1_{\mathbb{Z}}$ and $\nu =\operatorname{id}_{\mathbf{1}}$ one gets the category $\widetilde{\mathcal{H}}(\mathcal{C})$.
We first look at the case when $\mathcal{\mathcal{G}}=\mathbb{Z\times Z}$, $c=(1,0) $, $d=(0,1) $, $\nu =\operatorname{id}_{\mathbf{1}}$ and $\mathcal{C}={_{\Bbbk }\mathcal{M}}$.
If $M\in {_{\Bbbk }\mathcal{M}}$, a group homomorphism $f_{M}\colon
\mathbb{Z}
\times
\mathbb{Z}
\rightarrow \operatorname{Aut}_{\Bbbk }(M) $ is completely determined by $$\begin{gathered}
f_{M}((1,0)) =\alpha _{M}\qquad \text{and}\qquad
f_{M}((0,1)) =\beta _{M}^{-1}.\end{gathered}$$ Thus, an object in $\mathcal{H}(
\mathbb{Z}
\times
\mathbb{Z}
,{_{\Bbbk }\mathcal{M}})$ identifies with a triple $(
M,\alpha _{M},\beta _{M}) $, where $\alpha _{M},\beta _{M}\in \operatorname{Aut}_{\Bbbk }(M) $ and $\alpha _{M}\circ \beta _{M}=\beta
_{M}\circ \alpha _{M}$. For $( X,\alpha _{X},\beta _{X}) $, $( Y,\alpha _{Y},\beta _{Y}) $, $( Z,\alpha _{Z},\beta
_{Z}) $ objects in the category $\mathcal{H}^{(1,0),(0,1) ,1}(
\mathbb{Z}
\times
\mathbb{Z}
,{_{\Bbbk }\mathcal{M}})$, the associativity constraint in $\mathcal{H}^{(1,0) ,(0,1) ,1}(
\mathbb{Z}
\times
\mathbb{Z}
,{_{\Bbbk }\mathcal{M}})$ is given by $$\begin{gathered}
\big( \overline{a}^{c,d,\nu }\big) _{( X,\alpha _{X},\beta
_{X}) ,( Y,\alpha _{Y},\beta _{Y}) ,( Z,\alpha
_{Z},\beta _{Z}) }=a_{X,Y,Z}\circ \big[ ( \alpha _{X}\otimes
Y ) \otimes \beta _{Z}^{-1}\big] ,\end{gathered}$$ and the braiding is $$\begin{gathered}
\overline{\gamma }_{( X,\alpha _{X},\beta _{X}) ,( Y,\alpha
_{Y},\beta _{Y}) }^{{c,d,\nu}}=\tau \big[ \big( \alpha _{X}\beta
_{X}^{-1}\big) \otimes \big( \alpha _{Y}^{-1}\beta _{Y}\big) \big] ,\end{gathered}$$ where $\tau \colon X\otimes Y\rightarrow Y\otimes X$ denotes the usual flip in the category of linear spaces. Note that $\overline{\gamma }$ is a symmetric braiding. Being $\mathcal{H}^{(1,0) ,(0,1) ,1}
(\mathbb{Z}
\times
\mathbb{Z}
,{_{\Bbbk }\mathcal{M}})$ an additive braided monoidal category, all the concepts of algebra, Lie algebra and so on, can be introduced in this case.
By writing down the axioms for an algebra in $\mathcal{H}^{(1,0) ,(0,1) ,1}
(
\mathbb{Z}
\times
\mathbb{Z}
,{_{\Bbbk }\mathcal{M}})$ and discarding the invertibility of $\alpha $ and $\beta $ if not needed, we arrived at the following concept. A BiHom-associative algebra over $\Bbbk $ is a linear space $A$ endowed with a multiplication $\mu \colon A\otimes A\rightarrow A$, $\mu (a\otimes b)=ab$, and two commuting multiplicative linear maps $\alpha ,\beta \colon A\rightarrow A$ satisfying what we call the BiHom-associativity condition $$\begin{gathered}
\alpha (a)(bc)=(ab)\beta (c),\qquad \forall \, a,b,c\in A.\end{gathered}$$
Thus, a BiHom-associative algebra with *bijective* structure maps is exactly an algebra in $\mathcal{H}^{(1,0) ,(0,1) ,1}
(\mathbb{Z}
\times
\mathbb{Z}
,{_{\Bbbk }\mathcal{M}})$.
Obviously, a BiHom-associative algebra for which $\alpha =\beta $ is just a Hom-associative algebra.
The remarkable fact is that the twisting principle may be also applied: if $A
$ is an associative algebra and $\alpha ,\beta \colon A\rightarrow A$ are two commuting algebra maps, then $A$ with the new multiplication defined by $
a\ast b=\alpha (a)\beta (b)$ is a BiHom-associative algebra, called the *Yau twist* of $A$. As a matter of fact, although we arrived at the concept of BiHom-associative algebra via the categorical machinery presented above, it is the possibility of twisting the multiplication of an associative algebra by *two* commuting algebra endomorphisms that led us to believe that BiHom-associative algebras are interesting objects in their own. One can think of this as follows. Take again an associative algebra $A$ and $\alpha ,\beta \colon A\rightarrow A$ two commuting algebra endomorphisms; define a new multiplication on $A$ by $a\ast b=\alpha
(a)\beta (b)$. Then it is natural to ask the following question: what kind of structure is $( A,\ast )$? Example \[giacomoex\] in this paper shows that, in general, $( A,\ast) $ is *not* a Hom-associative algebra, so the theory of Hom-associative algebras is *not* general enough to cover this natural operation of twisting the multiplication of an associative algebra by *two* maps; but this operation fits in the framework of BiHom-associative algebras. The Yau twisting of an associative algebra by two maps should thus be considered as the “natural” example of a BiHom-associative algebra. We would like to emphasize that for this operation the two maps are *not* assumed to be bijective, so the resulting BiHom-associative algebra has possibly *non bijective* structure maps and as such it *cannot* be regarded, to our knowledge, as an algebra in a monoidal category.
Take now the group $\mathcal{G}$ to be arbitrary. It is natural to describe how an algebra in the monoidal category $\mathcal{H}^{c,d,\nu }(\mathcal{G},{_{\Bbbk }\mathcal{M}})$ looks like. By writing down the axioms, it turns out (see Claim \[Cl:Alg2\] and Remark \[remada\]) that an algebra in such a category is a BiHom-associative algebra with bijective structure maps (the associativity of the algebra in the category is equivalent to the BiHom-associativity condition) having some extra structure (like an action of the group on the algebra). So, morally, the group $\mathcal{G}=\mathbb{Z}\times \mathbb{Z}$ leads to BiHom-associative algebras but any other group would not lead to something like a “higher” structure than BiHom-associative algebras (for instance, one cannot have something like TriHom-associative algebras).
We initiate in this paper the study of what we will call BiHom-structures. The next structure we introduce is that of a BiHom-Lie algebra; for this, we use also a categorical approach. Unlike the Hom case, to obtain a BiHom-Lie algebra from a BiHom-associative algebra we need the structure maps $\alpha $ and $\beta $ to be bijective; the commutator is defined by the formula $[a,b]=ab-\alpha ^{-1}\beta (b)\alpha \beta ^{-1}(a)$. Nevertheless, just as in the Hom-case, the Yau twist works: if $( L,[ -]) $ is a Lie algebra over a field $\Bbbk $ and $\alpha ,\beta\colon L\rightarrow L$ are two commuting multiplicative linear maps and we define the linear map $ \{ - \} \colon L\otimes L\rightarrow L$, $ \{ a,b \} = [ \alpha ( a ) ,\beta ( b ) ]$, for all $a,b\in L$, then $L_{( \alpha ,\beta ) }:=(L, \{ -\}, \alpha ,
\beta )$ is a BiHom-Lie algebra, called the [*Yau twist*]{} of $( L,[ -] ) $.
We define representations of BiHom-associative algebras and BiHom-Lie algebras and find some of their basic properties. Then we introduce BiHom-coassociative coalgebras and BiHom-bialgebras together with some of the usual ingredients (comodules, duality, convolution product, primitive elements, module and comodule algebras). We define antipodes for a certain class of BiHom-bialgebras, called monoidal BiHom-bialgebras, leading thus to the concept of monoidal BiHom-Hopf algebras. We define smash products, as particular cases of twisted tensor products, introduced in turn as a particular case of twisting a BiHom-associative algebra by what we call a BiHom-pseudotwistor. We write down explicitly such a smash product, obtained from an action of a Yau twist of the quantum group $U_{q}(\mathfrak{sl}_{2})$ on a Yau twist of the quantum plane $\mathbb{A}_{q}^{2|0}$.
As a final remark, let us note that one could introduce a less restrictive concept of BiHom-associative algebra by dropping the assumptions that $\alpha $ and $\beta $ are multiplicative and/or that they commute (note that all the examples of $q$-deformations of Witt or Virasoro algebras are not multiplicative). Unfortunately, by dropping any of these assumptions, one loses the main class of examples, the Yau twists, in the sense that if $A$ is an associative algebra and $\alpha, \beta \colon A\rightarrow A$ are two arbitrary linear maps, and we define as before $a*b=\alpha (a)\beta (b)$, then $(A, *)$ in general is not a BiHom-associative algebra even in this more general sense.
The category $\boldsymbol{\mathcal{H}(\mathcal{G},\mathcal{C})}$
================================================================
Our aim in this section is to introduce so-called group Hom-categories; proofs of the results in this section may be found in [@giacomo].
Let $\mathcal{G}$ be a group and let $\mathcal{C}$ be a category. The *group Hom-category* $\mathcal{H}(\mathcal{G},\mathcal{C})$ associated to $\mathcal{G}$ and $\mathcal{C}$ is the category having as objects pairs $(A,f_{A})$, where $A\in \mathcal{C}$ and $f_{A}$ is a group homomorphism $\mathcal{G\rightarrow }\operatorname{Aut}_{\mathcal{C}}(A)$. A morphism $\xi \colon
(A,f_{A})\rightarrow (B,f_{B})$ in $\mathcal{H}(\mathcal{G},\mathcal{C})$ is a morphism $\xi \colon A\rightarrow B$ in $\mathcal{C}$ such that $f_{B}(g) \circ \xi =\xi \circ f_{A}(g)$, for all $g\in \mathcal{G}$.
A *monoidal category* (see [@Kassel Chapter XI]) is a category $\mathcal{C}$ endowed with an object $\mathbf{1}\in \mathcal{C}$ (called *unit*), a functor $\otimes \colon \mathcal{C}\times \mathcal{C}\rightarrow
\mathcal{C}$ (called *tensor product*) and functorial isomorphisms $a_{X,Y,Z}\colon (X\otimes Y)\otimes Z\rightarrow X\otimes (Y\otimes Z)$, $l_{X}\colon
\mathbf{1}\otimes X\rightarrow X$, $r_{X}\colon X\otimes \mathbf{1}\rightarrow X$, for every $X$, $Y$, $Z$ in $\mathcal{C}$. The functorial isomorphisms $a$ are called the *associativity constraints* and satisfy the pentagon axiom, that is $$\begin{gathered}
(U\otimes a_{V,W,X})\circ a_{U,V\otimes W,X}\circ (a_{U,V,W}\otimes
X)=a_{U,V,W\otimes X}\circ a_{U\otimes V,W,X} $$ holds true, for every $U$, $V$, $W$, $X$ in $\mathcal{C}$. The isomorphisms $l$ and $r $ are called the *unit constraints* and they obey the Triangle Axiom, that is $$\begin{gathered}
(V\otimes l_{W})\circ a_{V,\mathbf{1},W}=r_{V}\otimes W,\qquad \text{for every $V$, $W$ in $\mathcal{C}$}. $$
A *monoidal functor* $(F,\phi _{2},\phi _{0})\colon (\mathcal{C},\otimes ,\mathbf{1},a,l,r )\rightarrow (\mathcal{C}^{\prime }
,\otimes ^{\prime },\mathbf{1}^{\prime },a^{\prime },l^{\prime
},r^{\prime })$ between two monoidal categories consists of a functor $F\colon \mathcal{C}\rightarrow \mathcal{C}^{\prime }$, an isomorphism $
\phi _{2}(U,V)\colon F(U)\otimes ^{\prime }F(V)\rightarrow F(U\otimes V)$, natural in $U,V\in \mathcal{C}$, and an isomorphism $\phi _{0}\colon \mathbf{1}^{\prime
}\rightarrow F(\mathbf{1})$ such that the diagram $$\begin{gathered}
\xymatrixcolsep{67pt}\xymatrixrowsep{35pt}\xymatrix{ (F(U)\!\otimes'\!
F(V))\!\otimes'\! F(W) \ar[d]|{a'_{F(U),F(V),F(W)}} \ar[r]^{\phi_2(U,V)\otimes'
F(W)} & F(U\!\otimes\! V)\!\otimes'\! F(W) \ar[r]^{\phi_2(U\otimes V,W)} &
F((U\!\otimes\! V)\!\otimes\! W) \ar[d]|{F(a_{ U,V, W})} \\ F(U)\!\otimes'\!
(F(V)\!\otimes'\! F(W)) \ar[r]^{F(U)\otimes' \phi_2(V,W)} & F(U)\!\otimes'\!
F(V\!\otimes\! W) \ar[r]^{\phi_2(U,V\otimes W)} & F(U\!\otimes\! (V\!\otimes\! W)) }
$$ is commutative, and the following conditions are satisfied $$\begin{gathered}
{F(l_{U})}\circ {\phi _{2}(\mathbf{1},U)}\circ ({\phi _{0}\otimes }^{\prime }
{F(U)})={l}^{\prime }{_{F(U)}},
\qquad
{F(r_{U})}\circ {\phi _{2}(U,\mathbf{1})}\circ ({F(U)\otimes }^{\prime }{\phi _{0}})={r}^{\prime }{_{F(U)}}.\end{gathered}$$
\[Cl:monoidal\] Let $\mathcal{G}$ be a group and let $( \mathcal{C}
,\otimes ,\mathbf{1},a,l,r) $ be a monoidal category. Given any pair of objects $(A,f_{A}),(B,f_{B})\in \mathcal{H}(\mathcal{G},\mathcal{C})$, consider the map $f_{A}\otimes f_{B}\colon \mathcal{G}\rightarrow\operatorname{Aut}_{\mathcal{C}
}(A\otimes B)$ defined by setting $$\begin{gathered}
( f_{A}\otimes f_{B}) (g) =f_{A}(g)
\otimes f_{B}(g), $$ for all $g\in \mathcal{G}$. Then $f_{A}\otimes f_{B}$ is a group homomorphism and hence $$\begin{gathered}
( A\otimes B,f_{A}\otimes f_{B}) \in \mathcal{H}(\mathcal{G},\mathcal{C}).\end{gathered}$$ Moreover, if $\phi\colon (A,f_{A})\rightarrow (\tilde{A},f_{\tilde{A}})$ and $\xi
\colon (B,f_{B})\rightarrow (\tilde{B},f_{\tilde{B}})$ are morphisms in $\mathcal{H}(\mathcal{G},\mathcal{C})$, then $$\begin{gathered}
\phi \otimes \xi \colon \ ( A\otimes B,f_{A}\otimes f_{B} ) \rightarrow \big(\tilde{A}\otimes \tilde{B},f_{\tilde{A}}\otimes f_{\tilde{B}}\big)\end{gathered}$$ is a morphism in $\mathcal{H}(\mathcal{G},\mathcal{C})$.
Let $Z(\mathcal{G}) $ be the center of $\mathcal{G}$ and let $
c\in Z(\mathcal{G}) $. Then we can consider the functorial isomorphism $\varphi ( c) \colon \operatorname{Id}_{\mathcal{H}(\mathcal{G},\mathcal{C})
}\rightarrow \operatorname{Id}_{\mathcal{H}(\mathcal{G},\mathcal{C})}$ defined by setting $$\begin{gathered}
\varphi (c) (A,f_{A})=f_{A}(c), \qquad \text{for every $(A,f_{A})$ in $\mathcal{H}(\mathcal{G},\mathcal{C})$}.\end{gathered}$$ Also, let $\widehat{\operatorname{Id}_{\mathbf{1}}}\colon \mathcal{G}\rightarrow \operatorname{Aut}_{\mathcal{C}}( \mathbf{1}) $ denote the constant map equal to $\operatorname{Id}_{\mathbf{1}}$.
Let $c,d\in Z(\mathcal{G}) $ and let $\nu \in \operatorname{Aut}_{\mathcal{C}}( \mathbf{1}) $. We set $$\begin{gathered}
\overline{a}^{c,d,\nu }=a\circ \big[ \big( \varphi (c) \otimes
\operatorname{Id}_{\mathcal{H}(\mathcal{G},\mathcal{C})}\big) \otimes \varphi (
d ) \big],
\qquad
\overline{l}^{c,d,\nu }=\varphi \big( d^{-1}\big) \circ l\circ \big( \nu
\otimes \operatorname{Id}_{\mathcal{H}(\mathcal{G},\mathcal{C})}\big),
\\
\overline{r}^{c,d,\nu }=\varphi (c) \circ r\circ \big( \operatorname{Id}_{\mathcal{H}(\mathcal{G},\mathcal{C})}\otimes \nu \big) .\end{gathered}$$
\[Th:Monoidal\]In the setting of Claim [\[Cl:monoidal\]]{}, the category $$\begin{gathered}
\mathcal{H}^{{c,d,\nu}}(\mathcal{G},\mathcal{C})=\big( \mathcal{H}(\mathcal{G},
\mathcal{C}),\otimes ,\big( \mathbf{1,}\widehat{\operatorname{Id}_{\mathbf{1}}}
\big) ,\overline{a}^{c,d,\nu },\overline{l}^{c,d,\nu },\overline{r}
^{c,d,\nu }\big)\end{gathered}$$ is monoidal.
From now on, when $( \mathcal{C},\otimes ,\mathbf{1},a,l,r) $ is a monoidal category, $\mathcal{G}$ is a group, $c,d\in Z( \mathcal{G}
) $ and $\nu \in \operatorname{Aut}_{\mathcal{C}}( \mathbf{1}) $, we will indicate the monoidal category defined in Theorem \[Th:Monoidal\] by $\mathcal{H}^{c,d,\nu }(\mathcal{G},\mathcal{C})$. In the case when $c=d=
\mathbf{1}_{\mathcal{G}}$ and $\nu =\operatorname{Id}_{\mathbf{1}}$, we will simply write $\mathcal{H}(\mathcal{G},\mathcal{C})$.
\[thm:monoidaliso\]Let $( \mathcal{C},\otimes ,\mathbf{1}
,a,l,r) $ be a monoidal category and $\mathcal{G}$ a group. Then the identity functor $\mathcal{I}\colon \mathcal{H}^{c,d,\nu }(\mathcal{G},\mathcal{C})
\rightarrow \mathcal{H}(\mathcal{G},\mathcal{C})$ is a monoidal isomorphism via $$\begin{gathered}
\phi _{0}=\nu ^{-1}\colon \ \big(\mathbf{1},\widehat{\operatorname{Id}_{\mathbf{1}}}
\big)\rightarrow \big(\mathbf{1},\widehat{\operatorname{Id}_{\mathbf{1}}}\big)\qquad \text{and}
\qquad \phi _{2} ( (A,f_{A}),(B,f_{B}) ) =f_{A}\big(c^{-1}\big)\otimes
f_{B}(d),\end{gathered}$$ for every $(A,f_{A}),(B,f_{B})\in \mathcal{H}^{c,d,\nu }(\mathcal{G},\mathcal{C})$.
\[def braiding\] A *braided monoidal category* $(\mathcal{C},\otimes ,\mathbf{1},a,l,r,\gamma )$ is a monoidal category $(\mathcal{C},\otimes ,\mathbf{1,}a,l,r)$ equipped with a *braiding* $\gamma $, that is, an isomorphism $\gamma _{U,V}\colon U\otimes V\rightarrow
V\otimes U$, natural in $U,V\in \mathcal{C}$, satisfying, for all $U,V,W\in
\mathcal{C}$, the hexagon axioms $$\begin{gathered}
a_{V,W,U}\circ \gamma _{U,V\otimes W}\circ a_{U,V,W}=(V\otimes \gamma
_{U,W})\circ a_{V,U,W}\circ (\gamma _{U,V}\otimes W), \\
a_{W,U,V}^{-1}\circ \gamma _{U\otimes V,W}\circ a_{U,V,W}^{-1}=(\gamma
_{U,W}\otimes V)\circ a_{U,W,V}^{-1}\circ (U\otimes \gamma _{V,W}).
$$ A braided monoidal category is called *symmetric* if we further have $
\gamma _{V,U}\circ \gamma _{U,V}=\operatorname{Id}_{U\otimes V}$ for every $U,V\in
\mathcal{C}$. A *braided monoidal functor* is a monoidal functor $F\colon
\mathcal{C}\rightarrow \mathcal{C}^{\prime }$ such that $$\begin{gathered}
F( \gamma _{U,V}) \circ \phi _{2}(U,V)=\phi _{2}(V,U)\circ \gamma
_{F(U) ,F(V) }^{\prime }, \qquad \text{for every} \ \ U,
V\in {\mathcal{C}}. $$
\[cl:braid\]Let $\mathcal{G}$ be a group and let $( \mathcal{C},\otimes ,\mathbf{1},a,l,r,\gamma) $ be a braided monoidal category. Let $c,d\in Z(\mathcal{G}) $ and let $\nu \in \operatorname{Aut}_{\mathcal{C}}( \mathbf{1}) $. We will introduce a braided structure on the monoidal category $\mathcal{H}^{{c,d,\nu}}(\mathcal{G},\mathcal{C})$ by setting, for every $(A,f_{A})$ and $( B,f_{B}) $ in $\mathcal{H}(\mathcal{G},\mathcal{C})$, $$\begin{gathered}
\overline{\gamma }_{(A,f_{A}), ( B,f_{B} ) }^{{c,d,\nu}}=\gamma
_{A,B}\circ \big( f_{A}(cd)\otimes f_{B}\big(c^{-1}d^{-1}\big)\big) .\end{gathered}$$
\[Theo:braidcat\]In the setting of Claim [\[cl:braid\]]{}, the category $$\begin{gathered}
\big(\mathcal{H}(\mathcal{\mathcal{G}},\mathcal{C}),\otimes ,\big( \mathbf{1},
\widehat{\operatorname{Id}_{\mathbf{1}}}\big) ,\overline{a}^{{c,d,\nu}},
\overline{l}^{{c,d,\nu}},\overline{r}^{{c,d,\nu}},\overline{\gamma }
^{{c,d,\nu}}\big)\end{gathered}$$ is a braided monoidal category.
From now on, when $( \mathcal{C},\otimes ,\mathbf{1},a,l,r,\gamma
) $ is a braided monoidal category and $\mathcal{G}$ is a group, we will still denote the braided monoidal structure defined in Theorem \[Theo:braidcat\] with $\mathcal{H}^{{c,d,\nu}}(\mathcal{G},\mathcal{C})$. In the case when $c=d=\mathbf{1}_{\mathcal{G}}$ and $\nu =\operatorname{id}_{\mathbf{1}}$, we will simply write respectively $\mathcal{H}(\mathcal{G},\mathcal{C})$ instead of $\mathcal{H}^{{c,d,\nu}}(\mathcal{G},\mathcal{C})$ and $\gamma
_{(A,f_{A}),(B,f_{B})}$ instead of $\overline{\gamma }_{(A,f_{A}),(B,f_{B})}^{{c,d,\nu}}$.
\[teo:htuttiiso\]Let $\mathcal{G}$ be a group and let $( \mathcal{C}
,\otimes ,\mathbf{1},a,l,r,\gamma ) $ be a braided monoidal category. Then the identity functor $\mathcal{I}\colon \mathcal{H}^{c,d,\nu }(\mathcal{G},
\mathcal{C})\rightarrow \mathcal{H}(\mathcal{G},\mathcal{C})$ is a braided monoidal isomorphism via $$\begin{gathered}
\phi _{0}=\nu ^{-1}\colon \ \big(\mathbf{1},\widehat{\operatorname{Id}_{\mathbf{1}}}
\big)\rightarrow (\mathbf{1},\widehat{\operatorname{Id}_{\mathbf{1}}})\qquad \text{and}
\qquad\phi _{2} ( (A,f_{A}),(B,f_{B}) ) =f_{A}\big(c^{-1}\big)\otimes
f_{B}(d),\end{gathered}$$ for every $(A,f_{A}),(B,f_{B})\in \mathcal{H}^{c,d,\nu }(\mathcal{G},\mathcal{C})$.
Let $\mathcal{G}$ be a torsion-free abelian group. Corollary 4 in [@ABM-HomLie] states that, up to a braided monoidal category isomorphism, there is a unique braided monoidal structure (actually symmetric) on the category of representations over the group algebra $\Bbbk[\mathcal{G}]$, considered monoidal via a structure induced by that of vector spaces over the field $\Bbbk $. Thus Theorem \[teo:htuttiiso\] can be deduced from this result whenever $\mathcal{G}$ is a torsion-free abelian group. We should remark that this result in [@ABM-HomLie] stems from the fact that the third Harrison cohomology group $H_{\mathrm{Harr}}^{3}(\mathcal{G},\Bbbk ,\mathbb{G}_{m})$ has, in this case, just one element. If $\mathcal{G}$ is not a torsion-free abelian group then this might not happen. As one of the referees pointed out, in the case when $\Bbbk=\mathbb{C} $ and $\mathcal{G}=C_2$ then $H_{\mathrm{Harr}}^{3}(\mathcal{G},\Bbbk ,\mathbb{G}_{m})$ has exactly two elements and so in this case there are two distinct equivalence classes of braided monoidal structures on the category of representations over the group algebra $\Bbbk[\mathcal{G}]$, considered monoidal via a structure induced by that of vector spaces over the field $\Bbbk $. This does not contradict our Theorem \[teo:htuttiiso\]. In fact, there might exist braided monoidal structures different from the ones considered in the statement of Theorem \[teo:htuttiiso\].
\[Cl:alg\]Let $ ( \mathcal{C},\otimes ,\mathbf{1},a,l,r ) $ be a monoidal category and $\mathcal{G}$ a group, let $c,d\in Z ( \mathcal{G} ) $ and $\nu \in \operatorname{Aut}_{\mathcal{C}} ( \mathbf{1} ) $. A *unital algebra* in $\mathcal{H}^{c,d,\nu }(\mathcal{G},\mathcal{C})$ is a triple $( (A,f_{A}),\mu ,u) $ where
- $(A,f_{A})\in \mathcal{H}(\mathcal{G},\mathcal{C})$;
- $\mu \colon (A\otimes A, f_A\otimes f_A)\rightarrow (A, f_A)$ is a morphism in $\mathcal{H}(\mathcal{G},\mathcal{C})$;
- $u\colon (\mathbf{1},\widehat{\operatorname{Id}_{\mathbf{1}}})\rightarrow (A,
f_A)$ is a morphism in $\mathcal{H}(\mathcal{G},\mathcal{C})$;
- $\mu \circ ( \mu \otimes A ) =\mu \circ ( A\otimes
\mu ) \circ \overline{a}_{A,A,A}^{c,d,\nu }$;
- $\mu \circ ( u\otimes A ) \circ ( \overline{l}_{A}^{c,d,\nu } ) ^{-1}=\operatorname{Id}_{A}$;
- $\operatorname{Id}_{A}=\mu \circ ( A\otimes u ) \circ (
\overline{r}_{A}^{c,d,\nu } ) ^{-1}$.
Given a monoidal category $\mathcal{M}$, a quadruple $(A,\mu ,u,c)$ is called a *braided unital algebra* in $\mathcal{M}$ if (for simplicity, we will omit to write the associators):
- $(A,\mu ,u)$ is a unital algebra in $\mathcal{M}$;
- $(A,c)$ is a braided object in $\mathcal{M}$, i.e., $c\colon A\otimes
A\rightarrow A\otimes A$ is invertible and satisfies the Yang–Baxter equation $$\begin{gathered}
(c\otimes A) (A\otimes c) (c\otimes A)
=(A\otimes c) (c\otimes A) (A\otimes c) ;
$$
- the following conditions hold: $$\begin{gathered}
c(\mu \otimes A)=(A\otimes \mu )(c\otimes A)(A\otimes c), \qquad c(A\otimes \mu )=(\mu \otimes A)(A\otimes c) (c\otimes A),\\ c(u\otimes A)l_{A}^{-1}= ( A\otimes u ) r_{A}^{-1},\qquad
c(A\otimes u)r_{A}^{-1}= ( u\otimes A ) l_{A}^{-1}. $$
A braided unital algebra is called *symmetric* whenever $c^{2}=\operatorname{Id}_{A}$.
\[def:Lie\]Given an additive monoidal category $\mathcal{M}$, a *braided Lie algebra* in $\mathcal{M}$ consists of a triple $( L,c,[
-] \colon L\otimes L\rightarrow L) $ where $( L,c) $ is a braided object and the following equalities hold true: $$\begin{gathered}
[-] =-[-] \circ c \quad \text{(skew-symmetry)};\nonumber \\
[-] \circ ( L\otimes [-] ) \circ \big[
\operatorname{Id}_{L\otimes ( L\otimes L ) }+ ( L\otimes c )
a_{L,L,L} ( c\otimes L ) a_{L,L,L}^{-1}\nonumber\\
\qquad{} +a_{L,L,L} ( c\otimes
L ) a_{L,L,L}^{-1} ( L\otimes c ) \big] =0 \quad \text{(Jacobi condition)} ; \nonumber \\
c\circ ( L\otimes [-] ) a_{L,L,L}= ( [ -
] \otimes L ) a_{L,L,L}^{-1} ( L\otimes c )
a_{L,L,L} ( c\otimes L ) ; \label{cucu} \\
c\circ ( [-] \otimes L ) a_{L,L,L}^{-1}= (
L\otimes [-] ) a_{L,L,L} ( c\otimes L )
a_{L,L,L}^{-1}\left( L\otimes c\right) . \label{lala}\end{gathered}$$ Let $\mathcal{M}$ be an additive braided monoidal category. A *Lie algebra* in $\mathcal{M}$ consists of a pair $ ( L,[-]
\colon L\otimes L\rightarrow L ) $ such that $ ( L,c_{L,L},[-]
) $ is a braided Lie algebra in the additive monoidal category $\mathcal{M}$, where $c_{L,L}$ is the braiding $c$ of $\mathcal{M}$ evaluated on $L$ (note that in this case the conditions (\[cucu\]) and (\[lala\]) are automatically satisfied).
\[Cl:LIE\]Given a symmetric algebra $ ( A,\mu , u, c ) $, one has that $[-] :=\mu \circ ( \operatorname{Id}_{A\otimes
A}-c) $ defines a braided Lie algebra structure on $A$ (see [@GV Construction 2.16]).
In a symmetric monoidal category $( \mathcal{C},\otimes ,\mathbf{1},a,l,r,c)$, it is well known that any unital algebra $( A,\mu
,u) $ gives rise to a braided unital algebra $( A,\mu
,u,c_{A,A})$.
Generalized Hom-structures
==========================
Let $\Bbbk $ be a field and let ${_{\Bbbk }\mathcal{M}}$ be the category of linear spaces regarded as a braided monoidal category in the usual way. Then, for every group $\mathcal{G}$, the category $\mathcal{H}(\mathcal{G},{_{\Bbbk }\mathcal{M}})$ identifies with the category $\Bbbk [
\mathcal{G} ] $-$\text{Mod}$ of left modules over the group algebra $\Bbbk
[ \mathcal{G} ] $.
Let $c,d\in Z(\mathcal{G}) $ and $\nu $ an automorphism of $\Bbbk $ regarded as linear space over $\Bbbk $, that is $\nu $ is the multiplication by an element of $\Bbbk {\setminus} \{ 0 \} $ that we will also denote by $\nu $. Note that, given $X,Y,Z\in \Bbbk [ \mathcal{G} ] $-$\text{Mod}$, we have $$\begin{gathered}
\overline{a}_{X,Y,Z}^{c,d,\nu } ( ( x\otimes y ) \otimes
z ) =c\cdot x\otimes ( y\otimes d\cdot z ) , \qquad \text{for every} \ \
x\in X,y\in Y,z\in Z,
\\
\overline{l}_{X}^{c,d,\nu } ( t\otimes x ) =d^{-1}\cdot ( \nu
tx ) \qquad \text{and} \qquad \overline{r}_{X}^{c,d,\nu } ( x\otimes t )
=c\cdot ( \nu tx ) ,\\
\qquad{}\text{for every} \ \ t\in \Bbbk \ \ \text{and} \ \ x\in
X,\end{gathered}$$ so that $$\begin{gathered}
\big( \overline{l}_{A}^{c,d,\nu }\big) ^{-1} ( x ) = ( \nu
^{-1}\otimes d\cdot x ) \qquad \text{and} \qquad \big( \overline{r}^{c,d,\nu
}\big) ^{-1} ( x ) =\big( c^{-1}\cdot x\otimes \nu ^{-1}\big) , \\
\qquad{}
\text{for every} \ \ x\in X.\end{gathered}$$ The unit object of $\mathcal{H}^{c,d,\nu }(\mathcal{G},{_{\Bbbk }\mathcal{M}}
)$ is $ \{ 1_{\Bbbk } \}$ regarded as a left $\Bbbk [ \mathcal{G} ] $-module in the trivial way.
\[Cl:Alg2\]In view of \[Cl:alg\], a unital algebra in $\mathcal{H}^{c,d,\nu }(\mathcal{G},{_{\Bbbk }\mathcal{M}})$ is a triple $ ( (A,f_{A}),\mu ,u ) $, where
- $A\in \Bbbk [ \mathcal{G} ]$-$\text{Mod}$;
- $\mu \colon A\otimes A\rightarrow A$ is a morphism in $\Bbbk [
\mathcal{G} ] $-$\text{Mod}$, i.e., $g\cdot (ab)= ( g\cdot a ) (
g\cdot b ) $, for every $g\in \mathcal{G}$, $a,b\in A$;
- $u\colon \{ 1_{\Bbbk } \} \rightarrow A$ is a morphisms in $\Bbbk [ \mathcal{G}] $-$\text{Mod}$, i.e., $g\cdot u ( 1_{\Bbbk
} ) =u ( 1_{\Bbbk } ) $, for every $g\in \mathcal{G}$;
- $( x\cdot y) \cdot z=( c\cdot x) \cdot [
y\cdot ( d\cdot z) ] $, for every $x,y,z\in A$, which is equivalent to $$\begin{gathered}
( c\cdot x ) \cdot ( y\cdot z ) = ( x\cdot
y ) \cdot \big( d^{-1}\cdot z\big), \qquad \forall \, x,y,z\in A;\end{gathered}$$
- $u ( \nu ^{-1} ) \cdot ( d\cdot x ) =x$, for every $x\in A$;
- $ ( c^{-1}\cdot x ) \cdot u ( \nu ^{-1} ) =x$, for every $x\in A$.
Note that when $c=d=1_{\mathcal{G}}$ and $\nu =1_{\Bbbk }$, it turns out that $A$ is simply a $\Bbbk [ \mathcal{G} ]$-module algebra.
\[Ex:zetzet\]Let $M$ be a ${\Bbbk }$-linear space and $\mathcal{G}=
\mathbb{Z}\times \mathbb{Z}$. Then a group morphism $f_{M}\colon
\mathbb{Z}
\times
\mathbb{Z}
\rightarrow \operatorname{Aut}_{\Bbbk }(M) $ is completely determined by $$\begin{gathered}
f_{M}((1,0)) =\alpha _{M}\qquad \text{and}\qquad
f_{M}((0,1)) =\beta _{M}^{-1}.\end{gathered}$$ Thus an object in $\mathcal{H}(
\mathbb{Z}
\times
\mathbb{Z}
,{_{\Bbbk }\mathcal{M}})$ identifies with a triple $(
M,\alpha _{M},\beta _{M}) $, where $\alpha _{M},\beta _{M}\in \operatorname{Aut}_{\Bbbk }(M) $ and $\alpha _{M}\circ \beta _{M}=\beta
_{M}\circ \alpha _{M}$. Also, a morphism $f\colon ( M,\alpha _{M},\beta
_{M} ) \rightarrow ( N,\alpha _{N},\beta _{N} ) $ is just a linear map $f\colon M\rightarrow N$ such that $f\circ \alpha _{M}=\alpha _{N}\circ
f$ and $f\circ \beta _{M}=\beta _{N}\circ f$. Moreover, the tensor product, in the category, of the objects $ ( M,\alpha _{M},\beta _{M} ) $ and $ ( N,\alpha _{N},\beta _{N} ) $ is the object $(M\otimes
N,\alpha _{M}\otimes \alpha _{N},\beta _{M}\otimes \beta _{N})$.
We set $c=(1,0) $, $d=(0,1) $ and $\nu =1_{\Bbbk }$. For $ ( X,\alpha _{X},\beta _{X} ) $, $ ( Y,\alpha _{Y},\beta
_{Y} ) $, $ ( Z,\alpha _{Z},\beta _{Z} ) $ objects in $
\mathcal{H}^{(1,0) ,(0,1) ,1}(
\mathbb{Z}
\times
\mathbb{Z}
,{_{\Bbbk }\mathcal{M}})$, the associativity constraints in $
\mathcal{H}^{(1,0) ,(0,1) ,1}(
\mathbb{Z}
\times
\mathbb{Z}
,{_{\Bbbk }\mathcal{M}})$ are given by $$\begin{gathered}
\big( \overline{a}^{c,d,\nu }\big) _{( X,\alpha _{X},\beta
_{X}) ,( Y,\alpha _{Y},\beta _{Y}) ,( Z,\alpha
_{Z},\beta _{Z}) }\colon \ (X\otimes Y)\otimes Z\rightarrow X\otimes (Y\otimes
Z), \\
\big( \overline{a}^{c,d,\nu }\big) _{( X,\alpha _{X},\beta
_{X}) ,( Y,\alpha _{Y},\beta _{Y}) ,( Z,\alpha
_{Z},\beta _{Z}) }=a_{X,Y,Z}\circ \big[ ( \alpha _{X}\otimes
Y ) \otimes \beta _{Z}^{-1}\big] ,\end{gathered}$$ and the braiding is $$\begin{gathered}
\overline{\gamma }_{( X,\alpha _{X},\beta _{X}) ,( Y,\alpha
_{Y},\beta _{Y}) }^{{c,d,\nu}}=\tau \big[ \big( \alpha _{X}\beta
_{X}^{-1}\big) \otimes \big( \alpha _{Y}\beta _{Y}^{-1}\big) ^{-1}
\big] =\tau \big[ \big( \alpha _{X}\beta _{X}^{-1}\big) \otimes \big(
\alpha _{Y}^{-1}\beta _{Y}\big) \big] ,\end{gathered}$$ where $\tau \colon X\otimes Y\rightarrow Y\otimes X$ denotes the usual flip in the category of linear spaces. Note that $\overline{\gamma }$ is a symmetric braiding.
Then, in view of \[Cl:Alg2\], an algebra in $\mathcal{H}^{(1,0) ,(0,1) ,1}(
\mathbb{Z}
\times
\mathbb{Z}
,{_{\Bbbk }\mathcal{M}})$ is a triple $(( A,\alpha
,\beta ) ,\mu ,u) $, where
- $\alpha ,\beta \in \operatorname{Aut}_{\Bbbk } ( A ) $ and $\alpha \circ \beta =\beta \circ \alpha$;
- $\mu \colon ( A\otimes A,\alpha \otimes \alpha ,\beta \otimes \beta
) \rightarrow ( A,\alpha ,\beta ) $ is a morphism in $\Bbbk
[
\mathbb{Z}
\times
\mathbb{Z}
] $-$\text{Mod}$, i.e., $\alpha (a\cdot b)=\alpha ( a ) \cdot \alpha
(b) $ and $\beta (a\cdot b)=\beta (a) \cdot \beta
(b) $ for every $a,b\in A$;
- $u\colon \{ 1_{\Bbbk } \} \rightarrow ( A,\alpha ,\beta
) $ is a morphisms in $\Bbbk [
\mathbb{Z}
\times
\mathbb{Z}
] $-$\text{Mod}$, i.e., $\alpha ( u ( 1_{\Bbbk } ) )
=u ( 1_{\Bbbk } ) $ and $\beta ( u ( 1_{\Bbbk } )
) =u ( 1_{\Bbbk } )$;
- $\alpha (x) \cdot ( y\cdot z ) = (
x\cdot y ) \cdot \beta (z) $, for every $x,y,z\in A;$
- $u ( 1_{\Bbbk } ) \cdot ( \beta ^{-1}(x)
) =x$, for every $x\in A$, which is equivalent to $u( 1_{\Bbbk
} ) \cdot x=\beta (x) $, for every $x\in A$;
- $ ( \alpha ^{-1}(x) ) \cdot u ( 1_{\Bbbk
} ) =x$, for every $x\in A$, which is equivalent to $x\cdot u (
1_{\Bbbk } ) =\alpha (x) $, for every $x\in A$.
Inspired by Example \[Ex:zetzet\], we introduce the following concept.
Let ${\Bbbk}$ be a field. A *BiHom-associative algebra* over $\Bbbk $ is a 4-tuple $( A,\mu ,\alpha ,\beta )$, where $A$ is a $\Bbbk $-linear space, $\alpha \colon A\rightarrow A$, $\beta \colon A\rightarrow A$ and $\mu \colon A\otimes A\rightarrow A$ are linear maps, with notation $\mu
( a\otimes a^{\prime } ) =aa^{\prime }$, satisfying the following conditions, for all $a,a^{\prime },a^{\prime \prime }\in A$: $$\begin{gathered}
\alpha \circ \beta =\beta \circ \alpha , \\
\alpha ( aa^{\prime } ) =\alpha (a) \alpha (
a^{\prime } ) \qquad \text{and} \qquad \beta ( aa^{\prime } ) =\beta
(a) \beta ( a^{\prime } ) \quad \text{(multiplicativity)}, \\ \alpha (a) ( a^{\prime }a^{\prime \prime } ) = (
aa^{\prime } ) \beta ( a^{\prime \prime } ) \quad \text{(BiHom-associativity)}. $$
We call $\alpha $ and $\beta $ (in this order) the *structure maps* of $A$.
A morphism $f\colon (A, \mu _A , \alpha _A, \beta _A)\rightarrow (B, \mu _B ,
\alpha _B, \beta _B)$ of BiHom-associative algebras is a linear map $f\colon A\rightarrow B$ such that $\alpha _B\circ f=f\circ \alpha _A$, $\beta
_B\circ f=f\circ \beta _A$ and $f\circ \mu_A=\mu _B\circ (f\otimes f)$.
A BiHom-associative algebra $(A, \mu, \alpha, \beta )$ is called *unital* if there exists an element $1_A\in A$ (called a *unit*) such that $\alpha (1_A)=1_A$, $\beta (1_A)=1_A$ and $$\begin{gathered}
a1_A=\alpha (a) \qquad \text{and}\qquad 1_Aa=\beta (a), \qquad \forall \, a\in A.\end{gathered}$$
A morphism of unital BiHom-associative algebras $f\colon A\rightarrow B$ is called *unital* if $f(1_A)=1_B$.
A Hom-associative algebra $ ( A,\mu ,\alpha ) $ can be regarded as the BiHom-associative algebra $ ( A,\mu ,\alpha ,\alpha ) $.
\[remada\] A BiHom-associative algebra with *bijective* structure maps is exactly an algebra in $\mathcal{H}^{(1,0) ,(0,1) ,1}
(
\mathbb{Z}
\times
\mathbb{Z}
,{_{\Bbbk }\mathcal{M}})$. On the other hand, in the setting of Claim \[Cl:Alg2\], if we define the maps $\alpha , \beta \colon A\rightarrow A$ by $\alpha (a)=c\cdot a$ and $\beta (a)=d^{-1}\cdot a$, for all $a\in A$, the axiom $2)$ in Claim \[Cl:Alg2\] implies that $\alpha $ and $\beta $ are multiplicative and then the axiom $4)$ in Claim \[Cl:Alg2\] says that $(A, \mu , \alpha , \beta )$ is a BiHom-associative algebra.
We give now two families of examples of 2-dimensional unital BiHom-associative algebras, that are obtained by a computer algebra system. Let $\{e_1,e_2\}$ be a basis; for $i=1,2$ the maps $\alpha_i$, $\beta_i $ and the multiplication $\mu_i$ are defined by $$\begin{gathered}
\alpha_1(e_1)=e_1, \qquad \alpha_1(e_2)=\frac{2 a}{b-1}e_1-e_2, \\
\beta_1(e_1)=e_1, \qquad \beta_1(e_2)=-ae_1+b e_2, \\
\mu_1(e_1,e_1)=e_1, \qquad \mu_1(e_1,e_2)= -ae_1+b e_2, \\
\mu_1(e_2,e_1)=\frac{2 a}{b-1}e_1-e_2, \qquad \mu_1(e_2,e_2)=-\frac{
a^2(b-2)}{(b-1)^2}e_1+ae_2,\end{gathered}$$ and $$\begin{gathered}
\alpha_2(e_1)=e_1,\qquad \alpha_2(e_2)=\frac{b(1-a)}{a}e_1+a e_2, \\
\beta_2(e_1)=e_1,\qquad \beta_2(e_2)=be_1+(1-a) e_2, \\
\mu_2(e_1,e_1)=e_1, \qquad \mu_2(e_1,e_2)= be_1+(1-a) e_2, \\
\mu_2(e_2,e_1)=\frac{b(1-a)}{a}e_1+ae_2, \qquad \mu_2(e_2,e_2)=\frac{b}{a}e_2,\end{gathered}$$ where $a$, $b$ are parameters in ${\Bbbk}$, with $b\neq 1$ in the first case and $a\neq 0$ in the second. In both cases, the unit is $e_1$.
\[cl:YAU\] In view of Theorem \[thm:monoidaliso\], if $( A,\mu ,\alpha ,\beta
) $ is a BiHom-associative algebra, and $\alpha $ and $\beta $ are invertible, then $( A,\mu \circ ( \alpha ^{-1}\otimes \beta
^{-1}), \operatorname{Id}_{A},\operatorname{Id}_{A}) $ is a BiHom-associative algebra, i.e., the multiplication $\mu \circ ( \alpha
^{-1}\otimes \beta ^{-1}) $ is associative in the usual sense.
On the other hand, if $(A, \mu \colon A\otimes A\rightarrow A)$ is an associative algebra and $\alpha , \beta \colon A\rightarrow A$ are commuting algebra endomorphisms, then one can easily check that $ ( A,\mu
\circ ( \alpha \otimes \beta ) ,\alpha ,\beta ) $ is a BiHom-associative algebra, denoted by $A_{(\alpha ,\beta )}$ and called the *Yau twist* of $(A, \mu )$.
In view of Claim \[cl:YAU\], a BiHom-associative algebra with bijective structure maps is a Yau twist of an associative algebra.
The Yau twisting procedure for BiHom-associative algebras admits a more general form, which we state in the next result (the proof is straightforward and left to the reader).
\[yaugeneral\] Let $(D, \mu , \tilde{\alpha }, \tilde{\beta })$ be a BiHom-associative algebra and $\alpha , \beta \colon D\rightarrow D$ two multiplicative linear maps such that any two of the maps $\tilde{\alpha }$, $\tilde{\beta }$, $\alpha$, $\beta $ commute. Then $(D, \mu \circ (\alpha
\otimes \beta )$, $\tilde{\alpha }\circ \alpha , \tilde{\beta }\circ \beta )$ is also a BiHom-associative algebra, denoted by $D_{(\alpha , \beta )}$.
\[giacomoex\] We present an example of a BiHom-associative algebra that cannot be expressed as a Hom-associative algebra. Let $\Bbbk $ be a field and $A=\Bbbk
[ X ] $. Let $\alpha \colon A\rightarrow A$ be the algebra map defined by setting $\alpha ( X ) =X^{2}$ and let $\beta =\operatorname{Id}
_{\Bbbk [ X ] }$. Then we can consider the BiHom-associative algebra $A_{(\alpha ,\beta )}= ( A,\mu \circ ( \alpha \otimes \beta ) ,\alpha
,\beta ) $, where $\mu \colon A\otimes A\rightarrow A$ is the usual multiplication. For every $a,a^{\prime }\in A$ set $$\begin{gathered}
a\ast a^{\prime }=\mu \circ ( \alpha \otimes \beta ) (
a\otimes a^{\prime } ) =\alpha (a) a^{\prime }.\end{gathered}$$ Let us assume that there exists $\theta \in \mathrm{End} ( \Bbbk [ X
] ) $ such that $ ( A,\mu \circ ( \alpha \otimes \beta
) ,\theta ) $ is a Hom-associative algebra. Then we should have that $$\begin{gathered}
\theta (X) \ast ( X\ast X ) = ( X\ast X )
\ast \theta (X) . \label{form: gamma}\end{gathered}$$ Write $$\begin{gathered}
\theta (X) =\sum_{i=0}^{n}a_{i}X^{i}, \qquad \text{where $a_{i}\in
\Bbbk$ for every $i=0, 1,\ldots ,n$ and $a_{n}\neq 0$}.\end{gathered}$$ Since $$\begin{gathered}
X\ast X=\alpha (X) X=X^{3},\end{gathered}$$ (\[form: gamma\]) rewrites as $$\begin{gathered}
\sum_{i=0}^{n}a_{i}X^{i}\ast X^{3}=X^{3}\ast \sum_{i=0}^{n}a_{i}X^{i},\end{gathered}$$ and hence as $$\begin{gathered}
\sum_{i=0}^{n}a_{i}\alpha (X) ^{i}X^{3}=\alpha (X)
^{3}\sum_{i=0}^{n}a_{i}X^{i},\end{gathered}$$ i.e., $$\begin{gathered}
\sum_{i=0}^{n}a_{i}X^{2i+3}=\sum_{i=0}^{n}a_{i}X^{6+i},\end{gathered}$$ which implies that $$\begin{gathered}
2n+3=6+n, \qquad \text{i.e.}, \quad n=3, \quad \text{and hence}
\\
a_{0}X^{3}+a_{1}X^{5}+a_{2}X^{7}+a_{3}X^{9}=a_{0}X^{6}+a_{1}X^{7}+a_{2}X^{8}+a_{3}X^{9},\end{gathered}$$ so that $$\begin{gathered}
\theta (X) =a_{3}X^{3}.\end{gathered}$$ Let us set $c=a_{3}$ and let us check the equality $$\begin{gathered}
\theta \big( X^{2}\big) \ast ( X\ast X ) =\big( X^{2}\ast
X\big) \ast \theta (X) .\end{gathered}$$ The left-hand side is $$\begin{gathered}
\theta \big( X^{2}\big) \ast ( X\ast X ) =c^{2}X^{6}\ast
X^{3}=\alpha \big( c^{2}X^{6}\big) X^{3}=c^{2}X^{15}.\end{gathered}$$ The right-hand side is $$\begin{gathered}
\big( X^{2}\ast X\big) \ast \theta (X) =\big( \alpha \big(
X^{2}\big) X\big) \ast \theta (X) =X^{5}\ast \theta (
X ) =cX^{10}X^{3}=cX^{13}.\end{gathered}$$ Thus the equality does not hold.
\[remtensprod\] Given two algebras $ ( A,\mu _{A},1_{A} ) $ and $ ( B,\mu _{B},1_{B} ) $ in a braided monoidal category $ (
\mathcal{C},\otimes ,\mathbf{1},a,l,r,c ) $, it is well known that $
A\otimes B$ becomes also an algebra in the category, with multiplication $
\mu _{A\otimes B}$ defined by $$\begin{gathered}
\mu _{A\otimes B}=( \mu _{A}\otimes \mu _{B})\circ
a_{A,A,B\otimes B}^{-1} \circ ( A\otimes a_{A,B,B})\\
\hphantom{\mu _{A\otimes B}=}{}\circ (
A\otimes ( c_{B,A}\otimes B) ) \circ \big( A\otimes
a_{B,A,B}^{-1}\big) \circ a_{A,B,A\otimes B}.\end{gathered}$$
In the case of our category $\mathcal{H}^{c,d,\nu }(\mathcal{G},{_{\Bbbk }
\mathcal{M}})$, we have, for every $x,y\in A$, $x^{\prime },y^{\prime }\in B$: $$\begin{gathered}
\mu _{A\otimes B} ( ( x\otimes y ) \otimes ( x^{\prime
}\otimes y^{\prime } ) )
= ( ( \mu _{A}\otimes \mu _{B} ) \circ a_{A,A,B\otimes
B}^{-1}\circ ( A\otimes a_{A,B,B} ) \\
\qquad\quad{}
\circ ( A\otimes ( c_{B,A}\otimes B ) ) \circ \big(
A\otimes a_{B,A,B}^{-1}\big) \circ a_{A,B,A\otimes B}) ( (
x\otimes y ) \otimes ( x^{\prime }\otimes y^{\prime } )
) \\
\qquad {}
= \big( ( \mu _{A}\otimes \mu _{B} ) \circ a_{A,A,B\otimes
B}^{-1}\circ ( A\otimes a_{A,B,B} ) \circ ( A\otimes (
c_{B,A}\otimes B ) ) \\
\qquad\quad{}
\circ \big( A\otimes a_{B,A,B}^{-1}\big)\big) ( cx\otimes (
y\otimes ( dx^{\prime }\otimes dy^{\prime } ) ) ) \\
\qquad{}
=\big( ( \mu _{A}\otimes \mu _{B} ) \circ a_{A,A,B\otimes
B}^{-1}\circ ( A\otimes a_{A,B,B} ) \big) \big( cx\otimes
\big( \big( c^{-1}x^{\prime }\otimes dy\big) \otimes y^{\prime }\big)
\big) \\
\qquad{}
= \big( ( \mu _{A}\otimes \mu _{B} ) \circ a_{A,A,B\otimes
B}^{-1}\big) ( cx\otimes ( x^{\prime }\otimes ( dy\otimes
dy^{\prime } ) ) ) \\
\qquad {}
= ( \mu _{A}\otimes \mu _{B} ) ( ( x\otimes x^{\prime
} ) \otimes ( y\otimes y^{\prime } ) ) = ( x\cdot
x^{\prime } ) \otimes ( y\cdot y^{\prime } ) .\end{gathered}$$
In particular, if $(A,\alpha _{A},\beta _{A})$ and $(B,\alpha _{B},\beta
_{B})$ are two algebras in $\mathcal{H}^{(1,0) ,(0,1) ,1}(
\mathbb{Z}
\times
\mathbb{Z}
,{_{\Bbbk }\mathcal{M}})$, their braided tensor product $A
\underline{\otimes }B$ in the category is the algebra $(A\otimes B,\alpha
_{A}\otimes \alpha _{B},\beta _{A}\otimes \beta _{B})$, whose multiplication is given by $(a\otimes b)(a^{\prime }\otimes b^{\prime })=aa^{\prime
}\otimes bb^{\prime }$, for all $a,a^{\prime }\in A$ and $b,b^{\prime }\in B$.
If $(A, \mu _A, \alpha _A, \beta _A)$ and $(B, \mu _B, \alpha _B, \beta _B)$ are two BiHom-associative algebras over a field $\Bbbk$, then $(A\otimes B,
\mu _{A\otimes B}, \alpha _A\otimes \alpha _B, \beta _A\otimes \beta _B)$ is a BiHom-associative algebra (called the tensor product of $A$ and $B$), where $\mu _{A\otimes B}$ is the usual multiplication: $(a\otimes
b)(a^{\prime }\otimes b^{\prime })=aa^{\prime }\otimes bb^{\prime }$. If $A$ and $B$ are unital with units $1_A$ and respectively $1_B$ then $A\otimes B$ is also unital with unit $1_A\otimes 1_B$. This is consistent with Remark \[remtensprod\].
\[Ex:lie\]In view of Definition \[def:Lie\], a *Lie algebra* in $\mathcal{H}^{c,d,\nu }(\mathcal{G},{_{\Bbbk }\mathcal{M}})$ is a pair $ ( (L,f_{L}),[-] ) $, where
- $(L,f_{L})\in \Bbbk [ \mathcal{G} ]$-$\text{Mod}$;
- $[-] \colon L\otimes L\rightarrow L$ is a morphism in $\Bbbk
[ \mathcal{G}]$-$\text{Mod}$;
- $[-] =-[-] \circ $ $\overline{\gamma }_{L,L}$;
- $$\begin{gathered}
[-] \circ ( L\otimes [-] ) +
[-] \circ ( L\otimes [-] ) \circ (
L\otimes \overline{\gamma }_{L,L}) \overline{a}_{L,L,L}(
\overline{\gamma }_{L,L}\otimes L) \overline{a}_{L,L,L}^{-1}\\
\qquad{} +[-] \circ ( L\otimes [-] ) \overline{a}
_{L,L,L}( \overline{\gamma }_{L,L}\otimes L) \overline{a}
_{L,L,L}^{-1}( L\otimes \overline{\gamma }_{L,L}) =0,\end{gathered}$$
where $\overline{\gamma }_{L,L}=\tau \circ ( f_{L}(cd)\otimes
f_{L}(c^{-1}d^{-1}) ) $ and $\tau $ is the usual flip.
We will write down 4) explicitly. We have $$\begin{gathered}
\big( \big( L\otimes \overline{\gamma }_{L,L}\big) \overline{a}
_{L,L,L}\big( \overline{\gamma }_{L,L}\otimes L\big) \overline{a}
_{L,L,L}^{-1}\big) ( x\otimes ( y\otimes z ) ) \\
\qquad{}
=\big( L\otimes \overline{\gamma }_{L,L}\big) \overline{a}_{L,L,L}\big(
\overline{\gamma }_{L,L}\otimes L\big) \big( \big( c^{-1}x\otimes
y\big) \otimes d^{-1}z\big) \\
\qquad{}
=\big( L\otimes \overline{\gamma }_{L,L}\big) \overline{a}_{L,L,L}\big(
\big( c^{-1}d^{-1}y\otimes cdc^{-1}x\big) \otimes d^{-1}z\big) \\
\qquad{}=\big( L\otimes \overline{\gamma }_{L,L}\big) \big(
cc^{-1}d^{-1}y\otimes \big( cdc^{-1}x\otimes dd^{-1}z\big) \big) \\
\qquad{}
=cc^{-1}d^{-1}y\otimes \big( c^{-1}d^{-1}dd^{-1}z\otimes cdcdc^{-1}x\big)
\\
\qquad{}
=d^{-1}y\otimes \big( c^{-1}d^{-1}z\otimes dcdx\big),\end{gathered}$$ therefore $$\begin{gathered}
[-] \circ ( L\otimes [-] ) \big( \big(
L\otimes \overline{\gamma }_{L,L}\big) \overline{a}_{L,L,L}\big(
\overline{\gamma }_{L,L}\otimes L\big) \overline{a}_{L,L,L}^{-1}\big)
( x\otimes ( y\otimes z ) ) \\
\qquad{}
=\big[ d^{-1}y,\big[ c^{-1}d^{-1}z,cd^{2}x\big] \big],\end{gathered}$$ and $$\begin{gathered}
\big( \overline{a}_{L,L,L}\big( \overline{\gamma }_{L,L}\otimes L\big)
\overline{a}_{L,L,L}^{-1}\big( L\otimes \overline{\gamma }_{L,L}\big)
\big) ( x\otimes ( y\otimes z ) ) \\
\qquad{} =\overline{a}_{L,L,L}\big( \overline{\gamma }_{L,L}\otimes L\big)
\overline{a}_{L,L,L}^{-1}\big( x\otimes \big( c^{-1}d^{-1}z\otimes
cdy\big) \big) \\
\qquad{}
=\overline{a}_{L,L,L}\big( \overline{\gamma }_{L,L}\otimes L\big) \big(
c^{-1}x\otimes \big( c^{-1}d^{-1}z\otimes d^{-1}cdy\big) \big) \\
\qquad{}
=\overline{a}_{L,L,L}\big( \overline{\gamma }_{L,L}\otimes L\big) \big(
\big( c^{-1}x\otimes c^{-1}d^{-1}z\big) \otimes cy\big) \\
\qquad{}
=\overline{a}_{L,L,L}\big( \big( c^{-2}d^{-2}z\otimes cdc^{-1}x\big)
\otimes cy\big) \\
\qquad{}
=\big( \big( c^{-1}d^{-2}z\otimes dx\big) \otimes cdy\big),\end{gathered}$$ hence $$\begin{gathered}
[-] \circ ( L\otimes [-] ) \big(
\overline{a}_{L,L,L}\big( \overline{\gamma }_{L,L}\otimes L\big)
\overline{a}_{L,L,L}^{-1}\big( L\otimes \overline{\gamma }_{L,L}\big)
\big) ( x\otimes ( y\otimes z ) )
=\big[ c^{-1}d^{-2}z, [ dx,cdy ] \big].\end{gathered}$$ Thus 4) is equivalent to $$\begin{gathered}
[ x,[y,z] ] +\big[ d^{-1}y,\big[
c^{-1}d^{-1}z,cd^{2}x\big] \big] +\big[ c^{-1}d^{-2}z, [ dx,cdy] \big] =0, \qquad \text{for every} \ \ x,y,z\in L,\end{gathered}$$ which is equivalent to $$\begin{gathered}
\big[ d^{-2}x,\big[ d^{-1}y,cz\big] \big] +\big[ d^{-2}y,\big[
d^{-1}z,cx\big] \big] +\big[ d^{-2}z,\big[ d^{-1}x,cy\big] \big]
=0,\qquad \text{for every} \ \ x,y,z\in L.\end{gathered}$$
Thus, a Lie algebra in $\mathcal{H}^{c,d,\nu }(\mathcal{G},{_{\Bbbk }\mathcal{M}})$ is a pair $( L,[-]) $, where
- $L\in \Bbbk [ \mathcal{G} ]$-${\rm Mod}$;
- $g[x,y] =[gx,gy] $, for every $x,y\in L$;
- $[x,y] =- [ c^{-1}d^{-1}y,cdx ] $, for every $
x,y\in L$, i.e., $[ x,cdy] =-[ y,cdx] $, for every $x,y\in L$ (skew-symmetry);
- $[ d^{-2}x,[ d^{-1}y,cz] ] +[ d^{-2}y,[ d^{-1}z,cx] ] +[ d^{-2}z,[ d^{-1}x,cy] ] =0$, for every $x,y,z\in L$ (Jacobi condition).
In particular, a Lie algebra in $\mathcal{H}^{(1,0) ,(0,1) ,1}(
\mathbb{Z}
\times
\mathbb{Z}
,{_{\Bbbk }\mathcal{M}})$ is a pair $(( L,\alpha ,\beta) ,[-] ) $, where
- $\alpha ,\beta \in \operatorname{Aut}_{\Bbbk }(L) $ and $\alpha \circ \beta =\beta \circ \alpha$;
- $[-] \colon ( L\otimes L,\alpha \otimes \alpha ,\beta
\otimes \beta ) \rightarrow ( L,\alpha ,\beta ) $ is a morphism in $\Bbbk [
\mathbb{Z}
\times
\mathbb{Z}
]$-${\rm Mod}$, i.e., $\alpha [ a,b ] = [ \alpha (a)
,\alpha (b) ] $ and $\beta [a,b] = [ \beta
(a) ,\beta (b) ] $, for every $a,b\in L$;
- $[ a,\alpha \beta ^{-1}(b) ] =-[
b,\alpha \beta ^{-1}(a) ] $, for every $a, b\in L$, which is equivalent to $[ \beta (a) ,\alpha (b)
] =-[ \beta (b) ,\alpha (a)] $, for every $a, b\in L$;
- $[ \beta ^{2}x,[ \beta y,\alpha z] ] +[
\beta ^{2}y,[ \beta z,\alpha x] ] +[ \beta ^{2}z,[
\beta x,\alpha y] ] =0$, for every $x,y,z\in L$.
Inspired by Example \[Ex:lie\], we introduce the following concept.
A *BiHom-Lie algebra* over a field $\Bbbk $ is a 4-tuple $( L,[-] ,\alpha ,\beta ) $, where $L$ is a $\Bbbk $-linear space, $\alpha \colon L\rightarrow L$, $\beta \colon L\rightarrow L$ and $[-]
\colon L\otimes L\rightarrow L$ are linear maps, with notation $[-]
( a\otimes a^{\prime }) =[ a,a^{\prime }] $, satisfying the following conditions, for all $a,a^{\prime },a^{\prime \prime
}\in L$: $$\begin{gathered}
\alpha \circ \beta =\beta \circ \alpha , \\
\alpha ([ a^{\prime },a^{\prime \prime }])=[ \alpha (
a^{\prime }) ,\alpha ( a^{\prime \prime }) ]\qquad
\text{and} \qquad \beta ([ a^{\prime },a^{\prime \prime }])=[
\beta ( a^{\prime }) ,\beta ( a^{\prime \prime }) ], \\
[ \beta (a) ,\alpha ( a^{\prime } ) ] =-
[ \beta ( a^{\prime }) ,\alpha (a) ]
\qquad \text{(skew-symmetry)}, \\
\big[ \beta ^{2}(a) , [ \beta ( a^{\prime } )
,\alpha ( a^{\prime \prime } ) ] \big] +\big[ \beta
^{2} ( a^{\prime } ) , [ \beta ( a^{\prime \prime } )
,\alpha (a) ] \big] +\big[ \beta ^{2} ( a^{\prime
\prime } ) , [ \beta (a) ,\alpha ( a^{\prime
} ) ] \big] =0 \\
\qquad{} \text{(BiHom-Jacobi condition)}.\end{gathered}$$
We call $\alpha $ and $\beta $ (in this order) the *structure maps* of $L$. A morphism $f\colon ( L,[-] ,\alpha ,\beta )\rightarrow
( L^{\prime },[-]^{\prime },\alpha ^{\prime },\beta
^{\prime } )$ of BiHom-Lie algebras is a linear map $f\colon L\rightarrow
L^{\prime }$ such that $\alpha ^{\prime }\circ f=f\circ \alpha $, $\beta
^{\prime }\circ f=f\circ \beta $ and $f([x,y])=[f(x),
f(y)]^{\prime }$, for all $x, y\in L$.
Thus, a Lie algebra in $\mathcal{H}^{(1,0) ,(0,1) ,1}(
\mathbb{Z}
\times
\mathbb{Z}
,{_{\Bbbk }\mathcal{M}})$ is exactly a BiHom-Lie algebra with *bijective* structure maps.
Obviously, a Hom-Lie algebra $( L,[-] ,\alpha ) $ is a particular case of a BiHom-Lie algebra, namely $( L, [-]
,\alpha ,\alpha ) $. Conversely, a BiHom-Lie algebra $( L, [
-] ,\alpha ,\alpha ) $ with bijective $\alpha $ is the Hom-Lie algebra $( L,[-] ,\alpha) $.
In view of Claim \[Cl:LIE\], we have:
\[croset\] If $ ( A,\mu ,\alpha ,\beta
) $ is a BiHom-associative algebra with bijective $\alpha $ and $\beta
$, then, for every $a,a^{\prime }\in A$, we can set $$\begin{gathered}
[ a,a^{\prime }] =aa^{\prime }-\big(\alpha ^{-1}\beta (
a^{\prime } )\big)\big(\alpha \beta ^{-1}(a)\big) .\end{gathered}$$ Then $ ( A,[-] ,\alpha ,\beta ) $ is a BiHom-Lie algebra, denoted by $L(A)$.
The proofs of the following three results are straightforward and left to the reader.
Let $ ( L,[-] ) $ be an ordinary Lie algebra over a field $\Bbbk $ and let $\alpha ,\beta \colon L\rightarrow L$ two commuting linear maps such that $\alpha ( [ a,a^{\prime } ] ) = [
\alpha (a) ,\alpha ( a^{\prime } ) ]$ and $\beta
( [ a,a^{\prime } ] ) = [ \beta (a)
,\beta ( a^{\prime } ) ]$, for all $a,a^{\prime }\in L$. Define the linear map $ \{ - \} \colon L\otimes L\rightarrow L$, $$\begin{gathered}
\{ a,b \} = [ \alpha (a) ,\beta (b) ] ,\qquad \text{for all} \ \ a,b\in L.\end{gathered}$$ Then $L_{( \alpha ,\beta ) }:=(L, \{-\}, \alpha ,
\beta )$ is a BiHom-Lie algebra, called the *Yau twist* of $( L,[-]) $.
More generally, let $( L,[-] ,\alpha ,\beta ) $ be a BiHom-Lie algebra and $\alpha^{\prime }, \beta^{\prime }\colon L\rightarrow L$ linear maps such that $\alpha ^{\prime }([ a, b])=[ \alpha
^{\prime }(a) ,\alpha ^{\prime }(b) ]$ and $\beta ^{\prime }([ a, b])=[ \beta ^{\prime }(a)
,\beta ^{\prime }(b) ]$ for all $a, b\in L$, and any two of the maps $\alpha, \beta , \alpha ^{\prime }, \beta ^{\prime }$ commute. Then $( L,[-]_{(\alpha^{\prime },\beta^{\prime }) }:=[
-]\circ(\alpha^{\prime }\otimes\beta^{\prime }),
\alpha\circ\alpha^{\prime },\beta\circ\beta^{\prime }) $ is a BiHom-Lie algebra.
Let $( A,\mu ) $ be an associative algebra and $\alpha ,\beta
\colon A\rightarrow A$ two commuting algebra isomorphisms. Then $L( A_{(
\alpha ,\beta ) }) =L( A) _{( \alpha ,\beta
) }$, as BiHom-Lie algebras.
Let $\mathcal{G}$ be a group and $c,d\in Z(\mathcal{G}) $, $\nu \in \operatorname{Aut}_{\mathcal{C}} ( \mathbf{1} ) $. It is straightforward to prove that the category $\mathcal{H}^{c,d,\nu }(\mathcal{G},{_{\Bbbk }\mathcal{M}})$ fulfills the assumption of [@AM Theorem 6.4]. Hence, for any Lie algebra $ ( L,[-] ) $ in $\mathcal{H}^{c,d,\nu }(\mathcal{G},{_{\Bbbk }\mathcal{M}})$, we can consider the universal enveloping bialgebra $\overline{\mathcal{U}}( ( L,[-] ) ) $ as introduced in [@AM]. By [@AM Remark 6.5], $\overline{\mathcal{U}}( ( L,[-] ) ) $ as a bialgebra is a quotient of the tensor bialgebra $\overline{T}L$. The morphism giving the projection is induced by the canonical projection $p\colon TL\rightarrow \mathcal{U} ( L,[-]
) $ defining the universal enveloping algebra. At algebra level we have $$\begin{gathered}
\mathcal{U} ( L,[-] ) = \frac{TL}{ \big( [ x,y] -x\otimes y+\overline{\gamma }_{L,L}( x\otimes y) \,|\, x,y\in
L\big) } \\
\hphantom{\mathcal{U} ( L,[-] ) }{}
= \frac{TL}{\big( [x,y] -x\otimes y+\big(
f_{L}(c^{-1}d^{-1}) ( y ) \big) \otimes f_{L}(cd)(x)
\,|\, x,y\in L\big) }.\end{gathered}$$ By Theorem \[teo:htuttiiso\], the identity functor $\mathcal{I}\colon \mathcal{H}^{c,d,\nu }(\mathcal{G},{_{\Bbbk }\mathcal{M}})\rightarrow
\mathcal{H}(\mathcal{G},{_{\Bbbk }\mathcal{M}})$ is a braided monoidal isomorphism. Let $F\colon \mathcal{H}(\mathcal{G},{_{\Bbbk }\mathcal{M}})
\rightarrow {_{\Bbbk }\mathcal{M}}$ be the forgetful functor. Then $
F\circ \mathcal{I}$ is a monoidal functor $\mathcal{H}^{c,d,\nu }(\mathcal{G}
,{_{\Bbbk }\mathcal{M}})\rightarrow {_{\Bbbk }\mathcal{M}}$ to which we can apply [@AM Theorem 8.5] to get that $\mathcal{H}^{c,d,\nu }(\mathcal{G},{_{\Bbbk }\mathcal{M}})$ is what is called in [@AM] a Milnor–Moore category. This implies that, by [@AM Theorem 7.2], we have an isomorphism $ ( L,[-] ) \rightarrow \mathcal{P}\overline{\mathcal{U}}(
( L,[-] )) $, where $\mathcal{P}\overline{\mathcal{U}}( ( L,[-] ) ) $ denotes the primitive part of $\overline{\mathcal{U}}( ( L,[-]
) ) $. That is, half of the Milnor–Moore theorem holds.
The case $\mathcal{G}=\mathbb{Z}$ can be found in [@AM Remark 9.10].
In the particular case of a Lie algebra $ ( ( L,\alpha ,\beta
) ,[-] ) $ in $\mathcal{H}^{(1,0)
,(0,1) ,1}(\mathbb{Z}\times \mathbb{Z},{_{\Bbbk }
\mathcal{M}})$ we have that $$\begin{gathered}
\mathcal{U} ( L,[-] ) =\frac{TL}{ \big( [ x,y] -x\otimes y+ ( \alpha ^{-1}\beta ) ( y )
\otimes ( \alpha \beta ^{-1} ) (x) \,|\, x,y\in L\big) }.\end{gathered}$$
Enveloping algebras of Hom-Lie algebras where introduced in [@Y] (see also [@stef Section 8]).
Representations
===============
From now on, we will always work over a base field $\Bbbk$. All algebras, linear spaces etc. will be over $\Bbbk $; unadorned $\otimes $ means $\otimes_{\Bbbk}$. For a comultiplication $\Delta \colon C\rightarrow
C\otimes C$ on a linear space $C$, we use a Sweedler-type notation $\Delta
(c)=c_1\otimes c_2$, for $c\in C$. Unless otherwise specified, the (co)algebras ((co)associative or not) that will appear in what follows are *not* supposed to be (co)unital, and a multiplication $\mu \colon V\otimes
V\rightarrow V$ on a linear space $V$ is denoted by juxtaposition: $\mu
(v\otimes v^{\prime })=vv^{\prime }$. For the composition of two maps $f$ and $g$, we will write either $g\circ f$ or simply $gf$. For the identity map on a linear space $V$ we will use the notation $\operatorname{id}_V$.
Let $(A, \mu _A , \alpha _A, \beta _A)$ be a BiHom-associative algebra. A *left $A$-module* is a triple $(M, \alpha _M, \beta _M)$, where $M$ is a linear space, $\alpha _M, \beta _M\colon M \rightarrow M$ are linear maps and we have a linear map $A\otimes M\rightarrow M$, $a\otimes m\mapsto a\cdot m$, such that, for all $a, a^{\prime }\in A$, $m\in M$, we have $$\begin{gathered}
\alpha _M\circ \beta _M=\beta _M\circ \alpha _M, \\
\alpha _M(a\cdot m)=\alpha _A(a)\cdot \alpha _M(m), \label{ghommod11} \\
\beta _M(a\cdot m)=\beta _A(a)\cdot \beta _M(m), \label{ghommod12} \\
\alpha _A(a)\cdot (a^{\prime }\cdot m)=(aa^{\prime })\cdot \beta _M(m).
\label{ghommod2}\end{gathered}$$
If $(M, \alpha _M, \beta _M)$ and $(N, \alpha _N, \beta _N)$ are left $A$-modules (both $A$-actions denoted by $\cdot$), a morphism of left $A$-modules $f\colon M\rightarrow N$ is a linear map satisfying the conditions $\alpha _N\circ f=f\circ \alpha _M$, $\beta _N\circ f=f\circ \beta _M$ and $f(a\cdot m)=a\cdot f(m)$, for all $a\in A$ and $m\in M$.
If $(A, \mu _A, \alpha _A, \beta _A, 1_A)$ is a unital BiHom-associative algebra and $(M, \alpha _M, \beta _M)$ is a left $A$-module, them $M$ is called *unital* if $1_A\cdot m=\beta _M(m)$, for all $m\in M$.
If $(A, \mu , \alpha , \beta )$ is a BiHom-associative algebra, then $(A,
\alpha , \beta )$ is a left $A$-module with action defined by $a\cdot b=ab$, for all $a, b\in A$.
\[endo\] Let $(E, \mu , 1_E)$ be an associative unital algebra and $u,
v\in E$ two invertible elements such that $uv=vu$. Define the linear maps $\tilde{\alpha }, \tilde{\beta }\colon E\rightarrow E$, $\tilde{\alpha }(a)=uau^{-1} $, $\tilde{\beta }(a)=vav^{-1}$, for all $a\in E$, and the linear map $\tilde{\mu }\colon E\otimes E\rightarrow E$, $\tilde{\mu }(a\otimes
b):=a*b=uau^{-1}bv^{-1}$, for all $a, b\in E$. Then $(E, \tilde{\mu },
\tilde{\alpha }, \tilde{\beta })$ is a unital BiHom-associative algebra with unit $v$, denoted by $E(u, v)$.
Obviously $\tilde{\alpha }\circ \tilde{\beta }=\tilde{\beta }\circ \tilde{\alpha }$ because $uv=vu$. Then, for all $a, b, c\in E$: $$\begin{gathered}
\tilde{\alpha }(a)*\tilde{\alpha }(b) = \big(uau^{-1}\big)*\big(ubu^{-1}\big)
= uuau^{-1}u^{-1}ubu^{-1}v^{-1} \\
\hphantom{\tilde{\alpha }(a)*\tilde{\alpha }(b)}{}
=uuau^{-1}bu^{-1}v^{-1} =\tilde{\alpha }\big(uau^{-1}bv^{-1}\big) =\tilde{\alpha }(a*b),
\\
\tilde{\beta }(a)*\tilde{\beta }(b) = \big(vav^{-1}\big)*\big(vbv^{-1}\big) = uvav^{-1}u^{-1}vbv^{-1}v^{-1} \\
\hphantom{\tilde{\beta }(a)*\tilde{\beta }(b)}{}
= uvau^{-1}bv^{-1}v^{-1} =\tilde{\beta }\big(uau^{-1}bv^{-1}\big) =\tilde{\beta }(a*b),
\\
\tilde{\alpha }(a)*(b*c) = \big(uau^{-1}\big)*\big(ubu^{-1}cv^{-1}\big)
=uuau^{-1}u^{-1}ubu^{-1}cv^{-1}v^{-1} \\
\hphantom{\tilde{\alpha }(a)*(b*c)}{}
=uuau^{-1}bv^{-1}u^{-1}vcv^{-1}v^{-1}
=\big(uau^{-1}bv^{-1}\big)*\big(vcv^{-1}\big) =(a*b)*\tilde{\beta }(c),\end{gathered}$$ so $(E, \tilde{\mu }, \tilde{\alpha },
\tilde{\beta })$ is indeed a BiHom-associative algebra. To prove that $v$ is the unit, we compute $$\begin{gathered}
\tilde{\alpha }(v)=uvu^{-1}=v, \qquad
\tilde{\beta }(v)=vvv^{-1}=v, \\
a*v=uau^{-1}vv^{-1}=uau^{-1}=\tilde{\alpha }(a), \qquad
v*a=uvu^{-1}av^{-1}=vav^{-1}=\tilde{\beta }(a),\end{gathered}$$ finishing the proof.
Let $(A, \mu _A, \alpha _A, \beta _A)$ be a BiHom-associative algebra, $M$ a linear space and $\alpha _M, \beta _M\colon M\rightarrow M$ two commuting linear isomorphisms. Consider the associative unital algebra $E=\operatorname{End}(M)$ with its usual structure, denote $u:=\alpha _M$, $v:=\beta _M$, and construct the BiHom-associative algebra $(E, \tilde{\mu }, \tilde{\alpha }, \tilde{\beta }
)= \operatorname{End}(M)(\alpha _M, \beta _M)$ as in Lemma [\[endo\]]{}. Then setting a structure of a left $A$-module on $(M, \alpha _M, \beta _M)$ is equivalent to giving a morphism of BiHom-associative algebras $\varphi \colon (A, \mu _A,
\alpha _A, \beta _A)\rightarrow (E, \tilde{\mu }, \tilde{\alpha }, \tilde{\beta })$. If $A$ is moreover unital with unit $1_A$, then the module $(M,
\alpha _M, \beta _M)$ is unital if and only if the morphism $\varphi $ is unital.
The correspondence is given as follows: the module structure $A\otimes
M\rightarrow M$ is defined by setting $a\otimes m\mapsto a\cdot m$ if and only if $a\cdot m=\varphi (a)(m)$, for all $a\in A$, $m\in M$. It is easy to see that conditions (\[ghommod11\]) and (\[ghommod12\]) are equivalent to $\tilde{\alpha }\circ \varphi =\varphi \circ
\alpha _A$ and respectively $\tilde{\beta }\circ \varphi =\varphi \circ \beta _A$. We prove that, assuming (\[ghommod11\]) and (\[ghommod12\]), we have that (\[ghommod2\]) is equivalent to $\varphi \circ \mu _A=\tilde{\mu }\circ
(\varphi \otimes \varphi )$. Note first that (\[ghommod11\]) may be written as $\alpha _M\circ \varphi (a)=\varphi (\alpha _A(a))\circ \alpha _M$, for all $a\in A$, or equivalently $\alpha _M\circ \varphi (a)\circ \alpha
_M^{-1}=\varphi (\alpha _A(a))$, for all $a\in A$. Thus, for all $a, b\in A$, we have $$\begin{gathered}
\tilde{\mu }\circ (\varphi \otimes \varphi )(a\otimes b)=\varphi
(a)*\varphi (b)
=\alpha _M\circ \varphi (a)\circ \alpha _M^{-1}\circ \varphi (b)\circ
\beta _M^{-1}
=\varphi (\alpha _A(a))\circ \varphi (b)\circ \beta _M^{-1}.\end{gathered}$$ Hence, we have $$\begin{gathered}
\varphi \circ \mu _A=\tilde{\mu }\circ (\varphi \otimes \varphi)
\Longleftrightarrow \varphi (ab)=\varphi (a)*\varphi (b), \qquad \forall \, a, b\in A, \\
\hphantom{\varphi \circ \mu _A=\tilde{\mu }\circ (\varphi \otimes \varphi) }{}
\Longleftrightarrow \varphi (ab)(n)=(\varphi (a)*\varphi (b))(n),
\qquad \forall \, a, b\in A, n\in M, \\
\hphantom{\varphi \circ \mu _A=\tilde{\mu }\circ (\varphi \otimes \varphi) }{}
\Longleftrightarrow (ab)\cdot n=(\varphi (\alpha _A(a))\circ \varphi
(b)\circ \beta _M^{-1})(n), \qquad \forall \, a, b\in A, n\in M, \\
\hphantom{\varphi \circ \mu _A=\tilde{\mu }\circ (\varphi \otimes \varphi) }{}
\Longleftrightarrow (ab)\cdot \beta _M(m)=(\varphi (\alpha _A(a))\circ
\varphi (b))(m), \qquad \forall \, a, b\in A, m\in M, \\
\hphantom{\varphi \circ \mu _A=\tilde{\mu }\circ (\varphi \otimes \varphi) }{}
\Longleftrightarrow (ab)\cdot \beta _M(m)=\alpha _A(a)\cdot (b\cdot m),
\qquad \forall \, a, b\in A, m\in M,\end{gathered}$$ which is exactly (\[ghommod2\]).
Assume that $A$ is unital with unit $1_A$. The fact that $\varphi $ is unital is equivalent to $\varphi (1_A)=\beta _M$, which is equivalent to $1_A\cdot m=\beta _M(m)$, for all $m\in M$, which is equivalent to saying that the module $M$ is unital.
We recall the following concept from [@sheng] (see also [@said] on this subject).
Let $(L, [-] ,\alpha )$ be a Hom-Lie algebra. A *representation* of $L$ is a triple $(M, \rho , A)$, where $M$ is a linear space, $A\colon M\rightarrow M$ and $\rho \colon L\rightarrow \operatorname{End}(M)$ are linear maps such that, for all $x, y\in L$, the following conditions are satisfied: $$\begin{gathered}
\rho (\alpha (x))\circ A=A\circ \rho (x), \qquad
\rho ([x,y])\circ A=\rho (\alpha (x))\circ \rho (y)-\rho
(\alpha (y))\circ \rho (x).\end{gathered}$$
Let $(L, [-] ,\alpha )$ be a Hom-Lie algebra, $M$ a linear space, $A\colon M\rightarrow M$ and $\rho \colon L\rightarrow \operatorname{End}(M)$ linear maps such that $A$ is bijective. We can consider the Hom-associative algebra $\operatorname{End}(M)(A, A)$ as in Lemma \[endo\], and then the Hom-Lie algebra $L(\operatorname{End}(M)(A, A))$. Then one can check that $(M, \rho , A)$ is a representation of $L$ if and only if $\rho $ is a morphism of Hom-Lie algebras from $L$ to $L(\operatorname{End}(M)(A, A))$.
Inspired by this remark, we can introduce now the following concept:
Let $(L, [-] ,\alpha , \beta )$ be a BiHom-Lie algebra. A *representation* of $L$ is a 4-tuple $(M, \rho , \alpha _M, \beta _M
) $, where $M$ is a linear space, $\alpha _M, \beta _M\colon M\rightarrow M$ are two commuting linear maps and $\rho \colon L\rightarrow \operatorname{End}(M)$ is a linear map such that, for all $x, y\in L$, we have $$\begin{gathered}
\rho (\alpha (x))\circ \alpha _M=\alpha _M\circ \rho (x), \label{lierep1}
\\
\rho (\beta (x))\circ \beta _M=\beta _M\circ \rho (x), \label{lierep2} \\
\rho ( [\beta (x), y ])\circ \beta _M=\rho (\alpha \beta
(x))\circ \rho (y)- \rho (\beta (y))\circ \rho (\alpha (x)). \label{lierep3}\end{gathered}$$
A first indication that this is indeed the appropriate concept of representation for BiHom-Lie algebras is provided by the following result (extending the corresponding one for Hom-associative algebras in [@bakayoko]), whose proof is straightforward and left to the reader.
Let $(A, \mu _A, \alpha _A, \beta _A)$ be a BiHom-associative algebra with bijective structure maps and $(M, \alpha _M, \beta _M)$ a left $A$-module, with action $A\otimes M\rightarrow M$, $a\otimes m\mapsto a\cdot m$. Then we have a representation $(M, \rho , \alpha _M, \beta _M)$ of the BiHom-Lie algebra $L(A)$, where $\rho \colon L(A)\rightarrow \operatorname{End}(M)$ is the linear map defined by $\rho (a)(m)=a\cdot m$, for all $a\in A$, $m\in M$.
A second indication is provided by the fact that, under certain circumstances, we can construct the semidirect product (the Hom-case is done in [@sheng]).
Let $(L, [-] ,\alpha , \beta )$ be a BiHom-Lie algebra and $(M,
\rho , \alpha _M, \beta _M )$ a representation of $L$, with notation $\rho
(x)(a)=x\cdot a$, for all $x\in L$, $a\in M$. Assume that the maps $\alpha $ and $\beta _M$ are bijective. Then $L\ltimes M:=(L\oplus M, [-],
\alpha \oplus \alpha _M, \beta \oplus \beta _M)$ is a BiHom-Lie algebra $($called the semidirect product$)$, where $\alpha \oplus \alpha _M,
\beta \oplus \beta _M\colon L\oplus M \rightarrow L\oplus M$ are defined by $(\alpha \oplus \alpha _M)(x, a)=(\alpha (x), \alpha _M(a))$ and $(\beta
\oplus \beta _M)(x, a)=(\beta (x), \beta _M(a))$, and, for all $x, y\in L$ and $a, b\in M$, the bracket $[-]$ is defined by $$\begin{gathered}
[ (x, a), (y, b) ]=\big( [ x, y ], x\cdot b-\alpha
^{-1}\beta (y)\cdot\alpha _M\beta _M^{-1}(a)\big).\end{gathered}$$
Follows by a direct computation that is left to the reader.
Let $(L, [-] ,\alpha , \beta )$ be a BiHom-Lie algebra such that the map $\beta $ is surjective, $M$ a linear space, $\alpha _M, \beta
_M\colon M\rightarrow M$ two commuting linear isomorphisms and $\rho \colon L\rightarrow
\operatorname{End}(M)$ a linear map. Then $(M, \rho , \alpha _M, \beta _M )$ is a representation of $L$ if and only if $\rho $ is a morphism of BiHom-Lie algebras from $L$ to $L(\operatorname{End}(M)(\alpha _M, \beta _M))$.
Obviously, (\[lierep1\]) and (\[lierep2\]) are respectively equivalent to $\tilde{\alpha }\circ \rho =\rho \circ \alpha $ and $\tilde{\beta }\circ
\rho =\rho \circ \beta $, so we only need to prove that, assuming (\[lierep1\]) and (\[lierep2\]), (\[lierep3\]) is equivalent to $\rho ([x,y])= [\rho (x), \rho (y) ]$ for all $x, y\in L$. First we write down explicitly the bracket of $L(\operatorname{End}(M)(\alpha _M, \beta
_M))$. In view of Proposition \[croset\], this bracket looks as follows, for $f, g\in \operatorname{End}(M)$: $$\begin{gathered}
[f, g ] = f*g-\big(\tilde{\alpha }^{-1}\tilde{\beta }(g)\big)*\big(\tilde{\alpha }\tilde{\beta }^{-1}(f)\big) \\
\hphantom{[f, g]}{}
=f*g-\big(\tilde{\alpha }^{-1}\big(\beta _M\circ g\circ \beta _M^{-1}\big)\big)* \big(\tilde{\alpha }\big(\beta _M^{-1}\circ f\circ \beta _M\big)\big) \\
\hphantom{[f, g]}{}
= f*g-\big(\alpha _M^{-1}\circ \beta _M\circ g\circ \beta _M^{-1}\circ \alpha
_M\big)* \big(\alpha _M\circ \beta _M^{-1}\circ f\circ \beta _M\circ \alpha _M^{-1}\big)
\\
\hphantom{[f, g]}{}
=\alpha _M\circ f\circ \alpha _M^{-1}\circ g\circ \beta _M^{-1} \\
\hphantom{[f, g]=}{}- \alpha _M\circ \alpha _M^{-1}\circ \beta _M\circ g\circ \beta
_M^{-1}\circ \alpha _M\circ \alpha _M^{-1}\circ \alpha _M\circ \beta
_M^{-1}\circ f\circ \beta _M\circ \alpha _M^{-1}\circ \beta _M^{-1} \\
\hphantom{[f, g]}{}
= \alpha _M\circ f\circ \alpha _M^{-1}\circ g\circ \beta _M^{-1}- \beta
_M\circ g\circ \beta _M^{-1}\circ \alpha _M\circ \beta _M^{-1}\circ f\circ
\beta _M\circ \alpha _M^{-1}\circ \beta _M^{-1}.\end{gathered}$$ Let $x, y\in L$; we take $f=\rho (\beta (x))$, $g=\rho (y)$. We obtain $$\begin{gathered}
[\rho (\beta (x)), \rho (y) ]\circ \beta _M =
\alpha _M\circ \rho
(\beta (x))\circ \alpha _M^{-1} \circ \rho (y) \\
\hphantom{[\rho (\beta (x)), \rho (y) ]\circ \beta _M =}{}
- \beta _M\circ \rho (y)\circ \beta _M^{-1}\circ \alpha _M\circ \beta
_M^{-1}\circ \rho (\beta (x))\circ \beta _M\circ \alpha _M^{-1} \\
\hphantom{[\rho (\beta (x)), \rho (y) ]\circ \beta _M}{}
\overset{\eqref{lierep1}, \;\eqref{lierep2}}{=} \rho (\alpha \beta
(x))\circ \rho (y)- \rho (\beta (y))\circ \alpha _M\circ \rho (x)\circ
\alpha _M^{-1} \\
\hphantom{[\rho (\beta (x)), \rho (y) ]\circ \beta _M}{}
\overset{\eqref{lierep1}}{=} \rho (\alpha \beta (x))\circ \rho (y)- \rho
(\beta (y))\circ \rho (\alpha (x)),\end{gathered}$$ which is the right-hand side of (\[lierep3\]). So, we have that (\[lierep3\]) holds if and only if $\rho ([\beta (x), y])= [\rho (\beta (x)), \rho (y)]$ for all $x, y\in L$, which is equivalent to $\rho ([a,b])= [\rho (a), \rho (b)]$, for all $a,
b\in L$, because $\beta $ is surjective.
Let $(L, [-] ,\alpha , \beta )$ be a BiHom-Lie algebra and define the linear map $\operatorname{ad} \colon L\rightarrow \operatorname{End}(L)$, $\operatorname{ad} (x)(y)=[x,y]
$, for all $x, y\in L$. If the maps $\alpha $ and $\beta $ are bijective, then $(L, \operatorname{ad},\alpha , \beta )$ is a representation of $L$.
The conditions (\[lierep1\]) and (\[lierep2\]) are equivalent to $\alpha
([a,b])= [\alpha (a), \alpha (b) ]$ and $\beta ( [a, b])= [\beta (a), \beta (b)]$ for all $a, b\in L$, so we only need to prove (\[lierep3\]). Note first that the skew-symmetry condition implies $$\begin{gathered}
\operatorname{ad}(x)(y)=-\big[\alpha ^{-1}\beta (y), \alpha \beta ^{-1}(x)\big],
\qquad\forall \, x, y\in L.\end{gathered}$$ We compute the left-hand side of (\[lierep3\]) applied to $z\in L$: $$\begin{gathered}
(\operatorname{ad}( [\beta (x), y ])\circ \beta )(z) = \operatorname{ad}( [\beta (x), y ])(\beta (z))
= -\big[\alpha ^{-1}\beta ^2(z), \alpha \beta ^{-1}([\beta (x), y])\big] \\
\hphantom{(\operatorname{ad}( [\beta (x), y ])\circ \beta )(z)}{}
=-\big[\beta ^2(\alpha ^{-1}(z)), \big[\alpha (x), \alpha \beta ^{-1}(y)\big]\big] \\
\hphantom{(\operatorname{ad}( [\beta (x), y ])\circ \beta )(z)}{}
= -\big[\beta ^2(\alpha ^{-1}(z)), \big[\beta (\alpha \beta ^{-1}(x)),
\alpha (\beta ^{-1}(y))\big]\big].\end{gathered}$$ We compute the right-hand side of (\[lierep3\]) applied to $z\in L$: $$\begin{gathered}
(\operatorname{ad}(\alpha \beta (x))\circ \operatorname{ad}(y))(z)-(\operatorname{ad}(\beta (y))\circ
\operatorname{ad}(\alpha (x))(z)\\
\qquad{} = \operatorname{ad}(\alpha \beta (x))\big({-}\big[\alpha ^{-1}\beta (z), \alpha \beta ^{-1}(y)
\big]\big)- \operatorname{ad}(\beta (y))\big({-}\big[\alpha ^{-1}\beta (z), \alpha ^2\beta ^{-1}(x)
\big]\big) \\
\qquad{} = \big[\alpha ^{-1}\beta \big(\big[\alpha ^{-1}\beta (z), \alpha \beta
^{-1}(y)\big]\big), \alpha \beta ^{-1}\alpha \beta (x)\big] \\
\qquad\quad{}
- \big[\alpha ^{-1}\beta \big(\big[\alpha ^{-1}\beta (z), \alpha ^2\beta
^{-1}(x)\big]\big), \alpha \beta ^{-1}\beta (y)\big] \\
\qquad{}
= \big[\beta \big(\big[\alpha ^{-2}\beta (z), \beta ^{-1}(y)\big]\big), \alpha
^2(x)\big]- \big[\beta \big(\big[\alpha ^{-2}\beta (z), \alpha \beta ^{-1}(x)
\big]\big), \alpha (y)\big] \\
\qquad{}
\overset{\text{skew-symmetry}}{=} -\big[\beta \alpha (x), \big[\alpha ^{-1}\beta
(z), \alpha \beta ^{-1}(y)\big]\big]+\big[\beta (y), \big[\alpha
^{-1}\beta (z), \alpha ^2\beta ^{-1}(x)\big]\big] \\
\qquad{}
= [\beta \alpha (x), [y, z ] ]+\big[\beta (y), \big[
\beta \alpha ^{-1}(z), \alpha ^2\beta ^{-1}(x)\big]\big] \\
\qquad{}
= \big[\beta ^2\big(\alpha \beta ^{-1}(x)\big), \big[\beta \big(\beta ^{-1}(y)\big),
\alpha \big(\alpha ^{-1}(z)\big)\big]\big]\!
+\!\big[\beta ^2(\beta ^{-1}(y)), \big[\beta \big(\alpha ^{-1}(z)\big), \alpha
\big(\alpha \beta ^{-1}(x)\big)\big]\big],\end{gathered}$$ and (\[lierep3\]) holds because of the BiHom-Jacobi identity applied to the elements $a=\alpha \beta ^{-1}(x)$, $a^{\prime}=\beta^{-1}(y)$ and $a^{\prime
\prime}=\alpha^{-1}(z)$.
BiHom-coassociative coalgebras and BiHom-bialgebras
===================================================
We introduce now the dual concept to the one of BiHom-associative algebra.
A *BiHom-coassociative coalgebra* is a 4-tuple $(C, \Delta, \psi ,
\omega )$, in which $C$ is a linear space, $\psi , \omega \colon C\rightarrow C$ and $\Delta \colon C\rightarrow C\otimes C$ are linear maps, such that $$\begin{gathered}
\psi \circ \omega =\omega \circ \psi , \qquad
(\psi \otimes \psi )\circ \Delta = \Delta \circ \psi , \qquad
(\omega \otimes \omega )\circ \Delta = \Delta \circ \omega , \\
(\Delta \otimes \psi )\circ \Delta = (\omega \otimes \Delta )\circ \Delta .\end{gathered}$$
We call $\psi $ and $\omega $ (in this order) the *structure maps* of $C$.
A morphism $g\colon (C, \Delta _C , \psi _C, \omega _C)\rightarrow (D, \Delta _D ,
\psi _D, \omega _D)$ of BiHom-coassociative coalgebras is a linear map $g\colon C\rightarrow D$ such that $\psi _D\circ g=g\circ \psi _C$, $\omega _D\circ
g=g\circ \omega _C$ and $(g\otimes g)\circ \Delta _C=\Delta _D\circ g$.
A BiHom-coassociative coalgebra $(C, \Delta, \psi , \omega )$ is called *counital* if there exists a linear map $\varepsilon\colon C\rightarrow
\Bbbk$ (called a *counit*) such that $$\begin{gathered}
\varepsilon\circ \psi= \varepsilon, \qquad \varepsilon\circ \omega=
\varepsilon, \qquad
(\operatorname{id}_C\otimes \varepsilon) \circ \Delta=\omega \qquad \text{and} \qquad (
\varepsilon\otimes \operatorname{id}_C)\circ \Delta=\psi.\end{gathered}$$
A morphism of counital BiHom-coassociative coalgebras $g\colon C\rightarrow D$ is called *counital* if $\varepsilon_D\circ g=\varepsilon_C$, where $\varepsilon _C$ and $\varepsilon _D$ are the counits of $C$ and $D$, respectively.
If $(C, \Delta _C, \psi _C, \omega _C)$ and $(D, \Delta _D, \psi _D, \omega
_D)$ are two BiHom-coassociative coalgebras, then $(C\otimes D, \Delta
_{C\otimes D}, \psi _C\otimes \psi _D, \omega _C\otimes \omega _D)$ is also a BiHom-coassociative coalgebra (called the tensor product of $C$ and $D$), where $\Delta _{C\otimes D}\colon C\otimes D\rightarrow C\otimes D\otimes C\otimes
D$ is defined by $\Delta (c\otimes d)=c_1\otimes d_1\otimes c_2\otimes d_2$, for all $c\in C$, $d\in D$. If $C$ and $D$ are counital with counits $\varepsilon _C$ and respectively $\varepsilon _D$, then $C\otimes D$ is also counital with counit $\varepsilon _C\otimes \varepsilon _D$.
Let $(C,\Delta _{C},\psi _{C},\omega _{C})$ be a BiHom-coassociative coalgebra. A *right $C$-comodule* is a triple $(M,\psi _{M},\omega
_{M})$, where $M$ is a linear space, $\psi _{M},\omega _{M}\colon M\rightarrow M$ are linear maps and we have a linear map (called a coaction) $\rho
\colon M\rightarrow M\otimes C$, with notation $\rho (m)=m_{(0)}\otimes m_{(1)}$, for all $m\in M$, such that the following conditions are satisfied $$\begin{gathered}
\begin{split}
& \psi _{M}\circ \omega _{M}=\omega _{M}\circ \psi _{M}, \qquad (\psi _{M}\otimes \psi _{C})\circ \rho =\rho \circ \psi _{M},\qquad
(\omega _{M}\otimes \omega _{C})\circ \rho =\rho \circ \omega _{M},\\
& (\omega _{M}\otimes \Delta _{C})\circ \rho =(\rho \otimes \psi _{C})\circ
\rho . \end{split}\end{gathered}$$
If $(M, \psi _M, \omega _M)$ and $(N, \psi _N, \omega _N)$ are right $C$-comodules with coactions $\rho _M$ and respectively $\rho _N$, a morphism of right $C$-comodules $f\colon M\rightarrow N$ is a linear map satisfying the conditions $\psi _N\circ f=f\circ \psi _M$, $\omega _N\circ f=f\circ \omega
_M$ and $\rho _N\circ f=(f\otimes \operatorname{id}_C)\circ \rho _M$.
If $(C, \Delta _C, \psi _C, \omega _C, \varepsilon _C)$ is a counital BiHom-coassociative coalgebra and $(M, \psi _M, \omega _M)$ is a right $C$-comodule with coaction $\rho $, then $M$ is called *counital* if $(\operatorname{id}_M\otimes \varepsilon _C)\circ \rho =\omega _M$.
If $(C, \Delta, \psi , \omega )$ is a BiHom-coassociative coalgebra, then $(C, \psi , \omega )$ is a right $C$-comodule, with coaction $\rho =\Delta $.
We discuss now the duality between BiHom-associative and BiHom-coassociative structures.
Let $(C,\Delta, \psi , \omega )$ be a BiHom-coassociative coalgebra. Then its dual linear space is provided with a structure of BiHom-associative algebra $(C^*,\Delta^*,\omega^*,\psi^*)$, where $\Delta^*$, $\psi^*$, $\omega^*$ are the transpose maps. Moreover, the BiHom-associative algebra $C^*$ is unital whenever the BiHom-coassociative coalgebra $C$ is counital.
The product $\mu = \Delta^*$ is defined from $C^* \otimes C^*$ to $C^*$ by $$\begin{gathered}
(fg)(x) = \Delta^*(f,g)(x) = \langle \Delta(x),f \otimes g \rangle = (f
\otimes g)(\Delta(x)) = f(x_1)g(x_2), \qquad \forall \, x \in C,\end{gathered}$$ where $\langle \cdot,\cdot \rangle$ is the natural pairing between the linear space $C \otimes C$ and its dual linear space. For $f,g,h \in C^*$ and $x \in C$, we have $$\begin{gathered}
(fg) \psi^*(h)(x) = \langle (\Delta \otimes \psi) \circ \Delta(x),f \otimes
g \otimes h \rangle,\\
\omega^*(f)(gh)(x) = \langle (\omega
\otimes \Delta) \circ \Delta(x),f \otimes g \otimes h \rangle.\end{gathered}$$ Therefore, the BiHom-associativity condition $\mu \circ (\mu \otimes \psi^* - \omega^*
\otimes \mu) = 0$ follows from the BiHom-coassociativity condition $(\Delta \otimes
\psi - \omega \otimes \Delta) \circ \Delta = 0$.
Moreover, if $C$ has a counit $\varepsilon $ then for $f \in C^*$ and $x \in
C$ we have $$\begin{gathered}
(\varepsilon f)(x) = \varepsilon (x_1) f(x_2) = f(\varepsilon (x_1) x_2) =
f(\psi(x)) = \psi^*(f)(x), \\
(f \varepsilon )(x) = f(x_1) \varepsilon (x_2) = f(x_1 \varepsilon (x_2))
= f(\omega(x)) = \omega^*(f)(x),\end{gathered}$$ which shows that $\varepsilon $ is the unit of $C^*$.
The dual of a BiHom-associative algebra $(A,\mu,\alpha,\beta)$ is not always a BiHom-coassociative coalgebra, because $(A \otimes A)^* \supsetneq A^*
\otimes A^*$. Nevertheless, it is the case if the BiHom-associative algebra is finite-dimensional, since $(A\otimes A)^* = A^* \otimes A^*$ in this case.
More generally, we can define the finite dual of $A$ by $$\begin{gathered}
A^\circ = \{f \in A^*/ f(I)=0 \ \text{for some cof\/inite ideal $I$ of $A$}\},\end{gathered}$$ where a cofinite ideal $I$ is an ideal $I \subset A$ such that $A/I$ is finite-dimensional and where we say that $I$ is an ideal of $A$ if for $x
\in I$ and $y \in A$ we have $x y\in I$, $y x\in I$ and $\alpha (x)\in I$, $\beta (x)\in I$.
$A^\circ$ is a subspace of $A^*$ since it is closed under multiplication by scalars and the sum of two elements of $A^\circ$ is again in $A^\circ$ because the intersection of two cofinite ideals is again a cofinite ideal. If $A$ is finite-dimensional, of course $A^\circ = A^*$. As in the classical case, one can show that if $A$ and $B$ are two BiHom-associative algebras and $f \colon A \to B$ is a morphism of BiHom-associative algebras, then the dual map $f^* \colon B^* \to A^*$ satisfies $f^*(B^\circ)\subset A^\circ$.
Therefore, a similar proof to the one of the previous theorem leads to:
Let $(A,\mu,\alpha,\beta)$ be a BiHom-associative algebra. Then its finite dual is provided with a structure of BiHom-coassociative coalgebra $(A^\circ,\Delta,\beta^\circ,\alpha^\circ)$, where $\Delta = \mu^\circ =
\mu^*|_{A^\circ}$ and $\beta^\circ$, $\alpha^\circ$ are the transpose maps on $A^\circ$. Moreover, the BiHom-coassociative coalgebra is counital whenever $A $ is unital, with counit $\varepsilon \colon A^\circ \to \Bbbk $ defined by $\varepsilon (f) = f(1_A)$.
We can now define the notion of BiHom-bialgebra.
A *BiHom-bialgebra* is a 7-tuple $(H, \mu , \Delta, \alpha , \beta ,
\psi , \omega )$, with the property that $(H, \mu , \alpha , \beta )$ is a BiHom-associative algebra, $(H, \Delta , \psi , \omega )$ is a BiHom-coassociative coalgebra and moreover the following relations are satisfied, for all $h, h^{\prime }\in H$: $$\begin{gathered}
\Delta (hh^{\prime })=h_1h^{\prime }_1\otimes h_2h^{\prime }_2,
\label{ghombia2} \\
\alpha \circ \psi =\psi \circ \alpha , \qquad \alpha \circ \omega =\omega
\circ \alpha , \qquad \beta \circ \psi =\psi \circ \beta , \qquad \beta \circ
\omega =\omega \circ \beta , \nonumber\\
(\alpha \otimes \alpha )\circ \Delta =\Delta \circ \alpha , \qquad (\beta
\otimes \beta )\circ \Delta =\Delta \circ \beta , \nonumber\\
\psi (hh^{\prime })=\psi (h)\psi (h^{\prime }), \qquad \omega (hh^{\prime
})=\omega (h)\omega (h^{\prime }).\nonumber\end{gathered}$$
We say that $H$ is a unital and counital BiHom-bialgebra if, in addition, it admits a unit $1_H$ and a counit $\varepsilon _H$ such that $$\begin{gathered}
\Delta(1_H)=1_H\otimes 1_H,\qquad \varepsilon _H(1_H)=1, \qquad \psi
(1_H)=1_H, \qquad \omega (1_H)=1_H, \\
\varepsilon _H\circ \alpha =\varepsilon _H, \qquad \varepsilon _H\circ
\beta =\varepsilon _H, \qquad \varepsilon _H(hh^{\prime })=\varepsilon
_H(h)\varepsilon _H(h^{\prime }), \qquad \forall \,h, h^{\prime }\in H.\end{gathered}$$
Let us record the formula expressing the BiHom-coassociativity of $\Delta $: $$\begin{gathered}
\Delta (h_1)\otimes \psi (h_2)=\omega (h_1)\otimes \Delta (h_2),
\qquad \forall \,h\in H. \label{ghombia1}\end{gathered}$$
Obviously, a BiHom-bialgebra $(H, \mu , \Delta, \alpha , \beta , \psi ,
\omega )$ with $\alpha =\beta =\psi =\omega $ reduces to a Hom-bialgebra, as used for instance in [@mp1; @mp2], while a BiHom-bialgebra for which $\psi =\omega =\alpha ^{-1} =\beta ^{-1}$ reduces to a monoidal Hom-bialgebra, in the terminology of [@stef].
We see now that analogues of Yau’s twisting principle hold for the BiHom-structures we defined (proofs are straightforward and left to the reader):
\[yautwistdiverse\]
1. Let $(A, \mu )$ be an associative algebra and $
\alpha , \beta \colon A\rightarrow A$ two commuting algebra endomorphisms. Define a new multiplication $\mu _{(\alpha , \beta )}\colon A\otimes A\rightarrow A$, by $
\mu _{(\alpha , \beta )}:= \mu \circ (\alpha \otimes \beta )$. Then $(A, \mu
_{(\alpha , \beta )}, \alpha , \beta )$ is a BiHom-associative algebra, denoted by $A_{(\alpha , \beta )}$. If $A$ is unital with unit $1_A$, then $A_{(\alpha , \beta )}$ is also unital with unit $1_A$.
2. Let $(C, \Delta )$ be a coassociative coalgebra and $\psi , \omega
\colon C\rightarrow C$ two commuting coalgebra endomorphisms. Define a new comultiplication $\Delta _{(\psi , \omega ) }\colon C\rightarrow C\otimes C$, by $\Delta _{(\psi , \omega ) }:=(\omega \otimes \psi )\circ \Delta $. Then $(C,
\Delta _{(\psi , \omega ) }, \psi , \omega )$ is a BiHom-coassociative coalgebra, denoted by $C_{(\psi , \omega )}$. If $C$ is counital with counit $\varepsilon _C$, then $C_{(\psi , \omega )}$ is also counital with counit $\varepsilon _C$.
3. Let $(H, \mu , \Delta )$ be a bialgebra and $\alpha , \beta , \psi ,
\omega \colon H\rightarrow H$ bialgebra endomorphisms such that any two of them commute. If we define $\mu _{(\alpha , \beta )}$ and $\Delta _{(\psi ,
\omega ) }$ as in $(i)$ and $(ii)$, then $H_{(\alpha , \beta , \psi , \omega
)}:=(H, \mu _{(\alpha , \beta )}, \Delta _{(\psi , \omega ) }, \alpha ,
\beta , \psi , \omega )$ is a BiHom-bialgebra.
More generally, a BiHom-bialgebra $(H, \mu, \Delta , \alpha , \beta , \psi ,
\omega )$ and multiplicative and comultiplicative linear maps $
\alpha^{\prime }$, $\beta^{\prime }$, $\psi^{\prime }$, $\omega^{\prime }$ such that any two of the maps $\alpha$, $\beta$, $\psi$, $\omega$, $\alpha^{\prime }$, $\beta^{\prime }$, $\psi^{\prime }$, $\omega^{\prime }$ commute, give rise to a new BiHom-bialgebra $(H, \mu \circ(\alpha^{\prime }\otimes \beta^{\prime }),
(\omega ^{\prime }\otimes \psi ^{\prime })\circ \Delta ,
\alpha\circ\alpha^{\prime }, \beta\circ\beta^{\prime },
\psi\circ\psi^{\prime }, \omega\circ\omega^{\prime })$. Hence, if the maps $
\alpha$, $\beta$, $\psi$, $\omega$ are invertible, one can untwist the BiHom-bialgebra and get a bialgebra by taking $\alpha^{\prime }= \alpha^{-1}$, $\beta^{\prime}=\beta^{-1}$, $\psi^{\prime}=\psi^{-1}$, $\omega^{\prime}=\omega^{-1}$.
\[TwistLeftMod\] Let $(A, \mu _A)$ be an associative algebra and $\alpha
_A, \beta _A\colon A\rightarrow A$ two commuting algebra endomorphisms. Assume that $M$ is a left $A$-module, with action $A\otimes M\rightarrow M$, $
a\otimes m \mapsto a\cdot m$. Let $\alpha _M, \beta _M\colon M\rightarrow M$ be two commuting linear maps such that $\alpha _M(a\cdot m)=\alpha _A(a)\cdot
\alpha _M(m)$ and $\beta _M(a\cdot m)=\beta _A(a)\cdot \beta _M(m)$, for all $a\in A$, $m\in M$. Then $(M, \alpha _M, \beta _M)$ becomes a left module over $A_{(\alpha _A, \beta _A)}$, with action $A_{(\alpha _A, \beta
_A)}\otimes M\rightarrow M$, $a\otimes m\mapsto a\triangleright m:= \alpha
_A(a)\cdot \beta _M(m)$.
Let $(C, \Delta _C)$ be a coassociative coalgebra and $\psi _C, \omega
_C\colon C\rightarrow C$ two commuting coalgebra endomorphisms. Assume that $M$ is a right $C$-comodule, with coaction $\rho \colon M\rightarrow M\otimes C$, $\rho
(m)=m_{(0)}\otimes m_{(1)}$, for all $m\in M$. Let $\psi _M, \omega _M\colon
M\rightarrow M$ be two commuting linear maps such that $(\psi _M\otimes \psi
_C)\circ \rho =\rho \circ \psi _M$ and $(\omega _M\otimes \omega _C)\circ
\rho =\rho \circ \omega _M$. Then $(M, \psi _M, \omega _M)$ becomes a right comodule over the BiHom-coassociative coalgebra $C_{(\psi _C, \omega _C)}$, with coaction $$\begin{gathered}
M\rightarrow M\otimes C_{(\psi _C, \omega _C)}, \qquad m\mapsto
m_{\langle 0\rangle}\otimes m_{\langle 1\rangle }:= \omega _M(m_{(0)})\otimes \psi _C(m_{(1)}).\end{gathered}$$
We describe in what follows primitive elements of a BiHom-bialgebra.
Let $(H, \mu , \Delta, \alpha , \beta , \psi , \omega )$ be a unital and counital BiHom-bialgebra with a unit $1=\eta(1)$ and a counit $\varepsilon$. We assume that $\alpha $ and $\beta$ are bijective.
An element $x\in H$ is called *primitive* if $\Delta (x)=1\otimes x +
x\otimes 1$.
Let $x$ be a primitive element in $H$. Then $\varepsilon(x)1=\omega(x)-x=
\psi(x)-x$, and therefore $\omega(x)=\psi(x)$. Moreover, $\alpha^p\beta^q(x)$ is also a primitive element for any $p,q\in\mathbb{Z}$.
By the counit property, we have $\omega( x)=(\operatorname{id} _H\otimes
\varepsilon)(1\otimes x+x\otimes 1)=\varepsilon
(x)1+\varepsilon(1)x=\varepsilon (x)1+x$, and similarly $\psi(
x)=\varepsilon (x)1+x$.
Since $\alpha$ and $\beta$ are comultiplicative maps and $\alpha^p\beta^q(1)=1$, it follows that $\alpha^p\beta^q(x)$ is a primitive element whenever $x$ is a primitive element.
Let $(H, \mu , \Delta, \alpha , \beta , \psi , \omega )$ be a unital and counital BiHom-bialgebra, with unit $1=\eta(1)$ and counit $\varepsilon $. Assume that $\alpha $ and $\beta $ are bijective. If $x$ and $y$ are two primitive elements in $H$, then the commutator $[x,y]=x y -\alpha^{-1}\beta
(y) \alpha\beta^{-1}(x)$ is also a primitive element.
Consequently, the set of all primitive elements of $H$, denoted by $\operatorname{Prim}(H)$, has a structure of BiHom-Lie algebra.
We compute $$\begin{gathered}
\Delta (xy) = \Delta (x) \Delta (y)
=(1\otimes x+x\otimes 1)(1\otimes y+y\otimes 1) \\
\hphantom{\Delta (xy)}{}
= 1\otimes x y + \beta(y)\otimes \alpha(x)+\alpha(x)\otimes \beta(y) + x
y\otimes 1,
\\
\Delta \big(\alpha^{-1}\beta (y) \alpha\beta^{-1}(x)\big) = \Delta
\big(\alpha^{-1}\beta (y)\big) \Delta \big( \alpha\beta^{-1}(x)\big) \\
\hphantom{\Delta \big(\alpha^{-1}\beta (y) \alpha\beta^{-1}(x)\big)}{}
= \big(1\otimes \alpha^{-1}\beta (y)+\alpha^{-1}\beta (y)\otimes 1\big)\big(1\otimes
\alpha\beta^{-1}(x)+ \alpha\beta^{-1}(x)\otimes 1\big) \\
\hphantom{\Delta \big(\alpha^{-1}\beta (y) \alpha\beta^{-1}(x)\big)}{}
= 1\otimes \alpha^{-1}\beta (y) \alpha\beta^{-1}(x) +
\beta\big(\alpha\beta^{-1}(x)\big)\otimes \alpha\big(\alpha^{-1}\beta (y)\big) \\
\hphantom{\Delta \big(\alpha^{-1}\beta (y) \alpha\beta^{-1}(x)\big)=}{}
+\alpha\big(\alpha^{-1}\beta (y)\big)\otimes \beta\big( \alpha\beta^{-1}(x)\big) +
\alpha^{-1}\beta (y) \alpha\beta^{-1}(x)\otimes 1 \\
\hphantom{\Delta \big(\alpha^{-1}\beta (y) \alpha\beta^{-1}(x)\big)}{}
= 1\otimes \alpha^{-1}\beta (y) \alpha\beta^{-1}(x) + \alpha(x)\otimes
\beta (y) \\
\hphantom{\Delta \big(\alpha^{-1}\beta (y) \alpha\beta^{-1}(x)\big)=}{}
+\beta (y)\otimes \alpha(x) + \alpha^{-1}\beta (y)
\alpha\beta^{-1}(x)\otimes 1.\end{gathered}$$ Therefore, we have $$\begin{gathered}
\Delta([x,y])=\Delta (xy)-\Delta \big(\alpha^{-1}\beta (y) \alpha\beta^{-1}(x)\big)=1
\otimes[x,y]+[x,y]\otimes 1,\end{gathered}$$ which means that $Prim(H)$ is closed under the bracket multiplication $[\cdot,\cdot]$. Hence, $\operatorname{Prim}(H)$ is a BiHom-Lie algebra by Proposition \[croset\].
Now, we introduce the notion of $H$-module BiHom-algebra, where $H$ is a BiHom-bialgebra.
Let $(H, \mu _H, \Delta _H, \alpha _H, \beta _H, \psi _H, \omega _H)$ be a BiHom-bialgebra for which the maps $\alpha _H$, $\beta _H$, $\psi _H$, $\omega _H$ are bijective. A BiHom-associative algebra $(A, \mu _A, \alpha _A, \beta _A)$ is called a *left $H$-module BiHom-algebra* if $(A, \alpha _A,
\beta _A)$ is a left $H$-module, with action denoted by $H\otimes
A\rightarrow A$, $h\otimes a\mapsto h\cdot a$, such that the following condition is satisfied $$\begin{gathered}
h\cdot (aa^{\prime })=\big[\alpha _H^{-1}\big(\omega _H^{-1}(h_1)\big)\cdot a\big] [\beta
_H^{-1}\big(\psi _H^{-1}(h_2)\big)\cdot a^{\prime }], \qquad \forall \, h\in H, \quad a,
a^{\prime }\in A. \label{gmodalgcompat}\end{gathered}$$
This concept contains as particular cases the concepts of module algebras over a Hom-bialgebra, respectively monoidal Hom-bialgebra, introduced in [@yau1], respectively [@chenwangzhang].
The choice of (\[gmodalgcompat\]) is motivated by the following result, whose proof is also left to the reader:
\[gyaumodalg\] Let $(H, \mu _H, \Delta _H)$ be a bialgebra and $(A, \mu
_A)$ a left $H$-module algebra in the usual sense, with action denoted by $H\otimes A\rightarrow A$, $h\otimes a\mapsto h\cdot a$. Let $\alpha _H,
\beta _H, \psi _H, \omega _H\colon H\rightarrow H$ be bialgebra endomorphisms of $H $ such that any two of them commute; let $\alpha _A, \beta _A\colon A\rightarrow
A$ be two commuting algebra endomorphisms such that, for all $h\in H$ and $a\in A$, we have $$\begin{gathered}
\alpha _A(h\cdot a)=\alpha _H(h)\cdot \alpha _A(a)\qquad \text{and} \qquad \beta
_A(h\cdot a)=\beta _H(h)\cdot \beta _A(a).\end{gathered}$$ If we consider the BiHom-bialgebra $H_{(\alpha _H, \beta _H, \psi _H, \omega
_H)}$ and the BiHom-associative algebra $A_{(\alpha _A, \beta _A)}$ as defined before, then $A_{(\alpha _A, \beta _A)}$ is a left $H_{(\alpha _H,
\beta _H, \psi _H, \omega _H)}$-module BiHom-algebra in the above sense, with action $$\begin{gathered}
H_{(\alpha _H, \beta _H, \psi _H, \omega _H)}\otimes A_{(\alpha _A, \beta
_A)}\rightarrow A_{(\alpha _A, \beta _A)}, \qquad h\otimes a\mapsto
h\triangleright a:=\alpha _H(h)\cdot \beta _A(a).\end{gathered}$$
Monoidal BiHom-Hopf algebras and BiHom-Hopf algebras
====================================================
In this section, we introduce the concept of monoidal BiHom-Hopf algebra and discuss a possible generalization of Hom-Hopf algebras to BiHom-Hopf algebras.
We begin with a lemma whose proof is obvious.
\[subalg\] Let $(A,\mu ,\alpha ,\beta )$ be a BiHom-associative algebra. Define $\underline{A}:=\{a\in A/\alpha (a)=\beta (a)=a\}$. Then $(\underline{A},\mu )$ is an associative algebra. If $A$ is unital with unit $1_{A}$, then $1_{A}$ is also the unit of $\underline{A}$ $($in particular, it follows that the unit of a BiHom-associative algebra, if it exists, is unique$)$.
Let $ ( A,\mu ,\alpha,\beta ) $ be a BiHom-associative algebra and $(C, \Delta , \psi , \omega )$ a BiHom-coassociative coalgebra. Set, for $f,g\in \operatorname{Hom} ( C,A )$, $f \star g=\mu \circ ( f\otimes g
) \circ\Delta $. Define the linear maps $\phi , \gamma \colon \operatorname{Hom}(C,
A)\rightarrow \operatorname{Hom}(C, A)$ by $\phi(f)=\alpha\circ f\circ \omega$ and $\gamma(f)=\beta\circ f\circ \psi$, for all $f\in \operatorname{Hom}(C, A)$. Then $(\operatorname{Hom}( C,A ),\star,\phi,\gamma)$ is a BiHom-associative algebra.
Moreover, if $A$ is unital with unit $1_A$ and $C$ is counital with counit $\varepsilon$, then $\operatorname{Hom}( C,A)$ is a unital BiHom-asssociative algebra with unit $\eta \circ \varepsilon$, where we denote by $\eta $ the linear map $\eta \colon \Bbbk \rightarrow A$, $\eta (1)=1_A$.
In particular, if we denote by $\underline{\operatorname{Hom}}(C, A)$ the linear subspace of $\operatorname{Hom}(C, A)$ consisting of the linear maps $f\colon C\rightarrow A$ such that $\alpha \circ f\circ \omega =f$ and $\beta \circ f\circ \psi =f$, then $(\underline{\operatorname{Hom}}(C, A), \star , \eta \circ \varepsilon )$ is an associative unital algebra.
Let $f,g,h\in \operatorname{Hom}( C,A) $. We have $$\begin{gathered}
\phi (f)\star (g\star h) = \mu \circ ( \phi (f)\otimes (g\star
h)) \Delta =\mu \circ ( \phi (f)\otimes (\mu \circ (
g\otimes h ) \circ \Delta ) ) \Delta \\
\hphantom{\phi (f)\star (g\star h)}{}
= \mu \circ ( ( \alpha \otimes \mu ) \circ ( f\otimes
g\otimes h ) \circ ( \omega \otimes \Delta ) ) \Delta .\end{gathered}$$ Similarly, $$\begin{gathered}
(f\star g)\star \gamma (h)=\mu \circ ( ( \mu \otimes \beta )
\circ ( f\otimes g\otimes h ) \circ ( \Delta \otimes \psi
) ) \Delta .\end{gathered}$$ The BiHom-associativity of $\mu $ and the BiHom-coassociativity of $\Delta $ lead to the BiHom-associativity of the convolution product $\star $.
The map $\eta \circ \varepsilon $ is the unit for the convolution product. Indeed, for $f\in \operatorname{Hom}( C,A ) $ and $x\in C$, we have $$\begin{gathered}
\begin{split}
& (f\star (\eta \circ \varepsilon ))(x)=\mu \circ ( f\otimes \eta \circ
\varepsilon ) \circ \Delta (x)= \mu ( f(x_{1})\otimes \eta \circ
\varepsilon (x_{2}) ) = \varepsilon (x_{2})\mu ( f(x_{1})\otimes \eta (1) ) \\
& \hphantom{(f\star (\eta \circ \varepsilon ))(x)}{}
=
\varepsilon (x_{2}) ( \alpha \circ f ) (x_{1}) = ( \alpha
\circ f ) (x_{1}\varepsilon (x_{2})) =\alpha \circ f\circ \omega (x).
\end{split}\end{gathered}$$ A similar calculation shows that $(\eta \circ \varepsilon )\star f=\beta
\circ f\circ \psi $.
The last statement follows from Lemma \[subalg\].
Let $(H, \mu , \Delta, \alpha , \beta , \psi , \omega )$ be a unital and counital BiHom-bialgebra. We say that $H$ is a *monoidal BiHom-bialgebra* if $\alpha$, $\beta$, $\psi$, $\omega $ are bijective and $\omega =\alpha ^{-1}$ and $\psi =\beta ^{-1}$. We will refer to a monoidal BiHom-bialgebra as the $5$-tuple $(H, \mu , \Delta, \alpha , \beta )$.
If $(H, \mu , \Delta, \alpha , \beta )$ is a monoidal BiHom-bialgebra, we can consider the associative unital algebra $\underline{\operatorname{Hom}}(H, H)$, and since $\omega =\alpha ^{-1}$ and $\psi =\beta ^{-1}$, it follows that $\operatorname{id}_H\in \underline{\operatorname{Hom}}(H, H)$.
Let $(H, \mu , \Delta, \alpha , \beta )$ be a monoidal BiHom-bialgebra with a unit $1_H$ and a counit $\varepsilon _H$. A linear map $S\colon H\rightarrow H$ is called an *antipode* if $\alpha \circ
S=S\circ \alpha $ and $\beta \circ S=S\circ \beta $ (i.e., $S\in \underline{\operatorname{Hom}}(H, H)$) and $S$ is the convolution inverse of $\operatorname{id}_H$ in $\underline{\operatorname{Hom}}(H, H)$, that is $$\begin{gathered}
S(h_1)h_2=\varepsilon _H(h)1_H=h_1S(h_2), \qquad \forall \, h\in H.\end{gathered}$$ A *monoidal BiHom-Hopf algebra* is a monoidal BiHom-bialgebra endowed with an antipode.
Obviously, if the antipode exists, it is unique; we will refer to the monoidal BiHom-Hopf algebra as the $8$-tuple $(H, \mu , \Delta, \alpha , \beta , 1_H,
\varepsilon _H, S)$.
Let $(H, \mu , \Delta , 1_H, \varepsilon _H)$ be a Hopf algebra $($in the usual sense$)$ with antipode $S$. Let $\alpha , \beta \colon H\rightarrow H$ be two unital and counital commuting bialgebra automorphisms. Then $(H, \mu \circ
(\alpha \otimes \beta ), (\alpha ^{-1}\otimes \beta ^{-1})\circ \Delta ,
\alpha , \beta , 1_H, \varepsilon _H, S)$ is a monoidal BiHom-bialgebra.
A straightforward computation. Let us only note that $\alpha$, $\beta $ being bialgebra maps, they automatically commute with $S$.
We state now the basic properties of the antipode.
Let $(H, \mu , \Delta, \alpha , \beta , 1_H,
\varepsilon _H, S)$ be a monoidal BiHom-Hopf algebra. Then
1. $S(1_H)=1_H$ and $\varepsilon _H\circ S=\varepsilon _H$;
2. $S(\beta (a)\alpha (b))=S(\beta (b))S(\alpha (a))$, for all $a, b\in H$;
3. $\alpha (S(h)_1)\otimes \beta (S(h)_2)=\beta (S(h_2))\otimes \alpha
(S(h_1))$, for all $h\in H$.
\(i) By $\Delta (1_H)=1_H\otimes 1_H$ we obtain $S(1_H)1_H=\varepsilon
_H(1_H)1_H$, so $\alpha (S(1_H))=1_H$, and since $\alpha \circ S=S\circ
\alpha $ and $\alpha (1_H)=1_H$ we obtain $S(1_H)=1_H$. Then, if $h\in H$, we apply $\varepsilon _H$ to the equality $h_1S(h_2)=\varepsilon _H(h)1_H$, and we obtain $\varepsilon _H(h_1)\varepsilon _H(S(h_2)) =\varepsilon _H(h)$, so $\varepsilon _H(S(\varepsilon _H(h_1)h_2))=\varepsilon _H(h)$, hence $\varepsilon _H(S(\beta ^{-1}(h)))=\varepsilon _H(h)$, and since $S\circ
\beta =\beta \circ S$ and $\varepsilon _H\circ \beta =\varepsilon _H$ we obtain $\varepsilon _H\circ S=\varepsilon _H$.
\(ii) We define the linear maps $R, L, m\colon H\otimes H\rightarrow H$ by the formulae (for all $a, b\in H$): $$\begin{gathered}
R(a\otimes b)=S(\beta (b))S(\alpha (a)), \qquad
L(a\otimes b)=S(\beta (a)\alpha (b)), \qquad
m(a\otimes b)=\beta (a)\alpha (b).\end{gathered}$$ One can easily check that $R, L, m\in \underline{\operatorname{Hom}}(H\otimes H, H)$ (where $H\otimes H$ is the tensor product BiHom-coassociative coalgebra). Thus, to prove that $R=L$, it is enough to prove that $L$ (respectively $R$) is a left (respectively right) convolution inverse of $m$ in $\underline{\operatorname{Hom}}(H\otimes
H, H)$. We compute $$\begin{gathered}
(L\star m)(a\otimes b) = L(a_1\otimes b_1)m(a_2\otimes b_2) = S(\beta (a_1)\alpha (b_1))(\beta (a_2)\alpha (b_2)) \\
\hphantom{(L\star m)(a\otimes b)}{}
= S(\beta (a)_1\alpha (b)_1)(\beta (a)_2\alpha (b)_2)
= S((\beta (a)\alpha (b))_1)(\beta (a)\alpha (b))_2 \\
\hphantom{(L\star m)(a\otimes b)}{}
= \varepsilon _H(\beta (a)\alpha (b))1_H
= \varepsilon _H(a)\varepsilon _H(b)1_H,
\\
(m\star R)(a\otimes b) = m(a_1\otimes b_1)R(a_2\otimes b_2)
= (\beta (a_1)\alpha (b_1))(S(\beta (b_2))S(\alpha (a_2))) \\
\hphantom{(m\star R)(a\otimes b)}{}
= \alpha \big(\alpha ^{-1}\beta (a_1)b_1\big)(\beta (S(b_2))\alpha (S(a_2)))
= \big(\big(\alpha ^{-1}\beta (a_1)b_1\big)\beta (S(b_2)))\alpha \beta (S(a_2)\big) \\
\hphantom{(m\star R)(a\otimes b)}{}
= (\beta (a_1)(b_1S(b_2)))\alpha \beta (S(a_2))
= (\beta (a_1)\varepsilon _H(b)1_H)\alpha \beta (S(a_2)) \\
\hphantom{(m\star R)(a\otimes b)}{}
= \varepsilon _H(b)\alpha \beta (a_1)\alpha \beta (S(a_2))
= \varepsilon _H(b)\alpha \beta (a_1S(a_2)) \\
\hphantom{(m\star R)(a\otimes b)}{}
= \varepsilon _H(b)\alpha \beta (\varepsilon _H(a)1_H)
= \varepsilon _H(a)\varepsilon _H(b)1_H,\end{gathered}$$ finishing the proof.
\(iii) similar to the proof of (ii), by defining the linear maps $\mathcal{L}, \mathcal{R}, \delta \colon H\rightarrow H\otimes H$, $$\begin{gathered}
\mathcal{L}(h)=\alpha (S(h)_1)\otimes \beta (S(h)_2), \qquad\!
\mathcal{R}(h)=\beta (S(h_2))\otimes \alpha (S(h_1)), \qquad\!
\delta (h)=\alpha (h_1)\otimes \beta (h_2),\end{gathered}$$ for all $h\in H$, and proving that $\mathcal{L}$ (respectively $\mathcal{R}$) is a left (respectively right) convolution inverse of $\delta $ in $\underline{\operatorname{Hom}}(H, H\otimes H)$.
We had to restrict the definition of the antipode to the class of monoidal BiHom-bialgebras because, if $H$ is a Hopf algebra with antipode $S$ and we make an arbitrary Yau twist of $H$, then in general $S$ will not satisfy the defining property of an antipode for the Yau twist, as the next example shows.
Let $\Bbbk $ be a field and let $H=\Bbbk \left[ X\right] $, regarded as a Hopf algebra in the usual way. Let $\alpha \colon H\rightarrow H$ be the algebra map defined by setting $\alpha (X) =X^{2}$ and let $\beta =
\omega =\psi =\operatorname{Id}_{H}$. Then we can consider the BiHom-bialgebra $H_{(\alpha ,\beta ,\psi ,\omega )}:=(H,\mu _{(\alpha ,\beta )},\Delta
_{(\psi ,\omega )},\alpha ,\beta ,\psi ,\omega )$, where $\mu \colon H\otimes
H\rightarrow H$ is the usual multiplication and $\Delta \colon H\rightarrow
H\otimes H$ is the usual comultiplication. Moreover $H_{(\alpha ,\beta ,\psi
,\omega )}$ has unit $1_{H}=\eta _H\left( 1_{\Bbbk }\right) $ and counit $\varepsilon _{H}$ that coincide with the ones of $H$.
Assume that there exists a linear map $S\colon H\rightarrow H$ such that $S\star
\operatorname{Id}=\operatorname{Id}\star S=\eta _H\circ \varepsilon _H$, i.e., $$\begin{gathered}
\mu _{(\alpha ,\beta )}\circ ( S\otimes \operatorname{Id} ) \circ \Delta
_{(\psi ,\omega )}=\mu _{(\alpha ,\beta )}\circ ( \operatorname{Id}\otimes
S ) \circ \Delta _{(\psi ,\omega )}=\eta _{H}\circ \varepsilon _{H}.
\label{form:1}\end{gathered}$$ Then we compute $$\begin{gathered}
\big( \mu _{(\alpha ,\beta )}\circ ( \operatorname{Id}\otimes S ) \circ
\Delta _{(\psi ,\omega )}\big) (X) =\alpha (X)
S ( 1 ) +\alpha ( 1 ) S(X) =X^{2}S (
1 ) +S(X),
\\
( \mu _{(\alpha ,\beta )}\circ ( S\otimes \operatorname{Id} ) \circ
\Delta _{(\psi ,\omega )} ) (X) =\alpha ( S (
X ) ) 1+\alpha ( S ( 1 ) ) X,\end{gathered}$$ and $$\begin{gathered}
( \eta _{H}\circ \varepsilon _{H} ) (X) =01_{H},\end{gathered}$$ so that from (\[form:1\]) we get $$\begin{gathered}
S(X) =-X^{2}S(1) \label{form:2}\end{gathered}$$ and $$\begin{gathered}
\alpha ( S(X) ) =-\alpha ( S(1) ) X, \label{form:3}\end{gathered}$$ and hence $$\begin{gathered}
-\alpha ( S(1) ) X\overset{\eqref{form:3}}{=}\alpha ( S(X) ) \overset{\eqref{form:2}}{=}\alpha \big( {-}X^{2}S(1) ) \overset{\text{def.}\, \alpha }{=}-\alpha \big( X^{2}\big) \alpha ( S(1)
) \overset{\text{def.}\,\alpha }{=}-X^{4}\alpha ( S (1) ),\end{gathered}$$ so that we get $\alpha ( S(1) ) X=X^{4}\alpha (
S(1) )$, which implies that $\alpha ( S(
1) ) =0$.
On the other hand, we have $$\begin{gathered}
1= ( \eta _{H}\circ \varepsilon _{H} ) (1) \overset{\eqref{form:1}}{=}\big( \mu _{(\alpha ,\beta )}\circ (
S\otimes \operatorname{Id} ) \circ \Delta _{(\psi ,\omega )}\big) (
1 ) =\alpha ( S(1) ) 1=0,\end{gathered}$$ and this is a contradiction.
In view of all the above, we propose the following definition for what might be a BiHom-Hopf algebra, that is moreover invariant under Yau twisting:
Let $(H, \mu , \Delta, \alpha, \beta , \psi , \omega )$ be a unital and counital BiHom-bialgebra with a unit $1_H$ and a counit $\varepsilon _H$. A linear map $S\colon H\rightarrow H$ is called an *antipode* if it commutes with all the maps $\alpha$, $\beta$, $\psi$, $\omega $ and it satisfies the following relation: $$\begin{gathered}
\beta \psi (S(h_1))\alpha \omega (h_2)=\varepsilon _H(h)1_H= \beta \psi
(h_1)\alpha \omega (S(h_2)), \qquad \forall \, h\in H.\end{gathered}$$ A *BiHom-Hopf algebra* is a unital and counital BiHom-bialgebra with an antipode.
We hope to make a more detailed analysis of these structures in a forthcoming paper.
BiHom-pseudotwistors and BiHom-twisted tensor products
======================================================
Inspired by Proposition \[yaugeneral\], by the concept of pseudotwistor for associative algebras introduced in [@lpvo] and its generalization for Hom-associative algebras introduced in [@mp2], we arrive at the following concept and result:
\[generalpseudotwistor\] Let $(D, \mu , \tilde{\alpha }, \tilde{\beta })$ be a BiHom-associative algebra and $\alpha , \beta \colon D\rightarrow D$ two multiplicative linear maps such that any two of the maps $\tilde{\alpha }$, $\tilde{\beta }$, $\alpha$, $\beta $ commute. Let $T\colon D\otimes D\rightarrow
D\otimes D$ a linear map and assume that there exist two linear maps $\tilde{T}_1, \tilde{T}_2\colon D\otimes D\otimes D \rightarrow D\otimes D\otimes D$ such that the following relations hold: $$\begin{gathered}
(\alpha \otimes \alpha )\circ T=T\circ (\alpha \otimes \alpha ),
\label{ghommultT1} \\
(\beta \otimes \beta )\circ T=T\circ (\beta \otimes \beta ),
\label{ghommultT2} \\
(\tilde{\alpha }\otimes \tilde{\alpha })\circ T=T\circ (\tilde{\alpha }
\otimes \tilde{\alpha }), \label{ghommultT3} \\
(\tilde{\beta }\otimes \tilde{\beta })\circ T=T\circ (\tilde{\beta }
\otimes \tilde{\beta }), \label{ghommultT4} \\
T\circ (\tilde{\alpha }\otimes \mu )= (\tilde{\alpha }\otimes \mu )\circ
\tilde{T}_1\circ (T\otimes \operatorname{id}_D), \label{ghompstw1} \\
T\circ (\mu \otimes \tilde{\beta })= (\mu \otimes \tilde{\beta })\circ
\tilde{T}_2\circ (\operatorname{id}_D\otimes T), \label{ghompstw2} \\
\tilde{T}_1\circ (T\otimes \operatorname{id}_D)\circ (\alpha \otimes T)= \tilde{T}_2\circ
(\operatorname{id}_D\otimes T)\circ (T\otimes \beta ). \label{ghompstw3}\end{gathered}$$ Then $D^T_{\alpha , \beta }:=(D, \mu \circ T, \tilde{\alpha }\circ \alpha ,
\tilde{\beta }\circ \beta )$ is also a BiHom-associative algebra. The map $T$ is called an $(\alpha , \beta )$-BiHom-pseudotwistor and the two maps $\tilde{T}_1$, $\tilde{T}_2$ are called the companions of $T$. In the particular case $\alpha =\beta =\operatorname{id}_D$, we simply call $T$ a *BiHom-pseudotwistor* and we denote $D^T_{\alpha , \beta }$ by $D^T$.
The fact that $\tilde{\alpha }\circ \alpha $ and $\tilde{\beta }\circ \beta$ are multiplicative with respect to $\mu \circ T$ follows immediately from (\[ghommultT1\])–(\[ghommultT4\]) and the fact that $\alpha $, $\beta $, $\tilde{\alpha }$, $\tilde{\beta }$ are multiplicative with respect to $\mu $. Now we prove the BiHom-associativity of $\mu \circ T$: $$\begin{gathered}
(\mu \circ T)\circ ((\mu \circ T)\otimes (\tilde{\beta }\circ \beta ))
= \mu \circ T\circ (\mu \otimes \tilde{\beta })\circ (T\otimes \beta ) \\
\qquad {} \overset{\eqref{ghompstw2}}{=}\mu \circ (\mu \otimes \tilde{\beta })\circ
\tilde{T}_2\circ (\operatorname{id}_D\otimes T)\circ (T\otimes \beta ) \\
\qquad {}
\overset{\eqref{ghompstw3}}{=} \mu \circ (\mu \otimes \tilde{\beta })\circ
\tilde{T}_1\circ (T\otimes \operatorname{id}_D)\circ (\alpha \otimes T)
= \mu \circ (\tilde{\alpha }\otimes \mu )\circ \tilde{T}_1\circ (T\otimes
\operatorname{id}_D)\circ (\alpha \otimes T) \\
\qquad {}
\overset{\eqref{ghompstw1}}{=}\mu \circ T\circ (\tilde{\alpha }\otimes \mu
) \circ (\alpha \otimes T)
=(\mu \circ T)\circ ((\tilde{\alpha }\circ \alpha )\otimes (\mu \circ T)),\end{gathered}$$ finishing the proof.
Obviously, if $(D, \mu )$ is an associative algebra and $\tilde{\alpha }=\tilde{\beta }=\alpha =\beta =\operatorname{id}_D$, an $(\alpha , \beta )$-BiHom-pseudotwistor reduces to a pseudotwistor (as defined in [@lpvo]) and the BiHom-associative algebra $D^T_{\alpha , \beta }$ is actually associative.
We show now that Proposition \[yaugeneral\] is a particular case of Theorem \[generalpseudotwistor\].
Let $(D, \mu , \tilde{\alpha }, \tilde{\beta })$ be a BiHom-associative algebra and $\alpha , \beta \colon D\rightarrow D$ two multiplicative linear maps such that any two of the maps $\tilde{\alpha }$, $\tilde{\beta }$, $\alpha$, $\beta $ commute. Define the maps $$\begin{gathered}
T\colon \ D\otimes D\rightarrow D\otimes D, \qquad T=\alpha \otimes \beta, \\
\tilde{T}_1\colon \ D\otimes D\otimes D \rightarrow D\otimes D\otimes D, \qquad
\tilde{T}_1= \operatorname{id}_D\otimes \operatorname{id}_D\otimes \beta , \\
\tilde{T}_2\colon \ D\otimes D\otimes D \rightarrow D\otimes D\otimes D, \qquad
\tilde{T}_2=\alpha \otimes \operatorname{id}_D\otimes \operatorname{id}_D.\end{gathered}$$ Then $T$ is an $(\alpha , \beta )$-BiHom-pseudotwistor with companions $\tilde{T}_1$, $\tilde{T}_2$ and the BiHom-associative algebras $D^T_{\alpha
, \beta }$ and $D_{(\alpha , \beta )}$ coincide.
The conditions (\[ghommultT1\])–(\[ghommultT4\]) are obviously satisfied. We check (\[ghompstw1\]), for $a, b, c\in D$: $$\begin{gathered}
\big((\tilde{\alpha }\otimes \mu )\circ \tilde{T}_1\circ (T\otimes
\operatorname{id}_D)\big)(a\otimes b\otimes c) = \big((\tilde{\alpha }\otimes \mu )\circ \tilde{T}
_1\big)(\alpha (a)\otimes \beta (b)\otimes c) \\
\qquad {} = (\tilde{\alpha }\otimes \mu )(\alpha (a)\otimes \beta (b)\otimes \beta
(c))
= (\tilde{\alpha }\circ \alpha )(a)\otimes \beta (bc) \\
\qquad{}
= T(\tilde{\alpha }(a)\otimes bc)
= (T\circ (\tilde{\alpha }\otimes \mu ))(a\otimes b\otimes c).\end{gathered}$$ The condition (\[ghompstw2\]) is similar, so we check (\[ghompstw3\]): $$\begin{gathered}
\big(\tilde{T}_1\circ (T\otimes \operatorname{id}_D)\circ (\alpha \otimes T)\big)(a\otimes b\otimes
c) = \big(\tilde{T}_1\circ (T\otimes \operatorname{id}_D)\big)(\alpha (a)\otimes \alpha (b)\otimes
\beta (c)) \\
\qquad {} = \tilde{T}_1\big(\alpha ^2(a)\otimes \beta \alpha (b)\otimes \beta (c)\big)
= \alpha ^2(a)\otimes \beta \alpha (b)\otimes \beta ^2(c)
= \tilde{T}_2\big(\alpha (a)\otimes \alpha \beta (b)\otimes \beta ^2(c)\big) \\
\qquad{} = \big(\tilde{T}_2\circ (\operatorname{id}_D\otimes T)\big)(\alpha (a)\otimes \beta (b)\otimes
\beta (c))
= \big(\tilde{T}_2\circ (\operatorname{id}_D\otimes T\big)\circ (T\otimes \beta ))(a\otimes
b\otimes c).\end{gathered}$$ It is obvious that $D^T_{\alpha , \beta }$ and $D_{(\alpha , \beta )}$ coincide.
We consider the 2-dimensional BiHom-associative algebra $(D,\mu,\tilde{\alpha},\tilde{\beta})$ defined with respect to a basis $\mathcal{B}=\{e_1,e_2\}$ by $$\begin{gathered}
\mu(e_1,e_1)= \mu(e_1,e_2)=e_1,\qquad \mu(e_2,e_1)=\mu(e_2,e_2)=e_2, \\
\tilde{\alpha} (e_1 )=e_1, \qquad \tilde{\alpha}(e_2)= e_2, \qquad \tilde{\beta}
(e_1 )=e_1, \qquad \tilde{\beta}(e_2)= e_1.\end{gathered}$$ We have the following multiplicative linear maps $\alpha, \beta$ defined with respect to the basis $\mathcal{B}$ by $$\begin{gathered}
\alpha (e_1 )=e_1, \qquad \alpha(e_2)=a e_1+(1-a) e_2, \qquad
\beta (e_1 )=e_1, \qquad \beta(e_2)= b e_1+(1-b) e_2,\end{gathered}$$ where $a$, $b$ are parameters in $\Bbbk $. One can easily see that any two of the maps $\tilde{\alpha }$, $\tilde{\beta }$, $\alpha$, $\beta $ commute. By the previous proposition, we can construct the BiHom-associative algebra $
D_{(\alpha,\beta)}=(D, \mu _T=\mu \circ (\alpha \otimes \beta ), \alpha_T=
\tilde{\alpha }\circ \alpha , \beta_T=\tilde{\beta }\circ \beta )$ defined on the basis $\mathcal{B}$ by $$\begin{gathered}
\mu_T(e_1,e_1)=e_1,\qquad \mu_T(e_1,e_2)= e_1,\qquad \mu_T(e_2,e_1)=a e_1+
(1-a)e_2,\\ \mu_T(e_2,e_2)=a e_1+(1-a)e_2, \qquad
\alpha_T(e_1)=e_1, \qquad \alpha_T(e_2)=a e_1+(1-a) e_2,\\ \beta_T(e_1)=e_1, \qquad
\beta_T(e_2)= e_1.\end{gathered}$$
Let $(A, \mu _A)$, $(B, \mu _B)$ be two associative algebras. A *twisting map* between $A$ and $B$ is a linear map $R\colon B\otimes A \rightarrow A\otimes B$ satisfying the conditions $$\begin{gathered}
R\circ (\operatorname{id}_B\otimes \mu _A)=(\mu _A\otimes \operatorname{id}_B)\circ (\operatorname{id}_A\otimes R)\circ
(R\otimes \operatorname{id}_A), \label{twmap1} \\
R\circ (\mu _B\otimes \operatorname{id}_A)=(\operatorname{id}_A\otimes \mu _B)\circ (R\otimes \operatorname{id}_B)\circ
(\operatorname{id}_B\otimes R). \nonumber $$ If this is the case, the map $\mu _R=(\mu _A\otimes \mu _B)\circ
(\operatorname{id}_A\otimes R\otimes \operatorname{id}_B)$ is an associative product on $A\otimes B$; the associative algebra $(A\otimes B, \mu _R)$ is denoted by $A\otimes _RB$ and called the *twisted tensor product* of $A$ and $B$ afforded by $R$.
We introduce now twisted tensor products of BiHom-associative algebras.
Let $(A, \mu _A, \alpha _A, \beta _A)$ and $(B, \mu _B, \alpha _B, \beta _B)$ be two BiHom-associative algebras such that the maps $\alpha _A$, $\beta _A$, $\alpha _B$, $\beta _B$ are bijective. A linear map $R\colon B\otimes A \rightarrow
A\otimes B$ is called a *BiHom-twisting map* between $A$ and $B$ if the following conditions are satisfied $$\begin{gathered}
(\alpha _A\otimes \alpha _B)\circ R=R\circ (\alpha _B\otimes \alpha _A),
\label{ghomtwmap01} \\
(\beta _A\otimes \beta _B)\circ R=R\circ (\beta _B\otimes \beta _A),
\label{ghomtwmap02} \\
R\circ (\alpha _B\otimes \mu _A)=(\mu _A\otimes \beta _B)\circ
(\operatorname{id}_A\otimes R)\circ \big(\operatorname{id}_A\otimes \alpha _B\beta _B^{-1} \otimes \operatorname{id}_A\big)\circ
(R\otimes \operatorname{id}_A), \label{ghomtwmap1} \\
R\circ (\mu _B\otimes \beta _A)=(\alpha _A\otimes \mu _B)\circ (R\otimes
\operatorname{id}_B)\circ \big(\operatorname{id}_B\otimes \alpha _A^{-1}\beta _A \otimes \operatorname{id}_B\big) \circ
(\operatorname{id}_B\otimes R). \label{ghomtwmap2}\end{gathered}$$
If we use the standard Sweedler-type notation $R(b\otimes a)=a_R\otimes
b_R=a_r\otimes b_r$, for $a\in A$, $b\in B$, then the above conditions may be rewritten (for all $a, a^{\prime }\in A$ and $b, b^{\prime }\in B$) as follows $$\begin{gathered}
\alpha _A(a_R)\otimes \alpha _B(b_R)=\alpha _A(a)_R\otimes \alpha _B(b)_R,
\label{ghomsweed01} \\
\beta _A(a_R)\otimes \beta _B(b_R)=\beta _A(a)_R\otimes \beta _B(b)_R,
\label{ghomsweed02} \\
(aa^{\prime })_R\otimes \alpha _B(b)_R=a_Ra^{\prime }_r\otimes \beta
_B\big(\big[\alpha _B\beta _B^{-1}(b_R)\big]_r\big), \label{ghomsweed1} \\
\beta _A(a)_R\otimes (bb^{\prime })_R=\alpha _A([\alpha _A^{-1}\beta
_A(a_R)]_r)\otimes b_rb^{\prime }_R. \label{ghomsweed2}\end{gathered}$$
\[ghomttp\] Let $(A, \mu _A, \alpha _A, \beta _A)$ and $(B, \mu _B,
\alpha _B, \beta _B)$ be two BiHom-associative algebras with bijective structure maps, $R\colon B\otimes A \rightarrow A\otimes B$ a BiHom-twisting map. Define the linear map $$\begin{gathered}
T\colon \ (A\otimes B)\otimes (A\otimes B)\rightarrow (A\otimes B)\otimes
(A\otimes B), \\
T((a\otimes b)\otimes (a^{\prime }\otimes b^{\prime }))= (a\otimes
b_R)\otimes (a^{\prime }_R\otimes b^{\prime }).\end{gathered}$$ Then $T$ is a BiHom-pseudotwistor for the tensor product $(A\otimes B, \mu
_{A\otimes B}, \alpha _A\otimes \alpha _B, \beta _A\otimes \beta _B)$ of $A$ and $B$, with companions $$\begin{gathered}
\tilde{T}_1=\big(\operatorname{id}_A\otimes \alpha _B^{-1}\beta _B\otimes \operatorname{id}_A\otimes
\operatorname{id}_B\otimes \operatorname{id}_A\otimes \operatorname{id}_B\big)\circ T_{13} \\
\hphantom{\tilde{T}_1=}{} \circ (\operatorname{id}_A\otimes \alpha
_B\beta _B^{-1}\otimes \operatorname{id}_A\otimes \operatorname{id}_B\otimes \operatorname{id}_A\otimes \operatorname{id}_B), \\
\tilde{T}_2=\big(\operatorname{id}_A\otimes \operatorname{id}_B\otimes \operatorname{id}_A\otimes \operatorname{id}_B\otimes \alpha
_A\beta _A^{-1}\otimes \operatorname{id}_B\big)\circ T_{13} \\
\hphantom{\tilde{T}_2=}{}
\circ (\operatorname{id}_A\otimes \operatorname{id}_B\otimes
\operatorname{id}_A\otimes \operatorname{id}_B\otimes \alpha _A^{-1}\beta _A\otimes \operatorname{id}_B),\end{gathered}$$ where we use the standard notation for $T_{13}$. The BiHom-associative algebra $(A\otimes B)^T$ is denoted by $A\otimes _RB$ and is called the *BiHom-twisted tensor product* of $A$ and $B$; its multiplication is defined by $(a\otimes b)(a^{\prime }\otimes b^{\prime })=aa^{\prime
}_R\otimes b_Rb^{\prime }$, and the structure maps are $\alpha _A\otimes
\alpha _B$ and $\beta _A\otimes \beta _B$.
We begin by proving the following relation, for all $a\in A$, $b\in B$: $$\begin{gathered}
\alpha _B^{-1}\beta _B\big(\big[\alpha _B\beta _B^{-1}(b)\big]_R\big)\otimes a_R=
b_R\otimes \alpha _A\beta _A^{-1}\big(\big[\alpha _A^{-1}\beta _A(a)\big]_R\big).\label{helpful}\end{gathered}$$ This relation is equivalent to $$\begin{gathered}
\beta _B\big(\big[\alpha _B\beta _B^{-1}(b)\big]_R\big)\otimes \beta _A(a_R)= \alpha
_B(b_R)\otimes \alpha _A\big(\big[\alpha _A^{-1}\beta _A(a)\big]_R\big),\end{gathered}$$ which, by using (\[ghomsweed01\]) and (\[ghomsweed02\]), is equivalent to $$\begin{gathered}
\alpha _B(b)_R\otimes \beta _A(a)_R=\alpha _B(b)_R\otimes \beta _A(a)_R,\end{gathered}$$ which is obviously true.
We need to prove the relations (\[ghommultT1\])–(\[ghompstw3\]) (with $\tilde{\alpha }=\alpha _A\otimes \alpha _B$, $\tilde{\beta }=\beta _A\otimes
\beta _B$, $\alpha =\beta =\operatorname{id}_A\otimes \operatorname{id}_B$). We will prove only (\[ghompstw3\]), while (\[ghommultT1\])–(\[ghompstw2\]) are very easy and left to the reader. We compute ($r$ and $\mathcal{R}$ are two more copies of $R$) $$\begin{gathered}
\tilde{T}_1\circ (T\otimes \operatorname{id})\circ (\operatorname{id}\otimes T)(a\otimes
b\otimes a^{\prime }\otimes b^{\prime }\otimes a^{\prime \prime }\otimes
b^{\prime \prime }) =\tilde{T}_1(a\otimes b_r\otimes a^{\prime }_r\otimes b^{\prime }_R\otimes
a^{\prime \prime }_R\otimes b^{\prime \prime }) \\
\qquad {} = a\otimes \alpha _B^{-1}\beta _B\big(\big[\alpha _B\beta _B^{-1}(b_r)\big]_{\mathcal{R}}\big) \otimes a^{\prime }_r\otimes b^{\prime }_R\otimes (a^{\prime \prime }_R)_{\mathcal{R}}\otimes b^{\prime \prime },
\\
\tilde{T}_2\circ (\operatorname{id}\otimes T)\circ (T\otimes \operatorname{id})(a\otimes
b\otimes a^{\prime }\otimes b^{\prime }\otimes a^{\prime \prime }\otimes
b^{\prime \prime })
= \tilde{T}_2(a\otimes b_r\otimes a^{\prime }_r\otimes b^{\prime }_R\otimes
a^{\prime \prime }_R\otimes b^{\prime \prime }) \\
\qquad{} = a\otimes (b_r)_{\mathcal{R}}\otimes a^{\prime }_r\otimes b^{\prime
}_R\otimes \alpha _A \beta _A^{-1}\big(\big[\alpha _A^{-1}\beta _A(a^{\prime \prime
}_R)\big]_{\mathcal{R}}\big)\otimes b^{\prime \prime },\end{gathered}$$ and the two terms are equal because of the relation (\[helpful\]).
Let $(A, \mu _A, \alpha _A, \beta _A)$ and $(B, \mu _B, \alpha _B, \beta _B)$ be two BiHom-associative algebras with bijective structure maps. Then obviously the linear map $R\colon B\otimes A \rightarrow A\otimes B$, $R(b\otimes
a)=a\otimes b$, is a BiHom-twisting map and the BiHom-twisted tensor product $A\otimes _RB$ coincides with the ordinary tensor product $A\otimes B$.
\[gendefttp\] Let $(A, \mu _A)$ and $(B, \mu _B)$ be two associative algebras, $\alpha _A, \beta _A\colon A\rightarrow A$ two commuting algebra isomorphisms of $A$ and $\alpha _B, \beta _B\colon B\rightarrow B$ two commuting algebra isomorphisms of $B$. Let $P\colon B\otimes A\rightarrow A\otimes B$ be a twisting map satisfying the conditions $$\begin{gathered}
(\alpha _A\otimes \alpha _B)\circ P=P\circ (\alpha _B\otimes \alpha _A),
\label{cond1P} \\
(\beta _A\otimes \beta _B)\circ P=P\circ (\beta _B\otimes \beta _A).
\label{cond2P}\end{gathered}$$ Define the linear map $$\begin{gathered}
U\colon \ B\otimes A\rightarrow A\otimes B, \qquad U(b\otimes a)=\beta
_A^{-1}(\beta _A(a)_P)\otimes \alpha _B^{-1}(\alpha _B(b)_P).\end{gathered}$$ Then $U$ is a BiHom-twisting map between the BiHom-associative algebras $A_{(\alpha _A, \beta _A)}$ and $B_{(\alpha _B, \beta _B)}$ and the BiHom-associative algebras $A_{(\alpha _A, \beta _A)}\otimes _UB_{(\alpha
_B, \beta _B)}$ and $(A\otimes _PB)_{(\alpha _A\otimes \alpha _B, \beta
_A\otimes \beta _B)}$ coincide.
We only prove (\[ghomsweed1\]) for $U$ and leave the rest to the reader. We compute (by denoting by $p$ another copy of $P$ and by $u$ another copy of $U$) $$\begin{gathered}
(aa^{\prime })_U\otimes \alpha _B(b)_U = [\alpha _A(a)\beta _A(a^{\prime
})]_U\otimes \alpha _B(b)_U
= \beta _A^{-1}\big(\big[\beta _A\alpha _A(a)\beta _A^2(a^{\prime })\big]_P\big)\otimes
\alpha _B^{-1}(\alpha _B^2(b)_P) \\
\hphantom{(aa^{\prime })_U\otimes \alpha _B(b)_U}{}
\overset{\eqref{twmap1}}{=} \beta _A^{-1}(\beta _A\alpha _A(a)_P)\beta
_A^{-1}\big(\beta _A^2(a^{\prime })_p\big) \otimes \alpha _B^{-1}((\alpha
_B^2(b)_P)_p) \\
\hphantom{(aa^{\prime })_U\otimes \alpha _B(b)_U}{} = \beta _A^{-1}(\alpha _A(\beta _A(a))_P)\beta _A^{-1}\big(\beta _A^2(a^{\prime
})_p\big) \otimes \alpha _B^{-1}([\alpha _B(\alpha _B(b))_P]_p) \\
\hphantom{(aa^{\prime })_U\otimes \alpha _B(b)_U}{} \overset{\eqref{cond1P}}{=} \beta _A^{-1}\alpha _A(\beta _A(a)_P)\beta
_A^{-1}\big(\beta _A^2(a^{\prime })_p\big) \otimes \alpha _B^{-1}(\alpha _B(\alpha
_B(b)_P)_p),
\\
a_Ua^{\prime }_u\otimes \beta _B\big(\big[\alpha _B\beta _B^{-1}(b_U)\big]_u\big) = \alpha
_A(a_U)\beta _A(a^{\prime }_u)\otimes \beta _B\big(\big[\alpha _B\beta
_B^{-1}(b_U)\big]_u\big) \\
\hphantom{a_Ua^{\prime }_u\otimes \beta _B\big(\big[\alpha _B\beta _B^{-1}(b_U)\big]_u\big)}{}
= \alpha _A\beta _A^{-1}(\beta _A(a)_P)\beta _A(a^{\prime }_u)\otimes \beta
_B\big(\big[\beta _B^{-1}(\alpha _B(b)_P)\big]_u\big) \\
\hphantom{a_Ua^{\prime }_u\otimes \beta _B\big(\big[\alpha _B\beta _B^{-1}(b_U)\big]_u\big)}{}
= \alpha _A\beta _A^{-1}(\beta _A(a)_P)\beta _A(a^{\prime })_p\otimes
\alpha _B^{-1}\beta _B\big(\big[\alpha _B\beta _B^{-1}(\alpha _B(b)_P)\big]_p\big) \\
\hphantom{a_Ua^{\prime }_u\otimes \beta _B\big(\big[\alpha _B\beta _B^{-1}(b_U)\big]_u\big)}{}
= \alpha _A\beta _A^{-1}(\beta _A(a)_P)\beta _A(a^{\prime })_p\otimes
\alpha _B^{-1}\beta _B\big(\beta _B^{-1}(\alpha _B(\alpha _B(b)_P))_p\big) \\
\hphantom{a_Ua^{\prime }_u\otimes \beta _B\big(\big[\alpha _B\beta _B^{-1}(b_U)\big]_u\big)}{}
= \alpha _A\beta _A^{-1}(\beta _A(a)_P)\beta _A^{-1}\big(\beta _A^2(a^{\prime
})\big)_p\otimes \alpha _B^{-1}\beta _B\big(\beta _B^{-1}(\alpha _B(\alpha
_B(b)_P))_p\big) \\
\hphantom{a_Ua^{\prime }_u\otimes \beta _B\big(\big[\alpha _B\beta _B^{-1}(b_U)\big]_u\big)}{}
\overset{\eqref{cond2P}}{=} \alpha _A\beta _A^{-1}(\beta _A(a)_P)\beta
_A^{-1}\big(\beta _A^2(a^{\prime })_p\big) \otimes \alpha _B^{-1}(\alpha _B(\alpha
_B(b)_P)_p),\end{gathered}$$ finishing the proof.
BiHom-smash products
====================
We construct first a large family of BiHom-twisting maps.
\[BihomTwitMapR\] Let $(H, \mu _H, \Delta _H, \alpha _H, \beta _H, \psi
_H, \omega _H)$ be a BiHom-bialgebra, $(A, \mu _A, \alpha _A, \beta _A)$ a left $H$-module BiHom-algebra, with action denoted by $H\otimes A\rightarrow
A$, $h\otimes a\mapsto h\cdot a$, and assume that all structure maps $\alpha
_H, \beta _H, \psi _H, \omega _H, \alpha _A, \beta _A$ are bijective. Let $m, n, p\in \mathbb{Z}$. Define the linear map $$\begin{gathered}
R_{m, n, p}\colon \ H\otimes A\rightarrow A\otimes H, \qquad R_{m, n, p}(h\otimes
a)= \alpha _H^m\beta _H^n\omega _H^p(h_1) \cdot \beta _A^{-1}(a)\otimes \psi
_H^{-1}(h_2).\end{gathered}$$ Then $R_{m, n, p}$ is a BiHom-twisting map between $A$ and $H$.
The relations (\[ghomtwmap01\]) and (\[ghomtwmap02\]) are very easy to prove and left to the reader.
Proof of (\[ghomtwmap1\]): $$\begin{gathered}
(\mu _A\otimes \beta _H)\circ (\operatorname{id}_A\otimes R_{m, n, p})\circ
\big(\operatorname{id}_A\otimes \alpha _H\beta _H^{-1}\otimes \operatorname{id}_A\big) \circ (R_{m, n, p}\otimes
\operatorname{id}_A)(h\otimes a\otimes a^{\prime })\\
\qquad{} =(\mu _A\otimes \beta _H)\circ (\operatorname{id}_A\otimes R_{m, n, p})\big(\alpha _H^m\beta
_H^n\omega _H^p(h_1)\cdot \beta _A^{-1}(a)\otimes \alpha _H\beta _H^{-1}\psi
_H^{-1}(h_2)\otimes a^{\prime }\big) \\
\qquad{} = (\mu _A\otimes \beta _H)\big(\alpha _H^m\beta _H^n\omega _H^p(h_1)\cdot \beta
_A^{-1}(a)\otimes \alpha _H^m\beta _H^n\omega _H^p\big(\big[\alpha _H\beta
_H^{-1}\psi _H^{-1}(h_2)\big]_1\big)\cdot \beta _A^{-1}(a^{\prime }) \\
\qquad\quad {} \otimes \psi _H^{-1}\big(\big[\alpha _H\beta _H^{-1}\psi _H^{-1}(h_2)\big]_2\big)\big) \\
\qquad{} = (\mu _A\otimes \beta _H)\big(\alpha _H^m\beta _H^n\omega _H^p(h_1)\cdot \beta
_A^{-1}(a)\otimes \alpha _H^{m+1}\beta _H^{n-1}\omega _H^p\psi
_H^{-1}((h_2)_1)\cdot \beta _A^{-1}(a^{\prime }) \\
\qquad\quad{} \otimes \alpha _H\beta _H^{-1}\psi _H^{-2}((h_2)_2)\big) \\
\qquad{} = \big[\alpha _H^m\beta _H^n\omega _H^p(h_1)\cdot \beta _A^{-1}(a)\big] \big[\alpha
_H^{m+1}\beta _H^{n-1}\psi _H^{-1}\omega _H^p((h_2)_1)\cdot \beta
_A^{-1}(a^{\prime })\big]\otimes \alpha _H\psi _H^{-2}((h_2)_2) \\
\qquad{} \overset{(\ref{ghombia1})}{=} \big[\alpha _H^m\beta _H^n\omega
_H^{p-1}((h_1)_1) \cdot \beta _A^{-1}(a)\big] \big[\alpha _H^{m+1}\beta _H^{n-1}\psi
_H^{-1}\omega _H^p((h_1)_2)\cdot \beta _A^{-1}(a^{\prime })\big]\otimes \alpha
_H\psi _H^{-1}(h_2) \\
\qquad{} = \big[\alpha_H^{-1}\omega _H^{-1}\big(\alpha _H^{m+1}\beta _H^n\omega
_H^{p}((h_1)_1)\big) \cdot \beta _A^{-1}(a)\big] \big[\beta _H^{-1}\psi _H^{-1}\big(\alpha
_H^{m+1}\beta _H^{n}\omega _H^p((h_1)_2)\big)\cdot \beta _A^{-1}(a^{\prime })\big] \\
\qquad\quad{} \otimes \alpha _H\psi _H^{-1}(h_2) \\
\qquad{} = \big\{\alpha_H^{-1}\omega _H^{-1}\big(\big[\alpha _H^{m+1}\beta _H^n\omega
_H^{p}(h_1)\big]_1\big) \cdot \beta _A^{-1}(a)\big\} \big\{\beta _H^{-1}\psi _H^{-1}\big(\big[\alpha
_H^{m+1}\beta _H^{n}\omega _H^p(h_1)\big]_2\big)\cdot \beta _A^{-1}(a^{\prime })\big\} \\
\qquad\quad{}\otimes \alpha _H\psi _H^{-1}(h_2) \\
\qquad{} \overset{(\ref{gmodalgcompat})}{=} \alpha _H^{m+1}\beta _H^n\omega
_H^{p}(h_1)\cdot \beta _A^{-1}(aa^{\prime })\otimes \alpha _H\psi
_H^{-1}(h_2) = (R_{m, n, p}\circ (\alpha _H\otimes \mu _A))(h\otimes a\otimes a^{\prime}).\end{gathered}$$
Proof of (\[ghomtwmap2\]): $$\begin{gathered}
(\alpha _A\otimes \mu _H)\circ (R_{m, n, p}\otimes \operatorname{id}_H)\circ
\big(\operatorname{id}_H\otimes \alpha _A^{-1}\beta _A \otimes \operatorname{id}_H\big)\circ (\operatorname{id}_H\otimes R_{m, n,
p})(h\otimes h^{\prime }\otimes a)\\
\qquad{}
= (\alpha _A\otimes \mu _H)\circ (R_{m, n, p}\otimes \operatorname{id}_H)(h\otimes \alpha
_A^{-1}\beta _A \big(\alpha _H^m\beta _H^n\omega _H^p(h^{\prime }_1)\cdot \beta
_A^{-1}(a)\big)\otimes \psi _H^{-1}(h^{\prime }_2)) \\
\qquad {} = (\alpha _A\otimes \mu _H)\circ (R_{m, n, p}\otimes \operatorname{id}_H)\big(h\otimes \alpha
_H^{m-1}\beta _H^{n+1}\omega _H^p(h^{\prime }_1)\cdot \alpha
_A^{-1}(a)\otimes \psi _H^{-1}(h^{\prime }_2)\big) \\
\qquad {}
= (\alpha _A\otimes \mu _H)\big(\alpha _H^m\beta _H^n\omega _H^p(h_1)\cdot
\big(\alpha _H^{m-1}\beta _H^{n} \omega _H^p(h^{\prime }_1)\cdot \alpha
_A^{-1}\beta _A^{-1}(a)\big)\otimes \psi _H^{-1}(h_2)\otimes \psi
_H^{-1}(h^{\prime }_2)\big) \\
\qquad {}
= \alpha _H^{m+1}\beta _H^n\omega _H^p(h_1)\cdot \big(\alpha _H^{m}\beta _H^{n}
\omega _H^p(h^{\prime }_1)\cdot \beta _A^{-1}(a)\big)\otimes \psi
_H^{-1}(h_2h^{\prime }_2) \\
\qquad{} \overset{(\ref{ghommod2})}{=} \big\{\big[\alpha _H^{m}\beta _H^n\omega
_H^p(h_1)\big]\big[\alpha _H^{m}\beta _H^{n} \omega _H^p(h^{\prime }_1)\big]\big\}\cdot
a\otimes \psi _H^{-1}(h_2h^{\prime }_2) \\
\qquad{} = \alpha _H^{m}\beta _H^n\omega _H^p(h_1h^{\prime }_1)\cdot a\otimes \psi
_H^{-1}(h_2h^{\prime }_2) \\
\qquad{} \overset{(\ref{ghombia2})}{=} \alpha _H^{m}\beta _H^n\omega
_H^p((hh^{\prime })_1)\cdot a\otimes \psi _H^{-1}((hh^{\prime })_2)
= (R_{m, n, p}\circ (\mu _H\otimes \beta _A))(h\otimes h^{\prime }\otimes
a),\end{gathered}$$ finishing the proof.
Let $(H, \mu _H, \Delta _H, \alpha _H, \beta _H, \psi _H, \omega _H)$ be a BiHom-bialgebra and $(A, \mu _A, \alpha _A, \beta _A)$ a left $H$-module BiHom-algebra, with left $H$-module structure $H\otimes A\rightarrow A$, $h\otimes a\mapsto h\cdot a$, such that all structure maps $\alpha _H$, $\beta
_H$, $\psi _H$, $\omega _H$, $\alpha _A$, $\beta _A$ are bijective. Consider the BiHom-twisting map $$\begin{gathered}
R=R_{0, -1, -1}\colon \ H\otimes A\rightarrow A\otimes H, \qquad R(h\otimes
a)=\beta _H^{-1}\omega _H^{-1}(h_1) \cdot \beta _A^{-1}(a)\otimes \psi
_H^{-1}(h_2). \label{Rsmash}\end{gathered}$$ We denote the BiHom-associative algebra $A\otimes _RH$ by $A\# H$ (we denote $a\otimes h:=a\# h$, for $a\in A$, $h\in H$) and call it the *BiHom-smash product* of $A$ and $H$. Its structure maps are $\alpha
_A\otimes \alpha _H$ and $\beta _A\otimes \beta _H$, and its multiplication is $$\begin{gathered}
(a\# h)(a^{\prime }\# h^{\prime })=a\big(\beta _H^{-1}\omega _H^{-1}(h_1)
\cdot \beta _A^{-1}(a^{\prime })\big)\# \psi _H^{-1}(h_2)h^{\prime }.\end{gathered}$$
If $H$ is a Hom-bialgebra, i.e., $\alpha _H=\beta _H=\psi _H=\omega _H$, and $A$ is a Hom-associative algebra, the multiplication of $A\# H$ becomes $$\begin{gathered}
(a\# h)(a^{\prime }\# h^{\prime })=a\big(\alpha _H^{-2}(h_1) \cdot \alpha
_A^{-1}(a^{\prime })\big)\# \alpha_H^{-1}(h_2)h^{\prime },\end{gathered}$$ which is the formula introduced in [@mp2]. If $H$ is a monoidal Hom-bialgebra, i.e., $\psi _H=\omega _H=\alpha _H^{-1}=\beta _H^{-1}$, and $A$ is a Hom-associative algebra, the multiplication of $A\# H$ becomes $$\begin{gathered}
(a\# h)(a^{\prime }\# h^{\prime })=a\big(h_1 \cdot \alpha _A^{-1}(a^{\prime
})\big)\# \alpha_H(h_2)h^{\prime },\end{gathered}$$ which is the formula introduced in [@chenwangzhang], used also in [@LB] for defining the Radford biproduct for monoidal Hom-bialgebras.
In the same setting as in Proposition [\[gyaumodalg\]]{}, and assuming moreover that the maps $\alpha _A$ and $\beta _A$ are bijective, if we denote by $A\# H$ the usual smash product between $A$ and $H$, then $\alpha
_A\otimes \alpha _H$ and $\beta _A\otimes \beta _H$ are commuting algebra endomorphisms of $A\# H$ and the BiHom-associative algebras $(A\#
H)_{(\alpha _A\otimes \alpha _H, \beta _A\otimes \beta _H)}$ and $A_{(\alpha
_A, \beta _A)}\# H_{(\alpha _H, \beta _H, \psi _H, \omega _H)}$ coincide.
We will apply Proposition \[gendefttp\]. In our situation, we have the twisting map $P\colon H\otimes A \rightarrow A\otimes H$, $P(h\otimes a)=h_1\cdot
a\otimes h_2$, for which $A\# H=A\otimes _PH$. Obviously $P$ satisfies the conditions (\[cond1P\]) and (\[cond2P\]), so, by Proposition \[gendefttp\], we obtain the map $$\begin{gathered}
U\colon \ H\otimes A\rightarrow A\otimes H, \qquad U(h\otimes a)=\beta
_A^{-1}(\beta _A(a)_P)\otimes \alpha _H^{-1}(\alpha _H(h)_P),\end{gathered}$$ which is a BiHom-twisting map between $A_{(\alpha _A, \beta _A)}$ and $H_{(\alpha _H, \beta _H)}$ and we have $$\begin{gathered}
(A\# H)_{(\alpha _A\otimes \alpha _H, \beta _A\otimes \beta _H)}=
A_{(\alpha _A, \beta _A)}\otimes _UH_{(\alpha _H, \beta _H)}.\end{gathered}$$ Thus, the proof will be finished if we prove that the map $U$ coincides with the map $R$ affording the BiHom-smash product $A_{(\alpha _A, \beta _A)}\#
H_{(\alpha _H, \beta _H, \psi _H, \omega _H)}$. We compute $$\begin{gathered}
U(h\otimes a) = \beta _A^{-1}(\alpha _H(h)_1\cdot \beta _A(a))\otimes \alpha
_H^{-1}(\alpha _H(h)_2) \\
\hphantom{U(h\otimes a)}{}
=\beta _A^{-1}(\alpha _H(h_1)\cdot \beta _A(a))\otimes \alpha
_H^{-1}(\alpha _H(h_2))
=\alpha _H\beta _H^{-1}(h_1)\cdot a\otimes h_2,
\\
R(h\otimes a) = \beta _H^{-1}\omega _H^{-1}(\omega _H(h_1))\triangleright
\beta _A^{-1}(a) \otimes \psi _H^{-1}(\psi _H(h_2)) \\
\hphantom{R(h\otimes a)}{}
=\beta _H^{-1}(h_1)\triangleright \beta _A^{-1}(a)\otimes h_2
=\alpha _H\beta _H^{-1}(h_1)\cdot a\otimes h_2,\end{gathered}$$ finishing the proof.
We construct a class of examples of $U_{q}(\mathfrak{sl}_{2})_{(\alpha
,\beta ,\psi ,\omega )}$-module BiHom-algebra structures on $\mathbb{A}_{q,\alpha ,\beta }^{2|0}$, generalizing examples of $U_{q}(\mathfrak{sl}_{2})_{\alpha }$-module Hom-algebra structures on $\mathbb{A}_{q,\gamma
}^{2|0}$ given in [@homquantum3 Example 5.7] (here we take the base field $\Bbbk =\mathbb{C}$). The quantum group $U_{q}(\mathfrak{sl}_{2})$ is generated as a unital associative algebra by 4 generators $\{E,F,K,K^{-1}\}$ with relations $$\begin{gathered}
KK^{-1}=1=K^{-1}K, \qquad
KE=q^{2}EK,\qquad KF=q^{-2}FK, \qquad
EF-FE=\frac{K-K^{-1}}{q-q^{-1}},\end{gathered}$$ where $q\in \mathbb{C}$ with $q\neq 0$, $q\neq \pm 1$. The comultiplication is defined by $$\begin{gathered}
\Delta (E)=1\otimes E+E\otimes K, \qquad
\Delta (F)=K^{-1}\otimes F+F\otimes 1, \\
\Delta (K)=K\otimes K,\qquad \Delta \big(K^{-1}\big)=K^{-1}\otimes K^{-1}.\end{gathered}$$
We fix $\lambda_1,\lambda_2,\lambda_3,\lambda_4 \in \mathbb{C}$ some nonzero elements. The BiHom-bialgebra $U_q(\mathfrak{sl}_2)_{(\alpha,\beta,\psi,\omega )}= (U_q(\mathfrak{sl}_2),\mu_{(\alpha,\beta )},\Delta_{(\psi,\omega
)},\alpha,\beta,\psi,\omega)$ is defined (as in Proposition \[yautwistdiverse\](iii)) by $\mu_{(\alpha,\beta )}=\mu\circ (\alpha\otimes
\beta)$ and $\Delta_{(\psi,\omega )} =(\omega\otimes \psi)\circ\Delta$, where $\mu$ and $\Delta$ are respectively the multiplication and comultiplication of $U_q(\mathfrak{sl}_2)$ and $\alpha,\beta,\psi,\omega\colon U_q(\mathfrak{sl}_2)\rightarrow U_q(\mathfrak{sl}_2)$ are bialgebra morphisms such that $$\begin{gathered}
\alpha (E)=\lambda_1 E, \qquad \alpha (F)=\lambda_1^{-1} F, \qquad \alpha (K)=K, \qquad
\alpha \big(K^{-1}\big)=K^{-1}, \\
\beta (E)=\lambda_2 E, \qquad \beta (F)=\lambda_2^{-1} F, \qquad \beta (K)=K, \qquad
\beta \big(K^{-1}\big)=K^{-1}, \\
\psi (E)=\lambda_3 E, \qquad \psi (F)=\lambda_3^{-1} F, \qquad \psi (K)=K, \qquad \psi
\big(K^{-1}\big)=K^{-1}, \\
\omega (E)=\lambda_4 E, \qquad \omega (F)=\lambda_4^{-1} F, \qquad \omega (K)=K, \qquad
\omega \big(K^{-1}\big)=K^{-1}.\end{gathered}$$ Note that any two of the maps $\alpha$, $\beta$, $\psi$, $\omega$ commute.
Let $\mathbb{A}_{q}^{2|0}=k\langle x,y\rangle /(yx-q xy)$ be the quantum plane. We fix also some $\xi \in \mathbb{C}$, $\xi \neq 0$. The BiHom-quantum plane $\mathbb{A}_{q,\alpha,\beta}^{2|0}=(\mathbb{A}_{q}^{2|0},\mu_{\mathbb{A},\alpha_\mathbb{A},\beta_\mathbb{A}},\alpha_\mathbb{A},\beta_\mathbb{A})$ is the BiHom-associative algebra defined (as in Proposition \[yautwistdiverse\](i)) by $\mu_{\mathbb{A},\alpha_\mathbb{A},\beta_\mathbb{A}}=\mu_{\mathbb{A}}\circ(\alpha_\mathbb{A}\otimes\beta_\mathbb{A})$, where $\mu_\mathbb{A}$ is the multiplication of $\mathbb{A}_{q}^{2|0}$ and $\alpha_\mathbb{A},\beta_\mathbb{A}\colon \mathbb{A}_{q}^{2|0}\rightarrow \mathbb{A}_{q}^{2|0} $ are the (commuting) algebra morphisms such that $$\begin{gathered}
\alpha_\mathbb{A}(x)=\xi x,\qquad \alpha_\mathbb{A}(y)= \xi \lambda_1^{-1} y
\qquad \text{and} \qquad \beta_\mathbb{A}(x)=\xi x,\qquad \beta_\mathbb{A} (y)= \xi
\lambda_2^{-1} y.\end{gathered}$$
We consider $\mathbb{A}_{q}^{2|0}$ as a left $U_q(\mathfrak{sl}_2)$-module algebra as in [@homquantum3 Example 5.7] (we denote by $h\otimes
a\mapsto h\cdot a$ the $U_q(\mathfrak{sl}_2)$-action on $\mathbb{A}_{q}^{2|0} $). By the computations performed in [@homquantum3 Example 5.7] we know that $\alpha_\mathbb{A}(h\cdot a)=\alpha(h)\cdot
\alpha_\mathbb{A}( a)$ and $\beta_\mathbb{A}(h\cdot a)=\beta(h)\cdot \beta_\mathbb{A}( a)$, for all $h\in U_q(\mathfrak{sl}_2)$ and $a\in \mathbb{A}_{q}^{2|0}$. Then, according to Proposition \[gyaumodalg\], there exists a $U_q(\mathfrak{sl}_2)_{(\alpha,\beta,\psi,\omega )}$-module BiHom-algebra structure on $\mathbb{A}_{q,\alpha , \beta}^{2|0}$ defined by $$\begin{gathered}
\rho \colon \ U_q(\mathfrak{sl}_2)_{(\alpha,\beta,\psi,\omega )}\otimes \mathbb{A}_{q,\alpha , \beta}^{2|0} \rightarrow \mathbb{A}_{q,\alpha , \beta}^{2|0},
\qquad \rho (h\otimes a)=h\triangleright a=\alpha (h)\cdot \beta_\mathbb{A}(a).\end{gathered}$$ By using also the computations performed in [@homquantum3 Example 5.7] one can see that the map $\rho $ is given on generators by $$\begin{gathered}
\rho \big(E\otimes x^my^n \big)=[n]_q\xi^{m+n}\lambda_1\lambda_2^{-n}
x^{m+1}y^{n-1}, \\
\rho \big(F \otimes x^my^n \big)=[m]_q\xi^{m+n}\lambda_1^{-1}\lambda_2^{-n}
x^{m-1}y^{n+1}, \\
\rho\big(K^{\pm 1}\otimes P \big)=P\big(q^{\pm 1}\xi x,q^{\mp 1} \xi\lambda_2^{-1}y\big),\end{gathered}$$ for any monomial $x^my^n\in \mathbb{A}_{q}^{2|0}$, where $P=P(x,y)\in
\mathbb{A}_{q}^{2|0}$ and $[n]_q=\frac{q^n-q^{-n}}{q-q^{-1}}$.
Since $\xi \neq 0$ and $\lambda _i\neq 0$ for all $i=1, 2, 3, 4$, all the maps $\alpha$, $\beta$, $\psi$, $\omega$, $\alpha_\mathbb{A}$, $\beta_\mathbb{A}$ are bijective. According to Theorem \[BihomTwitMapR\], the map $R\colon U_q(\mathfrak{sl}_2)_{(\alpha,\beta,\psi,\omega )}\otimes \mathbb{A}_{q,\alpha,\beta}^{2|0} \rightarrow \mathbb{A}_{q,\alpha,\beta}^{2|0}\otimes
U_q(\mathfrak{sl}_2)_{(\alpha,\beta,\psi,\omega )} $ defined by leads to the smash product $\mathbb{A}_{q,\alpha,\beta}^{2|0}\# U_q(\mathfrak{sl}_2)_{(\alpha,\beta,\psi,\omega )} $ whose multiplication is defined by $$\begin{gathered}
(a\# h)(a'\# h')=a*\big(\beta ^{-1}\omega ^{-1}(h_{(1)})
\triangleright \beta _\mathbb{A}^{-1}(a')\big)\#
\psi ^{-1}(h_{(2)})\bullet h',\end{gathered}$$ where $h_{(1)}\otimes h_{(2)}=\Delta_{(\psi,\omega )}(h)$ and $*$ (respectively $\bullet $) is the multiplication of $\mathbb{A}_{q,\alpha,\beta}^{2|0}$ (respectively $U_q(\mathfrak{sl}_2)_{(\alpha,\beta,\psi,\omega )}$).
In particular, for any $G\in U_q(\mathfrak{sl}_2)$ and $m,n,r,s\in \mathbb{N}
$ we have $$\begin{gathered}
(x^my^n\# K^{\pm 1})(x^ry^s\# G)=q^{\pm r\mp s+n
r}\xi^{m+n+r+s}\lambda_1^{-n}\lambda_2^{-s}x^{m+r}y^{n+s} \# K ^{\pm
1}\beta(G), \\
(x^my^n\# E)(x^ry^s\# G)=q^{n
r}\xi^{m+n+r+s}\lambda_1^{-n+1}\lambda_2^{-s}x^{m+r}y^{n+s}\# E \beta(G) \\
\hphantom{(x^my^n\# E)(x^ry^s\# G)=}{}
+[s]_q q^{n (r+1)} \xi ^{m+n+r+s}\lambda_1^{1-n}\lambda_2^{-s}
x^{m+r+1}y^{n+s-1}\# K \beta(G), \\
(x^my^n\# F)(x^ry^s\# G)=q^{s-r+n
r}\xi^{m+n+r+s}\lambda_1^{-n-1}\lambda_2^{-s}x^{m+r}y^{n+s}\# F \beta(G) \\
\hphantom{(x^my^n\# F)(x^ry^s\# G)=}{} +[r]_q q^{n (r-1)} \xi^{m+n+r+s}
\lambda_1^{-n-1}\lambda_2^{-s}x^{m+r-1}y^{n+s+1}\# \beta (G),\end{gathered}$$ where $K ^{\pm 1}\beta(G)$, $E \beta(G)$ and $F\beta(G)$ are multiplications in $U_q(\mathfrak{sl}_2)$.
We introduce now the BiHom analogue of comodule Hom-algebras defined in [@yau2].
Let $(H, \mu _H, \Delta _H, \alpha _H, \beta _H, \psi _H, \omega _H)$ be a BiHom-bialgebra. A *right $H$-comodule BiHom-algebra* is a 7-tuple $(D, \mu _D, \alpha _D, \beta _D, \psi _D, \omega _D, \rho _D)$, where $(D,
\mu _D, \alpha _D, \beta _D)$ is a BiHom-associative algebra, $(D, \psi _D,
\omega _D)$ is a right $H$-comodule via the coaction $\rho _D\colon D\rightarrow
D\otimes H$ and moreover $\rho _D$ is a morphism of BiHom-associative algebras.
If $(H, \mu _H, \Delta _H, \alpha _H, \beta _H, \psi _H, \omega _H)$ is a BiHom-bialgebra, then we have the right $H$-comodule BiHom-algebra $(H, \mu
_H, \alpha _H, \beta _H, \psi _H, \omega _H, \Delta _H)$.
The next result generalizes Proposition 3.6 in [@mp2].
\[smashcomalg\] Let $(H, \mu _H, \Delta _H, \alpha _H, \beta _H, \psi
_H, \omega _H)$ be a BiHom-bialgebra and $(A, \mu _A, \alpha _A, \beta _A)$ a left $H$-module BiHom-algebra, with notation $H\otimes A\rightarrow A$, $h\otimes a\mapsto h\cdot a$, such that all structure maps $\alpha _H$, $\beta
_H$, $\psi _H$, $\omega _H$, $\alpha _A$, $\beta _A$ are bijective. Assume that there exist two more linear maps $\psi _A, \omega _A\colon A\rightarrow A$ such that any two of the maps $\alpha _A$, $\beta _A$, $\psi _A$, $\omega _A$ commute and moreover $$\begin{gathered}
\omega _A(aa^{\prime })=\omega _A(a)\omega _A(a^{\prime }), \qquad \forall
\, a, a^{\prime }\in A, \nonumber\\
\omega _A(h\cdot a)=\omega _H(h)\cdot \omega _A(a), \qquad \forall \, a\in
A,\quad h\in H. \label{inplus}\end{gathered}$$ Define the linear map $$\begin{gathered}
\rho _{A\# H}\colon \ A\# H\rightarrow (A\# H)\otimes H, \qquad \rho _{A\# H}(a\#
h)=(\omega _A(a)\# h_1)\otimes h_2.\end{gathered}$$ Then $(A\# H, \mu _{A\# H}, \alpha _A\otimes \alpha _H, \beta _A\otimes
\beta _H, \psi _A\otimes \psi _H, \omega _A\otimes \omega _H, \rho _{A\# H})$ is a right $H$-comodule BiHom-algebra.
We only prove that $\rho _{A\# H}$ is multiplicative and leave the other details to the reader: $$\begin{gathered}
\rho _{A\# H}((a\# h)(a^{\prime }\# h^{\prime })) = \omega _A\big(a\big(\beta
_H^{-1}\omega _H^{-1}(h_1) \cdot \beta _A^{-1}(a^{\prime })\big)\big)\# \big(\psi
_H^{-1}(h_2)h^{\prime }\big)_1\otimes \big(\psi _H^{-1}(h_2)h^{\prime }\big)_2 \\
\hphantom{\rho _{A\# H}((a\# h)(a^{\prime }\# h^{\prime }))}{}
= \omega _A(a) \omega _A\big(\beta _H^{-1}\omega _H^{-1}(h_1) \cdot \beta
_A^{-1}(a^{\prime })\big)\# \psi _H^{-1}((h_2)_1)h^{\prime }_1\otimes \psi
_H^{-1}((h_2)_2)h^{\prime }_2 \\
\hphantom{\rho _{A\# H}((a\# h)(a^{\prime }\# h^{\prime }))}{}
\overset{\eqref{inplus}}{=} \omega _A(a)\big(\beta _H^{-1}(h_1)\cdot \omega _A
\beta _A^{-1}(a^{\prime })\big)\# \psi _H^{-1}((h_2)_1)h^{\prime }_1\otimes \psi
_H^{-1}((h_2)_2)h^{\prime }_2 \\
\hphantom{\rho _{A\# H}((a\# h)(a^{\prime }\# h^{\prime }))}{}
\overset{\eqref{ghombia1}}{=} \omega _A(a)\big(\beta _H^{-1}\omega
_H^{-1}((h_1)_1)\cdot \omega _A \beta _A^{-1}(a^{\prime })\big)\# \psi
_H^{-1}((h_1)_2)h^{\prime }_1\otimes h_2h^{\prime }_2 \\
\hphantom{\rho _{A\# H}((a\# h)(a^{\prime }\# h^{\prime }))}{}
=\omega _A(a)\big(\beta _H^{-1}\omega _H^{-1}((h_1)_1)\cdot \beta
_A^{-1}\omega _A(a^{\prime })\big)\# \psi _H^{-1}((h_1)_2)h^{\prime }_1\otimes
h_2h^{\prime }_2 \\
\hphantom{\rho _{A\# H}((a\# h)(a^{\prime }\# h^{\prime }))}{}
=(\omega _A(a)\# h_1)(\omega _A(a^{\prime })\# h^{\prime }_1)\otimes
h_2h^{\prime }_2
=\rho _{A\# H}(a\# h)\rho _{A\# H}(a^{\prime }\# h^{\prime }),\end{gathered}$$ finishing the proof.
Let $(H, \mu _H, \Delta _H, \alpha _H, \beta _H, \psi _H, \omega _H)$ be a BiHom-bialgebra such that all structure maps are bijective. Denote by $A$ the linear space $H^*$. Then $A$ becomes a BiHom-associative algebra with multiplication and structure maps defined by $$\begin{gathered}
\begin{split}
& (f\bullet g)(h)=f\big(\alpha _H^{-1}\omega _H^{-1}(h_1)\big)g\big(\beta _H^{-1}\psi
_H^{-1}(h_2)\big), \\
& \alpha _A\colon \ H^*\rightarrow H^*, \qquad \alpha _A(f)(h)=f\big(\alpha _H^{-1}(h)\big),
\\
& \beta _A\colon \ H^*\rightarrow H^*, \qquad \beta _A(f)(h)=f\big(\beta _H^{-1}(h)\big),
\end{split}\end{gathered}$$ for all $f, g\in H^*$ and $h\in H$. Moreover, A becomes a left $H$-module BiHom-algebra, with action $$\begin{gathered}
\rightharpoonup \colon \ H\otimes H^*\rightarrow H^*,\qquad (h\rightharpoonup
f)(h^{\prime })=f(\alpha _H^{-1} \beta _H^{-1}(h^{\prime })h),\end{gathered}$$ for all $h, h^{\prime }\in H$ and $f\in H^*$. Obviously, $\alpha _A$ and $\beta _A$ are bijective maps. Define the linear map $$\begin{gathered}
\omega _A\colon \ H^*\rightarrow H^*, \qquad \omega _A(f)(h)=f\big(\omega _H^{-1}(h)\big),
\qquad \forall \, f\in H^*, \quad h\in H,\end{gathered}$$ and choose a linear map $\psi _A\colon H^*\rightarrow H^*$ that commutes with $\alpha _A$, $\beta _A$, $\omega _A$, for instance one can choose the map $\psi
_A $ defined by $\psi _A(f)(h)=f(\psi _H^{-1}(h))$, for all $f\in H^*$ and $h\in H$. Then one can check that the hypotheses of Proposition \[smashcomalg\] are satisfied, and consequently $H^*\# H$ becomes a right $H$-comodule BiHom-algebra.
Note also that, if $H$ is counital with counit $\varepsilon _H$ such that $\varepsilon _H\circ \alpha _H=\varepsilon _H$ and $\varepsilon _H\circ \beta
_H=\varepsilon _H$, then the BiHom-associative algebra $A=H^*$ is unital with unit $\varepsilon _H$.
Acknowledgements {#acknowledgements .unnumbered}
----------------
This paper was written while Claudia Menini was a member of GNSAGA. Florin Panaite was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0635, contract nr. 253/5.10.2011. Parts of this paper have been written while Florin Panaite was a visiting professor at University of Ferrara in September 2014, supported by INdAM, and a visiting scholar at the Erwin Schrodinger Institute in Vienna in July 2014 in the framework of the “Combinatorics, Geometry and Physics” programme; he would like to thank both these institutions for their warm hospitality.
The authors are grateful to the referees for their remarks and questions.
[99]{}
Aizawa N., Sato H., [$q$]{}-deformation of the [V]{}irasoro algebra with central extension, [*Phys. Lett. B*](http://dx.doi.org/10.1016/0370-2693(91)90671-C) **256** (1991), 185–190.
Aragón Periñán M.J., Calderón Martín A.J., On graded matrix Hom-algebras, [*Electron. J. Linear Algebra*](http://dx.doi.org/10.13001/1081-3810.1579) **24** (2012), 45–65.
Aragón Periñán M.J., Calderón Martín A.J., Split regular Hom-Lie algebras, *J. Lie Theory* **25** (2015), 875–888.
Ardizzoni A., Bulacu D., Menini C., Quasi-bialgebra structures and torsion-free abelian groups, *Bull. Math. Soc. Sci. Math. Roumanie (N.S.)* **56** (2013), 247–265, [arXiv:1302.2453](http://arxiv.org/abs/1302.2453).
Ardizzoni A., Menini C., [M]{}ilnor–[M]{}oore categories and monadic decomposition, [arXiv:1401.2037](http://arxiv.org/abs/1401.2037).
Bakayoko I., [$L$]{}-modules, [$L$]{}-comodules and [H]{}om-[L]{}ie quasi-bialgebras, *Afr. Diaspora J. Math.* **17** (2014), 49–64.
Benayadi S., Makhlouf A., Hom-[L]{}ie algebras with symmetric invariant nondegenerate bilinear forms, [*J. Geom. Phys.*](http://dx.doi.org/10.1016/j.geomphys.2013.10.010) **76** (2014), 38–60, [arXiv:1009.4226](http://arxiv.org/abs/1009.4226).
Caenepeel S., Goyvaerts I., Monoidal [H]{}om-[H]{}opf algebras, [*Comm. Algebra*](http://dx.doi.org/10.1080/00927872.2010.490800) **39** (2011), 2216–2240, [arXiv:0907.0187](http://arxiv.org/abs/0907.0187).
Cap A., Schichl H., Van[ž]{}ura J., On twisted tensor products of algebras, [*Comm. Algebra*](http://dx.doi.org/10.1080/00927879508825496) **23** (1995), 4701–4735.
Chaichian M., Kulish P., Lukierski J., [$q$]{}-deformed [J]{}acobi identity, [$q$]{}-oscillators and [$q$]{}-deformed infinite-dimensional algebras, [*Phys. Lett. B*](http://dx.doi.org/10.1016/0370-2693(90)91196-I) **237** (1990), 401–406.
Chen Y., Wang Z., Zhang L., Integrals for monoidal [H]{}om-[H]{}opf algebras and their applications, [*J. Math. Phys.*](http://dx.doi.org/10.1063/1.4813447) **54** (2013), 073515, 22 pages.
Curtright T.L., Zachos C.K., Deforming maps for quantum algebras, [*Phys. Lett. B*](http://dx.doi.org/10.1016/0370-2693(90)90845-W) **243** (1990), 237–244.
Goyvaerts I., Vercruysse J., Lie monads and dualities, [*J. Algebra*](http://dx.doi.org/10.1016/j.jalgebra.2014.05.021) **414** (2014), 120–158, [arXiv:1302.6869](http://arxiv.org/abs/1302.6869).
Graziani G., Group [H]{}om-categories, Thesis, University of Ferrara, Italy, 2013.
Hartwig J.T., Larsson D., Silvestrov S.D., Deformations of [L]{}ie algebras using [$\sigma$]{}-derivations, [*J. Algebra*](http://dx.doi.org/10.1016/j.jalgebra.2005.07.036) **295** (2006), 314–361, [math.QA/0408064](http://arxiv.org/abs/math.QA/0408064).
Hassanzadeh M., Shapiro I., Sütlü S., Cyclic homology for Hom-associative algebras, [*J. Geom. Phys.*](http://dx.doi.org/10.1016/j.geomphys.2015.07.026) **98** (2015), 40–56, [arXiv:1504.03019](http://arxiv.org/abs/1504.03019).
Kassel C., Quantum groups, [*Graduate Texts in Mathematics*](http://dx.doi.org/10.1007/978-1-4612-0783-2), Vol. 155, Springer-Verlag, New York, 1995.
Larsson D., Silvestrov S., Quasi-hom-[L]{}ie algebras, central extensions and 2-cocycle-like identities, [*J. Algebra*](http://dx.doi.org/10.1016/j.jalgebra.2005.02.032) **288** (2005), 321–344, [math.RA/0408061](http://arxiv.org/abs/math.RA/0408061).
Liu K.Q., Characterizations of the quantum [W]{}itt algebra, [*Lett. Math. Phys.*](http://dx.doi.org/10.1007/BF00420485) **24** (1992), 257–265.
Liu L., Shen B., Radford’s biproducts and [Y]{}etter–[D]{}rinfeld modules for monoidal [H]{}om-[H]{}opf algebras, [*J. Math. Phys.*](http://dx.doi.org/10.1063/1.4866760) **55** (2014), 031701, 16 pages.
L[ó]{}pez Pe[ñ]{}a J., Panaite F., Van Oystaeyen F., General twisting of algebras, [*Adv. Math.*](http://dx.doi.org/10.1016/j.aim.2006.10.003) **212** (2007), 315–337, [math.QA/0605086](http://arxiv.org/abs/math.QA/0605086).
Makhlouf A., Panaite F., Yetter–[D]{}rinfeld modules for [H]{}om-bialgebras, [*J. Math. Phys.*](http://dx.doi.org/10.1063/1.4858875) **55** (2014), 013501, 17 pages, [arXiv:1310.8323](http://arxiv.org/abs/1310.8323).
Makhlouf A., Panaite F., Twisting operators, twisted tensor products and smash products for [H]{}om-associative algebras, [*Glasg. Math. J.*](http://dx.doi.org/10.1017/S0017089515000294), [to]{} appear, [arXiv:1402.1893](http://arxiv.org/abs/1402.1893).
Makhlouf A., Silvestrov S., Hom-algebra structures, [*J. Gen. Lie Theory Appl.*](http://dx.doi.org/10.4303/jglta/S070206) **2** (2008), 51–64, [math.RA/0609501](http://arxiv.org/abs/math.RA/0609501).
Makhlouf A., Silvestrov S., Hom-[L]{}ie admissible [H]{}om-coalgebras and [H]{}om-[H]{}opf algebras, in [Generalized [L]{}ie Theory in Mathematics, Physics and Beyond](http://dx.doi.org/10.1007/978-3-540-85332-9_17), Editors S. Silvestrov, E. Paal, V. Abramov, A. Stolin, Springer-Verlag, Berlin, 2009, 189–206, [arXiv:0709.2413](http://arxiv.org/abs/0709.2413).
Makhlouf A., Silvestrov S., Hom-algebras and [H]{}om-coalgebras, [*J. Algebra Appl.*](http://dx.doi.org/10.1142/S0219498810004117) **9** (2010), 553–589, [arXiv:0811.0400](http://arxiv.org/abs/0811.0400).
Sheng Y., Representations of Hom-[L]{}ie algebras, [*Algebr. Represent. Theory*](http://dx.doi.org/10.1007/s10468-011-9280-8) **15** (2012), 1081–1098, [arXiv:1005.0140](http://arxiv.org/abs/1005.0140).
Van Daele A., Van Keer S., The [Y]{}ang–[B]{}axter and pentagon equation, *Compositio Math.* **91** (1994), 201–221.
Yau D., Enveloping algebras of [H]{}om-[L]{}ie algebras, [*J. Gen. Lie Theory Appl.*](http://dx.doi.org/10.4303/jglta/S070209) **2** (2008), 95–108, [arXiv:0709.0849](http://arxiv.org/abs/0709.0849).
Yau D., Hom-bialgebras and comodule [H]{}om-algebras, *Int. Electron. J. Algebra* **8** (2010), 45–64, [arXiv:0810.4866](http://arxiv.org/abs/0810.4866).
Yau D., Module [H]{}om-algebras, [arXiv:0812.4695](http://arxiv.org/abs/0812.4695).
Yau D., Hom-quantum groups III: Representations and module [H]{}om-algebras, [arXiv:0911.5402](http://arxiv.org/abs/0911.5402).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'This paper is a contribution to the development of the theory of representations of inverse semigroups in toposes. It continues the work initiated by Funk and Hofstra [@FH]. For the topos of sets, we show that torsion-free functors on Loganathan’s category $L(S)$ of an inverse semigroup $S$ are equivalent to a special class of non-strict representations of $S$, which we call connected. We show that the latter representations form a proper coreflective subcategory of the category of all non-strict representations of $S$. We describe the correspondence between directed and pullback preserving functors on $L(S)$ and transitive and effective representations of $S$, as well as between filtered such functors and universal representations introduced by Lawson, Margolis and Steinberg. We propose a definition of a universal representation, or, equivalently, an $S$-torsor, of an inverse semigroup $S$ in the topos of sheaves ${\mathsf{Sh}}(X)$ on a topological space $X$. We prove that the category of filtered functors from $L(S)$ to the topos ${\mathsf{Sh}}(X)$ is equivalent to the category of universal representations of $S$ in ${\mathsf{Sh}}(X)$. We finally propose a definition of an inverse semigroup action in an arbitrary Grothendieck topos, which arises from a functor on $L(S)$.'
address:
- 'Ganna Kudryavtseva, Faculty of Civil and Geodetic Engineering, University of Ljubljana, Jamova 2, 1000, Ljubljana, SLOVENIA; Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000, Ljubljana, SLOVENIA '
- 'Primož Škraba, Jožef Stefan Institute, Jamova 39, 1000, Ljubljana, SLOVENIA '
author:
- Ganna Kudryavtseva
- Primož Škraba
title: The principal bundles over an inverse semigroup
---
Introduction
============
The classifying topos ${\mathcal{B}}(S)$ of an inverse semigroup $S$ has recently begun to be investigated [@F; @FH; @FLS; @FS; @KL; @St]. This topos is by definition the presheaf topos over Loganathan’s category $L(S)$ of $S$. There are several equivalent characterizations of this topos, cf. [@F; @FS; @KL]. An immediate question one can ask about ${\mathcal{B}}(S)$ is “What does ${\mathcal{B}}(S)$ classify?” A direct application of well-known results [@MM Theorems VII.7.2, VII.9.1] of topos theory, provides the answer: for an arbitrary Grothendieck topos ${\mathcal E}$, the presheaf topos ${\mathcal{B}}(S)$ classifies filtered functors $L(S)\to {\mathcal E}$.
The category of geometric morphisms from ${\mathcal E}$ to ${\mathcal{B}}(S)$ is equivalent to the category of filtered functors $L(S)\to {\mathcal E}$.
The construction of the functors establishing the correspondence in the above theorem can be found in [@MM]. In particular, if $\gamma^*\colon {\mathcal{B}}(S)\to {\mathcal E}$ is the inverse image functor of a geometeric morphism, its composition with the Yoneda embedding $L(S)\to {\mathcal{B}}(S)$ is a filtered functor, and any such a functor is obtained this way.
This answer, however, is not quite satisfactory. We expect more structure given that for groups the category of filtered functors $G\to {\mathcal E}$ is equivalent to the category of $G$-[*torsors*]{}. The latter are just objects of ${\mathcal E}$ with a particular type of internal action of the group object obtained by applying the canonical constant sheaf functor $\Delta$ to $G$ [@MM VIII.2]. This naturally raises a question of how to define $S$-torsors, where $S$ is an inverse semigroup. The latter question was raised by Funk and Hofstra in [@FH], where in [@FH Theorem 3.9] they show that, for the topos of sets, $S$-torsors can be defined as (well-supported) transitive and free $S$-sets, where an inverse semigroup $S$-set is a homomorphism from $S$ to the symmetric inverse semigroup ${\mathcal{I}}(X)$. This description naturally generalizes the description of $G$-torsors in the topos of sets. It is mentioned (without providing details) by Lawson, Margolis and Steinberg [@LMS] that $S$-torsors in the topos of sets are precisely universal representations of $S$ defined and systematically studied in [@LMS].
Funk and Hofstra, in [@FH Definition 2.14], proposed a definition of an $S$-torsor, where $S$ is an inverse semigroup, in an arbitrary Grothendieck topos ${\mathcal E}$. They also state an equivalence of categories between $S$-torsors in ${\mathcal E}$ and filtered functors $L(S)\to {\mathcal E}$ ([@FH Theorem 3.10]). Their approach is based on internalizing $S$ in ${\mathcal E}$ as a semigroup (rather than as an inverse semigroup). Implicitly, [@FH Definition 2.14] considers actions (in an arbitrary Grothendieck topos) by partial bijections [@private_comm]. However, actions of by partial bijections in an arbitrary Grothendieck topos are not defined in [@FH], nor, to the best or our knowledge, anywhere else in the literature. Therefore, while the main theorem [@FH Theorem 3.10] is correct, the ommision of this definition along with other details, can make some of the definitions (e.g., [@FH Definition 2.14]) and proofs in [@FH] hard to follow or verify for the non-expert in topos theory. One of the main goals in the present paper is to try to make the constructions as detailed, simple and explicit as possible and particularly tailored to researchers in semigroup theory. Additionally, we provide a counter-example to a claim in [@FH Section 6] (see Section \[sec:fin\] for details).
The paper is structured as follows. Section \[sec:prel\] provides some preliminaries needed to read this paper (as well as suggestions of the literature for further reading). In Section \[sec:sets\], we focus on the topos of sets and describe (possibly non-strict) $S$-sets, where $S$ is an inverse semigroup, attached to classes of functors from $L(S)$ to this topos. Some of these results were already given in [@FH], but we provide detailed proofs. We introduce a class of [*connected*]{} non-strict $S$-sets and prove that they are in a categorical equivalence with torsion-free functors on $L(S)$ (Theorem \[th:equiv\]). We then show that connected non-strict $S$-sets form a proper coreflective subcategory of the category of all non-strict $S$-sets which corrects [@FH Proposition 3.6] (see Example \[ex:ce\] and Proposition \[prop:ce\]). We also discuss the connection of transitive and universal $S$-sets with appropriate classes of functors on $L(S)$. This in particular leads to a new perspective on the classical result due to Schein [@Sch] on transitive and effective representations of an inverse semigroup. In Section \[sec:bundles\], we define $S$-torsors in the topos of sheaves ${\mathsf{Sh}}(X)$ over a topological space $X$ and prove that these are categorically equivalent to filtered functors $L(S)\to {\mathsf{Sh}}(X)$ (Theorem \[th:sheaves\]). It follows that in the topos ${\mathsf{Sh}}(X)$ the classifying topos ${\mathcal B}(S)$ classifies universal $S$-bundles.This can be seen as an instance of [@FH Theorem 3.10], with more details which hopefully provide a better insight into why this works.
Finally, in Section \[sec:fin\], we outline an approach, which is substantially different from that used in [@FH], to the notion of an $S$-set in an arbitrary Grothendieck topos. We start from a functor on $L(S)$, construct an objects of action as a certain colimit (similarly as this is done for the topos of set) and then lift $S$ to the topos ${\mathcal H}$-class-wise, that is, we consider objects $\Delta H$, where $H$ is an ${\mathcal H}$-class of $S$ and $\Delta$ is the constant sheaf functor. It would be interesting to connect and compare this approach with the approach proposed in [@FH].
An important task which remains for future investigation is to further develop the general theory of actions of inverse semigroups by partial bijections in arbitrary Grothendieck toposes extending [@FH] and the present paper to the level of corresponding well-established theory of group actions [@MM V.6, VIII.2].
Preliminaries {#sec:prel}
=============
For more complete exposition on inverse semigroups, we refer the reader to [@L], on categories to [@Awo; @MacL], and on toposes to [@MM; @M].
Inverse semigroups and their representations {#subs:inv}
--------------------------------------------
Let $S$ be an inverse semigroup. By $E(S)$, we denote the semilattice of idempotents of $S$. For $s\in S$ we write ${\mathbf{d}}(s)=s^{-1}s$ and ${\mathbf{r}}(s)=ss^{-1}$. These idempotents are abstractions of the notions of the domain and the range idempotents, respectively, of a partial bijection. The natural partial order on $S$ is defined by $s\leq t$ if and only if $s=te$ for some $e\in E(S)$. For $X\subseteq S$, we write $$X^{\uparrow}=\{s\in S\colon s\geq x \text{ for some } x\in X\}.$$ The set $X^{\uparrow}$ is sometimes called the [*(upward) closure*]{} of $X$. The set $X$ is [*closed*]{} if $X^{\uparrow}=X$.
For a set $X$, let ${\mathcal I}(X)$ denote the [*symmetric inverse semigroup*]{} on $X$ which consists of all bijections between subsets of $X$ (we refer to such maps as [*partial bijections*]{}). If $s\in {\mathcal I}(X)$ we set ${\mathrm{dom}}(s)$ and ${\mathrm{im}}(s)$ to be the domain and the image of $s$.
A [*representation*]{} of an inverse semigroup $S$ on a set $X$, is an inverse semigroup homomorphism $\theta\colon S\to {\mathcal I}(X)$. Given a such a representation, we have a left action of $S$ on $X$ by partial bijections such that $s\cdot x$ is defined if and only if $x\in {\mathrm{dom}}(\theta(s))$ in which case $s\cdot x= \theta(s)(x)$. We say that $(X,\theta)$ is a [*left*]{} $S$-[*set*]{}. Where $\theta$ is clear, we will write $(X,\theta)$ as simply $X$. Unless otherwise stated, we assume that actions are left actions, and we refer to left $S$-sets as $S$-[*sets*]{}. Throughout the paper, we assume that the $S$-sets are [*effective*]{}, meaning that for every $x\in X$ there exists some $s\in S$ such that $s\cdot x$ is defined.
An $S$-set $(X,\mu)$ is called [*transitive*]{} if for any $x,y\in X$ there is $s\in S$ such that $\mu(s)(x)=y$. It is called [*free*]{}, if the equality $\mu(s)(x)=\mu(t)(x)$ implies that there is $c\leq s,t$ such that $\mu(c)(x)=\mu(s)(x)$. Finally, we call a transitive and free $S$-set an $S$-[*torsor*]{}.
Toposes in a nutshell
---------------------
By a [*topos*]{}, we restrict ourselves to a [*Grothendieck topos*]{}, that is, a category ${\mathcal{E}}$ that satisfies the [*Giraud’s axioms*]{}. We refer the reader, for example, to [@M 1.1] for a detailed introduction to the notion of a topos. For our purposes, we do not need to recount the definition of a topos. It is important however to mention the following examples of toposes:
1. The category ${\mathsf{Sets}}$ of sets and maps between sets.
2. The category ${\mathrm{Et}}(X)$ of étale spaces over a topological space $X$.
3. The category ${\mathcal B}({\mathcal{C}})$ of presheaves of sets $F\colon {\mathcal{C}}^{op} \to {\mathsf{Sets}}$ over a small category ${\mathcal{C}}$.
Let us look at these examples at greater detail. An étale space over a topological space $X$ is a triple $(E,p,X)$ where $E$ is a topological space and $p\colon E\to X$ is a local homeomorphism. A [*morphism*]{} $ (E,p,X) \to (G,q,X)$ between étale spaces is a continuous map $\alpha\colon E\to G$ such that $q\alpha=p$. Given the well known equivalence between étale spaces and sheaves, the topos ${\mathrm{Et}}(X)$ is equivalent to the topos ${\mathsf{Sh}}(X)$ of sheaves over $X$. From the topos of sheaves ${\mathsf{Sh}}(X)$ one can recover the frame of opens of $X$, and thus, if $X$ is a sober space, $X$ itself can also be recovered [@M; @MM]. It follows that a topos can be thought of as a generalization of a (sober) topological space. Bearing this in mind, it is useful (for example, to interpret the definition of a point of a topos) to consider the topos ${\mathsf{Sets}}$ as an analogue of a one-point space.
Turning to the third example, a [*presheaf of sets*]{} over a small category ${\mathcal{C}}$ is a contravariant functor $F$ from ${\mathcal{C}}$ to the category of sets ${\mathsf{Sets}}$, $F\colon {\mathcal{C}}^{op} \to {\mathsf{Sets}}$. If $\alpha\colon c\to d$ is a morphism in ${\mathcal{C}}$, then the map $F(\alpha)\colon F(d)\to F(c)$ is called the [*translation map*]{} along $\alpha$. Let $F,G\colon {\mathcal{C}}^{op} \to {\mathsf{Sets}}$ be presheaves of sets. By a morphism from $F$ to $G$, we mean a natural transformation $\pi$ from $F$ to $G$, that is, a collection of maps, $\pi_c\colon F(c)\to G(c)$, where $c$ runs through the objects of ${\mathcal{C}}$, which commute with the translation maps. The topos ${\mathcal B}({\mathcal{C}})$ is called [*the classifying topos*]{} of the small category ${\mathcal{C}}$.
For a detailed verification that each of our examples satisfies the Giraud’s axioms, we refer the reader to [@M].
The category of elements of a functor
-------------------------------------
Let ${\mathcal C}$ be a small category and $P\colon {\mathcal C}\to {\mathsf{Sets}}$ a covariant functor. The [*category of elements of*]{} $P$ is the category $\int_{\mathcal C}P$ whose objects are all pairs $(C,p)$ where $C$ is an object of ${\mathcal C}$ and $p\in P(C)$. Its morphisms $(C,p)\to (C',p')$ are those morphisms $u\colon C\to C'$ of ${\mathcal C}$ for which $P(u)(p)=p'$. The category of elements $\int_{\mathcal C} P$ of a contravariant functor $P\colon {\mathcal C}^{op}\to {\mathsf{Sets}}$ is defined similarly.
Filtered and directed categories and functors {#subs:2.5}
---------------------------------------------
A small category $I$ is called [*filtered*]{} if it satisfies the following axioms:
1. $I$ has at least one object.
2. For any two objects $i,j$ of $I$ there is a diagram $i\leftarrow k \to j$ in $I$, for some object $k$.
3. For any two parallel arrows $i\rightrightarrows j$ there exists a commutative diagram $k\to i\rightrightarrows j$ in $I$.
Equivalently, a small category $I$ is filtered if for any finite diagram in $I$ there is a cone on that diagram. A small category $I$ is called [*directed*]{} if it satisfies axioms (F1) and (F2) above.
A covariant functor $A\colon {\mathcal C}\to {\mathsf{Sets}}$ is called a [*filtered functor*]{} (resp. a [*directed functor*]{}) if its category of elements $\int_{\mathcal C} A$ is a filtered category (resp. a directed category).
Geometric morphisms
-------------------
Let ${\mathcal E}, {\mathcal F}$ be toposes. A [*geometric morphism*]{} $f\colon {\mathcal F} \to {\mathcal E}$ consists of a pair of functors $$f^*\colon {\mathcal E} \to {\mathcal F} \text{ and } f_*\colon {\mathcal F} \to {\mathcal E},$$ called the [*inverse image functor*]{} and the [*direct image functor*]{}, respectively, such that the following two axioms are satisfied:
1. $f^*$ is a left adjoint to $f_*$.
2. $f^*$ is left exact, that is, it commutes with finite limits.
Since $f^*$ is a left adjoint, it commutes with colimits (by the dual to the well-known RAPL theorem [@Awo]). It follows from the uniqueness of adjoints that a geometric morphism $f\colon {\mathcal F} \to {\mathcal E}$ is determined by its inverse image functor $f^*\colon {\mathcal E}\to {\mathcal F}$ which is required to commute with any colimits and finite limits.
Let $X,Y$ be topological spaces and $f\colon X\to Y$ a continuous map. This gives rise to a functor $f^*\colon {\mathrm{Et}}(Y)\to {\mathrm{Et}}(X)$, as follows. Let $(E,p,Y)$ be an étale space over $Y$ and put $$X\times_Y E =\{(x,e)\in X\times E\colon f(x)=p(e)\}.$$ Then the projection to the first coordinate $\pi_1\colon X\times_Y E \to X$ is a local homeomorphism. Indeed, assume that $(x,e)\in X\times_Y E$ and let $A$ be a neighborhood of $e$ such that $A$ is homeomorphic to $p(A)$. Then the set $$\{(x,t)\in X\times_Y E\colon t\in A\}$$ is homeomorphic to $f^{-1}(p(A))$ via $\pi_1$. The local homeomorphism $\pi_1$ is said to be obtained by [*pulling*]{} $p$ [*back along*]{} $f$. We set $$f^*(E,p,Y)=(X\times_Y E, \pi_1, X).$$ It is easy to see that $f^*$ preserves colimits and finite limits and thus gives rise to a geometric morphism $(f^*,f_*)$ from ${\mathrm{Et}}(X)$ to ${\mathrm{Et}}(Y)$. For sober spaces $X$ and $Y$ this construction gives rise to a bijective correspondence between continuous maps from $X$ to $Y$ and geometric morphisms from ${\mathrm{Et}}(X)$ to ${\mathrm{Et}}(Y)$. Thus, as toposes can be looked at as generalizations of topological spaces, geometric morphisms between toposes are generalizations of continuous maps.
The constant sheaf functor and the global section functor
---------------------------------------------------------
For any topos ${\mathcal E}$, there is a unique (up to isomorphism) geometric morphism $\gamma\colon {\mathcal E}\to {\mathsf{Sets}}$, given by $$\gamma^*(S)=\sum_{s\in S}1, \,\, \gamma_*(E)={\mathrm{Hom}}_{\mathcal E}(1,E),$$ where $1$ denotes the terminal object of ${\mathcal E}$. The inverse image part $\gamma^*$ of $\gamma$ is usually denoted by $\Delta$ and is called the [*constant sheaf functor*]{}, and the direct image part $\gamma_*$ is usually denoted by $\Gamma$ and is called the [*global section functor.*]{} This geometric morphism may be looked at as an analogue of the only continuous map from a topological space $X$ to a one-element topological space.
Filtered functors and geometric morphisms
-----------------------------------------
A [*point*]{} of a topos ${\mathcal E}$ is a geometric morphism $\gamma: {\mathsf{Sets}}\to {\mathcal E}$. This is parallel to looking at a point of a topological space $X$ as an inclusion of a one-element space into $X$. Note that such an inclusion $i\colon \{x\}\to X$ defines a filter $F$ in $X$ consisting of those $A\in X$ such that $i(x)\in A$. We now describe how this idea can be extended to a correspondence between points of the classifying topos of a category and filtered functors on this category.
Let ${\mathcal C}$ be a small category, and let $A\colon {\mathcal C}\to {\mathsf{Sets}}$ be a functor. We describe a construction to be found in [@MM] of a pair of adjoint functors $f^*\colon {\mathcal{B}}({\mathcal C})\to {\mathsf{Sets}}$ and $f_*\colon {\mathsf{Sets}}\to {\mathcal{B}}({\mathcal C})$. The functor $f_*$ is easier to define and thus we start from its description. We have $f_*=\underline{\mathrm{Hom}}_{\mathcal C}(A,-)$, where the latter is the presheaf defined for each set $R$ and $C\in {\mathcal C}$ by $$\underline{\mathrm{Hom}}_{\mathcal C}(A,R)(C)={\mathrm{Hom}}_{\mathsf{Sets}}(A(C),R).$$
For a presheaf $P\in {\mathcal{B}}({\mathcal C})$, we define $f^*(P)$ to be the colimit $$f^*(P)=\lim_{\longrightarrow}\left(\int_{\mathcal C}P\stackrel{\pi_1}{\to} {\mathcal C} \stackrel{A}{\to} {\mathsf{Sets}}\right),$$ where $\pi_1(C,p)=C$. This colimit is the set which we denote by $P\otimes_{\mathcal{C}}A$. It is the quotient of the set $\bigcup_{C\in {\mathcal{C}}}(P(C)\times A(C))$ by the equivalence relation $\sim$ generated by $$(pu,a')\sim (p,ua'), \, p\in P(C), u\colon C\to C', a'\in A(C'),$$ where we denote $pu=P(p)(u)$ and $ua'=A(u)(a')$. We denote the elements of $P\otimes_{\mathcal{C}}A$ by $p\otimes a$ and treat them as tensors where ${\mathcal{C}}$ ‘acts’ on $P$ on the right and on $A$ on the left.
The described adjoint pair $(f^*, f_*)$ is not in general a geometric morphism between toposes. By definition, it is a geometric morphism if and only if the tensor product functor $f^*$ is left exact. If this condition holds, the functor $A$ is called [*flat*]{}. Flat functors can be characterized precisely as filtered functors [@MM Theorem VII.6.3].
Let ${\mathsf{Filt}}({\mathcal C})$ denote the category of filtered functors ${\mathcal C}\to {\mathsf{Sets}}$, where morphisms are natural transformations, and ${\mathsf{Geom}}({\mathsf{Sets}}, {\mathcal{B}}({\mathcal C}))$ the category of geometric morphisms from ${\mathsf{Sets}}$ to the classifying topos ${\mathcal{B}}({\mathcal C})$ of ${\mathcal C}$ (or, equivalently the points of ${\mathcal{B}}({\mathcal C})$), where morphisms are natural transformations between the inverse image functors.
\[th:filt\] There is an equivalence of categories $${\mathsf{Filt}}({\mathcal C})\, \,{\mathrel{
\settowidth{\@tempdima}{$\scriptstyle\tau$}
\settowidth{\@tempdimb}{$\scriptstyle\rho$}
\ifdim\@tempdimb>\@tempdima \@tempdima=\@tempdimb\fi
\mathop{\vcenter{
\offinterlineskip\ialign{\hbox to\dimexpr\@tempdima+2em{##}\cr
\rightarrowfill\cr\noalign{\kern.3ex}
\leftarrowfill\cr}}}\limits^{\!\tau}_{\!\rho}}}\, \,{\mathsf{Geom}}({\mathsf{Sets}}, {\mathcal{B}}({\mathcal C}))$$ where the functors $\tau$ and $\rho$ are defined, for a filtered functor $A\colon {\mathcal C}\to {\mathsf{Sets}}$ and a point $f\in {\mathsf{Geom}}({\mathsf{Sets}}, {\mathcal{B}}({\mathcal C})) $, by $$\tau(A)^*=- \otimes_{\mathcal C} A, \,\, \tau(A)_*= \underline{\mathrm{Hom}}_{\mathcal C}(A,-),$$ $$\rho(f)=f^*\cdot {\mathbf{y}}\colon {\mathcal C}\to {\mathcal{B}}({\mathcal C})\to {\mathsf{Sets}},$$ where ${\mathbf{y}}$ denotes the Yoneda embedding of ${\mathcal C}$ into ${\mathcal{B}}({\mathcal C})$.
We remark that Theorem \[th:filt\] remains valid in a wider setting where the topos ${\mathsf{Sets}}$ is replaced by an arbitrary topos ${\mathcal E}$. To formulate this result, known as Diaconescu’s theorem, one needs a suitable definition of a filtered functor from a small category to a topos, such that being filtered is equivalent to being flat, cf. [@MM VII.8]. For our purposes, we will need filtered functors to the topos of sheaves over a topological space, which we discuss in Section \[sec:bundles\].
Principal group bundles and group torsors
-----------------------------------------
The connection between filtered functors on a small category and geometric morphisms to the classifying topos is a well known and fundamental result in topos theory. In the special case where the category is a group, denote it by $G$, it is known [@MM VIII.2] that the category of filtered functors $G\to {\mathcal E}$, where ${\mathcal E}$ is an arbitrary topos, is equivalent to the category of so-called $G$-torsors over ${\mathcal E}$. A $G$-[*torsor over*]{} ${\mathcal E}$ is an object $T$ of ${\mathcal E}$ equipped with an internal action (cf. [@MM V.6]) of a group object $\Delta G$ on it which satisfies some technical conditions. We remind the reader that $\Delta G$ is the value of the constant sheaf functor $\Delta\colon {\mathsf{Sets}}\to {\mathcal E}$ on $G$. Since $G$ has a structure of a group, $\Delta G$ inherits a structure of an internal group in ${\mathcal E}$. The aforementioned technical conditions arise as an abstraction of the well-known notion of a $G$-torsor in the topos of sheaves over a topological space $X$. In this setting, a $G$-torsor is just synonymous with a principal $G$-bundle. A [*principal*]{} $G$-[*bundle*]{} over a topological space $X$ can be characterized as an étale space $(E,p,X)$ with a continuous action $G\times E\to E$ over $X$ such that
1. For each point $x\in X$ the stalk $E_x=p^{-1}(x)$ is non-empty;
2. Stalks are invariant under the action;
3. The action map $G\times E_x \to E_x$ on each stalk $E_x$ is free meaning that $g\cdot x=x$ implies that $g$ is the identity element;
4. The action map $G\times E_x \to E_x$ on each stalk $E_x$ is free transitive, meaning that for every $a,b\in E_x$ there exists some $g\in G$ such that $g\cdot a=b$.
A $G$-torsor in the topos of sets is a set $X$ equipped with a free and transitive action of $G$ on it. Such a set is in a bijection with $G$ and the action is equivalent to the action of $G$ on itself by left translations (this is a direct consequence of the elementary fact that a transitive group action is equivalent to the left action on the set of cosets over the stabilizer of any point). In particular, up to isomorphism, there is only one $G$-torsor in the topos of sets.
The equivalence between filtered functors $G\to {\mathcal E}$ and $G$-torsors over ${\mathcal{E}}$, together with Theorem \[th:filt\], yields the result that the classifying topos ${\mathcal{B}}(G)$ classifies $G$-torsors in the sense that for an arbitrary topos ${\mathcal E}$, there is a categorical equivalence between geometric morphisms ${\mathcal E}\to {\mathcal{B}}(G)$ and $G$-torsors over ${\mathcal{E}}$. This parallels the topological result that the classifying space of $G$ classifies principal $G$-bundles.
Covariant functors $L(S)\to {\mathsf{Sets}}$ vs representations of $S$ in ${\mathsf{Sets}}$ {#sec:sets}
===========================================================================================
The relationship between various classes of (possibly non-strict) representations of an inverse semigroup $S$ in the topos of sets and covariant functors $L(S)\to {\mathsf{Sets}}$ was first observed and studied by Funk and Hofstra in [@FH]. In particular, they observe that filtered functors on $L(S)$ correspond to representations of $S$ which are transitive and free ([@FH Theorem 3.9], though these representations are wrongly referred to as another kind of representations). They also consider torsion-free and pullback preserving functors. In this section we prove that torsion-free functors on $L(S)$ correspond to a class of $S$-sets which we call [*connected*]{}. (This corrects an inaccuracy in [@FH Proposition 3.6].) We also put in correspondence directed functors on $L(S)$ and transitive effective representations of $S$, providing a different approach to the classical theory due to Schein [@Sch]. Finally, we explain that filtered functors on $L(S)$ correspond to a class of representations, called [*universal*]{}, which were introduced and studied by Lawson, Margolis and Steinberg in [@LMS].
Torsion-free functors and connected non-strict representations
--------------------------------------------------------------
A map $\varphi\colon S\to T$ between inverse semigroups is called a [*prehomomorphism*]{} if $\varphi(ab)\leq \varphi(a)\varphi(b)$ for any $a,b\in S$. A prehomomorphism $S\to {\mathcal{I}}(X)$ will be called a [*non-strict*]{} representation of $S$. Similarly as representations correspond to $S$-sets, non-strict representations correspond to [*non-strict*]{} $S$-[*sets*]{}[^1], where the latter means a set $X$ together with a partial map $S\times X \to X$, $(s,x)\mapsto s\cdot x$, where defined, such that if $st\cdot x$ is defined then $t\cdot x$ and $s\cdot (t\cdot x)$ are defined and $st\cdot x= s\cdot (t\cdot x)$. Just as $S$-sets, the non-strict $S$-sets we consider are effective.
The following constructions connecting non-strict $S$-sets and some covariant functors on $L(S)$ were introduced in [@FH]. We give here their slightly different but equivalent description. We also provide more details and notice the property of connectedness.
Let $(X,\mu)$ be a non-strict $S$-set where $(s,x)\mapsto \mu(s,x)=s\cdot x$, where defined. For each $e\in E(S)$ let $\Phi(X,\mu)(e)$ be the domain of the action of $e$, that is to say, $$\Phi(X,\mu)(e)=\{x\in X\colon e\cdot x\text{ is defined}\}.$$
If $(f,s)$ is an arrow in $L(S)$, we define $\Phi(X,\mu)(f,s)$ to be the map from $\Phi(X,\mu)({\mathbf{d}}(s))$ to $\Phi(X,\mu)(f)$ given by $x\mapsto s\cdot x$. Since $e\cdot x$ is defined and $e={\mathbf{d}}(s)=s^{-1}s$, we have that $s\cdot x$ is defined. Observe that $s \cdot x=(fs)\cdot x$, so that $f\cdot (s\cdot x)$ is defined. Thus $s\cdot x\in \Phi(X,\mu)(f)$. We have constructed the covariant functor $\Phi(X,\mu)$ on $L(S)$. We need to record that the functor $\Phi(X,\mu)$ has one important property. We first define this property.
Assume that $F\colon L(S)\to {\mathsf{Sets}}$ is a functor and put $\Psi(F)$ to be the colimit of the following composition of functors: $$E(S)\longrightarrow L(S)\stackrel{F}{\longrightarrow} {\mathsf{Sets}}.$$
This colimit is, by definition, equal to the quotient set $$\label{eq:colimit} \Psi(F)= \left(\bigcup_{e\in E(S)}\{e\}\times F(e)\right)/\sim,$$ where the equivalence $\sim$ on $\bigcup_{e\in E(S)}\{e\}\times F(e)$ is generated by $(e,x)\sim (e',F(e',e)(x))$. The functor $F$ is called [*torsion-free*]{} if $(e,x)\sim (e,y)$ implies that $x=y$.
The constructed functor $\Phi(X,\mu)\colon L(S)\to {\mathsf{Sets}}$ is torsion-free.
This follows from the definition of $\sim$ since $$\Phi(X,\mu)(e',e)(x)=e'\cdot x=e\cdot x=x$$ for any $e'\geq e$ in $E$ and any $x\in X$ such that $e\cdot x$ is defined.
We have therefore assigned to $(X,\mu)$ a torsion-free functor $\Phi(X,\mu)\colon L(S)\to {\mathsf{Sets}}$. We now describe the reverse direction. Assume that $F$ is a torsion-free functor $L(S)\to {\mathsf{Sets}}$. By $[e,x]$ we will denote the $\sim$-class of $(e,x)$. For $s\in S$ and $\alpha\in \Psi(F)$ we define $$\label{eq:action} s\circ\alpha=\left\lbrace\begin{array}{ll}[{\mathbf{r}}(s), F({\mathbf{r}}(s),s)(x)],& \text{ if } \alpha=[{\mathbf{d}}(s),x];\\ \text{undefined,} & \text{otherwise.} \end{array}\right.$$
If $\alpha\in \Psi(F)$ we define
$$\label{eq:connected}\pi_1(\alpha)=\{e\in E\colon \text{ there is some }(e,x)\in \alpha\}=\{e\in E\colon e\circ \alpha \text{ is defined}\}.$$
It follows that $s\circ \alpha$ is defined if and only if ${\mathbf{d}}(s)\in\pi_1(\alpha)$.
\[lem:lem3\]
1. The map $\alpha\mapsto s\circ\alpha$, given by , is injective on its domain.
2. The assignment defines on $\Psi(F)$ the structure of a non-strict $S$-set $(\Psi(F), \nu)$.
3. For any $\alpha\in \Psi(F)$ and $e,f\in \pi_1(\alpha)$, there are $$e=e_1,e_2,\dots, e_k=f$$ in $\pi_1(\alpha)$ such that $e_i\geq e_{i+1}$ or $e_i\leq e_{i+1}$ for all admissible $i$.
\(1) Follows from , since $F$ is torsion-free and thus all the translation maps are injective.
\(2) We use the fact that a map $\varphi\colon S\to T$ between inverse semigroups is a prehomomorphism if and only if $\varphi(st)=\varphi(s)\varphi(t)$ for any $s,t$ such that ${\mathbf{r}}(t)={\mathbf{d}}(s)$ and $\varphi(ef)\leq \varphi(e)\varphi(f)$ for any $e,f\in E(S)$. It is immediate from that both of these conditions hold for $\Psi(F)$.
\(3) Follows from the construction of $\Psi(F)$ and .
Since the set $\pi_1(\alpha)$ is expressable in terms of the action, as is given in , we can define a non-strict $S$-set $X$, $(s,x)\mapsto s\cdot x$, where defined, to be [*connected*]{} if for any $x\in X$ and any $e,f\in E$ such that $e\cdot x$ and $f\cdot x$ are defined, there is a sequence of idempotents $e=e_1,e_2,\dots, e_k=f$, called a [*connecting sequence over*]{} $x$, such that $e_i\cdot x$ is defined and $e_i\geq e_{i+1}$ or $e_i\leq e_{i+1}$ for all admissible $i$.
[*If $X$ is an $S$-set (that is, given by a homomomorphism), it is connected with $e,ef,f$ being a connecting sequence between $e$ and $f$ over any $x$ such that $e\cdot x$ and $f\cdot x$ are defined.* ]{}
[*If $S$ is a monoid, any non-strict $S$-set is connected with $e,1,f$ being a connecting sequence between $e$ and $f$, again over any $x$ such that $e\cdot x$ and $f\cdot x$ are defined.*]{}
It is not true that every non-strict $S$-set is connected, as the following example shows.
\[ex:ce\]
*Let $S=\{e,f,g\}$ be a three-element semilattice, given by the following Hasse diagram:*
\(e) [$e$]{}; (aux) \[node distance=0.6cm, right of=e\] ; (f) \[node distance=1.2cm, right of=e\] [$f$]{}; (g) \[node distance=1cm, below of=aux\] [$g$]{}; (e) edge node\[above\] (g) (f) edge node\[above\] (g);
Let $X=\{1,2\}$ and define the domains of action of $e$ and $f$ to be equal $\{1,2\}$, and the domain of action of $g$ to be equal $\{1\}$ (that is, $e$ and $f$ act by the identity map on $\{1,2\}$, and $g$ by the identity map on $\{1\}$). Thus $X$ becomes a non-strict $S$-set. It is however not connected, as both $e\cdot 2$ and $f\cdot 2$ are defined but there is no connecting sequence between $e$ and $f$ over $2$ as $g\cdot 2$ is undefined.
In view of Lemma \[lem:lem3\], it follows that the non-strict $S$-set from Example \[ex:ce\] can not be equal $\Psi(F)$ for any torsion-free functor $F$ on $L(S)$.
We now describe the correspondence between morphisms of non-strict $S$-sets and natural transformations of torsion-free functors on $L(S)$. Assume we are given non strict $S$-sets $(X,\mu)$, $(s,x)\mapsto s\cdot x$, where defined, and $(Y,\nu)$, $(s,x)\mapsto s\circ x$, where defined. A [*morphism*]{} from $(X,\mu)$ to $(Y,\nu)$ is a map $f\colon X\to Y$ such that if $s\cdot x$ is defined then $s\circ f(x)$ is also defined and $$f(s\cdot x) = s\circ (f(x)).$$
Let $f\colon (X,\mu)\to (Y,\nu)$ be a morphism, $e\in E$ and $x\in \Phi(X, \mu)(e)$. Then $f(x)\in \Phi(Y, \nu)(e)$ which defines a map $$\widetilde{f}_e\colon \Phi(X, \mu)(e)\to \Phi(Y, \nu)(e).$$ It is immediate that the maps $\widetilde{f}_e$ commute with the translation maps along any $(f,s)\in L(S)$ and thus define a natural transformation $\widetilde{f}\colon \Phi(X, \mu)\to \Phi(Y, \nu)$. We set $\Phi(f)=\widetilde{f}$.
In the reverse direction, let $F$ and $F'$ be torsion-free functors $L(S)\to {\mathsf{Sets}}$ and let $\alpha\colon F\to F'$ be a natural transformation. Let $\alpha(e)$ denote the component of $\alpha$ at $e$. Further, let $\sim$ denote the congruence on the set $\bigcup_{e\in E(S)}\{e\}\times F(e)$ which defines the set $\Psi(F)$, and $\sim'$ denote a similar congruence which defines the set $\Psi(F')$.
Let $x\in F(e)$, $y\in F(f)$ and $(e,x)\sim (f,y)$. Then $(e,\alpha(e)(x))\sim' (f,\alpha(f)(y))$.
Without loss of generality, we may assume that $e\leq f$ and that $y=F(f,e)(x)$. Since compotents of $\alpha$ commute with the translation maps, we can write $$\alpha(f)(y)=\alpha(f)(F(f,e)(x))=F'(f,e)(\alpha(e)(x)),$$ which yields that $(e,\alpha(e)(x))\sim' (f,\alpha(f)(y))$.
The proved lemma shows that the assignment $[e,x]\mapsto [e,\alpha(e)(x)]$ results in a well-defined map $$\widetilde{\alpha}\colon \Psi(F)\to \Psi(F').$$
The map $\widetilde{\alpha}$ is a morphism of non-strict $S$-sets.
Let the structure of an $S$-set on $\Psi(F)$ (defined in ) be given by $(s,\alpha)\mapsto s\circ \alpha$, where defined, and that on $\Psi(F')$ be given by $(s,\alpha)\mapsto s*\alpha$, where defined. Let $s\in S$ and assume that $s\circ [e,x]$ is defined. We may then assume that $e={\mathbf{d}}(s)$. Then $\widetilde{\alpha}([{\mathbf{d}}(s),x])=[{\mathbf{d}}(s),\alpha({\mathbf{d}}(s))(x)]$ showing that $s*\widetilde{\alpha}([{\mathbf{d}}(s),x])$ is defined, too. The proof is completed by the following calculation using the fact that components of $\alpha$ commute with the translation maps: $$\widetilde{\alpha}(s\circ ([{\mathbf{d}}(s),x]))=\widetilde{\alpha}([{\mathbf{r}}(s),F({\mathbf{r}}(s),s)(x)])=
[{\mathbf{r}}(s), \alpha({\mathbf{r}}(s))F({\mathbf{r}}(s),s)(x)];$$ $$s*\widetilde{\alpha}([{\mathbf{d}}(s),x])=s*[{\mathbf{d}}(s), \alpha({\mathbf{r}}(s))(x)]=[{\mathbf{r}}(s),F'({\mathbf{r}}(s),s)(\alpha({\mathbf{d}}(s),x))].$$
We set $\Psi(\alpha)=\widetilde{\alpha}$. Let ${\mathsf{Repr}}(S)$ denote the category of all non-strict $S$-sets, ${\mathsf{ConRepr}}(S)$ the category of all connected non-strict $S$-sets and ${\mathsf{TF}}(L(S))$ the category of torsion-free functors on $L(S)$.
It is routine to verify that the assignments $\Phi\colon {\mathsf{Repr}}(S)\to {\mathsf{TF}}(L(S))$ and $\Psi\colon {\mathsf{TF}}(L(S))\to {\mathsf{ConRepr}}(S)$ are functorial. We denote the restriction of the functor $\Phi$ to the category ${\mathsf{ConRepr}}(S)$ by $\Phi'$. We obtain the following result.
\[th:equiv\] There is an equivalence of categories $${\mathsf{ConRepr}}(S) \,\,
{\mathrel{
\settowidth{\@tempdima}{$\scriptstyle\Phi'$}
\settowidth{\@tempdimb}{$\scriptstyle\Psi$}
\ifdim\@tempdimb>\@tempdima \@tempdima=\@tempdimb\fi
\mathop{\vcenter{
\offinterlineskip\ialign{\hbox to\dimexpr\@tempdima+2em{##}\cr
\rightarrowfill\cr\noalign{\kern.3ex}
\leftarrowfill\cr}}}\limits^{\!\Phi'}_{\!\Psi}}} \,\, {\mathsf{TF}}(L(S)).$$
Let $F\colon L(S)\to {\mathsf{Sets}}$ be a torsion-free functor and show that $F$ is naturally isomorphic to the functor $\Phi\Psi(F)$. By construction, for $e\in E$ we have $$\Phi\Psi(F)(e)=\{[e,x]\colon e\cdot x \text{ is defined}\}.$$ Clearly, the maps $\tau_e\colon x\to [e,x]$, $x\in F(e)$, $e\in E$, are bijections. In addition, these maps commute with the translation maps because for any arrow $(f,s)$ in $L(S)$, we have $$[{\mathbf{d}}(s),x]\stackrel {\Phi\Psi(F)(f,s)}{\xrightarrow{\hspace*{1.2cm}}} [f, F(f,s)(x)].$$ It follows that we have constructed a natural isomorphism $\tau\colon F\to \Phi\Psi(F)$.
In the reverse direction, let $(X,\mu)$ be a connected non-strict $S$-set, where $(s,x)\mapsto s\cdot x$, where defined. The elements of the set $\Psi\Phi(X,\mu)$ are equivalence classes $[e,x]$ where $e\in E$ and $e\cdot x$ is defined. We define the map $$\label{eq:beta}
\beta_{\mu}\colon \Psi\Phi(X,\mu)\to X$$ by $[e,x]\mapsto x$. Let $x\in X$. Since $\mu$ is effective, there exists an $s\in S$ such that $s\cdot x$ is defined. But then ${\mathbf d}(s)\cdot x$ is defined, as well, which implies that the map $\beta_{\mu}$ is surjective. To show injectivity of $\beta_{\mu}$, we note that the condition $[e,x] \neq [f,x]$ is equivalent to the claim that there is no connecting sequence between $e$ and $f$, which does not happen as $\mu$ is connected. We obtain that the non-strict $S$-set $\Psi\Phi(X,\mu)$, $[e,x]\mapsto s\circ [e,x]$, where defined, is equivalent to $(X,\mu)$. Indeed, $s\circ [e,x]$ is defined if and only if $s\cdot x$ is defined, and in the case where $s\cdot x$ is defined we have the equality $$s\circ [e,x]=[{\mathbf{r}}(s), s\cdot x].$$ Moreover, this equivalence is natural in $(X,\mu)$.
\[cor:strict\] The category of $S$-sets is equivalent to the category of pull-back preserving functors on $L(S)$.
The statement follows from Theorem \[th:equiv\] using the facts that any $S$-set is connected, and that a non-strict $S$-set $(X,\mu)$ is strict if and only if $\mu(ef)=\mu(e)\mu(f)$ for any $e,f\in E(S)$.
We now establish a relationship between all non-strict $S$-sets and those of them which are connected. Recall that a subcategory ${\mathcal A}$ of a category ${\mathcal B}$ is called [*coreflective*]{} if the inclusion functor ${\mathrm{i}}\colon {\mathcal A}\to {\mathcal B}$ has a right adjoint. This adjoint is called a [*coreflector*]{}.
\[prop:ce\] The category ${\mathsf{ConRepr}}(S)$ is a coreflective subcategory of the category ${\mathsf{Repr}}(S)$. The coreflector is given by the functor $\Psi\Phi$.
Let $(X,\mu)$ be a non-strict $S$-set. Just as in the proof of Theorem \[th:equiv\], we have the map $\beta_{\mu}\colon \Psi\Phi(X,\mu)\to X$ given by . This map is surjective, and is injective if and only if $\mu$ is connected. We show that the functor $\Psi\Phi$ is a right adjoint to the functor ${\mathrm{i}}\colon {\mathsf{ConRepr}}(S) \to {\mathsf{Repr}}(S)$ where the maps $\beta_{\mu}$ are the components of the counit $\beta\colon i\circ \Psi\Phi \to {\mathrm{id}}_{{\mathsf{Repr}}(S)}$.
Let $(X,\mu)$ be any connected non-strict $S$-set, $(Y,\nu)$ be any non-strict $S$-set, and $$g\colon (X,\mu) \to (Y,\nu)$$ a morphism. To define the morphism $$f\colon (X,\mu) \to \Psi\Phi(Y,\nu),$$ let $x\in X$ and $e\in E(S)$ be such that $\mu(e)(x)$ is defined. Then it follows that $\nu(e)(f(x))$ is defined, as well. We set $$\label{eq:def_f}
f(x)=[e,g(x)]\in \Psi\Phi(Y,\nu).$$ For brevity, in this proof, we write $s\cdot x$ for $\mu(s)(x)$, $s\circ x$ for $\nu(s)(x)$ and $s*x$ for $\Psi\Phi(\nu)(s)(x)$. Note that if $h\cdot x$ is defined where $h\in E(S)$ then $h\circ f(x)$ is defined, and an induction shows that $(e,g(x))\sim (h,g(x))$ follows from $(e,x)\sim (h,x)$, where the latter equivalence holds because $\mu$ is connected. Therefore, the map $f$ is well-defined. Let us show that $f$ is a morphism of non-strict $S$-sets. Assume that $s\cdot x$ is defined. This is equivalent to that ${\mathbf{d}}(s)\cdot x$ is defined. It follows that $s*[{\mathbf{d}}(s),g(x)]$ is defined as well, and applying we have $$s*[{\mathbf{d}}(s),g(x)]=[{\mathbf{r}}(s),s\circ g(x)]=[{\mathbf{r}}(s),g(s\cdot x)].$$ On the other hand, $f(s\cdot x)=[{\mathbf{r}}(s),g(s\cdot x)]$ holds by . All that remains is to note that the equality $g=\beta_{\nu} f$ is a direct consequence of the definitions of $f$ and $\beta_{\nu}$.
Transitive representations of $S$ as directed functors on $L(S)$ {#sub:3.2}
----------------------------------------------------------------
\[prop:trans\] The equivalence in Corollary \[cor:strict\] restricts to an equivalence between the category of transitive $S$-sets and directed pullback preserving functors on $L(S)$.
Let $(X,\mu)$, $(s,x)\mapsto s\cdot x$, if defined, be a transitive $S$-set. We show that the functor $\Phi(X,\mu)$ is directed. Let $(e,x)$ and $(f,y)$ be objects of the category of elements $\int_{L(S)}\Phi(X,\mu)$ of $\Phi(X,\mu)$ and $s\in S$ be such that $s\cdot x=y$. We put $t=fse$ and observe that $t\cdot x=y$ and also ${\mathbf{d}}(t)\leq e$, ${\mathbf{r}}(t)\leq f$. Observe that $({\mathbf{d}}(t),x)$ is an object of the category $\int_{L(S)}\Phi(X,\mu)$. Since $e\cdot x=x$, the arrow $(e,{\mathbf{d}}(t))$ of $L(S)$ is an arrow from $({\mathbf{d}}(t),x)$ to $(e,x)$ of the category $\int_{L(S)}\Phi(X,\mu)$. Since $t\cdot x=y$, the arrow $(f,t)$ is an arrow from $({\mathbf{d}}(t),x)$ to $(f,y)$ of the category $\int_{L(S)}\Phi(X,\mu)$. It follows that the functor $\Phi(X,\mu)$ is directed.
In the reverse direction, assume that the functor $\Phi(X,\mu)$ is directed and let $x,y\in X$. Let $e,f\in E(S)$ be such that $e\cdot x$ and $f\cdot y$ are defined (such $e$ and $f$ exist since $X$ is effective: for some $s$ we have that $s\cdot x$ is defined, but then ${\mathbf d}(s)\cdot x$ is defined, as well). Since the category $\int_{L(S)}\Phi(X,\mu)$ is directed, there are $z\in X$ and $g\in E(S)$ such that $g\cdot z$ is defined, and in the category $\int_{L(S)}\Phi(X,\mu)$ there are arrows $(e,x)\leftarrow (g,z)\to (f,y)$. The arrow $(g,z)\to (f,y)$ is by definition an arrow $(f,s)$ in $L(S)$ from $g$ to $f$ such that $s\cdot z=y$. Likewise, the arrow $(g,z)\to (e,x)$ is an arrow $(e,t)$ in $L(S)$ from $g$ to $e$ such that $t\cdot z=x$. It follows that $st^{-1}\cdot x=y$ which implies that the action is transitive.
We now briefly recall the classical result due to Boris Schein [@Sch] (see also [@H; @LMS]) of the structure of transitive $S$-sets[^2].
An inverse subemigroup $H$ of $S$ is called [*closed*]{} if it is upward closed as a subset of $S$, i.e. $H^{\uparrow}=H$. Let $H$ be a closed inverse subsemigroup of $S$. A [*coset*]{} with respect to $H$ is a set $(xH)^{\uparrow}$ where ${\mathbf{d}}(x)\in H$. Let $X_H$ be the set of cosets with respect to $H$. Define the structure of an $S$-set on $X_H$ by putting $s\cdot (xH)^{\uparrow}$ is defined if and only if $(sxH)^{\uparrow}$ is a coset in which case $$\label{eq:structure} s\cdot (xH)^{\uparrow}=(sxH)^{\uparrow}.$$ The obtained $S$-set $X_H$ is transitive and any transitive $S$-set is equivalent to one so constructed.
[*It follows that Proposition \[prop:trans\] provides a link, which was not previously explicitly mentioned in the literature, between closed inverse subsemigroups of $S$ and directed and pullback preserving functors on $L(S)$.*]{}
Universal representations and filtered functors on $L(S)$ {#sub:3.3}
---------------------------------------------------------
Let $H$ be a closed inverse subsemigroup of $S$. Recall that a [*filter*]{} in a semilattice is an upward closed subset $F$ such that $a\wedge b\in F$ whenever $a,b\in F$. Since the meet in $E(S)$ coincides with the product of idempotents, it follows that $E(H)$ is a filter in $E(S)$. Since $H$ is closed, $H\supseteq E(H)^{\uparrow}$ always holds. On the other hand, for any filter $F$ in $E(H)$ we have that $F^{\uparrow}$ is a closed inverse subsemigroup of $S$.
An $S$-set $(X,\mu)$ is called [*universal*]{} [@LMS], if it is equivalent to a representation of $S$ on cosets with respect to a closed inverse subsemigroup $F^{\uparrow}$, where $F$ is a filter in $E(S)$. The following result is mentioned without proof in [@LMS]. We provide a proof for completeness.
\[prop:torsors\] An $S$-set $(X,\mu)$ is an $S$-torsor if and only if it is universal.
Let $(X,\mu)$, $(s,x)\mapsto s\cdot x$, where defined, be an $S$-set. Let $x\in X$ and put $$H=\{s\in S\colon s\cdot x \text{ is defined and } s\cdot x=x\}.$$ Then $H$ is a closed inverse subsemigroup of $S$, and $(X,\mu)$ is equivalent to the structure of an $S$-set, $(X_H,\nu)$, given in , on the set $X_H$ of cosets with respect to $H$. We may thus assume that $(X,\mu)=(X_H,\nu)$.
Assume that $(X_H,\nu)$ is an $S$-torsor. We show that $H=E(H)^{\uparrow}$. It is enough to verify that $H\subseteq E(H)^{\uparrow}$. Let $s\in H$. Since $(X_H,\nu)$ is free, the equalities $$s\cdot x = {\mathbf d}(s)\cdot x=x$$ imply that there is some $c\leq s, {\mathbf d}(s)$ such that $c\cdot x=x$. Therefore $c\in E(H)$ and $s\geq c$, so that we have the inclusion $H\subseteq E(H)^{\uparrow}$.
Conversely, assume that $(X_H,\mu)$ is universal and let $s,t\in S$ and $x\in X_H$ be such that $s\cdot x=t\cdot x$. Then there are some $e,f\in E(H)$ such that $s\geq e$, $t\geq f$ such that $e\cdot x$ and $f\cdot x$ are defined, and then of course $e\cdot x=f\cdot x=x$. We put $h=ef$. Then $s,t\geq h$ and $h\cdot x=x$, so that $(X_H,\mu)$ is an $S$-torsor.
The following result follows from Proposition \[prop:torsors\] and [@FH Proposition 3.9] stated there without proof.
\[prop:ff\] The equivalence in Proposition \[prop:trans\] restricts to an equivalence between the category of universal $S$-sets and the category of filtered functors on $L(S)$. Consequently, the category of points of the topos ${\mathcal B}(S)$ is equivalent to the category of universal $S$-sets.
Let $(X,\mu)$ be a universal $S$-set. Assume that we have two objects $(e,x)$ and $(f,y)$ and two arrows $$(e,x) {\mathrel{
\settowidth{\@tempdima}{$\scriptstyle(f,s)$}
\settowidth{\@tempdimb}{$\scriptstyle(f,t)$}
\ifdim\@tempdimb>\@tempdima \@tempdima=\@tempdimb\fi
\mathop{\vcenter{
\offinterlineskip\ialign{\hbox to\dimexpr\@tempdima+2em{##}\cr
\rightarrowfill\cr\noalign{\kern.3ex}
\rightarrowfill\cr}}}\limits^{\!(f,s)}_{\!(f,t)}}} (f,y)$$ in the category of elements $\int_{L(S)}\Phi(X,\mu)$. This implies that ${\mathbf d}(s)={\mathbf d}(t)=e$ and $s\cdot x=t\cdot x=y$. Since $(X,\mu)$ is free, there is $c\leq c,s$ such that $c\cdot x=y$. Since $c\leq s$ we have that $c=sg$ for some $g\in E(S)$ where we may assume that $g\leq e$. Then ${\mathbf d}(c)=g$. This and $c\leq t$ yield that $c=tg$. It follows that there is an arrow $$(g,x)\stackrel{(e,g)}{\longrightarrow} (e,x)$$ in the category $\int_{L(S)}\Phi(X,\mu)$. Since $(f,s)(e,g)=(f,c)=(f,t)(e,g)$ in $L(S)$, the diagram $$(g,x) \stackrel{(e,g)}{\longrightarrow}(e,x) {\mathrel{
\settowidth{\@tempdima}{$\scriptstyle(f,s)$}
\settowidth{\@tempdimb}{$\scriptstyle(f,t)$}
\ifdim\@tempdimb>\@tempdima \@tempdima=\@tempdimb\fi
\mathop{\vcenter{
\offinterlineskip\ialign{\hbox to\dimexpr\@tempdima+2em{##}\cr
\rightarrowfill\cr\noalign{\kern.3ex}
\rightarrowfill\cr}}}\limits^{\!(f,s)}_{\!(f,t)}}} (f,y)$$ is commutative. Therefore, the functor $\Phi(X,\mu)$ satisfies axiom (F3) from the definition of a filtered functor (see Subsection \[subs:2.5\]). It satisfies (F1) and (F2) due to Proposition \[prop:trans\], since universal $S$-sets are transitive.
Conversely, let $(X,\mu)$ be an $S$-set such that the functor $\Phi(X,\mu)$ is filtered. Assume that $s,t\in S$ and $x\in X$ are such that $s\cdot x=t\cdot x$. Let $e={\mathbf d}(s){\mathbf d}(t)$ and $h={\mathbf r}(s){\mathbf r}(t)$. Then $hse\cdot x=hte\cdot x$ and also ${\mathbf d}(hse)={\mathbf d}(hte)$, ${\mathbf r}(hse)={\mathbf r}(hte)$. We put $p={\mathbf d}(hse)$ and $q={\mathbf r}(hse)$. It follows that in the category $\int_{L(S)}\Phi(X,\mu)$ we have two parallel arrows $$(p,x) {\mathrel{
\settowidth{\@tempdima}{$\scriptstyle(q,hse)$}
\settowidth{\@tempdimb}{$\scriptstyle(q,hte)$}
\ifdim\@tempdimb>\@tempdima \@tempdima=\@tempdimb\fi
\mathop{\vcenter{
\offinterlineskip\ialign{\hbox to\dimexpr\@tempdima+2em{##}\cr
\rightarrowfill\cr\noalign{\kern.3ex}
\rightarrowfill\cr}}}\limits^{\!(q,hse)}_{\!(q,hte)}}} (q,y).$$ By axiom (F3), there is a commutative diagram $$(r,z) \stackrel{(p,a)}\longrightarrow (p,x) {\mathrel{
\settowidth{\@tempdima}{$\scriptstyle(q,hse)$}
\settowidth{\@tempdimb}{$\scriptstyle(q,hte)$}
\ifdim\@tempdimb>\@tempdima \@tempdima=\@tempdimb\fi
\mathop{\vcenter{
\offinterlineskip\ialign{\hbox to\dimexpr\@tempdima+2em{##}\cr
\rightarrowfill\cr\noalign{\kern.3ex}
\rightarrowfill\cr}}}\limits^{\!(q,hse)}_{\!(q,hte)}}} (q,y)$$ in the category $\int_{L(S)}\Phi(X,\mu)$. This means that $hsea=htea$. Then $hse{\mathbf r}(a)=hte{\mathbf r}(a)$ and also $$hse{\mathbf r}(a)\cdot y =hte{\mathbf r}(a)\cdot y = z,$$ which proves that $(X,\mu)$ is free. Applying Proposition \[prop:trans\] (noting that filtered functors preserve pullbacks) and Proposition \[prop:torsors\], we conclude that $(X,\mu)$ is an $S$-torsor.
Principal bundles over inverse semigroups {#sec:bundles}
=========================================
In this section, we obtain an equivalence between the category of universal representations of an inverse semigroup on étale spaces over a topological space $X$ and the category of principal $L(S)$-bundles over $X$. This extends the well known result for groups [@MM VIII.1, VIII.2], and also is an analogue of Proposition \[prop:ff\], if in the latter one replaces the topos of sets by the topos ${\mathsf{Sh}}(X)$.
The following definition is taken from [@M]. Let $X$ be a topological space and ${\mathcal C}$ a small category. A functor $E\colon {\mathcal C}\to {\mathsf{Sh}}(X)$ is called a ${\mathcal C}$-[*bundle*]{}. If $E\colon {\mathcal C}\to {\mathsf{Sh}}(X)$ is a ${\mathcal C}$-[*bundle*]{}, $\alpha\colon c\to d$ is an arrow in ${\mathcal C}$ and $y\in E(c)$, we put $$\alpha\cdot y=E(\alpha)(y)\in E(d).$$
A ${\mathcal C}$-bundle $E$ is called [*principal*]{}, if for each point $x\in X$ the following axioms are satisfied by the stalks $E(C)_x$:
1. (non-empty) There is an object $c$ of $C$ such that $E(c)_x\neq\varnothing$;
2. (transitive) For any $y\in E(c)_x$ and $z\in E(d)_x$, there are arrows $\alpha\colon b\to c$ and $\beta\colon b\to d$ for some object $b$ of $C$, and a point $w\in E(b)_x$, so that $\alpha\cdot w=y$ and $\beta\cdot w=z$.
3. (free) For any two parallel arrows $\alpha,\beta\colon c\rightrightarrows d$ and any $y\in E(c)_x$, for which $\alpha\cdot y=\beta\cdot y$, there exists an arrow $\gamma\colon b\to c$ and a point $z\in E(b)_x$ so that $\alpha\gamma=\beta\gamma$ and $\gamma\cdot z=y$.
Principal ${\mathcal C}$-bundles are known to coincide with filtered functors from ${\mathcal C}$ to the topos ${\mathsf{Sh}}(X)$. It is immediate that given a principal ${\mathcal C}$-bundle $E\colon {\mathcal C}\to {\mathsf{Sh}}(X)$ and $x\in X$, the induced restriction to stalk functor $E_x\colon {\mathcal C}\to {\mathsf{Sets}}$, $c\mapsto E(c)_x$, is a filtered functor.
For two principal ${\mathcal C}$-bundles $E$ and $E'$, a [*morphism*]{} from $E$ to $E'$ is simply a natural transformation $\varphi\colon E\to E'$, that is, a collection of sheaf maps $\varphi_c\colon E(c)\to E'(c)$, where $c$ runs through objects of ${\mathcal C}$, such that for each arrow $\alpha\colon c\to d$ in ${\mathcal C}$ and each $y\in E(c)$ we have that $\varphi_d(\alpha\cdot y)=\alpha\cdot\varphi_c(y)$. We have therefore defined the category ${\mathsf{Prin}}({\mathcal C}, X)$ of [*principal bundles*]{} over ${\mathcal C}$.
In the case where ${\mathcal C}$ is Loganathan’s category $L(S)$ for an inverse semigroup $S$, we will call a principal bundle over $L(S)$ a [*principal bundle over*]{} $S$ and the category ${\mathsf{Prin}}(L(S), X)$ the category of [*principal bundles over*]{} $S$. We will write ${\mathsf{Prin}}(S, X)$ for ${\mathsf{Prin}}(L(S), X)$.
We now define the notion of a [*universal*]{} $S$-[*set*]{} in the topos ${\mathsf{Sh}}(X)$. Let $\pi \colon E\to X$ be an étale space and assume that a structure of an $S$-set $(E,\mu)$, $(s,x)\mapsto s\cdot x$, if defined, is given on $E$ such that the following conditions are met:
1. (effective on each stalk) For any $x\in X$, there is at least one point $e\in E_x$ such that $s\cdot e$ is defined for some $s\in S$ (this in particular implies that all stalks are non-empty, that is, the map $\pi$ is surjective).
2. (domains are open) For any $s\in S$ the set $\{e\in E\colon s\cdot e \text{ is defined}\}$ is open.
3. (stalks are invariant) For any $s\in S$ and $e\in E$, if $s\cdot e$ is defined then $\pi(s\cdot e)=\pi(e)$.
4. (universal on stalks) For any $s\in S$ and $x\in X$, $(E_x,\mu|_{S\times E_x})$ (which is well-defined by (U3)) is a universal $S$-set.
5. (continuous) The partially defined map $S\times E \to E$, $(s,x)\mapsto s\cdot x$, is continuous ($S$ is considered as a discrete space and $S\times E$ as a product space).
It is easy to see that (U4) implies (U1), so that (U1) may be omitted from the above list.
Let $\pi \colon E\to X$, $\pi'\colon E'\to X$ be étale spaces and $(E,\mu)$, $(s,e)\mapsto s\cdot x$, if defined, $(E',\nu)$, $(s,e)\mapsto s\circ x$, if defined, be universal $S$-sets in the topos ${\mathsf{Sh}}(X)$. A morphism $$f\colon (E,\mu) \to (E',\nu)$$ is defined as a morphism $f\colon E\to E'$ of étale spaces (that is, a continuous map such that $\pi=\pi'f$, cf. [@MM]) which is simultaneously a morphism of $S$-sets (that is, if $s\cdot e$ is defined then $s\circ f(e)$ is defined and $f(s\cdot x)=s\circ f(x)$). We denote the category of universal $S$-sets in the topos ${\mathsf{Sh}}(X)$ by ${\mathsf{Univ}}(S,X)$.
\[th:sheaves\] There is an equivalence of categories $${\mathsf{Prin}}(S, X) \,\,
{\mathrel{
\settowidth{\@tempdima}{$\scriptstyle\tau$}
\settowidth{\@tempdimb}{$\scriptstyle\rho$}
\ifdim\@tempdimb>\@tempdima \@tempdima=\@tempdimb\fi
\mathop{\vcenter{
\offinterlineskip\ialign{\hbox to\dimexpr\@tempdima+2em{##}\cr
\rightarrowfill\cr\noalign{\kern.3ex}
\leftarrowfill\cr}}}\limits^{\!\tau}_{\!\rho}}} \,\, {\mathsf{Univ}}(S,X).$$
We begin with the construction of the functor $$\tau\colon {\mathsf{Prin}}(S, X)\to {\mathsf{Univ}}(S,X).$$ Let $E\colon L(S) \to {\mathsf{Sh}}(X)$ be a principal bundle over $X$ and $x\in X$. We first describe the colimit sheaf $\widetilde{E}\in {\mathsf{Sh}}(X)$. By definition, for each $x\in X$ we have a filtered functor $f_x\colon L(S)\to {\mathsf{Sets}}$ obtained by restricting $E$ to the stalks over $x$. We now apply the functor $\Psi$ to each $f_x$. Proposition \[prop:ff\] ensures us that each $\Psi(f_x)$ is a universal $S$-set. Note that each of the sets $\Psi(f_x)$ is non-empty by (PB1). As the stalks of the colimit sheaf are colimits of stalks, we may set $$\widetilde{E}=\bigcup_{x\in X} \Psi(f_x)$$ to be a disjoint union of all the sets $\Psi(f_x)$. The projection map $$p\colon \widetilde{E}\to X$$ given by $y\mapsto x$ if $y\in \Psi(f_x)$.
We may identify each space $E(e)$ with its image under the inclusion into $\widetilde{E}$. The colimit topology on $\widetilde{E}$ is the finest topology which makes all inclusion maps $E(e)\hookrightarrow \widetilde{E}$ continuous. The base of this topology is formed by the sets $A\subseteq \widetilde{E}$ such that $A\subseteq E(e)$ for some $e$. The structure of an $S$-set, $(s,y)\mapsto s*y$, where defined, on $\widetilde{E}$ is induced by the structures of $S$-sets on each on $\Psi(f_x)$, given by .
We now prove that for each $s\in S$ the set $$D_s=\{y\in \widetilde{E}\colon s*y \text{ is defined}\}$$ is open. Clearly, $D_s=D_{{\mathbf d}(s)}$. Let $y\in D_{{\mathbf d}(s)}$ and $A$ be a neighbourhood of $y$ in $\widetilde{E}$. Since the inclusion map $i\colon E_{{\mathbf{d}}(s)}\hookrightarrow \widetilde{E}$ is open, we have that $ii^{-1}(A)$ is a neighbourhood of $y$ in $\widetilde{E}$ which is contained in $D_{{\mathbf d}(s)}$. This implies that the set $D_{{\mathbf d}(s)}$ is open. Using the fact that the translation maps are continuous, it is routine to verify that the partially defined map $S\times E \to E$, $(s,x)\mapsto s\cdot x$, is continuous.
We now define $\tau$ on morphisms. Let $E$ and $E'$ be principal $S$-bundles and $\varphi\colon E\to E'$ be a natural transformation. The family of continuous maps $E(e)\to E'(e)$ for each $e\in E(S)$ and the construction of $\widetilde{E}$ yield a continuous map $\tau(\varphi)\colon \widetilde{E}\to \widetilde{E'}$ which obviously satisfies the definition of a morphism of universal representations.
We now turn to the construction of the functor $\rho\colon {\mathsf{Univ}}(S,X) \to {\mathsf{Prin}}(S, X)$. Let $p\colon \widetilde{E}\to X$ be an étale space and $(\widetilde{E},\mu)$, $(e,x)\mapsto e*x$, if defined, a structure of an $S$-set on $\widetilde{E}$ which satisfies (U1) – (U5). We fix $e\in E(S)$ and let $$E(e)=\{y\in \widetilde{E}\colon e*y\text{ is defined}\}.$$ Define $p_e\colon E(e)\to X$ to be the restriction of the map $p$ to $E(e)$. Clearly, $p_e$ is a local homeomorphism. By Proposition \[prop:ff\], for each $x\in X$, the restriction of $*$ to $E_x$ gives rise to a filtered functor $\Phi(\widetilde{E},\mu)_x\colon L(S)\to {\mathsf{Sets}}$ and it is routine to verify that these give rise to a filtered functor $\rho(p\colon \widetilde{E}\to X)\colon L(S)\to {\mathsf{Sh}}(X)$.
To define $\rho$ on morphisms, we observe that a morphism $\psi \colon E\to E'$ of universal representations yields a family of maps $E(e)\to E'(e)$ for each $e\in E(S)$. By construction, these maps are continuous and are components of a natural transformation from $\rho(E)$ to $\rho(E')$.
It follows that $S$-torsors in the topos ${\mathsf{Sh}}(X)$ can be defined as universal $S$-bundles.
As a direct consequence of Theorem \[th:equiv\] and an analogue of Theorem \[th:filt\] for the topos ${\mathsf{Sh}}(X)$, we obtain the following result.
The category of geometric morphisms ${\mathsf{Geom}}({\mathsf{Sh}}(X), {\mathcal{B}}(S))$ is equivalent to the category ${\mathsf{Univ}}(S,X)$ of universal $S$-bundles in the topos ${\mathsf{Sh}}(X)$.
*Let $S$ be an inverse semigroup. A [*filter*]{} in $S$ is a filter with respect to the natural partial order in $S$, that is, a nonempty subset $F$ of $S$ such that*
1. $a\in F$ and $b\geq a$ imply that $b\in S$;
2. if $a,b\in F$ then there is $c\in F$ such that $c\leq a,b$.
Let $E=E(S)$ and $\hat{E}$ denote the set of filters in $E$. A filter $F$ in $E$ defines a nonzero semilattice homomorphism, called a [*semi-character*]{}, $\varphi_F\colon E\to \{0,1\}$ such the inverse image of $1$ is $F$, and conversely, any nonzero semilattice homomorphism $\varphi\colon E\to \{0,1\}$ defines a filter $\varphi^{-1}(1)$ in $E$. These assignments are mutually inverse, so that the elements of $\hat{E}$ can be equivalently looked at as semi-characters. The space $\hat{E}$ is topologized as a subspace of the product space $\{0,1\}^E$ where $\{0,1\}$ is a discrete space. The space $\hat{E}$ is locally compact and is known as the [*filter space*]{} or the [*semi-character space*]{} of $S$.
Let ${\mathcal G}$ denote the set of filters in $S$. For $s\in S$ let $M(s)=\{F\in {\mathcal G}\colon s\in F\}.$ The set ${\mathcal G}$ is topologized by letting the sets $M(s)\cap M(s_1)^c \cap \dots \cap M(s_n)^c$, where $s,s_1,\dots s_n$, $n\geq 0$, to be a base of the topology. The connection between filters in $S$ and filters in $E$ was studied in detail in [@LMS]. If $F\in {\mathcal G}$, the assignment $$\mathrm{d}(F)=\{\mathbf{d}(a)\colon a\in F\}\in \hat{E}$$ defines a map $\mathrm{d}\colon {\mathcal G}\to \hat{E}$ which is a local homeomorphism (in fact, this map is equivalent to the domain map of the [*groupoid of filters*]{} of $S$).
We have an action of $S$ on each stalk ${\mathcal G}_F$ of the étale space $({\mathcal G}, {\mathrm{d}}, X)$ which is just the universal action of $S$ on the set of cosets with respect to the closed inverse subsemigroup $F^{\uparrow}$. It is routine to verify that these actions define on ${\mathcal G}$ the structure of a universal $S$-bundle which is natural to call the [*universal bundle associated to the domain map of the groupoid of filters of*]{} $S$.
Towards actions of inverse semigroups in an arbitrary topos {#sec:fin}
===========================================================
In [@FH Definition 2.14], Funk and Hofstra proposed a way to define a notion of a torsor for an arbitrary inverse semigroup $S$ in an arbitrary (Grothendieck) topos. Their definition [@FH Definition 2.14] is based on the concept of a semigroup $S$-set in an arbitrary topos: for an inverse semigroup $S$ they consider an internal semigroup $\Delta(S)$. In the topos of sets, a semigroup $S$-set $X$ is a (pre)homomorphism from $S$ to the partial transformation semigroup ${\mathcal{PT}}(X)$ on $X$. For $S$ inverse, only semigroup $S$-sets for which the action is by partial bijections should be considered. The diagrammatic definition of partial bijections is not written in [@FH], but can be done. Omitting the requirement of partial bijections leads to an incorrect claim in [@FH].
Section 6 of [@FH] discusses the actions of inverse semigroups in an arbitrary topos. However, the claim ‘If $T$ is inverse, then a $T$-set $T\to M(X)$ necessarily factors through $I(X)\subseteq M(X)$’ is incorrect (where $I(X)$ is ‘the object of partial bijections’). We now provide an example that, for the topos of sets where the meaning of ${\mathcal I}(X)$ is clear (the symmetric inverse semigroup on $X$), shows the claim to be incorrect.
\[ex:counterexample\][*Let $S$ be a linearly ordered set considered as a semilattice and let $|S|>1$. The map $\mu\colon S \to {\mathcal{PT}}(S)$ given by $x\mapsto \varphi_x$, where $\varphi_x(y)= x\wedge y$, $y\in S$ is a homomorphism, but this is not an inverse semigroup $S$-set as the action is not by partial bijections. It is also easy to check that $\mu$ is free and transitive according to [@FH Definition 2.8] and [@FH Definition 2.14].* ]{}
If $S$ is an inverse semigroup, then the internal semigroup $\Delta(S)$ can be readily endowed with the structure of an ‘internal inverse semigroup’ as $\Delta$ preserves the logic needed to express the fact of being an inverse semigroup (e.g. the varietal definition). Thus the approach taken in [@FH] of internalizing $S$ as a semigroup looks simpler than the possible internalizing it as an inverse semigroup. We, however, believe, that it is more natural, e.g., from the perspective of the cohomology theory [@Log], to keep the idempotents of $S$ and to internalize ${\mathcal H}$-classes of $S$ and the connection between them. This is the approach we outline below.
Let ${\mathcal{E}}$ be a (Grothendieck) topos and $S$ an inverse semigroup. We define an action of an inverse semigroup in an arbitrary topos which arises from a functor $L(S)\to {\mathcal{E}}$. Then classes of functors $L(S)\to {\mathcal{E}}$ (such as pullback preserving functors, torsion-free functors or filtered functors) can be connected with respective classes of actions of $S$ in ${\mathcal{E}}$. In particular, actions connected to filtered functors, can be naturally called $S$-torsors.
Let $e,f\in E(S)$ be such that $e\mathrel{\mathcal D} f$. By $H(e,f)$ be denote the ${\mathcal H}$-class of $S$ which consists of all $s\in S$ satisfying ${\mathbf{d}}(s)=f$ and ${\mathbf{r}}(s)=e$. Note that any ${\mathcal H}$-class of $S$ is of the form $H(e,f)$ for some $e\mathrel{\mathcal D} f$. We can bring all the sets $H(e,f)$ up to ${\mathcal{E}}$ by considering their images $\Delta H(e,f)$ under the constant sheaf functor $\Delta$.
Let $A\colon L(S)\to {\mathcal{E}}$ be a functor. The colimit construction, given in [@FH] for the topos of sets, extends to ${\mathcal{E}}$, and we construct the colimit object ${\mathcal X}$ of the composite of the functors $$E(S)\to L(S)\stackrel{A}{\to} {\mathcal{E}}.$$ In particular, all objects $A(e)$ are subobjects of ${\mathcal X}$. Therefore, a morphism $A(e)\to A(f)$ in ${\mathcal{E}}$ can be thought of as a ‘partial’ morphism of ${\mathcal X}$. Note that if $S$ is a monoid with unit $1$ then we have ${\mathcal X}=A(1)$. The restriction of $A$ to $E(S)$ gives us a functor $E(S)\to {\mathcal{E}}$. If $S$ is a group, this functor just selects an objects in ${\mathcal{E}}$, in particular, the object ${\mathcal X}$. Reasoning similarly as in [@MM p. 432], we see that for each ${\mathcal H}$-class $H(e,f)$ the functor $A$ gives rise to a map $$\begin{gathered}
\label{eq:manipulation}
H(e,f)\to {\mathrm{Hom}}_{{\mathcal{E}}}(A(f),A(e)) \simeq {\mathrm{Hom}}_{{\mathcal{E}}}(1, A(e)^{A(f)})\simeq \\ {\mathrm{Hom}}_{{\mathcal{E}}}(\Delta 1, A(e)^{A(f)})\simeq {\mathrm{Hom}}_{\mathsf{Sets}}(1, \Gamma(A(e)^{A(f)}))\simeq \Gamma(A(e)^{A(f)})\end{gathered}$$ (here $1$ denotes the terminal object of ${\mathcal E}$). We obtain the map $$\label{eq:manip1}
\Delta H(e,f) \to A(e)^{A(f)},$$ and applying the adjunction between product and exponentiation, $$\label{eq:manip2}
\Delta H(e,f) \times A(f)\to A(e).$$
We recall that every morphism in $L(S)$ is a composition of some $({\mathbf{r}}(s),s)$ and some $(e,f)$, where $e,f\in E(S)$. Therefore, a functor $A\colon L(S)\to {\mathcal E}$ is determined by its restriction to $E(S)$ and by translations along isomorphisms $({\mathbf{r}}(s),s)$. The restriction of $A$ to $E(S)$ in the group case degenerates to selecting an object in ${\mathcal E}$, so it is natural to keep this restriction as a part of the definition of an $S$-set associated to $A$ in ${\mathcal E}$. The translations along isomorphisms are internalized using , and .
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Andrej Bauer and Alex Simpson for useful discussions. We are also grateful to Jonathon Funk and Pieter Hofstra for helpful communication, to the referee for their comments, as well as the editor for facilitating a fruitful discussion.
[99]{}
S. Awodey, [*Category theory*]{}, Oxford University Press, 2006.
J. Funk, Semigroups and toposes, [*Semigroup Forum*]{} [**75**]{} (2007), no. 3, 481–520.
J. Funk, P. Hofstra, Topos theoretic aspects of semigroups actions, [*Theory Appl. Cat.*]{} [**24**]{} (2010), no.7, 117–147.
J. Funk, P. Hofstra, private communication.
J. Funk, M. V. Lawson, B. Steinberg, Characterizations of Morita equivalent inverse semigroups, [*J. Pure Appl. Algebra*]{} [**215**]{} (2011), 2262–2279.
J. Funk, B. Steinberg, The universal covering of an inverse semigroup, [*Appl. Categ. Structures*]{} [**18**]{} (2010), no. 2, 135–163.
J. M. Howie, [*Fundamentals of semigroup theory*]{}, The Clarendon Press, Oxford University Press, New York, 1995.
G. Kudryavtseva, M. V. Lawson. The classifying space of an inverse semigroup, [*Period. Math. Hungar.*]{}, [**70**]{} (1) (2015), 122–129.
M. V. Lawson, [*Inverse semigroups: the theory of partial symmetries*]{}, World Scientific Publishing Co., Inc., NJ, 1998.
M. V. Lawson, S. W. Margolis and B. Steinberg, The étale groupoid of an inverse semigroup as a groupoid of filters, [*J. Aust. Math. Soc.*]{} [**94**]{} (2014), 234–256.
J. Leech, $\mathcal{H}$-coextensions of monoids, [*Mem. Amer. Math. Soc.*]{} [**1**]{} (2), no. 157 (1975), 1–66.
M. Loganathan, Cohomology of inverse semigroups, [*J. Algebra*]{} [**70**]{} (1981), 375–393.
S. Mac Lane, [*Categories for the working mathematician,*]{} Grad. Texts in Math., vol. 5, Springer-Verlag, 1998.
S. Mac Lane, I. Moerdijk, [*Sheaves in geometry and logic. A first introduction to topos theory*]{}, Springer-Verlag, 1994.
I. Moerdijk, [*Classifying spaces and classifying topoi*]{}, Lecture Notes in Math 1616, Springer-Verlag, 1995.
B. M. Schein, Representations of generalized groups, [*Izv. Vyssh. Uchebn. Zaved Mat.*]{} [**28**]{} (3) (1962), 164–176 (in Russian).
B. Steinberg, Strong Morita equivalence of inverse semigroups, [*Houston J. Math.*]{} [**37**]{} (2011), 895–927.
[^1]: Note that in [@FH] non-strict $S$-sets are referred to as $S$-sets, and $S$-sets are referred to as strict $S$-sets.
[^2]: We recall our convention that all $S$-sets are effective.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Position-sensitive detectors for cold and ultra-cold neutrons (UCN) are in use in fundamental research. In particular, measuring the properties of the quantum states of bouncing neutrons requires micro-metric spatial resolution. To this end, a Charge Coupled Device (CCD) coated with a thin conversion layer that allows a real time detection of neutron hits is under development at LPSC. In this paper, we present the design and performance of a dedicated electronic board designed to read-out eight CCDs simultaneously and operating under vacuum.'
address:
- |
LPSC, Université Grenoble-Alpes, CNRS/IN2P3\
53, avenue des Martyrs, Grenoble, France
- |
ILL, Institut Laue Langevin\
71, avenue des Martyrs, Grenoble, France
author:
- 'O. Bourrion'
- 'B. Clement'
- 'D. Tourres'
- 'G. Pignol'
- 'Y. Xi'
- 'D. Rebreyend'
- 'V.V. Nesvizhevsky'
title: 'C2D8: An eight channel CCD readout electronics dedicated to low energy neutron detection.'
---
UCN, CCD readout, position sensitive detectors.
Introduction {#introSec}
============
Neutrons interact with matter mostly through strong nuclear interaction. When the neutron wavelength becomes commensurate with the inter-atomic spacing, only coherent scattering occurs. As a consequence, neutrons slowed down to kinetic energies below 100neV are totally reflected at any angle of incidence by most solid surfaces. These *ultra-cold neutrons* constitute a sensitive tool to study fundamental interactions and symmetries [@Dubbers2011]. In particular, ultra-cold neutrons are used to study gravity in a quantum context [@Nesvizhevsky2002; @Nesvizhevsky2010; @Jenke2011; @Pignol2015]. A neutron bouncing on top of an horizontal mirror realizes a simple one dimensional quantum well problem and the vertical motion of the neutron bouncer has discrete energy states. The wave functions associated to the stationary quantum states have a spatial extension governed by the parameter $z_0 = (\hbar^2/2m_n^2g)^{1/3} \approx 6$m. Therefore, observing the spatial structure of the quantum states requires position-sensitive neutron detectors with a micro-metric spatial resolution.
Semiconductor-based detectors, coated with adequate neutron converters, have been demonstrated to be well suited for this kind of measurements [@Jakubek2009; @Jakubek2009b; @Kawasaki2010; @Lauer2011]. The UCNBox (Ultra Cold Neutrons BOron piXels) detector has been recently developed as position sensitive sensor optimized to measure the wave-functions of the bouncing neutron in the GRANIT experiment [@Roulier2015]. The GRANIT facility uses a 30cm wide glass mirror as the surface where neutrons bounce. The setup is inside a vacuum chamber at $10^{-5}$mbar. The detector, composed of 8 Charge Coupled Devices (CCD), is designed to cover a sensitive area of $300 \, {\rm mm} \times 0.8 \, {\rm mm}$. Each CCD is an Hamamatsu S11071-1106N sensor (pixel size 14m $\times$ 14m and number of effective pixels $2048 \times 64$) coated with $^{10}$B, thanks to plasma assisted physical vapor deposition [@boronCCD].
Neutron capture on boron produces in most cases both a 1.5MeV $\alpha$ particle and a 0.8MeV $^{7}$Li nucleus. The CCD sensor is used as a pixelated silicon detector. The charge produced by the energy deposition migrates in the neighboring pixels allowing a precise reconstruction of the position using weighted average. To limit this migration of charges each CCD sensor must be read at approximately a 1Hz rate, with a dead-time as low as possible. In this project we aimed at dead-times below 1%. The UCN rate in the final experiment should not exceed 1Hz per sensor. Nevertheless, for calibration purposes, rates as high as 50Hz per sensor are desirable. Also, given the fact that no mechanical shutter system can be used to avoid neutron detection, the shortest possible readout time must be achieved for each CCD to avoid neutron detection during the CCD charge transfer, as it would corrupt the data. Consequently, the CCD must be read-out at the maximum speed specified by the manufacturer (10MHz), they can however be read one after another.
Typically, both detection-rate and number of hit pixels are low as all charges from one charged particle are collected within a $11\times11$ pixels matrix. For a maximum rate of 50Hz per sensor, only 5% of the sensor contains useful data. In normal conditions, this drops to 0.1%. This calls for the implementation of a data reduction system, that removes any pixel data below a discrimination threshold.
It must be noted, that an adjustable exposure time must be implemented to permit the use of bright light sources such as LED used to test and adjust CCD alignment.
Additionally, the readout system must be located very close to the sensor system in order to minimize signal integrity issues, but also to minimize the number of vacuum feed-through for the CCD signals (control and readout). This requirement, implies that the readout must be located inside the detector cell and thus withstand vacuum conditions. Consequently, special care must be taken on power usage and heat dissipation to ensure proper operation. This paper is organized as follows: section \[HardwareSec\] presents the hardware design, section \[FPGASec\] describes the firmware architecture. Eventually, a short summary is given in section \[SummarySec\].
Hardware description {#HardwareSec}
====================
To meet the requirements listed in section \[introSec\], we have opted for a solution based on two distinct modules: a front-end part composed of 8 boards, each holding a CCD; a back-end part for the control and readout circuits. As shown in figure \[c2d8Pic\], the front-end (FEB) and back-end boards are mounted on a common mechanical support. This support is designed such as to be able to finely adjust the position of each CCD board, thanks to dedicated screws. We have checked that this system allows for a relative alignment of the 8 CCDs within 10m, sufficient for our needs.
![Picture of the readout electronics mounted on its mechanical support. The electronic system is composed of two parts: the front-end composed of 8 carrier boards, each holding a CCD, and the back-end with the control and readout circuits. The boards are mounted on a mechanical support system equipped with adjustment screws to allow the adjustment of each CCD carrier board position. The four thermal sensors implemented on the board are indicated (T1 to T4). []{data-label="c2d8Pic"}](./figs/c2d8InstalledPic){width="90.00000%"}
Additionally, the electronics and the support system were designed to optimize the thermal coupling. Indeed, the support system is used to conduct a significant part of the heat flow to the vacuum vessel, while the remaining part of the heat is radiated in the chamber.
![Block diagram of the electronics system. Each FEB is connected to the back-end board with a Flexible Flat Cable (FFC). The back-end board is in charge of generating the CCD control signals (horizontal/ vertical shifts and reset gate), to perform the CCD signal digitization, to aggregate the data and to finally make them available for readout via a Universal Serial Bus (USB) interface. Each CCD signal digitization is done by a dedicated CCD signal processor [@ADDI7100].[]{data-label="elecDiag"}](./figs/hardBlockDiag){width="90.00000%"}
A block diagram of the electronics is shown in figure \[elecDiag\]. Each FEB is connected to the back-end board with a Flexible Flat Cable (FFC). The back-end board is in charge of generating the adequate CCD control signals (horizontal/ vertical shifts and reset gate); performing the CCD signal digitization; aggregating the data and finally making them available for readout via a Universal Serial Bus (USB) interface.
Each CCD signal digitization is carried-out by a dedicated CCD signal processor (Analog devices ADDI7100 [@ADDI7100]). This CCD processor can operate at 45MHz, which is significantly faster than the maximum readout speed of the Hamamatsu S11071-1106 CCD (10MHz), and has a digitization resolution of 12 bit with noise performance better than the CCD performance.
Indeed, the CCD processor is specified for having a system noise equivalent to 24.4e^-^ (0.8LSB rms with CDS gain set at +6dB for a typical CCD sensitivity of 8V/e- which corresponds to 30.5e^-^/ADU) while the typical CCD noise is composed of the readout noise (23e^-^ rms) and of the dark current integration (typically 50e^-^/pixel/s at 25C) resulting in a total of 73e^-^ for 1s of integration. The total expected system noise is thus dominated by the CCD and is about 77e^-^, which is compatible with the measurements that showed a system noise of 91.5$\pm$15e^-^.
The control signals required to read-out the CCD are generated by a Field Programmable Gate Array (FPGA). These signals, composed of Horizontal/Vertical (H/V) shifting signals and Reset Gate signals (RG), are amplified by dedicated drivers to accommodate the CCD load. The FPGA is also used to produce the signals necessary to operate the CCD signal processor, i.e. clamping and pre-blank signals, the correlated double sampler (CDS) signals and the serial control links.
The rationales for selecting the FPGA used in this design (Xilinx XC7A35-FGG484) were (i) its low power consumption; (ii) the possibility to precisely adjust the timing of the generated signals; (iii) the large amount of memory available. Indeed, the power had to be minimized by design as much as possible to permit the electronics operation under vacuum while avoiding a too fancy mechanical setup for thermalization (for instance: usage of the standby modes of the CCD processors during the exposure time). Additionally, this FPGA features high performance serializers in each of its input/output block. Thanks to these blocks, one can adjust output signals with a time resolution of about 2ns by using a high speed 480MHz clock (see section \[FPGASec\]). This makes it possible to conveniently set the CDS sampling times. The sizing of the memory was based on the criteria that its capacity should be at least half of the memory required to buffer the data generated by one CCD readout, i.e. $2048 \times 64 /2=65536$ 16-bits words, corresponding to more than 1Mbit of storage. This time equivalent buffering must be considered acknowledging the fact that, by specification, a new USB2 transaction can be placed every millisecond.
The total system power usage was measured to be 3W in full readout. In this budget, we estimate by combining specifications and measurements that about 140mW are used for each CCD (including about 32mW for the biasing and the line drivers losses); 600mW are used by the power converters (linear and switching) distributed on the board; about 1200mW are used by the FPGA. To asses the operating temperature of the system, dedicated measurements were performed under vacuum. A total of six temperatures were recorded over 25 hours, i.e. from the system power-up until system equilibrium (see figure \[ccdTemp\]). The temperature of the back-end electronics was recorded on four points thanks to the sensor circuits included in design (LM75B), see locations in figure \[elecDiag\] Additionally, the temperature of two CCD were recorded: one at the border and one in the middle of the detector plane with thermally coupled PT100 probes. We can see that the CCD highest temperature elevation from ambient is about 8C. The highest temperature elevation of 11C is measured for probe T2 which is located close to the FPGA.
![Plot of the system temperatures recorded from power-up to equilibrium. We can see that the CCD highest temperature elevation from ambient is about 8C. The highest temperature elevation of 11C is measured for probe T2 which is located close to the FPGA. []{data-label="ccdTemp"}](./figs_ext/CCDtemperature){width="80.00000%"}
Firmware description {#FPGASec}
====================
A block diagram of the FPGA firmware is shown in figure \[firmDiag\].
![Block diagram of the FPGA firmware. It is composed of six different blocks: clocking, USB interface, serializer, DAQ FIFO, DAQ manager and CCD timing. []{data-label="firmDiag"}](./figs/firmBlockDiag){width="90.00000%"}
It is composed of six different blocks: ‘clocking’, ‘USB interface’, ‘serializer’, ‘DAQ FIFO’, ‘DAQ manager’ and ‘CCD timing’. They are described hereafter.
The ‘clocking’ module is used to produce the CCD readout clock (10MHz) and the fast clocks required by the timing modules (respectively 80MHz and 240MHz). To achieve this, a Mixed Mode Clock Manager [@MMCM] (MMCM) uses the 50MHz reference clock, provided at the board level from a crystal oscillator, to perform the clock generation.
The ‘USB interface’ provides an interface between the USB micro-controller and the various configuration registers as well as an interface to read-out the acquisition FIFO (‘DAQ FIFO’).
The ‘DAQ FIFO’ is designed as a First Word Fall Through FIFO, it provides a buffering depth of 98305 words of 16-bit. This configuration was chosen to fully exploit the available memory in the selected FPGA and thus loosen the constraints on the acquisition software.
The ‘serializer’ module is used to interface the CCD processor serial configuration link, it is controlled either by the ‘USB interface’ or by the ‘DAQ manager’. When controlled by the ‘USB interface’, it allows the acquisition and control software to configure the CCD processor with the required parameters (gain settings, operation mode, ...). During acquisition mode, the ‘DAQ manager’ can use the ‘serializer’ to change the CCD processor operating mode (normal operation or full standby) of the active CCD. Hence, the CCD processors are in normal mode only when required, less than 1% of the time, and thus the power used is minimized.
The eight ‘CCD timing’ modules, one per CCD, are used to generate the CCD shifting signals (vertical, horizontal and reset gate) and the CCD processor control signals. These are the pedestal and data sampling control signals required for the correlated double sampler (respectively SHP and SHD), the preblank signal (PBLK) which is used to clear the processor output data during vertical shifting and the clamp optical black signal (CLOPB) used to remove residual offsets in the CCD processor signal chain. The CLOPB signal is supposed to be activated when the black pixels are being read-out. A detailed description of the ‘CCD timing’ module is given in section \[ccdTimingSec\].
The ‘DAQ manager’ module performs two main tasks. Its first role is to sequence the CCD acquisition by triggering the appropriate ‘CCD timing’ module. The CCD to read-out are individually selected by an eight bit selection mask (‘sel\_mask’). Its second purpose is to recover the data provided by the CCD processor associated with the CCD being accessed, to discriminate the data with respect to a threshold, to encapsulate the data and to eventually store them in the output buffer. More details about the ‘DAQ manager’ is given in section \[daqManagerSec\].
CCD timing module description {#ccdTimingSec}
-----------------------------
A block diagram detailing the internal architecture of the ‘CCD timing’ module is given in figure \[timingBlockDiag\].
![Block diagram of the CCD timing module.[]{data-label="timingBlockDiag"}](./figs/timingBlockDiag){width="90.00000%"}
The ‘horizontal timing’ module is in charge of controlling the four horizontal shifts (H shift) and the reset gate (RG) signals. Module operation is deactivated by the Horizontal Blank signal (HBLK), that is when the ‘timing core’ FSM is not in the horizontal shifting states (*preHor*, *validHor* and *postHor*). The horizontal shifting are done at the pixel clock speed, i.e. 10MHz. To cope with the various signal phases and widths required by the chosen CCD, the module is clocked eight times faster than the pixel clock.
The ‘vertical timing’ module controls the two vertical shifting signals. The module is designed to be able to shift several lines before signaling completion with *shiftDone*. The vertical shifting is requested by *startV* and the number of lines to move is determined by *shiftCount*. It may be noted that the CCD is operated in the Large Saturation Charge Mode [@LSCM].
The ‘timing core’ Finite State Machine (FSM) is used to coordinate the CCD control signals. It controls the horizontal and vertical timing modules and thus activates them when appropriate. The FSM is started either by the acquisition signal (*do\_acquire*) or by the flush request signal (*doFlush*), which is a delayed version of the *do\_acquire* signal. Note that by design, no data is recorded when the FSM is triggered by the *doFlush* signal. Given the fact that there is no mechanical shutter in the system, *doFlush* is used to implement an electronic shutter. Indeed, the effective exposure time of the CCD, is the amount of time elapsed between the last flush request and the new acquisition request.
Once started, the FSM moves to the *preVert* state, where it requests the removal of the optically covered lines to the ‘vertical timing’ module, thanks to the *startV* and *shiftCount* signals. As soon as the vertical shifting is done (*shiftDone*), the inner state machine loop processes each horizontal line (states *Vshift* to *postHor*). Likewise to the vertical shifting, optically covered pixels in the horizontal direction are removed before (*preH*) and after (*postH*) the optically effective zone (*validH*). Finally, when the full CCD processing is done, a *CCD\_done* signal is generated to inform the ‘DAQ manager’ module of the timing generation completion.
The PBLK signal, used to force the CCD processor data output at zero, is activated when the FSM is not moving on the inner loop, in other word, it is the complementary of HBLK. CLOPB, used to remove residual offsets in the CCD processor signal chain, is activated when reading the last optically covered pixels of each line (*postH* pixels). A *valid\_pixel* signal is constructed to ease the tasks of the acquisition manager module (see section \[daqManagerSec\]). This signal is activated only during the *validHor* state and if the line currently accessed is in the allowed range determined by *vertical start* and *vertical stop*. Indeed, the complete CCD sensitive zone can never be utilized for many reasons (alignment issues, exposed area, ...). The vertical selection was implemented to reduce the data volume by removing the meaningless lines.
The ‘ADC timing’ modules, that are used to precisely control the CCD processor sampling times, are activated at start-up time by *enable\_SHPD* and are free running. Each module is composed of a FSM operated at 80MHz and a fast Dual Data Rate serializer (OSERDES) operated at 240MHz. The FSM, which loops every eight 80MHz clock cycles (or steps), constructs and feeds the adequate 6 bit sub-step word to be serialized at 480Mbit/s at every steps. Thus, a tuning resolution of about 2ns (or 10MHz/48) can be reached. To set the start and stop times, 7 bit words are used. The three MSB of the word select during which step the signal must be activated, while the four LSB select at which sub-step the activation will take place.
Figure \[timingDiag\] gives an overview of the various signal timings involved in a CCD readout. The upper part of the diagram, depicts the behavior of the signals controlled by the ‘timing core’ inner FSM loop. The lower part of the diagram, shows the details of the ADC timing and the horizontal timing within a pixel clock cycle.
![Overview of the various signal timings involved in a CCD readout. The upper part of the diagram, depicts the behavior of the signals controlled by the ‘timing core’ inner FSM loop. The lower part of the diagram, shows the details of the ADC timing and the horizontal timing within a pixel clock cycle.[]{data-label="timingDiag"}](./figs/CCDtimings){width="90.00000%"}
DAQ manager module description {#daqManagerSec}
------------------------------
A block diagram of the ‘DAQ manager’ module is shown in figure \[daqDiag\]. The module is composed of two finite state machines (FSM): the ‘acquisition sequencer’ and the ‘data manager’.
![Block diagram of the ‘DAQ manager’ module. The module is composed of two finite state machines (FSM): the acquisition sequencer (l.h.s) and the data manager (r.h.s).[]{data-label="daqDiag"}](./figs/daqBlockDiag){width="90.00000%"}
The ‘acquisition sequencer’ FSM is restarted every second by a timer. Each CCD to be acquired (selection through the selection mask signal), is read-out one after another. At first, the ‘CCD processor’ is waken-up via a request to the ‘serializer’ module. This operation takes about 4ms. Then the ‘CCD timing’ module is started with the *do\_acquire* signal (duration of about 15ms) and the FSM moves directly in the *inter-CCD* waiting state. The waiting timer is set to 60ms, but given the fact that a CCD readout takes 15ms, the real waiting time with no operation is indeed 45ms. As soon as the CCD acquisition is done, a request is sent to the serializer to sleep the ‘CCD processor’.
The ‘data manager’ FSM is started at each CCD acquisition request. It first writes a data header, which contains useful information to determine if the data selection was used or not. If data selection is used the 15 LSB data only contain the CCD number, otherwise, they additionally contain the *vertical start* and *vertical stop* parameters. Indeed, when data selection is used, the valid pixel data words are not stored one after another but rather two data words per pixel, since their position in the payload do not reflect their coordinates anymore. In that latter case, the two words data results from the concatenation of the coordinate word composed of 17 bit and of the pixel digitized value (12 bit). Finally, once the full CCD payload is written in the memory, an End of Packet (EOP) is written in the FIFO memory. The EOP is composed of one or two words, again depending on the acquisition mode. In the full readout mode, the EOP word repeats the read-out CCD number, and the ‘vertical start’ and ‘vertical stop’ parameters. In the discrimination mode, the EOP two words contain the read-out CCD number and the number of pixels discriminated. Naturally, the use of the data selection mode makes sense only for situations where less than half of the pixels contains data above threshold.
Performances
============
Several tests of the detector were performed, first with $\alpha$ particles of varying energies to check the energy response of the system, then with a cold neutron beam at the PF1B beam line at the Institute Laue Langevin (ILL).
Particles are identified by looking at the pixel with the highest ADC value. On the $11\times11$ pixels matrix centered on that pixel, three quantities are reconstructed : the total sum of ADC values $\Sigma$ and the weighted averages $x$ and $y$ of the position in both directions. This procedure is then repeated on the next remaining highest ADC value pixels until all particles have been reconstructed
$\alpha$ measurement
--------------------
Energy measurement is not in itself necessary for the use of the UCNBox detector. Nevertheless, measuring the energy resolution and the linearity of the sensor is a test of the efficiency of the system. To this aim, an ^241^Am $\alpha$ source was used. The primary 5.48MeV particles were slowed down by a 12m aluminum foil. The energy was further decreased by changing the distance between the source and the sensor in air at 1bar, thus allowing to reach energies between 1 and 2MeV. This procedure widens significantly the energy distribution and reduces the precision of the measurement. For four different positions, the average energy was estimated using NIST tables [@NISTtable]. The resulting sum ADC spectra are presented in figure \[fig-calib\] with the resulting calibration curve. The response is found to be linear, within the precision of this measurement. The offset is set to zero, and the resulting fit gives a $\chi^2/Ndf = 2.13/3$, which is consistant with linearity. The slope translates into a collected charge of $7550\pm50$adu/MeV$ = 0.0369\pm0.0002$pC/MeV, whereas one would expect a created charge of $0.0443$pC/MeV in pure silicon. The difference can be accounted for by pair recombination within the CCD during the large exposition time (1s) and the clustering algorithm. It does not impact the final performance as $\alpha$ particles from 0.8MeV to 2MeV are clearly identified.
![Energy calibration and linearity: on the left the four energy spectra corresponding to different source-sensor distances and therefore to different peak energies; on the right the calibration curve obtained from the fit of the ADC peak.[]{data-label="fig-calib"}](./figs_ext/Calib_b_bw "fig:"){width="49.00000%"} ![Energy calibration and linearity: on the left the four energy spectra corresponding to different source-sensor distances and therefore to different peak energies; on the right the calibration curve obtained from the fit of the ADC peak.[]{data-label="fig-calib"}](./figs_ext/Calib_a_bw "fig:"){width="49.00000%"}
Neutron measurement
-------------------
The energy resolution was further investigated by exposing the detector to cold neutrons at the PF1B beam-line at ILL. For this experiment, the detector was in a dark room, not fully isolated from ambient light. Therefore the pixel threshold was set relatively high. Simulation of the attenuation of $\alpha$ particles produced in the boron layer was performed using SRIM [@SRIM]. The particle energy was determined using the previous calibration. The comparison between reconstructed data and simulation, shown in figure \[fig-neutron\], allows to extract an energy resolution of $58$keV. This value is sufficient for the purpose of the detector and will probably improve when lowering the pixel threshold.
![Energy spectrum of $\alpha$ particles produced in the boron layer by neutron capture. The blue lines are the result of SRIM simulations of the setup with (plain line) or without (dashed line) accounting for detector resolution.[]{data-label="fig-neutron"}](./figs_ext/EnergyAlpha){width="55.00000%"}
Summary {#SummarySec}
=======
To read-out a newly developed low energy neutron detector based on a set of 8 CCDs (sensitive area of $ \rm 300 \, mm \times 0.8 \, mm$), a dedicated electronics was designed. This electronics had to provide various features such as exposure time adjustment (0ms to 970ms), LED light calibration, embedded data-reduction, minimization of dead-time (< 1%) and low power usage (about 3W) to operate under vacuum. Additionally, the mechanical support had to provide a good thermal coupling to maintain the device at reasonable temperatures under vacuum and, at the same time, a precise adjustment mechanism to permit the relative height alignment of the CCDs. Using an $\alpha$ source of $^{241}$Am and a cold neutron beam, the performances of the full system (mechanical support, CCD and readout electronics) have been checked. In summary, this electronics is able to read-out simultaneously the eight CCDs at a rate of 1Hz and to meet all experimental requirements.
[99]{} D. Dubbers and M. G. Schmidt, Rev. Mod. Phys. [**83**]{} (2011) 1111.
V. V. Nesvizhevsky [*et al.*]{}, Nature [**415**]{} (2002) 297.
V. V. Nesvizhevsky, Physics Uspekhi [**53**]{} (2010) 645.
T. Jenke [*et al.*]{}, Nature Phys. [**7**]{} (2011) 468.
G. Pignol, Int. J. Mod. Phys. A [**24**]{} (2015) 1530048.
J. Jakubek [*et al.*]{}, Nucl. Instrum. Methods A [**600**]{} (2009) 651.
J. Jakubek [*et al.*]{}, Nucl. Instrum. Methods A [**607**]{} (2009) 45.
S. Kawasaki [*et al.*]{}, T. Sanuki c, S.Sonoda Nucl. Instrum. Methods A [**615**]{} (2010) 42.
T. Lauer [*et al.*]{}, Eur. Phys. J. A [**47**]{} (2011) 150.
D. Roulier [*et al.*]{}, Adv. High Energy Phys. [**2015**]{} (2015) 730437.
A. Bes et al., A position-sensitive neutron detector based on $^{10}$B coated CCD, in preparation.
[ADDI7100 datasheet on analog devices website.](http://www.analog.com/en/products/audio-video/cameracamcorder-analog-front-ends/addi7100.html)
UG472: 7 series FPGA Clocking ressources, Xilinx. [Image sensor handbook](http://www.hamamatsu.com/resources/pdf/ssd/e05_handbook_image_sensors.pdf), p25, Hamamatsu.
Berger, M.J., Coursey, J.S., Zucker, M.A., and Chang, J. (2005), ESTAR, PSTAR, and ASTAR: Computer Programs for Calculating Stopping-Power and Range Tables for Electrons, Protons, and Helium Ions(version 1.2.3). \[Online\] Available: <http://physics.nist.gov/Star>. National Institute of Standards and Technology, Gaithersburg, MD.
J. F. Ziegler, SRIM - The stopping and range of ions in matter (2010), [10.1016/j.nimb.2010.02.091](http://dx.doi.org/10.1016/j.nimb.2010.02.091) Nucl. Instrum. Methods Phys. Res. Sect. B, vol. 268, pp1818-1823
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A global picture of a random particle movement is given by the convex hull of the visited points. We obtained numerically the probability distributions of the volume and surface of the convex hulls of a selection of three types of self-avoiding random walks, namely the classical Self-Avoiding Walk, the Smart-Kinetic Self-Avoiding Walk, and the Loop-Erased Random Walk. To obtain a comprehensive description of the measured random quantities, we applied sophisticated large-deviation techniques, which allowed us to obtain the distributions over a large range of the support down to probabilities far smaller than $P = 10^{-100}$. We give an approximate closed form of the so-called large-deviation rate function $\Phi$ which generalizes above the upper critical dimension to the previously studied case of the standard random walk. Further we show correlations between the two observables also in the limits of atypical large or small values.'
author:
- Hendrik Schawe
- 'Alexander K. Hartmann'
- 'Satya N. Majumdar'
bibliography:
- 'lit.bib'
title: 'Large Deviations of Convex Hulls of Self-Avoiding Random Walks'
---
Introduction
============
The standard random walk is a simple Markovian process, which has a history as a model for diffusion. There are many exact results known [@hughes1996random]. If memory is added to the model, e.g., to interact with the past trajectory of the walk, analytic treatment becomes much harder. A class of self-interacting random walks that we will focus on in this study, are *self-avoiding* random walks, which live on a lattice and do not visit any site twice. This can be used to model systems with excluded volume, e.g., polymers whose single monomers can not occupy the same site at once [@Madras2013]. There are more applications which are not as obvious, e.g., a slight modification of the *Smart-Kinetic Self-Avoiding Walk* traces the perimeter of critical percolation clusters [@weinrib1985kinetic], while the *Loop-Erased Random Walk* can be used to study spanning trees [@manna1992spanning] (and vice versa [@Majumdar1992Exact]).
One of the central properties of random walk models is the exponent $\nu$, which characterizes the growth of the end-to-end distance $r$ with the number of steps $T$, i.e., $r \propto T^\nu$. While this has the value $\nu = 1/2$ for the standard random walk, its value is larger for the self-avoiding variations, which are effectively pushed away from their past trajectory. In two dimensions, this value (and other properties) can often be obtained by the correspondence to Schramm-Loewner evolution [@cardy2005sle; @lawler2002scaling; @Lawler2011; @Kennedy2015]. But between two dimensions and the upper critical dimension, above which the behavior is the same as the standard random walk, Monte Carlo simulations are used to obtain estimates for the exponent $\nu$.
Here we want to study the convex hulls of a selection of self-avoiding walk models featuring larger values of $\nu$. The convex hull allows one to obtain a global picture of the space occupied by a walk, without exposing all details of the walk. As an example, convex hulls are used to describe the home ranges of animals [@mohr1947; @worton1987; @boyle2009]. Namely, we will look at the *Smart-Kinetic Self-Avoiding Walk* (SKSAW), the classical *Self-Avoiding Walk* (SAW) and the *Loop-Erased Random Walk* (LERW), since they span a large range of $\nu$ values and are well established in the literature. About the convex hulls of standard random walks we already know plenty properties. The mean perimeter and area are known exactly since over 20 years [@Letac1980Expected; @Letac1993explicit] for large walk lengths $T$, i.e., the Brownian Motion limit. Since then simpler and more general methods were devised, which are based on using Cauchy’s formula with relates the support function of a curve to the perimeter and the area enclosed by the curve [@Majumdar2009Convex; @Majumdar2010Random]. More recently also the mean hypervolume and surface for arbitrary dimensions was calculated [@Eldan2014Volumetric]. For discrete-time random walks with jumps from an arbitrary distribution, the perimeter of the convex hull for finite (but large) walk lengths $T$ were computed explicitly [@grebenkov2017mean]. For the case of Gaussian jump lengths even an exact combinatorial formula for the volume in arbitrary dimensions is known [@kabluchko2016intrinsic]. For the variance there is an exact result for Brownian bridges [@Goldman1996]. Concerning the full distributions, no exact analytical results are available. Here sophisticated large-deviation simulations were used to numerically explore a large part of the full distribution, i.e., down to probabilities far smaller than $10^{-100}$ [@Claussen2015Convex; @Dewenter2016Convex; @schawe2017highdim].
Despite this increasing interest in the convex hulls of standard random walks, there seem to be no studies treating the convex hulls of self-avoiding walks. To fill this void, we use Markov chain Monte Carlo sampling to obtain the distributions of some quantities of interest over their whole support. To connect to previous studies [@Claussen2015Convex; @Dewenter2016Convex; @schawe2017highdim] we also compare the aforementioned variants to the standard random walk on a square lattice (LRW). We are mainly interested in the full distribution of the area $A$ and the perimeter $L$ of $d=2$ dimensional hulls for walks in the plane, since the effects of the self-interactions are stronger in lower dimensions. Though, we will also look into the volume $V$ in the $d=3$ dimensional case. In the past study on standard random walks [@schawe2017highdim] we found that the full distribution can be scaled to a universal distribution using only the exponent $\nu$ and the dimension for large walk lengths $T$. For the present case, where a walk might depend on its full history, one could expect a more complex behavior. Nevertheless, our results presented below show convincingly that also for self-interacting walks the distributions are universal and governed mainly by the exponent $\nu$, except for some finite-size effects, which are probably caused by the lattice structure. Further we use the distributions to obtain empirical large-deviation rate functions [@Touchette2009large], which suggests that a limiting rate function is mathematically well defined. We also give an estimate for the rate function, which is compatible with the known case of standard random walks and with all cases under scrutiny in this study.
Models and Methods {#sec:mm}
==================
This sections gives a short overview over the used models and methods, with references to literature more specialized on the corresponding subject. Where we deem adequate, also technical details applicable for this study are mentioned.
Sampling Scheme
---------------
To generate the whole distribution of the area or perimeter of the convex hull of a random walk over its full support, a sophisticated Markov chain Monte Carlo (MCMC) sampling scheme is applied [@Hartmann2002Sampling; @Hartmann2011]. The Markov chain is here a sequence of different walk configurations. The fundamental idea is to treat the observable $S$, i.e., the perimeter, area or volume, as the energy of a physical system which is coupled to a heat bath with adjustable “temperature” $\Theta$ and to sample its equilibrium distribution using the Markov chain. This can be easily done using the classical Metropolis algorithm [@metropolis1953equation]. Therefore the current walk configuration is changed a bit (the precise type of change is dependent on the type of walk, we are looking at and is explained in the following sections). The change is accepted with the acceptance probability $$\begin{aligned}
\label{eq:pacc}
p_\mathrm{acc} = \min\{1,\operatorname{e}^{-\Delta S / \Theta}\}
\end{aligned}$$ and rejected otherwise. The $\Theta$ will then bias the configuration towards specific ranges of $S$. Configurations at small and negative $\Theta$ will show larger than typical $S$, small and positive $\Theta$ show smaller than typical $S$ and large values independent of the sign sample configurations from the peak of the distribution. Fig. \[fig:saw\_temp\_cmp\] shows typical walk configurations of the self-avoiding walk at different values of $\Theta$.
In a second step, histograms of the equilibrium distribution $P_\Theta(S)$ are corrected for the bias introduced via $\Theta$ as follows. $$\begin{aligned}
P(S) = \operatorname{e}^{S / \Theta} Z(\Theta) P_\Theta(S)
\end{aligned}$$ The free parameter $Z(\Theta)$ can be obtained by enforcing continuity and normalization of the distribution. We do not present further details here, because the algorithm [@Hartmann2002Sampling] has been applied and explained in detail several times, also in a very general form [@Hartmann2014high]. In particular, the algorithm was already used successfully in other studies looking at the large deviation properties of convex hulls of random walks [@Claussen2015Convex; @Dewenter2016Convex].
![\[fig:saw\_temp\_cmp\] (color online) Typical SAW configurations with $T=200$ steps and their convex hulls at different temperatures $\Theta$. $\Theta = \pm \infty$ corresponds to a typical configuration without bias. ](SAW_temp_cmp){width="0.7\linewidth"}
Lattice Random Walk (LRW)
-------------------------
All of the self-interacting random walks, which are the focus of this study, are typically treated on a lattice. Hence, we will start by introducing the simple, i.e., non-interacting, isotropic random walk on a lattice. For simplicity we will use a square lattice with a lattice constant of $1$. A realization consists of $T$ randomly chosen discrete steps ${\bm{\delta}}_i$. Here we use steps between adjacent lattice sites, i.e., $d$-dimensional Cartesian base vectors ${\bm{e}}_i$, which are drawn uniformly from $\{\pm{\bm{e}}_i\}$. The realization can be defined as the tuple of the steps $({\bm{\delta}}_1, ..., {\bm{\delta}}_T)$ and the position at time $\tau$ as $$\begin{aligned}
{\bm{x}}(\tau) = {\bm{x}}_0 + \sum_{i=1}^\tau {\bm{\delta}}_i.
\end{aligned}$$ Here we set the start point ${\bm{x}}_0$ at the coordinate origin. The set of visited sites is therefore $\mathcal P = \{{\bm{x}}(0), ..., {\bm{x}}(T)\}.$
The central quantity of the LRW is the average end-to-end distance $$r = \sqrt{\langle ({\bm{x}} (T)- {\bm{x}}_0)^2 \rangle} \,,$$ where $\langle \ldots \rangle$ denotes the average over the disorder. It grows polynomially and is characterized by the exponent $\nu$ via $r \propto T^\nu$. For the LRW it is $\nu = 1/2$, which is typical for all diffusive processes.
As the change move for the Metropolis algorithm, we replace a randomly chosen ${\bm{\delta}}_i$ by a new randomly drawn displacement. Since our quantity of interest is the convex hull, i.e., a global property of the walk, we do not profit much from local moves, e.g., crankshaft moves. Thus we use this simple, global move.
Smart-Kinetic Self-Avoding Walk (SKSAW)\[sec:sksaw\]
----------------------------------------------------
The Smart-Kinetic Self-Avoiding Walk (SKSAW) [@weinrib1985kinetic; @kremer1985indefinitely] is probably the most naive approach to a self-avoiding walk. It grows on a lattice and never enters sites it already visited. Since it is possible to get trapped on an island inside already visited sites, this walk needs to be *smart* enough to never enter such traps.
In $d=2$ it is possible to avoid traps using just local information in constant time using the *winding angle* method [@kremer1985indefinitely]. In conjunction with hash table backed detection of occupied sites, a realization with $T$ steps can be constructed in time $\mathcal O(T)$.
This method will typically yield longer stretched walks than the LRW, due to the constraint that it needs to be self-avoiding. This can be characterized by the exponent $\nu$, which is larger than $1/2$ in $d=2$.
The sketch Fig. \[fig:saw\_prob\] shows that this ensemble does not contain every configuration with the same probability but prefers closely winded configurations. This is also visible in Fig. \[fig:SKSAW\]. This is characterized by the exponent $\nu=4/7$ [@Kennedy2015] which is larger than the $\nu$ for LRW, but smaller than for the SAW. Also note that it is conjectured that the upper critical dimension is $d=3$ [@kremer1985indefinitely], i.e., $\nu = 1/2$ for all $d \ge 3$ – possibly with logarithmic corrections in $d=3$. Therefore only $d=2$ is simulated in this study.
![\[fig:saw\_prob\] Decision tree visualizing the probability to arrive at certain configurations following the construction rules of the SKSAW. Not all possible configuration have the same probability, hence this rules define a different ensemble than SAW. ](saw_prob)
While it is easy to draw realizations from this ensemble uniformly, i.e., simple sampling, it is not so straight forward to apply the MCMC changes. If one just changes single steps like for the LRW, and accepts if it is self-avoiding or rejects if it is not, one will generate all self-avoiding walk configurations with equal probability. Our approach to generate realizations according to this ensemble handles the construction of the walk as a *black box*. It acts on the random numbers used to generate a realization from scratch. During the MCMC at each iteration one random number is replaced by a new random number and a SKSAW realization is regenerated from scratch using the modified random numbers [@Hartmann2014high]. This change is then accepted according to Eq. and undone otherwise.
Self-Avoiding Random Walk (SAW)
-------------------------------
While the above mentioned SKSAW does produce self-avoiding walks, SAW denotes another ensemble. The ensemble where realizations are drawn uniformly from the set of all self-avoiding configurations. It is not trivial to sample from this distribution efficiently. The black box method used for SKSAW is not feasible, since the construction of a SAW takes time exponential in the length with simple methods like dimerization [@dimerization; @Madras2013]. It is possible to perform changes directly on the walk configuration and accept them according to Eq. , but their rejection rate is typically quite high and the resulting configurations are very similar [@Madras2013], which makes this inefficient. The state of the art method to sample SAW is the *pivot algorithm* [@Madras2013]. It chooses a random point and uses it as the pivot for a random symmetry operation, i.e., rotation or mirroring. If the resulting configuration is not self avoiding, it is rejected. Otherwise we accept it with the temperature dependent acceptance probability Eq. .
As mentioned previously, the exponent $\nu=3/4$ [@lawler2002scaling] is larger than for the SKSAW. Since the upper critical dimension for SAW is $d=4$, this study will also look at $d=3$, where an exact value of $\nu$ is not known and the best estimate is $\nu = 0.587597(7)$ [@Clisby2010Accurate], though our focus is on $d=2$ for this type.
While there are highly efficient implementations of the pivot algorithm [@Clisby2010Accurate; @clisby2010efficient] the time complexity of the problem at hand is dominated by the time needed to construct the convex hull, thus we go with the simple hash table based $\mathcal O(T)$ approach [@Madras2013].
Loop-Erased Random Walk (LERW)
------------------------------
The LERW [@Lawler1980Self] uses a different approach to achieve the self-avoiding property. It is built as a simple LRW but each time a site is entered for the second time, the loop that is formed, i.e., all steps since the first entering of this site, is erased. While this ensures no crossings in the walk, the resulting ensemble is different from the SAW ensemble and the walks are longer stretched out, as characterized by the larger exponent $\nu = 4/5$ [@Lawler2011; @Guttmann1990Critical; @Majumdar1992Exact]. Similar to the SAW the upper critical dimension is $d=4$ and an estimate for $d=3$ is $\nu = 0.61576(2)$ [@Wilson2010].
For construction – similar to SKSAW – we need to keep all used random numbers and change them in the MCMC algorithm. This leads to a dramatically higher memory consumption than simple sampling, where each loop can be discarded as soon as it is closed.
Convex Hulls
------------
We will study the *convex hulls* $\mathcal C$ of the sites visited by the random walk $\mathcal P$. The convex hull of a point set $\mathcal P$ is the smallest polytope containing all Points $P_i \in \mathcal P$ and all line segments $(P_i, P_j)$. Some example hulls are shown in Fig. \[fig:rw\].
Convex hulls are one of the most basic concepts in computational geometry [^1] with noteworthy application in the construction of Voronoi diagrams and Delaunay triangulations [@brown1979Voronoi].
For point sets in the $d=2$ plane, we use Andrew’s *Monotone Chain* [@Andrew1979Another] algorithm for its simplicity and *Quickhull* [@Bykat1978Convex] as implemented by *qhull* [@Barber1996thequickhull] for $d=3$. Both algorithms have a time complexity of $\mathcal O(T \ln T)$. In $d=2$ Andrew’s Monotone Chain algorithm results in ordered points of the convex hull. Adjacent points $(i, j)$ in this ordering are the line segments of the convex hull. Quickhull results in the simplical facets of the convex hull.
To obtain the perimeter of a $d=2$ convex hull, we sum the lengths of its line segments $L_{ij}$. To calculate the area and the volume, we use the same fundamental idea. In both cases we subdivide the area/volume into simplexes, i.e., triangles for the area and tetrahedra for the volume. Therefore we choose an arbitrary fixed point $p_0$ inside of the convex hull and construct a simplex for each facet $f_m$, i.e., for $d=2$ each line segment of the hull $f_m = (i, j)$ forms a triangle $(i, j, p_0)$ and each triangular face $f_m = (i, j, k)$ of a $d=3$ dimensional polyhedron, forms a tetrahedron with $p_0$. The volume of a triangle is trivially $$\begin{aligned}
A_{ijp_0} = \frac{1}{2} \operatorname{dist}(f_m, p_0) L_{ij},
\end{aligned}$$ where $\operatorname{dist}(f_m, p_0)$ is the perpendicular distance from a facet $f_m$ to a point $p_0$. Since the union of all triangles built this way, is the whole polygon, the sum of their areas is the area of the polygon. Similar the volume of a polyhedron is the sum of the volumes of all tetrahedra constructed from its faces. The volume of the individual tetrahedra is given by $$\begin{aligned}
V_{ijkp_0} = \frac{1}{3} \operatorname{dist}(f_m, p_0) A_{ijk}.
\end{aligned}$$
For random walks on a lattice with $T$ steps of length $1$ in $d$ dimensions the maximum volume is $$\begin{aligned}
\label{eq:max}
S_\mathrm{max} = \frac{(T/{{\ensuremath{d_{\mathrm{e}}}}})^{{\ensuremath{d_{\mathrm{e}}}}}}{{{\ensuremath{d_{\mathrm{e}}}}}!}
\end{aligned}$$ for $T$ divisible by the effective dimension ${{\ensuremath{d_{\mathrm{e}}}}}$ of the observable, e.g., 2 for the area of a planar hull or 3 for the volume in three dimensions. For example, the configuration of maximum area corresponds to an L-shape, i.e., $A_\mathrm{max} = \frac{T^2}{8}$. This form can be derived by the general volume of an $d$-dimensional simplex defined by its $d+1$ vertices ${\bm{v}}_i$ [@Stein1966Volume] $$\begin{aligned}
V = \frac{1}{d!} \det{({\bm{v}}_1 - {\bm{v}}_0, \ldots, {\bm{v}}_d - {\bm{v}}_0)}.
\end{aligned}$$ Without loss of generality, we set ${\bm{v}}_0$ to be the coordinate origin. To achieve maximum volume all ${\bm{v}}_i, i>0$ need to be orthogonal and of equal length. Thus a random walk going $T/d$ steps along some base vector ${\bm{e}}_i$ and continuing with $T/d$ steps in direction ${\bm{e}}_{i+1}$ has a convex hull defined by the tetrahedron specified by ${\bm{v}}_i = \sum_{j=1}^{i} \frac{T}{d} {\bm{e}}_j$. The matrix $M = ({\bm{v}}_1, \ldots, {\bm{v}}_d)$ is thus triangular and its determinant is the product of its diagonal entries $M_{ii} = \frac{T}{d}$ which leads directly to Eq. . An exception occurs in $d=2$ where the perimeter is $L_\mathrm{max} = 2T$.
Results
=======
The focus of this work lies on $d=2$ dimensional SAW and LERW. The results for higher dimensions and for SKSAW are generated with less numerical accuracy. The LRW results also have a lower accuracy as their purpose is mainly to scrutinize the effect of the lattice structure underlying all considered walk types in comparison to the non-lattice results from [@schawe2017highdim]. Also not all combinations are simulated, but only those listed with a value in Table \[tab:kappa\].
The same raw data is evaluated for equidistant bins and logarithmic bins. And the respective variants are shown according to the scaling of the $x$-axis.
Correlations
------------
To get an intuition for how the configurations with atypical large areas $A$ or perimeters $L$ look like, we visualize the correlation between these two observables as scatter plots in Fig. \[fig:scatter\].
![\[fig:scatter\] (color online) The top row shows data from simulations biasing towards larger (and smaller) than typical perimeters $L$. The bottom row biases the area $A$. The left column shows data from LRW and the right from SAW both with $T=512$ steps. The results of simple sampling are shown in black. Note that only very narrow parts are covered by simple sampling for the LRW. ](scatter_SAWvsLRW)
Since the smallest possible SAW is an (almost) fully filled square, there can not be instances below some threshold, which explains the gaps on the left side of the scatter plots and of the distributions shown in the following section. In the center of the scatter plots, which is already in probability regions far beyond the capabilities of simple sampling methods, the behavior becomes strongly dependent on the bias.
If biasing for large perimeters (top) the area shows a non-monotonous behavior. First, somehow larger perimeters come along typically with larger areas for entropic reasons, i.e., there are less configurations which are long and thin, and more bulky, which have a larger area. Though, for the far right tail, the only configurations with extreme large perimeters are almost line like and have thus a very small area. Also note that the excluded volume effect of the SAW leads to overall larger areas at the same perimeters.
On the other hand, when biasing for large areas (bottom) the configurations with largest area, which are L-shaped (cf. Fig. \[fig:saw\_temp\_cmp\]), unavoidably have quite large perimeters, hence the scatter plots show an almost linear correlation between area and perimeter. Since the configurations of large areas naturally avoid self intersections, since steps on already visited points do not enlarge the convex hull, the differences between LRW and SAW diminish in the right tail. Note that with the large-area bias, no walks with the very extreme perimeters exist, for the reason already mentioned.
Note however that these scatter plots are very dependent on which observable we are biasing for. In principle we observe that small perimeters are strongly correlated with small areas while for large but not too large perimeters, there is a broad range of area sizes possible. For extremely large perimeters, the area must be small. For a comprehensive analysis, one would need a full two dimensional histogram, wich could be obtained using Wang Landau sampling, but which is beyond the scope of this study and would require a much larger numerical effort. Nevertheless, from looking at Fig. \[fig:scatter\] one can anticipate that the two dimensional histogram would exhibit a strong correlation for small values of $L$ and a broad scatter of the accessible values of $A$ for larger but not too large values of $L$.
Moments and Distributions
-------------------------
The distributions of the different walk types differ considerably. This can be observed in Fig. \[fig:compare\], where distributions of the area $A$ for all types with $T=1024$ steps are drawn. The main part of the distribution shifts to larger values for larger value of $\nu$ as expected and the probability of atypically large areas is boosted even more in the tails.
![\[fig:compare\] (color online) Distribution of all scrutinized walk types with $T=1024$ steps. The vertical line at $A_\mathrm{max}=131072$ denotes the maximum area (Eq. ), i.e., SAW and LERW are sampled across their full support and SKSAW and LRW are not. The inset shows the peak region. The gap on the left is due to excluded volume effects, i.e., there are no configurations with area below some threshold, since this would require self-intersection. ](compareTypes2D)
In the right tail, the distributions seem to bend down. Below, where we show results for different walk sizes $T$, we see that this is a finite-size effect of the lattice structure and the fixed step length. This can be seen also as follows: Since the lattice together with the fixed step length sets an upper bound on the area, the probability plummets near this bound for entropic reasons, i.e., there are for any walk length $T$ only 8 configurations with maximum area (due to symmetries) such that all self-avoiding types will meet at this point. (not visible because the bins are not fine enough)
This is supported from Ref. [@Claussen2015Convex] which shows that the distribution $P(A)$ for standard random walks with Gaussian jumps, i.e., without lattice or fixed step length, do not bend down and have an exponential right tail. We conclude that the deviation from this are thus caused by this difference.
First we will look at the rescaled means $\mu_S = {\ensuremath{\left<S\right>}} / T^{{{\ensuremath{d_{\mathrm{e}}}}}\nu}$, where $S$ is an observable and ${{\ensuremath{d_{\mathrm{e}}}}}$ its effective dimension, as introduced above in Eq. . The scaling is a combination of the scaling of the end-to-end distance $r \propto T^\nu$ and the typical scaling that a $d$-dimensional observable scales as $r^d$ with a characteristic length $r$.
![\[fig:means\] (color online) Scaled means $\mu_A = {\ensuremath{\left<A\right>}} / T^{2\nu}$ and $\mu_L = {\ensuremath{\left<L\right>}} / T^{\nu}$ for different walk types. The lines are fits to extrapolate the asymptotic values shown in Table \[tab:measuredMu\] according to Eq. . Errorbars of the values are smaller than the line of the fit and not shown for clarity. ](means)
Nevertheless, due to finite-size corrections, the ratios $\mu_S={\ensuremath{\left<S\right>}} / T^{{{\ensuremath{d_{\mathrm{e}}}}}\nu}$ will still depend on the walk length. Thus, the measured estimates $\mu_S=\mu_S(T)$ at specific walk lengths $T$ need to be extrapolated to get an estimate of the asymptotic value $\mu_S^\infty = \lim_{T\to \infty}\mu_S(T)$. For the extrapolation we use [@schawe2017highdim] $$\begin{aligned}
\label{eq:extrapolation}
\mu_S(T) = \mu_S^\infty + C_1 T^{-1/2} + C_2 T^{-1} + o(T^{-1}).
\end{aligned}$$ This choice is motivated by a large $T$ expansion for the area $A$ (${{\ensuremath{d_{\mathrm{e}}}}}=2$) of the convex hulls of standard random walks ($\nu=1/2$) with Gaussian jumps [@grebenkov2017mean] $$\begin{aligned}
\frac{{\ensuremath{\left<A\right>}}}{T} = \frac{\pi}{2} + \gamma \sqrt{8\pi}\, T^{-1/2} + \pi(1/4+\gamma^2)\, T^{-1}+ o(T^{-1}),
\end{aligned}$$ where the constant $\gamma= \zeta(1/2)/\sqrt{2\pi}=-0.58259\dots$. A natural guess for a generalization to oberservables of a different effective dimension ${{\ensuremath{d_{\mathrm{e}}}}}$ [@schawe2017highdim] and different walk types would be a similar behavior with different coefficients like Eq. .
Indeed, using this form to estimate the asymptotic means $\mu_S^\infty$ of the observable $S$ yields good fits, as visible in Fig. \[fig:means\]. In fact, for the fit quality we obtain $\chi_\mathrm{red}^2$ values between $0.4$ and $1.7$ (the fit ranges for SKSAW begin at $T=512$, for LRW, SAW and LERW at $T=128$, hinting at more severe corrections to scaling for the former). We assume that the scaling is thus valid for arbitrary random walk types. The resulting fit parameters are shown in Table \[tab:measuredMu\].
For standard random walks with Gaussian jumps the asymptotic means $\mu^\infty_{S,\mathrm{Gaussian}}$ are known [@Eldan2014Volumetric]. These results can be used to predict the corresponding values for LRW. First consider the following heuristic argument for a $d=2$ square lattice. On average a random walk takes the same amount of steps in $x$ and $y$ direction, such that on average two steps displace the walker by $\sqrt{2}$, i.e., the diagonal of a square. In contrast a Gaussian walker with variance $1$ will be displaced on average by $1$ every step. To make both types comparable, we can increase the lattice constant to $\sqrt{2}$, which leads to an average displacement of $1$ per step for the LRW. Using the same argumentation for higher dimensions, we can use the trivial scaling with the lattice constant $S^{{\ensuremath{d_{\mathrm{e}}}}}$ and the length of the diagonal of a unit hypercube $d^{1/2}$, to derive a general conversion: $$\begin{aligned}
\label{eq:lattice}
\mu_{S,\mathrm{LRW}}^\infty = \mu_{S,\mathrm{Gaussian}}^\infty / d^{{{\ensuremath{d_{\mathrm{e}}}}}/ 2}.
\end{aligned}$$ These known results are listed next to our measurements in Table \[tab:measuredMu\] and are within errorbars compatible with our measurements.
------------------------ -------------- ------------- ------------- -------------- --
\[0.05cm\] LRW (exact) $3.5449...$ $0.7854...$ $2.0944...$ $0.21440...$
LRW $3.5441(7)$ $0.7852(2)$ $2.0945(4)$ $0.21445(4)$
SKSAW $4.5355(12)$ $1.2642(5)$ - -
SAW $0.8233(7)$ $0.7714(1)$ $2.069(2)$ $0.1996(2)$
LERW $2.1060(3)$ $0.2300(1)$ $1.6436(2)$ $0.13908(3)$
------------------------ -------------- ------------- ------------- -------------- --
: \[tab:measuredMu\] Asymptotic mean values extrapolated from simulational data and the exactly known values for the standard random walk (LRW). The columns labeled with $\mu_L^\infty$ and $\mu_A^\infty$ are for $d=2$, those labeled with $\mu_{\partial V}^\infty$ and $\mu_V^\infty$ are for $d=3$. For $d=3$ we did not simulate the SKSAW, see Section \[sec:sksaw\]. Also SAW has lower accuracy because of fewer samples in $d=3$.
Since we have data for the whole distributions, a natural question is, whether this scaling does apply over the whole support of the distribution. There is evidence that this is true for the convex hulls of standard random walks [@Claussen2015Convex] in arbitrary dimensions [@schawe2017highdim]. That means the distributions of an observable $S$ for different walk lengths $T$ should collapse onto one universal function $$\begin{aligned}
\label{eq:scaling}
P(S) = T^{-{{\ensuremath{d_{\mathrm{e}}}}}\nu} \widetilde{P}(ST^{-{{\ensuremath{d_{\mathrm{e}}}}}\nu}).
\end{aligned}$$
Fig. \[fig:scaling\] shows the distributions of the $d=2$ area of all considered random walk types scaled according to Eq. . The curves collapse well in the peak region and in the intermediate right tail. In the far right tail clear deviations from a universal curve are obvious, which are the mentioned finite-size effects caused by the lattice.
The distributions look qualitatively similar, though with weaker finite size effects, i.e., a better collapse, for the perimeter $L$ (not shown). In $d=3$, where we have studied the volume, the results also look similar but exhibit stronger finite-size effects (not shown).
Using the full distributions at different values of the walk length $P_T$, we can test if it obeys the large deviation principle, i.e., if $\Phi$ exists, such that the distribution scales as $$\begin{aligned}
\label{eq:largeDev}
P_T \approx \operatorname{e}^{-T\Phi}
\end{aligned}$$ for large values of $T$ [@Touchette2009large]. To simplify comparison, the support of the rate function is usually normalized to $[0, 1]$. Here we achieve this by using the maximum Eq. . Solving Eq. for $\Phi$ results in $$\begin{aligned}
\Phi(S/S_\mathrm{max}) = -\frac{1}{T} \ln P(S/S_\mathrm{max}).
\end{aligned}$$ We plot this for a selection of our results in Fig. \[fig:rate\]. From these plots, $\Phi$ seems to approximately follow a power law in the intermediate right tail, while the finite-size effects caused by the lattice play a major role in the far right tail, which “bends up” consequently.
Assuming that the rate function behaves approximately as a power law, which seems consistent with our data shown in Fig. \[fig:rate\], i.e., $$\begin{aligned}
\Phi(s) \propto s^{\kappa},
\label{eq:power-law}
\end{aligned}$$ the exponent $\kappa$ can be estimated by combining the definition of $\Phi$ Eq. with the scaling assumption Eq. as follows, note that for clarity we use here $S_\mathrm{max} \propto T^{{{\ensuremath{d_{\mathrm{e}}}}}}$. $$\begin{aligned}
\exp{\ensuremath{\left(-T \Phi(S/T^{{{\ensuremath{d_{\mathrm{e}}}}}})\right)}} \sim \frac{1}{T^{\nu {{\ensuremath{d_{\mathrm{e}}}}}}} \widetilde{P}(S/T^{\nu {{\ensuremath{d_{\mathrm{e}}}}}})
\end{aligned}$$ The $1/T^{\nu {{\ensuremath{d_{\mathrm{e}}}}}}$ term on the right-hand side can be ignored next to the exponential. Since the right-hand side is a function of $S/T^{\nu {{\ensuremath{d_{\mathrm{e}}}}}}$, the left-hand side must also be only dependent on $S/T^{\nu {{\ensuremath{d_{\mathrm{e}}}}}}$. This can be achieved by assuming $-\nu {{\ensuremath{d_{\mathrm{e}}}}}\kappa + {{\ensuremath{d_{\mathrm{e}}}}}\kappa = 1$, as one can easily see by using Eq. (\[eq:power-law\]): $$\begin{aligned}
\intertext{Starting from the left-hand side}
& \exp{\ensuremath{\left(-T^{1} \Phi(S/T^{{{\ensuremath{d_{\mathrm{e}}}}}})\right)}} \\
\propto& \exp{\ensuremath{\left(-T^{1} {\ensuremath{\left(S/T^{{{\ensuremath{d_{\mathrm{e}}}}}}\right)}}^{\kappa}\right)}}\\
=& \exp{\ensuremath{\left(-T^{-\nu {{\ensuremath{d_{\mathrm{e}}}}}\kappa + {{\ensuremath{d_{\mathrm{e}}}}}\kappa} {\ensuremath{\left(S/T^{{{\ensuremath{d_{\mathrm{e}}}}}}\right)}}^{\kappa}\right)}}\\
=& \exp{\ensuremath{\left(-{\ensuremath{\left(S/T^{\nu {{\ensuremath{d_{\mathrm{e}}}}}}\right)}}^{\kappa}\right)}}
\end{aligned}$$ From this we can conclude $$\begin{aligned}
\label{eq:kappa}
\kappa = \frac{1}{d_{\mathrm{e}}(1-\nu)},
\end{aligned}$$ which simplifies to the case of the standard random walk above the critical dimension of the given walk type [@schawe2017highdim] $$\begin{aligned}
\kappa_g = \frac{2}{d_{\mathrm{e}}}.
\end{aligned}$$
To compare this crude estimate with the results of our simulations, we do a point-wise extrapolation of the empirical rate functions for fixed walk lengths $T$ as done before in [@Claussen2015Convex; @Dewenter2016Convex; @schawe2017highdim]. For the point-wise extrapolation, we use measurements $\Phi_T$ for multiple values of the walk length $T$ at fixed values of $S/S_\mathrm{max}$. Since our data are discrete due to binning, the values of $\Phi_T$ are obtained by cubic spline interpolation. With these data points, which can be thought of as vertical slices through the plots of Fig. \[fig:rate\], we extrapolate the $T\to\infty$ case with a fit to a power law with offset $$\begin{aligned}
\Phi = a T^b + \Phi_\infty.
\end{aligned}$$ The extrapolated values are marked with black dots in Fig. \[fig:rate\]. Since finite-size effects have major impact on the tails due to the lattice structure, we expect that our estimate is only valid for the intermediate right tail of our simulational data. To estimate sensible uncertainties, we fit different ranges of our data and give the center of the range of the obtained $\kappa$ as our estimate with an error including the extremes of the obtained $\kappa$. The black lines in Fig. \[fig:rate\] are our expected values, which are in all examples compatible with some range of our extrapolated data.
All exponents $\kappa$ we calculated, together with our expectations, are listed in Table \[tab:kappa\]. A more detailed discussion of the examples shown in Fig. \[fig:rate\] follows.
In Fig. \[fig:rate:LRW\] the LRW is shown, which is equivalent to Brownian motion in the large $T$ limit for which Ref. [@Claussen2015Convex; @schawe2017highdim] showed the rate function to behave like a power law with exponent $\kappa=1$ for the area in $d=2$. Using the above mentioned procedure we obtain $\kappa = 0.99(2)$ which is in perfect agreement with the expectation $\kappa = 1$.
Fig. \[fig:rate:SKSAW\] shows the same for the SKSAW. The obtained asymptotic rate function’s exponent $\kappa = 1.28(12)$ is compatible with our expectation, though the stronger finite-size effects, lead to larger uncertainties of our estimate.
Fig. \[fig:rate:SAW\] shows the same but for the volume of the SAW in $d=3$ dimensions. The finite-size effects are apparently stronger for the volume in $d=3$, as the slope of the right-tail rate function gets less steep with increasing system size.
Fig. \[fig:rate:LERW\] for the perimeter of a $d=2$ dimensional LERW. In contrast to the area and volume the far right tail of the perimeter seems to bend down instead of up, albeit slightly. Though in the intermediate right tail, the rate function seems to behave as expected.
--------------- --------------- ------------ --------------- ------------
\[0.1cm\] LRW $1 $ $0.99(2)$ $2 $ -
SKSAW $\frac{7}{6}$ $1.28(12)$ $\frac{7}{3}$ -
SAW $2 $ $2.2(4)$ $4 $ $4.11(14)$
SAW $d=3$ $0.809... $ $0.92(11)$ $1.214... $ -
LERW $\frac{5}{2}$ $2.57(24)$ $5 $ $4.82(19)$
LERW $d=3$ $0.867... $ $0.89(9)$ $1.299... $ -
--------------- --------------- ------------ --------------- ------------
: Comparison of expected and measured rate function exponent $\kappa$. The value is the center of multiple fit ranges and the error is chosen such that the largest and the smallest result is enclosed. []{data-label="tab:kappa"}
In general, our data supports the convergence to a limiting rate function, which, mathematically speaking, means that the *large-deviation principle* holds. This means that the distributions are somehow well behaved and might be accessible to analytical calculations. Though the estimate for what the rate function $\Phi$ actually is, can possibly be improved. However, since our estimate for $\kappa$ is always compatible with our measurements it appears plausible that also for interacting walks the distribution of the convex hulls is governed by the scaling behavior of the end-to-end distance, as given by the exponents $\nu$.
Conclusions {#sec:conclusion}
===========
We numerically studied the area and perimeter of the convex hulls of different types of self-avoiding random walks in the plane and to a lesser degree the volume of their convex hulls in $d=3$ dimensional space. By applying sophisticated large-deviation algorithms, we calculated the full distributions, down to extremely small probabilities like $10^{-400}$. We also obtained corresponding rate functions of these observables. Our data support a convergence of the rate functions, which means the large-deviation principle seems to hold. We observed a generalized scaling behavior, which was before established for standard random walks. Thus, although the self-avoiding types of walk exhibit a more complicated behavior as compared to standard random lattice walks, and although the limiting scaled distributions of their convex hull’s volume and surface look quite different for the various walk cases, in the end the convex hull behavior seems to be still governed by the single end-to-end distance scaling exponent $\nu$.
We also observed, rather expectedly, that the two observables area and perimter are highly correlated for small values. For large but not too large values of the perimeter, many different values of the area are possible, but statistically dominated by rather small values of the area. Extremly large values of the perimeter are only feasible with shrinking area.
Finally, we gave estimates for the large $T$ asymptotic mean values of the mentioned observables. These might be of interest for attempts to calculate these values analytically.
For future studies it could be interesting to look closer into the correlations between different observables that we briefly noticed. For a more throughout study, it would be useful to obtain full two-dimensional histograms.
This work was supported by the German Science Foundation (DFG) through the grant HA 3169/8-1. HS and AKH thank the LPTMS for hospitality and financial support during one and two-month visits, respectively, where considerable part of the projects were performed. The simulations were performed at the HPC clusters HERO and CARL, both located at the University of Oldenburg (Germany) and funded by the DFG through its Major Research Instrumentation Programme (INST 184/108-1 FUGG and INST 184/157-1 FUGG) and the Ministry of Science and Culture (MWK) of the Lower Saxony State. We also thank the GWDG Göttingen for providing computational resources.
[^1]: 3 of the first 4 examples for static problems of computational geometry in the Wikipedia can utilize convex hulls for their solution (<https://en.wikipedia.org/wiki/Computational_geometry>, 12.01.2018).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Many applications involving complex multi-task problems such as disaster relief, logistics and manufacturing necessitate the deployment and coordination of heterogeneous multi-agent systems due to the sheer number of tasks that must be executed simultaneously. A fundamental requirement for the successful coordination of such systems is leveraging the specialization of each agent within the team. This work presents a Receding Horizon Planning (RHP) framework aimed at scheduling tasks for heterogeneous multi-agent teams in a robust manner. In order to allow for the modular addition and removal of different types of agents to the team, the proposed framework accounts for the capabilities that each agent exhibits (e.g. quadrotors are agile and agnostic to rough terrain but are not suited to transport heavy payloads). An instantiation of the proposed RHP is developed and tested for a search and rescue scenario. Moreover, we present an abstracted search and rescue simulation environment, where a heterogeneous team of agents is deployed to simultaneously explore the environment, find and rescue trapped victims, and extinguish spreading fires as quickly as possible. We validate the effectiveness of our approach through extensive simulations comparing the presented framework with various planning horizons to a greedy task allocation scheme.'
address:
- 'Institute for Robotics and Intelligent Machines, Georgia Institute of Technology, Atlanta, GA 30332, USA (e-mail: {emamy, sean.t.wilson, magnus}@gatech.edu).'
- 'Siemens Corpororate Technology, Princeton, NJ 08540, USA (e-mail: {mathias.hakenberg, ulrich.muenz}@siemens.com)'
author:
- Yousef Emam
- Sean Wilson
- Mathias Hakenberg
- Ulrich Munz
- Magnus Egerstedt
bibliography:
- 'ifacconf.bib'
title: 'A Receding Horizon Scheduling Approach for Search & Rescue Scenarios'
---
Scheduling Algorithms, Optimization Problems, Multiagent Systems, Robotics, Search and Rescue.
Introduction {#sec:intro}
============
Multi-robot systems are well suited to solve complex tasks in dynamic and dangerous environments due to their redundancy, ability to operate in parallel, and system level fault tolerance to individual failure as highlighted in [@brambilla2013swarm; @sahin2004swarm]. Multi-Robot Task Allocation (MRTA) deals with the assignment of agents to tasks in order to achieve an overall system goal within the constraints of the deployment setting. Therefore, in order to leverage the potential multi-robot systems have to successfully operate in dynamic and dangerous environments to solve complex problems such as disaster response, search and rescue, environmental monitoring, and automated warehousing, effective methods for solving the MRTA problem are needed ([@gerkey2004formal]).
The MRTA problem becomes more complex when introducing morphological or behavioral heterogeneity within a deployed multi-robot system ([@swarmanoid2013]). However, this additional complexity comes with the benefit of improving the overall system efficiency by leveraging the strengths of individual robots within the collective. For example, in a Search and Rescue (SaR) scenario, quadrotors, which are quick and agile, are better suited for scouting and surveying while ground robots are better suited for debris clearing and resource extraction. By leveraging these strengths efficiently and allocating tasks appropriately the heterogeneous system could out perform a system comprised of only aerial or ground robots.
Moreover, in many scenarios the MRTA problem is accompanied by timing constraints where tasks must be performed sequentially, e.g. a robot must wait for a delivery before transporting the delivered package. Problems of this type are typically referred to as scheduling problems and involve an additional complexity. This class of problems can be solved with Mixed Integer Linear Programs (MILP) that attempt to schedule all the tasks at once, however, this approach suffers from an exponential complexity as noted in [@gombolay2013fast]. Additionally, when deploying a multi-agent system in dynamic environments, the system must be able to respond and reschedule tasks when unavoidable and inevitable environmental disturbances occur. Lastly and specific to the SaR operation following a natural disaster (e.g. wildfires, earthquakes, hurricanes) is that the team of agents must cover the targeted area and rescue victims amongst other tasks within a short-time window. This is due to the drastic decrease in the likelihood of victims surviving after $48$ hours as highlighted in [@48hours]. Therefore, the SaR problem can be cast as an instance of heterogeneous multi-agent system scheduling with the objective of minimizing the time of completion of all tasks (i.e. the makespan).
Inspired by the work in [@rhp], this paper proposes a Receding Horizon Planning framework to solve the heterogeneous MRTA scheduling problem and demonstrates its effectiveness in a SaR application. Inspired by Model Predictive Control, at fixed time intervals, the proposed framework detailed in Section \[sec:rhp\] schedules tasks for each agent up to a pre-defined time horizon and leverages a heuristic to estimate the cost to go for each schedule. As such, this framework does not suffer from the exponential complexity caused by the scheduling of the tasks and is robust to changes in the environment. Moreover, a specialized version of the framework for the SaR application is presented in Section \[sec:useCase\] along with a simulation environment for an abstracted SaR scenario. To validate the effectiveness of the proposed approach, Section \[sec:experiments\] presents extensive experimentation comparing the proposed Receding Horizon Planning based approach to a greedy scheduling scheme.
Literature Review {#sec:litReview}
=================
As mentioned in Section \[sec:intro\], the topic of coordination for multi-agent teams in SaR scenarios following natural disasters falls under the umbrella of Multi-Robot Task Allocation (MRTA). A comprehensive taxonomy of existing methods for MRTA can be found in [@gerkey2004formal]. As highlighted in [@khamis2015multi], most existing approaches can be categorized as decentralized or centralized approaches.
There are two main advantages to decentralized approaches: their robustness to varying team sizes and communication failures, and their scalability with respect to the size of the agent fleet thanks to the computational burden being shared amongst the agents. Moreover, market-based approaches such as [@coalitions; @dias2006market; @vig2006market], attempt to combine the benefits of centralized and decentralized methods by having the computational burden shared between a central entity and the remainder of the fleet. For example, in [@coalitions], the authors suggest a protocol where agents communicate their respective capabilities and use this information to form the coalitions in a decentralized fashion. As such, these methods are able to generate better solutions than fully decentralized approaches while maintaining a certain level of scalability. However, since SaR scenarios typically involve a bounded number of agents and little computational constraints, scalability with respect to the size of the team is of no concern; thus making fully-centralized approaches more suitable for this application.
Moreover, the topic of task allocation specifically pertaining to SaR scenarios is well-studied, most notably, by the participants of RoboCup SaR Agent Simulation competition as highlighted in [@Sheh2016]. The competition setup is as follows. A heterogeneous team is to be deployed to extinguish fires and rescue victims. Specifically, there are three types of agents: ambulances which rescue victims, fire brigades which extinguish fires and police units which remove the road blockades enabling the two other type of agents to reach their desired targets faster. The state-of-the-art task allocation strategy utilized by winning teams such as MRL in the competition is K-Means clustering of the Fires/Victims followed by a cluster to agent assignment using the Hungarian Algorithm which runs in polynomial time. We refer the reader to [@mrl19] for details.
Inspired by the RoboCup competition, the motivation behind the development of the new simulation environment presented in Section \[sec:useCase\] is two-fold. First, the proposed scenario can be seen as a generalized version of the competition’s scenario. Specifically, instead of having a fixed number of types of agents each associated with a single class of tasks (e.g. police units only capable of removing road blockades), through characterizing each agent through the capabilities it exhibits, the proposed simulation framework allows for the modular addition of agent types and the collaboration of a heterogeneous sub-team of agents in achieving a single task. For example, given any two agents and their potentially different capacities to transport water, in the proposed scenario, they can indeed collaborate to extinguish a target fire. Moreover, since not all SaR scenarios are identical in nature, the proposed simulation environment frames the problem as a dynamic set of pick and place tasks where victims and resources are to be delivered to target locations.
Based on this abstracted view of SaR problems, the Receding Horizon Planning framework presented in this paper aims to leverage the strengths of centralized scheduling approaches while keeping the problem size tractable and remaining robust to changes occurring in the environment. The robustness property is obtained through the repeated generation of schedules at fixed time intervals; whereas tractability of the problem size is obtained through only scheduling tasks up to a certain time horizon and leveraging a load-balancing Linear Program as a heuristic to estimate the cost-to-go. Consequently solutions produced by the proposed framework are not guaranteed global optimality since the schedules are of finite horizon. However, we present extensive empirical evidence demonstrating the effectiveness of the proposed approach in Section \[sec:experiments\]. In the next section, the proposed Receding Horizon Planning framework is introduced and presented in detail.
The Receding Horizon Planner {#sec:rhp}
============================
In this section, we present an extended version of the Receding Horizon Planner (RHP), first presented in [@rhp], aimed at solving the Single-Task robots, Single-Robot tasks, Time-extended Assignment (ST-SR-TA) problem, as defined in [@gerkey2004formal], for heterogeneous teams of agents. This problem class involves building a schedule of tasks for each agent that minimizes a given cost function and is strongly ${\mathcal}{N}{\mathcal}{P}$-hard as highlighted in [@brucker1999scheduling].
The brute-force approach for solving this class of problems is to enumerate all possible schedules and choose the one with the smallest associated cost. In its simplest form, the process of generating all possible schedules is done through iteratively assigning one of the remaining tasks to each agent’s schedule. As such, a set of partial schedules (i.e. schedules that do not include all tasks) will be generated to which the process is applied again. Similarly to Branch and Bound ([@lawler1966branch]), this process can be depicted as a tree-search where each partial schedule is associated to a node ${\mathcal}N_i$, and partial schedules generated through subsequent assignments are depicted as children of that node. However, given the set of tasks ${\mathcal}T$ and the set of agents ${\mathcal}A$, the number of possible schedules grows with $\bigO(|\mathcal{A}|^{|\mathcal{T}|})$, rendering the enumeration of all schedules intractable. A scalable alternative is the greedy approach, which solely considers the next best option given the current partial schedule. However, this approach suffers from a reduced performance of the overall system in terms of optimality.
The RHP is a task allocation scheme inspired by Model Predictive Control (MPC) which, at fixed time intervals, computes the optimal schedule for a limited number of tasks and leverages a heuristic to estimate the cost of executing the remaining tasks. Thus, the size of the optimization problem remains constant with respect to the number of tasks and can be adjusted to the computational resources available. With regard to this variable look-ahead time, the receding horizon approach is a superset of the greedy algorithms (zero look-ahead) and the full-blown optimization (infinite look-ahead).
Similarly to MPC, the number of assignments planned by the RHP is larger than the number of assignments that are executed. This creates an overlap between the consecutive optimization cycles, which reduces the loss in optimality due to the neglected future operations. The cyclic nature of this scheme allows for the incorporation of the current system state into the optimization. This feedback loop – as in classic control – provides robustness against disturbances and model deviations.
![Example decision tree generated by the Receding Horizon Planner.[]{data-label="fig:DecisionTree"}](figures/DecisionTree.pdf){width="0.7\columnwidth"}
As introduced in [@rhp], The RHP utilizes a branch-and-bound method to generate the optimal schedule up to the desired time-horizon. As illustrated by Figure \[fig:DecisionTree\], each node $\mathcal{N}_i$ in the search tree corresponds to a partial schedule and the addition of a new assignment of a task to an agent creates a new node. To enable an efficient exploration of the tree, the cost $J(\cdot)$ at node $\mathcal{N}_i$ is decomposed into an accumulated cost value $g(\cdot)$ and a remaining cost estimation $h(\cdot)$ $$J({\mathcal}N_i) = g({\mathcal}N_i) + h({\mathcal}N_i).
\label{eq:CostDecomposition}$$ The accumulated cost evaluates the already assigned operations and is a measure of the consumption of resources. Since each new task assignment increases the consumption of resources, the accumulated cost increases monotonically $$g({\mathcal}N_j) \geq g({\mathcal}N_i),
\label{eq:AccumulatedCost}$$ where ${\mathcal}{N}_j \in Children({\mathcal}{N}_i)$. The second summand in is a lower-bound estimate of the remaining efforts to reach the overall goal. As each new task assignment reduces the outstanding efforts the cost of the remaining tasks must decay $$h({\mathcal}N_j) \leq h({\mathcal}N_i).
\label{eq:RemainingCost}$$ Moreover, the estimate $h$ must provide a lower bound of the true remaining cost at each step, therefore satisfying $$g({\mathcal}N_j) - g({\mathcal}N_i) \geq h({\mathcal}N_i) - h({\mathcal}N_j).
\label{eq:LowerBoundCondition}$$ We will show later, how such a lower bound estimation can be obtained by relaxation of the integer constraints. If $h$ is chosen such that holds we can conclude that $$J({\mathcal}N_j) \geq J({\mathcal}N_i),
\label{eq:TotalCostEstimate}$$ the cost of each node increases as the tree grows. This allows to stop the further exploration of a branch if at any time $J({\mathcal}N_i) \geq J_{\text{opt}}$ (i.e. the cost of node ${\mathcal}N_i$ is larger than the cost of a known solution). This strategy is guaranteed to find the optimal solution on the tree.
To eliminate symmetries and thus reduce the number of nodes to be explored in a tree, only one resource (agent) is chosen for the set of offspring-nodes that are generated from any node in the tree. The agent is chosen as $$a^{\ast} = \operatorname*{arg\,min}_{t \in {\mathcal}T , a \in {\mathcal}A} \ (y_{ta} + T_{ta}).
\label{eq:NextAgent}$$ The indices $t$ and $a$ tally the available tasks ${\mathcal}T$ and agents ${\mathcal}A$ respectively. $T_{ta}$ is the duration of task $t$ if performed by agent $a$ and $y_{ta}$ is the potential start time for agent $a$ on task $t$. In other words, out of all agents, the RHP chooses the one with the earliest completion-time of all tasks. Once the agent is decided upon, all potential tasks that satisfy $$y_{ta} = \min_{t \in {\mathcal}T } (y_{ta^{\ast}} + T_{ta^{\ast}}),
\label{eq:NextStates}$$ are considered as next nodes. That is, we include all tasks that can be started, before the earliest task can be finished. This again reduces the search space in the tree exploration without affecting the optimality of the solution.
The main contribution of this paper is to extend the range of applications of the RHP to scenarios in which multiple agents, each exhibiting different capabilities, are needed to complete a single task or vice-versa. The wildfires in our SaR scenario represent the former type, where the combined effort of multiple agents is needed to extinguish a fire. The rescue operations represent the second type, where a single agent can carry multiple survivors.
The decision space in the tree search inherently includes the various agent capabilities and teaming scenarios, if this is properly described in the set of dispatching rules for each agent. These dispatching rules describe which possible tasks an agent can do, given its current location and occupation, and to what amount the agent can contribute to the overall task, i.e. its capacity. In the context of the SaR scenario, the capacity corresponds to the mundane load capacity for water or victims of each agent.
While the implementation of these dispatching rules is straightforward, the challenge for the tree search is again limiting the search space. To avoid the exploration of all possible combinations of capacities to complete a task, we make use of the heuristic $h(\cdot)$ to guide the tree search. To achieve this, we include a high level load balancing into the heuristic, which breaks the required effort for one large task down into a set of sub-tasks that can be handled by individual agents.
The general approach is to first group all agents with equivalent capabilities into different classes and then minimize the latest finishing time among all agents under the following constraints
(i) \[constraints:first\] The effort for each task is distributed among the different agent classes
(ii) \[constraints:second\] The efforts from (\[constraints:first\]) for each class are distributed among the individual member agents of this class
The detailed application to the SaR use-case is described below.
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Use Case: Search And Rescue {#sec:useCase}
===========================
Problem Setup
-------------
Inspired by the RoboCup Search and Rescue Agent competition, we setup the SaR scenario as follows. A fire breaks out in a forest near a city and starts spreading. Victims are hidden within the city and forest themselves. The objective is to locate and save all victims and extinguish all fires simultaneously using a heterogeneous team of agents as quickly as possible. Note that the fires also grow and spread, therefore solely focusing on rescuing the victims is commonly a sub-optimal strategy. Moreover, the victims are initially not visible to the agents. Therefore, exploring the map is also of paramount importance.
Each fire is represented as a circle of radius proportional to its health and requires a specific amount of water that is also proportional to its health. Upon reaching $100\%$ health, a fire then spread to a nearby territory which is illustrated through the creation of another circle in that location. In order to keep the problem setup as general as possible, we frame the scenario as a pick and place problem where agents need to repeatedly go back to the base and deliver water/victims to the fires/hospital. Moreover, we solely consider each agent’s capabilities when considering it for a given task. As such, we allow for the modular addition or removal of agents from the setup as discussed in the next subsection.
The Heterogeneous Team
----------------------
Teams deployed for SaR are typically heterogeneous due to the variety of the tasks at hand and terrains to be navigated. Therefore, we chose to create teams composed of $4$ types of agents: Ground Units, Helicopters, Drones and Autonomous Ground Vehicles (AGVs). It is important to note that our approach is agnostic to the specific types of agents and the number existing types. In fact, the proposed approach solely considers the number of agents of each type, and each type’s capabilities (as defined in [@gerkey2004formal]). We restrict the agent capabilities we consider to two categories. The first type deals with the mobility of the agents (e.g. what is the velocity of the agent when navigating in the forest?), and the second type considers what action the agent can perform once at the desired location (e.g. can the agent “pickup” a victim?). A tabulation of the capabilities of the $4$ types of agents is presented in Table \[tab:agentCaps\]. As such, each agent type’s specific capabilities can be accounted for explicitly by the proposed framework in the process of generating feasible schedules. Moreover, it is worth noting that additional capabilities can be modularly added since the framework solely considers the capabilities required by each task in the process of generating new nodes. In the next subsection, we present the load-balancing Linear Program used to estimate the cost-to-go.
Estimation of the cost-to-go
----------------------------
We formulate the cost-to-go required for the RHP as a Linear Program (LP), which added to the accumulated cost provides a lower-bound on the total cost of each node. The objective of the LP is to minimize the makespan $s$, which is the time of completion of the last task.
In order to compute the makespan, one must be able to estimate the time taken to complete each task (e.g. rescuing a victim). This is a non-trivial problem, since the time of completion of a task by an agent is dependant upon the previous assignment of the agent. This difficulty also arises in the travelling salesman problem, where the time to travel to a given city depends on the last destination of the salesman. In order to overcome this difficulty, we assume that the distances between the victims/fires are negligible compared to their distances to the base. Therefore by lower-bounding all distances between the targets and the base, we can obtain a “tight” lower-bound on the amount of time a given agent takes to complete a trip to any target $t_{Type(a)}$ given the agent’s velocity.
The decision variable for the LP are
- the number of assignments of task $t$ to the agents of class $c$, denoted by $n_{c,t}$
- the number of assignments of task $t$ to the individual agent $j$ denoted by $m_{j,t}$
- the total makespan denoted by $s$
With these constraints stacked into a vector $$x^{\text{T}} = [n_{c,t}\,,\,m_{j,t}\,,\, s]$$ the LP is formulated as follows
\[eq:heuristicLP\] $$\begin{aligned}
\min_{x} ~& \; \; [0\, \ldots\, 0\, 1] x \label{eq:lp:a}\\
\operatorname*{subject\,to}~~& \smashoperator{\sum_{c \in {\mathcal}C}} C_{c}^{(t)}n_{c,t} \geq R_{t} \: \forall t \in {\mathcal}T \label{eq:lp:b} \\
& y_j + \smashoperator{\sum_{t \in {\mathcal}T}} T_t^{(c)}m_{j,t} \leq s \:\: \forall j \in {\mathcal}A \label{eq:lp:c}\\
& \sum_{j \in \mathcal{A}}B_j^{(c)} m_{j,t} = n_{c,t} \:\: \forall c,t \in {\mathcal}C \otimes {\mathcal}T \label{eq:lp:d},\end{aligned}$$
where ${\mathcal}T$, ${\mathcal}C$ and ${\mathcal}A$ denote the set of all tasks, agent types and individual agents respectively. The program aims to minimize the makespan (i.e. the time of completion of the last task). Moreover, constraint ensures that the required effort for accomplishing task $t$ denoted by $R_{t}$ is matched by the agent fleet. The effort provided by the agent fleet is computed through summing the effort provided by each agent type. The capacity of agent type $c$ for task $t$ is denoted by $C_{c}^{(t)}$. Additionally, the makespan is computed in constraint , where $y_j$ denotes the completion of time of agent $j$’s current schedule and the execution time of task $t$ for class $c$ is denoted by $T_{t}^{(c)}$. Lastly, constraint ensures that the sum of contributions of each individual agent of a given type matches the total contribution of that type, where $B_{j}^{(c)}$ indicates if agent $j$ is of type $c$. In the next section, we present experimental results demonstrating the effectiveness of the SaR specialized Receding Horizon Planner presented in this section compared to a greedy scheduling approach.
---------------------- --------- ------- ------- ------
Types G. Unit Heli. Drone AGV
\[0.5ex\] Water Cap. 1 5 0 2
Rescue Cap. 1 4 0 4
Move Forest 0.1 0.5 0.40 0.20
Move City 0.1 0.5 0.40 0.00
---------------------- --------- ------- ------- ------
: The capabilities of the $4$ types of agents: Ground Units, Helicopters, Drones and AGVs. Note for example how AGVs are unable to navigate through the forest or how the capacity for carrying water units or victims vary depending on the type of agent.[]{data-label="tab:agentCaps"}
Experiments {#sec:experiments}
===========
In this section, we present empirical results validating the use of a receding horizon through simulations and experiments on the Robotarium, a remotely accessible swarm accessible testbed ([@robotarium]). The setup of the experiments is as follows. The team is composed of 6 ground units, 3 helicopter units, 14 drones and 3 AGVs. Moreover, in each experiment $10$ initially hidden victims are randomly placed in the forest and city along with $3$ fires that were also initialized at random positions. However, the number of total fires generated in a given experiment may differ depending on the planner being used. This is due to the fact that the fires grow and spread over time and therefore require even more resources to extinguish. This is an accurate depiction of many real-world SaR scenarios and serves to emphasize that the time of completion of tasks is an important measure in such scenarios. The path-action planning algorithm details the execution of the generated schedules and is implemented as follows. First, it generates each agent’s path to its corresponding task, then ensures that the agent takes the corresponding required action for the task if feasible. For example, once an agent tasked with rescuing a victim reaches its location, it will take the “pick-up” action if and only if the agent’s maximum capacity for the number of victims is not already reached. Since path planning is not the focus of this work, we assume all agents except the AGVs possess single-integrator dynamics and use proportional controllers to guide them to their targets. However, since the AGVs are presented as robots (GRITSBot X) in the Robotarium experiments and can indeed collide, we implemented multi-agent A\* to generate way-points the agents can follow to their targets. Moreover, since we do not explicitly check for collision-avoidance in the trajectories between way-points, we also utilize Control Barrier Functions (CBFs) to instantaneously ensure collision-avoidance at all times as described in [@AmesBarriers; @ames2014]. This is achieved through solving a Quadratic Program at each point in time that generates a minimally altered trajectory for the agents relative to their nominal trajectory to ensure collision-avoidance.
A run of $20$ simulated experiments with randomized initial conditions were run to compare several planning depths of the RHP and a greedy scheduling approach. The greedy scheduling scheme used for bench-marking the proposed RHP is a one-step look-ahead planning approach. Specifically, at each scheduling iteration, each idle agent is assigned to the task that it is closest to. The mean, median and variance of the makespans of each of the scheduling approaches over all experiments are presented in Table \[tab:results\]. As shown in the table, the mean makespan decreases significantly when the RHP is used. However, the rate of improvement decreases at higher planning depths, which highlights the trade-off between computing time and solution quality.
Moreover, to demonstrate the applicability of the RHP onto real systems, $10$ experiments were conducted on the Robotarium testbed comparing the RHP using a planning depth of $10$ to the greedy scheduler. The Robotarium’s robots (GRITSBot X) were used instead of the three simulated AGVs as depicted in Figure \[fig:robotariumExp\]. To obtain the desired frequency of operation on the Robotarium ($\sim100$ Hz), a separate computing node was used to run the scheduling algorithms, and the schedules were transmitted to the agents using a publisher-subscriber protocol. The results of the experiments are presented in Table \[tab:resultsR\]. Indeed, the use of the RHP reduces the makespan, thus validating the applicability of the proposed scheduling approach.
![Search and Rescue experiment on the Robotarium using 3 GRITSBot X as the AGVs. The description of all entities are as in Figure \[fig:sim\].[]{data-label="fig:robotariumExp"}](figures/roboExp.png){width="40.00000%"}
------------------ ------- -------- ----------
Types Mean Median $\sigma$
\[0.5ex\] Greedy 78.78 53.03 10.63
RHP10 42.54 40.34 9.54
RHP15 40.09 38.56 7.95
RHP20 39.87 36.84 9.39
------------------ ------- -------- ----------
: Results over $20$ simulations comparing the mean, median and standard deviation of the makespans of the Receding Horizon Planner with different planning depths and greedy scheduling scheme.[]{data-label="tab:results"}
------------------ ------- -------- ----------
Types Mean Median $\sigma$
\[0.5ex\] Greedy 46.70 45.06 9.64
RHP10 39.00 36.72 9.09
------------------ ------- -------- ----------
: Results over $10$ Robotarium experiments comparing the mean, median and variance of the makespans of the Receding Horizon Planner with planning depth $10$ and the greedy scheduling scheme.[]{data-label="tab:resultsR"}
Conclusion {#sec:conclusion}
==========
This paper introduces a task allocation framework capable of scheduling tasks for heterogeneous teams of agents in a manner that is tractable and robust to changes in the environment and in the agent fleet. This was achieved through repeatedly scheduling solely up to a fixed horizon and leveraging a load-balancing Linear Program for the estimation of the cost to go. Moreover, a simulation framework for an abstracted Search and Rescue scenario inspired by the RoboCup Search and Rescue Agent Simulation competition was presented along with a specialized formulation of the Receding Horizon Planning approach. Experimental results showcase the efficacy of the proposed scheduling method in extensive multi-agent simulations and experiments on the Robotarium.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
Aulchenko V.M.$^a$, Papushev P.A.$^a$, Sharafutdinov M.R.$^b$, Shekhtman L.I.$^a$[^1], Titov V.M.$^a$, Tolochko B.P.$^b$, Zhulanov V.V.$^a$\
Budker Institute of Nuclear Physics\
11 Lavrentiev Avenue, Novosibirsk 630090\
Russia. Fax: 7(383)3307163,\
e-mail: [email protected]\
Institute of Solid-State Chemistry and Mechano-Chemistry\
630090 Novosibirsk, Russian Federation
title: 'Development of one-coordinate gaseous detector for wide angle diffraction studies.'
---
Introduction. {#OD4-intro}
=============
Any detector for the studies of X-ray diffraction to large angles has to present essentially curved geometry with conversion, readout and possibly amplifying structure that surrounds the scattering source. The detector OD3 with angular aperture up to $30^{o}$ for powder diffraction experiments has been already designed and constructed in Budker Institute of Nuclear Physics [@OD3-descr]. This detector has been built on the basis of wire chamber. However further development of a detector with larger angular aperture and considerably curved cylindrical geometry is impossible on such basis. Thus in order to cover the wider range of diffraction angles the Gas Electron Multiplier (GEM) [@GEM-Sauli] was proposed to be used as the multiplying element.
GEM is a thin plastic foil double clad with copper layers from both sides and pierced with regular array of small diameter holes. Regular GEM is produced of 50$\mu$m thick kapton foil and has 140$\mu$m holes pitch and 80$\mu$m holes diameter. Gas amplification occurs in the GEM holes when high voltage is applied between the two foil sides. GEMs can be cascaded and in a triple-GEM cascade can provide stable gain up to and higher than $\sim10^5$ in a regular gas mixures like $Ar-CO_2$(70-30) [@GEM-highgain].
Flat and flexible amplifying structure of GEM allows to prepare arc-shaped GEM cascade that can surround the scattering source in a WAXS experiment. Such approach can solve the problem of large angular aperture for a gaseous detector.
The first measurements with the small prototype and simulations demonstrating feasibility of this approach were described elsewhere ( [@OD4-paper1], [@OD4-SNIC06]). This paper presents the first results obtained with full-size detector.
Detector design and experimental set-up.
========================================
The detector for WAXS studies based on cascaded GEM (OD4) is shown schematically in Fig. \[fig:OD4-design\]. X-rays from the scattering source get into the gas box through the Be window and are absorbed in 5.5mm thick drift gap between the drift cathode and the top GEM. The triple-GEM stack with GEM to GEM distance of 1.5mm is attached on top of the multi-strip PCB at a distance of 2.5mm. Drift cathode, GEMs and PCB have arc shape with the center at the scattering source. Strips of the PCB are positioned along radii of the circle with the center at the source. The PCB contains 2048 strips with the pitch of 0.2mm at the entrance side.
![Schematic view of OD4 design.[]{data-label="fig:OD4-design"}](Fig1.eps){width="60.00000%"}
The detector is intended to work with soft X-rays in the range of 5keV to 15kev and is filled with $Ar-CO_2$(3:1) mixture at atmospheric pressure.
The OD4 electronics is implemented on the basis of preamplifier-shaper chip IC31A [@BNL-TEC-PS] developed for the electronics of PHENIX detector at RHIC (BNL, USA). Each strip of the PCB is connected to an input of the preamplifier-shaper through the flexible cable. The outputs of preamplifiers are connected to the comparators with single and adjustable threshold and logical pulses after the comparators are counted by scalers.
At present the final electronics is not yet ready and only 32 strips in the center of the detector have been equipped with preamplifier-shapers and comparators. 64 strips from both sides of the equipped area have been connected to ground to ensure uniform field in the central region. The detector during assembling is shown in Fig. \[fig:OD4-view1\] where triple-GEM cascade installed on top of the PCB can be observed. Fig. \[fig:OD4-view2\] demonstrates assembled detector.
![Photo of the detector during assembly. Flexible kapton cables are connecting 160 strips to the feed-throughs in the central part of the detector. Triple-GEM cascade is installed on top of the PCB. []{data-label="fig:OD4-view1"}](OD4-view1.eps){width="70.00000%"}
![View of the assembled detector. []{data-label="fig:OD4-view2"}](OD4-view2.eps){width="70.00000%"}
The GEM electrodes and drift cathode have been powered through the single-line resistive divider that was adjusted to minimize transverse diffusion and have reasonable GEM transparency during electrons drift from drift gap to the PCB. The adjustment of the voltages across transfer gaps, induction gap and drift gap has been performed for the conditions when the voltages across GEMs provide total effective gain of the whole cascade around 10000.
For the measurements described in this paper the OD4 has been installed at one of synchrotron radiation lines at VEPP-3 electron ring. After monochromator the X-ray beam with 8.3 keV energy was collimated by 20$\mu$m slit. Effective beam size at the entrance window of the detector was 20$\mu$m in horizontal and $\sim$1mm in vertical. The detector was positioned with 3 stands that allow precise rotation around vertical axis, movement in horizontal and vertical directions. With these stands the strips equipped with the electronics could be aligned along the beam as the source was not at the focus of the detector(350mm).
Results and discussion
======================
When an X-ray photon is absorbed in the drift gap of the detector, after charge transport and amplification the final charge cluster occupies in average more than 1 strip. The charge distribution and its effect on spatial resolution was discussed in details in our previous paper( [@OD4-paper1]) where the results of simulations were compared to the measurements with the prototype of the present detector. However the important outcome of the charge distribution over several strips is that even if the detector is irradiated with the constant flux of photons, the counting rate will depend on the comparators threshold, gas gain and the flux distribution in space. Fig. \[fig:Fe55-count\] demonstrates the counting rate as a function of voltage at the resistive divider in 1 arbitrary channel, coincidence between 2,3 and 4 neighboring channels while the detector has been uniformly irradiated with $Fe^{55}$ 5.9keV photons. The comparators threshold has been close to 150mV in this measurement that, according to the electronic calibration (measured amplifier gain was 14mV/fC), corresponded to the gas gain of $\sim$300 for 5.9keV X-rays. Counting rate of a single channel is increasing in the whole range of voltages as more channels get hit with increasing gas gain. Above Vd=2750V most of the rate is produced by coincident hits of several channels.
However the counting rate dependence on voltage looks differently if the detector is irradiated with narrow X-ray beam. If all the photons are absorbed within one channel and no charge can come from the neighboring areas, the counting rate becomes constant when all the signals induced by absorbed photons become larger than the threshold. In Fig. \[fig:SR-count\] the counting rate vs voltage dependence for several comparator thresholds are presented. The 20$\mu$m beam of 8.3 keV photons hits the center of the channel in this measurement. The starting points of counting rate plateau give information about average signal value at a given voltage, thus the gain-voltage characteristic can be derived from this data. Table \[tab:Plato\] summarizes data on counting rate plateau starting voltages (50$\%$ level of plateau), corresponding threshold values and gain values, calculated using electronic calibration.
Plateau starting voltage, V Threshold, V Gain
----------------------------- -------------- ------ --
2420 0.3 450
2480 0.53 795
2500 0.77 1155
2520 1.0 1500
: Counting rate plateau starting voltage (50$\%$ level), corresponding comparator threshold and gain value, calculated from electronic calibration.[]{data-label="tab:Plato"}
This data is plotted in Fig. \[fig:OD4-gain\] with exponential fit through the experimental points. The gain-voltage dependence fits well to similar data from [@GEM-highgain] and [@OD4-paper1].
For the studies of spatial resolution the detector has been moved horizontally in such a way that 20$\mu$m wide X-ray beam has scanned the area of several detector channels. Counting rate as a function of beam position for each channel (channel response curve) has been used for characterization of the spatial resolution. The measurements have been performed at a different detector voltages and different comparator thresholds in order to observe the dependence of resolution on these parameters. An example of the set of channel response curves for Vd=2580V (V$_{GEM}\sim$360V, Gain$\sim$3000) and the threshold that provides 90$\%$ efficiency in the central channel is shown in Fig. \[fig:Channels\].
![Counting rate as a function of voltage at the resistive divider while the detector is uniformly irradiated by 5.9keV photons. The voltage across each GEM is shown at the top scale. The rate of single counts, double-, triple- and quadruple- coincidences is shown. []{data-label="fig:Fe55-count"}](Fe55-count.eps){width="60.00000%"}
![Counting rate as a function of voltage at the resistive divider while the detector is irradiated by thin beam of 8.3keV photons. The voltage across each GEM is shown at the top scale. Several dependences corresponding to different comparator thresholds are shown. []{data-label="fig:SR-count"}](SR-count.eps){width="70.00000%"}
![Gain as a function of voltage at the divider (bottom scale) and single GEM(top scale). Exponential fit is plotted through the experimental points. []{data-label="fig:OD4-gain"}](OD4-gain.eps){width="70.00000%"}
![Channel response curves for 3 channels scanned with 20$\mu$m wide X-ray beam. Voltage at the divider Vd=2580V, the comparator threshold provides 90$\%$ efficiency in the central channel. []{data-label="fig:Channels"}](Channels.eps){width="70.00000%"}
Different counting rate of the left and central channels can be explained by different gain in these areas. The left channel has slightly higher gain and thus reach full efficiency already at this voltage and threshold. The lower is the threshold the smaller signal can be detected and thus the channel response curve is becoming wider. On the other hand when the threshold is too high the efficiency starts to drop. Fig. \[fig:Resolution-Efficiency\] demonstrates the dependence of spatial resolution (FWHM of channel response curve) on the efficiency, derived from several measurements of channel response curves. All these measurements have been done at Vd=2580V.
Thus spatial resolution can be tuned in a wide range by the adjustment of the comparator threshold and/or the detector gain. At the level of 90$\%$ efficiency the resolution is close to 470$\mu$m and can be improved to FWHM$\sim$330$\mu$m at the expense of the efficiency that drops down to 50$\%$ in the latter case.
Spatial resolution obtained with 90$\%$ efficiency is good enough to allow separation of two diffraction spots at angular distance of 0.1 degree that corresponds to $\sim$0.6mm for this detector. The image of two such spots positioned symmetrically with respect to the central channel is calculated using the channel response curve from Fig. \[fig:Channels\] and shown in Fig. \[fig:Diffraction\].
High rate capability is one of the advantages of GEM based detectors over wire chambers. Smaller amplifier cells (140$\mu$m distance between holes in GEM) allow faster charge removal and thus produce lower space charge that affects the gain. Rate capability of cascaded GEMs was demonstrated up to the level of 10$^5$ Hz/mm$^2$ of 8 keV photons ( [@GEM-rate]). In OD4 we have been aiming to get the counting rate capability of up to 100kHz per 0.2mm wide channel.
The measurement of rate capability of OD4 was performed with 20$\mu$m wide beam of 8.3keV photons aligned at the center of a channel. The beam was attenuated with 50$\mu$m thick aluminum foils. In each subsequent measurement 1 foil was removed to increase the rate. Every time when the measurement with reduced number of foils was completed the additional normalizing measurement was performed with 12 foils. The normalizing measurement was necessary to correct the results for beam intensity variations that happened due to monochromator movements and electron beam instabilities. All the measurements have been done at Vd=2580V (gain$\sim$3000) and the comparator threshold adjusted to provide 90$\%$ efficiency of plateau level.
After the completion of all the measurements and correction on the normalizing data, the effective absorption of the foil has been calculated for each subsequent pair of measurements. This value has included the absorption by itself and possible additional rate reduction due to limited rate capability. Then the average effective foil absorption and its variance have been calculated using only the set of measurements where foil absorptions have been constant (at lower rates). The linear rate scale has been calculated using the rate value in the first measurement at the lowest rate and average foil absorption. The relative efficiency has been obtained as the ratio of the measured rate and the calculated linear rate. The variance of foil absorption value has been used to calculate the variance of the linear rate and the corresponding variance of the relative efficiency.
The result of this study is shown in Fig. \[fig:Rate\]. Within the measurement errors that have been mainly determined with the beam instabilities, the detector efficiency does not depend on the photons rate up to $\sim$100kHz/channel. Indeed we could expect this result because the effective area where charge is produced by the X-ray beam is spread over 0.2mm\*30mm=6mm$^2$ (0.2mm is channel width, 30mm is strip length) and maximum charge rate is equivalent to only $\sim$20kHz/mm$^2$ of 8.3keV photons at gain$\sim$3000.
Conclusion.
===========
Full size detector for WAXS studies has been assembled and tested at the synchrotron radiation line at VEPP-3. The detector has been partially equipped with electronics that included amplifier-shaper, comparator and scaler in each channel. OD4 has demonstrated stable performance with 8.3 keV photons in the range of gains from $\sim$500 to more than 10000.
Spatial resolution of the detector can be tuned with the comparator threshold and gas gain and it appeared to be simple function of efficiency when the latter is lower than 100$\%$. For 90$\%$ efficiency the spatial resolution is $\sim$470$\mu$m (FWHM of the channel response curve). Such resolution is enough to separate clearly the diffraction spots at an angular distance of 0.1 degree, that was initially established as a main requirement for OD4. The measurements of spatial resolution and efficiency in a wide range of gas gains and comparator thresholds have shown that the detector can work at rather low gains (below 1000) and by threshold adjustment the resolution and efficiency can be chosen at optimal level. We hope that for the operation at such gains the number of GEMs in the cascade can be reduced to 2 or even to 1 and this will be checked in future studies.
Rate capability of OD4 was tested up to the rate of $\sim$150kHz/channel. No significant degradation of efficiency has been detected.
The electronics that have been used for the measurements was not final. The boards that contain the amplifier-shapers and comparators will be connected to the motherboard that will collect data from all the 32-channel amplifier boards, control them and communicate with the computer. The first version of the final electronics including the motherboard and amplifier cards for total number of 256 channels will be ready during 2008.
[00]{}
V.M. Aulchenko, et al, Nucl. Instr. and Meth. A405 (1998), 269
F. Sauli, Nucl. Instr. and Meth. A386 (1997), 531.
S. Bachmann, et.al., Nucl. Instr. and Meth. A479(2002)294
V.M. Aulchenko, et al, Nucl. Instr. And Meth. A 575, n.1-2 (2007), 251.
Detectors for Time-resolved Studies at SR Beam, Aulchenko V.M., Bukin M.A., Papushev P.A., Shekhtman L.I., Titov V.M., Vasiljev A.V., Zhulanov V.V., in Proceedings of the SNIC Symposium, Stanford, California, 2006, edited by V. Luth, eConf C0604032 (2006), 0195.
A. Kandasamy, E. O$^\prime$Brien, P. O$^\prime$Connor, and W. Von Achen, A monolitic preamplifier-shaper for measurement energy loss and transition radiation, BNL-66629
A.Bressan, et.al., Nucl. Instr. and Meth., A 425 (1999) 262
![Spatial resolution (FWHM of channel response curve) as a function of efficiency. Voltage at the divider Vd=2580V. []{data-label="fig:Resolution-Efficiency"}](Resolution-Efficiency.eps){width="70.00000%"}
![Image of two diffraction spots separated by 0.1 degrees (0.6mm at 350mm distance to the scattering source). Image is calculated using channel response curve with 470$\mu$m FWHM.[]{data-label="fig:Diffraction"}](Diffraction.eps){width="70.00000%"}
![Rate capability of OD4. The measurement has been performed with 20$\mu$m wide X-ray beam aligned at the center of a channel.[]{data-label="fig:Rate"}](Rate.eps){width="70.00000%"}
[^1]: Corresponding author
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Andrew Larkoski,'
- 'Simone Marzani,'
- and Chang Wu
bibliography:
- 'pull.bib'
title: Safe Use of Jet Pull
---
Introduction {#sec:intro}
============
During this long shutdown phase, the experiments of the CERN Large Hadron Collider (LHC) are gearing up for the third run of the accelerator. While the increase in centre-of-mass energy will be modest, the path to discovery of new physics, which thus far has proven so elusive, will likely involve careful analyses of large dataset, in order to expose subtle deviations from Standard Model (SM) predictions. Together with the search for beyond the Standard Model (BSM) particles or interactions, careful studies of the Higgs sector will continue to constitute the second, but equally important, leg of the LHC physics program. In particular, pinning down the couplings of the Higgs boson to the fermions may lead to a deeper understanding of the flavour structure of the SM. In this context, both the ATLAS and CMS collaborations have reached the sought-for statistical significance for the decay of the Higgs into bottom quarks [@Aaboud:2018zhk; @Sirunyan:2018kst] in Run II data.
Typical events from proton-proton collisions at the LHC are filled with strongly-interacting particles, the dynamics of which is described by Quantum Chromo Dynamics (QCD). It follows that QCD radiation has a profound impact on both BSM and Higgs physics. The reason is twofold. Firstly, SM processes involving quarks and gluons often constitute the main background, which often dwarves the signal of interest by orders of magnitude. Furthermore, QCD radiation often accompanies the production of the particles of interest, and indeed it offers valuable handles to study them; e.g. Higgs production in association with jets. In our current study we concentrate on the latter issue, namely we discuss observables that by measuring QCD radiation in a signal event, provide us with information on the properties of the particle we are studying. In particular, we are interested in assessing the colour quantum numbers of a resonance decaying into quarks. This is of clear interest for BSM searches but it also provides a useful handle in distinguishing the decay of a colour singlet (such as the Higgs) into quarks from the decay of a colour octet (such as the gluon) in the same final state.
A powerful observable that is able to probe colour flow is jet pull, which was first proposed in Ref. [@Gallicchio:2010sw]. Since then, a number of experimental analyses has been devoted to this observable: from a pioneering measurement performed by the D$\emptyset$ collaboration at the Tevatron [@Abazov:2011vh], to two measurements performed by the ATLAS collaboration at the LHC, at centre-of-mass energy of 8 TeV [@Aad:2015lxa] and 13 TeV [@Aaboud:2018ibj]. Most of the measurements concentrate in a particular projection of the jet pull vector, the so-called pull angle, that would, in principle offer the best sensitivity. However, as the experimental uncertainties on the measurement grew smaller, it became apparent that general-purpose Monte Carlo parton showers struggled in modelling the pull angle distribution. In particular, it has been pointed out that the datapoints corresponding to the measurement of the pull angle in $W$ decay are almost equidistant from the result obtained from a standard Monte Carlo simulation and from a simulation where the $W$ is assumed to be a colour octet [@Aaboud:2018ibj].
In a previous Letter [@Larkoski:2019urm], we embarked in a detailed study of the pull angle distribution, with the hope that analytic resummation could shed light on those discrepancies. While our perturbative prediction, supplemented with an estimate of a non-perturbative contribution, could describe the experimental data, it still suffered from large theoretical uncertainties, rendering any firm conclusion difficult to draw. The main bottleneck of the theoretical calculation resides on the fact that the pull angle distribution is not infra-red and collinear (IRC) safe but only Sudakov safe [@Larkoski:2013paa; @Larkoski:2014wba; @Larkoski:2015lea]. Because the theoretical understanding of Sudakov safe observables is still in its infancy, it is not clear how theoretical accuracy can be achieved (and rigorously assessed) beyond the first order. Furthermore, while IRC safety ensures the presence of a kinematical region where non-perturbative effects are genuine power corrections, no such guarantee exists for Sudakov-safe observables and consequently, non-perturbative physics can contribute to the observable as an order-one effect. In this paper we overcome these difficulties by defining suitable projections of jet pull that share many of the desirable features of the pull angle, but at the same time are IRC safe. This enables us to perform perturbative calculations at a well-defined, and in principle improvable, accuracy.
The paper is organised as follows. In Section \[sec:obs\] we recall the definition of jet pull and we introduce the safe projections we want to study. Section \[sec:theory\] contains the all-order calculations for the observables of interest, while in Section \[sec:pheno\] we perform phenomenological studies, which include a comparison to the results obtained using Monte Carlo event generators. In Section \[sec:asym\] we exploit the theoretical understanding achieved so far to introduce novel asymmetry observables that aim to better probe colour flow in an infra-red and collinear safe way. Finally, we draw conclusions in Section \[sec:conclusions\] and outline our plan for future work on this topic.
Jet Pull {#sec:obs}
========
The pull vector is a two-dimensional jet shape defined as [@Gallicchio:2010sw] $$\label{pull-def}
\vec t= \frac{1}{p_{t}}\sum_{i\in J}p_{ti} |\vec{r}_i|^2 \hat r_i\,,$$ where the sum runs over all particles in a jet and $$\label{ri-def}
\vec{r}_i=(y_i-y_a, \phi_i-\phi_a), \quad \text{and} \quad \hat r_i= \frac{\vec{r}_i}{|\vec{r}_i|}.$$ The coordinates of the jet centre in the rapidity-azimuth plane are $(y_a,\phi_a)$ and $p_t$ is the jet transverse momentum. We are interested in measuring the pull of jet $a$ in the presence of a second jet $b$, that we center at $(y_b,\phi_b)$. To this purpose, we find useful to introduce the two unit vectors $$\begin{aligned}
\label{unit-vectors}
\hat{n}_\parallel&= \frac{1}{\sqrt{\Delta y^2 + \Delta \phi^2}} (\Delta y, \Delta \phi)=(\cos \beta, \sin \beta)
, \nonumber \\
\hat{n}_\perp&= \frac{1}{\sqrt{\Delta y^2 + \Delta \phi^2}} (-\Delta \phi, \Delta y)=(- \sin \beta, \cos \beta)
, \end{aligned}$$ where $\Delta y= y_b-y_a$ and $\Delta \phi= \phi_b-\phi_a$, as depicted in Fig. \[rap-azim\]. The angle $\beta$ has been introduced for future convenience. We now introduce two new observables that are defined as the projections of the pull vector in the two directions identified by the unit vectors above: $$\begin{aligned}
\label{pull-proj}
t_{\parallel}&= |\vec{t}\cdot \hat{n}_\parallel | \quad \text{and} \quad
t_{\perp}= |\vec{t}\cdot \hat{n}_\perp |.\end{aligned}$$ We will come back to the role of the absolute value in the expressions above in Sec. \[sec:asym\]. Furthermore, we note that the magnitude of the pull vector can be expressed as $$t=|\vec{t}|= \left | \frac{1}{p_{t}}\sum_{i\in J}p_{ti} |\vec{r}_i|^2 \hat r_i \right| = \sqrt{t_{\parallel}^2+t_{\perp}^2},$$ while the pull angle can be written as $$\theta_p= \cos^{-1} \frac{\vec{t}\cdot \hat{n}_\parallel}{t}.$$ It is easy to check that the pull magnitude $t$ and the two projections $t_{\parallel}$ and $t_{\perp}$ are IRC safe observables. However, this property is lost when considering the pull angle, essentially because $\theta_p$ does not vanish in the presence of a single soft emission because the ratio $t_\parallel / t$ is undetermined.
Our first aim in what follows is to obtain all-order predictions for the above safe observables at next-to-leading logarithmic accuracy. In Ref. [@Larkoski:2019urm] we have already performed a resummed calculation for the pull magnitude $t$, which then played the role of the IRC safe companion observable in the Sudakov safe calculation for $\theta_p$. However, in that calculation we have resorted to the collinear limit. Here, we want to relax this approximation and also consider contributions from soft emissions at wide angle, expressed as a power series in the jet radius $R$. Crucially, soft radiation at wide angle depends on the number of hard partonic legs present in the processes and on their kinematic configurations. Therefore, in order to perform our calculation we have to choose a process (or a class of processes) and fix the number of coloured legs.
In this paper we concentrate on measuring pull on one of the two jets originating from the hadronic, i.e. $b \bar b$, decay of a Higgs boson, while taking the other jet as reference. We point out that, as suggested in the original publication, pull can provide a valuable handle in distinguishing the above production of a Higgs boson from the dominant QCD background (specifically $g \to b \bar b$). Furthermore, this measurement can be also performed in the boosted regime, where the decay products are reconstructed into a single two-pronged jet. In this case, jet pull can be measured on one of the subjets.
We also advocate measuring jet pull in other Standard Model contexts. Measurements of the pull angle have been carried out by the D$\emptyset$ collaboration at the Tevatron [@Abazov:2011vh] and by the ATLAS collaboration at the LHC [@Aad:2015lxa; @Aaboud:2018ibj] (in their most recent analysis the ATLAS collaboration also measured the pull magnitude) in events featuring the production of a top and of an anti-top. The rich phenomenology of top decay allows for measuring jet pull in a singlet decay by looking, for instance, at the decay of the $W$ boson but also enables one to study more intricate colour correlations, by measuring the pull between one of the the $b$-jets and the incoming beam. Another interesting channel to consider is $Z$+jet production. This channel offers several interesting possibilities in the context of colour-flow measurements. For instance, by looking at the substructure of QCD jets, one can explore colour flow in higher-dimensional colour representation, see e.g. [@Bao:2019usu]. On the other hand, one can look at the hadronic decay of the $Z$ boson and measure colour flow between two jets (or subjets, if considering the boosted regime) originating from a colour singlet. This situation is very much analogous to what we discuss in this current work, but it features a higher rate at the LHC. Studies of colour flow in this context would provide a useful testing ground for an even more interesting Higgs and new physics programme.
Pull distributions at next-to-leading logarithmic accuracy {#sec:theory}
==========================================================
In this section we provide all-order calculations that resum large logarithms up to next-to-leading logarithmic accuracy (NLL) for the IRC safe projections of the pull vector considered in this study, namely $t$, ${t_\perp}$ and ${t_\parallel}$. These calculations can also be used as input for the Sudakov-safe determination of $\theta_p$.
Collinear emissions
-------------------
The NLL resummation of the pull vector in the collinear limit, was already performed in Ref. [@Larkoski:2019urm]. The all-order expression can be easily arrived at by noting that the pull vector is additive and recoil-free at leading power, essentially because of the quadratic dependence on $|\vec{r}_i|$ of Eq. (\[pull-def\]) [^1]. Furthermore, in the collinear limit the resummed cross section is universal and does not depend on the event surrounding the jet we are measuring. The resummed expression for the pull magnitude can be directly calculated from an infinite sum of emissions of energy fraction $z_i$ and (small) emission angles $\theta_i\ll R$ $$\begin{aligned}
\label{eq:t-resum}
\frac{1}{\sigma}\frac{d \sigma}{d t} &= \exp\left[
-\int_0^{R^2}\frac{d\theta^2}{\theta^2}\int_0^1 dz \int_0^{2\pi}\frac{d\phi}{2\pi}\frac{\alpha_s(z \theta p_t)}{2\pi}P_{gq}(z)
\right]
\nonumber \\
&
\hspace{1cm}
\times
\left[
\phantom{\delta\left(t- \sqrt{ \left(\sum_{i=1}^n z_i \theta_i^2\cos\phi_i\right)^2+ \left(\sum_{i=1}^n z_i \theta_i^2\sin\phi_i\right)^2} \right) }
\hspace{-8.4cm}
\sum_{n=0}^\infty\frac{1}{n!} \prod_{i=1}^n \int_0^{R^2}\frac{d\theta_i^2}{\theta_i^2}\int_0^1 dz_i \int_0^{2\pi}\frac{d\phi_i}{2\pi}\frac{\alpha_s (z_i \theta_i p_t)}{2\pi}P_{gq}(z_i)\right.\nonumber\\
&
\hspace{2cm}
\left.
\times\, \delta\left(t- \sqrt{ \left(\sum_{i=1}^n z_i \theta_i^2\cos\phi_i\right)^2+ \left(\sum_{i=1}^n z_i \theta_i^2\sin\phi_i\right)^2} \right)
\phantom{int_0^{2\pi}}
\hspace{-0.8cm}\right],\end{aligned}$$ where $R$ is the radius of the jet we are measuring pull on. For definiteness, we are going to define jets using the anti-$k_t$ algorithm [@Cacciari:2008gp]. The function $P_{gq}=C_F\frac{1+(1-z)^2}{z}$ represents the collinear splitting probability of a quark into a quark and a gluon and appears in the resummation formula because at NLL the parton originating a jet in $H\to b\bar b$ decay is always a quark. A more refined calculation, namely NLL$'$, would also account for the relative $\mathcal{O}({\alpha_s})$ probability of measuring pull on a gluon-initiated jets and would therefore would also feature the splitting probabilities $P_{gg}$ and $P_{qg}$. Furthermore, note that the argument of the running coupling, which must be evaluated at two-loop accuracy, is the transverse momentum of the emission relative to the hard quark. As already noticed in Ref. [@Larkoski:2019urm], the structure of the resummed results is akin to the well-known transverse-momentum resummation, e.g. [@Parisi:1979se; @Collins:1984kg], and consequently the sum over the emissions can be performed explicitly in the conjugate space of Fourier-Hankel moments: $$\begin{aligned}
\label{eq:coll-t-res-final}
\frac{1}{\sigma}\frac{d \sigma}{d t}&= \int_0^\infty db\, (bt)J_0(bt)e^{-2C_F {\mathcal{R}}_c(b)}, \end{aligned}$$ where $J_0(x)$ is the Bessel function and ${\mathcal{R}}_c(b)$ is the collinear radiator, which, at this accuracy, depends exclusively on the magnitude of the Fourier conjugate vector $b=|\vec b|$: $$\begin{aligned}
\label{coll-rad}
{\mathcal{R}}_c(b) = \int_0^{R^2}\frac{d\theta^2}{\theta^2}\int_0^1 dz \frac{\alpha_s(z \theta p_t)}{2\pi}\frac{P_{gq}(z)}{2C_F}\Theta \left( z \theta^2 -{\bar b}^{-1}\right),\end{aligned}$$ with $\bar b = b \frac{e^{\gamma_E}}{2}$. Explicit expressions for the NLL radiator will be reported in Section \[sec:resummed-formulae\].
The projections of the pull vector we are interested in can be found following the same steps. We have $$\begin{aligned}
\label{eq:tpp-resum}
\frac{1}{\sigma}\frac{d \sigma}{d {t_\perp}} &= \exp\left[
-\int_0^{R^2}\frac{d\theta^2}{\theta^2}\int_0^1 dz \int_0^{2\pi}\frac{d\phi}{2\pi}\frac{\alpha_s(z \theta p_t)}{2\pi}P_{gq}(z)
\right]
\nonumber \\
&
\hspace{1cm}
\times
\left[
\sum_{n=0}^\infty\frac{1}{n!} \prod_{i=1}^n \int_0^{R^2}\frac{d\theta_i^2}{\theta_i^2}\int_0^1 dz_i \int_0^{2\pi}\frac{d\phi_i}{2\pi}\frac{\alpha_s (z_i \theta_i p_t)}{2\pi}P_{gq}(z_i)\right.\nonumber\\
&
\hspace{2cm}
\left.
\times\,
\delta\left({t_\perp}- \left | \sum_{i=0}^n\left(-z_i \theta_i^2\cos\phi_i \sin \beta + z_i \theta_i^2\sin\phi_i\cos \beta \right) \right|\right)\right],\end{aligned}$$ where the $\delta$ function comes from the definition of the observable ${t_\perp}$ in Eq. (\[pull-proj\]). Note that in this case such constraint involves a one-dimensional sum, while the analogous term in the pull magnitude distribution, Eq (\[eq:t-resum\]), involved a vector sum. This situation presents strong similarities with the resummation of equivalent variables in the context of transverse-momentum resummation, such as $a_T$ and $\phi^*$ [@Banfi:2009dy; @Banfi:2011dx]. Thus, as in that case, the all-order sum can performed in a conjugate Fourier space. We obtain $$\begin{aligned}
\label{eq:coll-tpp-res-final}
\frac{1}{\sigma}\frac{d \sigma}{d {t_\perp}}&= \frac{2}{\pi }\int_0^\infty db\, \cos(b{t_\perp})e^{- 2C_F {\mathcal{R}}_c(b)},\end{aligned}$$ where the radiator in $b$ space is the same as the one obtained for the pull magnitude, Eq. (\[coll-rad\]) Finally, we find that, at this accuracy, the ${t_\parallel}$ and ${t_\perp}$ distributions share an identical collinear structure: $$\begin{aligned}
\label{eq:coll-res-final2}
\frac{1}{\sigma}\frac{d \sigma}{d {t_\parallel}}&= \frac{2}{\pi }\int_0^\infty db\, \cos(b{t_\parallel})e^{- 2C_F{\mathcal{R}}_c(b)}.\end{aligned}$$
Soft emissions at wide angle
----------------------------
We now focus our attention on the effect that soft emissions at wide angle have to the pull distributions. These contributions first appear at NLL and from general considerations we expect them to be suppressed in the small jet radius limit. However, unlike collinear radiation discussed above, the explicit form of soft contributions depends on the underlying hard processes we are considering. Physically, this comes about because soft gluons can attach to any hard parton, resulting in a potentially complicated pattern of colour correlations. In our current study, the situation is not too complicated because we are focusing on measuring pull on jets originating from a colour-singlet, while the colour structure is much richer when considering jets originating from higher-dimensional colour representations [@Bao:2019usu]. In particular, the hard process we are considering at Born level is $$q \bar q \to H(\to b \bar b) \; Z (\to l^+ l^-).$$
The soft contribution to the NLL radiator can be written as the sum over dipoles that can emit a soft gluon. In our case we only have two dipoles: the one formed by the initial-state partons and the one made up by the two bottom quarks, which we consider massless, therefore we have $$\begin{aligned}
\label{soft-radiator}
\mathcal{R}_s= -2{\bf T}_1 \cdot {\bf T}_2 {\mathcal{R}}_{12}-2 {\bf T}_a \cdot {\bf T}_b \widetilde{{\mathcal{R}}}_{ab},\end{aligned}$$ where $1,2$ refer to the initial state and $a,b$ to the final state. ${\bf T}_i$ are the colour insertion operators and the tilde on the second contribution indicates that we have subtracted the collinear contribution already included in ${\mathcal{R}}_c$. Because we are considering final-state jets produced by the decay of a singlet state, the colour algebra is trivial: $$\begin{aligned}
\label{colour-alg}
{\bf T}_1 + {\bf T}_2=0 \Rightarrow {\bf T}_1 \cdot {\bf T}_2= -\frac{1}{2} \left({\bf T}_1^2 +{\bf T}_2^2 \right)= - C_F, \nonumber \\
{\bf T}_a + {\bf T}_b=0 \Rightarrow {\bf T}_a \cdot {\bf T}_b= -\frac{1}{2} \left({\bf T}_a^2 +{\bf T}_b^2 \right)= - C_F, \nonumber \\\end{aligned}$$
We start by considering the contribution from the initial-state dipole. Indicating with $p_1$ and $p_2$ the momenta of the incoming quarks and with $k$ the momentum of the soft gluon, we have $$\begin{aligned}
\label{1gluon-IRS-begin}
{\mathcal{R}}_{12}= \int d k_t k_t d y \frac{d \phi}{2 \pi} \frac{{\alpha_s}(k_t)}{2 \pi} \frac{p_1\cdot p_2}{p_1 \cdot k \; p_2 \cdot k} \Theta_\text{jet} \Theta_\text{pull},\end{aligned}$$ where $\Theta_\text{jet}$ enforces the gluon to be recombined with one of the final-state partons (say parton $a$) to form the jet we are interested in, and $\Theta_\text{pull}$ enforces the gluon contribution to the observable of choice to be above a certain value.
The above integrals can be easily evaluated by introducing polar coordinates in the rapidity-azimuth plane: $$\begin{aligned}
\label{polar-coords}
y-y_a& = r \cos \alpha, \nonumber\\
\phi-\phi_a&= r \sin \alpha.\end{aligned}$$ With this choice of variables, the observables become $$\begin{aligned}
\label{1gluon-pull-proj}
t&=|\vec{t}|=z r^2, \nonumber\\
t_{\parallel}&= |\vec{t}\cdot \hat{n}_\parallel |= z r^2 | \cos (\alpha-\beta)|, \nonumber \\
t_{\perp}&= |\vec{t}\cdot \hat{n}_\perp |= z r^2 | \sin (\alpha-\beta)|,\end{aligned}$$ with $z = \frac{k_t}{p_{t}}$. The angle $\beta$ was introduced in Eq. (\[unit-vectors\]). Note that $\alpha-\beta$ is just the pull angle.
Thus, for the pull magnitude, we obtain $$\begin{aligned}
\label{1gluon-IRS-t}
{\mathcal{R}}_{12}= \int_0^1 \frac{dz}{z} \frac{{\alpha_s}(z p_t)}{\pi} \int_0^R d r r \int_0^{2 \pi} \frac{d \alpha}{2 \pi} \Theta(z r^2>t)=
R^2 \int_t^1 \frac{dz}{z} \frac{{\alpha_s}(z p_t)}{2\pi}+ \dots\end{aligned}$$ where the dots indicate subleading contributions. To NLL, the same expression also holds for $t_\parallel$ and $t_\perp$: $$\begin{aligned}
\label{1gluon-IRS-begin-final2}
{\mathcal{R}}_{12}&= \int_0^1 \frac{dz}{z} \frac{{\alpha_s}(z p_t)}{\pi} \int_0^R d r r \int_0^{2 \pi} \frac{d \alpha}{2 \pi} \Theta(z r^2 |\cos(\alpha-\beta)|>t_\parallel)=
R^2 \int_{t_\parallel}^1 \frac{dz}{z} \frac{{\alpha_s}(z p_t)}{2\pi}+ \dots\\
{\mathcal{R}}_{12}&= \int_0^1 \frac{dz}{z} \frac{{\alpha_s}(z p_t)}{\pi} \int_0^R d r r \int_0^{2 \pi} \frac{d \alpha}{2 \pi} \Theta(z r^2 |\sin(\alpha-\beta)|>t_\perp)=
R^2 \int_{t_\perp}^1 \frac{dz}{z} \frac{{\alpha_s}(z p_t)}{2\pi}+ \dots\end{aligned}$$ where again the dots indicate subleading contributions.
Thus far we have calculated the soft wide-angle contribution directly in momentum space. This is in principle sufficient at NLL accuracy we are working at. Nevertheless, in order to smoothly combine the soft contribution to the collinear one previously computed, we find convenient to perform the whole resummation in moment ($b$) space. Therefore to NLL we can write the soft contribution from the initial-state dipole as $$\begin{aligned}
\label{1gluon-IRS-begin-final}
{\mathcal{R}}_{12}&=R^2 \int_{1/\bar{b}}^1 \frac{dz}{z} \frac{{\alpha_s}(z p_t)}{2\pi}.\end{aligned}$$
Next we consider soft-wide angle emissions off the final-state $ab$ dipole. As in the previous case, we find convenient to express the phase-space integrals in polar coordinates. We have $$\begin{aligned}
\label{1gluon-FRS-begin}
{\mathcal{R}}_{ab}&= \int d k_t k_t d y \frac{d \phi}{2 \pi} \frac{{\alpha_s}(\kappa_{ab})}{2 \pi} \frac{p_a\cdot p_b}{p_a \cdot k \; p_b \cdot k} \Theta_\text{jet} \Theta_\text{pull} \nonumber\\
&= \int_0^1 \frac{dz}{z} \int_0^R d r \int_0^{2 \pi} \frac{d \alpha}{2 \pi} \frac{{\alpha_s}(\kappa_{ab})}{2 \pi}
\left[\frac{2}{r}+ \mathcal{A}(\alpha,\beta)+ \mathcal{B}(\alpha,\beta) r+ \dots \right]
\Theta_\text{pull}\end{aligned}$$ where the argument of the running coupling $\kappa_{ab}^2=\frac{2\; p_a \cdot k \; p_b \cdot k}{p_a\cdot p_b}$ is the transverse momentum of the gluon with respect to the dipole, in the dipole rest frame. We calculate this contribution as a power expansion in the jet radius $R$, which corresponds to expanding the integrand in powers of $r$. The first contribution within the square brackets is the soft and collinear piece, which we have already accounted for in ${\mathcal{R}}_c$. Therefore, we consider $$\begin{aligned}
\label{dipole-expanded}
\widetilde{{\mathcal{R}}}_{ab}&=
\int_0^1 \frac{dz}{z} \int_0^R d r \int_0^{2 \pi} \frac{d \alpha}{2 \pi} \frac{{\alpha_s}(\kappa_{ab})}{2 \pi}
\left[ \mathcal{A}(\alpha,\beta)+ \mathcal{B}(\alpha,\beta) r+ \dots \right]
\Theta_\text{pull}\end{aligned}$$ The first term above, namely $\mathcal{A}$ gives no contribution when we integrate over all possible angles. It would give rise to an $\mathcal{O}(R)$ correction if we impose further angular restrictions. We will come back to this observation in Section \[sec:asym\]. The $\mathcal{B}$ term gives rise to a contribution which is identical in all cases. Therefore, at NLL we have $$\begin{aligned}
\label{1gluon-FRS-la}
\widetilde{{\mathcal{R}}}_{ab}&= \frac{R^2}{4} \frac{\cosh \Delta y+ \cos \Delta \phi}{\cosh \Delta y- \cos \Delta \phi}\int_{1/\bar{b}}^1 \frac{dz}{z} \frac{{\alpha_s}(z p_t)}{2\pi} + \mathcal{O}(R^4).\end{aligned}$$ We remind the reader that explicit expressions for the NLL radiator will be reported in Section \[sec:resummed-formulae\].
Non-global logarithms {#sec:ngls}
---------------------
Jet pull is measured on an isolated jet and it is therefore a text-book example of a non-global observable [@Dasgupta:2001sh]. In this section we investigate the structure of non-global logarithms (NGLs) that affect the different projections of the pull vector. We focus on the final-state dipole $ab$ and we consider the double differential distribution in the pull magnitude and pull angle at $\mathcal{O}({\alpha_s}^2)$. To calculate the leading non-global logarithmic contribution to the pull vector, it suffices to consider correlated soft gluon emission from the dipole in which the two soft gluons have parametrically separated energies $k_h\gg k_s$, in the phase-space region where the harder gluon lies outside the measured jet, while the second one is inside. The matrix element for this non-global contribution can then be expressed as $$\begin{aligned}
\label{ngls-start}
\frac{d^2\sigma^\text{NG}}{dt\, d\theta_p} &= \frac{\alpha_s^2C_F C_A}{16\pi^4} \int_0^1 \frac{dk_{\perp h}}{k_{\perp h}}\int_{-\infty}^\infty dy_h \int_{-\pi}^\pi d\phi_h \int_0^1 \frac{dk_{\perp s}}{k_{\perp s}}\int_{-\infty}^\infty dy_s\int_{-\pi}^\pi d\phi_s \, \frac{2 p_a\cdot p_b}{(p_a\cdot k_h) (p_b\cdot k_h)}\nonumber\\
&
\hspace{3cm}
\times\frac{(p_a\cdot k_h)(p_b\cdot k_s)+(p_a\cdot k_s)(p_b\cdot k_h)-(p_a\cdot p_b)(k_h\cdot k_s)}{(p_a\cdot k_s)(p_b\cdot k_s)(k_h\cdot k_s)}\\
&
\hspace{3cm}
\times \Theta\left(R^2-(y_s-y_a)^2-(\phi_s-\phi_a)^2\right)\Theta\left((y_h-y_a)^2+(\phi_h-\phi_a)^2-R^2\right)\nonumber\\
&
\hspace{3cm}
\times\, \Theta(k_{\perp h}\cosh y_h - k_{\perp s}\cosh y_s)\,\delta\left(t-k_{\perp s}\left((y_s-y_a)^2+(\phi_s-\phi_a)^2\right)\right)\nonumber\\
&
\hspace{3cm}
\times\,\delta\left(\theta_p-\cos^{-1}\frac{(y_s-y_a)\cos\beta +(\phi_s-\phi_a)\sin \beta }{\sqrt{(y_s-y_a)^2+(\phi_s-\phi_a)^2}}\right)
\,.\nonumber\end{aligned}$$ Note that in the expression, the dependence on the perp magnitudes has been pulled out of all of the matrix elements and made explicit. The integral over $k_{\perp s}$ and $k_{\perp h}$ can easily performed. Furthermore, for compactness, we can shift the $y$ and $\phi$ coordinates to be measured with respect to the location of jet $a$, i.e. without loss of generality we can set $y_a=\phi_a=0$ in Eq. (\[ngls-start\]).
From this point, we will start approximating the integrals that remain. First, we only work to find the leading NGLs for $t\ll 1$. By the jet phase space constraints that remain, the relevant scaling is $ y_h\sim y_s \sim R \ll1$, by our assumption that the jet radii are small. Therefore, in the explicit logarithm in the integrals we can simply remove the hyperbolic cosine factors, as their contribution will be purely beyond leading NGL. Correspondingly, because $R\ll 1$, we can push the bounds of integration on $ \phi_s, \phi_h$ safely to infinity. The integrals then become $$\begin{aligned}
\frac{d^2\sigma^\text{NG}}{dt\, d\theta_p} &= \frac{\alpha_s^2C_F C_A}{16\pi^4} \, \frac{1}{t} \int_{-\infty}^\infty d y_h \int_{-\infty}^\infty d \phi_h \int_{-\infty}^\infty d y_s\int_{-\infty}^\infty d \phi_s \, \frac{2 p_a\cdot p_b}{(p_a\cdot k_h) (p_b\cdot k_h)}\\
&
\hspace{2cm}
\times\frac{(p_a\cdot k_h)(p_b\cdot k_s)+(p_a\cdot k_s)(p_b\cdot k_h)-(p_a\cdot p_b)(k_h\cdot k_s)}{(p_a\cdot k_s)(p_b\cdot k_s)(k_h\cdot k_s)}\,\log\frac{ y_s^2+\phi_s^2}{t}\nonumber\\
&
\hspace{2cm}
\times
\, \Theta\left(y_s^2+\phi_s^2-t\right)\, \Theta\left(R^2-y_s^2-\phi_s^2\right)\,\Theta\left(y_h^2+\phi_h^2-R^2\right)\nonumber\\
&
\hspace{2cm}
\times\,\delta\left(\theta_p-\cos^{-1}\frac{y_s \cos\beta +\phi_s \sin \beta }{\sqrt{y_s^2+\phi_s^2}}\right)
\,.\nonumber\end{aligned}$$ Similarly to the one-gluon dipoles previously discussed, the integrals are more easily performed in polar coordinates, see Eq. (\[polar-coords\]): $$\begin{aligned}
y_i& =r_i \cos \gamma_i\,,\nonumber\\
\phi_i&=r_i \sin\gamma_i\,.\end{aligned}$$ Then, the integrals become $$\begin{aligned}
\frac{d^2\sigma^\text{NG}}{dt\, d\theta_p} &= \frac{\alpha_s^2C_F C_A}{16\pi^4} \, \frac{1}{t} \int_0^\infty dr_h\, r_h \int_0^{2\pi} d\gamma_h \int_0^\infty dr_s\, r_s\int_0^{2\pi} d\gamma_s \, \frac{2 p_a\cdot p_b}{(p_a\cdot k_h) (p_b\cdot k_h)}\\
&
\hspace{2cm}
\times\frac{(p_a\cdot k_h)(p_b\cdot k_s)+(p_a\cdot k_s)(p_b\cdot k_h)-(p_a\cdot p_b)(k_h\cdot k_s)}{(p_a\cdot k_s)(p_b\cdot k_s)(k_h\cdot k_s)}\nonumber\\
&
\hspace{2cm}
\times
\,\log\frac{r_s^2}{t}\, \Theta\left(r_s^2-t\right)\, \Theta\left(R-r_s\right)\,\Theta\left(r_h-R\right)\,\delta\left(\theta_p-\gamma_s +\beta \right)
\,.\nonumber\end{aligned}$$ Now, we need to express the soft matrix element in these coordinates. Additionally, we work in the small jet radius limit, $R\ll1$, and note that the dominant contribution to the NGLs comes from the region of phase space in which $r_s\lesssim r_h \sim R$. We will thus expand the matrix element to first order in the $R \ll 1$ limit with this identified scaling. We find $$\begin{aligned}
\frac{d^2\sigma^\text{NG}}{dt\, d\theta_p} &= \left(\frac{\alpha_s}{2\pi}\right)^2 C_F C_A\,\frac{\pi}{3} \, \frac{\log\frac{R^2}{t}}{t}\biggl(1+\frac{24 (1-\log\, 2)}{\pi^2}R\\
&
\hspace{0.5cm}
\left.\, \times \frac{\sin \Delta \phi \; \sin(\theta_p+\beta)+\sinh \Delta y \;\cos(\theta_p+\beta)}{\cosh \Delta y-\cos \Delta \phi}+{\cal O}(R^2)\right)\,.\nonumber\end{aligned}$$ The first term in this expansion is the familiar expression for the narrow jet mass NGL matrix element. Note that this differs by a factor of $2\pi$ from the familiar expression for the jet mass NGLs; this factor is recovered when $\theta_p$ is integrated over. Furthermore, if we integrate over the full range for $\theta_p$, then the contribution which is linear in $R$ vanishes, leading to $$\label{ngls-final-as2}
\frac{d \sigma^\text{NG}}{dt} = \left(\frac{\alpha_s}{2\pi}\right)^2 C_F C_A\frac{2\pi^2}{3} \, \frac{\log\frac{R^2}{t}}{t}+{\cal O}(R^2)\,.$$ It is easy to verify that at NLL accuracy the same expression as Eq. (\[ngls-final-as2\]) holds for the projections ${t_\parallel}$ and ${t_\perp}$.
If we only to retain the leading $R$ term, then resummation of NGLs is analogous as the hemisphere mass originally studied in [@Dasgupta:2001sh]. We could, in principle, also include the $\mathcal{O}(R^2)$ corrections, as done in the global part. This would require evaluating the subsequent term in the small-$R$ expansion of Eq. (\[ngls-final-as2\]). Furthermore, we would also have to include the NGL contribution from initial-state radiation, as discussed, for instance in Ref. [@Dasgupta:2012hg], in the context of jet mass distributions. We leave this study for future work.
Resummed results {#sec:resummed-formulae}
----------------
We are now in a position to collect all the results derived so far and obtain a NLL resummed prediction for the safe projections of the pull vector we are considering. The all-order differential distribution can be written as: $$\begin{aligned}
\label{final-res-expr}
\frac{1}{\sigma}\frac{d \sigma}{d v} &=\int_0^\infty d b\, \mathcal{F}_v(b v) e^{-C_F \mathcal{R}(b) }\mathcal{S}^\text{NG}(b),\end{aligned}$$ with $$\begin{aligned}
\mathcal{F}_v(x)=
\begin{cases}
x J_0(x), & \text{if} \quad v=t,\\
\frac{2}{\pi}\cos (x), &\text{if} \quad v={t_\parallel}, \, {t_\perp}.
\end{cases}\end{aligned}$$ The resummed exponent $\mathcal{R}$ can be written in terms of leading (second line) and next-to-leading (third to fifth lines) contributions: $$\begin{aligned}
\label{radiator}
\mathcal{R}&=2\mathcal{R}_c+2\widetilde{\mathcal{R}}_{ab}+2\mathcal{R}_{12} \nonumber \\
&=
\frac{\left(1-2 \lambda \right )
\log \left(1-2\lambda \right)-2 \left ( 1-\lambda \right ) \log \left
(1-\lambda \right )}{2 \pi {\alpha_s}\beta_0^2}
\nonumber \\
&+
\frac{ B_q }{ \pi \beta_0} \log \left ( 1-\lambda \right )
+ \frac{ K}{4 \pi^2 \beta_0^2} \left [2 \log \left
(1-\lambda \right ) - \log \left (1-2 \lambda \right )\right ] \nonumber\\ &
+\frac{ \beta_1}{2 \pi \beta_0^3} \left [ \log \left (1-2\lambda \right )-2 \log
\left (1-\lambda \right ) + \frac{1}{2} \log^2 \left (1- 2 \lambda \right )
- \log^2 \left (1-\lambda \right ) \right ]
\nonumber\\
&+\frac{1}{\pi \beta_0}\log \frac{p_t R}{\mu_R}\left[\log(1-2 \lambda) -2 \log(1-\lambda)\right]
- \frac{R^2}{8 \pi \beta_0} \left[ 4+ \frac{\cosh \Delta y +\cos \Delta \phi}{\cosh \Delta y -\cos \Delta \phi} \right]\log (1-2 \lambda) \nonumber \\&+\mathcal{O}(R^4),\end{aligned}$$ with $\lambda= {\alpha_s}\beta_0 \log (\bar{b} R^2)$[^2] and ${\alpha_s}={\alpha_s}(\mu_R)$, where $\mu_R$ is the renormalisation scale, which we can vary around the hard scale $p_t$ in order to assess missing higher-order corrections. In the above results the $\beta$ function coefficients $\beta_0$ and $\beta_1$ are defined as $$\beta_0 = \frac{11 C_A - 2 n_f }{12 \pi}, \qquad \beta_1 = \frac{17 C_A^2 - 5 C_A n_f -3 C_F n_f}{24 \pi^2}\,,$$ and $$B_q=\frac{3}{4}, \qquad K = C_A \left (\frac{67}{18}- \frac{\pi^2}{6} \right ) - \frac{5}{9} n_f\,.$$
Finally, as already mentioned, in the small-$R$ limit, the non-global contribution can be taken equal to the hemisphere case. The resummation of NGLs can be performed in the large-$N_c$ limit exploiting a dipole cascade picture. We make use of the following parametrisation [@Dasgupta:2001sh]: $$\label{ngl-res}
\mathcal{S}^\text{NG}=\exp \left[-C_F C_A\frac{\pi^2}{3}\frac{1+(a \tau)^2}{1+(b \tau)^c} \tau^2 \right],$$ with $\tau=-\frac{1}{4 \pi \beta_0} \log(1- 2 \lambda)$, with $a=0.85 C_A, b=0.86 C_A$, and $c=1.33$.
Finally, we note that the above results are valid for jets defined with the anti-$k_t$ algorithm, which acts as a perfect cone in the soft limit [@Cacciari:2008gp]. Had we use a different clustering measure, such as Cambridge/Aachen [@Dokshitzer:1997in; @Wobisch:1998wt] or the $k_t$-algorithm [@Catani:1993hr; @Ellis:1993tq], nontrivial clustering logarithms would have modified both the global and non-global contributions to the resummed exponent [@Banfi:2005gj; @Delenda:2006nf; @Banfi:2010pa].
Towards phenomenology {#sec:pheno}
=====================
In the previous section, we have discussed all the theoretical ingredients that go into a NLL calculation for the jet pull projections considered in this paper. We now turn our attention towards some preliminary phenomenological studies. After discussing a simple model of non-perturbative corrections due to the hadronisation process, we move to compare our resummed results to the one obtained by a general purpose Monte Carlo event generator. While doing so, we also discuss the numerical impact of the various contributions that we have computed thus far. We postpone a more detailed phenomenological study, which would also include matching to fixed-order calculations, to future work and we look forward to comparison of our predictions to future experimental measurements.
Non-perturbative corrections {#sec:non-pert}
----------------------------
Because the pull vector is both an additive observable and recoil-free, corrections due to non-perturbative physics and hadronisation can be modelled by a shape function [@Korchemsky:1999kt; @Korchemsky:2000kp; @Bosch:2004th; @Hoang:2007vb; @Ligeti:2008ac]. This shape function is then convolved with the perturbative distribution to produce a non-perturbative distribution. The shape function depends on a dimensionful relative transverse-momentum scale $\epsilon$, and it has most of its support around $\epsilon = \Lambda_\text{QCD}$, the QCD scale. The shape function for the pull vector also has non-trivial azimuthal angle dependence, because non-perturbative emissions will be emitted in a preferential direction according to the dipole configuration.
In this section, we will construct a shape function for the pull vector, assuming that it exclusively has support at $\epsilon = \Lambda_\text{QCD}$. Further, we will assume that the dominant non-perturbative emission lies exactly at the boundary of the jet on which we measure the pull vector, and its azimuthal distribution about the jet axis is uniform. We will see that a non-uniform distribution of the pull vector is generated by a preferential emission of higher-energy non-perturbative emissions at small values of the pull angle.
To construct the shape function with these restrictions, we first note that the scale $\epsilon$ for an emission from a dipole with ends defined by the light-like directions $p_a$ and $p_b$ is $$\begin{aligned}
\epsilon = \Lambda_\text{QCD}= \sqrt{(k\cdot p_a)(k\cdot p_b)}\,,\end{aligned}$$ where $k$ is the four-momentum of the non-perturbative emission. The pull vector depends on the momentum transverse to the beam axis, $k_t$, and its value is constrained by the non-perturbative scale. Expressing the momentum $k$ as $$\begin{aligned}
k &= k_t(\cosh y,\cos\phi,\sin\phi,\sinh y)\,,\end{aligned}$$ we can express $k_t$ as $$\begin{aligned}
k_t = \frac{\Lambda_\text{QCD}}{\left(\cosh(y-y_a)-\cos(\phi-\phi_a)\right)^{1/2}\left(\cosh(y-y_b)-\cos(\phi-\phi_b)\right)^{1/2}}\,.\end{aligned}$$ Now, we expand this expression to second order in the jet radius $R$, fixing the angle between the non-perturbative emission and the jet axis $n_a$ to be $R$: $$\begin{aligned}
R^2 = (y - y_a)^2+(\phi - \phi_a)^2\,.\end{aligned}$$ We find $$\begin{aligned}
k_t = \frac{2\Lambda_\text{QCD}}{R}\frac{\sqrt{p_{ta}p_{tb}}}{m_H}+2\Lambda_\text{QCD}\frac{(p_{ta}p_{tb})^{3/2}}{m_H^3}\left[
\cos(\varphi+\beta)\sinh \Delta y+\sin(\varphi+\beta)\sin\Delta\phi
\right] + {\cal O}(R)\,.\end{aligned}$$ The relative rapidity $\Delta y$, azimuth $\Delta \phi$, and angle $\beta$ were defined in . The azimuthal angle $\varphi$ defines the angle about the jet axis $p_a$ with respect to $p_b$. Finally, we have introduced the transverse momentum of the ends of the dipole $p_{ta}$ and $p_{tb}$ and note that they are constrained by the Higgs mass: $$\begin{aligned}
m_H^2 = 2p_{ta}p_{tb}(\cosh\Delta y - \cos\Delta\phi)\,.\end{aligned}$$
With this construction, the shape function for the non-perturbative $k_t$ and azimuthal angle $\varphi$ is $$\begin{aligned}
&F(k_t,\varphi) \\
&= \frac{1}{2\pi}\delta\left(
k_t-\frac{2\Lambda_\text{QCD}}{R}\frac{\sqrt{p_{ta}p_{tb}}}{m_H}-2\Lambda_\text{QCD}\frac{(p_{ta}p_{tb})^{3/2}}{m_H^3}\left[
\cos(\varphi+\beta)\sinh \Delta y+\sin(\varphi+\beta)\sin\Delta\phi
\right]
\right)\,.\nonumber\end{aligned}$$ Given the perturbative pull vector distribution $\frac{1}{\sigma} \frac{d^2 \sigma^\text{pert}}{d \vec{t}^2}$, we now want to find the non-perturbative pull vector distribution $\frac{1}{\sigma} \frac{d^2 \sigma^\text{np}}{d \vec{t}^2}$ through convolution with the shape function. The contribution to pull from the non-perturbative emission that we identified in the rest frame of the Higgs boson will be $$\vec t_\text{np}(k_t, \varphi) = \frac{k_t R^2}{p_{ta}}\,(\cos\varphi,\sin\varphi)\,.$$ It then follows that the non-perturbative distribution of the pull vector is $$\begin{aligned}
\label{eq:NP-2D}
\frac{d^2 \sigma^\text{np}}{d \vec{t}^{\;2}}&= \int_0^\infty dk_t \int_0^{2\pi}d\varphi\, F(k_t,\varphi)\, \frac{d^2 \sigma^\text{pert}}{d \vec{t}^{\;2}}\left(
\vec t - \vec t_\text{np}(k_t, \varphi)
\right)\nonumber\\&
=\int_0^{2\pi}\frac{d\varphi}{2\pi} \,\frac{d^2 \sigma^\text{pert}}{d \vec{t}^{\;2}}\left(
\vec t - \vec t_\text{np}(k_t, \varphi)
\right)\,,\end{aligned}$$ where we leave the dependence on the non-perturbative transverse momentum $k_t$ implicit.
In order to understand the behaviour of the leading non-perturbative corrections, we expand the above expression in powers of $\Lambda_\text{QCD}$. Furthermore, we note that because of the particular choice of the reference frame we have used in this section, $\varphi=0$ corresponds to the line joining the two jet centres. Thus, we obtain $$\begin{aligned}
\label{eq:NP-2D-exp}
&\frac{d^2 \sigma^\text{np}}{d {t_\parallel}d {t_\perp}}=
\frac{d^2 \sigma^\text{pert}}{d {t_\parallel}d {t_\perp}}-\int_0^{2\pi}\frac{d\varphi}{2\pi} \,
\vec t_\text{np}(k_t, \varphi) \cdot {\bf} \nabla \left(\frac{d^2 \sigma^\text{pert}}{d {t_\parallel}d {t_\perp}}\right)+ \mathcal{O}\left(\frac{\Lambda_\text{QCD}^2}{m_H^2} \right)\\&
=\Bigg[ 1-\frac{\Lambda_\text{QCD}R^2\sqrt{p_{ta}p_{tb}^3}}{m_H^3\sqrt{\Delta y^2+\Delta\phi^2}}\Big(
\left(\Delta y\sinh \Delta y+\Delta\phi\sin\Delta\phi\right)\frac{\partial}{\partial {t_\parallel}}\nonumber \\ &\quad +\left(\Delta y\sin \Delta\phi-\Delta\phi\sinh\Delta y\right)\frac{\partial}{\partial {t_\perp}}
\Big) \Bigg]\frac{d^2 \sigma^\text{pert}}{d {t_\parallel}d {t_\perp}}.\nonumber\end{aligned}$$ Because of the derivative dependence in this non-perturbative correction, its effect can be included to lowest order in both $\Lambda_\text{QCD}$ and $\alpha_s$ with a shift of the appropriate argument of the perturbative cross section. For the cross sections of ${t_\parallel}$ and ${t_\perp}$ individually, we have $$\begin{aligned}
\label{np-shift-tpa-tpp}
\frac{d \sigma^\text{np}}{d {t_\parallel}} &=\frac{d \sigma^\text{pert}}{d {t_\parallel}}\left(
{t_\parallel}-\frac{\Lambda_\text{QCD}R^2\sqrt{p_{ta}p_{tb}^3}}{m_H^3\sqrt{\Delta y^2+\Delta\phi^2}}\left(\Delta y\sinh \Delta y+\Delta\phi\sin\Delta\phi\right)\right)+{\cal O}(\Lambda_\text{QCD}^2,\alpha_s)\,,\\
\frac{d \sigma^\text{np}}{d {t_\perp}} &=\frac{d \sigma^\text{pert}}{d {t_\perp}}\left(
{t_\perp}-\frac{\Lambda_\text{QCD}R^2\sqrt{p_{ta}p_{tb}^3}}{m_H^3\sqrt{\Delta y^2+\Delta\phi^2}}\left(\Delta y\sin \Delta\phi-\Delta\phi\sinh\Delta y\right)\right)+{\cal O}(\Lambda_\text{QCD}^2,\alpha_s)\,.\end{aligned}$$ The leading non-perturbative correction to the magnitude of the pull vector $t$ can be found by exploiting its relationship to ${t_\parallel}$ and ${t_\perp}$: $$\begin{aligned}
t = \sqrt{{t_\parallel}^2+{t_\perp}^2}\,.\end{aligned}$$ Then, we have that the pull magnitude distribution becomes $$\begin{aligned}
\label{np-shift-t}
\frac{d \sigma^\text{np}}{d t} = \frac{d \sigma^\text{pert}}{d t} \left(t-\frac{\Lambda_\text{QCD}R^2\sqrt{p_{ta}p_{tb}^3}}{m_H^3}\sqrt{\sinh^2\Delta y + \sin^2\Delta\phi}\right)+{\cal O}(\Lambda_\text{QCD}^2,\alpha_s)\,.\end{aligned}$$
Numerical studies
-----------------
We are now ready to perform some phenomenological studies of our results. From a technical point of view, we note that the integral over the Fourier variable $b$ which appears in the resummation formula, e.g. Eq. (\[final-res-expr\]), is ill-defined both at small and large $b$. The bad behaviour at small $b$, which corresponds to large values of the observables, is beyond the jurisdiction of the all-order calculations and it contributes to a region that would be dominated by fixed-order matrix elements. In order to address this issue, we adopt the standard procedure of $Q_T$ resummation [@Bozzi:2005wk] and we shift the argument of the logarithm in $b$-space by unity, i.e. $\log \bar b R^2 \to \log (1+ \bar b R^2)$. The resummed exponent is also ill-defined at large $b$ because of the presence of the QCD Landau pole which appears at $\lambda =\frac{1}{2}$. We circumvent this issue by further substituting the dependence on the variable $b$ in the resummed exponent with the so-called $b^*$ variable [@Collins:1984kg] $$b^{*}=\frac{b}{\sqrt{1+\frac{b^{2}}{b_\text{max}^{2}}}},$$ where $b_\text{max}$ is chosen in the vicinity of the Landau pole. Because $b^*\simeq b$ when $b\ll b_\text{max}$, the perturbative behaviour is unchanged, while the $b$ dependence of the resummed exponent is frozen as $b$ approaches the non-perturbative region, providing us with a prescription to deal with the Landau singularity.
We start by assessing the numerical impact of the different contributions that are included in our resummed results, namely collinear emissions, final-stare radiation (FSR), i.e. the $\mathcal{O}(R^2)$ contribution arising from the final-state dipole, initial-state radiation (ISR), and non-global logarithms. The results are show in Fig. \[theory-contributions\], on the left for the pull magnitude distribution and on the left for the ${t_\parallel}$ distribution (at NLL this is the same as ${t_\perp}$). The plots are for a representative phase-space point: $\Delta y=1$, $\Delta \phi=\frac{\pi}{6}$ and $p_t=\frac{m_H}{\sqrt{2(\cosh \Delta y-\cos \Delta \phi})}\simeq 110$ GeV, which corresponds to a symmetric decay of the Higgs boson. We note that the collinear approximation describes the two distributions well, down to values of the observables $\sim 10^{-3}$. Below that, in the Sudakov region, the impact of soft-emissions at large angle becomes sizeable. However, we note that finite $R$ corrections, which characterise FSR and ISR are not very large, due to the smallness of the jet radius parameter $R=0.4$, employed in this study. Perhaps surprising is the relativly large contribution due to non-global logarithms. By comparing the two distributions, $t$ and ${t_\parallel}$, we note that the former exhibits a Sudakov peak, while the latter appears to develop a plateau for ${t_\parallel}< 10^{-4}$. This behaviour is completely analogous to what is found when looking at $Q_T$ and $a_T$/$\phi^*$ distributions [@Banfi:2011dx]. Small values of $t$ or ${t_\parallel}$ can be obtained by soft/collinear emissions or by kinematical cancellations and the behaviour of ${t_\parallel}$ signals the fact that kinematical cancellation is the dominant mechanism and prevents the formation of the Sudakov peak, as opposed to what happens with $t$.
Next, in Fig. \[theory-uncertainties\] we show our final NLL predictions for $t$ (left) and ${t_\parallel}$ (right), with an estimate of the perturbative uncertainty, which we obtain by varying the renormalisation scale in the range $\frac{p_t}{2}\le \mu_R\le 2 p_t$. Furthermore, we also show the NLL calculation supplemented by our estimate of non-perturbative contributions due to the hadronisation process, i.e. Eqs. (\[np-shift-tpa-tpp\]) and (\[np-shift-t\]), using $\Lambda_\text{QCD}=1$ GeV. We note that because of the $R^2$ coefficient, the size of non-perturbative corrections is rather small. We expect that our simple implementation of non-perturbative corrections to fail in the peak (plateau) region, where one should retain more information about the shape function. Therefore, we only plot our NLL curves with non-perturbative corrections down to $t\sim 2\cdot 10^{-3}$ and ${t_\parallel}\sim 10^{-3}$, respectively.
In Fig. \[mc-comparisons\], we compare our results to those obtained with a general-purpose Monte Carlo event generator. We generate a single event $pp\to H Z$ at $\sqrt{s}=13$ TeV, with the Higgs decaying in $b \bar b$ and $Z$ leptonically, using MadGraph v2.6.6 [@Alwall:2014hca] and we then shower this event many times in Pythia v8.240 [@Sjostrand:2014zea]. FastJet v3.3.2 [@Cacciari:2011ma] is used to find jets and calculate the pull variables. The Monte Carlo results for $t$ and ${t_\parallel}$ are then compared to our NLL predictions, supplemented by the non-perturbative corrections. We find decent agreement between the Monte Carlo and our NLL prediction for $t$ and ${t_\parallel}$, supplemented by non-perturbative corrections. We note that the NLL and Monte Carlo predictions depart at the tail of the distributions. This effect is more noticeable for the pull magnitude and it signals the fact that the resummation alone is not enough to describe the distribution at large $t$ and matching to fixed-order is needed.
Finally, we expect additional non-perturbative contributions from the Underlying Event, due to multiple parton-parton interactions and pileup, due to multiple proton-proton interactions per bunch crossing. We have not included these effects in our studies, but we anticipate that their scaling with the jet radius will be the same as FSR, that we did calculate in this paper, albeit with a different, non-perturbative, coefficient.
Asymmetries {#sec:asym}
===========
The projections of the pull vector we have discussed thus far exhibit nice theoretical properties. In particular, as discussed at length, IRC safety ensures perturbative calculability, while non-perturbative contributions can be treated as (power) corrections. Furthermore, the particular definitions of the projections, see Eq. (\[pull-proj\]) resulted in observables that share many similarities in their all-order behaviour with variables that are among the most-studied in particle physics, such as the transverse momentum of a vector boson and its projections. However, we cannot fail to notice that presence of the absolute value in Eq. (\[pull-proj\]) leads to a loss of information. For instance, an emission in rapidity-azimuth region between the two jets and an emission outside, could potentially contribute to the same value of ${t_\perp}$ or ${t_\parallel}$. Therefore, in order to fully exploit the radiation pattern, we can construct asymmetric distributions by directly considering the projections of the pull vector along the two directions of interest, i.e. $\vec{t}\cdot \hat{n}_\parallel$ and $\vec{t}\cdot \hat{n}_\perp$. We note that the dot products, as opposed to ${t_\parallel}$ and ${t_\perp}$, are not positive-definite.
In Fig. \[mc-asymmetric-distributions\] we perform a Monte-Carlo study of these distributions for the colour singlet decay $H \to b \bar b $, using again the event generator Pythia v8.240, with the same kinematical settings of the previous section. For each distribution we show both parton-level and hadron-level results. We would expect the $\vec{t}\cdot \hat{n}_\perp$ to be roughly symmetric about zero, while the distribution of $\vec{t}\cdot \hat{n}_\parallel$ should be skewed in the direction of the colour-connected leg of the dipole, here the positive direction. The plots show that this is indeed the case. In order to emphasise these features even more, we can build the following asymmetry distributions $$\begin{aligned}
\label{asym_def}
\mathcal{A}_\parallel&=\frac{t_\parallel}{\sigma}\frac{d \sigma}{d t_\parallel}\Big|_{\vec{t}\cdot \hat{n}_\parallel>0}-\quad \frac{t_\parallel}{\sigma}\frac{d \sigma}{d t_\parallel}\Big|_{\vec{t}\cdot \hat{n}_\parallel<0}, \\
\mathcal{A}_\perp&=\frac{t_\perp}{\sigma}\frac{d \sigma}{d t_\perp}\Big|_{\vec{t}\cdot \hat{n}_\perp>0}-\quad \frac{t_\perp}{\sigma}\frac{d \sigma}{d t_\perp}\Big|_{\vec{t}\cdot \hat{n}_\perp<0}\end{aligned}$$ We expect $\mathcal{A}_\parallel$ to be more marked than $\mathcal{A}_\perp$ and this is indeed what is found in the simulations, as shown in Fig. \[mc-asymmetries\].
We note that the above asymmetries are still IRC safe and therefore can be calculated in perturbation theory. Indeed, we could argue that $\mathcal{A}_\parallel$ is essentially the IRC safe version of the pull angle distribution. The definitions of the asymmetries in Eq. (\[asym\_def\]) make explicit references to the sign of the scalar product which is used to project the pull vector. This constraint essentially introduces a new boundary in phase-space which renders the all-order structure of these observables richer. While we expect that this resummation can still be achieved, in this work we limit ourselves to analytically evaluate the asymmetries at fixed-order. The lowest-order contribution to the asymmetries originates from wide-angle soft emissions. In particular, we find that the contribution denoted by $\mathcal{A}$ in Eq. (\[dipole-expanded\]) does not vanish when we integrated separately over the $\vec{t}\cdot \hat{n}_i>0$ and $\vec{t}\cdot \hat{n}_i<0$ regions. We find $$\begin{aligned}
\mathcal{A}_\parallel&=\frac{{\alpha_s}C_F}{ \pi}\left[ \frac{4R}{\pi} \frac{\cos \beta \sinh \Delta y+ \sin \beta \sin \Delta \phi}{\cos \Delta \phi-\cosh \Delta y} + \mathcal{O} \left( R^3\right) \right]
+\mathcal{O} \left( {\alpha_s}^2\right)
, \\
\mathcal{A}_\perp&= \frac{{\alpha_s}C_F}{ \pi} \left[ \frac{4R}{\pi} \frac{\cos\beta \sin \Delta \phi-\sin \beta\sinh \Delta y}{\cos \Delta \phi-\cosh \Delta y}+ \mathcal{O} \left( R^3\right) \right]
+\mathcal{O} \left( {\alpha_s}^2\right).\end{aligned}$$ Interestingly, the asymmetries are sensitive to odd powers of the jet radius, in the small-$R$ expansion. This comes about because of the restrictions on the angular integrations imposed by the $\vec{t}\cdot \hat{n}_i>0$ and $\vec{t}\cdot \hat{n}_i<0$ constraints. We also point out that these asymmetries essentially depend on soft radiation, while collinear contributions cancel out. Soft radiation exhibit strong sensitivity to the pattern of colour correlations and therefore these observables can provide a valuable testing ground for Monte Carlo parton showers that attempt to go beyond the large-$N_c$ limit, e.g. [@Nagy:2019pjp; @Forshaw:2019ver].
Conclusions and Outlook {#sec:conclusions}
=======================
A detailed understanding of colour flow in hard scattering processes is of primary interest for LHC phenomenology for numerous reasons. First of all, it provides a valuable way of separating hadronic decay products of colour singlets, such as the Higgs or any other electroweak bosons, from the QCD background, often originating from gluon splittings. Furthermore, should new strongly-interacting states be discovered at the LHC, colour correlations can be used to characterise the colour representation these particles live in. However, precision studies of colour flow in hadron-hadron collisions are challenging because of the sensitivity to the soft and non-perturbative regimes of QCD. Therefore, it is important to devise observables that, while maintaining the desired sensitivity, offer theoretical robustness. In this context, infra-red and collinear safety is an important requirement because it ensures perturbative calculability, with dependence on non-perturbative corrections that is, at least parametrically, under control. Perturbative calculations for IRC safe observables can be used, in turn, to test the ability of general-purpose Monte Carlo event generators to correctly simulate colour flow in proton-proton collisions at hight energy.
In this study we have considered the observable jet pull, which has been introduced in order to probe colour flow between hard jets. Measurements of the pull angle have been advocated as sensitive probe of inter-jet radiation and have been performed at the Tevatron and the LHC. In particular, precision measurements by the ATLAS collaboration challenges the ability of general-purpose Monte Carlo event generators to correctly describe these distributions. In a previous Letter, we addressed the theoretical calculation of the pull angle distribution but we found difficult to draw firm theoretical conclusions due fact that the pull angle is not an IRC safe observable.
In this current paper, we have put forward novel observables that aim to probe colour flow in an efficient way, while featuring IRC safety. In particular, we have noticed that while the pull angle, i.e. the angle between the pull vector and the line joining the centres of the jets of interest, in the azimuth-rapidity plane, is not IRC safe, the projections of the pull vector along (${t_\parallel}$) and orthogonal to (${t_\perp}$) such an axis are. Therefore, these observables can be computed in perturbation theory. We have performed all-order calculations for these two projections and, for comparison, for the magnitude of the pull vector, considering the interesting case of a Higgs boson decaying into a pair of bottom quarks. Our results are valid to next-to-leading logarithmic accuracy, in the limit where the considered observable is small. In this context, besides collinear radiation, we have also investigated the structure of soft-emissions at wide angle and of non-global logarithms, expressing our results as a power series in the jet radius. Matching to fixed-order perturbation theory is possible but we have left it for future work. Furthermore, we have supplemented our results with an estimate of non-perturbative corrections arising from the hadronisation process and compared our results to simulations obtained with a Monte Carlo parton shower.
Finally, the theoretical understanding reached in this study has led us to the introduction of novel asymmetry distributions that measure the radiation pattern by looking at the difference between the jet pull vector pointing towards and away from the other jet of interest. In particular, the asymmetry distribution $\mathcal{A}_\parallel$ can be considered the IRC version of the pull angle distribution. We have pointed out that such asymmetries can have interesting applications both in the context of tagging colour singlets, such as $H\to b\bar b$ versus $g \to b \bar b$, and as a means to test how general-purpose Monte Carlo event generators probe soft emissions beyond the leading colour approximation. Therefore, we look forward to study these asymmetries in more detail in order to arrive to their all-order resummation.
We thank Andy Buckley and Giovanni Ridolfi for useful discussions, Ben Nachman, Matthew Schwartz, and Jesse Thaler for comments on the manuscript. This work is partly supported by the curiosity-driven grant “Using jets to challenge the Standard Model of particle physics" from Università di Genova.
[^1]: It would be interesting to study observables with a generalised $|\vec{r}_i|^\alpha$ dependence, perhaps employing different recombination schemes in the jet algorithm, such as winner-take-all [@Larkoski:2014uqa], in order to maintain the recoil-free property. We thank Jesse Thaler for pointing this out.
[^2]: Strictly speaking the jet radius dependence in argument of the logarithms only appears at this order in the soft-collinear contributions. However, we find that including it in the whole radiator, leads to better numerical stability. The difference between these choices is beyond NLL accuracy.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
The [[[Honey-Bee]{}]{}]{} game is a two-player board game that is played on a connected hexagonal colored grid or (in a generalized setting) on a connected graph with colored nodes. In a single move, a player calls a color and thereby conquers all the nodes of that color that are adjacent to his own current territory. Both players want to conquer the majority of the nodes. We show that winning the game is PSPACE-hard in general, NP-hard on series-parallel graphs, but easy on outerplanar graphs.
In the solitaire version, the goal of the single player is to conquer the entire graph with the minimum number of moves. The solitaire version is NP-hard on trees and split graphs, but can be solved in polynomial time on co-comparability graphs.
*Keywords:* combinatorial game; computational complexity; graph problem.
author:
- 'Rudolf Fleischer [^1]'
- 'Gerhard J. Woeginger [^2]'
date:
-
-
title: |
[**An Algorithmic Analysis of the\
Honey-Bee Game**]{}[^3]
---
Introduction {#s_intro}
============
The [[[Honey-Bee]{}]{}]{} game is a popular two-player board game that shows up in many different variants and at many different places on the web (the game is best be played on a computer). For a playable version we refer the reader for instance to Axel Born’s web-page [@Bor09]; see Fig. \[fig\_born\] for a screenshot. The playing field in [[[Honey-Bee]{}]{}]{} is a grid of hexagonal honey-comb cells that come in various colors; the coloring changes from game to game. The playing field may be arbitrarily shaped and may contain holes, but must always be connected. In the beginning of the game, each player controls a single cell in some corner of the playing field. Usually, the playing area is symmetric and the two players face each other from symmetrically opposing starting cells. In every move a player may call a color $c$, and thereby gains control over all connected regions of color $c$ that have a common border with the area already under his control. The only restriction on $c$ is that it cannot be one of the two colors used by the two players in their last move before the current move, respectively. A player wins when he controls the majority of all cells. On Born’s web-page [@Bor09] one can play against a computer, choosing from four different layouts for the playing field. The computer uses a simple greedy strategy: “Always call the color $c$ that maximizes the immediate gain.” This strategy is short-sighted and not very strong, and an alert human player usually beats the computer after a few practice matches.
(28000,18000)
In this paper we perform a complexity study of the [[[Honey-Bee]{}]{}]{} game when played by two players on some arbitrary connected graph instead of the hex-grid of the original game. We will show in Section \[s\_two\] that [[[Honey-Bee-2-Players]{}]{}]{} is NP-hard even on series-parallel graphs, and that it is PSPACE-complete in general. On outerplanar graphs, however, it is quite easy to compute a winning strategy.
In the *solitaire* (single-player) version of [[[Honey-Bee]{}]{}]{} the goal is to conquer the entire playing field as quickly as possible. Intuitively, a good strategy for the solitaire game will be close to a strong heuristic for the two-player game. For the solitaire version, our results draw a sharp separation line between easy and difficult cases. In particular, we show in Section \[s\_one\] that [[[Honey-Bee-Solitaire]{}]{}]{} is NP-hard for split graphs and for trees, but polynomial-time solvable on co-comparability graphs (which include interval graphs and permutation graphs). Thus, the complexity of the game is well-characterized for the class and subclasses of perfect graphs; see Fig. \[fig\_results\] for a summary of our results.
results.tex
Definitions {#s_def}
===========
We model [[[Honey-Bee]{}]{}]{} in the following graph-theoretic setting. The playing field is a connected, simple, loopless, undirected graph $G=(V,E)$. There is a set $C$ of $k$ colors, and every node $v\in V$ is colored by some color $col(v)\in C$; we stress that this coloring does not need to be proper, that is, there may be edges $[u,v]\in E$ with $col(u)=col(v)$. For a color $c\in C$, the subset $V_c\subseteq V$ contains the nodes of color $c$. For a node $v\in V$ and a color $c\in C$, we define the *color-$c$-neighborhood* $\Gamma(v,c)$ as the set of nodes in $V_c$ either adjacent to $v$ or connected to $v$ by a path of nodes of color $c$. Similarly, we denote by $\Gamma(W,c)=\bigcup_{w\in W}\Gamma(w,c)$ the color-$c$-neighborhood of a subset $W\subseteq V$. For a subset $W\subseteq V$ and a sequence $\gamma={\langle \gamma_1,\ldots,\gamma_b\rangle}$ of colors in $C$, we define a corresponding sequence of node sets $W_1=W$ and $W_{i+1}=W_i\cup \Gamma(W_i,\gamma_i)$, for $1\le i\le b$. We say that sequence $\gamma$ started on $W$ *conquers* the final node set $W_{b+1}$ in $b$ moves, and we denote this situation by $W\to_{\gamma}W_{b+1}$. The nodes in $V-W_{b+1}$ are called *free* nodes.
In the *solitaire* version of [[Honey-Bee]{}]{}, the goal is to conquer the entire playing field with the smallest possible number of moves. Note that [[[Honey-Bee-Solitaire]{}]{}]{} is trivial in the case of only two colors. But as we will see in Section \[s\_one\], the case of three colors can already be difficult.
In the *two-player* version of [[[Honey-Bee]{}]{}]{}, the two players $A$ and $B$ start from two distinct nodes $a_0$ and $b_0$ and then extend their regions step by step by alternately calling colors. Player $A$ makes the first move. One round of the game consists of a move of $A$ followed by a move of $B$. Consider a round, where at the beginning the two players control node sets $W_A$ and $W_B$, respectively. If player $A$ calls color $c$, then he extends his region $W_A$ to $W^\prime_A=W_A\cup(\Gamma(W_A,c)-W_B)$. If afterwards player $B$ calls color $d$, then he extends his region $W_B$ to $W^\prime_B=W_B\cup(\Gamma(W_B,c)-W^\prime_A)$. Note that once a player controls a node, he can never lose it again.
The game terminates as soon as one player controls more than half of all nodes. This player wins the game. To avoid draws, we require that the number of nodes is odd. There are three important rules that constrain the colors that a player is allowed to call.
1. A player must never call the color that has just been called by the other player.
2. A player must never call the color that he has called in his previous move.
3. A player must always call a color that strictly enlarges his territory, unless rules R1 and R2 prevent him from doing so.
r2.tex
What is the motivation for these three rules? Rule R1 is a technical condition that arises from the graphical implementation [@Bor09] of the game: Whenever a player calls a color $c$, his current territory is entirely recolored to color $c$. This makes it visually easier to recognize the territories controlled by both players. Rule R2 prevents the players from permanently blocking some color for the opponent. Fig. \[fig\_rule\_R2\] shows a situation where rule R2 actually prevents the game from stalling. Rule R3 is quite delicate, and is justified by situations as depicted in Fig. \[fig\_rule\_R3\]. Rule R3 guarantees that every game must terminate with either a win for player A or a win for player B. Note that rule R2 is redundant except in the case when a player has no move to gain territory (see Fig. \[fig\_rule\_R2\].
r3.tex
Note that [[[Honey-Bee-2-Players]{}]{}]{} is trivial in the case of only three colors: The players do not have the slightest freedom in choosing their next color, and always must call the unique color allowed by rules R1 and R2. However we will see in Section \[s\_two\] that the case of four colors can already be difficult.
Finally we observe that calling a color $c$ always conquers all connected components induced by $V_c$ that are adjacent to the current territory. Hence an equivalent definition of the game could use a graph with node weights (that specify the size of the corresponding connected component) and a *proper* coloring of the nodes. Any instance under the original definition can be transformed into an equivalent instance under the new definition by contracting each connected component of $V_c$, for some $c$, into a single node of weight $|V_c|$. However, we are interested in restrictions of the game to particular graph classes, some of which are not closed under edge contractions (as for instance the hex-grid graph of the original [[[Honey-Bee]{}]{}]{} game).
The Solitaire Game {#s_one}
==================
In this section we study the complexity of finding optimally short color sequences for [[[Honey-Bee-Solitaire]{}]{}]{}. We will show that this is easy for co-comparability graphs, while it is NP-hard for trees and split graphs. Since the family of co-comparability graphs contains interval graphs, permutation graphs, and co-graphs as sub-families, our positive result for co-comparability graphs implies all other positive results in Fig. \[fig\_results\].
A first straightforward observation is that [[[Honey-Bee-Solitaire]{}]{}]{} lies in NP: Any connected graph $G=(V,E)$ can be conquered in at most $|V|$ moves, and hence such a sequence of polynomially many moves can serve as an NP-certificate.
The Solitaire Game on Co-Comparability Graphs {#sec1:cocomparability}
---------------------------------------------
A *co-comparability graph* $G=(V,E)$ is an undirected graph whose nodes $V$correspond to the elements of some partial order $<$ and whose edges $E$ connect any two elements that are incomparable in that partial order, i.e., $[u,v]\in E$ if neither $u<v$ nor $v<u$ holds. For simplicity, we identify the nodes with the elements of the partial order. Golumbic [@GoRoUr83] showed that co-comparability graphs are exactly the intersection graphs of continuous real-valued functions over some interval $I$. If two function curves intersect, the corresponding elements are incomparable in the partial order; otherwise, the curve that lies complete above the other one corresponds to the larger element in the partial order. The function graph representation readily implies that the class of co-comparability graphs is closed under edge contractions. Therefore, we may w.l.o.g. restrict our analysis of [[[Honey-Bee-Solitaire]{}]{}]{} to co-comparability graphs with a proper node coloring, i.e., adjacent nodes have distinct colors (in the solitaire game we do not care about the weight of a node after an edge contraction). In this case, every color class is totally ordered because incomparable node pairs have been contracted.
Consider an instance of [[[Honey-Bee-Solitaire]{}]{}]{} with a minimal start node $v_0$ (in the partial order on $V$); a maximal start node could be handled similarly. The function graph representation implies the following observation.
\[obs\_smaller\] Conquering a node will simultaneously conquer all smaller nodes of the same color.
For any color $c$, let $Max(c)$ denote the largest node of color $c$. By Obs. \[obs\_smaller\], it suffices to find the shortest color sequence conquering all nodes $Max(c)$, for all colors $c$. We can do that by a simple shortest path computation. We assign every node $Max(c)$ weight $0$, and all other nodes weight $1$. Then we compute a shortest path (with respect to the node-weights) from $v_0$ to every node $Max(c)$ that is a *maximal element* in the partial order (which is actually exactly the set of all maximal elements). Let $OPT$ denote the smallest cost over all such paths.
For a color sequence $\gamma={\langle \gamma_1,\ldots,\gamma_b\rangle}$, we define the *length* of $\gamma$ as $|\gamma|=b$. We also define the *essential length* $ess(\gamma)$ of $\gamma$ as $|\gamma|$ minus the number of steps where $\gamma$ conquers a maximal node $Max(c)$ of some color class $c$. Obviously, $|\gamma|=ess(\gamma)+k$. Note that $OPT$ is the minimal essential cost of any color sequence conquering one of the maximal nodes.
\[thm\_opt\] The optimal solution for [[[Honey-Bee-Solitaire]{}]{}]{} has cost $OPT+k$.
Let $\gamma$ be a shortest color sequence conquering the entire graph starting at $v_0$. After conquering $v$, $\gamma$ only needs to conquer all free nodes $Max(c)$ to conquer the entire graph. Thus, $|\gamma| = ess(\gamma) + k \ge OPT+k$.
\[th\_cocomp\] [[[Honey-Bee-Solitaire]{}]{}]{} starting at an extremal node $v_0$ can be solved in polynomial time on co-comparability graphs.
Given the co-comparability graph $G$, we can compute the underlying partial order $<$ in polynomial time [@GoRoUr83]. Assigning the weights and solving one single source shortest path problem starting at $v_0$ also takes polynomial time.
We can also formulate this algorithm as a dynamic program. For any node $v$, let $D(v)$ denote the essential length of the shortest color sequence $\gamma$ that can conquer $v$ when starting at $v_0$. For any color $c$, let $min_v(c)$ denote the smallest node of color $c$ connected to $v$, if such nodes exist. Then we can compute $D(v)$ recursively as follows: $$D(v_0) = 0$$ and $$D(v) = \min_{c} (D(min_v(c)) + \delta_v) \>,$$ where $D(min_v(c))=\infty$ if $min_v(c)$ is undefined, and $\delta_v=0$ ($1$) if $v$ is (not) a maximal node for some color class.
Clearly, this dynamic program simulates the shortest path computation of our first algorithm and we have $OPT = \min_{v}(D(v)+k)$, where we minimize over all maximal nodes $v$. We now extend the dynamic program to the case that $v_0$ is not an extremal element. The problem is that we now must extend our territory in two directions. If we choose a move that makes good progress upwards it may make little progress downwards, or vice versa. In particular, the optimal strategy cannot be decomposed into two independent optimal strategies, one conquering upwards and one conquering downwards. Analogously to the algorithm above, for a clor $c$ define $Min(c)$ as the smallest node of color $c$, and $max_v(c)$ as the largest node of color $c$ connected to a node $v$.
Unfortunately, we must now redefine the essential length of a color sequence $\gamma$. In our original definition, we did not count coloring steps that conquered maximal elements of some color class. This is intuitively justified by the fact that these steps must be done by any color sequence conquering the entire graph at some time, therefore it is advantageous to do them as early as possible (which is guaranteed by giving these moves cost 0). But now we must also consider the minimal nodes of each color class. An optimal sequence conquering the entire graph will at some time have conquered a minimal node and a maximal node. Afterwards, it will only call extremal nodes for some color class. If both extremal nodes of a color class are still free, we only need *one* move to conquer both simultaneously. If one of them had been captured earlier, we still need to conquer the other one. This indicates that we should charge 1 for the first extremal node conquered while the second one should be charged 0, as before. If both nodes are conquered in the same move, we should also charge 0. Therefore, we now define the *essential length* $ess(\gamma)$ of $\gamma$ as $|\gamma|$ minus the number of steps where $\gamma$ conquers the second extremal node of some color class.
For a node $v$ below $v_0$ or incomparable to $v_0$ and a node $w$ above $v_0$ or incomparable to $v_0$ let $D(v,w)$ denote the essential length of the shortest color sequence $\gamma$ that can conquer $v$ and $w$ when starting at $v_0$. Note that we do not need to keep track of which first extremal nodes of a color class have been conquered because we can deduce this from the two nodes $v$ and $w$ currently under consideration. In particular, we can compute $D(v,w)$ recursively as follows: $$D(v_0,v_0) = 0$$ and $$D(v,w) = \min_{c} (D(v,min_w(c)) + \delta_w(v),
D(max_v(c),w) + \delta_v(w)) \>,$$ where $\delta_v(w)=0$ if and only if $w$ is an extremal node of some color class $c$ and the other extremal node of color class $c$ is either between $v$ and $w$, or incomparable to either $v$ or $w$, or both (it was either conquered earlier, or it will be conquered in this step); otherwise, $\delta_v(w)=1$. Obviously, $|\gamma|=ess(\gamma)+k$.
\[thm\_opt\_general\] The optimal solution for [[[Honey-Bee-Solitaire]{}]{}]{} has cost $\min_{v,w}(D(v,w)+k)$, where we minimize over all minimal nodes $v$ and all maximal nodes $w$.
Let $\gamma$ be a shortest color sequence conquering the entire graph starting at $v_0$. Let $v$ be the first minimal node conquered by $\gamma$ and $w$ the first maximal node. After conquering $v$ and $w$, $\gamma$ only needs to conquer all free nodes $Max(c)$ to conquer the entire graph. Thus, $|\gamma| \ge D(v,w) + k$.
\[th\_cocomp\_general\] [[[Honey-Bee-Solitaire]{}]{}]{} can be solved in polynomial time on co-comparability graphs.
The Solitaire Game on Split Graphs {#ss_split}
----------------------------------
A *split graph* is a graph whose node set can be partitioned into an induced clique and into an induced independent set. We will show that [[[Honey-Bee-Solitaire]{}]{}]{} is NP-hard on split graphs. Our reduction is from the NP-hard [Feedback Vertex Set]{} ([[FVS]{}]{}) problem in directed graphs; see for instance Garey and Johnson [@GaJo79].
\[thm\_split\] [[[Honey-Bee-Solitaire]{}]{}]{} on split graphs is NP-hard.
Consider an instance $(X,A,t)$ of [[FVS]{}]{}. To construct an instance $(V,E,b)$ of [[Honey-Bee-Solitaire]{}]{}, we first build a clique from the nodes in $X$ together with a new node $v_0$, the start node of [[Honey-Bee-Solitaire]{}]{}, where each node $x\in X+v_0$ has a different color $c_x$. Next, we build the independent set. For every arc $(x,y)\in A$, we introduce a corresponding node $v(x,y)$ of color $c_y$ which is only connected to node $x$ in the clique, i.e., it has degree one. Finally, we set $b=|X|+t$. We claim that the constructed instance of [[[Honey-Bee-Solitaire]{}]{}]{} has answer YES, if and only if the instance of [[[FVS]{}]{}]{} has answer YES.
Assume that the [[[FVS]{}]{}]{} instance has answer YES. Let $X^\prime$ be a smallest feedback set whose removal makes $(X,A)$ acyclic. Let $\pi$ be a topological order of the nodes in $X-X^\prime$, and let $\tau$ be an arbitrary ordering of the nodes in $X^\prime$. Consider the color sequence $\gamma$ of length $|X|+t$ that starts with $\tau$, followed by $\pi$, and followed by $\tau$ again. We claim that $\{v_0\}\to_{\gamma}V$. Indeed, $\gamma$ first runs through $\tau$ and $\pi$ and thereby conquers all clique nodes. Every independent set node $v(x,y)$ with $y\in X^\prime$ is conquered during the second transversal of $\tau$. Every independent set node $v(x,y)$ with $y\in X-X^\prime$ is conquered during the transversal of $\pi$, since $\pi$ first conquers $x$ with color $c_x$, and afterwards $v(x,y)$ with color $y$.
Next assume that the instance of [[[Honey-Bee-Solitaire]{}]{}]{} has answer YES. Let $\gamma$ be a color sequence of length $b=|X|+t$ conquering $V$. Define $X^\prime$ as the set of nodes $x$ such that color $c_x$ occurs at least twice in $\gamma$; clearly, $|X^\prime|\le t$. Consider an arc $(x,y)\in A$ with $x,y\in X-X^\prime$. Since $\gamma$ contains color $c_y$ only once, it must conquer node $v(x,y)$ of color $c_y$ after node $v(x)$ of color $c_x$. Hence, $\gamma$ induces a topological order of $X-X^\prime$.
The construction in the proof above uses linearly many colors. What about the case of few colors? On split graphs, [[[Honey-Bee-Solitaire]{}]{}]{} can always be solved by traversing the color set $C$ twice; the first traversal conquers all clique nodes, and the second traversal conquers all remaining free independent set nodes. Thus, every split graph can be completely conquered in at most $2|C|$ steps. If there are only few colors, we can simply check all color sequences of this length $2|C|$.
\[thm\_split\_const\] If the number of colors is bounded by a fixed constant, [[[Honey-Bee-Solitaire]{}]{}]{} on split graphs is polynomial-time solvable.
The Solitaire Game on Trees {#ss_tree}
---------------------------
In this section we will show that [[[Honey-Bee-Solitaire]{}]{}]{} is NP-hard on trees, even if there are at only three colors. We reduce [[[Honey-Bee-Solitaire]{}]{}]{} from a variant of the [Shortest Common Supersequence]{} ([[SCS]{}]{}) problem which is know to be NP-complete (see Middendorf [@Mid94]).
Middendorf’s hardness result also implies the hardness of the following variant of [[SCS]{}]{}:
\[thm\_mscs\] [[[MSCS]{}]{}]{} is NP-complete.
Here is a reduction from [[[SCS]{}]{}]{} to [[[MSCS]{}]{}]{}. Consider an arbitrary sequence $\tau$ with elements from $\{0,1\}$. We define $f(\tau)$ as the sequence we obtain from replacing every occurrence of the element 0 in $\tau$ by two consecutive elements 0 and 2. Now consider an instance $(\sigma_1,\ldots,\sigma_s,t)$ of [[SCS]{}]{}. We construct an instance $(\sigma_1^\prime,\ldots,\sigma_s^\prime,t^\prime)$ of [[[MSCS]{}]{}]{} by setting $\sigma^\prime_i=f(\sigma_i)$, for $1\le i\le s$. Then, for any sequence $\sigma$ with elements from $\{0,1\}$, $\sigma$ is a common supersequence of $\sigma_1,\ldots,\sigma_s$ if and only if $f(\sigma)$ is a common supersequence of $\sigma^\prime_1,\ldots,\sigma^\prime_s$. This implies the NP-hardness of [[MSCS]{}]{}.
\[thm\_tree\] [[[Honey-Bee-Solitaire]{}]{}]{} is NP-hard on trees, even in case of only three colors.
We reduce [[[MSCS]{}]{}]{} to [[[Honey-Bee-Solitaire]{}]{}]{} on trees. Consider an instance $(\sigma_1,\ldots,\sigma_s,t)$ of [[MSCS]{}]{}. We use color set $C=\{0,1,2\}$. We first construct a root $v_0$ of color $2$. Then we attach a path of length $|\sigma_i|$ to $v_0$ for each sequence $\sigma_i$, where an element $j$ is colored $j$. See the left half of Fig. \[fig\_sp\] for an example. Finally, we set $b=t$. It its straightforward to see that the constructed instance of [[[Honey-Bee-Solitaire]{}]{}]{} has answer YES if and only if the instance of [[[MSCS]{}]{}]{} has answer YES.
The Two-Player Game {#s_two}
===================
In this section we study the complexity of the two-player game. While on outerplanar graphs the players can compute their winning strategies in polynomial time, this problem is NP-hard for series-parallel graphs with four colors, and PSPACE-complete with four colors on arbitrary graphs. Our positive result for outerplanar graphs works for an arbitrary number of colors. Our negative results work for four colors, which is the strongest possible type of result (recall that instances with three colors are trivial to solve).
The Two-Player Game on Outer-Planar Graphs {#ss_outerplanar}
------------------------------------------
A graph is *outer-planar* if it contains neither $K_4$ nor $K_{2,3}$ as a minor. Outer-planar graphs have a planar embedding in which every node lies on the boundary of the so-called *outer face*. For example, every tree is an outer-planar graph.
Consider an outer-planar graph $G=(V,E)$ as an instance of [[[Honey-Bee-2-Players]{}]{}]{} with starting nodes $a_0$ and $b_0$ in $V$, respectively. The starting nodes divide the nodes on the boundary of the outer face $F$ into an upper chain $u_1,\ldots,u_s$ and a lower chain $\ell_1,\ldots,\ell_t$, where $u_1$ and $\ell_1$ are the two neighbors of $a_0$ on $F$, while $u_s$ and $\ell_t$ are the two neighbors of $b_0$ on $F$. We stress that this upper and lower chain are not necessarily disjoint (for instance, articulation nodes will occur in both chains).
Now consider an arbitrary situation in the middle of the game. Let $U$ (respectively $L$) denote the largest index $k$ such that player $A$ has conquered node $u_k$ (respectively node $\ell_k$). See Fig. \[fig\_outerplanar\] to illustrate these definitions and the following lemma.
\[thm\_outerplanar\_conquer\] Let $X$ denote the set of nodes among $u_1,\ldots,u_U$ and $\ell_1,\ldots,\ell_L$ that currently do neither belong to $A$ nor to $B$. Then no node in $X$ can have a neighbor among $u_{U+1},\ldots,u_s,b_0,\ell_t,\ldots,\ell_{L+1}$.
The existence of such a node in $X$ would lead to a $K_4$-minor in the outer-planar graph.
outerplanar.tex
\[thm\_outerplanar\] [[[Honey-Bee-2-Players]{}]{}]{} on outer-planar graphs is polynomial-time solvable.
The two indices $U$ and $L$ encode all necessary information on the future behavior of player $A$. Eventually, he will own all nodes $u_1,\ldots,u_U$ and $\ell_1,\ldots,\ell_L$, and the possible future expansions of his area beyond $u_U$ and $\ell_L$ only depend on $U$ and $L$. Symmetric observations hold true for player $B$.
As every game situation can be concisely described by just four indices, there is only a polynomial number $O(|V|^4)$ of relevant game situations. The rest is routine work in combinatorial game theory: We first determine the winner for every end-situation, and then by working backwards in time we can determine the winners for the remaining game situations.
The Two-Player Game on Series-Parallel Graphs {#ss_sp}
---------------------------------------------
A graph is *series-parallel* if it does not contain $K_4$ as a minor. Equivalently, a series-parallel graph can be constructed from a single edge by repeatedly doubling edges, or removing edges, or replacing edges by a path of two edges with a new node in the middle of the path. We stress that we do not know whether the two-player game on series-parallel graphs is contained in the class NP (and we actually see no reason why it should lie in NP); therefore the following theorem only states NP-hardness.
\[thm\_sp\] For four (or more) colors, problem [[[Honey-Bee-2-Players]{}]{}]{} on series-parallel graphs is NP-hard.
We use the color set $C=\{0,1,2,3\}$. A central feature of our construction is that player $B$ will have no real decision power, but will only follow the moves of player $A$: If player $A$ starts a round by calling color $0$ or $1$, then player $B$ must follow by calling the other color in $\{0,1\}$ (or waste his move). And if player $A$ starts a round by calling color $2$ or $3$, then player $B$ must call the other color in $\{2,3\}$ (or waste his move). In the even rounds the players will call the colors in $\{0,1\}$ and in the odd rounds they will call the colors in $\{2,3\}$. Both players are competing for a set of honey pots in the middle of the battlefield, and need to get there as quickly as possible. If a player deviates from the even-odd pattern indicated above, he might perhaps waste his move and delay the game by one round (in which neither player comes closer to the honey pots), but this remains without further impact on the outcome of the game.
The proof is by reduction from the supersequence problem [[[SCS]{}]{}]{} with binary sequences; see Section \[ss\_tree\]. Consider an instance $(\sigma_1,\ldots,\sigma_s,t)$ of [[[SCS]{}]{}]{}, and let $n$ denote the common length of all sequences $\sigma_i$. We first construct two start nodes $a_0$ and $b_0$ of colors $2$ and $3$, respectively. For each sequence $\sigma_i$ with $1\le i\le s$ we do the following:
- We construct a path $P_i$ that consists of $2n-1$ nodes and that is attached to $a_0$: The $n$ nodes with odd numbers mimic sequence $\sigma_i$, while the $n-1$ nodes with even numbers along the path all receive color $2$. The first node of $P_i$ is adjacent to $a_0$, and its last node is connected to a so-called honey pot $H_i$.
- The *honey pot* $H_i$ is a long path consisting of $4st$ nodes of color $3$. Intuitively, we may think of a honey pot as a single node of large weight, because conquering one of the nodes will simultaneously conquer the entire path.
- Every honey pot $H_i$ can also be reached from $b_0$ by another path $Q_i$ that consists of $2t-1$ nodes. Nodes with odd numbers get color $0$, and nodes with even numbers get color $3$. The first node of $Q_i$ is adjacent to $b_0$, and its last node is connected to $H_i$. Furthermore, we create for each odd-numbered node (of color $0$) a new twin node of color $1$ that has the same two neighbors as the color $0$ node. Note that for every path $Q_i$ there are $t$ twin pairs.
Finally we create a private honey pot $H_B$ for player $B$, that is connected to node $b_0$ and that consists of $4s(s-1)t+(2n-1)s$ nodes of color $2$. This completes the construction; see Fig. \[fig\_sp\] for an example.
Assume that the [[[SCS]{}]{}]{} instance has answer YES. During his first $2t-1$ steps, player $B$ can only conquer the paths $Q_i$ and his private honey pot $H_B$. At the same time, player $A$ can conquer all paths $P_i$ by calling color $2$ in his even moves and by following a shortest 0-1 supersequence in his odd moves. Then, in round $2t$ player $A$ will simultaneously conquer all the honey pots $H_i$ with $1\le i\le s$. This gives $A$ a territory of at least $1+(2n-1)s+4s^2t$ nodes, and $B$ a smaller territory of at most $1+(3t-1)s+4s(s-1)t+(2n-1)s$ nodes. Hence $A$ can enforce a win.
Next assume that player $A$ has a winning strategy. Player $B$ can always conquer his starting node $b_0$ and his private honey pot $H_B$. If $B$ also manages to conquer one of the pots $H_i$, then he gets a territory of at least $1+4s(s-1)t+(2n-1)s+4st$ nodes and surely wins the game. Hence player $A$ can only win if he conquers all $s$ honey pots $H_i$. To reach them before player $B$ does, player $A$ must conquer them within his first $2t$ moves. In every odd round, player $A$ will call a color $0$ or $1$ and player $B$ will call the other color in $\{0,1\}$. Hence, in the even rounds, colors $0$ and $1$ are forbidden for player $A$, and the only reasonable move is to call color $2$. Note that the slightest deviation of these forced moves would give player $B$ a deadly advantage. In order to win, the odd moves of player $A$ must induce a supersequence of length at most $t$ for all sequences $\sigma_i$. Therefore, the [[[SCS]{}]{}]{} instance has answer YES.
sp.tex
The Two-Player Game on Arbitrary Graphs {#ss_pspace}
---------------------------------------
In this section we will show that problem [[[Honey-Bee-2-Players]{}]{}]{} is PSPACE-complete on arbitrary graphs. Our reduction is from the PSPACE-complete [Quantified Boolean Formula]{} ([[QBF]{}]{}) problem; see for instance Garey & Johnson [@GaJo79].
\[thm\_pspace\] For four (or more) colors, problem [[[Honey-Bee-2-Players]{}]{}]{} on arbitrary graphs is PSPACE-complete.
We reduce from [[[QBF]{}]{}]{}. Let $F=\exists x_1\forall x_2\cdots\exists x_{2n-1}\forall x_{2n}
\bigwedge_j C_j$ be an instance of [[QBF]{}]{}. We construct a bee graph $G_F=(V,E)$ with four colors (white, light-gray, dark-gray, and black) such that player $A$ has a winning strategy if and only if $F$ is true. Let $a_0$ (colored light-gray) and $b_0$ (colored dark-gray) denote the start nodes of players $A$ and $B$, respectively.
Each player controls a *pseudo-path*, that is, a path where some nodes may be duplicated as parallel nodes in a diamond-shaped structure; see Fig. \[fig\_var\]. A so-called *choice pair* consists of a node on a pseudo-path together with some duplicated node in parallel. The start nodes are at one end of the respective pseudo-paths, and the players can conquer the nodes on their own path without interference from the other player. However, they must do so in a timely manner because either path ends at a humongous *honey pot*, denoted respectively by $H_A$ and $H_B$. A honey pot is a large clique of identically-colored nodes (we may think of it as a single node of large weight, because conquering one node will simultaneously conquer the entire clique). Both honey pots have the same size but different colors, namely black ($H_A$) and white ($H_B$), and they are connected to each other by an edge. Consequently, both players must rush along their pseudo-paths as quickly as possible to reach their honey pot before the opponent can reach it and to prevent the opponent from winning by conquering both honey pots. The last nodes before the honey pots are denoted by $a_f$ and $b_f$, respectively. They separate the last variable gadgets (described below) from the honey pots.
var.tex
Fig. \[fig\_var\] shows an overview of the pseudo-paths and one *variable gadget* in detail. A variable gadget is a part of the two pseudo-paths corresponding to a pair of variables $\exists x_{2i-1} \forall x_{2i}$, for some $i\ge1$. For player $A$, the gadget starts at node $a_{i-1}$ with a choice pair $a_{2i-1}^F$ and $a_{2i-1}^T$, colored white and black, respectively. The first node conquered by $A$ will determine the truth value for variable $x_{2i-1}$. In the same round, player $B$ has a choice on his pseudo-path $P_B$ between nodes $b_{2i-1}^F$ and $b_{2i-1}^T$. Since these nodes have the same color as $A$’s choices in the same round, $B$ actually does not have a choice but must select the other color not chosen by $A$.
Three rounds later, player $B$ has a choice pair $b_{2i}^F$ and $b_{2i}^T$, assigning a truth value to variable $x_{2i}$. In the next step (which is in the next round), player $A$ has a choice pair $a_{2i}^F$ and $a_{2i}^T$ with the same colors as $B$’s choice pair for $x_{2i}$. Again, this means that $A$ does not really have a choice but must select the color not chosen by $B$ in the previous step. Since we want $A$ to conquer those clauses containing a literal set to true by player $B$, the colors in $B$’s choice pair have been switched, i.e., $b_{2i}^F$ is black and $b_{2i}^T$ is white.
Note that all the nodes $a_0,a_1,\ldots,a_n$ are light-gray and all the nodes $b_0,b_1,\ldots,b_n$ are dark-gray. This allows us to concatenate as many variable gadgets as needed. Further note that $a_f$ is white, while $b_f$ is light-gray.
The *clause* gadgets are very simple. Each clause $C_j$ corresponds to a small honey pot $H_j$ of color white. The size of the small honey pots is smaller than the size of the large honey pots $H_A$ and $H_B$, but large enough such that player $A$ loses if he misses one of them. Player $A$ should conquer $H_j$ if and only if $C_j$ is true in the assignment chosen by the players while conquering their respective pseudo-paths. We could connect $a_{2i-1}^T$ directly with $H_j$ if $C_j$ contains literal $x_{2i-1}$, however then player $A$ could in subsequent rounds shortcut his pseudo-path by entering variable gadgets for the other variables in $C_j$ from $H_j$. To prevent this from happening, we place waiting gadgets between the variable gadgets and the clauses.
Let $a_{k}^\star$ denote the node on $P_A$ right after the choice pair $a_k^F$ and $a_k^T$, for $k=1,\ldots,2n$; similarly, $b_k^\star$ are the nodes on $P_B$ right after $B$’s choice pairs. A *waiting gadget* $W_k$ consists of two copies $W_k^F$ and $W_k^T$ of the sub-path of $P_A$ starting at $a_k^\star$ and ending at $a_n$, see Fig. \[fig\_wait\]. If clause $C_j$ contains literal $x_k$, $H_j$ is connected to the node $w_n^T$ corresponding to $a_n$ in $W_k^T$; if $C_j$ contains literal $\overline{x_k}$, $H_j$ is connected to the node $w_n^F$ corresponding to $a_n$ in $W_k^F$. If $k=2i-1$ (i.e., we have an existential variable $x_{2i-1}$ whose value is assigned by player $A$), then $a_{2i-1}^F$ and $b_{2i-1}^F$ are connected to $w_{2i-1}^{\star F}$, and $a_{2i-1}^T$ and $b_{2i-1}^T$ are connected to $w_{2i-1}^{\star T}$. If $k=2i$ (i.e., we have a universal variable $x_{2i}$ whose value is assigned by player $B$), then $a_{2i}^F$ and $b_{2i}^\star$ are connected to $w_{2i}^{\star F}$, and $a_{2i}^T$ and $b_{2i-1}^\star$ are connected to $w_{2i}^{\star T}$.
wait.tex
Finally, we connect $b_f$ with all clause honey pots $H_j$ to give player $B$ the opportunity to conquer all those clauses that contain no true literal. This completes the construction of $G_F$. Fig. \[fig\_example\] shows the complete graph $G_F$ for a small example formula $F$.
We claim that player $A$ has a winning strategy on $G_F$ if and only if formula $F$ is true. It is easy to verify that player $A$ can indeed win if $F$ is true. All he has to do is to conquer those nodes in his existential choice pairs corresponding to the variable values in a satisfying assignment for $F$. For the existential variables, he has full control to select any value, and for the universal variables he must pick the opposite color as selected by player $B$ in the previous step, which corresponds to setting the variable to exactly the value that player $B$ has selected. Hence player $B$ can block a move of player $A$ by appropriately selecting a value for a universal variable. Note that no other blocking moves of player $B$ are advantageous: If $B$ blocks $A$’s next move by choosing a color that does not make progress on his own pseudo-path, then $A$ will simply make an arbitrary waiting move and then in the next round $B$ cannot block $A$ again. When player $A$ conquers node $a_n$, he will simultaneously conquer the last nodes in all waiting gadgets corresponding to true literals. Since every clause contains a true literal for a satisfying assignment, player $A$ can then in the next round conquer $a_f$ together with all clause honey pots (which all have color white). Player $B$ will respond by conquering $b_f$, and the game ends with both players conquering their own large honey pots $H_A$ and $H_B$, respectively. Since player $A$ got all clause honey pots, he wins.
To make this argument work, we must carefully chose the sizes of the honey pots. Each pseudo-path contains $9n+1$ nodes, of which at most $n$ can be conquered by the other player. The waiting gadgets contain two paths of length $9k+6$ for existential variables and $9k+1$ for universal variables. At the end, player $A$ will have conquered one of the two paths completely and maybe some parts of the sibling path, that is, we do not know exactly the final owner of less than $n^2$ nodes. The clause honey pots should be large enough to absorb this fuzzyness, which means it is sufficient to give them $2n^2$ nodes. The honey pots $H_A$ and $H_B$ should be large enough to punish any foul play by the players, that is, when they do not strictly follow their pseudo-paths. It is sufficient to give them $2n^3$ nodes.
To see that $F$ is true if player $A$ has a winning strategy note that player $A$ must strictly follow his pseudo-path, as otherwise player $B$ could beat him by reaching the large honey pots first. Thus player $A$’s strategy induces a truth assignment for the existential variables. Similarly, player $B$’s strategy induces a truth assignment for the universal variables. Player $A$ can only win if he also conquers all clause honey pots, and hence the players must haven chosen truth values that make at least one literal per clause true. This means that formula $F$ is satisfiable.
Conclusions {#s_conclusion}
===========
We have modeled the Honey Bee game as a combinatorial game on colored graphs. For the solitaire version, we have analyzed the complexity on many classes of perfect graphs. For the two player version, we have shown that even the highly restricted case of series-parallel graphs is hard to tackle. Our results draw a clear separating line between easy and hard variants of these problems.
Acknowledgements
================
Part of this research was done while G. Woeginger visited Fudan University in 2009.
[4]{}
A. Born. Flash application for the computer game *“Biene” (Honey-Bee)*, 2009.\
<http://www.ursulinen.asn-graz.ac.at/Bugs/htm/games/biene.htm>.
M. R. Garey and D. S. Johnson. . W. H. Freeman and Company, New York, 1979.
M. C. Golumbic, D. Rotem, and J. Urrutia. Comparability graphs and intersection graphs. , 43(1):37–46, 1983.
M. Middendorf. More on the complexity of common superstring and supersequence problems. , 125:205–228, 1994.
[^1]: School of Computer Science, IIPL, Fudan University, Shanghai 200433, China. Email: [[email protected]]{}.
[^2]: [[email protected]]{}. Department of Mathematics and Computer Science, TU Eindhoven, P.O. Box 513, 5600 MB Eindhoven, Netherlands.
[^3]: RF acknowledges support by the National Natural Science Foundation of China (No. 60973026), the Shanghai Leading Academic Discipline Project (project number B114), the Shanghai Committee of Science and Technology of China (09DZ2272800), and the Robert Bosch Foundation (Science Bridge China 32.5.8003.0040.0). GJW acknowledges support by the Netherlands Organisation for Scientific Research (NWO grant 639.033.403), and by BSIK grant 03018 (BRICKS: Basic Research in Informatics for Creating the Knowledge Society).
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Isaac Goldstein[^1]'
- 'Tsvi Kopelowitz[^2]'
- 'Moshe Lewenstein[^3]'
- Ely Porat
bibliography:
- 'ms.bib'
title: 'How Hard is it to Find (Honest) Witnesses?'
---
Appendix {#appendix .unnumbered}
========
[^1]: This research is supported by the Adams Foundation of the Israel Academy of Sciences and Humanities
[^2]: This research is supported by NSF grants CCF-1217338, CNS-1318294, and CCF-1514383
[^3]: This research is supported by a BSF grant 2010437 and a GIF grant 1147/2011.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'This paper presents a systematic approach for implementing a class of nonlinear signal processing systems as a distributed web service, which in turn is used to solve optimization problems in a distributed, asynchronous fashion. As opposed to requiring a specialized server, the presented approach requires only the use of a commodity database back-end as a central resource, as might typically be used to serve data for websites having large numbers of concurrent users. In this sense the presented approach leverages not only the scalability and robustness of various database systems in sharing variables asynchronously between workers, but also critically it leverages the tools of signal processing in determining how the optimization algorithm might be organized and distributed among various heterogeneous workers. A publicly-accessible implementation is also presented, utilizing Firebase as a back-end server, and illustrating the use of the presented approach in solving various optimization problems commonly arising in the context of signal processing.'
author:
-
bibliography:
- 'refs.bib'
title: 'Web Services for Asynchronous, Distributed Optimization Using Conservative Signal Processing'
---
Introduction {#sec:intro}
============
In designing and implementing signal processing systems, a general implementation strategy is to (1) begin with a set of desired equations to be satisfied, (2) represent these equations as a graphical structure, (3) distribute state throughout the graph, e.g. introducing scalar or vector delay elements, and (4) determine a protocol for exchanging state, resulting in an algorithm or iteration satisfying the original equations. Various specific methods consistent with this general approach have been described formally, e.g. in [@BaranLahlouICASSP15; @Crochiere].
Consistent with these steps, the issue of distributing state is perhaps the most central issue in effectively distributing algorithms in general, including distributing algorithms across multiple heterogeneous processing nodes. As has been discussed in [@NikkilArvind], this observation suggests opportunity in utilizing the general strategy of specifying algorithms first using a declarative language, which after determining a protocol for distributing and exchanging state, would be decomposed as an ensemble of distributed programs and implemented on processing nodes using imperative languages.
The formal approaches used in implementing signal processing systems form a broad and concrete class of examples that are consistent with this general strategy, with state-free signal-flow diagrams being a declarative representation, and with an eventual arrangement of run-loops being imperative. Drawing upon this, the results outlined in [@tbaran-phd; @BLOpt; @LahlouBaranLinProg] describe a straightforward method for implementing a variety of *optimization algorithms* by casting them as signal processing systems, in turn leveraging the various common associated implementation strategies in distributing and transferring state.
The intent of this paper is to describe a distributed web service for solving optimization problems that results as a consequence of the way of thinking described in [@BLOpt; @LahlouBaranLinProg; @tbaran-phd]. The service is freely accessible online as part of the general site “Signal Processing Conservation”[@spconservation], which provides a general overview and examples of the use of conservation principles in signal processing systems, importantly also describing the mathematical foundation underlying [@BLOpt; @LahlouBaranLinProg; @tbaran-phd]. The portion of the site containing the web service for optimization discussed in this paper is available at http://optimization.spconservation.org, which we refer to herein as “O-SPC”.
The architecture of O-SPC in particular is built on Firebase [@firebase] and utilizes the service primarily as a high-performance back-end for asynchronous representational state transfer between browser-based clients, e.g. as opposed to as a centralized resource for coordinating data processing as with [@parameterServer]. In this sense, O-SPC represents an example of how the thinking described in [@BLOpt; @LahlouBaranLinProg; @tbaran-phd] can be used to create a performant system operating in the somewhat extreme case where numerical computation is distributed entirely to the extremities of the graph. The considerations described in this paper would similarly apply to the creation of a web-based optimization service utilizing an alternative key-value store system, e.g. MongoDB [@MongoDB] or Redis [@Redis], or any number of relational database systems. In each of these cases, the performance of the distributed system would be able to draw upon the particular strengths of the data store being utilized.
We begin in Section \[sec:sig-flow\] by specifying the targeted class of signal processing systems and reviewing their utility as optimization algorithms. In Section \[sec:opt.spc\] we focus on general considerations regarding their distributed implementation as a web service, consistent with the architecture of O-SPC. In Section \[sec:examples\], we collate the numerical experiments referenced throughout and provide concluding remarks.
{width="\textwidth"}
Signal processing systems and optimization {#sec:sig-flow}
==========================================
The general framework presented in [@BLOpt] facilitates the construction of distributed, asynchronous signal processing systems for solving optimization problems by analyzing the structure of the optimization problem itself and without relying on any existing non-distributed and/or synchronous methods. Therefore using this framework, signal processing systems, and by extension optimization algorithms, may be generated that might not be readily derived by conventional techniques. In the remainder of this section we briefly review the key steps in casting optimization algorithms as signal processing systems.
A conservative signal processing system is one for which the variables available for interconnection between subsystems admit an organization adhering to an indefinite quadratic form of a particular class that is invariant to the evolution of the system [@tbaran-phd]. The utility of conservation principles in [@BLOpt] is twofold: (1) in defining the primal optimization problem in Fig. \[fig:signal-flow\](a) and its dual so that the joint feasibility conditions depicted in Fig. \[fig:signal-flow\](b) serve as sufficient conditions for stationarity, and (2) in transforming said conditions into the algebraic form illustrated in Fig. \[fig:signal-flow\](c) where $R:\Rn \to \Rn$ and $H\colon \R^{N-K}\to\R^{N-K}$ are orthogonal matrices, $m \colon \Rk \to \Rk$ is a generally nonlinear map, and $\e\in\R^{N-K}$ is a system bias. The maps $m$ and $H$ as well as the bias $\e$ are associated with the set constraints $\mathcal{A}_{k}$ and objective functionals $\widehat{Q}_{k}$ in Fig. \[fig:signal-flow\](a) defined on the decision variables $a_k$ while $R$ is given by $$R = \begin{bmatrix} I & -A^{T} \\ A & I \end{bmatrix}^{2}\begin{bmatrix}\left(I + A^{T}A \right)^{-1} & 0 \\ 0 & \left(I + AA^{T} \right)^{-1} \end{bmatrix} \label{eq:Q}$$ where $A$ represents the aggregate linear equality constraints $A_\ell$ involving only the primal decision variables. Without loss of generality, the system in Fig. \[fig:signal-flow\](c) may be recast into an equivalent system, in the sense that a solution to one yields a solution to both, of the form $$\begin{aligned}
\c^\star = m(\d^\star) & \text{ and } & \d^\star = G\c^\star + \f \label{eq:c-and-d}
\end{aligned}$$ where $\c^\star,\d^\star \in \Rk$ denote a solution, $G:\Rk \to \Rk$ is an orthogonal matrix, and $\f \in \Rk$ is a system bias. Figure \[fig:signal-flow\](d) illustrates the reduced system utilized in [@BLOpt2] where the algebraic loops have been broken by inserting state/memory into the system.
The precompute required to assemble a signal processing system of the presented class is analytic and involves purely linear operations. In particular, aside from the computation of $R$ in , the algebraic reduction of $(R,H,\e)$ to $(G,\f)$ corresponds to identifying the intersection of affine subspaces and thus can be expressed in closed form. The postcompute associated with recovering the solution to the optimization problem given a solution $(\c^\star,\d^\star)$ to is also linear. For example, let $\a_j$ denote a primal decision variable and assume the precompute retains the system variables $\c_j$ and $\d_j$ associated with $\a_j$. Then, the value $\a_j^\star$ at a stationary point $\a^\star$ of the problem is $$\begin{aligned}
\label{eq:readout}
\a^{\star}_{j} = \frac{1}{2}\left(\d^\star_j + \c^\star_j\right) & \text{ or } & \a^{\star}_{j} = \frac{1}{2}\left(\d^\star_j - \c^\star_j\right)
\end{aligned}$$ depending on whether $\a_j$ is an input to or output from $A$, respectively.
In the context of numerically solving by generating state sequences $\cn$ and $\dn$, we refer to an *iterative solver* as a system implementation in which the processing directly yields the next state values and an *incremental solver* as one in which the processing yields values to be added to the current state in order to produce the next. We refer to either solver as being *filtered* when additional processing is used to produce the next state value as an affine combination of the current state value and the state value produced by the unfiltered solver. A sufficient condition under which the state sequences converge to a solution $(\c^\star,\d^\star)$ of that encompasses the numerical examples presented in this paper (provided that we appropriately implement the filtered solvers) is that the nonlinear map $m$ be non-expansive, i.e. $m$ must satisfy $$\begin{aligned}
\label{eq:passive-everywhere}
\forall\,\u,\v\in\Rk, & \left\|m(\v) - m(\u)\right\|_2 \leq \left\|\v - \u\right\|_2. &
\end{aligned}$$ Convergence is in particular in the Euclidean sense for synchronous implementations and in mean square for stochastic/asynchronous implementations; we refer to [@convergence] for a complete treatment. For the purpose of illustration and not by limitation, we assume hereon that $m$ is a coordinatewise nonlinearity, this assumption is true for all numerical examples in this paper. The handling of general nonlinearities follows in an analogous way.
Implementation as a web service {#sec:opt.spc}
===============================
We now overview the operating principle behind O-SPC which we believe suggests opportunity in designing future systems in this way. Referring to the website content, several optimization problems frequently occurring in signal processing contexts have been assembled into an examples library, including those discussed in Section \[sec:examples\].
In the remainder of this section we present the details associated with two distributed and four non-distributed solvers used to obtain a solution $(\c^\star,\d^\star)$ to . We comment upfront, however, that the specific form of the solvers presented in this section differ from the implementations in O-SPC in that they have been adapted here for the purpose of clarity rather than computational efficiency.
{width="\textwidth"}
Distributed implementations {#subsec:distributed}
---------------------------
A longstanding approach to efficiently solving a large class of numerical problems is to recast any problem of the class into a fixed representation to which a set of generic tools may be immediately applied. In this spirit, a signal processing system conforming to is automatically synthesized once the parameters of a problem have been specified, from which several solvers corresponding to various distributions of state and processing instructions may be applied.
Consistent with the general implementation strategy discussed in Section \[sec:intro\], Fig. \[fig:distributed-solvers\] illustrates the organization of the set of equations in into a generic key-value store, e.g. a non-relational database, for implementation on the graph depicted in Fig. \[fig:signal-flow\](e). The protocol for state transfer consists of workers asynchronously accessing the database to retrieve a subset of the computation and the associated signals to be processed, processing these signals, and asynchronously writing the result back into the database. This strategy represents a form of object-oriented signal processing wherein the objects contain data in the form of the signals to be processed and methods in the form of processing instructions.
Screen captures from the O-SPC application interface are provided in Fig. \[fig:distributed-solvers\] for a non-negative least squares problem, depicting the dashboard through which distributed workers can be controlled. Through the dashboard interface, metaparameters for the problem can be set, in turn generating a corresponding uniform resource locator (URL) through which workers can attach to the problem instance to perform computation. For worker devices with integrated cameras, a quick response (QR) code is also dynamically generated. The particular solution depicted in Fig. \[fig:distributed-solvers\] was obtained using $24$ distributed workers. Analytics regarding the computational platforms of the connected workers, as well as the individual contributions to the overall optimization progress, are provided via dynamically-generated graphs.
It is worth noting that nearly any computational resource equipped with network access and a basic JavaScript engine may be utilized as a worker on O-SPC. For example, a heterogeneous set of workers might include modern web browsers on mobile, tablet and desktop machines as well as JavaScript enabled microcontrollers [@tessel; @Espruino].
The worker initialization and processing instructions for two distributed solvers are summarized in column 3 of Fig. \[fig:distributed-solvers\]. Specifically, each worker, independent of any and all other workers, performs the following steps *ad infinitum* to implement an iterative filtered solver:
1. generate a random integer $j\in\{1,\dots,K\}$ corresponding to the state variables $\c_j$ and $\d_j$ to be processed;
2. read the current state of the vector $\c$ as well as the object [varj]{} consisting of a characterization of the nonlinearity $m_j$ labeled [m]{}, the value of $\f_j$ labeled [f]{}, and the row vector $\g^{(j)}$ corresponding to the $j^{\text{th}}$ row of $G$ labeled [Grow]{};
3. generate the intermediary state value $\d_j$ as $$\begin{aligned}
\label{eq:dist-filt-pre}
\d_{j} \leftarrow g^{(j)}_{1}\c_{1} + \cdots + g^{(j)}_{K}\c_{K} + \f_{j}
\end{aligned}$$
4. generate the new state value $\c_j$ as $$\begin{aligned}
\label{eq:dist-filt}
\c_{j} \leftarrow \rho m_{j}\left( \d_{j} \right) + (1-\rho)\c_{j}
\end{aligned}$$ where the filtering parameter $\rho$ is a metaparameter obtained during the worker initialization phase;
5. asynchronously write the new state value $\c_j$ into the $j^{\text{th}}$ position of $\c$ in the database.
For iterative solvers, the state variable $\d$ does not need to be explicitly stored in the database. Indeed, once the partial solution $\c^\star$ is identified, the state vector $\d^\star$ may be generated using and thus the original optimization problem is effectively solved. Referring again to Fig. \[fig:distributed-solvers\], the processing procedure for an iterative solver corresponds to modifying the instructions outlined above by setting $\rho = 1$ in .
We call special attention to the fact that no attempt is made at the algorithm level to regulate global task allocation among the workers nor to enforce concurrency of any form. Specifically, the data requests and updates are respectively executed using asynchronous read and write operations with no concept of precedent or preference among the workers. For example, if multiple workers request data associated with the same state variable $\c_j$ and each experiences a different latency (and thus each possibly retrieves different state vectors $\c$) then the database records the updates in the order they are received irrespective of the order of the read operations.
Referring to Fig. \[fig:signal-flow\](e), the database might simultaneously contain numerous active problem instances. Workers may be added or removed at any time (including changing problem instances) without any form of coordination since workers are never assigned responsibility for any particular part of the workload. In this sense, O-SPC facilitates the time-varying allocation of compute resources in order to adaptively respond to real-time constraints, time-varying network congestion, and resource outages. Another advantage to utilizing the presented approach for solving optimization problems in practice is the ability to update the portion of the database (and by extension the signal processing structure as well) associated with measurements and/or observations as new data becomes available. The response of the system is then to transform the state of the database associated with the current solution toward the new fixed-point or invariant state corresponding to the new solution. Consequently, the distributed solvers summarized in Fig. \[fig:distributed-solvers\] are sufficient for solving a broad class of optimization problems over delay or disruption tolerant networks and further do not rely critically upon the availability or synchronization of any particular compute resources.
Non-distributed implementations {#subsec:local}
-------------------------------
The toolset in O-SPC also provides support for four local or non-distributed solver types which organize and implement the associated signal processing system using a single JavaScript enabled web browser as the compute engine. We define an asynchronous implementation protocol in this setting as one for which the behavior of the system state is that of coordinate-wise discrete-time sample-and-hold elements triggered by discrete-time Bernoulli processes.
More formally, let $\{\mathcal{I}_{n}\}_{n=1}^{\infty}$ denote a sequence of randomly generated subsets of $\{1,\dots,K\}$ such that for every value of $n$ each $i\in\{1,\dots,K\}$ is included in $\mathcal{I}_{n}$ with probability $p$ and not included with probability $1-p$ independently and independent of $n$. Further, denote $\mathcal{I}^{c}_{n}$ as the set compliment of $\mathcal{I}_{n}$, i.e. $\mathcal{I}^c_n = \{i\in\{1,\dots,K\} \colon i \not \in \mathcal{I}_{n}\}$, and let $I_{\mathcal{I}_n}$ denote the diagonal matrix with ones on the diagonal entries indicated by the index set $\mathcal{I}_n$ and zeros elsewhere. Then, the update procedure for the state sequence $\dn$ given by $$\begin{aligned}
\d^n = I_{\mathcal{I}_{n}}\left(Gm\left(\d^{n-1}\right)+\f\right) + I_{\mathcal{I}_n^c}\d^{n-1}, & & n \in \N, \label{eq:local-iter}
\end{aligned}$$ corresponds to the iterative solver for the signal processing system depicted in Fig. \[fig:signal-flow\](d). Reorganizing the computation and modifying the initial conditions such that the first difference of the signals rather than the signals themselves are being processed results in the incremental solver processing procedure where $\c^{n} = m(\d^{n-1})$ and $$\begin{aligned}
\d^n = \d^{n-1} + GI_{\mathcal{I}_n}\left(m\left(\d^{n-1}\right) - \c^{n-1} \right), & & n \in \N.\label{eq:local-incr}
\end{aligned}$$
The system initialization and processing procedure for the local solvers discussed hereto and their filtered counterparts are summarized in Fig. \[fig:local-solvers\]. In addition, screen captures from O-SPC illustrate the solution obtained to a sparse signal recovery problem wherein the signal processing system was implemented using the incremental filtered solver with $\rho=0.5$ and $p=0.25$.
{width="7.1in"}
Numerical examples {#sec:examples}
==================
Mathematical optimization typically manifests itself in signal processing applications as either a design tool for optimal parameter selection or a processing stage in the signal chain itself. In this section we provide context and commentary for examples of these types depicted in Figs. \[fig:distributed-solvers\] and \[fig:local-solvers\]. In addition, we present a third and final example related to error correction in transform coding theory solved using O-SPC. The obtained solutions agree with those generated by CVX [@cvx]. We conclude with a discussion of the relationships between the specific signal processing systems associated with the examples.
Sparse signal recovery
----------------------
A well-established approach to recovering a sparse signal measured through an underdetermined linear system that has potentially been corrupted by noise is to solve the LASSO or basis pursuit denoising problem. In particular, this recovery formulation is posed as a regularized least squares problem of the form $$\begin{aligned}
\label{eq:ssr}
\displaystyle \minimize_{\x} & \frac{\rho}{2}\|A\x - \y\|_2^2 + \|\x\|_1
\end{aligned}$$ where $A\in\R^{m\times n}$ is the linear measurement system, $\y\in\R^m$ is a vector of measurements, $\rho > 0$ scales the objective function, and $\x\in\R^{n}$ is the desired sparse vector. We draw $A$ at random from a Gaussian ensemble to ensure it satisfies the restricted isometry property with high probability [@cs]. The solution depicted in Fig. \[fig:local-solvers\], solved using O-SPC, corresponds to $(m,n)=(60,128)$. Problems sizes of the order $(m,n) = (2400,5120)$ were additionally solved, i.e. where $A$ has $\approx12$ million non-zero entries.
Non-negative least squares
--------------------------
The non-negative least squares problem, which is commonly used as a subroutine in solving more general non-negative tensor factorization problems, is formulated as the quadratic program $$\begin{aligned}
\label{eq:nnls}
\displaystyle \minimize_{\x } & \frac{1}{2}\|A\x - \y\|_2^2 & \text{s.t. } \x \geq \underline{0}
\end{aligned}$$ where $A\in\R^{m\times n}$ is a general linear system, $\y\in\R^m$ is a vector of observations, and $\x\in\R^{n}$ is the desired non-negative vector. For the example depicted in Fig. \[fig:distributed-solvers\], $m$ and $n$ were respectively selected to be $128$ and $60$. Constrained least squares problems such as have immediate application to system design in a number of ways. For example, in the context of filter design, facilitates the design of filters including peak-constrained least squares filters [@lsfilter] with additional non-negativity constraints on the filter taps enabling their use on implementation technologies with unsigned number systems.
Error correction decoding
-------------------------
Let $A\in\R^{m\times n}$ denote a linear codebook, i.e. with each column of $A$ denoting a codeword, and consider the recovery of a plaintext vector $\x \in \R^{n}$ from a cyphertext vector $\y \in \R^{m}$ which has been additively corrupted by a $p$-sparse noise vector $\z \in \R^{m}$ according to $\y = A\x + \z$. We cast the recovery procedure as the problem $$\begin{aligned}
\label{eq:ecd}
\displaystyle \minimize_{\x} & \left\|A\x - \y \right\|_{1},
\end{aligned}$$ hence decoding a given cyphertext vector in this way corresponds to solving a linear program since may be recast as the standard basis pursuit problem. Furthermore, is guaranteed to identify the correct plaintext vector $\x^\star$ so long as $A$ and the triple $(n,m,p)$ satisfy the conditions provided in [@tao]. Figure \[fig:decoder-example\] depicts the solution obtained from the numerical experiment outlined in [@tao] using the distributed iterative filtered solver presented in this paper where we specifically select the transmitted plaintext vector to be binary valued and round the decoded plaintext vector for further noise suppression. The solver was implemented using $50$ distributed workers. Note that the plaintext vector obtained using is indeed the synthetic plaintext vector before transmission.
![An illustration of the computed solution of obtained using $50$ distributed workers implementing an iterative filtered solver with parameter $\rho = 0.75$. A breakdown of the workers computational platform distribution and individual contributions to the overall iteration count is also depicted.[]{data-label="fig:decoder-example"}](decoder-example.eps){width="4.5in"}
Comments on the example signal processing systems
-------------------------------------------------
The optimization problems - were specifically chosen to underscore the flexibility and generality of the framework in [@BLOpt] with respect to the implementation paradigm discussed in this paper. In particular, for the same linear system $A$ and observation vector $\y$, the signal processing system associated with these three problems differ only in the analytic form of the nonlinearity $m(\cdot)$ used in defining the transformed stationarity conditions . The coordinatewise nonlinearity $m_{\eqref{eq:ssr}}: \R \to \R$ associated with the sparse signal recovery problem is given by $$\begin{aligned}
m_{\eqref{eq:ssr}}(x) = \left\{\begin{array}{lr}-x,& |x| \leq 1 \\ x - 2\text{ sign}(x), & |x| > 1\end{array}\right.,
\end{aligned}$$ whereas the coordinatewise nonlinearities $m_{\eqref{eq:nnls}}: \R \to \R$ and $m_{\eqref{eq:ecd}}:\R \to \R$ respectively associated with the non-negative least squares problem and the error correction decoding problem are given by $m_{\eqref{eq:nnls}}(x) = |x|$ and $m_{\eqref{eq:ecd}}(x) = m_{\eqref{eq:ssr}}(-x)$. Each of these nonlinearities (as scalar operators or stacked into an operator from $\Rk$ into itself) satisfy the sufficient condition for convergence in and thus, for example, the filtered solvers discussed in Subsections \[subsec:distributed\] and \[subsec:local\] may be directly utilized to solve the corresponding problems. We conclude with a remark on the similarity of the complexity associated with solving and in the sense of identifying fixed-points of the algebraic system due in part to the relationship between $m_{\eqref{eq:ssr}}$ and $m_{\eqref{eq:ecd}}$. This similarity may not be readily apparent from the optimization problem statements since is a linear program while is convex quadratic, but can be leveraged to efficiently solve both problem instances without replicating the entire problem in the database.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The preliminary results of a three-site CCD photometric campaign are reported. The $\delta$ Scuti variable V650 Tauri belonging to the Pleiades cluster was observed photometrically for 14 days on three continents during 2008 November. An overall run of 164 hr of data was collected. At least five significant frequencies for V650 Tauri have been detected.'
author:
- 'L. Fox Machado'
- 'R. Michel'
- 'M. Alvarez'
- 'C. Zurita'
- 'J.N. Fu'
title: CCD Photometry of the Pleiades Delta Scuti Star V650 Tauri
---
[ address=[Observatorio Astronómico Nacional, Instituto de Astronomía, Universidad Nacional Autónoma de México, A.P. 877, Ensenada, BC 22860, Mexico]{} ]{}
[ address=[Observatorio Astronómico Nacional, Instituto de Astronomía, Universidad Nacional Autónoma de México, A.P. 877, Ensenada, BC 22860, Mexico]{} ]{}
[ address=[Observatorio Astronómico Nacional, Instituto de Astronomía, Universidad Nacional Autónoma de México, A.P. 877, Ensenada, BC 22860, Mexico]{} ]{}
[ address=[Instituto de Astrofísica de Canarias, C/Vía Láctea s/n, E-38205, La Laguna, Tenerife, Spain]{} ]{}
[ address=[Department of Astronomy, Beijing Normal University, Beijing 100875, China]{} ]{}
Introduction
============
$\delta$ Scuti variables are stars with masses between 1.5 and 2.5 $M_{\odot}$ located at the intersection of the classical Cepheid instability strip with the main sequence. These variables are thought to be excellent laboratories for probing the internal structure of intermediate mass stars. Intents of modelling $\delta$ Scuti stars belonging to open clusters have been performed recently (e.g. \[1\], \[2\], \[3\]). Although the constraints imposed by the cluster parameters have proved to be very useful when modelling an ensemble of $\delta$ Scuti stars, more detected frequencies in individual stars would improve current seismic studies.
The target star V650 Tau (HD 23643, $V=7^{\rm m}.79$, A7) was identified as a short-period pulsating variable by Breger (1972). Intensive observations performed by the STEPHI network in November 1990, revealed four frequency peaks in V650 Tau \[4\]. One-site CCD photometric observations carried out by \[5\] in November-December 1993, confirmed the results obtained by the STEPHI campaign. Since then, no new observations of V650 Tauri have been performed.
The present paper provides preliminary observational results of a three-site campaign on V650 Tauri in 2008.
Observations and data reduction
===============================
Three observatories were involved in the observational campaign. They are listed in Table \[tab:tel\] together with the telescopes and instruments used. Table \[tab:log\] gives the log of observations. A total amount of 164 hours of useful data were obtained from the three sites.
The observations were obtained through a Johnson $V$ filter except at the SPM observatory where a Strömgren $y$ filter was used. Table \[tab:stars\] shows the main observational parameters corresponding to the target and comparison stars as taken from the SIMBAD database operated by CDS (Centre de Données astronomique de Strasbourg). Two comparison stars have been used during the observations depending on the constraints set by the field of view of the CCD’s and sizes of the telescopes. The first one, HD 23605 ($V=6^{\rm m}.99$, F5), is a suitable comparison star considering its brightness and spectral type. However, this star could not be observed neither at Teide nor at San Pedro Martir observatory because it is so bright that at these telescopes the CCD detectors saturated in a few seconds of exposure time. Rather at these sites we observed the comparison star, HD 23653 ($V=7^{\rm m}.71$, K0), since its magnitude is similar to that of the target star. Figure \[fig:field\] shows a typical image of the CCD’s field of view ($20' \times 20'$) at the 0.50m telescope of the Xing Long observatory.
Sky flats, dark and bias exposures were taken every night at all sites. All data were calibrated and reduced using standard IRAF routines. Aperture photometry was implemented to extract the instrumental magnitudes of the stars. The differential magnitudes were normalized by subtracting the mean of differential magnitudes for each night. In Figure \[fig:curves\] the entire light curves V650 Tau - Comp 2 are presented.
Observatory Telescope Instrument Observers
--------------------------------------------- ----------- --------------- -------------
Observatorio del Teide (OT, Spain) 0.80m 2048x2048 CCD CZ
Observatorio San Pedro Mártir (SPM, Mexico) 0.84m 1024x1024 CCD LFM, RM, MA
Xing Long Station (XL, China) 0.50m 1024x1024 CCD JNF
: List of instruments and telescopes involved in the campaign. Observer’s abbreviations correspond to initial of the co-authors.[]{data-label="tab:tel"}
-------- ----------- ---------- ------- ------- -------
Day Date 2008 HJD SPM XL OT
2454774+
01 Nov 11 07 0.70 5.70
02 Nov 12 08 3.58 8.80
03 Nov 13 09 4.76 -
04 Nov 14 10 9.68 1.08
05 Nov 15 11 9.42 -
06 Nov 16 12 9.79 - 8.17
07 Nov 17 13 7.64 - -
08 Nov 18 14 10.15 10.28 -
09 Nov 19 15 10.26 10.19 -
10 Nov 20 16 10.33 10.59 -
11 Nov 21 17 7.22 10.36 -
12 Nov 22 18 - 10.66 -
13 Nov 23 19 - - -
14 Nov 24 20 - 4.98 -
Total observing time SPM XL OT
Nov 11 Nov 24 164.34 83.52 57.06 23.75
-------- ----------- ---------- ------- ------- -------
: Log of observations. Observing time is expressed in hours.
\[tab:log\]
![Image of the CCD field-of-view ($20' \times 20'$) of the Xing Long Observatory. The positions of the target and comparison stars are indicated in the figure. North is up and East is left.[]{data-label="fig:field"}](fox2_fig1.eps){width="7cm"}
-------------- ------- ---- ------ --------- --------- -------------------------- ---------
Star HD ST $V$ $B-V$ $U-B$ $v \sin i$ $\beta$
$(\mathrm{km\, s}^{-1})$
V650 Tau 23643 A7 7.79 $+$0.25 $+$0.14 219 2.823
Comparison 1 23605 F5 6.99 $+$0.50 $+$0.09 - 2.653
Comparison 2 23653 K0 7.71 $+$1.27 $+$1.12 - -
-------------- ------- ---- ------ --------- --------- -------------------------- ---------
: Observational properties of the stars observed in the campaign.
\[tab:stars\]
Spectral analysis
=================
The period analysis has been performed by means of standard Fourier analysis and least-squares fitting. In particular, the amplitude spectra of the differential time series were obtained by means of Period04 package \[6\], which considers Fourier as well as multiple least-squares algorithms. This computer package allows to fit all the frequencies simultaneously in the magnitude domain.
The amplitude spectrum of the differential light curve V650 Tauri - Comparison 2 (see Fig \[fig:curves\]) is shown in Figure \[fig:spec\]. As can be seen, V650 Tauri presents high-amplitude peaks distributed between 17 c/d and 35 c/d.
The frequencies have been extracted by means of standard prewhitening method. In order to decide which of the detected peaks in the amplitude spectrum can be regarded as intrinsic to the star we follow Breger’s criterion given by \[7\], where it was shown that the signal-to-noise ratio (in amplitude) should be at least 4 in order to ensure that the extracted frequency is significant.
The frequencies, amplitudes and phases are listed in Table \[tab:frec\]. Five significant frequencies have been detected in V650 Tauri. Among these only the first two frequencies (35.66 c/d and 17.04 c/d) are similar to that found by Kim & Lee (1996). A detailed analysis of these observations will be given in a forthcoming paper \[8\].
![Differential light curve V650 Tau $-$ Comparison 2.[]{data-label="fig:curves"}](fox2_fig2.ps){width="7.3cm"}
--------- -------- -------------------- -------
Freq. A $\varphi$/($2\pi$) $S/N$
(c/d) (mmag)
32.6565 4.43 0.39 13.4
17.0356 2.65 0.09 10.0
35.5970 2.49 0.25 8.6
31.6285 1.91 0.85 6.4
34.6409 1.41 0.04 4.5
--------- -------- -------------------- -------
: Frequency peaks detected in the light curve V650 Tauri - Comparison 2. S/N is the signal-to-noise ratio in amplitude after the prewhitening process.[]{data-label="tab:frec"}
![Amplitude spectrum derived from the light curve V650 Tau - Comparison 2. The amplitude is in mag and the frequencies in c/d.[]{data-label="fig:spec"}](fox2_fig3.eps){width="7.7cm"}
This work has received financial support from the UNAM under grants PAPIIT IN108106 and IN114309. Special thanks are given to the technical staff and night assistants of the San Pedro Mártir, Teide and Xing Long Observatories. This research has made use of the SIMBAD database operated at CDS, Strasbourg (France).
[9]{}
L. Fox Machado, et al., 2006, A&A, 446, 611. M.M. Hernández, et al., 1998, A&A, 338, 511. E. Michel, et al., 1999, A&A, 342, 253. J.A. Belmonte, et al., in: Weiss W.W., Baglin, A. (eds.) IAU Coll. 137, Inside the stars, p. 347 S.-L. Kim, S.-W. Lee, A&A, 310, 831 P. Lenz, M. Breger, 2005, CoAst, 146, 53 M. Breger, et al., 1993, A&A, 271, 482. L. Fox Machado, 2009, et al., AJ, submitted.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'F. Arrigoni Battaia'
- 'Chian-Chou Chen'
- 'M. Fumagalli'
- Zheng Cai
- 'G. Calistro Rivera'
- Jiachuan Xu
- 'I. Smail'
- 'J. X. Prochaska'
- Yujin Yang
- 'C. De Breuck'
bibliography:
- 'allrefs.bib'
date: 'Received —; accepted —'
subtitle: 'The environment and the powering of an Enormous Lyman-Alpha Nebula'
title: 'Overdensity of submillimeter galaxies around the $z\simeq 2.3$ MAMMOTH-1 nebula'
---
Introduction
============
In the present-day Universe, giant elliptical galaxies are found at the centers of massive clusters. Being characterized by old, and coeval stellar populations, these central galaxies must have formed the bulk of their stars in exceptional star-forming events at early epochs, or must have accreted several coeval galaxies (e.g., @Kauffmann96). Indeed, the current hierarchical structure formation model predicts that these central galaxies merge with several nearby satellite galaxies to build up their stellar mass (e.g., @West1994). This violent merging process is thought to take place in the highest density peaks in the early Universe, in the so-called protoclusters. Despite that a lot of effort has been put in characterizing overdensities of galaxies at high-redshift, there is still an open debate on which is the best technique to map protoclusters and on which systems represent the nurseries of present-day massive clusters, and thus the site of formation of elliptical galaxies (e.g., @steidel00 [@Venemans2007; @Dannerbauer2014; @Orsi2016; @Cai2017a; @Miller2018; @Oteo2018]).
To date, high-redshift radio galaxies (HzRGs) are one of the best candidates for pinpointing the location of these extremely dense environments (@mileyd08). This result is supported by the rarity of these systems, by Ly$\alpha$ emitter (LAEs) overdensities near them, and in some cases by overdensities in submillimeter observations (@Stevens2003 [@Humphrey2011; @Rigby2014; @Zeballos2018]). Being the host of an active galactic nucleus (AGN) and characterized by intense radio emission, HzRGs are also known for their associated giant Ly$\alpha$ nebulae on hundreds of kpc scales, suggesting the presence of a large amount of gas in these systems (e.g., @rvr+03). This Ly$\alpha$ emission is a complex result of AGN ionization, jet-ambient gas interaction, and intense star formation (@VillarMartin2003 [@Vernet2017]). Despite these pieces of evidence for protoclusters around HzRGs, we have recently discovered enormous Ly$\alpha$ nebulae (ELANe; @Cai2016), more extended than those around HzRGs, and in even more extreme environments at $z\sim2-3$ (@hennawi+15 [@Cai2016; @FAB2018]).
The ELANe – with observed Ly$\alpha$ surface brightness SB$_{\rm Ly\alpha}\gtrsim10^{-17}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$ on $\gtrsim 100$ kpc, maximum extents of $>250$ kpc and Ly$\alpha$ luminosities of $L_{\rm Ly\alpha}>10^{44}$ erg s$^{-1}$ – represent the extrema of known radio-quiet Ly$\alpha$ nebulosities. Indeed, previously well studied radio-quiet Ly$\alpha$ nebulae at $z\sim2-6$, [*a.k.a*]{} Ly$\alpha$ blobs (LABs; e.g., @steidel00 [@matsuda04; @Yang2010; @Matsuda2011; @Prescott2015; @Geach2016; @Umehata2017]), are characterized by smaller luminosities $L_{\rm Ly\alpha}\sim10^{43-44}$ erg s$^{-1}$, and smaller extents (50-120 kpc) down to similar surface brightness levels (SB$_{\rm Ly\alpha}\sim10^{-18}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$). While most of the LABs have a powering mechanism that is still debated (e.g., @Mori2004 [@Dijkstra2009_hd; @Rosdahl12; @Overzier2013; @fab+15a; @Prescott2015b; @Geach2016]), ELANe are usually explained by photoionization and/or feedback activity of the associated quasars and companions (@cantalupo14 [@hennawi+15; @Cai2016; @FAB2018]).
The current sample of ELANe still comprises only a handful of objects (@hennawi+15 [@Cai2016; @FAB2018; @Cai2018]). All these ELANe are associated with local overdensities of AGN, with up to 4 known quasars sitting at the same redshift of the extended Ly$\alpha$ emission for the ELAN Jackpot (@hennawi+15). Given the current clustering estimates for AGN, the probability of finding a multiple AGN system is very low, $\approx 10^{-7}$ for a quadrupole AGN system (@hennawi+15). This occurrence makes a compelling case that these nebulosities are sitting in very dense environments. This working hypothesis is further strengthen by the detection of a large number of associated LAEs on small (@FAB2018) and on large scales (@hennawi+15 [@Cai2016]). Such overdensities of LAEs are comparable or even higher than in the case of HzRGs and LABs (@hennawi+15 [@Cai2016; @FAB2018]).
Most of the known ELANe (@cantalupo14 [@hennawi+15; @fab+15b; @FAB2018]) show (i) at least one bright type-1 quasar embedded in the extended emission, (ii) non-detections in $\lambda1640$Å and $\lambda1549$Å down to sensitive SB limits ($\sim10^{-18}-10^{-19}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$), and (iii) relatively quiescent kinematics for the Ly$\alpha$ emission (FWHM$\simeq600$ km s$^{-1}$) with a single peaked Ly$\alpha$ line down to the current resolution of the instrument used.
Notwithstanding these results, the ELANe and their environment have been currently studied only in unobscured tracers, possibly resulting in a biased vision of the phenomenon. A complete view of these systems requires a multiwavelength dataset. In particular, submillimeter galaxies (SMGs; @Smail1997) have been shown to be linked to merger events (e.g., @Engel2010 [@Ivison2012; @Alaghband-Zadeh2012; @Fu2013; @TC2015; @Oteo2016]) and to be good tracers of protoclusters (e.g., @Smail2014 [@Casey2016; @Hung2016; @Wang2016; @Oteo2018; @Miller2018]). For these reasons, and to directly test whether our newly discovered ELANe could be powered by intense obscured star-formation, we have initiated a submillimeter campaign with the James Clerk Maxwell Telescope (JCMT) and the Atacama Pathfinder EXperiment (APEX) telescopes to map the obscured star-forming activity (if any) associated with these rare systems and their environment.
Here we report the results of our observations of the ELAN MAMMOTH-1 at $z=2.319$ (@Cai2016) using the Submillimetre Common-User Bolometer Array 2 (SCUBA-2; @Holland2013) on JCMT. This ELAN has been discovered close to the density peak of the large-scale structure BOSS1441 (@Cai2017a). BOSS1441 has been identified thanks to a group of strong IGM Ly$\alpha$ absorption systems (@Cai2017a). Follow-up narrow-band imaging, together with spectroscopic observations have constrained the Lyman-$\alpha$ Emitters (LAEs, i.e. sources with rest-frame equivalent width EW$_0^{Ly\alpha}>20$Å) in this field (@Cai2017a). With a LAE density of $\approx12\times$ that in random fields in a (15 cMpc)$^3$ volume, BOSS1441 is one of the most overdense fields discovered to date.
The ELAN MAMMOTH-1 is unique compared to the other few ELAN so far discovered, showing (i) only a relatively faint source ($i=24.2$) embedded in it, (ii) extended emission ($\gtrsim30$ kpc) in $\lambda1640$Å and $\lambda1549$Å and (iii) double-peaked line profiles with velocity offsets of $\approx700$ km s$^{-1}$ for Ly$\alpha$, , and . In light of these evidences, @Cai2016 explained this ELAN as circumgalactic/intergalactic gas powered by photoionization or shocks due to a galactic outflow, most likely powered by an enshrouded AGN. With the SCUBA-2 data we can start to better constrain the nature of this powering source.
This work is structured as follows. In Section \[sec:obs\], we describe our observations and data reduction. In Section \[catalogs\] we present the catalogs at 450 and 850 $\mu$m, along with reliability and completeness tests. In Section \[NC\], we describe how we determined the pure source number counts, estimated the underlying counts model through Monte Carlo simulations, and how we used this models to get the true counts. The same Monte Carlo simulations allowed us to assess the flux boosting (Section \[fluxBoost\]) and the positional uncertainties (Section \[pos\_err\]) inherent to our observations. In Section \[sec:results\] we show (i) the true number counts and compare them to number counts in blank fields, and (ii) the location of the discovered submillimeter sources in comparison to known LAEs. We then discuss our overall detections and the counterpart of the ELAN MAMMOTH-1 in Section \[sec:disc\], and we summarize our results in Section \[sec:summ\].
Throughout this paper, we adopt the cosmological parameters $H_0=70$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_M =0.3$ and $\Omega_{\Lambda}=0.7$. In this cosmology, 1 corresponds to about 8.2 physical kpc at $z=2.319$. All distances reported in this work are proper.
Observations and Data Reduction {#sec:obs}
===============================
The SCUBA-2 observations for the MAMMOTH-1 field were conducted at JCMT during flexible observing in 2018 January 16, 17, and 18 (program ID: M17BP024) under good weather conditions (band 1 and 2, $\tau_{225{\rm GHz}}\leq 0.07$). The observations were performed with a Daisy pattern covering $\simeq13.7\arcmin$ in diameter, and were centered at the location of the ELAN MAMMOTH-1 as indicated in @Cai2016. Note however that the exact coordinate of the ELAN MAMMOTH-1 have been refined to be R.A. $=$ 14:41:24.456, and Dec. $=$ +40:03:09.45. To facilitate the scheduling we divided the observations in 6 scans/cycles of about 30 minutes, for a total of 3 hours.\
The data reduction follows closely the procedures detailed in @TC2013a. In short, the data were reduced using the Dynamic Iterative Map Maker (DIMM) included in the SMURF package from the STARLINK software (@Jenness2011 [@Chapin2013]). The standard configuration file dimmconfig\_blank\_field.lis was adopted for our science purposes. Data were reduced for each scan and the MOSAIC\_JCMT\_IMAGES recipe in PICARD, the Pipeline for Combining and Analyzing Reduced Data (@Jenness2008), was used to coadd the reduced scans into the final maps.
The final maps were applied standard matched filter to increase the point source detectability, using the PICARD recipe SCUBA2\_MATCHED\_FILTER. Standard flux conversion factors (FCFs; 491 Jy pW$^{-1}$ for 450 $\mu$m and 537 Jy pW$^{-1}$ for 850 $\mu$m) with 10% upward corrections were adopted for flux calibration. The relative calibration accuracy is shown to be stable and good to 10% at 450 $\mu$m and 5% at 850 $\mu$m (@Dempsey2013).
The final central noise level for our data is 0.88 mJy/beam and 5.4 mJy/beam, respectively at 850 $\mu$m and 450 $\mu$m. In the reminder of this work we focused on the regions of the data characterized by a noise level less than three times the central noise. We refer to this area as effective area. In Figure \[BOSS1441\] we overlay the field-of-view (corresponding to the effective area) of our [SCUBA-2]{} observations (dashed red) on the overdensity of LAEs known from the work of @Cai2017a (green contours). In Table \[obs\] we summarize the center, the effective area, and the central noise ($\sigma_{\rm CN}$) of our observations.
![The galaxy overdensity BOSS1441 at $z=2.32\pm0.02$ (@Cai2017a). The density contours (green) for LAEs are shown in steps of 0.1 galaxies per arcmin$^2$, with the inner density peak of 1.0 per arcmin$^2$. The density contours are shown with increasing thickness for increasing galaxy number density. The brown crosses indicate the positions of known QSOs in the redshift range $2.30\leq z < 2.34$, and thus likely within the overdensity. We also highlight the position of the ELAN MAMMOTH-1 (dotted crosshair), and the effective area of our SCUBA-2 observations (red dashed contour).[]{data-label="BOSS1441"}](Fig_1.pdf){width="1.0\columnwidth"}
Source Extraction and Catalogues {#catalogs}
================================
To extract the detections from both maps, we proceed following @TC2013a. We first extracted sources with a peak S/N $>2$ within the effective area of our observations (see Table \[obs\]). Specifically, our algorithm for source extraction finds the maximum pixel within the selected region, takes the position and the information of the peak, and subtracts a scaled PSF centered at such position[^1]. The process has been iterated until the peak S/N went below 2.0. These sources constituted the preliminary catalogs at 850 and 450 $\mu$m. We then cross-checked the two catalogs to find counterparts in the other band. We considered a source as a counterpart if its position at 450 $\mu$m lays within the 850 $\mu$m beam.
The final catalogs were built by keeping every $> 4\sigma$ source in the preliminary catalogs, but also every $>3\sigma$ source characterized by a $>3\sigma$ counterpart in the other band. Overall, we discovered 27 sources at 850 $\mu$m and 14 sources at 450 $\mu$m. In Tables \[850\] and \[450\] we list the information for these sources. Figure \[Maps\] shows the final S/N maps at 850 and 450 $\mu$m for the targeted field with the discovered sources over-plotted.
{width="18.5cm"}
Reliability of Source Extraction {#jackknife}
--------------------------------
To determine the number of spurious sources that could affect our catalogues, we proceeded as follows. First, we applied the source extraction algorithm to the inverted maps. We found 2 and 1 detections at $>4\sigma$ at 850 and 450 $\mu$m, respectively. Second, we constructed true noise maps, we applied the source extraction algorithm, and checked the number of detections with $>4\sigma$. To obtain true noise maps we used the jackknife resampling technique. Specifically, we subtracted two maps obtained by co-adding roughly half of the data for each band. In this way, any real source in the maps is subtracted irrespective of its significance. The residual maps are thus source-free noise maps. To account for the difference in exposure time, we scaled these true noise maps by a factor of $\sqrt{t1\times t2}/(t1+t2)$, with t1 and t2 being the exposure time of each pixel from the two maps. These jackknife maps are characterized by a central noise of 0.88 and 5.39 mJy/beam, respectively at 850 $\mu$m and 450 $\mu$m, in agreement with the noise in the science data. By applying the source extraction algorithm to these maps, we found 1 and 4 detections at $>4\sigma$ at $850$ and $450$ $\mu$m, respectively. We thus expect a similar number of spurious sources in our $>4\sigma$ source catalogs.
Further, we tested the number of spurious detections for the $3\sigma$ sources identified as having a counterpart in the other bandpass (lower portion of Table \[850\] and \[450\]) by using once again the jackknife maps. Specifically, from this maps we extracted sources between $3$ and $4\sigma$, and cross-correlated them with the detections in the real data at the other wavelength. We found that none of such spurious sources matched a detection in the real maps. We then repeated the test $1000$ times by randomizing the position of the spurious sources within the effective area of our observations. On average we found $0.3$ and $0.7$ spurious sources at $850$ and $450$ $\mu$m, indicating that sources selected at the $>3\sigma$ level in both bandpasses are even more reliable than $>4\sigma$ sources selected in only one bandpass.
Overall these tests suggest that – most likely – the sources at $450$ $\mu$m without a detection at $850$ $\mu$m are spurious for our observations. For the sake of completeness, we decided to list all the sources in our catalogs. As it will be clear from our analysis, our conclusions are not affected.
Completeness Tests
------------------
We tested at which flux our data can be considered complete. We proceeded as follows. We took the true noise maps introduced in section \[jackknife\], and populated them with mock sources of a given flux and placed at random positions. We have then extracted the sources, considering them as recovered if the detection is above 4$\sigma$ and within the beam area. Specifically, we injected sources with flux from 0.1 to 25.1 mJy (0.1 to 80.1 mJy) with a step of 0.5 mJy (1.0 mJy) for 850 (450) $\mu$m. For each step in flux we iterate the extraction by introducing 1000 sources. To fully characterize the completeness in the whole extent of the maps, we repeated this procedure for areas of the images characterized by different depths, i.e. $<3\sigma_{\rm CN}$, $2\sigma_{\rm CN}<\sigma<3\sigma_{\rm CN}$, $1.5\sigma_{\rm CN}<\sigma<2\sigma_{\rm CN}$, and $\sigma<1.5\sigma_{\rm CN}$ (see Fig. \[Maps\]). Figure \[completeness\] shows the results of the tests. For the whole area with $<3\sigma_{\rm CN}$, the $50\%$ completeness is at 5.8 and 37 mJy at 850 and 450 $\mu$m, respectively, and the $80\%$ is at 6.8 and 44 mJy, respectively. As expected the central portion of the maps with $\sigma<1.5\sigma_{\rm CN}$ have a better sensitivity, with the $50\%$ completeness being around 4.8 and 30 mJy at 850 and 450 $\mu$m, respectively, and the $80\%$ being around 5.3 and 32 mJy, respectively.
![Top: completeness at $850$ $\mu$m versus flux for different portions of the map, i.e. $2\sigma_{\rm CN}<\sigma<3\sigma_{\rm CN}$, $1.5\sigma_{\rm CN}<\sigma<2\sigma_{\rm CN}$, $\sigma<1.5\sigma_{\rm CN}$, and $<3\sigma_{\rm CN}$ (see Fig. \[Maps\]). Bottom: same as above, but fot the $450$ $\mu$m dataset.[]{data-label="completeness"}](Fig_3a.pdf "fig:"){width="0.98\columnwidth"}\
![Top: completeness at $850$ $\mu$m versus flux for different portions of the map, i.e. $2\sigma_{\rm CN}<\sigma<3\sigma_{\rm CN}$, $1.5\sigma_{\rm CN}<\sigma<2\sigma_{\rm CN}$, $\sigma<1.5\sigma_{\rm CN}$, and $<3\sigma_{\rm CN}$ (see Fig. \[Maps\]). Bottom: same as above, but fot the $450$ $\mu$m dataset.[]{data-label="completeness"}](Fig_3b.pdf "fig:"){width="0.98\columnwidth"}
Number Counts {#NC}
=============
In this section we determine the pure source number counts at 850 and 450 $\mu$m around the ELAN MAMMOTH-1, and estimate the underlying counts models for each wavelength. A precise measurement of the galaxy number counts needs an accurate estimate of the number of spurious sources contaminating the counts. For this purpose, we followed the procedure in @TC2013a [@TC2013b], and use the jackknife maps produced in Section \[jackknife\] to assess how many spourious sources affect the counts. As a first step, in Figure \[histos\] we show the S/N histograms of the true noise maps (orange shading) and the signal maps (blue shading with black edge). The excess signals with respect to the pure noise distribution are from real astronomical sources. On the other hand, the negative excesses are due to the negative throughs of the matched-filter PSF. From these histograms it is well evident that the 450 $\mu$m data are less sensitive and more affected by the presence of spurious sources (as already noted in Section \[jackknife\]).
![Normalized histograms of the S/N values for the pixels within the portions of the 850 and 450 $\mu$m maps characterized by less than three times the central noise. The orange and blue histograms indicate the distributions of the jackknife maps, and of the data, respectively. The jackknife maps represents the pixel noise distributions which dominates at low S/N (see Section \[jackknife\] for details). The data (especially at 850 $\mu$m) shows excesses at high S/N where sources contribute to the distributions. The matched filter technique introduces residual troughs around bright detections, which are visible here as negative excesses.[]{data-label="histos"}](Fig_4.pdf){width="0.95\columnwidth"}
In contrast to what done with the catalogs in Section \[catalogs\] – where we have selected only detections with S/N$>4$ or with S/N$>3$ at both wavelengths – we lowered our detection threshold to $2\sigma$. Indeed, as the positional information is not relevant for number counts analyses, the detection threshold can be lowered to explore statistically significant positive excesses (e.g., @TC2013b). We thus use the preliminary catalogues produced in Section \[catalogs\], and additionally ran the source extraction algorithm on the true noise maps down to S/N$=2$.
The pure source differential number counts are then obtained as follows. First, for each extracted source in the signal map we calculated the number density by inverting the detectable area, which is the portion of the field-of-view with noise level low enough to allow the detection of the source. Second, we obtained the raw number counts by summing up the number densities of the sources within each flux bin. Finally, to get to the pure source differential number counts, we subtracted the number counts similarly obtained from the true noise maps, if any, from the counts obtained from the signal maps. Figure \[counts\_MCMC\] shows the obtained pure source differential number counts (black data points) for both 850 (top panel) and 450 $\mu$m (lower panel).
![Pure source differential number counts (black data-point) at 850 and 450 $\mu$m around the ELAN MAMMOTH-1 compared to the simulated mean counts (red dashed line). The yellow shadings represent the 90% confidence range obtained from 500 realizations of the blue dot-dashed curves. The blue dot-dashed curves are the final adopted underlying models for the Monte Carlo simulations (see section \[NC\]), and represent the true number counts. The dashed vertical lines indicate the mean $4\sigma$ within the effective area. The horizontal errorbars for the data-points indicate the width of each flux bin.[]{data-label="counts_MCMC"}](Fig_5.pdf){width="0.98\columnwidth"}
To obtain the underlying counts models we ran Monte Carlo simulations following e.g., @TC2013a [@TC2013b]. First, we create a simulated image by randomly injecting mock sources onto the jackknife maps. The mock sources are drawn from an assumed model and convolved with the PSFs. For the counts models we use a broken power-law of the form
$$\label{eqn:diff_counts}
\frac{dN}{dS} = \left\{
\begin{array}{l l}
{N_0}\left(\frac{S}{S_0}\right)^{-\alpha} & \quad \text{if $S \leq S_0$}\\
{N_0}\left(\frac{S}{S_0}\right)^{-\beta} & \quad \text{if $S > S_0$}\\
\end{array} \right.,$$
and started from a fit to the observed counts. As faintest fluxes for our models, we use the fluxes at which the integrated flux density agrees with the values for the extragalactic background light (EBL; e.g., @Puget1996).
After obtaining a mock map, we ran the source extraction algorithm and computed the number counts in exactly the same way as done with the real data. We then calculated the ratio between the recovered counts and the input model, which reflects the Eddington bias (@Eddington1913), and then applied this ratio to the observed counts to correct for this bias. A $\chi^2$ fit is performed to the corrected observed counts using the broken power-law to get the normalization and power-law indices, which are then used in the next iteration. This iterative process was terminated once the input model agreed with the corrected counts at the $1\sigma$ level. Given the low number of data-points at 450 $\mu$m, we only fitted the normalization and the bright-end slope at this wavelength. We fixed $S_0$ and $\alpha$ to the values in @TC2013b.
To test the reliability of our results, we have then created 500 realizations of simulated maps using as input the model curves determined through the Monte Carlo simulations, and calculated the pure source number counts for each of them. In Figure \[counts\_MCMC\] we show the results of the Monte Carlo simulations and compare them to the data. We give the derived underlying counts models (blue dot-dashed lines), the mean counts (red dashed lines), and the 90% confidence range of the 500 realization (yellow). The 500 realizations well match the pure source number counts within the uncertainties. We can then apply the ratio between the mean number counts and the input model to correct our data, and thus obtain the true differential number counts (see Section \[true\_number\_counts\]). Table \[MC\_input\] summarizes the parameters of the obtained underlying count models at both 850 and 450 $\mu$m.
Flux boosting estimates {#fluxBoost}
=======================
With the Monte Carlo simulations we found a systematic flux/count boost, which we characterized by comparing the flux of the injected mock sources with the detections. In particular, we selected the brightest input source located within the beam area of each of the $>3\sigma$ recovered sources, and computed the flux ratio. In Figure \[Flux\_boosting\] we show this test as a function of S/N for both wavelengths. We plot the mean (red) and the median (yellow) values of the flux boosting, together with the $1\sigma$ ranges (blue) relative to the mean values. At S/N$=4$, the estimated median flux boosting is 2.0 and 1.5 at 450 and 850 $\mu$m, respectively. This values are in agreement within the uncertainties with similar previous studies conducted with SCUBA-2 (e.g., @Casey2013 [@TC2013a]). We then corrected the observed fluxes for the catalogs obtained in Section \[catalogs\] using the median curves, and listed the de-boosted fluxes in Tables \[450\] and \[850\]. This flux boost is usually found in previous SCUBA studies (e.g., @Wang2017) and it is ascribed to the so-called Eddington bias (@Eddington1913).
![Ratio between the fluxes of the detected sources and the injected sources from the 500 realizations of the estimated underlying counts models (section \[NC\]) as a function of the S/N of the detections. The gray dots are $\sim100,000$ simulated data-points. We show the mean (red) and median (yellow) values of the flux ratio in different S/N bins. The blue dashed curves enclose the $1\sigma$ range relative to the mean curve. The test is shown for both 850 (top) and 450 $\mu$m (bottom).[]{data-label="Flux_boosting"}](Fig_6.pdf){width="0.98\columnwidth"}
Positional uncertainties {#pos_err}
========================
Using the same Monte Carlo simulations and the same algorithm to find counterparts in the injected and recovered catalogs, we can estimate the positional offset between the location of the injected and the recovered sources. Figure \[pos\_off\] shows this test at both 450 and 850 $\mu$m. At S/N$\lesssim5$, there is a large scatter, suggesting positional uncertainties of the order of $\gtrsim1.7$ and $\gtrsim2.2$ arcsec, respectively for 450 and 850 $\mu$m. At larger S/N the uncertainty is lower, down to 1 arcsec for sources as strong as the brightest objects in our 850 $\mu$m catalog (S/N$\approx16$). At 450 $\mu$m – characterized by a $\approx 1.5\times$ smaller beam – the positional uncertainties are slightly smaller. These results well agrees – within the uncertainties – with the predicted positional offset based on the LESS sample (dashed black line; equation B22 in @Ivison2007). Based on this test, we can then assign the mean value of the offsets as positional uncertainty to the detections listed in Tables \[450\] and \[850\].
![Positional offset between the detected sources and the injected sources from the 500 realizations of the estimated underlying counts models (section \[NC\]) as a function of the S/N of the detections. The gray dots are $\sim100,000$ simulated data-points. We show the mean (red) and median (yellow) values of the positional offsets in different S/N bins. The blue dashed curves enclose the $1\sigma$ range relative to the mean curve. The test is shown for both 850 (top) and 450 $\mu$m (bottom). The dot-dashed black lines indicate the predictions from @Ivison2007 based on the LESS sample.[]{data-label="pos_off"}](Fig_7.pdf){width="0.98\columnwidth"}
Results {#sec:results}
=======
True Number Counts {#true_number_counts}
------------------
{width="95.00000%"}
Figure \[true\_counts\] presents the corrected differential and cumulative number counts at both 450 and 850 $\mu$m for the effective area of our observations, together with the derived underlying broken power-law (bpl) models (blue dot-dashed lines). As explained in Section \[NC\], these true number counts have been obtained by dividing the pure source counts by the ratio between the mean number counts and the input models. We list the values of our corrected data-points in Table \[table:counts\].
We then compare our data-points with the most comprehensive literature studies for blank fields at both 450 and 850 $\mu$m (@TC2013b [@Casey2013; @Geach2013; @Wang2017; @Zavala2017; @Geach2017]). In Figure \[true\_counts\] we plot the fit – Schechter (Sc.) or broken power-law (bpl)[^2] – from those works. Our 450 $\mu$m data well agree with these literature curves[^3], while the 850 $\mu$m data-points are above these current expectations for blank fields. Especially the more robust data at about 5 and 7 mJy are clearly suggesting the presence of higher number counts with respect to the literature values.
To quantify this overdensity of counts at 850 $\mu$m, we fit our corrected differential number counts with the functions from each of the literature works allowing only the normalization to vary. The difference in counts is then estimated through the ratio between the normalizations. Specifically,
- @TC2013b quoted a best fit with a broken power-law function of the form shown in eqn. \[eqn:diff\_counts\], with[^4] $N_0=120^{+65}_{-45}$ mJy$^{-1}$ deg$^{-2}$, $S_0=6.21$ mJy, $\alpha=2.27$, $\beta=3.71$;
- @Casey2013 and @Geach2017 reported a Schechter function of the form $$\label{eqn:Schechter1}
\frac{dN}{dS} = \frac{N_0}{S_0}\left(\frac{S}{S_0}\right)^{\gamma}{\rm exp}\left(-\frac{S}{S_0}\right),$$ with $N_0=(3.3\pm1.4)\times10^3$ deg$^{-2}$, $S_0=3.7$ mJy, $\gamma=1.4$, and[^5] $N_0=4550\pm546$ deg$^{-2}$, $S_0=3.40\pm0.21$ mJy, $\gamma=1.97\pm0.08$, respectively;
- @Zavala2017 preferred a Schechter function of the form $$\label{eqn:Schechter1}
\frac{dN}{dS} = \frac{N_0}{S_0}\left(\frac{S}{S_0}\right)^{1-\gamma}{\rm exp}\left(-\frac{S}{S_0}\right),$$ with $N_0=8300\pm300$ deg$^{-2}$, $S_0=2.3$ mJy, $\gamma=2.6$ for all their data.
We show the results of the fit with free normalizations $N_0$ in Figure \[fit\_with\_Lit\], and we list in Table \[table:overdensity\] the ratio between the derived normalizations needed to match our data and the literature values. From these ratios it is clear that the probed effective area is indeed overdense with respect to blank fields. On average, around the ELAN MAMMOTH-1 there are $4.0\pm1.3$ times more counts than in blank fields. In this mean estimate we do *not* include the ratio with respect to @Zavala2017 because this work does not cover effectively the sources bright-end (their last bin is at 4.9 mJy), probably biasing their fit. We however report the comparison with this work for completeness.
![Top: fit of the SCUBA-2 MAMMOTH-1 true differential number counts at 850 $\mu$m using the functions given in literature works for blank fields (@TC2013b [@Casey2013; @Zavala2017; @Geach2017]) with $N_0$ free to vary. Bottom: the SCUBA-2 MAMMOTH-1 true cumulative number counts at 850 $\mu$m compared to the fit models obtained in the top panel. In both panels, the blue dot-dashed curve is our true number counts curve from Table \[MC\_input\]. All the literature models need a significant increase of their normalization parameter $N_0$ to fit our data at $850~\mu$m, revealing that the covered effective area is overdense with respect to blank fields. We list the values in Table \[table:overdensity\].[]{data-label="fit_with_Lit"}](Fig_9.pdf){width="0.95\columnwidth"}
Position of the catalog sources within the LAE overdensity {#Pos}
----------------------------------------------------------
Even though the association to the BOSS1441 overdensity of the sources listed in the SCUBA-2 catalogs has to be confirmed spectroscopically, we can still search for LAE counterparts to our submillimeter detections, if any. In Figure \[Comp\_with\_LAEs\], we show the location of the 450 (yellow squares) and 850 $\mu$m (blue circles, with fluxes) catalog sources along with (i) the position of known LAEs (black circles; @Cai2017a), (ii) the position of known QSOs at $2.30\leq z<2.34$ (brown crosses; @Cai2017a), and (iii) the LAEs density contours (green; @Cai2017a). From this figure it is clear that only 2 sources out of the 27 850 $\mu$m detections could be considered to be possibly associated with a LAE from the catalogue of @Cai2017a. This 2 sources (highlighted in orange) are (i) MAM-850.14 close to the ELAN MAMMOTH-1 (see Section \[MAMMOTH-1\] for a discussion), and (ii) MAM-850.16 close to a LAE at RA=220.3906 and Dec=40.0286, with rest-frame equivalent width $EW_0=25.16\pm0.01\AA$, which is actually a $z\simeq2.3$ QSO.
The other 25 LAEs lay at a separation greater than the 850 $\mu$m beam from any of our detections. Given the large offsets, the positional uncertainties presented in Section \[pos\_err\] are not affecting the lack of association between LAEs and our submillimeter detections. If future follow-up studies confirm the association of most of the SCUBA-2 sources with the BOSS1441 overdensity, the lack of submillimeter flux at the location of LAEs is consistent with the usual finding that most of the strongly Ly$\alpha$ emitting galaxies are relatively devoid of dust (e.g., @Ono2010 [@Hayes2013; @Sobral2018]).
In addition, the brightest detections at 850 $\mu$m, MAM-850.1 and MAM-850.2 ($f_{850}^{\rm Deboosted}=18.3\pm2.8$ mJy and $f_{850}^{\rm Deboosted}=16.3\pm2.7$ mJy) lay intriguingly close to the peak of the LAEs overdensity. Their observed ratios between 450 and 850 $\mu$m suggest that these two bright detections are unlikely to be low redshift sources. Therefore, they probably are associated with the protocluster given the rare alignment with the peak of the LAEs overdensity.
{width="95.00000%"}
Discussion {#sec:disc}
==========
BOSS1441: a rich and diverse protocluster {#disc:rich}
-----------------------------------------
In the previous sections we have demonstrated the presence of a $\sim 4$ times higher density fluctuation compared to blank fields at 850 $\mu$m. In addition, we found that the brightest of our detections are located at the peak of the LAEs overdensity. This unique alignments suggest that most of the SCUBA-2 detections are likely associated with the BOSS1441 overdensity (and the ELAN MAMMOTH-1), rather than being intervening.
To test this, we search the available multiwavelength catalogs and build the spectral energy distributions (SEDs) for all the sources in our sample. We thus look for counterparts in the [*AllWISE*]{} Source Catalog[^6] (@Wright2010) at 3.4, 4.6, 12.1, 22.2 $\mu$m (W1, W2, W3, W4), and in the Faint Images of the Radio Sky at Twenty-cm (FIRST) Survey at 1.4GHz (@Becker1994). This portion of the sky has not been covered by the [*Herschel*]{} telescope and thus our SCUBA-2 observations are key in covering the far-infrared portion of the SED. To match the different catalogs, we look for counterparts within a 850 $\mu$m beam, and select the closest source. We found that 8 of our detections have a counterpart in [*AllWISE*]{}, while none has been detected in FIRST down to the catalog detection limit at each source position ($\approx0.95$ mJy). To estimate the likelihood of false match for the WISE counterparts, we use the $p$-value as defined in @Downes1986
$$p = 1 - \exp(-\pi n \theta^{2}),$$
where $n$ is the [*AllWISE*]{} source density within the effective area, $n\simeq0.00134$ sources/arcsec$^2$, and $\theta$ is the angular separation between the [*AllWISE*]{} source and the SCUBA-2 detection[^7]. A value of $p<0.05$ usually makes a counterpart reliable (e.g., @Ivison2002 [@TC2016]), while $0.05<p<0.1$ makes it tentative (e.g., @Chapin2009). Of the eight counterparts we found that only four are robust, i.e. MAM-850.8, MAM-850.18, MAM-850.26 and MAM-850.27, while the others are tentative. However, we have shown in Section \[Pos\] that MAM-850.16 is likely associated with a quasar at $z=2.30$. As the quasar is detected by WISE and no other close WISE detections are present, we consider this match secure. Further, the lack of available radio and/or high resolution submm data prevents us to perform a robust identification of counterparts in our recently obtained LBT/LBC $U$, $V$ and $i$ band images (@Cai2017a) and in our UKIRT/WIRCAM $J$, $H$, and $K$ band images (Xu et al. in prep.). We summarize the sources with multiwavelength detections in Table \[SED\_values\_ALL\] , and display for illustration purposes the SEDs of the five sources with robust counterparts in [*AllWISE*]{} in Figure \[SEDs\_ALL\].
We leave a detailed classification of our detections to future studies encompassing the whole protocluster extent, and better covering the electromagnetic spectrum. However, we used the average SED template for SMGs obtained from 99 sources in the ALESS survey (@daCunha2015)[^8] to compute a rough estimate for the far-infrared (FIR) luminosity $L_{\rm FIR}$ for each source assuming they are all SMGs at $z=2.32$ (redshift of BOSS1441). After normalizing the SED template to our SCUBA-2 observations, we found $L_{\rm FIR}\geq4.8\times10^{12}$ L$_{\odot}$ for each of the sources[^9].
Next, we compared the volume density implied by our observations with expectations from the current luminosity function of SMGs. Assuming the effective area of our observations ($\sim336.9$ Mpc$^2$) and the distance interval spanned in redshift by the protocluster ($z=2.3-2.34$; $\sim34.9$ Mpc$^{-3}$), the comoving volume targeted by our observations is about $11800$ Mpc$^3$. If all (75%) of the detections at 850$\mu$m belongs to the protocluster, their volume density would then be $2.3\times10^{-3}$ Mpc$^{-3}$ ($1.7\times10^{-3}$ Mpc$^{-3}$). These values are a factor of $>30$ above the volume density expected from the current luminosity function of SMGs with $L_{\rm FIR}\geq4.8\times10^{12}$ L$_{\odot}$ ($\sim5\times10^{-5}$ Mpc$^{-3}$; @Casey2014). Therefore, BOSS1441 is a potentially SMG-rich volume.
We further noticed that the two brightest sources, MAM-850.1 and MAM-850.2, seem to depart from the templates at 850$\mu$m, showing higher fluxes than expected. This deviation could be explained by allowing a different (higher) redshift, or most probably by the fact that single-dish submillimeter sources as bright as MAM-850.1 and MAM-850.2 are usually a blend of $\geq2$ SMGs once observed with interferometers (@Karim2013 [@Simpson2015]). This explanation is compelling as the two sources sit at the peak of the overdensity. They could thus be groups of interacting galaxies, pinpointing the core of the protocluster.
Extremely compact ($20\arcsec$-$40\arcsec$) protocluster cores made of several ($>10$) starbursting galaxies at $z\sim4$ have been recently discovered by @Miller2018 and @Oteo2018. These structures have a global $L_{\rm FIR}\approx10^{14}$ L$_{\odot}$ (@Miller2018). The BOSS1441 protocluster might thus have similar, but scaled down central structures. We can compare this central portion of the overdensity with other known protoclusters at $z\sim2$ (@Dannerbauer2014 [@Casey2015; @Kato2016]). These studies found that spheres with 1 Mpc radius centered at the protocluster core enclose a star-formation rate density of ${\rm SFRD}\sim1000-1500$ M$_{\odot}$ yr$^{-1}$ Mpc$^{-3}$. Following the approach in those works, we centered a 1 Mpc sphere at the position of MAM-850.2, which is the closest SMG to the peak of the LAE overdensity. Within this sphere we potentially found six detections (MAM-850.1,MAM-850.2,MAM-850.12,MAM-850.14,MAM-850.16,MAM-850.21), which add up to a total star formation rate of ${\rm SFR}=9100$ M$_{\odot}$ yr$^{-1}$, translating to ${\rm SFRD}\approx2200$ M$_{\odot}$ yr$^{-1}$ Mpc$^{-3}$ after dividing by the sphere volume[^10]. As our detections are candidate SMGs, this value represents an upper limit. Subtracting the field average value as done in @Kato2016 for $z=2.3$, and assuming that only 75% of our detections are within the sphere, we obtained ${\rm SFRD}\approx1200$ M$_{\odot}$ yr$^{-1}$ Mpc$^{-3}$. We conservatively conclude that ${\rm SFRD}=1200^{+1000}_{-1100}$ M$_{\odot}$ yr$^{-1}$ Mpc$^{-3}$, with the lower limit given by the very unlike case that none of the detections (apart the ELAN MAMMOTH-1 counterpart) are within the sphere. This SFRD value is in agreement with values for protoclusters in the literature (e.g. see Fig. 5 in @Kato2016).
Spectroscopic and interferometric follow-ups are needed to confirm the redshift of our sources, and to unveil their nature. Our analysis however suggests that BOSS1441 is a rich protocluster hosting several LAEs, one ELAN, and likely several SMGs.
The Counterpart of the ELAN MAMMOTH-1 {#MAMMOTH-1}
-------------------------------------
@Cai2016 reported the presence of a continuum source at $z=2.319\pm0.004$, named source B, and of a $z=0.16$ AGN, both at the location of the peak of the ELAN MAMMOTH-1. Source B has been invoked as the powering source of the extended Ly$\alpha$ emission. At a separation of $4.58\arcsec$ from source B and at $2.75\arcsec$ from the $z=0.16$ AGN, our SCUBA-2 observations resulted in a bright detection at 850 $\mu$m, MAM-850.14, with flux density of $f_{850}=4.57\pm0.93$ mJy ($f_{850}^{Deboosted}=2.83\pm1.03$ mJy), and a $3\sigma$ upper limit of $f_{450}<16.65$ mJy at 450 $\mu$m. The non-detection at 450 $\mu$m suggests that the emission at 850 $\mu$m is associated with the $z=2.319$ source. Indeed, a $z=0.16$ AGN with such a detection at 850 $\mu$m should have a much brighter dust thermal emission at 450 $\mu$m. Specifically, if we assume a modified black-body for optically thin thermal dust emission, a dust temperature of $T_{\rm dust}=45$ K, and the largely used emissivity index $\beta=1.5$ (e.g., @Casey2012), we find that a $z=0.16$ AGN should have $f_{450}=35.3$ mJy (or $21.8$ mJy for the deboosted flux). These fluxes would be detected at high significance ($\gtrsim4\sigma$) even in our shallow 450 $\mu$m data at the location of the ELAN MAMMOTH-1.
To constrain the nature of source B, we compiled all its data available from the literature, and compared them to known SED. We summarize all the available observations in Table \[SED\_values\] [^11], while we plot them in Figure \[SEDsourceB\].
To compare these data-points to known SEDs, we fixed the redshift of the source to the redshift $z=2.319$ determined from the line emission (@Cai2016), and we fitted the data leaving the normalization free. We first take in consideration the average SED for SMGs obtained from the 99 sources in ALESS (@daCunha2015), and all the available average SEDs from that publication. The left panel of Figure \[SEDsourceB\] shows this test, highlighting the shortage of emission at the WISE bands for these SED templates (we plot only two to avoid confusion) in comparison to the source B’s data-points. None of the average templates in @daCunha2015 match the W1,W2,W3 data-points, with the SED with $A_V<1$ (yellow) giving the closer values, though differing still significantly. We then follow the same procedure with the template SED of the local starburst galaxy M82 (@Silva1998; solid black line in Figure \[SEDsourceB\]). This template match significantly better the observations, with only the W3 data-point underestimated. Most likely a hotter dust component powered by an AGN (e.g., @Silva2004 [@Fritz2006]) would allow a better match of the data of source B. Indeed @Cai2016 demonstrate that only hard-ioninzing sources – most likely an AGN or a wind – could power the and emission in this object.
To test this interpretation further, we used the publicly available SED fitting code, [*AGNfitter*]{} (@CalistroRivera2016), which adopts a fully Bayesian Markov Chain Monte Carlo method to model the SEDs of galaxies and AGN. [*AGNfitter*]{} fits simultaneously the sub-mm to UV photometry decomposing the SED into four physically motivated components: the AGN accretion disk emission (Big Blue Bump), the hot dust emission from the obscuring structure around the accretion disk (torus), the cold dust emission from star-forming regions and the stellar populations of the host galaxy[^12]. Details on the specific models are presented by @CalistroRivera2016 and references therein. The right panel of Figure \[SEDsourceB\] shows the best fit (in gray) produced by [*AGNfitter*]{}, with each component highlighted by a different color. The W3 data-point is now well covered by a composition of the AGN-powered hot dust, star emission and starformation-powered cold dust.
This analysis thus suggests that source B is an enshrouded strong starbursting galaxy, likely hosting an obscured AGN. Using the output from [*AGNfitter*]{}, we can separate the far-infrared (FIR) luminosity $L_{\rm FIR}$ (rest-frame 8-1000 $\mu$m) due to the AGN and to star-formation (SF). We find $L_{\rm FIR}^{\rm AGN}=8.0^{+1.2}_{-6.3}\times10^{11}$ L$_{\odot}$ and $L_{\rm FIR}^{\rm SF}=2.4^{+7.4}_{-2.1}\times10^{12}$ L$_{\odot}$, respectively. Source B thus meets the generally used criteria to define an UltraLuminous InfraRed Galaxy (ULIRG; L$_{\rm 8-1000\mu m}>10^{12}$ L$_{\odot}$; e.g., @SandersMirabel1996). Following the classical conversion in @kennicutt98 and considering only the star-formation powered emission, one would then obtain a star formation rate of SFR $=400^{+1300}_{-400}$ M$_{\odot}$ yr$^{-1}$.
In agreement with the observations and analysis in @Cai2016, our analysis thus suggests a strong similarity between the source B embedded within the ELAN MAMMOTH-1 and the ULIRG sample hosting AGN activity in @Harrison2012. Source B – with its obscured AGN and starburst – can thus easily power the surrounding ELAN, and thus the outflow resulting in the velocity offset of 700 km s$^{-1}$ between the two spectral components in Ly$\alpha$, , and (@Cai2016). The very broad \[\] emission presented for the targets in @Harrison2012 however extends to lower distances (15 kpc) with respect to the rest-frame UV lines seen in the ELAN MAMMOTH-1 ($\gtrsim30$ kpc; @Cai2016). As the ELAN MAMMOTH-1 hosts an ULIRG we thus expect to see broad \[\] emission in its central portion down to similar depths.
Finally, [*AGNfitter*]{} estimated the stellar mass of source B to be log$(M_{\rm star}/{\rm M_{\odot}})=11.4^{+0.3}_{-0.2}$. By inverting the halo mass $M_{\rm halo}$ - $M_{\rm star}$ relation in @Moster2013, we derived that – if source B is a central galaxy – the ELAN MAMMOTH-1 is hosted by a very massive halo of log$(M_{\rm halo}/{\rm M_{\odot}})=15.2^{+1.4}_{-1.6}$. Given the large uncertainties this result has to be confirmed. However, it certainly highlights the peculiarity of the halo hosting the ELAN MAMMOTH-1, indicating that it sits at the high-mass end of the halo population at this redshift. We further note that the stellar mass of source B is intriguingly close to current estimates for the stellar mass of host galaxies of HzRGs (log$(M_{\rm star}/{\rm M_{\odot}})\simeq 11 - 11.5$; @Seymour2007 [@DeBreuck2010]). HzRGs are currently thought to reside in massive halos of mass log$(M_{\rm halo}/{\rm M_{\odot}})\approx 13$ (e.g., @Stevens2003). The halo hosting the ELAN MAMMOTH-1 could thus be similarly massive or exceed such halos.
{width="100.00000%"}
Summary {#sec:summ}
=======
We are conducting a survey of all the known ELANe (@cantalupo14 [@hennawi+15; @Cai2016; @FAB2018]) with the JCMT and APEX telescopes to assess the presence of starburst activity in these systems and their environments. In this work we focused on the SCUBA-2/JCMT data at 450 and 850 $\mu$m obtained for an effective area of $\sim127$ arcmin$^2$ around the ELAN MAMMOTH-1 at $z=2.319$ (@Cai2016), and thus targeting the known peak area of the LAE overdensity BOSS1441 (@Cai2017a). Thanks to this dataset we found that
1. the 850 $\mu$m source counts are $4.0\pm1.3$ times higher than in blank fields (@TC2013b [@Casey2013; @Geach2017]), confirming also in obscured tracers the presence of an overdensity. Intriguingly, the two brightest submillimeter detections, MAM-850.1 and MAM-850.2, are located at the peak of the LAE overdensity, possibly pinpointing the core of the protocluster and multiple mergers/interactions (e.g., @Miller2018). The association of the discovered submillimeter sources with BOSS1441 needs however a spectroscopic confirmation.
2. the continuum source at the center of the ELAN MAMMOTH-1, source B (@Cai2016), is associated to a strong detection at 850 $\mu$m, MAM-850.14, with flux density of $f_{850}=4.6\pm0.9$ mJy ($f_{850}^{\rm Deboosted}=2.8\pm1.0$ mJy) and a $3\sigma$ upper limit of $f_{450}<16.6$ mJy at 450 $\mu$m. Together with the data from the literature, the SED of source B agrees with a strongly starbursting galaxy hosting an obscured AGN, and having a FIR luminosity of $L_{\rm FIR}^{\rm SF}=2.4^{+7.4}_{-2.1}\times10^{12}$ L$_{\odot}$. Source B is thus an ULIRG with a star-formation rate of SFR $=400^{+1300}_{-400}$ M$_{\odot}$ yr$^{-1}$ assuming the classical @kennicutt98 calibration. Such a source – containing both an AGN and a violent starburst – is able to power the hard photoionization plus outflow scenario depicted in @Cai2016.
The acquisition of wide-field multiwavelength data (X-ray, UV, optical, submillimeter, radio) is key in painting a coherent and detailed picture of a protocluster, and ultimately to understand the assembly of massive galaxies within the cosmic nurseries of the soon-to-be large clusters. The results of this pilot project are encouraging and reflect the importance of such a multi-wavelength approach in fully comprehending the ELAN phenomenon and the environment in which they reside.
We thank the referee Yuichi Matsuda for his careful read of the manuscript. The James Clerk Maxwell Telescope is operated by the East Asian Observatory on behalf of The National Astronomical Observatory of Japan; Academia Sinica Institute of Astronomy and Astrophysics; the Korea Astronomy and Space Science Institute; the Operation, Maintenance and Upgrading Fund for Astronomical Telescopes and Facility Instruments, budgeted from the Ministry of Finance (MOF) of China and administrated by the Chinese Academy of Sciences (CAS), as well as the National Key R&D Program of China (No. 2017YFA0402700). Additional funding support is provided by the Science and Technology Facilities Council of the United Kingdom and participating universities in the United Kingdom and Canada. This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, and NEOWISE, which is a project of the Jet Propulsion Laboratory/California Institute of Technology. WISE and NEOWISE are funded by the National Aeronautics and Space Administration. M.F. acknowledges support by the Science and Technology Facilities Council \[grant number ST/P000541/1\]. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 757535). I.R.S. acknowledges support from the ERC Advanced Grant [*DUSTYGAL*]{} (321334) and STFC (ST/P000541/1). Y.Y.’s research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2016R1C1B2007782). The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.
[*AllWISE*]{} counterparts to our SCUBA-2 detections
====================================================
In this appendix we show the multiwavelength dataset for [*AllWISE*]{} counterparts to our SCUBA-2 detections. Specifically, in Table \[SED\_values\_ALL\] we list (i) the likelihood of false match for each [*AllWISE*]{} counterpart, i.e. the $p$-value (see Section \[disc:rich\]), (ii) the magnitudes from the [*AllWISE*]{} catalog, (iii) the flux from the FIRST survey, and (iv) the rough estimate of $L_{\rm FIR}$ for each source. The four sources with $p<0.05$ are considered robust, while the others are tentative. We further consider robust the match with MAM-850.16 has it is clearly associated with a quasar at $z=2.30$ (see Section \[Pos\] for more details). Finally, for illustration purposes, in Figure \[SEDs\_ALL\] we show the SED of the five sources with robust [*AllWISE*]{} counterparts along with the template SED of M82 (@Silva1998; black line), of the average ALESS SMGs (dashed brown line), and the average ALESS SMGs with $A_V<1$, and $A_V \geq 3$ (@daCunha2015; yellow). All the template SEDs have been normalized to the SCUBA-2 data assuming $z=2.32$. We will perform a more detailed analysis of the SED in future works encompassing the full extent of the protocluster, and covering a broader range of the electromagnetic spectrum.
{width="90.00000%"}
[^1]: As PSFs for our observations, we adopt the PSFs at 850 and 450 $\mu$m generated by @TC2013b (see their Figure 2).
[^2]: If a work presented both functions for their fits, we selected their Schechter fit. Our results do not depend on this choice.
[^3]: We remind the reader that – as already noted in @Casey2013 – the equation (1) of @Geach2013 should be written as $dN/dS=(N'/S')(S/S')^{1-\alpha}\exp(-S/S')$, and the best-fitting parameter $N'$ for this 450 $\mu$m data should be quoted as $N'=4900\pm1040$ deg$^{-2}$ mJy$^{-1}$ rather than $N'=490\pm1040$ deg$^{-2}$ mJy$^{-1}$.
[^4]: For all the fits in the literature, we report only the errors on the parameter $N_0$.
[^5]: @Geach2017 only showed a Schechter fit to their data in their Figure 15. Here we thus report the values for a Schechter fit to their data.
[^6]: <http://wise2.ipac.caltech.edu/docs/release/allwise/>
[^7]: As the positional uncertainty is small for the 450 $\mu$m band, for our sources we adopt the coordinates at 450 $\mu$m if available.
[^8]: @daCunha2015 provides average SEDs made in bins of redshifts, observed ALMA 870 $\mu$m flux, average V-band attenuation $A_V$, total dust luminosity (<http://astronomy.swinburne.edu.au/~ecunha/ecunha/SED_Templates.html>).
[^9]: We estimated the FIR luminosity $L_{\rm FIR}$ for each source as frequently done by integrating the luminosity in the rest-frame range 8-1000 $\mu$m.
[^10]: We convert $L_{\rm FIR}$ to SFR using the classical conversion in @kennicutt98.
[^11]: The $U$, $V$, $i$-band photometry here reported is slightly (within errors) different from @Cai2016 because of the image degradation applied to the data to match the UKIRT observations (Xu et al. in prep.).
[^12]: [*AGNfitter*]{} does not currently cover the radio portion of the spectrum. This does not affect our results as we do not have stringent limits at the radio wavelengths.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
A new kinematic and dynamic study of the halo of the giant elliptical galaxy NGC 5128 is presented. From a spectroscopically confirmed sample of $340$ globular clusters and $780$ planetary nebulae, the rotation amplitude, rotation axis, velocity dispersion, and total dynamical mass are determined for the halo of NGC 5128. The globular cluster kinematics were searched for both radial dependence and metallicity dependence by subdividing the globular cluster sample into 158 metal-rich (\[Fe/H\]$ > -1.0$) and 178 metal-poor (\[Fe/H\]$ <
-1.0$) globular clusters. Our results show that the kinematics of the metal-rich and metal-poor subpopulations are quite similar: over a projected radius of $0-50$ kpc, the mean rotation amplitudes are $47\pm15$ and $31\pm14$ km s$^{-1}$ for the metal-rich and metal-poor populations, respectively. There is a indication within $0-5$ kpc that the metal-poor clusters have a lower rotation signal than in the outer regions of the galaxy. The rotation axis shows an interesting twist at 5 kpc, agreeing with the zero-velocity curve presented by Peng and coworkers. Within 5 kpc, both metal-rich and metal-poor populations have a rotation axis nearly parallel to the north-south direction, which is $0^o$, while beyond 5 kpc the rotation axis twists $\sim180 ^o$. The velocity dispersion displays a steady increase with galactocentric radius for both metallicity populations, with means of $111\pm6$ and $117\pm6$ km s$^{-1}$ within a projected radius of 15 kpc for the metal-rich and metal-poor populations; however, the outermost regions suffer from low number statistics and spatial biases. The planetary nebula kinematics are slightly different. Out to a projected radius of $90$ kpc from the center of NGC 5128, the planetary nebulae have a higher rotation amplitude of $76\pm6$ km s$^{-1}$, and a rotation axis of $170\pm5 ^o$ east of north, with no significant radial deviation in either determined quantity. The velocity dispersion decreases with galactocentric distance. The total mass of NGC 5128 is found using the tracer mass estimator, described by Evans et al., to determine the mass supported by internal random motions and the spherical component of the Jeans equation to determine the mass supported by rotation. We find a total mass of $1.0\pm0.2 \times 10^{12}$ $M_{\odot}$ from the planetary nebula data extending to a projected radius of 90 kpc. The similar kinematics of the metal-rich and metal-poor globular clusters allow us to combine the two subpopulations to determine an independent estimate of the total mass, giving $1.3\pm0.5 \times 10^{12}$ $M_{\odot}$ out to a projected radius of 50 kpc. Lastly, we publish a new and homogeneous catalog of known globular clusters in NGC 5128. This catalog combines all previous definitive cluster identifications from radial velocity studies and $HST$ imaging studies, as well as 80 new globular clusters with radial velocities from a study of M.A. Beasley et al. (in preparation).
author:
- 'Kristin A. Woodley'
- 'William E. Harris'
- 'Michael A. Beasley'
- 'Eric W. Peng'
- 'Terry J. Bridges'
- 'Duncan A. Forbes'
- 'Gretchen L. H. Harris'
title: The Kinematics and Dynamics of the Globular Clusters and Planetary Nebulae of NGC 5128
---
Introduction {#sec:intro}
============
Globular clusters (GCs), as single-age, single-metallicity objects, are excellent tracers of the formation history of their host galaxies, through their dynamics, kinematics, metallicities, and ages. For most galaxies within $\simeq 20$ Mpc, GCs can be identified through photometry and image morphology, from which follow-up radial velocity studies can be carried out with multi-object spectroscopy on 4 and 8 m class telescopes. The ability to target hundreds of objects in a single field has vastly increased the observed samples of GCs confirmed in many galaxies, providing the necessary basis for detailed kinematic and age studies.
Another benefit of using GCs as a kinematic tracer is that they provide a useful, independent basis for comparison with results from planetary nebulae (PNe). This is particularly true given the current debate surrounding the use of PN velocities and the implications for low dark matter halos. For example, [@romanowsky03] reported, based on PN velocities, that three low-luminosity ellipticals revealed declining velocity dispersion profiles and little or no dark matter. However, subsequent simulations of merger-remnant ellipticals suggested that the radial anisotropy of intermediate-age PNe could give rise to the observed profiles within standard halos of dark matter [@dekel05; @mamon05]. Flattening of the galaxy along the line of sight is another possible explanation. One of the ellipticals studied by [@romanowsky03], NGC 3379, has also been investigated using GC kinematics [@pierce06; @bergond06]; both studies found evidence of a dark matter halo. But the two studies of NGC 3379 suffered from small number statistics. Clearly there is a need to directly compare PN and GC kinematics for the same elliptical with sufficiently large numbers of tracer objects.
Previous velocity-based studies of globular cluster systems (GCSs) have shown an intriguing variety of results in their overall kinematics. Prominent recent examples include the following:
1\. [@cote01] performed a kinematic analysis of the GCS in M87 (NGC 4486), the cD galaxy in the Virgo Cluster. With a sample of $280$ GCs, they showed that the entire GCS rotates on an axis matching the photometric [*minor*]{} axis of the galaxy, except for the inner metal-poor sample. Inside the onset of the cD envelope, the metal-poor clusters appear to rotate around the [*major*]{} axis of the galaxy instead. This study also found evidence for an increase in velocity dispersion, $\sigma_v$, with radius, due to the larger scale Virgo Cluster mass distribution. They also showed no strong evidence for a difference in $\sigma_v$ between the metal-poor and metal-rich groups. Using a Virgo mass model, they investigated anisotropy and found that as a whole, the GCS had isotropy, but considered separately, the metal-poor and metal-rich subpopulations had slight anisotropy.
2\. [@cote03] performed a dynamical analysis for M49 (NGC 4472), the other supergiant member of the Virgo Cluster. Using over $260$ GCs, they found that the metal-rich population shows no strong evidence for rotation, while the metal-poor population does rotate about the minor axis of the galaxy. In addition, they found that the metal-poor clusters had an overall higher dispersion than the metal-rich population.
3\. [@richtler04] determined the kinematics for the GCS in NGC 1399, the brightest elliptical in the Fornax Cluster of galaxies. Armed with a sample of over $460$ GCs, they found a marginal rotation signal for the entire GC sample and the outer metal-poor sample, while no rotation was seen for the metal-rich subpopulation. Their projected velocity dispersion showed no radial trend within their determined uncertainty, but the metal-poor clusters had a higher dispersion than the metal-rich clusters. The kinematics of the PNe in NGC 1399 have been most recently studied by [@saglia00] and [@napolitano02] using a small sample of 37 PNe from [@arnaboldi94]. These studies do not indicate any significant deviations in velocity dispersion with radius for the PNe, yet interestingly, [@napolitano02] found stong rotation for the PNe in the inner region of the galaxy.
4\. [@pff04II] studied the GC kinematics of the giant elliptical NGC 5128 from a total of $215$ GCs. Their results showed definite rotation signals in both metallicity groups beyond a 5 kpc distance from the center of the galaxy, as well as similar velocity dispersions in both the metal-poor and the metal-rich populations within a 20 kpc projected radius. A similar PN kinematic study in NGC 5128 [@peng04] showed that the PNe population is rotating around a twisted axis that turns just beyond the 5 kpc distance.
These few detailed kinematic studies of GCSs in elliptical galaxies, show results that appear to differ on a galaxy-by-galaxy basis, without any clear global trends. The age distributions of the GC populations in these same galaxies tend to show a consistent pattern in which the blue, or metal-poor, population is found to be universally old. The red, or metal-rich, population has also been shown to be old in the study of [@strader05]. Their study found that [*both*]{} the metal-poor and metal-rich GCs in a sample of eight galaxies ranging from dwarf to massive ellipticals, all have ages as old as their Galactic GC counterparts. Conversely, in a small sample of studies, the red GC population has also been found to be 2-4 Gyr younger than the metal-poor GC population, and with a wider spread in determined ages [see the studies of @pff04II; @puzia05 among others], although this is not yet a well-established trend. The GCs in NGC 5128 appear to be old, with an intermediate-age population [@pff04II; @beasley06].
NGC 5128 (Centaurus A), the central giant in the Centaurus group of galaxies at a distance of only $\sim4$ Mpc, is a prime candidate for both kinematic and age studies. Its GCS has a specific frequency of $S_N \simeq 2.2\pm0.6$ [@harris06], toward the low end of the giant elliptical range but about twice as high as in typical disk galaxies. Its optical features show faint isophotal shells located in the halo [e.g. @peng02], the prominent dust lane in the inner 5 kpc, the presence of gas, and star formation, all of which suggest that NGC 5128 could be a merger product. [@baade54] first suggested that NGC 5128 could be the result of a merger between two galaxies, a spiral and an elliptical. This idea was followed by the general formation mechanism of disk-disk mergers proposed by [@toomre72]. [@bhh03] found that the metallicity distribution function of the halo field stars could be reproduced by a gas-free (”dry”) disk-disk merger scenario. Recent numerical simulations by [@bekki06] also demonstrate that the PN kinematics observed in NGC 5128 [@peng04] can be reproduced relatively well from a merger of unequal-mass disk galaxies (with one galaxy half the mass of the other) colliding on a highly inclined orbital configuration.
Alternatively, much of NGC 5128 could be a ”red and dead” galaxy, passively evolving since its initial formation as a large seed galaxy [@w06], while undergoing later minor mergers and satellite accretions. Evidence consistent with this scenario is in the halo population of stars in NGC 5128, which have a mean age of $8^{+3.0}_{-3.5}$ Gyr [@rejkuba05]. Its metal-poor GC ages have also been shown to have ages similar to Milky Way GCs, while the metal-rich population appears younger [@pff04II]. Our new spectroscopic study [@beasley06] suggests that NGC 5128 has a trimodal distribution of cluster ages: $\sim50\%$ of metal-rich clusters have ages of $6-8$ Gyr, only a small handful of metal-rich clusters have ages of $1-3$ Gyr, and a large fraction of both metal-rich and metal-poor clusters have ages of $\simeq 12$ Gyr. Lastly, the halo kinematics of NGC 5128 have also been recently shown to match the surrounding satellite galaxies in the low-density Centaurus group, suggesting that NGC 5128 acts like an inner component to its galaxy group [@w06]. Kinematic and age studies with large number of GCs are thus starting to help disentangle the formation of this giant elliptical.
The confirmed GC population in NGC 5128 is now large enough to allow a new kinematic analysis subdivided to explore radial and metallicity dependence, while avoiding small number statistics in almost all regions of the galaxy. The analysis presented here complements the detailed age distribution study provided by [@beasley06]. The results provide a broader picture of the formation scenario of NGC 5128.
The sections of this paper are divided as follows: § \[sec:cat\] contains the full catalog of NGC 5128 GCs with known photometry and radial velocities, § \[sec:kin\] contains the kinematic analysis of GCSs, § \[sec:kin\_PN\] contains the kinematic analysis of the PN population, § \[sec:dyn\] contains the discussion of the dynamical mass of NGC 5128, and § \[sec:concl\] contains our final discussion, as well as concluding remarks.
The Catalog of Globular Clusters in NGC 5128 {#sec:cat}
============================================
Finding GCs in NGC 5128 is challenging. This process begins with photometric surveys of the many thousands of objects projected onto the NGC 5128 field [e.g. @rejkuba01; @hhg04II]; only a few percent of these are the GCs that we seek. This daunting task is made difficult by the galactic latitude of NGC 5128 ($b=19^o$), which means that many foreground stars are present in the field of NGC 5128. Background galaxies are another major contaminant in the field, forcing the use of search criteria such as magnitude, color, and object morphology to help build the necessary candidate list of GCs. Confirmation of such candidate objects can then be done with spectroscopic radial velocity measurements. The GCs in NGC 5128 have radial velocities in the range $v_r = 200-1000$ km s$^{-1}$, while most foreground stars have $v_r < 200$ km s$^{-1}$. Background galaxies have radial velocities of many thousands of km s$^{-1}$ and can easily be eliminated [see the recent studies of @pff04I; @whh05].
Over the past quarter century, there have been seven distinct radial velocity studies to identify GCs in NGC 5128 [@vhh81; @hhvh84; @hhh86; @hghh92; @pff04I; @whh05; @beasley06]. Although [@hghh92] was not a radial velocity study itself, but rather a CCD photometric study of previously confirmed GCs, it does include the GCs determined spectroscopically from [@sharples88]. A recent study with measured GC radial velocities published by [@rejkuba07] has confirmed two new GCs, HCH15 and R122, included in our catalog.
Within this catalog are 80 new GCs with radial velocity measurements from [@beasley06]. The combination of these studies now leads to a confirmed population of 342 GCs.[^1] Also included in our catalog are the new GC candidates from $HST$ STIS imaging from [@hhhm02], labelled C100-C106, and from $HST$ ACS imaging from [@harris06], labelled C111-C179. All these previous studies have their own internal numbering systems, which makes the cluster identifications somewhat confusing at this point. Here we define a new, homogeneous listing combining all this material and with a single numbering system.
Our catalog of the GCs of NGC 5128 is given in Table \[tab:cat\_GC\]. In successive columns, the Table gives the new cluster name in order of increasing right ascension; the previous names of the cluster in the literature; right ascension and declination (J2000); the projected radius from the center of NGC 5128 in arcminutes; the $U$, $B$, $V$, $R$, and $I$ photometric indices and their measured uncertainties; the $C$, $M$, and $T_1$ photometric indices and their uncertainties; the colors $U-B$, $B-V$, $V-R$, $V-I$, $M-T_1$, $C-M$, and $C-T_1$; and, lastly, the weighted mean velocity $v_r$ and its associated uncertainty from all previous studies. All $UBVRI$ photometry is from the imaging survey described in [@pff04I]. The $CMT_1$ data are from [@hhg04II].
The mean velocities are weighted averages with weights on each individual measurement equal to $\varepsilon_{v}^{-2}$ where $\varepsilon_{v}$ is the quoted velocity uncertainty from each study. The uncertainty in the mean velocity is then $<\varepsilon_v> = (\sum \varepsilon_{i}^{-2})^{-1/2}$. There are no individual uncertainties supplied for the velocities for clusters studied by [@hhh86], but their study reports that the mean velocity uncertainty for clusters with $R_{gc} < 11'$ is 25 km s$^{-1}$ and for $R_{gc} > 11'$ is 44 km s$^{-1}$. We have adopted these values accordingly for their clusters.[^2]
The study by [@hghh92] also does not report velocity uncertainties; however, these clusters have all been recently measured by [@pff04I]. The rms scatter of the [@hghh92] values from theirs was 58 km s$^{-1}$. This value has been adopted as the velocity uncertainty of the [@hghh92] clusters in the weighted means.
The weighted mean velocity of cluster C10 does not include the measured value determined by [@pff04I] which is significantly different from other measurements. Also, cluster C27 does not include the measurement of $v_r = 1932\pm203$ km s$^{-1}$ from [@beasley06], indicating that this object is a galaxy. We include C27 as a GC, but with caution.
In the weighted velocity calculations, the velocities and uncertainties of the 27 GCs from [@rejkuba07] have been rounded to the nearest whole number, with any velocity uncertainty below 1 km s$^{-1}$ rounded up to a value of 1.
Lastly, the GC pff\_gc-089 overlaps the previously existing confirmed cluster, C49, within a 0.5” radius; pff\_gc-089 is therefore removed from the catalog of confirmed GCs.
The data in Table \[tab:cat\_GC\] provide the basis for the kinematic study presented in this paper. We use them to derive the rotation amplitude, rotation axis and velocity dispersion in the full catalog of clusters, as well as for the metal-poor (\[Fe/H\] $< -1$) and metal-rich (\[Fe/H\] $> -1$) subpopulations. For this purpose, we define the metallicity of the GCs by transforming the dereddened colors $(C - T_1)_o$ to \[Fe/H\] through the standard conversion [@harris02], calibrated through Milky Way cluster data. A foreground reddening value of $E(B - V) =
0.11$ for NGC 5128, corresponding to $E(C - T_1) = 0.22$, has been adopted. The division of \[Fe/H\] = -1 between metal-rich and metal-poor GCs has been shown as a good split between the two metallicity populations from \[Fe/H\] values converted from $C - T_1$ in [@whh05] and [@hhg04II] for NGC 5128. If no $C$ and/or $T_1$ values are available for the cluster, it is classified as metal-rich or metal-poor through a transformation from $(U-B)_o$ to \[Fe/H\] from [@reed94].
In Figure \[fig:position\] we show the spatial distributions of all the GCs from Table \[tab:cat\_GC\] ([*left*]{}) and the distribution of the known PNe ([*right*]{}). Both systems are spatially biased to the major axis of the galaxy because this is where most of the GC and PN searches have concentrated.
Kinematics of the Globular Cluster System {#sec:kin}
=========================================
Velocity Field {#sec:velfield}
--------------
For the present discussion we adopt a distance of $3.9$ Mpc for NGC 5128. This value is based on four stellar standard candles that each have internal precisions near $\pm 0.2$ mag: the PN luminosity function, the tip of the old-halo red giant branch, the long-period variables, and the Cepheids [@hhp99; @rejkuba04; @ferr06].
HCH15 and R122 have not been included in our kinematic study, as our study was completed before publication of these velocities. The weighted velocities used in this kinematic study do not include the most recent 25 velocities of previously known GCs published in [@rejkuba07]. Note that the velocities published in Table \[tab:cat\_GC\] do, however, include the [@rejkuba07] velocities in the quoted final weighted radial velocities for completeness.
The velocity distribution of the entire sample of 340 is shown in Figure \[fig:gausfit\_all\] ([*top left*]{}), binned in 50 km s$^{-1}$ intervals. A fit with a single Gaussian yields a mean velocity of $546\pm7$ km s$^{-1}$, nicely matching the known systemic velocity of $541\pm7$ km s$^{-1}$ [@hui95]. There is a slight asymmetry at the low-velocity end that is likely due to contamination by a few metal-poor Milky Way halo stars (also seen in the metal-poor subpopulation in the bottom left panel, which has a mean velocity determined by the Gaussian fit as $532\pm13$ km s$^{-1}$).
Selecting the clusters with radial velocity uncertainties less than 50 km s$^{-1}$ leaves 226 clusters, plotted in Fig. \[fig:gausfit\_all\] ([*top right*]{}). The close fit to a single Gaussian is consistent with an isotropic distribution of orbits; the mean velocity is $554\pm5$ km s$^{-1}$. The metal-rich population, with a mean velocity determined by the Gaussian fit of $565\pm11$ km s$^{-1}$, is plotted in the bottom right panel, and also shows no strong asymmetries.
Looking closer at the metal-poor velocity asymmetry, we note that the 15 metal-poor clusters between 250 and 300 km s$^{-1}$ (in the region where contamination by Milky Way field stars could occur) are balanced by only two GCs at the high-velocity end on reflection across the systemic velocity. The same velocity regions in the metal-rich population are nearly equally balanced with four clusters between 250 and 300 km s$^{-1}$ with three clusters at the reflected high-velocity range. Interestingly, the four metal-rich clusters between 250 and 300 km s$^{-1}$ have projected radii $> 17$ kpc even though the metal-rich population is more centrally concentrated than the metal-poor [see @pff04II; @whh05 among others]. The metal-poor clusters between 250 and 300 km s$^{-1}$, conversely, are more evenly distributed, with five clusters between projected radii of 5 and 10 kpc, five clusters between 10 and 20 kpc, and five clusters beyond 20 kpc from the center of NGC 5128. Some of these low-velocity, metal-poor objects could be foreground stars with velocities in the realm of GCs in NGC 5128 ($v_r \gtrsim 250$ km s$^{-1}$). However, with only 340 GCs currently confirmed within $\sim45$’ from the center of NGC 5128, out of an estimated $\simeq
1500$ total clusters within 25’ [@harris06], these metal-poor, low-velocity objects could simply be part of a very incomplete GC sample that is also spatially biased. This potential bias is clearly shown in Figure \[fig:gc\_thetar\], which shows the projected radial distribution as a function of azimuthal angle for our GC sample. Beyond 12 kpc, the two ”voids” coincide with the photometric minor axis of the galaxy, attributed at least partly to incomplete cluster surveys in these regions. These objects should, therefore, not be dropped from the GC catalog without further spectroscopic analysis.
Rotation Amplitude, Rotation Axis, and Velocity Dispersion {#sec:kin_gc}
----------------------------------------------------------
### Mathematic and Analytic Description {#sec:math}
We determine the rotation amplitude and axis of the GCS of NGC 5128 from $$\label{eqn:kin}
v_r(\Theta) = v_{sys} + \Omega R sin(\Theta - \Theta_o)$$ [see @cote01; @richtler04; @w06]. In Equation \[eqn:kin\], $v_r$ is the observed radial velocity of the GCs in the system, $v_{sys}$ is the galaxy’s systemic velocity, $R$ is the projected radial distance of each GC from the center of the system assuming a distance of 3.9 Mpc to NGC 5128, and $\Theta$ is the projected azimuthal angle of the GC measured in degrees east of north. The systemic velocity of NGC 5128 is held constant at $v_{sys}=541$ km s$^{-1}$ [@hui95] for all kinematic calculations. The rotation axis of the GCs, $\Theta_o$, and the product $\Omega R$, the rotation amplitude of the GCs in the system, are the values obtained from the numerical solution. We use a Marquardt-Levenberg non-linear fitting routine [@press92].
Eqn. \[eqn:kin\] assumes spherical symmetry. While this may be a decent assumption for the inner 12 kpc region [it has a low ellipticity of $\sim0.2$; @peng04], true ellipticities for the outer regions of the system are not well known because of the sample bias (see Fig. \[fig:gc\_thetar\]). Future studies to remove these biases are vital to obtaining a sound kinematic solution for the entire system. Eqn. \[eqn:kin\] also assumes that $\Omega$ is only a function of the projected radius and that the rotation axis lies in the plane of the sky. It is not entirely clear how these assumptions, discussed thoroughly in [@cote01], apply to the GC and PN systems of NGC 5128. The $\Omega$ we solve for is, therefore, only a lower limit to the true $\Omega$ if the true rotation axis is not in the plane of the sky.
The projected velocity dispersion is also calculated from the normal condition, $$\label{eqn:veldisp}
\sigma_{v}^{2} = \sum_{i=1}^{N}\frac{(v_{f_i} - v_{sys})^2}{N}$$ where $N$ is the number of clusters in the sample, $v_{f_i}$ is the GC’s radial velocity [*after subtraction of the rotational component determined with Eqn. \[eqn:kin\]*]{}, and $\sigma_v$ is the projected velocity dispersion.
The GCs were assigned individual weights in the sums that combine in quadrature the individual observational uncertainty, $\varepsilon_v$, in $v_r$ and the random velocity component, $\varepsilon_{random}$, of the GCS. The dominance of the latter is evident by the large dispersion in the GC velocities in the kinematic fitting (see Figure \[fig:kin\_plot\]). In other words, the clusters have individual weights, $\omega_i = (\varepsilon_v^2 +
\varepsilon_{random}^2)^{-1}$; the main purpose of this is to assign a bit more importance to the clusters with more securely measured velocities. This random velocity term dominates in nearly every case, leaving the GCs with very similar base weights in the kinematic fitting.
The three kinematic parameters - rotation amplitude, rotation axis, and velocity dispersion - are determined with three different binning methods. The first involves binning the GCs in radially projected circular annuli from the center of NGC 5128. The chosen bins keep a minimum of 15 clusters in each, ranging as high as 124 clusters. The bins are 0-5, 5-10, 10-15, 15-25, and 25-50 kpc. Also, we include 0-50 kpc to determine the overall kinematics of the system.
The second method adopts bins with equal numbers of clusters. The entire population of clusters had nine bins of 38 clusters each, the metal-poor clusters had nine bins of 20 clusters, and the metal-rich clusters had eight bins of 20 clusters. The base weighting is applied to the clusters in both the first and second binning methods.
The third method uses an exponential weighting function, outlined in [@bergond06], to generate a smoothed profile. This method determines each kinematic parameter at the radial position, $R$, of every GC in the entire sample by exponentially weighting all other GCs surrounding that position based on their radial separation, $R - R_i$, following
$$\label{eqn:exp_wei}
w_i(R) = \frac{1}{\sigma_R} exp[\frac{-(R - R_i)^2}{2 \sigma_{R}^{2}}].$$
In Equation \[eqn:exp\_wei\], $w_i$ is the determined weight on each GC in the sample, and $\sigma_R$ is the half-width of the window size. For this study, $\sigma_R$ is incrementally varied in a linear fashion for the total sample from $\sigma_R
= 1.0$ kpc at the radius of the innermost GC in the sample out to $\sigma_R = 4.5$ kpc at the radius of the outermost GC, where the population is lowest. The metal-poor population was given a half-width window of $\sigma_R = 1.0-6.5$ kpc, and the metal-rich population was given a half-width window of $\sigma_R = 2-5.3$ kpc, again from the innermost to outermost cluster. The progressive radial increase in $\sigma_R$ ensured that each point $R$ had roughly equal total weights.
### Rotation Amplitude of the Globular Cluster System {#sec:rotamp}
The kinematic parameters were determined for the entire sample of 340 GCs, as well as the subpopulations of 178 metal-poor and 158 metal-rich GCs (four clusters have unknown metallicity). The kinematic results for the entire population of GCs are shown in Table \[tab:all\_GC\], reproduced almost in full from [@w06], while the results for the metal-poor and metal-rich clusters are shown in Tables \[tab:MP\_GC\] & \[tab:MR\_GC\], respectively. The columns give the radial bin, the mean projected radius in the bin, the radius of the outermost cluster, the number of clusters in the bin, the rotation amplitude, the rotation axis, and the velocity dispersion, with associated uncertainties. These are followed by the mass correction, the pressure-supported mass, the rotationally supported mass, and the total mass in units of solar mass (see § \[sec:dyn\] for the mass discussion). The results for the alternate two methods, using an equal number of GCs per bin and the exponentially weighted GCs, are not shown in tabular form but are included in all of the figures.
Figure \[fig:kin\_plot\] shows the sine fit of Eqn. \[eqn:kin\] for the total population and for the metal-poor and metal-rich subpopulations. All three populations show rotation about a similar axis. As discussed in § \[sec:velfield\], the metal-poor population has more members with low velocities ($V_r \leq 300$ km s$^{-1}$) than the metal-rich population, suggesting possible contamination of Milky Way foreground stars in the sample.
Figures \[fig:rotamp\_final\] & \[fig:rotamp\_metal\] show the rotation amplitude results for the entire population and for the metal-poor and metal-rich subpopulations, respectively. The three kinematic methods, described in Section \[sec:math\], appear to agree relatively well for all three populations of clusters. While there appears to be no extreme difference in rotation amplitude between the cluster populations, the metal-poor subpopulation of clusters has lower rotation in the inner 5 kpc of NGC 5128 than the metal-rich subpopulation. The weighted average of the 0-5 kpc radial bin and the innermost equal-numbered bin, shows that the entire population has a rotation amplitude of $\Omega R = 31\pm17$ km s$^{-1}$, while the metal-poor population has $\Omega R = 17\pm26$ km s$^{-1}$ and the metal-rich population has $\Omega R = 57\pm22$ km s$^{-1}$. [@pff04II] show in their study that the metal-poor population has very little rotation in the central regions, completely consistent with our findings. The rotation amplitude does not appear to differ between the two populations outside of 5 kpc.
### Rotation Axis of the Globular Cluster System {#sec:rotaxis}
The results of the rotation axis solutions are shown in Figures \[fig:rotaxis\_final\] & \[fig:rotaxis\_metal\], again for the entire population and for the metal-poor and metal-rich subpopulations. The solution for $\Theta_0$ agrees well for all three kinematic methods and all subgroups. The inner 5 kpc region has a different rotation axis than the outer regions, demonstrated clearly in all three binning methods. The innermost bin yields weighted averages of $\Theta_o = 369\pm24 ^o$, $\Theta_o
= 25\pm55 ^o$, and $\Theta_o = 352\pm18 ^o$, all of which are equal within their uncertainties. Beyond 5 kpc, the rotation axes for all three populations are in even closer agreement, with averages of $\Theta_o = 189\pm6 ^o$, $\Theta_o = 199\pm7 ^o$, and $\Theta_o = 196\pm7 ^o$ for the entire population, the metal-poor subpopulation, and the metal-rich subpopulation, respectively. The position angle of the photometric major axis of NGC 5128 is $\Theta = 35^o$ and $215^o$ east of north and the photometric minor axis is $\Theta = 119^o$ and $299^o$ east of north [@dufour79]. It appears the GCS is rotating about an axis similar to the photometric major axis for the full extent of the galaxy, with a possible axial twist or counterrotation within 5 kpc.
### Velocity Dispersion of the Globular Cluster System {#sec:veldisp}
Figures \[fig:veldisp\_final\] & \[fig:veldisp\_metal\] show the velocity dispersion for the entire population and for the metal-poor and metal-rich subpopulations. Our results for $\sigma_v$ show no significant differences between the metallicity subpopulations. All three show a relatively flat velocity dispersion ($\sigma_v = 119\pm4$, $\sigma_v = 117\pm6$, and $\sigma_v = 111\pm6$ km s$^{-1}$ within 15 kpc of the center of NGC 5128 for the entire population and for the metal-poor and metal-rich subpopulations, respectively). These results match the previous study of NGC 5128 by [@pff04II], whose determined velocity dispersion for the GCs within 20 kpc ranged between 75 and 150 km s$^{-1}$. At a larger radius, we find that $\sigma_v$ then slowly increases to $\sigma_v > 150$ km s$^{-1}$ towards the outer regions of the halo for all populations. The velocity dispersion of the metal-rich GCs, interestingly, appears higher than that of the metal-poor GCs in the outer regions (although still consistent within the determined uncertainties). In most previous studies, the velocity dispersion of the metal-poor GCs usually appears higher than that of the metal-rich GCs, if there is a notable velocity dispersion difference between the subpopulations [see the studies of @cote03; @richtler04 as examples].
To explore the cause of the distinct rise past 15 kpc a bit further, we have plotted the actual velocity histograms in Figure \[fig:vf\_histo\] for the metal-poor and metal-rich subgroups, subdivided further into inner ($R < 15$ kpc) and outer ($R > 15$ kpc) regions. In the inner 15 kpc, both samples show histograms strongly peaked near $v_f = 0$ and with at least roughly Gaussian falloff to both high and low velocities. By contrast, the histograms for the outer regions ($15-50$ kpc) are noticeably flatter, so that the clusters with larger velocity residuals have relatively more importance to the formal value of $\sigma_v$. Nominally, the flatter shape of the velocity distribution would mean that the outer-halo clusters display anisotropy in the direction of a bias towards more circular orbits. However, such a conclusion would be premature at this point for two reasons. First, the sample size in the outer regions is still too small to lead to high significance, and a direct comparison between the inner and outer histograms (through a Kolmogorov-Smirnov test) does not show a statistically significant difference between them larger than the 70% level. Second, the outer samples may still be spatially biased in favor of objects along the major axis of the halo, as discussed above, and this bias sets in strongly for $R > 12$ kpc (see Fig. \[fig:gc\_thetar\]), very near where we have set the radial divisions in this Figure. This type of velocity distribution can also arise from the accretions of satellite galaxies with their own small numbers of GCs [@bekki03]. We will need to have a larger sample of the outer-halo clusters, and one in which these potential sample biases have been removed, before we can draw any firmer conclusions. However, it needs to be explicitly stated that the outermost point in the kinematic plots for the GCs representing $25-50$ kpc suffers from very high spatial biases and low number statistics ($< 40$ GCs) and covers a large radial interval. The rise in velocity dispersion could be driven purely by systematic effects resulting from the radial gradient of the number density of GCs in this outermost bin [@napolitano01].
Kinematics of the Planetary Nebula System of NGC 5128 {#sec:kin_PN}
=====================================================
NGC 5128 has a large number, 780, of identified PNe with measured radial velocity from the studies of [@peng04] and [@hui95]; these PNe are projected out to 90 kpc assuming a distance to NGC 5128 of 3.9 Mpc. Since these are also old objects, it is of obvious interest to compare them with the GCS. The PNe also have the advantage of giving us the best available look at the kinematics of the halo field stars.
The PN kinematic results are listed in Table \[tab:PN\], with the same columns as Table \[tab:all\_GC\]. The results are shown in Figures \[fig:rotamp\_final\], \[fig:rotaxis\_final\], & \[fig:veldisp\_final\] for the rotation amplitude, rotation axis, and velocity dispersion, respectively. The spatial distribution of the known PNe is, like the GCS, biased toward the major axis at large radii (see Fig. \[fig:position\]). Nevertheless, their kinematics closely resemble the GCs.
The kinematics of the PN system are very consistent among all three binning methods. The rotation amplitude and rotation axis show little radial trend, while the velocity dispersion appears relatively flat within the first 15 kpc at $\sigma_v = 122\pm7$ km s$^{-1}$ and then slowly [*decreases*]{} to $\sigma_v \simeq 85$ km s$^{-1}$ at large galactocentric radius. [@pff04I] show that the velocity dispersion of the PNe drops from a central value of 140 to 75 km s$^{-1}$ in the outer regions of the galaxy, consistent with the findings of this study. Their velocity field analysis led to the discovery of a ”zero-velocity curve” located between the photometric minor axis, $119\pm5 ^o$ east of north [@dufour79], and the north-south direction, for the innermost region of the galaxy. Just beyond 5 kpc, the zero-velocity curve turns and follows a straight line at a $7^o$ angle from the photometric major axis, $35^o$ east of north [@dufour79] [see Figure 7 of @pff04II].
This study does not show a strong change in rotation axis for the PNe in the innermost regions of the galaxy. However, it clearly shows in all three GC populations a significant change in the rotation axis just beyond 5 kpc from the center of the galaxy. A change in axis of 5 kpc outward ($\sim180 ^o$ for the entire population and metal-poor subpopulation and $\sim160 ^o$ for the metal-rich subpopulation) has been found, as discussed in § \[sec:rotaxis\]. Similarly, [@pff04II] show from their sample of 215 clusters that a clear sign of rotation beyond 5 kpc about a misaligned axis appears particularly in their metal-rich subpopulation. The kinematics of the GCs in this study matches the line of zero velocity relatively nicely. Within 5 kpc the rotation axis of the GCS is nearly-parallel to the north-south direction, and beyond 5 kpc the rotation axis is near $200 ^o$ east of north, which is only $\sim10^o$ from the zero-velocity curve.
However, the velocity field of NGC 5128 is complex and not entirely captured by these approximate solutions. The two-dimensional velocity field shown in [@peng04] (see their Figure 7), shows that the photometric major axis (which happens to be very close to our maximum rotation as discussed above) is only $7^o$ from the line of zero velocity. This could lead to a very asymmetric velocity profile that may not be well fit by the sine curve described in Equation \[eqn:kin\]. Biased kinematics, especially the rotation axis, may develop from the sine fit that could lead to a higher estimated velocity dispersion.
[@hui95] similarly studied the kinematics of the PN system in NGC 5128 with a sample of 433 PNe. They obtain a rotation axis of $344\pm10^o$. Our result of $170\pm5^o$ east of north is consistent with their findings on comparing their sine curve fit of their PN data in their Figure 11 to our corresponding fit shown in Fig. \[fig:kin\_plot\] for the GCS, which shares a similar axis to our PN sample (note that in their study, $\phi = 0 ^o$ corresponds to our $\Theta = 305^o$ east of north). Our fits both correspond to a positive rotation amplitude for a rotation axis near $170^o$ east of north and a negative rotation axis near $350^o$ east of north. Therefore, the rotation axis quoted in [@hui95] of $344\pm10^o$ corresponds to a [*negative*]{} rotation amplitude of approximately $70-75$ km s$^{-1}$ (taken from their Figure 11), nicely matching our result of a positive $76\pm6$ km s$^{-1}$ about an axis of $170\pm5^o$ east of north.
Dynamics of NGC 5128 {#sec:dyn}
====================
Both GCs [@cote01; @larsen02; @evans03; @cote03; @beasley04; @pff04II among others] and PNe [@ciardullo93; @hui95; @arnaboldi98; @peng04 among others] can be used to estimate the total dynamical mass of their host galaxies. A variety of tools are in use including derived mass models, the virial mass estimator [@bahcall81], the projected mass estimator [@heisler85], and the tracer mass estimator [@evans03].
NGC 5128 does not have a large X-ray halo [detected by @kraft03; @osullivan01 the latter reporting a measurement of log $L_x = 40.10$ erg s$^{-1}$], such as is evident in other giant ellipticals such as M87 [@cote01] or NGC 4649 [@bridges06]. Thus it is difficult to model the dark matter profile of NGC 5128 with [*a priori*]{} constraints. Without such a mass model, we turn to the tracer mass estimator for the dynamical mass determination. The tracer mass estimator has the distinct advantage over the virial and projected mass estimators that the tracer population does not have to follow the dark matter density in the galaxy - an extremely useful feature for stellar subsystems such as GCs and PNe that might, in principle, have significantly different radial distributions (see [@evans03] for extensive discussion). Below, we determine the mass of NGC 5128 using the tracer populations of GCs and PNe (our mass estimates do not include stellar kinematics in the inner regions).
Mass Determination {#sec:mass}
------------------
The mass contributed by the random internal motion of the galaxy (pressure-supported mass) is determined from the tracer mass estimator as $$\label{eqn:tme}
M_{p} = \frac{C}{GN} \sum_{i}(v_{f_i} - v_{sys})^2R_i$$ where $N$ is the number of objects in the sample and $v_{f_i}$ is the radial velocity of the tracer object [*with the rotation component removed*]{}. For an isotropic population of tracer objects, assumed in this study, the value of $C$ is $$\label{eqn:C}
C = \frac{4(\alpha + \gamma)(4 - \alpha -\gamma)(1-(\frac{r_{in}}{r_{out}})^{(3-\gamma)})}{\pi(3-\gamma)(1-(\frac{r_{in}}{r_{out}})^{(4-\alpha-\gamma)})}$$ where $r_{in}$ and $r_{out}$ are the three-dimensional radii corresponding to the two-dimensional projected radii of the innermost, $R_{in}$, and outermost, $R_{out}$, tracers in the sample. The parameter $\alpha$ is set to zero for an isothermal halo potential in which the system has a flat rotation curve at large distances. Finally, $\gamma$ is the slope of the volume density distribution, which goes as $r^{-\gamma}$, and is found by determining the surface density slope of the sample and deprojecting the slope to three-dimensions. The tracer mass estimator uses a sample of tracer objects defined between $r_{in}$ and $r_{out}$, yet it is important to emphasize that it determines the [*total*]{} enclosed mass within $r_{out}$.
There is also a contribution to the total mass by the rotational component, as determined in § \[sec:rotamp\] for the GCs and § \[sec:kin\_PN\] for the PNe. This mass component is determined from the rotational component of the Jeans equation, $$\label{eqn:rje}
M_{r} = \frac{R_{out}v^{2}_{max}}{G}$$ where $R_{out}$ is the outermost tracer projected radius in the sample and $v_{max}$ is the rotation amplitude. Therefore, the total mass of NGC 5128, $M_t$, is determined by the addition of the mass components supported by rotation, $M_r$, and random internal motion, $M_p$, $$\label{eqn:total_mass}
M_{t} = M_{p} + M_{r}.$$
In the determination of the pressure-supported mass, one must estimate values for $r_{in}$ and $r_{out}$ knowing $R_{in}$ and $R_{out}$. [@evans03] suggest that $r_{in}\simeq
R_{in}$ and $r_{out}\simeq R_{out}$ for distributions taken over a wide angle. However, in this study the inner and outer radii of the chosen bins are at intermediate radial values within the distribution. Their assumption would therefore lead to an underestimate of the determined mass, since the true $r_{out}$ can be quite a bit larger than the projected $R_{out}$. To correct for this contributed uncertainty, distributions of sample tracer populations were generated through Monte Carlo simulations. In the simulations, 340 GCs were randomly placed in a spherically symmetric system extending out to 50 kpc with an $r^{-2}$ projected density, while 780 PNe were placed in the same environment extending out to 90 kpc. From the generated distributions, the value of $C$ in Eqn. \[eqn:C\] was determined for both the real and projected positions of the tracer populations in each designated radial bin. This correction factor, listed in Tables \[tab:MP\_GC\]-\[tab:PN\] as $M_{corr}$, multiplies the pressure-supported mass from Eqn. \[eqn:tme\]. The same correction was applied to the full GC sample and the corresponding $M_{corr}$ values are listed in Table 1 of [@w06]. These values are generally small, but in the worst case they triple $M_p$.
Surface Density Profiles {#sec:surden}
------------------------
In Eqn. \[eqn:C\], the value of $\gamma$ is determined for the tracer populations by deprojecting the slope of the surface density profile to three-dimensions. Figure \[fig:gc\_surden\] shows the surface density profiles for the entire, metal-poor, and metal-rich GC populations, along with the PN profile. The populations were binned, following [@maiz05], into circular annuli of equal numbers of objects, providing the same statistical weight to each bin (although spatial biases may still affect the GC population in the outer regions along the major axis; see Fig. \[fig:gc\_thetar\]). In the inner 5 kpc of all tracer populations, incompleteness due to the obscuration of the dust lane is evident by the flattening of the surface density profile. The innermost objects were, therefore, excluded from the surface density profile fittings. Outside of 5 kpc, the surface densities fit well to power laws, leading to $\gamma = 3.65\pm0.17$, $3.49\pm0.34$, $3.37\pm0.30$, and $3.47\pm0.12$ for the entire GC population, the metal-poor and metal-rich subpopulations of GCs, and the PNe in NGC 5128, respectively. These are all very similar within their uncertainties.
Mass Results {#sec:mass_results}
------------
The similar kinematics we find between the metal-poor and metal-rich subpopulations of GCs in this study strongly justifies the combining of the two populations for the mass determination performed in [@w06]. The GC population provides a [*total*]{} mass estimate of $(1.3\pm0.5) \times 10^{12}$ $M_{\odot}$ from 340 clusters out to a projected radius of 50 kpc. Removing the GCs in our sample with $v_r
\leq 300$ km s$^{-1}$, which will remove all possible contamination from foreground stars, discussed in § \[sec:velfield\], leads to a total mass of $(1.0\pm0.4) \times 10^{12}$ $M_{\odot}$. This mass agrees nicely with our mass determined from our entire GC sample. The PN population provides a total mass of $(1.0\pm0.2) \times 10^{12}$ $M_{\odot}$ from 780 PNe out to 90 kpc in projected radius, agreeing with the GC value within the uncertainty.
We are also able to generate a mass profile of NGC 5128 from the total GC population and the PNe, shown in Figure \[fig:mass\]. The tracer mass estimator determines the [*total*]{} enclosed mass for NGC 5128 within the outermost radius of a given tracer sample. It calculates this total mass using a sample of objects defined within the radial range defined by the sample’s inner and outermost radii. It is therefore possible to use a unique set of tracer objects, denoted by the radial bin range, listed in the first column of Tables \[tab:all\_GC\]-\[tab:PN\], to determine a mass profile from independent mass estimates. The independent binning, leads to sample sizes in the mass determination, in some cases generating higher uncertainties in the total enclosed mass. The most certain mass is the one determined from the full sample of tracers.
In the mass determinations above, we have implicity assumed isotropy for the velocity distributions. But the possibility exists that the PNe (for example) might have radial anisotropy which would produce their gradually falling $\sigma_v(R)$ curve. Replacing Equation \[eqn:C\] in the tracer mass estimator by $$\label{eqn:C_aniso}
C = \frac{16(\alpha + \gamma - 2\beta)(4 - \alpha
-\gamma)(1-(\frac{r_{in}}{r_{out}})^{(3-\gamma)})}{\pi(4 - 3\beta)(3-\gamma)(1-(\frac{r_{in}}{r_{out}})^{(4-\alpha-\gamma)})}$$ which includes the anisotropy parameter, $\beta$, from [@evans03], we find that the mass estimate from the PNe can be forced to agree with the mass estimate from the GCS for a nominal $\beta = 0.8$. For perfect isotropy, $\beta = 0$. This would mean roughly 2:1 radial anisotropy for the PNe in the outer halo. However, we find that any $\beta$ in the wide range of $-10 \leq \beta \leq 1$ would still keep the two methods in agreement within their internal uncertainties, so we are not yet in a position to tightly constrain any anisotropy. It is possible that the GCs may also have anisotropy; it may therefore be too simplistic to find a range of $\beta$ for the PNe for which the masses of the PNe and GCs agree. However, the GCs are likelier to be nearly isotropic than the PNe; the GCs are older, ”hotter” subsystems of the halo. In other studies, the isotropy of the GCS orbits has also been shown to be a good assumption from mass profiles of elliptical galaxies using X-ray observations [@cote01; @cote03; @bridges06 among others].
Both mass estimates can be compared to previous studies. First, we note the total mass determined from the PN data with that of [@peng04]. While the rotationally supported mass was determined here with different values of the mean rotational velocity, they calculated the pressure supported mass using the identical tracer mass estimator technique with exactly the same PN population. The total mass estimate given by [@peng04] is $(5.3\pm0.5) \times 10^{11}$ $M_{\odot}$. Subtracting their rotationally supported mass leaves a pressure supported mass of $\sim 3.4 \times
10^{11}$ $M_{\odot}$, quite different from our $(8.46\pm1.72) \times 10^{11}$ $M_{\odot}$. Recalculating our pressure supported mass estimate with $\gamma = 2.54$, which was used in [@peng04], we are able to reproduce their mass estimate within the uncertainty. The values of $\gamma$ differ between the two studies simply because the $\gamma$ used in [@peng04] was the inverse of the surface density slope instead of the inverse of the volume density slope. Using the correct value of $\gamma = 3.54$, their pressure supported mass estimate would increase to $8.7 \times
10^{11} M_{\odot}$, matching the mass found in this study.
Second, we compare our total mass determined using the GC population with that from [@pff04II]. Using 215 GCs out to 40 kpc, they found a pressuresupported mass of $(3.4\pm0.8) \times 10^{11}$ $M_{\odot}$, again much different from our pressure supported mass of $(1.26\pm0.47) \times 10^{12}$ $M_{\odot}$ using 340 clusters out to 50 kpc. The large difference can again be attributed to their using $\gamma = 2.72$ instead of deprojecting their surface density slope to $\gamma = 3.72$. Using the correct value of $\gamma$, we find a pressure-supported mass of $7.5 \times 10^{11}$ $M_{\odot}$ using the same 215 clusters they used in their study. This corrected estimate is closer to the pressure supported mass determined in our study, but it is not necessarily expected to agree with our result, as our sample contains $130$ more GCs and uses a slightly different $\gamma$ that we have independently redetermined.
Third, the mass determined by the H $_I$ shell study of NGC 5128 by [@schiminovich94] found a mass of $2 \times 10^{11}$ $M_{\odot}$ [*within 15 kpc*]{} assuming a distance of 3.5 Mpc. With the distance of 3.9 Mpc used in this study, the mass determined in their study would increase to $2.2 \times 10^{11}$ $M_{\odot}$, which is $30\%$ smaller than our total mass of $3.89\pm0.94 \times
10^{11}$ $M_{\odot}$ within 15 kpc.
Lastly, we compare our determined mass to a recent study by [@samurovic06]. [@samurovic06] determined a total mass of NGC 5128 using GCs, PNe, and an X-ray data technique. The galaxy mass determined from the GC and PN data was obtained using the tracer mass estimator and the spherical Jeans equation, as performed in our study. However, [@samurovic06] used the volume density slopes determined by [@peng04] and [@pff04II], for the PN and GC data, respectively. They obtained mass estimates for NGC 5128 similar to those of [@peng04] and [@pff04II], discussed above, using an identical PN sample and slightly increased GC sample. They also included an X-ray-modelling mass estimate for NGC 5128 from which they obtained masses of $(7.0\pm0.8) \times 10^{11}$ $M_{\odot}$ out to 50’ and $(11.6\pm1.0) \times 10^{11}$ $M_{\odot}$ out to 80’. This mass estimate is similar to our PNe estimate out to the same radial extent, but the author cautions that it is an overestimate of the true mass of NGC 5128 resulting from a lack of hydrostatic equilibrium in the outer region of the galaxy.
We note here that the mass estimates obtained are higher than those from [@hui95], and [@peng04] derived from the PNe using a two-component mass model, as well as [@samurovic06], using an X-ray modelling technique. This discrepancy is not fully understood and possibly lies in the assumptions that go into the mass estimators with a spatially biased sample. We intend to pursue this issue further with an upcoming larger sample of GCs with less spatial biases.
The mass of NGC 5128 that we find appears to be in the range of other giant elliptical galaxies, such as NGC 1399 [$\sim 2 \times 10^{12}$ $M_{\odot}$ out to 50 kpc; @richtler04], M49 [$\sim 2 \times 10^{12}$ $M_{\odot}$ out to their kinematically studied radius of 35 kpc; @cote03], and M87 [$\sim 9 \times 10^{11}$ $M_{\odot}$ at 20 kpc, the onset of the projected cD envelope; @cote01]. Clearly, it is legitimate to say that NGC 5128 is the largest, most massive galaxy in the neighborhood of the Local Group, and one that can be talked about in the same category as these other giants that reside in larger clusters.
The sample biases mentioned above in our currently available set of both GCs and PNe place limitations on how much we can reasonably interpret the kinematic and dynamic data. We are currently carrying out a set of new spectroscopic programs to increase the tracer sample size and to remove the sample biases, leading to a more complete analysis of the halo velocity field.
Discussion and Conclusions {#sec:concl}
==========================
Angular momentum is an essential quantity for characterizing the sizes, shapes, and formation of galaxies and is often represented as the dimensionless spin parameter,
$$\label{eqn:spin}
\Lambda = \frac{J |E|^{1/2}}{G M^{5/2}}$$
where $J$ is the angular momentum, $E$ is the binding energy, and $M$ is the mass of the galaxy. The spin parameter is representative of a galaxy’s angular momentum compared to the amount of angular momentum needed for pure rotational support: the lower the $\Lambda$-value, the less rotation and rotational support within the galaxy. For an elliptical galaxy in gravitational equilibrium, the spin parameter simplifies to $\Lambda
\sim 0.3 <(\Omega R / \sigma_v)>$ [@fall79], yielding $\Lambda =
0.10$ with $(\Omega R / \sigma_v) = 0.33$ for the entire population of GCs in NGC 5128.
Table \[tab:spin\] shows the spin parameter for four giant galaxies with large GCS kinematic studies, M87, M49, NGC 1399, and NGC 5128. The table columns give the galaxy name, the rotation amplitude, the projected velocity dispersion, and the ratio of the rotation amplitude to the velocity dispersion, followed by the spin parameter. These quantities are shown for the metal-poor and metal-rich populations. What is clearly evident in Table \[tab:spin\] is the strong galaxy to galaxy differences between these four galaxies, already hinted at in § \[sec:intro\]. Though the sample is still quite small, no obvious pattern emerges. There is an indication of metal-poor and metal-rich GCSs having similar spin parameters within the same galaxy. M49 is the only galaxy studied here where this may not be the case. Although the metal-poor and metal-rich cluster spin parameters are consistent within the uncertainties, the metal-rich cluster spin parameter of M49 is also consistent with zero.
In the monolithic collapse scenario, [@peebles69] describes the angular momentum within the galaxy as attributed to the tidal torque transferred from neighbouring proto-galaxies during formation. In this scenario, [@efstathiou79] found that a spin parameter of $\Lambda = 0.06$ for elliptical systems is expected from simulations of the collapse of an isolated protogalactic cloud. But NGC 5128, among many other giant elliptical galaxies, is not in isolation, and therefore not necessarily expected to reproduce such a low spin parameter. Also, the internal rotation axis changes at 5 kpc are not easily explained with only the monolithic collapse scenario. In the monolithic collapse model, the inner regions would be expected to have more pronounced rotations. Yet all four of the galaxies with major kinematic studies presented here do not show a higher rotational signal in the inner regions. In fact, for NGC 1399, the outer region ($R > 6'$) indicates rotation in the metal-poor population that is not evident in the inner regions. Also, a slightly lower rotational signal is present in the inner regions of NGC 5128 for the metal-poor population than in the outer regions.
Hierarchical clustering of cold dark matter also relies on angular momentum in a galaxy being produced by gravitational tidal torques during the growth of initial perturbations. [@sugerman00] have demonstrated that the tidal torque theory predicts an increase in angular momentum during the collapse, and with time, the increase in angular momentum slows. Accretion of satellites and/or merger events is therefore a possible culprit for moving the angular momentum outward, as major mergers of disks and bulges suggest that angular momentum resides largely in the outer regions of the galaxy [@barnes92; @hernquist93].
Alternatively, [@vitvitska02] examine the change in spin parameter in a scenario where the angular momentum in a galaxy is built up by mass accretion. Their results show that the spin parameter changes sharply in major merger events in the galaxy and steadily decreases with small satellite accretion events. They also show that the spin parameter for a galaxy with a major merger after a redshift of $z = 3$ should be notably larger than a galaxy that did not undergo such a major merger. Their study obtains an average of $\Lambda = 0.045$ from $\Lambda$CDM $N$-body simulations for galaxies with halo masses of $(1.1-1.5) \times 10^{12} h^{-1}$ $M_{\odot}$ with $h = 0.7$.
NGC 1399 and M49, with their weak rotation signals, are consistent with the model predictions discussed above, whereby their major formation events could have occurred at early times and with perhaps only minor accretions happening since then. However, NGC 5128 and M87 have spin parameters 2-3 times larger than predicted by the model averages. For NGC 5128, this relatively large rotation (which is nearly independent of both metallicity and radius) may, perhaps, be connected with its history within the Centaurus group environment. The rotation speed and rotation axis for its extended group of satellite galaxies are nearly identical to the NGC 5128 halo [@w06], much as if the accretion events experienced by the central giant have been taking place preferentially along the main axis of the entire group and in the same orientation. The GCS age distribution discussed by [@beasley06] and the mean age for the halo field stars [@rejkuba05] strongly suggest that a high fraction of the stellar population in NGC 5128 formed long ago, with particularly large bursts between 8 and 12 Gyr. Even if the galaxy underwent a significant merger perhaps a few Gyr ago (the traces of which now appear in the halo arcs and shells), the stars in it may already have been old at the time of the merger. Although a very few younger GCs have formed since then, these make up a small minority of what is present, at least for the $R > 5$ kpc halo outside the bulge region that now contains gas and dust.
The situation for M87, with its even larger rotation signature, may require a different sort of individual history. Of the four galaxies compared here, it is at the dynamical center of the richest environment (Virgo), has the most extensive cD-type envelope, and sits within the most massive, extended, and dynamically evolved potential well. A single relatively recent major merger could in principle have caused its present high rotation, but the lack of distinctive tidal features does not necessarily favor such an interpretation and would at least suggest that such a merger should have been with another large elliptical and nondissipative galaxy. [@cote01] discuss an interpretation - at least partially resembling what we suggest for NGC 5128 - that stellar material ”is gradually infalling onto M87 along the so-called principal axis of the Virgo Cluster.”
In conclusion, we have presented a kinematic study of NGC 5128 that makes it now comparable to recent studies of the other giant galaxies, M87, M49, and NGC 1399. Using $340$ GCs ($158$ metal-rich and $178$ metal-poor GCs), we have calculated the rotation amplitude, rotation axis, and velocity dispersion and have searched for radial and metallicity dependences. Our findings show that both metallicity populations rotate with little dependence on projected radius, with $\Omega R = 40\pm10$, 31$\pm$14, and 47$\pm$15 km s$^{-1}$ for the total, metal-poor, and metal-rich populations, respectively. Perhaps the inner 5 kpc shows a slower rotation of the metal-poor population, but more clusters would be needed to confirm this finding. The rotation axis is 189$\pm$12$^o$, 177$\pm$22$^o$, and 202$\pm$15$^o$ east of north for the total, metal-poor, and metal-rich populations out to a 50 kpc projected radius, assuming the velocity field is best fit by a sine curve. The rotation axis does change at 5 kpc, following the zero-velocity curve proposed by [@peng04] or possibly full-on counterrotation. A study with more GCs and lower uncertainties is needed to see what is happening in the innermost 5 kpc of NGC 5128. The velocity dispersion shows a modest increase with galactocentric radius, although the outer regions (especially the metal-rich population) have less reliable statistics; this increase could be driven purely by statistical effects. We find the velocity dispersion we find 123$\pm$5, 117$\pm$7, and 129$\pm$9 km s$^{-1}$ for the total, metal-poor, and metal-rich populations, respectively.
The PN data are also used to determine the kinematics of the halo of NGC 5128. These show results that are encouragingly similar to those of the GC data, except that no rotation axis change is noted with radius, and a [*decrease*]{} in velocity dispersion is found with radius, possibly indicating a difference in orbital anisotropy compared with the GCs. A very similar effect has been noted for the Leo elliptical NGC 3379, although with a much smaller data sample [@romanowsky03; @bergond06; @pierce06]. We also determine the total dynamical mass using both the GCs and the PNe by separately calculating the pressure supported mass with the tracer mass estimator and the rotationally supported mass using the spherical component of the Jeans equation. The total mass is $(1.3\pm0.5) \times 10^{12}$ $M_{\odot}$ from the GC population out to a projected radius of 50 kpc, or $(1.0\pm0.2) \times 10^{12}$ $M_{\odot}$ out to 90 kpc from the PNe.
Overall, we have enough evidence to cautiously conclude that a major episode of star formation occurred about $8-10$ Gyr ago (corresponding to a redshift z = 1.2 - 1.8) and this may have been when the bulk of the visible galaxy was built. We still do not know just why the most metal-poor clusters show up in such relatively large numbers and appear to have ages of $10-12$ Gyr, but this is a common issue in all big galaxies.
This kinematic study and the age study of [@beasley06] on the NGC 5128 cluster system indicate that additional spectroscopic studies to build up both the radial velocity database and age distribution can lead to rich dividends. Large GC samples are clearly needed to remove the current sample biases and to fully understand the complex kinematics and history of this giant elliptical galaxy. It seems clear as well that each galaxy needs to be individually studied to fully understand the different galaxy formation histories. We are continuing these studies particularly for NGC 5128, with the eventual aim of at least doubling the total GC sample size in this unique system.
Acknowledgements: WEH and GLHH acknowledge financial support from NSERC through operating research grants. DAF thanks the ARC for financial support.
Arnaboldi, M., Freeman, K.C., Capaccioli, M., & Ford, H. 1994, ESO Messanger, 76, 40 Arnaboldi, M., Freeman, K.C., Gerhard, O., Matthias, M., Kudritzki, R.P. Méndez, R.H., Capaccioli, M., & Ford, H. 1998, , 507, 759 Baade, W., & Minkowski, R. 1954, , 119, 215 Bahcall, J.N., & Tremaine, S. 1981, , 244, 805 Barnes, J.E. 1992, , 393, 484 Beasley, M.A., Forbes, D.A., Brodie, J.P., & Kissler-Patig, M. 2004, , 347, 1150 Beasley, M.A., Bridges, T., Peng, E., Harris, W.E., Harris, G.L.H., Forbes, D.A., & Mackie, G. 2006, in preparation Bekki, K., Harris, W.E., & Harris, G.L.H. 2003, , 338, 587 Bekki, K., Forbes, D.A., Beasley, M.A., & Couch, W.J. 2003, , 334, 1334 Bekki, K., & Peng, E.W. 2006, , 370, 1737 Bergond, G., Zepf, S.E., Romanowsky, A.J., Sharples, R.M., & Rhode, K.L. 2006, , 448, 155 Bridges, T., Gebhardt, K., Sharples, R., Faifer, F.R., Forte, J.C., Beasley, M.A., Zepf, Z.E., Forbes, D.A., Pierce, M. 2006, , 373, 157 Ciardullo, R., Jacoby, G.H., & Dejonghe, H.B. 1993, , 414, 454 Côté, P., McLaughlin, D.E., Hanes, D.A., Bridges, T.J., Geisler, D., Merritt, D., Hesser, J.E., Harris, G.L.H., & Lee, M.G. 2001, , 559, 828 Côté, P., McLaughlin, D., Cohen, J.G., & Blakesless, J.P. 2003, , 591, 850 Dekel, A., Stoehr, F., Mamon, G.A., Cox, T.J., Novak, G.S., & Primak, J.R., 2005, Nature, 437, 707 Dufour, R.J., Harvel, C.A., Martins, D.M., Schiffer, F.H., Talent, D.L., Wells, D.C., van den Bergh, S., & Talbot, R.J. 1979, , 84, 284 Efstathiou, G., & Jones, B.J.T. 1979, , 186, 133 Evans, N.W., Wilkinson, M.I., Perrett, K.M., & Bridges, T.J. 2003, , 583, 752 Fall, S.M. 1979, Rev. Mod. Phys., 51, 21 Ferrarese, L., Mould, J.R., Stetson, P.B., Tonry, J.L., Blakeslee, J.P., & Ajhar, E.A. 2006, ApJ, in press (astro-ph/0605707) Harris, G.L.H., Geisler, D., Harris, H.C., & Hesser, J.E. 1992, , 104, 613 Harris, G.L.H., Harris, W.E., & Poole, G. 1999, AJ, 117, 855 Harris, W. E., & Harris, G. L. H. 2002, , 123, 3108 Harris, W.E., Harris, G.L.H., Holland, S.T., & McLaughlin, D.E. 2002, , 124, 1435 Harris, G.L.H., Harris, W.E., & Geisler, D. 2004, , 128, 723 Harris, W.E., Harris, G.L.H., Barmby, P., McLaughlin, D.E., & Forbes, D. 2006, , 132, 2187 Heisler, J., Tremaine, S., & Bahcall, J.N. 1985, , 298, 8 Hernquist, I. 1993, , 409, 548 Hesser, J.E., Harris, H.C., van den Bergh, S., & Harris, G.L.H. 1984, , 276,491 Hesser, J.E., Harris, H.C., & Harris, G.L.H. 1986, , 303, L51 Hui, X., Ford, H.C., Freeman, K.C., & Dopita, M.A. 1995, , 449, 592 Kraft, R.P., Váquez, S.E., Forman, W.R., Jones, C., Murray, S.S., Hardcastle, M.J., Worrall, D.M., Churazov, E. 2003, , 592, 129 Larsen, S.S., Brodie, J.P., Beasley, M.A., & Forbes, D.A. 2002, , 124, 828 Lee, H., & Worthey, G. 2005, , 160, 176 Maíz Apellániz, J. & Úbeda, L. 2005, , 629, 7873 Mamon, G.A., & [Ł]{}okas, E.L. 2005, , 362, 95 Napolitano, N.R., Arnaboldi, M., Freeman, K.C., & Capaccioli, M. 2001, , 377, 784 Napolitano, N.R., Arnaboldi, M., & Capaccioli, M. 2002, , 383, 791 O’Sullivan, E., Forbes, D.A., & Ponman, T.J. 2001,, 328, 461 Peebles, P.J.E. 1969, , 155, 393 Peng, E.W., Ford, H.C., Freeman, K.C., & White, R.L. 2002, , 124, 3144 Peng, E.W., Ford, H.C., & Freeman, K.C. 2004, , 150, 367 Peng, E.W., Ford, H.C., & Freeman, K. 2004, , 602, 685 Peng, E.W., Ford, H.C., & Freeman, K. 2004, , 602, 705 Pierce, M., Beasley, M.A., Forbes, D.A., Bridges, T., Gebhardt, K., Faifer, F.R., Forte, J.C., Zepf, S.E., Sharples, R., Hanes, D.A., & Proctor, R. 2006, , 366, 1253 Press, W.H., Teukolsky, S.A., Vetterling, W.T., & Flannery, B.P. 1992, Numerical Recipes in Fortran (2nd Ed., Cambridge: Camdridge University Press) Puzia, T.H., Kissler-Patig, M., Thomas, D., Maraston, C., Saglia, R.P., Bender, R., Goudfrooij, P., & Hempel, M. 2005, , 439, 997 Reed, L.G., Harris, G.L.H., & Harris, W.E. 1994, , 107, 555 Rejkuba, M. 2001, , 269, 812 Rejkuba, M. 2004, , 413, 903 Rejkuba, M., Greggio, L., Harris, W.E., Harris, G.L.H., & Peng, E.W. 2005, , 631, 262 Rejkuba, M., Dubath, P., Minniti, D., & Meylan, G. 2007, , in press (astro-ph/0703385) Richtler, T., Dirsch, B., Gebhardt, K., Geisler, D., Hilker, M., Alonso, M. V., Forte, J. C,. and Grebel, E. K., Infante, L., Larsen, S., Minniti, D., & Rejkuba, M. 2004, , 127, 2094 Romanowsky, A.J., Douglas, N.G., Arnaboldi, M., Kuijken, K., Merrifield, M.R., Napolitano, N.R., Capaccioli, M., & Freeman, K.C., 2003, Science, 301, 1696 Saglia, R.P., Kronawitter, A., Gerhard, O., & Bender, R. 2000, , 119, 153 Samurović, S. 2006, Serb. Astron. J., 173, 35 Schiminovich, D., van Gorkom, J.H., van der Hulst, J.M., & Kasow, S. 1994, , 423, L101 Sharples, R. 1988, IAU Symp. 126: The Harlow-Sharpley Symposium on Globular Cluster Systems in Galaxies, 126, 245 Strader, J., Brodie, J.P., Cenarro, A.J., Beasley, M.A., & Forbes, D.A. 2005, , 130, 1315 Sugerman, B., Summers, F.J., & Kamionkowski, M. 2000, , 311, 762 Thomas, D., Maraston, C., & Korn, A. 2004, , 351, 19 Toomre, A., & Toomre, J. 1972, , 178, 623 van den Bergh, S., Hesser, J.E., & Harris, G.L.H. 1981, , 86, 24 Vitvitska, M., Klypin, A.A., Kravtsov, A.V., Wechsler, R.H., Primack, J.R., & Bullock, J.S. 2002, , 581, 799 Woodley, K.A., Harris, W.E., & Harris, G.L.H. 2005, , 129, 2654 Woodley, K.A. 2006, , 132, 2424
[llllrcccccccccccccccccccccccl]{}
GC0001 & pff\_gc-028 & 13 20 01.16 &-42 56 51.5& 6.46 & 20.11& 19.68& 18.75& 18.20& 17.60& 0.07& 0.02& 0.01& 0.01& 0.03& 19.92& 19.04& 18.22& 0.01& 0.01& 0.01& 0.43& 0.93& 0.55& 1.15& 0.82& 0.88& 1.70& 524$\pm$16\
GC0002 & HH-048 & 13 22 45.36 &-43 07 08.8& 30.26 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & 856$\pm$50\
GC0003 & HH-080 & 13 23 38.33 &-42 46 22.8& 24.84 & - & - & - & - & - & - & - & - & - & - & 19.45& 18.72& 17.98& 0.02& 0.01& 0.01& - & - & - & - & 0.74& 0.72& 1.46& 497$\pm$33\
GC0004 & HGHH-40/HH-044/C40 & 13 23 42.37 &-43 09 37.8& 21.02 & 19.37& 19.01& 18.12& 17.59& 17.03& 0.04& 0.01& 0.01& 0.01& 0.03& 20.12& 19.28& 18.49& 0.02& 0.01& 0.01& 0.35& 0.89& 0.53& 1.10& 0.79& 0.84& 1.63& 365$\pm$24\
GC0005 & HGHH-01/C1 & 13 23 44.19 &-43 11 11.8& 21.41 & - & - & - & - & - & - & - & - & - & - & 18.73& 18.00& 17.25& 0.06& 0.03& 0.03& - & - & - & - & 0.76& 0.72& 1.48& 642$\pm$1\
GC0006 & pff\_gc-001 & 13 23 49.62 &-43 14 32.0& 22.36 & 20.39& 19.91& 18.91& 18.36& 17.60& 0.09& 0.02& 0.01& 0.01& 0.03& 20.22& 19.25& 18.38& 0.02& 0.01& 0.01& 0.47& 1.01& 0.55& 1.31& 0.87& 0.97& 1.84& 711$\pm$35\
GC0007 & AAT301956 & 13 23 54.52 &-43 20 01.1& 25.41 & 21.22& 21.13& 20.42& 19.98& 19.43& 0.18& 0.06& 0.02& 0.02& 0.04& - & - & - & - & - & - & 0.09& 0.71& 0.44& 0.98& - & - & - & 287$\pm$162\
GC0008 & HH-099 & 13 23 56.70 &-42 59 59.8& 16.66 & - & - & - & - & - & - & - & - & - & - & 21.72& 20.90& 20.01& 0.09& 0.09& 0.06& - & - & - & - & 0.89& 0.82& 1.71& 798$\pm$49\
GC0009 & AAT101931 & 13 23 58.58 &-42 57 17.0& 16.73 & 20.58& 20.57& 19.88& 19.43& 18.96& 0.06& 0.02& 0.01& 0.01& 0.03& 20.60& 20.09& 19.45& 0.01& 0.01& - & 0.02& 0.68& 0.45& 0.92& 0.64& 0.51& 1.15& 590$\pm$144\
GC0010 & AAT101906 & 13 23 58.76 &-43 01 35.2& 16.25 & 20.45& 19.90& 18.89& 18.28& 17.57& 0.06& 0.02& 0.01& 0.01& 0.03& 20.23& 19.11& 18.28& 0.02& 0.03& 0.02& 0.54& 1.01& 0.61& 1.32& 0.83& 1.12& 1.95& 511$\pm$31\
GC0011 & pff\_gc-002 & 13 23 59.51 &-43 17 29.1& 22.94 & 20.70& 20.43& 19.55& 19.02& 18.44& 0.07& 0.02& 0.01& 0.01& 0.03& 20.61& 19.83& 19.07& 0.03& 0.03& 0.01& 0.27& 0.88& 0.53& 1.11& 0.76& 0.78& 1.55& 653$\pm$37\
GC0012 & AAT102120 & 13 23 59.61 &-42 55 19.4& 17.11 & 20.60& 20.59& 19.92& 19.49& 19.03& 0.06& 0.02& 0.01& 0.01& 0.03& 20.61& 20.13& 19.49& 0.01& 0.01& 0.01& 0.02& 0.67& 0.44& 0.89& 0.64& 0.48& 1.12& 293$\pm$83\
GC0013 & HH-001 & 13 24 02.67 &-42 48 32.2& 20.00 & - & - & - & - & - & - & - & - & - & - & 22.52& 21.20& 19.89& 0.08& 0.05& 0.03& - & - & - & - & 1.31& 1.32& 2.63& 775$\pm$74\
GC0014 & pff\_gc-003 & 13 24 03.23 &-43 28 13.9& 31.17 & 20.39& 20.18& 19.31& 18.80& 18.21& 0.15& 0.02& 0.01& 0.01& 0.03& 20.39& 19.64& 18.80& 0.02& 0.02& 0.02& 0.21& 0.87& 0.51& 1.09& 0.84& 0.76& 1.59& 704$\pm$22\
GC0015 & pff\_gc-004 & 13 24 03.74 &-43 35 53.4& 37.98 & 21.12& 20.84& 20.01& 19.49& 18.92& 0.28& 0.03& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.28& 0.83& 0.52& 1.09& - & - & - & 571$\pm$38\
GC0016 & AAT103195 & 13 24 05.98 &-43 03 54.7& 15.18 & 21.57& 21.10& 20.13& 19.58& 18.98& 0.14& 0.03& 0.01& 0.01& 0.03& 21.34& 20.39& 19.50& 0.01& 0.01& 0.01& 0.47& 0.97& 0.56& 1.15& 0.89& 0.95& 1.84& 277$\pm$67\
GC0017 & AAT304867 & 13 24 08.62 & -43 16 27.1& 21.04 & 22.12& 21.15& 19.91& 19.20& 18.54& 0.25& 0.03& 0.01& 0.01& 0.03& 21.62& 20.28& 19.18& 0.01& - & - & 0.97& 1.24& 0.71& 1.36& 1.10& 1.34& 2.44& 305$\pm$56\
GC0018 & AAT305341 & 13 24 10.97 &-43 12 52.8& 18.27 & 22.13& 21.54& 20.51& 19.88& 19.19& 0.23& 0.04& 0.01& 0.01& 0.03& 21.87& 20.80& 19.90& 0.01& 0.01& 0.01& 0.59& 1.03& 0.63& 1.31& 0.89& 1.07& 1.97& 440$\pm$118\
GC0019 & HHH86-28/C28 & 13 24 18.06 &-42 49 01.1& 17.57 & 19.52& 19.33& 18.50& 17.99& 17.45& 0.03& 0.01& 0.01& 0.01& 0.03& 19.46& 18.75& 18.03& 0.02& 0.01& 0.01& 0.19& 0.83& 0.52& 1.05& 0.72& 0.71& 1.43& 461$\pm$22\
GC0020 & pff\_gc-005 & 13 24 18.92 &-43 14 30.1& 18.33 & 21.89& 21.31& 20.29& 19.71& 19.07& 0.18& 0.04& 0.01& 0.01& 0.03& 21.63& 20.64& 19.72& 0.02& 0.02& 0.01& 0.58& 1.02& 0.58& 1.22& 0.91& 1.00& 1.91& 750$\pm$33\
GC0021 & WHH-1/HH-096 & 13 24 21.40 &-43 02 36.8& 12.19 & 19.14& 18.85& 18.01& 17.48& 16.98& 0.03& 0.01& 0.01& 0.01& 0.03& 19.11& 18.33& 17.63& 0.02& 0.02& 0.03& 0.29& 0.84& 0.53& 1.03& 0.70& 0.78& 1.48& 583$\pm$29\
GC0022 & pff\_gc-006 & 13 24 23.72 &-43 07 52.1& 13.48 & 20.22& 20.00& 19.22& 18.70& 18.20& 0.05& 0.02& 0.01& 0.01& 0.03& 20.10& 19.46& 18.73& 0.01& 0.01& 0.01& 0.21& 0.78& 0.52& 1.02& 0.73& 0.65& 1.37& 644$\pm$28\
GC0023 & WHH-2 & 13 24 23.98 &-42 54 10.7& 13.56 & 20.57& 20.45& 19.67& 19.17& 18.65& 0.06& 0.02& 0.01& 0.01& 0.03& 20.54& 19.94& 19.27& 0.03& 0.03& 0.02& 0.12& 0.78& 0.50& 1.02& 0.68& 0.59& 1.27& 582$\pm$81\
GC0024 & pff\_gc-007 & 13 24 24.15 &-42 54 20.6& 13.45 & 21.85& 21.16& 20.12& 19.53& 18.82& 0.17& 0.03& 0.01& 0.01& 0.03& 21.54& 20.50& 19.60& 0.02& 0.02& 0.01& 0.69& 1.04& 0.59& 1.30& 0.90& 1.04& 1.94& 617$\pm$25\
GC0025 & AAT308432 & 13 24 25.55 &-43 21 35.6& 23.38 & 20.98& 20.43& 19.40& 18.80& 18.11& 0.27& 0.02& 0.01& 0.01& 0.03& 20.83& 19.78& 18.84& 0.03& 0.01& 0.02& 0.54& 1.04& 0.60& 1.29& 0.94& 1.05& 1.99& 835$\pm$83\
GC0026 & C111 & 13 24 26.97 &-43 17 20.0& 19.62 & - & - & - & - & - & - & - & - & - & - & 22.54& 22.25& 21.36& 0.04& 0.04& 0.04& - & - & - & - & 0.89& 0.29& 1.17& -\
GC0027 & AAT106695 & 13 24 28.18 &-42 53 04.3& 13.54 & 20.95& 20.75& 19.63& 18.96& 18.30& 0.09& 0.03& 0.01& 0.01& 0.03& 21.16& 20.17& 19.19& 0.06& 0.06& 0.04& 0.20& 1.12& 0.67& 1.33& 0.98& 0.99& 1.96& 835$\pm$83\
GC0028 & AAT106880 & 13 24 28.44 &-42 57 52.9& 11.30 & 21.11& 20.94& 20.14& 19.61& 19.11& 0.10& 0.03& 0.01& 0.01& 0.03& 21.06& 20.43& 19.65& 0.01& 0.01& 0.01& 0.18& 0.80& 0.53& 1.03& 0.79& 0.63& 1.41& 558$\pm$98\
GC0029 & pff\_gc-008 & 13 24 29.20 &-43 21 56.5& 23.38 & 21.40& 20.93& 19.94& 19.41& 18.75& 0.35& 0.03& 0.01& 0.01& 0.03& 21.24& 20.31& 19.43& 0.02& 0.01& 0.01& 0.47& 0.99& 0.54& 1.19& 0.87& 0.93& 1.81& 466$\pm$38\
GC0030 & AAT107060 & 13 24 29.23 &-43 08 36.6& 13.02 & 21.81& 21.26& 20.26& 19.63& 19.00& 0.19& 0.04& 0.01& 0.01& 0.03& 21.60& 20.62& 19.69& 0.01& 0.01& 0.01& 0.55& 1.01& 0.63& 1.25& 0.93& 0.99& 1.92& 600$\pm$58\
GC0031 & AAT107145 & 13 24 29.73 &-43 02 06.5& 10.62 & 21.20& 20.93& 20.08& 19.50& 18.97& 0.11& 0.03& 0.01& 0.01& 0.03& 21.16& 20.33& 19.55& 0.03& 0.02& 0.04& 0.27& 0.85& 0.58& 1.11& 0.78& 0.82& 1.60& 595$\pm$202\
GC0032 & pff\_gc-009 & 13 24 31.35 &-43 11 26.7& 14.55 & 20.82& 20.58& 19.77& 19.25& 18.70& 0.07& 0.02& 0.01& 0.01& 0.03& 20.73& 20.05& 19.26& 0.03& 0.02& 0.01& 0.23& 0.81& 0.52& 1.07& 0.80& 0.68& 1.47& 683$\pm$38\
GC0033 & WHH-3 & 13 24 32.17 &-43 10 56.9& 14.10 & 20.56& 20.32& 19.49& 18.97& 18.43& 0.06& 0.02& 0.01& 0.01& 0.03& 20.46& 19.76& 18.99& 0.02& 0.02& 0.01& 0.25& 0.83& 0.52& 1.06& 0.76& 0.71& 1.47& 709$\pm$54\
GC0034 & C112 & 13 24 32.66 &-43 18 48.8& 20.32 & - & - & - & - & - & - & - & - & - & - & 22.98& 22.22& 21.31& 0.03& 0.02& 0.02& - & - & - & - & 0.91& 0.75& 1.67& -\
GC0035 & pff\_gc-010 & 13 24 33.09 &-43 18 44.8& 20.23 & 20.45& 20.36& 19.68& 19.26& 18.84& 0.06& 0.02& 0.01& 0.01& 0.03& 20.45& 19.91& 19.24& 0.01& 0.01& 0.01& 0.09& 0.68& 0.42& 0.85& 0.67& 0.54& 1.21& 344$\pm$58\
GC0036 & AAT107977 & 13 24 34.63 &-43 12 50.5& 15.18 & 21.73& 21.18& 20.18& 19.57& 18.89& 0.17& 0.04& 0.01& 0.01& 0.03& 21.52& 20.54& 19.59& 0.02& 0.02& 0.01& 0.54& 1.01& 0.60& 1.28& 0.95& 0.98& 1.93& 517$\pm$123\
GC0037 & pff\_gc-011 & 13 24 36.87 &-43 19 16.2& 20.36 & 20.01& 19.82& 19.03& 18.53& 18.00& 0.13& 0.02& 0.01& 0.01& 0.03& 19.95& 19.33& 18.55& 0.01& 0.01& 0.01& 0.18& 0.79& 0.50& 1.04& 0.78& 0.62& 1.40& 616$\pm$41\
GC0038 & C113 & 13 24 37.75 &-43 16 26.5& 17.80 & - & - & - & - & - & - & - & - & - & - & 20.50& 19.87& 19.12& 0.02& 0.02& 0.01& - & - & - & - & 0.75& 0.63& 1.38& -\
GC0039 & pff\_gc-012 & 13 24 38.77 &-43 06 26.6& 10.38 & 21.24& 20.51& 19.45& 18.83& 18.15& 0.13& 0.02& 0.01& 0.01& 0.03& 20.92& 19.78& 18.85& 0.01& 0.01& 0.01& 0.73& 1.06& 0.62& 1.30& 0.93& 1.14& 2.07& 573$\pm$21\
GC0040 & HGHH-41/C41 & 13 24 38.98 &-43 20 06.4& 20.94 & 20.20& 19.59& 18.59& 17.94& 17.32& 0.06& 0.02& 0.01& 0.01& 0.03& 19.95& 18.92& 17.97& 0.02& 0.01& 0.01& 0.61& 1.00& 0.65& 1.27& 0.95& 1.03& 1.98& 363$\pm$1\
GC0041 &HGHH-29/C29 & 13 24 40.39 &-43 18 05.3& 19.01 & 19.77& 19.15& 18.15& 17.54& 16.89& 0.04& 0.01& 0.01& 0.01& 0.03& 19.46& 18.37& 17.53& 0.04& 0.03& 0.02& 0.62& 1.00& 0.61& 1.26& 0.84& 1.09& 1.92& 726$\pm$1\
GC0042 &pff\_gc-013 & 13 24 40.42 &-43 35 04.9& 35.01 & 20.06& 19.83& 19.00& 18.49& 17.98& 0.15& 0.02& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.24& 0.83& 0.51& 1.03& - & - & - & 727$\pm$31\
GC0043 &C114 & 13 24 40.48 &-42 53 35.3& 11.46 & - & - & - & - & - & - & - & - & - & - & 23.46& 22.39& 21.42& 0.05& 0.04& 0.03& - & - & - & - & 0.97& 1.07& 2.04& -\
GC0044 & WHH-4/HH-024 & 13 24 40.60 &-43 13 18.1& 14.89 & 20.71& 20.12& 19.12& 18.50& 17.85& 0.07& 0.02& 0.01& 0.01& 0.03& 20.46& 19.40& 18.53& 0.03& 0.02& 0.01& 0.60& 1.00& 0.62& 1.27& 0.87& 1.06& 1.93& 688$\pm$25\
GC0045 &pff\_gc-014 & 13 24 41.05 &-42 59 48.4& 8.62 & 21.60& 21.17& 20.23& 19.67& 18.94& 0.26& 0.06& 0.02& 0.02& 0.03& 21.47& 20.56& 19.76& 0.04& 0.03& 0.04& 0.43& 0.94& 0.57& 1.30& 0.80& 0.91& 1.71& 690$\pm$34\
GC0046 &pff\_gc-015 & 13 24 41.20 &-43 01 45.6& 8.51 & 20.27& 20.01& 19.22& 18.74& 18.15& 0.06& 0.02& 0.01& 0.01& 0.03& 20.12& 19.40& 18.70& 0.05& 0.04& 0.05& 0.26& 0.79& 0.47& 1.06& 0.70& 0.72& 1.42& 533$\pm$25\
GC0047 &AAT109380 & 13 24 43.58 &-43 08 43.2& 11.05 & 19.64& 19.52& 18.80& 18.32& 17.87& 0.04& 0.01& 0.01& 0.01& 0.03& 19.60& 19.02& 18.30& 0.01& - & - & 0.12& 0.73& 0.48& 0.93& 0.72& 0.59& 1.31& 465$\pm$38\
GC0048 &pff\_gc-016 & 13 24 43.60 &-42 53 07.3& 11.36 & 21.38& 20.87& 19.85& 19.26& 18.58& 0.12& 0.03& 0.01& 0.01& 0.03& 21.19& 20.21& 19.33& 0.02& 0.02& 0.01& 0.51& 1.02& 0.60& 1.27& 0.88& 0.98& 1.85& 505$\pm$17\
GC0049 &pff\_gc-017 & 13 24 43.63 &-42 58 16.4& 8.54 & 20.54& 20.32& 19.51& 18.99& 18.47& 0.07& 0.02& 0.01& 0.01& 0.03& 20.44& 19.78& 19.01& 0.01& 0.01& 0.01& 0.22& 0.81& 0.52& 1.04& 0.77& 0.66& 1.43& 963$\pm$50\
GC0050 & WHH-5 & 13 24 44.58 &-43 02 47.3& 8.04 & 20.47& 20.03& 19.06& 18.49& 17.87& 0.08& 0.02& 0.01& 0.01& 0.03& 20.29& 19.19& 18.90& 0.05& 0.04& 0.04& 0.44& 0.97& 0.57& 1.19& 0.29& 1.09& 1.39& 671$\pm$41\
GC0051 & HH-054 & 13 24 45.22 &-43 23 19.2& 23.49 & - & - & - & - & - & - & - & - & - & - & 20.92& 20.08& 19.25& 0.05& 0.03& 0.04& - & - & - & - & 0.82& 0.84& 1.66& 998$\pm$250\
GC0052 &AAT109711 & 13 24 45.35 &-42 59 33.5& 7.89 & 19.88& 19.87& 19.30& 18.90& 18.33& 0.06& 0.02& 0.01& 0.01& 0.03& 19.84& 19.48& 18.91& 0.02& 0.02& 0.04& 0.02& 0.56& 0.40& 0.98& 0.58& 0.36& 0.94& 276$\pm$59\
GC0053 &AAT109788 & 13 24 45.78 &-43 02 24.5& 7.75 & 21.82& 21.02& 19.89& 19.23& 18.56& 0.25& 0.04& 0.01& 0.01& 0.03& 21.39& 20.17& 19.43& 0.02& 0.02& 0.03& 0.80& 1.14& 0.65& 1.33& 0.74& 1.22& 1.96& 527$\pm$30\
GC0054 &HGHH-G019/G19 & 13 24 46.46 &-43 04 11.6& 8.12 & 20.16& 19.95& 19.16& 18.64& 18.13& 0.06& 0.02& 0.01& 0.01& 0.03& 20.06& 19.38& 18.64& 0.04& 0.03& 0.03& 0.21& 0.79& 0.52& 1.03& 0.75& 0.67& 1.42& 710$\pm$22\
GC0055 &HGHH-09/C9 & 13 24 46.72 &-43 01 18.4& 7.48 & 20.00& 19.76& 18.95& 18.40& 17.90& 0.05& 0.02& 0.01& 0.01& 0.03& 19.87& 19.13& 18.42& 0.04& 0.04& 0.04& 0.24& 0.81& 0.56& 1.05& 0.70& 0.74& 1.45& 501$\pm$21\
GC0056 &pff\_gc-018 & 13 24 47.10 &-43 06 01.7& 8.87 & 20.90& 20.03& 18.91& 18.25& 17.54& 0.10& 0.02& 0.01& 0.01& 0.03& 20.45& 19.23& 18.28& 0.02& 0.01& 0.01& 0.87& 1.12& 0.66& 1.37& 0.95& 1.22& 2.17& 534$\pm$15\
GC0057 &HGHH-G277/G277 & 13 24 47.37 &-42 58 29.8& 7.82 & 20.26& 19.96& 19.10& 18.56& 18.02& 0.06& 0.02& 0.01& 0.01& 0.03& 20.14& 19.37& 18.61& 0.01& 0.01& 0.01& 0.30& 0.87& 0.53& 1.08& 0.76& 0.77& 1.53& 714$\pm$33\
GC0058 & WHH-6 & 13 24 47.37 &-42 57 51.2& 8.06 & 21.33& 20.75& 19.73& 19.12& 18.44& 0.15& 0.03& 0.01& 0.01& 0.03& 21.08& 20.07& 19.15& 0.01& 0.01& 0.01& 0.58& 1.02& 0.62& 1.29& 0.93& 1.00& 1.93& 685$\pm$43\
GC0059 &AAT110138 & 13 24 47.61 &-43 10 48.5& 12.12 & 20.87& 20.53& 19.67& 19.11& 18.57& 0.10& 0.03& 0.01& 0.01& 0.03& 20.75& 19.98& 19.16& 0.03& 0.02& 0.01& 0.34& 0.87& 0.56& 1.10& 0.82& 0.77& 1.59& 358$\pm$137\
GC0060 &HHH86-10/C10 & 13 24 48.06 &-43 08 14.2& 10.13 & 19.87& 19.44& 18.49& 17.91& 17.33& 0.05& 0.01& 0.01& 0.01& 0.03& 19.66& 18.75& 17.93& 0.03& 0.03& 0.01& 0.44& 0.95& 0.58& 1.16& 0.81& 0.91& 1.73& 829$\pm$22\
GC0061 &C115 & 13 24 48.71 &-42 52 35.5& 11.12 & - & - & - & - & - & - & - & - & - & - & 20.88& 20.23& 19.51& 0.02& 0.02& 0.01& - & - & - & - & 0.73& 0.65& 1.37& -\
GC0062 &pff\_gc-019 & 13 24 48.97 &-42 57 48.4& 7.81 & 20.10& 19.87& 19.05& 18.53& 18.00& 0.05& 0.02& 0.01& 0.01& 0.03& 20.00& 19.31& 18.57& 0.03& 0.03& 0.01& 0.23& 0.81& 0.53& 1.05& 0.74& 0.69& 1.43& 607$\pm$27\
GC0063 &AAT110410 & 13 24 49.38 &-43 08 17.7& 10.00 & 21.47& 21.18& 20.34& 19.79& 19.23& 0.16& 0.04& 0.02& 0.01& 0.03& 21.37& 20.66& 19.81& 0.03& 0.02& 0.01& 0.29& 0.84& 0.55& 1.11& 0.85& 0.71& 1.56& 580$\pm$81\
GC0064 &AAT110551 & 13 24 50.09 &-43 07 36.2& 9.42 & 21.43& 21.08& 20.44& 20.04& 19.49& 0.16& 0.04& 0.02& 0.02& 0.03& 21.19& 20.70& 20.03& 0.02& 0.02& 0.01& 0.35& 0.64& 0.41& 0.96& 0.67& 0.49& 1.16& 615$\pm$76\
GC0065 &pff\_gc-020 & 13 24 50.48 &-42 59 49.0& 6.92 & 20.57& 20.33& 19.53& 19.00& 18.47& 0.09& 0.03& 0.01& 0.01& 0.03& 20.42& 19.58& 19.68& 0.05& 0.04& 0.04& 0.24& 0.80& 0.53& 1.06&-0.11& 0.85& 0.74& 331$\pm$47\
GC0066 &HGHH-42/C42 & 13 24 50.87 &-43 01 22.9& 6.72 & 20.53& 19.91& 18.86& 18.24& 17.53& 0.09& 0.02& 0.01& 0.01& 0.03& 20.21& 19.11& 18.21& 0.04& 0.03& 0.05& 0.62& 1.05& 0.63& 1.33& 0.90& 1.10& 2.00& 552$\pm$12\
GC0067 &VHH81-02/C2 & 13 24 51.49 &-43 12 11.1& 12.86 & 19.64& 19.33& 18.50& 18.01& 17.42& 0.04& 0.01& 0.01& 0.01& 0.03& 19.48& 18.72& 17.94& 0.04& 0.03& 0.01& 0.31& 0.83& 0.49& 1.07& 0.78& 0.77& 1.55& 628$\pm$22\
GC0068 &C100 & 13 24 51.80 &-43 04 33.7& 7.33 & - & - & 20.08& - & - & - & - & - & - & - & 20.44& 19.72& 19.03& 0.08& 0.08& 0.10& - & - & - & 1.28& 0.69& 0.72& 1.41& -\
GC0069 &AAT111033 & 13 24 52.98 &-43 11 55.8& 12.50 & 21.17& 20.93& 20.12& 19.66& 19.13& 0.11& 0.03& 0.01& 0.01& 0.03& 21.10& 20.40& 19.64& 0.01& 0.01& 0.01& 0.24& 0.81& 0.46& 0.99& 0.76& 0.70& 1.47& 302$\pm$166\
GC0070 &HGHH-G302/G302 & 13 24 53.29 &-43 04 34.8& 7.15 & 20.30& 20.01& 19.20& 18.69& 18.16& 0.07& 0.02& 0.01& 0.01& 0.03& 20.18& 19.50& 18.73& 0.02& 0.01& 0.01& 0.28& 0.81& 0.52& 1.04& 0.77& 0.68& 1.45& 558$\pm$43\
GC0071 &AAT111185 & 13 24 54.00 &-43 04 24.4& 6.96 & 20.92& 20.76& 19.92& 19.41& 18.92& 0.11& 0.03& 0.01& 0.01& 0.03& 20.90& 20.28& 19.37& 0.02& 0.02& 0.02& 0.16& 0.84& 0.51& 1.00& 0.91& 0.62& 1.53& 466$\pm$87\
GC0072 &pff\_gc-021 & 13 24 54.18 &-42 54 50.4& 8.78 & 20.21& 20.03& 19.25& 18.75& 18.24& 0.06& 0.02& 0.01& 0.01& 0.03& 20.13& 19.52& 18.81& 0.03& 0.02& 0.01& 0.18& 0.78& 0.50& 1.01& 0.70& 0.62& 1.32& 594$\pm$50\
GC0073 &pff\_gc-022 & 13 24 54.33 &-43 03 15.5& 6.44 & 20.72& 20.58& 19.84& 19.37& 18.90& 0.11& 0.03& 0.01& 0.01& 0.03& 20.69& 20.09& 19.37& 0.03& 0.02& 0.02& 0.14& 0.74& 0.48& 0.95& 0.72& 0.60& 1.32& 619$\pm$44\
GC0074 &HHH86-30/C30 & 13 24 54.35 &-42 53 24.8& 9.84 & 18.76& 18.24& 17.25& 16.67& 16.02& 0.03& 0.01& 0.01& 0.01& 0.03& 18.47& 17.49& 16.68& 0.03& 0.04& 0.02& 0.52& 0.98& 0.58& 1.23& 0.81& 0.98& 1.79& 778$\pm$13\
GC0075 &AAT111296 & 13 24 54.49 &-43 05 34.7& 7.50 & 21.70& 20.99& 19.96& 19.31& 18.61& 0.23& 0.04& 0.01& 0.01& 0.03& 21.40& 20.28& 19.33& 0.02& 0.01& 0.01& 0.71& 1.03& 0.65& 1.35& 0.96& 1.11& 2.07& 695$\pm$45\
GC0076 &pff\_gc-023 & 13 24 54.55 &-42 48 58.7& 13.59 & 20.55& 20.30& 19.44& 18.90& 18.37& 0.06& 0.02& 0.01& 0.01& 0.03& 20.48& 19.74& 18.97& 0.01& 0.01& 0.01& 0.25& 0.86& 0.54& 1.08& 0.77& 0.74& 1.51& 457$\pm$31\
GC0077 &HGHH-11/C11 & 13 24 54.73 &-43 01 21.7& 6.01 & 19.55& 18.96& 17.91& 17.26& 16.61& 0.05& 0.01& 0.01& 0.01& 0.03& 19.21& 17.98& 17.20& 0.06& 0.05& 0.06& 0.59& 1.05& 0.66& 1.30& 0.79& 1.23& 2.01& 753$\pm$1\
GC0078 &AAT111406 & 13 24 55.29 &-43 03 15.6& 6.28 & 21.84& 21.15& 20.13& 19.50& 18.80& 0.30& 0.05& 0.02& 0.01& 0.03& 21.66& 20.70& 18.80& 0.03& 0.02& 0.01& 0.69& 1.03& 0.63& 1.33& 1.90& 0.96& 2.86& 669$\pm$77\
GC0079 &C116 & 13 24 55.46 &-43 09 58.5& 10.61 & - & - & - & - & - & - & - & - & - & - & 23.80& 22.74& 21.84& 0.06& 0.02& 0.03& - & - & - & - & 0.90& 1.07& 1.97& -\
GC0080 & HH-052 & 13 24 55.71 &-43 22 48.4& 22.43 & - & - & - & - & - & - & - & - & - & - & 22.58& 21.61& 20.42& 0.05& 0.03& 0.03& - & - & - & - & 1.19& 0.97& 2.16& 921$\pm$146\
GC0081 &pff\_gc-024 & 13 24 55.71 &-43 20 39.1& 20.36 & 21.07& 20.73& 19.84& 19.31& 18.73& 0.28& 0.03& 0.01& 0.01& 0.03& 21.00& 20.17& 19.34& 0.02& 0.01& 0.01& 0.34& 0.89& 0.53& 1.11& 0.83& 0.83& 1.66& 279$\pm$38\
GC0082 &C117 & 13 24 56.06 &-42 54 29.6& 8.80 & - & - & - & - & - & - & - & - & - & - & 21.27& 20.18& 19.26& 0.01& 0.02& 0.01& - & - & - & - & 0.92& 1.09& 2.01& -\
GC0083 &AAT111563 & 13 24 56.08 &-43 10 16.4& 10.79 & 21.34& 21.03& 20.45& 20.05& 19.62& 0.15& 0.04& 0.02& 0.02& 0.03& 21.14& 20.73& 20.05& 0.02& 0.01& 0.02& 0.31& 0.58& 0.41& 0.83& 0.68& 0.41& 1.09& 649$\pm$102\
GC0084 &HGHH-G279 & 13 24 56.27 &-43 03 23.4& 6.15 & 19.97& 19.91& 19.45& 19.17& 18.88& 0.06& 0.02& 0.01& 0.01& 0.03& 19.76& 19.38& 19.83& 0.01& 0.01& 0.01& 0.06& 0.45& 0.29& 0.57&-0.45& 0.38&-0.07& 366$\pm$34\
GC0085 &C118 & 13 24 57.17 &-43 08 42.6& 9.39 & - & - & - & - & - & - & - & - & - & - & 22.60& 21.76& 20.67& 0.04& 0.03& 0.02& - & - & - & - & 1.08& 0.84& 1.92& -\
GC0086 &HGHH-31/C31 & 13 24 57.44 &-43 01 08.1& 5.52 & 20.09& 19.42& 18.38& 17.75& 17.06& 0.08& 0.02& 0.01& 0.01& 0.03& 19.73& 18.59& 17.71& 0.04& 0.03& 0.04& 0.67& 1.04& 0.62& 1.31& 0.88& 1.14& 2.02& 690$\pm$18\
GC0087 &HGHH-G369 & 13 24 57.52 &-42 59 23.3& 5.78 & 19.82& 19.58& 18.74& 18.22& 17.69& 0.06& 0.02& 0.01& 0.01& 0.03& 19.67& 18.96& 18.24& 0.04& 0.03& 0.05& 0.24& 0.83& 0.53& 1.06& 0.72& 0.71& 1.44& 512$\pm$17\
GC0088 &pff\_gc-025 & 13 24 57.56 &-43 05 32.8& 7.04 & 21.50& 21.02& 20.02& 19.39& 18.67& 0.20& 0.04& 0.02& 0.01& 0.03& 21.41& 20.41& 19.45& 0.04& 0.03& 0.02& 0.47& 1.00& 0.63& 1.35& 0.96& 1.00& 1.97& 923$\pm$30\
GC0089 &C119 & 13 24 57.69 &-42 55 48.4& 7.64 & - & - & - & - & - & - & - & - & - & - & 21.16& 21.02& 20.67& 0.03& 0.03& 0.02& - & - & - & - & 0.35& 0.14& 0.49& -\
GC0090 &C120 & 13 24 57.95 &-42 52 04.9& 10.56 & - & - & - & - & - & - & - & - & - & - & 23.26& 22.34& 21.43& 0.04& 0.04& 0.03& - & - & - & - & 0.90& 0.92& 1.83& -\
GC0091 &VHH81-03/C3 & 13 24 58.21 &-42 56 10.0& 7.33 & 19.34& 18.73& 17.71& 17.10& 16.44& 0.04& 0.01& 0.01& 0.01& 0.03& 19.02& 17.88& 17.08& 0.04& 0.03& 0.02& 0.61& 1.02& 0.61& 1.26& 0.80& 1.14& 1.94& 562$\pm$2\
GC0092 &C121 & 13 24 58.42 &-43 08 21.2& 8.97 & - & - & - & - & - & - & - & - & - & - & 23.53& 23.01& 22.45& 0.05& 0.04& 0.09& - & - & - & - & 0.56& 0.52& 1.08& -\
GC0093 &pff\_gc-026 & 13 24 58.45 &-42 42 53.3& 19.02 & 20.44& 20.33& 19.52& 19.07& 18.50& 0.08& 0.02& 0.01& 0.01& 0.03& 20.47& 19.82& 19.09& 0.02& 0.01& 0.01& 0.12& 0.81& 0.45& 1.02& 0.73& 0.65& 1.37& 490$\pm$67\
GC0094 &C122 & 13 24 59.01 &-43 08 21.4& 8.91 & - & - & - & - & - & - & - & - & - & - & - & - & 22.33& - & - & - & - & - & - & - & - & - & - & -\
GC0095 &C123 & 13 24 59.92 &-43 09 08.6& 9.46 & - & - & - & - & - & - & - & - & - & - & 21.88& 21.10& 20.24& 0.02& 0.01& 0.01& - & - & - & - & 0.86& 0.78& 1.64& -\
GC0096 &AAT112158 & 13 25 00.15 &-42 54 09.0& 8.61 & 21.99& 21.43& 20.43& 19.86& 19.23& 0.25& 0.05& 0.02& 0.01& 0.03& 21.75& 20.78& 19.88& 0.01& 0.01& 0.01& 0.55& 1.00& 0.57& 1.20& 0.90& 0.96& 1.87& 699$\pm$43\
GC0097 &C124 & 13 25 00.37 &-43 10 46.9& 10.85 & - & - & - & - & - & - & - & - & - & - & 23.07& 22.36& 21.57& 0.03& 0.02& 0.03& - & - & - & - & 0.78& 0.71& 1.50& -\
GC0098 &pff\_gc-027 & 13 25 00.64 &-43 05 30.3& 6.58 & 21.22& 20.75& 19.74& 19.13& 18.45& 0.17& 0.04& 0.01& 0.01& 0.03& 21.05& 20.06& 19.11& 0.01& 0.01& 0.01& 0.47& 1.01& 0.61& 1.29& 0.95& 0.99& 1.94& 524$\pm$34\
GC0099 &C125 & 13 25 00.83 &-43 11 10.6& 11.16 & - & - & - & - & - & - & - & - & - & - & 22.34& 21.55& 20.66& 0.03& 0.02& 0.02& - & - & - & - & 0.89& 0.78& 1.67& -\
GC0100 &C126 & 13 25 00.91 &-43 09 14.5& 9.45 & - & - & - & - & - & - & - & - & - & - & 24.02& 23.54& 22.66& 0.04& 0.05& 0.08& - & - & - & - & 0.88& 0.48& 1.35& -\
GC0101 &C127 & 13 25 01.32 &-43 08 43.4& 8.97 & - & - & - & - & - & - & - & - & - & - & 23.21& 22.42& 21.62& 0.04& 0.02& 0.03& - & - & - & - & 0.80& 0.79& 1.59& -\
GC0102 &C128 & 13 25 01.46 &-43 08 33.0& 8.81 & - & - & - & - & - & - & - & - & - & - & 23.08& 22.04& 20.95& 0.05& 0.03& 0.02& - & - & - & - & 1.09& 1.04& 2.13& -\
GC0103 &pff\_gc-029 & 13 25 01.60 &-42 54 40.9& 8.03 & 19.97& 19.75& 19.37& 19.06& 18.71& 0.05& 0.02& 0.01& 0.01& 0.03& 19.72& 19.50& 19.10& 0.03& 0.03& 0.01& 0.22& 0.38& 0.31& 0.65& 0.40& 0.22& 0.62& 570$\pm$38\
GC0104 &C129 & 13 25 01.63 &-42 50 51.3& 11.34 & - & - & - & - & - & - & - & - & - & - & 22.59& 21.71& 20.83& 0.04& 0.02& 0.02& - & - & - & - & 0.88& 0.88& 1.76& -\
GC0105 &pff\_gc-030 & 13 25 01.73 &-43 00 09.9& 4.83 & 21.68& 20.94& 19.89& 19.26& 18.54& 0.36& 0.06& 0.02& 0.02& 0.03& 21.21& 19.93& 20.43& 0.05& 0.04& 0.02& 0.73& 1.05& 0.64& 1.35&-0.50& 1.29& 0.78& 357$\pm$29\
GC0106 &HGHH-04/C4 & 13 25 01.83 &-43 09 25.4& 9.52 & 19.10& 18.86& 18.04& 17.50& 16.98& 0.03& 0.01& 0.01& 0.01& 0.03& 18.95& 18.24& 17.50& 0.04& 0.03& 0.01& 0.23& 0.82& 0.54& 1.06& 0.74& 0.71& 1.45& 689$\pm$16\
GC0107 &C130 & 13 25 01.86 &-42 52 27.8& 9.88 & - & - & - & - & - & - & - & - & - & - & 21.22& 20.58& 19.85& 0.01& 0.01& 0.01& - & - & - & - & 0.73& 0.64& 1.37& -\
GC0108 &pff\_gc-031 & 13 25 02.76 &-43 11 21.2& 11.17 & 20.51& 20.29& 19.48& 18.99& 18.42& 0.07& 0.02& 0.01& 0.01& 0.03& 20.43& 19.75& 18.95& 0.03& 0.02& 0.01& 0.23& 0.81& 0.49& 1.06& 0.80& 0.68& 1.48& 444$\pm$33\
GC0109 &HGHH-G176 & 13 25 03.13 &-42 56 25.1& 6.51 & 20.47& 19.94& 18.93& 18.32& 17.67& 0.09& 0.02& 0.01& 0.01& 0.03& 20.24& 19.23& 18.36& 0.01& 0.02& 0.01& 0.54& 1.01& 0.61& 1.26& 0.87& 1.01& 1.88& 551$\pm$13\
GC0110 &HGHH-G066 & 13 25 03.18 &-43 03 02.5& 4.85 & 20.26& 19.67& 18.68& 18.15& 17.44& 0.09& 0.02& 0.01& 0.01& 0.03& 19.94& 18.96& 18.05& 0.01& 0.01& 0.02& 0.59& 0.99& 0.52& 1.24& 0.91& 0.98& 1.89& 576$\pm$8\
GC0111 &pff\_gc-032 & 13 25 03.24 &-42 57 40.5& 5.65 & 20.92& 20.59& 19.73& 19.18& 18.61& 0.14& 0.04& 0.01& 0.01& 0.03& 20.81& 20.05& 19.22& 0.01& 0.02& 0.01& 0.33& 0.86& 0.55& 1.12& 0.83& 0.76& 1.59& 648$\pm$29\
GC0112 &pff\_gc-033 & 13 25 03.34 &-43 15 27.4& 14.98 & 21.10& 20.49& 19.46& 18.89& 18.21& 0.33& 0.03& 0.01& 0.01& 0.03& 20.83& 19.79& 18.90& 0.02& 0.02& 0.01& 0.61& 1.02& 0.57& 1.25& 0.89& 1.04& 1.93& 531$\pm$18\
GC0113 &HHH86-32/C32 & 13 25 03.37 &-42 50 46.2& 11.28 & 20.22& 19.50& 18.44& 17.81& 17.15& 0.05& 0.01& 0.01& 0.01& 0.03& 19.86& 18.71& 17.85& 0.03& 0.03& 0.01& 0.72& 1.06& 0.63& 1.29& 0.85& 1.15& 2.01& 718$\pm$11\
GC0114 &pff\_gc-034 & 13 25 03.37 &-43 11 39.6& 11.41 & 20.98& 20.70& 19.91& 19.35& 18.84& 0.09& 0.03& 0.01& 0.01& 0.03& 20.89& 20.19& 19.38& 0.03& 0.02& 0.01& 0.28& 0.80& 0.56& 1.06& 0.81& 0.69& 1.50& 605$\pm$46\
GC0115 &C131 & 13 25 03.67 &-42 51 21.7& 10.72 & - & - & - & - & - & - & - & - & - & - & 21.57& 20.98& 20.20& 0.02& 0.02& 0.01& - & - & - & - & 0.78& 0.59& 1.36& -\
GC0116 &AAT112752 & 13 25 04.12 &-43 00 19.6& 4.37 & 20.51& 20.16& 19.34& 18.88& 18.44& 0.15& 0.04& 0.02& 0.01& 0.03& 20.35& 19.58& 18.87& 0.02& 0.02& 0.03& 0.35& 0.82& 0.46& 0.90& 0.72& 0.76& 1.48& 679$\pm$82\
GC0117 &pff\_gc-035 & 13 25 04.48 &-43 10 48.4& 10.54 & 21.32& 20.73& 19.71& 19.07& 18.44& 0.14& 0.03& 0.01& 0.01& 0.03& 21.13& 20.07& 19.13& 0.03& 0.02& 0.01& 0.59& 1.02& 0.64& 1.27& 0.94& 1.06& 2.00& 627$\pm$22\
GC0118 &AAT112964 & 13 25 04.61 &-43 07 21.7& 7.50 & 20.45& 20.39& 19.61& 19.13& 18.68& 0.08& 0.03& 0.01& 0.01& 0.03& 20.47& 19.90& 19.11& 0.01& 0.01& 0.01& 0.06& 0.78& 0.49& 0.94& 0.78& 0.57& 1.35& 456$\pm$118\
GC0119 &HGHH-43/C43 & 13 25 04.81 &-43 09 38.8& 9.47 & 19.74& 19.45& 18.60& 18.07& 17.53& 0.04& 0.01& 0.01& 0.01& 0.03& 19.59& 18.85& 18.07& 0.03& 0.02& 0.01& 0.29& 0.85& 0.53& 1.07& 0.78& 0.74& 1.52& 518$\pm$25\
GC0120 & WHH-7 & 13 25 05.02 &-42 57 15.0& 5.68 & 18.96& 18.39& 17.41& 16.81& 16.19& 0.04& 0.01& 0.01& 0.01& 0.03& 18.69& 17.60& 16.83& 0.02& 0.03& 0.01& 0.57& 0.98& 0.60& 1.22& 0.77& 1.09& 1.86& 722$\pm$20\
GC0121 &HGHH-G035 & 13 25 05.29 &-42 58 05.8& 5.09 & 21.02& 20.59& 19.62& 19.01& 18.41& 0.17& 0.04& 0.01& 0.01& 0.03& 20.87& 19.95& 19.08& 0.01& 0.02& 0.01& 0.43& 0.97& 0.61& 1.21& 0.87& 0.92& 1.80& 776$\pm$26\
GC0122 &pff\_gc-036 & 13 25 05.46 &-43 14 02.6& 13.52 & 20.09& 19.83& 19.04& 18.52& 17.91& 0.05& 0.02& 0.01& 0.01& 0.03& 19.96& 19.27& 18.49& 0.02& 0.02& 0.01& 0.26& 0.79& 0.52& 1.13& 0.78& 0.69& 1.47& 666$\pm$30\
GC0123 &HGHH-12/C12/R281 & 13 25 05.72 &-43 10 30.7& 10.18 & - & - & - & - & - & - & - & - & - & - & 19.34& 18.23& 17.36& 0.04& 0.03& 0.02& - & - & - & - & 0.87& 1.11& 1.98& 440$\pm$1\
GC0124 &HGHH-G342 & 13 25 05.83 &-42 59 00.6& 4.52 & 19.65& 19.14& 18.18& 17.57& 16.96& 0.08& 0.02& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.51& 0.96& 0.61& 1.22& - & - & - & 553$\pm$21\
GC0125 &HHH86-13/C13 & 13 25 06.25 &-43 15 11.6& 14.58 & 20.02& 19.60& 18.68& 18.13& 17.51& 0.04& 0.01& 0.01& 0.01& 0.03& 19.82& 18.90& 18.12& 0.03& 0.03& 0.02& 0.41& 0.93& 0.55& 1.16& 0.78& 0.92& 1.70& 601$\pm$12\
GC0126 & R276 & 13 25 07.33 &-43 08 29.6& 8.23 & 21.93& 21.62& 20.78& 20.26& 19.73& 0.26& 0.07& 0.02& 0.02& 0.04& 21.88& 21.16& 20.32& 0.03& 0.03& 0.02& 0.31& 0.83& 0.52& 1.05& 0.84& 0.71& 1.55& 550$\pm$28\
GC0127 &AAT113428 & 13 25 07.33 &-43 06 20.6& 6.38 & 21.25& 20.92& 20.04& 19.49& 18.93& 0.16& 0.04& 0.01& 0.01& 0.03& 21.14& 20.36& 19.51& 0.02& 0.01& 0.01& 0.33& 0.88& 0.54& 1.11& 0.85& 0.78& 1.63& 657$\pm$67\
GC0128 &pff\_gc-037 & 13 25 07.48 &-43 12 29.4& 11.93 & 21.60& 21.29& 20.42& 19.91& 19.33& 0.55& 0.05& 0.02& 0.01& 0.03& 21.46& 20.73& 19.90& 0.02& 0.01& 0.01& 0.31& 0.87& 0.50& 1.08& 0.83& 0.73& 1.56& 554$\pm$23\
GC0129 & WHH-8 & 13 25 07.62 &-43 01 15.2& 3.66 & 19.83& 19.16& 18.12& 17.46& 16.82& 0.13& 0.03& 0.01& 0.01& 0.03& 19.48& 18.32& 17.45& 0.03& 0.03& 0.04& 0.67& 1.04& 0.66& 1.30& 0.87& 1.16& 2.03& 690$\pm$32\
GC0130 & WHH-9 & 13 25 08.51 &-43 02 57.4& 3.93 & 20.59& 19.95& 18.91& 18.37& 17.64& 0.17& 0.03& 0.01& 0.01& 0.03& 20.28& 19.17& 18.30& 0.05& 0.04& 0.05& 0.64& 1.04& 0.54& 1.27& 0.87& 1.11& 1.98& 315$\pm$100\
GC0131 &C132 & 13 25 08.79 &-43 09 09.6& 8.72 & - & - & - & - & - & - & - & - & - & - & 20.38& 19.69& 18.90& 0.02& 0.02& 0.01& - & - & - & - & 0.79& 0.69& 1.48& -\
GC0132 & R271 & 13 25 08.81 &-43 09 09.5& 8.72 & 20.49& 20.23& 19.42& 18.91& 18.39& 0.08& 0.02& 0.01& 0.01& 0.03& 20.38& 19.69& 18.90& 0.02& 0.02& 0.01& 0.26& 0.80& 0.52& 1.03& 0.79& 0.69& 1.48& 436$\pm$45\
GC0133 &pff\_gc-038 & 13 25 08.82 &-43 04 14.9& 4.63 & 20.83& 20.29& 19.29& 18.69& 18.03& 0.17& 0.04& 0.01& 0.01& 0.03& 20.58& 19.57& 18.71& 0.03& 0.03& 0.03& 0.54& 1.00& 0.61& 1.26& 0.87& 1.01& 1.88& 431$\pm$19\
GC0134 &pff\_gc-039 & 13 25 09.10 &-42 24 00.9& 37.29 & 20.01& 19.77& 18.95& 18.44& 17.91& 0.17& 0.02& 0.01& 0.01& 0.03& 19.92& 19.22& 18.50& 0.01& 0.01& 0.01& 0.23& 0.82& 0.51& 1.04& 0.72& 0.70& 1.42& 387$\pm$27\
GC0135 & K-029 & 13 25 09.19 &-42 58 59.2& 4.00 & 19.13& 18.65& 17.71& 17.13& 16.55& 0.08& 0.02& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.48& 0.94& 0.58& 1.16& - & - & - & 677$\pm$44\
GC0136 &pff\_gc-040 & 13 25 09.54 &-42 55 18.5& 6.71 & 20.09& 19.89& 19.12& 18.60& 18.11& 0.06& 0.02& 0.01& 0.01& 0.03& 20.00& 19.37& 18.66& 0.02& 0.02& 0.01& 0.20& 0.76& 0.52& 1.01& 0.71& 0.62& 1.34& 504$\pm$35\
GC0137 &HGHH-G329 & 13 25 10.20 &-43 02 06.7& 3.33 & 21.73& 20.89& 19.71& 19.07& 18.38& 0.69& 0.11& 0.03& 0.02& 0.03& 21.34& 20.04& 19.07& 0.02& 0.01& 0.03& 0.84& 1.18& 0.64& 1.33& 0.97& 1.29& 2.26& 502$\pm$16\
GC0138 & K-033 & 13 25 10.25 &-42 55 09.5& 6.78 & 21.07& 20.47& 19.46& 18.90& 18.21& 0.13& 0.03& 0.01& 0.01& 0.03& 20.78& 19.77& 18.86& 0.01& 0.01& 0.01& 0.60& 1.01& 0.57& 1.25& 0.91& 1.02& 1.93& 582$\pm$33\
GC0139 & K-034 & 13 25 10.27 &-42 53 33.1& 8.23 & 19.33& 18.79& 17.80& 17.26& 16.55& 0.04& 0.01& 0.01& 0.01& 0.03& 19.14& 18.03& 17.25& 0.03& 0.03& 0.02& 0.55& 0.99& 0.54& 1.25& 0.79& 1.11& 1.89& 464$\pm$42\
GC0140 &HHH86-14/C14 & 13 25 10.49 &-42 44 52.6& 16.57 & 19.23& 18.84& 17.94& 17.40& 16.79& 0.03& 0.01& 0.01& 0.01& 0.03& 19.06& 18.16& 17.41& 0.01& 0.02& 0.01& 0.39& 0.90& 0.54& 1.15& 0.75& 0.90& 1.65& 705$\pm$10\
GC0141 &AAT113992 & 13 25 10.51 &-43 03 24.0& 3.85 & 21.95& 21.39& 20.40& 19.79& 19.19& 0.58& 0.11& 0.03& 0.03& 0.04& 21.77& 20.78& 19.82& 0.03& 0.03& 0.01& 0.57& 0.99& 0.61& 1.20& 0.97& 0.99& 1.95& 648$\pm$58\
GC0142 &C133 & 13 25 11.05 &-43 01 32.3& 3.05 & - & - & - & - & - & - & - & - & - & - & - & - & 19.30& - & - & - & - & - & - & - & - & - & - & -\
GC0143 &HGHH-G348 & 13 25 11.10 &-42 58 03.0& 4.32 & 19.90& 19.66& 18.92& 18.46& 18.03& 0.10& 0.03& 0.01& 0.01& 0.03& 19.80& 19.15& 18.45& 0.01& 0.01& - & 0.24& 0.74& 0.46& 0.89& 0.70& 0.65& 1.35& 416$\pm$30\
GC0144 &pff\_gc-041 & 13 25 11.17 &-43 03 09.6& 3.62 & 20.57& 20.32& 19.51& 18.99& 18.48& 0.19& 0.05& 0.02& 0.02& 0.03& 20.48& 19.77& 19.01& 0.02& 0.02& 0.02& 0.24& 0.81& 0.52& 1.03& 0.76& 0.71& 1.47& 456$\pm$29\
GC0145 &HGHH-G327 & 13 25 11.98 &-43 04 19.3& 4.27 & 21.46& 20.83& 19.78& 19.16& 18.56& 0.34& 0.06& 0.02& 0.02& 0.03& 21.16& 20.11& 19.19& 0.01& 0.01& 0.01& 0.63& 1.05& 0.61& 1.22& 0.92& 1.05& 1.97& 608$\pm$19\
GC0146 &HGHH-G081 & 13 25 12.11 &-42 57 25.2& 4.68 & 19.90& 19.52& 18.62& 18.05& 17.49& 0.08& 0.02& 0.01& 0.01& 0.03& 19.72& 18.88& 18.10& 0.03& 0.03& 0.01& 0.38& 0.90& 0.57& 1.13& 0.77& 0.85& 1.62& 536$\pm$43\
GC0147 &pff\_gc-042 & 13 25 12.21 &-43 16 33.9& 15.67 & 19.91& 19.74& 18.97& 18.47& 17.91& 0.12& 0.02& 0.01& 0.01& 0.03& 19.82& 19.19& 18.47& 0.02& 0.02& 0.01& 0.17& 0.78& 0.50& 1.05& 0.72& 0.63& 1.35& 627$\pm$23\
GC0148 &AAT114302 & 13 25 12.34 &-42 58 07.7& 4.11 & 21.87& 21.39& 20.44& 19.90& 19.35& 0.53& 0.11& 0.03& 0.03& 0.04& 21.64& 20.76& 19.97& 0.02& 0.01& 0.01& 0.47& 0.95& 0.55& 1.09& 0.79& 0.88& 1.67& 754$\pm$143\
GC0149 &pff\_gc-043 & 13 25 12.45 &-43 14 07.4& 13.27 & 20.51& 20.22& 19.40& 18.89& 18.33& 0.20& 0.02& 0.01& 0.01& 0.03& 20.38& 19.67& 18.90& 0.03& 0.02& 0.01& 0.29& 0.82& 0.52& 1.07& 0.77& 0.70& 1.48& 527$\pm$33\
GC0150 & WHH-10 & 13 25 12.84 &-42 56 59.8& 4.95 & 20.62& 20.36& 19.57& 19.07& 18.48& 0.12& 0.03& 0.01& 0.01& 0.03& 20.43& 19.90& 19.10& 0.03& 0.02& 0.01& 0.26& 0.79& 0.50& 1.09& 0.80& 0.54& 1.33& 664$\pm$141\
GC0151 & R261 & 13 25 12.90 &-43 07 59.1& 7.35 & 19.65& 19.15& 18.20& 17.60& 17.03& 0.05& 0.01& 0.01& 0.01& 0.03& 19.42& 18.49& 17.62& 0.02& 0.01& 0.01& 0.51& 0.94& 0.60& 1.17& 0.88& 0.93& 1.81& 615$\pm$4\
GC0152 &pff\_gc-044 & 13 25 13.19 &-43 16 35.6& 15.67 & 20.29& 20.11& 19.37& 18.87& 18.33& 0.16& 0.02& 0.01& 0.01& 0.03& 20.19& 19.59& 18.87& 0.02& 0.02& 0.01& 0.17& 0.74& 0.50& 1.04& 0.72& 0.60& 1.32& 568$\pm$53\
GC0153 &C134 & 13 25 13.20 &-43 02 31.3& 2.97 & - & - & - & - & - & - & - & - & - & - & 22.45& 21.64& 20.71& 0.07& 0.05& 0.07& - & - & - & - & 0.93& 0.80& 1.74& -\
GC0154 &pff\_gc-045 & 13 25 13.31 &-42 52 12.4& 9.31 & 21.57& 20.86& 19.82& 19.21& 18.53& 0.19& 0.04& 0.01& 0.01& 0.03& 21.38& 20.32& 19.40& 0.07& 0.06& 0.04& 0.71& 1.04& 0.62& 1.29& 0.93& 1.06& 1.99& 474$\pm$23\
GC0155 & R259 & 13 25 13.88 &-43 07 32.5& 6.87 & 22.13& 21.70& 20.70& 20.11& 19.48& 0.37& 0.08& 0.02& 0.02& 0.03& 21.96& 21.07& 20.10& 0.02& 0.01& 0.01& 0.43& 1.00& 0.60& 1.23& 0.96& 0.89& 1.85& 628$\pm$48\
GC0156 &HGHH-G271 & 13 25 13.95 &-42 57 42.6& 4.25 & 19.45& 19.36& 18.72& 18.32& 17.90& 0.07& 0.02& 0.01& 0.01& 0.03& 19.40& 18.92& 18.30& 0.01& 0.01& - & 0.09& 0.63& 0.40& 0.82& 0.62& 0.47& 1.09& 353$\pm$27\
GC0157 &C135 & 13 25 14.07 &-43 00 51.8& 2.49 & - & - & - & - & - & - & - & - & - & - & - & - & 19.90& - & - & - & - & - & - & - & - & - & - & -\
GC0158 &C136 & 13 25 14.07 &-43 03 35.0& 3.47 & - & - & - & - & - & - & - & - & - & - & 22.48& 21.50& 21.19& 0.04& 0.02& 0.02& - & - & - & - & 0.30& 0.99& 1.29& -\
GC0159 & WHH-11/K-051 & 13 25 14.24 &-43 07 23.5& 6.71 & 21.26& 20.63& 19.55& 18.90& 18.21& 0.18& 0.04& 0.01& 0.01& 0.03& 21.02& 19.95& 18.95& 0.03& 0.02& 0.01& 0.64& 1.07& 0.65& 1.34& 1.00& 1.06& 2.07& 582$\pm$30\
GC0160 &pff\_gc-046 & 13 25 14.83 &-43 41 10.6& 40.10 & 19.91& 19.72& 18.93& 18.43& 17.95& 0.11& 0.02& 0.01& 0.01& 0.03& 19.87& 19.14& 18.37& 0.01& - & 0.01& 0.20& 0.79& 0.50& 0.98& 0.77& 0.73& 1.50& 532$\pm$21\
GC0161 &pff\_gc-047 & 13 25 15.12 &-42 50 30.4& 10.88 & 20.92& 20.72& 19.87& 19.37& 18.87& 0.09& 0.03& 0.01& 0.01& 0.03& 20.87& 20.22& 19.41& 0.02& 0.02& 0.01& 0.20& 0.85& 0.50& 0.99& 0.81& 0.65& 1.46& 717$\pm$55\
GC0162 &AAT114769 & 13 25 15.12 &-42 57 45.7& 4.08 & 21.20& 21.00& 20.25& 19.76& 19.25& 0.30& 0.09& 0.03& 0.03& 0.04& 21.11& 20.52& 19.80& 0.02& 0.02& 0.01& 0.20& 0.75& 0.49& 1.00& 0.71& 0.59& 1.30& 650$\pm$169\
GC0163 & R257 & 13 25 15.24 &-43 08 39.2& 7.84 & 22.29& 21.77& 20.78& 20.18& 19.58& 0.35& 0.07& 0.02& 0.02& 0.03& 22.02& 21.09& 20.16& 0.02& 0.02& 0.02& 0.51& 1.00& 0.60& 1.20& 0.92& 0.93& 1.85& 339$\pm$51\
GC0164 &pff\_gc-048 & 13 25 15.79 &-42 49 15.1& 12.09 & 20.92& 20.67& 19.85& 19.30& 18.78& 0.09& 0.03& 0.01& 0.01& 0.03& 20.85& 20.13& 19.41& 0.02& 0.02& 0.02& 0.25& 0.82& 0.55& 1.07& 0.71& 0.72& 1.44& 535$\pm$53\
GC0165 &AAT114913 & 13 25 15.93 &-43 06 03.3& 5.35 & 21.69& 21.33& 20.39& 19.88& 19.29& 0.34& 0.08& 0.02& 0.02& 0.04& 21.60& 20.72& 19.86& 0.02& 0.02& 0.02& 0.36& 0.94& 0.52& 1.10& 0.86& 0.88& 1.74& 563$\pm$63\
GC0166 &pff\_gc-049 & 13 25 16.06 &-43 05 06.5& 4.49 & 20.74& 20.45& 19.59& 19.05& 18.50& 0.15& 0.04& 0.01& 0.01& 0.03& 20.64& 19.85& 19.07& 0.03& 0.03& 0.03& 0.29& 0.86& 0.54& 1.09& 0.78& 0.79& 1.57& 674$\pm$32\
GC0167 &C137 & 13 25 16.06 &-43 02 19.3& 2.42 & - & - & - & - & - & - & - & - & - & - & - & - & 19.04& - & - & - & - & - & - & - & - & - & - & -\
GC0168 &VHH81-05/C5 & 13 25 16.12 &-42 52 58.2& 8.44 & 18.70& 18.49& 17.68& 17.13& 16.68& 0.03& 0.01& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.21& 0.81& 0.55& 1.00& - & - & - & 555$\pm$3\
GC0169 & HCH01 & 13 25 16.22 &-42 59 43.4& 2.52 & 18.69& 18.36& 17.47& 16.94& 16.40& 0.52& 0.15& 0.04& 0.03& 0.04& - & - & - & - & - & - & 0.33& 0.89& 0.53& 1.07& - & - & - & 649$\pm$45\
GC0170 &HHH86-33/C33 & 13 25 16.26 &-42 50 53.3& 10.47 & 19.58& 19.34& 18.50& 18.02& 17.50& 0.04& 0.01& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.24& 0.84& 0.48& 1.00& - & - & - & 596$\pm$18\
GC0171 &AAT114993 & 13 25 16.44 &-43 03 33.1& 3.16 & 22.49& 21.50& 20.44& 19.82& 19.32& 1.41& 0.19&.382& 0.308& 0.224& 21.95& 20.81& 19.83& 0.02& 0.02& 0.03& 0.99& 1.06& 0.62& 1.12& 0.98& 1.14& 2.12& 352$\pm$136\
GC0172 & HCH02 & 13 25 16.69 &-43 02 08.7& 2.23 & 19.27& 18.87& 17.93& 17.37& 16.76& 0.98& 0.26& 0.07& 0.05& 0.05& - & - & - & - & - & - & 0.40& 0.94& 0.56& 1.17& - & - & - & 300$\pm$2\
GC0173 &pff\_gc-050 & 13 25 16.73 &-42 50 18.4& 11.02 & 21.63& 21.28& 20.39& 19.86& 19.27& 0.16& 0.04& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.35& 0.89& 0.54& 1.12& - & - & - & 688$\pm$70\
GC0174 &C138 & 13 25 16.91 &-43 03 08.0& 2.79 & - & - & - & - & - & - & - & - & - & - & 21.53& 20.79& 19.97& 0.02& 0.02& 0.03& - & - & - & - & 0.81& 0.74& 1.55& -\
GC0175 & R254 & 13 25 16.96 &-43 09 28.0& 8.54 & 20.52& 20.26& 19.43& 18.91& 18.33& 0.09& 0.03& 0.01& 0.01& 0.03& 20.42& 19.65& 18.91& 0.04& 0.03& 0.03& 0.27& 0.82& 0.52& 1.10& 0.73& 0.77& 1.51& 561$\pm$27\
GC0176 &C139 & 13 25 17.06 &-43 02 44.6& 2.50 & - & - & - & - & - & - & - & - & - & - & 20.73& 19.62& 18.86& 0.03& 0.02& 0.04& - & - & - & - & 0.76& 1.11& 1.87& -\
GC0177 &HGHH-G219 & 13 25 17.31 &-42 58 46.6& 3.03 & 20.69& 19.92& 18.84& 18.20& 17.55& 0.43& 0.08& 0.02& 0.02& 0.03& - & - & - & - & - & - & 0.77& 1.09& 0.63& 1.29& - & - & - & 535$\pm$24\
GC0178 & R253 & 13 25 17.33 &-43 08 39.0& 7.74 & 20.75& 20.49& 19.67& 19.14& 18.64& 0.09& 0.03& 0.01& 0.01& 0.03& 20.68& 20.00 & 19.47& 0.03& 0.03& 0.05& 0.26& 0.81& 0.53& 1.04& 0.53& 0.69& 1.22& 486$\pm$52\
GC0179 &C140 & 13 25 17.42 &-43 03 25.2& 2.94 & - & - & - & - & - & - & - & - & - & - & 21.40& 20.65& 19.83& 0.03& 0.02& 0.03& - & - & - & - & 0.82& 0.75& 1.57& -\
GC0180 &C141 & 13 25 18.14 &-43 02 50.9& 2.43 & - & - & - & - & - & - & - & - & - & - & 22.23& 21.55& 20.95& 0.06& 0.05& 0.06& - & - & - & - & 0.60& 0.68& 1.29& -\
GC0181 & WHH-12 & 13 25 18.27 &-42 53 04.8& 8.25 & 20.36& 20.00& 19.07& 18.47& 17.97& 0.07& 0.02& 0.01& 0.01& 0.03& 20.23& 19.35& 18.71& 0.04& 0.04& 0.01& 0.36& 0.93& 0.60& 1.10& 0.64& 0.89& 1.52& 558$\pm$99\
GC0182 &AAT115339 & 13 25 18.44 &-43 04 09.8& 3.45 & 21.19& 20.91& 20.11& 19.53& 19.01& 0.39& 0.10& 0.03& 0.03& 0.04& 21.11& 20.38& 19.56& 0.02& 0.01& 0.02& 0.27& 0.81& 0.58& 1.09& 0.81& 0.73& 1.55& 618$\pm$108\
GC0183 &C142 & 13 25 18.50 &-43 01 16.4& 1.67 & - & - & - & - & - & - & - & - & - & - & - & - & 17.64& - & - & - & - & - & - & - & - & - & - & -\
GC0184 &AAT320656 & 13 25 19.13 &-43 12 03.8& 11.03 & 20.61& 20.50& 19.81& 19.35& 18.89& 0.08& 0.03& 0.01& 0.01& 0.03& 20.57& 20.01& 19.35& 0.02& 0.01& 0.02& 0.10& 0.70& 0.46& 0.92& 0.66& 0.56& 1.22& 453$\pm$233\
GC0185 &HGHH-G331 & 13 25 19.50 &-43 02 28.4& 1.99 & 20.00& 19.60& 18.92& 18.47& 17.93& 0.55& 0.14& 0.05& 0.04& 0.05& - & - & - & - & - & - & 0.40& 0.68& 0.44& 0.99& - & - & - & 371$\pm$22\
GC0186 &AAT115561 & 13 25 19.83 &-42 58 27.2& 3.05 & 22.13& 21.47& 20.50& 19.92& 19.29& 1.44& 0.26& 0.07& 0.06& 0.06& - & 20.98& 20.15& 1.00& 0.04& 0.02& 0.66& 0.98& 0.57& 1.20& 0.83& - & - & 514$\pm$46\
GC0187 & R247 & 13 25 19.99 &-43 07 44.1& 6.73 & 21.86& 21.20& 20.11& 19.40& 18.62& 0.28& 0.05& 0.02& 0.01& 0.03& 21.73& 20.65& 19.75& 0.05& 0.04& 0.04& 0.66& 1.10& 0.70& 1.49& 0.90& 1.08& 1.98& 662$\pm$48\
GC0188 &AAT115605 & 13 25 20.44 &-42 54 08.5& 7.13 & 21.44& 21.22& 20.47& 20.00& 19.50& 0.18& 0.05& 0.02& 0.02& 0.03& 21.36& 20.76& 20.11& 0.01& 0.03& 0.01& 0.22& 0.75& 0.47& 0.97& 0.66& 0.60& 1.25& 713$\pm$131\
GC0189 &AAT115679 & 13 25 20.72 &-43 06 35.9& 5.60 & 21.93& 21.49& 20.44& 19.80& 19.15& 0.40& 0.09& 0.02& 0.02& 0.03& 21.71& 20.79& 19.84& 0.02& 0.02& 0.03& 0.44& 1.05& 0.64& 1.30& 0.95& 0.92& 1.87& 511$\pm$165\
GC0190 & WHH-13/HH-090 & 13 25 21.29 &-42 49 17.7& 11.91 & 20.75& 20.22& 19.20& 18.61& 17.95& 0.08& 0.02& 0.01& 0.01& 0.03& 20.59& 19.52& 18.74& 0.03& 0.03& 0.02&0.53& 1.02& 0.59& 1.25& 0.78& 1.08& 1.85& 444$\pm$33\
GC0191 &AAT321194 & 13 25 21.32 &-43 23 59.3& 22.87 & 21.46& 21.03& 20.06& 19.48& 18.82& 0.38& 0.03& 0.01& 0.01& 0.03& 21.33& 20.42& 19.53& 0.03& 0.02& 0.01& 0.43& 0.97& 0.58& 1.24& 0.90& 0.90& 1.80& 254$\pm$169\
GC0192 &HGHH-06/C6 & 13 25 22.19 &-43 02 45.6& 1.89 & 18.65& 18.17& 17.21& 16.61& 16.03& 0.24& 0.06& 0.02& 0.01& 0.03& - & - & - & - & - & - & 0.48& 0.96& 0.60& 1.18& - & - & - & 855$\pm$2\
GC0193 &pff\_gc-051 & 13 25 22.35 &-43 15 00.1& 13.89 & 21.15& 20.73& 19.79& 19.20& 18.61& 0.10& 0.02& 0.01& 0.01& 0.03& 21.00& 20.10& 19.22& 0.03& 0.03& 0.02& 0.42& 0.94& 0.59& 1.18& 0.88& 0.90& 1.78& 468$\pm$39\
GC0194 &C143 & 13 25 23.20 &-43 03 12.9& 2.22 & - & - & - & - & - & - & - & - & - & - & 21.87& 21.35& 20.52& 0.05& 0.04& 0.06& - & - & - & - & 0.82& 0.52& 1.34& -\
GC0195 &AAT116025 & 13 25 23.46 &-42 53 26.2& 7.75 & 21.23& 20.88& 20.00& 19.48& 18.85& 0.14& 0.04& 0.01& 0.01& 0.03& 21.16& 20.32& 19.62& 0.04& 0.05& 0.01& 0.35& 0.88& 0.51& 1.15& 0.70& 0.84& 1.54& 545$\pm$64\
GC0196 &AAT116220 & 13 25 24.40 &-43 07 58.9& 6.86 & 20.76& 20.39& 19.47& 18.91& 18.36& 0.11& 0.03& 0.01& 0.01& 0.03& 20.62& 19.72& 18.92& 0.03& 0.03& 0.03& 0.37& 0.92& 0.56& 1.11& 0.80& 0.90& 1.70& 524$\pm$41\
GC0197 &AAT116385 & 13 25 25.39 &-42 58 21.5& 2.82 & 21.71& 21.23& 20.26& 19.59& 18.99& 1.22& 0.26& 0.07& 0.05& 0.06& 21.58& 20.68& 19.94& 0.02& 0.02& 0.01& 0.48& 0.97& 0.66& 1.27& 0.74& 0.90& 1.64& 550$\pm$40\
GC0198 & WHH-14 & 13 25 25.49 &-42 56 31.2& 4.64 & 20.99& 20.83& 20.06& 19.56& 19.05& 0.18& 0.05& 0.02& 0.02& 0.03& 20.98& 20.36& 19.68& 0.02& 0.03& 0.01& 0.15& 0.77& 0.50& 1.01& 0.68& 0.62& 1.30& 461$\pm$75\
GC0199 &AAT204119 & 13 25 25.70 &-42 37 40.9& 23.47 & 20.59& 20.49& 19.81& 19.35& 18.93& 0.18& 0.02& 0.01& 0.01& 0.03& 20.60& 20.04& 19.42& 0.01& - & - & 0.10& 0.67& 0.47& 0.88& 0.62& 0.56& 1.18& 404$\pm$74\
GC0200 &pff\_gc-052 & 13 25 25.75 &-43 05 16.5& 4.14 & 20.88& 20.70& 19.88& 19.39& 18.86& 0.19& 0.06& 0.02& 0.02& 0.03& 20.86& 20.17& 19.42& 0.03& 0.03& 0.03& 0.18& 0.82& 0.48& 1.01& 0.75& 0.69& 1.44& 462$\pm$52\
GC0201 &HGHH-46/C46 & 13 25 25.97 &-43 03 25.7& 2.30 & 19.74& 19.71& 19.10& 18.74& 18.30& 0.37& 0.12& 0.04& 0.04& 0.05& 19.77& 19.30& 18.64& 0.01& 0.02& 0.22& 0.03& 0.61& 0.36& 0.80& 0.66& 0.47& 1.13& 508$\pm$19\
GC0202 &C144 & 13 25 26.28 &-43 04 38.5& 3.50 & - & - & - & - & - & - & - & - & - & - & 23.22& 22.36& 21.96& 0.06& 0.05& 0.05& - & - & - & - & 0.41& 0.85& 1.26& -\
GC0203 &AAT116531 & 13 25 26.75 &-43 08 53.4& 7.74 & 20.92& 20.65& 19.83& 19.34& 18.83& 0.10& 0.03& 0.01& 0.01& 0.03& 20.77& 19.95& 19.94& 0.07& 0.06& 0.02& 0.27& 0.82& 0.49& 1.00& 0.01& 0.82& 0.83& 267$\pm$71\
GC0204 & WHH-15 & 13 25 26.78 &-42 52 39.9& 8.48 & 20.73& 20.56& 19.81& 19.36& 18.87& 0.09& 0.03& 0.01& 0.01& 0.03& 20.68& 20.80& 19.53& 0.03& 0.02& 0.01& 0.16& 0.75& 0.46& 0.94& 1.27&-0.12& 1.15& 513$\pm$53\
GC0205 & R235 & 13 25 26.82 &-43 09 40.5& 8.53 & 20.55& 20.30& 19.53& 19.02& 18.51& 0.08& 0.02& 0.01& 0.01& 0.03& 20.42& 19.73& 19.01& 0.04& 0.03& 0.04& 0.25& 0.77& 0.51& 1.03& 0.72& 0.69& 1.41& 498$\pm$28\
GC0206 & WHH-16/K-102 & 13 25 27.97 &-43 04 02.2& 2.89 & 20.90& 20.23& 19.18& 18.56& 17.90& 0.56& 0.11& 0.03& 0.02& 0.04& 20.64& 19.52& 18.60& 0.04& 0.03& 0.03& 0.66& 1.06& 0.61& 1.28& 0.93& 1.11& 2.04& 661$\pm$47\
GC0207 & HCH13 & 13 25 28.69 &-43 02 55.0& 1.78 & 21.32& 20.82& 19.89& 19.30& 18.63& 2.53& 0.55& 0.15& 0.12& 0.12& - & - & - & - & - & - & 0.50& 0.93& 0.59& 1.26& - & - & - & 641$\pm$24\
GC0208 &C145 & 13 25 28.81 &-43 04 21.6& 3.22 & - & - & - & - & - & - & - & - & - & - & 19.32& 18.54& 17.81& 0.01& 0.01& 0.02& - & - & - & - & 0.73& 0.78& 1.51& -\
GC0209 & WHH-17 & 13 25 29.25 &-42 57 47.1& 3.38 & 19.68& 19.39& 18.57& 18.06& 17.48& 0.15& 0.04& 0.02& 0.01& 0.03& 19.54& 18.77& 18.13& 0.02& 0.03& - & 0.29& 0.82& 0.51& 1.09& 0.65& 0.76& 1.41& 619$\pm$45\
GC0210 &AAT116969 & 13 25 29.41 &-42 53 25.6& 7.73 & 21.55& 21.28& 20.46& 20.10& 19.48& 0.21& 0.06& 0.02& 0.02& 0.03& 21.45& 20.77& 20.10& 0.03& 0.02& 0.01& 0.27& 0.82& 0.35& 0.98& 0.68& 0.68& 1.36& 446$\pm$61\
GC0211 &HGHH-G169 & 13 25 29.43 &-42 58 09.9& 3.00 & 21.39& 20.58& 19.49& 18.92& 18.12& 0.81& 0.13& 0.03& 0.03& 0.04& 21.00& 19.80& 18.94& 0.01& 0.02& 0.01& 0.81& 1.10& 0.57& 1.37& 0.86& 1.21& 2.07& 643$\pm$24\
GC0212 &pff\_gc-053 & 13 25 29.62 &-42 54 44.5& 6.42 & 20.91& 20.29& 19.26& 18.73& 17.98& 0.14& 0.03& 0.01& 0.01& 0.03& 20.65& 19.54& 18.78& 0.03& 0.04& 0.01& 0.62& 1.03& 0.53& 1.29& 0.76& 1.11& 1.86& 439$\pm$20\
GC0213 & R229 & 13 25 29.74 &-43 11 42.8& 10.57 & 20.70& 20.49& 19.72& 19.20& 18.69& 0.08& 0.02& 0.01& 0.01& 0.03& 20.62& 19.94& 19.22& 0.04& 0.03& 0.04& 0.21& 0.77& 0.52& 1.03& 0.73& 0.68& 1.41& 517$\pm$66\
GC0214 &HCH15 & 13 25 29.80 &-43 00 07.0& 1.10 & 19.27& 18.87& 17.93& 17.37& 16.76& 0.98& 0.26& 0.07& 0.05& 0.05& - & - & - & - & - & - & 0.04& 0.94& 0.56& 1.17& - & - & - & 519$\pm$1\
GC0215 &C146 & 13 25 29.87 &-43 05 09.2& 4.03 & - & - & - & - & - & - & - & - & - & - & 21.71& 20.78& 19.94& 0.03& 0.02& 0.02& - & - & - & - & 0.84& 0.93& 1.77& -\
GC0216 & WHH-18 & 13 25 30.07 &-42 56 46.9& 4.39 & 19.85& 19.36& 18.36& 17.85& 17.16& 0.09& 0.02& 0.01& 0.01& 0.03& 19.63& 18.60& 17.88& 0.01& 0.03& - & 0.50& 0.99& 0.51& 1.20& 0.72& 1.03& 1.75& 752$\pm$25\
GC0217 &pff\_gc-054 & 13 25 30.28 &-43 41 53.6& 40.75 & 20.65& 20.60& 19.99& 19.57& 19.09& 0.19& 0.03& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.05& 0.62& 0.41& 0.89& - & - & - & 297$\pm$40\
GC0218 & HCH16 & 13 25 30.29 &-42 59 34.8& 1.64 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & 458$\pm$2\
GC0219 &HHH86-15/C15/R226 & 13 25 30.41 &-43 11 49.6& 10.69 & 20.11& 19.56& 18.56& 17.94& 17.35& 0.06& 0.02& 0.01& 0.01& 0.03& 19.80& 18.68& 17.92& 0.05& 0.05& 0.05& 0.55& 1.00& 0.62& 1.21& 0.76& 1.12& 1.88& 644$\pm$1\
GC0220 &C147 & 13 25 30.65 &-43 03 47.1& 2.70 & - & - & - & - & - & - & - & - & - & - & 21.58& 20.88& 20.05& 0.05& 0.05& 0.03& - & - & - & - & 0.83& 0.69& 1.52& -\
GC0221 &pff\_gc-055 & 13 25 30.72 &-42 48 13.4& 12.94 & 20.87& 20.68& 19.92& 19.44& 18.97& 0.08& 0.03& 0.01& 0.01& 0.03& 20.83& 20.20& 19.54& 0.02& 0.02& 0.02& 0.19& 0.76& 0.48& 0.96& 0.66& 0.62& 1.28& 485$\pm$47\
GC0222 &AAT205071 & 13 25 30.74 &-42 30 16.0& 30.89 & 19.65& 19.61& 18.86& 18.37& 17.96& 0.09& 0.01& 0.01& 0.01& 0.03& 19.68& 19.10& 18.50& 0.02& 0.02& - & 0.04& 0.74& 0.49& 0.90& 0.60& 0.58& 1.18& 288$\pm$61\
GC0223 & WHH-19 & 13 25 31.03 &-42 50 14.9& 10.92 & 18.47& 18.16& 17.29& 16.78& 16.24& 0.02& 0.01& 0.01& 0.01& 0.03& 18.08& 17.67& 16.91& 0.06& 0.06& 0.02& 0.31& 0.87& 0.51& 1.05& 0.76& 0.42& 1.18& 451$\pm$40\
GC0224 &AAT117287 & 13 25 31.08 &-43 04 17.0& 3.20 & 22.20& 21.44& 20.36& 19.82& 19.13& 1.22& 0.20& 0.05& 0.04& 0.05& 21.82& 20.56& 20.45& 0.04& 0.03& 0.03& 0.76& 1.08& 0.54& 1.23& 0.11& 1.27& 1.37& 554$\pm$60\
GC0225 &HGHH-G292 & 13 25 31.48 &-42 58 08.3& 3.09 & 21.77& 21.05& 19.97& 19.33& 18.64& 1.04& 0.18& 0.05& 0.03& 0.04& 21.48& 20.85& 19.53& 0.01& 0.02& 0.01& 0.72& 1.08& 0.64& 1.34& 1.33& 0.63& 1.95& 655$\pm$43\
GC0226 & HCH18 & 13 25 31.60 &-43 00 02.8& 1.32 & 21.06& 20.99& 19.93& 19.20& 18.43& 2.42& 0.76& 0.19& 0.13& 0.12& - & - & - & - & - & - & 0.07& 1.06& 0.73& 1.50& - & - & - & 455$\pm$1\
GC0227 &AAT117322 & 13 25 31.73 &-42 55 15.7& 5.93 & 21.14& 20.91& 20.08& 19.55& 19.07& 0.19& 0.05& 0.02& 0.02& 0.03& 21.07& 20.38& 19.75& 0.04& 0.05& 0.01& 0.22& 0.84& 0.52& 1.01& 0.63& 0.69& 1.32& 636$\pm$133\
GC0228 &HGHH-44/C44 & 13 25 31.73 &-43 19 22.6& 18.25 & 19.76& 19.50& 18.69& 18.14& 17.66& 0.04& 0.01& 0.01& 0.01& 0.03& 19.59& 18.85& 18.15& 0.03& 0.02& 0.01& 0.26& 0.80& 0.55& 1.03& 0.70& 0.74& 1.44& 505$\pm$1\
GC0229 &C148 & 13 25 31.75 &-43 05 46.0& 4.68 & - & - & - & - & - & - & - & - & - & - & 21.25& 20.79& 20.21& 0.07& 0.06& 0.05& - & - & - & - & 0.58& 0.46& 1.04& -\
GC0230 & R224/C149 & 13 25 32.33 &-43 07 17.1& 6.20 & 21.07& 20.90& 20.15& 19.64& 19.19& 0.15& 0.04& 0.02& 0.02& 0.03& 21.02& 20.42& 19.68& 0.03& 0.02& 0.02& 0.16& 0.76& 0.50& 0.96& 0.74& 0.60& 1.34& 389$\pm$45\
GC0231 &HGHH-G359 & 13 25 32.42 &-42 58 50.2& 2.47 & 20.44& 19.86& 18.86& 18.24& 17.64& 0.45& 0.10& 0.03& 0.02& 0.04& 20.13& 19.09& 18.33& 0.01& 0.01& - & 0.58& 1.00& 0.62& 1.22& 0.76& 1.04& 1.80& 489$\pm$34\
GC0232 &pff\_gc-056 & 13 25 32.80 &-42 56 24.4& 4.83 & 19.62& 19.43& 18.64& 18.15& 17.64& 0.06& 0.02& 0.01& 0.01& 0.03& 19.51& 18.82& 18.22& 0.03& 0.04& - & 0.19& 0.79& 0.48& 1.00& 0.60& 0.69& 1.29& 306$\pm$27\
GC0233 & R223 & 13 25 32.80 &-43 07 02.2& 5.97 & 20.08& 19.67& 18.77& 18.23& 17.64& 0.07& 0.02& 0.01& 0.01& 0.03& 19.89& 18.98& 18.19& 0.03& 0.03& 0.03& 0.41& 0.91& 0.54& 1.13& 0.79& 0.92& 1.71& 776$\pm$1\
GC0234 & K-131 & 13 25 32.88 &-43 04 29.2& 3.48 & 21.23& 20.44& 19.37& 18.77& 18.12& 0.38& 0.07& 0.02& 0.02& 0.03& 20.84& 19.65& 18.74& 0.03& 0.02& 0.03& 0.79& 1.08& 0.59& 1.24& 0.91& 1.20& 2.10& 639$\pm$44\
GC0235 &pff\_gc-057 & 13 25 33.17 &-42 59 03.2& 2.33 & 22.21& 21.38& 20.35& 19.75& 18.95& 2.11& 0.37& 0.09& 0.07& 0.07& 21.67& 20.75& 19.86& 0.03& 0.02& 0.01& 0.82& 1.04& 0.60& 1.40& 0.90& 0.91& 1.81& 515$\pm$12\
GC0236 &C150 & 13 25 33.82 &-43 02 49.6& 2.03 & - & - & - & - & - & - & - & - & - & - & - & - & 19.78& - & - & - & - & - & - & - & - & - & - & -\
GC0237 &C151 & 13 25 33.93 &-43 03 51.4& 2.94 & - & - & - & - & - & - & - & - & - & - & 22.27& 20.90& 19.95& 0.03& 0.01& 0.03& - & - & - & - & 0.95& 1.37& 2.32& -\
GC0238 & MAGJM-11 & 13 25 33.94 &-42 59 39.4& 1.89 & 19.80& 19.64& 18.87& 18.44& 17.83& 0.80& 0.25& 0.07& 0.06& 0.07& - & - & - & - & - & - & 0.16& 0.77& 0.43& 1.04& - & - & - & 444$\pm$17\
GC0239 &HGHH-G206 & 13 25 34.10 &-42 59 00.7& 2.44 & 20.68& 20.09& 19.10& 18.58& 17.91& 0.56& 0.12& 0.03& 0.03& 0.04& 20.37& 19.36& 18.61& 0.02& 0.03& 0.01& 0.58& 0.99& 0.52& 1.19& 0.75& 1.00& 1.75& 600$\pm$24\
GC0240 &HGHH-45/C45 & 13 25 34.25 &-42 56 59.1& 4.33 & 19.99& 19.81& 19.03& 18.60& 18.05& 0.10& 0.03& 0.01& 0.01& 0.03& 19.91& 19.17& 18.67& 0.08& 0.07& 0.02& 0.18& 0.78& 0.43& 0.98& 0.50& 0.75& 1.25& 612$\pm$34\
GC0241 & WHH-20 & 13 25 34.36 &-42 51 05.9& 10.12 & 20.30& 19.93& 19.01& 18.52& 17.90& 0.06& 0.02& 0.01& 0.01& 0.03& 20.11& 19.25& 18.59& 0.04& 0.05& 0.01& 0.37& 0.92& 0.49& 1.10& 0.66& 0.86& 1.52& 259$\pm$33\
GC0242 &C152 & 13 25 34.64 &-43 03 16.4& 2.48 & - & - & - & - & - & - & - & - & - & - & - & - & 17.82& - & - & - & - & - & - & - & - & - & - & -\
GC0243 &C153 & 13 25 34.64 &-43 03 27.8& 2.65 & - & - & - & - & - & - & - & - & - & - & - & - & 18.23& - & - & - & - & - & - & - & - & - & - & -\
GC0244 & HCH21 & 13 25 34.65 &-43 03 27.7& 2.65 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & 663$\pm$2\
GC0245 &C154 & 13 25 34.71 &-43 03 30.2& 2.69 & - & - & - & - & - & - & - & - & - & - & - & - & 19.58& - & - & - & - & - & - & - & - & - & - & -\
GC0246 &HHH86-16/C16 & 13 25 35.00 &-42 36 05.0& 25.10 & 20.01& 19.57& 18.59& 18.00& 17.41& 0.12& 0.01& 0.01& 0.01& 0.03& 19.85& 18.83& 18.12& 0.03& 0.03& 0.01& 0.44& 0.98& 0.59& 1.19& 0.70& 1.02& 1.73& 538$\pm$16\
GC0247 &pff\_gc-058 & 13 25 35.12 &-42 56 45.3& 4.60 & 19.67& 19.47& 18.68& 18.19& 17.70& 0.07& 0.02& 0.01& 0.01& 0.03& 19.55& 18.85& 18.29& 0.03& 0.03& 0.01& 0.20& 0.78& 0.49& 0.99& 0.56& 0.70& 1.26& 365$\pm$22\
GC0248 & K-144 & 13 25 35.16 &-42 53 01.0& 8.25 & 22.10& 21.31& 20.25& 19.63& 18.94& 0.33& 0.05& 0.02& 0.01& 0.03& 21.72& 20.44& 19.75& 0.01& 0.01& 0.01& 0.79& 1.06& 0.62& 1.31& 0.69& 1.28& 1.97& 593$\pm$43\
GC0249 & WHH-21 & 13 25 35.22 &-43 12 01.5& 10.97 & 19.64& 19.60& 19.00& 18.63& 18.25& 0.04& 0.01& 0.01& 0.01& 0.03& 19.61& 19.15& 18.62& 0.02& 0.01& 0.02& 0.04& 0.60& 0.37& 0.75& 0.53& 0.46& 0.99& 243$\pm$61\
GC0250 & WHH-22 & 13 25 35.31 &-43 05 29.0& 4.56 & - & - & - & - & - & - & - & - & - & - & 19.66& 18.80& 18.03& 0.04& 0.04& 0.04& - & - & - & - & 0.77& 0.86& 1.63& 492$\pm$126\
GC0251 & MAGJM-08 & 13 25 35.50 &-42 59 35.2& 2.12 & 20.76& 20.18& 19.17& 18.68& 17.94& 1.15& 0.24& 0.06& 0.05& 0.05& - & - & - & - & - & - & 0.58& 1.01& 0.49& 1.22& - & - & - & 701$\pm$28\
GC0252 & R215 & 13 25 35.64 &-43 08 36.8& 7.61 & 22.06& 21.68& 20.80& 20.27& 19.70& 0.27& 0.06& 0.02& 0.02& 0.03& 21.98& 21.18& 20.35& 0.03& 0.03& 0.04& 0.38& 0.87& 0.53& 1.11& 0.83& 0.79& 1.63& 543$\pm$57\
GC0253 & R213 & 13 25 35.93 &-43 07 27.9& 6.50 & 22.70& 22.03& 20.92& 20.35& 19.62& 0.56& 0.10& 0.03& 0.02& 0.03& 22.34& 21.26& 20.34& 0.03& 0.01& 0.02& 0.67& 1.11& 0.58& 1.30& 0.92& 1.08& 2.00& 532$\pm$52\
GC0254 &AAT117899 & 13 25 36.05 &-42 53 40.3& 7.63 & 21.70& 21.17& 20.17& 19.62& 18.97& 0.23& 0.05& 0.02& 0.01& 0.03& 21.51& 20.51& 19.70& 0.03& 0.03& 0.01& 0.53& 1.01& 0.54& 1.20& 0.81& 1.01& 1.81& 609$\pm$116\
GC0255 &C155 & 13 25 36.47 &-43 08 03.5& 7.10 & - & - & - & - & - & - & - & - & - & - & 22.98& 22.21& 21.36& 0.06& 0.05& 0.05& - & - & - & - & 0.85& 0.77& 1.63& -\
GC0256 & MAGJM-06 & 13 25 36.69 &-42 59 59.2& 2.02 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & 443$\pm$15\
GC0257 &AAT118198 & 13 25 37.47 &-43 05 44.9& 4.94 & 21.49& 20.68& 19.64& 18.98& 18.29& 0.31& 0.05& 0.02& 0.01& 0.03& 21.14& 19.94& 19.03& 0.04& 0.02& 0.03& 0.81& 1.04& 0.66& 1.35& 0.91& 1.20& 2.11& 575$\pm$36\
GC0258 &AAT118314 & 13 25 38.03 &-43 16 59.2& 15.95 & 20.69& 20.62& 19.89& 19.45& 18.98& 0.08& 0.03& 0.01& 0.01& 0.03& 20.67& 20.13& 19.42& 0.01& 0.01& 0.01& 0.08& 0.72& 0.44& 0.92& 0.71& 0.53& 1.24& 257$\pm$59\
GC0259 & R209 & 13 25 38.13 &-43 13 02.2& 12.04 & 21.92& 21.51& 20.62& 20.05& 19.43& 0.20& 0.05& 0.02& 0.01& 0.03& 21.91& 20.95& 20.13& 0.04& 0.04& 0.04& 0.41& 0.90& 0.57& 1.19& 0.83& 0.96& 1.79& 558$\pm$40\
GC0260 &C156 & 13 25 38.43 &-43 05 02.6& 4.37 & - & - & - & - & - & - & - & - & - & - & 19.76& 18.55& 17.65& 0.02& 0.02& 0.03& - & - & - & - & 0.90& 1.21& 2.11& -\
GC0261 &C157 & 13 25 38.45 &-43 03 28.9& 3.06 & - & - & - & - & - & - & - & - & - & - & - & - & 19.21& - & - & - & - & - & - & - & - & - & - & -\
GC0262 &HGHH-G268 & 13 25 38.61 &-42 59 19.6& 2.71 & 20.23& 19.83& 18.93& 18.36& 17.79& 0.34& 0.08& 0.03& 0.02& 0.04& 20.05& 19.15& 18.44& 0.02& 0.02& - & 0.40& 0.91& 0.56& 1.13& 0.71& 0.91& 1.62& 436$\pm$43\
GC0263 &C158 & 13 25 38.76 &-43 05 34.5& 4.88 & - & - & - & - & - & - & - & - & - & - & 21.67& 20.86& 20.06& 0.05& 0.04& 0.05& - & - & - & - & 0.80& 0.81& 1.61& -\
GC0264 &C159 & 13 25 39.17 &-43 04 33.8& 4.02 & - & - & - & - & - & - & - & - & - & - & 21.91& 20.82& 19.93& 0.02& 0.02& 0.02& - & - & - & - & 0.90& 1.09& 1.99& -\
GC0265 & MAGJM-01 & 13 25 39.33 &-43 00 48.8& 2.17 & 19.90& 19.52& 18.61& 18.06& 17.50& 0.32& 0.08& 0.02& 0.02& 0.03& - & - & - & - & - & - & 0.37& 0.91& 0.55& 1.11& - & - & - & 645$\pm$36\
GC0266 &pff\_gc-059 & 13 25 39.65 &-43 04 01.4& 3.62 & 21.53& 20.97& 19.96& 19.35& 18.73& 0.55& 0.11& 0.03& 0.02& 0.04& 21.42& 20.33& 19.48& 0.05& 0.04& 0.04& 0.56& 1.01& 0.61& 1.23& 0.86& 1.09& 1.94& 525$\pm$27\
GC0267 &HGHH-17/C17 & 13 25 39.73 &-42 55 59.2& 5.62 & 18.82& 18.51& 17.63& 17.10& 16.57& 0.04& 0.01& 0.01& 0.01& 0.03& 18.61& 17.72& 17.19& 0.04& 0.05& 0.01& 0.32& 0.88& 0.53& 1.06& 0.53& 0.89& 1.42& 782$\pm$2\
GC0268 &HHH86-18/C18/K-163 & 13 25 39.88 &-43 05 01.9& 4.49 & 18.79& 18.42& 17.53& 16.97& 16.43& 0.04& 0.01& 0.01& 0.01& 0.03& 18.49& 17.51& 16.89& 0.05& 0.04& 0.05& 0.38& 0.89& 0.56& 1.10& 0.62& 0.99& 1.60& 480$\pm$2\
GC0269 &C160 & 13 25 40.09 &-43 03 07.1& 3.01 & - & - & - & - & - & - & - & - & - & - & - & - & 19.99& - & - & - & - & - & - & - & - & - & - & -\
GC0270 &C101 & 13 25 40.47 &-42 56 02.7& 5.65 & - & - & 20.34& - & - & - & - & - & - & - & 21.84& 20.90& 20.08& 0.03& 0.04& 0.01& - & - & - & - & 0.82& 0.94& 1.76& -\
GC0271 & R204/C161 & 13 25 40.52 &-43 07 17.9& 6.59 & 21.58& 20.84& 19.80& 19.16& 18.41& 0.21& 0.04& 0.01& 0.01& 0.03& 21.21& 20.15& 19.26& 0.04& 0.03& 0.03& 0.74& 1.05& 0.63& 1.39& 0.89& 1.06& 1.95& 425$\pm$32\
GC0272 &HGHH-34/C34 & 13 25 40.61 &-43 21 13.6& 20.22 & 19.37& 19.01& 18.12& 17.59& 17.03& 0.04& 0.01& 0.01& 0.01& 0.03& 19.64& 18.54& 17.76& 0.04& 0.03& 0.02& 0.35& 0.89& 0.53& 1.10& 0.77& 1.10& 1.87& 669$\pm$18\
GC0273 &C162 & 13 25 40.87 &-43 05 00.4& 4.56 & - & - & - & - & - & - & - & - & - & - & 21.95& 21.41& 20.91& 0.06& 0.05& 0.01& - & - & - & - & 0.51& 0.54& 1.05& -\
GC0274 & R203 & 13 25 40.90 &-43 08 16.0& 7.52 & 20.66& 20.56& 19.84& 19.36& 18.88& 0.09& 0.03& 0.01& 0.01& 0.03& 20.62& 20.08& 19.38& 0.04& 0.03& 0.03& 0.09& 0.73& 0.48& 0.96& 0.70& 0.54& 1.24& 455$\pm$35\
GC0275 &AAT118874 & 13 25 41.36 &-42 58 08.9& 3.91 & 21.55& 21.02& 19.99& 19.43& 18.81& 0.44& 0.09& 0.03& 0.02& 0.04& 21.49& 20.30& 19.57& 0.05& 0.02& - & 0.53& 1.03& 0.56& 1.18& 0.73& 1.19& 1.92& 570$\pm$82\
GC0276 &C163 & 13 25 41.63 &-43 03 45.8& 3.66 & - & - & - & - & - & - & - & - & - & - & 21.98& 21.25& 20.43& 0.05& 0.04& 0.04& - & - & - & - & 0.81& 0.73& 1.55& -\
GC0277 & R202 & 13 25 42.00 &-43 10 42.2& 9.91 & 20.34& 20.08& 19.26& 18.72& 18.22& 0.06& 0.02& 0.01& 0.01& 0.03& 20.35& 19.74& 18.94& 0.07& 0.03& 0.06& 0.27& 0.81& 0.54& 1.05& 0.80& 0.61& 1.41& 286$\pm$45\
GC0278 &C164 & 13 25 42.09 &-43 03 19.5& 3.43 & - & - & - & - & - & - & - & - & - & - & - & - & 19.60& - & - & - & - & - & - & - & - & - & - & -\
GC0279 &HGHH-G370 & 13 25 42.25 &-42 59 17.0& 3.26 & 19.61& 19.29& 18.39& 17.86& 17.32& 0.17& 0.05& 0.02& 0.01& 0.03& 19.45& 18.59& 17.93& 0.01& 0.03& 0.01& 0.32& 0.90& 0.54& 1.08& 0.66& 0.86& 1.52& 507$\pm$34\
GC0280 &pff\_gc-060 & 13 25 42.43 &-42 59 02.6& 3.43 & 20.11& 19.74& 18.81& 18.24& 17.66& 0.20& 0.05& 0.02& 0.01& 0.03& 19.96& 19.02& 18.33& 0.02& 0.03& 0.01& 0.37& 0.93& 0.57& 1.15& 0.69& 0.95& 1.63& 890$\pm$19\
GC0281 &AAT119058 & 13 25 42.53 &-43 03 41.5& 3.73 & 21.85& 21.42& 20.39& 19.77& 19.18& 0.60& 0.13& 0.04& 0.03& 0.04& 21.69& 20.74& 19.86& 0.04& 0.03& 0.04& 0.43& 1.03& 0.62& 1.21& 0.88& 0.95& 1.83& 385$\pm$87\
GC0282 &pff\_gc-061 & 13 25 42.62 &-42 45 10.9& 16.20 & 21.59& 21.15& 20.25& 19.69& 19.12& 0.14& 0.03& 0.01& 0.01& 0.03& 21.42& 20.55& 19.75& 0.01& 0.01& 0.01& 0.44& 0.90& 0.55& 1.13& 0.80& 0.87& 1.67& 500$\pm$48\
GC0283 &pff\_gc-062 & 13 25 43.23 &-42 58 37.4& 3.81 & 21.13& 20.46& 19.42& 18.83& 18.18& 0.34& 0.06& 0.02& 0.02& 0.03& 20.79& 19.71& 18.92& 0.01& 0.01& 0.01& 0.67& 1.04& 0.59& 1.24& 0.80& 1.08& 1.88& 697$\pm$37\
GC0284 &HGHH-19/C19 & 13 25 43.40 &-43 07 22.8& 6.87 & 19.37& 19.01& 18.12& 17.59& 17.03& 0.04& 0.01& 0.01& 0.01& 0.03& 18.96& 17.83& 18.64& 0.05& 0.04& 0.01& 0.35& 0.89& 0.53& 1.10&-0.80& 1.13& 0.32& 632$\pm$10\
GC0285 &C165 & 13 25 43.43 &-43 04 56.5& 4.77 & - & - & - & - & - & - & - & - & - & - & 20.10& 18.98& 18.17& 0.05& 0.04& 0.05& - & - & - & - & 0.81& 1.12& 1.93& -\
GC0286 &pff\_gc-063 & 13 25 43.80 &-43 07 54.9& 7.39 & 20.66& 20.55& 19.83& 19.37& 18.89& 0.08& 0.03& 0.01& 0.01& 0.03& 20.63& 20.13& 19.45& 0.04& 0.03& 0.03& 0.11& 0.72& 0.46& 0.94& 0.67& 0.50& 1.18& 554$\pm$75\
GC0287 &pff\_gc-064 & 13 25 43.90 &-42 50 42.7& 10.85 & 21.16& 20.67& 19.66& 19.07& 18.41& 0.11& 0.03& 0.01& 0.01& 0.03& 20.96& 19.97& 19.20& 0.02& 0.03& 0.01& 0.50& 1.01& 0.59& 1.25& 0.76& 0.99& 1.76& 560$\pm$24\
GC0288 &HGHH-35/C35 & 13 25 44.21 &-42 58 59.4& 3.72 & 19.92& 19.51& 18.58& 18.01& 17.43& 0.13& 0.03& 0.01& 0.01& 0.03& 19.74& 18.79& 18.11& 0.02& 0.02& 0.01& 0.41& 0.93& 0.57& 1.14& 0.68& 0.94& 1.63& 544$\pm$13\
GC0289 &C166 & 13 25 44.90 &-43 04 21.1& 4.50 & - & - & - & - & - & - & - & - & - & - & - & - & 20.72& - & - & - & - & - & - & - & - & - & - & -\
GC0290 &AAT208065 & 13 25 45.77 &-42 34 18.0& 27.05 & 21.79& 21.45& 20.09& 19.40& 18.71& 0.49& 0.05& 0.01& 0.01& 0.03& 22.06& 20.70& 19.66& 0.04& 0.03& 0.02& 0.33& 1.36& 0.69& 1.39& 1.04& 1.36& 2.40& 836$\pm$41\
GC0291 & WHH-23 & 13 25 45.90 &-42 57 20.2& 5.07 & 19.26& 19.12& 18.37& 17.89& 17.41& 0.05& 0.02& 0.01& 0.01& 0.03& 19.13& 18.50& 18.00& 0.03& 0.05& 0.01& 0.14& 0.75& 0.48& 0.96& 0.50& 0.64& 1.14& 286$\pm$63\
GC0292 &C167 & 13 25 45.97 &-43 06 45.4& 6.54 & - & - & - & - & - & - & - & - & - & - & - & - & 20.41& - & - & - & - & - & - & - & - & - & - & -\
GC0293 & WHH-24 & 13 25 46.00 &-42 56 53.0& 5.43 & 20.64& 20.43& 19.63& 19.12& 18.62& 0.12& 0.04& 0.01& 0.01& 0.03& 20.54& 19.91& 19.25& 0.02& 0.03& 0.01& 0.21& 0.80& 0.51& 1.01& 0.66& 0.63& 1.29& 566$\pm$48\
GC0294 &AAT119596 & 13 25 46.06 &-43 08 24.5& 8.01 & 21.19& 20.69& 19.70& 19.10& 18.51& 0.12& 0.03& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.50& 1.00& 0.60& 1.19& - & - & - & 573$\pm$53\
GC0295 &HGHH-G378 & 13 25 46.13 &-43 01 22.9& 3.39 & 21.30& 20.59& 19.53& 18.86& 18.22& 0.55& 0.10& 0.03& 0.02& 0.03& - & - & - & - & - & - & 0.71& 1.06& 0.67& 1.31& - & - & - & 487$\pm$31\
GC0296 &AAT208206 & 13 25 46.49 &-42 34 54.1& 26.47 & 20.08& 20.10& 19.47& 19.07& 18.69& 0.12& 0.02& 0.01& 0.01& 0.03& 20.13& 19.73& 19.19& 0.01& 0.01& 0.01&-0.02& 0.63& 0.40& 0.78& 0.55& 0.40& 0.94& 293$\pm$115\
GC0297 &HGHH-G284 & 13 25 46.59 &-42 57 03.0& 5.37 & 21.47& 20.89& 19.87& 19.28& 18.63& 0.26& 0.05& 0.02& 0.01& 0.03& 21.26& 20.20& 19.41& 0.01& 0.01& 0.01& 0.57& 1.03& 0.58& 1.24& 0.79& 1.06& 1.85& 479$\pm$12\
GC0298 &AAT119697 & 13 25 46.68 &-42 53 48.6& 8.12 & 21.61& 21.06& 20.05& 19.49& 18.87& 0.23& 0.05& 0.02& 0.01& 0.03& 21.42& 20.42& 19.65& 0.02& 0.02& 0.01& 0.54& 1.01& 0.57& 1.18& 0.77& 1.01& 1.78& 633$\pm$108\
GC0299 &pff\_gc-065 & 13 25 46.92 &-43 08 06.6& 7.81 & 20.60& 20.17& 19.23& 18.66& 18.08& 0.08& 0.02& 0.01& 0.01& 0.03& 20.43& 19.55& 18.71& 0.01& 0.01& 0.01& 0.42& 0.94& 0.57& 1.15& 0.83& 0.88& 1.72& 450$\pm$44\
GC0300 &HGHH-G204 & 13 25 46.99 &-43 02 05.4& 3.66 & 19.35& 19.12& 18.33& 17.83& 17.31& 0.12& 0.04& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.23& 0.79& 0.51& 1.02& - & - & - & 690$\pm$18\
GC0301 &pff\_gc-066 & 13 25 47.14 &-43 06 08.8& 6.14 & 21.21& 20.82& 19.92& 19.37& 18.80& 0.15& 0.04& 0.01& 0.01& 0.03& 21.07& 20.21& 19.44& 0.01& 0.01& 0.01& 0.39& 0.91& 0.55& 1.11& 0.77& 0.86& 1.63& 530$\pm$44\
GC0302 &HGHH-G357 & 13 25 47.78 &-43 00 43.4& 3.71 & 20.87& 20.55& 19.71& 19.24& 18.71& 0.26& 0.07& 0.02& 0.02& 0.04& 20.71& 19.96& 19.31& 0.01& 0.02& 0.01& 0.33& 0.83& 0.47& 1.00& 0.66& 0.75& 1.41& 664$\pm$31\
GC0303 &AAT119894 & 13 25 47.92 &-42 55 52.4& 6.45 & 21.45& 21.30& 20.25& 19.59& 18.92& 0.23& 0.07& 0.02& 0.02& 0.03& 21.68& 20.75& 19.85& 0.05& 0.04& 0.03& 0.15& 1.05& 0.66& 1.33& 0.90& 0.93& 1.84& 448$\pm$90\
GC0304 &C168 & 13 25 48.46 &-43 07 12.5& 7.16 & - & - & - & - & - & - & - & - & - & - & - & - & 19.71& - & - & - & - & - & - & - & - & - & - & -\
GC0305 &HGHH-G251 & 13 25 48.54 &-42 57 41.2& 5.16 & 20.50& 19.93& 18.93& 18.33& 17.69& 0.13& 0.03& 0.01& 0.01& 0.03& 20.23& 19.22& 18.46& 0.02& 0.01& 0.01& 0.57& 1.00& 0.60& 1.24& 0.76& 1.01& 1.77& 574$\pm$27\
GC0306 &pff\_gc-067 & 13 25 48.77 &-43 11 38.7& 11.19 & 20.51& 20.35& 19.60& 19.08& 18.60& 0.07& 0.02& 0.01& 0.01& 0.03& 20.47& 19.87& 19.14& 0.01& 0.01& 0.01& 0.16& 0.75& 0.52& 1.00& 0.74& 0.60& 1.33& 632$\pm$43\
GC0307 &pff\_gc-068 & 13 25 49.27 &-43 02 20.4& 4.13 & 20.73& 20.41& 19.55& 18.99& 18.48& 0.20& 0.05& 0.02& 0.02& 0.03& 20.65& 20.17& 19.13& 0.03& 0.04& 0.01& 0.32& 0.85& 0.56& 1.08& 1.04& 0.48& 1.52& 393$\pm$26\
GC0308 &HGHH-20/C20 & 13 25 49.69 &-42 54 49.3& 7.50 & 19.18& 18.83& 17.95& 17.43& 16.88& 0.04& 0.01& 0.01& 0.01& 0.03& 19.01& 18.19& 17.54& 0.03& 0.02& 0.01& 0.35& 0.88& 0.53& 1.07& 0.65& 0.82& 1.47& 744$\pm$9\
GC0309 &AAT120259 & 13 25 49.73 &-43 05 04.7& 5.64 & 21.95& 21.16& 20.12& 19.51& 18.80& 0.35& 0.06& 0.02& 0.01& 0.03& 21.57& 20.48& 19.56& 0.01& 0.01& 0.01& 0.79& 1.04& 0.62& 1.32& 0.92& 1.09& 2.01& 477$\pm$54\
GC0310 &HGHH-48/C48 & 13 25 49.82 &-42 50 15.3& 11.62 & 19.63& 19.46& 18.67& 18.16& 17.67& 0.04& 0.01& 0.01& 0.01& 0.03& 19.56& 18.92& 18.28& 0.02& 0.02& 0.01& 0.17& 0.79& 0.51& 1.01& 0.64& 0.64& 1.28& 547$\pm$19\
GC0311 &pff\_gc-069 & 13 25 49.93 &-42 40 08.2& 21.40 & 21.55& 20.90& 19.87& 19.26& 18.62& 0.41& 0.03& 0.01& 0.01& 0.03& 21.29& 20.24& 19.39& 0.01& 0.01& 0.01& 0.64& 1.03& 0.61& 1.25& 0.85& 1.05& 1.90& 518$\pm$31\
GC0312 &HGHH-47/C47 & 13 25 49.95 &-42 52 09.4& 9.87 & 19.80& 19.55& 18.69& 18.18& 17.65& 0.05& 0.02& 0.01& 0.01& 0.03& 19.70& 18.97& 18.30& 0.03& 0.02& 0.01& 0.26& 0.86& 0.51& 1.04& 0.67& 0.73& 1.40& 589$\pm$22\
GC0313 &AAT120336 & 13 25 50.22 &-43 06 08.5& 6.48 & 21.66& 21.14& 20.17& 19.57& 18.96& 0.23& 0.05& 0.02& 0.01& 0.03& 21.48& 20.50& 19.67& 0.02& 0.01& 0.01& 0.52& 0.97& 0.60& 1.21& 0.83& 0.97& 1.81& 452$\pm$67\
GC0314 & WHH-25 & 13 25 50.34 &-43 04 08.2& 5.12 & 21.33& 20.93& 19.97& 19.42& 18.83& 0.22& 0.05& 0.02& 0.01& 0.03& 21.19& 20.27& 19.57& 0.02& 0.02& 0.01& 0.40& 0.95& 0.55& 1.14& 0.71& 0.92& 1.62& 525$\pm$50\
GC0315 &AAT120355 & 13 25 50.37 &-43 00 32.6& 4.20 & 21.26& 21.14& 20.42& 20.02& 19.58& 0.28& 0.09& 0.03& 0.03& 0.04& 21.32& 20.75& 20.17& 0.03& 0.03& 0.01& 0.12& 0.72& 0.39& 0.84& 0.59& 0.56& 1.15& 548$\pm$77\
GC0316 &pff\_gc-070 & 13 25 50.40 &-42 58 02.3& 5.20 & 20.39& 20.23& 19.42& 18.94& 18.45& 0.12& 0.04& 0.01& 0.01& 0.03& 20.35& 19.67& 19.06& 0.01& 0.01& 0.01& 0.16& 0.81& 0.48& 0.97& 0.61& 0.68& 1.29& 556$\pm$49\
GC0317 &C169 & 13 25 51.01 &-42 55 36.3& 7.00 & - & - & - & - & - & - & - & - & - & - & 21.60& 21.00& 20.39& 0.03& 0.03& 0.02& - & - & - & - & 0.62& 0.60& 1.21& -\
GC0318 &AAT120515 & 13 25 51.31 &-42 59 29.4& 4.64 & 21.11& 20.81& 19.92& 19.38& 18.81& 0.21& 0.06& 0.02& 0.02& 0.03& 21.04& 20.24& 19.46& 0.04& 0.04& 0.01& 0.30& 0.90& 0.53& 1.11& 0.77& 0.81& 1.58& 461$\pm$141\
GC0319 &pff\_gc-071 & 13 25 51.54 &-42 59 46.8& 4.58 & 20.79& 20.59& 19.82& 19.34& 18.83& 0.16& 0.05& 0.02& 0.02& 0.03& 20.71& 20.07& 19.45& 0.01& 0.01& 0.01& 0.20& 0.77& 0.48& 0.99& 0.62& 0.64& 1.27& 475$\pm$41\
GC0320 &C102 & 13 25 52.07 &-42 59 14.4& 4.87 & - & - & 21.43& - & - & - & - & - & - & - & 22.71& 21.87& 21.04& 0.02& 0.02& 0.02& - & - & - & - & 0.83& 0.84& 1.67& -\
GC0321 &pff\_gc-072 & 13 25 52.14 &-42 58 30.2& 5.20 & 20.68& 20.45& 19.62& 19.12& 18.54& 0.15& 0.04& 0.02& 0.01& 0.03& 20.64& 19.92& 19.25& 0.02& 0.02& 0.01& 0.23& 0.83& 0.51& 1.08& 0.67& 0.72& 1.39& 504$\pm$22\
GC0322 &HGHH-21/C21 & 13 25 52.74 &-43 05 46.4& 6.52 & 19.16& 18.76& 17.87& 17.32& 16.77& 0.04& 0.01& 0.01& 0.01& 0.03& 18.97& 18.17& 17.40& 0.03& 0.02& 0.01& 0.40& 0.89& 0.55& 1.11& 0.77& 0.80& 1.58& 462$\pm$2\
GC0323 &pff\_gc-073 & 13 25 52.78 &-42 58 41.7& 5.21 & 21.36& 20.94& 19.97& 19.38& 18.74& 0.25& 0.06& 0.02& 0.02& 0.03& 21.25& 20.29& 19.50& 0.02& 0.01& 0.01& 0.42& 0.98& 0.58& 1.22& 0.79& 0.97& 1.76& 401$\pm$29\
GC0324 &HGHH-G256 & 13 25 52.88 &-43 02 00.0& 4.70 & 20.69& 20.18& 19.00& 18.35& 17.65& 0.18& 0.04& 0.01& 0.01& 0.03& 20.50& 19.31& 18.44& 0.03& 0.04& 0.01& 0.51& 1.17& 0.65& 1.36& 0.87& 1.20& 2.06& 495$\pm$18\
GC0325 &AAT209412 & 13 25 53.30 &-42 30 52.7& 30.63 & 21.94& 21.23& 20.22& 19.60& 18.97& 0.52& 0.04& 0.01& 0.01& 0.03& 21.69& 20.64& 19.76& 0.01& 0.01& 0.01& 0.71& 1.01& 0.62& 1.25& 0.88& 1.05& 1.92& 998$\pm$135\
GC0326 &pff\_gc-074 & 13 25 53.37 &-42 51 12.4& 11.00 & 21.16& 20.83& 20.00& 19.48& 18.90& 0.11& 0.03& 0.01& 0.01& 0.03& 21.04& 20.27& 19.56& 0.01& 0.01& 0.01& 0.33& 0.83& 0.52& 1.10& 0.71& 0.77& 1.48& 471$\pm$24\
GC0327 &pff\_gc-075 & 13 25 53.50 &-43 03 56.6& 5.50 & 20.96& 20.45& 19.48& 18.90& 18.28& 0.14& 0.03& 0.01& 0.01& 0.03& 20.78& 19.76& 19.01& 0.02& 0.03& 0.01& 0.51& 0.97& 0.58& 1.20& 0.75& 1.02& 1.77& 748$\pm$18\
GC0328 &HGHH-22/C22 & 13 25 53.57 &-42 59 07.6& 5.16 & 19.39& 19.06& 18.15& 17.62& 17.07& 0.05& 0.02& 0.01& 0.01& 0.03& 19.21& 18.34& 17.70& 0.02& 0.03& - & 0.34& 0.90& 0.54& 1.09& 0.64& 0.87& 1.52& 578$\pm$1\
GC0329 &pff\_gc-076 & 13 25 53.75 &-43 19 48.6& 19.26 & 20.67& 20.06& 19.07& 18.41& 17.78& 0.08& 0.02& 0.01& 0.01& 0.03& 20.44& 19.44& 18.52& 0.02& 0.01& 0.01& 0.62& 0.99& 0.66& 1.29& 0.92& 1.00& 1.92& 368$\pm$15\
GC0330 &AAT120976 & 13 25 54.28 &-42 56 20.6& 6.84 & 21.51& 21.23& 20.38& 19.84& 19.27& 0.23& 0.06& 0.02& 0.02& 0.03& 21.50& 20.73& 19.99& 0.03& 0.02& 0.02& 0.28& 0.85& 0.54& 1.11& 0.74& 0.77& 1.51& 595$\pm$69\
GC0331 &AAT328533 & 13 25 54.39 &-43 18 40.1& 18.19 & 21.79& 21.26& 20.31& 19.74& 19.09& 0.16& 0.04& 0.01& 0.01& 0.03& 21.62& 20.64& 19.77& 0.01& 0.01& 0.01& 0.53& 0.95& 0.58& 1.23& 0.87& 0.98& 1.85& 577$\pm$89\
GC0332 &HGHH-23/C23 & 13 25 54.58 &-42 59 25.4& 5.22 & 18.92& 18.29& 17.22& 16.62& 15.95& 0.04& 0.01& 0.01& 0.01& 0.03& 18.59& 17.44& 16.69& 0.01& 0.03& - & 0.63& 1.07& 0.60& 1.28& 0.75& 1.15& 1.90& 674$\pm$1\
GC0333 &C103 & 13 25 54.98 &-42 59 15.4& 5.36 & - & - & 18.88& - & - & - & - & - & - & - & 20.43& 20.44& 18.44& 0.01& 0.02& 0.02& - & - & - & - & 2.00&-0.01& 1.99& -\
GC0334 &C170 & 13 25 56.11 &-42 56 12.9& 7.17 & - & - & - & - & - & - & - & - & - & - & 23.31& 22.83& 22.16& 0.05& 0.04& 0.06& - & - & - & - & 0.67& 0.48& 1.15& -\
GC0335 &AAT121367 & 13 25 56.26 &-43 01 32.9& 5.25 & 21.06& 20.92& 20.24& 19.82& 19.39& 0.25& 0.08& 0.03& 0.03& 0.04& 21.01& 20.45& 19.85& 0.01& 0.01& 0.01& 0.13& 0.68& 0.42& 0.85& 0.60& 0.56& 1.15& 438$\pm$80\
GC0336 & WHH-26 & 13 25 56.59 &-42 51 46.6& 10.76 & 20.50& 20.02& 19.05& 18.43& 17.81& 0.07& 0.02& 0.01& 0.01& 0.03& 20.33& 19.36& 18.56& 0.02& 0.01& 0.01& 0.48& 0.97& 0.62& 1.24& 0.79& 0.97& 1.76& 412$\pm$36\
GC0337 &AAT329209 & 13 25 57.28 &-43 41 09.0& 40.37 & 20.87& 20.70& 19.87& 19.40& 18.90& 0.21& 0.02& 0.01& 0.01& 0.03& 20.78& 20.12& 19.35& 0.01& - & 0.01& 0.17& 0.82& 0.47& 0.97& 0.77& 0.66& 1.43& 601$\pm$65\
GC0338 &C171 & 13 25 57.78 &-42 55 36.1& 7.82 & - & - & - & - & - & - & - & - & - & - & 22.25& 21.45& 20.67& 0.04& 0.03& 0.02& - & - & - & - & 0.78& 0.79& 1.58& -\
GC0339 &C172 & 13 25 57.95 &-42 53 04.3& 9.79 & - & - & - & - & - & - & - & - & - & - & 22.15& 21.45& 20.91& 0.02& 0.01& 0.01& - & - & - & - & 0.55& 0.70& 1.24& -\
GC0340 &pff\_gc-077 & 13 25 58.15 &-42 31 38.2& 30.03 & 20.57& 20.32& 19.50& 18.96& 18.47& 0.16& 0.02& 0.01& 0.01& 0.03& 20.55& 19.83& 19.09& 0.02& 0.01& 0.01& 0.24& 0.83& 0.53& 1.03& 0.74& 0.71& 1.45& 675$\pm$41\
GC0341 &pff\_gc-078 & 13 25 58.47 &-43 08 06.3& 8.96 & 20.93& 20.34& 19.30& 18.69& 18.02& 0.10& 0.02& 0.01& 0.01& 0.03& 20.70& 19.68& 18.75& 0.02& 0.01& 0.01& 0.59& 1.03& 0.62& 1.28& 0.93& 1.02& 1.95& 545$\pm$18\
GC0342 &HGHH-G143 & 13 25 58.69 &-43 07 11.0& 8.29 & 20.31& 20.02& 19.21& 18.68& 18.18& 0.06& 0.02& 0.01& 0.01& 0.03& 20.22& 19.52& 18.77& 0.02& 0.01& 0.01& 0.29& 0.81& 0.53& 1.03& 0.75& 0.70& 1.45& 503$\pm$38\
GC0343 &pff\_gc-079 & 13 25 58.91 &-42 53 18.9& 9.70 & 20.74& 20.43& 19.58& 19.06& 18.52& 0.10& 0.03& 0.01& 0.01& 0.03& 20.62& 19.86& 19.14& 0.02& 0.01& 0.01& 0.30& 0.86& 0.51& 1.06& 0.72& 0.77& 1.48& 410$\pm$21\
GC0344 &AAT121826/C104 & 13 25 59.49 &-42 55 30.8& 8.10 & 20.91& 20.65& 19.81& 19.28& 18.73& 0.13& 0.03& 0.01& 0.01& 0.03& 20.81& 20.15& 19.40& 0.02& 0.02& 0.01& 0.26& 0.83& 0.53& 1.08& 0.74& 0.66& 1.40& 448$\pm$98\
GC0345 &pff\_gc-080 & 13 25 59.55 &-42 32 39.4& 29.08 & 21.10& 20.80& 19.96& 19.40& 18.84& 0.25& 0.03& 0.01& 0.01& 0.03& 21.07& 20.32& 19.56& 0.02& 0.01& 0.01& 0.29& 0.85& 0.56& 1.11& 0.75& 0.75& 1.51& 598$\pm$56\
GC0346 &C173 & 13 25 59.57 &-42 55 01.5& 8.46 & - & - & - & - & - & - & - & - & - & - & 23.15& 22.00& 21.06& 0.04& 0.03& 0.02& - & - & - & - & 0.94& 1.15& 2.09& -\
GC0347 &C174 & 13 25 59.63 &-42 55 15.7& 8.30 & - & - & - & - & - & - & - & - & - & - & 23.24& 22.42& 21.42& 0.04& 0.04& 0.03& - & - & - & - & 0.99& 0.82& 1.81& -\
GC0348 &pff\_gc-081 & 13 26 00.15 &-42 49 00.7& 13.51 & 20.43& 20.17& 19.34& 18.80& 18.28& 0.07& 0.02& 0.01& 0.01& 0.03& 20.36& 19.61& 18.93& 0.02& 0.01& 0.01& 0.26& 0.83& 0.53& 1.05& 0.69& 0.75& 1.43& 304$\pm$31\
GC0349 & K-217 & 13 26 00.81 &-43 09 40.1& 10.46 & 21.63& 21.09& 20.09& 19.54& 18.91& 0.16& 0.03& 0.01& 0.01& 0.03& 21.42& 20.48& 19.57& 0.01& 0.01& 0.01& 0.54& 0.99& 0.56& 1.19& 0.91& 0.94& 1.85& 315$\pm$157\
GC0350 &C175 & 13 26 00.93 &-42 58 28.9& 6.65 & - & - & - & - & - & - & - & - & - & - & 23.51& 22.53& 21.84& 0.06& 0.04& 0.03& - & - & - & - & 0.69& 0.97& 1.67& -\
GC0351 &pff\_gc-082 & 13 26 00.98 &-42 22 03.4& 39.56 & 20.83& 20.17& 19.16& 18.58& 17.89& 0.21& 0.02& 0.01& 0.01& 0.03& 20.51& 19.53& 18.69& 0.02& 0.01& 0.01& 0.66& 1.01& 0.58& 1.27& 0.84& 0.98& 1.82& 573$\pm$19\
GC0352 &AAT122146 & 13 26 01.00 &-43 06 55.3& 8.40 & 21.14& 20.90& 20.05& 19.53& 19.04& 0.11& 0.03& 0.01& 0.01& 0.03& 21.09& 20.38& 19.58& 0.01& 0.01& 0.01& 0.24& 0.85& 0.52& 1.01& 0.80& 0.71& 1.51& 517$\pm$99\
GC0353 &HGHH-G221 & 13 26 01.11 &-42 55 13.5& 8.52 & 20.87& 20.24& 19.30& 18.75& 18.13& 0.10& 0.02& 0.01& 0.01& 0.03& 20.54& 19.60& 18.83& 0.01& 0.01& 0.01& 0.62& 0.94& 0.55& 1.16& 0.77& 0.94& 1.72& 390$\pm$15\
GC0354 &AAT329848 & 13 26 01.29 &-43 34 15.5& 33.68 & 19.44& 19.19& 18.39& 17.89& 17.37& 0.08& 0.01& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.25& 0.79& 0.50& 1.03& - & - & - & 558$\pm$46\
GC0355 &pff\_gc-083 & 13 26 01.83 &-42 58 15.0& 6.89 & 21.23& 20.80& 19.86& 19.30& 18.72& 0.16& 0.04& 0.01& 0.01& 0.03& 21.12& 20.21& 19.44& 0.02& 0.02& 0.01& 0.42& 0.95& 0.55& 1.13& 0.77& 0.91& 1.68& 458$\pm$31\
GC0356 &pff\_gc-084 & 13 26 02.25 &-43 08 55.6& 10.03 & 21.31& 20.80& 19.84& 19.24& 18.63& 0.12& 0.03& 0.01& 0.01& 0.03& 21.13& 20.19& 19.32& 0.02& 0.01& 0.01& 0.51& 0.97& 0.59& 1.21& 0.87& 0.93& 1.80& 458$\pm$38\
GC0357 &AAT122445 & 13 26 02.43 &-43 00 11.7& 6.43 & 21.13& 20.99& 20.28& 19.81& 19.42& 0.15& 0.05& 0.02& 0.02& 0.03& 21.08& 20.50& 19.92& 0.01& 0.02& 0.01& 0.14& 0.71& 0.47& 0.86& 0.58& 0.58& 1.16& 342$\pm$98\
GC0358 &C176 & 13 26 02.79 &-42 57 05.0& 7.61 & - & - & - & - & - & - & - & - & - & - & 22.35& 21.92& 21.30& 0.02& 0.02& 0.03& - & - & - & - & 0.62& 0.43& 1.05& -\
GC0359 &HGHH-25/C25 & 13 26 02.85 &-42 56 57.0& 7.69 & 20.17& 19.56& 18.49& 17.85& 17.17& 0.07& 0.02& 0.01& 0.01& 0.03& 19.92& 18.83& 17.97& 0.03& 0.02& 0.01& 0.61& 1.07& 0.64& 1.32& 0.86& 1.09& 1.95& 703$\pm$9\
GC0360 &AAT122526 & 13 26 02.90 &-43 05 43.0& 7.90 & 21.89& 21.35& 20.36& 19.73& 19.05& 0.24& 0.05& 0.02& 0.01& 0.03& 21.72& 20.72& 19.81& 0.02& 0.01& 0.01& 0.54& 1.00& 0.62& 1.31& 0.90& 1.00& 1.90& 506$\pm$50\
GC0361 &C177 & 13 26 03.20 &-42 54 30.1& 9.30 & - & - & - & - & - & - & - & - & - & - & 22.95& 22.11& 21.09& 0.11& 0.07& 0.05& - & - & - & - & 1.02& 0.84& 1.87& -\
GC0362 &C178 & 13 26 03.85 &-42 56 45.3& 7.95 & - & - & - & - & - & - & - & - & - & - & 23.02& 22.32& 21.53& 0.04& 0.04& 0.03& - & - & - & - & 0.79& 0.69& 1.48& -\
GC0363 &HGHH-G293/G293 & 13 26 04.20 &-42 55 44.7& 8.60 & 20.11& 19.90& 19.10& 18.61& 18.11& 0.06& 0.02& 0.01& 0.01& 0.03& 20.04& 19.36& 18.69& 0.01& 0.01& 0.01& 0.21& 0.80& 0.49& 0.99& 0.67& 0.69& 1.35& 581$\pm$15\
GC0364 &AAT122808 & 13 26 04.61 &-43 09 10.2& 10.49 & 21.05& 20.87& 20.12& 19.66& 19.18& 0.10& 0.03& 0.01& 0.01& 0.03& 21.00& 20.45& 19.69& 0.01& 0.01& 0.01& 0.18& 0.75& 0.46& 0.94& 0.76& 0.55& 1.31& 264$\pm$131\
GC0365 &AAT122794 & 13 26 04.69 &-42 47 35.1& 15.16 & 21.89& 21.15& 20.00& 19.29& 18.69& 0.21& 0.04& 0.01& 0.01& 0.03& 21.60& 20.37& 19.38& 0.01& 0.01& 0.01& 0.74& 1.15& 0.71& 1.31& 0.98& 1.23& 2.22& 336$\pm$161\
GC0366 &C105 & 13 26 05.12 &-42 55 37.0& 8.80 & - & - & 22.01& - & - & - & - & - & - & - & 23.68& 22.58& 21.76& 0.07& 0.04& 0.04& - & - & - & - & 0.82& 1.10& 1.92& -\
GC0367 &HGHH-07/C7 & 13 26 05.41 &-42 56 32.4& 8.30 & 18.38& 18.03& 17.17& 16.65& 16.08& 0.03& 0.01& 0.01& 0.01& 0.03& 18.18& 17.33& 16.64& 0.03& 0.03& 0.01& 0.35& 0.86& 0.53& 1.09& 0.68& 0.85& 1.53& 595$\pm$1\
GC0368 &C106 & 13 26 06.15 &-42 56 45.4& 8.32 & - & - & 21.28& - & - & - & - & - & - & - & 22.54& 21.46& 20.66& 0.03& 0.01& 0.01& - & - & - & - & 0.80& 1.08& 1.88& -\
GC0369 &pff\_gc-085 & 13 26 06.42 &-43 00 38.1& 7.11 & 20.39& 20.20& 19.43& 18.99& 18.47& 0.07& 0.02& 0.01& 0.01& 0.03& 20.29& 19.64& 19.03& 0.02& 0.02& 0.01& 0.20& 0.76& 0.44& 0.96& 0.61& 0.65& 1.26& 548$\pm$31\
GC0370 &pff\_gc-086 & 13 26 06.55 &-43 06 14.5& 8.75 & 21.48& 20.93& 19.95& 19.37& 18.76& 0.16& 0.03& 0.01& 0.01& 0.03& 21.29& 20.32& 19.45& 0.02& 0.01& 0.01& 0.56& 0.97& 0.59& 1.20& 0.87& 0.97& 1.84& 440$\pm$26\
GC0371 &pff\_gc-087 & 13 26 06.87 &-42 33 17.3& 28.77 & 20.26& 19.96& 19.11& 18.56& 18.05& 0.14& 0.02& 0.01& 0.01& 0.03& 20.17& 19.42& 18.70& 0.02& 0.01& 0.01& 0.30& 0.86& 0.55& 1.05& 0.73& 0.75& 1.47& 830$\pm$29\
GC0372 &HGHH-G170 & 13 26 06.93 &-42 57 35.1& 8.02 & 20.78& 20.21& 19.22& 18.65& 17.97& 0.11& 0.02& 0.01& 0.01& 0.03& 20.57& 19.56& 18.73& 0.02& 0.02& 0.01& 0.57& 0.99& 0.57& 1.25& 0.83& 1.01& 1.84& 636$\pm$27\
GC0373 &AAT123188 & 13 26 06.94 &-43 07 52.6& 9.85 & 21.70& 21.30& 20.40& 19.85& 19.27& 0.17& 0.04& 0.01& 0.01& 0.03& 21.57& 20.71& 19.94& 0.02& 0.01& 0.01& 0.40& 0.90& 0.56& 1.13& 0.77& 0.86& 1.63& 364$\pm$56\
GC0374 &HGHH-36/C36/R113 & 13 26 07.73 &-42 52 00.3& 11.72 & 19.42& 19.19& 18.35& 17.81& 17.33& 0.04& 0.01& 0.01& 0.01& 0.03& 19.32& 18.61& 17.94& 0.02& 0.01& 0.01& 0.23& 0.84& 0.54& 1.03& 0.67& 0.71& 1.38& 703$\pm$1\
GC0375 &AAT123453 & 13 26 08.38 &-42 59 18.9& 7.67 & 21.29& 20.95& 20.09& 19.55& 19.03& 0.17& 0.04& 0.02& 0.01& 0.03& 21.20& 20.40& 19.67& 0.03& 0.02& 0.01& 0.34& 0.86& 0.54& 1.06& 0.73& 0.80& 1.53& 257$\pm$154\
GC0376 &pff\_gc-088 & 13 26 08.86 &-43 01 21.4& 7.54 & 19.99& 19.79& 18.99& 18.46& 18.00& 0.05& 0.02& 0.01& 0.01& 0.03& 19.86& 19.14& 18.59& 0.03& 0.03& 0.01& 0.20& 0.80& 0.53& 0.99& 0.55& 0.72& 1.27& 554$\pm$29\
GC0377 &AAT123656 & 13 26 09.61 &-43 07 05.9& 9.71 & 21.77& 21.13& 20.08& 19.46& 18.79& 0.19& 0.04& 0.01& 0.01& 0.03& 21.57& 20.51& 19.59& 0.03& 0.02& 0.02& 0.64& 1.05& 0.63& 1.29& 0.92& 1.07& 1.99& 380$\pm$93\
GC0378 & R111 & 13 26 09.71 &-42 50 29.5& 13.14 & 22.07& 21.57& 20.59& 19.91& 19.09& 0.28& 0.06& 0.02& 0.02& 0.03& 21.96& 21.03& 20.20& 0.03& 0.03& 0.02& 0.51& 0.97& 0.68& 1.50& 0.82& 0.94& 1.76& 717$\pm$48\
GC0379 &C179 & 13 26 09.87 &-42 56 36.0& 8.96 & - & - & - & - & - & - & - & - & - & - & 22.13& 21.71& 21.04& 0.03& 0.03& 0.01& - & - & - & - & 0.67& 0.42& 1.09& -\
GC0380 &HGHH-37/C37/R116 & 13 26 10.58 &-42 53 42.7& 10.81 & 19.86& 19.38& 18.43& 17.87& 17.26& 0.04& 0.01& 0.01& 0.01& 0.03& 19.65& 18.72& 17.96& 0.01& 0.01& 0.01& 0.48& 0.95& 0.56& 1.17& 0.76& 0.93& 1.69& 612$\pm$1\
GC0381 & WHH-27 & 13 26 12.82 &-43 09 09.2& 11.51 & 20.27& 19.68& 18.66& 18.06& 17.42& 0.05& 0.01& 0.01& 0.01& 0.03& 20.02& 19.03& 18.13& 0.02& 0.01& 0.01& 0.59& 1.02& 0.60& 1.24& 0.90& 0.99& 1.89& 545$\pm$60\
GC0382 & WHH-28 & 13 26 14.18 &-43 08 30.4& 11.25 & 20.21& 19.98& 19.17& 18.65& 18.14& 0.05& 0.02& 0.01& 0.01& 0.03& 20.13& 19.48& 18.71& 0.02& 0.01& 0.01& 0.23& 0.81& 0.52& 1.03& 0.77& 0.65& 1.41& 506$\pm$123\
GC0383 &HHH86-26/C26 & 13 26 15.27 &-42 48 29.4& 15.36 & 19.99& 19.26& 18.13& 17.47& 16.77& 0.05& 0.01& 0.01& 0.01& 0.03& 19.66& 18.45& 17.59& 0.04& 0.03& 0.02& 0.73& 1.13& 0.66& 1.35& 0.86& 1.21& 2.07& 377$\pm$14\
GC0384 &R122 & 13 26 15.95 &-42 55 00.5& 10.76 & 19.02& 18.78& 18.02& 17.58& 17.14& 0.03& 0.01& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.24& 0.76& 0.44& 0.88& - & - & - & 588$\pm$2\
GC0385 &AAT125079 & 13 26 17.29 &-43 06 39.3& 10.62 & 21.13& 20.90& 20.09& 19.59& 19.06& 0.10& 0.03& 0.01& 0.01& 0.03& 21.07& 20.41& 19.64& 0.02& 0.01& 0.01& 0.23& 0.81& 0.51& 1.03& 0.77& 0.66& 1.43& 513$\pm$183\
GC0386 &C50/K-233 & 13 26 19.66 &-43 03 18.6& 9.76 & 20.09& 19.68& 18.74& 18.17& 17.57& 0.05& 0.02& 0.01& 0.01& 0.03& 19.91& 18.92& 18.28& 0.04& 0.04& 0.01& 0.41& 0.93& 0.58& 1.17& 0.64& 1.00& 1.63& 615$\pm$58\
GC0387 &C49/pff\_gc-089 & 13 26 20.20 &-43 10 35.7& 13.48 & 19.90& 19.73& 18.95& 18.45& 17.95& 0.12& 0.02& 0.01& 0.01& 0.03& 19.85& 19.30& 18.53& 0.03& 0.02& 0.01& 0.17& 0.78& 0.50& 1.00& 0.77& 0.56& 1.32& 538$\pm$30\
GC0388 &pff\_gc-090 & 13 26 20.53 &-43 03 18.5& 9.91 & 20.90& 20.34& 19.33& 18.74& 18.10& 0.10& 0.02& 0.01& 0.01& 0.03& 20.70& 19.61& 18.85& 0.03& 0.03& 0.01& 0.56& 1.00& 0.60& 1.24& 0.75& 1.09& 1.85& 486$\pm$26\
GC0389 &AAT215171 & 13 26 20.66 &-42 38 32.0& 24.60 & 21.03& 20.63& 19.69& 19.11& 18.53& 0.27& 0.03& 0.01& 0.01& 0.03& 20.97& 20.06& 19.26& 0.02& 0.02& 0.01& 0.40& 0.94& 0.58& 1.15& 0.80& 0.90& 1.70& 527$\pm$48\
GC0390 & R107 & 13 26 21.11 &-42 48 41.1& 15.84 & 21.53& 20.82& 19.75& 19.13& 18.48& 0.16& 0.03& 0.01& 0.01& 0.03& 21.23& 20.16& 19.30& 0.05& 0.03& 0.02& 0.71& 1.07& 0.62& 1.27& 0.86& 1.06& 1.92& 405$\pm$28\
GC0391 &pff\_gc-091 & 13 26 21.14 &-43 42 24.6& 42.41 & 20.42& 20.13& 19.29& 18.77& 18.21& 0.15& 0.02& 0.01& 0.01& 0.03& 20.30& 19.56& 18.72& 0.01& 0.01& 0.01& 0.29& 0.84& 0.52& 1.08& 0.84& 0.74& 1.58& 623$\pm$37\
GC0392 &pff\_gc-092 & 13 26 21.31 &-42 57 19.1& 10.53 & 21.36& 20.86& 19.92& 19.35& 18.76& 0.14& 0.03& 0.01& 0.01& 0.03& 21.18& 20.27& 19.46& 0.02& 0.02& 0.01& 0.50& 0.93& 0.57& 1.16& 0.81& 0.92& 1.73& 462$\pm$27\
GC0393 & R117 & 13 26 21.99 &-42 53 45.5& 12.38 & 20.52& 20.41& 19.68& 19.23& 18.75& 0.07& 0.02& 0.01& 0.01& 0.03& 20.50& 19.98& 19.32& 0.02& 0.02& 0.01& 0.10& 0.73& 0.45& 0.93& 0.66& 0.53& 1.18& 484$\pm$26\
GC0394 & R118 & 13 26 22.01 &-42 54 26.5& 11.99 & 20.73& 20.60& 19.87& 19.38& 18.93& 0.08& 0.03& 0.01& 0.01& 0.03& 20.72& 20.19& 19.50& 0.03& 0.02& 0.01& 0.13& 0.73& 0.49& 0.95& 0.70& 0.52& 1.22& 440$\pm$73\
GC0395 & WHH-29 & 13 26 22.08 &-43 09 10.7& 12.79 & 20.97& 20.70& 19.82& 19.28& 18.73& 0.08& 0.02& 0.01& 0.01& 0.03& 20.92& 20.17& 19.38& 0.03& 0.02& 0.01& 0.27& 0.88& 0.54& 1.09& 0.79& 0.75& 1.54& 505$\pm$78\
GC0396 &pff\_gc-093 & 13 26 22.65 &-42 46 49.8& 17.50 & 20.58& 20.39& 19.62& 19.13& 18.66& 0.08& 0.03& 0.01& 0.01& 0.03& 20.52& 19.91& 19.23& 0.02& 0.01& 0.01& 0.20& 0.77& 0.48& 0.96& 0.69& 0.61& 1.30& 576$\pm$42\
GC0397 & WHH-30 & 13 26 23.60 &-43 03 43.9& 10.55 & 20.56& 20.25& 19.38& 18.83& 18.30& 0.10& 0.03& 0.01& 0.01& 0.03& 20.45& 19.62& 18.96& 0.03& 0.03& 0.01& 0.31& 0.87& 0.55& 1.08& 0.65& 0.83& 1.49& 470$\pm$66\
GC0398 &pff\_gc-094 & 13 26 23.66 &-43 00 45.6& 10.25 & 20.99& 20.77& 19.97& 19.49& 18.93& 0.09& 0.03& 0.01& 0.01& 0.03& 20.92& 20.24& 19.57& 0.03& 0.02& 0.01& 0.22& 0.80& 0.47& 1.04& 0.67& 0.68& 1.35& 334$\pm$64\
GC0399 &HHH86-38/C38/R123 & 13 26 23.78 &-42 54 01.1& 12.50 & 19.67& 19.30& 18.41& 17.87& 17.32& 0.04& 0.01& 0.01& 0.01& 0.03& 19.51& 18.68& 17.96& 0.02& 0.02& 0.01& 0.37& 0.89& 0.54& 1.09& 0.72& 0.83& 1.54& 405$\pm$1\
GC0400 &HGHH-51 & 13 26 23.86 &-42 47 17.1& 17.26 & 19.63& 19.34& 18.47& 17.93& 17.38& 0.05& 0.02& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.29& 0.87& 0.54& 1.09& - & - & - & 343$\pm$43\
GC0401 &AAT335187 & 13 26 23.95 &-43 17 44.4& 19.53 & 21.19& 21.11& 20.46& 20.04& 19.60& 0.10& 0.03& 0.01& 0.01& 0.03& 21.20& 20.72& 20.08& 0.01& 0.01& 0.01& 0.08& 0.65& 0.42& 0.86& 0.64& 0.49& 1.12& 277$\pm$158\
GC0402 &pff\_gc-095 & 13 26 25.50 &-42 57 06.2& 11.33 & 21.13& 20.60& 19.60& 19.02& 18.40& 0.11& 0.02& 0.01& 0.01& 0.03& 20.97& 19.97& 19.17& 0.04& 0.03& 0.02& 0.53& 1.00& 0.58& 1.20& 0.80& 0.99& 1.79& 405$\pm$18\
GC0403 & R124 & 13 26 28.87 &-42 52 36.4& 14.08 & 20.32& 20.17& 19.42& 18.96& 18.43& 0.05& 0.02& 0.01& 0.01& 0.03& 20.28& 19.69& 19.02& 0.02& 0.02& 0.01& 0.15& 0.75& 0.46& 0.99& 0.68& 0.58& 1.26& 541$\pm$35\
GC0404 &pff\_gc-096 & 13 26 30.29 &-42 34 41.7& 28.83 & 21.24& 20.73& 19.73& 19.10& 18.42& 0.31& 0.03& 0.01& 0.01& 0.03& 21.08& 20.09& 19.24& 0.01& 0.01& 0.01& 0.52& 1.00& 0.63& 1.30& 0.85& 0.99& 1.83& 532$\pm$25\
GC0405 & R105 & 13 26 33.55 &-42 51 00.9& 15.74 & 21.28& 20.78& 19.86& 19.28& 18.70& 0.11& 0.03& 0.01& 0.01& 0.03& 21.18& 20.25& 19.46& 0.02& 0.01& 0.01& 0.50& 0.92& 0.58& 1.15& 0.79& 0.93& 1.72& 534$\pm$34\
GC0406 &HGHH-27/C27 & 13 26 37.99 &-42 45 49.9& 20.00 & 19.60& 19.39& 18.60& 18.11& 17.62& 0.03& 0.01& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.21& 0.78& 0.49& 0.98& - & - & - & 492$\pm$37\
GC0407 & WHH-31 & 13 26 41.43 &-43 11 25.0& 16.96 & 20.76& 20.44& 19.50& 18.94& 18.36& 0.07& 0.02& 0.01& 0.01& 0.03& 20.67& 19.89& 19.05& 0.03& 0.02& 0.02& 0.32& 0.94& 0.56& 1.14& 0.85& 0.77& 1.62& 573$\pm$67\
GC0408 &HHH86-39/C39 & 13 26 42.03 &-43 07 44.8& 15.12 & 18.72& 18.33& 17.43& 16.91& 16.42& 0.02& 0.01& 0.01& 0.01& 0.03& 18.57& 17.73& 16.92& 0.01& 0.01& - & 0.39& 0.89& 0.53& 1.01& 0.81& 0.83& 1.65& 271$\pm$20\
GC0409 &pff\_gc-097 & 13 26 45.40 &-43 26 34.1& 29.13 & 19.75& 19.51& 18.72& 18.23& 17.76& 0.10& 0.01& 0.01& 0.01& 0.03& 19.66& 19.04& 18.28& 0.02& 0.01& 0.01& 0.24& 0.79& 0.49& 0.96& 0.76& 0.62& 1.38& 599$\pm$30\
GC0410 & HH-017 & 13 26 49.31 &-43 04 57.8& 15.41 & 23.21& 22.54& 20.80& 19.87& 19.06& 0.67& 0.11& 0.02& 0.01& 0.03& 23.32& 21.63& 20.13& 0.05& 0.03& 0.02& 0.67& 1.75& 0.93& 1.73& 1.50& 1.68& 3.19& 839$\pm$63\
GC0411 &pff\_gc-098 & 13 26 53.94 &-43 19 17.7& 24.05 & 19.54& 19.17& 18.28& 17.73& 17.16& 0.09& 0.01& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.37& 0.89& 0.55& 1.12& - & - & - & 631$\pm$18\
GC0412 &AAT222977 & 13 26 58.91 &-42 38 53.4& 27.82 & 21.18& 21.02& 20.34& 19.84& 19.38& 0.28& 0.03& 0.01& 0.01& 0.03& 21.16& 20.64& 20.02& 0.02& 0.01& 0.01& 0.17& 0.68& 0.50& 0.96& 0.62& 0.52& 1.14& 655$\pm$112\
GC0413 & HH-060 & 13 26 59.78 &-42 55 26.5& 17.79 & - & - & - & - & - & - & - & - & - & - & 23.41& 21.69& 20.14& 0.05& 0.04& 0.02& - & - & - & - & 1.55& 1.72& 3.26& 811$\pm$32\
GC0414 &pff\_gc-099 & 13 26 59.82 &-42 32 40.5& 33.09 & 20.96& 20.72& 19.94& 19.40& 18.92& 0.23& 0.03& 0.01& 0.01& 0.03& 20.92& 20.29& 19.59& 0.02& 0.01& 0.02& 0.25& 0.78& 0.53& 1.02& 0.70& 0.63& 1.33& 425$\pm$34\
GC0415 &pff\_gc-100 & 13 27 03.41 &-42 27 17.2& 38.12 & 19.92& 19.37& 18.41& 17.81& 17.17& 0.11& 0.01& 0.01& 0.01& 0.03& - & - & - & - & - & - & 0.56& 0.96& 0.60& 1.24& - & - & - & 513$\pm$10\
GC0416 &pff\_gc-101 & 13 27 21.56 &-42 38 41.4& 30.63 & 19.89& 19.51& 18.72& 18.20& 17.74& 0.10& 0.01& 0.01& 0.01& 0.03& 19.74& 19.02& 18.39& 0.01& 0.01& - & 0.38& 0.79& 0.51& 0.97& 0.63& 0.72& 1.35& 263$\pm$16\
GC0417 &pff\_gc-102 & 13 28 18.45 &-42 33 12.5& 41.90 & 20.38& 20.11& 19.29& 18.76& 18.24& 0.14& 0.02& 0.01& 0.01& 0.03& 20.28& 19.59& 18.94& 0.04& 0.04& 0.01& 0.27& 0.83& 0.52& 1.04& 0.65& 0.70& 1.34& 428$\pm$40\
[rrrrrrrrrrr]{}
0-50 & 12.9 & 48.8 & 340 & 40$\pm$10 & 189$\pm$12 & 123$\pm$5 & 1.0 & 125.8$\pm$46.5 & 3.1$\pm$1.3 & 128.9$\pm$46.5\
0-5 & 3.64 & 4.96 & 54 & 24$\pm$21 & 334$\pm$59 & 120$\pm$12 & - & - & - & -\
5-10 & 7.65 & 9.96 & 124 & 43$\pm$15 & 195$\pm$20 & 112$\pm$8 & 2.3 & 37.4$\pm$6.4 & 0.4$\pm$0.3 & 37.8$\pm$6.4\
10-15 & 12.4 & 14.9 & 68 & 83$\pm$25 & 195$\pm$12 & 105$\pm$10 & 1.6 & 36.5$\pm$9.3 & 2.4$\pm$1.5 & 38.9$\pm$9.4\
15-25 & 19.0 & 24.3 & 56 & 35$\pm$26 & 184$\pm$34 & 147$\pm$16 & 1.3 & 89.5$\pm$27.4 & 0.7$\pm$1.0 & 90.2$\pm$27.4\
25-50 & 34.7 & 48.8 & 39 & 96$\pm$45 & 169$\pm$17 & 148$\pm$21 & 1.0 & 184.7$\pm$84.5 & 10.4$\pm$9.8 & 195.1$\pm$85.1\
[rrrrrrrrrrr]{}
0-50 & 13.8 & 48.1 & 178 & 31$\pm$14 & 177$\pm$22 & 117$\pm$7 & 1.0 & 94.5$\pm$58.5 & 1.1$\pm$1.0 & 95.6 $\pm$58.6\
0-5 & 3.84 & 4.96 & 22 & 16$\pm$38 & 373$\pm$120& 99$\pm$19 & - & - & - & -\
5-10 & 7.76 & 9.96 & 66 & 30$\pm$21 & 223$\pm$39 & 116$\pm$12 & 2.1 & 38.9$\pm$12.4 & 0.2$\pm$0.3 & 39.1$\pm$12.4\
10-15 & 12.7 & 14.8 & 41 & 97$\pm$38 & 199$\pm$13 & 102$\pm$15 & 1.5 & 31.4$\pm$15.1 & 3.3$\pm$2.5 & 34.6$\pm$15.3\
15-25 & 18.8 & 24.3 & 26 & 45$\pm$31 & 139$\pm$49 & 112$\pm$20 & 1.3 & 59.7$\pm$32.7 & 1.2$\pm$1.6 & 60.8$\pm$32.7\
25-50 & 36.7 & 48.1 & 23 & 61$\pm$43 & 137$\pm$50 & 141$\pm$24 & 1.0 & 122.4$\pm$82.9 & 4.2$\pm$5.9 & 126.7$\pm$83.1\
[rrrrrrrrrrr]{}
0-50 & 12.0 & 48.8 & 158 & 47$\pm$15 & 202$\pm$15 & 129$\pm$9 & 1.0 & 116.5$\pm$73.8 & 4.3$\pm$2.3 & 120.8$\pm$73.9\
0-5 & 3.61 & 4.98 & 29 & 44$\pm$27 & 308$\pm$47 & 134$\pm$23 & - & - & - & -\
5-10 & 7.52 & 9.93 & 58 & 65$\pm$22 & 180$\pm$19 & 105$\pm$11 & 1.5 & 17.4$\pm$7.7 & 1.0$\pm$0.7 & 18.3$\pm$7.7\
10-15 & 12.0 & 14.9 & 27 & 74$\pm$34 & 186$\pm$22 & 108$\pm$17 & 1.2 & 23.8$\pm$11.3 & 1.9$\pm$1.8 & 25.7$\pm$13.3\
15-25 & 19.3 & 24.3 & 30 & 50$\pm$36 & 223$\pm$32 & 168$\pm$24 & 1.1 & 72.1$\pm$46.2 & 1.2$\pm$1.7 & 73.3$\pm$46.3\
25-50 & 31.7 & 48.8 & 15 & 102$\pm$96 & 191$\pm$21 & 146$\pm$65 & 1.0 & 219.1$\pm$174.6 & 11.8$\pm$22.2 & 230.9$\pm$176.0\
[rrrrrrrrrrr]{}
0-90 &14.1 & 88.2 & 780 & 76$\pm$6 & 170$\pm$5 & 118$\pm$13 & 1.0 & 84.6$\pm$17.2 &11.9$\pm$1.9 & 96.5$\pm$17.3\
0-5 & 3.3 & 4.99 & 184 & 75$\pm$16 & 179$\pm$11 & 131$\pm$7 & - & - & - & -\
5-10 & 7.6 & 9.98 & 211 & 82$\pm$12 & 177$\pm$8 & 120$\pm$6 & 3.0 & 43.9$\pm$3.6 & 1.6$\pm$0.5 & 45.4$\pm$3.6\
10-15 &11.8 & 14.8 & 138 & 76$\pm$15 & 171$\pm$10 & 118$\pm$7 & 2.0 & 42.3$\pm$5.0 & 2.0$\pm$0.8 & 44.2$\pm$5.1\
15-20 &17.4 & 20.0 & 71 & 96$\pm$24 & 169$\pm$12 & 116$\pm$10 & 1.7 & 47.7$\pm$5.6 & 4.3$\pm$2.1 & 52.0$\pm$6.9\
20-30 &24.6 & 30.0 & 87 & 76$\pm$17 & 157$\pm$17 & 108$\pm$8 & 1.5 & 53.2$\pm$8.4 & 4.0$\pm$1.8 & 57.2$\pm$8.6\
30-40 &34.6 & 39.8 & 50 & 44$\pm$15 & 132$\pm$38 & 87$\pm$9 & 1.3 & 40.7$\pm$7.4 & 1.8$\pm$1.2 & 42.3$\pm$7.5\
40-80 &48.7 & 71.2 & 36 & 61$\pm$45 & 183$\pm$22 & 85$\pm$11 & 1.0 & 48.6$\pm$11.5 & 6.2$\pm$9.1 & 54.8$\pm$14.7\
[lrrrrrrrr]{}
NGC 5128 & 31$\pm$14 & 117$\pm$7 & 0.26$\pm$0.12 & 0.08$\pm$0.04 & 47$\pm$15 & 129$\pm$9 & 0.36$\pm$0.11 & 0.11$\pm$0.03\
M87 &186$^{+58}_{-41}$ &397$^{+36}_{-14}$ & 0.47$^{+0.13}_{-0.11}$& 0.14$^{+0.04}_{-0.03}$ &155$^{+53}_{-37}$ &365$^{+38}_{-18}$ & 0.43$^{+0.14}_{-0.12}$ & 0.13$^{+0.04}_{-0.04}$\
M49 & 93$^{+69}_{-37}$ &342$^{+33}_{-18}$ & 0.27$^{+0.19}_{-0.11}$& 0.08$^{+0.06}_{-0.03}$ &-26$^{+64}_{-79}$ &265$^{+34}_{-13}$ & 0.10$^{+0.27}_{-0.25}$ & 0.03$^{+0.08}_{-0.08}$\
NGC 1399 & 15$\pm$26 &291$\pm$14 & 0.05$\pm$0.09 & 0.02$\pm$0.03 & 7$\pm$24 & 255$\pm$13 & 0.03$\pm$0.09 & 0.01$\pm$0.03\
![Sine curve fit for the GCs in NGC 5128 ([*circles*]{}) with a fixed systemic velocity of $v_{sys} = 541$ km s$^{-1}$, for 0-50 kpc from the center of NGC 5128. The top panel shows all 340 GCs with rotation amplitude $\Omega R = 40\pm10 $ km s$^{-1}$ and rotation axis $\Theta_o = 189\pm12^{o}$ east of north. The middle panel shows the 178 metal-poor clusters with $\Omega R = 31\pm14$ km s$^{-1}$ and $\Theta_o = 177\pm22^{o}$ east of north, and the bottom panel shows the 158 metal-rich clusters with $\Omega R = 47\pm15$ km s$^{-1}$ and $\Theta_o = 202\pm15^{o}$ east of north. The squares represent the weighted velocities in $72^{o}$ bins.[]{data-label="fig:kin_plot"}](f4.eps)
[^1]: The confirmed GC list in [@w06] containing 343 GCs has been reduced to 340 based on recent spectroscopic and imaging studies. Object 304867 with a high radial velocity of $305\pm56$ km s$^{-1}$ appears to be an M-type star based on the strong molecular bands in its spectrum, see [@beasley06] for further discussion. Objects pff\_gc-010 and 114993 are also rejected as GCs because of their starlike appearance under $HST$ ACS imaging; see [@harris06]. However, newly confirmed GCs HCH15 and R122 have now been added [@rejkuba07].
[^2]: [@hhh86] report C32 at a distance of $R_{gc} = 10.8'$, but more recently [@pff04I] claim a distance of $R_{gc} =
11.25'$, so it has an adopted uncertainty of 44 km s$^{-1}$ in the weighted mean.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present spectral data cubes of the [\[CI\]]{} 809GHz, [$\rm ^{12}CO$]{} 115GHz, [$\rm ^{13}CO$]{} 110GHz and HI 1.4GHz line emission from an $\sim 1$ square degree region along the $l = 328^{\circ}$ () sightline in the Galactic Plane. Emission arises principally from gas in three spiral arm crossings along the sight line. The distribution of the emission in the CO and [\[CI\]]{} lines is found to be similar, with the [\[CI\]]{} slightly more extended, and both are enveloped in extensive HI. Spectral line ratios per voxel in the data cubes are found to be similar across the entire extent of the Galaxy. However, towards the edges of the molecular clouds the [\[CI\]]{}/[$\rm ^{13}CO$]{} and [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{} line ratios rise by $\sim 50$%, and the [\[CI\]]{}/[HI]{} ratio falls by $\sim 10$%. We attribute this to these sightlines passing predominantly through the surfaces of photodissociation regions (PDRs), where the carbon is found mainly as C or C$^+$, while the H$_2$ is mostly molecular, and the proportion of atomic gas also increases. We undertake modelling of the PDR emission from low density molecular clouds excited by average interstellar radiation fields and cosmic-ray ionization to quantify this comparison, finding that depletion of sulfur and reduced PAH abundance is needed to match line fluxes and ratios. Roughly one-third of the molecular gas along the sightline is found to be associated with this surface region, where the carbon is largely not to be found in CO. $\sim 10\%$ of the atomic hydrogen along the sightline is cold gas within PDRs.'
author:
- 'Michael G. Burton, Michael C.B. Ashley, Catherine Braiding, Matthew Freeman'
- Craig Kulesa
- 'Mark G. Wolfire'
- 'David J. Hollenbach'
- Gavin Rowell
- James Lau
title: |
Extended Carbon Line Emission in the Galaxy:\
Searching for Dark Molecular Gas along the G328 Sightline
---
Introduction {#sec:intro}
============
One of the basic activities of a spiral galaxy like our own Milky Way is the continual collection of diffuse and fragmented gas and dust clouds into giant clouds of molecules, which in turn produce stars [e.g. @2014prpl.conf..125M]. This takes place as part of a cycle of matter between the stars and the gas, driven by energy flows arising from radiation and mass loss from stars, and the dynamical motions of the gas in the gravitating interstellar medium. Cloud formation occurs as the gas cools and the density rises, first as atomic clouds, then as molecular clouds. Clouds coalesce, cooling is enhanced due to molecule formation as radiation is shielded from their interior, and star formation initiated as gravitational collapse is triggered inside.
Determination of the spatial structure and the kinematics of the gas as it transitions between the phases of the gas is needed to discern how clouds are formed. While the hydrogen gas can be seen from the atomic phase through its principal 21cm emission line, its measurement does not distinguish between the warm and cold phases, only providing the total column density of the atomic gas. In the molecular phase the bulk component – hydrogen molecules – cannot be seen at all, remaining unexcited in their ground state at the typical $10-20$K temperatures found in molecular clouds.
Trace species in the gas are thus needed with emission lines that are sensitive to the excitation conditions in the gas, to follow the thermal and chemical transitions that occur. After helium (which is inert) and oxygen (whose key lines are blocked by the Earth’s atmosphere), carbon is the next most abundant element in the Universe. Carbon can be found in ionized (C$^+$), atomic (C) and molecular (CO – carbon monoxide) forms in the bulk of the dense interstellar medium. All of these species are readily excited in the prevailing conditions, with emission lines produced in the terahertz portion of the spectrum for C and C$^+$ ($0.5 - 1.9$THz), and the millimeter portion for CO (3mm $\equiv 0.1$ THz).
CO is readily measured from good observatory sites. C and C$^+$ are, however, virtually unobservable from all but the driest sites on the surface of our planet. We have established a new observatory at Ridge A, near the summit of the Antarctic plateau, to open up the terahertz spectrum for observation in order to obtain the wide-field, high resolution images of the carbon lines needed to pursue this science [@2012SPIE.8444E..1RA; @2013IAUS..288..256K]. The measurement of the key diagnostic lines from all these species in the atomic and molecular phases of interstellar medium we call “following the galactic carbon trail”.
In this paper we present wide-field data from the Mopra telescope in Australia, obtained as part of the Southern Galactic Plane CO survey [@2013PASA...30...44B; @2015PASA...32...20B] and the HEAT (High Elevation Antarctic Terahertz) telescope at Ridge A in Antarctica of atomic carbon, aimed at pursuing this objective. The data covers roughly 1 square degree along the G328 sightline through the Galactic plane (i.e. $l = 328^{\circ}$). In particular, we have examined the data set to determine whether there is evidence for dark molecular gas present. This is defined here as regions exhibiting [\[CI\]]{} emission but without corresponding CO line emission. Practically, since there are no clear regions where [\[CI\]]{} is present but CO is not detected in the moderate beam sizes used (2 arcmin for the [\[CI\]]{}), we search for regions of emission where the \[C/CO\] abundance may be enhanced.
Such regions of gas may be expected in the surfaces of molecular clouds. These are photodissociation regions [PDRs; e.g. @1985ApJ...291..722T; @1990ApJ...365..620B; @1992ApJ...399..563B; @1993ApJ...402..195W], and within extinctions of $A_V \lesssim 1$mags. from the atomic surface, self-shielding may allow significant columns of H$_2$ to exist. However the CO abundance will be greatly reduced from the cloud interior, being photodissociated by the far-UV radiation that heats and drives the chemistry inside the PDR. Since CO is the normal tracer used to indicate the presence of molecular gas, such gas is “dark” to standard survey techniques. In these molecular cloud envelopes the carbon will instead be found as either C or C$^+$, and so be amenable to detection through THz frequency observations.
@2010ApJ...716.1191W model the fraction of such a dark component that exists in giant molecular clouds, suggesting that it comprises about one-third of the molecular gas, in good agreement with the estimates of the dark gas fraction from gamma-ray observations [@2005Sci...307.1292G]. In these and previous cloud models [e.g. @1988ApJ...334..771V] the [\[CI\]]{} emission arises mostly from gas where the hydrogen exists as H$_2$ but without significant CO present. Thus [\[CI\]]{} line emission can be used to trace the dark molecular gas.
Furthermore, small molecular clouds, here defined as those with total column densities such that $A_V \lesssim 1\, {\rm mag.} \equiv {\rm N_{H_2}} = 10^{21} {\rm cm}^{-2}$, would not be found at all in CO surveys. A population of such clouds could remain unseen without a corresponding carbon-line survey to detect their presence.
There is also the possibility of dark atomic gas, where 21cm HI is optically thick and so its line intensity under-estimates the gas column. Analysis of the Planck satellite data , comparing the optical depth of the dust emission at 353GHz to the column of the HI derived from its 21cm line emission, suggests that such clouds may be widespread across the Galaxy. This can occur when the atomic gas is cold ($T_S < 80$K), through absorption of background continuum. @2015ApJ...798....6F examine this further for atomic clouds within a few hundred parsecs of the Sun, but out of the Galactic plane. They suggest that, if the column density derived from the dust is indeed all atomic, then there are extensive regions where $T_S$ is below 35K, with optical depths $\tau_{\rm HI}$ as high as 3.0; i.e. dark atomic gas[^1]. Carbon would exist as either C or C$^+$ in such atomic clouds (but not as CO), and so emission from the THz frequency lines of these species might be used as a tracer of such gas. This is especially so in the Galactic plane where there is also extensive molecular gas, making it difficult to separate out the dust associated with the atomic component.
By mapping the distribution of carbon along the Galactic plane we can therefore estimate where the dark gas may be distributed and how much of it exists within the interstellar medium. In this paper we concentrate on an analysis of the carbon emission associated with the molecular gas on the Galactic sightline that passes through the G328 sector, also comparing it the [$\rm ^{12}CO$]{}, [$\rm ^{13}CO$]{}, [$\rm C^{18}O$]{} and HI line emission. We describe the observations in §\[sec:obs\] and present the data in §\[sec:results\]. Examining the variation of \[C/CO\] at the edges of emission features, where the fluxes are smallest, presents several challenges so we undertake the analysis through a variety of techniques: flux images (§\[sec:moment\]), histograms of line ratio distributions (§\[sec:histograms\]), averaged line profiles (§\[sec:profiles\]) and a relatively new technique: saturation-hue plots (§\[sec:evans\]). We undertake PDR modelling for the molecular environment appropriate for the bulk of the data set in §\[sec:pdr\], presenting several Figures showing line fluxes, optical depths and line ratios as a function of depth into the PDR in §\[sec:pdrmodelplots\]. We then compare these to the data in §\[sec:discuss\] and summarize the results in §\[sec:summary\]. This is Paper II. Paper I [@2014ApJ...782...72B] analyzed this same data set, but just for a narrow velocity range associated with a single molecular filament.
Observations and Data Reduction {#sec:obs}
===============================
The data analyzed here comes from three separate spectral line surveys conducted along the southern Galactic plane, of THz-band [\[CI\]]{}, millimetre-band CO and centimeter-band HI line emission. The 62cm HEAT (High Elevation Antarctic Terahertz) telescope has imaged the [\[CI\]]{} 809.3GHz line ($\rm ^3P_2 - ^3P_1$) from Ridge A, Antarctica, yielding data cubes with $2'$ and 0.7km/s resolution [@2013IAUS..288..256K]. [$\rm ^{12}CO$]{}, [$\rm ^{13}CO$]{} and [$\rm C^{18}O$]{} J=1–0 emission lines (115.3, 110.2 & 109.8GHz, respectively) were imaged as part of the 22m Mopra telescope Southern Galactic Plane CO Survey [@2013PASA...30...44B; @2015PASA...32...20B], with $0.6'$ and 0.1km/s resolution. The 1.420GHz HI line comes from archival data in the Parkes–ATCA Southern Galactic Plane Survey [the SGPS; @2005ApJS..158..178M] and provides $2'$ and 3km/s resolution. We will generally refer to these lines as simply [\[CI\]]{}, [$\rm ^{12}CO$]{}, [$\rm ^{13}CO$]{}, [$\rm C^{18}O$]{} & HI, respectively, in this paper, though sometimes will refer to the [\[CI\]]{} line as [\[CI\]]{} 2–1 to distinguish it from the \[CI\] $\rm ^3P_1 - ^3P_0$ (1–0) line at 492GHz. We will also include discussion of the [\[CII\]]{} $\rm ^3P_{3/2} - ^3P_{1/2}$ 1.90THz line, referred to here simply as [\[CII\]]{} (and also commonly known as the 158$\mu$m line).
The contiguous data set obtained for the C, CO and H lines shown here covers $l=327.8$–$328.7^{\circ}, b=-0.4$ to $+0.4^{\circ}$. This data set was first presented in @2014ApJ...782...72B [hereafter Paper I], where an analysis of the emission from a filamentary feature, approximately 75pc long and 5pc wide, and with only a narrow 2 km/s FWHM in all these spectral lines, was presented, hypothesized to represent a giant molecular cloud in the process of formation. In this paper we now examine the data from the complete range of velocities covered by these observations with the HEAT telescope, taking in all of the emission from across the Galaxy along this sight line.
The Mopra CO and SGPS HI cubes were re-binned to the same voxel resolution as the HEAT CI cubes (both in position and velocity; i.e. $2'$ and 0.7km/s). All cubes were then smoothed with a $2'$ FWHM Gaussian. The analysis presented here concentrates on examining variations seen in the [\[CI\]]{}/[$\rm ^{13}CO$]{} line ratio in the data cubes. The [$\rm ^{13}CO$]{} line is used in preference to the brighter [$\rm ^{12}CO$]{} line in order to minimize complications in interpretation caused by optical depth in the [$\rm ^{12}CO$]{} line, as the [$\rm ^{13}CO$]{} line is generally optically thin in the Mopra CO survey .
The full velocity range of the \[CI\] emission extends from $-120$ to 0 km/s. Three principal features are seen in its averaged spectral profile (see Fig. \[fig:profiles\], where the [\[CI\]]{} line is shown together with the CO and HI profiles). Fig. \[fig:galaxymodel\] shows a schematic model of the Galaxy [adapted from @2014AJ....148....5V] to aid in the interpretation of these profiles. The three features correspond to spiral arm crossings at $-100$ to $-85$ km/s, $-80$ to $-65$ km/s and $-55$ to $-40$ km/s, respectively. In turn, they equate predominantly to emission from the Norma spiral arm near- and far- crossings, and the Scutum-Crux near-crossing along the G328 sight line, at distances of $\sim$6, 10 & 3.5kpc from Sun. In addition, the velocity range from $-80$ to $-76$ km/s corresponds to the quiescent filament analyzed in Paper I, at $\sim 5$kpc distance. The data set is split into these four velocity ranges for the analysis described below. There is also a weaker feature at $-20$ km/s arising from the Scutum-Crux far-arm crossing, $\sim 13$kpc distant.
A visual rendering of the data cubes for G328 in the four principal lines studied here (HI, [\[CI\]]{}, [$\rm ^{13}CO$]{}, [$\rm ^{12}CO$]{}) is also shown in Fig. \[fig:3dview\]. Here the velocity dimension is extended along the long axis, and provides a proxy for distance from the Sun (with the proviso of the near-far ambiguity). The three features corresponding to the spiral arm crossings are readily apparent (Norma near-, Norma far- and Scutum-Crux near-, going from left to right). The greater extent of the atomic hydrogen is clear, enveloping the carbon and carbon monoxide emitting gas. The carbon also gives the impression of being more extended than the carbon monoxide, in-between the distributions for the atomic and the fully molecular gas. However care needs to be given towards making such an interpretation, as the relative optical depths of the emitting species, as well as the display scales chosen, can result in mis-leading visual appearances. In the rest of this Paper we concentrate on a quantitative analysis of the relative distributions of the hydrogen, carbon and carbon monoxide emission to ascertain whether the impression given here is correct, and to explain why and when such a separation can occur.
Results {#sec:results}
=======
In this section we first consider the integrated flux images for the CO and [\[CI\]]{} lines (i.e. zeroth moment images) in order to compare their morphologies in each of the four velocity ranges. While these show suggestions of a more extended distribution for the [\[CI\]]{} emission than the CO this is not conclusive, so we then consider other measures that can probe their relative distributions. This includes histograms of the distribution of lines ratios and filtering the data to examine averaged profiles from voxels with different values of the [\[CI\]]{}/CO line ratio. We also present a new method of presentation (Evans plots) which applies a 2D color table (hue and saturation) in order to visualize two variables of interest in such a way that pixels with low signal to noise do not dominate a display of the ratio of two intensities.
Moment Images of Line Flux Distributions {#sec:moment}
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Integrated flux (Moment 0) images over the four velocity ranges are shown in Fig. \[fig:momentimages120\] for both the [$\rm ^{13}CO$]{} and [\[CI\]]{} lines, accompanied by an image of the [$\rm ^{13}CO$]{} overlaid with [\[CI\]]{} contours. These images trace the locations of the molecular clouds within three spiral arm crossings along the G328 sight line (Norma far-, Norma near- and Scutum-Crux near-), and so are all quite different. The head of the filamentary structure studied in Paper I is to the left of centre of the lowest panel in this Figure ($\sim 328.4^{\circ}, -0.1^{\circ}$)[^2].
The [$\rm ^{13}CO$]{} and [\[CI\]]{} images clearly look similar in each of the velocity ranges. However their overlay in the third column suggests that the [\[CI\]]{} is, in general, smoother and more extended than in the [$\rm ^{13}CO$]{} images. To check that this conclusion is not an artifact of the different original resolutions of the [$\rm ^{13}CO$]{} and [\[CI\]]{} data sets the analysis was also repeated with the cubes smoothed by a $200''$ Gaussian rather than $120''$. The same result still holds (though is not shown here), of the [\[CI\]]{} being smoother and more extended than the [$\rm ^{13}CO$]{}, though it is not especially striking when seen at this lower angular resolution.
This is as expected if the carbon is wrapped around the molecular gas, as well as being contained within it. Such behaviour would occur if the outer portion of each molecular cloud was lacking in CO. However, while suggestive this result cannot be regarded as definitive from these images, so we now examine other metrics to probe the relative distributions of the [\[CI\]]{} and CO emitting gas.
Histograms of Line Ratio Distributions {#sec:histograms}
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We show in Fig. \[fig:histograms\] histograms of selected ratios from the [\[CI\]]{}, [$\rm ^{12}CO$]{}, [$\rm ^{13}CO$]{}, [$\rm C^{18}O$]{} and HI lines in the data. These distributions are determined per voxel (i.e. per $2'$ spatial pixel, 0.7 km/s velocity channel). To be included here the data for a [\[CI\]]{} voxel needed to be $> 3\sigma$, whereas for the three CO lines we show histograms for thresholds of $1\sigma$ and $3\sigma$. These $\sigma$ values are determined from the standard deviation of voxels in the continuum portion of the data cubes, $-120$ to $-100$km/s. For the HI data, for which $\sigma$ is not well defined since there are no clear regions of line-free emission in the spectra, but whose signal-to-noise (S/N) is much higher than that of the [\[CI\]]{} and CO, the threshold was set to an arbitrary value. The number of voxels passing these threshold criteria is listed in Table \[tab:voxels\], and some statistics on the resulting histograms in Fig. \[fig:histograms\] are listed in Table \[tab:voxelstats\]. These are given for both the $1 \sigma$ and $3 \sigma$ thresholds applied to the CO fluxes. The mean values and standard deviations for the ratios [$\rm ^{12}CO$]{}/[\[CI\]]{}, [$\rm ^{13}CO$]{}/[\[CI\]]{} and [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{} are $6.8 \pm 1.6, 1.4 \pm 0.5 \, {\rm and} \, 5.5 \pm 2.5$, respectively, when the $1 \sigma$ threshold for CO is applied.
The intent here is to search for regions of strong [\[CI\]]{} and weak (or absent) CO emission, hence the $1\sigma$ threshold limit applied to the CO data. There is limited dynamic range in the data set, so repeating this analysis with a $3\sigma$ cut-off for CO yields few voxels where the [\[CI\]]{}/[$\rm ^{13}CO$]{} ratio is “high” (as defined below). We do, however, list metrics in Table \[tab:voxelstats\] for the corresponding distributions when a $3 \sigma$ threshold for the CO lines is applied. The analysis discussed below yields consistent results with the $1 \sigma$ threshold when this is done, though from far fewer data points.
For [$\rm ^{12}CO$]{}/[\[CI\]]{} the distribution is roughly Gaussian, however for all the other line ratios shown in Fig. \[fig:histograms\] this is not the case. A significant non-Gaussian tail exists for larger values of the other line ratios. This highlights excess voxels where the line ratio is typically about twice the mean value. The tail is especially evident for the [\[CI\]]{}/[$\rm ^{13}CO$]{} ratio, which has a mean value of $\sim 0.8$ and “high” values extending to $\sim 2$. That the tail is also sensitive to the value of the threshold value used to select the CO lines is also apparent from the (dotted line) overlays in Fig. \[fig:histograms\], where the corresponding distribution for [\[CI\]]{}/[$\rm ^{13}CO$]{} for the $3 \sigma$ threshold is almost Gaussian in form. This, of course, means that care is needed in interpreting the data in the tail, as noise fluctuations from voxels with weak fluxes will cause some [$\rm ^{13}CO$]{} values to be “low”, and hence the corresponding [\[CI\]]{}/[$\rm ^{13}CO$]{} values to be “high”. Nevertheless, by averaging over voxels where the line ratios are high the S/N can be improved, allowing us to examine where such voxels tend to be distributed. We undertake such an analysis in the following section (§\[sec:profiles\]).
Fig. \[fig:ratiovsflux\] presents a series of scatter plots showing the relationship between the line ratios ([\[CI\]]{}/[$\rm ^{13}CO$]{}, [\[CI\]]{}/[$\rm ^{12}CO$]{} and [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{}) and the fluxes of the CO and [\[CI\]]{} lines, for every voxel in the datacube, where the $3 \sigma$ threshold is applied for the [\[CI\]]{} flux, and $1 \sigma$ for [$\rm ^{12}CO$]{} and [$\rm ^{13}CO$]{}. The top panels show the line ratios as a function of the [$\rm ^{12}CO$]{} and [$\rm ^{13}CO$]{} fluxes, the bottom line as a function of [\[CI\]]{} flux. As these fluxes increase, the line ratios tend towards roughly constant values, given by [\[CI\]]{}/[$\rm ^{13}CO$]{}$\sim 0.5$, [\[CI\]]{}/[$\rm ^{12}CO$]{}$\sim 0.15$ and [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{}$\sim 3$. In contrast, as the fluxes decrease, the typical values of these line ratios tend to increase. Of course, care needs to be given to such an interpretation, for noise fluctuations will result in higher derived line ratios for the lowest fluxes when the flux threshold applied is also the denominator in the ratio (as it is for the plots in the top line).[^3] Nevertheless, the median values of the [\[CI\]]{}/CO lines ratios are clearly rising as the CO line fluxes decrease. The overplotted lines in Fig. \[fig:ratiovsflux\] show the weighted minimum least squares linear fits to the data[^4]. The weak, but clearly negative slopes, to the fits plotted against the CO flux, as well as to [\[CI\]]{}/[$\rm ^{13}CO$]{} vs. [\[CI\]]{}, quantify this relationship; e.g. when the [$\rm ^{13}CO$]{} flux is 1K in a voxel, the [\[CI\]]{}/[$\rm ^{13}CO$]{} ratio is $\sim 50\%$ higher than when it is 3K, and for a [\[CI\]]{} flux of 0.5K in a voxel, the [\[CI\]]{}/[$\rm ^{13}CO$]{} ratio is $\sim 40\%$ higher than when it is 1.5K.
The [$\rm ^{13}CO$]{} line flux, being generally optically thin, provides a measure of the product $N_{\rm CO} f_{\rm CO}$, the column density of the CO molecules times their filling factor in the beam, whereas for the [\[CI\]]{} flux it is $N_{\rm C} f_{\rm C}$[^5]. While for any given voxel these factors cannot be separated, statistically, as the flux decreases there will be a greater proportion of lower column density regions associated with the corresponding voxels. We deduce that for these voxels the ratio of the columns of gas in C to CO tends to rise, although the total column in both species is lower.
Averaged Line Profiles {#sec:profiles}
----------------------
We pick a dividing ratio between “high” and “normal” of 1.0 for [\[CI\]]{}/[$\rm ^{13}CO$]{} from inspection of the histogram in Fig. \[fig:histograms\], and then split the data set into two cubes for those voxels that are “high” and those that are “normal”. Visual inspection of the resulting cubes suggests that “high” ratio values are generally found around the edges of CO emission regions. However, care is needed over any interpretation of high [\[CI\]]{}/[$\rm ^{13}CO$]{} existing at the peripheries of molecular clouds because these are also the regions where the S/N is lowest since the flux is least. In particular, the $1\sigma$ threshold limit applied to the selection of CO voxels exacerbates the selection of such voxels at the edges of emitting clouds.
### Splitting the Data based on the [\[CI\]]{}/[$\rm ^{13}CO$]{} Line Ratio {#sec:split}
We thus apply a mask to separate the voxels with “high” and “normal” ratios, and average the resulting cubes to improve the resulting S/N (data from $\sim19,000$ and $\sim55,000$ voxels are averaged, respectively). The results are displayed in Fig. \[fig:avprofiles\]. Here are shown the average profiles for [$\rm ^{13}CO$]{} (in red) and [\[CI\]]{} (green) in the data cube for all voxels that have “high” and “normal” [\[CI\]]{}/[$\rm ^{13}CO$]{} ratios (i.e. $>$ and $< 1.0$, respectively, as well as passing the flux thresholds discussed in §\[sec:histograms\]). Profiles for voxels with “normal” ratios are drawn with the solid lines, while those for “high” [\[CI\]]{}/[$\rm ^{13}CO$]{} ratios are shown with the dotted lines. It is clear that “normal” ratios generally correspond to brighter fluxes for both [\[CI\]]{} and [$\rm ^{13}CO$]{}, as well as having the [$\rm ^{13}CO$]{} line flux about 50% stronger than the [\[CI\]]{} flux (i.e. red $>$ green). Profiles for “high” [\[CI\]]{}/[$\rm ^{13}CO$]{} voxels, on the other hand, show weaker fluxes on average, as well as (by design) [\[CI\]]{} fluxes that are (slightly) higher than [$\rm ^{13}CO$]{}. This result applies across all velocities, and in particular in each of the three principal spiral arm crossings along the G328 sightline. Since the resulting profiles have been averaged across hundreds of voxels meeting the selection criteria their S/N is good. We can conclude that [\[CI\]]{}/[$\rm ^{13}CO$]{} is higher in pixels where the [\[CI\]]{} and [$\rm ^{13}CO$]{} line fluxes are lowest; i.e. as would occur preferentially on the edges of molecular clouds.
There is also the possibility that high ratios occur in voxels which have low optical depth (or column density), for, as we discuss in §\[sec:pdr\], clouds which only extend for $A_v \sim 1-2$mags. have a large proportion of their gas in the surface layer (i.e. $A_v < 1$). Here the carbon will predominantly be found as C–atoms (both singly-ionised and neutral). However, as the [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{} line ratios shown in Fig. \[fig:ratiovsflux\] illustrate, their typical flux ratio is $\sim 5$ and virtually all voxels have \[[$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{}\] $< 10$; i.e. the [$\rm ^{12}CO$]{} remains strongly optically thick.[^6] Thus, this possibility (low optical depth) could only apply to relatively few voxels in the data set, at most.
### Line Profiles and Ratios as a Function of Velocity
We now examine the CO, [\[CI\]]{} and [HI]{} lines profiles and ratios as a function of velocity, comparing the two groups (“normal” and “high” \[CI\]/$^{13}$CO ratios) with each other.
Fig. \[fig:lineprofiles\] shows the mean profiles for the lines, and Fig. \[fig:lineratios\] for their line ratios, when the data has been selected so that the [\[CI\]]{}/[$\rm ^{13}CO$]{} ratio is “high” (i.e. $> 1.0$) or “normal” (i.e. $< 1.0$) (as well as thresholded, as before[^7]). The behaviour is similar for the three velocity ranges (i.e. spiral arm crossings). The [$\rm ^{12}CO$]{} and [\[CI\]]{} fluxes are slightly brighter, on average, when the [\[CI\]]{}/[$\rm ^{13}CO$]{} ratio is “normal”, whereas the [$\rm ^{13}CO$]{} fluxes are significantly stronger for “normal” ratios. The [HI]{} flux is, however, is slightly smaller for “normal” [\[CI\]]{}/[$\rm ^{13}CO$]{} ratios.
For the plots showing line ratios (Fig. \[fig:lineratios\]), the [\[CI\]]{}/[$\rm ^{13}CO$]{} ratio is significantly larger for the “high” voxels (as it should be, by design). The [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{} ratio is also similarly higher. There is, however, little difference in the value of the [\[CI\]]{}/[$\rm ^{12}CO$]{} ratio between the “high” and the “normal” voxels. None of these line ratios depend on the velocity, either. They are constant, only dependent on the relative value of [$\rm ^{13}CO$]{} line flux; i.e. higher [\[CI\]]{}/[$\rm ^{13}CO$]{} and [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{} ratios for voxels with lower [$\rm ^{13}CO$]{} fluxes, and vice-versa.
The [\[CI\]]{}/[HI]{} ratio, however, [*decreases*]{} going from “normal” to “high” [\[CI\]]{}/[$\rm ^{13}CO$]{} values, by about $\sim 10-20\%$. This indicates that there is slightly more atomic gas associated with the “high” [\[CI\]]{}/[$\rm ^{13}CO$]{} ratio gas. Also noticeable is the velocity range for the narrow, quiescent filament ($-80$ to $-76$km/s) discussed in Paper I, where the [\[CI\]]{}/[HI]{} ratio is about 40% higher where [\[CI\]]{}/[$\rm ^{13}CO$]{} is “normal”, indicating substantively more atomic gas associated with this filament than for other molecular gas.
The conclusion is that high [\[CI\]]{}/[$\rm ^{13}CO$]{} and [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{} ratios occur when the [$\rm ^{13}CO$]{} flux is weak – i.e. on the edges of the various molecular emission features. At these same locations the [$\rm ^{12}CO$]{} emission is not, however, significantly weaker, on average. This can be attributed to optical depth; the [$\rm ^{12}CO$]{} line remains optically thick, even when relatively less of the species is present, whereas the same sightline samples all of the [$\rm ^{13}CO$]{} in the gas, if it passes through the edges of clouds. Since the [$\rm ^{13}CO$]{} flux is lower here then, as all the CO is being sampled, its total column must also be lower. In these same sightlines we find that the [\[CI\]]{} flux is only slightly lower, however, and hence so is the column of carbon. Thus, elevated carbon to carbon monoxide column density ratios are measured towards the edges of all emission features, i.e. at the edges of molecular clouds.
Furthermore, the lower [\[CI\]]{}/[HI]{} ratios in this same gas indicates that this material is associated with relatively more atomic gas at molecular cloud edges. This is as expected for sightlines passing preferentially through surface layers ($A_V < 1$) of PDRs, where CO should be photodissociated into C and C$^+$ and a significant amount of H$_2$ is photodissociated into H.
Thus, the relative column of C to CO along these sightlines is increased in comparison to sightlines passing through the interiors of molecular clouds. This is also the observational signature we have defined for detecting “dark gas”. This amounts to a statistical detection of regions where the carbon (C) is enhanced in abundance compared to the carbon monoxide (CO). These must preferentially contain the surface regions of molecular clouds since they pass through their edges. Finally, for the quiescent filament, which we hypothesized to be undergoing the process of molecular cloud formation in Paper I, the increased [HI]{} abundance associated with the cloud edges is consistent with the assertion that it is condensing out of the atomic substrate.
Hue – Saturation Plots to Examine Variations in Line Ratios {#sec:evans}
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We present here another technique to examine the significance of the increased [\[CI\]]{}/[$\rm ^{13}CO$]{} ratios at the edges of the molecular clouds – hue–saturation plots.
Hue / Saturation plots[^8] provide a way of visualizing two variables of interest, one of which provides some measure of “importance”, and the other its value. This is done by assigning the hue of a color to the value and the saturation (or intensity) of the color to the “importance”. Hue is cyclic, representing the angle around a color wheel running through the spectrum from red to purple (and back to red again); i.e. with range from $0^{\circ}$ up to $360^{\circ}$. Low importance causes the color to fade into grey regardless of what the value actually is. Such plots are useful for examining the ratio of two variables when the overall S/N is limited. It avoids the eye being drawn to seemingly high line ratios in regions of low S/N data.
We have constructed such Evans plots to examine the variation of the line ratios by using hue (i.e. color) to denote the value of the line ratio, and saturation (i.e. intensity) to denote its S/N. Hue is ranged from 0 to $300^{\circ}$ in these plots to avoid wrapping, i.e. from red for the lowest ratio displayed to purple for the highest. Saturation is ranged from 0 to 1, with 1 normalized to the value of the highest S/N. Hue-Saturation-Lightness values (the latter, which represents the perceived luminance of the system, is always set to a fixed value of 0.5 here) are then converted to red-green-blue for display (using IDL library routines which transform between the HSL and RGB color systems).
Two Evans plots are shown in Figs. \[fig:evans\_v8076\] – \[fig:evans\_v100\] for the velocity ranges of $\rm (-80,-76)$ and $(-100,-85)$ km/s; i.e. the filament and the Norma near- spiral arm crossing (the other two spiral arm velocity ranges show similar characteristics and are included in the Appendix; see Figs. \[fig:evans\_v80\] – \[fig:evans\_v55\]). For each velocity range three plots are displayed, [\[CI\]]{}/[$\rm ^{13}CO$]{}, [\[CI\]]{}/[$\rm ^{12}CO$]{} and [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{}, so the behaviour of all three lines can be compared. The Evans plots are on the left and their corresponding 2D color tables on the right.
Examination of these images (and in comparison to the flux images of Fig. \[fig:momentimages120\]) shows that:
1. For [\[CI\]]{}/[$\rm ^{13}CO$]{}, the highest S/N regions tend to be surrounded by higher-ratio, but lower S/N regions.
2. For [\[CI\]]{}/[$\rm ^{12}CO$]{}, on the other hand, there are regions of differing line ratio across the fields, but with no particular tendency for high ratio regions to surround low ratio regions, or vice-versa. There is also no tendency for lower S/N regions to have different ratios than higher S/N regions.
3. For [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{}, the line ratios are generally lower within the emission regions where there is more CO, and higher at their edges.
Taking these results in reverse order, we can interpret them as follows:
Since [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{} variations measure optical depth changes, we deduce that the optical depth is highest (and so the line ratio is lowest) along the sight lines passing through cores of clouds. It is lowest (with the highest ratios) for sight lines that pass through the edges of clouds.
[\[CI\]]{}/[$\rm ^{12}CO$]{} ratio variations trace changes in the relative [\[CI\]]{} flux between molecular clouds, since [$\rm ^{12}CO$]{} is, in general, optically thick and so relatively constant.
Higher line ratios for [\[CI\]]{}/[$\rm ^{13}CO$]{} are seen for sight lines that do not pass through the cores of GMCs, but rather tend to pass through their peripheries. Since this ratio is sensitive to all the C and CO gas along each sight line, as both lines are largely optically thin, then along them the C abundance must be elevated compared to that of the CO. This is as expected for the surface layers of molecular clouds, where for the first optical depth CO will generally be photodissociated, and so the carbon can only be seen as [\[CI\]]{} (and [\[CII\]]{}), rather than as CO. This is likely mostly dark molecular gas, with $\rm H_2$ existing in the surface layer, and the carbon largely in atomic form rather than molecular.
In the PDR models discussed in the following section the [\[CI\]]{} peaks in gas which is molecular rather than atomic. It may, however, also represent atomic gas, in particular if there is [\[CII\]]{} present. As determined in Paper I, the column density of [HI]{} is typically $\rm \sim 10^{21}\,cm^{-2}$ in each voxel. Future measurements, at the same resolution, of the 1.9 THz [\[CII\]]{} line, which will co-exist with much of the [HI]{}, would allow the amount of C$^+$ and C associated with the atomic gas to be quantified.
There are also specific regions which stand out in these plots. For instance, at $(328.65,-0.25)$ in Fig. \[fig:evans\_v8076\] both [\[CI\]]{}/[$\rm ^{13}CO$]{} and [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{} are measured to be about twice as high as other regions within the filament, while [\[CI\]]{}/[$\rm ^{12}CO$]{} is unchanged. This can be ascribed to a correspondingly lower optical depth in the CO.
Photo Dissociation Region (PDR) Models {#sec:pdr}
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We have examined a range of cloud properties using a modified form of the [@2010ApJ...716.1191W] photodissociation region code to calculate column densities and line emission to compare with the data. The model consists of a slab of gas of total optical depth $A_{V,{\rm tot}}$ illuminated by the interstellar radiation field on two sides. The gas temperature and the abundances of atomic/molecular species are calculated as a function of optical depth, $A_V$, under the assumptions of thermal balance, chemical equilibrium and constant thermal pressure. For details on the chemistry and thermal processes we refer to @1985ApJ...291..722T, @2006ApJ...644..283K, [@2010ApJ...716.1191W] and [@2012ApJ...754..105H].
We have made a number of modifications to the code used in [@2014ApJ...782...72B]. First we have explicitly included ${\rm ^{13}C}$ chemistry and line transfer in order to model the [$\rm ^{13}CO$]{} observations. A number of researchers have presented their results on ${\rm ^{13}C}$ chemistry in PDRs including , and [@2014MNRAS.445.4055S]. Here we use the fit by to the temperature dependence of the important fractionation reaction ${\rm ^{13}C^+} + {\rm ^{12}CO}\rightarrow {\rm ^{12}C^+} + {\rm ^{13}CO} +34.8 {\rm K}$, and use a ${\rm ^{12}C/^{13}C}$ ratio of 50 as appropriate for the inner Galaxy [@2005ApJ...634.1126M]. Second, we found that a deeper penetration of the external radiation field, compared to our previous models, is required to accurately fit the [\[CI\]]{}/[$\rm ^{13}CO$]{} line ratios. Similar results have been reported by [@2015MNRAS.448.1607G] and naturally arise in a turbulent gas in which lines of sight pass through low column density material. We approximate this process in our models by changing the angle dependence of the incident radiation field from isotropic to normally incident. For a gas layer at optical depth $A_V$, the resulting FUV field strength is $e^{1.8 A_V}$ times larger for the normally incident case compared to the isotropic case [see discussion in @2010ApJ...716.1191W]. We used the shielding functions for [$\rm ^{12}CO$]{} and [$\rm ^{13}CO$]{} given by which accounts for self-shielding, shielding of [$\rm ^{12}CO$]{} and [$\rm ^{13}CO$]{} by H$_2$, and shielding of [$\rm ^{13}CO$]{} by [$\rm ^{12}CO$]{}.
Additional changes involve abundances of metals and Polycyclic Aromatic Hydrocarbons (PAHs). [@1985ApJ...291..722T] noted that an important production reaction for C was charge exchange with ${\rm S}$ (${\rm C^+ + S \rightarrow C + S^+}$). In addition, it has since been realized that ion recombination on PAHs can be important in increasing the abundance of neutrals (and also decreasing the abundance of free electrons). We find that a decrease in both the sulfur abundance and PAH abundance is required to match the [$\rm ^{13}CO$]{} to [\[CI\]]{} line ratio. Our previous models used a sulfur abundance of ${\rm S/H} = 2.8\times 10^{-5}$ based on the analysis of . However, a comprehensive study by [@2009ApJ...700.1299J] finds a sulfur abundance in depleted lines of sight in the solar neighborhood to be ${\rm S/H} = 3.5\times 10^{-6}$. In a recent study of the [\[CI\]]{} line emission towards the Taurus molecular Cloud, [@2014ApJ...795...26O] find a sulfur depletion factor of at least 50 is required to match the observations. In light of the more direct observational estimate of the S abundance and the likely presence of PAHs, we limit the sulfur depletion to the [@2009ApJ...700.1299J] value. Since we are modelling clouds in the inner Galaxy we multiply by a factor of two all gas phase metal abundances, dust abundances and PAH abundances to account for the increased metallicity there. This results in a gas phase sulfur abundance of $\rm S/H = 7\times10^{-6}$ (see Table \[tab:pdrmodel\]).
Even using the depleted sulfur abundance we find that we still need to reduce the Galactic PAH abundance by a factor of two. Reducing the PAH abundance decreases the rate of $\rm S^+$ recombination on PAHs and thus decreases the S abundance in the $\rm C^+/C$ zone. This further reduces the neutral carbon abundance. @2012ApJ...754..105H used $\rm PAH/H = 2\times10^{-7}$ for a local abundance. Accounting for higher metallicities in the inner Galaxy would result in $\rm PAH/H = 4\times10^{-7}$, however we reduce this by a factor of two to arrive at a model abundance of $\rm PAH/H = 2\times10^{-7}$. There is still considerable uncertainty in both PAH rates and PAH abundances and a factor of two decrease is not unreasonable.
Our “standard” model parameters are listed in Table \[tab:pdrmodel\], with some predicted lines fluxes and column densities then listed in Table \[tab:pdrmodeloutput\]. For this standard model we adopt a radiation field of $G_0 = 5.1$ (with half of this incident on each side; $G_0$ is the interstellar field strength in units of Habing fields with $G_0 = 1.7$ representing an average field strength in the ISM; @1968BAN....19..421H [@1978ApJS...36..595D]). This field is roughly consistent with the interstellar field expected at a Galactocentric radius of 5kpc [@2003ApJ...587..278W]. We also use a constant thermal pressure of $P_{\rm th}/k=2\times 10^4$ K ${\rm cm^{-3}}$. The typical density (when $T_{\rm gas} \sim 30$K) is then $n_{\rm H_2} \sim 600$cm$^{-3}$. Models are computed as a function of total optical extinction through the cloud from $A_{V,{\rm tot}}=0.5$ mags. for ‘thin’ clouds, up to $A_{V,{\rm tot}}=6$ mags., where $A_{V,{\rm tot}}=[N({\rm H~I})+2N({\rm H_2})]/2.0\times 10^{21}$ ${\rm cm^{-2}}$. We demonstrate in §\[sec:pdrmodelplots\] the effects of varying our standard parameters.
We have adopted a primary cosmic ray ionization rate of $\rm \zeta _{crp}=2\times 10^{-16}\,s^{-1}$, consistent with rates for low column density diffuse clouds determined from observations of H$_3^+$ [@2012ApJ...745...91I], OH$^+$ and H$_3$O$^+$ [@2012ApJ...754..105H; @2015ApJ...800...40I]. Furthermore, our modelling of the MeV–GeV cosmic-ray fluxes from the known supernova remnants (G329.7+00.4, G327.4+01.0, G327.2-00.1, G327.1-01.1, G328.4+00.2 and G327.4+00.4) and Fermi-LAT GeV gamma-ray sources (3FGL J1554.4-5315c and 3FGL J1552.8-5330) in the region suggests no enhancement of the ionization rate beyond the galactic average (i.e. $\rm \zeta_{crp}=2\times 10^{-16}\,s^{-1}$). There is some evidence that the ionization rate is depth dependent in clouds, varying from $\rm \sim 2\times 10^{-16} \,s^{-1}$ in low $A_V$ clouds to values of $\rm \sim 2\times 10^{-17} \,s^{-1}$ in the centers of molecular clouds . As discussed in Paper I, the model has a high cosmic-ray ionization rate but equally good fits to the data can be obtained with the low cosmic ray rate. However, we find that the higher cosmic ray rates produce a slightly higher [\[CI\]]{} line intensity since the higher rates allow neutral C to extend deeper into the cloud.
In addition to providing results from these models for the three principal lines observed in the data presented here (i.e. [$\rm ^{12}CO$]{} & [$\rm ^{13}CO$]{} 1–0 and [\[CI\]]{} 2–1) we also include the results for [\[CI\]]{} 1–0 and [\[CII\]]{} lines emitted at 492GHz and 1.9THz, respectively. This is so that they may also be applied towards interpreting measurements made with sub-mm telescopes in Chile such as Nanten2, APEX and ALMA [e.g. @2008stt..conf..488G; @2014ApJ...797L..17L], as well as THz frequency telescopes. These latter include airborne and space facilities as well as future telescopes under development for Antarctica such as STO–2 [@2010SPIE.7733E..0NW] and DATE5 [@2013RAA....13.1493Y].
PDR Model Comparison {#sec:pdrmodelplots}
--------------------
We present here the results for a representative model using our ‘standard’ parameter set with $A_{V,\rm tot}=6$ mags. The abundance of C$^+$, C, [$\rm ^{12}CO$]{} and [$\rm ^{13}CO$]{}, as well as of H$_2$, as a function of the optical extinction, $A_V$, from the cloud surface is shown in Fig. \[fig:xh2covavplt\]. Both the ionized and neutral carbon are seen to be confined to the surface of the PDR, with the C$^+$ peaking at the front surface, and mostly disappearing by $A_V \sim 0.6$ mags, where the C peaks. Further in to the cloud the carbon is converted into CO, with [$\rm ^{13}CO$]{} rising rapidly between $A_V\sim 0.5 - 1$ mags. followed by a slow rise to $A_V \sim 1.8$ mags. At this point photo-desorption of CO from the cold grains diminishes due to dust attenuation of the FUV and freeze-out of the CO onto grain mantles begins, with the CO abundance dropping by a factor of $\sim 3$ at $A_V \sim 3$ mags. (at the cloud center). The hydrogen gas, on the other hand, is converted almost entirely into H$_2$ by $A_V = 0.1$ mags. so the C is mostly found in regions where the hydrogen is molecular H$_2$ rather than atomic H.
The optical depth of the [\[CI\]]{}, [\[CII\]]{} and CO lines as function of $A_V$ in this representative model is shown in Fig. \[fig:tauci13covavplt\]. The total extinction through the model cloud is $A_{V,{\rm tot}} = 6$ mags., and lines here are plotted to cloud center. Thus, the total optical depth in the sightline through the cloud is twice that shown at $A_V = 3$ in the Figure. [\[CII\]]{} rises rapidly until $A_V \sim 0.5$ and then remains constant, as by then all the C$^+$ has been converted to C or CO. The [\[CI\]]{} lines rise rapidly to $A_V\sim 0.8$ past the abundance peak, and then only gradually thereafter as the [\[CI\]]{} abundance falls. The [$\rm ^{13}CO$]{} continues to rise within the cloud but starts to flatten near $A_V\sim 2.5$ due to [$\rm ^{13}CO$]{} freeze-out. All the associated lines for these three species remain optically thin to cloud center. Through the full cloud extent (i.e. $A_V = 6$) the [\[CII\]]{} and [\[CI\]]{} 2–1 lines remain thin, while the [\[CI\]]{} 1–0 and [$\rm ^{13}CO$]{} lines are marginally thick. [$\rm ^{12}CO$]{} is strongly optically thick beyond $A_V \sim 0.8$.
The gas temperature, $T_{\rm gas}$, and dust temperature, $T_{\rm dust}$, through the PDR are shown as a function of the extinction for the representative model in Fig. \[fig:tgasplt2\]. The gas temperature averages around 30K within the first magnitude of the extinction, though rises slightly to $\sim 37$K for $A_V < 0.1$, at the very front surface of the cloud. The temperature rise is due to molecular hydrogen formation. The collisional rate coefficient of molecular hydrogen exciting C$^+$ is about half that of atomic hydrogen exciting C$^+$[^9]. Since the C$^+$ is the dominant coolant, the lower excitation rate leads to warmer gas. At deeper layers, the temperature falls due to extinction of FUV radiation field resulting in a lower photoelectric heating rate. At $A_V > 2$, the photodesorption of molecules from grain surfaces falls sufficiently so that CO stays frozen onto grains. The freeze-out of the dominant coolants cause the temperature to rise slightly at cloud center.
The integrated line intensities for the [\[CI\]]{}, [\[CII\]]{} and CO lines through the PDR as a function of column density (i.e. $A_V$) for the representative model are shown in Fig. \[fig:ci13coTkmvAv\]. The intensity plotted is the total intensity emitted through one side of the cloud, including the emission from both the near and far sides (i.e., the emission generated at the far side of the cloud that passes through the cloud center and emerges at the near side of the cloud). These fluxes increase with depth into the PDR, until the carbon has been converted into CO, and the CO then frozen-out onto grains. The model line fluxes for the [\[CI\]]{} 2–1, [$\rm ^{12}CO$]{} and [$\rm ^{13}CO$]{} lines are typically 6–12 times those measured in the individual voxels. However, the model uses a FWHM of 4 km/s, a factor 6 greater than the voxel width of 0.7 km/s used in the data analysis, and so is consistent with flux filling factors in the 2 arcmin beam of between $\sim 50$ and 100%.
In Fig. \[fig:ci13covAv3\] are shown plots of the [\[CI\]]{} 2–1 / [$\rm ^{13}CO$]{} line ratio as a function of optical extinction, $A_{V,{\rm tot}}$, through molecular clouds for a variety of values of the input parameters. For each curve we keep the parameters fixed for the ‘standard’ model, and vary only the parameter that is listed in the figure legend. The $\rm PAH/H$ curve is for a PAH abundance that is twice as high as for the standard model while the $\rm S/H$ curve is for a sulfur abundance that is twice the abundance used in our previous modelling [@1999ApJ...527..795K] and eight times larger than in our standard case. The $G_0$ curve is for a radiation field that is the local Galactic field, and the thermal pressure curve is a factor of 3 lower than the standard model. The parameters range roughly between those expected for nearby clouds and those in the inner Galaxy. The higher PAH/H and S/H curves increase the [\[CI\]]{}/[$\rm ^{13}CO$]{} ratio by increasing the C abundance as described in §\[sec:pdr\]. The lower $G_0$ model produces a lower [\[CI\]]{}/[$\rm ^{13}CO$]{} ratio due to a lower heating rate in the [\[CI\]]{} region. For the low pressure models, the lower density pushes the molecular transition deeper into the cloud, thereby reducing the [$\rm ^{13}CO$]{} column for the low $A_{V, \rm{tot}}$ models.
The line ratio is seen to decrease as the cloud column density increases, since the [\[CI\]]{} emission is confined to the front surface, $A_V < 0.6$ mags. (see Fig. \[fig:ci13coTkmvAv\]), whereas the [$\rm ^{13}CO$]{} emission extends deeper into the cloud. Except for the low thermal pressure model, beyond $A_{V{\rm tot}} \sim 4$ mags. the line ratio remains constant, indicating that neither line is contributing significant additional line flux for thicker clouds. All models obtain the observed [\[CI\]]{}/[$\rm ^{13}CO$]{} ratios of $< 2$ for sufficiently thick clouds. The lowest allowed column density cloud is $A_{V,{\rm tot}}\sim 1$, for the low FUV field model.
The dependence of the [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{} line ratio as a function of the total optical depth through the cloud, $A_{V,{\rm tot}}$ is shown in Fig. \[fig:co12co13vAv\]. Again, the line ratio is high for low optical depths, but rapidly decreases as $A_V$ rises, plateauing beyond $\sim 4$ mags., similarly to [\[CI\]]{}/[$\rm ^{13}CO$]{}. In this case, however, this is caused by the freeze-out of the CO molecules onto dust grains.
The dependence of these two line ratios ([$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{} and [\[CI\]]{} 2–1/[$\rm ^{13}CO$]{}) are also shown as a function of the [$\rm ^{13}CO$]{} line flux in Figs. \[fig:co12co13vco13\] and \[fig:cico13vco13\]. Similar behaviour to the plots against extinction, $A_V$, are exhibited, with the line ratios being high for small values of the [$\rm ^{13}CO$]{} line flux. This reflects, correspondingly, the CO optical depth being lower for thin clouds (i.e. $A_v < 2$), and the [\[CI\]]{} emission arising from the same region, near to the cloud surface. In practice, however, observing such high ratio but low intensity gas in our data set will be difficult since one or both lines may not be detectable. We also caution that low intensity gas could have low ratios, if the beam filling factor is small. In principle, however, it is possible to have gas with [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{} and [\[CI\]]{}/[$\rm ^{13}CO$]{} ratios much higher than those observed in the G328 data set presented here. They would also be more readily detectable with higher angular resolution measurements, able to separate such regions out from high column density sightlines.
We also calculated the ‘dark gas’ contribution in the standard model. Using the definitions in [@2010ApJ...716.1191W], we assume the dark gas layer begins when the gas is half molecular (i.e., $2n({\rm H_2})=0.5n$) and ends when the optical depth in the ${\rm CO\, (1-0)}$ transition equals one. We find the line-of-sight dark gas fraction is 27% of the total column density of this $A_{V,tot} = 6$mags. slab. Since the ${\rm H_2}$ forms essentially at the cloud surface ($A_V \sim 0.1$), the dark gas fraction of the total molecular column density is also about 27%. We can also estimate the fraction of dark gas in a spherical cloud by converting the $A_V$ steps to radius steps ($\delta r = \delta A_V(r) \, 2.0 \times 10^{21}/n(r)$, where $\delta r$ and $\delta A_V$ are the distances between radius and $A_V$ grid points, respectively). The mass at radius $r = \Sigma \delta r$ is found by summing the density in spherical shells $M(r)= \Sigma n(r)4\pi r^2 \delta r$ (H ${\rm cm^{-3}}$). We find a fraction of dark gas compared to the total molecular gas is $\sim 67$% with a similar fraction when the total (molecular plus atomic) mass is included. The dark gas fraction is a factor of two higher than that found by [@2010ApJ...716.1191W] for local Galactic GMCs. However, the mean area-averaged column density for the model cloud is found to be $\bar{A_V} = 2.7$. This is smaller than that for GMCs [${\bar A_V}\sim 8$; @1987ApJ...319..730S] and the dark gas fraction is predicted to increase as the mean column density decreases [@2010ApJ...716.1191W].
Finally, we have estimated the optical depth of the (cold) atomic gas in the PDR through application of the radiative transfer equation $N = T_{S} \Delta V X (1 - e{^{-\tau}})$, where $N$ is the column density of atomic hydrogen in the PDR model (Table \[tab:pdrmodeloutput\]), $T_S$ its spin temperature (i.e. as at the front of the PDR; 35K from Fig. \[fig:tgasplt2\]), $\Delta V$ the FWHM line width (Table \[tab:pdrmodel\]), and $X$ the HI X-factor ($\rm 1.8 \times 10^{18} \, cm^{-2} \, K^{-1} \,km^{-1} \,s$; ). This yields $\tau \sim 0.3$ for the atomic gas in the PDR. Following @1999ApJ...517..209G, the antenna temperature for the 21cm HI line emission from this gas is then given $N \ = T_{A} \, \Delta V X \frac{\tau}{1 - e{^{-\tau}}}$, which yields $T_A \sim 8$K. This can be compared to the typical brightness temperatures observed along the G328 sightline, 90–100K (Fig. \[fig:profiles\]), which are dominated by hydrogen in (warmer) atomic gas clouds rather than in PDRs. From their ratio we estimate that the fraction of atomic hydrogen found in PDRs along the sight line is $\sim 10\%$, with the remaining $\sim 90\%$ being in atomic clouds. Furthermore, we may also estimate the amount of “dark” atomic gas in the PDR by calculating the column derived by applying the optically thin limit in the above formula, and comparing this to the actual column from the model. We find that $\sim 15\%$ of the atomic gas in the PDR is dark; i.e. a column of H amounting to $\sim \rm 10^{19} \, cm^{-2}$. This is also a negligible fraction of the total column of atomic gas along the G328 sightline, around $\sim 1\%$ of it.
Discussion {#sec:discuss}
==========
The models presented above in §\[sec:pdrmodelplots\] show that, in the low–FUV field PDR environment typical of the diffuse molecular environment, that ionized C$^+$ is found within $A_V < 0.5$ mags. of the cloud surface. Neutral C extends as far as $A_V \sim 2$mags. into the PDR, but peaks in abundance at $A_V \sim 0.5$mags. Molecular CO exists deeper in than $A_V \sim 0.5$mags. Freeze-out of CO begins by $A_V \sim 2$mags. with only one-third of the molecule remaining in the gas-phase by $A_V \sim 3$mags.
Given the different extents where these species are found inside a PDR, the measured line ratios, such as [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{} and [\[CI\]]{}/[$\rm ^{13}CO$]{}, will depend on the column density of the PDR sampled by the line of sight. In particular, those sight lines that either only pass through their edges, or include low column density PDRs (i.e. only extending for 1–2 mags. rather than the more typical 8 mags. for GMCs) will display elevated values of these ratios.
Optical depth, however, also needs to be accounted for in determining line fluxes, and hence line ratios. [\[CI\]]{}, [\[CII\]]{} and [$\rm ^{13}CO$]{} remain optically thin within these emitting regions. However, the [$\rm ^{12}CO$]{} line rapidly becomes optically thick further into the molecular gas from the position where it is first encountered.
Thus [\[CII\]]{} line flux rise until a depth of $A_V \sim 0.5$ mags. is reached into a PDR (and then remains flat as the sightline penetrates further), while [\[CI\]]{} rises until a depth of $A_V \sim 1$ mags. In both cases a flattening in flux with further depth is because the species are not found further into cloud. For the [$\rm ^{12}CO$]{} line flux, in contrast, while it rises rapidly between $A_V = 0.5$ and 1 mags., it then saturates as it becomes optically thick. Only modest rises in flux then occur going deeper. On the other hand, [$\rm ^{13}CO$]{} line fluxes continue to rise steadily with increasing depth into a PDR until CO freezes out. Representative line fluxes and column densities, for a PDR with a total column of $A_V = 6$ mags., are listed for the standard model parameters in Table \[tab:pdrmodeloutput\]. The column of C and C$^+$ are both approximately one-third that of the total column of gas-phase CO found though the PDR.
Comparison with the flux measurements in Fig. \[fig:lineprofiles\] shows that these model predictions are consistent with the data for [$\rm ^{12}CO$]{}, [$\rm ^{13}CO$]{} and [\[CI\]]{} 2–1, if the average beam filling factor in the $2'$ voxel is $\sim 50\%$ (and taking into account the difference between the model 4 km/s FWHM and the voxel channel width of 0.7 km/s). The corresponding [\[CI\]]{} 1–0 and [\[CII\]]{} fluxes are thus predicted to be of order 7 and 1Kkm/s, respectively, if observed with the same spectral and spatial resolution.
The models show that the [\[CI\]]{} 2–1/[$\rm ^{13}CO$]{} and [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{} line ratios are constant for clouds with total extinctions $A_{V,{\rm tot}} > 2-4$ mags. (dependent on the particular model), with ratios of $\sim 1$ and $\sim 4$, respectively. For lower $A_{V,{\rm tot}}$ these line ratios rise, as the relative line fluxes are sensitive to the depth of the PDR. Small $A_{V,{\rm tot}}$ corresponds to PDRs which are either relatively thin, or for sight lines which are predominantly passing through the surface layers of PDRs. This latter case would also correspond to sight lines passing through the edges of molecular clouds rather than through their cores. The ratios can rise to high values ($>3$ and $>15$, respectively) for small $A_{V,{\rm tot}}$, but then the corresponding line fluxes are themselves small as only a limited column of gas is being sampled. No data with such high ratios was seen, though possibly with higher angular resolution (e.g. as with ALMA), when beam dilution of any such smaller clouds that might exist would be much reduced, then higher ratios might be found in some gas.
On the other hand, limb brightening would result in larger line fluxes. Since the measured line fluxes are actually smaller when [\[CI\]]{}/[$\rm ^{13}CO$]{} is larger (see Figs. \[fig:ratiovsflux\] and \[fig:lineratios\]), this implies that any limb brightening that occurs is overwhelmed by smaller beam filling factors.
The typical [\[CI\]]{}/[$\rm ^{13}CO$]{} line ratios measured here are $\sim 0.7$, rising to $\sim 1.3$ (Fig. \[fig:lineratios\]). The latter are found where the fluxes are, in general, weaker (Fig. \[fig:ratiovsflux\]). While for individual pixels the effects of weak intrinsic flux or beam dilution cannot be disentangled, the modelling shows that if the [$\rm ^{13}CO$]{} is intrinsically weak then the [\[CI\]]{}/[$\rm ^{13}CO$]{} ratio is higher (Fig. \[fig:cico13vco13\]). This is consistent with the associated data being dominated by voxels with intrinsically weaker flux rather than by beam dilution.
Typical [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{} line ratios observed are $\sim 5$, rising up to $\sim 15$ (Fig. \[fig:lineratios\]). As for [\[CI\]]{}/[$\rm ^{13}CO$]{}, the latter generally occur where [$\rm ^{13}CO$]{} fluxes are lower. And similarly (Fig. \[fig:co12co13vco13\]), these values are consistent with the higher ratios being associated with sight lines passing through lower column density gas (and/or just the front surfaces of PDRs).
To summarize, the line fluxes for [$\rm ^{12}CO$]{}, [$\rm ^{13}CO$]{} and [\[CI\]]{} presented here for the G328 sightline of the Galaxy are consistent with being produced in low density ($n_{\rm H_2} \sim 600$cm$^{-3}$) photodissociation regions being excited by far–UV radiation fields with strengths around the typical average interstellar value ($G_0 \sim 1.7 - 5.1$). Larger line ratios of [\[CI\]]{}/[$\rm ^{13}CO$]{} are found, in general, associated with smaller values of the [$\rm ^{13}CO$]{} line flux. These smaller line fluxes are, in turn, generally associated with being emitted by gas with lower intrinsic line fluxes, rather than arising in sight lines with more beam dilution (which is found to average around 50%). Such regions are associated with the front surfaces of PDRs, within an optical extinction of 2 mags. from the atomic hydrogen interface. Here carbon can be found in ionized (C$^+$), neutral (C) or molecular (CO) form, rather than being dominated by CO at greater depths. Such regions are also associated with dark molecular gas, molecular clouds (i.e. H$_2$) where CO does not provide the dominant tracer for the gas.
Summary {#sec:summary}
=======
We have presented data cubes showing the distribution of the [\[CI\]]{} 2–1, [$\rm ^{12}CO$]{}, [$\rm ^{13}CO$]{} and [$\rm C^{18}O$]{} 1–0, and HI line emission, over a $\sim 1^{\circ}$ region of the Galactic plane along a sightline towards G328, with angular and spectral resolutions of $\sim 1'$ and $\sim 1$ km/s. The [\[CI\]]{} data comes from a new telescope, HEAT, sited on the summit of the Antarctic plateau at the driest location on the Earth, where the THz atmospheric windows are opened for observation. The CO data was taken with the Mopra telescope, and the HI data with the Parkes and ATCA telescopes, all located in Australia.
Complex morphology is evident in all species, with the [\[CI\]]{} and CO line emission extending over 120 km/s in extent, and arising principally from molecular cloud complexes in three spiral arm crossings along the sight line. The distribution in these atomic and molecular species is very similar in both angular and spectral dimensions, and is encompassed by more extensive HI line emission. However close examination of the [\[CI\]]{} and CO emission shows the latter to be slightly more extended spatially.
Average line ratios for [$\rm ^{12}CO$]{}/[\[CI\]]{}, [$\rm ^{13}CO$]{}/[\[CI\]]{} and [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{} are found to $6.8 \pm 1.6, 1.4 \pm 0.5 \, {\rm and} \, 5.5 \pm 2.5$, respectively, across all voxels in the data cubes when resampled to the same resolution. These ratios are relatively constant with velocity, across the three spiral arms crossed along the sight line. However, at the edges of the emission features the [\[CI\]]{}/[$\rm ^{13}CO$]{} and [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{} ratios are typically found to be 50% larger, with [\[CI\]]{}/[HI]{} ratios around 10% lower. This is attributed to relatively more C than CO at the edges of molecular clouds, with the [$\rm ^{13}CO$]{} intensity providing the best measure for the column of CO, rather than the optically thick [$\rm ^{12}CO$]{}. On the other hand, the relative amount of atomic gas is seen to rise at the edges of the clouds.
PDR models were constructed for diffuse molecular gas exposed to average interstellar radiation fields to explore the behaviour of the [$\rm ^{12}CO$]{}, [$\rm ^{13}CO$]{}, [\[CI\]]{} 1–0 and 2–1, and [\[CII\]]{} line emission with increasing extinction, $A_V$, into molecular clouds from their front (atomic) surface. Charge exchange reactions between C$^+$ and both S and PAHs were also found to be important to match the data, and require significant depletions of the S and PAHs. The models then reproduce the broad behaviour seen in the data, in particular the line fluxes (with average beam filling factors of $\sim 50$%) and the increased line ratios for [\[CI\]]{}/[$\rm ^{13}CO$]{} and [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{} in the surface layers, i.e. when $A_V < 2$mags. The highest values of the [\[CI\]]{}/[$\rm ^{13}CO$]{} ratio measured require sightlines passing through clouds that have extinctions $A_V < 1$ from their exciting sources. These may arise from sightlines that only pass through the edges of clouds, and/or small clouds whose total column density extends for $A_V \sim 1$. The [$\rm ^{13}CO$]{} and [\[CI\]]{} lines remain marginally optically thin through the PDR, and so provide direct probes of the amount of CO and C present, respectively.
Roughly one-third of the molecular gas along the sightline is estimated by these models to be associated with dark molecular gas, and about two-thirds in total when accounting for a spherical geometry. This is gas where CO line emission does not provide a full tracer for the column of molecular gas that is present. Considering the column of cold atomic gas in the front surface of the PDR, we also estimate that it contains $\sim 10\%$ of the total atomic gas along each sight line (the rest being in atomic gas clouds). Of the atomic gas within the PDR, $\sim 15\%$ of it is dark, i.e. the fraction not measured through use of the standard optically thin assumption for determining column densities. This is only $\sim 1\%$ of the total atomic column along sight line, however.
The modelling would also be aided by measurements of the [\[CI\]]{} 1–0 line emission at 492 GHz, whose ratio with the [\[CI\]]{} 2–1 line we have observed at 809GHz provides a probe that is sensitive to the gas temperature, predicted here to be $\sim 30$K for $A_V < 1$. Such measurements could be made from the Atacama plateau in Chile, where the the superior angular resolution obtainable with larger telescopes such as Nanten2, APEX and ALMA would allow for a better comparison of the relative distributions of the C and CO, and so in determining more clearly the form and extent of the carbon envelope around molecular clouds.
A full quantification of the amount of dark molecular present also requires measurement of the [\[CII\]]{} line emission, which is confined within $A_V < 0.5$ of the PDR surface. Some airborne and space measurements currently for such [\[CII\]]{} now exist . We provide predictions of its intensity and distribution with extinction into molecular clouds, to aid in their interpretation and for future observations made with THz facilities in Antarctica, such as the STO–2 long duration balloon-borne telescope scheduled for launch in 2016 and China’s proposed 5m DATE5 telescope for Dome A on the summit of the Antarctic plateau.
Acknowledgements {#acknowledgements .unnumbered}
================
Funding for the HEAT telescope is provided by the US National Science Foundation under grant number PLR-0944335. PLATO–R was funded by Astronomy Australia Limited, as well as the University of New South Wales, as an initiative of the Australian Government being conducted as part of the Super Science Initiative and financed from the Education Investment Fund. Support for the UNSW program in Antarctica is also provided by the Australian Antarctic Division. Logistical support for HEAT and PLATO–R is provided by the United States Antarctic Program.
The Mopra radio telescope is part of the Australia Telescope National Facility. Operations support was provided by the University of New South Wales and the University of Adelaide. The UNSW Digital Filter Bank used for the observations with Mopra was provided with financial support from the Australian Research Council (ARC), UNSW, Sydney and Monash universities. We also acknowledge ARC support through Discovery Project DP120101585. M.G.W. and D.J.H. were supported in part by NSF grant AST–1411827.
M.G.B. thanks the Dublin Institute for Advanced Studies (DIAS) in Ireland and the University of Leeds in the UK for their hospitality, during which much of the analysis for this paper was undertaken.
[*Facilities:*]{} , , , .
Ashley, M. C. B., Augarten, Y., Bonner, C. S., et al. 2012, , 8444, 84441R
Barinovs, [Ğ]{}., van Hemert, M. C., Krems, R., & Dalgarno, A. 2005, , 620, 537
Braiding, C., Burton, M. G., Blackwell, R., et al. 2015, , 32, e020
Brand, J., & Blitz, L. 1993, , 275, 67
Burton, M. G., Ashley, M. C. B., Braiding, C., et al. 2014, , 782, 72
Burton, M. G., Braiding, C., Glück, C., et al. 2013, , 30, 44
Burton, M. G., Hollenbach, D. J., & Tielens, A. G. G. M. 1990, , 365, 620
Burton, M. G., Hollenbach, D. J., & Tielens, A. G. G. 1992, , 399, 563
Dickey, J. M., & Lockman, F. J. 1990, , 28, 215
Draine, B. T. 1978, , 36, 595
Flower, D. 1990, Cambridge Astrophysics Series, Cambridge: University Press, 1990,
Fukui, Y., Torii, K., Onishi, T., et al. 2015, , 798, 6
Glover, S. C. O., Clark, P. C., Micic, M., & Molina, F. 2015, , 448, 1607
Goldsmith, P. F., & Langer, W. D. 1999, , 517, 209
Graf, U. U., Honingh, C. E., Jacobs, K., et al. 2008, Nineteenth International Symposium on Space Terahertz Technology, 488
Grenier, I. A., Casandjian, J.-M., & Terrier, R. 2005, Science, 307, 1292
Habing, H. J. 1968, , 19, 421
Hollenbach, D., Kaufman, M. J., Neufeld, D., Wolfire, M., & Goicoechea, J. R. 2012, , 754, 105
Indriolo, N., Neufeld, D. A., Gerin, M., et al. 2015, , 800, 40
Indriolo, N., & McCall, B. J. 2012, , 745, 91
Jenkins, E. B. 2009, , 700, 1299
Kaufman, M. J., Wolfire, M. G., & Hollenbach, D. J. 2006, , 644, 283
Kaufman, M. J., Wolfire, M. G., Hollenbach, D. J., & Luhman, M. L. 1999, , 527, 795
Kulesa, C. A., Ashley, M. C. B., Augarten, Y., et al. 2013, IAU Symposium, 288, 256
Liszt, H. S. 2007, , 476, 291
Langer, W. D., Velusamy, T., Pineda, J. L., Willacy, K., & Goldsmith, P. F. 2014, , 561, A122
Lee, M.–Y., Stanimorović, S., Murray, C.E. Heiles, C. & Miller, J. 2015, arXiv:1504.070405
Lo, N., Cunningham, M. R., Jones, P. A., et al. 2014, , 797, L17
McClure-Griffiths, N. M., & Dickey, J. M. 2007, , 671, 427
McClure-Griffiths, N. M., Dickey, J. M., Gaensler, B. M., et al. 2005, , 158, 178
McElroy, D., Walsh, C., Markwick, A. J., et al. 2013, , 550, AA36
Milam, S. N., Savage, C., Brewster, M. A., Ziurys, L. M., & Wyckoff, S. 2005, , 634, 1126
Molinari, S., Bally, J., Glover, S., et al. 2014, Protostars and Planets VI, 125
Orr, M. E., Pineda, J. L., & Goldsmith, P. F. 2014, , 795, 26
Padovani, M., Galli, D., & Glassgold, A. E. 2009, , 501, 619
P[é]{}rez-Beaupuits, J. P., Wiesemeyer, H., Ossenkopf, V., et al. 2012, , 542, L13
Pineda, J. L., Langer, W. D., & Goldsmith, P. F. 2014, , 570, A121
Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2011, , 536, A19
Rimmer, P. B., Herbst, E., Morata, O., & Roueff, E. 2012, , 537, A7
R[ö]{}llig, M., & Ossenkopf, V. 2013, , 550, AA56
Savage, B. D., & Sembach, K. R. 1996, , 34, 279
Schuppan, F., Becker, J. K., Black, J. H., & Casanova, S. 2012, , 541, A126
Solomon, P. M., Rivolo, A. R., Barrett, J., & Yahil, A. 1987, , 319, 730
Sz[ű]{}cs, L., Glover, S. C. O., & Klessen, R. S. 2014, , 445, 4055
Tielens, A. G. G. M., & Hollenbach, D. 1985, , 291, 722
Vall[é]{}e, J. P. 2014, , 148, 5
van Dishoeck, E. F. & Black, J. H. 1988, , 334, 771
Velusamy, T., Langer, W. D., Goldsmith, P.F. & Pineda, J. L. 2015, , in press
Velusamy, T., Langer, W. D., Pineda, J. L., et al. 2010, , 521, L18
Visser, R., van Dishoeck, E. F., & Black, J. H. 2009, , 503, 323
Walker, C., Kulesa, C., Bernasconi, P., et al. 2010, , 7733, 77330N
Wiesenfeld, L. & Goldsmith, P. F. 2014, , 780, 183
Wolfire, M. G., Hollenbach, D., & McKee, C. F. 2010, , 716, 1191
Wolfire, M. G., Hollenbach, D., & Tielens, A. G. G. M. 1993, , 402, 195
Wolfire, M. G., McKee, C. F., Hollenbach, D., & Tielens, A. G. G. M. 2003, , 587, 278
Yang, J., Zuo, Y.-X., Lou, Z., et al. 2013, Research in Astronomy and Astrophysics, 13, 1493
Tables {#tables .unnumbered}
======
[ccccccc]{} [$\rm ^{12}CO$]{} & 204,350 & 1 & 0.9 & 90,059 & 3 & 3\
[$\rm ^{13}CO$]{} & 148,623 & 1 & 0.3 & 38,379 & 3 & 0.8\
[$\rm C^{18}O$]{} & 91,875 & 1 & 0.2 & 2,783 & 3 & 0.6\
[\[CI\]]{} & 83,857 & 3 & 0.4\
HI & 457,746 & Arbitrary & 40\
[ccccccccccc]{} [$\rm ^{12}CO$]{}/[\[CI\]]{}& 83,545 & 6.8 & 6.7 & 18 & 1.6 & 72,112 & 7.1 & 7.0 & 18 & 1.5\
[\[CI\]]{}/[$\rm ^{12}CO$]{}& 83,545 & 0.16 & 0.15 & 0.7 & 0.04 & 72,112 & 0.15 & 0.14 & 0.34 & 0.03\
[$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{}& 115,687 & 5.5 & 5.0 & 25 & 2.5 & 36,301 & 4.3 & 4.2 & 10 & 1.2\
[\[CI\]]{}/[$\rm ^{13}CO$]{}& 74,063 & 0.84 & 0.76 & 3.0 & 0.33 & 36,751 & 0.63 & 0.61 & 1.6 & 0.14\
[$\rm ^{13}CO$]{}/[\[CI\]]{}& 74,063 & 1.4 & 1.3 & 4.1 & 0.5 & 36,751 & 1.7 & 1.6 & 4.1 & 0.4\
[$\rm ^{12}CO$]{}/[$\rm C^{18}O$]{}& 46,750 & 10 & 8.7 & 62 & 6.1 & 1,715 & 7.8 & 7.0 & 22 & 3.2\
[$\rm ^{12}CO$]{}/HI & 203,197 & 0.03 & 0.02 & 0.34 & 0.02 & 89,828 & 0.05 & 0.04 & 0.34& 0.02\
[\[CI\]]{}/HI & 83,588 & 0.007 & 0.006& 0.047& 0.003\
[$\rm ^{13}CO$]{}/HI & 142,961& 0.007 & 0.005& 0.12 & 0.006 & 38,181 & 0.007 & 0.012 & 0.12 & 0.008\
[lc]{} $G_0$ & 5.1 Habings\
$A_{V,{\rm tot}}$ & 6 mags.\
& (unless otherwise specified)\
$P_{\rm th} / k$ & $\rm 2.0 \times 10^4$ K ${\rm cm^{-3}}$\
$\Delta V$ (FWHM) & 4.0 km s$^{-1}$\
$\zeta_{\rm crp}$ & $2.0 \times 10^{-16}$ ${\rm s^{-1}}$\
${\rm [^{12}C/^{13}C}]$ isotopologue ratio & $50$\
$\rm [C/H]$ abundance & $3.2\times 10^{-4}$\
$\rm [S/H]$ abundance & $7.0\times 10^{-6}$\
$\rm [PAH/H]$ abundance & $2.0\times 10^{-7}$\
[lc]{} $I(^{12}$CO 1–0) & 50 K km/s\
$I(^{13}$CO 1–0) & 12 K km/s\
$I$(\[CI\] 1–0) & 14 K km/s\
$I$(\[CI\] 2–1) & 5 K km/s\
$I$(\[CII\]) & 2 K km/s\
$I$(HI) & 32 K km/s\
N($^{12}$CO) & $\rm 9.5 \times 10^{17}\,cm^{-2}$\
N($^{13}$CO) & $\rm 2.1 \times 10^{16}\,cm^{-2}$\
N(C) & $\rm 3.3 \times 10^{17}\,cm^{-2}$\
N(C$^+$) & $\rm 2.9 \times 10^{17}\,cm^{-2}$\
N(H) & $\rm 7.1 \times 10^{19}\,cm^{-2}$\
$T_{\rm gas}$ & 18 K\
$T_{\rm dust}$ & 9 K\
$T_{\rm gas}(\tau_{\rm CO} = 1)$ & 28 K\
$T_{\rm dust}(\tau_{\rm CO} = 1)$ & 14 K\
$n_{\rm H_2}(\tau_{\rm CO} = 1)$ & 580 cm$^{-3}$\
Figures {#figures .unnumbered}
=======
![Line profiles of the mean integrated emission from the entire region covered by the $\sim 1^{\circ} \times 1^{\circ}$ region of the HEAT G328 data cube. Lines are [$\rm ^{12}CO$]{}/5 (blue), [$\rm ^{13}CO$]{} (red), [$\rm C^{18}O$]{} (magenta), [\[CI\]]{} (green) and HI/150 (black). Three principal spiral arm crossings can be seen in the CO and \[CI\] data between $-100$ to $-85$ km/s (Norma; near-portion of arm), $-80$ to $-65$ km/s (Norma; far-portion of arm) and $-55$ to $-40$ km/s (Scutum–Crux; near-portion), respectively. The weaker feature at $-20$ km/s corresponds to the far-portion of the Scutum-Crux arm. \[fig:profiles\]](g328_extendedanalysis_smoothed_3b-eps-converted-to.pdf)
![Schematic of a four-arm model of the Galaxy to aid in visualization of the data along the G328 sight line. The model uses the parameters from @2014AJ....148....5V with the spiral arms shown by color lines: Perseus (turquoise), Sagittarius (green), Scutum-Crux (red) and Norma (black). Their extent is indicated by the corresponding dashed lines. The G328 sightline from the Sun is indicated by the wedge, with the Solar Circle (at $R_{\odot} = 8$kpc) shown by the dashed purple line (the cross indicates the Galactic center). The color shading within the wedge shows the expected line of sight radial velocities ($V_{\rm LSR}$) using the galactic rotation model of @2007ApJ...671..427M for the Fourth Quadrant, matched to the model for the outer Galaxy (i.e. positive velocities, when $R > R_{\odot}$). The short orange arc indicates the locus for the tangent point (where radial velocities are at their most negative). The spatial scale along the axes is in kpc, and the numbers along the wedge show the distance from the Sun, also in kpc. \[fig:galaxymodel\]](galaxy_model_328_zoom-eps-converted-to.pdf)
![Renderings of the G328 data cubes for [$\rm ^{12}CO$]{} (blue), [$\rm ^{13}CO$]{} (red), [\[CI\]]{} (green) and HI (yellow). Galactic longitude and latitude are along the short axes, with velocity along the long axis (with the most negative velocities to left). The three principal arm crossings along the sight line can be distinguished: Norma near-, Norma far- and Scutum-Crux near-, going from left to right. These are also shown as moment images in Fig. \[fig:momentimages120\]. The relative extent of the atomic and molecular gas can be gauged, with the atomic hydrogen enveloping both the carbon and carbon monoxide emitting-gas. \[fig:3dview\]](cubess120volb.jpeg)
![Across: (i) Column 1: [$\rm ^{13}CO$]{} moment 0 images (i.e. integrated flux), overlaid with their own contours (in black, at every 10% of the peak value), (ii) Column 2: [\[CI\]]{} moment 0 images, overlaid with their own contours (in red, at every 10% of the peak value) and (iii) Column 3: [$\rm ^{13}CO$]{} images overlaid with red [\[CI\]]{} contours. Down: Velocity ranges for (a) $-100$ to $-85$ km/s (Norma near), (b) $-80$ to $-65$ km/s (Norma far), (c) $-55$ to $-40$ km/s (Scutum-Crux near) and (d) $-80$ to $-76$ km/s (the filament studied in Paper I). The scale in the color bar refers to the flux in image in K km/s. The original datacubes were all first smoothed with a $120''$ FWHM Gaussian. \[fig:momentimages120\]](G328_v-100_-85_13COimage-eps-converted-to.pdf "fig:") ![Across: (i) Column 1: [$\rm ^{13}CO$]{} moment 0 images (i.e. integrated flux), overlaid with their own contours (in black, at every 10% of the peak value), (ii) Column 2: [\[CI\]]{} moment 0 images, overlaid with their own contours (in red, at every 10% of the peak value) and (iii) Column 3: [$\rm ^{13}CO$]{} images overlaid with red [\[CI\]]{} contours. Down: Velocity ranges for (a) $-100$ to $-85$ km/s (Norma near), (b) $-80$ to $-65$ km/s (Norma far), (c) $-55$ to $-40$ km/s (Scutum-Crux near) and (d) $-80$ to $-76$ km/s (the filament studied in Paper I). The scale in the color bar refers to the flux in image in K km/s. The original datacubes were all first smoothed with a $120''$ FWHM Gaussian. \[fig:momentimages120\]](G328_v-100_-85_CIimage-eps-converted-to.pdf "fig:") ![Across: (i) Column 1: [$\rm ^{13}CO$]{} moment 0 images (i.e. integrated flux), overlaid with their own contours (in black, at every 10% of the peak value), (ii) Column 2: [\[CI\]]{} moment 0 images, overlaid with their own contours (in red, at every 10% of the peak value) and (iii) Column 3: [$\rm ^{13}CO$]{} images overlaid with red [\[CI\]]{} contours. Down: Velocity ranges for (a) $-100$ to $-85$ km/s (Norma near), (b) $-80$ to $-65$ km/s (Norma far), (c) $-55$ to $-40$ km/s (Scutum-Crux near) and (d) $-80$ to $-76$ km/s (the filament studied in Paper I). The scale in the color bar refers to the flux in image in K km/s. The original datacubes were all first smoothed with a $120''$ FWHM Gaussian. \[fig:momentimages120\]](G328_v-100_-85_13COimage_CIcontours-eps-converted-to.pdf "fig:")\
![Across: (i) Column 1: [$\rm ^{13}CO$]{} moment 0 images (i.e. integrated flux), overlaid with their own contours (in black, at every 10% of the peak value), (ii) Column 2: [\[CI\]]{} moment 0 images, overlaid with their own contours (in red, at every 10% of the peak value) and (iii) Column 3: [$\rm ^{13}CO$]{} images overlaid with red [\[CI\]]{} contours. Down: Velocity ranges for (a) $-100$ to $-85$ km/s (Norma near), (b) $-80$ to $-65$ km/s (Norma far), (c) $-55$ to $-40$ km/s (Scutum-Crux near) and (d) $-80$ to $-76$ km/s (the filament studied in Paper I). The scale in the color bar refers to the flux in image in K km/s. The original datacubes were all first smoothed with a $120''$ FWHM Gaussian. \[fig:momentimages120\]](G328_v-80_-65_13COimage-eps-converted-to.pdf "fig:") ![Across: (i) Column 1: [$\rm ^{13}CO$]{} moment 0 images (i.e. integrated flux), overlaid with their own contours (in black, at every 10% of the peak value), (ii) Column 2: [\[CI\]]{} moment 0 images, overlaid with their own contours (in red, at every 10% of the peak value) and (iii) Column 3: [$\rm ^{13}CO$]{} images overlaid with red [\[CI\]]{} contours. Down: Velocity ranges for (a) $-100$ to $-85$ km/s (Norma near), (b) $-80$ to $-65$ km/s (Norma far), (c) $-55$ to $-40$ km/s (Scutum-Crux near) and (d) $-80$ to $-76$ km/s (the filament studied in Paper I). The scale in the color bar refers to the flux in image in K km/s. The original datacubes were all first smoothed with a $120''$ FWHM Gaussian. \[fig:momentimages120\]](G328_v-80_-65_CIimage-eps-converted-to.pdf "fig:") ![Across: (i) Column 1: [$\rm ^{13}CO$]{} moment 0 images (i.e. integrated flux), overlaid with their own contours (in black, at every 10% of the peak value), (ii) Column 2: [\[CI\]]{} moment 0 images, overlaid with their own contours (in red, at every 10% of the peak value) and (iii) Column 3: [$\rm ^{13}CO$]{} images overlaid with red [\[CI\]]{} contours. Down: Velocity ranges for (a) $-100$ to $-85$ km/s (Norma near), (b) $-80$ to $-65$ km/s (Norma far), (c) $-55$ to $-40$ km/s (Scutum-Crux near) and (d) $-80$ to $-76$ km/s (the filament studied in Paper I). The scale in the color bar refers to the flux in image in K km/s. The original datacubes were all first smoothed with a $120''$ FWHM Gaussian. \[fig:momentimages120\]](G328_v-80_-65_13COimage_CIcontours-eps-converted-to.pdf "fig:")\
![Across: (i) Column 1: [$\rm ^{13}CO$]{} moment 0 images (i.e. integrated flux), overlaid with their own contours (in black, at every 10% of the peak value), (ii) Column 2: [\[CI\]]{} moment 0 images, overlaid with their own contours (in red, at every 10% of the peak value) and (iii) Column 3: [$\rm ^{13}CO$]{} images overlaid with red [\[CI\]]{} contours. Down: Velocity ranges for (a) $-100$ to $-85$ km/s (Norma near), (b) $-80$ to $-65$ km/s (Norma far), (c) $-55$ to $-40$ km/s (Scutum-Crux near) and (d) $-80$ to $-76$ km/s (the filament studied in Paper I). The scale in the color bar refers to the flux in image in K km/s. The original datacubes were all first smoothed with a $120''$ FWHM Gaussian. \[fig:momentimages120\]](G328_v-55_-40_13COimage-eps-converted-to.pdf "fig:") ![Across: (i) Column 1: [$\rm ^{13}CO$]{} moment 0 images (i.e. integrated flux), overlaid with their own contours (in black, at every 10% of the peak value), (ii) Column 2: [\[CI\]]{} moment 0 images, overlaid with their own contours (in red, at every 10% of the peak value) and (iii) Column 3: [$\rm ^{13}CO$]{} images overlaid with red [\[CI\]]{} contours. Down: Velocity ranges for (a) $-100$ to $-85$ km/s (Norma near), (b) $-80$ to $-65$ km/s (Norma far), (c) $-55$ to $-40$ km/s (Scutum-Crux near) and (d) $-80$ to $-76$ km/s (the filament studied in Paper I). The scale in the color bar refers to the flux in image in K km/s. The original datacubes were all first smoothed with a $120''$ FWHM Gaussian. \[fig:momentimages120\]](G328_v-55_-40_CIimage-eps-converted-to.pdf "fig:") ![Across: (i) Column 1: [$\rm ^{13}CO$]{} moment 0 images (i.e. integrated flux), overlaid with their own contours (in black, at every 10% of the peak value), (ii) Column 2: [\[CI\]]{} moment 0 images, overlaid with their own contours (in red, at every 10% of the peak value) and (iii) Column 3: [$\rm ^{13}CO$]{} images overlaid with red [\[CI\]]{} contours. Down: Velocity ranges for (a) $-100$ to $-85$ km/s (Norma near), (b) $-80$ to $-65$ km/s (Norma far), (c) $-55$ to $-40$ km/s (Scutum-Crux near) and (d) $-80$ to $-76$ km/s (the filament studied in Paper I). The scale in the color bar refers to the flux in image in K km/s. The original datacubes were all first smoothed with a $120''$ FWHM Gaussian. \[fig:momentimages120\]](G328_v-55_-40_13COimage_CIcontours-eps-converted-to.pdf "fig:")\
![Across: (i) Column 1: [$\rm ^{13}CO$]{} moment 0 images (i.e. integrated flux), overlaid with their own contours (in black, at every 10% of the peak value), (ii) Column 2: [\[CI\]]{} moment 0 images, overlaid with their own contours (in red, at every 10% of the peak value) and (iii) Column 3: [$\rm ^{13}CO$]{} images overlaid with red [\[CI\]]{} contours. Down: Velocity ranges for (a) $-100$ to $-85$ km/s (Norma near), (b) $-80$ to $-65$ km/s (Norma far), (c) $-55$ to $-40$ km/s (Scutum-Crux near) and (d) $-80$ to $-76$ km/s (the filament studied in Paper I). The scale in the color bar refers to the flux in image in K km/s. The original datacubes were all first smoothed with a $120''$ FWHM Gaussian. \[fig:momentimages120\]](G328_v-80_-76_13COimage-eps-converted-to.pdf "fig:") ![Across: (i) Column 1: [$\rm ^{13}CO$]{} moment 0 images (i.e. integrated flux), overlaid with their own contours (in black, at every 10% of the peak value), (ii) Column 2: [\[CI\]]{} moment 0 images, overlaid with their own contours (in red, at every 10% of the peak value) and (iii) Column 3: [$\rm ^{13}CO$]{} images overlaid with red [\[CI\]]{} contours. Down: Velocity ranges for (a) $-100$ to $-85$ km/s (Norma near), (b) $-80$ to $-65$ km/s (Norma far), (c) $-55$ to $-40$ km/s (Scutum-Crux near) and (d) $-80$ to $-76$ km/s (the filament studied in Paper I). The scale in the color bar refers to the flux in image in K km/s. The original datacubes were all first smoothed with a $120''$ FWHM Gaussian. \[fig:momentimages120\]](G328_v-80_-76_CIimage-eps-converted-to.pdf "fig:") ![Across: (i) Column 1: [$\rm ^{13}CO$]{} moment 0 images (i.e. integrated flux), overlaid with their own contours (in black, at every 10% of the peak value), (ii) Column 2: [\[CI\]]{} moment 0 images, overlaid with their own contours (in red, at every 10% of the peak value) and (iii) Column 3: [$\rm ^{13}CO$]{} images overlaid with red [\[CI\]]{} contours. Down: Velocity ranges for (a) $-100$ to $-85$ km/s (Norma near), (b) $-80$ to $-65$ km/s (Norma far), (c) $-55$ to $-40$ km/s (Scutum-Crux near) and (d) $-80$ to $-76$ km/s (the filament studied in Paper I). The scale in the color bar refers to the flux in image in K km/s. The original datacubes were all first smoothed with a $120''$ FWHM Gaussian. \[fig:momentimages120\]](G328_v-80_-76_13COimage_CIcontours-eps-converted-to.pdf "fig:")\
![Normalized histograms of the distributions of selected line ratios from the data set ([$\rm ^{12}CO$]{}/[\[CI\]]{}, [\[CI\]]{}/[$\rm ^{12}CO$]{}, [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{}, [$\rm ^{13}CO$]{}/[\[CI\]]{}, [\[CI\]]{}/[$\rm ^{13}CO$]{}, [$\rm ^{12}CO$]{}/[$\rm C^{18}O$]{}, [$\rm ^{12}CO$]{}/HI, [\[CI\]]{}/HI & [\[CI\]]{}/[$\rm ^{13}CO$]{}, respectively, as labelled) determined on a voxel-by-voxel basis (i.e. pixel-velocity channel). To be included the \[CI\] flux needed to be $> 3\sigma$ per voxel but the fluxes for each of the 3 CO lines ([$\rm ^{12}CO$]{}, [$\rm ^{13}CO$]{}, [$\rm C^{18}O$]{}) only needed to be $> 1\sigma$ in the voxel (see Table \[tab:voxels\]). Overlaid as dotted lines are the corresponding distributions if a $3 \sigma$ threshold is applied for the three CO lines instead. Statistics relating to the distributions are listed in Table \[tab:voxelstats\]. \[fig:histograms\]](lineratio_histogram-eps-converted-to.pdf)
![Distributions of various line ratios vs. line intensities. $1\sigma$ thresholds are applied to CO, $3\sigma$ for \[CI\] (see text). Shown are, on top row, then second row: (a) \[CI\]/$^{13}$CO vs. $^{13}$CO, (b) \[CI\]/$^{12}$CO vs. $^{12}$CO, (c) $^{12}$CO/$^{13}$CO vs. $^{13}$CO, (d) \[CI\]/$^{13}$CO vs. \[CI\], (e) \[CI\]/$^{12}$CO vs. \[CI\] and (f) $^{12}$CO/$^{13}$CO vs. \[CI\]. Ten adjacent points have been averaged together in making these plots in order to improve their clarity and error bars have also not been shown. The lines show the best linear fit (weighted, minimum least squares) to the data; the coefficients (A, B), with [$Ratio = A \times Flux + B$]{} are as follows: $(0.8, -0.1), (0.2, -0.004), (5, -0.7), (0.7, -0.2), (0.1, 0.02)\, {\rm and}\, (5, -1.4)$, respectively \[and, as discussed in the text, error bars are lower for the smallest ratios when the flux is low, hence the linear fits are weighted to pass through these points rather than the bulk of the points, which have lower S/N\]. The number of points fitted in each plot is evident from Table \[tab:voxelstats\]. \[fig:ratiovsflux\]](g328_extendedanalysis_smoothed_12a_1sig_nsum10_fit-eps-converted-to.pdf)
![Averaged line profiles for [$\rm ^{13}CO$]{} and [\[CI\]]{} over the entire data cube for all voxels that have “high” ($>1.0$) and “normal” ($<1.0$) [\[CI\]]{}/[$\rm ^{13}CO$]{} line ratios. [$\rm ^{13}CO$]{} is shown in red and [\[CI\]]{} in green. Voxels with “normal” ratio values are the solid lines and those with “high” ratios are dotted. Clearly those regions with “high” ratios also have lower fluxes, on average. \[fig:avprofiles\]](g328_extendedanalysis_smoothed_13c-eps-converted-to.pdf)
![Mean line brightness per velocity channel for the [$\rm ^{12}CO$]{}/5 (blue), [$\rm ^{13}CO$]{} (red), [\[CI\]]{} (green) and [HI]{}/150 (black) lines. Solid curves are for voxels where [\[CI\]]{}/[$\rm ^{13}CO$]{} $< 1.0$ (i.e. “normal”) and dotted for voxels where [\[CI\]]{}/[$\rm ^{13}CO$]{} $> 1.0$ (i.e. “high”). The velocity ranges shown are chosen for the three spiral arm crossings discussed in §\[sec:obs\] (i.e. $-100$ to $-85$ km/s, $-80$ to $-65$ km/s and $-55$ to $-40$ km/s, with the narrow filament being contained within the first of these). The flux in a voxel needs to exceed the thresholds in Table \[tab:voxels\] to be included in this analysis. Profiles are then averaged over all the voxels which exceed the thresholds, and placed into the appropriate line ratio range (i.e. “high”, “normal”) for display. Voxels that do not meet the threshold criteria are excluded from the averages shown here. The normalization for each velocity channel is thus different than in Figs. \[fig:profiles\] and \[fig:avprofiles\]. \[fig:lineprofiles\]](g328_extendedanalysis_smoothed_13d-eps-converted-to.pdf)
![As for Fig. \[fig:lineprofiles\], but showing the averaged line ratios for (i)[\[CI\]]{}/[$\rm ^{13}CO$]{} (in red), (ii)[\[CI\]]{}/[$\rm ^{12}CO$]{} (in blue), (iii) \[[$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{}\] /10 (in magenta) and (iv) [\[CI\]]{}/[HI]{} $\times 100$ (in black), instead of the line intensities, for the same velocity ranges. As before, solid curves are for voxels where [\[CI\]]{}/[$\rm ^{13}CO$]{} $< 1.0$ (i.e. “normal”) and dotted for voxels where [\[CI\]]{}/[$\rm ^{13}CO$]{} $> 1.0$ (i.e. “high”). It is clear that for the “high” voxels the [\[CI\]]{}/[$\rm ^{13}CO$]{} and [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{} ratios nearly double, [\[CI\]]{}/[$\rm ^{12}CO$]{} is unchanged and [\[CI\]]{}/[HI]{} decreases by $\sim 10\%$ (and noticeably more so in the velocity range of the narrow filament, $-80$ to $-76$ km/s). \[fig:lineratios\]](g328_extendedanalysis_smoothed_13e-eps-converted-to.pdf)
![Evans plots for the velocity range $-80$ to $-76$ km/s corresponding to the filament (left) and their corresponding 2D color tables (right). Shown are, from top to bottom, [\[CI\]]{}/[$\rm ^{13}CO$]{}, [\[CI\]]{}/[$\rm ^{12}CO$]{} and [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{}. Color (hue) denotes the value of the line ratio $x-$axis of the color table), and the saturation (intensity) its S/N, with low S/N pixels fading to grey ($y-$axis of the color table). \[fig:evans\_v8076\]](g328_extendedanalysis_smoothed_17_1-eps-converted-to.pdf)
![Evans plots for the velocity range $-100$ to $-85$ km/s corresponding to the Norma near-spiral arm crossing (left) and their corresponding 2D color tables (right). Other details are as for Fig. \[fig:evans\_v8076\]. \[fig:evans\_v100\]](g328_extendedanalysis_smoothed_17_2-eps-converted-to.pdf)
![Abundance of H$_2 / 10^3$ (black dash), C$^+$ (yellow dash-dot-dot), C (green dot-dash), [$\rm ^{12}CO$]{} (blue long-dash), and [$\rm ^{13}CO$]{}$\times 10$ (red solid), as a function of optical depth $A_V$ (in magnitudes) for the standard cloud model. \[fig:xh2covavplt\]](xh2covavplt-eps-converted-to.pdf)
![Optical depth of the emission lines for [\[CI\]]{} 1–0 (brown) and 2–1 (green), [$\rm ^{12}CO$]{}/15 (blue) and [$\rm ^{13}CO$]{} (red) 1–0, and [\[CII\]]{} (yellow), as a function of $A_V$ for the standard model. The total extinction through the cloud is $A_{V,{\rm tot}} = 6$ mags., and lines are plotted to the cloud center. The total optical depth in the line through the cloud is thus twice that shown at $A_V = 3$. \[fig:tauci13covavplt\]](tauci13covavplt-eps-converted-to.pdf)
![The gas temperature, $T_{\rm gas}$, and dust temperature $T_{\rm dust}$ as a function of $A_V$ for the standard model. \[fig:tgasplt2\]](tgasplt2-eps-converted-to.pdf)
![Integrated intensities, in K km/s, of the emission lines for [$\rm ^{12}CO$]{}/3 (blue), [$\rm ^{13}CO$]{} (red) 1–0, [\[CI\]]{} 1–0 (brown), [\[CI\]]{} 2–1 (green) and [\[CII\]]{} (yellow), as a function of optical extinction, $A_V$, into the PDR, for the standard model. The total extinction in magnitudes through the cloud is $A_{V,{\rm tot}} = 6$. The intensity plotted is the total intensity emitted through one side of the cloud including the emission from both the near and far sides. \[fig:ci13coTkmvAv\]](ci13coTkmvAv-eps-converted-to.pdf)
![The [\[CI\]]{} 2–1 / [$\rm ^{13}CO$]{} 1–0 line ratio as a function of total optical extinction through the cloud, $A_{V\rm ,tot}$, for a range of model parameter values, as indicated in the legend. \[fig:ci13covAv3\]](ci13covAv3-eps-converted-to.pdf)
![The [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{} J=1–0 line ratio as a function of total optical extinction through the cloud, $A_{V,{\rm tot}}$, for a variety of model parameters as indicated in the legend. \[fig:co12co13vAv\]](co12co13vAv-eps-converted-to.pdf)
![[$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{} J=1–0 line ratio as a function of the [$\rm ^{13}CO$]{} line flux for a variety of model parameters, as indicated in the legend. \[fig:co12co13vco13\]](co12co13vco13-eps-converted-to.pdf)
![[\[CI\]]{} 2–1 / [$\rm ^{13}CO$]{} line ratio as a function of the [$\rm ^{13}CO$]{} line flux for a variety of model parameters, as indicated in the legend. \[fig:cico13vco13\]](cico13vco13-eps-converted-to.pdf)
Appendix
========
Two Evans plots are shown in Figs. \[fig:evans\_v80\] – \[fig:evans\_v55\] for the velocity ranges of $\rm (-80,-65)$ and $(-55,-40)$ km/s; associated with Norma far- and Scutum-Crux near- spiral arm crossings. For each velocity range three plots are displayed, [\[CI\]]{}/[$\rm ^{13}CO$]{}, [\[CI\]]{}/[$\rm ^{12}CO$]{} and [$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{}, so the behaviour of all three lines can be compared. The Evans plots are on the left and their corresponding 2D color tables on the right.
![Evans plots for the velocity range $-80$ to $-65$ km/s (left) and their corresponding 2D color tables (right). Shown are \[CI\]/13CO, \[CI\]/12CO and 12CO/13CO. Color (hue) denotes the value of the line ratio, and the saturation (intensity) its S/N. This velocity range corresponds to the Norma far-spiral arm crossing. \[fig:evans\_v80\]](g328_extendedanalysis_smoothed_17_3-eps-converted-to.pdf)
![Evans plots for the velocity range $-55$ to $-40$ km/s (left) and their corresponding 2D color tables (right). Shown are \[CI\]/13CO, \[CI\]/12CO and 12CO/13CO. Color (hue) denotes the value of the line ratio, and the saturation (intensity) its S/N. This velocity range corresponds to the Scutum-Crux near-spiral arm crossing. \[fig:evans\_v55\]](g328_extendedanalysis_smoothed_17_4-eps-converted-to.pdf)
[^1]: We note that @Lee2015 argue that these results are incorrect and that the HI is not optically thick in the Perseus region that they investigated, though the two studies do probe the ISM on very different spatial scales.
[^2]: The full filament can be seen in CO and HI in Fig. 1 of Paper I.
[^3]: Note also that the curvature seen here as the lower boundary in the first two plots in Fig. \[fig:ratiovsflux\] represents the ratio when the threshold value is applied to the numerator; i.e. the [\[CI\]]{} line.
[^4]: The weighted fit parameters are listed in the caption to Fig. \[fig:ratiovsflux\]. These linear fits are strongly weighted towards lower ratios for the lower fluxes as the S/N is highest for such data points. Hence the fit lines fall below the bulk of the data points.
[^5]: Note that we are implicitly assuming that the [\[CI\]]{} emission is also optically thin. This is consistent with the results of the PDR models we present later, and was also the case for the analysis given in Paper I.
[^6]: $\tau \sim 5$ when \[[$\rm ^{12}CO$]{}/[$\rm ^{13}CO$]{}\] = 10, for an isotope ratio \[$^{12}$C/$^{13}$C\] of 50.
[^7]: The profiles have been constructed for each voxel meeting the criteria, so the normalization is different for each velocity channel; hence the different appearance from Fig. \[fig:avprofiles\].
[^8]: These are sometimes known as Evans plots; see the description given at http://www.ncl.ucar.edu/Applications/evans.shtml.
[^9]: We note that the collision rates we use for C$^+$ with H$_2$ are within a few percent of the recent rates of para–H$_2$ with C$^+$ in @2014ApJ...780..183W. Adopting these rates results in only a few percent decrease in the gas temperature. We also note that our collision rate of H with C$^+$ [from @1990mcim.book.....F] is slightly higher than that in @2005ApJ...620..537B. Adopting this more recent rate would further increase the gas temperature in the atomic H layer but since collisions with atomic hydrogen dominate in only a narrow region at the outer edge of the cloud this makes a negligible difference to our results.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study the polymer system consisting of two polymer chains situated in a fractal container that belongs to the three–dimensional Sierpinski Gasket (3D SG) family of fractals. Each 3D SG fractal has four fractal impenetrable 2D surfaces, which are, in fact, 2D SG fractals. The two-polymer system is modelled by two interacting self-avoiding walks (SAWs), one of them representing a 3D floating polymer, while the other corresponds to a chain adhered to one of the four 2D SG boundaries. We assume that the studied system is immersed in a poor solvent inducing the intra-chain interactions. For the inter-chain interactions we propose two models: in the first model (ASAWs) the SAW chains are mutually avoiding, whereas in the second model (CSAWs) chains can cross each other. By applying an exact Renormalization Group (RG) method, we establish the relevant phase diagrams for $b=2,3$ and $b=4$ members of the 3D SG fractal family for the model with avoiding SAWs, and for $b=2$ and $b=3$ fractals for the model with crossing SAWs. Also, at the appropriate transition fixed points we calculate the contact critical exponents, associated with the number of contacts between monomers of different chains. Throughout the paper we compare results obtained for the two models and discuss the impact of the topology of the underlying lattices on emerging phase diagrams.'
address:
- ' Faculty of Natural Sciences and Mathematics, University of Kragujevac, 34000 Kragujevac, Serbia'
- 'Faculty of Physics, University of Belgrade, P.O.Box 368, 11001 Belgrade, Serbia'
author:
- 'I Živić, S Elezović-Hadžić and S Milošević'
title: 'Critical behavior of interacting two-polymer system in a fractal solvent: an exact renormalization group approach'
---
Introduction {#uvod}
============
The self–avoiding walk (SAW) is well-disposed as a standard lattice model for a flexible linear polymer chain in various types of solvents [@vc]. In this model, the monomers that comprise a polymer chain are related to the steps of a random walk that must not contain self-intersections, while the surrounding solvent is represented by an underlying lattice. In a good solvent, with each step of the SAW we associate the same weight factor $x$, while in a poor solvent, when two non-consecutive monomers of a polymer chain become nearest neighbors, we introduce the additional statistical factor $u$, which corresponds to the intra-chain energy $\epsilon_u<0$. Even though an isolated polymer chain is difficult to observe experimentally, numerous studies of the single-chain statistics have been upheld as a requisite step towards understanding the statistics of many-chain systems. A natural extension of a single polymer concept is a model of two interacting linear polymers, which may be relevant to perceive behavior of multicomponent polymer solutions [@palissetto]. To study the critical properties of the two-polymer system we shall apply the following two models: The first is the model of two mutually avoiding self-avoiding walks (ASAWs), whose paths on a lattice cannot cross each other, and the second is the model of two mutually crossing self-avoiding walks (CSAWs), that is, the case of the two SAWs whose paths can intersect each other. Various types of models with two avoiding SAWs have been successfully applied in the studies of phase transition of diblock copolymers [@stella1; @stella2], as well as in the studies of unzipping double-stranded DNA molecules [@dna1; @dna2; @dna3; @dna4]. On the other hand, the model with two crossing SAWs was applied for studying the collapse transition of two-chain interacting system on three- and four-simplex lattice [@ks93; @kspa1], and Euclidean lattices [@leoni], as well as to study two randomly interacting directed polymers on diamond hierarchical lattice [@bata; @haddad2].
In this paper we apply both ASAWs and CSAWs model to study the two-polymer system that displays both intra- and inter-chain interactions, on the three-dimensional (3D) fractal lattices, which belong to the Sierpinski gasket (SG) family of fractals. We assume that one of the two polymers is a floating chain in the bulk of a 3D SG fractal, while the other is a polymer chain that stays affixed to one of the four boundary surfaces (being actually 2D SG fractals) [@ZivicJSTAT]. In the ASAWs model we assume that two SAWs are in contact when they approach each other at the distance which is equal to a lattice constant, and in this situation we ascribe the contributing contact energy $\epsilon_v$ to the total model energy. Similarly, in the CSAWs model we assume that each crossing between two SAW paths corresponds to a contact of two monomers that belong to different polymer chains, and therefore we associate the contact energy $\epsilon_c$ with such a crossing. Since in both models, one of two polymers is adhered to one of the fractal boundary surfaces, and because its monomers take effect of surface interacting points for the bulk floating polymer chain, the proposed models may also be of interest for the problem of surface-interacting polymer chain in homogeneous [@r1; @r2; @r3] and disordered media [@ustenko]. The main goal of this study is to establish phase diagrams in the space of interaction parameters (which consists of the intra- and inter-chain interaction energy parameters), for both models, as well as to calculate the contact critical exponents that describe behavior of numbers of monomer-monomer contacts between two polymer chains.
This paper is organized as follows. In section \[ASAWs\] of the paper, we first describe the 3D SG fractals for general scaling parameter $b$, as well as the ASAWs model. Then, we present the general framework of an exact renormalization group method, within the model, and elaborate on the phase diagrams, obtained for the fractals designated by $b=2,3$ and $b=4$. We also display our findings for the contact exponents (associated with the number of contacts between the two SAWs). The CSAWs model is described in section \[CSAWs\]. Again, by applying an exact RG method, which is in the latter case technically more complicated, phase diagrams and contact exponents for $b=2$ and 3 SG fractals are obtained, and discussed. Brief summary and the concomitant conclusion are presented in section \[sumiranje\]. Explicit form of the RG equations for particular fractals are given in appendices.
The model of two evading self-avoiding walks {#ASAWs}
============================================
In this section we are going to apply the renormalization group (RG) method to the model of two mutually avoiding self-avoiding walks on the 3D SG family of fractals. First, we give a summary of the basic properties of these fractals. We start with recalling the fact that each member of the 3D SG fractal family is labeled by an integer $b\ge 2$ and can be constructed in stages. At the first stage ($r=1$) of the construction there is a tetrahedron of base $b$ that contains $b(b+1)(b+2)/6$ upward oriented unit tetrahedrons. The subsequent fractal stages are constructed recursively, so that the complete self-similar fractal lattice can be obtained as the result of an infinite iterative process of successive $(r\to r+1)$ enlarging the fractal structure $b$ times, and replacing the smallest parts of enlarged structure with the initial ($r=1$) structure. Fractal dimension $d_f$ of the 3D SG fractal is equal to $d_f^{3D}={{\ln [{{b(b+1)(b+2)}/6}}]/{\ln
b}}$. Each of the four boundaries of the 3D SG fractal is itself a 2D SG fractal, with the fractal dimension $d_f^{2D}=\ln[b(b+1)/2]/\ln b\>$.
In the terminology that applies to the SAW, we assign the weight $x_3$ to a step of the SAW in the bulk (3D SG fractal), which represents a floating polymer (we mark it by $P_3$), and the weight $x_2$ to a step of the SAW performed on one of the fractal boundaries (2D SG fractal), which represents a 2D surface-adhered polymer (marked by $P_2$), whose monomers act as interacting counterparts for monomers of the 3D polymer chain. To describe the intra-chain interaction of $P_3$ chain, we introduce the Boltzmann factor $u=e^{-\epsilon_u/k_BT}$, where $\epsilon_u<0$ is the interaction energy of two non-consecutive neighboring monomers of $P_3$.
In ASAWs model the two SAWs, that represent polymer chains, must not intersect each other. We assume that monomers, belonging to different chains, interact when they reach a distance which is equal to a fractal lattice constant, and to a such mutual position of $P_3$ and $P_2$ monomers we associate the weight factor $v=e^{-\epsilon_v/k_BT}$ (see figure \[fig:interakcije\](a)), where $\epsilon_v\leq0$ is the appropriate inter-chain interaction energy.
![The structure of the three-dimensional SG fractal, for $b=2$, at the first stage of construction, with an example of the bulk polymer chain ($P_3$) depicted by green line and the surface-adhered polymer chain ($P_2$) depicted by yellow line. The shaded area represents the adhering surface (the two-dimensional SG fractal). The intra-chain interactions $u$, for the $P_3$ polymer, are depicted by blue bonds. In the ASAWs model (a) the SAW paths, representing $P_3$ and $P_2$ polymers, cannot intersect each other, and two SAWs interact when approach each other at a distance which is equal to a lattice constant (red bonds, weighted with $v$). On the other hand, in the case of CSAWs model (b), the polymers $P_3$ and $P_2$ are cross-linked at the two sites, so that each contact contributes the weight factor $w$, while the red bonds (marked by $t$) correspond to the interactions between those monomers which are nearest neighbors to the cross-linked points. The two depicted examples for ASAWs (a) and CSAWs models (b), contribute the weights $x_3^{5}x_2^{3}u^4v^{12}$ and $x_3^{4}x_2^{3}w^2t^3$, respectively.[]{data-label="fig:interakcije"}](figure1.eps)
To describe exactly all possible configurations of the two-chain polymer system within the adopted model, we need four restricted partition functions $A^{(r)}$, $B^{(r)}$, $C^{(r)}$ and $D^{(r)}$, which are defined as $$\begin{aligned}
\fl A^{(r)}=\sum_{N_3,L} {\mathcal A}^{(r)}(N_3,L) x_3^{N_3} u^L,\quad && B^{(r)}=\sum_{N_3,L} {\mathcal B}^{(r)}(N_3,L) x_3^{N_3} u^L,\quad \nonumber\\
\fl C^{(r)}=\sum_{N_2} {\mathcal C}^{(r)}(N_2) x_2^{N_2},\quad && D^{(r)}= \sum_{N_2,N_3,L,M}{\mathcal D}^{(r)}(N_2,N_3,L,M) x_2^{N_2}x_3^{N_3}u^Lv^M, %\nonumber\end{aligned}$$ where ${\mathcal A}^{(r)}$, ${\mathcal B}^{(r)}$, ${\mathcal C}^{(r)}$, and ${\mathcal D}^{(r)}$ represent the numbers of particular configurations, consisting of one or two SAW strands on the $r$-th fractal structure (see figure \[fig:RGparametri\]). For instance, ${\mathcal D}^{(r)}(N_2,N_3,L,M)$ is the number of configurations consisting of $N_3$-step $P_3$ chain with $L$ pairs of non-consecutive nearest-neighbor monomers, and $N_2$-step $P_2$ chain, such that there are $M$ contacts between these two chains. The recursive nature of the fractal construction implies the following recursion relations for restricted partition functions $$\begin{aligned}
A'&=&\sum_{N_{A},N_{B}} a(N_{A},N_{B})\, A^{N_{A}}
B^{N_{B}}\,,
\label{eq:RGA}\\
B'&=&\sum_{N_{A},N_{B}} b(N_{A},N_{B})\, A^{N_{A}}
B^{N_{B}}\,,
\label{eq:RGB}\\
C'&=&\sum_{N_C} c(N_C)\, C^{N_C}\,,
\label{eq:RGC}\\
D'&=& \sum_{N_{A},N_{B},N_{C},N_{D}}
d(N_{A},N_{B},N_{C},N_{D})\,
A^{N_A}B^{N_B}C^{N_C}
D^{N_{D}}
\>,
\label{eq:RGA4}\end{aligned}$$ where we have used the prime symbol as a superscripts for $(r+1)$-th restricted partition functions and no indices for the $r$-th order partition functions. These relations can be considered as the RG equations for the problem under study, with the initial conditions $$\label{initalmodel2}
A^{(0)}=x_3\,,\quad B^{(0)}=x_3^2u^4\,,\quad C^{(0)}=x_2\,,\quad
\,D^{(0)}=x_3x_2v^4\,,$$ which correspond to the unit tetrahedron[^1].
![Schematic depiction of restricted generating functions used in the description of all possible two-SAW configurations, within the $r$-th stage of the 3D SG fractal structure, for ASAWs model. The 3D floating chain is depicted by green line, while the 2D surface-adhered chain is depicted by the yellow one. The interior details of the $r$-th stage fractal structure, as well as details of the chains, are not shown (for the chains, they are manifested by the wiggles of the SAW paths). The functions $A^{(r)}$, $B^{(r)}$, and $C^{(r)}$, describe one-polymer configurations (they are the same for both ASAWs and CSAWs models), while the function $D^{(r)}$ depicts the inter-chain configurations of ASAWs model.[]{data-label="fig:RGparametri"}](figure2.eps)
Equation (\[eq:RGC\]), alone, describes a single SAW on 2D SG fractal, whereas (\[eq:RGA\]) and (\[eq:RGB\]) are RG equations for a single SAW on 3D SG fractal. Critical properties of the SAW, based on the analysis of these equations, have been well established previously, and here we recall their basic properties relevant for the present work.
First, we describe the behavior of a single 2D SG chain. The RG equation (\[eq:RGC\]), for any $b$, has only one non-trivial fixed point $C^*$, corresponding to the extended polymer phase [@dhar78; @EKM], that is, the 2D SG chain is always swollen, and it cannot be in the compact phase. The corresponding eigenvalue $\lambda_{\nu_2}$ of (\[eq:RGC\]) is larger than 1, and determines the value of the critical exponent $\nu_2=\ln b/\ln
\lambda_{\nu_2}$, that governs the behavior of the mean end-to-end distance of 2D SG chain $\langle R\rangle\sim {\langle
N_2\rangle}^ {\nu_2}$, where $\langle N_2\rangle$ is the average number of 2D SG SAW steps.
In what follows we provide short summary of the results concerning the critical behavior of a solitary 3D SG chain. Depending on the value of the intra-chain interaction parameter $u$, a single 3D SG chain can be found in three phases: extended chain (for $u<u_\theta$), $\theta$-chain (when $u=u_\theta$) and globule ($u>u_\theta$). These phases (for arbitrary $b$) are described by the fixed points $(A_E,B_E)$, $(A_\theta,B_\theta)$ and $(A_G,B_G)$, respectively [@DharVannimenus; @Knezevic; @EZM]. The mean end-to-end distance $\langle R\rangle\sim {\langle
N_3\rangle}^ {\nu_3}$ of SAW on 3D SG fractal, is equal to $\nu_3=\ln b/\ln \lambda_{\nu_3}$, where $\lambda_{\nu_3}$ is the largest eigenvalue of the linearized RG equations (\[eq:RGA\]) and (\[eq:RGB\]), at the corresponding fixed point. For each 3D SG fractal, the following relationship $\nu_3^E>\nu_3^\theta>\nu_3^G$, is valid, where $\nu_3^E$, $\nu_3^\theta$ and $\nu_3^G$ are the end-to-end distance critical exponents in extended, $\theta$ and globule phase, respectively.
The interacting configurations of $P_2$ and $P_3$ chains are described with the restricted partition function $D^{(r)}$. The mean number of contacts between $P_2$ and $P_3$, on the $r$-th stage of fractal construction, is equal to $$\label{eq:srednjeMASAWs}
\langle M^{(r)}\rangle= {1\over D^{(r)}}\sum_{N_2,N_3,L,M}M{\mathcal D}^{(r)} x_2^{N_2}x_3^{N_3}u^L v^M =
{v\over D^{(r)}}
\frac{\partial D^{(r)}}{\partial v}
\>.$$ On the other hand, taking into account the function dependance $D^{(r+1)}=D^{(r+1)}(A^{(r)},B^{(r)},C^{(r)},D^{(r)})$, and the fact that $A^{(r)}$, $B^{(r)}$, and $C^{(r)}$ do not depend on the interaction parameter $v$, we have $$\label{dda}
\frac{\partial D^{(r+1)}}{\partial v}=\frac{\partial D^{(r+1)}}{\partial D^{(r)}}\frac{\partial D^{(r)}}{\partial v}\, ,$$ from which follows that, in the vicinity of the transition fixed point $(A^*, B^*, C^*, D^*)$ of the two-polymer system, the mean number of contacts $\langle M^{(r)}\rangle$, for large $r$, behaves as $\langle M^{(r)}\rangle\sim \lambda_{D}^r
\label{eq:lambdaD}$, where $$\label{svvrednost2}
\lambda_{D}={\left(\frac{\partial D^{(r+1)}}{\partial D^{(r)}}\right)}^*\>,$$ is relevant eigenvalue of RG equation (\[eq:RGA4\]), calculated at the transition fixed point. Knowing that $\langle {N_3^{(r)}}\rangle\sim \lambda_{\nu_3}^r$, one obtains $\ln \langle M^{(r)}\rangle/{\ln \langle N_3^{(r)}\rangle}\sim {\ln\lambda_{D}}/{\ln\lambda_{\nu_3}}$, [*i.e.*]{} the following scaling relation is satisfied $$\label{asawfi}
\langle M^{(r)}\rangle\sim \langle N_3^{(r)}\rangle^{\phi}\>,$$ where $$\phi=\frac{{\ln\lambda_{D}}}{\ln\lambda_{\nu_3}}\, , \label{eq:skaliranje}$$ is so-called contact critical exponent.
To establish the exact forms of RG equations, for each fractal, one needs to find the coefficients $a$, $b$, $c$, and $d$, that appear in (\[eq:RGA\])–(\[eq:RGA4\]). Using the computer facilities, by direct enumeration and classification of all possible SAW configurations on the first stage of fractal construction, it is feasible to find these coefficients for fractals labelled by $b=2,3$ and 4 (see \[app:ASAWsRG\]). Precise numerical analysis of the obtained RG equations (for $b=2,3$, and 4) reveals that two-polymer system can reside in several phases, depending on the values of the interaction parameters $u$ and $v$. In particular, for each value of $u$, there is a critical value $v=v_c(u)$, such that for $v<v_c(u)$ the two chains exist almost independently in the solution. This is indicated by the fact that $(A^*,B^*)$, and $C^*$ retain their fixed values that correspond to the solitary chain on 3D SG, and 2D SG, respectively (see table \[tab:avoiding\]), and confirmed by calculating the mean number of contacts $\langle
M^{(r)}\rangle$ between the chains, which quickly approaches some constant value as $r\to\infty$.
[lclllllll]{} $b$ & $v$&$A^*$&$B^*$&$C^*$ & $D^*$ &$\lambda_{\nu_3}$ & $\lambda_{D}$&$\phi$\
\
&$v<v_c(u)$&0.4294&0.0499&0.6180&0.1165&2.7965 &$<1$ &–\
2 & $v=v_c(u)$& 0.4294&0.0499&0.6180&2.3303 &2.7965 &2.0904&0.7170\
& $v>v_c(u)$ &0.4294&0.0499&0&3.0887 &2.7965 &2.9537&1.0532\
&$v<v_c(u)$& 0.3420&0.0239&0.5511&0.0779&5.3620 &$<1$ &–\
3 &$v=v_c(u)$& 0.3420&0.0239&0.5511&1.5388 &5.3620 & 2.7879&0.6105\
&$v>v_c(u)$ & 0.3420&0.0239&0&2.8591 &5.3620 & 4.6651&0.9171\
&$v<v_c(u)$&0.2899&0.0122&0.5063& 0.0580 &8.6911 &$<1$ &–\
4 &$v=v_c(u)$& 0.2899&0.0122&0.5063&1.2051 &8.6911 &3.4427&0.5717\
&$v>v_c(u)$ & 0.2899&0.0122&0&2.0837 &8.6911 &8.4170&0.9852\
\
&$v<v_c(u_\theta)$&1/3&1/3&0.6180& 0.0613&100/27 &$<1$ &–\
2& $v=v_c(u_\theta)$& 1/3&1/3&0.6180&0.6180 &100/27 &1.8526&0.4709\
&$v>v_c(u_\theta)$& 1/3&1/3&0&0.8229 &100/27 &2.4514&0.6848\
&$v<v_c(u_\theta)$&0.2071&0.4307&0.5511&0.0211 &8.7231 &$<1$ &–\
3&$v=v_c(u_\theta)$& 0.2071&0.4307&0.5511&0.5773 &8.7231 & 2.4203&0.4081\
&$v>v_c(u_\theta)$& 0.2071&0.4307&0&1.2781 &8.7231 & 3.7800&0.6139\
&$v<v_c(u_\theta)$&0.1918&0.3393&0.5063& 0.0180 &15.424 &$<1$ &–\
4 &$v=v_c(u_\theta)$& 0.1918&0.3393&0.5063&0.5758 &15.424 &3.5367&0.4617\
& $v>v_c(u_\theta)$& 0.1918&0.3393&0&0.7984 &15.424 &5.9331&0.6508\
\
&$v<v_c(u)$& 0&$22^{-1/3}$&0.6180& 0 &4&$<1$ &–\
2&$v=v_c(u)$& 0&$22^{-1/3}$&0.6180&0.7637 &4 &2.4163&0.6364\
&$v>v_c(u)$& 0&$22^{-1/3}$&0&0.7637 &4 &2.4163&0.6364\
&$v<v_c(u)$&0&$\infty$ &0.5511& 0 &9.772 &$<1$ &–\
3&$v=v_c(u)$& 0&$\infty$ &0.5511&$\infty$ &9.772 & 2& 0.3041\
&$v>v_c(u)$& 0&$\infty$ &0&$\infty$ &9.772 & 2& 0.3041\
&$v<v_c(u)$&0&$22^{-1/3}$&0.5063& 0&16 &$<1$ &–\
4 &$v=v_c(u)$ & 0&$22^{-1/3}$&0.5063&0.7637 &16 &5.8387&0.6364\
&$v>v_c(u)$& 0&$22^{-1/3}$&0&0.7637 &16 &5.8387&0.6364\
For $v=v_c(u)$ fixed values $(A^*,B^*)$ and $C^*$ remain the same as for $v<v_c(u)$, but $D^*$ becomes larger, and $\langle
M^{(r)}\rangle$ increases with $r$, obeying the scaling relation (\[asawfi\]), with eigenvalue $\lambda_D$ being larger than one. Although there are large number of contacts between them, both chains also have large parts that are not interconnected. Even for $v>v_c(u)$ fixed value $(A^*,B^*)$ does not change, but then $C^*$ becomes equal to zero, meaning that the whole $P_2$ chain is covered with the $P_3$ chain (which still has a lot of monomers in the bulk, far from the boundary in which $P_2$ lies). The regions $v>v_c(u)$ and $v<v_c(u)$ of the phase plane $u-v$, as well as the critical line $v_c(u)$, are additionally partitioned by the vertical line $u=u_\theta$, so that each part obtained in such a way is characterized by different fixed point $(A^*,B^*,C^*,D^*)$, corresponding to different phase. Coordinates of all fixed points are given in table \[tab:avoiding\], whereas in figure \[fig:FDAvoiding\] one can see obtained phase diagrams for $b=2$, 3, and 4 SG fractals.
![Phase diagrams obtained for the model of two avoiding SAWs on 3D SG fractals with $b=2,3$ and 4. The solid vertical line $u=u_\theta$ divides the $u-v$ plane in two areas, corresponding to the phases in which the 3D SAW is either extended ($u<u_\theta$) or collapsed ($u>u_\theta$). Each of these two areas is additionally partitioned by the critical line $v=v_c(u)$. For $v<v_c(u)$ the two polymers are segregated one from another, and the system exists either as “3D extended SAW + 2D SAW" for $u<u_\theta$, or “3D globule + 2D SAW" for $u>u_\theta$, whereas for $u=u_\theta$ precisely, $\theta$-chain coexists with 2D SAW. For $v\geq v_c(u)$ the mean number $\langle M\rangle$ of contacts between the two SAWs scales with the mean length $\langle
N_3\rangle$ of the 3D SAW as $\langle M\rangle\sim \langle
N_3\rangle^{\phi}$. Depending on the value of $u$, critical exponent $\phi$ has different values, which are presented within the corresponding areas.[]{data-label="fig:FDAvoiding"}](figure3.eps)
Shape of the line $v_c(u)$, as well as values of the exponent $\phi$, show that the interplay between the intra- and inter-chain interactions in the system under study is quite subtle. In the $b=3$ case, $v_c(u)$ decreases monotonically with $u$, meaning that stronger monomer-monomer attraction within the $P_3$ chain eases its attaching to the $P_2$ chain. However, for $b=2$ and 4 fractals this is correct only for values of $u$ up to $u_\theta$, where $v_c(u)$ has its minimum. For larger values of $u$, function $v_c(u)$ monotonically increases with $u$, [*i.e.*]{} for $u>u_\theta$ intra-chain prevails inter-chain interaction and hinders attaching. Different behavior of the system on fractals with $b=2,4$ and $b=3$ is due to the peculiar topology of these lattices. Namely, although for $u>u_\theta$ polymer is in globular phase, compactness of that structure is not always the same. Only on $b=2$ SG the globule is completely compact, [*i.e.*]{} its fractal dimension $d_f^G$ is equal to the fractal dimension $d_f^{3D}$ of the lattice [@DharVannimenus]. In the $b=3$ and 4 cases $d_f^G<d_f^{3D}$, but this quasi-compactness is much more pronounced in the $b=3$ case [@Knezevic; @EZM], which brings about different behavior of the system on the $b=3$ SG fractal. Concerning the exponent $\phi$, which can be taken as a measure of interconnection between the two chains, one can notice that it has different values on the critical line $v_c(u)$ and in the region $v>v_c(u)$, and in addition depends on intra-chain interaction parameter $u$ (see table \[tab:avoiding\] and figure \[fig:FDAvoiding\]). For each of the three studied fractals, in the range $v>v_c(u)$ the inequality $\phi(u<u_\theta)>\phi(u=u_\theta)>\phi(u>u_\theta)$ is satisfied. Such inequality could have been expected, since it means that when $P_3$ chain completely covers $P_2$ chain, the number of contacts between them is smaller if structure of the $P_3$ chain is more compact. On the line $v_c(u)$, however, chain $P_2$ is only partially covered with $P_3$, so that some similar conclusion is not plausible, which is indeed in accord with the calculated values of $\phi$. Besides, it is interesting that for $b=2$ and 4, the smallest value of $\phi$ is obtained for $u=u_\theta$, which is not the case for $b=3$ fractal.
Finally, one should note that in the case $b=3$, for the globular state of a solitary 3D chain ($u>u_\theta$), the coordinates of the corresponding fixed point are $A_G=0$ and $B_G=\infty$. Furthermore, a numerical analysis of function $D^{(r)}$, in the range $v\ge v_c(u)$ reveals that $D^*=\infty$. Nevertheless, the relation $\langle M^{(r)}\rangle\sim\lambda_D^r$ and formula (\[eq:skaliranje\]) are applicable, but with different meaning of $\lambda_D$. For the globule state of $b=3$ fractal, it was demonstrated [@Knezevic] that equations (\[eq:Ab3\]) and (\[eq:Bb3\]) in the vicinity of the corresponding fixed point $(0,\infty)$ have the following approximate form $$\label{knez1}
A^{(r+1)}= 320 (A^{(r)})^3 (B^{(r)})^6\, , \quad B^{(r+1)}= 4308 (A^{(r)})^2 (B^{(r)})^8\>,$$ from which it follows $\lambda_{\nu_3}=\frac{\sqrt{73}+11}2=9.772$ and $\nu_3^G=\ln 3/\ln 9.772=0.4819$. Besides, for $v\ge v_c(u)$, the inequality $D^{(r)}\ll B^{(r)}$ is valid, so that RG equation (\[eq:A4b3\]) obtains the approximate form $$\label{jedn2}
D^{(r+1)}=320 A^{(r)} (B^{(r)})^6 C^{(r)} (D^{(r)})^2\, .$$ Then, from equations (\[eq:srednjeMASAWs\]) and (\[dda\]), follows $
\langle M^{(r+1)}\rangle=2\frac v{D^{(r)}}\frac{\partial D^{(r)}}{\partial v}=2\langle M^{(r)}\rangle
$, implying that $
\langle M^{(r)}\rangle\sim \lambda_D^r$ (for large $r$), with $\lambda_D=2$. Finally, from (\[eq:skaliranje\]), one obtains $\phi={\ln2}/{\ln9.772}=0.3041$.
The model of crossing walks {#CSAWs}
===========================
In order to describe the physical situation when closer contact between the two polymers is possible, in this section we analyze the CSAWs model in which chains $P_2$ and $P_3$ can cross each other [@ZivicJSTAT]. If we assume that chains interact only at the crossing sites, and, similarly as in the ASAWs case, introduce the weight factor $w=e^{-\epsilon_c/k_BT}$, where $\epsilon_c\leq0$ is the energy of two monomers in contact, it turns out that the two chains cannot exist independently, even for extremely weak attraction ($|\epsilon_c|\ll k_BT$). Therefore, we define additional weight factor $t=e^{-\epsilon_t/k_BT}$, where $\epsilon_t>0$ is the energy associated with two sites, visited by different SAWs, and both neighbouring a crossing site (see figure \[fig:interakcije\](b)), so that unbinding transition can occur. To describe exactly all possible configurations of the two-chain polymer system, within this model we need to introduce nine restricted partition functions: $A^{(r)}$, $B^{(r)}$, $C^{(r)}$, $A_1^{(r)}$, $A_2^{(r)}$, $A_3^{(r)}$, $A_4^{(r)}$, $B_1^{(r)}$, and $B_2^{(r)}$.
![The six restricted generating functions used in the description of all possible inter-chain configurations for the CSAWs model of the two-polymer system, within the $r$-th stage of 3D SG fractal structure. The 3D chain is depicted by green line, while the 2D surface-adhered chain is depicted by yellow line.[]{data-label="figure4"}](figure4.eps)
Functions $A^{(r)}$, $B^{(r)}$ and $C^{(r)}$, which correspond to one-polymer configurations are the same as in the ASAWs model (see figure \[fig:RGparametri\], and RG relations (\[eq:RGA\]) and (\[eq:RGB\])), whereas the remaining six functions, which describe the inter-chain configurations, are depicted in figure \[figure4\], and they are defined as $$\begin{aligned}
A_i^{(r)}&=& \sum_{N_2,N_3,L,M,K}{\mathcal A}_i^{(r)}(N_2,N_3,L,M,K) x_2^{N_2}x_3^{N_3}u^L w^M t^K\, , \quad i=1,2,3,4\, ,\nonumber\\
B_i^{(r)}&=&\sum_{N_2,N_3,L,M,K}{\mathcal B}_i^{(r)}(N_2,N_3,L,M,K) x_2^{N_2}x_3^{N_3}u^L w^M t^K\, , \quad i=1,2\, ,\nonumber\end{aligned}$$ where ${\mathcal A}_i^{(r)}$ and ${\mathcal B}_i^{(r)}$ are the numbers of particular two-polymer configurations on the $r$-th fractal structure. For instance, ${\mathcal A}_4^{(r)}(N_2,N_3,L,M,K)$ is the number of configurations in which the $N_3$-step $P_3$ chain (with $L$ intra-chain contacts) and $N_2$-step $P_2$ chain (with different entering end exiting vertices from $P_3$ chain) cross $M$ times and have $K$ pairs of sites belonging to different chains and neighboring the crossing sites. Functions $A_i^{(r)}$ and $B_i^{(r)}$ satisfy the following recursion relations $$\begin{aligned}
A'_i&=& \sum_{\cal{N}}
a_i({\cal{N}})\,
A^{N_A}B^{N_B}C^{N_C}
\prod_{j=1}^{4} A_{j}^{N_{A_j}}
\prod_{k=1}^2B_{k}^{N_{B_k}}\,,\quad i=1,2,3,4\>,
\label{eq:RGAi}\\
B'_i&=& \sum_{\cal{N}} b_i({\cal{N}})\,
A^{N_A}B^{N_B}C^{N_C}
\prod_{j=1}^{4} A_{j}^{N_{A_j}}\prod_{k=1}^2
B_{k}^{N_{B_k}}\,,\quad i=1,2\>, \label{eq:RGBi}\end{aligned}$$ where $\cal{N}$ denotes the set of numbers ${\cal{N}}=\{N_{A},N_{B},N_{C},N_{A_1},N_{A_2},N_{A_3,}N_{A_4},
N_{B_1},N_{B_2}\}$, and where we have used the prime symbol as a superscript for $(r+1)$-th restricted partition functions and no indices for the $r$-th order partition functions. The above set of relations (\[eq:RGAi\])–(\[eq:RGBi\]), together with the previously introduced relations (\[eq:RGA\])–(\[eq:RGC\]) for the functions $A$, $B$, and $C$, can be considered as the system of RG equations for the problem under study, with the initial conditions $$\begin{aligned}
&&A^{(0)}=x_3\,,\quad B^{(0)}=x_3^2u^4\,,\quad C^{(0)}=x_2\,,\nonumber\\
&& A_1^{(0)}=x_3x_2w^2\,,\quad
A_2^{(0)}=A_3^{(0)}=x_3x_2wt\,,\quad
A_4^{(0)}=x_3x_2\,, \label{pocuslovi}\\
&& B_1^{(0)}=B_2^{(0)}=x_3^2x_2w^2u^4\,,\nonumber\end{aligned}$$ corresponding to the unit tetrahedron. Because the number of all possible configurations is extremely large, we have been able to find explicit form of the RG equations (\[eq:RGAi\])–(\[eq:RGBi\]) only for $b=2$ and $b=3$ SG fractals (see \[app:CSAWsRG\]). For both cases numerical analysis shows that, for each considered value of $t$, there is a critical line $w_c(u,t)$ dividing the $u-w$ plane into regions where the two polymers are either segregated ($w<w_c(u,t)$) or entangled ($w\geq w_c(u,t)$). Depending on the value of the intra-chain interaction parameter $u$, the area $w\leq w_c(u,t)$ is further partitioned into smaller regions, corresponding to various phases of the system (see figure \[fig:fdCSAWs\]).
![Phase diagrams in the space of interaction parameters for CSAWs model in the case of $b=2$ and $b=3$ SG fractal, for $t=0.5$. In both cases the critical line $w=w_c(u,t)$ separates the $u-w$ plane into the area $w>w_c(u,t)$ of entangled phase and area $w<w_c(u,t)$, in which the two chains are segregated. The latter area is divided by vertical line $u=u_\theta$ into regions, corresponding to three segregated phases: (i) 2D chain (always extended) and extended 3D chain ($u<u_\theta$), (ii) 2D chain and 3D $\theta$-chain ($u=u_\theta$), and (iii) 2D chain and 3D globule ($u>u_\theta$). One should observe that there appears the multi-critical point (full red circle) at the crossing of the $\theta$–line and the critical line $w=w_c(u,t)$. For other values of $t$ ($0<t<1$), the critical line $w_c(u,t)$ also monotonically decreases, for both $b=2$ and $b=3$ fractals.[]{data-label="fig:fdCSAWs"}](figure5.eps)
To each of these area different fixed point of the general type $$\label{fpgen}
(A^*,B^*,C^*,A_1^*,A_2^*,A_3^*,A_4^*,B_1^*,B_2^*)\>,$$ pertains. We describe general features of the fixed points and the corresponding phases in the three following subsections.
Weak self-attraction of the 3D chain
------------------------------------
For each value of $0<t<1$, and small values of the interaction parameter $1\leq u<u_\theta$, there is some critical value $w=w_c(u,t)$ such that
- For $w<w_c(u,t)$ the fixed point of the form $$\label{fp1}
(A_E,B_E,C^*,0,0,0,A_{4}^*,0,0)\>,$$ is reached. This point corresponds to the phase in which 2D chain and extended 3D chain are segregated, since as it is approached, after some number $r\gg1$ of RG iterations, the average number of contacts between the two chains, quickly becomes constant. Values of $A_E$ and $B_E$ are fixed values of the RG parameters for the solitary extended chain on 3D SG fractal, and they are presented in table \[tab:CSAWs\], together with the values of $C^*$, corresponding to 2D chain, which can exist only in extended state. RG fixed point value $A_4^*$ is equal to $0.1165$ and $0.0779$, for $b=2$ and 3 respectively, and they coincide with the values of $D^*$ for $v<v_c(u<u_\theta)$ case in the ASAWs model (see table \[tab:avoiding\]).
- For $w=w_c(u,t)$ one obtains the symmetrical fixed point $$\label{fp2}
(A_E,B_E,C^*,A_EC^*,A_EC^*,A_EC^*,A_EC^*,B_EC^*,B_EC^*)\>,$$ which appears to be a tricritical fixed point. As one approaches this fixed point, the average number of contacts ${\langle M^{(r)}\rangle}$ becomes infinitely large (although large parts of $P_2$ and $P_3$ are not in contact), and it turns out that it scales with the average length ${\langle {N_3}^{(r)}\rangle}$ of the 3D chain, according to the power law $$\label{ficsaw}
{\langle M^{(r)}\rangle}\sim \langle
{N_3}^{(r)}\rangle^{\varphi}\>.$$ To calculate the contact critical exponent $\varphi$, within the CSAWs model, we find the average number of contacts between two chains at the $r$th stage of fractal construction, through the formula $$\begin{aligned}
\langle M^{(r)}\rangle&=& {\sum_{N_2,N_3,L,M,K}M\left(\sum_{i=1}^4{\mathcal A}_i^{(r)}+{\sum_{j=1}^2{\mathcal B}_i^{(r)}}\right) x_2^{N_2}x_3^{N_3}u^L w^M t^K\over \sum_{i=1}^4 A_i^{(r)}+\sum_{j=1}^2 B_j^{(r)}}\nonumber\\
&=& {w\over \sum_{i=1}^4 A_i^{(r)}+\sum_{j=1}^2 B_j^{(r)}}
\left(\sum_{i=1}^{4}\frac{\partial A_i^{(r)}}{\partial w}+
\sum_{j=1}^{2}\frac{\partial B_j^{(r)}}{\partial w}\right)\nonumber\\
&=&
{w\over \sum_{i=1}^6 X_i^{(r)}}
\sum_{i=1}^{6}\frac{\partial X_i^{(r)}}{\partial w}
\>,\end{aligned}$$ where $X_i=A_i$ ($i=1,2,3,4$), $X_5=B_1$, and $X_6=B_2$. Since $$\label{nov1}
\frac{\partial X_i^{(r+1)}}{\partial w}=\sum_{j=1}^6 \frac{\partial X_i^{(r+1)}}{\partial X_j^{(r)}}\frac{\partial X_j^{(r)}}{\partial w}\>,\quad i=1,\ldots,6\>,$$ one expects, for large $r$, that $\frac{\partial X_i^{(r)}}{\partial w}$ behaves as $\lambda_\varphi^r$, where $\lambda_\varphi$ is the largest relevant solution of the eigenvalue equation $$\label{csawlamdafi}
\mbox{det}\left| \left({\partial X^{(r+1)}_i\over \partial X^{(r)}_j}
\right)^{*}-
\lambda_\varphi\,\delta_{ij} \right|=0\>,$$ where the asterisk means that the derivatives should be taken at the tricritical fixed point. From here follows $\langle M^{(r)}\rangle\sim \lambda_\varphi^r$, which together with $\langle N_3^{(r)}\rangle\sim \lambda_{\nu_3}^r$ (where $\lambda_{\nu_3}$, as before, is the largest eigenvalue of the linearized RG equations for the bulk parameters $A$ and $B$), and (\[ficsaw\]), gives $$\label{fie2}
\varphi=\frac{\ln\lambda_{\varphi}}
{\ln\lambda_{\nu_3}}\>.$$
- For larger values of the interaction parameter $w>w_c(u,t)$, the RG parameters flow towards the fixed point $$\label{fp5}
(0,0,0,A_1^*=C^*,0,0,0,0,0)\>,$$ which describes the entangled phase of the two polymers, in which $P_3$ chain is completely attracted to $P_2$ chain.
[lllllllllll]{} $b$ & $A^*$&$B^*$&$C^*$ & $A_1^*$ & $A_2^*$ &$A_3^*$&$A_4^*$ &$B_1^*$ & $B_2^*$&$\varphi$\
\
2 & 0.4294&0.0499&0.6180&0.2654 &0.2654&0.2654&0.2654&0.0308 &0.0308&0.5428\
3 & 0.3420&0.0239&0.5511& 0.1884& 0.1884& 0.1884& 0.1884&0.0131&0.0131&0.4973\
\
2& 1/3 & 1/3&0.6180& 0.0510&0&0&0.0613 &0.2365&0.2362&0.6714\
3& 0.2071 & 0.4307 &0.5511&0.0810&0.0310&0.0250&0.0270 &0.3130&0.3150&0.6226\
\
2 & 0 & 0.3569 & 0.6180&0&0&0&0&0.2206& 0.2206&0.6261\
3 & 0 & $\infty$ &0.5511& 0&0&0&0&$\infty$&$\infty$&0.6073\
Critical self-attraction of the 3D chain
----------------------------------------
For $u=u_\theta$ the solitary 3D chain is in the state of the $\theta$-chain, for which $(A^*,B^*)=(A_\theta,B_\theta)$, whereas the two-polymer system can be in the following phases:
- For $w<w_c(u_\theta,t)$ the corresponding fixed point is of the form $$\label{fpt1}
(A_\theta,B_\theta,C^*,0,0,0,A_4^*,0,0)\>.$$ This is the case when the 3D $\theta$-chain is segregated from the 2D chain chain. Fixed point values of $A_4^*$ are: 0.0613 for $b=2$, and 0.0211 for $b=3$ fractal, equal to $D^*$ for the corresponding cases $v<v_c(u_\theta)$ of the ASAWs model.
- When $w=w_c(u_\theta,t)$, the RG parameters tend to the fixed point $$\label{fpt2}
(A_\theta,B_\theta,C^*,A_1^*,A_2^*,A_3^*,A_4^*,B_1^*,
B_2^*)\>,$$ which corresponds to the phase in which chains are not segregated anymore, but they are not yet completely entangled. In contrast to the $w=w_c(u<u_\theta,t)$ case, for which symmetrical fixed point is obtained, values of $A_i$ $(i=1,2,3,4)$, as well as $B_1$ and $B_2$, are not mutually equal ($A_i\neq A_\theta C^*$, $B_i\neq
B_\theta C^*$). The scaling relation $\langle M^{(r)}\rangle\sim
\langle {N_3}^{(r)}\rangle^{\varphi}$ is satisfied, with $\varphi$ given by (\[fie2\]).
- For $w>w_c(u_\theta,t)$ the RG parameters flow towards the entangled fixed point (\[fp5\]).
Strong self-attraction of the 3D chain
--------------------------------------
When self-attraction of the 3D polymer is strong ($u>u_\theta$), depending on the values of inter-chain interaction parameters, the following phases are possible:
- For $w<w_c(u,t)$ the chains are segregated. Due to the large compactness of the 3D chain, with $(A^*,B^*)=(0,B_G)$, none of the configurations $A_1,A_2,A_3,A_4,B_1,B_2$ can be accomplished, and the corresponding fixed point is $$\label{fpg1}
(0,B_G,C^*,0,0,0,0,0,0)\, .$$ The chains are completely separated.
- When attraction between the chains is critical, $w=w_c(u,t)$, the chains are partially entangled, and the fixed point $$\label{fpg2}
(0,B_G,C^*,0,0,0,0,B_G C^*,B_G C^*)\>,$$ is attained. In this case the interaction between chains is sufficiently strong to connect them, but not strong enough to destroy the compactness of the 3D globule, so that all $A_i^*=0$. Again, the scaling relation $\langle M^{(r)}\rangle\sim \langle
{N_3}^{(r)}\rangle^{\varphi}$ is satisfied for both $b=2$ and 3, with $\varphi$ given by (\[fie2\]). However, while in the case $b=2$ the coordinates of corresponding fixed point have definite values (and $\lambda_\varphi$ can be directly calculated from linearizied RG equations for $A_i$ and $B_i$), in the $b=3$ case some fixed point coordinates diverge, and calculation of $\lambda_\varphi$ requires an additional effort. To be more specific, a numerical analysis of RG equations (\[eq:RGAi\]) and (\[eq:RGBi\]) reveals that $A_i^{(r)}\approx A^{(r)}C^*\to 0$, $B_i^{(r)}\approx B^{(r)}C^*\to \infty$. In this situation the appropriate eigenvalue $\lambda_\varphi$ can also be calculated, using the following transformation. If we write the relation (\[nov1\]) in the form $$\label{nov2}
\fl{1\over X_i^{(r+1)}}\frac{\partial X_i^{(r+1)}}{\partial w}=\sum_{j=1}^6 \left({X_j^{(r)}\over X_i^{(r+1)}}\frac{\partial X_i^{(r+1)}}{\partial X_j^{(r)}}\right){1\over X_j^{(r)}}\frac{\partial X_j^{(r)}}{\partial w}\>,\quad i=1,\ldots,6\>,$$ it can be shown that, when we keep only dominant terms in the RG equations, and in the derivatives $\frac{\partial
X_i^{(r+1)}}{\partial X_j^{(r)}}$, then, the matrix elements ${\left(\frac{X_j^{(r)}}{X_i^{(r+1)}}\frac{\partial
X_i^{(r+1)}}{\partial X_j^{(r)}}\right)}^*$ of the new eigenvalue problem are either equal to zero or to some finite constants (depending on $C^*$), from which we find $\lambda_\varphi=3.9919$, and therefrom $\varphi={\ln\lambda_\varphi}/{\ln\lambda_{\nu_3}}=0.6073$.
- Strong inter-chain attraction $w>w_c(u,t)$ destroys the globule and completely attaches the 3D chain to the 2D chain. This entangled phase is again characterized by the fixed point (\[fp5\]).
In table \[tab:CSAWs\] we have presented the numerical results for the crossover fixed points and the corresponding values of the contact exponent $\varphi$, obtained for the unbinding transitions from entangled two-polymer phase to segregated phases of 2D and 3D chains on the $b=2$ and $b=3$ 3D SG fractals. These values are correct for all studied cases of $t$ in the interval $(0,1)$. Varying the parameter $t$ in this interval changes only the particular values of $w_c(u,t)$, but, for both $b=2$ and $b=3$, the function $w_c(u,t)$ for fixed $t$ is monotonically decreasing function (see figure \[fig:fdCSAWs\]). Dependence of $w_c(u,t)$ on $t$, when $u$ is fixed, is presented in figure \[fig:wcODt\], for several values of $u$.
![Critical value of the inter-chain interaction parameter $w_c(u,t)$, depicted as a function of $t$, for three different values of intra-chain interaction parameter $u$, in the cases of $b=2$ and $b=3$ 3D SG fractals. []{data-label="fig:wcODt"}](figure6.eps)
As one can see, the limiting values $t=0$ and $t=1$ are also included in this figure. However, in these cases different fixed points, from those obtained for $0<t<1$, can be reached, which is expounded in the following paragraphs.
First, we analyze the value $t=0$, which represents the limiting case, within the CSAWs model, when the energy $\varepsilon_t$ (corresponding to the repelling of two different chain monomers, placed at sites which are nearest neighbours to a crossing site) is infinitely large. Starting with the initial values (\[pocuslovi\]), it can be shown that, in the case of the $b=3$ fractal, the same fixed points of the RG equations (\[eq:RGAi\]) and (\[eq:RGBi\]), as for $0<t<1$ are reached. However, for the $b=2$ fractal, it can be seen, from the explicit form of the RG equations (\[eq:b2jednacinaA1\])–(\[eq:b2jednacinaB2\]), that $t=0$ leads to $A_2^{(r)}=A_3^{(r)}=0$, for every $r$, $x_2$, $x_3$, $u$ and $w$. This is due to the topology of this fractal, and a consequence is that the fixed point $(A_E,B_E,C^*,C^*,0,0,A_4^*,A_4^*,0)$ corresponds to the critical values $w=w_c(1\leq u< u_\theta,t=0)$. The coordinates of this fixed point $A_E=A^*$, $B_E=B^*$ and $C^*$ are given in the part “extended 3D chain ($u<u_\theta$)" of the table \[tab:CSAWs\], while $A_4^*=0.1164$, and the concomitant critical exponent is $\varphi=0.8439$. The remaining fixed points are the same as for $0<t<1$.
The second limiting value ($t=1$) corresponds to the case $\varepsilon_t=0$ (when there is no repelling interaction). In this case, for all $u$, the critical value of the interaction parameter $w$ is equal to $w_c(u,t=1)=1$. This means that the chains can not be segregated, even for extremely small attraction between them. For both fractals $b=2$ and $b=3$, the fixed points that pertain to the critical value $w_c$, for $u\neq u_\theta$, are the same as for $0<t<1$. For $u=u_\theta$ the symmetrical fixed point is reached, $(A_\theta,B_\theta,C^*,A_\theta
C^*,A_\theta C^*,A_\theta C^*,A_\theta C^*,B_\theta C^*,B_\theta
C^*)$, in contrast to the case $0<t<1$. Values of $A_\theta=A^*$, $B_\theta=B^*$ and $C^*$ can be found in the middle part of the table \[tab:CSAWs\], while the values of the contact critical exponents are $\varphi(b=2)=0.6102$, and $\varphi(b=3)=0.5907$.
Finally, one should mention that recently, using the Monte Carlo renormalization group (MCRG) method, the contact exponent $\varphi$ was calculated for $b$ up to 40, for the case when the intra-chain interactions within the 3D chain are negligible, $u\to
0$ [@ZivicJSTAT]. Comparing the reported MCRG data $\varphi^{MC}(b=2)=0.5440\pm0.0056$ and $\varphi^{MC}(b=3)=0.4969\pm0.0024$ with our exact findings $\varphi(b=2)=0.5428$ and $\varphi(b=3)=0.4973$, we can see that MCRG data are in excellent agreement with our exact findings. In [@ZivicJSTAT] it was also demonstrated that $\varphi^{MC}$, as a function of the scaling parameter $b$, continues decreasing with increasing $b$, and, it seems that in the fractal-to-Euclidean crossover region (*i.e.* in the limit $b\to\infty$) it goes to the proposed zero Euclidean value. The inequality $\varphi(b=2)>\varphi(b=3)$ is satisfied not only for weak intra-chain interactions ($u<u_\theta$), but also for $u\geq
u_\theta$, as can be seen in table \[tab:CSAWs\]. Unfortunately, in the range $u\geq u_\theta$, the MCRG calculation of $\varphi$ is not feasible, so that prediction for the large $b$ behavior of $\varphi(u\geq u_\theta)$, only on the bases of our exact data, is not possible.
Summary and conclusion {#sumiranje}
======================
In this paper we have studied a system of two interacting chemically different polymer chains in a poor solvent. Such a situation can be modelled by two avoiding self-avoiding walks (ASAWs), as well as by two crossing self-avoiding walks (CSAWs). We assume that polymers are situated in fractal containers modelled by members of 3D SG fractal family, which are labelled by an integer $b$ ($2\le b<\infty$). We adopt that the first polymer ($P_3$) is a floating chain in the bulk of 3D SG fractal, while the second ($P_2$) is stuck to one of the four boundaries of the 3D SG fractal, which appears to be a 2D SG fractal. To take into account the intra-chain interaction of $P_3$ polymer we have introduced the interaction parameter $u={\mathrm{
e}}^{-\varepsilon_{u}/k_BT}$, where $\varepsilon_{u}<0$ is the energy corresponding to interaction between two nonconsecutive neighboring monomers within the chain. In the case of ASAWs model, the two SAW paths cannot intersect each other, and we assume that two polymers interact when they approach a distance equal to a lattice constant. We associate the weight factor $v=e^{-\epsilon_v/k_BT}$ with each such contact, where $\epsilon_v<0$ is the appropriate energy of inter-chain interaction. On the other hand, in the case of CSAWs model, in order to describe the inter-chain interactions of $P_3$ and $P_2$, we have introduced the parameters $w={\mathrm{e}}^{-\varepsilon_{c}/k_BT}$ and $t={\mathrm{e}}^{-\varepsilon_t/k_BT}$, where $\epsilon_c<0$ is the energy corresponding to each crossing of SAWs, while $\epsilon_t>0$ is the energy associated with a pair of sites, visited by different SAWs, which are nearest neighbors to a crossing site.
To obtain the phase diagrams and the contact critical exponents between the two polymers, we have applied an exact RG method for the 3D SG fractals labelled by $b=2,3$ and 4, in the case of ASAWs model, and for fractals $b=2$ and 3, in the case of CSAWs model. In both models, for various values of intra-chain interaction parameter $u$, a solitary 3D floating polymer chain can be found in one of the three possible phases (extended, $\theta$-phase, or globule phase), whereas a solitary 2D chain is always extended. Depending on the values of the inter-chain interaction parameters ($v$ in the case of ASAWs model, and $w$ and $t$ in the case of CSAWs model), the system can be either in the segregated phase, when the chains can be considered as almost independent, or in phases in which the number of contacts between the chains is comparable with their length (entangled phases). For both models, there is a critical line in the plane of the interaction parameters ($v_c(u)$ for ASAWs, and, $w_c(u,t)$, with fixed $t$, for CSAWs model), which divides it into areas corresponding to segregated and entangled phases. In the case of the ASAWs model, for $v\geq v_c(u)$, the average number $\langle M\rangle$ of contacts between the two polymers scales with the average length $\langle N_3\rangle$ of the 3D chain as $\langle M\rangle \sim
{\langle N_3\rangle}^\phi$. Different values of the contact exponent $\phi$ correspond to $v=v_c(u)$ and $v>v_c(u)$, and, in addition, $\phi$ also depends on the strength of the intra-chain interaction parameter $u$. However, in all entangled phases large parts of the 3D chain remain in the bulk, beyond the scope of the inter-chain interaction, since the prohibition of crossings between two chains hinders their complete entanglement, even for extremely large values of $v$. On the contrary, in CSAWs model for $w>w_c(u,t)$ the two chains are completely entangled, while they only partially cover each other at the critical line $w=w_c(u,t)$, where the scaling relation $\langle M\rangle \sim {\langle
N_3\rangle}^\varphi$ is satisfied, and where $\varphi$ takes different values in the intra-chain interaction regions $u<u_\theta$, $u=u_\theta$, and $u>u_\theta$.
In the end, we would like to point out that for ASAWs model, in the space of interaction parameters, the arrangement of possible phases is approximately the same, as in the case of the surface-interacting polymer chain in a poor solvent in Euclidean spaces [@r1; @r2; @r3]. On the other hand, the obtained phase diagrams for CSAWs model, resemble the phase diagrams of the same surface-interacting chain problem, in fractal containers [@EZM]. This similarity is not surprising, since in both models studied, one of the two interacting polymers is adhered to one of four fractal surfaces, and its monomers appear as a part of interacting surface (in the surface-interacting polymer problem). Here we may conclude that, our findings should be useful in making the corresponding 3D models of the system of several interacting polymer chains in porous media. Besides, our results may serve inspiring in advancing theories of mutually interacting polymers, as well as for polymers interacting with boundary surfaces of homogeneous 3D lattices, in which case so far (to the best of our knowledge) an exact approach has not been yet made.
Renormalization group equations for the ASAWs model \[app:ASAWsRG\]
===================================================================
In this Appendix we give explicit RG equations for the model in which two chains avoid each other, for the cases $b=2$, and $b=3$ of 3D SG fractals. Equations for “bulk" parameters $A$ and $B$, as well as for the “surface" parameter $C$, were found in earlier works, and we give them here only for the sake of completeness.
First, we give the RG equations for $b=2$ 3D SG fractal $$\begin{aligned}
A'&=&A^2 + 2\,A^3 + 2\,A^4 + 4\,A^3\,B + 6\,A^2\,B^2\, , \label{eq:Ab2}\\
B'&=&A^4 + 4\,A^3\,B + 22\,B^4\, ,\label{eq:Bb2}\\
C'&=&C^2 + C^3\, ,\label{eq:Cb2}\\
D'&=& 2\,{ D}^3\,B + 6\,{ D}^2\,B^2 + 2\,A^2\, D\,C +
A^2\,C^2 + A\, D\,C^2\, . \label{eq:Db2}\end{aligned}$$ We note that first three equations were established for the first time in [@dhar78].
Next, we present RG equations for the $b=3$ case $$\begin{aligned}
\fl A'&=&A^3+6 A^4+16 A^5+34 A^6+76 A^7+112 A^8+112 A^9+ 64
A^{10}+ 8 A^4 B+ 36 A^5 B\nonumber\\ \fl
&+& 140 A^6 B+292 A^7 B +424 A^8 B+ 332 A^9 B+12 A^3
B^2+12 A^4
B^2+ 118 A^5 B^2\nonumber\\ \fl
&+& 380 A^6 B^2+ 806 A^7 B^2+664 A^8 B^2+72 A^4
B^3 +352 A^5 B^3+704 A^6 B^3+ 1728 A^7 B^3
\nonumber\\ \fl
&+& 344 A^4 B^4+ 1568 A^5 B^4+848
A^6B^4+264 A^4 B^5+3192 A^5 B^5+ 320 A^3 B^6\, , \label{eq:Ab3} \\
\fl B'&=&A^6+12 A^7+40 A^8+60 A^9+32 A^{10}+28 A^6 B + 88 A^7
B+224 A^8 B+160 A^9 B\nonumber\\
\fl &+& 40 A^6 B^2+496 A^7 B^2 +596 A^8 B^2 + 176 A^5
B^3 +768 A^6
B^3+ 1056 A^7 B^3+ 88 A^3 B^4\nonumber\\ \fl
&+& 264 A^5 B^4 + 2534 A^6 B^4+
1152 A^4 B^5+1888 A^5 B^5\nonumber\\ \fl &+& 5808 A^4 B^6+1936
A^3 B^7+ 4308 A^2 B^8 \, , \label{eq:Bb3}\\
\fl C'&=&C^3+3 C^4+C^5+2C^6\, , \label{eq:Cb3}
\\
\fl D'&=&2A^6 D^3 + 4 A^7 D^3 + 4 A^6 D^4 +
2 A^5 D^5 + 28 A^6 D^3 B + 4 A^4 D^4 B +
14 A^5 D^4 B \nonumber
\\ \fl&+& 4 A^4 D^5 B +
8 A^4 D^3 B^2 + 56 A^5 D^3 B^2 +
44 A^4 D^4 B^2 + 4 A^3 D^5 B^2 \nonumber\\
\fl &+&
144 A^4 D^3 B^3 + 36 A^3 D^4 B^3 +
12 A^2 D^5 B^3 + 72 A^3 D^3 B^4 +
132 A^2 D^4 B^4 \nonumber\\
\fl&+& 264 A^2 D^3 B^5 +
12 A^6 D^2 C + 18 A^7 D^2 C +
2 A^4 D^3 C + 8 A^5 D^3 C + 8 A^6 D^3 C \nonumber
\\ \fl &+& 4 A^5 D^4 C + 2 A^4 D^5 C +
16 A^4 D^2 B C + 48 A^5 D^2 B C +
64 A^6 D^2 B C \nonumber\\
\fl &+& 4 A^2 D^3 B C +
8 A^4 D^3 B C + 48 A^5 D^3 B C +
4 A^3 D^4 B C + 8 A^4 D^4 B C \nonumber\\
\fl &+&
8 A^3 D^5 B C + 12 A^2 D^2 B^2 C +
36 A^4 D^2 B^2 C + 162 A^5 D^2 B^2 C + 24 A^3 D^3 B^2 C\nonumber\\
\fl &+& 28 A^4 D^3 B^2 C +
44 A^3 D^4 B^2 C + 8 A^2 D^5 B^2 C + 96 A^3 D^2 B^3 C + 64 A^4 D^2 B^3 C \nonumber\\
\fl &+& 152 A^3 D^3 B^3 C + 24 A^2 D^4 B^3 C +
24 A D^5 B^3 C + 512 A^3 D^2 B^4 C \nonumber\\
\fl &+&88 A D^4 B^4 C + 264 A^2 D^2 B^5 C +
320 A D^2 B^6 C + 6 A^4 D C^2 + 16 A^5 D C^2 \nonumber\\
\fl &+& 28 A^6 D C^2 + 12 A^7 D C^2 +
12 A^5 D^2 C^2 + 18 A^6 D^2 C^2 + 2 A^3 D^3 C^2 \nonumber\\\fl&+&
6 A^4 D^3 C^2 +
8 A^5 D^3 C^2 + 4 A^4 D B C^2 + 64 A^5 D B C^2 + 56 A^6 D B C^2\nonumber\\
\fl &+&
12 A^3 D^2 B C^2 + 30 A^4 D^2 B C^2 +
36 A^5 D^2 B C^2 + 2 A D^3 B C^2 +6 A^3 D^3 B C^2 \nonumber\\
\fl &+& 24 A^4 D^3 B C^2+
100 A^4 D B^2 C^2 + 88 A^5 D B^2 C^2 +
6 A D^2 B^2 C^2 \nonumber\\
\fl &+& 18 A^3 D^2 B^2 C^2 +
56 A^4 D^2 B^2 C^2 + 4 A^2 D^3 B^2 C^2 +
20 A^3 D^3 B^2 C^2 \nonumber\\
\fl &+& 160 A^4 D B^3 C^2+
36 A^2 D^2 B^3 C^2 + 32 A^2 D^3 B^3 C^2 +256 A^3 D B^4 C^2 \nonumber\\
\fl &+& 132 A^2 D^2 B^4 C^2 +
44 A D^3 B^4 C^2 + A^3 C^3 + 6 A^4 C^3 + 10 A^5 C^3 + 10 A^6 C^3 \nonumber\\
\fl &+& 6 A^7 C^3 + 8 A^3 D C^3 + 10 A^4 D C^3 +
14 A^5 D C^3 + 10 A^6 D C^3 \nonumber\\
\fl &+&4 A^4 D^2 C^3 +
6 A^5 D^2 C^3 + 2 A^3 D^3 C^3 +
6 A^4 D^3 C^3 + 8 A^4 B C^3 + 16 A^5 B C^3\nonumber\\
\fl &+& 20 A^6 B C^3 +
28 A^4 D B C^3 + 36 A^5 D B C^3 +
8 A^3 D^2 B C^3 + 12 A^4 D^2 B C^3 \nonumber\\
\fl &+&
4 A^2 D^3 B C^3 + 16 A^3 D^3 B C^3 + 12 A^3 B^2 C^3 +
18 A^5 B^2 C^3 + 12 A^2 D B^2 C^3 \nonumber\\
\fl &+& 44 A^4 D B^2 C^3 +
12 A^3 D^2 B^2 C^3 + 20 A^2 D^3 B^2 C^3 +
24 A^4 B^3 C^3\nonumber\\
\fl &+& 48 A^3 D B^3 C^3 + 24 A^2 D^2 B^3 C^3 + 24 A D^3 B^3 C^3
+ 3 A^3 C^4 + 10 A^4 C^4 \nonumber\\
\fl &+& 10 A^5 C^4 + 4 A^6 C^4 +A^2 D C^4
+ 2 A^3 D C^4 + 2 A^4 D C^4 +
12 A^4 B C^4 \nonumber\\
\fl &+& 8 A^5 B C^4 + 4 A^3 D B C^4 + 18 A^3 B^2 C^4 + 6 A^2 D B^2 C^4 + 2 A^2 D C^5\nonumber\\\fl&+&
4 A^3 D C^5 +
4 A^4 D C^5 + 8 A^3 D B C^5 + 12 A^2 D B^2 C^5 \> . \label{eq:A4b3}\end{aligned}$$ Equations (\[eq:Ab3\]) and (\[eq:Bb3\]) were found in [@Knezevic], and (\[eq:Cb3\]) in [@EKM].
For the $b=4$ case, equations are too cumbersome to be quoted here, and, they are available upon request to the authors.
Renormalization group equations for the CSAWs model \[app:CSAWsRG\]
===================================================================
It can be shown, via direct computer enumeration of the corresponding paths within the generator of the $b=2$ 3D SG fractal, that RG parameters $A_1, A_2, A_3, A_4, B_1$, and $B_2$ fulfil the following recursion relations $$\begin{aligned}
%
\fl A'_1&=&A_1^2 + A_1^3 + A{A_2^2} + 2A^2A_2A_3 + A{A_3^2} +
2AA_1{A_3^2} + 2AA_2A_3B_1 + 2AA_2A_3B_2 \nonumber\\
\fl &+& 4A^2A_1B_2 + 4A^2B_1B_2 +
4AA_1B_1B_2 + 2A^2{B_2^2}+ 2AA_1{B_2^2} +
{A_2^2}C + A{A_3^2}C\>, \label{eq:b2jednacinaA1}\\
%
\fl A'_2&=&AA_1A_2 + A_2^3 + A^2A_1A_3 + AA_2A_3^2 +
A^2A_3A_4 + AA_2A_4^2 + 2AA_2A_4B+ AA_3A_4C\nonumber\\
\fl &+& A^2A_3B_1+
AA_1A_3B_1 + 2AA_2BB_1 +
AA_2B_1^2 + A^2A_3B_2 +4AA_2BB_2 +A^2A_3C
\nonumber\\
\fl &+& 2AA_2B_1B_2 + 3AA_2B_2^2 + AA_2C +
A_1A_2C + AA_1A_3B_2 +
2AA_2A_4B_2 \, ,\\
\fl A'_3&=&A^2A_1A_2 + AA_1A_3 + AA_1^2\, A_3 +
AA_2^2A_3 + A^2A_2A_4 + 2A_3^3B+
4AA_3BB_2+ AA_3C^2
\nonumber\\ \fl&+& 2AA_3A_4B+
A^2A_2B_1 + AA_1A_2B_1 + 2AA_3BB_1 +
2A_3A_4BB_1+AA_2A_4C \nonumber\\
\fl&+& A^2A_2B_2+4A_3A_4BB_2+A^2A_2C + AA_3C +
AA_1A_3C + AA_1A_2B_2 \, ,\\
\fl A'_4&=&2A^2A_2A_3 + 2AA_2^2A_4 + 2AA_2^2B + 2AA_3^2\,B +
2A_4^3B + 6A_4^2B^2 + 2A_3^2BB_1
\nonumber\\ \fl&+& 2AA_2^2B_2 +
4A_3^2BB_2 + 2AA_2A_3C + 2A^2A_4C + A^2C^2 + AA_4C^2\, , \label{eq:b2jednacinaa4}
\\
\fl B'_1&=&A^2A_1^2 + AA_2^2A_4 + AA_2^2B + AA_3^2B + A_3^2A_4B +
2A^2A_1B_1 + AA_1^2B_1+ 8BB_2^3\nonumber\\ \fl&+& 6B^2B_1^2+
2BB_1^3 + 2AA_2^2B_2 + 8B^2B_1B_2 +
4BB_1^2B_2+ 8B^2B_2^2 + 8BB_1B_2^2 \, ,
\\
\fl B'_2&=&A^2A_2A_3 + AA_1A_2A_3 + AA_2^2B + AA_3^2B + A_3^2 A_4B
+ AA_2^2B_1+10BB_1B_2^2 + 6BB_2^3\nonumber\\ \fl&+& 2A^2A_1B_2+
AA_1^2B_2 + AA_2^2B_2 + 12B^2B_1B_2 + 6BB_1^2B_2 +
10B^2B_2^2 \, .
\label{eq:b2jednacinaB2}\end{aligned}$$ One can check, by inserting $A_1=A_2=A_3=B_1=B_2=0$ and $A_4=D$, into equation (\[eq:b2jednacinaa4\]) for the function $A_4$, that RG equation (\[eq:Db2\]) for the function $D$ in the case of the ASAWs model is recovered. This is not surprising, since it follows from the definitions of $A_4$ and $D$, and it is certainly also correct for the $b=3$ fractal equations. However, here we do not quote the $b=3$ RG equations because they are extremely intricate. For instance, equation for the parameter $A_1$ has 2753 terms, and it is similar for the remaining $A_i$ and $B_i$ equations.
References {#references .unnumbered}
==========
[10]{}
Vanderzande C, 1998 [*Lattice Models of Polymers*]{}, Cambridge: Cambrige University Press
Pelissetto A and Vicari E, 2006 [*Phys. Rev.*]{} E [**73**]{} 051802
Orlandini E, Seno F and Stella A L, 2000 [*Phys. Rev. Lett.*]{} [**84**]{} 294
Baiesi M, Carlon E, Orlandini E and Stella A L, 2001 [[*Phys. Rev.*]{} E]{} [**63**]{} 041801
Marenduzzo D, Bhattacharjee S M, Maritan A, Orlandini E and Seno F, 2002 [*Phys. Rev. Lett.*]{} [**88**]{} 028102
Kapri R and Bhattacharjee S M, 2006 [*J. Phys.: Condens. Matter*]{} [**18**]{} S215
Giri D and Kumar S, 2006 [[*Phys. Rev.*]{} E]{} [**73**]{} 050903(R)
Kapri R and Bhattacharjee S M, 2007 [*Phys. Rev. Lett.*]{} [**98**]{} 098101
Kumar S and Singh Y, 1993 [*J. Phys. A: Math. Gen.*]{} [**26**]{} L987
Kumar S and Singh Y, 2001 [*Physica*]{} A [**293**]{} 345
Leoni P, Vanderzande C and Vandeurzen J, 2001 [*J. Phys. A: Math. Gen.*]{} [**34**]{} 9777
Mukherji S and Bhattacharjee S M 1995 [*Phys. Rev.*]{} E [**52**]{} 1930
Haddad T A S, Andrade R F S and Salinas S R, 2004 [*J. Phys. A: Math. Gen.*]{} [**37**]{} 1499
Živi' c I, 2007 [*J. Stat. Mech.*]{} P02005
Singh Y, Giri D and Kumar S, 2001 [*J. Phys. A: Math. Gen.*]{} [**34**]{} L67
Rajesh R, Dhar D, Giri D, Kumar S and Singh Y, 2002 [*Phys. Rev.*]{} E [**65**]{} 056124
Owczarek A L, Rechnitzer A, Krawczyk J and Prellberg T, 2007 [*J. Phys. A: Math. Gen.*]{} [**40**]{} 13257
Usatenko Z and Sommer J-U, 2007 [*J. Stat. Mech.*]{} P10006
Dhar D, 1978 [*J. Math. Phys.*]{} [**19**]{} 5
Elezović S, Knežević M and Milošević S, 1987 [*J. Phys. A: Math. Gen.*]{} [**20**]{} 1215
Dhar D and Vannimenus J, 1987 [*J. Phys. A: Math. Gen.*]{} [**20**]{} 199
Kneževi' c M and Vannimenus V, 1987 [*J. Phys. A: Math. Gen.*]{} [**20**]{} L969
Elezović-Hadžić S, Živić I and Milošević S, 2003 [*J. Phys. A: Math. Gen.*]{} [**36**]{} 1213
[^1]: Such initial conditions imply that a SAW can traverse unit tetrahedrons along only one, or two nonconsecutive edges. This restriction of the standard SAW model does not alter the critical behavior of the system.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- Masashi KOSUDA
title: |
**Partition Algebra,\
Its Characterization and Representations**
---
[**Abstract**]{}
In this note we give representations for the partition algebra $A_{3}(Q)$ in Young’s seminormal form. For this purpose, we also give characterizations of $A_{n}(Q)$ and $A_{n-\frac{1}{2}}(Q)$.
Introduction
============
Definition of the partition algebra
-----------------------------------
Let $M = \{1, 2, \ldots, n\}$ be a set of $n$ symbols and $F = \{1', \ldots, n'\}$ another set of $n$ symbols. We assume that the elements of $M$ and $F$ are ordered by $1<2<\cdots <n$ and $1'<2'<\cdots<n'$ respectively. Consider the following set of set partitions: $$\begin{aligned}
\Sigma_n^1
&=&
\{\{T_1, \ldots, T_s\}\ |\ s=1,2, \ldots \ ,\nonumber\\
& & \quad T_j(\neq\emptyset)\subset M\cup F\
(j = 1, 2, \ldots, s),\\
& & \quad \cup T_j = M\cup F,\quad T_i\cap T_j = \emptyset \mbox{ if }
i\neq j\nonumber\}.\end{aligned}$$ We call an element $w$ of $\Sigma_n^1$ [*a seat-plan*]{} and each element of $w$ a [*part*]{} of $w$. It is easy to see that the number of seat-plans is equal to $B_{2n}$, the Bell number.
For $w\in\Sigma_n^1$ consider a rectangle with $n$ marked points on the bottom and the same $n$ on the top as in Figure \[fig:plan\].
![A seat-plan of $\Sigma_5$[]{data-label="fig:plan"}](1.eps)
The $n$ marked points on the top are labeled by $1, 2, \ldots n$ from left to right. Similarly, the $n$ marked points on the bottom are labeled by $1', 2', \ldots, n'$. If $w$ consists of $s$ parts, then put $s$ shaded circles in the middle of the rectangle so that they have no intersections. Then we join the $2n$ marked points and the $s$ circles with $2n$ shaded bands so that each shaded circle represent a part of $w$.
Using these diagrams, for $w_1,w_2\in\Sigma_n^1$, an arbitrary pair of seat-plans, we can define a product $w_1 w_2$. The product is obtained by placing $w_1$ on $w_2$, gluing the corresponding boundaries and shrinking half along the vertical axis. We then have a new diagram possibly containing some shaded regions which are not connected to the boundaries. If the resulting diagram has $p$ such regions, then the product is defined by the diagram with such region removed and multiplied by $Q^p$. Here $Q$ is an indeterminate. (It is easily checked that the product defined above is closed in the linear span of the set of seat-plans $\Sigma_n^1$ over $\mathbb{Z}[Q]$.) For example, if $$w_1 = \{\{1,1',4'\}, \{2,5\},\{3,4\}, \{2'\},\{3',5'\}\}\in\Sigma_5^1$$ and $$w_2 = \{\{1,1',3',4'\}, \{2\}, \{3,5\}, \{4\}, \{2',5'\}\}\in\Sigma_5^1,$$ then we have $$w_1w_2 =
Q^2\{\{ 1,1',3',4'\}, \{2,5\}, \{3,4\}, \{2',5'\}\}
\in \mathbb{Z}[Q]\Sigma_5^1$$ as in Figure \[fig:prod\].
![The product of seat-plans[]{data-label="fig:prod"}](2.eps)
By this product, the set of linear combinations of the elements of $\Sigma_n^1$ over $\mathbb{Z}[Q]$ makes an algebra $A_{n}(Q)$ called the [*partition algebra*]{}. The identity of $A_{n}(Q)$ is a diagram which corresponds to the partition $$1 = \{\{1, 1'\}, \{2, 2'\}, \ldots, \{n, n'\}\}.$$ We put $A_{0}(Q) = A_{1}(Q)= \mathbb{Z}[Q]$. We can define $A_{n}(Q)$ more rigorously in terms of the set partitions (See P. P. Maritin’s paper [@Ma2]).
Next we define special elements $s_i, f_i$ ($1\leq i \leq n-1$) and $e_i$ ($1\leq i\leq n$) of $\Sigma_n^1$ by $$\begin{aligned}
s_i &=& \{\{1,1'\},\ldots, \{i-1,(i-1)'\},
\{i+2, (i+2)'\},\ldots, \{n, n'\},\\
& &\quad \{i, (i+1)'\}, \{i+1, i'\}\}\\
f_i &=& \{\{1,1'\},\ldots, \{i-1,(i-1)'\},
\{i+2, (i+2)'\},\ldots, \{n, n'\},\\
& &\quad \{i, i+1, i', (i+1)'\}\}\\
e_i &=& \{\{1,1'\},\ldots, \{i-1,(i-1)'\}, \{i\}, \{i'\}
\{i+1, (i+1)'\},\ldots, \{n, n'\}\}.\end{aligned}$$ The diagrams of these special elements are illustrated by the figures in Figure \[fig:gen\]. Note that in the picture of $e_i$, there exist “a male” only part and “a female” only part. We call such a part “defective” (see Section 3.1).
![Special elements[]{data-label="fig:gen"}](3.eps)
We easily find that they satisfy the following basic relations. $$\begin{array}{rcl}\tag{$R0$}
f_{i+1} &=& s_is_{i+1}f_is_{i+1}s_i \quad (i = 1, 2, \ldots, n-2),\\
e_{i+1} &=& s_ie_{i}s_i \quad (i = 1, 2, \cdots, n-1)
\end{array}$$ $$\begin{array}{rcl}\tag{$R1$}
s_i^2 &=& 1 \quad (i = 1, 2, \ldots, n-1),\\
s_is_{i+1}s_i &=& s_{i+1}s_is_{i+1} \quad (i = 1, 2, \ldots, n-2),\\
s_is_j &=& s_js_i \quad (|i-j|\geq 2),
\end{array}$$ $$f_i^2 = f_i,\ f_if_j = f_jf_i,\tag{$R2$}$$ $$f_is_i = s_if_i=f_i,\tag{$R3$}$$ $$f_is_j = s_jf_i \quad (|i -j|\geq 2),\tag{$R4$}$$ $$e_{i}^2 = Qe_i ,\tag{$E1$}$$ $$s_ie_{i}e_{i+1}
= e_ie_{i+1}s_i
= e_ie_{i+1}
\quad (i = 1, 2, \ldots, n-1),\tag{$E2$}$$ $$e_is_j = s_je_i
\quad(j-i\geq1,\ i-j\geq 2),
\quad e_ie_j
= e_je_i,\tag{$E3$}$$ $$\begin{array}{rcl}\tag{$E4$}
e_if_ie_i = e_i
& e_{i+1}f_{i}e_{i+1} = e_{i+1}
& (i = 1, 2, \ldots, n-1),\\
f_ie_{i}f_i = f_{i},
&f_{i}e_{i+1}f_{i} = f_{i}
&(i = 1, 2, \ldots, n-1),
\end{array}$$ $$e_if_j = f_je_i
\quad(j-i\geq1,\ i-j\geq 2).\tag{$E5$}$$ Here we make a remark on the special elements above.
\[rem:gen\] The relation $(R0)$ implies that the special elements $\{f_i\}$ and $\{e_i\}$ are generated by $f=f_1$, $e = e_1$ and $s_1,\ldots, s_{n-1}$.
In this note, firstly we show that the special elements and the basic relations ($R0$)-($R4$) and ($E1$)-($E5$) above characterize the partition algebra $A_{n}(Q)$, [*i.e.*]{} the special elements generate $A_{n}(Q)$, and all the possible relations in $A_{n}(Q)$ are obtained from the basic relations. By Remark \[rem:gen\], the basic relations will be translated into the relations among the symbols $f$, $e$ and $s_i$s. Characterizations will be stated by these symbols.
Characterization for $A_{n}(Q)$
-------------------------------
Since generators $\{s_i\ | \ 1\leq i\leq n-1\}$ of the partition algebra $A_{n}(Q)$ satisfy the relations of the symmetric group ${\mathfrak S}_n$, we can understand that $f_i$ and $e_i$ are “conjugate” to $f$ and $e$ respectively.
Hence the basic relations $(R2)$-$(R4)$ and $(E1)$-$(E5)$ among the special elements are translated into the relations $(R2')$-$(R4')$ and $(E1')$-$(E5')$ among the generators as follows.
\[th:main\] The partition algebra $A_{n}(Q)$ is characterized by the generators $$f, e, s_1, s_2, \ldots, s_{n-1},$$ and the relations $$\begin{array}{rcl}\tag{$R1$}
s_i^2 &=& 1 \quad (i = 1, 2, \ldots, n-1),\\
s_is_{i+1}s_i &=& s_{i+1}s_is_{i+1} \quad (i = 1, 2, \ldots, n-2),\\
s_is_j &=& s_js_i \quad (|i-j|\geq 2, \ i, j =1,2, \ldots, n-1),
\end{array}$$ $$f^2 = f,\ fs_2fs_2 = s_2fs_2f,\ fs_2s_1s_3s_2fs_2s_1s_3s_2 = s_2s_1s_3s_2fs_2s_1s_3s_2f,\tag{$R2'$}$$ $$fs_1 = s_1f=f,\tag{$R3'$}$$ $$fs_i = s_if \quad (i = 3, 4, \ldots, n-1),\tag{$R4'$}$$ $$e^2 = Qe,\tag{$E1'$}$$ $$es_1es_1 = s_1es_1e = es_1e,\tag{$E2'$}$$ $$es_i=s_ie \quad (i = 2, 3, \ldots, n-1),\tag{$E3'$}$$ $$efe = e,\ fef = f,\tag{$E4'$}$$ $$fs_2s_1es_1s_2
= s_2s_1es_1s_2f.\tag{$E5'$}$$
In Sections 2-4 we prove this theorem not using the generators and the relations in the theorem but using the special elements and the basic relations $(R0)$-$(R4)$ and $(E1)$-$(E5)$.
The partition algebras $A_n(Q)$ were introduced in early 1990s by Martin [@Ma1; @Ma2] and Jones [@Jo] independently and have been studied, for example, in the papers [@Ma3; @DW; @HR]. The theorem above has already shown in the paper [@HR]. Here we give another poof defining a “standard” expression of a word of the special elements of $A_{n}(Q)$ according to the papers [@Ko1; @Ko3; @Ko4]. From this standard expression, we will find that the partition algebra $A_{n}(Q)$ is cellular in the sense of Graham and Lehrer [@GL]. Thus, applying the general representation of cellular algebras to the partition algebras, we will get a description of the irreducible modules of $A_{n}(Q)$ for any field of arbitrary characteristic. (For the cell representations, we also refer the paper [@KL].)
Further, we can make the character table of $A_{n}(Q)$ using the standard expressions. These topics will be studied in near future. For the present we refer the notes [@Ko3; @Na1] and the results about the partition algebras [@DW; @Xi].
Local moves deduced from the basic relations
============================================
Let $${\cal L}_n^1 =
\{
s_1, s_2, \ldots, s_{n-1},
f_1, f_2, \ldots, f_{n-1},
e_1, e_2, \ldots, e_{n}
\}$$ be the set of symbols whose words satisfy the basic relations $(R0)$-$(R4)$ and $(E1)$-$(E5)$. There are many relations among these symbols which are deduced from the basic relations. These relations are pictorially expressed as local moves. Among them, we frequently use relations $f_{i+1}s_is_{i+1}=s_is_{i+1}f_i$ ($R0$), $f_is_{i+1}f_i = f_if_{i+1}$ ($R2''$) and $e_is_i = e_if_ie_{i+1} =s_ie_{i+1}$ ($E4''$) as in Figure \[fig:fss\],\[fig:fsf\] and \[fig:efe\] respectively. The latter two relations are deduced from the relations ($R0$)-($R3$) and ($R0$), ($R3$), ($E4$) respectively.
![$f_{i+1}s_is_{i+1}=s_is_{i+1}f_i$ ($R0$)[]{data-label="fig:fss"}](4.eps)
![$f_is_{i+1}f_i = f_if_{i+1}$ ($R2''$)[]{data-label="fig:fsf"}](5.eps)
![$e_is_{i}=e_if_ie_{i+1} = s_ie_{i+1}$ ($E4''$)[]{data-label="fig:efe"}](6.eps)
As we noted in the previous paper [@Ko4], these basic relations are invariant under the transpositions of indices $i\leftrightarrow n-i+1$ as well as the $\mathbb{Z}[Q]$-linear involution $*$ defined by $(xy)^{*} = y^*x^*$ ($x, y\in A_{n}(Q)$). This implies that if one local move is allowed then other three moves —obtained by reflections with respect to the vertical and the horizontal lines and their composition— are also allowed.
Further, we note that if we put $$e^{[r]} = f_1f_2\cdots f_{r-1}e_1e_2\cdots e_rf_1f_2\cdots f_{r-1}$$ then we can check that $e^{[r]}$, $f$ and $s_i$ ($1\leq i\leq n-1$) satisfy the defining relations of $P_{n,r}(Q)$, the $r$-modular party algebra, defined in the paper [@Ko4]. This means that the local moves shown in the paper [@Ko4] also hold in $A_n(Q)$ (in fact, these local moves are more easily verified in $A_n(Q)$). Some of them are pictorially expressed in Figures \[fig:dpjrE\],\[fig:roeE\],\[fig:dpsE\] and \[fig:dpeE\].
![Defective part jump rope ($R13'$)[]{data-label="fig:dpjrE"}](7.eps)
![Removal (addition) of excrescences ($R14'$)[]{data-label="fig:roeE"}](8.eps)
![Defective part shift ($R16'$)[]{data-label="fig:dpsE"}](9.eps)
![Defective part exchange ($R17'$)[]{data-label="fig:dpeE"}](10.eps)
Standard expressions of seat-plans
==================================
In this section, for a seat-plan $w$ of $\Sigma_n^1$, we define a [*basic expression*]{}, as a word in the alphabet $\Gamma_n^1$. Then we define more general forms called [*crank form expression*]{}s. As a special type of the crank form expression, we define the [*standard expression*]{}. In the next section, we show that any two crank form expressions of a seat-plan will be moved to each other by using the basic relations $(R0)$-$(R4)$ and $(E1)$-$(E5)$ finite times. Consequently, we find that any seat-plan can be moved to its standard expression. To define these expressions, we introduce some terminologies.
Propagating number
------------------
Let $w = \{T_1, T_2, \ldots, T_s\}$ be a seat-plan of $A_{n}(Q)$. For a part $T_i\in w$, the intersection with $M$, or $T_i^M = M\cap T_i$, is called the [*upper part*]{} of $T_i$. Similarly, $T_i^F = F\cap T_i$ is called the [*lower part*]{} of $T_i$. If $T_i^M\neq\emptyset$ and $T_i^F\neq\emptyset$ hold simultaneously, $T_i$ is called [*propagating*]{}, otherwise, it is called [*non-propagating*]{}, or [*defective*]{}. Let $\pi(w) := \{T\in w\ |\ T:\mbox{propagating}\}$ be the set of propagating parts. The number of propagating parts $|\pi(w)|$ is called the [*propagating number*]{} (of $w$). If $T_i\in\pi(w)$ then the upper \[resp. lower\] part $T^M_i$ \[resp. $T^F_i$\] of $T_i$ is also called [*propagating*]{}. If $T_i\in w\setminus\pi(w)$ and $T^M_i = T_i$ \[resp. $T^F_i = T_i$\], then $T^M_i$ \[resp. $T^F_i$\] is called [*defective*]{}.
For example, in Figure \[fig:plan\], $\pi(w) = \{T_1, T_4\}$. Hence $|\pi(w)| = 2$. On the other hand $T_2$, $T_3$ and $T_5$ are defective. The upper and the lower propagating parts are $\{1\}$, $\{5\}$ and $\{1',2',4'\}$, $\{3'\}$ respectively. The upper defective parts are $T_2$ and $T_3$. The lower defective part is $T_5$.
A basic expression of a seat-plan
---------------------------------
For a part $T_i\in w$, define $t(T_i)$ by $$t(T_i) =
\left\{
\begin{array}{ll}
1&\mbox{if $T_i$ is propagating,}\\
0&\mbox{if $T_i$ is defective.}
\end{array}
\right.$$ Similarly we define $t(T_i^M)$ \[resp. $t(T_i^F)$\] to be $1$ or $0$ in accordance with that $T_i^M$ \[resp. $T_i^F$\] is propagating or not.
Using the terminologies above, first we define a [*basic expression*]{} of an seat-plan. Let $w\in\Sigma_n^1$ be a seat-plan and $\rho_w = (T_1, \ldots, T_s)$ be an arbitrary sequence of all parts of $w$. For the sequence $\rho_w$, we define the sequence of the upper \[resp. lower\] parts $\mathbb{M} = \mathbb{M}(\rho_w) = (T^M_{i_1}, \ldots, T^M_{i_u})$ ($i_1<\cdots<i_u$, $u\leq s$) \[resp. $\mathbb{F} = \mathbb{F}(\rho_w) = (T^F_{j_1}, \ldots, T^F_{j_v})$ ($j_1<\cdots<j_v$, $v\leq s$)\] omitting empty parts.
Using these data, we define [*cranks*]{} $C_{\mathbb{M}}[i]$, $C^*_{\mathbb{F}}[i]$ and $C^{\mathbb{M}}_{\mathbb{F}}[\sigma])$ as products of the generators as in Figure \[fig:mcrank\], \[fig:fcrank\] and \[fig:midcrank\] respectively. Here $\sigma$ is a word in the alphabet $\{s_1,\ldots, s_{|\pi(w)|-1}\}$.
![$C_{\mathbb{M}}[l]$[]{data-label="fig:mcrank"}](11.eps)
![$C^*_{\mathbb{F}}[l]$[]{data-label="fig:fcrank"}](12.eps)
![$C^{\mathbb{M}}_{\mathbb{F}}[\sigma]$[]{data-label="fig:midcrank"}](13.eps)
Further we define the “product of cranks” ${C}[\mathbb{M}]$ and ${C}[\mathbb{F}]$ by $${C}[\mathbb{M}] =
C_{\mathbb{M}}[1]C_{\mathbb{M}}[2]\cdots C_{\mathbb{M}}[u-1]$$ and $${C}^*[\mathbb{F}] =
C^*_{\mathbb{F}}[v-1]\cdots C^*_{\mathbb{F}}[2]C^*_{\mathbb{F}}[1]$$ respectively. We note that $C_{\mathbb{M}}[l]$ \[resp. $C^*_{\mathbb{F}}[l]$\] is defined by a composition $\mathbb{E} = (E_1,\ldots, E_s)$ of $n$ whose components have labels either “propagating” or “defective”. For example if $\mathbb{M} = (2,1,2,2,3)$, $(t(M_i))_{1\leq i \leq 5} = (0,1,0,1,1)$, $\mathbb{F} = (3,4,3)$, $(t(F_i))_{i=1,2,3} = (1,1,1)$ and $\sigma=(1,2)(2,3)\in\mathfrak{S}_3$, then the product of cranks ${C}[\mathbb{M}]C^{\mathbb{M}}_{\mathbb{F}}[\sigma]C^*[\mathbb{F}]$ is presented as in Figure \[fig:crank\].
![Product of cranks[]{data-label="fig:crank"}](14.eps)
Let $\overline{\mathbb M}$ be the sequence of $n$ symbols obtained from $\mathbb{M} = \mathbb{M}(\rho_w)$ by arranging all elements of $T^M_{i_k}$s in accordance with the sequence $\mathbb{M}$ so that all elements of each $T^M_{i_k}$ are increasingly lined up from left to right. For example, if $\mathbb{M} = (\{3,1,7\},\{6,4\},\{5,2\})$, then $\overline{\mathbb{M}} = (1,3,7,4,6,2,5)$. Similarly $\overline{\mathbb{F}}$ is defined from $\mathbb{F} = \mathbb{F}(\rho_w)$.
Then the following product becomes an expression of a seat-plan $w$. $${\cal C}(\mathbb{M}, id, \mathbb{F})
= x_{\overline{\mathbb{M}}}{C}[\mathbb{M}]
C^{\mathbb{M}}_{\mathbb{F}}[id]
C^*[\mathbb{F}]x^*_{\overline{\mathbb{F}}}.$$ Here $x_{\overline{\mathbb{M}}}$ \[resp. $x^*_{\overline{\mathbb{F}}}$\] is a permutation which maps $j$ to the number in the $j$-th coordinate of $\overline{\mathbb{M}}$. \[resp. the number written in the $j$-th coordinate of $\overline{\mathbb{F}}$ to $j'$\]. We call this expression a [*basic expression*]{} of $w$. We note that for a seat-plan $w$ there are several ways to choose $\rho_w$, a sequence of the parts of $w$. [*i.e.*]{} Several basic expressions can be defined for one seat-plan.
The standard expressions of a seat-plan
---------------------------------------
Our claim is that any basic expression of a seat-plan $w$ can be moved to a special expression called the [*standard expression*]{} by using the relations $(R0)$-$(R4)$ and $(E1)$-$(E5)$ finite times. In order to show this claim, next we introduce the notion of a [*crank form expression*]{} of $w$.
Consider the propagating parts $\pi(w)=\{T_{i_1},\ldots, T_{i_p}\}\ (p = |\pi(w)|)$ of $w$. Let $(M_1, \ldots, M_p)$ be a sequence of the upper parts of $\pi(w)$ and $(F_1, \ldots, F_p)$ the one of the lower parts. Then there exists a permutation $\sigma\in{\mathfrak S_p}$ such that $\{M_{\sigma(k)}\sqcup F_{k}\ |\ k = 1, \ldots, p \} = \pi(w)$. As is well known, a permutation $\sigma$ of degree $p$ is presented by $p$-strings braid which connects the lower $k$-th point to the upper $\sigma(k)$-th point.
Now we define a [*crank form expression*]{} of $w$. Let $\mathbb{M} = (M_{1}, \ldots, M_{u})$ \[resp. $\mathbb{F} = (F_1, \ldots, F_v)$\] be any fixed sequence of the upper \[resp. lower\] parts of $w$ (whose empty parts are omitted and propagating parts are specified). From the sequences $\mathbb{M}$ and $\mathbb{F}$, we obtain products of cranks $C[\mathbb{M}]$ and $C^*[\mathbb{F}]$. Further, from $\pi(\mathbb{M})$ and $\pi(\mathbb{F})$, we obtain a permutation $\sigma\in{\mathfrak S}_p$ such that $\{M_{i_{\sigma(k)}}\sqcup F_{j_{k}}\ ;\ k = 1, \ldots, p \}
= \pi(w)$. Then the product $${\cal C}(\mathbb{M},\sigma,\mathbb{F})
=
x_{\mathbb{\overline{M}}}
{C}[\mathbb{M}]C^{\mathbb{M}}_{\mathbb{F}}[\sigma]C^*[\mathbb{F}]
x^*_{\overline{F}}$$ becomes a presentation of $w$. We call this presentation a [*crank form expression of $w$ defined by ${\mathbb M}$ and ${\mathbb F}$*]{}. If a crank form expression is made from sequences $(M_1, \ldots, M_u)$ and $(F_1, \ldots, F_v)$ such that
1. $M_1, \ldots, M_p$ and $F_1, \ldots, F_p$ are propagating,
2. $M_{p+1}, \ldots, M_{u}$ and $F_{p+1}, \ldots, F_{v}$ are defective.
then we call it [*in normal form*]{}.
Finally, we define the [*standard expression*]{} of $w$, as a special expression of crank form expressions in normal form by properly choosing the sequences $(M_1, \ldots, M_u)$ and $(F_1, \ldots, F_v)$. For this purpose first we sort the parts $T_1, \ldots, T_s$ of $w$ so that they satisfy:
1. $\pi(w) = \{T_1, T_2, \ldots, T_p\},$
2. $\{T_i\ |\ i = p+1, p+2, \ldots, u\}$ is the set of all upper defective parts,
3. $\{T_i\ |\ i = u+1, u+2, \ldots, u+(v-p)\}$ is the set of all lower defective parts.
For an ordered set $E$, let $\min E$ be the minimum element in $E$. Let $T_1, T_{2}, \ldots, T_{p}$ be the parts of $\pi(w)$. Define $(M_1, M_{2}, \ldots, M_{p})$ so that they satisfy $$\{M_1, M_{2}, \ldots, M_{p}\}
= \{T_1^M, T_{2}^M, \ldots, T_{p}^M\}$$ and $$\min M_1 <\min M_{2} <\cdots <\min M_{p}.$$ Similarly $(F_1, F_{2}, \ldots, F_{p})$ are defined using the lower parts of $\pi(w)$. In such a method, the sequences of the upper parts $(M_1,\ldots, M_p)$ and the lower parts $(F_1, \ldots, F_p)$ are uniquely defined from a seat-plan $w$.
Now we define $(M_{p+1}, \ldots, M_{u})$ so that they satisfy $$\{M_{p+1}, M_{p+2}, \ldots, M_{u}\}
= \{T_{p+1}, T_{p+2}, \ldots, T_{u}\}$$ and $$\min M_{p+1} <\min M_{p+2} <\cdots <\min M_{u}.$$ Similarly we define $(F_{p+1}, \ldots, F_{v})$ so that they satisfy $$\{F_{p+1}, F_{p+2}, \ldots F_{v}\}
= \{T_{u+1}, T_{u+2}, \ldots, T_{u+(v-p)}\}$$ and $$\min F_{p+1} <\min F_{p+2} <\cdots <\min F_{v}.$$ Using these upper and lower sequences defined above, we can obtain a crank from expression in normal form called the [*standard expression*]{} of $w$.
Proof of Theorem 1.2
====================
In the previous section, we have defined the standard expression of a word in the alphabet ${\cal L}_n^1$ as a special expression of the crank form expressions in normal form. In this section, first we show that any two crank form expressions of a seat-plan $w$ are transformed to each other by finitely using the local moves shown in Section 2. Then we show that any word in the alphabet ${\cal L}_n^1$ is moved to a scalar multiple of one of the crank form expressions. Thus we can find that any word in the alphabet ${\cal L}_n^1$ is reduced to a scalar multiple of a the standard expression. Since the set of seat-plans makes a basis of $A_{n}(Q)$ and since every seat-plan has its standard expression, this proves that the partition algebra $A_{n}(Q)$ is characterized by the generators and relations in Theorem \[th:main\].
First we show that any two crank form expressions are transformed to each other. For $w\in\Sigma_n^1$, let ${\mathbb M} = (M_1, \ldots, M_u)$ and ${\mathbb F} = (F_1, \ldots, F_v)$ be sequences of the upper and the lower parts of $w$ respectively. Assume that the subsequence $\pi(\mathbb{M}) = (M_{i_1}, \ldots, M_{i_p})$ ($i_1<\cdots <i_p$) of $\mathbb{M}$ is the sequence of the upper propagating parts and $\pi(\mathbb{F}) = (F_{j_1}, \ldots, F_{j_p})$ ($j_1<\cdots <j_p$) is that of the lower propagating parts. Then there exists a permutation $\sigma$ of degree $p = |\pi(w)|$ which specifies how the propagating parts of $w$ are recovered from $\pi(\mathbb{M})$ and $\pi(\mathbb{F})$. Let $\mathbb{E} = (E_1, \ldots, E_s)$ be a sequence of the upper or lower parts. Suppose that $\tau\in{\mathfrak S}_s$ acts on $\mathbb{E}$ by $\tau\mathbb{E} = (E_{\tau^{-1}(1)}, \ldots, E_{\tau^{-1}(s)})$. Then the following lemma holds.
\[lem:defect\] Let ${\mathbb M} = (M_1, \ldots, M_u)$ and ${\mathbb F} = (F_1, \ldots, F_v)$ be sequences of the upper and the lower (non-empty) parts of a seat-plan respectively. If $M_i$ \[resp. $F_i$\] is defective and $\sigma_i = (i,i+1)$, the $i$-th adjacent transposition, then the crank form expression ${\cal C}(\mathbb{M}, \sigma, \mathbb{F})$ is moved to another crank form expression ${\cal C}(\sigma_i\mathbb{M}, \sigma, \mathbb{F})$ \[resp. ${\cal C}(\mathbb{M}, \sigma, \sigma_i\mathbb{F})$ \].
We consider the case $M_i$ is defective. In case $F_i$ is defective, the similar proof will hold. Let $P_{\mathbb{M},i}\in{\mathfrak S}_n$ be a permutation defined by $$P_{\mathbb{M},i}(x):=
\left\{
\begin{array}{ll}
x + |M_{i+1}|&
\mbox{if}\ \sum_{j=1}^{i-1}|M_{j}| <x \leq \sum_{j=1}^{i}|M_{j}|,\\
x - |M_{i}|&
\mbox{if}\ \sum_{j=1}^{i}|M_{j}| <x \leq \sum_{j=1}^{i+1}|M_{j}|,\\
x&
\mbox{otherwise}.
\end{array}
\right.$$ Then we find that $x_{\overline{\mathbb{M}}}P^{-1}_{\mathbb{M},i}$ maps $j$ to the $j$-th coordinate of $\overline{\sigma_i\mathbb{M}}$. Hence we have $x_{\overline{\mathbb{M}}}P^{-1}_{\mathbb{M},i}
= x_{\overline{\sigma_i\mathbb{M}}}$. (For the definition of $\overline{\mathbb{M}}$, see Section 3.2.)
On the other hand, since $M_i$ is defective, we have $P_{\mathbb{M},i}C[\mathbb{M}] = C[\sigma_i\mathbb{M}]$ by removing an excrescence of $M_i$ and iteratively using “defective part exchange” ($R17'$) in Figure \[fig:dpeE\] (if $M_{i+1}$ is defective) or iteratively using “defective part shift” ($R16'$) in Figure \[fig:dpsE\] (if $M_{i+1}$ is propagating), and then adding an excrescence to $M_i$ just moved. Thus we obtain $$\begin{aligned}
{\cal C}(\mathbb{M}, \sigma, \mathbb{F})
&=& x_{\overline{\mathbb{M}}}
C[\mathbb{M}]C^{\mathbb{M}}_{\mathbb{F}}[\sigma]C^*[\mathbb{F}]
x^*_{\overline{\mathbb{F}}}\\
&=& (x_{\overline{\mathbb{M}}}P^{-1}_{\mathbb{M},i})
(P_{\mathbb{M},i}C[\mathbb{M}])
C^{\mathbb{M}}_{\mathbb{F}}[\sigma]C^*[\mathbb{F}]
x^*_{\overline{\mathbb{F}}}\\
&=& x_{\overline{\sigma_i\mathbb{M}}}
C[\sigma_i\mathbb{M}]
C^{\mathbb{M}}_{\mathbb{F}}[\sigma]C^*[\mathbb{F}]
x^*_{\overline{\mathbb{F}}}\\
&=&{\cal C}(\sigma_i\mathbb{M}, \sigma, \mathbb{F}).\end{aligned}$$
\[rem:defect2\] Lemma \[lem:defect\] also holds if $M_{i+1}$ \[resp. $F_{i+1}$\] is defective.
By Lemma \[lem:defect\] and Remark \[rem:defect2\] we may assume that any crank form expression is given in normal form.
\[lem:crex\] Let ${\cal C}(\mathbb{M}, \sigma, \mathbb{F})$ be a crank form expression of $w$ in normal form. If $M_i$ and $M_{i+1}$ are propagating then ${\cal C}(\mathbb{M},\sigma,\mathbb{F})$ is moved to another crank form expression ${\cal C}(\sigma_i\mathbb{M}, \sigma_i\sigma,\mathbb{F})$ in normal form. Similarly if $F_i$ and $F_{i+1}$ are propagating, then ${\cal C}(\mathbb{M},\sigma,\mathbb{F})$ is moved to ${\cal C}(\mathbb{M},\sigma\sigma_i,\sigma_i\mathbb{F})$.
Let $C_{\mathbb{M}}[i]$ and $C_{\mathbb{M}}[i+1]$ be $i$-th and $(i+1)$-st cranks of $C[\mathbb{M}]$. By Figure \[fig:crex\], we have $$P_{\mathbb{M},i}C_{\mathbb{M}}[i]C_{\mathbb{M}}[i+1]
=C_{\sigma_i\mathbb{M}}[i]C_{\sigma_i\mathbb{M}}[i+1]\sigma_i.$$
![Crank form exchange[]{data-label="fig:crex"}](15.eps)
Thus we obtain $$\begin{aligned}
{\cal C}(\mathbb{M}, \sigma, \mathbb{F})
&=&
x_{\overline{\mathbb{M}}}
C[\mathbb{M}]
C^{\mathbb{M}}_{\mathbb{F}}[\sigma]C^*[\mathbb{F}]
x^*_{\overline{\mathbb{F}}}\\
&=&
(x_{\overline{\mathbb{M}}}P^{-1}_{\mathbb{M},i})
(P_{\mathbb{M},i}C[\mathbb{M}])
C^{\mathbb{M}}_{\mathbb{F}}[\sigma]C^*[\mathbb{F}]
y^*_{\overline{\mathbb{F}}}\\
&=&
x_{\overline{\sigma_i\mathbb{M}}}
C[\sigma_i\mathbb{M}]C^{\mathbb{M}}_{\mathbb{F}}[\sigma_i\sigma]
C^*[\mathbb{F}]
x^*_{\overline{\mathbb{F}}}\\
&=& {\cal C}(\sigma_i\mathbb{M}, \sigma_i\sigma, \mathbb{F}).\end{aligned}$$
By Lemma \[lem:defect\], Remark \[rem:defect2\] and Lemma \[lem:crex\] we obtain the following.
\[prop:normalize\] A crank form expression of a seat-plan is moved to its standard expression.
Now we prove that any word in the alphabet ${\cal L}_n^1$ is moved to a crank form expression. By the above proposition, we will find that any word can be moved to its standard expression.
If ${\cal C}(\mathbb{M},\sigma,\mathbb{F})$ is the standard expression of a seat-plan $w$, then $s_i{\cal C}(\mathbb{M},\sigma,\mathbb{F})$ is moved to a crank form expression of $s_iw$.
If $i$ and $i+1$ are both included one of the (upper) parts of $w$, say $M_k$, then we have $$\sum_{j=1}^{k-1}|M_j|
<x^{-1}_{\overline{\mathbb{M}}}(i) < x^{-1}_{\overline{\mathbb{M}}}(i+1)
= x^{-1}_{\overline{\mathbb{M}}}(i)+1
\leq\sum_{j=1}^{k}|M_j|$$ and $$(x^{-1}_{\overline{\mathbb{M}}}(i), x^{-1}_{\overline{\mathbb{M}}}(i+1))
C_{\mathbb{M}}[k]
= (x^{-1}_{\overline{\mathbb{M}}}(i), x^{-1}_{\overline{\mathbb{M}}}(i)+1)
C_{\mathbb{M}}[k]
= {\cal C}_{\mathbb{M}}[k].$$ Since $$s_ix_{\overline{\mathbb{M}}} = (i,i+1)x_{\overline{\mathbb{M}}}
= x_{\overline{\mathbb{M}}}
(x^{-1}_{\overline{\mathbb{M}}}(i), x^{-1}_{\overline{\mathbb{M}}}(i+1)),$$ we find that $s_i{\cal C}(\mathbb{M}, \sigma, \mathbb{F})
= {\cal C}(\mathbb{M}, \sigma, \mathbb{F})$ is a crank form expression.
If $i$ is included in $M_j$ and $i+1$ is included in $M_k$ ($j\neq k$), then we have $s_ix_{\overline{\mathbb{M}}} = x_{\overline{\mathbb{M}'}}$. Here $\mathbb{M}'$ is the sequence of the upper parts obtained from $\mathbb{M}=(M_1,\ldots, M_u)$ by replacing $M_j$ with $M_j' = (M_j\setminus\{i\})\cup\{i+1\}$ and $M_k$ with $M_k' = (M_k\setminus\{i+1\})\cup\{i\}$.
Hence we find that $s_i{\cal C}(\mathbb{M},\sigma,\mathbb{F})$ is moved to ${\cal C}(\mathbb{M}',\sigma,\mathbb{F})$, a crank form expression. In particular this expression again becomes the standard expression, unless $k = j+1$, $t(M_j) = t(M_{j+1})$, and $i = \min M_j$, $i+1 = \min M_{j+1}$.
If ${\cal C}(\mathbb{M},\sigma,\mathbb{F})$ is the standard expression of a seat-plan $w$, then $f{\cal C}(\mathbb{M},\sigma,\mathbb{F})$ is moved to a crank form expression of $fw$.
First consider the case $\{1, 2\}\subset M_k$ for some $k$. In this case, there exists an integer $i$ such that $i = x^{-1}_{\overline{\mathbb{M}}}(1)$ and $i+1 = x^{-1}_{\overline{\mathbb{M}}}(2)$. Hence in this case we have $fx_{\overline{\mathbb{M}}} = x_{\overline{\mathbb{M}}}f_i$ and $f_i{\cal C}_{\mathbb{M}}[k] = {\cal C}_{\mathbb{M}}[k]$. Thus we obtain $f{\cal C}(\mathbb{M}, \sigma, \mathbb{F})
= {\cal C}(\mathbb{M}, \sigma, \mathbb{F})$.
Next consider the case $1\in M_j$ and $2\in M_k$ ($j\neq k$). In the following we assume that $M_j$ and $M_k$ are both propagating. Even if either $M_j$ or $M_k$ or both of them are defective, the similar proof will hold. Proposition \[prop:normalize\] implies that the standard expression ${\cal C}(\mathbb{M}, \sigma, \mathbb{F})$ is moved to a crank form expression ${\cal C}(\mathbb{M}', id, \mathbb{F}')$ so that the first and the second components of $\mathbb{M}'$ are $M_j$ and $M_k$ respectively and the first and the second components of $\mathbb{F}'$ are jointed to $M_j$ and $M_k$ respectively. Using the relations ($R2''$), ($R2$) and ($R12''$), we find that the first and the second components of $\mathbb{M}'$ and those of $\mathbb{F}'$ are merged by the action of $f$. For example, if $|M_j| = 5$ and $|M_k| = 4$ then we have Figure \[fig:fwE\].
![Action of $f$ on $w$[]{data-label="fig:fwE"}](16.eps)
The merged propagating parts will be moved to a crank form expression ${\cal C}(\mathbb{M}'', id, \mathbb{F}'')$ by “bumping” as in Figure \[fig:bump\]. Here $\mathbb{M}''$ \[resp. $\mathbb{F}''$\] is a sequence of upper \[resp. lower\] parts obtained from $\mathbb{M}$ \[resp. $\mathbb{F}$\] by merging the first two components.
![Bumping[]{data-label="fig:bump"}](17.eps)
If ${\cal C}(\mathbb{M},\sigma,\mathbb{F})$ is the standard expression of a seat-plan $w$, then $e{\cal C}(\mathbb{M},\sigma,\mathbb{F})$ is moved to a crank form expression of $ew$.
By the same argument in the previous proposition, we may assume that ${\cal C}(\mathbb{M},\sigma,\mathbb{F})$ is moved to a crank form expression $${\cal C}(\mathbb{M}'', id, \mathbb{F}'')$$ such that the first component $M_1''$ of $\mathbb{M}''$ contains $\{1\}$.
First consider the case $|M_1''|>1$. In this case, it is easy to check that $e{\cal C}(\mathbb{M}'', id, \mathbb{F}'')$ is again a crank form expression of $ew$ as it is.
Next consider the case $|M_1''| = 1$. If $M_1''$ is defective, then we have a scalar multiple of a crank form expression $e{\cal C}(\mathbb{M}'', id, \mathbb{F}'') =
Q{\cal C}(\mathbb{M}'', id, \mathbb{F}'')$. If $M_1''$ is propagating, then applying “addition of excrescences ($R14'$)” and “bumping” in Figures \[fig:roeE\] and \[fig:bump\] we can move $e{\cal C}(\mathbb{M}'', id, \mathbb{F}'')$ to a crank form expression.
Let $\widetilde{A_{n}(Q)}$ be the associative algebra over $\mathbb{Z}[Q]$ abstractly defined by the generators and the relations in Theorem 1.2. So there exists a surjective morphism $\psi$ from $\widetilde{A_{n}(Q)}$ to $A_{n}(Q)$. As we noted in Section 1, we may assume that $\widetilde{A_{n}(Q)}$ is generated by the alphabets ${\cal L}_n^1$ which satisfy the relations ($R0$)-($R4$) and ($E1$)-($E5$). Here we note that the “geometrical moves” we have shown previously can be applied to any algebra which satisfies the relations ($R0$)-($R4$) and ($E1$)-($E5$). Hence if we associate the alphabets in ${\cal L}_n^1$ with the diagrams in Figure \[fig:gen\], then we can apply the notion of [*basic expressions*]{}, [*crank form expressions*]{} and [*standard expressions*]{} to the words in the alphabets ${\cal L}_n^1$ of $\widetilde{A_{n}(Q)}$.
Let $w$ be a word in the alphabet ${\cal L}_n^1$ of $\widetilde{A_{n}(Q)}$. Suppose that $w$ is presented in a standard expression. Then by Proposition 4.5-4.7, $s_iw$, $fw$ and $ew$ are all moved to (possibly scalar multiples of) crank form expressions. By Proposition 4.4, they are moved to the standard expressions. Since $s_i$ ($1\leq i\leq n-1$), $f$, and $e$ are crank form expressions as they are, by induction on the lengths of the words in the alphabets ${\cal L}_n$, any word turn out to be equal to (a scalar multiple) of the standard expression of a seat-plan $w$ of $\Sigma_n^1$. Hence we have $$\mbox{rank}\ \widetilde{A_{n}(Q)} \leq |\Sigma_n^1|.$$
As Tanabe showed in [@Ta], $\Sigma_n^1$ makes a basis of ${\mbox{${\mathbb C}$}}\otimes A_{n}(k) = {\mbox{${\mathbb C}$}}\otimes \psi(\widetilde{A_{n}(k)})$ if $k\geq n$. Hence $\mbox{rank}\ {\mbox{${\mathbb C}$}}\otimes A_{n}(z) = |\Sigma_n^1|$ holds as far as $z$ takes any integer value $k\geq n$. This implies that $\psi$ is an isomorphism and we find that the generators and the relations in Theorem 1.2 characterize the partition algebra $A_{n}(Q)$.
Definition of $A_{n-\frac{1}{2}}(Q)$, a subalgebra of $A_n(Q)$ {#sec:5-1}
==============================================================
In this section, we consider a subalgebra $A_{n-\frac{1}{2}}(Q)$ of $A_n(Q)$ generated by the special elements $s_1, \ldots, s_{n-2}$, $f_1, \ldots, f_{n-1}$ and $e_1,\ldots, e_{n-1}$. As we have noted in Remark \[rem:gen\], $\{f_i\}$ ($1\leq n-2$) and $\{e_i\}$ ($1\leq n-1$) are written as products of $f=f_1$, $e = e_1$ and $s_1,\ldots, s_{n-2}$. The special element $f_{n-1}$, however, can not be expressed as a product of other special elements in $A_{n-\frac{1}{2}}(Q)$, since we deleted $s_{n-1}$ from the generators of $A_n(Q)$. Hence $A_{n-\frac{1}{2}}(Q)$ can be defined as a subalgebra of $A_n(Q)$ generated by the following elements: $s_1, \ldots, s_{n-2}$, $f = f_1$, $f_* = f_{n-1}$ and $e = e_1$. We can obtain the defining relations among these generators just as in the case of $A_n(Q)$.
\[def:half-int-alg\] Let $\mathbb{Z}$ be the ring of rational integers and $Q$ the indeterminate. We put ${A}_{\frac{1}{2}}(Q) = \mathbb{Z}[Q]\cdot 1$. For an integer $n\geq2$, ${A}_{n-\frac{1}{2}}(Q)$ is characterized by the generators $$e, f, s_1, s_2, \ldots, s_{n-2}, f_{*} \mbox{(if $n>2$)}$$ and the relations ($R0$), ($R1'$)-($R4'$) and ($E1'$)-($E5'$) omitting the ones which involve $s_{n-1}$ and adding the following relations: $$\begin{gathered}
f_{*}s_{n-2}s_{n-3}\cdots s_3s_2s_1
s_2s_3\cdots s_{n-3}s_{n-2}f_{*} \nonumber\\
\quad =\, f_{*}s_{n-2}s_{n-3}\cdots s_3s_2f
s_2s_3\cdots s_{n-3}s_{n-2}\phantom{,} \tag{$R2^*$} \\
\quad =\, s_{n-2}s_{n-3}\cdots s_3s_2f
s_2s_3\cdots s_{n-3}s_{n-2}f_{*},\nonumber\end{gathered}$$ $$\tag{$R4^*$}
ff_{*} = f_{*}f,\quad
ef_{*} = f_{*}e,\quad
f_{*}s_i = s_if_{*}\ \mbox{($1\leq i \leq n-3$)},$$ $$\tag{$E4^*$}
\begin{array}{rcl}
f_{*}s_{n-2}s_{n-3}\cdots s_1es_1\cdots s_{n-3}s_{n-2}f_{*}
&=& f_{*},\\
es_{1}s_{2}\cdots s_{n-2}f_{*}s_{n-2}\cdots s_{2}s_{1}e
&=& e.
\end{array}$$ We understand $A_{1+\frac{1}{2}}(Q) = A_{2-\frac{1}{2}}(Q)$ is defined by the generators $1$, $e$ and $f$ with the relations $e^2 = Qe$, $f^2 = f$, $efe =e$, $fef = f$. (Hence, $A_{2-\frac{1}{2}}(Q)$ is a rank 5 module with a basis $\{1, e, f, ef, fe\}$.)
The relations ($R2^*$) correspond to the relations $f_{n-1}s_{n-2}f_{n-1} = f_{n-1}f_{n-2} = f_{n-2}f_{n-1}$. We deduce $f_{*}s_{n-2}f_{*} = f_{*}s_{n-2}f_{*}s_{n-2}
= f_{*}s_{n-2}f_{*}s_{n-2}$ from ($R2^*$).
First we note that all the generators of $A_{n-\frac{1}{2}}(Q)$ have the part which contains $n$ and $n'$ simultaneously.
We consider the transpositions of indices $i\leftrightarrow n-i+1$. These transpositions make $A_{n-\frac{1}{2}}(Q)$ a subalgebra of $A_{n}(Q)$ generated by $${\cal L}_{n-\frac{1}{2}}^1 = \{f_1, \ldots, f_{n-1},
s_2, \ldots, s_{n-1}, e_2, \ldots, e_n\}.$$ By the relation ($R0$), $A_{n-\frac{1}{2}}(Q)$ is actually generated by letters $\{f_1$, $f_2$, $e_2$ and $s_2,\ldots, s_{n-1}\}$. Each of these generators has a part which includes $\{1, 1'\}$. In the following in this section, we suppose that $A_{n-\frac{1}{2}}(Q)$ is generated by the letters in ${\cal L}^1_{n-\frac{1}{2}}$. The $\mathbb{Z}[Q]$ bases of $A_{n-\frac{1}{2}}(Q)$ consist of $\Sigma^1_{n-\frac{1}{2}}$ a subset of seat-plans in $\Sigma^1_n$ which have at least one propagating part which contains $1$ and $1'$ simultaneously. In the diagram of the standard expression of a seat-plan of $\Sigma^1_{n-\frac{1}{2}}$, the vertices $1$ and $1'$ are joined by a vertical line. Shrinking this vertical line to one vertex, we have one to one correspondences between $\Sigma^1_{n-\frac{1}{2}}$ and the set of the set-partitions of order $2n-1$. (Hence we find $|\Sigma^1_{n-\frac{1}{2}}| = B_{2n-1}$, the Bell number.)
Under this preparation, we prove the theorem. Since the relations in the theorem allow us to use all the required local moves, we can show just in the course of the arguments of Section 4 that any word in the alphabet ${\cal L}^1_{n-\frac{1}{2}}$ is equal to (possibly a scalar multiple of) a standard expression in the abstract algebra $\widetilde{A_{n-\frac{1}{2}}(Q)}$.
Hence we have $$\mbox{rank}\ \widetilde{A_{n-\frac{1}{2}}}(Q)
\leq |\Sigma_{n-\frac{1}{2}}^1|.$$
As Murtin and Rollet showed in [@MR], $\Sigma_{n-\frac{1}{2}}^1$ makes a basis of ${\mbox{${\mathbb C}$}}\otimes A_{n-\frac{1}{2}}(k)
= {\mbox{${\mathbb C}$}}\otimes \psi(\widetilde{A_{n-\frac{1}{2}}(k)})$ if $k>n$. Hence $\mbox{rank}\ {\mbox{${\mathbb C}$}}\otimes A_{n}(z) = |\Sigma_{n-\frac{1}{2}}^1|$ holds as far as $z$ takes any integer value $k> n$. This implies that $\psi$ is an isomorphism and we find that the generators and the relations in the theorem characterize the subalgebra $A_{n-\frac{1}{2}}(Q)$.
Bratteli diagram of the partition algebras {#sec:bra}
==========================================
In this section, we get back to the original definition of $A_{n-\frac{1}{2}}(Q)$. ([*i. e.*]{} $A_{n-\frac{1}{2}}(Q)$ is generated by $s_1, \ldots, s_{n-2}$, $f_1, \ldots, f_{n-1}$ and $e_1,\ldots, e_{n-1}$.) Since, $A_{n-\frac{1}{2}}(Q)$ contains all the generators of ${A}_{n-1}(Q)$, it becomes a subalgebra of ${A}_{n-\frac{1}{2}}(Q)$. Hence we obtain the sequence of inclusions $A_0(Q)\subset A_\frac{1}{2}(Q) \subset \cdots \subset A_{i-\frac{1}{2}}(Q)
\subset A_{i}(Q)\subset A_{i+\frac{1}{2}}(Q)\subset \cdots$.
First we define a graph $\Gamma_n$ \[resp. $\Gamma_{n+\frac{1}{2}}$\] for a non-negative integer $n\in\mathbb{Z}_{\geq 0}$. Then we define the sets of [*tableaux*]{} as sets of paths on this graph. Figure \[fig:brad\] will help the reader to understand the recipe.
![$\Gamma_4$[]{data-label="fig:brad"}](18.eps)
For the moment, we assume that $Q$ is a sufficiently large integer. Let $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_l)$ be a partition. For this $\lambda$, define $$\begin{aligned}
\widetilde{\lambda}
&=& (Q-|\lambda|, \lambda_1, \lambda_2, \ldots, \lambda_l)\\
\big[\mbox{resp.}\ \widehat{\lambda}
&=& (Q-1-|\lambda|, \lambda_1, \lambda_2, \ldots, \lambda_l)\big]\end{aligned}$$ to be a partition of size $Q$ \[resp. $Q-1$\]. Pictorially, $\widetilde{\lambda}$ \[resp. $\widehat{\lambda}$\] is obtained by adding $Q-|\lambda|$ \[resp. $Q-1-|\lambda|$\] boxes on the top of $\lambda$.
Let $
P_{\leq i}
= \bigcup_{j=0}^i\{ \lambda\ |\ \lambda\vdash j\}
$ be a set of Young diagrams of size less than or equal to $i$. We define ${\mbox{\boldmath $\Lambda$}}_i$ and ${\mbox{\boldmath $\Lambda$}}_{i+\frac{1}{2}}$ to be $${\mbox{\boldmath $\Lambda$}}_i = \{\widetilde{\lambda}\ |\ \lambda\in P_{\leq i}\}
\mbox{ and }
{\mbox{\boldmath $\Lambda$}}_{i+\frac{1}{2}} = \{\widehat{\lambda}\ |\ \lambda\in P_{\leq i}\},$$ which are set of Young diagrams of size $Q$ and $Q-1$ respectively.
Under these preparations we define a graph $\Gamma_n$ \[resp. $\Gamma_{n+\frac{1}{2}}$\] which consists of the vertices labeled by: $$\left(
\bigsqcup_{i=0,1, \ldots, n-1}
({\mbox{\boldmath $\Lambda$}}_i \sqcup {\mbox{\boldmath $\Lambda$}}_{i+\frac{1}{2}})
\right)
\bigsqcup{\mbox{\boldmath $\Lambda$}}_n
\quad
\left[ \mbox{resp.}\
\left(
\bigsqcup_{i=0,1, \ldots, n}
({\mbox{\boldmath $\Lambda$}}_i \sqcup {\mbox{\boldmath $\Lambda$}}_{i+\frac{1}{2}})
\right)
\right]$$ and the edges joined by either of the following rule:
- join $\widetilde{\lambda}\in{\mbox{\boldmath $\Lambda$}}_{i}$ and $\widehat{\mu}\in{\mbox{\boldmath $\Lambda$}}_{i+\frac{1}{2}}$ if $\widehat{\mu}$ is obtained from $\widetilde{\lambda}$ by removing a box ($i = 0, 1, 2, \ldots n-1$) \[resp. ($i=0, 1, 2, \dots, n$)\],
- join $\widehat{\mu}\in{\mbox{\boldmath $\Lambda$}}_{i-\frac{1}{2}}$ and $\widetilde{\lambda}\in{\mbox{\boldmath $\Lambda$}}_i$ if $\widetilde{\lambda}$ is obtained from $\widehat{\mu}$ by adding a box ($i = 1, 2, \ldots n$).
For a pair of Young diagrams $({\mbox{\boldmath $\alpha$}}, {\mbox{\boldmath $\beta$}})$, if ${\mbox{\boldmath $\beta$}}$ is obtained from ${\mbox{\boldmath $\alpha$}}$ by one of the method above, we write this as ${\mbox{\boldmath $\alpha$}}\smile{\mbox{\boldmath $\beta$}}$.
Finally, we define the sets of the tableaux. For a half integer $n\in\frac{1}{2}\mathbb{Z}$ and ${\mbox{\boldmath $\alpha$}}\in{\mbox{\boldmath $\Lambda$}}_n$, we define ${\mathbb T}({\mbox{\boldmath $\alpha$}})$, [*tableaux of shape ${\mbox{\boldmath $\alpha$}}$*]{}, to be $$\begin{aligned}
{\mathbb T}({\mbox{\boldmath $\alpha$}})&=&
\{P = ({\mbox{\boldmath $\alpha$}}^{(0)}, {\mbox{\boldmath $\alpha$}}^{(1/2)}, \ldots, {\mbox{\boldmath $\alpha$}}^{(n)})\ |
\ {\mbox{\boldmath $\alpha$}}^{(j)} \in {\mbox{\boldmath $\Lambda$}}_j\ (j = 0, 1/2, \ldots, n),\\
& & \quad{\mbox{\boldmath $\alpha$}}^{(n)} = {\mbox{\boldmath $\alpha$}},
{\mbox{\boldmath $\alpha$}}^{(j)}\smile{\mbox{\boldmath $\alpha$}}^{(j+1/2)}
\ (j = 0, 1/2, \ldots, n-1/2)\}.\end{aligned}$$
Construction of representation {#sec:rep}
==============================
Now we have defined the sets of tableaux, we define linear transformations among the tableaux.
Let ${\mathbb Q}$ be the field of rational numbers and $K_0 = {\mathbb Q}(Q)$ its extension. In the following, the linear transformations are defined over $K_0$. If they preserve the relations defined in the previous sections, they define representations of ${A}_n = {A}_n(Q)\otimes K_0$. Similar methods are used for example in the references [@AK; @GHJ; @Mu; @W1; @W2; @Ko2].
Let ${\mathbb V}({\mbox{\boldmath $\alpha$}})
= \oplus_{P \in {\mathbb T}({\mbox{\boldmath $\alpha$}}) }K_0 v_P$ be a vector space over $K_0$ with the standard basis $\{v_P|P\in {\mathbb T}({\mbox{\boldmath $\alpha$}})\}$.
For generators $e_i$, $f_i$ and $s_i$ of ${A}_n$, we define linear maps $\rho_{{\mbox{\boldmath $\alpha$}}}(e_i)$, $\rho_{{\mbox{\boldmath $\alpha$}}}(f_i)$ and $\rho_{{\mbox{\boldmath $\alpha$}}}(s_i)$ on ${\mathbb V}({\mbox{\boldmath $\alpha$}})$ giving the matrices $E_i$ $F_i$ and $M_i$ respectively with respect to the basis $\{ v_P | P\in {\mathbb T}({\mbox{\boldmath $\alpha$}}) \}$.
Definition of $\rho_{{\mbox{\boldmath $\alpha$}}}(e_i)$
-------------------------------------------------------
Firstly, we define a linear map for $e_i$.
For a tableaux $P = ({\mbox{\boldmath $\alpha$}}^{(0)}, {\mbox{\boldmath $\alpha$}}^{(1/2)}, \ldots, {\mbox{\boldmath $\alpha$}}^{(n)})$ of ${\mathbb T}({\mbox{\boldmath $\alpha$}})$, we define $\rho_{{\mbox{\boldmath $\alpha$}}}(e_i)(v_P)
= \sum_{Q \in {\mathbb T}({\mbox{\boldmath $\alpha$}})}(E_i)_{QP}v_Q$. Let $Q = ({\mbox{\boldmath $\alpha$}}^{\prime(0)}, {\mbox{\boldmath $\alpha$}}^{\prime(1/2)},
\ldots, {\mbox{\boldmath $\alpha$}}^{\prime(n)})$.
If there is an $i_0 \in \{1/2, 1, \ldots, n-1/2 \} \setminus \{i-1/2\}$ such that ${\mbox{\boldmath $\alpha$}}^{(i_0)}\neq {\mbox{\boldmath $\alpha$}}^{\prime(i_0)}$, then we put $$(E_i)_{QP} = 0.$$ In the following, we consider the case that ${\mbox{\boldmath $\alpha$}}^{(i_0)} = {\mbox{\boldmath $\alpha$}}^{\prime(i_0)}$ for $i_0\in\{0, 1/2, 1, \ldots, n-1/2\}\setminus\{i-1/2\}$.
If ${\mbox{\boldmath $\alpha$}}^{(i-1)}$ and ${\mbox{\boldmath $\alpha$}}^{(i)}$ are not labeled by the same Young diagram, then we put $$(E_i)_{QP} = 0.$$
We consider the case ${\mbox{\boldmath $\alpha$}}^{(i-1)}$ and ${\mbox{\boldmath $\alpha$}}^{(i)}$ have the same label $\widetilde{\lambda}$. In this case, the possible vertices as ${\mbox{\boldmath $\alpha$}}^{(i-1/2)}$ have labels $\{\widetilde{\lambda}^{-}_{(s)}\}$, which are obtained by removing one box from $\widetilde{\lambda}$. Let $\{Q_s\}$ be the set of tableaux obtained from $P$ by replacing ${\mbox{\boldmath $\alpha$}}^{(i-1/2)}$ with $\widetilde{\lambda}^{-}_{(s)}$.
Then we define $(E_i)_{QP}$ to be $$(E_i)_{Q_sP}
= \frac{h(\widetilde{\lambda})}{h(\widetilde{\lambda}^{-}_{(s)})}.$$ Here $h(\lambda)$ is the product of hook lengths defined by $$h(\lambda) = \prod_{x\in\lambda} h_{\lambda}(x)$$ and $h_{\lambda}(x)$ is the [*hook-length*]{} at $x\in\lambda$.
Note that the matrix $E_i$ is determined by the label $\widetilde{\lambda}$ itself not by the vertex at which the tableau $P$ goes through. In other words, if another vertex in different level, say $i'$, has the same label $\widetilde{\lambda}$, then $E_{i'}$ becomes the same matrix.
Let $v(\lambda^-_{(s)}, \lambda)$ be the standard vector which corresponds to a tableau whose $(i-1)$-st, $(i-1/2)$-th and $i$-th coordinate $({\mbox{\boldmath $\alpha$}}^{(i-1)}, {\mbox{\boldmath $\alpha$}}^{(i-1/2)}, {\mbox{\boldmath $\alpha$}}^{(i)})$ are labeled by $(\lambda, \lambda^-_{(s)}, \lambda)$. Then for a tableau $P$ which goes through $\lambda$ at the $(i-1)$-st and the $i$-th coordinate of $P$, we have $$\rho(e_i)(v_P)
=
\sum_{s'} \frac{h(\lambda)}{h(\lambda^{-}_{(s')})}v(\lambda^-_{(s')}, \lambda).$$ Here $\lambda^{-}_{(s')}$ runs through Young diagrams obtained from $\lambda$ by removing one box.
![Representation spaces for $\rho(e_i)$[]{data-label="fig:repE4a"}](19.eps)
![Representation spaces for $\rho(e_i)$[]{data-label="fig:repE4b"}](20.eps)
Suppose that tableaux $\{p_r\}$ goes through paths in pictures illustrated in Figure \[fig:repE4a\] or \[fig:repE4b\]. Then we have $$\begin{aligned}
\rho(e_i)(v_0) &=&
\frac{h(\widetilde{\emptyset})}{h(\widehat{\emptyset})}v_0 = Qv_0,\\
\rho(e_i)(v_1\ v_2) &=& (v_1\ v_2)
\begin{pmatrix}
h(\widetilde{{\mbox{\tiny\yng(1)}}})/h(\widehat{\emptyset})
&h(\widetilde{{\mbox{\tiny\yng(1)}}})/h(\widehat{\emptyset})\\
h(\widetilde{{\mbox{\tiny\yng(1)}}})/h(\widehat{{\mbox{\tiny\yng(1)}}})
&h(\widetilde{{\mbox{\tiny\yng(1)}}})/h(\widehat{{\mbox{\tiny\yng(1)}}})
\end{pmatrix}\\
&=& (v_1\ v_2)
\begin{pmatrix}
\frac{Q}{Q-1} &\frac{Q}{Q-1}\\
\frac{Q(Q-2)}{Q-1} &\frac{Q(Q-2)}{Q-1}
\end{pmatrix}
\\
\rho(e_i)(v_3\ v_4) &=& (v_3\ v_4)
\begin{pmatrix}
h(\widetilde{{\mbox{\tiny\yng(2)}}})/h(\widehat{{\mbox{\tiny\yng(1)}}})
&h(\widetilde{{\mbox{\tiny\yng(2)}}})/h(\widehat{{\mbox{\tiny\yng(1)}}})\\
h(\widetilde{{\mbox{\tiny\yng(2)}}})/h(\widehat{{\mbox{\tiny\yng(2)}}})
&h(\widetilde{{\mbox{\tiny\yng(2)}}})/h(\widehat{{\mbox{\tiny\yng(2)}}})
\end{pmatrix},\\
&=& (v_3\ v_4)
\begin{pmatrix}
\frac{2(Q-2)}{Q-3} &\frac{2(Q-2)}{Q-3}\\
\frac{(Q-1)(Q-4)}{Q-3} &\frac{(Q-1)(Q-4)}{Q-3}
\end{pmatrix}\\
\rho(e_i)(v_5\ v_6) &=& (v_5\ v_6)
\begin{pmatrix}
h(\widetilde{{\mbox{\tiny\yng(1,1)}}})/h(\widehat{{\mbox{\tiny\yng(1)}}})
&h(\widetilde{{\mbox{\tiny\yng(1,1)}}})/h(\widehat{{\mbox{\tiny\yng(1)}}})\\
h(\widetilde{{\mbox{\tiny\yng(1,1)}}})/h(\widehat{{\mbox{\tiny\yng(1,1)}}})
&h(\widetilde{{\mbox{\tiny\yng(1,1)}}})/h(\widehat{{\mbox{\tiny\yng(1,1)}}})
\end{pmatrix},\\
&=& (v_5\ v_6)
\begin{pmatrix}
\frac{2Q}{Q-1} &\frac{2Q}{Q-1}\\
\frac{Q(Q-3)}{Q-1} &\frac{Q(Q-3)}{Q-1}
\end{pmatrix}.\end{aligned}$$ Here $v_i$ is the standard vector which corresponds to $p_i$. Similarly for the bases $\langle v_7, v_8\rangle$, $\langle v_9, v_{10}, v_{11}\rangle$ and $\langle v_{12}, v_{13}\rangle$, we have the following matrices respectively: $$\begin{aligned}
&&\begin{pmatrix}
\frac{3(Q-4)}{Q-5} &\frac{3(Q-4)}{Q-5}\\
\frac{(Q-2)(Q-6)}{Q-5} &\frac{(Q-2)(Q-6)}{Q-5}
\end{pmatrix},\\
&&
\begin{pmatrix}
\frac{3(Q-1)}{2(Q-2)} &\frac{3(Q-1)}{2(Q-2)} &\frac{3(Q-1)}{2(Q-2)}\\
\frac{3(Q-3)}{2(Q-4)} &\frac{3(Q-3)}{2(Q-4)} &\frac{3(Q-3)}{2(Q-4)}\\
\frac{(Q-1)(Q-3)(Q-5)}{(Q-2)(Q-4)}&
\frac{(Q-1)(Q-3)(Q-5)}{(Q-2)(Q-4)}&
\frac{(Q-1)(Q-3)(Q-5)}{(Q-2)(Q-4)}
\end{pmatrix},\\
&&
\begin{pmatrix}
\frac{3Q}{Q-1} &\frac{3Q}{Q-1}\\
\frac{Q(Q-4)}{Q-1} &\frac{Q(Q-4)}{Q-1}
\end{pmatrix}.\end{aligned}$$
Definition of $\rho_{{\mbox{\boldmath $\alpha$}}}(f_i)$
-------------------------------------------------------
Next, we define a linear map for $f_i$.
For a tableaux $P = ({\mbox{\boldmath $\alpha$}}^{(0)}, {\mbox{\boldmath $\alpha$}}^{(1/2)}, \ldots, {\mbox{\boldmath $\alpha$}}^{(n)})$ of ${\mathbb T}({\mbox{\boldmath $\alpha$}})$, we define $\rho_{{\mbox{\boldmath $\alpha$}}}(f_i)(v_P)
= \sum_{Q \in {\mathbb T}({\mbox{\boldmath $\alpha$}})}(F_i)_{QP}v_Q$. Let $Q = ({\mbox{\boldmath $\alpha$}}^{\prime(0)}, {\mbox{\boldmath $\alpha$}}^{\prime(1/2)},
\ldots, {\mbox{\boldmath $\alpha$}}^{\prime(n)})$.
If there is an $i_0 \in \{1/2, 1, \ldots, n-1/2 \} \setminus \{i\}$ such that ${\mbox{\boldmath $\alpha$}}^{(i_0)}\neq {\mbox{\boldmath $\alpha$}}^{\prime(i_0)}$, then we put $$(F_i)_{QP} = 0.$$ In the following, we consider the case that ${\mbox{\boldmath $\alpha$}}^{(i_0)} = {\mbox{\boldmath $\alpha$}}^{\prime(i_0)}$ for $i_0\in\{0, 1/2, 1, \ldots, n-1/2\}\setminus\{i\}$.
If ${\mbox{\boldmath $\alpha$}}^{(i-1/2)}$ and ${\mbox{\boldmath $\alpha$}}^{(i+1/2)}$ are not labeled by the same Young diagram, then we put $$(F_i)_{QP} = 0.$$
We consider the case ${\mbox{\boldmath $\alpha$}}^{(i-1/2)}$ and ${\mbox{\boldmath $\alpha$}}^{(i+1/2)}$ have the same label $\widehat{\mu}$. In this case, the possible vertices as ${\mbox{\boldmath $\alpha$}}^{(i)}$ have labels $\{\widehat{\mu}^{+}_{(r)}\}$, which are obtained by adding one box to $\widetilde{\mu}$. Suppose that ${\mbox{\boldmath $\alpha$}}^{(i)}$, the $i$-th coordinate of $P$, has its label $\widetilde{\mu}^{+}_{(r_0)}$. Let $Q$ be a tableau obtained from $P$ by replacing ${\mbox{\boldmath $\alpha$}}^{(i)}$ with one of $\{\widehat{\mu}^{+}_{(r)}\}$.
Then we define $(F_i)_{QP}$ to be $$(F_i)_{Q_rP} = \frac{h(\widehat{\mu})}{h(\widehat{\mu}^{+}_{(r_0)})}.$$
Let $v(\mu^+_{(r)}, \mu)$ be the standard vector which corresponds to a tableau whose $(i-1/2)$-th, $i$-th and $(i+1/2)$-th coordinate $({\mbox{\boldmath $\alpha$}}^{(i-1/2)}, {\mbox{\boldmath $\alpha$}}^{(i)}, {\mbox{\boldmath $\alpha$}}^{(i+1/2)})$ are labeled by $(\mu, \mu^+_{(r)}, \mu)$. Then for a tableau $P$ which goes through $\mu$ at the $(i-1/2)$-th and the $(i+1/2)$-th coordinate of $P$, we have $$\rho(f_i)(v_P)
=
\sum_{r} \frac{h(\mu)}{h(\mu^{+}_{(r_0)})}v(\mu^+_{(r)}, \mu).$$ Here $\mu^{+}_{(r)}$ runs through Young diagrams obtained from $\mu$ by adding one box.
![Representation spaces for $\rho(f_i)$[]{data-label="fig:repF4"}](21.eps)
Suppose that tableau $\{q_r\}$ go through paths in the picture illustrated in Figure \[fig:repF4\]. Then we have $$\rho(f_i)(v_0\ v_1)
= (v_0\ v_1)
\begin{pmatrix}
h(\widehat{\emptyset})/h(\widetilde{\emptyset})
&h(\widehat{\emptyset})/h(\widetilde{{\mbox{\tiny\yng(1)}}})\\
h(\widehat{\emptyset})/h(\widetilde{\emptyset})
&h(\widehat{\emptyset})/h(\widetilde{{\mbox{\tiny\yng(1)}}})
\end{pmatrix}
= (v_0\ v_1)
\begin{pmatrix}
\frac{1}{Q} &\frac{Q-1}{Q}\\
\frac{1}{Q} &\frac{Q-1}{Q}
\end{pmatrix}$$ and $$\begin{aligned}
\rho(f_i)(v_2\ v_{3}\ v_{4})
&=& (v_2\ v_{3}\ v_{4})
\begin{pmatrix}
h(\widehat{{\mbox{\tiny\yng(1)}}})/h(\widetilde{{\mbox{\tiny\yng(1)}}})
&h(\widehat{{\mbox{\tiny\yng(1)}}})/h(\widetilde{{\mbox{\tiny\yng(2)}}})
&h(\widehat{{\mbox{\tiny\yng(1)}}})/h(\widetilde{{\mbox{\tiny\yng(1,1)}}})\\
h(\widehat{{\mbox{\tiny\yng(1)}}})/h(\widetilde{{\mbox{\tiny\yng(1)}}})
&h(\widehat{{\mbox{\tiny\yng(1)}}})/h(\widetilde{{\mbox{\tiny\yng(2)}}})
&h(\widehat{{\mbox{\tiny\yng(1)}}})/h(\widetilde{{\mbox{\tiny\yng(1,1)}}})\\
h(\widehat{{\mbox{\tiny\yng(1)}}})/h(\widetilde{{\mbox{\tiny\yng(1)}}})
&h(\widehat{{\mbox{\tiny\yng(1)}}})/h(\widetilde{{\mbox{\tiny\yng(2)}}})
&h(\widehat{{\mbox{\tiny\yng(1)}}})/h(\widetilde{{\mbox{\tiny\yng(1,1)}}})
\end{pmatrix}\\
&=& (v_2\ v_{3}\ v_{4})
\begin{pmatrix}
\frac{Q-1}{Q(Q-2)} &\frac{Q-3}{2(Q-2)} &\frac{Q-1}{2Q}\\
\frac{Q-1}{Q(Q-2)} &\frac{Q-3}{2(Q-2)} &\frac{Q-1}{2Q}\\
\frac{Q-1}{Q(Q-2)} &\frac{Q-3}{2(Q-2)} &\frac{Q-1}{2Q}
\end{pmatrix}.\end{aligned}$$ Here $v_i$ is the standard vector which corresponds to $q_i$. Similarly, for the bases $\langle v_5, v_6, v_7\rangle$ and $\langle v_8, v_{9}, v_{10}\rangle$ we have the following matrices respectively: $$\begin{pmatrix}
\frac{Q-3}{(Q-1)(Q-4)} &\frac{Q-5}{3(Q-4)} &\frac{2(Q-2)}{3(Q-1)}\\
\frac{Q-3}{(Q-1)(Q-4)} &\frac{Q-5}{3(Q-4)} &\frac{2(Q-2)}{3(Q-1)}\\
\frac{Q-3}{(Q-1)(Q-4)} &\frac{Q-5}{3(Q-4)} &\frac{2(Q-2)}{3(Q-1)}
\end{pmatrix},\quad
\begin{pmatrix}
\frac{Q-1}{Q(Q-3)} &\frac{2(Q-4)}{3(Q-3)} &\frac{Q-1}{3Q}\\
\frac{Q-1}{Q(Q-3)} &\frac{2(Q-4)}{3(Q-3)} &\frac{Q-1}{3Q}\\
\frac{Q-1}{Q(Q-3)} &\frac{2(Q-4)}{3(Q-3)} &\frac{Q-1}{3Q}
\end{pmatrix}.$$
Definition of $\rho_{{\mbox{\boldmath $\alpha$}}}(s_i)$
-------------------------------------------------------
Finally, we define linear maps for $s_i$. Unfortunately, we do not have uniform description for $\rho_{{\mbox{\boldmath $\alpha$}}}(s_i)$, except for “non-reductive” paths. So first we define $\rho_{{\mbox{\boldmath $\alpha$}}}(s_i)$ for the non-reductive paths. Then we define $\rho_{{\mbox{\boldmath $\alpha$}}}(s_1)$ and $\rho_{{\mbox{\boldmath $\alpha$}}}(s_2)$ for “reductive” paths one by one.
### Non-Reductive Case {#non-reductive-case .unnumbered}
In the following, we use notation $\mu{\mbox{$\vartriangleleft$}}\lambda$ if a Young diagram $\lambda$ is obtained from a Young diagram $\mu$ by adding one box.
For $1\leq j\leq i$, let ${\nu}$, ${\mu}$, ${\lambda}$ be Young diagrams of size $j-1$, $j$ and $j+1$ respectively such that $\nu{\mbox{$\vartriangleleft$}}\mu{\mbox{$\vartriangleleft$}}\lambda$. If a tableau $P$ of $\mathbb{T}({\mbox{\boldmath $\alpha$}})$ goes through $\widetilde{\nu}$, $\widetilde{\mu}$ and $\widetilde{\lambda}$ at the $(i-2)$-nd, the $(i-1)$-st and the $i$-th coordinate, then $P$ goes through $\widehat{\nu}$ and $\widehat{\mu}$ at the $(i-3/2)$-th and the $(i-1/2)$-th coordinate. We call such a tableau [*non-reductive*]{} at $i$. If a tableau $P$ does not satisfy the property above, then we call $P$ [*reductive*]{} at $i$.
Recall that if $\nu{\mbox{$\vartriangleleft$}}\mu{\mbox{$\vartriangleleft$}}\lambda$, then we can define the [*axial distance*]{} $d = d(\nu, \mu, \lambda)$. Namely, if $\mu$ differs from $\nu$ in the $r_0$-th row and the $c_0$-th column only, and $\lambda$ differs from $\mu$ in the $r_1$-th row and the $c_1$-th column only, then $d = d(\nu, \mu, \lambda)$ is defined by $$d = d(\nu, \mu, \lambda) = (c_1 - r_1) - (c_0 - r_0)
= \left\{
\begin{array}{ll}
h_{\lambda}(r_1, c_0) -1 & \mbox{ if } r_0 \geq r_1,\\
1 - h_{\lambda}(r_0, c_1) & \mbox{ if } r_0 < r_1.
\end{array}
\right.$$ Here $h_{\lambda}(i,j)$ is the [*hook-length*]{} at $(i,j)$ in $\lambda$.
If $|d|\geq 2$, then there is a unique Young diagram $\mu^{\prime}\neq \mu$ which satisfies $\nu{\mbox{$\vartriangleleft$}}\mu^{\prime}{\mbox{$\vartriangleleft$}}\lambda$. Let $P'$ be a tableau of shape ${\mbox{\boldmath $\alpha$}}$ which are obtained from $P$ by replacing $(i-1)$-st and $(i-1/2)$-th coordinates of $P$ with $\widetilde{\mu'}$ and $\widehat{\mu'}$ respectively. For the standard vectors $v_P$ and $v_{P'}$ which correspond to $P$ and $P'$, we define the linear map $\rho_{{\mbox{\boldmath $\alpha$}}}(s_i)$ as follows: $$\label{eq:non-red}
\rho_{{\mbox{\boldmath $\alpha$}}}(s_i)\ :\ (v_{P}, v_{P'})
\longmapsto (v_{P}, v_{P'})
\left(
\begin{array}{cc}
1/d & \left(1-1/d^2\right)/c\\
c & -1/d
\end{array}
\right),$$ where we can arbitrarily chose $c\in K_0\setminus\{0\}$. If we put $$\label{eq:ad}
a_d = 1/d\quad \mbox{and}\quad b_d = 1-a_d^2,$$ then the matrix in the expression is written as follows: $$\left(
\begin{array}{rr}
a_{d} & b_{d}/c\\
c & -a_{d}
\end{array}
\right).$$ If $|d_1|=1$, then there does not exist a distinct Young diagram $\mu^{\prime}$ which satisfies $\nu{\mbox{$\vartriangleright$}}\mu^{\prime}{\mbox{$\vartriangleright$}}\lambda$ other than $\mu$. In this case, we define $\rho_{{\mbox{\boldmath $\alpha$}}}(s_i)$ to be $$\rho_{{\mbox{\boldmath $\alpha$}}}(s_i)\ :\ v_{P}
\longmapsto a_d v_{P}.$$ Here $a_d$ is the one defined by .
Suppose that a tableau $p_1$ of $\mathbb{T}({\mbox{\boldmath $\alpha$}})$ goes through $\widetilde{\emptyset}$, $\widetilde{{\mbox{\tiny\yng(1)}}}$ and $\widetilde{{\mbox{\tiny\yng(2)}}}$ at the 0-th, the 1-st and the 2-nd coordinates respectively, then for the standard vector $u_1$ which corresponds to $p_1$ we have $$\rho_{{\mbox{\boldmath $\alpha$}}}(s_1) u_1 = u_1.$$ For the standard vector $v_2$ which corresponds to $p_2$, a tableau of $\mathbb{T}({\mbox{\boldmath $\alpha$}})$ which goes through $\widetilde{\emptyset}$, $\widetilde{{\mbox{\tiny\yng(1)}}}$ and $\widetilde{{\mbox{\tiny\yng(1,1)}}}$ at the 0-th, the 1-st and the 2-nd coordinates respectively, we have $$\rho_{{\mbox{\boldmath $\alpha$}}}(s_1) u_2 = -u_2.$$
Let $\lambda^{(1)} = (3)$, $\lambda^{(2)} = (2,1)$ and $\lambda^{(3)} = (1,1,1)$ be partitions of 3. Suppose that tableaux $q_1$ and $q_2$ of $\mathbb{T}({\mbox{\boldmath $\alpha$}})$ both go through $\widetilde{{\mbox{\tiny\yng(1)}}}$ and $\widetilde{{\mbox{\tiny\yng(2)}}}$ at the 1-st and the 2-nd coordinates respectively, and tableaux $q_3$ and $q_4$ of $\mathbb{T}({\mbox{\boldmath $\alpha$}})$ both go through $\widetilde{{\mbox{\tiny\yng(1)}}}$ and $\widetilde{{\mbox{\tiny\yng(1,1)}}}$ at the 1-st and the 2-nd coordinates respectively. Further, the tableaux $q_1$, $q_2$, $q_3$ and $q_4$ go through $\widetilde{\lambda^{(1)}}$, $\widetilde{\lambda^{(2)}}$, $\widetilde{\lambda^{(2)}}$ and $\widetilde{\lambda^{(3)}}$ at the 3-rd coordinates respectively. Then we have $$\rho_{{\mbox{\boldmath $\alpha$}}}(s_2) (v_1\ v_2\ v_3\ v_4)
= (v_1\ v_2\ v_3\ v_4)
\begin{pmatrix}
1 & 0 & 0 & 0\\
0 & -1/2& 3/(4c) & 0\\
0 & c & 1/2 & 0\\
0 & 0 & 0 & -1
\end{pmatrix}.$$ Here $v_i$ is the standard vector which corresponds to $q_i$.
### Reductive Case {#reductive-case .unnumbered}
Consider the case a tableau $P$ is reductive at $i$. So far, we do not have uniform description for $\rho_{{\mbox{\boldmath $\alpha$}}}(s_i)$. So we define $\rho_{{\mbox{\boldmath $\alpha$}}}(s_1)$ and $\rho_{{\mbox{\boldmath $\alpha$}}}(s_2)$ one by one.
First we define $\rho_{{\mbox{\boldmath $\alpha$}}}(s_1)$. For tableaux $p_1$ and $p_2$ of $\mathbb{T}({\mbox{\boldmath $\alpha$}})$ which go through $(\widetilde{\emptyset}, \widehat{\emptyset},
\widetilde{\emptyset}, \widehat{\emptyset}, \widetilde{\emptyset})$ and $(\widetilde{\emptyset}, \widehat{\emptyset},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset}, \widetilde{\emptyset})$ at the 0-th, the $1-\frac{1}{2}$-th, the 1-st, the $2-\frac{1}{2}$-th and the 2-nd coordinate respectively, let $u_1$ and $u_2$ be the corresponding standard vectors. Then we define $\rho_{{\mbox{\boldmath $\alpha$}}}(s_1)(u_1\ u_2)$ by $$\rho_{{\mbox{\boldmath $\alpha$}}}(s_1)(u_1\ u_2)
= (u_1\ u_2)
\begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix}.$$
For tableaux $p_3$, $p_4$ and $p_5$ of $\mathbb{T}({\mbox{\boldmath $\alpha$}})$ which go through $(\widetilde{\emptyset}, \widehat{\emptyset},
\widetilde{\emptyset}, \widehat{\emptyset}, \widetilde{{\mbox{\tiny\yng(1)}}})$, $(\widetilde{\emptyset}, \widehat{\emptyset},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset}, \widetilde{{\mbox{\tiny\yng(1)}}})$ and $(\widetilde{\emptyset}, \widehat{\emptyset},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}}, \widetilde{{\mbox{\tiny\yng(1)}}})$ at the 0-th, the $1-\frac{1}{2}$-th, the 1-st, the $2-\frac{1}{2}$-th and the 2-nd coordinate respectively, let $u_3$, $u_4$ and $u_5$ be the corresponding standard vectors. Then we define $\rho_{{\mbox{\boldmath $\alpha$}}}(s_1)(u_1\ u_2\ u_3)$ by $$\rho_{{\mbox{\boldmath $\alpha$}}}(s_1)(u_1\ u_2\ u_3)
= (u_1\ u_2\ u_3)
\begin{pmatrix}
0 & 1 & 1\\
\frac{1}{Q-1} & \frac{Q-2}{Q-1} & \frac{-1}{Q-1}\\
\frac{Q-2}{Q-1} & -\frac{Q-2}{Q-1} & \frac{1}{Q-1}
\end{pmatrix}.$$
Next we define $\rho_{{\mbox{\boldmath $\alpha$}}}(s_2)$. In the following, we write $$p = (\lambda^{(1)}, \lambda^{(2)}, \lambda^{(3)},
\lambda^{(4)}, \lambda^{(5)})$$ to mean the tableau $p$ goes through $\lambda^{(1)}$, $\lambda^{(2)}$, $\lambda^{(3)}$, $\lambda^{(4)}$, $\lambda^{(5)}$ at the 1-st, the $(2-\frac{1}{2})$-th, the 2-nd, the $(3-\frac{1}{2})$-th and the 3-rd coordinates respectively.
Suppose that $$\begin{array}{lr}
\begin{array}{rcl}
q_1 &=& (\widetilde{\emptyset}, \widehat{\emptyset},
\widetilde{\emptyset}, \widehat{\emptyset}, \widetilde{\emptyset}),\\
q_2 &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset},
\widetilde{\emptyset}, \widehat{\emptyset}, \widetilde{\emptyset}),\\
q_3 &=& (\widetilde{\emptyset}, \widehat{\emptyset},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset}, \widetilde{\emptyset}),
\end{array}
&
\begin{array}{rcl}
q_4 &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset}, \widetilde{\emptyset}),\\
q_5 &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset}, \widetilde{\emptyset}).
\end{array}
\end{array}$$ Then for the standard vectors $(v_j)_{j=1}^5$ which correspond to $(q_j)_{j=1}^5$ we define $\rho_{{\mbox{\boldmath $\alpha$}}}(s_2)(v_1\ v_2\ v_3\ v_4\ v_5)$ by $$\rho_{{\mbox{\boldmath $\alpha$}}}(s_2)(v_1\ v_2\ v_3\ v_4\ v_5)
= (v_1\ v_2\ v_3\ v_4\ v_5)
\begin{pmatrix}
1 & 0 & 0& 0 &0\\
0 & 0 & 0& 1 &1\\
0 & 0 & 1& 0 &0\\
0 & \frac{1}{Q-1} & 0& \frac{Q-2}{Q-1} &\frac{-1}{Q-1}\\
0 & \frac{Q-2}{Q-1} & 0& -\frac{Q-2}{Q-1} &\frac{1}{Q-1}
\end{pmatrix}.$$ Assume that $$\begin{array}{lr}
\begin{array}{rcl}
q_6 &=& (\widetilde{\emptyset}, \widehat{\emptyset},
\widetilde{\emptyset}, \widehat{\emptyset}, \widetilde{{\mbox{\tiny\yng(1)}}}),\\
q_7 &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset},
\widetilde{\emptyset}, \widehat{\emptyset}, \widetilde{{\mbox{\tiny\yng(1)}}}),\\
q_8 &=& (\widetilde{\emptyset}, \widehat{\emptyset},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset}, \widetilde{{\mbox{\tiny\yng(1)}}}),\\
q_9 &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset}, \widetilde{{\mbox{\tiny\yng(1)}}}),\\
q_{10} &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset}, \widetilde{{\mbox{\tiny\yng(1)}}}),
\end{array}
&
\begin{array}{rcl}
q_{11} &=& (\widetilde{\emptyset}, \widehat{\emptyset},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}}, \widetilde{{\mbox{\tiny\yng(1)}}}),\\
q_{12} &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}}, \widetilde{{\mbox{\tiny\yng(1)}}}),\\
q_{13} &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}}, \widetilde{{\mbox{\tiny\yng(1)}}}),\\
q_{14} &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}},
\widetilde{{\mbox{\tiny\yng(2)}}}, \widehat{{\mbox{\tiny\yng(1)}}}, \widetilde{{\mbox{\tiny\yng(1)}}}),\\
q_{15} &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}},
\widetilde{{\mbox{\tiny\yng(1,1)}}}, \widehat{{\mbox{\tiny\yng(1)}}}, \widetilde{{\mbox{\tiny\yng(1)}}}).
\end{array}
\end{array}$$ Then for the standard vectors $(v_j)_{j=6}^{15}$ which correspond to $(q_j)_{j=6}^{15}$ we define $\rho_{{\mbox{\boldmath $\alpha$}}}(s_2)(v_6\ v_8\ v_{11})$ and $\rho_{{\mbox{\boldmath $\alpha$}}}(s_2)(v_7\ v_9\ v_{10} \ v_{12}\ v_{13}\ v_{14}\ v_{15})$ by $$\rho_{{\mbox{\boldmath $\alpha$}}}(s_2)(v_6\ v_8\ v_{11})
= (v_6\ v_8\ v_{11})
\begin{pmatrix}
0&1&1\\
\noalign{\medskip}
\frac{1}{(Q-1)}
&\frac{Q-2}{Q-1}
&\frac{-1}{(Q-1)}\\
\noalign{\medskip}
\frac{Q-2}{Q-1}
&-\frac{Q-2}{Q-1}
&\frac{1}{Q-1}
\end{pmatrix}$$ and $$\rho_{{\mbox{\boldmath $\alpha$}}}(s_2)(v_7\ v_9\ v_{10}
\ v_{12}\ v_{13}\ v_{14}\ v_{15})
= (v_7\ v_9\ v_{10}
\ v_{12}\ v_{13}\ v_{14}\ v_{15})M_i$$ Here the matrix $M_{i}$ is $$\begin{pmatrix}
\noalign{\medskip}
\frac{1}{Q-1}
&\frac{Q-2}{Q-1}
&\frac{-1}{Q-1}
&\frac{-1}{Q-1}
&\frac{1}{(Q-1)(Q-2)}
&\frac{(Q-1)(Q-2)-2}{2(Q-1)(Q-2)}
&-1/2\\
\noalign{\medskip}
\frac{Q-2}{(Q-1)^{2}}
&\frac{Q^2-3Q+3}{(Q-1)^2}
&\frac{1}{(Q-1)^2}
&\frac{1}{(Q-1)^2}
&\frac{-1}{(Q-1)^2(Q-2)}
&\frac{-Q(Q-3)}{2(Q-1)^2(Q-2)}
&\frac{1}{2(Q-1)}\\
\noalign{\medskip}
\frac{-(Q-2)}{(Q-1)^2}
&\frac{Q-2}{(Q-1)^2}
&\frac{-1}{(Q-1)^2}
&\frac{Q(Q-2)}{(Q-1)^2}
&\frac{1}{(Q-1)^2(Q-2)}
&\frac{Q(Q-3)}{2(Q-1)^2(Q-2)}
&\frac{-1}{2(Q-1)}\\
\noalign{\medskip}
\frac{-(Q-2)}{(Q-1)^2}
&\frac{Q-2}{(Q-1)^2}
&\frac{Q(Q-2)}{(Q-1)^2}
&\frac{-1}{(Q-1)^2}
&\frac{1}{(Q-1)^2(Q-2)}
&\frac {Q(Q-3)}{2(Q-1)^2(Q-2)}
&\frac{-1}{2(Q-1)}\\
\noalign{\medskip}
\frac{Q-2}{(Q-1)^2}
&\frac{-(Q-2)}{(Q-1)^2}
&\frac{1}{(Q-1)^2}
&\frac{1}{(Q-1)^2}
&\frac{(Q-1)^2(Q-2)-1}{(Q-1)^2(Q-2)}
&\frac{-Q(Q-3)}{2(Q-1)^2(Q-2)}
&\frac{1}{2(Q-1)}\\
\noalign{\medskip}
\frac{Q-2}{Q-1}
&\frac{-(Q-2)}{Q-1}
&\frac{1}{Q-1}
&\frac{1}{Q-1}
&\frac{-1}{(Q-1)(Q-2)}
&\frac{Q^2-3Q+4}{2(Q-1)(Q-2)}
&1/2\\
\noalign{\medskip}
\frac{-(Q-2)}{Q-1}
&\frac{Q-2}{Q-1}
&\frac{-1}{Q-1}
&\frac{-1}{Q-1}
&{\frac{1}{(Q-1)(Q-2)}}
&\frac{Q(Q-3)}{2(Q-1)(Q-2)}
&1/2
\end{pmatrix}.$$ Next assume that $$\begin{array}{lr}
\begin{array}{rcl}
q_{16} &=& (\widetilde{\emptyset}, \widehat{\emptyset},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}}, \widetilde{{\mbox{\tiny\yng(2)}}}),\\
q_{17} &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}}, \widetilde{{\mbox{\tiny\yng(2)}}}),\\
q_{18} &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}}, \widetilde{{\mbox{\tiny\yng(2)}}}),
\end{array}
&
\begin{array}{rcl}
q_{19} &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}},
\widetilde{{\mbox{\tiny\yng(2)}}}, \widehat{{\mbox{\tiny\yng(1)}}}, \widetilde{{\mbox{\tiny\yng(2)}}}),\\
q_{20} &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}},
\widetilde{{\mbox{\tiny\yng(1,1)}}}, \widehat{{\mbox{\tiny\yng(1)}}}, \widetilde{{\mbox{\tiny\yng(2)}}}),\\
q_{21} &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}},
\widetilde{{\mbox{\tiny\yng(2)}}}, \widehat{{\mbox{\tiny\yng(2)}}}, \widetilde{{\mbox{\tiny\yng(2)}}}).
\end{array}
\end{array}$$ Then for the standard vectors $(v_j)_{j=16}^{21}$ which correspond to $(q_j)_{j=16}^{21}$ we define $\rho_{{\mbox{\boldmath $\alpha$}}}(s_2)(v_{16}\ v_{17}\ v_{18}\ v_{19}\ v_{20}\ v_{21}
)$ by $$\rho_{{\mbox{\boldmath $\alpha$}}}(s_2)(v_{16}\ v_{17}\ v_{18}\ v_{19}\ v_{20}\ v_{21})
= (v_{16}\ v_{17}\ v_{18}\ v_{19}\ v_{20}\ v_{21})M_i.$$ Here the matrix $M_i$ is $$\begin{aligned}
\begin{pmatrix}
1
&0
&0
&0
&0
&0\\
\noalign{\medskip}0
&\frac{-1}{(Q-1)}
&\frac{1}{(Q-1)(Q-2)}
&\frac{Q(Q-3)}{2(Q-1)(Q-2)}
&-1/2
&0\\
\noalign{\medskip}0
&\frac{1}{(Q-1)}
&\frac{-1}{(Q-1)(Q-2)}
&\frac{{Q}^{2}-3\,Q+4}{2(Q-1)(Q-2)}
&1/2
&1\\
\noalign{\medskip}0
&1
&\frac{{Q}^{2}-3\,Q+4}{Q(Q-3)(Q-2)}
&\frac{(Q-1)( Q-4)}{2( Q-2)(Q-3)}
&\frac{(Q-1)(Q-4)}{2Q(Q-3)}
&\frac{-1}{(Q-3)}\\
\noalign{\medskip}0
&-1
&\frac{1}{(Q-2)}
&\frac {Q-4}{2(Q-2)}
&1/2
&\frac{-1}{(Q-1)}\\
\noalign{\medskip}0
&0
&\frac{(Q-1)^{2}(Q-4)}{Q(Q-3)(Q-2)}
&\frac{-(Q-1)(Q-4)}{2(Q-2)(Q-3)}
&\frac{-(Q-1)(Q-4)}{2Q(Q-3)}
&\frac{1}{(Q-3)}
\end{pmatrix}.\end{aligned}$$
Finally assume that $$\begin{array}{lr}
\begin{array}{rcl}
q_{22} &=& (\widetilde{\emptyset}, \widehat{\emptyset},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}}, \widetilde{{\mbox{\tiny\yng(1,1)}}}),\\
q_{23} &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{\emptyset},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}}, \widetilde{{\mbox{\tiny\yng(1,1)}}}),\\
q_{24} &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}},
\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}}, \widetilde{{\mbox{\tiny\yng(1,1)}}}),
\end{array}
&
\begin{array}{rcl}
q_{25} &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}},
\widetilde{{\mbox{\tiny\yng(2)}}}, \widehat{{\mbox{\tiny\yng(1)}}}, \widetilde{{\mbox{\tiny\yng(1,1)}}}),\\
q_{26} &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}},
\widetilde{{\mbox{\tiny\yng(1,1)}}}, \widehat{{\mbox{\tiny\yng(1)}}}, \widetilde{{\mbox{\tiny\yng(1,1)}}}),\\
q_{27} &=& (\widetilde{{\mbox{\tiny\yng(1)}}}, \widehat{{\mbox{\tiny\yng(1)}}},
\widetilde{{\mbox{\tiny\yng(2)}}}, \widehat{{\mbox{\tiny\yng(1,1)}}}, \widetilde{{\mbox{\tiny\yng(1,1)}}}).
\end{array}
\end{array}$$ Then for the standard vectors $(v_j)_{j=22}^{27}$ which correspond to $(q_j)_{j=22}^{27}$ we define $\rho_{{\mbox{\boldmath $\alpha$}}}(s_2)(v_{22}\ v_{23}\ v_{24}\ v_{25}\ v_{26}\ v_{27}
)$ by $$\rho_{{\mbox{\boldmath $\alpha$}}}(s_2)(v_{22}\ v_{23}\ v_{24}\ v_{25}\ v_{26}\ v_{27})
= (v_{22}\ v_{23}\ v_{24}\ v_{25}\ v_{26}\ v_{27})M_i.$$ Here the matrix $M_i$ is $$\begin{aligned}
\begin{pmatrix}
-1
&0
&0
&0
&0
&0\\
\noalign{\medskip}
0
&\frac{1}{(Q-1)}
&\frac{-1}{(Q-1)(Q-2)}
&\frac{-Q(Q-3)}{2(Q-1)( Q-2 )}
&1/2
&0\\
\noalign{\medskip}
0
&\frac{-1}{(Q-1)}
&\frac{1}{(Q-1)(Q-2)}
&\frac{Q(Q-3)}{2(Q-1)(Q-2)}
&1/2
&1\\
\noalign{\medskip}
0
&-1
&\frac{1}{(Q-2)}
&\frac{Q-4}{2(Q-2)}
&1/2
&\frac{-1}{(Q-3)}
\\
\noalign{\medskip}
0
&1
&\frac{1}{(Q-2)}
&\frac{Q(Q-3)}{2(Q-1)(Q-2)}
&\frac{Q-3}{2(Q-1)}
&\frac{-1}{(Q-1)}\\
\noalign{\medskip}
0
&0
&\frac{Q-3}{Q-2}
&\frac{-Q(Q-3)}{2(Q-1)(Q-2)}
&\frac{-(Q-3)}{2(Q-1)}
&\frac{1}{(Q-1)}
\end{pmatrix}.\end{aligned}$$
Discussion {#sec:dec}
==========
In the previous section, we gave linear maps $\rho_{{\mbox{\boldmath $\alpha$}}}(e_i)$ and $\rho_{{\mbox{\boldmath $\alpha$}}}(f_i)$ for all the tableaux on $\Gamma_n$. and defined $\rho_{{\mbox{\boldmath $\alpha$}}}(s_i)$ for non-reductive tableaux on $\Gamma_n$. We also defined $\rho_{{\mbox{\boldmath $\alpha$}}}(s_1)$ and $\rho_{{\mbox{\boldmath $\alpha$}}}(s_2)$ for the reductive tableaux on $\Gamma_n$. (So far, we have further obtained $\rho_{{\mbox{\boldmath $\alpha$}}}(s_3)$ for almost all reductive tableaux on $\Gamma_4$.) These linear maps preserve the relations in Theorem 1.2 and Theorem 5.3. Hence they give representations of $A_n(Q)$ for all ${\mbox{\boldmath $\alpha$}}\in{\mbox{\boldmath $\Lambda$}}_n$ ($n = 2-\frac{1}{2}, 2, 3-\frac{1}{2}, 3, 4-\frac{1}{2}$) and for almost all ${\mbox{\boldmath $\alpha$}}\in{\mbox{\boldmath $\Lambda$}}_4$.
These representations also coincide with the ones calculated through the Murphy’s operators which are introduced in the paper [@HR] and programmed by Naruse. Moreover, the traces of the representation matrices above coincide with the “characters” which is defined by Naruse in the paper [@Na1]. This means that the representations we have presented in this note will be irreducible and define Young’s seminormal form representations of the partition algebras $A_n(Q)$.
[99]{} S. Ariki and K. Koike, A Hecke algebra of $({\mathbb Z}/r{\mathbb Z})\wr {\mathfrak S}_n$ and construction of its irreducible representations, [*Adv. in Math.*]{} **106** (1994), 216–243.
W. Doran IV and D. Wales, The partition algebra revisited, [*J. Algebra*]{} **231** (2000), 265–330.
F. M. Goodman, P. de la Harpe and V. F. R. Jones, [Coxeter Graphs and Towers of Algebras]{}, Springer-Verlag, New York, 1989.
J. J. Graham and G. I. Lehrer, Cellular algebras, [*Invent. Math.*]{} **123** (1996), 1–34.
T. Halverson and A. Ram, Partition algebras, [*European J. Combin.*]{} **26** (2005), 869–921.
V. F. R. Jones, The potts model and the symmetric group, [*Subfactors (Kyuzeso, 1993)*]{} 259–267, World Sci. Publishing, River Edge, NJ, 1994.
D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, [*Invent. Math.*]{} **53** (1979), 165–184.
M. Kosuda, Characterization for the party algebras, [*Ryukyu Math. J.*]{} **13** (2000), 7–22.
M. Kosuda, Irreducible representations of the party algebra, [*Osaka J. Math.*]{} **43** (2006), 431–474.
M. Kosuda, The standard expression for the party algebra, [*Sūrikaisekikenkyūsho Kōkyūroku*]{} **No. 1497** (2006), 52–69.
M. Kosuda, Characterization for the modular party algebra, [*J. Knot Theory Ramifications*]{} **17** (2008), 939–960.
P. Martin, Representations of graph Temperley-Lieb algebras, [*Publ. Res. Inst. Math. Sci.*]{} **26** (1990), 485–503.
P. Martin, Temperley-Lieb algebras for non-planar statistical mechanics – The partition algebra construction, [*J. Knot Theory Ramifications*]{} **3** (1994), 51–82.
P. Martin, The structure of the partition algebras, [*J. Algebra*]{} **183** (1996), 319–358.
P. Martin, and G. Rollet, The potts model representation and a Robinson Schensted correspondence for the partition algebra [*Compositio Math.*]{} **112** (1998), 2337–254.
J. Murakami, The representations of the $q$-analogue of Brauer’s centralizer algebras and the Kauffman polynomial of links, [*Publ. Res. Inst. Math. Sci.*]{} **26** (1990), 935–945.
H. Naruse, Characters of the party algebras, [*manuscript for Workshop on Cellular and Diagram Algebras in Mathematics and Physics*]{}, Oxford, 2005.
G. C. Shephard and J. A. Todd, Finite unitary reflection groups, [*Canad. J. Math.*]{} **6** (1954), 274–304.
K. Tanabe, On the centralizer algebra of the unitary reflection group $G(m, p, n)$, [*Nagoya. Math. J.*]{} **148** (1997), 113–126.
H. Wenzl, On the structure of Brauer’s centralizer algebras, [*Ann. of Math.*]{} **128** (1988), 173–193.
H. Wenzl, Hecke algebras of type $A_n$ and subfactors, [*Invent. Math.*]{} **92** (1988), 349–383.
C. C. Xi, Partition algebras are cellular, [*Compositio. Math.*]{} **119** (1999), 99–109.
Department of Mathematical Sciences\
Faculty of Science\
University of the Ryukyus\
Nishihara-cho, Okinawa 903-0213\
JAPAN\
\
[email protected]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we give a very simple proof of the main result of Dafni (Canad Math Bull 45:46–59, 2002) concerning with weak$^*$-convergence in the local Hardy space $h^1({{\mathbb R}}^d)$.'
address:
- 'High School for Gifted Students, Hanoi National University of Education, 136 Xuan Thuy, Hanoi, Vietnam'
- 'Department of Natural Science and Technology, Tay Nguyen University, Daklak, Vietnam.'
- 'Department of Mathematics, University of Quy Nhon, 170 An Duong Vuong, Quy Nhon, Binh Dinh, Vietnam'
author:
- Ha Duy HUNG
- Duong Quoc Huy
- 'Luong Dang Ky $^*$'
title: 'A note on weak$^*$-convergence in $h^1({{\mathbb R}}^d)$'
---
[^1]
\[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\]
Introduction
============
A famous and classical result of Fefferman [@Fef] states that the John-Nirenberg space $BMO(\mathbb R^d)$ is the dual of the Hardy space $H^1(\mathbb R^d)$. It is also well-known that $H^1(\mathbb R^d)$ is one of the few examples of separable, nonreflexive Banach space which is a dual space. In fact, let $C_c({{\mathbb R}}^d)$ be the space of all continuous functions with compact support and denote by $VMO(\mathbb R^d)$ the closure of $C_c({{\mathbb R}}^d)$ in $BMO(\mathbb R^d)$, Coifman and Weiss showed in [@CW] that $H^1(\mathbb R^d)$ is the dual space of $VMO(\mathbb R^d)$, which gives to $H^1(\mathbb R^d)$ a richer structure than $L^1(\mathbb R^d)$. For example, the classical Riesz transforms $\nabla (-\Delta)^{-1/2}$ are not bounded on $L^1(\mathbb R^d)$, but are bounded on $H^1(\mathbb R^d)$. In addition, the weak$^*$-convergence is true in $H^1(\mathbb R^d)$ (see [@JJ]), which is useful in the application of Hardy spaces to compensated compactness (see [@CLMS]) and in studying the endpoint estimates for commutators of singular integral operators (see [@Ky1; @Ky2; @Ky3]). Recently, Dafni showed in [@Da] that the local Hardy space $h^1({{\mathbb R}}^d)$ of Goldberg [@Go] is in fact the dual space of $vmo({{\mathbb R}}^d)$ the closure of $C_c({{\mathbb R}}^d)$ in $bmo(\mathbb R^d)$. Moreover, the weak$^*$-convergence is true in $h^1(\mathbb R^d)$. More precisely, in [@Da], the author proved:
\[Dafni 1\] The space $h^1({{\mathbb R}}^d)$ is the dual of the space $vmo({{\mathbb R}}^d)$.
\[Dafni 2\] Suppose that $\{f_n\}_{n=1}^\infty$ is a bounded sequence in $h^1(\mathbb R^d)$, and that $\lim_{n\to\infty} f_n(x) = f(x)$ for almost every $x\in\mathbb R^d$. Then, $f\in h^1(\mathbb R^d)$ and $\{f_n\}_{n= 1}^\infty$ weak$^*$-converges to $f$, that is, for every $\phi\in vmo(\mathbb R^d)$, we have $$\lim_{n\to\infty} \int_{\mathbb R^d} f_n(x) \phi(x)dx = \int_{\mathbb R^d} f(x) \phi(x) dx.$$
The aim of the present paper is to give very simple proofs of the two above theorems. It should be pointed out that our method is different from that of Dafni and it can be generalized to the setting of spaces of homogeneous type (see [@Ky4]).
To this end, we first recall some definitions of the function spaces. As usual, $\mathcal S(\mathbb R^d)$ denotes the Schwartz class of test functions on $\mathbb R^d$. The subspace $\mathcal A$ of $\mathcal S(\mathbb R^d)$ is then defined by $$\mathcal A=\Big\{\phi\in \mathcal S(\mathbb R^d): |\phi(x)|+ |\nabla\phi(x)|\leq (1+ |x|^2)^{-(d+1)}\Big\},$$ where $\nabla= (\partial/\partial x_1,..., \partial/\partial x_d)$ denotes the gradient. We define $$\mathfrak M f(x):= \sup\limits_{\phi\in\mathcal A}\sup\limits_{|y-x|<t}|f*\phi_t(y)|\quad\mbox{and}\quad \mathfrak mf(x):= \sup\limits_{\phi\in\mathcal A}\sup\limits_{|y-x|<t<1}|f*\phi_t(y)|,$$ where $\phi_t(\cdot)= t^{-d}\phi(t^{-1}\cdot)$. The space $H^1(\mathbb R^d)$ is the space of all integrable functions $f$ such that $\mathfrak M f\in L^1(\mathbb R^d)$ equipped with the norm $\|f\|_{H^1}= \|\mathfrak M f\|_{L^1}$. The space $h^1(\mathbb R^d)$ denotes the space of all integrable functions $f$ such that $\mathfrak m f\in L^1(\mathbb R^d)$ equipped with the norm $\|f\|_{h^1}= \|\mathfrak m f\|_{L^1}$.
We remark that the local real Hardy space $h^1(\mathbb R^d)$, first introduced by Goldberg [@Go], is larger than $H^1(\mathbb R^d)$ and allows more flexibility, since global cancellation conditions are not necessary. For example, the Schwartz class $\mathcal S({{\mathbb R}}^d)$ is contained in $h^1(\mathbb R^d)$ but not in $H^1(\mathbb R^d)$, and multiplication by cutoff functions preserves $h^1(\mathbb R^d)$ but not $H^1(\mathbb R^d)$. Thus it makes $h^1(\mathbb R^d)$ more suitable for working in domains and on manifolds.
It is well-known (see [@Fef]) that the dual space of $H^1(\mathbb R^d)$ is $BMO(\mathbb R^d)$ the space of all locally integrable functions $f$ with $$\|f\|_{BMO}:=\sup\limits_{B}\frac{1}{|B|}\int_B \Big|f(x)-\frac{1}{|B|}\int_B f(y) dy\Big|dx<\infty,$$ where the supremum is taken over all balls $B\subset {{\mathbb R}}^d$. It was also shown in [@Go] that the dual space of $h^1(\mathbb R^d)$ can be identified with the space $bmo(\mathbb R^d)$, consisting of locally integrable functions $f$ with $$\|f\|_{bmo}:= \sup\limits_{|B|\leq 1}\frac{1}{|B|}\int_B \Big|f(x)-\frac{1}{|B|}\int_B f(y) dy\Big|dx+ \sup\limits_{|B|\geq 1}\frac{1}{|B|}\int_B |f(x)|dx<\infty,$$ where the supremums are taken over all balls $B\subset {{\mathbb R}}^d$.
It is clear that, for any $f\in H^1({{\mathbb R}}^d)$ and $g\in bmo({{\mathbb R}}^d)$, $$\|f\|_{h^1} \leq \|f\|_{H^1}\quad\mbox{and}\quad \|g\|_{BMO} \leq \|g\|_{bmo}.$$
Recall that the space $VMO(\mathbb R^d)$ (resp., $vmo(\mathbb R^d)$) is the closure of $C_c(\mathbb R^d)$ in $(BMO(\mathbb R^d),\|\cdot\|_{BMO})$ (resp., $(bmo(\mathbb R^d),\|\cdot\|_{bmo})$). The following theorem is due to Coifman and Weiss [@CW].
\[Coifman-Weiss\] The space $H^1({{\mathbb R}}^d)$ is the dual of the space $VMO({{\mathbb R}}^d)$.
Throughout the whole paper, $C$ denotes a positive geometric constant which is independent of the main parameters, but may change from line to line.
Proof of Theorems \[Dafni 1\] and \[Dafni 2\]
=============================================
In this section, we fix $\varphi\in C_c({{\mathbb R}}^d)$ with supp $\varphi\subset B(0,1)$ and $\int_{{{\mathbb R}}^d} \varphi(x) dx=1$. Let $\psi:= \varphi*\varphi$. The following lemma is due to Goldberg [@Go].
\[Golberg\] There exists a positive constant $C=C(d,\varphi)$ such that
[i)]{} for any $f\in L^1({{\mathbb R}}^d)$, $$\|\varphi*f\|_{h^1}\leq C \|f\|_{L^1};$$
[ii)]{} for any $g\in h^1({{\mathbb R}}^d)$, $$\|g- \psi*g\|_{H^1}\leq C \|g\|_{h^1}.$$
As a consequence of Lemma \[Golberg\](ii), for any $\phi\in C_c({{\mathbb R}}^d)$, $$\label{from big bmo to small bmo}
\|\phi - \overline{\psi}*\phi\|_{bmo} \leq C \|\phi\|_{BMO},$$ here and hereafter, $\overline{\psi}(x):= \psi(-x)$ for all $x\in {{\mathbb R}}^d$.
Since $vmo(\mathbb R^d)$ is a subspace of $bmo(\mathbb R^d)$, which is the dual space of $h^1(\mathbb R^d)$, every function $f$ in $h^1(\mathbb R^d)$ determines a bounded linear functional on $vmo(\mathbb R^d)$ of norm bounded by $\|f\|_{h^1}$.
Conversely, given a bounded linear functional $L$ on $vmo(\mathbb R^d)$. Then, $$|L(\phi)|\leq \|L\| \|\phi\|_{vmo}\leq \|L\| \|\phi\|_{L^\infty}$$ for all $\phi\in C_c(\mathbb R^d)$. This implies (see [@Ro]) that there exists a finite signed Radon measure $\mu$ on ${{\mathbb R}}^d$ such that, for any $\phi\in C_c(\mathbb R^d)$, $$L(\phi)= \int_{{{\mathbb R}}^d} \phi(x) d\mu(x),$$ moreover, the total variation of $\mu$, $|\mu|({{\mathbb R}}^d)$, is bounded by $\|L\|$. Therefore, $$\label{Dafni 1, 1}
\|\psi*\mu\|_{h^1} = \|\varphi*(\varphi*\mu)\|_{h^1} \leq C \|\varphi*\mu\|_{L^1}\leq C |\mu|({{\mathbb R}}^d)\leq C \|L\|$$ by Lemma \[Golberg\]. On the other hand, by (\[from big bmo to small bmo\]), we have $$\begin{aligned}
|(L-\psi*L)(\phi)|=|L(\phi - \overline{\psi}*\phi)|&\leq& \|L\| \|\phi - \overline{\psi}*\phi\|_{vmo}\\
&\leq& C \|L\| \|\phi\|_{BMO}\end{aligned}$$ for all $\phi\in C_c({{\mathbb R}}^d)$. Consequently, by Theorem \[Coifman-Weiss\], there exists a function $h$ belongs $H^1({{\mathbb R}}^d)$ such that $\|h\|_{H^1}\leq C \|L\|$ and $$(L-\psi*L)(\phi)= \int_{{{\mathbb R}}^d} h(x) \phi(x) dx$$ for all $\phi\in C_c({{\mathbb R}}^d)$. This, together with (\[Dafni 1, 1\]), allows us to conclude that $$L(\phi)= \int_{{{\mathbb R}}^d} f(x) \phi(x) dx$$ for all $\phi\in C_c({{\mathbb R}}^d)$, where $f:= h+ \psi*\mu\in h^1({{\mathbb R}}^d)$ satisfying $\|f\|_{h^1}\leq \|h\|_{H^1} + \|\psi*\mu\|_{h^1}\leq C \|L\|$. The proof of Theorem \[Dafni 1\] is thus completed.
Let $\{f_{n_k}\}_{k=1}^\infty$ be an arbitrary subsequence of $\{f_n\}_{n=1}^\infty$. As $\{f_{n_k}\}_{k=1}^\infty$ is a bounded sequence in $h^1({{\mathbb R}}^d)$, by Theorem \[Dafni 1\] and the Banach-Alaoglu theorem, there exists a subsequence $\{f_{n_{k_j}}\}_{j=1}^\infty$ of $\{f_{n_k}\}_{k=1}^\infty$ such that $\{f_{n_{k_j}}\}_{j=1}^\infty$ weak$^*$-converges to $g$ for some $g\in h^1({{\mathbb R}}^d)$. Therefore, for any $x\in {{\mathbb R}}^d$, $$\lim_{j\to\infty} \int_{{{\mathbb R}}^d} f_{n_{k_j}}(y) \psi(x-y) dy = \int_{{{\mathbb R}}^d} g(y) \psi(x-y) dy.$$ This implies that $\lim_{j\to\infty}[f_{n_{k_j}}(x) - (f_{n_{k_j}}*\psi)(x)]= f(x) - (g*\psi)(x)$ for almost every $x\in\mathbb R^d$. Hence, by Lemma \[Golberg\](ii) and the Jones-Journé’s theorem (see [@JJ]), $$\|f- g*\psi\|_{H^1}\leq \sup_{j\geq 1}\|f_{n_{k_j}} - f_{n_{k_j}}*\psi\|_{H^1}\leq C \sup_{j\geq 1} \|f_{n_{k_j}}\|_{h^1}<\infty,$$ moreover, $$\lim_{j\to\infty} \int_{\mathbb R^d} [f_{n_{k_j}}(x) - (f_{n_{k_j}}*\psi)(x)]\phi(x)dx = \int_{\mathbb R^d} [f(x) - (g*\psi)(x)]\phi(x) dx$$ for all $\phi\in C_c(\mathbb R^d)$. As a consequence, we obtain that $$\begin{aligned}
\|f\|_{h^1} \leq \|f- g*\psi\|_{h^1} + \|g*\psi\|_{h^1}&\leq& \|f- g*\psi\|_{H^1} + C\|g\|_{h^1}\\
&\leq& C \sup_{j\geq 1} \|f_{n_{k_j}}\|_{h^1}<\infty,\end{aligned}$$ moreover, $$\begin{aligned}
&&\lim_{j\to\infty} \int_{\mathbb R^d} f_{n_{k_j}}(x)\phi(x)dx \\
&=& \lim_{j\to\infty} \int_{\mathbb R^d} [f_{n_{k_j}}(x) - (f_{n_{k_j}}*\psi)(x)]\phi(x)dx + \lim_{j\to\infty} \int_{\mathbb R^d} f_{n_{k_j}}(x) (\overline\psi*\phi)(x)dx\\
&=& \int_{\mathbb R^d} [f(x) - (g*\psi)(x)]\phi(x) dx + \int_{\mathbb R^d} g(x) (\overline\psi*\phi)(x)dx\\
&=& \int_{\mathbb R^d} f(x)\phi(x)dx\end{aligned}$$ since $\{f_{n_{k_j}}\}_{j=1}^\infty$ weak$^*$-converges to $g$ in $h^1(\mathbb R^d)$. This, by $\{f_{n_k}\}_{k=1}^\infty$ be an arbitrary subsequence of $\{f_n\}_{n=1}^\infty$, allows us to complete the proof of Theorem \[Dafni 2\].
[MTW1]{}
R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645.
R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces. J. Math. Pures Appl. (9) 72 (1993), no. 3, 247–286.
G. Dafni, Local VMO and weak convergence in $h^1$. Canad. Math. Bull. 45 (2002), no. 1, 46–59.
C. Fefferman, Characterizations of bounded mean oscillation. Bull. Amer. Math. Soc. 77 (1971), no. 4, 587–588.
D. Goldberg, A local version of real Hardy spaces. Duke J. Math. 46 (1979), 27–42.
P. W. Jones and J-L. Journé, On weak convergence in $H^1({\bf R}^d)$. Proc. Amer. Math. Soc. 120 (1994), no. 1, 137–138.
L. D. Ky, Bilinear decompositions and commutators of singular integral operators. Trans. Amer. Math. Soc. 365 (2013), no. 6, 2931–2958.
L. D. Ky, On weak$^*$-convergence in $H^1_L(\mathbb R^d)$. Potential Anal. 39 (2013), no. 4, 355–368.
L. D. Ky, Endpoint estimates for commutators of singular integrals related to Schrödinger operators, Rev. Mat. Iberoam. (to appear) or arXiv:1203.6335.
L. D. Ky, On weak$^*$-convergence in $H^1_\rho(\mathcal X)$, preprint.
H. L. Royden, Real analysis. Third edition. Macmillan Publishing Company, New York, 1988.
[^1]: The paper was completed when the third author was visiting to Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the VIASM for financial support and hospitality.\
$^{*}$Corresponding author: Luong Dang Ky
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
In this paper, we apply the Hausdorff measure of noncompactness to obtain the necessary and sufficient conditions for certain matrix operators on the Fibonacci difference sequence spaces $\ell_{p}(\widehat{F})$ and $\ell
_{\infty}(\widehat{F})$ to be compact, where $1\leq p<\infty$.
address:
- 'DEPARTMENT OF MATHEMATICS, DÜZCE UNIVERSITY, 81620, DÜZCE, TURKEY'
- 'DEPARTMENT OF MATHEMATICS, SAKARYA UNIVERSITY, 54187, SAKARYA, TURKEY'
- 'DEPARTMENT OF MATHEMATICS, ALIGARH MUSLIM UNIVERSITY, 202002, ALIGARH, INDIA'
author:
- Emrah Evren KARA
- Metİn Başarir
- 'M. Mursaleen'
title: Compactness of matrix operators on some sequence spaces derived by Fibonacci numbers
---
**Introduction and preliminaries**
==================================
Let $\mathbb{N}
=\{0,1,2,...\}$ and $\mathbb{R}
$ be the set of all real numbers. We shall write $\lim_{k}$, $\sup_{k}$, $\inf_{k}$ and $\sum_{k}$ instead of $\lim_{k\rightarrow \infty}$, $\sup_{k\in\mathbb{N}
}$, $\inf_{k\in\mathbb{N}
}$ and $\sum_{k=0}^{\infty}$, respectively. Let $\omega$ be the vector space of all real sequences $x=(x_{k})_{k\in\mathbb{N}
}$. By the term $\mathit{sequence}$ $\mathit{space}$, we shall mean any linear subspace of $\omega$. Let $\varphi,$ $\ell_{\infty},$ $c$ and $c_{0}$ denote the sets of all finite, bounded, convergent and null sequences, respectively. We write $\ell_{p}=\{x\in \omega:\sum_{k}\left \vert
x_{k}\right \vert ^{p}<\infty \}$ for $1\leq p<\infty.$ Also, we shall use the conventions that $e=(1,1,...)$ and $e^{(n)}$ is the sequence whose only non-zero term is $1$ in the $n^{\text{th}}$ place for each $n\in\mathbb{N}
$. For any sequence $x=(x_{k})$, let $x^{[n]}=\sum_{k=0}^{n}x_{k}e^{(k)}$ be its $n$-section. Morever, we write $bs$ and $cs$ for the sets of sequences with bounded and convergent partial sums, respectively.
The $\mathit{Fibonacci}$ $\mathit{numbers}$ are the sequence of numbers $\{f_{n}\}_{n=0}^{\infty}$ defined by the linear recurrence equations $$f_{0}=f_{1}=1\text{ and }f_{n}=f_{n-1}+f_{n-2}\text{; \ }n\geq2.$$
Fibonacci numbers have many interesting properties and applications in arts, sciences and architecture. For example, the ratio sequences of Fibonacci numbers converge to the golden ratio which is important in sciences and arts. Also, some basic properties of Fibonacci numbers can be found in \[2\].
A $\mathit{B-space}$ ** is a complete normed space. A topological sequence space in which all coordinate functionals $\pi_{k}$, $\pi
_{k}(x)=x_{k}$, are continuous is called a $\mathit{K-}$ $\mathit{space}$. A $\mathit{BK-}$ $\mathit{space}$ is defined as a $K-$ space which is also a $B-$ space, that is, a $BK-$ space is a Banach space with continuous coordinates. A $BK-$ space $X\supset \varphi$ is said to have $AK$ if every sequence $x=(x_{k})\in X$ has a unuqiue representation $x=\sum_{k}x_{k}e^{(k)}$. For example, the space $\ell_{p}$ $(1\leq p<\infty)$ is $BK-$ space with $\left \Vert x\right \Vert _{p}=\left( \sum_{k}\left \vert x_{k}\right \vert
^{p}\right) ^{1/p}$ and $c_{0}$, $c$ and $\ell_{\infty}$ are $BK-$ spaces with $\left \Vert x\right \Vert _{\infty}=\sup_{k}\left \vert x_{k}\right \vert $. Further, the $BK-$ spaces $c_{0}$ and $\ell_{p}$ have $AK$, where $1\leq
p<\infty$ (cf. \[3,4\].
A sequence $(b_{n})$ in a normed space $X$ is called a $\mathit{Schauder}$ ** $\mathit{basis}$ for $X$ if for every $x\in X$ there is a unique sequence $(\alpha_{n})$ of scalars such that $x=\sum_{n}\alpha_{n}b_{n}$, i.e., $\lim_{m}\left \Vert x-\sum_{n=0}^{m}\alpha_{n}b_{n}\right \Vert =0.$
The $\beta$-dual of a sequence space $X$ is defined by $$X^{\beta}=\left \{ a=(a_{k})\in \omega:ax=(a_{k}x_{k})\in cs\text{ for all
}x=(x_{k})\in X\right \} .$$
Let $A=(a_{nk})_{n,k=0}^{\infty}$ be an infinite matrix of real numbers $a_{nk}$, where $n,k\in\mathbb{N}
$. We write $A_{n}$ for the sequence in the $n^{\text{th}}$ row of $A$, that is $A_{n}=(a_{nk})_{k=0}^{\infty}$ for every $n\in\mathbb{N}
$. In addition, if $x=(x_{k})_{k=0}^{\infty}\in \omega$ then we define the $A$*-*$\mathit{transform}$ $\mathit{of}$ $x$ as the sequence $Ax=\left \{ A_{n}(x)\right \} _{n=0}^{\infty}$, where $$A_{n}(x)={\displaystyle \sum_{k=0}^{\infty}}
a_{nk}x_{k};\text{ \ \ }\left( n\in\mathbb{N}
\right) \tag{1.1}$$ provided the series on the right side converges for each $n\in\mathbb{N}
.$
For arbitrary subsets $X$ and $Y$ of $\omega$, we write $\left( X,Y\right) $ for the class of all infinite matrices that map $X$ into $Y$. Thus, $A\in \left( X,Y\right) $ if and only if $A_{n}\in X^{\beta}$ for all $n\in\mathbb{N}
$ and $Ax\in Y$ for all $x\in X$.
The matrix domain $X_{A}$ of an infinite matrix $A$ in sequence space $X$ is defined by$$X_{A}=\left \{ x=\left( x_{k}\right) \in \omega:Ax\in X\right \} \tag{1.2}$$ which is a sequence space.
Let $\Delta$ denotes the matrix $\Delta=(\Delta_{nk})$ defined by$$\Delta_{nk}=\left \{
\begin{array}
[c]{cc}(-1)^{n-k} & (n-1\leq k\leq n)\\
0 & (0\leq k<n-1)\text{ or\ }(k>n)
\end{array}
\right.$$ or$$\Delta_{nk}=\left \{
\begin{array}
[c]{cc}(-1)^{n-k} & (n\leq k\leq n+1)\\
0 & (0\leq k<n)\text{ or\ }(k>n+1).
\end{array}
\right.$$
In the literature, the matrix domain $\lambda_{\Delta}$ is called the $\mathit{difference}$ ** $\mathit{sequence}$ ** $\mathit{space}$ whenever $\lambda$ is a normed or paranormed sequence space. The idea of difference sequence space was introduced by Kizmaz \[5\]. In 1981, Kizmaz \[5\] defined the sequence spaces$$X(\Delta)=\left \{ x=(x_{k})\in \omega:(x_{k}-x_{k+1})\in X\right \}$$ for $X=\ell_{\infty}$, $c$ and $c_{0}$. The difference space $bv_{p}$, consisting of all sequnces $(x_{k})$ such that $(x_{k}-x_{k-1})$ is in the sequence space $\ell_{p}$, was studied in the case $0<p<1$ by Altay and Başar \[6\] and in the case $1\leq p\leq \infty$ by Başar and Altay \[7\] and Çolak et al. \[8\]. The paranormed difference sequence space$$\Delta \lambda(p)=\{x=\left( x_{k}\right) \in \omega:(x_{k}-x_{k+1})\in
\lambda(p)\}$$ was examined by Ahmad and Mursaleen \[9\] and Malkowsky \[10\], where $\lambda(p)$ is any of the paranormed spaces $\ell_{\infty}(p)$, $c(p)$ and $c_{0}(p)$ defined by Simons \[11\] and Maddox \[12\].
Recently, Başar et al. \[13\] have defined the sequence spaces $bv(u,p)$ and $bv_{\infty}(u,p)$ by$$bv(u,p)=\{x=\left( x_{k}\right) \in \omega:\sum_{k}\left \vert u_{k}(x_{k}-x_{k-1})\right \vert ^{p_{k}}<\infty \}$$ and $$bv_{\infty}(u,p)=\{x=\left( x_{k}\right) \in \omega:\sup_{k\in\mathbb{N}
}\left \vert u_{k}(x_{k}-x_{k-1})\right \vert ^{p_{k}}<\infty \},$$ where $u=(u_{k})$ is an arbitrary fixed sequence and $0<p_{k}\leq H<\infty$ for all $k\in\mathbb{N}
.$ Also in \[14-23\], authors studied some difference sequence spaces.
Let $S_{X}$ denote the unit sphere in a normed linear space $X$. If $X\supset \varphi$ is a $BK$ space and $a=(a_{k})\in \omega$, then we write $$\left \Vert a\right \Vert _{X}^{\ast}=\sup_{x\in S_{X}}\left \vert \sum
\limits_{k}a_{k}x_{k}\right \vert \tag{1.3}$$ provided the expression on the right side is defined and finite which is the case whenever $a\in X^{\beta}$.
The following results are very important in our study.
\[[\[3, Theorem 1.29\]]{}\]Let $1<p<\infty$ and $q=p/(p-1)$. Then, we have $\ell_{\infty}^{\beta}=\ell_{1}$, $\ell_{1}^{\beta}=\ell_{\infty}$ and $\ell_{p}^{\beta}=\ell_{q}$. Furthermore, let $X$ denote any of the spaces $\ell_{\infty},$ $\ell_{1}$ or $\ell_{p}$. Then, we have $\left \Vert
a\right \Vert _{X}^{\ast}=\left \Vert a\right \Vert _{X^{\beta}}$ for all $a\in
X^{\beta}$, where $\left \Vert .\right \Vert _{X^{\beta}}$ is the natural norm on the dual space $X^{\beta}$.
\[[\[3, Theorem 1.23 (a)\]]{}\]Let $X$ and $Y$ be $BK$-spaces. Then we have $(X,Y)\subset B(X,Y)$, that is, every matrix $A\in(X,Y)$ defines a linear operator $L_{A}\in B(X,Y)$ by $L_{A}(x)=Ax$ for all $x\in X$, where $B(X,Y)$ denotes the set all bounded (continuous) linear operators $L:X\rightarrow Y.$
\[[\[3, Lemma 2.2\]]{}\]Let $X\supset \phi$ be $BK$-space and $Y$ be any of the spaces $c_{0},$ $c$ or $\ell_{\infty}$. If $A\in(X,Y)$, then $$\left \Vert L_{A}\right \Vert =\left \Vert A\right \Vert _{(X,\ell_{\infty})}=\underset{n}{\sup}\left \Vert A_{n}\right \Vert _{X}^{\ast}<\infty.$$
By $M_{X},$ we denote the collection of all bounded subsets of a metric space $\left( X,d\right) .$ If $Q\in M_{X},$ then the *Hausdorff measure of noncompactness* of the set $Q,$ denoted by $\chi \left( Q\right) ,$ is defined by $$\chi \left( Q\right) :=\inf \left \{ \varepsilon>0:Q\subset \underset
{i=1}{\overset{n}{\cup}}B\left( x_{i},r_{i}\right) ,\text{ }x_{i}\in
X,\text{ }r_{i}<\varepsilon \text{ }\left( i=1,2,...,n\right) ,\text{ }n\in\mathbb{N}
-\{0\} \right \} .$$ The function $\chi:M_{X}\rightarrow \left[ 0,\infty \right) $ is called the $\mathit{Hausdorff}$ ** $\mathit{measure}$ ** $\mathit{of}$ ** $\mathit{noncompactness}$.
The basic properties of the Hausdorff measure of noncompactness can be found in \[3\]
The following result gives an estimate for the Hausdorff measure of noncompactness in the $BK$ space $\ell_{p}$ for $1\leq p<\infty.$
\[[\[24, Theorem 2.8\]]{}\]Let $1\leq p<\infty$ and $Q\in M_{\ell_{p}}.$ If $P_{m}:\ell_{p}\rightarrow \ell_{p}$ $(m\in\mathbb{N}
)$ is the operator defined by $P_{m}(x)=(x_{0},x_{1},...,x_{m},0,0,...)$ for all $x=(x_{k})\in \ell_{p}$, then we have$$\chi(Q)=\lim_{m\rightarrow \infty}\left( \sup_{x\in Q}\left \Vert
(I-P_{m})(x)\right \Vert _{\ell_{p}}\right) ,$$ where $I$ is the identity operator on $\ell_{p}.$
Let $X$ and $Y$ be Banach spaces. Then, a linear operator $L:X\rightarrow Y$ is said to be $\mathit{compact}$ if the domain of $L$ is all of $X$ and $L(Q)$ is a totally bounded subset of $Y$ for every $Q\in M_{X}$. Equivalently, we say that $L$ is compact if its domain is all of $X$ and for every bounded sequence $\left( x_{n}\right) $ in $X,$ the sequence $\left( L\left(
x_{n}\right) \right) $ has a convergent subsequence in $Y.$
The idea of compact operators between Banach spaces is closely related to the Hausdorff measure of noncompactness, and it can be given as follows:
Let $X$ and $Y$ be Banach spaces and $L\in B(X,Y)$. Then, the Hausdorff measure of noncompactness of $L$, is denoted by $\left \Vert L\right \Vert
_{\chi}$, can be given by$$\left \Vert L\right \Vert _{\chi}=\chi(L(S_{X})) \tag{1.4}$$ and we have $$L\text{ is compact if and only if }\left \Vert L\right \Vert _{\chi}=0.
\tag{1.5}$$
The Hausdorff measure of noncompactness has various applications in the theory of sequence spaces, one of them is to obtain necessary and sufficient conditions for matrix operators between $BK$ spaces to be compact. Recently, several authors have studied compact operators on the sequence spaces and given very important results related to the Hausdorff measure of noncompactness of a linear operator. For example \[25-38\].
In this paper, we derive some identities for the Hausdorff measure of noncompactness on the Fibonacci difference sequence spaces $\ell_{p}(\widehat{F})$ and $\ell_{\infty}(\widehat{F})$ defined by Kara \[1\]. We also apply the Hausdorff measure of noncompactness to obtain the necessary and sufficient conditions for such operators to be compact.
** The Fibonacci Difference Sequence Spaces** $\ell
_{p}(\widehat{F})$ **and** $\ell_{\infty}(\widehat{F})$
=======================================================
Throughout, let $1\leq p\leq \infty$ and $q$ denote the conjugate of $p$, that is, $q=p/(p-1)$ for $1<p<\infty$, $q=\infty$ for $p=1$ or $q=1$ for $p=\infty.$
Recently, Kara\[1\] has defined the Fibonacci difference sequence spaces $\ell_{p}(\widehat{F})$ and $\ell_{\infty}(\widehat{F})$ by$$\ell_{p}(\widehat{F})=\left \{ x=(x_{n})\in \omega:{\displaystyle \sum \limits_{n}}
\left \vert \frac{f_{n}}{f_{n+1}}x_{n}-\frac{f_{n+1}}{f_{n}}x_{n-1}\right \vert
^{p}<\infty \right \} ;\text{ }1\leq p<\infty$$ and$$\ell_{\infty}(\widehat{F})=\left \{ x=(x_{n})\in \omega:\sup_{n\in\mathbb{N}
}\left \vert \frac{f_{n}}{f_{n+1}}x_{n}-\frac{f_{n+1}}{f_{n}}x_{n-1}\right \vert
<\infty \right \} .\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$$
With the notation of (1.2), the sequence spaces $\ell_{p}(\widehat{F})$ and $\ell_{\infty}(\widehat{F})$ may be redefined by $$\ell_{p}(\widehat{F})=(\ell_{p})_{\widehat{F}}\text{ }(1\leq p<\infty)\text{
\ and \ }\ell_{\infty}(\widehat{F})=(\ell_{\infty})_{\widehat{F}}, \tag{2.1}$$ where the matrix $\widehat{F}=(\widehat{f}_{nk})$ is defined by$$\widehat{f}_{nk}=\left \{
\begin{array}
[c]{cc}-\frac{f_{n+1}}{f_{n}} & (k=n-1)\\
\frac{f_{n}}{f_{n+1}} & (k=n)\\
0 & (0\leq k<n-1)\text{ or }(k>n)
\end{array}
\right. ;\text{ }(n,k\in\mathbb{N}
). \tag{2.2}$$ Further, it is clear that the spaces $\ell_{p}(\widehat{F})$ and $\ell
_{\infty}(\widehat{F})$ are $BK$ spaces with the norms given by $$\left \Vert x\right \Vert _{\ell_{p}(\widehat{F})}=\left( \sum_{n}\left \vert
y_{n}(x)\right \vert ^{p}\right) ^{1/p}\text{ ; \ }(1\leq p<\infty)\text{
\ and }\left \Vert x\right \Vert _{\ell_{\infty}(\widehat{F})}=\sup_{n\in\mathbb{N}
}\left \vert y_{n}(x)\right \vert , \tag{2.3}$$ where the sequence $y=(y_{n})=(\widehat{F}_{n}(x))$ is the $\widehat{F}$-transform of a sequence $x=(x_{n})$, i.e., $$y_{n}=\widehat{F}_{n}(x)=\left \{
\begin{array}
[c]{cc}\frac{f_{0}}{f_{1}}x_{0}=x_{0}\text{ \ \ \ \ } & (n=0)\\
\frac{f_{n}}{f_{n+1}}x_{n}-\frac{f_{n+1}}{f_{n}}x_{n-1} & (n\geq1)
\end{array}
\right. \text{ };\text{ }(n\in\mathbb{N}
). \tag{2.4}$$
Moreover, it is obvious by (2.2) that $\widehat{F}$ is a triangle. Thus, it has a unique inverse $\widehat{F}^{-1}$ which is also a triangle and the entries of $\widehat{F}^{-1}$ are given by$$\widehat{f}_{nk}^{-1}=\left \{
\begin{array}
[c]{cc}\frac{f_{n+1}^{2}}{f_{k}f_{k+1}} & (0\leq k\leq n)\\
0 & (k>n)
\end{array}
\right. \tag{2.5}$$ for all $n,k\in\mathbb{N}
$. Therefore, we have by (2.4) that$$x_{n}=\sum_{k=0}^{n}\frac{f_{n+1}^{2}}{f_{k}f_{k+1}}y_{k}\text{ ; }(n\in\mathbb{N}
). \tag{2.6}$$
In \[1\], the $\beta$-duals of the sequence spaces $\ell_{p}(\widehat{F})$ $(1\leq p<\infty)$ and $\ell_{\infty}(\widehat{F})$ have been determined and some related matrix classes characterized. Now, by taking into account that the inverse of $\widehat{F}$ is given by (2.5), we have the following lemma which is immediate by \[1, Theorem 4.6\].
Let $1\leq p\leq \infty$. If $a=(a_{k})\in \{ \ell_{p}(\widehat{F})\}^{\beta}$, then $\bar{a}=(\bar{a}_{k})\in \ell_{q}$ and we have $$\sum \limits_{k}a_{k}x_{k}=\sum \limits_{k}\bar{a}_{k}y_{k} \tag{2.7}$$ for all $x=(x_{k})\in \ell_{p}(\widehat{F})$ with $y=\widehat{F}x$, where $$\bar{a}_{k}={\displaystyle \sum \limits_{j=k}^{\infty}}
\frac{f_{j+1}^{2}}{f_{k}f_{k+1}}a_{j}\text{ ; }(k\in\mathbb{N}
). \tag{2.8}$$
Now, we prove the following results which will be needed in the sequel.
Let $1<p<\infty,$ $q=p/(p-1)$ and $\bar{a}=(\bar{a}_{k})$ be the sequence defined by (2.8) Then, we have
\(a) If $a=(a_{k})\in \{ \ell_{\infty}(\widehat{F})\}^{\beta}$, then $\left \Vert
a\right \Vert _{\ell_{\infty}(\widehat{F})}^{\ast}=\sum_{k}\left \vert \bar
{a}_{k}\right \vert <\infty.$
\(b) If $a=(a_{k})\in \{ \ell_{1}(\widehat{F})\}^{\beta}$, then $\left \Vert
a\right \Vert _{\ell_{1}(\widehat{F})}^{\ast}=\sup_{k}\left \vert \bar{a}_{k}\right \vert <\infty.$
\(c) If $a=(a_{k})\in \{ \ell_{p}(\widehat{F})\}^{\beta}$, then $\left \Vert
a\right \Vert _{\ell_{p}(\widehat{F})}^{\ast}=\left( \sum_{k}\left \vert
\bar{a}_{k}\right \vert ^{q}\right) ^{1/q}<\infty.$
\(a) Let $a=(a_{k})\in \{ \ell_{\infty}(\widehat{F})\}^{\beta}$. Then, it follows by Lemma 2.1 that $\bar{a}=(\bar{a}_{k})\in \ell_{1}$ and the equality (2.7) holds for all sequences $x=(x_{k})\in \ell_{\infty}(\widehat{F})$ and $y=(y_{k})\in \ell_{\infty}$ which are connected by the relation $y=\widehat
{F}x$. Further, we have by (2.3) that $x\in S_{\ell_{\infty}(\widehat{F})}$ if and only if $y\in S_{\ell_{\infty}}$. Therefore, we derive from (1.3) and (2.7) that$$\left \Vert a\right \Vert _{\ell_{\infty}(\widehat{F})}^{\ast}=\sup_{x\in
S_{\ell_{\infty}(\widehat{F})}}\left \vert \sum \limits_{k}a_{k}x_{k}\right \vert
=\sup_{y\in S_{\ell_{\infty}}}\left \vert \sum \limits_{k}\bar{a}_{k}y_{k}\right \vert =\left \Vert \bar{a}\right \Vert _{\ell_{\infty}}^{\ast}.$$ Hence, by using Lemma 1.1, we have that$$\left \Vert a\right \Vert _{\ell_{\infty}(\widehat{F})}^{\ast}=\left \Vert
\bar{a}\right \Vert _{\ell_{\infty}}^{\ast}=\left \Vert \bar{a}\right \Vert
_{\ell_{1}}.$$ This completes the proof of part (a).
Since parts (b) and (c) can also be proved by analogy with part (a), we leave the detailed proof to the reader.
Throughout this paper, if $A=(a_{nk})$ is an infinite matrix, we define the associated matrix defined $\bar{A}=(\bar{a}_{nk})$ by $$\bar{a}_{nk}={\displaystyle \sum \limits_{j=k}^{\infty}}
\frac{f_{j+1}^{2}}{f_{k}f_{k+1}}a_{nj}\text{; }(n,k\in\mathbb{N}
) \tag{2.9}$$ provided the series on the right side converges for all $n,k\in\mathbb{N}
$ which is the case whenever $A_{n}\in \{ \ell_{p}(\widehat{F})\}^{\beta}$ for all $n\in\mathbb{N}
$, where $1\leq p\leq \infty$. Then, we have:
Let $1\leq p\leq \infty$, $X$ be a sequence space and $A=(a_{nk})$ be an infinite matrix. If $A\in(\ell_{p}(\widehat{F}),X)$, then $\bar{A}\in(\ell
_{p},X)$ such that $Ax=\bar{A}y$ for all $x\in \ell_{p}(\widehat{F})$ with $y=\widehat{F}x,$ where $\bar{A}=(\bar{a}_{nk})$ is the associated matrix defined by (2.9).
Suppose that $A\in(\ell_{p}(\widehat{F}),X)$ and let $x\in \ell_{p}(\widehat
{F})$. Then $A_{n}\in \{ \ell_{p}(\widehat{F})\}^{\beta}$ for all $n\in\mathbb{N}
$. Thus, it follows by Lemma 2.1 that $\bar{A}_{n}\in \ell_{q}$ for all $n\in\mathbb{N}
$ and the equality $Ax=\bar{A}y$ holds which yields that $\bar{A}y\in X$, where $y=\widehat{F}x.$ Since every $y\in \ell_{p}$ is the assocaited sequence of some $x\in \ell_{p}(\widehat{F})$, we obtain that $\bar{A}\in(\ell_{p},X)$. This concludes the proof.
Let $1\leq p<\infty.$ If $A\in(\ell_{1}(\widehat{F}),\ell_{p})$, then $$\left \Vert L_{A}\right \Vert =\left \Vert A\right \Vert _{(\ell_{1}(\widehat
{F}),\ell_{p})}=\sup_{k}\left(
{\displaystyle \sum \limits_{n}}
\left \vert \bar{a}_{nk}\right \vert ^{p}\right) ^{1/p}<\infty.$$
The proof is elementary and left to the reader.
**Compact Operators on the spaces** $\ell_{p}(\widehat{F})$** and** $\ell_{\infty}(\widehat{F})$
================================================================================================
In this section, we give some classes of compact operators on the spaces $\ell_{p}(\widehat{F})$ and $\ell_{\infty}(\widehat{F}),$ where $1\leq
p<\infty$.
The following lemma gives necessary and sufficient conditions for a matrix transformation from a $BK$ space $X$ to $c_{0}$, $c$ and $\ell_{\infty}$ to be compact (the only sufficient condition for $\ell_{\infty}$).
\[[\[25, Theorem 3.7\]]{}\]Let $X\supset \varphi$ be a BK space. Then, we have
\(a) If $A\in(X,\ell_{\infty})$, then$$0\leq \left \Vert L_{A}\right \Vert _{\chi}\leq \underset{n}{\lim \sup}\left \Vert
A_{n}\right \Vert _{X}^{\ast}.$$
\(b) If $A\in(X,c_{0})$, then$$\left \Vert L_{A}\right \Vert _{\chi}=\underset{n}{\lim \sup}\left \Vert
A_{n}\right \Vert _{X}^{\ast}.$$
\(c) If $X$ has $AK$ or $X=\ell_{\infty}$ and $A\in(X,c)$, then $$\frac{1}{2}.\underset{n}{\lim \sup}\left \Vert A_{n}-\alpha \right \Vert
_{X}^{\ast}\leq \left \Vert L_{A}\right \Vert _{\chi}\leq \underset{n}{\lim \sup
}\left \Vert A_{n}-\alpha \right \Vert _{X}^{\ast},$$ where $\alpha=(\alpha_{k})$ with $\alpha_{k}=\lim_{n}a_{nk}$ for all $k\in\mathbb{N}
$.
Now, let $A=(a_{nk})$ be an infinite matrix and $\bar{A}=(\bar{a}_{nk})$ the associated matrix defined by (2.9). Then, by combining Lemmas 2.2, 2.3 and 3.1, we have the following result:
Let $1<p<\infty$ and $q=p/(p-1)$. Then we have:
\(a) If $A\in(\ell_{p}(\widehat{F}),\ell_{\infty})$, then$$0\leq \left \Vert L_{A}\right \Vert _{\chi}\leq \underset{n}{\lim \sup}\left(
{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}\right \vert ^{q}\right) ^{1/q} \tag{3.1}$$ and $$L_{A}\text{ is compact if }\underset{n}{\lim}\left(
{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}\right \vert ^{q}\right) ^{1/q}=0. \tag{3.2}$$
\(b) If $A\in(\ell_{p}(\widehat{F}),c_{0})$, then $$\left \Vert L_{A}\right \Vert _{\chi}=\underset{n}{\lim \sup}\left(
{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}\right \vert ^{q}\right) ^{1/q} \tag{3.3}$$ and $$L_{A}\text{ is compact if and only if }\underset{n}{\lim}\left(
{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}\right \vert ^{q}\right) ^{1/q}=0. \tag{3.4}$$
\(c) If $A\in(\ell_{p}(\widehat{F}),c)$, then$$\frac{1}{2}.\underset{n}{\lim \sup}\left(
{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}-\bar{\alpha}_{k}\right \vert ^{q}\right) ^{1/q}\leq \left \Vert L_{A}\right \Vert _{\chi}\leq \underset{n}{\lim \sup}\left(
{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}-\bar{\alpha}_{k}\right \vert ^{q}\right) ^{1/q}
\tag{3.5}$$ and $$L_{A}\text{ is compact if and only if }\underset{n}{\lim}\left(
{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}-\bar{\alpha}_{k}\right \vert ^{q}\right) ^{1/q}=0,
\tag{3.6}$$ where $\bar{\alpha}=(\bar{\alpha}_{k})$ with $\bar{\alpha}_{k}=\lim_{n}\bar
{a}_{nk}$ for all $k\in\mathbb{N}
$.
It is obvious that (3.2), (3.4) and (3.6) are respectively obtained from (3.1), (3.3) and (3.5) by using (1.5). Thus, we have to proof (3.1), (3.3) and (3.5).
Let $A\in(\ell_{p}(\widehat{F}),\ell_{\infty})$ or $A\in(\ell_{p}(\widehat
{F}),c_{0}).$ Since $A_{n}\in \{ \ell_{p}(\widehat{F})\}^{\beta}$ for all $n\in\mathbb{N}
$, we have from Lemma 2.2(c) that$$\left \Vert A_{n}\right \Vert _{\ell_{p}(\widehat{F})}^{\ast}=\left \Vert \bar
{A}_{n}\right \Vert _{\ell_{p}}^{\ast}=\left(
{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}\right \vert ^{q}\right) ^{1/q} \tag{3.7}$$ for all $n\in\mathbb{N}
$. Hence, by using the equation in (3.7), we get (3.1) and (3.3) from parts (a) and (b) of Lemma 3.1, respectively.
To prove (3.5), we have $A\in(\ell_{p}(\widehat{F}),c)$ and hence $\bar{A}\in(\ell_{p},c)$ by Lemma 2.3. Therefore, it follows by part (c) of Lemma 3.1 with Lemma 1.1 that $$\frac{1}{2}.\underset{n}{\lim \sup}\left \Vert \bar{A}_{n}-\bar{\alpha
}\right \Vert _{\ell_{q}}\leq \left \Vert L_{\bar{A}}\right \Vert _{\chi}\leq \underset{n}{\lim \sup}\left \Vert \bar{A}_{n}-\bar{\alpha}\right \Vert
_{\ell_{q}}, \tag{3.8}$$ where $\bar{\alpha}=(\bar{\alpha}_{k})$ with $\bar{\alpha}_{k}=\lim_{n}\bar
{a}_{nk}$ for all $k\in\mathbb{N}
$.
Now, let us write $S=S_{\ell_{p}(\widehat{F})}$ and $\bar{S}=S_{\ell_{p}},$ for short. Then, we obtain by (1.4) and Lemma 1.2 that$$\left \Vert L_{A}\right \Vert _{\chi}=\chi(L_{A}(S))=\chi(AS) \tag{3.9}$$ and$$\left \Vert L_{\bar{A}}\right \Vert _{\chi}=\chi(L_{\bar{A}}(\bar{S}))=\chi
(\bar{A}\bar{S}). \tag{3.10}$$
Further, we have by (2.3) that $x\in S$ if and only if $y\in \bar{S}$ and since $Ax=\bar{A}y$ by Lemma 2.3, we deduce that $AS=\bar{A}\bar{S}.$ This leads us with (3.9) and (3.10) to the consequence that $\left \Vert L_{A}\right \Vert
_{\chi}=\left \Vert L_{\bar{A}}\right \Vert _{\chi}.$ Hence, we get (3.5) from (3.8). This completes the proof.
The conclusions of Theorem 3.2 still hold for $\ell_{1}(\widehat{F})$ or $\ell_{\infty}(\widehat{F})$ instead of $\ell_{p}(\widehat{F})$ with $q=1,$ and on replacing the summations over $k$ by the supremums over $k$ in the case $\ell_{1}(\widehat{F})$. Then, we have the following results:
Let $\bar{A}=(\bar{a}_{nk})$ be the associated matrix defined by (2.9). Then we have
\(a) If $A\in(\ell_{\infty}(\widehat{F}),\ell_{\infty})$, then $$0\leq \left \Vert L_{A}\right \Vert _{\chi}\leq \underset{n}{\lim \sup}{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}\right \vert$$ and$$L_{A}\text{ is compact if }\underset{n}{\lim}{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}\right \vert =0.$$
\(b) If $A\in(\ell_{\infty}(\widehat{F}),c_{0})$, then $$\left \Vert L_{A}\right \Vert _{\chi}=\underset{n}{\lim \sup}{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}\right \vert$$ and$$L_{A}\text{ is compact if and only if }\underset{n}{\lim}{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}\right \vert =0.$$
\(c) If $A\in(\ell_{\infty}(\widehat{F}),c)$, then$$\frac{1}{2}.\underset{n}{\lim \sup}{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}-\bar{a}_{k}\right \vert \leq \left \Vert L_{A}\right \Vert
_{\chi}\leq \underset{n}{\lim \sup}{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}-\bar{a}_{k}\right \vert$$ and$$L_{A}\text{ is compact if and only if }\underset{n}{\lim}{\displaystyle \sum \limits_{k}}
\left \vert \bar{a}_{nk}-\bar{a}_{k}\right \vert =0,$$ where $\bar{\alpha}=(\bar{\alpha}_{k})$ with $\bar{\alpha}_{k}=\lim_{n}\bar
{a}_{nk}$ for all $k\in\mathbb{N}
$.
Let $\bar{A}=(\bar{a}_{nk})$ be the associated matrix defined by (2.9). Then we have
\(a) If $A\in(\ell_{1}(\widehat{F}),\ell_{\infty})$, then$$0\leq \left \Vert L_{A}\right \Vert _{\chi}\leq \underset{n}{\lim \sup}\left(
\sup_{k}\left \vert \bar{a}_{nk}\right \vert \right)$$ and$$L_{A}\text{ is compact if }\underset{n}{\lim}\left( \sup_{k}\left \vert
\bar{a}_{nk}\right \vert \right) =0.$$
\(b) If $A\in(\ell_{1}(\widehat{F}),c_{0})$, then $$\left \Vert L_{A}\right \Vert _{\chi}=\underset{n}{\lim \sup}\left( \sup
_{k}\left \vert \bar{a}_{nk}\right \vert \right)$$ and$$L_{A}\text{ is compact if and only if }\underset{n}{\lim}\left( \sup
_{k}\left \vert \bar{a}_{nk}\right \vert \right) =0.$$
\(c) If $A\in(\ell_{1}(\widehat{F}),c)$, then$$\frac{1}{2}.\underset{n}{\lim \sup}\left( \sup_{k}\left \vert \bar{a}_{nk}-\bar{a}_{k}\right \vert \right) \leq \left \Vert L_{A}\right \Vert _{\chi}\leq \underset{n}{\lim \sup}\left( \sup_{k}\left \vert \bar{a}_{nk}-\bar{a}_{k}\right \vert \right)$$ and$$L_{A}\text{ is compact if and only if }\underset{n}{\lim}\left( \sup
_{k}\left \vert \bar{a}_{nk}-\bar{a}_{k}\right \vert \right) =0,$$ where $\bar{\alpha}=(\bar{\alpha}_{k})$ with $\bar{\alpha}_{k}=\lim_{n}\bar
{a}_{nk}$ for all $k\in\mathbb{N}
$.
Morever, as an immediate consequence of Theorem 3.3, we have the following corollary.
If either $A\in(\ell_{\infty}(\widehat{F}),c_{0})$ or $A\in(\ell_{\infty
}(\widehat{F}),c)$, then the operator $L_{A}$ is compact.
Let $A\in(\ell_{\infty}(\widehat{F}),c_{0}).$ Then, we have by Lemma 2.3 that $\bar{A}\in(\ell_{\infty},c_{0})$ which implies that $\lim_{n}\left( \sum
_{k}\left \vert \bar{a}_{nk}\right \vert \right) =0,$ \[39\]. This leads us with Theorem 3.3(b) to the consequence that $L_{A}$ is compact. Similarly, if $A\in(\ell_{\infty}(\widehat{F}),c)$ then $\bar{A}\in(\ell_{\infty},c)$ and hence $\lim_{n}\left( \sum_{k}\left \vert \bar{a}_{nk}-\bar{a}_{k}\right \vert
\right) =0$, where $\bar{\alpha}=(\bar{\alpha}_{k})$ with $\bar{\alpha}_{k}=\lim_{n}\bar{a}_{nk}$ for all $k\in\mathbb{N}
$. Therefore, we deduce from Theorem 3.3(c) that $L_{A}$ is compact.
Throughout, let $\mathcal{F}_{m}$ $(m\in\mathbb{N}
)$ be the subcollection of $\mathcal{F}$ consisting of all nonempty and finite subsets of $\mathbb{N}
$ with elements that are greater than $m$, that is$$\mathcal{F}_{m}=\left \{ N\in\mathcal{F}:n>m\text{ for all }n\in\mathbb{N}
\right \} ;\text{ }(m\in\mathbb{N}
).$$
The next lemma \[25, Theorem 3.11\] gives necessary and sufficient conditions for a matrix transformation from a $BK$ space to $\ell_{1}$ to be compact.
Let $X\supset \varphi$ be a $BK$ space. If $A\in(X,\ell_{1})$, then $$\lim_{m}\left( \sup_{N\in\mathcal{F}_{m}}\left \Vert
{\displaystyle \sum \limits_{n\in N}}
A_{n}\right \Vert _{X}^{\ast}\right) \leq \left \Vert L_{A}\right \Vert _{\chi
}\leq4.\lim_{m}\left( \sup_{N\in\mathcal{F}_{m}}\left \Vert
{\displaystyle \sum \limits_{n\in N}}
A_{n}\right \Vert _{X}^{\ast}\right) .$$
Now, we prove the following result:
Let $1<p<\infty$ and $q=p/(p-1)$. If $A\in(\ell_{p}(\widehat{F}),\ell_{1})$, then$$\lim_{m}\left \Vert A\right \Vert _{(\ell_{p}(\widehat{F}),\ell_{1})}^{(m)}\leq \left \Vert L_{A}\right \Vert _{\chi}\leq4.\lim_{m}\left \Vert A\right \Vert
_{(\ell_{p}(\widehat{F}),\ell_{1})}^{(m)} \tag{3.11}$$ and$$L_{A}\text{ is compact if and only if }\lim_{m}\left \Vert A\right \Vert
_{(\ell_{p}(\widehat{F}),\ell_{1})}^{(m)}=0, \tag{3.12}$$ where $$\left \Vert A\right \Vert _{(\ell_{p}(\widehat{F}),\ell_{1})}^{(m)}=\sup_{N\in\mathcal{F}_{m}}\left(
{\displaystyle \sum \limits_{k}}
\left \vert
{\displaystyle \sum \limits_{n\in N}}
\bar{a}_{nk}\right \vert ^{q}\right) ^{1/q};\text{ \ }(m\in\mathbb{N}
).$$
It is obvious that (3.11) is obtained by combining Lemmas 2.2(c), 2.3 and 3.6. Also, by using (1.5), we get (3.12) from (3.11).
Let $1\leq p<\infty$. If $A\in(\ell_{1}(\widehat{F}),\ell_{p})$, then$$\left \Vert L_{A}\right \Vert _{\chi}=\lim_{m}\left( \sup_{k}\left(
{\displaystyle \sum \limits_{n=m}^{\infty}}
\left \vert \bar{a}_{nk}\right \vert ^{p}\right) ^{1/p}\right) \tag{3.13}$$ and$$L_{A}\text{ is compact if and only if }\lim_{m}\left( \sup_{k}\left(
{\displaystyle \sum \limits_{n=m}^{\infty}}
\left \vert \bar{a}_{nk}\right \vert ^{p}\right) ^{1/p}\right) =0, \tag{3.14}$$
Let us remark that the limit in (3.13) exists by Lemma 2.4.
Now, we write $S=S_{\ell_{1}(\widehat{F})}$. Then, we have by Lemma 1.2 that $L_{A}(S)=AS\in M_{\ell_{p}}$. Thus, it follows from (1.4) and Lemma 1.4 that$$\left \Vert L_{A}\right \Vert _{\chi}=\chi(AS)=\lim_{m}\left( \sup_{x\in
S}\left \Vert (I-P_{m})(Ax)\right \Vert _{\ell_{p}}\right) , \tag{3.15}$$ where $P_{m}:\ell_{p}\rightarrow \ell_{p}$ $(m\in\mathbb{N}
)$ is the operator defined by $P_{m}(x)=(x_{0},x_{1},...,x_{m},0,0,...)$ for all $x=(x_{k})\in \ell_{p}$ and $I$ is the identity operator on $\ell_{p}$.
On the other hand, let $x\in \ell_{1}(\widehat{F})$ be given. Then $y\in
\ell_{1}$ and since $A\in(\ell_{1}(\widehat{F}),\ell_{p})$, we obtain from Lemma 2.3 that $\bar{A}\in(\ell_{1},\ell_{p})$ and $Ax=\bar{A}y.$ Thus, we have for every $m\in\mathbb{N}
$ that$$\begin{aligned}
\left \Vert (I-P_{m})(Ax)\right \Vert _{\ell_{p}} & =\left \Vert (I-P_{m})(\bar{A}y)\right \Vert _{\ell_{p}}\\
& =\left(
{\displaystyle \sum \limits_{n=m+1}^{\infty}}
\left \vert \bar{A}_{n}(y)\right \vert ^{p}\right) ^{1/p}\\
& =\left(
{\displaystyle \sum \limits_{n=m+1}^{\infty}}
\left \vert
{\displaystyle \sum \limits_{k}}
\bar{a}_{nk}y_{k}\right \vert ^{p}\right) ^{1/p}\\
& \leq{\displaystyle \sum \limits_{k}}
\left(
{\displaystyle \sum \limits_{n=m+1}^{\infty}}
\left \vert \bar{a}_{nk}y_{k}\right \vert ^{p}\right) ^{1/p}\\
& \leq \left \Vert y\right \Vert _{\ell_{1}}\left( \sup_{k}\left(
{\displaystyle \sum \limits_{n=m+1}^{\infty}}
\left \vert \bar{a}_{nk}\right \vert ^{p}\right) ^{1/p}\right) \\
& =\left \Vert x\right \Vert _{\ell_{1}(\widehat{F})}\left( \sup_{k}\left(
{\displaystyle \sum \limits_{n=m+1}^{\infty}}
\left \vert \bar{a}_{nk}\right \vert ^{p}\right) ^{1/p}\right) .\end{aligned}$$ This yields that $$\sup_{x\in S}\left \Vert (I-P_{m})(Ax)\right \Vert _{\ell_{p}}\leq \sup
_{k}\left(
{\displaystyle \sum \limits_{n=m+1}^{\infty}}
\left \vert \bar{a}_{nk}\right \vert ^{p}\right) ^{1/p}\text{; \ }(m\in\mathbb{N}
).$$ Therefore, we deduce from (3.15) that $$\left \Vert L_{A}\right \Vert _{\chi}\leq \lim_{m}\left( \sup_{k}\left(
{\displaystyle \sum \limits_{n=m+1}^{\infty}}
\left \vert \bar{a}_{nk}\right \vert ^{p}\right) ^{1/p}\right) . \tag{3.16}$$
To prove the converse inequality, let $c^{(k)}\in \ell_{1}(\widehat{F})$ be such that $\widehat{F}c^{(k)}=e^{(k)}$ $(k\in\mathbb{N}
)$, that is, $e^{(k)}$ is the $\widehat{F}$-transform of $c^{(k)}$ for each $k\in\mathbb{N}
$. Then, we have by Lemma 2.3 that $Ac^{(k)}=\bar{A}e^{(k)}$ for every $k\in\mathbb{N}
$.
Now, let $U=\{c^{(k)}:$ $k\in\mathbb{N}
\}$. Then $U\subset S$ and hence $AU\subset AS$ which implies that $\chi(AU)\leq \chi(AS)=\left \Vert L_{A}\right \Vert _{\chi}$.
Further, it follows by applying Lemma 1.4 that $$\begin{aligned}
\chi(AU) & =\lim_{m}\left( \sup_{k}\left( \left(
{\displaystyle \sum \limits_{n=m+1}^{\infty}}
\left \vert A_{n}(c^{(k)})\right \vert ^{p}\right) ^{1/p}\right) \right) \\
& =\lim_{m}\left( \sup_{k}\left(
{\displaystyle \sum \limits_{n=m+1}^{\infty}}
\left \vert \bar{a}_{nk}\right \vert ^{p}\right) ^{1/p}\right) .\end{aligned}$$
Thus, we obtain that $$\lim_{m}\left( \sup_{k}\left(
{\displaystyle \sum \limits_{n=m+1}^{\infty}}
\left \vert \bar{a}_{nk}\right \vert ^{p}\right) ^{1/p}\right) \leq \left \Vert
L_{A}\right \Vert _{\chi}. \tag{3.17}$$
Hence, we get (3.13) by combining (3.16) and (3.17). This completes the proof, since (3.14) is immediate by (1.5) and (3.13).
[99]{}
E.E. Kara, Some topological and geometrical properties of new Banach sequence spaces, J. Inequal. Appl. 2013 (38) (2013), 15 pages.
T. Koshy, *Fibonacci and Lucas Numbers with Applications*, Wiley, 2001.
E. Malkowsky, V. Rakočević, An introduction into the theory of sequence spaces and measure of noncompactness, Zbornik radova, Matematicki inst. SANU, Belgrade, **9**(17) (2000) 143-234.
A. Wilansky, *Summability through Functional Analysis*, North-Holland Mathematics Studies 85, Elsevier Science Publishers, Amsterdam: New York: Oxford, 1984.
H. Kizmaz, On certain sequence spaces, Canad. Math. Bull. **24**(2) (1981) 169–176.
B. Altay, F. Başar, The matrix domain and the fine spectrum of the difference operator $\Delta$ on the sequence space $\ell_{p},$ $(0<p<1)$, Commun. Math. Anal. **2**(2) (2007) 1-11.
F. Başar, B. Altay, On the space of sequences of $p$-bounded variation and related matrix mappings, Ukrainian Math. J. **55** (2003) 136–147.
R. Çolak, M. Et, E. Malkowsky, Some Topics of Sequence Spaces, Firat University Press, ISBN: 975-394-0386-6, pp.1-63, 2004.
Z.U. Ahmad, M. Mursaleen, Köthe–Toeplitz duals of some new sequence spaces and their matrix maps, Pub. de l’Inst. Math. Beograd, **42** (56) (1987) 57–61.
E. Malkowsky, Absolute and ordinary Köthe–Toeplitz duals of some sets of sequences and matrix transformations, Pub. de l’Inst. Math. Beograd, **46**(60) (1989) 97–103.
S. Simons, The sequence spaces $\ell(p_{v})$ and $m(p_{v})$, Proc. London Math. Soc. **3** (15) (1965) 422–436.
I.J. Maddox, Continuous and Köthe–Toeplitz duals of certain sequence spaces, Proc. Camb. Phil. Soc. **65** (1965) 431–435.
B. Altay, F. Başar, M. Mursaleen, Some generalizations of the space $bv_{p}$ of $p$-bounded variation sequences, Nonlinear Anal. TMA*,* **68** (2008) 273–28.
B. Choudhary, S.K. Mishra, A note on Köthe–Toeplitz duals of certain sequence spaces and their matrix transformations, Int. J. Math. Math. Sci. 18(4) (1995) 681-688.
M.A. Sarigöl, On difference sequence spaces, J. Karadeniz Tech. Univ. Fac. Arts Sci. Ser. Math.-Phys., **10** (1987) 63–71.
M. Et, On some difference sequence spaces, Turkish J. Math.7 (1993) 18–24.
M. Mursaleen, Generalized spaces of difference sequences, J. Math. Anal. Appl. **203**(3) (1996), pp. 738–745.
S.K. Mishra, Matrix maps involving certain sequence spaces, Indian J. Pure Appl. Math. 24(2) (1993) 125–132.
A.K. Gaur, M. Mursaleen, Difference sequence spaces, Int. J. Math. Math. Sci. **21**(4) (1998) 701–706.
E. Malkowsky, M. Mursaleen, S. Suantai, The dual spaces of sets of difference sequences of order m and matrix transformations, Acta Math. Sinica (Engl. Ser.) **23**(3)(2007) 521-532.
M. Mursaleen, A.K. Noman, On some new difference sequence spaces of non-absolute type, Math. Comput. Modelling **52** (2010) 603-617.
M. Kirişçi, F. Başar, Some new sequence spaces derived by the domain of generalized difference matrix, Comput. Math. Appl. **60** (2010) 1299-1309.
A. Sönmez, Some new sequence spaces derived by the domain of the triple band matrix, Comput. Math. Appl. ** 62(2) (2011) 641-650.
V. Rakočević, Measures of noncompactness and some applications, Filomat, **12** (2) (1998) 87-120.
M. Mursaleen, A.K. Noman, Compactness by the Hausdorff measure of noncompactness, Nonlinear Anal. TMA, **73** (8) (2010) 2541–2557.
M. Mursaleen, A.K. Noman, The Hausdorff measure of noncompactness of matrix operators on some $BK$ spaces, Operators and Matrices, **5**(3) (2011) 473–486.
E.E. Kara, M. Başarir, On some Euler $B^{(m)}$ difference sequence spaces and compact operators, J. Math. Anal. Appl. **379** (2011) 499–511.
M. Mursaleen, V. Karakaya, H. Polat, N. Şimşek, Measure of noncompactness of matrix operators on some difference sequence spaces of weighted means, Comput. Math. Appl. **62** (2011) 814–820.
M. Başarir, E.E. Kara, On the $B$-difference sequence space derived by generalized weighted mean and compact operators, J. Math. anal. Appl. **391** (2012) 67-81.
F. Başar, E. Malkowsky, The characterization of compact operators on spaces of strongly summable and bounded sequences, Appl. Math. Comput. **217** (2011) 5199–5207.
M. Mursaleen, S.A. Mohiuddine, Applications of measures of noncompactness to the infinite system of differential equations in $\ell_{p}$ spaces, Nonlinear Anal. **75** (2012) 2111–2115.
M. Mursaleen, A.K. Noman, On $\sigma$-conservative matrices and compact operators on the space $V_{\sigma}$, Appl. Math. Lett. **24** (2011) 1554–1560.
B. de Malafosse, V. Rakočević, Applications of measure of noncompactness in operators on the spaces $s_{\alpha}$, $s_{\alpha}^{0}$, $s_{\alpha}^{c}$, $\ell_{\alpha}^{p},$ J. Math. Anal. Appl. **323**(1) (2006), 131–145.
M. Başarir, E.E. Kara, On compact operators on the Riesz $B^{(m)}$-difference sequence spaces, Iran. J. Sci. Technol. **35**(A4) (2011) 279–285.
M. Mursaleen, A.K. Noman, Compactness of matrix operators on some new difference sequence spaces, Linear Algebra Appl. **436**(1) (2012) 41–52.
M. Başarir, E.E. Kara, On some difference sequence spaces of weighted means and compact operators, Ann. Funct. Anal. **2** (2011) 114-129.
M. Başarir, E.E. Kara, On compact operators on the Riesz $B^{(m)}$-difference sequence spaces II, Iran. J. Sci. Technol. **36**(A3) (2012) (Special Issue-Mathematics), 371-376.
M. Mursaleen, A.K. Noman, Applications of Hausdorff measure of noncompactness in the spaces of generalized means, Math. Inequal. Appl. **16** (2013) 207-220.
M. Stieglitz, H. Tietz, Matrix transformationen von folgenräumen eine ergebnisübersicht, Math. Z. **154** (1977) 1–16.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present new results regarding the features of high energy photon emission by an electron beam of 178GeV penetrating a 1.5cm thick single Si crystal aligned at the Strings-Of-Strings(SOS) orientation. This concerns a special case of coherent bremsstrahlung where the electron interacts with the strong fields of successive atomic strings in a plane and for which the largest enhancement of the highest energy photons is expected. The polarization of the resulting photon beam was measured by the asymmetry of e$^+$e$^-$ pair production in an aligned diamond crystal analyzer. By the selection of a single pair the energy and the polarization of individual photons could be measured in an the environment of multiple photons produced in the radiator crystal. Photons in the high energy region show less than 20% linear polarization at the 90% confidence level.'
bibliography:
- 'na59-sos.bib'
title: Measurement of Coherent Emission and Linear Polarization of Photons by Electrons in the Strong Fields of Aligned Crystals
---
\[sec:intro\]Introduction
=========================
Interest in the generation of intense, highly polarized high energy photon beams [@proposal; @e159] comes in part from the need to investigate the polarized photo-production mechanisms. For example, the so-called “spin crisis of the nucleon” and its connection to the gluon polarization has attracted much attention [@compass]. Future experiments will require intense photon beams with a high degree of polarization. The radiation emitted by electrons passing through oriented single crystals is important for these purposes. The coherent bremsstrahlung(CB) of high energy unpolarized electrons is a well established and widely applied technique for producing intense photon beams with a high degree of linear polarization. The coherence arises in this case due to crystal effects which become pronounced when the electron incidence angle with respect to a major plane is small. The resulting CB radiation differs from incoherent bremsstrahlung(ICB) in an amorphous target in that the cross section is substantially enhanced and relatively sharp coherent peaks appear in the photon spectrum. The position of these peaks can be tuned by adjusting the electron beam incidence angle with respect to the major plane of the lattice.
There is another less well known method of producing greater enhancement as well as a harder photon spectrum than the CB case. This is achieved by selecting a very specific electron incident angle with respect to the crystal. If the electron beam is incident very close to the plane (within the planar channelling critical angle) and also closely well aligned to a major axis (but beyond the axial channelling critical angle), then the electron interacts dominantly with successive atomic strings in the plane. This orientation had been aptly described by the term “String-Of-Strings"(SOS) by Lindhard, a pioneer of beam-crystal phenomena [@lindhard]. The NA43 Collaboration has studied the radiation emitted by electrons incident in the SOS orientation. The reference [@kirsebom01] and references therein are an account of this study, as well as many other related effects. There remained the issue of the polarisation of the SOS radiation. Polarisation measurements have been reported [@kirsebom99] which could be consistent with substantial polarisation of the hard component of SOS radiation. However the ability to distinguish clearly a single photon spectrum from the total radiated energy spectrum was not yet developed for that measurement. In this paper, a new study of SOS-produced high energy photon beams is reported in which we were able to study the beam on a photon-by-photon basis, and measure both the enhancement and the linear polarisation as a function of photon energy.
\[theory\]Theoretical Description
=================================
The CB mechanism produces linearly polarized photons in a selected energy region when the crystal type, its orientation with respect to the electron beam, and the electron energy are appropriately chosen. In the so-called point effect(PE) orientation of the crystal the direction of the electron beam has a small angle with respect to a chosen crystallographic plane and a relatively large angle with the crystallographic axes that are in that plane. For this PE orientation of the single crystal only one reciprocal lattice vector contributes to the CB cross section. The CB radiation from a crystal aligned in this configuration is more intense than the ICB radiation in amorphous media and a high degree of linear polarization can be achieved [@termisha]. The PE orientation of the crystal was used in a previous NA59 experiment, where a large linear polarization of high energy photons was measured. The photons had been produced by an unpolarized electron beam. The conversion of the linear polarisation to circular polarization induced by a birefringent effect in an aligned single crystal was also studied [@na59-1; @na59-2].
The character of the radiation, including its linear polarization, is changed when the direction of the electron (i) has a small angle with a crystallographic axis and (ii) is parallel with the plane that is formed by the atomic strings along the chosen axes. This is the so-called SOS orientation. It produces a harder photon spectrum than the CB case because the coherent radiation arises from successive scattering off the axial potential, which is deeper than the planar potential. The radiation phenomena in single crystals aligned in SOS mode have been under active theoretical investigation since the NA43 collaboration discovered, for the first time, two distinct photon peaks, one in the low energy region and one in the high energy region of the radiated energy spectrum for about 150GeV electrons traversing a diamond crystal [@new-effect]. It was established that the hard photon peak was a single photon peak. However, the radiated photons were generally emitted with significant multiplicity in such a way that a hard photon would be accompanied by a few low energy photons. It will be seen later that two different mechanisms are responsible for the soft and the hard photons. In the former case, it is planar channelling(PC) radiation, while in the latter case, it is SOS radiation.
The issue of the polarisation of SOS radiation also came into question. Early experiments with electron beams of up to 10GeV in single crystals showed a smaller linear polarization of the more intense radiation in the SOS orientation than in the PE orientation (see [@saenz] and references therein). The first measurements of linear polarization for high energy photons ($E_{\gamma} \approx 50-150$GeV) were consistent with a high degree of linear polarization of the radiated photons [@kirsebom99]. At this stage the theoretical prediction of the SOS hard photon polarisation was unresolved. However, it was clear that the photons emitted by the PC mechanism would be linearly polarised. This experiment therefore could not be considered conclusive, as the polarimeter recorded the integral polarisation for a given radiated energy, which was likely to have a multi-photon character. The extent to which pile-up from the low energy photons perturbed the high energy part of the total radiated energy spectrum was not resolved. These results therefore required more theoretical and experimental investigation.
A theory of photon emission by electrons along the SOS orientation of single crystals has since been developed. The theory takes into account the change of the effective electron mass in the fields due to the crystallographic planes and the crossing of the atomic strings [@bks]. The authors show that the SOS specific potential affects the high energy photon emission and also gives an additional contribution in the low energy region of the spectrum. In Refs. [@simon; @strakh] the linear polarization of the emitted photons was derived and analysed for different beam energies and crystal orientations. The predicted linear polarization of hard photons produced using the SOS orientation of the crystal is small compared to the comparable case using the PE orientation of the crystal. On the other hand, the additional soft photons produced with SOS orientation of the crystal are predicted to exhibit a high degree of polarization.
The emission mechanism of the high energy photons is CB connected to the periodic structure of the crystal [@termisha].
The peak energy of the CB photons, $E_\gamma$, is determined from the condition ( the system of units used here has $\hbar={\rm c}=1$ ), $$\frac{1}{|q_{\Vert}|} = 2 \lambda_c \gamma \frac{E_0-E_\gamma}{E_\gamma}~,$$ where $|q_{\Vert}|$ is the component of the momentum recoil, $\mathbf{q}$, parallel to the initial electron velocity and the other symbols have their usual meanings. Recall, in a crystal possible values of $\mathbf{q}$ are discrete: $\mathbf{q}=\mathbf{g}$ [@termisha], where $\mathbf{g}$ is a reciprocal lattice vector. The minimal reciprocal lattice vector giving rise to the main CB peak in both the PE and the SOS orientations is given by $$|g_{\Vert}|_{min} = \frac{2\pi}{d}\Theta.$$
For the PE orientation, $d$ is the interplanar distance and $\Theta=\psi$, the electron incident angle with respect to the plane. For the SOS orientation $d$ is the spacing between the axes (strings) forming the planes, and $\Theta=\theta$, the electron incident angle with respect to the axis. The position of the hard photon peak can be selected by simultaneous solution of the last two equations, $$\Theta =\frac{d}{4\pi\gamma\lambda_c}\frac{E_{\gamma}}{E_0-E_{\gamma}}.$$
With the appropriate choice of $\theta=\Theta$ the intensity of the SOS radiation may exceed the ICB radiation by an order of magnitude.
When a thin silicon crystal is used with an electron beam of energy $E_0
=178$GeV incident along the SOS orientation, within the $(110)$ plane and with an angle of $\theta=0.3$ mrad to the $<100>$ axis, the hard photon peak position is expected at $E_{\gamma}=129$GeV.
In the current experiment, a 1.5cm thick silicon crystal was used in the SOS orientation with the electron beam ($E_0 =178$GeV) incident within the $(110)$ plane with an angle of $\theta=0.3$ mrad to the $<100>$ axis. This gives the hard photon peak position at $x_{max}=0.725$. This corresponds to the photon energy $E_{\gamma}=129$GeV. Under this condition the radiation is expected to be enhanced by about a factor 30 with respect to the ICB for a randomly oriented crystalline Si target.
The coherence length determines the effective longitudinal dimension of the interaction region for the phase coherence of the radiation process: $$l_{coh} = \frac{1}{|q_{\Vert}|}.$$
The radiation spectrum with the crystal aligned in SOS orientation has in addition to the CB radiation a strong component at a low energy which is characteristic of PC radiation. As the electron direction lines up with a crystallographic plane in the SOS orientation, the planar channelling condition is fulfilled. For channelling radiation the coherence length is much longer than the interatomic distances and the long range motion, characteristic of planar channelled electrons, becomes dominant over short range variations with the emission of low energy photons. Theoretical calculations [@strakh; @armen] predict a more intense soft photon contribution with a high degree of linear polarization of up to 70%.
The simulation of the enhancements of both the low energy and the high energy components of the radiation emission for the SOS orientation under conditions applicable to this experiment are presented in Fig. \[F:Strak-1b\].
![\[F:Strak-1b\] Photon power yield per unit of thickness, $E_\gamma d^2N/dE_\gamma dl$, for a thin silicon crystal in the SOS orientation for a $E_0 =178$GeV electron beam incident within the $(110)$ plane and at an angle of $\theta=0.3$mrad to the $<100>$ axis. At low energy the PC radiation dominates and at high energies the SOS radiation peaks. The solid curve represents the total of the contributions from (green dash-dotted)ICB, (blue dotted)PC, and (red dashed) SOS radiation. vThe insert is a logarithmic representation and shows the flat incoherent contribution and the enhancement with a factor of about 30 for SOS radiation at 129GeV.](na59-sos-fig1)
\[setup\]Experimental Setup
===========================
The NA59 experiment was performed in the North Area of the CERN SPS, where unpolarized electron beams with energies above 100GeV are available. We used a beam of 178GeV electrons with angular divergence of $\sigma_{x'}=48\,\mu$rad and $\sigma_{y'}=35\,\mu$rad in the horizontal and vertical plane, respectively.
The experimental setup shown in Fig. \[F:setup\] was also used to investigate the linear polarization of CB and birefringence in aligned single crystals [@na59-1; @na59-2]. This setup is ideally suited for detailed studies of the photon radiation and pair production processes in aligned crystals.
The main components of the experimental setup are: two goniometers with crystals mounted inside vacuum chambers, a pair spectrometer, an electron tagging system, a segmented leadglass calorimeter, wire chambers, and plastic scintillators. In more detail a 1.5cm thick Si crystal can be rotated in the first goniometer with 2$\mu$rad precision and serves as radiator. A multi-tile synthetic diamond crystal on the first goniometer can be rotated with 20$\mu$rad precision and is used as the analyzer of the linear polarization of the photon beam.
The photon tagging system consists of a dipole magnet B8, wire chamber dwc0, and scintillators T1 and T2. Given the geometrical acceptances and the magnetic field, the system, tags the radiated energy between 10% and 90% of the electron beam energy. Drift chambers dch1up, dch2up, and delay wire chamber dwc3 define the incident and the exit angle of the electron at the radiator.
The e$^+$e$^-$ pair spectrometer consists of dipole magnet Trim 6 and of drift chambers dch05, dch1, dch2, and dch3. The drift chambers measure the horizontal and vertical positions of the passing charged particles with 100$\mu$m precision. Together with the magnetic field in the dipole this gives a momentum resolution of $\sigma_p/p^2=0.0012$ with $p$ in units ofGeV/c. The pair spectrometer enables the measurement of the energy of a high energy photon, $E_\gamma$, in a multi-photon environment. Signals from the plastic scintillators S1, S2, S3, T1, T2, S11 and veto detector ScVT provide several dedicated triggers.
The total radiated energy $E_{tot}$ is measured in a 12-segment array of leadglass calorimeter with a thickness of 24.6 radiation lengths and a resolution of $\sigma_E=0.115~\sqrt{E}$ with $E$ in units ofGeV. A central element of this leadglass array is used to map and to align the crystals with the electron beam.
A detailed description of the NA59 experimental apparatus can be found in reference [@na59-1].
Results and Discussion
======================
The experiment can be divided in two parts: (A) production of the photon beam by the photon radiation of the 178GeV electron beam in the Si radiator oriented in the SOS mode and (B) measurement of the linear polarization by using diamond crystals as analyzers. Prior to the experiment Monte Carlo(MC) simulations were used to estimate the photon yield, the radiated energy, and the linear polarization of the photon beam and we optimized the orientation of the crystal radiator. The MC calculations also included the crystal analyzer to estimate the asymmetry of the e$^+$e$^-$ pair production. The simulations further included the angular divergence of the electron beam, the spread of 1% in the beam energy, and the generation of the electromagnetic shower that develops in oriented crystals. To optimize the processing time of the MC simulation, energy cuts of 5GeV for electrons and of 500MeV for photons were applied.
Photon Beam
-----------
We used a beam angle of $\theta=0.3$mrad to the $\langle 100 \rangle$ axis in the $(110)$ plane of the 1.5cm thick Si crystal which is the optimal angle for a high energy SOS photon peak at 129GeV (see Fig. \[F:Strak-1b\]). As is mentioned above, the radiation probability with a thin radiator is expected to be 30 times larger at that energy than the Bethe-Heitler(ICB) prediction for randomly oriented crystalline Si.
![\[F:sps\] Photon power yield, $E_\gamma dN/dE_\gamma$, as a function of the energy $E_\gamma$ of individual photons radiated by an electron beam of 178GeV in the SOS-aligned 1.5cm Si crystal. The black crosses are the measurements with the pair spectrometer, the vertical lines represent the errors including the uncertainty in the acceptance of the spectrometer. The (red solid) histogram represent the MC prediction for our experimental conditions. The (green dotted) represent the small contribution due to incoherent interactions. For completeness, we also show the theoretical predictions if the experimental effects are ignored (blue dashed).](na59-sos-fig3)
However, there are several consequences for the photon spectrum due to the use of a 1.5 cm thick crystal For the chosen orientation of the Si crystal, the emission of mainly low energy photons from planar coherent bremsstrahlung (PC) results in a total average photon multiplicity above 15. And the most probable radiative energy loss of the 178 GeV electrons is expected to be 80%. The beam energy decreases significantly as the electrons traverse the crystal. The peak energy of both SOS and PC radiation also decreases with the decrease in electron energy. Consequently, the SOS radiation spectrum is not peaked at the energy for a thin radiator, but becomes a smooth energy distribution. Clearly, many electrons may pass through the crystal without emitting SOS radiation and still lose a large fraction of their energy due to PC and ICB. Hard photons emitted in the first part of the crystal that convert in the later part do not contribute anymore to the high energy part of the photon spectrum. A full Monte Carlo calculation is necessary to propagate the predicted photon yield with a thin crystal, as shown in Fig. \[F:Strak-1b\] for 178 GeV electrons, to the current case with a 1.5cm thick crystal.
This has been implemented for the measured photon spectrum shown in Fig. \[F:sps\]. We see that the measured SOS photon spectrum shows a smoothly decreasing distribution. The low energy region of the photon spectrum is especially saturated, due to the abundant production of low energy photons. Above 25GeV however, there is satisfactory agreement with the theoretical Monte Carlo prediction, which includes the effects mentioned above.
The enhancement of the emission probability compared to the ICB prediction is given in Fig. \[F:enh\] as a function of the total radiated energy as measured in the calorimeter. The maximal enhancement is about a factor of 18 at 150GeV and corresponds well with the predicted maximum of about 20 at 148GeV. This is a multi-photon spectrum measured with the photon calorimeter. The peak of radiated energy is situated at 150GeV, which means that each electron lost about 80% of its initial energy due to the large thickness of the radiator. This means that the effective radiation length of the oriented single crystal is several times shorter in comparison with the amorphous target. The low energy region is depleted due to the pile-up of several photons.
![\[F:enh\] Enhancement of the intensity with respect to the Bethe-Heitler(ICB) prediction for randomly oriented polycrystalline Si as a function of the total radiated energy $E_{tot}$ in the SOS-aligned Si crystal by 178GeV electrons. The black crosses are the measurements and the red histogram represent the MC prediction.](na59-sos-fig4)
The expected linear polarization is shown in Fig. \[F:SOS-pol\] as a function of photon energy. It is well known that channelling radiation in single crystals is linearly polarized [@Adishchev; @Vorobyov] and the low energy photons up to 70GeV are also predicted to be linearly polarized in the MC simulations. High energy photons are predicted with an insignificant polarization.
Asymmetry Measurement
---------------------
In this work, the photon polarization is always expressed using the Stoke’s parametrization with the Landau convention, where the total elliptical polarization is decomposed into two independent linear components and a circular component. In mathematical terms, one writes:
$$P_{\hbox {linear}}=\sqrt{\eta _{1}^{2}+\eta _{3}^{2}},
\quad \; P_{\hbox {circular}}=\sqrt{\eta _{2}^{2}},
\quad \; P_{\hbox {total}}=\sqrt{P_{\hbox {linear}}^{2}+P_{\hbox
{circular}}^{2}} \quad .
\label{eq:pol-def}$$
![\[F:SOS-pol\] Expected linear polarization as a function of the energy $E_\gamma$ of the photons produced in the SOS-aligned Si crystal by 178GeV electrons.](na59-sos-fig5)
The radiator angular settings were chosen to have the total linear polarization from the SOS radiation purely along $\eta _{3}$, that is $\eta _{1}=0$. The $\eta _{2}$ component is also zero because the electron beam is unpolarized. The expected $\eta _{3}$ component of the polarization shown is in Fig. \[F:SOS-pol\].
In order to determine the linear polarization of the photon beam the method proposed in reference [@barbiellini] with an oriented crystal was chosen. This method of measurement of the linear polarization of high energy photons is based on coherent e$^+$e$^-$ pair production(CPP) in single crystals which depends on the orientation of the reciprocal lattice vector and the linear polarization vector. Thus, the dependence of the CPP cross section on the linear polarization of the photon beam makes an oriented single crystal suitable as an efficient polarimeter for high energy photons.
The basic characteristic of the polarimeter is the analyzing power $R$ of the analyzer crystal [@barbiellini]. By choosing the appropriate crystal type and its orientation a maximal analyzing power can be obtained. The relevant experimental quantity is the asymmetry $A$ of the cross sections $\sigma (\gamma \rightarrow e^+e^-)$ for parallel and perpendicular polarization, where the polarization direction is defined with respect to a particular crystallographic plane of the [*analyzer*]{} crystal. This asymmetry is related to the linear polarization of the photon beam, $P_{\rm linear}$, through: $$A \equiv \frac{\sigma (\gamma _{\perp }\rightarrow e^{+}e^{-})-\sigma
(\gamma _{\parallel }\rightarrow e^{+}e^{-})}{\sigma (\gamma _{\perp }
\rightarrow e^{+}e^{-})+\sigma (\gamma _{\parallel }\rightarrow
e^{+}e^{-})}
=R \times P_{\rm linear}.
\label{eq:asym}$$ The analyzing power $R$ corresponds to the asymmetry expected for photons that are 100% linearly polarized perpendicular to the chosen crystallographic plane.
Denoting the number of e$^+$e$^-$ pairs produced in perpendicular and parallel cases by $p_{1}$ and $p_{2}$, and the number of the normalisation events in each case by $n_{1}$ and $n_{2}$, respectively, the measured asymmetry can be written as: $$A=\frac{p_{\perp }/n{\perp } - p_{\parallel }/n_{\parallel }}{p_{\perp
}/n_{\perp } + p_{\parallel }/n_{\parallel }},
\label{eq:asy-meas}$$ where $p$ and $n$ are acquired simultaneously and therefore correlated. Further details of this method, as well as refinements to enhance the analyzing power $R$ by using kinematic cuts on the pair spectra, may be found in reference [@na59-1].
The existence of a strong anisotropy for the channelling of the e$^+$e$^-$ pairs during their formation is the reason for the polarization dependent CPP cross section of photons passing through oriented crystals. This means that perfect alignment along a crystallographic axis is not an efficient analyzer orientation due to the approximate cylindrical symmetry of the crystal around atomic strings. However, for small angles of the photon beam with respect to the crystallographic symmetry directions the conditions for the formation of the e$^+$e$^-$ pairs prove to be very anisotropic. As it turns out, the orientations with the highest analyzing power are those where the e$^+$e$^-$ pair formation zone is not only highly anisotropic but also inhomogeneous with maximal fluctuations of the crystal potential along the electron path. At the crystallographic axes the potential is largest and so are the fluctuations. These conditions are related to the ones of the SOS orientation: (i) a small angle to a crystallographic axis to enhance the pair production (PP) process by the large fluctuations and (ii) a smaller angle to the crystallographic plane to have a long but still anisotropic formation zone for CPP.
In the NA59 experiment we used a multi-tile synthetic diamond crystal as an analyzer oriented with the photon beam at 6.2 mrad to the axis and at 465$\mu$rad from the $(110)$ plane. This configuration is predicted to have a maximal analyzing power for a photon energy of 125GeV as is shown in Fig. \[F:anpow\]. The predicted analyzing power in the high energy peak region is about 30%.
![\[F:anpow\] Analyzing power $R$ with the aligned diamond crystal as a function of the photon energy $E_\gamma$ (black curve) for an ideal photon beam without angular divergence and (red curve) for the Monte Carlo simulation of photons with the beam conditions in the NA59 experiment.](na59-sos-fig6)
The measured asymmetry and the predicted asymmetry are shown in Fig. \[F:asy\]. One can see that the measured asymmetry is consistent with zero over the whole photon energy range. For the photon energy range of 100-155GeV we find less than 5% polarization at 0.9 confidence level. The null result is expected to be reliable as the correct operation of the polarimeter had been confirmed in the same beam-time in measurements of the polarisation of CB radiation [@na59-1]. Note, that the expected asymmetry is small, especially in the high energy range of 120-140GeV, where the analyzing power is large, see Fig. (\[F:anpow\]). This corresponds to the expected small linear polarization in the high energy range, see Fig. (\[F:SOS-pol\]).
![\[F:asy\] Asymmetry of the e$^+$e$^-$ pair production in the aligned diamond crystal as a function of the photon energy $E_\gamma$ which is measured to determine the $P_1$ component of the photon polarization in the SOS-aligned Si crystal by 178GeV electrons. The black crosses are the measurements and the red histogram represent the MC prediction.](na59-sos-fig7)
In contrast to the result of a previous experiment [@kirsebom99], our results are consistent with calculations that predict negligible polarization in the high energy photon peak for the SOS orientation. The analyzing power of the diamond analyzer crystal in the previous experiment’s [@kirsebom99] setup peaked in the photon energy range of 20-40GeV where a high degree of linear polarization is expected. But in the high energy photon region we expect a small analyzing power of about 2-3%, also following recent calculation [@simon; @strakh]. The constant asymmetry measured in a previous experiment [@kirsebom99] over the whole range of total radiated energy may therefore not be due to the contribution of the high energy photons.
From Fig. \[F:SOS-pol\] one can expect a large linear polarization for photons in the low energy range of 20-50GeV. However, the analyzing power was optimized for an photon energy of 125GeV and is small in the region where we expect a large polarization. A different choice of orientation of the analyzer crystal can move the analyzing power peak to the low energy range and may be used to measure the linear polarization in the low energy range.
Conclusion
==========
We have performed an investigation of both enhancement and polarisation of photons emitted in the so called SOS radiation. This is a special case of coherent bremsstrahlung for multi-hundred GeV electrons incident on oriented crystalline targets, which provides some advantages comparing with other types of CB orientations. The experimental set-up had the capacity to deal with the relatively high photon multiplicity and single photon spectra were measured. This is very important in view of the fact that there are several production mechanisms for the multiphotons, which have different radiation characteristics.
We have confirmed the single photon nature of the hard photon peak produced in SOS radiation.
The issue of the polarisation of the SOS photons had previously not been conclusively settled. Earlier results in a previous experiment [@kirsebom99] had indicated that a large polarization might be obtained for the high energy SOS photons. Our experimental results show that the high energy photons emitted by electrons passing through the Si crystal radiator oriented in the SOS mode have a linear polarization smaller than 20% at a confidence level of 90%.
Since the previous experiments, the theoretical situation for the polarisation of hard SOS photons has also become clearer. Our results therefore also confirm recent calculations which predict that the linear polarization of high energy photons created in SOS orientation of the crystal is small compared to the polarization obtained with the PE orientation.
Photon emission by electrons traversing single crystals oriented in the SOS orientation has interesting peculiarities since three different radiation processes are involved: (1) incoherent bremsstrahlung, (2) channelling radiation, and (3) coherent bremsstrahlung induced the periodic structure of the atomic strings in the crystal that are crossed by the electron. The calculations presented here have taken these three processes into account, and predict around a 5% polarization for the high energy SOS photons. This prediction is consistent with our null polarization asymmetry measurement for the single photons with energies above 100GeV.
We dedicate this work to the memory of Friedel Sellschop. We express our gratitude to CNRS, Grenoble for the crystal alignment and Messers DeBeers Corporation for providing the high quality synthetic diamonds. We are grateful for the help and support of N. Doble, K. Elsener and H. Wahl. It is a pleasure to thank the technical staff of the participating laboratories and universities for their efforts in the construction and operation of the experiment.
This research was partially supported by the Illinois Consortium for Accelerator Research, agreement number 228-1001. UIU acknowledges support from the Danish Natural Science research council, STENO grant no J1-00-0568.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Semi-supervised and unsupervised systems provide operators with invaluable support and can tremendously reduce the operators’ load. In the light of the necessity to process large volumes of video data and provide autonomous decisions, this work proposes new learning algorithms for activity analysis in video. The activities and behaviours are described by a dynamic topic model. Two novel learning algorithms based on the expectation maximisation approach and variational Bayes inference are proposed. Theoretical derivations of the posterior estimates of model parameters are given. The designed learning algorithms are compared with the Gibbs sampling inference scheme introduced earlier in the literature. A detailed comparison of the learning algorithms is presented on real video data. We also propose an anomaly localisation procedure, elegantly embedded in the topic modeling framework. It is shown that the developed learning algorithms can achieve $95\%$ success rate. The proposed framework can be applied to a number of areas, including transportation systems, security and surveillance.'
author:
- 'Olga Isupova, Danil Kuzin, and Lyudmila Mihaylova, [^1]'
bibliography:
- 'TNNLS-2016-P-6800-R1-Biblist.bib'
title: Learning Methods for Dynamic Topic Modeling in Automated Behaviour Analysis
---
behaviour analysis, unsupervised learning, learning dynamic topic models, variational Bayesian approach, expectation maximisation, video analytics
Introduction
============
analysis is an important area in intelligent video surveillance, where abnormal behaviour detection is a difficult problem. One of the challenges in this field is informality of the problem formulation. Due to the broad scope of applications and desired objectives there is no unique way, in which normal or abnormal behaviour can be described. In general, the objective is to detect unusual events and inform in due course a human operator about them.
This paper considers a probabilistic framework for anomaly detection, where less probable events are labelled as abnormal. We propose two learning algorithms and an anomaly localisation procedure for spatial detection of abnormal behaviours.
Related work
------------
There is a wealth of methods for abnormal behaviour detection, for example, pattern-based methods [@Raghavendra2011; @Yen13; @Ouivirach13]. These methods extract explicit patterns from data and use them as behaviour templates for decision making. In [@Raghavendra2011] the sum of the visual features of a reference frame is treated as a normal behaviour template. Another common approach for representing normal templates is using clusters of visual features [@Yen13; @Ouivirach13]. Visual features can range from raw intensity values of pixels to complex features that exploit the data nature [@Zhou2016].
In the testing stage new observations are compared with the extracted patterns. The comparison is based on some similarity measure between observations, e.g., the Jensen-Shannon divergence in [@Su2014] or the $Z$-score value in [@Yen13; @Ouivirach13]. If the distance between the new observation and any of the normal patterns is larger than a threshold, then the observation is classified as abnormal.
Abnormal behaviour detection can be considered as a classification problem. It is difficult in advance to collect and label all kind of abnormalities. Therefore, only one class label can be expected and one-class classifiers are applied to abnormal behaviour detection: e.g., a one-class Support Vector Machine [@Cheng2013], a support vector data description algorithm [@Liu2010], a neural network approach [@Maddalena2013], a level set method [@Osher1988] for normal data boundary determination [@Ding2015].
Another class of methods rely on the estimation of probability distributions of the visual data. These estimated distributions are then used in the decision making process. Different kinds of probability estimation algorithms are proposed in the literature, e.g., based on non-parametric sample histograms [@Adam2008], Gaussian distribution modelling [@Basharat2008]. Spatio-temporal motion data dependency is modelled as a coupled Hidden Markov Model in [@Kratz2009]. Auto-regressive process modelling based on self-organised maps is proposed in [@Brighenti2011].
An efficient approach is to seek for feature sets that tend to appear together. These feature sets form typical activities or behaviours in the scene. Topic modeling [@Hofmann99; @Blei03LDA] is an approach to find such kinds of statistical regularities in a form of probability distributions. The approach can be applied for abnormal behaviour detection, e.g., [@Mehran09; @Li2008; @Varadarajan2009]. A number of variations of the conventional topic models for abnormal behaviour detection have been recently proposed: clustering of activity distributions [@Wang09], modelling temporal dependencies among activities [@Hospedales2011], a continuous model for an object velocity [@Jeong14].
Within the probabilistic modelling approach [@Jeong14; @Li2008; @Mehran09; @Basharat2008; @Wang09; @Kratz2009] the decision about abnormality is mainly made by computing likelihood of a new observation. The comparison of the different abnormality measures based on the likelihood estimation is provided in [@Varadarajan2009].
Topic modeling is originally developed for text mining [@Hofmann99; @Blei03LDA]. It aims to find latent variables called *“topics”* given the collection of unlabelled text *documents* consisted of *words*. In probabilistic topic modeling documents are represented as a mixture of topics, where each topic is assumed to be a distribution over words.
There are two main types of topic models: Probabilistic Latent Semantic Analysis (PLSA) [@Hofmann99] and Latent Dirichlet Allocation (LDA) [@Blei03LDA]. The former considers the problem from the frequentist perspective while the later studies it within the Bayesian approach. The main learning techniques proposed for these models include maximum likelihood estimation via the Expectation-Maximisation (EM) algorithm [@Hofmann99], variational Bayes inference [@Blei03LDA], Gibbs sampling [@Griffiths2004] and Maximum a Posteriori (MAP) estimation [@Chien2008].
Contributions
-------------
In this paper inspired by ideas from [@Hospedales2011] we propose an unsupervised learning framework based on a Markov Clustering Topic Model for behaviour analysis and anomaly detection. It groups possible topic mixtures of visual documents and forms a Markov chain for the groups.
The key contributions of this work consist in developing new learning algorithms, namely MAP estimation using the EM-algorithm and variational Bayes inference for the Markov Clustering Topic Model (MCTM), and in proposing an anomaly localisation procedure that follows concepts of probabilistic topic modeling. We derive the likelihood expressions as a normality measure of newly observed data. The developed learning algorithms are compared with the Gibbs sampling scheme proposed in [@Hospedales2011]. A comprehensive analysis of the algorithms is presented over real video sequences. The empirical results show that the proposed methods provide more accurate results than the Gibbs sampling scheme in terms of anomaly detection performance.
Our preliminary results with the EM-algorithm for behaviour analysis are published in [@Isupova2015]. In contrast to [@Isupova2015] we now consider a fully Bayesian framework, where we propose the EM-algorithm for MAP estimates rather than the maximum likelihood ones. We also propose here a novel learning algorithm based on variational Bayes inference and a novel anomaly localisation procedure. The experiments are performed on more challenging datasets in comparison to [@Isupova2015].
The rest of the paper is organised as follows. Section \[sec:features\] describes the overall structure of visual documents and visual words. Section \[sec:model\] introduces the dynamic topic model. The new learning algorithms are presented in Section \[sec:inference\], where the proposed MAP estimation via the EM-algorithm and variational Bayes algorithm are introduced first and then the Gibbs sampling scheme is reviewed. The methods are given with a detailed discussion about their similarities and differences. The anomaly detection procedure is presented in Section \[sec:abnormality\]. The learning algorithms are evaluated with real data in Section \[sec:experiments\] and Section \[sec:conlusion\] concludes the paper.
Video analytics within the topic modeling framework {#sec:features}
===================================================
Video analytics tasks can be formulated within the framework of topics modeling. This requires a definition of visual documents and visual words, e.g., as in [@Wang09; @Hospedales2011]. The whole video sequence is divided into non-overlapping short clips. These clips are treated as visual documents. Each frame is divided next into grid cells of pixels. Motion detection is applied to each of the cells. The cells where motion is detected are called moving cells. For each of the moving cells the motion direction is determined. This direction is further quantised into four dominant ones - up, left, down, right (see Figure \[fig:feature\_extraction\]). The position of the moving cell and the quantised direction of its motion form a visual word.
Each of the visual documents is then represented as a sequence of visual words’ identifiers, where identifiers are obtained by some ordering of a set of unique words. This discrete representation of the input data can be processed by topic modeling methods.
The Markov Clustering Topic Model for behavioural analysis {#sec:model}
==========================================================
Motivation
----------
In topic modeling there are two main kinds of distributions — the distributions over words, which correspond to topics, and the distributions over topics, which characterise the documents. The relationship between documents and words is then represented via latent low-dimensional entities called topics. Having only an unlabelled collection of documents, topic modeling methods restore a hidden structure of data, i.e., the distributions over words and the distributions over topics.
Consider a set of distributions over topics and a topic distribution for each document is chosen from this set. If the cardinality of the set of distributions over topics is less than the number of documents, then documents are clustered into groups such that documents have the same topic distribution within a group. A unique distribution over topics is called a *behaviour* in this work. Therefore, each document corresponds to one behaviour. In topic modeling a document is fully described by a corresponding distribution over topics, which means in this case a document is fully described by a corresponding behaviour.
There are a number of applications where we can observe documents clustered into groups with the same distribution over topics. Let us consider some examples from video analytics where a visual word corresponds to a motion within a tiny cell. As topics represent words that statistically often appear together, in video analytics applications topics define some motion patterns in local areas.
Let us consider a road junction regulated by traffic lights. A general motion on the junction is the same with the same traffic light regime. Therefore, the documents associated to the same traffic light regimes have the same distributions over topics, i.e., they correspond to the same behaviours.
Another example is a video stream generated by a video surveillance camera from a train station. Here it is also possible to distinguish several types of general motion within the camera scene: getting off and on a train and waiting for it. These types of motion correspond to behaviours, where the different visual documents showing different instances of the same behaviour have very similar motion structures, i.e., the same topic distribution.
Each action in real life lasts for some time, e.g., a traffic light regime stays the same and people get on and off a train for several seconds. Moreover, often these different types of motion or behaviours follow a cycle and their changes occur in some order. These insights motivate to model a sequence of behaviours as a Markov chain, so that the behaviours remain the same during some documents and change in a predefined order. The model that has these described properties is called a Markov Clustering Topic Model (MCTM) in [@Hospedales2011]. The next section formally formulates the model.
Model formulation
-----------------
This section starts from the introduction of the main notations used through the paper. Denote by $\mathcal{X}$ the vocabulary of all visual words, by $\mathcal{Y}$ the set of all topics and by $\mathcal{Z}$ the set of all behaviours, $x$, $y$ and $z$ are used for elements from these sets, respectively. When an additional element of a set is required it is denoted with a prime, e.g., $z'$ is another element from $\mathcal{Z}$.
Let $\mathbf{x}_t = \{x_{i, t}\}_{i = 1}^{N_t}$ be a set of words for the document $t$, where $N_t$ is the length of the document $t$. Let $\mathbf{x}_{1:T_{tr}} = \{\mathbf{x}_t\}_{t = 1}^{T_{tr}}$ denote a set of all words for the whole dataset, where $T_{tr}$ is the number of documents in the dataset. Similarly, denote by $\mathbf{y}_t = \{y_{i, t}\}_{i = 1}^{N_t}$ and $\mathbf{y}_{1:T_{tr}} = \{\mathbf{y}_t\}_{t = 1}^{T_{tr}}$ a set of topics for the document $t$ and a set of all topics for the whole dataset, respectively. Let $\mathbf{z}_{1:T_{tr}} = \{z_t\}_{t = 1}^{T_{tr}}$ be a set of all behaviours for all documents.
Note that $x$, $y$ and $z$ without subscript denote possible values for a word, topic and behaviour from $\mathcal{X}$, $\mathcal{Y}$ and $\mathcal{Z}$, respectively, while the symbols with subscript denote word, topic and behaviour assignments in particular places in a dataset.
Here, $\boldsymbol{\Phi}$ is a matrix corresponding to the distributions over words given the topics, $\boldsymbol{\Theta}$ is a matrix corresponding to the distributions over topics given behaviours. For a Markov chain of behaviours a vector $\boldsymbol{\pi}$ for a behaviour distribution for the first document and a matrix $\boldsymbol{\Xi}$ for transition probability distributions between the behaviours are introduced: $$\begin{aligned}
\boldsymbol{\Phi} &= \{\phi_{x, y}\}_{x \in \mathcal{X}, y \in \mathcal{Y}}, &\phi_{x, y} &= p(x | y), &\boldsymbol{\phi}_y &= \{\phi_{x, y}\}_{x \in \mathcal{X}};\\
\boldsymbol{\Theta} &= \{\theta_{y, z}\}_{y \in \mathcal{Y}, z \in \mathcal{Z}}, &\theta_{y, z} &= p(y | z), &\boldsymbol{\theta_z} &= \{\theta_{y, z}\}_{y \in \mathcal{Y}};\\
\boldsymbol{\pi} &= \{\pi_z\}_{z \in \mathcal{Z}}, &\pi_z &= p(z); \\
\boldsymbol{\Xi} &= \{\xi_{z', z}\}_{z' \in \mathcal{Z}, z \in \mathcal{Z}}, &\xi_{z', z} &= p(z' | z), &\boldsymbol{\xi}_z &= \{\xi_{z', z}\}_{z' \in \mathcal{Z}},\end{aligned}$$ where the matrices $\boldsymbol{\Phi}$, $\boldsymbol{\Theta}$ and $\boldsymbol{\Xi}$ and the vector $\boldsymbol{\pi}$ are formed as follows. An element of a matrix on the $i$-th row and $j$-th column is a probability of the $i$-th element given the $j$-th one, e.g., $\phi_{x, y}$ is a probability of the word $x$ in the topic $y$. The columns of the matrices are then distributions for corresponding elements, e.g., $\boldsymbol\theta_z$ is a distribution over topics for the behaviour $z$. Elements of the vector $\boldsymbol\pi$ are probabilities of behaviours to be chosen by the first document. All these distributions are categorical.
The introduced distributions form a set $$\boldsymbol{\Omega} = \{\boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\pi}, \boldsymbol{\Xi}\}$$ of model parameters and they are estimated during a learning procedure.
Prior distributions are imposed to all the parameters. Conjugate Dirichlet distributions are used: $$\begin{aligned}
\boldsymbol{\phi}_y &\sim Dir(\boldsymbol{\phi}_y | \boldsymbol{\beta}), &\forall y \in \mathcal{Y};\\
\boldsymbol{\theta}_z &\sim Dir(\boldsymbol{\theta}_z | \boldsymbol{\alpha}), &\forall z \in \mathcal{Z};\\
\boldsymbol{\pi} &\sim Dir(\boldsymbol{\pi} | \boldsymbol{\eta}); \\
\boldsymbol{\xi}_z &\sim Dir(\boldsymbol{\xi}_z | \boldsymbol{\gamma}), &\forall z \in \mathcal{Z},\end{aligned}$$ where $Dir(\cdot)$ is a Dirichlet distribution and $\boldsymbol{\beta}$, $\boldsymbol{\alpha}$, $\boldsymbol{\eta}$ and $\boldsymbol{\gamma}$ are the corresponding hyperparameters. As topics and behaviours are not known a priori and will be specified via the learning procedure, it is impossible to distinguish two topics or two behaviours in advance. This is the reason why all the prior distributions are the same for all topics and all behaviours.
![Graphical representation of the Markov Clustering Topic Model.[]{data-label="fig:graph_model"}](TNNLS-2016-P-6800-R1-MCTM_graph_model)
The generative process for the model is as follows. All the parameters are drawn from the corresponding prior Dirichlet distributions. At each time moment $t$ a behaviour $z_t$ is chosen first for a visual document. The behaviour is sampled using the matrix $\boldsymbol{\Xi}$ according to the behaviour chosen for the previous document. For the first document the behaviour is sampled using the vector $\boldsymbol\pi$.
Once the behaviour is selected, the procedure of choosing visual words repeats for the number of times equal to the length of the current document $N_t$. The procedure consists of two steps — sampling a topic $y_{i, t}$ using the matrix $\boldsymbol\Theta$ according to the chosen behaviour $z_t$ followed by sampling a word $x_{i, t}$ using the matrix $\boldsymbol\Phi$ according to the chosen topic $y_{i, t}$ for each token $i \in \{1, \dotsc, N_t\}$, where a token is a particular place inside a document where a word is assigned. The generative process is summarised in Algorithm \[alg:generative\_em\]. The graphical model, showing the relationships between the variables, can be found in Figure \[fig:graph\_model\].
The number of clips – $T_{tr}$, the length of each clip – $N_t$ $\forall t = \{1, \dotsc, T_{tr}\}$, the hyperparameters – $\boldsymbol{\beta}$, $\boldsymbol{\alpha}$, $\boldsymbol{\eta}$, $\boldsymbol{\gamma}$; The dataset $\textbf{x}_{1:T_{tr}} = \{x_{1, 1}, \dotsc, x_{i, t}, \dotsc, x_{N_{T_{tr}}, T_{tr}}\}$; draw a word distribution for the topic $y$: $$\boldsymbol\phi_y \sim Dir(\boldsymbol\phi_y | \boldsymbol\beta);$$ draw a topic distribution for behaviour $z$: $$\boldsymbol\theta_z \sim Dir(\boldsymbol\theta_z | \boldsymbol\alpha);$$ draw a transition distribution for behaviour $z$: $$\boldsymbol\xi_z \sim Dir(\boldsymbol\xi_z | \boldsymbol\gamma);$$ draw a behaviour probability distribution for the initial document $$\boldsymbol\pi \sim Dir(\boldsymbol\phi | \boldsymbol\eta);$$
draw a behaviour for the document from the initial distribution: $z_t \sim Cat(z_t | \boldsymbol{\pi})$; draw a behaviour for the document based on the behaviour of the previous document: $z_t \sim Cat(z_t | \boldsymbol{\xi}_{z_{t-1}})$; \[alg\_step:gen\_behaviour\] draw a topic for the token $i$ based on the chosen behaviour: $y_{i, t} \sim Cat(y_{i, t} | \boldsymbol{\theta}_{z_t}$); \[alg\_step:gen\_topic\] draw a visual word for the token $i$ based on the chosen topic: $x_{i, t} \sim Cat(x_{i, t} | \boldsymbol{\phi}_{y_{i, t}})$; \[alg\_step:gen\_word\]
The full likelihood of the observed variables $\mathbf{x}_{1:T_{tr}}$, the hidden variables $\mathbf{y}_{1:T_{tr}}$ and $\mathbf{z}_{1:T_{tr}}$ and the set of parameters $\boldsymbol\Omega$ can be written then as follows:
$$\begin{aligned}
&p(\mathbf{x}_{1:T_{tr}}, \mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}, \boldsymbol\Omega | \boldsymbol\beta, \boldsymbol\alpha, \boldsymbol\eta, \boldsymbol\gamma) = \nonumber\\
& \underbrace{p(\boldsymbol\pi | \boldsymbol\eta) \, p(\boldsymbol\Xi | \boldsymbol\gamma) \, p(\boldsymbol\Theta | \boldsymbol\alpha) \, p(\boldsymbol\Phi | \boldsymbol\beta)}_{\text{Priors}} \times \nonumber\\
\label{eq:full_likelihood}
& \underbrace{p(z_1 | \boldsymbol\pi) \left[\prod\limits_{t = 2}^{T_{tr}} p(z_t | z_{t-1}, \boldsymbol\Xi) \right] \prod\limits_{t = 1}^{T_{tr}} \prod\limits_{i = 1}^{N_t} p(x_{i, t} | y_{i, t}, \boldsymbol\Phi) p(y_{i, t} | z_t, \boldsymbol\Theta)}_{\text{Likelihood}}\end{aligned}$$
In [@Hospedales2011] Gibbs sampling is implemented for parameters learning in the MCTM. We propose two new learning algorithms: based on an EM-algorithm for the MAP estimates of the parameters and based on variational Bayes inference to estimate posterior distributions of the parameters. We introduce the proposed learning algorithms below and briefly review the Gibbs sampling scheme.
Parameters learning {#sec:inference}
===================
Learning: EM-algorithm scheme {#sec:em}
-----------------------------
We propose a learning algorithm for MAP estimates of the parameters based on the Expectation-Maximisation algorithm [@Dempster77]. The algorithm consists of repeating E and M-steps. Conventionally, the EM-algorithm is applied to get maximum likelihood estimates. In that case the M-step is: $$\mathcal{Q}(\boldsymbol\Omega, \boldsymbol\Omega^{\text{old}}) \longrightarrow \max\limits_{\boldsymbol\Omega},$$ where $\boldsymbol\Omega^{\text{old}}$ denotes the set of parameters obtained at the previous iteration and $\mathcal{Q}(\boldsymbol\Omega, \boldsymbol\Omega^{\text{old}})$ is the expected logarithm of the full likelihood function of the observed and hidden variables: $$\begin{gathered}
\mathcal{Q}(\boldsymbol\Omega, \boldsymbol\Omega^{\text{old}}) = \\
\mathbb{E}_{p(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}})} \log p(\mathbf{x}_{1:T_{tr}}, \mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}} | \boldsymbol\Omega).\end{gathered}$$ The subscript of the expectation sign means the distribution, with respect to which the expectation is calculated. During the E-step the posterior distribution of the hidden variables is estimated given the current estimates of the parameters.
In this paper the EM-algorithm is applied to get MAP estimates instead of traditional maximum likelihood ones. The M-step is modified in this case as: $$\label{eq:M_step_functional}
\mathcal{Q}(\boldsymbol\Omega, \boldsymbol\Omega^{\text{old}}) + \log p(\boldsymbol\Omega | \boldsymbol\beta, \boldsymbol\alpha, \boldsymbol\eta, \boldsymbol\gamma) \longrightarrow \max\limits_{\boldsymbol\Omega},$$ where $p(\boldsymbol\Omega | \boldsymbol\beta, \boldsymbol\alpha, \boldsymbol\eta, \boldsymbol\gamma)$ is the prior distribution of the parameters.
As the hidden variables are discrete, the expectation converts to a sum of all possible values for the whole set of the hidden variables $\{\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}\}$. The substitution of the likelihood expression from (\[eq:full\_likelihood\]) into (\[eq:M\_step\_functional\]) allows to marginalise some hidden variables from the sum. The remaining distributions that are required for computing the $\mathcal{Q}$-function are as follows:
- $p(z_1 = z | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}})$ — the posterior distribution of a behaviour for the first document;
- $p(z_t = z', z_{t-1} = z | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}})$ — the posterior distribution of two behaviours for successive documents;
- $p(y_{i, t} = y | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}})$ — the posterior distribution of a topic assignment for a given token;
- $p(y_{i, t} = y, z_t = z | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}})$ — the joint posterior distribution of a topic and behaviour assignments for a given token.
With the fixed current values for these posterior distributions the estimates of the parameters that maximise the required functional of the M-step (\[eq:M\_step\_functional\]) can be computed as: $$\begin{aligned}
\label{eq:M:phi}
\widehat{\phi}_{x, y}^{\, \text{EM}} &= \dfrac{\left(\beta_x + \hat{n}_{x, y}^{\,\text{EM}} - 1\right)_+}{\sum\limits_{x' \in \mathcal{X}} \left(\beta_{x'} + \hat{n}_{x', y}^{\,\text{EM}} - 1\right)_{+}}, &\forall x \in \mathcal{X}, y \in \mathcal{Y};\\
\label{eq:M:theta}
\widehat{\theta}_{y, z}^{\, \text{EM}} &= \dfrac{\left(\alpha_y + \hat{n}_{y, z}^{\,\text{EM}} - 1\right)_+}{\sum\limits_{y' \in \mathcal{Y}} \left(\alpha_{y'} + \hat{n}_{y', z}^{\,\text{EM}} - 1\right)_+}, &\forall y \in \mathcal{Y}, z \in \mathcal{Z};\\
\label{eq:M_psi_k,l}
\widehat{\xi}_{z', z}^{\, \text{EM}} &= \dfrac{\left(\gamma_{z'} + \hat{n}_{z', z}^{\, \text{EM}} - 1\right)_+}{\sum\limits_{\check{z} \in \mathcal{Z}} \left(\gamma_{\check{z}} + \hat{n}_{\check{z}, z}^{\, \text{EM}} - 1\right)_{+}}, &\forall z', z \in \mathcal{Z};\\
\label{eq:M:pi}
\widehat{\pi}_z^{\, \text{EM}} &= \dfrac{\left(\eta_z + \hat{n}_{z}^{\, \text{EM}} - 1\right)_+}{\sum\limits_{z' \in \mathcal{Z}} \left(\eta_{z'} + \hat{n}_{z'}^{\, \text{EM}} - 1\right)_+}, &\forall z \in \mathcal{Z},\end{aligned}$$ where $(a)_+ \stackrel{\text{def}}{=} \max(a, 0)$ [@Vorontsov2014ARTMArticle]; $\beta_x$, $\alpha_y$ and $\gamma_{z'}$ are the elements of the hyperparameter vectors $\boldsymbol\beta$, $\boldsymbol\alpha$ and $\boldsymbol\gamma$, respectively; and $\hat{n}_{x, y}^{\,\text{EM}} = \sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} p(y_{i, t} = y | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}}) \mathbb{I}(x_{i, t} = x)$ is the expected number of times, when the word $x$ is associated to the topic $y$, where $\mathbb{I}(\cdot)$ is the indicator function; $\hat{n}_{y, z}^{\,\text{EM}} = \sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} p(y_{i,t} = y, z_t = z | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}})$ is the expected number of times, when the topic $y$ is associated to the behaviour $z$; $\hat{n}_{z}^{\, \text{EM}} = p(z_1 = z | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}})$ is the “expected number of times”, when the behaviour $z$ is associated to the first document, in this case the “expected number” is just a probability, the notation is used for the similarity with the rest of the parameters; $\hat{n}_{z', z}^{\, \text{EM}} = \sum\limits_{t = 2}^{T_{tr}} p(z_t = z', z_{t - 1} = z| \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}})$ is the expected number of times, when the behaviour $z$ is followed by the behaviour $z'$.
During the E-step with the fixed current estimates of the parameters $\boldsymbol\Omega^{\text{old}}$, the updated values for the posterior distributions of the hidden variables should be computed. The derivation of the updated formulae for these distributions is similar to the Baum-Welch forward-backward algorithm [@Murphy2012], where the EM-algorithm is applied to the maximum likelihood estimates for a Hidden Markov Model (HMM). This similarity appears because the generative model can be viewed as extension of a HMM.
For effective computation of the required posterior distributions the additional variables $\acute{\alpha}_z(t)$ and $\acute{\beta}_z(t)$ are introduced. A dynamic programming technique is applied for computation of these variables. Having the updated values for $\acute{\alpha}_z(t)$ and $\acute{\beta}_z(t)$ one can update the required posterior distributions of the hidden variables. The E-step is then formulated as follows (for simplification of notation the superscript “old” for the parameters variables is omitted inside the formulae):
$$\begin{aligned}
\label{eq:E:alpha}
&\begin{cases}
\begin{aligned}
&\acute{\alpha}_z(t) = \prod\limits_{i = 1}^{N_t} \sum\limits_{y \in \mathcal{Y}} \phi_{x_{i, t}, y} \, \theta_{y, z} \times\\
&\quad\sum\limits_{z'\in \mathcal{Z}} \acute{\alpha}_{z'}(t-1) \xi_{z, \tilde{z}}, \,\text{if}\, t \geq 2;
\end{aligned}\\
\acute{\alpha}_{z}(1) = \pi_z \prod\limits_{i = 1}^{N_1} \sum\limits_{y\in \mathcal{Y}} \phi_{x_{i, 1}, y} \, \theta_{y, z};
\end{cases}\\
\label{eq:E:beta}
&\begin{cases}
\begin{aligned}
&\acute{\beta}_{z}(t) = \sum\limits_{z' \in \mathcal{Z}} \acute{\beta}_{z'}(t+1) \xi_{z', z} \times \\
&\quad \prod\limits_{i = 1}^{N_{t+1}} \sum\limits_{y \in \mathcal{Y}} \phi_{x_{i, t+1}, y} \, \theta_{y, z'} , \,\text{if}\, t < T_{tr};
\end{aligned}\\
\acute{\beta}_{z}(T_{tr}) = 1;
\end{cases}\\
\label{eq:E:normalisation_const}
&K = \sum\limits_{z \in \mathcal{Z}} \acute{\alpha}_{z}(1) \acute{\beta}_{z}(1);\\
\label{eq:E:z_t}
&p(z_1 | \textbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}}) = \dfrac{\acute{\alpha}_{z_1}(1) \acute{\beta}_{z_1}(1)}{K}; \\
\label{eq:E:z_t, z_t-1}
&\begin{aligned}
&p(z_t, z_{t-1} | \textbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}}) = \dfrac{\acute{\alpha}_{z_{t-1}}(t-1) \acute{\beta}_{z_t}(t) \xi_{z_t, z_{t-1}}}{K} \times\\
&\quad \prod\limits_{i = 1}^{N_t} \sum\limits_{y \in \mathcal{Y}} \phi_{x_{i, t}, y} \theta_{y, z_t};
\end{aligned}\\
\label{eq:E:y_i,t, z_t}
&\begin{cases}
\begin{aligned}
&p(y_{i, t}, z_t | \textbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}})
= \dfrac{\phi_{x_{i, t}, y_{i, t}} \theta_{y_{i, t}, z_t} \acute{\beta}_{z_t}(t)}{K} \times\\
&\quad\sum\limits_{z' \in \mathcal{Z}} \acute{\alpha}_{z'}(t-1) \xi_{z_t, z'} \prod\limits_{\substack{j = 1 \\ j \neq i}}^{N_t} \sum\limits_{y' \in \mathcal{Y}} {\phi_{x_{j, t}, y'} \theta_{y', z_t}},\,\text{if}\, t \geq 2;
\end{aligned} \\
\begin{aligned}
&p(y_{i, 1}, z_1 | \textbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}})
= \dfrac{\phi_{x_{i, 1}, y_{i, 1}} \theta_{y_{i, 1}, z_1} \acute{\beta}_{z_1}(1)}{K} \times\\
&\quad \pi_{z_1} \prod\limits_{\substack{j = 1 \\ j \neq i}}^{N_1} \sum\limits_{y' \in \mathcal{Y}} {\phi_{x_{j, 1}, y'} \theta_{y', z_1}};
\end{aligned}
\end{cases} \\
\label{eq:E:y_i,t}
&\begin{aligned}
p(y_{i, t} | \textbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}}) &= \sum\limits_{z \in \mathcal{Z}} p(y_{i, t}, z | \textbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{\text{old}}),
\end{aligned}\end{aligned}$$
where $K$ is a normalisation constant for all the posterior distributions of the hidden variables.
Starting with some random initialisation of the parameter estimates, the EM-algorithm iterates the E and M-steps until convergence. The obtained estimates of the parameters are used for further analysis.
Learning: Variational Bayes scheme {#sec:vb}
----------------------------------
We also propose a learning algorithm based on the variational Bayes (VB) approach [@Jordan1999] to find approximated posterior distributions for both the hidden variables and the parameters.
In the VB inference scheme the true posterior distribution, in this case the distribution of the parameters and the hidden variables $p(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}, \boldsymbol\Omega | \mathbf{x}_{1:T_{tr}}, \boldsymbol\eta, \boldsymbol\gamma, \boldsymbol\alpha, \boldsymbol\beta)$, is approximated with a factorised distribution — $q(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}, \boldsymbol\Omega)$. The approximation is made to minimise the Kullback-Leibler divergence between the factorised distribution and true one. We factorise the distribution in order to separate the hidden variables and the parameters: $$\begin{gathered}
\label{eq:vb_factorisation}
\hat{q}(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}, \boldsymbol\Omega) = \hat{q}(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}) \hat{q}(\boldsymbol\Omega) \stackrel{\text{def}}{=} \\
\operatorname*{argmin}\mathrm{KL} \left(q(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}) q(\boldsymbol\Omega) || \right.\\
\left.p(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}, \boldsymbol\Omega | \mathbf{x}_{1:T_{tr}}, \boldsymbol\eta, \boldsymbol\gamma, \boldsymbol\alpha, \boldsymbol\beta)\right),\end{gathered}$$ where $\mathrm{KL}$ denotes the Kullback-Leibler divergence. The minimisation of the Kullback-Leibler divergence is equivalent to the maximisation of the evidence lower bound (ELBO). The maximisation is done by coordinate ascent [@Jordan1999].
During the update of the parameters the approximated distribution $q(\boldsymbol\Omega)$ is further factorised: $$\label{eq:q_param_factorisation}
q(\boldsymbol\Omega) = q(\boldsymbol\pi) q(\boldsymbol\Xi) q(\boldsymbol\Theta) q(\boldsymbol\Phi).$$ Note that this factorisation of approximated parameter distributions is a corollary of our model and not an assumption.
The iterative process of updating the approximated distributions of the parameters and the hidden variables can be formulated as an EM-like algorithm, where during the E-step the approximated distributions of the hidden variables are updated and during the M-step the approximated distributions of the parameters are updated.
The M-like step is as follows: $$\begin{aligned}
\label{eq:VB:beta}
&\begin{cases}
q(\boldsymbol\Phi) = \prod\limits_{y \in \mathcal{Y}} Dir\left(\boldsymbol\phi_y; \tilde{\boldsymbol\beta}_y\right),\\
\tilde{\beta}_{x, y} = \beta_x + \hat{n}_{x, y}^{\, \text{VB}}, &\forall x \in \mathcal{X}, y \in \mathcal{Y};
\end{cases}\\
&\begin{cases}
q(\boldsymbol\Theta) = \prod\limits_{z \in \mathcal{Z}} Dir(\boldsymbol\theta_z; \tilde{\boldsymbol\alpha}_z),\\
\tilde{\alpha}_{y, z} = \alpha_y + \hat{n}_{y, z}^{\, \text{VB}}, &\forall y \in \mathcal{Y}, z \in \mathcal{Z};
\end{cases}\\
&\begin{cases}
q(\boldsymbol\pi) = Dir(\boldsymbol\pi; \tilde{\boldsymbol\eta}),\\
\tilde{\eta}_z = \eta_z + \hat{n}_z^{\,\text{VB}}, &\forall z \in \mathcal{Z};
\end{cases}\\
\label{eq:VB:gamma}
&\begin{cases}
q(\boldsymbol\Xi) = \prod\limits_{z \in \mathcal{Z}} Dir(\boldsymbol\xi_{z}; \tilde{\boldsymbol\gamma}_z),\\
\tilde{\gamma}_{z', z} = \gamma_{z'} + \hat{n}_{z', z}^{\, \text{VB}}, &\forall z', z \in \mathcal{Z},
\end{cases}\end{aligned}$$ where $\tilde{\boldsymbol\beta}_y$, $\tilde{\boldsymbol\alpha}_z$, $\tilde{\boldsymbol\eta}$ and $\tilde{\boldsymbol\gamma}_z$ are updated hyperparameters of the corresponding posterior Dirichlet distributions; and $\hat{n}_{x, y}^{\, \text{VB}} = \sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \mathbb{I}(x_{i, t} = x) q(y_{i, t} = y)$ is the expected number of times, when the word $x$ is associated with the topic $y$. Here and below the expected number is computed with respect to the approximated posterior distributions of the hidden variables; $\hat{n}_{y, z}^{\, \text{VB}} = \sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} q(y_{i, t} = y, z_t = z)$ is the expected number of times, when the topic $y$ is associated with the behaviour $z$; $\hat{n}_z^{\,\text{VB}} = q(z_1 = z)$ is the “expected number” of times, when the behaviour $z$ is associated to the first document; $\hat{n}_{z', z}^{\, \text{VB}} = \sum\limits_{t = 2}^{T_{tr}} q(z_t = z', z_{t-1} = z)$ is the expected number of times, when the behaviour $z$ is followed by the behaviour $z'$.
The following additional variables are introduced for the E-like step: $$\begin{aligned}
\label{eq:VB:introduced_pi}
\tilde{\pi}_z &= \exp\left(\psi\left(\tilde{\eta}_z\right) - \psi\left(\sum\limits_{z' \in \mathcal{Z}} \tilde{\eta}_{z'}\right)\right)\\
\tilde{\xi}_{\tilde{z}, z} &= \exp\left(\psi\left(\tilde{\gamma}_{\tilde{z}, z}\right) - \psi\left(\sum\limits_{z' \in \mathcal{Z}} \tilde{\gamma}_{z', z}\right)\right);\\
\tilde{\phi}_{x, y} &= \exp\left(\psi\left(\tilde{\beta}_{x, y} \right) - \psi\left(\sum\limits_{x' \in \mathcal{X}} \tilde{\beta}_{x', y} \right)\right);\\
\label{eq:VB:introduced_theta}
\tilde{\theta}_{y, z} &= \exp\left(\psi\left(\tilde{\alpha}_{y, z}\right) - \psi\left(\sum\limits_{y' \in \mathcal{Y}}\tilde{\alpha}_{y', z}\right)\right),\end{aligned}$$ where $\psi(\cdot)$ is the digamma function.
Using these additional notations, the E-like step is formulated the same as the E-step of the EM-algorithm, replacing everywhere the estimates of the parameters with the corresponding tilde introduced notation and true posterior distributions of the hidden variables with the corresponding approximated ones in (\[eq:E:alpha\]) – (\[eq:E:y\_i,t\]).
The point estimates of the parameters can be obtained by expected values of the posterior approximated distributions. An expected value for a Dirichlet distribution (a posterior distribution for all the parameters) is a normalised vector of hyperparameters. Using the expressions for the hyperparameters from (\[eq:VB:beta\]) – (\[eq:VB:gamma\]), the final parameters estimates can be obtained by: $$\begin{aligned}
\label{eq:VB:phi}
\widehat{\phi}_{x, y}^{\,\text{VB}} &= \dfrac{\beta_x + \hat{n}_{x, y}^{\, \text{VB}}}{\sum\limits_{x' \in \mathcal{X}} \left(\beta_{x'} + \hat{n}_{x', y}^{\, \text{VB}}\right)}, &\forall x \in \mathcal{X}, y \in \mathcal{Y};\\
\label{eq:VB:theta}
\widehat{\theta}_{y, z}^{\,\text{VB}} &= \dfrac{\alpha_y + \hat{n}_{y, z}^{\, \text{VB}}}{\sum\limits_{y' \in \mathcal{Y}} \left(\alpha_{y'} + \hat{n}_{y', z}^{\, \text{VB}}\right)}, &\forall y \in \mathcal{Y}, z \in \mathcal{Z};\\
\label{eq:VB:xi}
\widehat{\xi}_{z', z}^{\,\text{VB}} &= \dfrac{\gamma_{z'} + \hat{n}_{z', z}^{\, \text{VB}}}{\sum\limits_{\check{z} \in \mathcal{Z}} \left(\gamma_{\check{z}} + \hat{n}_{\check{z}, z}^{\, \text{VB}}\right)}, &\forall z', z \in \mathcal{Z};\\
\label{eq:VB:pi}
\widehat{\pi}_{z}^{\,\text{VB}} &= \dfrac{\eta_z + \hat{n}_z^{\,\text{VB}}}{\sum\limits_{z' \in \mathcal{Z}} \left(\eta_{z'} + \hat{n}_{z'}^{\,\text{VB}}\right)}, &\forall z \in \mathcal{Z}.\end{aligned}$$
Learning: Gibbs sampling algorithm {#sec:gibbs}
----------------------------------
In [@Hospedales2011] the collapsed version of Gibbs sampling (GS) is used for parameter learning in the MCTM. The Markov chain is built to sample only the hidden variables $y_{i, t}$ and $z_t$, while the parameters $\boldsymbol{\Phi}$, $\boldsymbol{\Theta}$ and $\boldsymbol{\Xi}$ are integrated out (note that the distribution for the initial behaviour choice $\boldsymbol\pi$ is not considered in [@Hospedales2011]).
During the burn-in stage the hidden topic and behaviour assignments to each token in the dataset are drawn from the conditional distributions given all the remaining variables. Following the Markov Chain Monte Carlo framework it would draw samples from the posterior distribution $p(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\beta, \boldsymbol\alpha, \boldsymbol\eta, \boldsymbol\gamma)$. From the whole sample for $\{\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}\}$ the parameters can be estimated by [@Griffiths2004]: $$\begin{aligned}
\label{eq:GS:phi}
\widehat{\phi}_{x, y}^{\,\text{GS}} &= \dfrac{\hat{n}_{x, y}^{\,\text{GS}} + \beta_x}{\sum\limits_{x' \in \mathcal{X}} \left(\hat{n}_{x', y}^{\,\text{GS}} + \beta_{x'}\right)}, &\forall x \in \mathcal{X}, y \in \mathcal{Y};\\
\widehat{\theta}_{y, z}^{\,\text{GS}} &= \dfrac{\hat{n}_{y, z}^{\,\text{GS}} + \alpha_y}{\sum\limits_{y' \in \mathcal{Y}} \left(\hat{n}_{y', z}^{\,\text{GS}} + \alpha_{y'} \right)}, &\forall y \in \mathcal{Y}, z \in \mathcal{Z};\\
\label{eq:GS:xi}
\widehat{\xi}_{z', z}^{\,\text{GS}} &= \dfrac{\hat{n}_{z', z}^{\,\text{GS}} + \gamma_{z'}}{\sum\limits_{\check{z} \in \mathcal{Z}} \left(\hat{n}_{\check{z}, z}^{\,\text{GS}} + \gamma_{\check{z}} \right)}, &\forall z', z \in \mathcal{Z},\end{aligned}$$ where $\hat{n}_{x, y}^{\,\text{GS}}$ is the count for the number of times when the word $x$ is associated with the topic $y$, $\hat{n}_{y, z}^{\,\text{GS}}$ is the count for the topic $y$ and the behaviour $z$ pair, $\hat{n}_{z', z}^{\,\text{GS}}$ is the count for the number of times when the behaviour $z$ is followed by the behaviour $z'$.
Similarities and differences of the learning algorithms {#sec:comparison}
-------------------------------------------------------
The point parameter estimates for all three learning algorithms (\[eq:M:phi\]) – (\[eq:M:pi\]), (\[eq:VB:phi\]) – (\[eq:VB:pi\]) and (\[eq:GS:phi\]) – (\[eq:GS:xi\]) have a similar form. The EM-algorithm estimates differ up to the hyperparameters reassignment — adding one to all the hyperparameters in the VB or GS algorithms ends up with the same final equations for the parameters estimates in the EM-algorithm. We explore this in the experimental part. This “-1” term in the EM-algorithm formulae (\[eq:M:phi\]) – (\[eq:M\_psi\_k,l\]) occurs because it uses modes of the posterior distributions while the point estimates obtained by the VB and GS algorithms are means of the corresponding posterior distributions. For a Dirichlet distribution, which is a posterior distribution for all the parameters, mode and mean expressions differ by this “-1” term.
The main differences of the methods consist in the ways the counts $n_{x, y}$, $n_{y, z}$ and $n_{z', z}$ are estimated. In the GS algorithm they are calculated by a single sample from the posterior distribution of the hidden variables $p(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\beta, \boldsymbol\alpha, \boldsymbol\gamma)$. In the EM-algorithm the counts are computed as expected numbers of the corresponding events with respect to the posterior distributions of the hidden variables. In the VB algorithm the counts are computed in the same way as in the EM-algorithm up to replacing the true posterior distributions with the approximated ones.
Our observations for the dynamic topic model confirm the comparison results for the vanilla PLSA and LDA models provided in [@Asuncion2009].
Anomaly detection {#sec:abnormality}
=================
This paper presents on-line anomaly detection with the MCTM in video streams. The decision making procedure is divided into two stages. At a learning stage the parameters are estimated using $T_{tr}$ visual documents by one of the learning algorithms, presented in Section \[sec:inference\]. After that during a testing stage a decision about abnormality of new upcoming testing documents is made comparing a marginal likelihood of each document with a threshold. The likelihood is computed using the parameters obtained during the learning stage. The threshold is a parameter of the method and can be set empirically, for example, to label 2% of the testing data as abnormal. This paper presents a comparison of the algorithms (Section \[sec:experiments\]) using the measure independent of threshold value selection.
We also propose an anomaly localisation procedure during the testing stage for those visual documents that are labelled as abnormal. This procedure is designed to provide spatial information about anomalies, while documents labelled as abnormal provide temporal detection. The following sections introduce both the anomaly detection procedure on a document level and the anomaly localisation procedure within a video frame.
Abnormal documents detection
----------------------------
The marginal likelihood of a new visual document $\mathbf{x}_{t+1}$ given all the previous data $\mathbf{x}_{1:t}$ can be used as a normality measure of the document [@Hospedales2011]: $$\begin{gathered}
\label{eq:online_likelihood_integral}
p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}) = \\
\iiint p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}, \boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi}) p(\boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi} | \mathbf{x}_{1:t}) \mathrm{d}\boldsymbol\Phi \mathrm{d}\boldsymbol\Theta \mathrm{d} \boldsymbol{\Xi}.\end{gathered}$$
If the likelihood value is small it means that the current document cannot be fitted to the learnt behaviours and topics, which represent typical motion patterns. Therefore, this is an indication for an abnormal event in this document. The decision about abnormality of a document is then made by comparing the marginal likelihood of the document with the threshold.
In real world applications it is essential to detect anomalies as soon as possible. Hence an approximation of the integral in (\[eq:online\_likelihood\_integral\]) is used for efficient computation. The first approximation is based on the assumption that the training dataset is representative for parameter learning, which means that the posterior probability of the parameters would not change if there is more observed data: $$p(\boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi} | \mathbf{x}_{1:t}) \approx p(\boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi} | \mathbf{x}_{1:Tr}) \quad \forall t \geq T_{tr}.$$
The marginal likelihood can be then approximated as $$\begin{gathered}
\label{eq:online_likelihood_integral_train}
\iiint p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}, \boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi}) p(\boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi} | \mathbf{x}_{1:t}) \mathrm{d}\boldsymbol\Phi \mathrm{d}\boldsymbol\Theta \mathrm{d} \boldsymbol{\Xi} \approx \\
\iiint p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}, \boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi}) p(\boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi} | \mathbf{x}_{1:T_{tr}}) \mathrm{d}\boldsymbol\Phi \mathrm{d}\boldsymbol\Theta \mathrm{d} \boldsymbol{\Xi}.\end{gathered}$$
Depending on the algorithm used for learning the integral in (\[eq:online\_likelihood\_integral\_train\]) can be further approximated in different ways. We consider two types of approximation.
### Plug-in approximation
The point estimates of the parameters can be plug-in in the integral (\[eq:online\_likelihood\_integral\_train\]) for approximation: $$\begin{gathered}
\label{eq:online_likelihood_plugin}
\iiint p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}, \boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi}) p(\boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi} | \mathbf{x}_{1:Tr}) \mathrm{d}\boldsymbol\Phi \mathrm{d}\boldsymbol\Theta \mathrm{d} \boldsymbol{\Xi} \approx\\
\iiint p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}, \boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi}) \delta_{\hat{\boldsymbol{\Phi}}}(\boldsymbol{\Phi}) \delta_{\hat{\boldsymbol{\Theta}}}(\boldsymbol{\Theta}), \delta_{\hat{\boldsymbol{\Xi}}}(\boldsymbol{\Xi}) \mathrm{d}\boldsymbol\Phi \mathrm{d}\boldsymbol\Theta \mathrm{d} \boldsymbol{\Xi} = \\
p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}, \hat{\boldsymbol{\Phi}}, \hat{\boldsymbol{\Theta}}, \hat{\boldsymbol{\Xi}}),\end{gathered}$$ where $\delta_{a}(\cdot)$ is the delta-function with the centre in $a$; $\hat{\boldsymbol{\Phi}}$, $\hat{\boldsymbol{\Theta}}$, $\hat{\boldsymbol{\Xi}}$ are point estimates of the parameters, which can be computed by any of the considered learning algorithms using (\[eq:M:phi\]) – (\[eq:M\_psi\_k,l\]), (\[eq:VB:phi\]) – (\[eq:VB:xi\]) or (\[eq:GS:phi\]) – (\[eq:GS:xi\]).
The product and sum rules, the conditional independence equations from the generative model are then applied and the final formula for the plug-in approximation is as follows: $$\begin{gathered}
\label{eq:online_likelihood_final}
p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}) \approx p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}, \hat{\boldsymbol{\Phi}}, \hat{\boldsymbol{\Theta}}, \hat{\boldsymbol{\Xi}}) =\\
\sum\limits_{z_t}\sum\limits_{z_{t+1}} \left[ p(\mathbf{x}_{t+1} | z_{t+1}, \hat{\boldsymbol{\Phi}}, \hat{\boldsymbol{\Theta}}) \times \right.\\
\left. p(z_{t+1} | z_t, \hat{\boldsymbol{\Xi}}) p(z_t | \mathbf{x}_{1:t}, \hat{\boldsymbol{\Phi}}, \hat{\boldsymbol{\Theta}}, \hat{\boldsymbol{\Xi}})\right],\end{gathered}$$ where the predictive probability of the behaviour for the current document, given the observed data up to the current document, can be computed via the recursive formula: $$\begin{gathered}
\label{eq:predictive_behaviour}
p(z_{t} | \mathbf{x}_{1:t}, \hat{\boldsymbol{\Phi}}, \hat{\boldsymbol{\Theta}}, \hat{\boldsymbol{\Xi}}) = \\
\sum_{z_{t-1}} \dfrac{p(\mathbf{x}_{t} | z_{t}, \hat{\boldsymbol{\Phi}}, \hat{\boldsymbol{\Theta}}) p(z_{t} | z_{t-1}, \hat{\boldsymbol{\Xi}}) p(z_{t-1} | \mathbf{x}_{1:t-1}, \hat{\boldsymbol{\Phi}}, \hat{\boldsymbol{\Theta}}, \hat{\boldsymbol{\Xi}})}{p(\mathbf{x}_{t} | \mathbf{x}_{1:t-1}, \hat{\boldsymbol{\Phi}}, \hat{\boldsymbol{\Theta}}, \hat{\boldsymbol{\Xi}})}.\end{gathered}$$
The point estimates can be computed for all three learning algorithms, therefore a normality measure based on the plug-in approximation of the marginal likelihood is applicable for all of them.
### Monte Carlo approximation
If samples $\{\boldsymbol{\Phi}^{s}, \boldsymbol{\Theta}^{s}, \boldsymbol{\Xi}^{s}\}$ from the posterior distribution $p(\boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi} | \mathbf{x}_{1:T_{tr}})$ of the parameters can be obtained, the integral (\[eq:online\_likelihood\_integral\_train\]) is further approximated by the Monte Carlo method: $$\begin{gathered}
\iiint p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}, \boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi}) p(\boldsymbol{\Phi}, \boldsymbol{\Theta}, \boldsymbol{\Xi} | \mathbf{x}_{1:T_{tr}}) \mathrm{d}\boldsymbol\Phi \mathrm{d}\boldsymbol\Theta \mathrm{d} \boldsymbol{\Xi} \approx\\
\dfrac{1}{S} \sum\limits_{s = 1}^{S} p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}, \boldsymbol{\Phi}^{s}, \boldsymbol{\Theta}^{s}, \boldsymbol{\Xi}^{s}),\end{gathered}$$ where $S$ is the number of samples. These samples can be obtained (i) from the approximated posterior distributions $q(\boldsymbol{\Phi})$, $q(\boldsymbol{\Theta})$, and $q(\boldsymbol{\Xi})$ of the parameters, computed by the VB learning algorithm, or (ii) from the independent samples of the GS scheme. For the conditional likelihood $p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}, \boldsymbol{\Phi}^{s}, \boldsymbol{\Theta}^{s}, \boldsymbol{\Xi}^{s})$ the formula (\[eq:online\_likelihood\_final\]) is valid.
Note that for the approximated posterior distribution of the parameters, i.e., the output of the VB learning algorithm, the integral (\[eq:online\_likelihood\_integral\_train\]) can be resolved analytically, but it would be computationally infeasible. This is the reason why the Monte Carlo approximation is used in this case.
Finally, in order to compare documents of different lengths the normalised likelihood is used as a normality measure $s$: $$s(\mathbf{x}_{t+1}) = \dfrac{1}{N_{t+1}} p(\mathbf{x}_{t+1} | \mathbf{x}_{1:t}).$$
Localisation of anomalies {#sec:localisation}
-------------------------
The topic modeling approach allows to compute a likelihood function not only of the whole document but of an individual word within the document too. Recall that the visual word contains the information about a location in the frame. We propose to use the location information from the least probable words (e.g., 10 words with the least likelihood values) to localise anomalies in the frame. Note, we do not require anything additional to a topic model, e.g., modelling regional information explicitly as in [@Haines2010] or comparing a test document with training ones as in [@Pathak2015]. Instead, the proposed anomaly localisation procedure is general and can be applied in any topic modeling based method, where spatial information is encoded to visual words.
The marginal likelihood of a word can be computed in a similar way to the likelihood of the whole document. For the point estimates of the parameters and plug-in approximation of the integral it is: $$p(x_{i, t+1} | \mathbf{x}_{1:t}) \approx
p(x_{i, t+1} | \mathbf{x}_{1:t}, \hat{\boldsymbol{\Phi}}, \hat{\boldsymbol{\Theta}}, \hat{\boldsymbol{\Xi}}).$$ For the samples from the posterior distributions of the parameters and the Monte Carlo integral approximation it is: $$p(x_{i, t+1} | \mathbf{x}_{1:t}) \approx
\dfrac{1}{S} \sum\limits_{s = 1}^{S} p(x_{i, t+1} | \mathbf{x}_{1:t}, \boldsymbol{\Phi}^{s}, \boldsymbol{\Theta}^{s}, \boldsymbol{\Xi}^{s}).$$
Performance validation {#sec:experiments}
======================
We compare the two proposed learning algorithms, based on EM and VB, with the GS algorithm, proposed in [@Hospedales2011], on two real datasets.
Setup
-----
The performance of the algorithms is compared on the QMUL street intersection data [@Hospedales2011] and Idiap traffic junction data [@Varadarajan2009]. Both datasets are $45$-minutes video sequences, captured busy traffic road junctions, where we use a $5$-minute video sequence as a training dataset and others as a testing one. The documents that have less than $20$ visual words are discarded from consideration. In practice these documents can be classified to be normal by default as there is no enough information to make a decision. The frame size for both datasets is $288 \times 360$. Sample frames are presented in Figure \[fig:sample\_frames\].
The size of grid cells is set to $8 \times 8$ pixels for spatial quantisation of the local motion for visual word determination. Non-overlapping clips with a one second length are treated as visual documents.
We also study the influence of the hyperparameters on the learning algorithms. In all the experiments we use the symmetric hyperparameters: $\boldsymbol\alpha = \{\alpha, \dotsc, \alpha\}$, $\boldsymbol\beta = \{\beta, \dotsc, \beta\}$, $\boldsymbol\gamma = \{\gamma, \dotsc, \gamma\}$ and $\boldsymbol\eta = \{\eta, \dotsc, \eta\}$. The three groups of the hyperparameters settings are compared: $\{\alpha = 1, \beta = 1, \gamma = 1, \eta = 1\}$ (referred as “prior type 1”), $\{\alpha = 8 , \beta = 0.05, \gamma = 1, \eta = 1\}$ (“prior type H”) and $\{\alpha = 9, \beta = 1.05, \gamma = 2, \eta = 2\}$ (“prior type H+1”). Note that the first group corresponds to the case when in the EM-algorithm learning scheme the prior components are cancelled out, i.e., the MAP estimates in this case are equal to the maximum likelihood ones. The equations for the point estimates in the EM learning algorithm with the prior type H+1 of the hyperparameters settings are equal to the equations for the point estimates in the VB and GS learning algorithms with the prior type H of the settings. The corresponding Dirichlet distributions with all used parameters are presented in Figure \[fig:dirichlet\_pdf\].
Note that parameter learning is an ill-posed problem in topic modeling [@Vorontsov2014ARTMArticle]. This means there is no unique solution for parameter estimates. We use $20$ Monte Carlo runs for all the learning algorithms with different random initialisations resulting with different solutions. The mean results among these runs are presented below for comparison.
All three algorithms are run with three different groups of hyperparameters settings. The number of topics and behaviours is set to $8$ and $4$, respectively, for the QMUL dataset, $10$ and $3$ are used for the corresponding values for the Idiap dataset. The EM and VB algorithms are run for $100$ iterations. The GS algorithm is run for $500$ burn-in iterations and independent samples are taken with a $100$ iterations delay after the burn-in period.
Performance measure
-------------------
Anomaly detection performance of the algorithms depends on threshold selection. To make a fair comparison of the different learning algorithms we use a performance measure, which is independent of threshold selection.
In binary classification the following measures [@Murphy2012] are used: $\text{TP}$ — true positive, a number of documents, which are correctly detected as positive (abnormal in our case); $\text{TN}$ — true negative, a number of documents, which are correctly detected as negative (normal in our case); $\text{FP}$ — false positive, a number of documents, which are incorrectly detected as positive, when they are negative; $\text{FN}$ — false negative, a number of documents, which are incorrectly detected as negative, when they are positive; $\text{precision} = \dfrac{\text{TP}}{\text{TP} + \text{FP}}$ — a fraction of correct detections among all documents labelled as abnormal by an algorithm; $\text{recall} = \dfrac{\text{TP}}{\text{TP} + \text{FN}}$ — a fraction of correct detections among all truly abnormal documents.
The area under the precision-recall curve is used as a performance measure in this paper. This measure is more informative for detection of rare events than the popular area under the receiver operating characteristic (ROC) curve [@Murphy2012].
Parameter learning
------------------
We visualise the learnt behaviours for the qualitative assessment of the proposed framework (Figures \[fig:qmul\_behaviours\] and \[fig:idiap\_behaviours\]). For illustrative purposes we consider one run of the EM learning algorithm with the prior type H+1 of the hyperparameters settings.
The behaviours learnt for the QMUL data are shown in Figure \[fig:qmul\_behaviours\] (for visualisation words representing $50\%$ of probability mass of a behaviour are used). One can notice that the algorithm correctly recognises the motion patterns in the data. The general motion of the scene follows a cycle: a vertical traffic flow (the first behaviour in Figure \[fig:qmul\_behav\_1\]), when cars move downward and upward on the road; left and right turns (the fourth behaviour in Figure \[fig:qmul\_behav\_4\]): some cars moving on the “vertical” road turn to the perpendicular road at the end of the vertical traffic flow; a left traffic flow (the second behaviour in Figure \[fig:qmul\_behav\_2\]), when cars move from right to left on the “horizontal” road; and a right traffic flow (the third behaviour in Figure \[fig:qmul\_behav\_3\]), when cars move from left to right on the “horizontal” road. Note that the ordering numbers of behaviours correspond to their internal representation in the algorithm. The transition probability matrix $\boldsymbol{\Xi}$ is used to recognise the correct behaviours order in the data.
Figure \[fig:idiap\_behaviours\] presents the behaviours learnt for the Idiap data. In this case the learnt behaviours have also a clear semantic meaning. The scene motion follows a cycle: a pedestrian flow (the first behaviour in Figure \[fig:idiap\_behav\_1\]), when cars stop in front of the stop line and pedestrians cross the road; a downward traffic flow (the third behaviour in Figure \[fig:idiap\_behav\_3\]), when cars move downward along the road; an upward traffic flow (the second behaviour in Figure \[fig:idiap\_behav\_2\]), when cars from left and right sides move upward on the road.
Anomaly detection {#anomaly-detection}
-----------------
In this section the anomaly detection performance achieved by all three learning algorithms is compared. The datasets contain the number of abnormal events, such as jaywalking, car moving on the opposite lane, disruption of the traffic flow (see examples in Figure \[fig:sample\_abnormalities\]).
For the EM learning algorithm the plug-in approximation of the marginal likelihood is used for anomaly detection. For both the VB and GS learning algorithms both the plug-in and Monte Carlo approximations of the likelihood are used. Note that for the GS algorithm samples are obtained during the learning stage, $5$ and $100$ independent samples are taken. For the VB learning algorithm samples are obtained after the learning stage from the posterior distributions, parameters of which are learnt. This means that the number of samples that are used for anomaly detection does not influence on the computational cost of learning. We test the Monte Carlo approximation of the marginal likelihood with $5$ and $100$ samples for the VB learning algorithm.
As a result, we have $21$ methods to compare: obtained by three learning algorithms, three different groups of hyperparameters settings, one type of marginal likelihood approximation for the EM learning algorithm, two types of marginal likelihood approximation for the VB and GS learning algorithms, where two Monte Carlo approximations are used with $5$ and $100$ samples. The list of methods references can be found in Table \[tab:methods\_references\].
Note that we achieve a very fast decision making performance in our framework. Indeed, anomaly detection is made for approximately $0.0044$ sec per visual document by the plug-in approximation of the marginal likelihood, for $0.0177$ sec per document by the Monte Carlo approximation with $5$ samples and for $0.3331$ sec per document by the Monte Carlo approximation with $100$ samples[^2].
Reference Learning algorithm Hyper-parameters settings Marginal likelihood approximation Number of posterior samples
--------------- -------------------- --------------------------- ----------------------------------- -----------------------------
EM 1 p EM type 1 Plug-in —
EM H p EM type H Plug-in —
EM H+1 p EM type H+1 Plug-in —
VB 1 p VB type 1 Plug-in —
VB 1 mc 5 VB type 1 Monte Carlo 5
VB 1 mc 100 VB type 1 Monte Carlo 100
VB H p VB type H Plug-in —
VB H mc 5 VB type H Monte Carlo 5
VB H mc 100 VB type H Monte Carlo 100
VB H+1 p VB type H+1 Plug-in —
VB H+1 mc 5 VB type H+1 Monte Carlo 5
VB H+1 mc 100 VB type H+1 Monte Carlo 100
GS 1 p GS type 1 Plug-in —
GS 1 mc 5 GS type 1 Monte Carlo 5
GS 1 mc 100 GS type 1 Monte Carlo 100
GS H p GS type H Plug-in —
GS H mc 5 GS type H Monte Carlo 5
GS H mc 100 GS type H Monte Carlo 100
GS H+1 p GS type H+1 Plug-in —
GS H+1 mc 5 GS type H+1 Monte Carlo 5
GS H+1 mc 100 GS type H+1 Monte Carlo 100
: Methods references[]{data-label="tab:methods_references"}
The mean areas under precision-recall curves for anomaly detection for all $21$ compared methods can be found in Figure \[fig:mean\_results\]. Below we examine the results with respect to hyperparameters sensitivity, an influence of the likelihood approximation on the final performance, we also compare the learning algorithms and discuss anomaly localisation results.
### Hyperparameters sensitivity
This section presents sensitivity analysis of the anomaly detection methods with respect to changes of the hyperparameters.
The analysis of the mean areas under curves (Figure \[fig:mean\_results\]) suggests that the hyperparameters almost do not influence on the results of the EM learning algorithm, while there is a significant dependence between hyperparameters changes and results of the VB and GS learning algorithms. These conclusions are confirmed by examination of the individual runs of the algorithms. For example, Figure \[fig:hyperparam\_sensitivity\] presents the precision-recall curves for all $20$ runs with different initialisations of $4$ methods for the QMUL data: the VB learning algorithm using the plug-in approximation of the marginal likelihood with the prior types 1 and H of the hyperparameters settings and the EM learning algorithm with the same prior groups of the hyperparameters settings. One can notice that the variance of the curves for the VB learning algorithm with the prior type 1 is larger than the corresponding variance with the prior type H, while the similar variances for the EM learning algorithm are very close to each other.
Note that the results of the EM learning algorithm with the prior type 1 do not significantly differ from the results with the other priors, despite of the fact that the prior type 1 actually cancels out the prior influence on the parameters estimates and equates the MAP and maximum likelihood estimates. We can conclude that the choice of the hyperparameters settings is not a problem for the EM learning algorithm and we can even simplify the derivations considering only the maximum likelihood estimates without the prior influence.
The VB and GS learning algorithms require a proper choice of the hyperparameters settings as they can significantly change the anomaly detection performance. This choice can be performed empirically or with the type II maximum likelihood approach [@Murphy2012].
### Marginal likelihood approximation influence
In this section the influence of the type of the marginal likelihood approximation on the anomaly detection results is studied.
The average results for both datasets (Figure \[fig:mean\_results\]) demonstrate that the type of the marginal likelihood approximation does not influence remarkably on anomaly detection performance. As the plug-in approximation requires less computational resources both in terms of time and memory (as there is no need to sample and store posterior samples and average among them) this type of approximation is recommended to be used for anomaly detection in the proposed framework.
### Learning algorithms comparison
This section compares the anomaly detection performance obtained by three learning algorithms.
The best results in terms of a mean area under a precision-recall curve are obtained by the EM learning algorithm, the worst results are obtained by the GS learning algorithm (Figure \[fig:mean\_results\] and Table \[tab:mean\_area\]). In Table \[tab:mean\_area\] for each learning algorithm the group of hyperparameters settings and the type of marginal likelihood approximation is chosen to have the maximum of the mean area under curves, where a mean is taken over independent runs of the same method and maximum is taken among different settings for the same learning algorithm.
Dataset EM VB GS
--------- -------- -------- --------
QMUL 0.3166 0.3155 0.2970
Idiap 0.3759 0.3729 0.3673
: Mean area under precision-recall curves[]{data-label="tab:mean_area"}
Figure \[fig:best\_worst\_curves\] presents the best and the worst precision-recall curves (in terms of the area under them) for the individual runs of the learning algorithms. The figure shows that among the individual runs the EM learning algorithm also demonstrates the most accurate results. Although, the minimum area under the precision-recall curve for the EM learning algorithm is less than the area under the corresponding curve for the VB algorithm. It means that the variance among the individual curves for the EM learning algorithm is larger in comparison with the VB learning algorithm.
The variance of the precision-recall curves for both VB and GS learning algorithms is relatively small. However, the VB learning algorithm has the curves higher than the curves obtained by the GS learning algorithm. It can be confirmed by examination of the best and worst precision-recall curves (Figure \[fig:best\_worst\_curves\]) and the mean values of the area under curves (Figure \[fig:mean\_results\] and Table \[tab:mean\_area\]).
We also present the results of classification accuracy, i.e., the fraction of the correctly classified documents, for anomaly detection, which can be achieved with some fixed threshold. The best classification accuracy for the EM learning algorithm in both datasets can be found in Table \[tab:accuracy\].
Dataset Accuracy
--------- ----------
QMUL 0.9544
Idiap 0.8891
: Best classification accuracy for the EM learning algorithm[]{data-label="tab:accuracy"}
### Anomaly localisation
We apply the proposed method for anomaly localisation, presented in Section \[sec:localisation\], and get promising results. We demonstrate the localisation results for the EM learning algorithm with the prior type H+1 on both datasets in Figure \[fig:abnormality\_localisation\]. The red rectangle is manually set to locate the abnormal events within the frame, the arrows correspond to the visual words with the smallest marginal likelihood computed by the algorithm. It can be seen that the abnormal events correctly localised by the proposed method.
For quantitative evaluation we analyse $10$ abnormal events ($5$ from each dataset). For each clip for a given number $N_{\text{top}}$ of the least probable words, we measure the recall: $\text{recall} = \dfrac{\text{TP}}{N_{\text{an}}}$, where $N_{\text{an}}$ is the maximum possible number of abnormal words among $N_{\text{top}}$, i.e., $N_{\text{an}} = N_{\text{top}}$ if $N_{\text{top}} \leq N_{\text{total an}}$, where $N_{\text{total an}}$ is the total number of abnormal words, and $N_{\text{an}} = N_{\text{total an}}$ if $N_{\text{top}} > N_{\text{total an}}$. Figure \[fig:quantitative\_localisation\] presents the mean results for all events. One can notice, for example, that when the localisation procedure can possibly detect $45\%$ of the total number of abnormal words, it correctly finds $\approx 90\%$ of them.
![Recall results of the proposed anomaly localisation procedure[]{data-label="fig:quantitative_localisation"}](TNNLS-2016-P-6800-R1-localisation_quantitative){width="0.9\columnwidth"}
Conclusions {#sec:conlusion}
===========
This paper presents two learning algorithms for the dynamic topic model for behaviour analysis in video: the EM-algorithm is developed for the MAP estimates of the model parameters and a variational Bayes inference algorithm is developed for calculating the posterior distributions of them. A detailed comparison of these proposed learning algorithms with the Gibbs sampling based algorithm developed in [@Hospedales2011] is presented. The differences and the similarities of the theoretical aspects for all three learning algorithms are well emphasised. The empirical comparison is performed for abnormal behaviour detection using two unlabelled real video datasets. Both proposed learning algorithms demonstrate more accurate results than the algorithm proposed in [@Hospedales2011] in terms of anomaly detection performance.
The EM learning algorithm demonstrates the best results in terms of the mean values of the performance measure, obtained by the independent runs of the algorithm with different random initialisations. Although, it is noticed that the variance among the precision-recall curves of the individual runs is relatively high. The variational Bayes learning algorithm shows the smaller variance among the precision-recall curves than the EM-algorithm. The results show that the VB algorithm answers are more robust to different initialisation values. However, it is shown that the results of the algorithm are significantly influenced by the choice of the hyperparameters. The hyperparameters require additional tuning before the algorithm can be applied to data. Note that the results of the EM learning algorithm only slightly depend on the choice of the hyperparameters settings. Moreover, the hyperparameters can be even set in such a way as the EM algorithm is applied to obtain the maximum likelihood estimates instead of the maximum a posteriori ones. Both proposed learning algorithms — EM and VB — provide more accurate results in comparison to the Gibbs sampling based algorithm.
We also demonstrate that consideration of marginal likelihoods of visual words rather than visual documents can provide satisfactory results about locations of anomalies within a frame. In our best knowledge the proposed localisation procedure is the first general approach in probabilistic topic modeling that requires only presence of spatial information encoded in visual words.
EM-algorithm derivations
========================
This Appendix presents the details of the proposed EM learning algorithm derivation. The objective function in the EM-algorithm is: $$\begin{aligned}
&\mathcal{Q}(\boldsymbol\Omega, \boldsymbol\Omega^{\text{old}}) + \log p(\boldsymbol\Omega | \boldsymbol\beta, \boldsymbol\alpha, \boldsymbol\eta, \boldsymbol\gamma) = \nonumber\\
&\sum_{\mathbf{y}_{1:T_{tr}}} \sum_{\mathbf{z}_{1:T_{tr}}} \left( p(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old}) \times \right.\nonumber\\
&\left. \log{p(\mathbf{x}_{1:T_{tr}}, \mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}} | \boldsymbol\Omega, \boldsymbol\alpha, \boldsymbol\beta, \boldsymbol\gamma, \boldsymbol\eta)} \vphantom{p(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old})} \right) + \nonumber\\
&+ \log p(\boldsymbol\Omega | \boldsymbol\beta, \boldsymbol\alpha, \boldsymbol\eta, \boldsymbol\gamma) = \nonumber\\
&= Const + \sum_{z_1 \in \mathcal{Z}} \left( \log{\pi_{z_1}} \, p(z_1 | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old}) \right) + \nonumber\\
&\sum_{t = 2}^{T_{tr}} \sum_{z_t \in \mathcal{Z}} \sum_{z_{t-1} \in \mathcal{Z}} \left( \log{\xi_{z_t, z_{t-1}}} \, p(z_t, z_{t-1} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old}) \right) + \nonumber\\
&\sum_{t = 1}^{T_{tr}} \sum_{i = 1}^{N_t} \sum_{y_{i, t} \in \mathcal{Y}} \left( \log{\phi_{x_{i, t}, y_{i, t}}} \, p(y_{i, t} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old}) \right) + \nonumber\\
&+ \sum_{t = 1}^{T_{tr}} \sum_{i = 1}^{N_t} \sum_{z_t \in \mathcal{Z}} \sum_{y_{i, t} \in \mathcal{Y}} \left( \log{\theta_{y_{i, t}, z_t}}\, p(y_{i, t}, z_t | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old}) \right) + \nonumber\\
&\sum\limits_{z \in \mathcal{Z}} (\eta_z - 1) \log\pi_{z} + \sum\limits_{z \in \mathcal{Z}}\sum\limits_{z' \in \mathcal{Z}} (\gamma_z - 1) \log\xi_{z, z'} + \nonumber\\
\label{eq:em_maximised_function}
&\sum\limits_{z \in \mathcal{Z}} \sum\limits_{y \in \mathcal{Y}} (\alpha_y - 1) \log\theta_{y, z} + \sum\limits_{y \in \mathcal{Y}} \sum\limits_{x \in \mathcal{X}} (\beta_x - 1) \log\phi_{x, y}\end{aligned}$$
On the M-step the function (\[eq:em\_maximised\_function\]) is maximised with respect to the parameters $\boldsymbol\Omega$ with fixed values for $p(z_1 | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old})$, $p(z_t, z_{t-1} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old})$, $p(y_{i, t} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old})$, $p(y_{i, t}, z_t | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old})$. The optimisation problem can be solved separately for each parameter, which leads to the equations (\[eq:M:phi\]) – (\[eq:M\_psi\_k,l\]).
On the E-step for the efficient implementation the forward-backward steps are developed for the auxiliary variables $\acute{\alpha}_z(t)$ and $\acute{\beta}_z(t)$: $$\begin{gathered}
\label{eq:alpha_def}
\acute{\alpha}_z(t) \stackrel{\text{def}}{=} p(\mathbf{x}_1, \dotsc, \mathbf{x}_t, z_t = z | \boldsymbol\Omega^{Old}) = \\
\sum\limits_{\mathbf{z}_{1:t-1}} \pi^{Old}_{z_1} \left[ \prod_{\acute{t} = 2}^{t-1} \xi^{Old}_{z_{\acute{t}}, z_{\acute{t}-1}} \right] \left[\prod_{\acute{t} = 1}^{t-1} \prod_{\vphantom{\acute{t}} i = 1}^{N_{\acute{t}}} \sum\limits_{\vphantom{\acute{t}} y \in \mathbf{Y}} \phi^{Old}_{x_{i, \acute{t}}, y} \theta^{Old}_{y, z_{\acute{t}}}\right] \times \\
\xi^{Old}_{z_t = k, z_{t-1}} \prod\limits_{i = 1}^{N_t} \sum\limits_{y \in \mathcal{Y}} \phi^{Old}_{x_{i, t}, y} \theta^{Old}_{y, z_t = z}.\end{gathered}$$ Reorganisation of the terms in (\[eq:alpha\_def\]) leads to the recursive expressions (\[eq:E:alpha\]).
Similarly for $\acute{\beta}_z(t)$: $$\begin{gathered}
\label{eq:beta_def}
\acute{\beta}_k(t) \stackrel{\text{def}}{=} p(\mathbf{x}_{t+1}, \dotsc, \mathbf{x}_{T_{tr}} | z_t = z, \boldsymbol\Omega^{Old}) = \\
\sum\limits_{\mathbf{z}_{t+1 : T_{tr}}} \xi^{Old}_{z_{t+1}, z_t = z} \left[\prod\limits_{\acute{t} = t+2}^{T_{tr}} \xi^{Old}_{z_{\acute{t}}, z_{\acute{t}-1}} \right] \prod\limits_{\acute{t} = t+1}^{T_{tr}} \prod\limits_{\vphantom{\acute{t}} i = 1}^{N_{\acute{t}}} \sum\limits_{\vphantom{\acute{t}} y \in \mathcal{Y}} \phi^{Old}_{x_{i, \acute{t}}, y} \theta^{Old}_{y, z_{\acute{t}}}.\end{gathered}$$ The recursive formula (\[eq:E:beta\]) is obtained by interchanging the terms in (\[eq:beta\_def\]).
The required posterior of the hidden variables terms $p(z_1 | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old})$, $p(z_t, z_{t-1} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old})$, $p(y_{i, t} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old})$, $p(y_{i, t}, z_t | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega^{Old})$ are then expressed via the axillary variables $\acute{\alpha}_z(t)$ and $\acute{\beta}_z(t)$, which leads to (\[eq:E:z\_t\]) – (\[eq:E:y\_i,t\]).
VB algorithm derivations
========================
This Appendix presents the details of the proposed variational Bayes inference derivation. We have separated the parameters and the hidden variables. Let us consider the update formula of the variational Bayes inference scheme [@Murphy2012] for the parameters: $$\begin{aligned}
\label{eq:q_param_full}
&\log q(\boldsymbol\Omega) = Const + \nonumber\\
&\mathbb{E}_{q(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}})} \log p(\mathbf{x}_{1:T_{tr}}, \mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}, \boldsymbol\Omega | \boldsymbol\eta, \boldsymbol\gamma, \boldsymbol\alpha, \boldsymbol\beta)= \nonumber\\
&Const + \mathbb{E}_{q(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}})} \left( \sum\limits_{z \in \mathcal{Z}} \left(\eta_z - 1 \right) \log \pi_z + \right.\nonumber\\
&\sum\limits_{z \in \mathcal{Z}} \sum\limits_{\tilde{z} \in \mathcal{Z}} \left(\gamma_{\tilde{z}} - 1\right) \log \xi_{\tilde{z}, z} + \sum\limits_{z \in \mathcal{Z}} \sum\limits_{y \in \mathcal{Y}} \left(\alpha_y - 1\right) \log \theta_{y, z} + \nonumber\\
&\sum\limits_{y \in \mathcal{Y}} \sum\limits_{x \in \mathcal{X}} \left(\beta_x - 1 \right) \log \phi_{x, y} + \sum\limits_{z \in \mathcal{Z}} \mathbb{I}(z_1 = z) \log \pi_z + \nonumber\\
&\sum\limits_{t = 2}^{T_{tr}} \sum\limits_{z \in \mathcal{Z}} \sum\limits_{\tilde{z} \in \mathcal{Z}} \mathbb{I}(z_t = \tilde{z}) \mathbb{I}(z_{t-1} = z) \log\xi_{\tilde{z}, z} + \nonumber\\
&\sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \sum\limits_{y \in \mathcal{Y}} \mathbb{I}\left(y_{i, t} = y\right) \log \phi_{x_{i, t}, y} + \nonumber\\
&\left. \sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \sum\limits_{z \in \mathcal{Z}} \sum\limits_{y \in \mathcal{Y}} \mathbb{I}(y_{i, t} = y) \mathbb{I}(z_t = z) \log \theta_{y, z} \right)\end{aligned}$$
One can notice that $\log q(\boldsymbol\Omega)$ is further factorised as in (\[eq:q\_param\_factorisation\]). Now each factorisation term can be considered independently. Derivations of the equations (\[eq:VB:beta\]) – (\[eq:VB:gamma\]) are very similar to each other. We provide the derivation only of the term $q(\boldsymbol{\Phi})$: $$\begin{aligned}
&\log q(\boldsymbol\Phi) = Const + \nonumber\\
&\mathbb{E}_{q(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}})} \left(\vphantom{\sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \sum\limits_{y \in \mathcal{Y}}}\sum\limits_{y \in \mathcal{Y}} \sum\limits_{x \in \mathcal{X}} \left(\beta_x - 1 \right) \log \phi_{x, y} + \right. \nonumber\\
&\left.\sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \sum\limits_{y \in \mathcal{Y}} \mathbb{I}\left( y_{i, t} = y \right) \log \phi_{x_{i, t}, y} \right) = \nonumber\\
&Const + \sum\limits_{y \in \mathcal{Y}} \sum\limits_{x \in \mathcal{X}} \left(\beta_x - 1 \right) \log \phi_{x, y} + \nonumber\\
&\sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \sum\limits_{y \in \mathcal{Y}} \log \phi_{x_{i, t}, y} \underbrace{\mathbb{E}_{q(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}})}\left( \mathbb{I}\left(y_{i, t} = y\right)\right)}_{q(y_{i, t} = y)} = \nonumber\\
&Const + \nonumber\\
\label{eq:log_q_phi}
&\sum\limits_{y \in \mathcal{Y}} \sum\limits_{x \in \mathcal{X}} \log \phi_{x, y} \left( \beta_x - 1 + \sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \mathbb{I}(x_{i, t} = x) q(y_{i,t} = y) \right)\end{aligned}$$ It can be noticed from (\[eq:log\_q\_phi\]) that the distribution of $\boldsymbol\Phi$ is a product of the Dirichlet distributions (\[eq:VB:beta\]).
The update formula in the variational Bayes inference scheme for the hidden variables is as follows: $$\begin{aligned}
&\log q(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}) = Const + \nonumber\\
&\mathbb{E}_{q(\boldsymbol\pi)q(\boldsymbol\Xi)q(\boldsymbol\Theta)q(\boldsymbol\Phi)} \log p(\mathbf{x}_{1:T_{tr}}, \mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}, \boldsymbol\Omega | \boldsymbol\eta, \boldsymbol\gamma, \boldsymbol\alpha, \boldsymbol\beta) = \nonumber\\
&Const + \sum\limits_{z \in \mathcal{Z}} \mathbb{I}\left(z_1 = z\right) \mathbb{E}_{q(\boldsymbol\pi)} \log \pi_z + \nonumber\\
&\sum\limits_{t = 2}^{T_{tr}} \sum\limits_{z \in \mathcal{Z}} \sum\limits_{\tilde{z} \in \mathcal{Z}} \mathbb{I}\left(z_t = \tilde{z}\right) \mathbb{I}\left(z_{t-1} = z\right) \mathbb{E}_{q(\boldsymbol\Xi)} \log \xi_{\tilde{z}, z} + \nonumber\\
&\sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \sum\limits_{y \in \mathcal{Y}} \mathbb{I}\left(y_{i, t} = y\right) \mathbb{E}_{q(\boldsymbol\Phi)} \log \phi_{x_{i, t}, y} + \nonumber\\
\label{eq:log_q_yz_beginning}
&\sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \sum\limits_{z \in \mathcal{Z}} \sum\limits_{y \in \mathcal{Y}} \mathbb{I}\left(y_{i, t} = y \right) \mathbb{I} \left(z_t = z\right) \mathbb{E}_{q(\boldsymbol\Theta)} \log \theta_{y, z}\end{aligned}$$ We know from the parameters update (\[eq:VB:beta\]) – (\[eq:VB:gamma\]) that their distributions are Dirichlet. Therefore, $\mathbb{E}_{q(\boldsymbol\pi)}\log \pi_z = \psi\left(\tilde{\eta}_z\right) - \psi\left(\sum_{z' \in \mathcal{Z}} \tilde{\eta}_{z'}\right)$ and similarly for all the other expected value expressions.
Using the introduced notations (\[eq:VB:introduced\_pi\]) – (\[eq:VB:introduced\_theta\]) the update formula (\[eq:log\_q\_yz\_beginning\]) for the hidden variables can be then expressed as: $$\begin{aligned}
&\log q(\mathbf{y}_{1:{T_{tr}}}, \mathbf{z}_{1:T_{tr}}) = Const +
\sum\limits_{z \in \mathcal{Z}} \mathbb{I}\left(z_1 = z\right) \log \tilde{\pi}_z + \nonumber\\
&\sum\limits_{t = 2}^{T_{tr}} \sum\limits_{z \in \mathcal{Z}} \sum\limits_{\tilde{z} \in \mathcal{Z}} \mathbb{I}\left(z_t = \tilde{z}\right) \mathbb{I}\left(z_{t-1} = z\right) \log \tilde{\xi}_{\tilde{z}, z} + \nonumber\\
&\sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \sum\limits_{y \in \mathcal{Y}} \mathbb{I}\left(y_{i, t} = y\right) \log\tilde{\phi}_{x_{i, t}, y} + \nonumber\\
&\sum\limits_{t = 1}^{T_{tr}} \sum\limits_{i = 1}^{N_t} \sum\limits_{z \in \mathcal{Z}} \sum\limits_{y \in \mathcal{Z}} \mathbb{I} \left(y_{i, t} = y\right) \mathbb{I}\left(z_t = z\right) \log\tilde{\theta}_{y, z}\end{aligned}$$
The approximated distribution of the hidden variables is then: $$\begin{gathered}
\label{eq:q_yz}
q(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}}) = \\
\dfrac{1}{\tilde{K}} \tilde{\pi}_{z_1} \left[\prod\limits_{t = 2}^{T_{tr}} \tilde{\xi}_{z_t, z_{t-1}}\right] \prod\limits_{t = 1}^{T_{tr}} \prod\limits_{i = 1}^{N_t} \tilde{\phi}_{x_{i, t}, y_{i, t}} \tilde{\theta}_{y_{i, t}, z_t},\end{gathered}$$ where $\tilde{K}$ is a normalisation constant. Note that the expression of the true posterior distribution of the hidden variables is the same up to replacing the true parameters variables with the corresponding tilde variables: $$\begin{gathered}
p(\mathbf{y}_{1:T_{tr}}, \mathbf{z}_{1:T_{tr}} | \mathbf{x}_{1:T_{tr}}, \boldsymbol\Omega) = \\
\dfrac{1}{K} \pi_{z_1} \left[\prod\limits_{t = 2}^{T_{tr}} \xi_{z_t, z_{t-1}} \right] \prod\limits_{t = 1}^{T_{tr}} \prod\limits_{i = 1}^{N_t} \phi_{x_{i, t}, y_{i, t}} \theta_{y_{i, t}, z}\end{gathered}$$ Therefore, to compute the required expressions of the hidden variables $q(z_1 = z)$, $q(z_{t-1} = z, z_t = z')$, $q(y_{i, t} = y, z_t = z)$ and $q(y_{i, t} = y)$ one can use the same forward-backward procedure and update formula as in the E-step of the EM-algorithm replacing all the parameter variables with the corresponding introduced tilde variables.
Acknowledgments {#acknowledgments .unnumbered}
===============
Olga Isupova and Lyudmila Mihaylova would like to thank the support from the EC Seventh Framework Programme \[FP7 2013-2017\] TRAcking in compleX sensor systems (TRAX) Grant agreement no.: 607400. Lyudmila Mihaylova also acknowledges the UK Engineering and Physical Sciences Research Council (EPSRC) for the support via the Bayesian Tracking and Reasoning over Time (BTaRoT) grant EP/K021516/1.
[Olga Isupova]{} is a PhD student at the Department of Automatic Control and Systems Engineering at the University of Sheffield and an Early Stage Researcher in the FP7 Programme TRAX. She received the Specialist (eq. to M.Sc.) degree in Applied Mathematics and Computer Science, 2012, from Lomonosov Moscow State University, Moscow, Russia. Her research is on machine learning, Bayesian nonparametrics, anomaly detection.
[Danil Kuzin]{} is a PhD student at the Department of Automatic Control and Systems Engineering at the University of Sheffield and an engineer at the Rinicom, Ltd. He received the Specialist degree in Applied Mathematics and Computer Science, 2012, from Lomonosov Moscow State University, Moscow, Russia. His research is mainly in sparse modelling for video. His other research interests include nonparametric Bayes and deep reinforcement learning.
\[[{width="1in" height="1.25in"}]{}\] [Lyudmila Mihaylova]{} (M’98, SM’2008) is Professor of Signal Processing and Control at the Department of Automatic Control and Systems Engineering at the University of Sheffield, United Kingdom. Her research is in the areas of machine learning and autonomous systems with various applications such as navigation, surveillance and sensor network systems. She has given a number of talks and tutorials, including the plenary talk for the IEEE Sensor Data Fusion 2015 (Germany), invited talks University of California, Los Angeles, IPAMI Traffic Workshop 2016 (USA), IET ICWMMN 2013 in Beijing, China. Prof. Mihaylova is an Associate Editor of the IEEE Transactions on Aerospace and Electronic Systems and of the Elsevier Signal Processing Journal. She was elected in March 2016 as a president of the International Society of Information Fusion (ISIF). She is on the board of Directors of ISIF and a Senior IEEE member. She was the general co-chair IET Data Fusion $\&$ Target Tracking 2014 and 2012 Conferences, Program co-chair for the 19th International Conference on Information Fusion, 2016, academic chair of Fusion 2010 conference.
[^1]: O.Isupova, D.Kuzin, L.Mihaylova are with the Department of Automatic Control and Systems Engineering, The University of Sheffield, Sheffield, UK e-mail: [email protected], [email protected], [email protected]
[^2]: The computational time is provided for a laptop computer with i7-4702HQ CPU with 2.20GHz, 16 GB RAM using Matlab R2015a implementation.
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} |
---
abstract: 'Using 3D spectroscopy with a scanning Fabry–Perot interferometer, we study the ionized gas kinematics in 59 nearby dwarf galaxies. Combining our results with data from literature, we provide a global relation between the gas velocity dispersion $\sigma$ and the star formation rate (SFR) and luminosity for galaxies in a very broad range of star formation rates SFR=$0.001-300\,M_\odot\,{\rm yr}^{-1}$. We find that the $SFR - \sigma$ relation for the combined sample of dwarf galaxies, star forming, local luminous, and ultra-luminous infrared galaxies can be fitted as $\sigma\propto SFR^{5.3\pm0.2}$. This implies that the slope of the $L-\sigma$ relation inferred from the sample of rotation supported disc galaxies (including mergers) is similar to the $L-\sigma$ relation of individual giant HII regions. We present arguments that the velocity dispersion of the ionized gas does not reflect the virial motions in the gravitational potential of dwarf galaxies, and instead is mainly determined by the energy injected into the interstellar medium by the ongoing star formation.'
author:
- |
Alexei V. Moiseev$^1$[^1], Anton V. Tikhonov$^2$[^2], and Anatoly Klypin$^3$\
$^1$Special Astrophysical Observatory, Russian Academy of Sciences[^3], 369167 Nizhnii Arkhyz, Karachaevo-Cherkesskaya Republic, Russia\
$^2$St. Petersburg State University, Universitetskii pr. 28, 198504 St. Petersburg, Stary Peterhof, Russia\
$^3$New Mexico State University, USA
date: 'Accepted .... Received ....'
title: 'What controls the ionized gas turbulent motions in dwarf galaxies?'
---
\[firstpage\]
galaxies: dwarf – galaxies: kinematics and dynamics – galaxies: ISM – ISM: bubbles.
Introduction
============
The nature of high-velocity turbulent motions of ionized gas in giant star-forming regions and dwarf galaxies has been studied for almost half a century, starting with @SmithWeedman1970, who showed that the velocity dispersion, $\sigma$ (the rms velocity along line-of-sight) in giant star forming regions of the M33 and M101 galaxies, determined from the width of the integral emission line profiles is about a few tens of ${\,\mbox{km}\,\mbox{s}^{-1}}$. A close correlation was later found between the rms velocity $\sigma$, the diameter, and the total Balmer line luminosity (in the $\mbox{H}\beta$ emission line) of the emitting nebula [@Melnick1977; @TerlevichMelnick1981]. Similar trends were found for individual HII regions and for dwarf irregular (dIrr) and blue compact (BCDG or HII) galaxies. Because of the tightness of the observed $L(\mbox{H}\beta)-\sigma$ relation of star-bursting compact galaxies, it was proposed as an independent indicator of the cosmological distance [e.g. @Melnick1987; @Melnick2000; @Chavez2012; @Koulouridis2013; @Chavez2014].
Relation between the emission-line luminosity $L$ and ionized gas $\sigma$ is also important for understanding how star formation affects motion of gas and how star formation is regulated by different stellar feedback processes. Although the existence of a close correlation between these quantities has been known for a long time, the origin of luminosity–velocity dispersion relation in HII galaxies and giant HII regions remains unclear [@TerlevichMelnick1981; @ChuKennicutt1994; @Scalo1999; @Bordalo2009; @MoisLoz2012]. The following factors may affect the width of the observed ionized hydrogen lines:
1. Thermal broadening, which amounts to $\sigma_{th}\approx8-10{\,\mbox{km}\,\mbox{s}^{-1}}$ for typical electron temperatures in the HII regions 7000–10000 K.
2. Turbulent motions determined by the combined influence of young stellar clusters on the interstellar medium (ISM).
3. Gravitational broadening, caused by virial motions of gas clouds in galaxy gravitational potential.
4. Non-virial gravitational motions: ISM turbulence associated with tidal interactions, galaxy merging and external gas accretion (e.g. numerical simulations by [@Bournaud2011], observational constrains in [@Arribas2014] and references therein.)
When presenting and discussing the $L-\sigma$ relation, we will always assume, except in specially mentioned cases, that $L$ is the total luminosity of a galaxy or a HII region in the line, and $\sigma$ is the average (luminosity-weighed) velocity dispersion of ionized gas.
It is often assumed that the gravitational effects are the dominante factor in giant HII regions and HII galaxies [e.g., @TerlevichMelnick1981; @Tenorio-Tagle1993; @Melnick1987]. The main argument was that the velocity dispersion of ionized gas is largely determined by the wind of stars that participate in virial motions with the characteristic velocity $\sigma_{\rm stars}$. Then $\sigma$ and $\sigma_{\rm stars}$ are mainly controlled by the gravitational potential of the object, and the $L-\sigma$ relation is similar to the Faber-Jackson relation for elliptical galaxies: $L\propto\sigma_{\rm stars}^4$. Different studies of the ionized gas kinematics in giant extragalactic HII regions give a value between $\sim3$ and $\sim7$ for the exponent in the $L-\sigma$ relation [@Blasco-Herrera2010; @Blasco-Herrera2013]. @Chavez2014 present a detailed analysis of this relation for a sample of 128 local compact HII galaxies. They demonstrate that adding the second (the size of HII regions) or even the third (the emission line equivalent width $EW$ or the continuum color and metallicity) parameters significantly improve the correlation. They also argue in favor of the gravitational origin of the ionized gas velocity dispersion $\sigma$. They also emphasize that in order the $L-\sigma$ to be tight, the regions of recent bursts of star formation should be gravitationally bound, compact, massive, and have strong emission lines with $EW(H\beta)>50$Åwith pure Guassian proile without any evidences of multiplicity or rotation.
A different view on the $L-\sigma$ relation was developed beginning with @GallagherHunter1983, who suggested that the processes related with the energy of embedded OB stars drive the ionized gas velocity dispersion in giant HII regions on scales smaller than 0.5 kpc, while properties of larger supergiant HII complexes agree with a gravitationally driven $\sigma$. Based on Fabry–Perot interferometric observations of nearby galaxies @Arsenault1988 also concluded that effects of stellar wind and turbulence are more important for the kinematics of giant HII regions compared with virial motions. There are different ways of how a young stellar population affects the surrounding ISM. According @MacLowKlessen2004, [see also @Lopez2014 and references therein] the main mechanisms are: protostellar outflows, stellar winds and ionizing radiation pressure of massive stars, supernovae (SN) explosions, the dust-processed infrared radiation, and warm and hot gas pressure. The contribution of different factors changes with spatial and density scale. For instance, @Lopez2014 found that warm ionized gas dominates over the other terms of pressure in all considered HII regions in Large and Small Magellanic Clouds. Numerical simulations also demonstrate that the radiation pressure is a very important feedback mechanism for models of formation of galaxies compared with the hot gas contribution (heated by SNs and stellar winds) [e.g., @Hopkins2012; @Ceverino2014; @Trujillo2015]. Note that together with the chaotic gas motions, responsible for the Gaussian emission line profile, the effects related with individual expanding shells may lead to the appearance of non-symmetric features such as wings and peaks in the line profile [see examples in @Melnick1999; @BordaloTelles2011].
Some recent observations also provide evidence that energy injected into the interstellar medium by the ongoing star formation process is the main factor affecting gas turbulent motions. For example, @Green2010 show that in a wide range of galaxy luminosities the ionized gas rms velocity $\sigma$ is determined by the star formation rate (SFR), which is proportional to the luminosity, and does not correlate with the galaxy mass. Earlier, @Dib2006 showed that $\sigma$ for neutral gas also depends on the SFR. However other numerical simulations with higher resolution in a stratified ISM suggested that this trend is absent if the gas surface density increases with the SN rate [@Joung2009]. On the other hand, the galaxy-scale simulations that included stellar feedback [@Hopkins2012] clearly demonstrate that the average velocity dispersion of the gas (in all cold, warm and hot phases) increases with the total SFR. Also, @Dopita2008 argues that kinetic energy of ionized gas in the star formation regions is proportional to the local SFR, integrated over the duration of the burst. @MoisLoz2012 demonstrate a close correlation between the two-dimensional distribution of the radial velocity dispersion of ionized gas and the surface brightness in the line for a sample of nearby dwarf galaxies: most of the regions with the highest velocity dispersion belong to a low-brightness diffuse background surrounding large HII-regions.Recent simulations of multiple SN explosions were able to reproduce the diagrams ‘ intensity – velocity dispersion’ observed in these galaxies when a realistic spatial resolution was used [@Vasiliev2015].
However, some other recent studies of high-redshift galaxies contradict @Green2010. For instance, integral-field data by @Genzel2011 demonstrate a very weak correlation of $\sigma$ and SFR density for star-forming clumps in galaxies at $z=2.2-2.4$. Further @Wisnioski2012 and @Swinbank2012 using observations with similar technique argue that such clumps follow the same $L-\sigma$ relation as gravitational bounded local giant HII regions according @TerlevichMelnick1981 and related studies.
To summarize, the question of the nature of high-velocity turbulent motions of ionized gas in dwarf galaxies still remains open. One of the problems is the lack of sufficiently uniform observational data. Recent results by @BordaloTelles2011 fill this gap to a certain extent. They give a uniform set of high-resolution spectroscopy observations for 118 star formation regions in HII galaxies. @Chavez2014 presented similar data for 128 HII galaxies selected from the SDSS. They also discussed the impact of other factors (gas metallicity, ionization state, the history of star formation) on the $L-\sigma$ relation.
However, most of the results are based on slit spectroscopy. The distribution of ionized gas in dwarf galaxies has a complex irregular morphology, and the two-dimensional velocity dispersion maps, obtained using the 3D spectroscopy can provide the most complete information about gas turbulence.
The second important issue is related with the fact that the luminosity-weighted velocity dispersion has become a widely used value to characterize the turbulent gas motions [in samples of distant and nearby galaxies [@Ostlin2001; @Green2010; @Davies2011; @Blasco-Herrera2013; @Arribas2014].]{} How does the $L-\sigma$ relation work for galaxies of different types and luminosities? Do galaxies obey the same scaling relation as the HII regions and star forming clumps?
In this paper we analyse ionized gas turbulent motions in 59 galaxies observed at the 6-m Big Telescope Alt-azimuthal (BTA) of Special Astrophysical Observatory of the Russian Academy of Sciences (SAO RAS) using a scanning Fabry–Perot interferometer (FPI). Most of our sample consists of dwarf galaxies of the Local Universe. Using this unique material, we were able to extend the $L-\sigma$ relation for objects much weaker than in the samples described earlier.
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Observations and data reduction {#obs}
===============================
Sample of galaxies
------------------
Our sample is based on the archive data of observations at the 6-m telescope of SAO RAS. It consists of several subsamples of galaxies studied within the framework of several observational programs:
- The Local Volume (LV) subsample: 36 nearby dwarf galaxies with galactocentric distances $D<15$ Mpc. These are mostly low-luminosity objects with the average absolute magnitude $M_B\approx-14$. The ionized gas velocity dispersion maps of 10 galaxies from this subsample were already reported by @MoisLoz2012.
- The XMD subsample of 9 more distant ($D=14-86$ Mpc) and bright ($M_B\approx-15$) dwarf galaxies with extremely low metallicity, and 2 low surface brightness companion galaxies. Their internal kinematics were described and analysed in @Moiseev2010.
- The BCDG subsample: 12 blue compact dwarf galaxies at distances $D=10-78$ Mpc; all are brighter than $M_B=-15$. Some of them (e.g., Mrk 297 or III Zw 102) are sometimes included in BCDG samples [see @Cairos2001], though they cannot be formally called dwarf galaxies since their absolute magnitude is brighter than $M_B=-20$ and the amplitude of the rotation curve exceeds $140{\,\mbox{km}\,\mbox{s}^{-1}}$. The observational data for 3 of these galaxies were presented by @delgado.
The full sample consists of 59 galaxies, and covers a wide range of luminosities from $M_B=-11$ to $M_B=-21$. The overwhelming majority (51/59=86 per cents) of them are dwarf galaxies with the absolute magnitudes $M_B>-18$.
The observations were carried out in the prime focus of the 6-m telescope of SAO RAS using a scanning FPI mounted inside the SCORPIO focal reducer [@AfanasievMoiseev2005]. In 2014 its new version SCORPIO-2 [@AfanasievMoiseev2011] was used. The operating spectral range around the H$\alpha$ line was cut using a narrow bandpass filter. About $\sim2/3$ of the observations were performed using the FPI501 interferometer, providing in the H$\alpha$ emission line a free spectral range between the neighboring interference orders $\Delta\lambda=13$Å and spectral resolution ($FWHM$ of the instrumental profile) of about $0.8$Å ($35{\,\mbox{km}\,\mbox{s}^{-1}}$), sampled by 0.36Å per channel. After November 2009 we used a new FPI751 interferometer, which has $\Delta\lambda=8.7$Å and a spectral resolution of $0.4$Å ($18{\,\mbox{km}\,\mbox{s}^{-1}}$), sampled by 0.21Å per channel. The Mrk 33 galaxy was observed in the \[NII\]$\lambda6583$ line, all the others – in the line.
In 2003-2014, the detectors were CCD EEV 42-40 and E2V 42-90, given the image scale of $0.71\,\mbox{arcsec}\, \mbox{pixel}^{-1}$ in the on-chip binned $4\times4$ mode. In 2002 the TK1024 CCD was used, yielding the scale of $0.56\,\mbox{arcsec}\,\mbox{pixel}^{-1}$ in the $2\times2$ binning mode.
During the scanning process we have consistently obtained 36 interferograms (40 for FPI751) at different distances between the FPI plates. The seeing on different nights varied from 1 to 4 arcsec. The reduction of observational data was performed using the software package running in the IDL environment [@Moiseev02ifp; @MoiseevEgorov2008]. Following the primary reduction, airglow lines subtraction, photometric and seeing corrections using the reference stars and wavelength calibration, the observational data were combined into the data cubes, where each pixel in the field of view contains a 36- or 40-channel spectrum.
The log of observations is shown in Table \[tab\_obs\], listing the galaxy name; the date of observations; the interferometer type; exposure time; the resulting angular resolution after smoothing the reduced data cubes with a two-dimensional Gaussian to increase the signal-to-noise ratio in the areas of low surface brightness. Only the information for the Local Volume galaxies and the BCDG subsamples is shown; see the log of observations in @Moiseev2010 for the XMD subsample.
[lcccc]{} Name & Date & FPI & Exp. time & Ang.resol.\
& & & time (s) & (arcsec)\
\
CGCG269-049& 06.02.2010 & FPI751 & $150\times40$& 4.4\
DDO 53 & 26.02.2009 & FPI501 & $200\times36$& 3.3\
DDO 68 & 30.12.2006 & FPI501 & $240\times36$& 2.7\
DDO 99 & 26.02.2009 & FPI501 & $180\times36$& 3.8\
DDO 125 & 18.05.2005 & FPI501 & $180\times36$& 3.0\
DDO 190 & 04.03.2009 & FPI501 & $100\times36$& 3.3\
IC 10 & 08.09.2005 & FPI501 & $300\times36$& 1.2\
IC 1613 & 12.09.2001 & FPI501 & $200\times36$& 2.2\
KK 149 & 05.03.2009 & FPI501 & $150\times36$& 2.8\
KKH 12 & 23.08.2004 & FPI501 & $120\times36$& 2.7\
KKH 34 & 12.11.2009 & FPI751 & $230\times40$ & 3.4\
KKR 56 & 20.05.2010 & FPI751 & $150\times40$ & 3.3\
UGC 231 & 11.11.2009 & FPI751 & $200\times40$ & 2.4\
UGC 891 & 11.11.2009 & FPI751 & $200\times40$ & 3.1\
UGC 1281 & 14.08.2009 & FPI751 & $110\times40$ & 2.7\
UGC 1501 & 10.11.2009 & FPI751 & $200\times40$ & 2.3\
UGC 1924 & 11.11.2009 & FPI751 & $180\times40$ & 2.3\
UGC 3476 & 02.11.2010 & FPI751 & $220\times40$ & 2.4\
UGC 3672 & 12.11.2009 & FPI751 & $160\times40$ & 3.1\
UGC 5221 & 16.12.2014 & FPI751 & $160\times40$ & 2.0\
UGC 5423 & 26.02.2009 & FPI501 & $180\times36$ & 3.5\
UGC 5427 & 04.03.2009 & FPI501 & $180\times36$ & 3.7\
UGC 8638 & 24.02.2009 & FPI501 & $150\times36$ & 3.9\
UGC 6456 & 29.11.2002 & FPI501 & $300\times36$& 2.2\
UGC 7611 & 19.05.2010 & FPI751 & $ 160\times40$ &3.5\
UGC 8508 & 16.05.2005 & FPI501 & $200\times36$& 3.0\
UGC 11425 & 14.08.2009 & FPI751 & $140\times40$ & 3.0\
UGC 11583 & 10.11.2009 & FPI751 & $220\times40$ & 1.9\
UGC 12713 & 16.05.2005 & FPI501 & $200\times36$& 3.0\
UGC 2455 & 07.10.2010 & FPI751 & $240\times40$& 2.3\
UGC 7047 & 17-18.03.2012 & FPI751 & $360\times40$& 2.2\
UGC 7648/51 & 08.02.2011 & FPI751 & $160\times40$& 2.8\
UGC 8313 & 19.03.2012 & FPI751 & $450\times40$& 2.7\
UGCA 92 & 10.11.2009 & FPI751 & $180\times40$& 2.5\
UGCA 292 & 07.02.2010 & FPI751 & $120\times40$ & 3.6\
\
II Zw 40 & 01.12.2003 & FPI501 & $120\times36$& 1.9\
II Zw 70 & 30.01.2004 & FPI501 & $180\times36$& 2.5\
III Zw 107 & 01.12.2006 & FPI501 & $100\times36$& 1.9\
III Zw 102 & 30.11.2003 & FPI501 & $180\times36$& 2.7\
Mrk 5 & 30.01.2004 & FPI501 & $240\times36$& 2.6\
Mrk 33 & 30.01.2004 & FPI501 & $150\times36$& 2.3\
Mrk 35 & 01.12.2003 & FPI501 & $150\times36$& 2.1\
Mrk 36 & 29.11.2003 & FPI501 & $160\times36$& 1.6\
Mrk 297 & 13.08.2009 & FPI751 & $144\times40$& 3.5\
Mrk 324 & 30.11.2003 & FPI501 & $120\times36$& 2.8\
Mrk 370 & 30.11.2003 & FPI501 & $120\times36$& 2.6\
Mrk 600 & 01.12.2006 & FPI501 & $120\times36$& 2.2\
Construction of maps and measuring velocity dispersion
------------------------------------------------------
We define the velocity dispersion of ionized gas $\sigma$ as the standard deviation of the Gaussian profile fitted the H$\alpha$ emission line after accounting for the FPI instrumental profile and subtracting the contribution of thermal broadening in the HII regions. The procedure to measure $\sigma$ is described in detail in @MoiseevEgorov2008. In short, the observed profiles of the H$\alpha$ line were fitted by the Voigt function, which is a convolution of Lorentzian and Gaussian functions corresponding to the FPI instrumental profile and broadening of observed emission lines respectively. The FWHM of instrumental profile was estimated each night from Lorentzian fitting of the He-Ne-Ar calibration lamp emission scanned with FPI [@MoiseevEgorov2008]. The results of profile fitting were used to construct two-dimensional line-of-sight velocity fields of ionized gas, maps of line-of-sight velocity dispersion, free from the instrumental profile ($\sigma_{obs}$) influence, and also the images of galaxies in the H$\alpha$ emission line and in the continuum.
The accuracy of velocity dispersion was estimated from the measurements of the S/N using the relations given in Figure 5 of @MoiseevEgorov2008. On the $\sigma$ maps we masked out the regions with a weak signal, where the formal error of velocity dispersion measurements exceeded $10-12{\,\mbox{km}\,\mbox{s}^{-1}}$ (which corresponds to $S/N\leq5$). Emission line intensity maps were constructed even for regions where the signal-to-noise ratio was smaller: $S/N\approx2-3$.
The correction from the measured $\sigma_{i, obs}$ to the final $\sigma_i$, where $i$ corresponds to the pixel number, was done according to the relation [@Rozas2000]: $$\sigma_i^2=\sigma_{i,obs}^2-\sigma_{N}^2-\sigma_{tr}^2, \label{eq1}$$ where $\sigma_{N}\approx3{\,\mbox{km}\,\mbox{s}^{-1}}$ and $\sigma_{tr}\approx9.1{\,\mbox{km}\,\mbox{s}^{-1}}$ correspond to the natural width of the emission line and its thermal broadening at the temperature of $10^4$ K.
In most of the objects the emission line spectrum is very well described by a single-component Voigt profile. Only a few galaxies have areas where the emission line profile has a complex (usually two-component) structure, showing the presence of expanding shells around the regions of star formation or possible supernova remnants. In addition to the previously reported cases of UGC 8508, UGCA 92 [@MoisLoz2012], and SBS 0335-052E [@Moiseev2010], compact regions with a two-component profile were found in UGC 260, UGC 1281, UGC 7047 and UGC 7651.
The mean velocity dispersion for the whole galaxy, weighted with intensity, was calculated as: $$\sigma= \frac{ \sum\sigma_i I_i}{\sum I_i}, \label{eq2}$$ where $I_i$ is the observed emission line in the $i$-th pixel.
[lcccclcr]{} Name & D & $M_B$ & $M_K$ &$\log\,L_{H\alpha}$& $\sigma$ & $i$ &$V_{max}$\
& Mpc & & & $\mbox{erg}\,\mbox{s}^{-1}$ & ${\,\mbox{km}\,\mbox{s}^{-1}}$ & deg. & ${\,\mbox{km}\,\mbox{s}^{-1}}$\
\
CGCG 269-049 & 4.59 & -13.11 & -15.46 & 37.24 & $13.6\pm 1.9$ & 43 & 9.8\
DDO 53 & 3.56 & -13.37 & -15.00 & 38.93 & $21.0\pm 1.8$ & 27 & 16.4\
DDO 68 & 9.80 & -15.27 & -17.15 & 39.33 & $19.9\pm 3.9$ & 65 & 57.2\
DDO 99 & 2.64 & -13.52 & -15.26 & 38.44 & $19.2\pm 2.4$ & 52 & 11.7\
DDO 125 & 2.74 & -14.33 & -16.97 & 38.28 & $16.3\pm 2.3$ & 63 & 17.8\
DDO 190 & 2.80 & -14.19 & -16.52 & 38.44 & $18.5\pm 2.9$ & 60 & 24.7\
IC 10 & 0.66 & -15.99 & -17.90 & 40.73 & $17.6\pm 0.7$ & 31 & 52.8$^*$\
IC 1613 & 0.73 & -14.54 & -16.90 & 38.43 & $25.9\pm 1.3$ & 22 & 26.7$^*$\
KK 149 & 8.90 & -14.85 & -17.20 & 38.58 & $19.0\pm 4.1$ & 58 & 26.2\
KKH 12 & 3.00 & -13.03 & -15.89 & 38.65 & $17.9\pm 4.5$ & 90 & 19.5$^*$\
KKH 34 & 4.61 & -12.30 & -14.65 & 37.18 & $11.6\pm 6.6$ & 55 & 12.8$^*$\
KKR 56 & 5.90 & -14.39 & -16.74 & 38.27 & $17.8\pm 6.3$ & – & –\
UGC 231 & 12.82 & -18.38 & -19.98 & 39.75 & $18.0\pm 3.8$ & 90 & 92.8\
UGC 891 & 9.38 & -15.90 & 0.00 & 38.94 & $15.4\pm 5.9$ & 65 & 60.0\
UGC 1281 & 4.97 & -16.07 & -15.51 & 39.07 & $16.7\pm 5.0$ & 90 & 56.4\
UGC 1501 & 4.97 & -16.52 & -18.22 & 39.52 & $16.6\pm 2.4$ & 75 & 47.5\
UGC 1924 & 9.86 & -15.80 & -17.41 & 38.66 & $14.5\pm 5.4$ & 90 & 50.6\
UGC 2455 & 7.80 & -18.14 & -20.00 & 40.71 & $18.3\pm 2.6$ & 51 & 47.9\
UGC 3476 & 7.00 & -14.27 & -16.62 & 39.22 & $16.5\pm 2.3$ & 90 & 47.3$^*$\
UGC 3672 & 15.10 & -13.89 & 0.00 & 39.30 & $18.3\pm 3.6$ & 56 & 67.8\
UGC 5221 & 3.56 & -17.09 & -20.27 & 40.02 & $18.1\pm 1.6$ & 61 & 58.5$^*$\
UGC 5423 & 8.71 & -15.62 & -17.74 & 39.20 & $22.0\pm 2.2$ & 56 & 24.8\
UGC 5427 & 7.10 & -14.48 & -15.50 & 38.75 & $21.3\pm 4.8$ & 55 & 54.1\
UGC 6456 & 4.34 & -14.03 & -15.72 & 39.23 & $17.9\pm 1.3$ & 66 & 15.0\
UGC 7047 & 4.31 & -15.07 & -17.42 & 39.25 & $15.3\pm 1.5$ & 46 & 37.5\
UGC 7611 & 9.59 & -17.73 & -20.86 & 40.26 & $23.1\pm 2.5$ & 77 & 51.5\
UGC 7648 & 5.80 & -16.72 & -18.26 & 40.01 & $18.9\pm 1.8$ & 55 & 78.1$^*$\
UGC 7651 & 5.80 & -19.42 & -21.50 & 40.93 & $22.8\pm 1.8$ & 47 & 129.2\
UGC 8313 & 9.20 & -15.22 & -17.94 & 39.58 & $21.9\pm 2.0$ & 77 & 45.0\
UGC 8508 & 2.69 & -13.09 & -15.58 & 38.43 & $13.3\pm 2.4$ & 51 & 32.6\
UGC 8638 & 4.27 & -13.74 & -16.63 & 38.66 & $16.1\pm 1.9$ & 49 & 18.2$^*$\
UGC 11425 & 3.60 & -14.32 & -15.57 & 38.49 & $14.4\pm 0.0$ & 35 & 37.1\
UGC 11583 & 5.90 & -14.32 & -16.67 & 38.35 & $14.6\pm 4.5$ & 90 & 46.7\
UGC 12713 & 12.20 & -15.95 & -16.83 & 39.45 & $18.6\pm 2.2$ & 72 & 44.9$^*$\
UGCA 92 & 3.01 & -15.59 & -16.56 & 39.44 & $16.5\pm 3.1$ & 56 & 39.7$^*$\
UGCA 292 & 3.62 & -11.79 & -13.56 & 38.44 & $11.8\pm 1.7$ & 45 & 21.9\
\
HS 0822+3542 & 13.50 & -12.90 & – & 39.24 & $19.4\pm 0.7$ & 31 & 12.3\
HS 2236+1344 & 86.40 & -17.04 & – & 41.02 & $28.0\pm 0.9$ & 35 & 21.8\
SAO 0822+3545 & 13.50 & -13.26 & – & 37.96 & $17.9\pm 5.4$ & 63 & 12.4\
SBS 0335-052E & 53.80 & -16.87 & -18.44 & 41.02 & $30.6\pm 1.6$ & 37 & 28.2\
SBS 0335-052W & 53.80 & -14.68 & -16.12 & 39.70 & $20.2\pm 2.7$ & 37 & 12.4\
SBS 1116+517 & 23.10 & -14.73 & – & 39.97 & $27.4\pm 1.5$ & 50 & 10.4\
SBS 1159+545 & 52.20 & -14.65 & – & 40.24 & $22.9\pm 1.3$ & 66 & 9.1\
SDSS J1044+03 & 53.80 & -16.19 & – & 40.69 & $28.7\pm 1.9$ & 51 & 9.0\
UGC 772 & 16.30 & -14.88 & – & 39.33 & $22.0\pm 2.9$ & 40 & 38.5\
UGC 993 & 40.30 & -17.72 & – & 40.55 & $22.1\pm 2.5$ & 69 & 48.5\
Anon J0125+07 & 40.30 & -16.20 & – & 39.46 & $25.0\pm 5.0$ & 64 & 22.9\
\
II Zw 40 & 9.69 & -18.29 & -17.89 & 41.14 & $32.5\pm 1.2$ & 60 & 50.5\
II Zw 70 & 19.12 & -16.56 & -18.61 & 40.47 & $25.7\pm 1.2$ & 76 & 35.0\
III Zw 102 & 22.71 & -19.24 & -22.91 & 40.85 & $31.7\pm 2.6$ & 60 & 108.4\
III Zw 107 & 78.09 & -19.53 & -22.04 & 41.37 & $41.1\pm 1.4$ & 51 & 25.1\
Mrk 5 & 13.96 & -15.47 & -18.04 & 39.55 & $18.0\pm 1.6$ & 48 & 33.5$^*$\
Mrk 33 & 22.30 & -18.28 & -21.31 & 40.98 & $37.7\pm 2.4$ & 47 & 32.6\
Mrk 35 & 15.60 & -17.76 & -20.14 & 40.43 & $27.7\pm 1.2$ & 27 & 63.6\
Mrk 36 & 10.43 & -14.71 & -16.14 & 39.91 & $23.8\pm 1.2$ & 47 & 18.8\
Mrk 297 & 65.10 & -21.16 & -23.49 & 41.65 & $36.7\pm 1.7$ & 40 & 121.8\
Mrk 324 & 22.43 & -16.44 & -18.95 & 39.68 & $28.8\pm 5.1$ & 38 & 108.0$^*$\
Mrk 370 & 10.85 & -16.83 & -19.51 & 40.12 & $24.9\pm 2.0$ & 45 & 53.5\
Mrk 600 & 12.81 & -15.43 & -17.43 & 39.78 & $18.7\pm 1.6$ & 59 & 35.5$^*$\
\
Figure \[fig1\_1\] presents the results of our observations with the scanning FPI: the image in the line, the velocity field, and the velocity dispersion, corrected for the thermal broadening and natural width according eq.(\[eq1\]). The velocity fields usually have more points than the velocity dispersion maps since, at the same $S/N$ level,the velocity is measured with a higher accuracy than the line width.
{width="\textwidth"}
{width="\textwidth"}
{width="\textwidth"}
$L-\sigma$ relation {#sec3}
===================
The calculated kinematic parameters of the galaxies together with their adopted distances, absolute magnitudes in the $B$ and $K_s$-bands and total luminosity are given in the Table \[tab\_dat\]. We use the distances, luminosities in the $B$ and $K_s$-bands, and in the emission line for the nearby galaxies provided by the LVG database[^4] [@Kaisina2012]. All luminosities are corrected for the internal ($A^i_B$) and Galactic ($A^g_B$) extinction, according to the values given in this database[^5]
For nearby galaxies not listed in the LVG database (UGC 231, UGC 891, UGC 1924, UGC 3672), the distances were adopted from @Karachentsev2004. For UGC 8313 we use data from @Tully1988. Apparent magnitudes of these galaxies were adopted from RC3 [@RC3]; the $m_K$ values were retrieved from the 2MASS, $A^i_B$ – according to the relation of Verheijen (2001). The flux in the was taken from @vanZee2000 and @Kennicutt2008. Distances and luminosities for the XMD sample were taken – as per @Moiseev2010, and for the BCDG – from @Cairos2001.
When estimating $L_{H\alpha}$, we take into account the fact that in observations with narrow-band filters authors measure the flux in +\[NII\]. The contribution of the nitrogen lines \[NII\]$\lambda\lambda6548,6584$ was determined by an empirical correlation, linking the \[NII\]/ ratio with $M_B$ [@Kennicutt2008; @Lee2009]. For the most of considered galaxies, fainter than $M_B=-18$ this ratio is small (\[NII\]$/H\alpha<0.2$).
The inclination angles $i$ and maximal rotational velocities $V_{\rm max}$ for the XMD sample are based on the results of @Moiseev2010. For most of Local Volume galaxies the data are taken from @Moiseev2014. The kinematic parameters of III Zw 102 and UGC 8313 were presented by @Moiseev2008 and @Voigtlaender2015. For other galaxies $i$ and $V_{\rm max}$ were derived from the ionized gas velocity field in the same manner as described in @Moiseev2014. In cases, where rotation curves never clearly come to a plateau, or velocity field of ionized gas are dominated by non-circular motions (objects marked by asterisk in the Table \[tab\_dat\]), we use estimates from the HyperLeda [@Makarov2014].
The velocity fields shown in Figure \[fig1\_1\] reveal a component related with regular rotating discs in a majority of the sample (61 per cents). The fraction of rotation dominated galaxies significantly changes in the different subsamples. In the Local Volume subsample the most of galaxies (72 per cents or 26 objects) have disc-like ionized gas kinematics. Even among the remaining objects, in some dwarf galaxies (DDO 53, UGC 6456, UGC 8638) appears a component corresponding to a circular rotation. However, in these galaxiers non-circular velocities have larger amplitude [@Moiseev2014]. In these low-mass galaxies the -emitted gas is observed only in the central region, where amplitude of their rotation curve is about 5–10${\,\mbox{km}\,\mbox{s}^{-1}}$. In contrast with the LV subsample, only half of XMD galaxies (55 per cents or 6 objects) demonstrate disc-like rotation gradient, including cases of merger remains (HS 0822+3542 and UGC 772) which reveal two rotating discs with different orientation of spins. This is not surprising because analysis of @Moiseev2010 provides the evidence for the crucial role of interaction-induced star formation among galaxies in this subsample. The similar situation is also true for BCDG subsample, where only 33 per cents of objects (namely III Zw 102, Mrk 33, Mrk 35 and Mrk 370) show disc-dominated rotation in their velocity fields.
The maps of velocity dispersion $\sigma$ clearly demonstrate that in the centre of star forming regions the velocity dispersion of ionized gas has a minimum, whereas $\sigma$ increases towards periphery. Such a feature in the distribution of the ionized gas velocity dispersion was noted earlier in a number of studies [e.g. @Moiseev2010; @Marino2013] and was discussed in @MoisLoz2012, where such behavior is attributed to the influence of young stellar groups on the surrounding gas.
![Examples of “cores” (regions with very large $H\alpha$ fluxes, dark gray on the maps) and “diffuse” regions (low fluxes, light gray) in DDO53, DDO99 and UGC 5423 galaxies. The isophotes of the images are overlayed. The horizontal bar shows the linear scale of 1 kpc. To the right of each map we plot a histogram of the surface brightness distribution of the image pixels in . The “core” and “diffuse” components are filled with the same shades of gray.[]{data-label="fig_diff"}](DDO53.eps "fig:"){width="50.00000%"} ![Examples of “cores” (regions with very large $H\alpha$ fluxes, dark gray on the maps) and “diffuse” regions (low fluxes, light gray) in DDO53, DDO99 and UGC 5423 galaxies. The isophotes of the images are overlayed. The horizontal bar shows the linear scale of 1 kpc. To the right of each map we plot a histogram of the surface brightness distribution of the image pixels in . The “core” and “diffuse” components are filled with the same shades of gray.[]{data-label="fig_diff"}](DDO99.eps "fig:"){width="50.00000%"} ![Examples of “cores” (regions with very large $H\alpha$ fluxes, dark gray on the maps) and “diffuse” regions (low fluxes, light gray) in DDO53, DDO99 and UGC 5423 galaxies. The isophotes of the images are overlayed. The horizontal bar shows the linear scale of 1 kpc. To the right of each map we plot a histogram of the surface brightness distribution of the image pixels in . The “core” and “diffuse” components are filled with the same shades of gray.[]{data-label="fig_diff"}](U5423.eps "fig:"){width="50.00000%"}
Figure \[fig\_lum1\] shows different variants of $L-\sigma$ dependence for the galaxies in our sample. The literature most often discusses the dependence of $\sigma$ on $L_{H\alpha}$ (or $L_{H\beta}$) , shown in the upper panels. The top left figure separately shows all the three subsamples of galaxies we have observed. It is clear that they form a common sequence without any significant systematic offsets. Given that, the galaxies from the LV in general have a lower luminosity than the XMD and BCDG. For completeness, we present the same data for the nine bright BCDGs from @Ostlin1999 [@Ostlin2001] and twelve starburst galaxies from @Blasco-Herrera2013 since the measurement technique used by these authors is completely analogous to ours – averaging of the map of velocity dispersion, obtained with a scanning FPI. Their measurements complement our $L_{H\alpha}-\sigma$ sequence in the direction of higher luminosities. Note that we have been able to significantly continue the dependence on $L_{H\alpha}-\sigma$ towards the dwarf galaxies, up to $L_{H\alpha}\propto10^{37}\,\mbox{erg}\,\mbox{s}^{-1}$, while the vast majority of papers [@TerlevichMelnick1981; @BordaloTelles2011] considers the HII galaxies with $L_{H\alpha}>10^{39}\,\mbox{erg}\,\mbox{s}^{-1}$ (with the exception of papers devoted to various HII regions in the interiors of large galaxies: @Arsenault1988 [@Wisnioski2012], and others). Not to crowd the figures, further on we shall not show separately subsamples nor depict the error bars.
The upper right panel shows the distribution of $L_{H\alpha}-\sigma$ in a more traditional form, on a logarithmic scale along both axes. It can be clearly seen that the dependence between the logarithms of luminosity and dispersion is almost exactly linear with a rather small scatter. This fact was noted in many studies, but, as already mentioned in the Introduction, in the case of giant HII regions it was usually associated with the virial ratio, i.e. by the fact that the ionized gas velocity dispersion is definitely related with stellar velocity dispersion and is controlled by the total mass of the system. However, the luminosity in the Balmer emission lines is not a unique function of mass, while it is determined by the number of young OB-stars [@Kennicutt1998 and references therein]. While for any reasonable initial stellar mass function, the total stellar mass is determined by the more numerous but less massive stars that can not ionize the surrounding gas. Also the mass of the dark matter correlates exactly with this stellar mass through the Tully-Fisher relation. If $\sigma$ is defined by the mass of the system, it should better correlate not with $L_{H\alpha}$, but with the luminosity in broader spectral bands, or other parameters that are directly related to the mass.
But this is not observed. Figure \[fig\_lum1\] shows the $L-\sigma$ dependence, built for $M_B$. In the optical $B$-band the older stellar population has a larger contribution, compared to the luminosity. However, the point spread here is larger, and the correlation coefficient is smaller: $r=0.68$ versus $r=0.77$ in the case of of $L_{H\alpha}$. The $K$-band luminosity is directly related with the total mass of stellar population of the galaxy, moreover, the effect of interstellar reddening is much smaller here. However, the point spread on the $M_K-\sigma$ diagram is same with $M_B$ ($r=0.71$, Figure \[fig\_lum1\] bottom right).
We also use a parameter related to the dynamical mass of the system – the amplitude of the rotation velocity, $V_{\rm max}$. Left panel in Figure \[fig\_lum2\] indicates that ionized gas velocity dispersion does not depend on the amplitude of the rotation velocity $V_{\rm max}$ with the correlation coefficient being very low ($r=0.14$). For galaxies with $V_{\rm max}=10-50{\,\mbox{km}\,\mbox{s}^{-1}}$ the gas velocity dispersion is remarkably large. On average $\sigma\approx 20{\,\mbox{km}\,\mbox{s}^{-1}}$ with substantial galaxy-to-galaxy variations. @Green2010 came to a similar conclusion that $\sigma$ is not related to the total stellar mass for more distant galaxies with violent star formation.
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Another argument against a direct relation of $\sigma$ with the galaxy mass is that there should be observed a systematic departure from the general trend for the galaxies in the transient unrelaxed state, such as merger and interacting. We also attribute to clear mergers such systems as Mrk 297 (two clearly kinematically decoupled components); UGC 993, HS 2236+1344 and SBS 1116+517 , where two rotating discs can be identified in the velocity fields [@Moiseev2010], and a tight pair UGC 7648/7651 (NGC 4485/4490) distorted by interaction. As ‘possible interaction’ galaxies we classify UGC 772, UGC 3672, SBS 0335-052W/E and SBS 1159+545 [see the arguments in @Moiseev2010] and the galaxies with kinematically decoupled polar components: Mrk 33, Mrk 370 [@Moiseev2011] and IIIZw102 [@Moiseev2008]. However, in Figure \[fig\_lum2\] all these galaxies follow a general relationship, just like isolated galaxy UGC 7611 (NGC 4460), where the ionized gas is associated with the circumnuclear star burst and galactic wind outflow [@Moiseev2010_4460; @Moiseev2014].
Discussed above features of the $L-\sigma$ and $V_{\rm max}-\sigma$ correlations suggest that the relation between $\sigma$ and $L_{H\alpha}$ (i.e. the current SFR) is primary. In this case the correlations of velocity dispersion with $M_B$, $M_L$, with the stellar and total mass are secondary, being the consequences of other scaling relations in the galaxies. Indeed, the more massive and bright galaxies with ongoing star formation as a rule tend to have a greater luminosity in the Balmer lines as well.
Line width $\sigma$ in dense and diffuse gas
--------------------------------------------
@Tenorio-Tagle1993 used an analytical model to support the idea that $\sigma$ of ionized gas, observed in giant star-forming regions and HII-galaxies is determined by the mass of these objects. The authors conclude that $\sigma$ in the regions of greatest brightness (“the kinematic cores of HII-regions”) is directly related to the mass and size of the star forming region. At the same time, bright HII-regions surrounded by the low-brightness coronae of ionized gas with a larger velocity dispersion. Such structure occurs as a result of the influence of young stellar groups on the ISM. But if the velocity dispersion in the centres of star-forming regions determined by the virial motions, then this value – $\sigma({\rm core})$ should be better correlated with the parameters related with mass ($M_B$, $M_K$), rather than with $L_{H\alpha}$, which is controlled by the number of ionizing photons from young massive stars. On the other hand, the average velocity dispersion of diffuse environment should show a clearer (than on the average for the entire galaxy) relation with $L_{H\alpha}$, since not only the number of Lyman quanta, but also the kinetic energy output of the winds of young stars and supernovae is directly scaled with total number of OB stars [see Figure 1 in @Dopita2008]. While the gas velocity dispersion is proportional to the square root of the kinetic energy of turbulent motions.
We separate the velocity dispersion maps into cores of HII regions and a diffuse component. For that we use a histogram of the brightness distribution in the emission line for each galaxy. We consider as belonging to “cores” all the pixels, which contain the top 20 per cents of the total luminosity, while the same 20 percent of the luminosity in the low flux pixels are assigned to the diffuse component. Typical examples of galaxy map separation into two components are shown in Figure \[fig\_diff\] that also presents the corresponding histograms of intensity of the surface brightness in , explaining the method of separation (the bright and faint tail of the distribution of surface brightness). We can see that “cores” really correspond to the very centres of bright HII regions, while pixels marked as “diffuse” correspond to the envelopes surrounding the regions of star formation.
For each of the components we calculate the flux-weighted velocity dispersion. Figure \[fig\_lum3\] shows the corresponding relations. It is clear that the gas velocity dispersion of the cores $\sigma({\rm core})$ and of the diffuse component $\sigma({\rm diffuse})$ are very similar. This is also confirmed by the correlation coefficients with respective luminosities that are almost the same, or even lower than those for the $\sigma$ of the entire disc.
Therefore, the separation into the central and diffuse components shows that the velocity dispersion at the centres of HII regions, as well as the diffuse component are primarily determined by luminosity, i.e. by the number of young massive stars. We have performed this analysis for various “core/diffuse” separation criteria. The general conclusion is the same as for the 20 percent criterion described here.
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Global $L-\sigma$ and SFR-$\sigma$ relations
============================================
As discussed in the Introduction, the $L-\sigma$ relation has been extensively studied, mainly for intergalactic and extragalactic HII regions and compact HII galaxies. All the results agree that there is a power-law relation: $L\propto\sigma^\alpha$. However, the exponent $\alpha$ measured in different studies considerably varies from $\alpha=3.15$ [@Roy1986] to 6.6 [@Hippelein1986]. For discussion see also @Blasco-Herrera2010 [@BordaloTelles2011; @Wisnioski2012]. Recently @Chavez2014 obtained $\alpha=4.65\pm0.14$ using a large sample of compact HII galaxies restricted by velocity dispersion ($\log\sigma<1.8{\,\mbox{km}\,\mbox{s}^{-1}}$) and equivalent width ($EW(H_\beta)>50$Å in order to minimize the contribution of rotation-supported system and/or objects with a complex emission lines structure. @Swinbank2012 suggested $\alpha=3.8$ which provides a common fit to the both local HII regions and starforming clumps in high-redshift galaxies. It is possible that different flux limits used in observations to measure $\sigma$ for different types of objects (e.g., individual HII regions, dwarf emission galaxies, objects at large redshift) make some contribution to the uncertainty with $\alpha$. Potential biases of selecting bright $H\alpha$ regions in the long-slit spectroscopy as compared with the full 3D spectroscopy may also contribute to the disagreements. Indeed, Figure \[fig\_lum3\] shows that the velocity width $\sigma({\rm core})$ in regions with high $H\alpha$ fluxes is systematically smaller than $\sigma({\rm diffuse})$ in regions with low fluxes. If one uses the spectra that are not deep enough, the measured mean velocity dispersion will be underestimated. Another possible reason for the disagreement is that the slope of the $L(\sigma)$ relation changes with the luminosity and with the age of the most recent burst of star formation episode.
Figure \[fig\_lum4\] combines our measurements with results by other authors who used similar techniques to estimate the ionized gas $\sigma$, i.e. 3D-spectroscopic observations (integral-field or scanning FPI) and calculation of the flux-weighted velocity dispersion for the whole galaxy instead individual HII regions: 9 blue compact galaxies from [@Ostlin2001], 11 starburst galaxies selected from the SDSS for $z<0.03$ [@Blasco-Herrera2013], 65 star-forming disc galaxies at $z\sim0.1$ [@Green2010] and 57 local luminous and ultra-luminous infrared galaxies (U/LIRGSs) without evidences of active galactic nucleus [@Arribas2014]. The published data on high-redshift systems were not included because the beam smearing of the velocity gradient can produce a bias in the estimate of the velocity dispersion [@Davies2011].
The figure shows, that for a wide range of luminosities $L_{H\alpha} = 10^{37}-10^{43.5} \,\mbox{erg}\,\mbox{s}^{-1}$, there is a tight correlation between $L_{H\alpha}$ and $\sigma$. Using the least square fit we derive the slope $\alpha=5.0\pm0.2$ for the present $L-\sigma$ relation [^6] This is why a linear approximation of the $\log L_{H\alpha}-\log \sigma$ relation yields substantially different slopes for large and small luminosities.
Another way of presenting our results is to relate $L_{H\alpha}$ luminosity to the star formation rate SFR. It is known that the $L_{H\alpha}$ luminosity is almost exactly proportional to the rate of ongoing star formation (SFR) of young massive stars [@Kennicutt1998]. However, a comparison of SFRs estimated separately from $H\alpha$ data and *GALEX* far-ultraviolet observations shows that for the low luminosities this ratio is broken due to the relative scarcity of massive stars in dwarf galaxies, in relation with their initial mass function. @Lee2009 show that SFR$\propto L_{H\alpha}^{0.62}$ for $L_{H\alpha}<2.5\times10^{39}\,\mbox{erg}\,\mbox{s}^{-1}$.
The right panel of Figure \[fig\_lum4\] presents the SFR-$\sigma$ relation when we use @Lee2009 conversion equations. The full range of the SFR in considered objects is $0.001-300\,M_\odot\,{\rm yr}^{-1}$. For the U/LIRGSs sample we accepted $SFR$ calculated from the near-infrared (IR) luminosity using an equation from @Kennicutt1998, because a large uncertainties with reddening correction of $L_{H\alpha}$ in these “dusty” systems [@Arribas2014]. Indeed, the resulting scatter of points corresponded to the high $\sigma$ for the $SFR$ seems to be smaller than for the $H\alpha$ luminosities. Note that the combination of the above factors (non-linearity of $SFR-L_{H\alpha}$ relation for the faint dwarf galaxies, and IR-based $SFR$ for the most luminous ULIRGs) leads a linear relation fitted as $SFR\propto\sigma^{5.3\pm0.2}$.
Using their data for the U/LIRGSs sample, @Arribas2014 fitted the similar relation as $\sigma\propto SFR^{0.12\pm0.03}$ that corresponds to the slope $\alpha=8.3\pm2.1$. This value is significantly larger than our measurements derived for the full sample included both faint and ultra-luminous galaxies. @Arribas2014 suggested that relatively weak dependency of $\sigma$ on the total $SFR$, inferred from their fit, is related with the fact that star formation is not a dominant source driving the ionized gas velocity dispersion. Instead, they present arguments in support the scenario where gravitational energy associated with interaction and mergers has a significant contribution to the gas turbulence for the $SFR >10\,M_\odot\,{\rm
yr}^{-1}$. However, our results clearly demonstrate the same tendency $SFR-\sigma$ for the dwarf galaxies with and without interaction as well as for U/LIRGs, which appear to have a larger fraction of ongoing mergers. This fact may indicate that the role of interactions in driving ionized gas turbulence in U/LIRGs was overestimated.
The other intriguing fact is that the slope of the $L-\sigma$ relation inferred from our analysis ($\alpha\approx5.0-5.3$) is near the value $\alpha=4.7$ obtained by @Chavez2014. From the one hand, @Chavez2014 considered individual giant HII regions and avoided rotation-support system. From the other hand, the most of galaxies presented on Figure \[fig\_lum4\] reveal a domination of circular rotation discs in their kinematics. In previous sections we presented arguments that the ongoing star formation is a dominant driver of $\sigma$ calculated for the whole galactic disc. Why are the galaxies of very different types and luminosities follow a similar $L-\sigma$ relation with the systems where “the main mechanism of line broadening is linked to the gravitational potential of the young massive cluster” [@Chavez2014]? Note that the range of luminosities on the relation shown on the Fig. \[fig\_lum4\] is twice larger as compared with $L-\sigma$ considered in previous works for compact HII galaxies.
Discussion
==========
@Green2010 convincingly show that in a wide range of galaxy luminosities, including the objects at $z=1-3$, the mean velocity dispersion $\sigma$ is determined only by the star formation rate (i.e. by the luminosity) and does not correlate with mass. In this case, $\sigma$ is characteristic of the energy injected in the ISM by stellar winds, supernova explosions, and stellar radiation. For the velocity dispersion of neutral gas, a similar conclusion – weak correlation with galaxy mass – was previously drawn by @Dib2006. Further, the analysis of the shape of integrated HI profiles by @Stilp2013 shows that the velocity dispersion of a broad component of the HI line in dwarf galaxies is defined by the SFR$/M_{HI}$ ratio, although the dependence on the galaxy mass is also present.
Previously we found that there is a close relationship between the two-dimensional distributions of the line-of-sight velocity dispersions of the ionized gas and the local luminosity [@MoisLoz2012]. Specifically, we found that most of the areas with the highest velocity dispersion belong to the diffuse low brightness gas, surrounding the star forming regions. This contadicts the idea that $\sigma$ is determined mainly by the distribution of mass in the of galaxy. In this case one would have expected a different pattern – the maximum of velocity dispersion at the centres of star-forming regions and a decreasing $\sigma $ with the distance from the centre.
In the present paper we discuss different types of correlations of $\sigma$ with integral parameters of galaxies – the amplitude of the rotation curve, and luminosities in different bands. We find that the relationship with the parameters characterizing the mass of the galaxy is considerably less distinct than the one with the ongoing star formation, determined by the luminosity. Moreover, the current SFR determines the magnitude of supersonic turbulent motions of gas not only in the starburst galaxies, but also in objects with a very low SFR up to $10^{-3}\,M_\odot\,{\rm yr}^{-1}$.
We analyse two-dimensional velocity fields of ionized gas using observations at the 6-m BTA telescope. This allows us to confidently measure the mean $\sigma$ across a galaxy, and study details of its distribution inside and outside of star-forming regions. Our new data significantly extend the published $L-\sigma$ relations to the low mass galaxies and provide the observational evidence that the star formation determines the velocity dispersion of the ionized gas:
- The ionized gas velocity dispersion, luminosity-averaged across the galaxy, is better correlated with the luminosity in the line than in the broad $B$ and $K$-bands. There is almost no correlation of the velocity dispersion$\sigma$ with the rotation velocity of galaxy.
- The gas velocity dispersion $\sigma$ in the cores of star-forming regions is nearly the same as $\sigma$ in the duffuse component with low fluxes.
- There a common $SFR-\sigma$ relation for the local galaxies in a very broad range of luminosities $L_{H\alpha} = 10^{37}-10^{43.5}$ that corresponds $SFR=0.001-300\,M_\odot\,{\rm yr}^{-1}$. The fit of this relation $\sigma\propto SFR^\alpha$ provides the slope $\alpha=5.3\pm0.2$.
We therefore conclude that velocity of turbulent motions of ionized gas in galaxies is defined mainly by the energy that is transferred to the interstellar medium from young stellar populations in the form of ionizing radiation pressure, and by the winds of young stars and supernova explosions. We believe that this conclusion is important for both simulations of galaxy formation and for interpretation of the apparent emission line widths in galaxies affected by various processes (e.g., star formation, merging, and virial motions).
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
We are very grateful to the anonymous referee, and to Santiago Arribas, Eduardo Telles and Roberto Terlevich for their constructive comments and suggestions that helped us to improve and clarify our result. We also thank Sean Markert for discussions and comments. Our observations were done with the 6-m telescope of the Special Astrophysical Observatory of the Russian Academy of Sciences. We grateful to the staff of the Observatory and specially Victor Afanasiev for his great contribution to spectroscopy at the 6-m telescope. The observations were carried out with the financial support of the Ministry of Education and Science of the Russian Federation (agreement No. 14.619.21.0004, project ID RFMEFI61914X0004). We have used the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under the contract with the National Aeronautics and Space Administration. We acknowledge the usage of the HyperLeda database (http://leda.univ-lyon1.fr). This work was supported by the Ministry of Education and Science of the Russian Federation (project 8523) and by the Research Program OFN-17 of the Division of Physics, Russian Academy of Sciences. AM is also grateful for the financial support of the non-profit “Dynasty” Foundation. AK acknowledge support of NASA and NSF grants to NMSU.
Afanasiev V. L., Moiseev A. V., 2005, Astronomy Letters, 31, 194
Afanasiev V. L., Moiseev A. V., 2011, Baltic Astronomy, 20, 486
Arribas S., Colina L., Bellocchi E., Maiolino R., Villar-Martín M., 2014, A&A, 568, A14
Arsenault, R., Roy, J.-R. 1988, A&A, 201, 199
Blasco-Herrera J., Fathi K., Beckman J. et al., 2010, MNRAS, 407, 2519
Blasco-Herrera J., Fathi K., [Ö]{}stlin G., Font J., Beckman J. E., 2013, MNRAS, 435, 1958
Bournaud F., Chapon D., Teyssier R., 2011, ApJ, 730, 4
Bordalo V., Plana H., Telles E., 2009, ApJ, 696, 1668
Bordalo V., Telles E., 2011, ApJ, 735, 52
Burkert A., 2006, Comptes Rendus - Physique, 7, 433
Cairós L. M., Vílchez J. M., González-Pérez J. N., Iglesias-Páramo J., Caon N., 2001, ApJS, 133, 321
Ceverino D., Klypin A., Klimek E. S. et al., 2014, [MNRAS]{}, 442, 1545
Ch[á]{}vez R., Terlevich E., Terlevich R. et al. 2012, MNRAS, 425, L56
Ch[á]{}vez R., Terlevich R., Terlevich E. et al., 2014, MNRAS, 442, 3565
Chu Y.-H., Kennicutt R. C. Jr., 1994, Ap&SS, 216, 253
Davies R., F[ö]{}rster Schreiber N. M., Cresci G., et al., 2011, ApJ, 741, 69
de Vaucouleurs G., de Vaucouleurs A., Corwin H. G. Jr. et al., 1991, Third Reference Catalogue of Bright Galaxies, Version 3.9 (New York: Springer)
Dib S., Bell E., Burkert A., 2006, ApJ, 638, 797
Dopita M. A., 2008, [*in Massive Stars as Cosmic Engines*]{}, Proceedings of the International Astronomical Union, IAU Symposium, 250, 367
Gallagher J. S., Hunter D. A., 1983, ApJ, 274, 141
Genzel R., Newman S., Jones T. et al., 2011, ApJ, 733, 101
Green A. W., Glazebrook K., McGregor P. J. et al., 2010, Nature, [467]{}, 684
Hippelein H. H., 1986, A&A, 160, 374
Hopkins P.F., Quataert E., Murray N., 2012, MNRAS, 421, 3488
Joung M. R., Mac Low M.-M., Bryan G. L., 2009, ApJ, 704, 137
Kaisina, E. I., Makarov, D. I., Karachentsev I. D., Kaisin S. S., 2012, Astrophys. Bull., 67, 115
Karachentsev I. D., Karachentseva V. E., Huchtmeier W. K., Makarov D. I., 2004, AJ, 127, 2031
Kennicutt Jr. R. C., 1998, ARA&A, 36, 189
Kennicutt R. C. Jr., Lee J. C., Funes S. J., José G., Sakai S., Akiyama S., 2008, ApJS, 178, 247
Koulouridis E., Plionis M., Chávez R. et al., 2013, A&A, 544, 13
Lopez L. A., Krumholz M. R., Bolatto A. D., Prochaska J. X., Ramirez-Ruiz E., Castro D., 2014, ApJ, 795, 121
Lee J. C., Gil de Paz, A., Tremonti Ch. et al. 2009, ApJ, 706, 599
Mac Low M.-M., Klessen R. S., 2004, Reviews of Modern Physics, 76, 125
Makarov D., Prugniel P., Terekhova N., Courtois H., Vauglin I., 2013, MNRAS, 428, 476
Marino A., Plana H., Rampazzo R. et al., 2013, MNRAS, 428, 476
Martínez-Delgado I., Tenorio-Tagle G., Muñoz-Tuñón C. et al., 2007, AJ, [133]{}, 2892
Melnick J., 1977, ApJ, 213,15
Melnick J., Moles M., Terlevich R., Garcia-Pelayo J-M., 1987, MNRAS, 226, 849
Melnick J., Tenorio-Tagle G., Terlevich R., 1999, MNRAS, 302, 677
Melnick J., Terlevich R., Terlevich E., 2000, MNRAS, 311, 629
Moiseev A.V., 2002, Bull. Spec. Astrophys. Obs. 54, 74
Moiseev A.V., 2008, Astrophys. Bull., 63, 201
Moiseev A.V., 2011, EAS Publications Series, 48, 115
Moiseev A.V., 2014, Astrophys. Bull., 69, 1
Moiseev A.V., Egorov O.V., 2008, Astrophys. Bull., 63, 181
Moiseev A. V., Lozinskaya T. A., 2012, MNRAS, 423, 1831
Moiseev A.V., Karachentsev I.D., Kaisin S.S., 2010a, MNRAS, 403, 1849
Moiseev A.V., Pustilnik S. A., Kniazev A. Y., 2010b, MNRAS, 405, 2453
Muñoz-Tuñón C., Tenorio-Tagle G., Castañeda H.O., Terlevich R., 1996, AJ, 112, 1636
stlin G., Amram P., Masegosa J., Bergvall N., Boulesteix J., 1999, A&AS, 137, 419
stlin G., Amram P., Bergvall N., Masegosa J., Boulesteix J., M[á]{}rquez I., 2001, A&A, 374, 800
Roy J.-R., Arsenault R., Joncas G., 1986, ApJ, 300, 624
Rozas M., Zurita A., Beckman J. E., Pérez D., 2000, A&AS, 142, 259
Scalo J., Chappell D., 1999, [MNRAS]{}, 310, 1
Smith M. G., Weedman D. W., 1970, ApJ 161, 33
Stilp A. M., Dalcanton J. J., Warren S. R., Skillman, E., Ott J., Koribalski B., 2013, ApJ, 765, 136
Swinbank M., Smail I., Sobral D., Theuns T., Best Ph.n., Geach J.E., 2012, ApJ, 760, 130
Tenorio-Tagle G., Muñoz-Tuñón C., Cox P. D, 1993, ApJ, 418 , 767
Terlevich R., Melnick J., 1981, MNRAS, [195]{}, 839
Trujillo-Gomez S., Klypin A., Col[í]{}n P., et al., 2015, [MNRAS]{}, 446, 1140
Tully R. B., 1988, Nearby Galaxy Catalog, Cambridge University Press, Cambridge: 1988
van Zee L., 2000, AJ, 119, 2757
Vasiliev E. O., Moiseev A. V., Shchekinov Y. A., 2015, Baltic Astronomy, accepted; arXiv:1411.1269
Voigtlaender P., Bomans D.J., Moiseev A.V., 2015, A&A, sumbitted.
Wisnioski E., Glazebrook K., Blake , Ch., et al., Ch. 2012, MNRAS, 422, 3339
\[lastpage\]
[^1]: [email protected]
[^2]: Deceased
[^3]: The system of Russian Academy of Sciences institutes was liquidated on Sep 2013
[^4]: http://lv.sao.ru/lvgdb/
[^5]: The absorption in the line was assumed to be $0.538(A^i_B+A^g_B)$, and in the $K_s$-band: $0.085(A^i_B+A^g_B)$.
[^6]: Here $\sigma$ is considered as an independent variable in agreement with previous studies as it discussed in @BordaloTelles2011. The plots show that there is a sort of kink at $\sim10^{41}\,\mbox{erg}\,\mbox{s}^{-1}$, corresponding to $\sigma\sim30{\,\mbox{km}\,\mbox{s}^{-1}}$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we design Dirichlet-Neumann boundary feedback controllers for the Korteweg-de Vries (KdV) equation that act at the right endpoint of the domain. The length of the domain is allowed to be critical. Constructing backstepping controllers that act at the right endpoint of the domain is more challenging than its left endpoint counterpart. The standard application of the backstepping method fails, because corresponding kernel models become overdetermined. In order to deal with this difficulty, we introduce the *pseudo-backstepping* method, which uses a *pseudo-kernel* that satisfies all but one desirable boundary condition. Moreover, various norms of the pseudo-kernel can be controlled through a parameter in one of its boundary conditions. We prove that the boundary controllers constructed via this pseudo-kernel still exponentially stabilize the system with the cost of a low exponential rate of decay. We show that a single Dirichlet controller is sufficient for exponential stabilization with a slower rate of decay. We also consider a second order feedback law acting at the right Dirichlet boundary condition. We show that this approach works if the main equation includes only the third order term, while the same problem remains open if the main equation involves the first order and/or the nonlinear term(s). At the end of the paper, we give numerical simulations to illustrate the main result.'
address: 'Department of Mathematics, Izmir Institute of Technology, Urla, Izmir, TURKEY'
author:
- 'Türker Özsarı^\*^ & Ahmet Batal'
bibliography:
- 'myreferences.bib'
title: 'Pseudo-backstepping and its application to the control of Korteweg-de Vries equation from the right endpoint on a finite domain'
---
\#1
Introduction
============
This article is devoted to the study of the boundary feedback stabilization of the Korteweg-de Vries (KdV) equation on a bounded domain $\Omega=(0,L)\subset \mathbb{R}$. The linear version of the model under consideration is given by $$\label{KdVBurgers}
\begin{cases}
\displaystyle u_{t} + u_{x} + u_{xxx} =0 & \text { in } \Omega\times \mathbb{R_+},\\
u(0,t) = 0, u(L,t) = U(t), u_{x}(L,t)=V(t) & \text { in } \mathbb{R_+},\\
u(x,0)=u_0(x) & \text { in } \Omega,
\end{cases}$$ whereas the nonlinear version of this model is written with the main equation in replaced with $$\label{nonlinearKdV}u_t+u_x+u_{xxx}+uu_x=0.$$
In , $u=u(x,t)$ is a real valued function that can model the evolution of the amplitude of a weakly nonlinear shallow dispersive wave in space and time [@KdVPaper]. The inputs $U(t)=U(u(t,\cdot))$ and $V(t)=V(u(t,\cdot))$ at the right endpoint of the boundary are feedbacks. The goal is to choose these boundary feedbacks so that the solutions of and decay to zero as $t\rightarrow \infty$, at an exponential rate in the mean-square sense.
Controlling the behavior of solutions of evolution equations is an important topic, and many approaches have been proposed. One method is to use local or global interior controllers. Another method is to use external (boundary) controllers, especially in those models where it is difficult to access the domain. Using feedback type controls is a common tactic to stabilize the solutions. However, non-feedback type controls (open loop control systems) are also used for steering solutions to or near a desired state. Exact, null, or approximate controllability models have been developed for almost all well-known PDEs.
Exact boundary controllability of linear and nonlinear KdV equations with the same type of boundary conditions as in was studied by [@Rosier1997], [@Cor2004], [@Zhang99], [@Glass08], [@Cerpa07], [@Cerpa09], [@RosZha09], and [@Glass10]. In these papers, the boundary inputs are chosen in advance to steer solutions to a desired final state at a given time. This results in an open loop model. In contrast, the boundary inputs in our model depend on the solution itself, and is therefore closed loop.
Stabilization of solutions of the KdV equation with a localised interior damping was achieved by [@Perla2002], [@Pazo05], [@Mass07], and [@Balogh2000]. There are also some results achieving stabilization of the KdV equation by using predetermined local boundary feedbacks; see for instance [@Liu2002], and [@Jia2016].
Motivation
----------
and with homogeneous boundary conditions ($U=V\equiv 0$) are both dissipative since $\frac{d}{dt}\|u(t)\|_{L^2(\Omega)}^2\le 0.$ However, this does not always guarantee exponential decay. It is well-known that if $\displaystyle L\in \mathcal{N}\equiv \left\{2\pi\sqrt{\frac{k^2+kl+l^2}{3}}, k,l\in \mathbb{N}\right\}$ (so called *critical lengths* for KdV), then the solution does not need to decay to zero at all. For example, if $L=2\pi$, $u=1-\cos(x)$ is a (time independent) solution of on $\Omega=(0,2\pi)$, but its $L^2-$norm is constant in $t$. On the other hand, if $L$ is not critical, one can show the exponential stabilization of solutions for under homogeneous boundary conditions; see for example [@Perla2002 Theorem 2.1].
Recently, [@Cerpa2013] studied the boundary feedback stabilization of the KdV equation with the boundary conditions $$\label{BCforKdVLeft}
u(0,t) = U(t), u_{x}(L,t) = 0, u(L,t) = 0$$ by using the back-stepping technique (see for example [@KrsBook]). [@Cerpa2013] proved that given any $r>0$, there corresponds a smooth kernel $k=k(x,y)$ such that the boundary feedback controller $U(t)=U(u(t,\cdot))=\int_0^Lk(0,y)u(y,t)dy$ steers the solution of the linear KdV equation to zero with the decay rate estimate $\|u(t)\|_{L^2(\Omega)}\lesssim \|u_0\|_{L^2(\Omega)}e^{-r t}.$ Moreover, the same result also holds true for the nonlinear KdV equation provided that $u_0$ is sufficiently small in the $L^2-$sense. Here, $k=k(x,y)$ is an appropriately chosen kernel function satisfying a third order PDE model on a triangular domain that involves three boundary conditions. In [@Cerpa2013], the control acts on the Dirichlet boundary condition at the *left* endpoint of the domain. However, the situation is very different if the control acts at the *right* endpoint of the domain, because then the kernel of the backstepping controller has to satisfy an overdetermined PDE model whose solution may or may not exist. Therefore, the problem of finding backstepping controllers acting at the *right* endpoint of the domain is interesting.
Coron & Lü [@Cor14] studied this problem with a single controller acting from the Neumann boundary condition on domains of *uncritical* lengths. They prove the rapid exponential stabilization of solutions for the KdV equation under a smallness assumption on the initial datum. The method of [@Cor14] is based on using a rough kernel function in the backstepping integral transformation. The construction of the rough kernel relies on the exact controllability of the linear KdV equation by the Neumann boundary control acting at the right endpoint of the domain. However, the exact controllability was proved only for the domains of uncritical lengths. On the other hand, the exponential decay of solutions for the linearized KdV equation holds even without adding any control to the system when the length of the domain does not belong to the set of critical lengths [@Perla2002]. Therefore, the following remains as an important problem:
\[mainprob\] Let $L>0$ (not necessarily uncritical). Can you find a kernel $k=k(x,y)$ such that the solution of and satisfies $$\label{decayest}\|u(\cdot,t)\|_{L^2(\Omega)}=\mathcal{O}(e^{-rt})$$ for some $r>0$ with boundary feedback controllers given by $$\label{controller}
U(t)=\int_0^Lk(L,y)u(y,t)dy \text{ and } V(t)=\int_0^Lk_x(L,y)u(y,t)dy\,?$$
A stronger version of the above problem is the following:
\[mainprob1\] Given $r>0$, can you find a kernel $k=k(x,y)$ such that the solution of and satisfies the $L^2-$decay estimate with the boundary feedback controllers given in ?
This paper and the method proposed address only Problem \[mainprob\], and the latter problem still remains open for domains of critical length.
In order to understand the nature of the problem and the difficulty here, let us consider the linearised KdV equation in . A backstepping controller for this linear model is generally constructed by using a transformation given by $$\label{transform}w(x,t)\equiv u(x,t)-\int_0^xk(x,y)u(y,t)dy,$$ where the unknown kernel function $k(x,y)$ is chosen in such a way that if $u$ is a solution of with the boundary feedback controllers given in , then $w$ is a solution of the damped homogeneous initial-boundary value problem (so called “*target system*”) $$\label{HomKdVBurgers}
\begin{cases}
\displaystyle w_{t} + w_{x} + w_{xxx} + \lambda w = 0 & \text { in } \Omega\times \mathbb{R_+},\\
w(0,t) = w(L,t) = w_{x}(L,t) = 0 & \text { in } \mathbb{R_+},\\
w(x,0)=w_0(x)\equiv u_0-\int_0^xk(x,y)u_0(y)dy & \text { in } \Omega.
\end{cases}$$ The reason is that the solution of satisfies $\|w(t)\|_{L^2(\Omega)}=O(e^{-\lambda t})$, and if the given transformation is invertible, one can hope to get a similar decay property for $u$.
The essence of the back-stepping algorithm is to find an appropriate kernel function $k$ which serves the purpose. In order to do this, one simply assumes that $u$ solves and plugs in $u(x,t)-\int_0^xk(x,y)u(y,t)dy$ into the main equation in wherever one sees $w$. This gives a set of sufficient conditions that the kernel has to satisfy. Note that $w$ satisfies the given homogeneous boundary conditions $w(0,t)=w(L,t)=w_x(L,t)=0$ by the transformation in and the choice of the feedback controllers in . In order for the main equation in to be satisfied, one can impose a few conditions on $k$. Indeed, computing the derivative of $w$ with respect to the temporal and spatial derivatives and putting these together, we obtain the following: $$\begin{gathered}
\nonumber w_{t}(x,t) + w_{x}(x,t) + w_{xxx}(x,t) + \lambda w(x,t)= k_y(x,0)u_x(0,t)\\
-\int_{0}^{x}u(y,t)\left[k_{xxx}(x,y) + k_{x}(x,y) + k_{yyy}(x,y) + k_{y}(x,y) + \lambda k(x,y)\right]dy \label{23} \\
\nonumber -k(x,0)u_{xx}(0,t) - u_{x}(x,t)\left[k_{y}(x,x) + k_{x}(x,x) + 2\frac{d}{dx}k(x,x)\right] \\
\nonumber + u(x,t)\left[\lambda - k_{xx}(x,x) + k_{yy}(x,x) - \frac{d}{dx}k_{x}(x,x) - \frac{d^{2}}{dx^{2}}k(x,x)\right].\end{gathered}$$ The above equation is the same as that of the target system if $k$ solves the third order partial differential equation together with the set of boundary conditions given by $$\begin{aligned}
\label{kEq}
% \nonumber to remove numbering (before each equation)
\nonumber k_{xxx} + k_{yyy} + k_{y} + k_{x} &=& -\lambda k, \\
k(x,x) = k(x,0)=k_y(x,0) &=& 0, \\
\nonumber k_{x}(x,x) &=& \frac{\lambda}{3}x,\end{aligned}$$ where the PDE model is considered on the triangular spatial domain $\mathcal{T}\equiv \{(x,y)\in \mathbb{R}^2\,|\,x\in [0,L], y\in [0,x]\}\,\,\text{(see Figure \ref{regT} below)}.$
![Triangular region $T$ for $L=2\pi$[]{data-label="regT"}](RecT)
In order to solve the problem , one generally first applies a change of variables. Here, an appropriate choice would be to define $t\equiv y$, $s\equiv x-y$, and $G(s,t)\equiv k(x,y)$. Then, $G$ satisfies the boundary value problem given by $$\begin{aligned}
% \nonumber to remove numbering (before each equation)
\label{Geq}G_{ttt} - 3G_{stt}+ 3G_{sst} + G_{t} &=& -\lambda G, \\
\label{Geqb}G(s,0) = G_t(s,0) = G(0,t) &=& 0, \\
\label{Geqc}G_s(0,t) &=& \frac{\lambda}{3}t\end{aligned}$$ on the triangular domain $\mathcal{T}_{0}\equiv \left\{ (s,t) \,|\, t \in [0,L], s \in [0,L-t]\right\}\,\,\text{(see Figure \ref{regT0} below)}.$
Unfortunately, it is not easy to decide whether - has a solution. Note that there is also a mismatch between the boundary conditions $ G_t(s,0)=0$ and $G_s(0,t) = \displaystyle \frac{\lambda}{3}t$ in the sense that $G_{ts}(0,0)=0\neq G_{st}(0,0)=\displaystyle \frac{\lambda}{3}.$ Hence, the standard back-stepping algorithm fails because it enforces us to solve an overdetermined singular PDE model. This issue does not arise if one tries to control the system from the left endpoint of the domain as in [@Cerpa2013].
![Triangular region $T_0$ for $L=2\pi$[]{data-label="regT0"}](RecT0)
The adverse effect of the nonhomogeneous boundary condition in the kernel PDE model was eliminated by expanding the domain from a triangle into a rectangle in [@Cor14]. However, this approach brings a dirac delta term to the right hand side of the main equation; see the kernel model in [@Cor14 Section 1]. The cost of this is that the constructed kernel cannot be expected to be very smooth. However, the higher regularity is crucial to rigorously justify the calculations in that show the equivalence of the original plant and the exponentially stable target system. Therefore, we rely on a different idea based on constructing an imperfect but smooth kernel. The details of this construction are given below.
Pseudo-backstepping
-------------------
We introduce a new backstepping technique which eliminates the difficulties explained in the previous section. In the standard backstepping method, the plant model is transformed into the most desirable (e.g., exponentially stable) target system with a transformation as in . This is called forward transformation. The target system is then transformed back into the plant model via an inverse transformation, generally in the form $$\label{backwardt}u(x,t)=w(x,t)+\int_0^xp(x,y)w(y,t)dy.$$ This is called backward transformation. A combination of these two steps allows one to conclude that the plant is stable if and only if the target system is stable in the same sense (see Figure \[Backstepping\]).
![Standard back-stepping[]{data-label="Backstepping"}](Backstepping)
Unfortunately, applying this algorithm to our problem forces kernels $p$ and $k$ to be solutions of overdetermined boundary value problems, and thus the method fails.
Our strategy uses a pseudo-kernel which is chosen as a solution of a corrected version of the gain control PDE given by: $$\begin{aligned}
% \nonumber to remove numbering (before each equation)
\label{Geqeps1}\tilde{G}_{ttt} - 3\tilde{G}_{stt}+ 3\tilde{G}_{sst} + \tilde{G}_{t} &=& -\lambda \tilde{G}, \\
\label{Geqeps2}\tilde{G}(s,0) = \tilde{G}(0,t) &=& 0, \\
\label{Geqeps3}\tilde{G}_s(0,t) &=& \frac{{\lambda}}{3}t\end{aligned}$$ on the triangular domain $\mathcal{T}_{0}$. Unlike in the previous model -, here the boundary condition $\tilde{G}_t(s,0)=0$ is completely disregarded. One advantage of using this modified model is that we can solve it. Another is that, even though the boundary condition $\tilde{G}_t(s,0)=0$ is disregarded, we can control the size of this boundary condition by choosing ${\lambda}$ sufficiently small. The cost of using a pseudo-kernel is that the target system changes, (see the modified target system in ), which causes a slower rate of decay. Nevertheless, this new method (henceforth referred to as *pseudo-backstepping*) allows us to obtain physically reasonable exponential decay rates for some choice of $\lambda$ (see Table \[table1\] for sample decay rates for some values of $\lambda$ on a domain of length $L=2\pi$).
Another aspect of our method is that instead of using a concrete backward transformation as in , we rely on the existence of an abstract inverse transformation that maps the solution of the modified target system back into the original plant. The existence of such a transformation is proved via succession (see Lemma \[inverselem\] below). This type of backward transformation was previously used in the stabilization of the heat equation with a localized source of instability [@Liu03]. We do not search for an inverse of type to avoid a highly overdetermined system that would result from computing the temporal and spatial derivatives of the given transformation and finding the conditions that $p$ has to satisfy.
![Pseudo-backstepping[]{data-label="pseudo-bs"}](pseudo-bs)
Main results
------------
Applying the pseudo-backstepping method explained above to the linearized and nonlinear KdV models given in and , we are able to prove the following wellposedness and stabilization theorems:
\[Linthm0\] Let $T>0$, $u_0\in L^2(\Omega)$ and $$\label{controllers}U(t) = \int_0^L\tilde{k}(L,y)u(y,t)dy,\,V(t)= \int_0^L\tilde{k}_x(L,y)u(y,t)dy,$$ where $\tilde{k}$ is a smooth kernel given by . Then, has a unique solution $u\in C([0,T];L^2(\Omega))\cap L^2(0,T;H^1(\Omega))$ satisfying also $u_x\in C([0,L];L^2(0,T)).$ Moreover, the same result also holds true for the nonlinear KdV equation if $\|u_0\|_{L^2(\Omega)}$ is sufficiently small.
Indeed, our analysis in this paper also shows that if $u_0\in H^3(\Omega)$ and satisfies the compatibility conditions $$\label{compcond}u_0(0)=0, u_0(L)=\int_0^L\tilde{k}(L,y)u_0(y)dy, u_0'(L)=\int_0^L\tilde{k}_x(L,y)u_0(y)dy,$$ then the solution of or the local solution of satisfies $u\in C([0,T];H^3(\Omega))\cap L^2(0,T;H^4(\Omega)).$ One can also interpolate to get regularity in the fractional spaces. For example, let $u_0\in H^s(\Omega)$ ($s\in [0,3]$) so that it satisfies the compatibility conditions $u_0(0)=0, u_0(L)=\int_0^L\tilde{k}(L,y)u_0(y)dy$ if $s\in [0,3/2]$ and the compatibility conditions if $s\in (3/2,3].$ Then, the solution of or the local solution of satisfies $u\in C([0,T];H^s(\Omega))\cap L^2(0,T;H^{s+1}(\Omega)).$
\[Linthm\] Let $u_0\in L^2(\Omega)$. Then, for sufficiently small $\lambda>0$, one has $\alpha=\lambda-\frac{1}{2}\|\tilde{k}_y(\cdot,0)\|_{L^2(\Omega)}^2>0$, where $\tilde{k}$ is given by , and the corresponding solution of with the boundary feedback controllers satisfies $ \|u(t)\|_{L^2(\Omega)}\lesssim \|u_0\|_{L^2(\Omega)}e^{-\alpha t}.$ Moreover, the same decay property is also true for the nonlinear KdV equation if $\|u_0\|_{L^2(\Omega)}$ is sufficiently small.
The proof of Theorem \[Linthm\] is given in the next section. Table \[table1\] gives some examples where exponential stabilization can be achieved. For example, when $\lambda=0.03$, the decay rate is approximately of order $\mathcal{O}(e^{-0.18 t})$ on a domain of length $L=2\pi$. The exponential decay rate is substantially small (see Table \[table1\]) relative to the decay rates one can get by controlling the equation from the left end-point with the same type of boundary conditions. Indeed, what matters is is not where the controller is located,but rather the number of boundary conditions specified on the opposite side of the boundary. For example, if one specified two boundary conditions at the left and only one boundary condition at the right, then it would be easier to control from right and more difficult to control from left, in contrast to the problem studied in this paper.
$\lambda$ $\alpha=\lambda-\frac{1}{2}\|\tilde{k}_y(\cdot,0)\|_{L^2(\Omega)}^2$
----------- ----------------------------------------------------------------------
0.01
0.02
0.03
0.04
0.05
0.10
1
: \[table1\]Numerical experiments on a domain of critical length $L=2\pi$
Stabilization
=============
In this section, we prove Theorem \[Linthm\]. First, we prove the existence of the pseudo-kernel and the abstract inverse transformation. Secondly, by using the multiplier method, we obtain the stabilization for suitable $\lambda.$ The multiplier method is applied only formally, but the calculations can be justified by a standard density argument and the regularity results proved in the next section.
Linearised model {#ProofofThm}
----------------
The sought-after solution of - can be constructed by applying the successive approximations technique to the integral equation $$\label{GepsInt}
\tilde{G}(s,t) = \frac{{\lambda}}{3}st+\frac{1}{3}\int_0^t\int_0^s\int_0^\omega (-\tilde{G}_{ttt} + 3\tilde{G}_{stt} - \tilde{G}_{t} -\lambda \tilde{G})(\xi,\eta)d\xi d\omega d\eta.$$ Indeed, we have the following lemma.
\[lemback\] There exists a $C^\infty$-function $\tilde{G}$ such that $\tilde{G}$ solves the integral equation as well as the boundary value problem given in -.
Let $P$ be defined by $$\label{aP}
(P f)(s,t) = \frac{1}{3}\int_0^t\int_0^s\int_0^\omega (-f_{ttt} + 3f_{stt} - f_{t} -\lambda f)(\xi,\eta)d\xi d\omega d\eta.$$ By , we need to solve the equation $\tilde{G}(s,t)=\frac{{\lambda}}{3}st+P\tilde{G}(s,t).$ Define $\tilde{G}^0\equiv 0,$ $\displaystyle\tilde{G}^1(s,t)=\frac{{\lambda}}{3}st,$ and $\tilde{G}^{n+1}=\tilde{G}^1+P \tilde{G}^n.$ Then for $n\geq 1$, $\tilde{G}^{n+1}-\tilde{G}^{n} = P(\tilde{G}^{n}-\tilde{G}^{n-1}).$ So if we define $H^0(s,t)=st$ and $H^{n+1}=PH^n$, we get $H^n=\frac{3}{\lambda}(\tilde{G}^{n+1}-\tilde{G}^{n}).$ Moreover, for $j>i,$ $$\label{aCauchy}
\tilde{G}^j-\tilde{G}^i= \sum_{n=i}^{n=j-1}\tilde{G}^{n+1}-\tilde{G}^{n}=\frac{\lambda}{3}\sum_{n=i}^{n=j-1}H^{n}.$$ Let $\| \cdot \|_{\infty}$ denote the supremum norm of a function on the triangle $T_0$. It follows from that to prove $\tilde{G}_n$ (and its partial derivatives) is Cauchy with respect to the norm $\| \cdot \|_{\infty}$ it is enough to show $H^n$ (and its partial derivatives) is an absolutely summable sequence with respect to the same norm.
To show $H^n$’s are absolutely summable, let us first write $P$ as the sum of four operators $P= P_{-2}+P_{-1}+P_0+P_1,$ where $$P_{-2}f= \frac{1}{3}\int_0^t\int_0^s\int_0^\omega -f_{ttt}(\xi,\eta) d\xi d\omega'd\eta,\,P_{-1}f= \int_0^t\int_0^s\int_0^\omega f_{stt}(\xi,\eta) d\xi d\omega'd\eta,$$ $$P_{0}f= \frac{1}{3}\int_0^t\int_0^s\int_0^\omega -f_{t}(\xi,\eta) d\xi d\omega'd\eta,\,P_{1}f= \frac{1}{3}\int_0^t\int_0^s\int_0^\omega -\lambda f(\xi,\eta) d\xi d\omega'd\eta.$$ Then $$\label{aproduct}
H^n=P^nH^0=(P_{-2}+P_{-1}+ P_0+P_1)^nst=\sum_{r=1}^{4^n}R_{r,n}st$$ where $R_{r,n}:=P_{j_{r,n}}P_{j_{r,n-1}}\cdot\cdot\cdot P_{j_{r,1}}$, $j_{r,i} \in \{-2,-1,0,1\}$. Observe that for positive integers $m$ and nonnegative integers $k$ $$\label{aPi}
P_{-1}s^m t^k= c_{-1}s^{m+1}t^{k-1} \; \text{and} \; P_{i}s^m t^k= c_{i}s^{m+2} t^{k+i} \; \text{for} \; i=-2,0,1,$$ where $$\label{acm2}
c_{-2}=
\begin{cases}
0 & \text { if } k\leq 2,\\
-\frac{k(k-1)}{3(m+1)(m+2)} & \text { if } k > 2,\\
\end{cases}$$ $$\label{acm1}
c_{-1}=
\begin{cases}
0 & \text { if } k\leq 1,\\
\frac{k}{(m+1)} & \text { if } k > 1,\\
\end{cases}$$ $$\label{ac0}
c_{0}=-\frac{1}{3(m+1)(m+2)},$$ $$\label{ac1}
c_{1}=-\frac{\lambda}{3(m+1)(m+2)(k+1)}.$$ Let $\sigma=\sigma(n,r)=\sum_{i=1}^n j_{r,i}$. From - one can easily see that for each $n$ and $r$ $$\label{amonomials}
R_{r,n}st=
\begin{cases}
0 & \text { if } \sigma <-1,\\
C_{r,n}s^\beta t^{\sigma+1} & \text { if } \sigma \geq -1\\
\end{cases}$$ where $n+1\leq \beta\leq 2n+1$ and $C_{r,n}$ is a constant which only depends on $n$ and $r$.
Let $\tilde{\lambda}=\max\{1,\lambda\}$. We claim that for each $n$ and $r$, $$\label{aclaim}
|C_{r,n}|\leq \frac{\tilde{\lambda}^n}{(n+1)!(\sigma+1)!}.$$ Taking $m=1$, $k=1$ in -, one can check that the claim holds for $n=1$. Suppose it holds for $n=\ell-1$ and for all $r \in \{1,2,.. ,4^{\ell -1}\}$. Then for $n=\ell$ and $r^* \in \{1,2,.. ,4^{\ell}\}$, using and , we obtain $R_{r^*,\ell}st=P_i R_{r,\ell-1}st= C_{r,\ell-1}P_i s^\beta t^{\sigma+1}=C_{r,\ell-1}c_i s^{\beta^*} t^{\sigma^*+1}$ for some $i\in\{-2,-1,0,1\}$ and $r \in \{1,2,.. ,4^{\ell -1}\}$, where $\beta^*$ is either $\beta+1$ or $\beta+2$, $\sigma^*=\sigma +i$. By the induction assumption $C_{r,\ell-1}\leq \frac{\tilde{\lambda}^{\ell-1}}{\ell!(\sigma+1)!}.$ Moreover - and the fact that $\beta\geq \ell$ imply $|c_i|\leq \frac{\sigma+1}{\ell+1}$ for $i=-1,-2$, $|c_0| <\frac{1}{\ell+1}$, and $|c_1|< \frac{\lambda}{(\sigma+2)(\ell+1)}$. Hence for each $i\in \{-2,-1,0,1\}$ we get $|C_{r^*,\ell}|= |C_{r,(\ell-1)}c_i| \leq \frac{\tilde{\lambda}^{\ell}}{(\ell+1)!(\sigma+i+1)!}=\frac{\tilde{\lambda}^{\ell}}{(\ell+1)!(\sigma^*+1)!}$, which proves that the claim holds for $n=\ell$ as well.
By , , and the fact that $0\leq s, t \leq L$ in the triangle $T_0$, we obtain $$\label{Hnest0}\|H^n\|_{\infty}\leq \frac{4^n\tilde{\lambda}^nL^{3n+2}}{(n+1)!}$$ which is summable. Moreover, since $H^n$ is a linear combination of $4^n$ monomials of the form $s^\beta t^{\sigma+1}$ with $\beta\leq 2n+1$ and $\sigma\leq n$, any partial derivative $\partial^a_s \partial^b_t H^n$ of $H^n$ will be absolutely less than $$\label{Hnest}\displaystyle\frac{(2n+1)^a (n+1)^b 4^n\tilde{\lambda}^nL^{3n+2-a-b}}{(n+1)!}$$ which is also summable.
Now, we define the pseudo-kernel by $$\label{ktilde}\tilde{k}(x,y):=\tilde{G}(x-y,y)$$ and consider the transformation given by $$\label{mod-transform}\tilde{w}(x,t)\equiv u(x,t)-\int_0^x\tilde{k}(x,y)u(y,t)dy.$$
![Pseudo-kernel $\tilde{k}$ when $\lambda=0.01$ ($L=2\pi$)[]{data-label="Impk"}](Impk)
![Control effort at the Dirichlet b.c. for different $\lambda$ ($L=2\pi$)[]{data-label="effortk1y"}](effortk1y)
![Control effort at the Neumann b.c. for different $\lambda$ ($L=2\pi$)[]{data-label="effortky1y"}](effortky1y)
Note that we have $\tilde{u}_x(0,t)=\tilde{w}_x(0,t)$ by the boundary conditions of $\tilde{k}$. Using this fact, we can rewrite the modified target system as $$\label{HomKdVBurgers-1}
\begin{cases}
\displaystyle \tilde{w}_{t} +\tilde{ w}_{x} + \tilde{w}_{xxx} + \lambda \tilde{w} = \tilde{k}_y(x,0)\tilde{w}_x(0,t) & \text { in } \Omega\times \mathbb{R_+},\\
\tilde{w}(0,t) = \tilde{w}(L,t) = \tilde{w}_{x}(L,t) = 0 & \text { in } \mathbb{R_+},\\
\tilde{w}(x,0)=\tilde{w}_0(x):= u_0-\int_0^x\tilde{k}(x,y)u_0(y)dy & \text { in } \Omega.
\end{cases}$$ Multiplying the above model by $\tilde{w}$ and integrating over $(0,1)$, using the Cauchy-Schwarz inequality, we obtain $$\begin{gathered}
\label{HomKdVBurgers-2}
\frac{1}{2}\frac{d}{dt}\|\tilde{w}(t)\|_{L^2(\Omega)}^2+\lambda\|\tilde{w}(t)\|_{L^2(\Omega)}^2
\le -\frac{1}{2}|\tilde{w}_x(0,t)|^2+\int_0^L\tilde{k}_y(x,0)\tilde{w}_x(0,t)\tilde{w}(x,t)dx\\
\le \cancel{-\frac{1}{2}|\tilde{w}_x(0,t)|^2}+\cancel{\frac{1}{2}|\tilde{w}_x(0,t)|^2}+\frac{1}{2}\left(\int_0^L|\tilde{k}_y(x,0)||\tilde{w}(x,t)|dx\right)^2.\end{gathered}$$ Since $\tilde{k}$ is smooth on the compact set $\mathcal{T}$, we have $$\label{HomKdVBurgers-3}
\frac{1}{2}\frac{d}{dt}\|\tilde{w}(t)\|_{L^2(\Omega)}^2+ \left(\lambda-\frac{1}{2}\|\tilde{k}_y(\cdot,0)\|_{L^2(\Omega)}^2\right)\|\tilde{w}(t)\|_{L^2(\Omega)}^2\le 0.$$ It follows that $$\label{HomKdVBurgers-4}
\|\tilde{w}(t)\|_{L^2(\Omega)}^2\le \|\tilde{w}_0\|_{L^2(\Omega)}^2e^{-2{\alpha}t} ,$$ where ${\alpha}\equiv \lambda-\frac{1}{2}\|\tilde{k}_y(\cdot,0)\|_{L^2(\Omega)}^2.$ The graph of the function $\tilde{k}_y(\cdot,0)$ is depicted in Figure \[ky0tilde\] on a domain of length $L=2\pi$.
![Pseudo-kernel $\tilde{k}$ when $\lambda=0.03$ ($L=2\pi$)[]{data-label="ky0tilde"}](ky0tilde)
By taking $L^2(\Omega)$ norms of both sides of (with $t=0$) and using the Cauchy-Schwarz inequality, we get $$\label{w0u0} \|\tilde{w}_0\|_{L^2(\Omega)}\le \left(1+\|\tilde{k}\|_{L^2(T)}\right) \|u_0\|_{L^2(\Omega)}.$$
Let $K:H^l(\Omega)\rightarrow H^l(\Omega)$ ($l\ge 0$) be the integral operator defined by $(K\varphi)(x):=\int_0^x\tilde{k}(x,y)\varphi(y)dy.$ It is not difficult to prove that the operator $I-K$ is invertible from $H^l(\Omega)\rightarrow H^l(\Omega)$ (for $l\ge 0$) with a bounded inverse. This is proved in a general setting in the lemma below:
\[inverselem\] $I-K$ is invertible with a bounded inverse from $H^l(\Omega)\rightarrow H^l(\Omega)$ ($l\ge 0$).
The above lemma can be expressed in a sharper form. Indeed, the proof below shows that $\Phi$ is a bounded operator from $L^2(\Omega)\rightarrow H^l(\Omega)$ ($l=0,1,2$) and it is a bounded operator from $H^{l-2}(\Omega)\rightarrow H^{l}(\Omega)$ ($l> 2$).
The above lemma can be proven by slightly modifying the proof of [@Liu03 Lemma 2.4]. However, we will still give a brief proof here since we will need to refer to some crucial details of the proof of this lemma later in the proofs of the stabilization and well-posedness results.
To this end, let us first consider the case $l=0$ and let $\psi=(I-K)\varphi$ for some $\varphi\in L^2(\Omega)$. The idea is to first write $\psi=\varphi-v$ where $v=K\varphi.$ Note that then, $$\psi(x)=\varphi(x)-[K\varphi](x)=(\psi(x)+v(x))-\int_0^x\tilde{k}(x,y)(\psi(y)+v(y))dy.$$ This gives $$v(x)=\int_0^x\tilde{k}(x,y)\psi(y)dy+\int_0^x\tilde{k}(x,y)v(y)dy.$$ Given a fixed $\psi$, one can solve this equation via succession (see [@Liu03 Lemma 2.4] for the details of the succession argument). This implicitly defines a linear operator $\Phi:\psi\mapsto v$ on $L^2(\Omega)$ with the property that $\Phi$ is bounded, i.e., there exists $C_0>0$ such that $$\label{l0}\|v\|_{L^2(\Omega)}\le C_0\|\psi\|_{L^2(\Omega)},$$ where $C_0$ depends only on $\|\tilde{k}\|_{L^\infty(\mathcal{T})}$. But then, $\varphi$ is simply equal to $(I+\Phi)\psi$, and therefore $(I-K)^{-1}$ exists, equals $I+\Phi$, and is bounded. By differentiating and using the smoothness of $\tilde{k}$, $(I-K)^{-1}$ extends to a linear bounded operator also on Sobolev spaces $H^l(\Omega)$ ($l\ge 1$). Indeed, since $\tilde{k}(x,x)=0$, we have $$\label{vxx}v_x(x)=\int_0^x\tilde{k}_x(x,y)(\psi(y)+v(y))dy,$$ which implies $\|v_x\|_{L^2(\Omega)}\le \|\tilde{k}_x\|_{L^2(\mathcal{T})}\left(\|\psi\|_{L^2(\Omega)}+\|v\|_{L^2(\Omega)}\right).$ Hence, using , we have $$\label{l1}\|v\|_{H^1(\Omega)}\le C_1\|\psi\|_{L^2(\Omega)},$$ where $C_1$ depends on $\|\tilde{k}_x\|_{L^2(\mathcal{T})}$ and $C_0$. This shows that $\Phi$ is bounded from $L^2(\Omega)$ into $H^1(\Omega)$, a fortiori bounded from $H^1(\Omega)$ into $H^1(\Omega)$. Now for $l=2$, using $k_x(x,x)=\frac{\lambda}{3}x$, $(\partial_x^2v)(x)=\frac{\lambda}{3}x(\psi(x)+v(x))+\int_0^x(\partial_x^2\tilde{k})(x,y)(\psi(y)+v(y))dy.$ Taking $L^2(\Omega)$ norms of both sides and using the previous inequalities, we get $\|v\|_{H^2(\Omega)}\le C_2\|\psi\|_{L^2(\Omega)},$ where $C_2$ depends on $\|\partial_x^2\tilde{k}\|_{L^2(\mathcal{T})}$, $C_1$, and $\lambda$. This shows that $\Phi$ is bounded from $L^2(\Omega)$ into $H^2(\Omega)$, a fortiori bounded from $H^1(\Omega)$ or $H^2(\Omega)$ into $H^2\Omega)$. Proceeding in the same fashion, one can show that $\|v\|_{H^3(\Omega)}\le C_3\|\psi\|_{H^1(\Omega)},$ where $C_3$ is a fixed constant depending on various norms of $\tilde{k}$. More generally, $\|v\|_{H^l(\Omega)}\le C_l\|\psi\|_{H^{l-2}(\Omega)},$ where $l> 2$ and $C_l$ depends on various norms of $\tilde{k}$. Hence, for $l> 2$, $\Phi$ is a bounded operator from $H^{l-2}(\Omega)$ into $H^l(\Omega)$, and a fortiori bounded from $H^{l}(\Omega)$ into $H^l(\Omega)$.
Another important estimate that follows from via is that $$\label{linftyrem}\|v_x\|_{L^\infty(\Omega)}\le C\|\psi\|_{L^2(\Omega)}$$ for some $C>0$ that depends on $\tilde{k}$.
From the above lemma, it follows in particular that $u(x,t)=[(I-K)^{-1}\tilde{w}](x,t)$, and moreover $$\label{ulessw}\|u(t)\|_{L^2(\Omega)}\le \|(I-K)^{-1}\|_{B[L^2(\Omega)]}\cdot \|\tilde{w}(t)\|_{L^2(\Omega)},$$ where $\|\cdot\|_{B[L^2(\Omega)}$ is the operator norm of $(I-K)^{-1}$ from $L^2(\Omega)$ into $L^2(\Omega)$.
Combining with and , we conclude that
$$\label{linshot}\|u(t)\|_{L^2(\Omega)}\le \left(1+\|\tilde{k}\|_{L^2(T)}\right)\|(I-K)^{-1}\|_{B[L^2(\Omega)]}\,\|u_0\|_{L^2(\Omega)}e^{-\alpha t}.$$
We can prove that the parameter $\alpha$ in the above estimate is positive if $\lambda$ is sufficiently small. Indeed, we have the following lemma.
\[alemlambda\]For a given $L$, there exists sufficiently small $\lambda$ such that $\alpha=\lambda-\frac{1}{2}\|k_y(\cdot, 0)\|^2_{L^2}>0.$
Taking the partial derivative of both sides of with respect to $t$ and taking $i=0$ we see that $\tilde{G}^j_t(s,t)=\frac{\lambda}{3}\sum_{n=0}^{j-1}H^n_t(s,t).$ Passing to the limit we obtain $\tilde{G}_t(s,t)=\frac{\lambda}{3}\sum_{n=0}^{\infty}H^n_t(s,t).$ Note that for $\lambda<1$, $\tilde{\lambda}=1$. Therefore by the summation term is absolutely less than some constant $M$ that only depends on $L$. Hence we get $\|\tilde{G}_t\|_{\infty}\leq\frac{\lambda M}{3}.$ Since $k_y(x,0)=\tilde{G}_t(s,0)$, in particular we have $\|k_y(\cdot, 0)\|^2_{L^2}\leq L \|k_y(\cdot, 0)\|^2_{\infty}\leq L \|\tilde{G}_t\|^2_{\infty}\leq \frac{\lambda^2 M^2 L}{9}.$ As a result, $\alpha=\lambda-\frac{1}{2}\|k_y(\cdot, 0)\|^2_{L^2}\geq \lambda-\frac{\lambda^2 M^2 L}{18}=\lambda^2(\frac{1}{\lambda}-\frac{ M^2 L}{18})$ which is positive for sufficiently small $\lambda$.
The inequality together with Lemma \[alemlambda\] proves the linear part of Theorem \[Linthm\].
Nonlinear model {#ProofofThm2}
---------------
In this section, we consider the nonlinear KdV model with the feedback controllers given in . By using the transformation given in , we obtain the following PDE from , noting that $\tilde{k}(x,x)=0$: $$\label{ch414}
\tilde{w}_{t} + \tilde{w}_{x} + \tilde{w}_{xxx} + \lambda \tilde{w}
= \tilde{k}_y(\cdot,0)\tilde{w}_x(0,\cdot)-(I-K)[\left(\tilde{w}+ v\right)\left(\tilde{w}_{x} + v_x\right)]$$ with homogeneous boundary conditions $$\tilde{w}(0,t) = 0 \; , \; \tilde{w}(L,t) = 0, \quad \textrm{and} \quad \tilde{w}_{x}(L,t) = 0,$$ where $v(x,t)=[\Phi\tilde{w}](x,t)$, with $\Phi$ being the linear operator defined in Section \[ProofofThm\] in the proof of Lemma \[inverselem\]. Multiplying by $\tilde{w}(x,t)$ and integrating over $\Omega=(0,L)$, we obtain $$\begin{gathered}
\label{ch416}
\int_{0}^{L}\tilde{w}(x,t)\tilde{w}_{t}(x,t)dx = \int_0^L\tilde{k}_y(x,0)\tilde{w}_x(0,t)\tilde{w}(x,t)dx-\int_{0}^{L}\tilde{w}(x,t)\tilde{w}_{x}(x,t)dx \\
- \int_{0}^{L}\tilde{w}(x,t)\tilde{w}_{xxx}(x,t)dx
-\lambda\int_{0}^{L}\tilde{w}^{2}(x,t)dx - \int_{0}^{L}\tilde{w}^2(x,t)\tilde{w}_x(x,t)dx - \int_{0}^{L}\tilde{w}^2(x,t)v_x(x,t)dx \\
-\int_{0}^{L}\tilde{w}(x,t)\tilde{w}_x(x,t)v(x,t)dx-\int_{0}^{L}\tilde{w}(x,t)v(x,t)v_x(x,t)dx\\
+\int_0^L\left(\int_0^x\tilde{k}(x,y)\tilde{w}(y,t)\tilde{w}_y(y,t)dy\right)\tilde{w}(x,t)dx
+\int_0^L\left(\int_0^x\tilde{k}(x,y)\tilde{w}(y,t)\tilde{v}_y(y,t)dy\right)\tilde{w}(x,t)dx\\
+\int_0^L\left(\int_0^x\tilde{k}(x,y)\tilde{v}(y,t)\tilde{w}_y(y,t)dy\right)\tilde{w}(x,t)dx
+\int_0^L\left(\int_0^x\tilde{k}(x,y)\tilde{v}(y,t)\tilde{v}_y(y,t)dy\right)\tilde{w}(x,t)dx.\end{gathered}$$
We estimate the last four terms at the right hand side of as follows: $$\begin{gathered}
\label{four-1}
\int_0^L\left(\int_0^x\tilde{k}(x,y)\tilde{w}(y,t)\tilde{w}_y(y,t)dy\right)\tilde{w}(x,t)dx = \frac{1}{2}\int_0^L\left(\int_0^x\tilde{k}(x,y)\frac{\partial}{\partial y}\tilde{w}^2(y,t)dy\right)\tilde{w}(x,t)dx\\
=\frac{1}{2}\int_0^L\left.\tilde{k}(x,y)\tilde{w}^2(y,t)\right|_{0}^x\tilde{w}(x,t)dx- \frac{1}{2}\int_0^L\left(\int_0^x\tilde{k}_y(x,y)\tilde{w}^2(y,t)dy\right)\tilde{w}(x,t)dx\\
\le \frac{\sqrt{L}}{2}\|\tilde{k}_y\|_{L^\infty(T_0)}\|\tilde{w}(t)\|_{L^2(\Omega)}^3,\end{gathered}$$
$$\int_0^L\left(\int_0^x\tilde{k}(x,y)\tilde{w}(y,t){v}_y(y,t)dy\right)\tilde{w}(x,t)dx \le \|\tilde{k}\|_{L^2(T_0)}\|v_x(t)\|_{L^\infty(\Omega)}\|\tilde{w}(t)\|_{L^2(\Omega)}^2,$$
$$\begin{gathered}
\int_0^L\left(\int_0^x\tilde{k}(x,y){v}(y,t)\tilde{w}_y(y,t)dy\right)\tilde{w}(x,t)dx \\
=\int_0^L\left.\tilde{k}(x,y){v}(y,t)\tilde{w}(y,t)\right|_{0}^x\tilde{w}(x,t)dx- \int_0^L\left(\int_0^x\tilde{k}_y(x,y){v}(y,t)\tilde{w}(y,t)dy\right)\tilde{w}(x,t)dx\\
- \int_0^L\left(\int_0^x\tilde{k}(x,y){v}_y(y,t)\tilde{w}(y,t)dy\right)\tilde{w}(x,t)dx\le \sqrt{L}\|\tilde{k}_y\|_{L^\infty(T_0)}\|v(t)\|_{L^2(\Omega)}\|\tilde{w}(t)\|_{L^2(\Omega)}^2\\
+\|\tilde{k}\|_{L^2(T_0)}\|v_x(t)\|_{L^\infty(\Omega)}\|\tilde{w}(t)\|_{L^2(\Omega)}^2,\end{gathered}$$
$$\begin{gathered}
\label{four-4}
\int_0^L\left(\int_0^x\tilde{k}(x,y)v(y,t)v_y(y,t)dy\right)\tilde{w}(x,t)dx = \frac{1}{2}\int_0^L\left(\int_0^x\tilde{k}(x,y)\frac{\partial}{\partial y}v^2(y,t)dy\right)\tilde{w}(x,t)dx\\
=\frac{1}{2}\int_0^L\left.\tilde{k}(x,y)v^2(y,t)\right|_{0}^x\tilde{w}(x,t)dx- \frac{1}{2}\int_0^L\left(\int_0^x\tilde{k}_y(x,y)v^2(y,t)dy\right)\tilde{w}(x,t)dx\\
\le \frac{\sqrt{L}}{2}\|\tilde{k}_y\|_{L^\infty(T_0)}\|v(t)\|_{L^2(\Omega)}^2\|\tilde{w}(t)\|_{L^2(\Omega)}.\end{gathered}$$
Now, estimating the other terms using integration by parts and the Cauchy-Schwarz inequality, and combining these with -, it follows that $$\begin{gathered}
\label{ch417}
\frac{1}{2}\frac{d}{dt}\|\tilde{w}(t)\|_{L^2(\Omega)}^2 + \left(\lambda-\frac{1}{2}\|\tilde{k}_y(\cdot,0)\|_{L^2(\Omega)}^2\right)\|\tilde{w}(t)\|_{L^2(\Omega)}^2 \\
\le \left(\frac{3}{2}+2\|\tilde{k}\|_{L^2(T_0)}\right)\|\tilde{w}(t)\|_{L^2(\Omega)}^2\|v_x(t)\|_{L^\infty(\Omega)}+\|\tilde{w}(t)\|_{L^2(\Omega)}\|v(t)\|_{L^2(\Omega)}\|v_x(t)\|_{L^\infty(\Omega)}\\
+\frac{\sqrt{L}}{2}\|\tilde{k}_y\|_{L^\infty(T_0)}\|\tilde{w}(t)\|_{L^2(\Omega)}^3
+\sqrt{L}\|\tilde{k}_y\|_{L^\infty(T_0)}\|v(t)\|_{L^2(\Omega)}\|\tilde{w}(t)\|_{L^2(\Omega)}^2\\
+\frac{\sqrt{L}}{2}\|\tilde{k}_y\|_{L^\infty(T_0)}\|v(t)\|_{L^2(\Omega)}^2\|\tilde{w}(t)\|_{L^2(\Omega)}.\end{gathered}$$
Using and , we deduce the following inequality: $$\label{bernoulli}
y'+2\alpha y-cy^\frac{3}{2}\le 0,$$ where $y(t)\equiv \|\tilde{w}(t)\|_{L^2(\Omega)}^2$, and $c$ is a constant that depends on $L$ and various norms of $\tilde{k}$. Solving the inequality and assuming $\displaystyle \|\tilde{w}_0\|_{L^2(\Omega)}<\frac{\alpha}{c}$, we get $$\label{Nonlinw}
\|\tilde{w}(t)\|_{L^2(\Omega)}^2=y(t)\le \frac{1}{\left[\left(\frac{1}{\|\tilde{w}_0\|_{L^2(\Omega)}}-\frac{c}{2\alpha}\right)e^{\alpha t}+\frac{c}{2\alpha}\right]^2}<\frac{1}{\left[\frac{e^{\alpha t}}{2\|\tilde{w}_0\|_{L^2(\Omega)}}\right]^2}.$$ Recall that $\|\tilde{w}_0\|_{L^2(\Omega)}\lesssim\|u_0\|_{L^2(\Omega)}$. Combining this with and , we deduce $$\|u(t)\|_{L^2(\Omega)}\lesssim \|u_0\|_{L^2(\Omega)}e^{-\alpha t}, \text{ for } t\ge 0.$$ Hence, the proof of Theorem \[Linthm\] for the nonlinear KdV equation is also complete. Note that the smallness assumption on the initial datum $\tilde{w}_0$ implies a smallness assumption on $u_0$ due to the fact that we also have $\|u_0\|_{L^2(\Omega)}\lesssim\|\tilde{w}_0\|_{L^2(\Omega)}$ thanks to Lemma \[inverselem\].
Well-posedness
==============
In this section, we prove the well-posedness of the PDE models studied in the previous sections. For simplicity, we assume $L=1$ throughout this section. This assumption has no consequence as far as wellposedness is concerned, and all results proved here are also true for any $L>0$. Thanks to Lemma \[lemback\], it is enough to prove the well-posedness of the respective modified target systems in order to obtain well-posedness of and .
Linearised model {#linearised-model}
----------------
Consider the following linear KdV equation with homogeneous boundary conditions. $$\label{KdV-wp}
\begin{cases}
\displaystyle y_{t} +y_{x} + y_{xxx} + {\lambda} y = a(x)y_x(0,\cdot) & \text { in } \Omega\times \mathbb{R_+},\\
{y}(0,t) = {y}(1,t) = {y}_{x}(1,t) = 0 & \text { in } \mathbb{R_+},\\
{y}(x,0)=y_0\in L^2(\Omega) & \text { in } \Omega.
\end{cases}$$ We have the following result.
\[wellposednessprop1\]
1. Let $T'>0$ be arbitrary and $y_0,a\in L^2(\Omega)$. Then, there exists $T\in (0,T')$ independent of the size of $y_0$ such that has a unique local solution $y\in C([0,T];L^2(\Omega))\cap L^2(0,T;H^1(\Omega))$ satisfying also $y_x\in C([0,1];L^2(0,T)).$ Moreover, if $a\in L^\infty(\Omega)$, then $y$ extends as a global solution. In other words, $T$ can be taken as $T'$.
2. Let $a\in H^1(\Omega)$ and let $y_0\in H^3(\Omega)$ satisfy the compatibility conditions $y_0(0)=y_0(1)=y_0'(1)=0.$ Then, the (local/global) solution in part (i) enjoys the extra regularity $y\in C([0,T];H^3(\Omega))\cap L^2(0,T;H^4(\Omega)).$
**Step 1 - Local wellposedness:** Let us define the linear operator $A:D(A)\subset L^2(\Omega)\rightarrow L^2(\Omega)$ by $A\varphi := -\varphi'-\varphi''',$ where $D(A):=\{\varphi\in H^3(\Omega):\varphi(0)=\varphi(1)=\varphi'(1)=0\}.$ Then, the initial boundary value problem can be rewritten in the abstract operator theoretic form $$\label{KdV-wp-abs}
\begin{cases}
\displaystyle \dot{y} = Ay+Fy,\\
{y}(0)=y_0,
\end{cases}$$ where $F\varphi:=-\lambda \varphi + a(\cdot)\gamma_1^0\varphi$. Here, $\gamma_1^0$ is the first order trace operator at the left endpoint, i.e., $\gamma_1^0\varphi:=\varphi'(0)$. This operator is well-defined for $\varphi\in H^{\frac{3}{2}+\epsilon}(\Omega)\supset D(A)$.
It is not difficult to see that the adjoint of $A$ is defined by $A^*\varphi:= \varphi'+\varphi'''$ with $D(A^*):=\{\varphi\in H^3(\Omega):\varphi(0)=\varphi(1)=\varphi'(0)=0\}.$
$A$ is a densely defined closed operator, and moreover, $A$ and $A^*$ are dissipative [@Rosier1997 Proposition 3.1]. Therefore, $A$ generates a strongly continuous semigroup of contractions $\displaystyle\{S(t)\}_{t\ge 0}$ on $L^2(\Omega)$ [@Pazy Corollary I.4.4]. Now we construct the operator $$\label{soloperator}y=[\Psi z](t):= S(t)y_0+\int_0^tS(t-s)Fz(s)ds.$$
Let us define the space (see e.g., [@BSZ03]) $$\label{ourspace}Y_T:=\{z\in C([0,T];L^2(\Omega))\cap L^2(0,T;H^1(\Omega))\,|\,z_x\in C([0,1];L^2(0,T))\}$$ equipped with the norm $\|z\|_{Y_T}:=\left(\|z\|_{C([0,T];L^2(\Omega))}^2+\|z\|_{L^2(0,T;H^1(\Omega))}^2+\|z_x\|_{C([0,1];L^2(0,T))}^2\right)^{\frac{1}{2}}.$ Then, for $z\in Y_T$, by using the semigroup estimates [@BSZ03 Prop 2.1, Prop 2.4, Prop 2.16, Prop 2.17], we have $$\begin{gathered}
\label{estimate01}\|y\|_{Y_T}=\|\Psi z\|_{Y_T}\le \|S(t)y_0\|_{Y_T}+\left\|\int_0^tS(t-s)Fz(s)ds\right\|_{Y_T}\\
\le c_0(1+T)^\frac{1}{2}\|y_0\|_{L^2(\Omega)}+c_1(1+T)^\frac{1}{2}\left\|-\lambda z+az_x(0,\cdot)\right\|_{L^1(0,T;L^2(\Omega))}\\
\le c_0(1+T)^\frac{1}{2}\|y_0\|_{L^2(\Omega)}+c_1(1+T)^\frac{1}{2}\sqrt{T}(1+\|a\|_{L^2(\Omega)})\|z\|_{Y_T}\,,\end{gathered}$$ where $c_0$ and $c_1$ are positive constants which do not depend on the varying parameters. It follows that $\Psi$ maps $Y_T$ into itself. Now, let $z_1,z_2\in Y_T$ and $y_1=\Psi z_1$, $y_2=\Psi z_2$. By using similar arguments, we have $$\|y_1-y_2\|_{Y_T}=\|\Psi z_1-\Psi z_2\|_{Y_T}\le c_1(1+T)^\frac{1}{2}\sqrt{T}(1+\|a\|_{L^2(\Omega)})\|z_1-z_2\|_{Y_T}.$$ Let $T\in (0,T')$ be such that $0< (1+T)^\frac{1}{2}\sqrt{T}< \left(\frac{1}{c_1\left(1+\|a\|_{L^2(\Omega)}\right)}\right).$ Then, $\Psi$ is a contraction on $Y_T$, and this gives us a unique local solution $y\in Y_T$. Here, the size of $T$ is independent of the size of the initial datum. This contrasts with the corresponding nonlinear model in which the size of $T$ is related to the size of the initial datum.
**Step 2 - Global wellposedness:** Let $T_{\max}\le T'$ be the maximal time of existence for the local solution found in Step 1 in the sense that $y\in Y_T$ for all $T<T_{\max}$. In order to prove that $y$ is global, and deduce that $T$ can be taken as $T'$, it is enough to show that $\displaystyle \lim_{T\rightarrow T_{\max}^-}\|y\|_{Y_T}<\infty.$ This will be proved via multipliers, which will be done only formally, but the calculations can always be justified by a density argument which relies on the regularity result in part (ii) of this proposition. To this end, we multiply by $y$ and integrate over $\Omega$ to obtain $$\label{Iden01}
\frac{1}{2}\frac{d}{dt}\|y(t)\|_{L^2(\Omega)}^2+\frac{1}{2}|y_x(0,t)|^2 + \lambda \|y(t)\|_{L^2(\Omega)}^2 = \int_0^1 a(x)y_x(0,t)y(x,t)dx.$$ Using $\epsilon$-Young’s inequality with $\displaystyle\epsilon=\frac{1}{4}$, we have $$\label{Iden02}
\frac{1}{2}\frac{d}{dt}\|y(t)\|_{L^2(\Omega)}^2+\frac{1}{4}|y_x(0,t)|^2 + \lambda \|y(t)\|_{L^2(\Omega)}^2 \le \|a(x)\|_{L^\infty(\Omega)}^2\|y(t)\|_{L^2(\Omega)}^2.$$ Integrating the above inequality over $(0,t)$, we get $$\label{Iden03}
\|y(t)\|_{L^2(\Omega)}^2+\int_0^t|y_x(0,t)|^2dt + \le 2\|y_0\|_{L^2(\Omega)}^2+4(\|a(x)\|_{L^\infty(\Omega)}^2-\lambda)\int_0^t\|y(s)\|_{L^2(\Omega)}^2ds.$$ Let $E_0(t):=\|y(t)\|_{L^2(\Omega)}^2+\int_0^t|y_x(0,t)|^2dt.$ Then, from , we get $$E_0(t)\le 2\|y_0\|_{L^2(\Omega)}^2+4\left|\|a(x)\|_{L^\infty}^2-\lambda\right|\int_0^tE_0(s)ds.$$ Now, thanks to the Gronwall’s lemma, we have $$\label{Iden04}
E_0(t)=\|y(t)\|_{L^2(\Omega)}^2+\int_0^t|y_x(0,t)|^2dt \le 2\|y_0\|_{L^2(\Omega)}^2e^{4\left|\|a(x)\|_{L^\infty}^2-\lambda\right|t},\, t\in [0,T_{\max}).$$ We in particular deduce that $$\label{Firstimpest}
\lim_{T\rightarrow T_{\max}^-}\|y\|_{C([0,T];L^2(\Omega))}\le\sqrt{2}\|y_0\|_{L^2(\Omega)}e^{2\left|\|a(x)\|_{L^\infty}^2-\lambda\right|T_{max}}<\infty.$$ By using , we also deduce that $$\label{Iden05}
\lim_{T\rightarrow T_{\max}^-}\|y\|_{L^2(0,T;L^2(\Omega))}\le \sqrt{2T_{max}}\|y_0\|_{L^2(\Omega)}e^{2\left|\|a(x)\|_{L^\infty}^2-\lambda\right|T_{max}}.$$ Secondly, we multiply by $xy$ and integrate over $\Omega\times (0,t)$ and get $$\begin{gathered}
\label{Iden06}
\int_{0}^1xy^2(x,s)dx+3\int_0^t\int_0^1y_x^2(x,s)dxds+\lambda\int_0^t\int_0^1xy^2(x,s)dxds\\
= \int_0^1xy_0^2(x)dx+\int_0^t\int_0^1y^2(x,s)dxds+ \int_0^t\int_0^1xay_x(0,s)y(x,s)dxds.\end{gathered}$$ From the above identity, it follows that
$$\label{Iden07}
\|y_x\|_{L^2(0,t;L^2(\Omega))}^2\le \frac{1}{3}\|y_0\|_{L^2(\Omega)}^2+\left(\frac{1}{2}+\frac{\|a\|_{L^\infty(\Omega)}^2}{18}\right)\int_0^tE_0(s)ds.$$
Combining the above inequality with , we deduce that
$$\begin{gathered}
\label{Iden08}
\lim_{T\rightarrow T_{\max}^-}\|y_x\|_{L^2(0,T;L^2(\Omega))}\\
\le \frac{1}{\sqrt{3}}\|y_0\|_{L^2(\Omega)}+\left(\frac{1}{\sqrt{2}}+\frac{\|a\|_{L^\infty(\Omega)}}{3\sqrt{2}}\right)\sqrt{2T_{\max}}\left(\|y_0\|_{L^2(\Omega)}e^{2\left|\|a(x)\|_{L^\infty}^2-\lambda\right|T_{max}}\right).\end{gathered}$$
Using and , we deduce that $$\begin{gathered}
\label{Iden09}
\lim_{T\rightarrow T_{\max}^-}\|y\|_{L^2(0,T;H^1(\Omega))}\\
\le \frac{1}{\sqrt{3}}\|y_0\|_{L^2(\Omega)}+\left(1+\frac{1}{\sqrt{2}}+\frac{\|a\|_{L^\infty(\Omega)}}{3\sqrt{2}}\right)\sqrt{2T_{\max}}\left(\|y_0\|_{L^2(\Omega)}e^{2\left|\|a(x)\|_{L^\infty}^2-\lambda\right|T_{max}}\right)<\infty.\end{gathered}$$
Since $y$ is the fixed point in , we have $$y=S(t)y_0+\int_0^tS(t-s)Fy(s)ds.$$ Using [@BSZ03 Prop 2.16 and Prop 2.17], we know that the semigroup enjoys the properties $$\label{semi-prop1}
\sup_{x\in \Omega}\left\|\partial_x [S(t)y_0](x)\right\|_{L^2(0,T)}\le c_2\|y_0\|_{L^2(\Omega)}$$ and $$\label{semi-prop2}
\sup_{x\in \Omega}\left\|\partial_x \left[\int_0^tS(t-s)Fy(s)ds\right](x)\right\|_{L^2(0,T)}\le c_3\int_0^T\left\|[Fy](\cdot,t)\right\|_{L^2(\Omega)}dt$$ for some $c_2,c_3>0.$ From the definition of $Fy$ we have $$\|[Fy](\cdot,t)\|_{L^2(\Omega)}\le \lambda\|y(\cdot,t)\|_{L^2(\Omega)}+\|a\|_{L^2(\Omega)}|y_x(0,t)|.$$ Therefore, by and the Cauchy-Schwarz inequality, we have the estimate $$\begin{gathered}
\label{semi-prop3}\int_0^T\left\|[Fy](\cdot,t)\right\|_{L^2(\Omega)}dt\le \lambda\int_0^T\sqrt{E_0(t)}dt+\|a\|_{L^2(\Omega)}\sqrt{T}\sqrt{E_0(T)}\\
\le \left(\lambda T_{\max}+\|a\|_{L^2(\Omega)}\sqrt{T_{\max}}\right)\sqrt{2}\|y_0\|_{L^2(\Omega)}e^{2\left|\|a(x)\|_{L^\infty}^2-\lambda\right|T_{max}}.\end{gathered}$$ Now, it follows from - that $$\begin{gathered}
\lim_{T\rightarrow T_{\max}^-}\|y_x\|_{C([0,1];L^2(0,T))}\\
\le c_2\|y_0\|_{L^2(\Omega)}+c_3\left(\lambda T_{\max}+\|a\|_{L^2(\Omega)}\sqrt{T_{\max}}\right)\sqrt{2}\|y_0\|_{L^2(\Omega)}e^{2\left|\|a(x)\|_{L^\infty}^2-\lambda\right|T_{max}}<\infty.\end{gathered}$$
**Step 3 - Regularity:** Regarding the regular solutions, assume that $y_0\in D(A)$ and consider the following problem: $$\label{KdV-wp-regular}
\begin{cases}
\displaystyle q_{t} +q_{x} + q_{xxx} + {\lambda} q = a(\cdot)q_x(0,\cdot) & \text { in } \Omega\times (0,T),\\
{q}(0,t) = {q}(1,t) = {q}_{x}(1,t) = 0 & \text { in } (0,T),\\
{q}(x,0)=q_0\equiv -y_0'(x)-y_0'''(x)-\lambda y_0(x)+y_0'(0)a(x) & \text { in } \Omega.
\end{cases}$$ Note that $q_0\in L^2(\Omega)$, and we can solve in $Y_T$ as before. Now, we set $y(x,t):=y_0(x)+\int_0^tq(x,s)ds.$ Then, $$\begin{gathered}
\label{regular01}
y_t(x,t)+y_x(x,t)+y_{xxx}(x,t)+\lambda y(x,t)-a(x)y_x(0,t)
=q(x,t)+y_0'(x)+y_0'''(x)+\lambda y_0-y_0'(0)a(x)\\
+\int_0^t\left(q_x(x,s)+q_{xxx}(x,s)+\lambda q(x,t)-a(x)q_x(0,s)\right)ds=0,\end{gathered}$$ and moreover $y(x,0)=y_0$ and $y(0,t)=y(1,t)=y_x(1,t)=0.$ Therefore, $y$ solves . Writing $$y_{xxx}(x,t)=-q(x,t)-y_x(x,t)-\lambda y(x,t)+a(x)y_x(0,t)$$ and taking $L^2(\Omega)$ norms of both sides we get $$\|\partial_x^3 y(t)\|_{L^2(\Omega)}\le \|q(t)\|_{L^2(\Omega)}+\|\partial_x y(t)\|_{L^2(\Omega)}+\lambda\|y(t)\|_{L^2(\Omega)}+|y_x(0,t)|\|a\|_{L^2(\Omega)}.$$ Recall that we have the Gargliardo-Nirenberg inequalities $$\|\partial_x y(t)\|_{L^2(\Omega)}\lesssim \|y\|_{L^2(\Omega)}^\frac{2}{3}\|\partial_x^3y\|_{L^2(\Omega)}^\frac{1}{3}\text{ and }\|\partial_x^2 y(t)\|_{L^2(\Omega)}\lesssim \|y\|_{L^2(\Omega)}^\frac{1}{3}\|\partial_x^3y\|_{L^2(\Omega)}^\frac{2}{3},$$ and the trace inequality (remember that $y_x(1,t)=0$): $|y_x(0,t)|\le \|\partial_x^2 y\|_{L^2(\Omega)}.$
Using these estimates, we get $\|\partial_x^3 y(t)\|_{L^2(\Omega)}\lesssim \|q(t)\|_{L^2(\Omega)}+\|y(t)\|_{L^2(\Omega)}.$ By taking the sup norm with respect to the temporal variable, we deduce that $y\in C([0,T];H^3(\Omega)).$
Similarly, writing out $\partial_x^4y(x,t)=-q_x(x,t)-y_{xx}(x,t)-\lambda y_x(x,t)+a'(x)y_x(0,t),$ using the Gagliardo-Nirenberg and trace inequality, we get $\|\partial_x^4 y(t)\|_{L^2(\Omega)}\lesssim \|q_x(t)\|_{L^2(\Omega)}+\|y_x(t)\|_{L^2(\Omega)}.$ Taking $L^2(0,T)$ norms of both sides we deduce that $y\in L^2(0,T;H^4(\Omega)).$
Global well-posedness of the linearized model now follows from the Proposition \[wellposednessprop1\] that we have just proved.
One can interpolate between part (i) and part (ii) of the above proposition with respect to the smoothness of initial data and get the corresponding well-posedness and regularity result in fractional spaces. For example, let $y_0\in H^s(\Omega)$ ($s\in [0,3]$) so that it satisfies the compatibility conditions $y_0(0)=y_0(1)=0$ if $s\in [0,3/2]$ and the compatibility conditions $y_0(0)=y_0(1)=y_0'(1)=0$ if $s\in (3/2,3].$ Then, with $a=a(x)$ sufficiently smooth, one has $$y\in Y_T^s:= \{\psi\in C([0,T];H^s(\Omega))\cap L^2(0,T;H^{s+1}(\Omega))\,|\,\psi_x\in C([0,1];L^2(0,T))\}.$$ The arguments in Step 3 of the proof of the above proposition can be easily extended to the nonhomogeneous equation $\displaystyle y_{t} +y_{x} + y_{xxx} + {\lambda} y = a(x)y_x(0,\cdot)+f.$ One can first study this equation with $s=0$, $f\in L^1(0,T;L^2(\Omega)),$ and secondly with $s=3$, $f\in W^{1,1}(0,T;L^2(\Omega)).$ Then, by interpolation, for $s\in (0,3)$, one can get $y\in Y_T^s$ if $f\in W^{s/3,1}(0,T;L^2(\Omega))$. Moreover, the following estimates are true: $$\label{YsT}
\|y\|_{Y_{T}^s}\lesssim \|y_0\|_{H^s(\Omega)}+\|f\|_{W^{s/3,1}(0,T;L^2(\Omega))},$$ and for $s=3$, $$\label{YsT3}
\|y_t\|_{Y_{T}}\lesssim \|y_0\|_{H^3(\Omega)}+\|f\|_{W^{1,1}(0,T;L^2(\Omega))}.$$
Nonlinear model {#nonlinear-model}
---------------
Consider the following nonlinear KdV equation with homogeneous boundary conditions. $$\label{nonlinKdV-wp}
\begin{cases}
\displaystyle y_{t} +y_{x} + y_{xxx} + {\lambda} y = a(x)y_x(0,\cdot) -(I-K)[\left(y+v\right)\left(y_{x} + v_x\right)] & \text { in } \Omega\times \mathbb{R_+},\\
{y}(0,t) = {y}(1,t) = {y}_{x}(1,t) = 0 & \text { in } \mathbb{R_+},\\
{y}(x,0)=y_0\in L^2(\Omega) & \text { in } \Omega,
\end{cases}$$ where $v=\Phi(y)$, $\Phi$ being the linear operator defined in Section \[ProofofThm\] in the proof of Lemma \[inverselem\].
\[wellposednessprop2\]
1. Let $T'>0$ be arbitrary and $y_0,a\in L^2(\Omega)$. Then, there exists $T\in (0,T')$ depending on the size of $y_0$ such that has a unique local solution $y\in C([0,T];L^2(\Omega))\cap L^2(0,T;H^1(\Omega))$ satisfying also $y_x\in C([0,1];L^2(0,T)).$ Moreover, if $a\in L^\infty(\Omega)$ and $\|y_0\|_{L^2(\Omega)}$ is sufficiently small, then $y$ extends as a global solution. In other words, $T$ can be taken as $T'$.
2. Let $a\in H^1(\Omega)$ and let $y_0\in H^3(\Omega)$ satisfy the computability conditions $y_0(0)=y_0(1)=y_0'(1)=0.$ Then, the local solution in part (i) enjoys the extra regularity $y\in C([0,T];H^3(\Omega))\cap L^2(0,T;H^4(\Omega)).$
**Step 1 - Local wellposedness:** At first, we set a nonlinear operator $\Upsilon$ as follows: $$\label{soloperator2}y=[\Upsilon z](t):= S(t)y_0+\int_0^tS(t-s)Fz(s)ds,$$ where $Fz:= -\lambda z+ a(\cdot)z_x(0,\cdot)-(I-K)[\left(z+v\right)\left(z_{x} + v_x\right)]$ with $v=\Phi(z).$ Here, we consider $\Upsilon$ on a set given by $S_{T,r}:=\{z\in Y_T,\,\|z\|_{Y_T}\le r\},$ where $Y_T$ is as in . The parameters $T,r>0$ will be determined later. $S_{T,r}$ is a complete metric subspace of $Y_T$ with respect to the metric induced by the norm of $Y_T$. Since $v=\Phi z$, due to we have $$\label{vzrel2}
\|v\|_{C([0,T];L^2(\Omega))}\le C_0\|z(t)\|_{C([0,T];L^2(\Omega))}.$$ Similarly, using we deduce $$\label{vzrel4}
\|v\|_{L^2(0,T);H^1(\Omega))}\le C_1\|z(t)\|_{L^2(0,T);L^2(\Omega))}.$$ Finally,
$$\begin{gathered}
\label{vzrel5}
\|v_x(x)\|_{L^2(0,T)}^2=\int_0^T\left|\int_0^x\tilde{k}_x(x,y)z(y,t)dy\right|^2dt
\le \left(\int_0^1|\tilde{k}_x(x,y)|^2dy\right)\|z\|_{L^2(0,T);L^2(\Omega))}^2\,,
\end{gathered}$$
from which it follows that
$$\label{vzrel6}
\sup_{x\in (0,1)}\|v_x(x)\|_{L^2(0,T)}\\
\le \|z\|_{L^2(0,T);L^2(\Omega))}\sup_{x\in (0,1)}\left(\int_0^1|\tilde{k}_x(x,y)|^2dy\right)^\frac{1}{2}.$$
Combining , , and , we have $$\label{vzrel7}
\|v\|_{Y_T}\le c_{\tilde{k}}\|z\|_{Y_T}\,,$$ where $c_{\tilde{k}}>0$ is a constant which only depends on various finite norms of $\tilde{k}$. Taking the $Y_{T}$ norm of both sides of , using the same semigroup estimates on $Y_T$ and the boundedness of $I-K$, we obtain $$\begin{gathered}
\label{vzrel8}
\|\Upsilon z\|_{Y_T}\le c_0\|y_0\|_{Y_T}+c_1\int_0^{T}\left\|[Fz](\cdot,s)\right\|_{L^2(\Omega)}ds\\
\le c_0\|y_0\|_{Y_T}+c_1\int_0^{T}\left\|a(\cdot)z_x(0,s)-\lambda z-(I-K)[(z+v)(z_x+v_x)]\,\right \|_{L^2(\Omega)}ds\\
\le c_0\|y_0\|_{Y_T}+c_1\int_0^{T}\left[\left\|a(\cdot)z_x(0,s)-\lambda z\right\|_{L^2(\Omega)}+\left\|zz_x\right \|_{L^2(\Omega)}+\left\|zv_x\right \|_{L^2(\Omega)}+\left\|vz_x\right \|_{L^2(\Omega)}+\left\|vv_x\right \|_{L^2(\Omega)}\right]ds\\
\le c_0\|y_0\|_{Y_T}
+c_1\left[(1+T)^\frac{1}{2}\sqrt{T}(1+\|a\|_{L^2(\Omega)})\|z\|_{Y_T}+(T^\frac{1}{2}+T^\frac{1}{3})
\left(\|z\|_{Y_T}^2+2\|z\|_{Y_T}\|v\|_{Y_T}+\|v\|_{_{Y_T}}^2\right)\right]\\
\le c_0\|y_0\|_{Y_T}+c_1\left[(1+T)^\frac{1}{2}\sqrt{T}(1+\|a\|_{L^2(\Omega)})+(1+3c_{\tilde{k}})(T^\frac{1}{2}+T^\frac{1}{3})
\|z\|_{Y_T}\right]\|z\|_{Y_T},
\end{gathered}$$ where the fourth inequality follows from [@BSZ03 Lemma 3.1]. Let us set $r=2c_0\|y_0\|_{Y_T}$, and choose $T>0$ to be small enough that $$c_1\left[(1+T)^\frac{1}{2}\sqrt{T}(1+\|a\|_{L^2(\Omega)})+(1+3c_{\tilde{k}})(T^\frac{1}{2}+T^\frac{1}{3})
r\right]\le\frac{1}{2}.$$ With such choice of $(r,T)$, we get $\|\Upsilon z\|_{Y_T}\le r$ for all $z\in S_{T,r}$. Therefore, $\Upsilon$ is a map from $S_{T,r}$ into $S_{T,r}$.
Now, we claim that $\Upsilon$ is indeed a contraction on $S_{T,r}$ if $T$ is sufficiently small. In order to see this, let $z,z'\in S_{T,r}.$ Then, similar to , we have $$\begin{gathered}
\label{vzrel9}
\|\Upsilon z-\Upsilon z'\|_{Y_T}\le c_1\int_0^{T}\left\|[Fz-Fz'](\cdot,s)\right\|_{L^2(\Omega)}ds\\
\le c_1\int_0^{T}\left\|a(\cdot)(z_x(0,s)-z'_x(0,s))-\lambda (z-z')\right\|_{L^2(\Omega)}ds\\
+c_1\int_0^T\left[\left\|zz_x-z'z'_x\right \|_{L^2(\Omega)}+\left\|zv_x-z'v'_x\right \|_{L^2(\Omega)}+\left\|vz_x-v'z'_x\right \|_{L^2(\Omega)}+\left\|vv_x-v'v'_x\right \|_{L^2(\Omega)}\right]ds\\
\le c_1(1+T)^\frac{1}{2}\sqrt{T}(1+\|a\|_{L^2(\Omega)})\|z-z'\|_{Y_T}+c_1(T^\frac{1}{2}+T^\frac{1}{3})(\|z\|_{Y_T}+\|z'\|_T)\|z-z'\|_{Y_T}\\
+c_1(T^\frac{1}{2}+T^\frac{1}{3})(\|z'\|_{Y_T}\|v-v'\|_{Y_T}+\|v\|_T\|z-z'\|_{Y_T})+c_1(T^\frac{1}{2}+T^\frac{1}{3})(\|z\|_{Y_T}\|v-v'\|_{Y_T}+\|v\|_{Y_T}\|z-z'\|_{Y_T})\\
+c_1(T^\frac{1}{2}+T^\frac{1}{3})(\|v\|_{Y_T}+\|v'\|_{Y_T})\|v-v'\|_{Y_T}.
\end{gathered}$$
Now, using , for the same $r$ as before, but choosing $T$ smaller if necessary, we obtain $$\|\Upsilon z-\Upsilon z'\|_{Y_T}\le\rho \|z-z'\|_{Y_T}$$ for some $\rho\in (0,1)$. Then, by the Banach contraction theorem, we get the existence and uniqueness of a local solution in $S_{T,r}$.
**Step 2 - Regularity:** Let $y_0\in D(A)$. We define the closed space $$B_{T,r}:=\{(\psi,\varphi)\in Y_T^3\times Y_T\,|\,\psi\in C([0,T];L^2(\Omega))\cap L^2(0,T;H^1(\Omega)), \varphi=\psi_t, \|\psi\|_{Y_T^3}+\|\varphi\|_{Y_T}\le r\}.$$ Now, given $(z,\tilde{z})\in B_{T,r}$ let $q$ be a solution of
$$\label{nonlinKdV-wp-q-2}
\begin{cases}
\displaystyle q_{t} +q_{x} + q_{xxx} + {\lambda} q \\
= a(x)q_x(0,\cdot) - (I-K)[(\tilde{z}+\tilde{v})(z_x+v_{x})-(z+v)(\tilde{z}_{x}+\tilde{v}_{x})] & \text { in } \Omega\times (0,T),\\
{q}(0,t) = {q}(1,t) = {q}_{x}(1,t) = 0 & \text { in } (0,T),\\
{q}(x,0)= q_0:=-y_0'-y_0'''-\lambda y_0+a(x)y_0'(0)-(y_0+v_0)(y_0'+v_0') & \text { in } \Omega,
\end{cases}$$
where $v=\Phi(z)$, $v_0=\Phi(y_0)$, $\tilde{v}=\Phi(\tilde{z})$. Set $y=y_0+\int_0^tqds$. Then, $y_t=q$ and $y$ solves
$$\label{nonlinKdV-wp-q-1}
\begin{cases}
\displaystyle y_{t} +y_{x} + y_{xxx} + {\lambda} y = a(x)y_x(0,\cdot) - (I-K)[(z+v)(z_x+v_x)] & \text { in } \Omega\times (0,T),\\
{y}(0,t) = {y}(1,t) = {y}_{x}(1,t) = 0 & \text { in } (0,T),\\
{y}(x,0)=y_0 & \text { in } \Omega.
\end{cases}$$
We set an operator $\Theta: (z,\tilde{z})\mapsto (y,q)$ associated with the system of equations given by -. One can show that for suitable $r$ and small $T$, the operator $\Theta$ maps $B_{T,r}$ onto itself in a contractive manner. This can be done by obtaining the same type of estimates given in Step 1 for both the solution of and . Therefore, it has a unique fixed point whose first component is the regular local solution we are looking for.
**Step 3 - Global solutions:** Global wellposedness in $Y_T$ with small initial datum follows directly from the stabilization estimate proved in Section \[ProofofThm2\].
Global well-posedness of the nonlinear modified target system now follows from the Proposition \[wellposednessprop2\] that we just proved.
Using a single controller
=========================
Smaller decay rate
------------------
Although we studied the model with two controls at the right hand side, it is also possible to use only one control. For example [@Cor14] proves exponential stability with the control acting only from the Neumann boundary condition when $L$ is not of critical length. When $L$ is not restricted to uncritical lengths, we can still obtain exponential stability with a single Dirichlet control rather than a Neumann control by using the pseudo-backstepping method above. However, this causes a smaller rate of decay. Consider for instance the plant $$\label{singlecontrol}
\begin{cases}
\displaystyle u_{t} + u_{x} + u_{xxx} =0 & \text { in } \Omega\times \mathbb{R_+},\\
u(0,t) = 0, u(L,t) = U(t), u_{x}(L,t)=0 & \text { in } \mathbb{R_+},\\
u(x,0)=u_0(x) & \text { in } \Omega.
\end{cases}$$ Then the backstepping transformation gives the following target system $$\label{HomKdVBurgers-single-1}
\begin{cases}
\displaystyle \tilde{w}_{t} +\tilde{ w}_{x} + \tilde{w}_{xxx} + \lambda \tilde{w} = \tilde{k}_y(x,0)\tilde{w}_x(0,t) & \text { in } \Omega\times \mathbb{R_+},\\
\tilde{w}(0,t) = \tilde{w}(L,t) = 0, \tilde{w}_{x}(L,t) = -\int_0^L\tilde{k}_x(L,y)u(y,t)dy & \text { in } \mathbb{R_+},\\
\tilde{w}(x,0)=\tilde{w}_0(x):= u_0-\int_0^x\tilde{k}(x,y)u_0(y)dy & \text { in } \Omega.
\end{cases}$$ If we multiply the above system by $\tilde{w}$, integrate over $(0,L)$, and use integration by parts, the Cauchy-Schwarz inequality, and boundary conditions we obtain $$\label{HomKdVBurgers-single-2}
\frac{1}{2}\frac{d}{dt}\|\tilde{w}(t)\|_{L^2(\Omega)}^2+ \left(\lambda-\frac{1}{2}\|\tilde{k}_y(\cdot,0)\|_{L^2(\Omega)}^2-\frac{1}{2}\|\tilde{k}_x(L,\cdot )\|_{L^2(\Omega)}^2\|(I-K)^{-1}\|_{B[L^2(\Omega)]}^2 \right)\|\tilde{w}(t)\|_{L^2(\Omega)}^2\le 0.$$ Comparing and we see that we still achieve where $\alpha= \lambda-\frac{1}{2}\|\tilde{k}_y(\cdot,0)\|_{L^2(\Omega)}^2$ is replaced by $\beta=\left(\lambda-\frac{1}{2}\|\tilde{k}_y(\cdot,0)\|_{L^2(\Omega)}^2-\frac{1}{2}\|\tilde{k}_x(L,\cdot )\|_{L^2(\Omega)}^2\|(I-K)^{-1}\|_{B[L^2(\Omega)]}^2 \right)$. Recall that in Lemma \[alemlambda\] we showed $\|\tilde{k}_y(\cdot,0)\|_{L^2(\Omega)} \sim \lambda $. One can also get $\|\tilde{k}_x(L,\cdot )\|_{L^2(\Omega)} \sim \lambda $ by using similar arguments. Moreover, using the calculations in [@Liu03] we deduce that $\|(I-K)^{-1}\|_{B[L^2(\Omega)]}\sim 1+\lambda e^{C\lambda}$ where $C>0$ depends only on $L$. Hence positivity of $\beta$ is guaranteed for a sufficiently small choice of $\lambda$. As a result, the decay rate decreases while the exponential stability still holds.
Glass and Guerrero [@Glass10] proved that is exactly controllable if and only if $L$ does not belong to a set of critical lengths $\mathcal{O}$, defined by them, which is different than $\mathcal{N}$. They showed that if $L\in \mathcal{O}$, then the following problem has a nontrivial solution: $$\label{phieq1}
\begin{cases}
\varphi'''+\varphi'=\lambda \varphi & \text { in } (0,L),\\
\varphi(0) = \varphi(L) = \varphi'(0)=\varphi''(L)=0.
\end{cases}$$ Moreover, it was found that for any $u_0\in L^2(\Omega)$ and control input $U\in L^2(0,T)$, the function $$\label{ftcons}
e^{-\lambda t}\int_0^Lu(x,t;U)\varphi(x)dx$$ remains constant in time, where $u=u(x,t;U)$ is the corresponding trajectory for with control input $U$. Regarding the same KdV system, we construct a boundary feedback which yields stabilizability with a certain decay rate for all $L>0$ including $L\in \mathcal{O}$. This shows that for a certain control input $U$, the integral $\int_0^Lu(x,t;U)\varphi(x)dx$ decays in time because $\left|\int_0^Lu(x,t;U)\varphi(x)dx\right|\le \|u(t;U)\|_{L^2(\Omega)}\|\varphi\|_{L^2(\Omega)}$, where $\|u(t;U)\|_{L^2(\Omega)}$ decays exponentially. Therefore, since is invariant in time, it follows that $|e^{-\lambda t}|$ must increase, which is only possible if $Re(\lambda)<0$. However, if $Re(\lambda)<0$, then $|e^{-\lambda t}|\rightarrow \infty$. Since is valid for all control inputs, we conclude in particular that $\int_0^Lu(x,t;0)\varphi(x)dx\rightarrow 0$ $(U(t)\equiv 0$). This shows that any uncontrollable trajectory corresponding to some initial state $u_0$ with no feedback is attracted to the orthogonal complement of the span of the set consisting of the real and imaginary parts of the nontrivial solutions of .
Regarding the nonlinear KdV equation with same boundary conditions, Glass and Guerrero [@Glass10] obtained the exact controllability given that the domain is of uncritical length and the initial and final states are small. This was not an if and only if statement unlike the linear problem. Therefore, we do not know whether the exact controllability on domains of critical lengths is true with the control acting at the right Dirichlet b.c. On the other hand, our feedback control design can be easily extended to the nonlinear system for small data as in Section 2.2. The nonlinear case is quite interesting because maybe the nonlinear term $uu_x$ is creating a further stability effect on the solutions, which might drive them to zero by themselves. As far as we know, the exact controllability for the (nonlinear) KdV equation as well as the decay of solutions to zero by themselves in the presence of the nonlinear term $uu_x$ remain as open problems on critical length domains, see for instance Cerpa [@cer14].
A second order feedback law
---------------------------
In this section, we check whether it is possible to stabilize the solutions of the KdV equation by using the feedback law $u(L,t)=U(t)$ with the input $U(t)=u_{xx}(L,t)$. In order to gain some intuition regarding this problem, let us consider the linearized model with the input $U(t)=u_{xx}(L,t)$. Multiplying the main equation in by $u$ and integrating over $\Omega$ by using the given boundary conditions, one obtains $$\label{newL2iden}
\frac{d}{dt}\|u(t)\|_{L^2(\Omega)}^2 = -\frac{3}{2}|u(L,t)|^2-\frac{1}{2}|u_x(0,t)|^2\le 0.$$ The inequality shows that $\|u(t)\|_{L^2(\Omega)}$ is non-increasing, but it is not clear whether it decays to zero. In order to better understand the behavior of the solution, one generally must study the spectral properties of the corresponding evolution. Regarding , one can study the operator $$Au=-u'''-u',\,\,\,D(A)=\{u\in H^3(\Omega)\,|\,u(0)=u'(L)=u(L)-u''(L)=0\}.$$ However, it is quite difficult to analyze the eigenvalues of this operator. This is because the characteristic equation corresponding to the eigenvalue problem $Au=\lambda u$ takes the form $r^3+r+\lambda =0$, which is not easy to study. Therefore, we will consider this problem on the rather simplified model given below, neglecting the first order term $u_x$: $$\label{simplemodel}
\begin{cases}
\displaystyle u_{t} + u_{xxx} =0 & \text { in } \Omega\times \mathbb{R_+},\\
u(0,t) = 0, u(L,t) = u_{xx}(L,t), u_{x}(L,t)=0 & \text { in } \mathbb{R_+},\\
u(x,0)=u_0(x) & \text { in } \Omega,
\end{cases}$$ where the inequality takes the form $$\label{newL2iden2}
\frac{d}{dt}\|u(t)\|_{L^2(\Omega)}^2 = -|u(L,t)|^2-\frac{1}{2}|u_x(0,t)|^2\le 0.$$
Spectral properties {#spectral-properties .unnumbered}
-------------------
The operator which generates the evolution corresponding to is a third order dissipative differential operator given by $$Au=-u''',\,\,\,D(A)=\{u\in H^3(\Omega)\,|\,u(0)=u'(L)=u(L)-u''(L)=0\},$$ which has compact resolvent and spectrum involving countably many eigenvalues $\{\lambda_k\}_{k\in \mathbb{Z}}$ satisfying $\text{Re}\lambda_k\le 0$. Moreover, these eigenvalues satisfy the properties given in the lemma below, whose proof uses the approach presented in [@Zhang001 Prop 2.2] and [@Zhang002 Prop 3.1].
$\lambda_k = -\frac{8\pi^3}{3\sqrt{3}L^3}|k|^3 $ as $|k|\rightarrow \infty$. Moreover, $\exists \eta<0$ s.t. $\text{Re}\lambda_k<\eta$ $\forall k\in \mathbb{Z}$.
Let $\lambda$ be an eigenvalue of $A$. Then, $\text{Re}\lambda \le 0$, and we can assume wlog that $\text{Im}\lambda\le 0$ since $\bar{\lambda}$ is also an eigenvalue of $A$. Let us first see that $\text{Re}\lambda$ cannot be equal to zero. To this end, let $i\xi$ be an eigenvalue with $\xi\in \mathbb{R}$ and $u$ be the corresponding eigenvector. Then, we note that $$\text{Re}(Au,u)_{L^2(\Omega)}=-|u|^2(L)-\frac{1}{2}|u'(0)|^2=\text{Re}(i\xi\|u\|_{L^2(\Omega)}^2)=0.$$ We get $u(L)=u'(0)=0$, which implies together with other boundary conditions $u\equiv 0$. This contradicts the fact that $u$ was an eigenvector. Hence, $\text{Re}\lambda<0$.
Let $r_i$ $(i=0,1,2)$ be the three roots of the characteristic equation $r^3+\lambda=0$ corresponding to the ode $$\label{Aulambdau}
Au=\lambda u,$$with $r_1$ being the root in the first quadrant. Note that we have $r_i=\alpha^i r_1$, $\alpha=e^{\frac{2\pi i}{3}}$. The solution of is $u(x)=\sum_{i=0}^2c_ie^{r_i x}$ where $\{c_i, i=0,1,2\}$, due to given boundary conditions, satisfy the system of equations $$\begin{cases} \sum_{i=0}^2c_i =0 \\ \sum_{i=0}^2c_ir_ie^{r_i L} =0 \\ \sum_{i=0}^2 c_i(1-r_i^2e^{r_i L})=0, \end{cases}$$ which has a nontrivial solution if $$\sum_{i=0}^2a_{ij}r_i(1-r_j^2)e^{(r_i+r_j)L}=0,$$ where $a_{12}=a_{20}=a_{01}=1$ and $a_{21}=a_{02}=a_{10}=-1$. Multiplying the above equation by $e^{-r_0L}$ and neglecting the relatively small terms which involve $e^{(r_1+r_2-r_0)L}$, we get $$(1+\alpha)(1+\alpha^2r_0^2)e^{r_2L}+(1+\alpha r_0^2)e^{r_1L}=0.$$ Now, we see that we can further neglect the terms $(1+\alpha)e^{r_2L}$ and $e^{r_1L}$ since they are much smaller than $(1+\alpha)\alpha^2r_0^2e^{r_2L}$ and $\alpha r_0^2e^{r_1L}$, respectively. Therefore, asymptotically, we get $$(1+\alpha)\alpha e^{r_2L}+e^{r_1L}=0.$$ Observe that $\alpha(1+\alpha)=-1$. Therefore, the asymptotic relation is reduced to $$e^{(r_2-r_1)L}=1,$$ from which, together with the definition of $\alpha,r_1$, and $r_2$, it follows that $\lambda_k=-r_{0,k}^3=-\frac{8\pi^3}{3\sqrt{3}L^3}|k|^3$ asymptotically. This property combined with the fact that $\text{Re}\lambda_k<0$ proves the second part of the lemma.
It is easy to see that $A^*$ is defined by $$A^*w=w''',\,\,\,D(A^*)=\{w\in H^3(\Omega)\,|\,w(0)=w'(0)=w(L)+w''(L)=0\}.$$ Now, the following result follows classically from the spectral properties of $A$ above.
([@Zhang001 Prop 2.1, 2.2], [@Zhang002 Prop 3.2]) $A$ is a discrete spectral operator, and all but a finite number of eigenvalues of $A$ correspond to one dimensional projections $E(\lambda;T)$. Both $A$ and $A^*$ have complete sets of eigenvectors, $\{\phi_k\}_{k\in \mathbb{Z}}$ and $\{\psi_j\}_{j\in \mathbb{Z}}$, respectively, satisfying $(\phi_k,\psi_j)_{L^2(\Omega)}=\delta_{kj}$ and forming dual Riesz bases for $L^2(\Omega)$.
A special multiplier and stabilization {#a-special-multiplier-and-stabilization .unnumbered}
--------------------------------------
We set $Y:=\sum_{k}Y_k$ where $Y_k$ is defined by $$Y_k(u)=(u,\psi_k)_{L^2(\Omega)}\psi_k.$$ Then $Y$ is bounded and positive definite by the uniform $\ell^2$ convergence property of $\{\phi_k\}_{k\in \mathbb{Z}}$ and $\{\psi_j\}_{j\in \mathbb{Z}}$. Second, we define the symmetric, bounded, nonnegative operator $X=\sum_{k}\xi_kY_k$, where $\xi_k=-\frac{1}{2\text{Re}\lambda_k}$, which satisfies the additional property $$A^*X+XA+Y=0.$$ Using $Xu$ as a multiplier, we compute $$\label{XYcomp}
\frac{d}{dt}(Xu,u)_{L^2(\Omega)} = (\frac{d}{dt}Xu,u)_{L^2(\Omega)}+(Xu,u_t)_{L^2(\Omega)}
=(XAu,u)_{L^2(\Omega)}+(Xu,Au)_{L^2(\Omega)}
=-(Yu,u)_{L^2(\Omega)}.$$
together with imply that $$\label{gronprep1}
\frac{d}{dt}((I+X)u,u)_{L^2(\Omega)}\le -(Yu,u)_{L^2(\Omega)}.$$ Now, integrating in time and using the positive definiteness of $(I+X)$ and $Y$, applying Gronwall’s inequality, we obtain the exponential decay of solutions in $L^2(\Omega)$, and the following theorem follows.
\[specthm\] Let $u$ be a solution of the linearized KdV equation in . Then, there exists some $\gamma>0$ independent of $u_0$ such that $$\|u(t)\|_{L^2(\Omega)} \lesssim \|u_0\|_{L^2(\Omega)}e^{-\gamma t}$$ for $t\ge 0$.
Note that Theorem \[specthm\] was proved for the simplified linearized model . The situation is more challenging for more general models involving other terms such as $u_x$ and/or the nonlinear term $uu_x$. Moreover, the approaches of [@Zhang001] and [@Zhang002] do not seem to directly apply to these more general problems under the boundary conditions $u(0)=u'(L)=u(L)-u''(L)=0$. This is due to the fact that the eigenvalue analysis gets much more challenging with a more complicated third order characteristic equation. In order to simplify the eigenvalue analysis, one can still use the simpler operator $Au=-u'''$ treating $-u_x$ and/or $uu_x$ as source term(s). But then, one needs the adjoint operator $A^*$ to satisfy very desirable boundary conditions so that the trace terms are cancelled out when one applies the special multiplier. However, this does not become the case with the given boundary conditions in the model. This issue is not present with the boundary conditions used in [@Zhang001] and [@Zhang002]. Therefore, the case of more general equations with the first order term $u_x$ and/or the nonlinear term $uu_x$ remain interesting open problems.
Numerical simulations
=====================
We modify the finite difference scheme given in [@Pazato] to fit it into the present situation, where we have first order trace terms in the main equations of the target systems and inhomogeneous boundary inputs of feedback type in the original plant. We numerically solve the KdV equation both in the controlled and uncontrolled cases. We are also able to verify our main result numerically. First, we simulate an uncontrolled solution of the KdV equation and then we simulate the controlled solution. From our simulations, one can see that the boundary controllers constructed using a pseudo-kernel effectively stabilize the solutions with a suitable choice of $\lambda$. The calculations are performed in Wolfram Mathematica^^11.
For simplicity, we consider only the linearised problem. The nonlinear problem can be treated in a similar way by including an additional fixed point argument to the algorithm we describe here. We use the notation given in [@Pazato]. To this end, we set the discrete space $$X_J:=\{\tilde{w}=(\tilde{w}_0,\tilde{w}_1,...,\tilde{w}_J)\in \mathbb{R}^{J+1}\,|\,\tilde{w}_0=\tilde{w}_{J-1}=\tilde{w}_J=0\},$$ and the difference operators $\displaystyle (D^+\tilde{w})_j:=\frac{\tilde{w}_{j+1}-\tilde{w}_j}{\delta x}$, $\displaystyle (D^-\tilde{w})_j:=\frac{\tilde{w}_{j}-\tilde{w}_{j-1}}{\delta x}$ for $j=1,...,J-1$, and $\displaystyle D=\frac{1}{2}(D^++D^-)$. We will call the space and time steps $\delta x$ and $\delta t$ for $j=0,...,J,$ and $n=0,1,...,N$, respectively. Using this notation, the numerical approximation of the linearised target system takes the form $$\begin{aligned}
% \nonumber to remove numbering (before each equation)
\label{wjn1}\frac{\tilde{w}_{j}^{n+1}-\tilde{w}_j^n}{\delta t}+(\mathcal{A}\tilde{w}^{n+1})_j+\lambda \tilde{w}_j^{n+1}&=& \tilde{k}_y(x_j,0)\frac{\tilde{w}_{1}^{n}}{\delta x},\hspace{.1in} j=1,...,J-1\\
\label{wjn2}\tilde{w}_0=\tilde{w}_{J-1}=\tilde{w}_J &=& 0, \\
\label{wjn3} \tilde{w}_0 &=&\int_{x_{j-\frac{1}{2}}}^{x_{j^+\frac{1}{2}}}\tilde{w}_0(x)dx,\hspace{.1in} j=1,...,J-1,\end{aligned}$$ where $x_{j\mp\frac{1}{2}}=(j\mp\frac{1}{2})\delta x$, $x_j=j\delta x$. The $(J-1)\times (J-1)$ matrix $\mathcal{A}$ approximates $\tilde{w}_x+\tilde{w}_{xxx}$ and is defined by $\mathcal{A}:=D^+D^+D^-+D$. Let us set $\mathcal{C}:=(1+\delta t\lambda)I+\delta t A$. Then, from the main equation, we obtain $\tilde{w}_{j}^{n+1}=\mathcal{C}^{-1}\left(\tilde{w}_j^n+\frac{\delta t}{\delta x}\tilde{k}_y(x_j,0)\tilde{w}_{1}^{n}\right)$ for $j=1,...,J-1$.
In order to approximate the solution of the original plant with feedback controllers, we use the succession idea in the proof of Lemma \[inverselem\]. Note that given $\tilde{w}$, $v$ is the fixed point of the equation $v=K(\tilde{w}+v)$. For numerical purposes, let $m$ denote the number of iterations in the succession and set $v^0=\mathcal{K}\tilde{w}$, $v^{k}:=\mathcal{K}(\tilde{w}+v^{k-1})$ for $1\le k\le m$, where $\mathcal{K}$ is the numerical approximation of the integral in the definition of $K$. Then, $v^m$ is an approximation of $v=\Phi(\tilde{w})$, and one gets an approximation of the original plant by setting $u(x_j,t_n):=\tilde{w}(x_j,t_n)+v^m(x_j,t_n)$.
On a domain of critical length, one can find time-independent solutions, as we mentioned in the introduction. Figure \[uncont-sol\] below shows such a solution on a domain of length $L=2\pi$ whose $L^2$-norm is preserved in time.\
![Uncontrolled solution with initial datum $u_0=1-\cos(x)$ on a domain of length $2\pi$.[]{data-label="uncont-sol"}](uncont-sol)
If one applies the boundary controllers constructed with the same initial profile that the uncontrolled solution has in Figure \[uncont-sol\], then the new solution will decay to zero as we illustrate in Figure \[controlled\].
![Controlled solution with initial datum $u_0=1-\cos(x)$, $\lambda=0.03$, with a controller using the pseudo-kernel $\tilde{k}$ on a domain of length $2\pi$.[]{data-label="controlled"}](controlled)
Figure \[u1t\] shows the controller behavior on the Dirichlet boundary condition at the right endpoint. As one can see, less control is needed as the wave gets supressed.\
![Dirichlet controller at the right endpoint ($\lambda=0.03$)[]{data-label="u1t"}](u1t "fig:")\
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to express our gratitude to the anonymous referee whose valuable insights significantly improved the quality of this article. We would also like to thank Katherine H. Willcox from Izmir Institute of Technology for her English editing of this paper.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present an exact collapsing solution to 2+1 gravity with a negative cosmological constant minimally coupled to a massless scalar field, which exhibits physical properties making it a candidate critical solution. We discuss its global causal structure and its symmetries in relation with those of the corresponding continously self-similar solution derived in the $\Lambda=0$ case. Linear perturbations on this background lead to approximate black hole solutions. The critical exponent is found to be $\gamma = 2/5$.'
author:
- |
Gérard Clément$^{a}$ [^1] and Alessandro Fabbri$^{b}$ [^2]\
\
[$^{(a)}$Laboratoire de Physique Théorique LAPTH (CNRS),]{}\
[B.P.110, F-74941 Annecy-le-Vieux cedex, France]{}\
[$^{(b)}$Dipartimento di Fisica dell’Università di Bologna]{}\
[and INFN sezione di Bologna,]{}\
[Via Irnerio 46, 40126 Bologna, Italy]{}
date: 'March 23, 2001'
title: |
**Analytical treatment of critical collapse in 2+1 dimensional AdS spacetime:\
a toy model**
---
Introduction
============
Since its discovery, the BTZ black hole solution \[1\] of 2+1 dimensional AdS gravity has attracted much interest because it represents a simplified context in which to study the classical and quantum properties of black holes. A line of approach which has been opened only recently [@CP; @HO; @gar; @burko] concerns black hole formation through collapse of matter configurations coupled to 2+1 gravity with a negative cosmological constant. As first discovered in four dimensions by Choptuik [@chop], collapsing configurations which lie at the threshold of black hole formation exhibit properties, such as universality, power-law scaling of the black hole mass, and continuous or discrete self-similarity, which are characteristic of critical phenomena [@gund]. In the case of a spherically symmetric massless, minimally coupled scalar field, a class of analytical continously self-similar (CSS) solutions was first given by Roberts [@rob; @brady; @oshiro]. These include critical solutions, lying at the threshold between black holes and naked singularities, and characterized by the presence of null central singularities. Linear perturbations of these solutions [@fro; @hay] lead to approximate black hole solutions with a spacelike central singularity.
Numerical simulations of circularly symmetric scalar field collapse in 2+1 dimensional AdS spacetime were recently performed by Pretorius and Choptuik [@CP] and Husain and Olivier [@HO]. Both groups observed critical collapse, which was determined in [@CP] to be continuously self-similar near $r=0$. In [@gar], Garfinkle has found a one-parameter family of exact CSS solutions of 2+1 gravity without cosmological constant, and argued that one of these solutions should give the behaviour of the full critical solution ($\Lambda\neq 0$) near the singularity.
The purpose of this paper is to present a new CSS solution to the field equations with $\L=0$ which can be extended to a threshold solution of the full $\L \neq 0$ equations. The new $\L
= 0$ solution is derived in Sect. 3. It presents a null central singularity and, besides being CSS, possesses four Killing vectors. In Sect. 4 we address the extension of this CSS solution to a quasi-CSS solution of the full $\L < 0$ problem, and show that the requirement of maximal symmetry selects a unique extension. This inherits the null central singularity of the $\L = 0$ solution, and has the correct AdS boundary at spatial infinity. Finally, we perform in Sect. 5 the linear perturbation analysis in this background, find that it does lead to black hole formation, and determine the critical exponent.
CSS solutions
=============
The Einstein equations for cosmological gravity coupled to a massless scalar field in (2+1) dimensions are G\_ - Łg\_ = T\_, with the stress-energy tensor for the scalar field T\_ = \_\_- 12 g\_ \^\_. The signature of the metric is (+ - -), and the cosmological constant $\L$ is negative for AdS spacetime, $\L = -l^{-2}$. Static solutions of these equations include the BTZ black hole solutions [@BTZ] with a vanishing scalar field $\phi = 0$, and singular solutions when a non-trivial scalar field is coupled with the positive sign for the gravitational constant $\k$ [@sigcosm].
We shall use for radial collapse the convenient parametrisation of the rotationally symmetric line element in terms of null coordinates $(u,v)$: ds\^2 = e\^[2]{}dudv - r\^2d\^2, with metric functions $\s(u,v)$ and $r(u,v)$. The corresponding Einstein equations and scalar field equation are & r\_[,uv]{} = r e\^[2]{}, &\
& 2\_[,uv]{} = e\^[2]{} - \_[,u]{}\_[,v]{}, &\
& 2\_[,u]{} r\_[,u]{} -r\_[,uu]{} = r\_[,u]{}\^2, &\
& 2\_[,v]{} r\_[,v]{} -r\_[,vv]{} = r\_[,v]{}\^2, &\
& 2r\_[,uv]{} + r\_[,u]{}\_[,v]{} + r\_[,v]{}\_[,u]{} = 0 & .
From the Einstein equations, the Ricci scalar is R = -6Ł+ 4e\^[-2]{}\_[,u]{}\_[,v]{}. It follows from (\[ric1\]) and (\[ein2\]) that the behavior of the solutions near the singularity is governed by the equations (\[ein1\])-(\[phi\]) with vanishing cosmological constant $\L = 0$ (see also [@burko]). Assuming $\L = 0$, Garfinkle has found [@gar] the following family of exact CSS solutions to these equations ds\^2 & = & -A()\^[c\^2]{}dudv - 14(v+u)\^2d\^2,\
& = & -2c( + ), depending on an arbitrary constant $c$ and a scale $A > 0$. In (\[gar1\]), $u$ is retarded time, and $-v$ is advanced time. These solutions are continuously self-similar with homothetic vector $(u\part_u + v\part_v)$. An equivalent form of these CSS solutions, obtained by making the transformation -u = (-|[u]{})\^[2q]{}, v = (|[v]{})\^[2q]{} (1/2q = 1 - c\^2) to the barred null coordinates $(\bar{u},\bar{v})$, is ds\^2 & = & -|[A]{}(|[v]{}\^q + (-|[u]{})\^q)\^[2(2q-1)/q]{}d|[u]{}d|[v]{} - 14(|[v]{}\^[2q]{} - (-|[u]{})\^[2q]{})\^2d\^2,\
& = & -2c(|[v]{}\^q + (-|[u]{})\^q). The corresponding Ricci scalar is R = (|[v]{}\^q + (-|[u]{})\^q)\^[2(1-3q)/q]{}(-|[u]{})\^[q-1]{} (|[v]{})\^[q-1]{}.
Garfinkle suggested that the line element (\[gar1\]) describes critical collapse in the sector $r = -(u+v)/2 \ge
0$, near the future point singularity $r = 0$ (where the Ricci scalar behaves, for $v \propto u$, as $u^{-2}$). The corresponding Penrose diagram (Fig. 1) is a triangle bounded by past null infinity $u \to -\infty$, the other null side $v = 0$, and the central regular timelike line $r = 0$. For $\k c^2 \ge 1$ ($q < 0$), the Ricci scalar R \~(|[v]{})\^[q-1]{} \~(v)\^[(q-1)/2q]{} is regular near $v = 0$, which moreover turns out to be at infinite geodesic distance. To show this, we consider the geodesic equation (e\^[2]{}) = -2rr\_[,u]{}\^2 = -2l\^2r\^[-3]{}r\_[,u]{} ($l$ constant) near $v = 0$, $u$ constant, which gives $v \propto (ls)^{4q}$ for $l \neq 0$, or $s^{2q}$ for $l=0$, so that in all cases the affine parameter $s \to \infty$ for $v \to 0$, and the spacetime is geodesically complete. For $\k c^2 < 1$ ($q
> 0$), we see from (\[ric2\]) that the null line $v = 0$ is a curvature singularity if $\k c^2 < 1/2$ ($q < 1$). If $1/2 \le \k c^2 < 1$ ($q \ge 1$), the surface $v = 0$ is regular. However, as discussed by Garfinkle, the metric (\[gar2\]) can be extended through this surface only for $q = n$, where $n$ is a positive integer. For $n$ even, the extended spacetime is made of two symmetrical triangles joined along the null side $\bar{v} = 0$, and has two coordinate singularities $r = 0$, one timelike ($\bar{u} - \bar{v} = 0$) and one spacelike ($\bar{u} + \bar{v} = 0$), but no curvature singularity. For $n$ odd, one of the $r = 0$ sides becomes a future spacelike curvature singularity ($e^{2\s} = 0$), similar to that of Brady’s supercritical solutions for scalar field collapse in (3+1) dimensions [@brady], except for the fact that in the present case the singularity is not hidden behind a spacelike apparent horizon (Fig. 2).
Let us point out that, besides the solutions (\[gar1\]), the system (\[ein1\])-(\[phi\]) also admits for $\L
= 0$ another family of CSS solutions ds\^2 & = & A()\^[c\^2]{}dudv - 14(v+u)\^2d\^2,\
& = & |[A]{}(|[v]{}\^q - (-|[u]{})\^q)\^[2(2q-1)/q]{}d|[u]{}d|[v]{} - 14(|[v]{}\^[2q]{} - (-|[u]{})\^[2q]{})\^2d\^2, with $\phi = -2c\ln(\sqrt{v} - \sqrt{-u})$, and we choose $A > 0$ and consider the sector $0 \le v \le -u$. These solutions have a future spacelike central ($r = 0$) curvature singularity at $(-\bar{u})^q =
\bar{v}^q$ (where the Ricci scalar (\[ric2\]) diverges) for all $q < 0$ or $q > 0$ (implying $q
> 1/2$). For $q < 0$, the Penrose diagram is a triangle bounded by past null infinities $\bar{u} \to
-\infty$ and $\bar{v} = 0$ (which is at infinite geodesic distance). For $q
> 0$, geodesics terminate at $\bar{v} = 0$, unless $q = n$ integer. For $n$ even, the extended spacetime has two central curvature singularities $r = 0$, one spacelike and the other timelike. The extended spacetime for $n$ odd is more realistic. In this case the extension from $\bar{v}
> 0$ to $\bar{v} < 0$ amounts to replacing (\[gar3\]) with $A > 0$ by the original Garfinkle solution (\[gar1\]) with $A > 0$, the resulting Penrose diagram being that of Fig. 2.
A new CSS solution for $\L = 0$
===============================
Among the one-parameter ($c$ or $q$) family of CSS solutions (\[gar1\]), the special solution, corresponding to $\k c^2 = 1$, ds\^2 = A( + )\^4 - 14(v+u)\^2d\^2, is singled out by the fact that the transformation (\[q\]) breaks down for this value. The transformation appropriate to this case, -u = 2e\^[-U]{}, v = 2e\^V = 2e\^[U-2T]{} (with $T \ge U$ for $u+v \le 0$) transforms the solution (\[gar4\]) to ds\^2 & = & e\^[-2U]{}\[-4A(1 + e\^[U-T]{})\^4dUdV - (1 - e\^[2(U-T)]{})\^2 d\^2\],\
& = & U - 2(1 + e\^[U-T]{}) (we use from now on units such that $\k = 1$, and have dropped an irrelevant additive constant from $\phi$).
Starting from this special CSS solution of the Garfinkle class, we now derive, by a limiting process, a new CSS solution which, as we shall see, exhibits a null singularity. We translate $T$ to $T-T_0$, and take the late-time limit $T_0 \to
-\infty$, leading to the new CSS solution (written for $A = -1/2$) ds\^2 = e\^[-2U]{}(2dUdV - d\^2), = U, with a very simple form which is reminiscent of the Hayward critical solution for scalar field collapse in 3+1 dimensions [@hay], ds\^2 = e\^[2]{}(2d\^2 - 2d\^2 - d\^2), = . The transformation |[u]{} = -e\^[-2U]{}, |[v]{} = V leads from (\[crit0\]) to the even more simple form of this solution ds\^2 = d|[u]{}d|[v]{} + |[u]{}d\^2, = -12(-|[u]{}), which is reminiscent of the other form of the Hayward solution ds\^2 = 2d|[u]{}d|[v]{} + |[u]{}|[v]{}d\^2, = -12(-|[u]{}/|[v]{}).
The solution (\[crit0\]) or (\[crit1\]) is continuously self-similar, with homothetic vector K = \_U = -2|[u]{}\_[|[u]{}]{}. It also has a high degree of symmetry, with 4 Killing vectors L\_1 & = & \_U + 2V\_V + \_,\
L\_2 & = & \_V + U\_,\
L\_3 & = & \_V,\
L\_4 & = & \_, generating the solvable Lie algebra & \[L\_1, L\_2\] = L\_4-L\_2, & \[L\_2,L\_3\] = 0,\
& \[L\_1, L\_3\] = -2L\_3, & \[L\_2,L\_4\] = -L\_3,\
& \[L\_1, L\_4\] = -L\_4, & \[L\_3,L\_4\] = 0.
The Ricci scalar (\[ric1\]) is identically zero for the solution (\[crit0\]), for which the sole nonvanishing Ricci tensor component is $R_{UU} = 1$. It follows that this metric is devoid of curvature singularity. However there is an obvious coordinate singularity at $U \to +\infty$, or $\bar{u} = 0$ (where $r=0$). To determine the nature of this singularity, we study geodesic motion in the spacetime (\[crit1\]). The geodesic equations are integrated by = , |[u]{} = l, + l = , where $\pi$ and $l$ are the constants of the motion associated with the Killing vectors $L_3$ and $L_4$, and the sign of $\varepsilon$ depends on that of $ds^2$ along the geodesic. The null line $\bar{u} = 0$ can be reached only by those geodesics with $\pi \neq 0$. Then, the third equation (\[geos\]) integrates to |[v]{} = |[u]{} - + [const.]{} = |[u]{} - (-|[u]{}) + [const.]{}. It follows that nonradial geodesics ($l \neq 0$) terminate at $\bar{u} = 0, \bar{v} \to +\infty$, while radial geodesics ($l =
0$), which behave as in cylindrical Minkowski space, can be continued through the null line $\bar{u} = 0$ to $\bar{u} \to
+\infty$ . So in this sense only the endpoint $\bar{v} \to
+\infty$ of the null line $\bar{u} = 0$ is singular. However formal analytic continuation of the metric (\[crit1\]) from $\bar{u} < 0$ to $\bar{u} > 0$ involves a change of signature from (+ - -) to (+ - +), leading to the appearance of closed timelike curves. So the null line $\bar{u} = 0$ corresponds to a singularity in the causal structure of the spacetime, analogous to the central singularity in the causal structure of the BTZ black holes [@BTZ]. The resulting Penrose diagram, reminiscent of that of the Hayward critical solution [@hay], is a diamond bound by three lines at null infinity ($\bar{v} = -\infty, \bar{u} = -\infty, \bar{v} =
+\infty$) and the null singularity $\bar{u} = 0$ (Fig. 3).
Extending the new solution to $\L \neq 0$
=========================================
In the preceding section we have found an exact solution for scalar field collapse with $\L = 0$, which presents a central null singularity. This property makes it a candidate threshold solution, lying at the boundary between naked singularities and black holes. However black holes exist only for $\L < 0$, so the solution (\[crit1\]) can only represent the behavior of the true threshold solution near the central singularity, where the cosmological constant can be neglected. This hypothetical $\L < 0$ solution cannot be self-similar, essentially because the scale is fixed preferentially by the cosmological constant [@CP]. So what we need is to find some other way to extend (\[crit1\]) to a solution of the full system of Einstein equations with $\L < 0$.
A first possible approach is to expand this solution in powers of $\L$, with the zeroth order given by the CSS solution (\[crit1\]). In the parametrisation (\[an\]), this zeroth order is (dropping the bars in (\[crit1\]) ) r\_0 = (-u)\^[1/2]{}, \_0 = 0, \_0 = -12 |u|. We look for an approximate solution to first order in $\L$ of the form r = (-u)\^[1/2]{} + Łr\_1, \_ = Ł\_1, = -12 |u| + Ł\_1 , \[lmk\]with the boundary condition that the fonctions $r_1$, $\sigma_1$ and $\phi_1$ vanish on the central singularity $u = 0$ . Eq (\[ein1\]) gives \[cmm\] r\_1 = (-u)\^[1/2]{}(13uv + f(u)), with $f(0) = 0$. Then, the linearized Eq. (\[ein4\]) gives 2r\_0\^[1/2]{}(r\_0\^[1/2]{}[\_1]{}\_[,v]{})\_[,u]{} = -[r\_1]{}\_[,v]{}[\_0]{}\_[,u]{} = 16(-u)\^[1/2]{}, which is solved by \[zzw\] \_1 = (1[15]{}uv + g(u)). The linearized Eq. (\[ein2\]) 2[\_1]{}\_[,uv]{} = 1 - [\_0]{}\_[,u]{}[\_1]{}\_[,v]{} = then gives \[nxn\] \_1 = 4[15]{}uv + h(u). Finally Eq. (\[ein2\]) leads to the relation between the arbitrary functions $f$, $g$, $h$ uf”(u) + f’(u) = g’(u) + h’(u).
Not only does this first order solution break the continuous self-similarity generated by (\[K\]), as expected, but it also breaks the isometry group generated by the Killings (\[kill\]) down to $U(1)$ (generated by $L_4 = \part_{\theta}$), except in the special case $f = g = h = 0$, where the Killing subalgebra $(L_1, L_4)$ remains. This suggests looking for an exact $\L < 0$ extension of the $\L = 0$ CSS solution of the form ds\^2 = e\^[2(x)]{}dudv + u\^2(x)d\^2, = -12|u| + (x), with $x = uv$. This will automatically preserve to all orders the Killing subalgebra $(L_1,
L_4)$. Inserting this ansatz into the field equations (\[ein1\])-(\[phi\]) leads to the system x” + 32’ & = & e\^[2]{},\
2(x” + ’) + ’(x’-12) & = & e\^[2]{},\
x\^2(-” + 2’’- ’\^2) + x(- ’ + (’+’)) & = & 0\
- ” + 2’’- ’\^2 & = & 0\
2x(’)’ + 52’ & = & 12’. ($' = d/dx$). The unique, maximally symmetric extension of the CSS solution (\[crit1\]) reducing to (\[crit1\]) near $u = 0$ is the solution of the system (\[einx1\])-(\[phix\]) with the boundary conditions (0) = 1, (0) = 0, (0) = 0.
The comparison of (\[einx3\]) and (\[einx4\]) yields = e\^[+]{}. The combination $(\ref{einx1}) + x(\ref{einx4})$ then gives, together with (\[rhospsi\]), x(2’\^2 + 2’’ - ’\^2) + 32(’ + ’) = e\^[2]{}. The third independent equation is for instance (\[einx2\]): 2(x” + ’) + ’(x’ - 12) = e\^[2]{}. Using these last two equations with the boundary conditions (\[boundx\]), one can in principle write down series expansions for $\s(x)$ and $\psi(x)$. Another simple relation, deriving from (\[einx4\]) and (\[rhospsi\]), is ” + ” - ’\^2 + 2’\^2 = 0.
We are interested in the behavior of this extended solution in the sector $u < 0$, $v > 0$, i.e. $x < 0$. In this sector, Eqs. (\[einx1\]), (\[phix\]) and (\[einx2\]) can be integrated to (-x)\^[3/2]{}’ & = & \_x\^0 (-x)\^[1/2]{}e\^[2]{}dx,\
(-x)\^[5/4]{}’ & = & 14 \_x\^0 (-x)\^[1/4]{}’ dx,\
-x’ & = & 12 \_x\^0 ( e\^[2]{} + ’(12 - x’))dx . As long as $\rho > 0$, Eq. (\[sum1\]) (with $x < 0$, $\L < 0$) implies $\rho' < 0$, so that $\rho(x)$ decreases to 1 when $x$ increases to 0. It then follows from (\[sum2\]) that $\psi' <0$. Also, (\[sum2\]) can be integrated by parts to x’ = 14 - \_x\^0 (-x)\^[-3/4]{}dx, showing that $x\psi' < 1/4$. It then follows from (\[sum3\]) that $\s' < 0$. So, as $x$ decreases, the functions $\rho$ and $e^{2\s}$ increase and possibly go to infinity for a finite value $x = x_1$. If this is the case, the behavior of these functions near $x_1$ must be & = & \_1(1[|[x]{}]{} + 1[4x\_1]{} - + ... )\
e\^[2]{} & = & (1 + + ...)\
& = & \_1 + - (|[x]{}) + ... ($\bar{x} = x - x_1$).
These expectations are borne out by the actual numerical solution of the system x” + ’ &=& - e\^[2]{},\
-”+4’’ &=& ’\^2 +\^2’\^2, \[sist\] (this last equation comes from (\[einx4\]) where $\psi'$ is given by derivation of (\[rhospsi\]) ) where we have set $\L=-2$, with the boundary counditions $\rho(0)=1$, $\rho'(0)=-2/3$ (see eqs. (\[cmm\]) and (\[lmk\]) ), $\sigma(0)=0$. The plots of the functions $ \rho (x)$ , $\sigma (x)$ and $\psi' (x)$ are given in Figs. (4,5,6,). The value of $x_1$ is found to be approximately $-1.94$ (i.e. $\L x_1 = +3.88$).
The coordinate transformation[^3] u = Ł\^[-1]{}e\^[-|[U]{}]{}, v = e\^[|[V]{}]{} (|[U]{} = |[T]{} - |[R]{}, |[V]{} = |[T]{}+|[R]{}) leads to $x = \L^{-1} e^{2\bar{R}}$ and, on account of (\[anext\]) and (\[rhospsi\]), to the form of the metric ds\^2 = -Ł\^[-1]{}e\^[2((|[R]{})+|[R]{})]{}(d|[U]{}d|[V]{} - e\^[2(|[R]{}) -|[V]{}]{}d\^2). Near the spacelike boundary $\bar{R} = \bar{R}_1$ of the spacetime, the collapsing metric and scalar field behave, from (\[bound1\]), as ds\^2 -Ł\^[-1]{}(|[R]{}\_1-|[R]{})\^[-2]{}(d|[T]{}\^2-d|[R]{}\^2-e\^[|[T]{}\_1 -|[T]{}]{}d\^2), = \_1 + |[T]{}/2 ($\bar{R}-\bar{R}\simeq \bar{x}/2x_1$). This metric is asymptotically AdS, as may be shown by making the further coordinate transformation, |[R]{}-|[R]{}\_1 = -2/XT, |[T]{}-|[T]{}\_1 = 2(T/2), leading to ds\^2 -Ł\^[-1]{}(X\^2 dT\^2 - - X\^2d\^2 ), = \_1 + (T/2). The next-to-leading terms in the metric containing logarithms, this asymptotic behavior differs from that of BTZ black holes.
It follows from this discussion that the Penrose diagram of the $\L < 0$ threshold solution in the sector $v > 0$, $u < 0$ is a triangle bounded by the null line $v = 0$, the null causal singularity $u = 0$, and the spacelike AdS boundary $X \to \infty$. The null singularity $u = 0$ remains naked, i.e. is not hidden behind a trapping horizon, which would correspond to \_v r = -(-u)\^[3/2]{}’(x) = 0, because $\rho' <0$ (as discussed above) implies that the only solution of this equation is $u = 0$.
For the sake of completeness, let us also discuss the behavior of the solution of the system (\[einx1\])-(\[phix\]) in the sector $x > 0$. In this case, one can write down integro-differential equations similar to (\[sum1\])-(\[sum3\]), from which one again derives that $\rho' < 0$, $\psi' < 0$ and $\s'< 0$. It follows that the metric function $e^{2\s}$ decreases as $x$ increases, eventually vanishing for a finite value $x = x_0$, corresponding to a spacelike curvature singularity (this has been confirmed numerically). The behavior of the solution near this singularity is found to be (x\_0-x), (x\_0-x), (x\_0-x) (= 3 - 1), and the coordinate transformation $u = e^U, v = e^V (x = e^{2T})$ leads to the form of the metric near the singularity ds\^2 (T\_0-T)\^[\^2]{}(dT\^2-dR\^2) +e\^[R\_0-R]{}(T\_0-T)\^2d\^2.
Perturbations
=============
To check whether the quasi-CSS solution (\[anext\]) of the full $\L \neq 0$ problem determined in the preceding section is indeed a threshold solution, we now study linear perturbations of this solution. Our treatment will follow the analysis of perturbations of critical solutions in the case of scalar field collapse in 3+1 dimensions [@fro; @hay].
The relevant time parameter in critical collapse being the retarded time $U = -(1/2)\ln(-u)$ (the “scaling variable” of [@fro]), we expand these perturbations in modes proportional to $e^{kU} = (-u)^{-k/2}$, with $k$ a complex constant. We recall that only the modes with $Re\ k>0$ grow as $U \to +\infty$ ($u \to
-0$) and lead to black hole formation, whereas those with $Re\ k<0$ decay and are irrelevant. The other relevant variable is the “spatial” coordinate $x =
uv$, and the perturbations are decomposed as r & = & (-u)\^[1/2]{}((x) + (-u)\^[-k/2]{}(x)),\
& = & -12|u| + (x) + (-u)\^[-k/2]{}(x),\
& = & (x) + (-u)\^[-k/2]{}(x). Then, the Einstein equations (\[ein1\])-(\[phi\]) are linearized in $\tilde{r}$, $\tilde{\phi}$, $\tilde{\sigma}$, using \_[,u]{} =-(-u)\^[[-k/2]{}-1]{}(x’-2), \_[,v]{} =-(-u)\^[[-k/2]{}+1]{}’ . The resulting equations are homogeneous in $u$, which drops out, and the linearized system reduces to & & x” + ([-k/2]{}+3/2)’ = e\^[2]{}( + 2),\
& & 2x” +([-k]{}+2)’ = Łe\^[2]{} - (2x’-1/2)’ + (k/2)’,\
& & -(-k+1)x’ + ((-k+1)x’-(k\^2-1)/4) + x’ - k(x’ + /2) =\
& & -(x’ - k(1/2-x’)) + (1/4-x’),\
& & 2(’’ + ’’) - ” = ’(2’ + ’),\
& & 2x” + (2x’ + (-k+5/2))’ - (k/2)’ + (2x’-1/2)’\
& & + (2x” + ([-k/2]{}+5/2)’) = 0.
What is the number of the independent constants for this system? The perturbed Klein-Gordon equation (\[pphi\]) is clearly redundant, while Eqs. (\[pein3\]) and (\[pein4\]) are constraints. So, as in the (3+1)-dimensional case [@fro; @hay], the order of the system is four, and the general solution depends on four integration constants. However, one of these four independent solutions corresponds to a gauge mode and is irrelevant. The parametrisation (\[anext\]) is invariant under infinitesimal coordinate transformations $v \to v + f(v)$. For $f(v) =
-\alpha v^{1+k/2}$, these lead to $x \to x - \alpha (-u)^{-k/2}(-x)^{1+k/2}$, giving rise to the gauge mode \_k(x) & = & (-x)\^[1+k/2]{}’(x),\
\_k(x) & = & (-x)\^[1+k/2]{}’(x),\
\_k(x) & = & , which solves identically the system (\[pein1\])-(\[pphi\]). So, up to gauge transformations, the general solution of this system depends only on three independent constants.
These will be determined, together with the possible values of $k$ (the eigenfrequencies) by enforcing appropriate and reasonable boundary conditions. We shall use here the “weak boundary conditions” of [@hay] on the boundaries $u = 0$ and $x = x_1$ ($X \to \infty$) \_[u 0]{}r\^[-1]{} 0, \_[x x\_1]{}r 0, together with the condition (0) = 0, which guarantees that the singularity of the perturbed solution starts smoothly from that of the unperturbed one. On the third boundary $v = 0$, we shall impose a stronger condition by requiring that the perturbations are analytic in $v$, in order for the perturbed solution to be extendible beyond $v=0$ to negative values of $v$ at finite $u$.
First, we consider the region $x\to 0$ where, according to Eqs. (\[zwx\]), (\[cmm\]), (\[zzw\]) and (\[nxn\]), \[wxc\] 1 + x, e\^[2]{}1+x, x. Let us assume a power-law behavior r(x) \~a (-x)\^[p]{} where $p$ is a constant to be determined. Then Eqs. (\[pein1\]), (\[pein2\]) and (\[pein4\]) can be approximated near $x = 0$ as & & x” + ([-k/2]{}+3/2)’ Ł,\
& & x” + ([-k/2]{}+1)’ 14’\
& & 2’’ - ” 2’’. Eliminating the functions $\tilde{\s}$ and $\tilde{\phi}$ between these three equations and using Eq. (\[wxc\]), we obtain the fourth-order equation 4x\^2”” + (-4k + 13)x”’ + (k/2-1)(2k - 5)” 0, which implies the power-law behavior (\[pow\]) with the exponent $p$ constrained by p(p-1)(p-k/2-3/4)(p-k/2-1)=0. Obviously the root $p = k/2 + 1$ corresponds to the gauge mode (\[gauge\]) and must be discarded as irrelevant. As a consequence the general solution near $x = 0$ can be given in terms of three independent constants as & &r(x) \~A +B(-x) +ŁC(-x)\^[3/4 +k/2]{},\
& &(x) \~- 2 + Ł\^[-1]{}2 - (k + )(-x)\^[-1/4 +k/2]{},\
& & (x) \~2 - Ł\^[-1]{}2 + (k + )(-x)\^[-1/4 +k/2]{}.\[pti\] Let us note that this solution remains valid in the limit $\L \to 0$, leading to the limiting solution $\tilde{r} \sim A +
B(-x)$ (with $B = 0$ for $k \neq 3$), which could also be obtained directly by solving the equation $\tilde{r}'' = 0$ which results from (\[pein4\]) in the limit $\L
\to 0$, together with the stronger condition (from Eq. (\[pein1\])) $(k-3)\tilde{r}' = 0$.
Now we enforce the boundary conditions at $x = 0$. For $k > 0$, $\tilde{r}$ is dominated by its first constant term in (\[rti\]), so that the condition (\[bound0\]) can only be satisfied for $u \to 0$ if A = 0. Then, for $k
> 1/2$, $\tilde{r}$ is dominated by its second term $-Bx$, leading to a perturbation $(-u)^{1/2-k/2}\tilde{r}(x)$ which blows up as $u \to 0$ and violates (\[bound\]) unless \[rang\] k 3. Then we impose the condition of analyticity in $v$ at fixed $u$. This is satisfied if \[unn\] k = 2n - 3/2, where $n$ is a positive integer. Combining eqs. (\[rang\]) and (\[unn\]) we find that $k$ has only two positive eigenvalues k = 1/2, k = 5/2. However, in the above analysis we have disregarded the fact that $k = 1/2$ is a double root of the secular equation (\[sec\]). For $k = 1/2$ the correct behavior of the general solution near $x = 0$ is & &r(x) \~A +B(-x) +ŁC(-x)|x|,\
& &(x) \~- 2 - Ł\^[-1]{} 4 - 4 - 4 |x|\
& & (x) \~4 + Ł\^[-1]{} 4 + 4 + 4 |x| , which satisfies the condition of analyticity only if $C = 0$.
At the AdS boundary ($x\to x_1$) the leading behaviour of the background is, from Eqs. (\[bound1\]), \[cdfg\] , e\^[2]{}(), \_1. We again assume a power-law behavior \~b|[x]{}\^q ($\bar{x} = x-x_1$). Then Eq. (\[pein2\]), where $\tilde{\phi}$ can be neglected, gives q(q-1) = 2, i.e. $q = -1$ or $q = 2$. Then, Eq. (\[pein1\]) reduces near $\bar{x}
= 0$ to ” - 2|[x]{}\^[-2]{} 4b\_1|[x]{}\^[q-3]{}. If $q = -1$, the behavior of the solution is governed by the right-hand side, i.e. $\tilde{r} \propto \bar{x}^{-2}$, which violates the boundary condition (\[bound\]) for $x \to x_1$. So the behavior $\tilde{\s} \sim b\bar{x}^{-1}$ must be excluded, which fixes another integration constant $D=0$ (where $D$ is a linear combination of $B$ and $C$). Then, the generic behavior of the solution of Eq. (\[asr\]) with $q = 2$ is governed by that for the homogeneous equation, i.e. r \~. This is consistent with the boundary condition (\[bound\]), and is an admissible small perturbation if its amplitude is small enough, $E \ll \rho_1$.
For $k = 1/2$, we have seen that two of the three integration constants in (\[rtii\])-(\[ptii\]) are fixed ($A = C = 0$) by condition (\[bound0\]) and the analyticity condition, while the weak boundary condition at the AdS boundary fixes a third constant $D=0$. However this is impossible, as the perturbation amplitude must remain as a free parameter. So the mode $k = 1/2$ cannot satisfy all our boundary conditions, and we are left with a single eigenmode, k = 5/2, completely determined up to an arbitrary amplitude by the two conditions $A=D=0$.
The corresponding perturbed metric function $r$ behaves near $x = 0$ as r (-u)\^[1/2]{}\[1 + Łx - (-u)\^[-5/4]{}Bx\]. For $B < 0$, the central singularity $r = 0$ is approximately given by (-u)\^[1/4]{} -Bv. Our boundary conditions guarantee that it starts at $u=v=0$ (as for the unperturbed solution) and then becomes spacelike in the region $v>0$. This singularity is hidden behind a trapping horizon (defined by Eq. (\[trap\])) which, near $x = 0$, is null, (-u)\^[5/4]{}= (a null trapping horizon was also found in [@hay]). Let us point out the crucial role played by the cosmological constant $\Lambda$ in the formation of this trapping horizon. For $\L = 0$, $\rho(x) = 1$, while, as discussed after Eq. (\[pti\]), the perturbation $\tilde{r}$ with the boundary condition (\[bound0\]) vanishes for $\L = 0$, so that the perturbed radial function $r$ is (as in [@gar]) identical to the CSS one, and the trapping horizon does not exist. Near the AdS boundary $x \to x_1$, it follows from (\[cdfg\]) and (\[asol\]) that both the central singularity and the trapping horizon are tangent to the null line (-u)\^[5/4]{} = -E ()\^[-1/2]{}.
Thus, perturbations of the quasi-CSS solution lead to black hole formation, showing that this solution is indeed a threshold solution, and is a candidate to describe critical collapse. Near-critical collapse is characterized by a critical exponent $\gamma$, defined by the scaling relation $Q
\propto |p - p^*|^{s\gamma}$, for a quantity $Q$ with dimension $s$ depending on a parameter $p$ (with $p = p^*$ for the critical solution). Choosing for $Q$ the radius $r_{AH}$ of the apparent horizon, and identifying $p - p^*$ with the perturbation amplitude $B$, we obtain from (\[trap1\]) r\_[AH]{} ()\^[2/5]{}, leading to the value of the critical exponent $\gamma = 2/5$, in agreement with the renormalization group argument [@kha] leading to $\gamma = 1/k$.
Conclusion
==========
We have discussed in detail the causal structure of the Garfinkle CSS solutions (\[gar1\]) to the $\L= 0$ Einstein-scalar field equations. From a special solution of this class, we have derived by a limiting process a new CSS solution, which we have extended to a unique solution of the full $\L < 0$ equations, describing collapse of the scalar field onto a null central singularity. This is not a curvature singularity (all the curvature invariants remain finite), but a singularity in the causal structure similar to that of the BTZ black hole. Finally, we have analyzed linear perturbations of the $\L < 0$ solution, found a single eigenmode $k = 5/2$, checked that this mode does indeed give rise to black holes, and determined the critical exponent $\gamma = 2/5$.
For comparison, Choptuik and Pretorius [@CP] derived, by analysing the observed scaling behavior of the maximum scalar curvature, the value $1.15 < \gamma < 1.25$ for the critical exponent. This value is different from the value $\gamma\sim 0.81$ obtained in the numerical analysis of Husain and Olivier [@HO] from the scaling behavior of the apparent horizon radius. Our value $\gamma = 0.4$, while significantly smaller than these two conflicting estimates, is of the order of the theoretical value $\gamma = 1/2$ derived either from the analysis of dust-ring collapse [@ps], of black hole formation from point particle collisions [@bir], or of the $J = 0$ to $J \neq 0$ transition of the BTZ black hole [@kg].
It is worth mentioning here that, even though they were obtained for a vanishing cosmological constant and thus solve the $\L\neq 0$ equations only near the singularity, the Garfinkle CSS solutions are, for the particular value (chosen in order to better fit the numerical curves) $c = (7/8)^{1/2} \simeq 0.935$, in good agreement [@gar] with the numerical results of [@CP] at an intermediate time. The fact that this value is close to 1 suggests that the $c = 1$ CSS solution (\[gar5\]) approximately describes near-critical collapse at intermediate times. If this the case, then it would not be surprising if its late-time limit, our new CSS solution Eq. (\[crit0\]), gives a good description of exactly critical collapse near the singularity. A fuller understanding of the relationship between the numerically observed near-critical collapse and these various $\L = 0$ CSS solutions could be achieved by extending them to $\L < 0$, as done in the present work for the special solution (\[crit1\]).
[20]{}
M. Bañados, C. Teitelboim and J. Zanelli, [*Phys. Rev. Lett.*]{} [**69**]{}, 1849 (1992); M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, [*Phys. Rev.*]{} D [**48**]{}, 1506 (1993)
F. Pretorius and M.W. Choptuik, [*Phys. Rev.*]{} D [**62**]{}, 124012 (2000)
V. Husain and M. Olivier, [*Class. Quant. Grav.*]{} [**18**]{}, L1 (2001)
D. Garfinkle, [*Pys. Rev.*]{} D [**63**]{}, 044007 (2001)
L. Burko, [*Phys. Rev.*]{} D [**62**]{}, 127503 (2000)
M. Choptuik, [*Phys. Rev. Lett.*]{} [**70**]{}, 2980 (1993)
C. Gundlach, [*Living Rev. Rel.*]{} [**2**]{}, 4 (1999)
M.D. Roberts, [*Gen. Rel. Grav.*]{} [**21**]{}, 907 (1989)
P.R. Brady, [*Class. Quantum Grav.*]{} [**11**]{}, 1255 (1994)
Y. Oshiro, K. Nakamura and A. Tomimatsu, [*Progr. Theor. Phys.*]{} [**91**]{}, 1265 (1994)
A.V. Frolov, [*Phys. Rev.* ]{} D [**56**]{}, 6433 (1997)
S.A. Hayward, [*Class. Quantum Grav.*]{} [**17**]{}, 4021 (2000)
G. Clément and A. Fabbri, [*Class. Quantum Grav.*]{} [**17**]{}, 2537 (2000)
T. Koike, T. Hara and S. Adachi, [*Phys. Rev. Lett.*]{} [**74**]{}, 5170 (1995)
Y. Peleg and A. Steif, [*Phys. Rev.*]{} D [**51**]{}, 3992 (1995)
D. Birmingham and S. Sen, [*Phys. Rev. Lett.*]{} [**84**]{}, 1074 (2000) ; D. Birmingham, I. Sachs and S. Sen, “Exact results for the BTZ black hole”, hep-th/0102155
J.P. Krisch and E.N. Glass, “Critical exponents for Schwarzschild-Kerr and BTZ systems”, gr-qc/0102075
[^1]: Email: [email protected]
[^2]: Email: [email protected]
[^3]: We have taken care that in (\[anext\]) $u$ has the dimension of a length squared while $v$ is dimensionless.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We consider asymptotically anti-de Sitter spacetimes in three-dimensional topologically massive gravity with a negative cosmological constant, for all values of the mass parameter $\mu$ ($\mu\neq0$). We provide consistent boundary conditions that accommodate the recent solutions considered in the literature, which may have a slower fall-off than the one relevant for General Relativity. These conditions are such that the asymptotic symmetry is in all cases the conformal group, in the sense that they are invariant under asymptotic conformal transformations and that the corresponding Virasoro generators are finite. It is found in particular that at the chiral point $|\mu l|=1$ (where $l$ is the anti-de Sitter radius), one must allow for logarithmic terms (absent for General Relativity) in the asymptotic behavior of the metric in order to accommodate the new solutions present in topologically massive gravity, and that these logarithmic terms make *both* sets of Virasoro generators nonzero even though one of the central charges vanishes.'
author:
- 'Marc Henneaux$^{1,2}$, Cristián Martínez$^{1,3}$, Ricardo Troncoso$^{1,3}$'
title: |
Asymptotically anti-de Sitter spacetimes\
in topologically massive gravity
---
Introduction
============
Following the lead of [@LSStrominger], topologically massive gravity with a negative cosmological constant [@Deser:1982vy] has received a great deal of renewed interest in the last year. The theory is described by the action [@Footnote] $$I[e] =2\int\ \left[ e^{a}\left( d\omega_{a}+\frac{1}{2}\epsilon_{abc}%
\omega^{b}\omega^{c}\right) +\frac{1}{6}\frac{1}{\ell^{2}}\epsilon_{abc}%
e^{a}e^{b}e^{c} \right]
+\frac{1}{\mu}\int \omega^{a}\left( d\omega_{a}+\frac{1}{3}%
\epsilon_{abc}\omega^{b}\omega^{c}\right) \label{actionTMG}%$$ so that the field equations read$$G^{\mu}{}_{\sigma}-\frac{1}{l^{2}}\delta_{\sigma}^{\mu}-\frac{1}{\mu}C^{\mu}%
{}_{\sigma}=0,\label{eom}%$$ where $\mu\neq0$ is the mass parameter, $l$ is the AdS radius, and $C^{\mu}%
{}_{\sigma}:=\epsilon^{\mu\nu\rho}\nabla_{\nu}\left( R_{\rho\sigma}-\frac
{1}{4}g_{\rho\sigma}R\right) $, stands for the Cotton tensor. The case $|\mu
l|=1$ is known as the chiral point and has been advocated in [@LSStrominger] to enjoy remarkable properties.
In the absence of the topological mass term, the relevant asymptotic behavior of the metric is given by [@Brown-Henneaux] $$%
\begin{array}
[c]{lll}%
\Delta g_{rr} & = & f_{rr}r^{-4}+O(r^{-5})\;,\\[2mm]%
\Delta g_{rm} & = & f_{rm}r^{-3}+O(r^{-4})\;,\\[1mm]%
\Delta g_{mn} & = & f_{mn}+O(r^{-1})\;.
\end{array}
\label{Standard-Asympt}%$$ Here $f_{\mu\nu}=f_{\mu\nu}(t,\phi)$, and the indices have been split as $\mu=(r,m)$, where $m$ includes the time and the angle. We have also decomposed the metric as $g_{\mu\nu}=\bar{g}_{\mu\nu}+\Delta g_{\mu\nu} $, where $\Delta g_{\mu\nu}$ is the deviation from the AdS metric, $$d\bar{s}^{2}=-(1+r^{2}/l^{2})dt^{2}+(1+r^{2}/l^{2})^{-1}dr^{2}+r^{2}d\phi
^{2}\;.$$ The boundary conditions (\[Standard-Asympt\]) fulfill the following three crucial consistency requirements explicitly spelled out in [@Henneaux-Teitelboim]:
- They are invariant under the anti-de Sitter group.
- They decay sufficiently slowly to the exact anti-de Sitter metric at infinity so as to contain the asymptotically anti-de Sitter" solutions of the theory of physical interest (in this case, the BTZ black holes [@BTZ]).
- But at the same time, the fall-off is sufficiently fast so as to yield finite charges.
It was actually found in [@Brown-Henneaux] that the asymptotic conditions (\[Standard-Asympt\]) are invariant not just under $SO(2,2)$ but under the bigger infinite-dimensional conformal group in two dimensions. The Poisson brackets algebra of the corresponding charges (given by surface integrals at infinity) gives two copies of the Virasoro algebra with a central charge equal to $c=3l/(2G)$.
If one changes the theory, the asymptotic behavior of the physically interesting solutions might be different and the asymptotic conditions might therefore have to be modified in order to accommodate the new solutions of physical interest. This was investigated at length in [@Henneaux:2002wm; @HMTZ2; @HMTZ3] for anti-de Sitter gravity with scalar fields in any number of dimensions (see also [@Hertog-Maeda; @Marolf]). It was found that the standard anti-de Sitter boundary conditions indeed had to be relaxed in that case, but that the charges remained finite thanks to a delicate cancellation of divergences between the relaxed terms in the metric and contributions from the scalar fields.
The same phenomenon occurs if one modifies the action of pure Einstein gravity by the topological mass term, as in (\[actionTMG\]) above. Indeed, as observed in [@Grumiller-Johansson-Asympt], the metric could acquire then a slower decay to the anti-de Sitter metric at infinity for a class of physically interesting linearized solutions. For a generic value of $\mu l>-1
$, an exact asymptotically AdS solution describing a chiral pp-wave was found in [@DS], and further developed in [@OST], whose metric reads $$ds^{2}=l^{2}\frac{dr^{2}}{r^{2}}-r^{2}dx^{+}dx^{-}+F(x^{-})r^{1-\mu l}\left(
dx^{-}\right) ^{2}\ , \label{pp-wave-mu}%$$ where $F(x^{-})$ is an arbitrary function and $x^{\pm}=\frac{t}{l}\pm\phi$. This solution is to be compared with the AdS metric written in the same coordinates, $$d\bar{s}^{2}=\displaystyle\left( 1+\displaystyle\frac{r^{2}}{l^{2}%
}\right) ^{-1}dr^{2}-\frac{l^{2}}{4}(dx^{+2}\!+dx^{-2})-\left(
\frac{l^{2}}{2}+r^{2}\right) dx^{+}dx^{-},$$ and one sees that the $F(x^{-})r^{1-\mu l}$ term spoils the asymptotic behavior (\[Standard-Asympt\]).
The purpose of this note is to provide a consistent set of new boundary conditions that accommodate these solutions with slower decay at infinity and that are yet compatible with the full conformal symmetry, for all values of the mass parameter. It turns out that the analysis carries many features in common with the scalar case studied previously. (The asymptotic study of topologically massive gravity has been carried out recently in [@Grumiller-Johansson2] at the chiral point. While we agree with the asymptotic form of the metric and the symmetries given in that paper, we do find however that *both* set of Virasoro generators are generically nonzero, a fact that shows that the theory with these boundary conditions cannot be chiral [@StromingerAugust].)
Because the computations are rather cumbersome, and because the logic follows the scalar case situation, we shall, in this note, only report the results and discuss some of their properties. The full details will be provided elsewhere [@HMTfuture].
Range $0<|\mu l|<1$ of the mass parameter
=========================================
We first consider the most intricate case, which occurs when the mass parameter $\mu$ fulfills the condition $0<|\mu l|<1$.
Asymptotic conditions
---------------------
We have found that there are two consistent sets of boundary conditions fulfilling the three consistency requirements repeated in the introduction. The existing solutions given in the literature fulfill one or the other set of boundary conditions. We shall first give the boundary conditions and we shall then explain how one verifies that they are indeed consistent.
*Negative chirality.* The boundary conditions are in that case $$%
\begin{array}
[c]{lll}%
\Delta g_{rr} & = & f_{rr}r^{-4}+\cdot\cdot\cdot\\
\Delta g_{r+} & = & f_{r+}r^{-3}+\cdot\cdot\cdot\\
\Delta g_{r-} & = & h_{r-}\ r^{-2-\mu l}+f_{r-}r^{-3}+\cdot\cdot\cdot\\
\Delta g_{++} & = & f_{++}+\cdot\cdot\cdot\\
\Delta g_{+-} & = & f_{+-}+\cdot\cdot\cdot\\
\Delta g_{--} & = & h_{--}\ r^{1-\mu l}+f_{--}+\cdot\cdot\cdot
\end{array}
\label{Asympt relaxed metric mu Neg}%$$ where $f_{\mu\nu}$ and $h_{\mu\nu}$ depend only on $x^{+}$ and $x^{-}$ and not on $r$. We use the convention that the $f$-terms are the standard deviations from AdS already encountered in (\[Standard-Asympt\]), while the $h$-terms represent the relaxed terms that need to be included in order to accommodate the solutions of the topologically massive theory with slower fall-off. We see that only the negative chirality $h$-terms $h_{r-}$ and $h_{--}$ are present, hence the terminology.
*Positive chirality.* The boundary conditions are in that case $$%
\begin{array}
[c]{lll}%
\Delta g_{rr} & = & f_{rr}r^{-4}+\cdot\cdot\cdot\\
\Delta g_{r+} & = & h_{r+}\ r^{-2+\mu l}+f_{r+}r^{-3}+\cdot\cdot\cdot\\
\Delta g_{r-} & = & f_{r-}r^{-3}+\cdot\cdot\cdot\\
\Delta g_{++} & = & h_{++}\ r^{1+\mu l}+f_{++}+\cdot\cdot\cdot\\
\Delta g_{+-} & = & f_{+-}+\cdot\cdot\cdot\\
\Delta g_{--} & = & f_{--}+\cdot\cdot\cdot
\end{array}
\label{Asympt relaxed metric mu Pos}%$$ with only the positive chirality $h$-terms $h_{r+}$ and $h_{++}$.
Although the known solutions [@DS; @OST] are of a given chirality and hence completely covered by the above boundary conditions, one might try to be more general and include both chiralities simultaneously. This cannot be done, however, in a manner that is compatible with the other consistency requirements as it will be explained below.
Asymptotic symmetry
-------------------
One easily verifies that both sets of asymptotic conditions are invariant under diffeomorphisms that behave at infinity as $$\begin{aligned}
\eta^{+} & =T^{+}+\frac{l^{2}}{2r^{2}}\partial_{-}^{2}T^{-}+\cdot\cdot
\cdot\nonumber\\
\eta^{-} & =T^{-}+\frac{l^{2}}{2r^{2}}\partial_{+}^{2}T^{+}+\cdot\cdot
\cdot\label{Asympt KV}\\
\eta^{r} & =-\frac{r}{2}\left( \partial_{+}T^{+}+\partial_{-}T^{-}\right)
+\cdot\cdot\cdot\nonumber\end{aligned}$$ where $T^{\pm}=T^{\pm}(x^{\pm})$. The $\cdots$ terms are of lowest order and do not contribute to the surface integrals. Hence, the boundary conditions are invariant under the full conformal group in two dimensions, generated by $T^{+}(x^{+})$ and $T^{-}(x^{-})$.
Surface integrals
-----------------
We shall compute the conserved (Virasoro) charges within the canonical formalism, à la Regge-Teitelboim" [@Regge-Teitelboim]. The canonical analysis of topologically massive gravity has been performed in [@Deser-Xiang; @Carlip]. As noticed in [@Giacomini-Troncoso-Willison], there is a useful choice of variables allowing one to write topologically massive gravity with a cosmological constant as a Chern-Simons theory. In this case, since the action is already written in first order, the Hamiltonian formalism can be readily done once the torsion constraint is incorporated as an additional constraint [@Carlip]. This enables one to skip the standard and somewhat awkward procedure associated with higher order derivatives.
The charges that generate the diffeomorphisms (\[Asympt KV\]) take the form [@Regge-Teitelboim] $$H[\eta]=\hbox{``Bulk piece"}+Q_{+}[T^{+}]+Q_{-}[T^{-}] \,, \label{generator}%$$ where the bulk piece is a linear combinations of the constraints with coefficients involving $\eta^{+},\eta^{-}$, and $\eta^{r}$, which has been explicitly worked out in [@Carlip], and where $Q_{+}[T^{+}]$ and $Q_{-}[T^{-}]$ are surface integrals at infinity that involve only the asymptotic form of the vector field $\eta^{+},\eta^{-}$, and $\eta^{r}$. On shell, the bulk piece vanishes and $H[\eta]$ reduces to $Q_{+}[T^{+}]+Q_{-}[T^{-}]$.
Evaluating the variation of the bulk piece in (\[generator\]) under the asymptotic conditions (\[Asympt relaxed metric mu Neg\]) or (\[Asympt relaxed metric mu Pos\]), one obtains that the surface terms at infinity should obey:$$\delta Q_{\pm}[T^{\pm}]=\left( 1\pm\frac{1}{\mu l}\right) \delta Q_{\pm}%
^{0}[T^{\pm}] \,,$$ where $$\delta Q_{\pm}^{0}[T^{\pm}]:=\frac{2}{l} \int T^{\pm}\delta f_{\pm\pm}%
d\phi\ ,$$ is exactly the same expression as that valid for the standard asymptotic behavior. The Virasoro charges are then easily integrated to yield $$\label{once}Q_{\pm}[T^{\pm}]=\frac{2}{l}\left( 1\pm\frac{1}{\mu l}\right)
\int T^{\pm}f_{\pm\pm}d\phi\$$ (up to additive constants). The details will be given in [@HMTfuture]. What happens is that the diverging pieces associated with the slower fall-off $h_{--}$ or $h_{++}$ disappear in $\delta Q_{\pm}[T^{\pm}]$ so that $Q_{\pm}$ is given by (\[once\]), and hence the charges acquire no correction involving the terms associated with the relaxed behavior. One can then view $h_{--}$ (or $h_{++}$), which cannot be gauged away, as defining a kind of hair.“ This situation is analogous to the one found for a scalar field with mass $m$ in the range $m_{ \rm BF}^{2}<m^{2}<m_{\rm BF}^{2}+1/l^{2}$ (where $m_{\rm BF}$ is the Breitenlohner-Freedman bound [@BF]). There are then two possible admissible behaviors (two branches”) for the scalar field, and the analysis proceeds as here when only the branch with slower behavior is switched on [@HMTZ3][^1].
Under an asymptotic conformal transformation (\[Asympt KV\]), $f_{++}$ and $f_{--}$ are straightforwardly found to transform as $$\begin{aligned}
\delta_{\eta}f_{++} & =2f_{++}\partial_{+}T^{+}+T^{-}\partial_{-}%
f_{++}+T^{+}\partial_{+}f_{++}
-l^{2}\left( \partial_{+}T^{+}+\partial_{+}^{3}T^{+}\right)
\!/2\ ,\label{deltaf++}\\
\delta_{\eta}f_{--} & =2f_{--}\partial_{-}T^{-}+T^{-}\partial_{-}%
f_{--}+T^{+}\partial_{+}f_{--}
-l^{2}\left( \partial_{-}T^{-}+\partial_{-}^{3}T^{-}\right) \!/2\ .
\label{deltaf--}%\end{aligned}$$ On shell, one verifies that $$\partial_{+}f_{--}=0=\partial_{-}f_{++}%$$ and so (\[deltaf++\]) and (\[deltaf–\]) reduce to $$\begin{aligned}
\delta_{\eta}f_{++} & =2f_{++}\partial_{+}T^{+}+T^{+}\partial_{+}%
f_{++}-\frac{l^{2}}{2}\left( \partial_{+}T^{+}+\partial_{+}^{3}T^{+}\right)
,\label{delta2h++}\\
\delta_{\eta}f_{--} & =2f_{--}\partial_{-}T^{-}+T^{-}\partial_{-}%
f_{--}-\frac{l^{2}}{2}\left( \partial_{-}T^{-}+\partial_{-}^{3}T^{-}\right)
. \label{delta2h--}%\end{aligned}$$ As $\beta_{+}\,\delta_{\eta}\int f_{++}Y^{+}d\phi\sim\lbrack Q_{+}%
(Y^{+}),Q_{+}(T^{+})+Q_{-}(T^{-})]$ (with $\beta_{\pm}=2l^{-1}\left( 1\pm(\mu
l)^{-1}\right) $) and $\beta_{-}\,\delta_{\eta}\int f_{--}Y^{-}d\phi
\sim\lbrack Q_{+}(Y^{+}),Q_{+}(T^{+})+Q_{-}(T^{-})]$ (with $Y^{+}$ and $Y^{-}$ the asymptotic conformal transformation associated with a second spacetime diffeomorphism $\xi^{+}$, $\xi^{-}$ and $\xi^{r}$), one can easily infer from (\[delta2h++\]) and (\[delta2h–\]) that $Q_{+}(Y^{+})$ and $Q_{-}(T^{-})$ commute with each other and each fulfills the Virasoro algebra with central charges $$c_{\pm}=\left( 1\pm\frac{1}{\mu l}\right) \,c$$ (see [@Brown-Henneaux2] for general theorems).
Range $|\mu l|>1$ of the mass parameter
=======================================
Take for definiteness $\mu l$ positive and hence $>1$. Solving the equations starting from infinity shows that again, one should expect both chiralities to be present, taking exactly the same form as (\[Asympt relaxed metric mu Neg\]) and (\[Asympt relaxed metric mu Pos\]) above. However, the positive chirality blows up at infinity ($\Delta g_{++}$ dominates the background) and the space is not asymptotically of constant curvature. So, if $h_{++}\not =0$, the space is not asymptotically anti-de Sitter. For this reason, one must set $h_{++}=0$. But the other $h_{--}$-term is subdominant with respect to $f_{--}$, so that the asymptotic negative chirality behavior reproduces (\[Standard-Asympt\]). The same analysis holds when $\mu l$ is negative (with an interchange of the roles of the two chiralities). Therefore, the behavior of the metric can be taken to be (\[Standard-Asympt\]). The asymptotic derivation of the charges and the central charges proceeds then straightforwardly (no divergence to be canceled) and yields $$Q_{\pm}[T^{\pm}]=\frac{2}{l}\left( 1\pm\frac{1}{\mu l}\right) \int T^{\pm
}f_{\pm\pm}d\phi\$$ with central charges $$c_{\pm}=\left( 1\pm\frac{1}{\mu l}\right) \,c\ .$$
The chiral point
================
Asymptotic behavior
-------------------
Hereafter we only consider $\mu l=1$, since the case of $\mu l=-1$ just corresponds to the interchange $x^{+}\longleftrightarrow x^{-}$.
In the case of $\mu l=1$, the appropriate asymptotic behavior for $\Delta
g_{\mu\nu}$ reads $$%
\begin{array}
[c]{lll}%
\Delta g_{rr} & = & f_{rr}r^{-4}+\cdot\cdot\cdot\\
\Delta g_{r+} & = & f_{r+}r^{-3}+\cdot\cdot\cdot\\
\Delta g_{r-} & = & h_{r-}\ r^{-3}\ln\left( r\right) +f_{r-}r^{-3}%
+\cdot\cdot\cdot\\
\Delta g_{++} & = & f_{++}+\cdot\cdot\cdot\\
\Delta g_{+-} & = & f_{+-}+\cdot\cdot\cdot\\
\Delta g_{--} & = & h_{--}\;\ln\left( r\right) +f_{--}+\cdot\cdot\cdot
\end{array}
\label{Asympt relaxed metric}%$$ where $f_{\mu\nu}$ and $h_{--}$ depend only on $x^{\pm}=\frac{t}{l}\pm\phi$. This behavior accommodates the known solutions with constant curvature at infinity [@DS; @OST; @Gaston], whose metric is given by $$ds^{2}=l^{2}\frac{dr^{2}}{r^{2}}-r^{2}dx^{+}dx^{-}+F(x^{-})\log(r)\left(
dx^{-}\right) ^{2}\ . \label{pp-wave chiral}%$$ with $F(x^{-})$ being an arbitrary function.
Asymptotic symmetry
-------------------
Just as for $\mu l\not =1$, the asymptotic conditions are invariant under diffeomorphisms that behave at infinity as in Eq. (\[Asympt KV\]), where the $\cdots$ terms are again of lowest order and do not contribute to the surface integrals. Hence, the boundary conditions are invariant under the conformal group in two dimensions, generated by $T^{+}(x^{+})$ and $T^{-}(x^{-})$.
Under the action of the Virasoro symmetry, one obtains$$\delta_{\eta}h_{--}=2h_{--}\partial_{-}T^{-}+T^{-}\partial_{-}h_{--}%
+T^{+}\partial_{+}h_{--}\ \label{deltah--chiral}%$$ and $$\delta_{\eta}f_{++} =2f_{++}\partial_{+}T^{+}+T^{-}\partial_{-}%
f_{++}+T^{+}\partial_{+}f_{++}
-l^{2}\left( \partial_{+}T^{+}+\partial_{+}^{3}T^{+}\right) \! /2 .
\label{deltaf--chiral}%$$
The field equations are easily verified to imply that$$\partial_{-}f_{++}=0\text{, and }\partial_{+}h_{--}=0\text{.}%$$ Note that this time the equations do not impose $\partial_{+}f_{--}=0$ and furthermore, the transformation rule of $f_{--}$ also differs from the one found off the chiral point.
Conserved **charges for** $\mu l=1$
-----------------------------------
Evaluating the variation of the surface charges using the expressions of [@Carlip] with the asymptotic conditions (\[Asympt relaxed metric\]) one obtains:$$\delta Q_{+}=\frac{4}{l}\int T^{+}\delta f_{++}d\phi\ ,\text{ and }\delta
Q_{-}=\frac{2}{l}\int T^{-}\delta h_{--}d\phi\ .$$ This implies (up to additive constants) $$Q_{+}=\frac{4}{l}\int T^{+}f_{++}d\phi\ ,\text{ and }Q_{-}=\frac{2}{l}\int
T^{-}h_{--}d\phi\ .$$ The crucial new feature found here, apparently overlooked in the previous literature, is that $Q_{-}[T^{-}]$ does not vanish identically. Rather, the relaxation term $h_{--}$ does contribute to it. This behavior is somehow similar to what occurs for scalar fields that saturates the BF bound. One may verify explicitly that on definite solutions, $Q_{-}[T^{-}]$ is not zero [@HMTfuture]. Indeed, for the metric (\[pp-wave chiral\]), $h_{--}=F(x^{-})$ which in general does not vanish.
From the variations (\[deltah–chiral\]) and (\[deltaf–chiral\]) of $h_{--}$ and $f_{++}$ and the asymptotic field equations, one finds that both $Q_{+}[T^{+}]$ and $Q_{-}[T^{-}]$ fulfill the Virasoro algebra with the central charge $$c_{+}=2\,c\,,\;\;\;\;c_{-}=0\,.$$ Even though $Q_{-}[T^{-}]$ does not vanish, the central charge $c_{-}$ is zero because the inhomogeneous terms $-l^{2}\left( \partial_{-}T^{-}+\partial
_{-}^{3}T^{-}\right) /2$ are absent from $\delta_{\eta}h_{--}$.
In this paper we have exhibited the boundary conditions appropriate to accommodate the solutions of topologically massive gravity found in the literature with a slower decay at infinity than the one for pure standard gravity discussed in [@Brown-Henneaux]. These boundary conditions fulfill the consistency conditions listed in the introduction. The analysis proceeds very much as in the case of anti-de Sitter gravity coupled to a scalar field [@Henneaux:2002wm; @HMTZ2; @HMTZ3] and the results turn out to be comparable.
A question not addressed here is the exact physical relevance of the new solutions which the more liberal boundary conditions enable one to consider. One might question whether they should be included [@StromingerAugust] and it is not clear what one loses if one does not include them, i.e., if one sticks to the more restrictive boundary conditions of [@Brown-Henneaux]. Note that for the scalar field, the softening of the boundary conditions leads to interesting developments. In particular, this enlarges the space of admissible solutions to include hairy black holes [@Henneaux:2002wm; @Hertog-Maeda; @MTZ], solitons and instantons [@GMT].
We have also shown that with the new boundary conditions, the Virasoro generators with both chiralities are actually nonzero at the chiral point (while one chiral set of them does vanish under the boundary conditions of [@Brown-Henneaux]). The corresponding central charge vanishes, however. This puzzling fact should be understood from the point of view of conformal field theory.
A longer version of this paper, with detailed proofs and more information on the charges of various solutions is in preparation [@HMTfuture].
*Note added*: After this paper was posted on the arXiv, we received comments by various colleagues (i) confirming the intriguing result established above that at the chiral point, the left-moving generators have a zero central charge even though they are not identically zero; and (ii) investigating the interpretation of this result in terms of a dual logarithmic CFT. We thank A. Strominger and D. Grumiller for kindly providing us this information prior to publication
*Acknowledgments.* We thank G. Compère, G. Giribet, D. Grumiller, N. Johansson, A. Schwimmer and A. Strominger for useful discussions and enlightening comments. This research is partially funded by FONDECYT grants 1061291, 1071125, 1085322, 7080044, 1095098. The work of MH is partially supported by IISN - Belgium (conventions 4.4511.06 and 4.4514.08) and by the Belgian Federal Science Policy Office through the Interuniversity Attraction Pole P6/11. C. M. and R. T. wish to thank the kind hospitality at the Physique théorique et mathématique at the Université Libre de Bruxelles and the International Solvay Institutes. The Centro de Estudios Científicos (CECS) is funded by the Chilean Government through the Millennium Science Initiative and the Centers of Excellence Base Financing Program of Conicyt. CECS is also supported by a group of private companies which at present includes Antofagasta Minerals, Arauco, Empresas CMPC, Indura, Naviera Ultragas and Telefónica del Sur. CIN is funded by Conicyt and the Gobierno Regional de Los Ríos.
[99]{}
W. Li, W. Song and A. Strominger, JHEP **0804**, 082 (2008).
S. Deser, R. Jackiw and S. Templeton, Phys. Rev. Lett. **48** (1982) 975; S. Deser, R. Jackiw and S. Templeton, Annals Phys. **140**, 372 (1982); \[Erratum-ibid. **185**, 406 (1988)\]; **281**, 409 (2000); S. Deser, Cosmological Topological Supergravity in Quantum Theory of Gravity: Essays in honor of the 60th birthday of Bryce S. DeWitt. Edited by S.M. Christensen. Published by Adam Hilger Ltd., Bristol, England 1984, p.374.
Here we are using the conventions of [@Carlip]: $16\pi
G=1$, a metric of signature $(-++)$, and a cosmological constant $\Lambda=-1/l^{2}$, and set $\epsilon_{012}=1$ (so $\epsilon^{012}=-1$).
S. Carlip, JHEP **0810**, 078 (2008).
J. D. Brown and M. Henneaux, Commun. Math. Phys. **104**, 207 (1986).
M. Henneaux and C. Teitelboim, Commun. Math. Phys. **98**, 391 (1985).
M. Bañados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. **69**, 1849 (1992); M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, Phys. Rev. D **48**, 1506 (1993).
M. Henneaux, C. Martínez, R. Troncoso and J. Zanelli, Phys. Rev. D **65**, 104007 (2002).
M. Henneaux, C. Martínez, R. Troncoso and J. Zanelli, Phys. Rev. D **70**, 044034 (2004).
M. Henneaux, C. Martínez, R. Troncoso and J. Zanelli, Annals Phys. **322**, 824 (2007).
T. Hertog and K. Maeda, JHEP **0407**, 051 (2004).
A. J. Amsel and D. Marolf, Phys. Rev. D **74**, 064006 (2006) \[Erratum-ibid. D **75**, 029901 (2007)\].
D. Grumiller and N. Johansson, JHEP **0807**, 134 (2008); S. Carlip, S. Deser, A. Waldron and D. K. Wise, Class. Quant. Grav. [**26**]{}, 075008 (2009); G. Giribet, M. Kleban and M. Porrati, JHEP **0810**, 045 (2008).
T. Dereli and O. Sarioglu, Phys. Rev. D **64**, 027501 (2001); E. Ayón-Beato and M. Hassaïne, Annals Phys. **317**, 175 (2005).
S. Olmez, O. Sarioglu and B. Tekin, Class. Quant. Grav. **22**, 4355 (2005); E. Ayón-Beato and M. Hassaïne, Phys. Rev. D **73**, 104001 (2006); S. Carlip, S. Deser, A. Waldron and D. K. Wise, Phys. Lett. B **666**, 272 (2008); G. W. Gibbons, C. N. Pope and E. Sezgin, Class. Quant. Grav. **25**, 205005 (2008).
D. Grumiller and N. Johansson, Int. J. Mod. Phys. D [**17**]{}, 2367 (2008).
A. Strominger, “A Simple Proof of the Chiral Gravity Conjecture,” arXiv:0808.0506 \[hep-th\].
M. Henneaux, C. Martínez and R. Troncoso, in preparation.
T. Regge and C. Teitelboim, Annals Phys. **88**, 286 (1974).
S. Deser and X. Xiang, Phys. Lett. B **263**, 39 (1991); I. L. Buchbinder, S. L. Lyahovich and V. A. Krychtin, Class. Quant. Grav. **10**, 2083 (1993); K. Hotta, Y. Hyakutake, T. Kubota and H. Tanida, JHEP **0807**, 066 (2008); M. i. Park, JHEP **0809**, 084 (2008); D. Grumiller, R. Jackiw and N. Johansson, Canonical analysis of cosmological topologically massive gravity at the chiral point, arXiv:0806.4185 \[hep-th\]; M. Blagojevic and B. Cvetkovic, Canonical structure of topologically massive gravity with a cosmological constant, arXiv:0812.4742 \[gr-qc\].
A. Giacomini, R. Troncoso and S. Willison, Class. Quant. Grav. **24**, 2845 (2007).
P. Breitenlohner and D. Z. Freedman Phys. Lett. B **115**, 197 (1982); Annals Phys. **144**, 249 (1982).
J. D. Brown and M. Henneaux, J. Math. Phys. **27**, 489 (1986).
G. Clement, Class. Quant. Grav. [**11**]{}, L115 (1994); A. Garbarz, G. Giribet and Y. Vasquez, Phys. Rev. D [**79**]{}, 044036 (2009).
C. Martínez, R. Troncoso and J. Zanelli, Phys. Rev. D **70**, 084035 (2004); C. Martínez, J. P. Staforelli and R. Troncoso, Phys. Rev. D **74**, 044028 (2006); C. Martínez and R. Troncoso, Phys. Rev. D **74**, 064007 (2006).
J. Gegenberg, C. Martínez and R. Troncoso, Phys. Rev. D **67**, 084007 (2003); E. Radu and E. Winstanley, Phys. Rev. D **72**, 024017 (2005); T. Hertog and G. T. Horowitz, JHEP **0407**, 073 (2004); T. Hertog and G. T. Horowitz, Phys. Rev. Lett. **94**, 221301 (2005).
[^1]: In the scalar field case, one can switch on simultaneously the two branches in a manner compatible with AdS asymptotics [@Henneaux:2002wm; @HMTZ2; @HMTZ3; @Hertog-Maeda]. If one tries to do this here, i.e., allows both positive and negative chiralities simultaneously, one finds that the charges are integrable only if there is a relationship $h_{++}=h_{++}(h_{--})$ between the two chiralities. This is analogous to what happens for the scalar field. Contrary to the scalar field case, however, no such relationship $h_{++}=h_{++}(h_{--})$ is preserved by both the right and left copies of the Virasoro algebra. Depending on how one chooses the relationship $h_{++}=h_{++}(h_{--})$, the asymptotic symmetry is reduced to one copy of the Virasoro algebra (right or left) times $L_{0}$ (left or right). This is the difficulty mentioned above in trying to include simultaneously both chiralities. The details will be given in [@HMTfuture].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Macedo and Guedes showed recently how to solve a system of coupled harmonic oscillators with time dependent parameters \[[ J. Math. Phys.]{} [**53**]{}, 052101 (2012)\]. We show here that the way in which they get rid of the time dependent masses is incorrect and some terms lack in the transformed Hamiltonian. We also show a correct way of eliminating from the Hamiltonian the time dependent masses.'
author:
- 'H.M. Moya-Cessa'
- 'J. Récamier'
title: 'Time-dependent coupled harmonic oscillators: Comment'
---
Macedo and Guedes [@Macedo2012] have studied the interaction of two coupled time dependent oscillators [@Macedo2012] and have used a time dependent transformation that takes the two different time dependent masses of the single oscillators to an effective single time dependent mass. We show here that what they do is wrong and show a correct form of eliminating the masses.
First, consider the Hamiltonian for a single time dependent harmonic oscillator [@Guasti2003; @JPA; @Ramos2018] $${H}=\frac{ {P}^2}{2M(t)}+\frac{M(t) {X}^2}{2},$$ with constant frequency equal to one. Following Macedo and Guedes [@Macedo2012; @Macedo2014] we could do $x=M(t) X$ and $p=\frac{1}{M(t)}P$, such that we would obtain a Hamiltonian that looks time independent $${H}=\frac{ {p}^2}{2}+\frac{ {x}^2}{2},$$ but is time dependent since the new position and momentum operators are explicitly time dependent. In fact, eliminating the time dependent mass is not a an simple task, as we show here for the case of two coupled time dependent oscillators Hamiltonian.
Given the Hamiltonian [@Macedo2012; @Macedo2014] $${H}(t)=\frac{ {p}_{1x}^2}{2m_1(t)}+\frac{ {p}_{2x}^2}{2m_2(t)}+\frac{m_1(t)\omega_1^2(t) {x}_1^2}{2}+\frac{m_2(t)\omega_2^2(t) {x}_2^2}{2}+\frac{k(t)( {x}_2- {x}_1)^2}{2},$$ and the equation $$i\frac{\partial |\psi(t)\rangle}{\partial t}= {H}(t)|\psi(t)\rangle$$ by doing the squeezing [@Yuen; @Caves; @Vidiella] unitary transformation $T_u|\psi(t)\rangle=|\phi(t)\rangle$, with $${T}_{u}=e^{i\frac{ u_1(t)}{2 }( {x}_1
{p}_{1x}+ {p}_{1x} {x}_1)}e^{i\frac{ u_2(t)}{2 }( {x}_2 {p}_{2x}+ {p}_{2x} {x}_2)}$$ that produces $${T}_{u} {x}_1 {T}^{\dagger}_{u}= {x}_1e^{u_1(t)}, \qquad {T}_{u} {x}_2 {T}^{\dagger}_{u}= {x}_2e^{u_2(t)}$$ $${T}_{u} {p}_{1x} {T}^{\dagger}_{u}= {p}_{1x}e^{-u_1(t)}, \qquad {T}_{u} {p}_{2x} {T}^{\dagger}_{u}= {p}_{2x}e^{-u_2(t)},$$ we obtain $$i\frac{\partial {T}^{\dagger}_{u}}{\partial t}|\phi(t)\rangle+i {T}^{\dagger}_{u}\frac{\partial |\phi(t)\rangle}{\partial t}= {H}(t) {T}^{\dagger}_{u}|\phi(t)\rangle.$$ with $$\frac{\partial {T}^{\dagger}_{u}}{\partial t}=-i {T}^{\dagger}_{u}[\frac{\dot{u}_1}{2}( {x}_1
{p}_{1x}+ {p}_{1x} {x}_1)+\frac{\dot{u}_2}{2}( {x}_2
{p}_{2x}+ {p}_{2x} {x}_2)],$$ that, by substituting in the equation above and multiplying by $ {T}_u$ by the left gives $$\begin{aligned}
i\frac{\partial |\phi(t)\rangle}{\partial t}+[\frac{\dot{u}_1}{2}( {x}_1
{p}_{1x}+ {p}_{1x} {x}_1)+\frac{\dot{u}_2}{2}( {x}_2
{p}_{2x}+ {p}_{2x} {x}_2)]|\phi(t)\rangle= {T}_u {H}(t) {T}^{\dagger}_{u}|\phi(t)\rangle.\end{aligned}$$ or $$\begin{aligned}
i\frac{\partial |\phi(t)\rangle}{\partial t}= {\tilde{H}}(t)|\phi(t)\rangle.\label{phi}\end{aligned}$$ with $$\begin{aligned}
{\tilde{H}}(t)&=&-\frac{\dot{u}_1}{2}( {x}_1
{p}_{1x}+ {p}_{1x} {x}_1)-\frac{\dot{u}_2}{2}( {x}_2
{p}_{2x}+ {p}_{2x} {x}_2)\\ \nonumber&+&
\frac{ {p}_{1x}^2e^{-2u_1(t)}}{2m_1(t)}+\frac{ {p}_{2x}^2e^{-2u_2(t)}}{2m_2(t)}+\frac{m_1(t)\omega_1^2(t) {x}_1^2e^{2u_1(t)}}{2}+\frac{m_2(t)\omega_2^2(t) {x}_2^2e^{2u_2(t)}}{2}\\ \nonumber &+&\frac{k(t)( {x}_2e^{u_2(t)}- {x}_1e^{u_1(t)})^2}{2}.\end{aligned}$$ With the choice $u_j=\ln \sqrt{\frac{m(t)}{m_j(t)}}$ we obtain the first part of the Hamiltonian (6) in Macedo and Guedes. However, note that the first term on the r.h.s. of equation (10) is lacking in their equation (6). The parameter $m(t)$ is an arbitrary function of time that we could set as $m(t)=\sqrt{m_1(t)m_2(t)}$ to obtain a result more similar to the one obtained by Macedo and Guedes. Here we prefer to set it to one, i.e., $u_j=-\frac{1}{2}\ln {m_j}$, then we get $$\begin{aligned}
{\tilde{H}}(t)&=&-\frac{\dot{u}_1}{2}( {x}_1
{p}_{1x}+ {p}_{1x} {x}_1)-\frac{\dot{u}_2}{2}( {x}_2
{p}_{2x}+ {p}_{2x} {x}_2)\\ \nonumber&+&
\frac{ {p}_{1x}^2}{2}+\frac{ {p}_{2x}^2}{2}+\frac{\omega_1^2(t) {x}_1^2}{2}+\frac{\omega_2^2(t) {x}_2^2}{2}+\frac{k(t)( {x}_2e^{u_2(t)}- {x}_1e^{u_1(t)})^2}{2},\end{aligned}$$ that may be rewritten to give $$\begin{aligned}
{\tilde{H}}(t)=
\frac{ ({p}_{1x}-\frac{\dot{u}_1}{2}x_1)^2}{2}+\frac{ ({p}_{2x}-\frac{\dot{u}_2}{2}x_2)^2}{2}+\frac{(\omega_1^2-\frac{\dot{u}_1^2}{4}) {x}_1^2}{2}+\frac{(\omega_2^2-\frac{\dot{u}_2^2}{4}) {x}_2^2}{2}+\frac{k(t)( {x}_2e^{u_2}- {x}_1e^{u_1})^2}{2}.\end{aligned}$$ By doing the transformation $R|\phi(t)\rangle=|\xi(t)\rangle$, with $R=\exp\{-i(\frac{\dot{u}_1}{4}x_1^2+\frac{\dot{u}_2}{4}x_2^2)\}$ and, inserting it into equation (\[phi\]), we obtain $$\begin{aligned}
i\frac{\partial (R^{\dagger} |\xi(t)\rangle)}{\partial t}= {\tilde{H}}(t)R^{\dagger}|\xi(t)\rangle,\label{xi}\end{aligned}$$ that is rewritten as $$\begin{aligned}
i\frac{\partial |\xi(t)\rangle}{\partial t}=\left[\frac{\ddot{u}_1}{4}x_1^2+\frac{\ddot{u}_2}{4}x_2^2+ R{\tilde{H}}(t)R^{\dagger}\right]|\xi(t)\rangle,\label{xi2}\end{aligned}$$ that finally yields $$\begin{aligned}
i\frac{\partial |\xi(t)\rangle}{\partial t}=\frac{1}{2}\left[{ {p}_{1x}^2}+{ {p}_{2x}^2}+{(\omega_1^2-\frac{\dot{u}_1^2-2\ddot{u}_1}{4}) {x}_1^2}+{(\omega_2^2-\frac{\dot{u}_2^2-2\ddot{u}_2}{4}) {x}_2^2}+{k(t)( {x}_2e^{u_2}- {x}_1e^{u_1})^2}\right]|\xi(t)\rangle.\label{xi3}\end{aligned}$$ In Ref. [@Urzua] it has been shown how this equation may be solved for time dependent arbitrary parameters.
In conclusion, we have shown how to eliminate the time dependent masses from the two coupled, time dependent, harmonic oscillators Hamiltonian.
[9]{}
Macedo D.X. and Guedes, I. [ J. Math. Phys.]{} [**53**]{}, 052101 (2012). Fernández Guasti M. and Moya-Cessa H. [ Phys. Rev. A]{} [**67**]{}, 063803 (2003). Fernández Guasti M. and Moya-Cessa H. [ J. Phys. A]{} [**36**]{}, 2069-2076 (2003). Ramos-Prieto I., Espinosa-Zuñiga A., Fernández-Guasti M. and Moya-Cessa H.M. [Mod. Phys. Lett. B]{} **32**, 1850235 (2018). Macedo D.X. and Guedes I. [ Int. J. Mod. Phys. E ]{} [**23**]{}, 1450048 (2014). Yuen H.P. [Phys. Rev. A]{} [**13** ]{}, 2226-2243 (1976). Caves C.M. [Phys. Rev. D]{} [**23**]{}, 1693-1708 (1981). Moya-Cessa H. and Vidiella-Barranco A. [ J. Mod. Optics]{} [**39**]{}, 2481-2499 (1992). Urzúa A.R., Ramos-Prieto I., Fernández-Guasti M. and Moya-Cessa H.M. Quant. Rep. [**1**]{}, 82-90 (2019).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Although recent measurements of the shower profiles of ultra-high energy cosmic rays suggest that they are largely initiated by heavy nuclei, such conclusions rely on hadronic interaction models which have large uncertainties. We investigate an alternative test of cosmic ray composition which is based on the observation of ultra-high energy photons produced through cosmic ray interactions with diffuse low energy photon backgrounds during intergalactic propagation. We show that if the ultra-high energy cosmic rays are dominated by heavy nuclei, the flux of these photons is suppressed by approximately an order of magnitude relative to the proton-dominated case. Future observations by the Pierre Auger Observatory may be able to use this observable to constrain the composition of the primaries, thus providing an important cross-check of hadronic interaction models.'
author:
- Dan Hooper
- 'Andrew M. Taylor'
- Subir Sarkar
title: 'Cosmogenic photons as a test of ultra-high energy cosmic ray composition'
---
Despite considerable experimental and theoretical effort, the chemical composition of the ultra-high energy cosmic rays (UHECRs) remains ambiguous. Recent measurements of air shower profiles by the Pierre Auger Observatory (PAO) suggest that UHECRs are increasingly dominated by heavy nuclei at energies above $10^{18.5}$ eV [@xmax; @heavy]. Uncertainties at such high energies in the hadronic interaction models used to interpret the data [@hadronic], however, can undermine this conclusion [@uncertain]. In this paper, we discuss a complementary observation that, without relying on hadronic interaction models, can be used to constrain the chemical composition of the UHECRs.
Protons with energy above $\sim$$10^{19.6}$ eV (the “GZK cutoff” energy) interact efficiently with the cosmic microwave (and infrared) background, producing charged and neutral pions [@gzk] whose decays yield potentially observable fluxes of UHE neutrinos and photons. The detection of these “cosmogenic neutrinos” [@cosmogenic] is a key target for present [@limits] and planned high energy neutrino telescopes [@interest]. The PAO has placed stringent limits on the fraction of UHECRs that are photons [@augersdphotonfractionlimit] and is expected to ultimately reach the level of sensitivity required to detect the cosmogenic photon flux [@Risse:2007sd].
If UHECRs are largely heavy or intermediate mass nuclei, however, they will interact with radiation backgrounds primarily through photo-disintegration, breaking up into lighter nuclei and nucleons. As these nucleons are often below the energy threshold for pion-production, fewer UHE neutrinos and photons are produced. This leads to significant suppression of the cosmogenic neutrino flux [@neutrinos]. We describe here how a heavy chemical composition of the UHECR spectrum also leads to suppression of the cosmogenic photon flux. Thus as the PAO’s sensitivity to UHE photons increases, this will provide a new probe of the composition of UHECRs.
Following our previous work [@previous], we simulate the intergalactic propagation of UHECRs by an analytically validated Monte Carlo method, including the effects of photo-pion and pair production as well as photo-disintegration (for related work see Ref. [@other]). We assume that the UHECR sources are homogeneously distributed and that they produce protons or nuclei with a power-law spectrum up to a maximum energy, above which the flux is exponentially suppressed: ${\rm d}N/{\rm d}E \propto E^{-\alpha} \exp(-E/E_{\rm
max,Z})$. To maintain consistency with our previous work, we express the maximum energy in terms of the quantity $E_{\rm max, Z} \equiv
E_{\rm max} \times (26/Z)$, where $Z$ is the electric charge of the cosmic ray nucleus.
If the UHECRs are all protons, a good fit to the observed cosmic ray spectrum above $10^{19}$ eV can be found for an injected spectrum with a spectral index in the range $\alpha \approx 1.6-2.4$, and $E_{\rm
max} \sim (1-5) \times 10^{21}$ eV; a similar range of spectral indices can also provide reasonable fits for heavy or intermediate mass UHECRs [@spectrum]. Henceforth we set $\alpha=2.0$, although our results depend only weakly on the precise value [@alphaprecise]. For iron, silicon, or nitrogen nuclei, we find that the observed spectrum requires $E_{\rm max} \gtrsim 10^{20}$, $10^{20.5}$, or $10^{21}$ eV, respectively. We do not consider values of $E_{\rm max}$ greater than $10^{22}$ eV since there is no plausible astrophysical source which can even contain such high energy particles [@Hillas:1985is].
Our Monte Carlo code tracks the propagation of each individual UHE nucleus, nucleon, photon, and electron down to an energy of $10^{18}$ eV. As they propagate, UHE photons produce $e^- e^+$ pairs through interactions with the cosmic radio (and microwave) background at a rate given by $$\begin{aligned}
R (E_{\gamma}) = \frac{2 m^2_e}{E^2_\gamma} \int \frac{1}{\epsilon^2}
\frac{{\rm d}n}{{\rm d}\epsilon} {\rm d}\epsilon
\int_0^{E_\gamma \epsilon/m_e} \epsilon^\prime \sigma_{\gamma\gamma}
(E_\gamma, \epsilon^\prime) {\rm d} \epsilon^\prime, \end{aligned}$$ where $E_{\gamma}$ is the energy of the propagating photon, $\epsilon$ is the energy of the background photon, ${\rm d}n/{\rm d}\epsilon$ describes the background photon distribution, and $\sigma_{\gamma\gamma} (E_\gamma, \epsilon)$ is the cross-section for pair production. At energies above $10^{18}$ eV, the interaction length of a photon is comparable to or shorter than that of UHE protons and nuclei, [*viz.*]{} $\sim 1-10$ Mpc.
In each collision, the incoming photon transfers a significant fraction of its energy to an outgoing electron or positron (a plot showing this quantity for different center-of-mass energies is shown in, [*e.g.*]{}, Ref. [@Taylor:2009iw]). For a $10^{19}$ eV ($10^{20}$ eV) photon scattering off of a $10^{-6}$ eV radio photon, for example, more than 90% (97%) of the energy is transferred to the highly boosted outgoing $e^-$/$e^+$.
UHE electrons and positrons produced in this manner can subsequently regenerate an UHE photon through inverse Compton scattering with CMB photons at a rate given by $$\begin{aligned}
R (E_e) = \frac{2 m^2_e}{E^2_e} \int \frac{1}{\epsilon^2}
\frac{{\rm d}n}{{\rm d}\epsilon} {\rm d} \epsilon
\int_0^{4E_e\epsilon/m_e} \epsilon^\prime \sigma_{e \gamma}
(E_e, \epsilon^\prime){\rm d}\epsilon^\prime.\end{aligned}$$ Each collision transfers the bulk of the initial particle energy into the photon. We follow the development of the resulting electromagnetic cascade following the technique described in Ref. [@Taylor:2008jz] (see also Ref. [@Eungwanichayapant:2009bi]).
Electrons and positrons can also lose energy through synchrotron radiation in magnetic fields. Whether typical UHE electrons lose a substantial fraction of their energy before inverse Compton scattering depends on the relative energy densities of the extragalactic magnetic field and the cosmic radio background. Competing with this effect is the fact that UHE nuclei and protons will also be deflected by magnetic fields, increasing their energy losses during propagation. Taken together, we find that the presence of nano-Gauss scale extragalactic magnetic fields increases slightly the resulting fraction of UHECRs that are photons at energies $\sim 10^{18}$ eV, and decreases the photon fraction at energies $> 10^{19}$ eV.
For the cosmic radio background we adopt the two extreme possibilities. The first is the estimate from observations given in Ref. [@Clark:1970] which may well be contaminated by foreground synchrotron emission from cosmic ray electrons in the galactic halo; hence, following Ref. [@Keshet:2004dr], we consider this to represent an upper limit. We also present results for the case in which only the radio component of the cosmic microwave background contributes, representing a lower limit. We consider two specific extragalactic magnetic field strengths, ranging from the observational upper limit of $\sim$$10^{-9}$ G to (negligibly) weak values of $3 \times 10^{-12}$ G [@Taylor:2008jz]. These field strengths bound the range of possible effects that extragalactic magnetic fields may have on the results.
We show in Fig. \[window\] the photon fraction of UHECRs at Earth in different models of the primary composition for the case of weak extragalactic magnetic fields ($<3~$pG). If the primaries are largely protons, then the UHE photon fraction at $10^{19}$ eV ranges from $\sim 10^{-4}$ for $E_{\rm max}=10^{21}$ eV to $\sim 10^{-3}$ for $E_{\rm max}=10^{22}$ eV. The bands shown in the figure represent the variation resulting from the range of radio backgrounds considered. For comparison, we show the upper limits on the photon fraction set by the PAO [@augersdphotonfractionlimit] as well as its projected reach (after 20 years of observation) [@Risse:2007sd]. We see that proton dominated UHECR will likely provide a detectable photon fraction so long as $E_{\rm max}$ is not too close to the GZK cutoff.
The situation is very different if the UHECRs are mostly heavy or intermediate mass nuclei. Generally speaking, this leads to approximately an order of magnitude suppression of the photon fraction. If, for example, the UHECR sources inject only iron nuclei (as shown in the lower frames of Fig. \[window\]), the photon fraction never exceeds $\sim 3\times 10^{-4}$, and is thus beyond the reach of the PAO. For intermediate mass nuclei at source, the photon fraction is less suppressed, but is still considerably lower than for the all-proton case. Note that all of the models considered here are consistent with the cascade limit on the GeV-TeV photon flux and with bounds on the cosmogenic neutrino flux [@evolution].
In Fig. \[window2\], we show the photon fraction of UHECRs at Earth in different models of the primary composition for the case of 0.3 nG extragalactic magnetic fields. The effect of the presence of such a strong extragalactic magnetic field is to increase the photon fraction at energies near $\sim 10^{18}$ eV, and decrease it above $10^{19}$ eV, as was previously suggested in Ref. [@Taylor:2008jz].
{width="0.325\linewidth"} {width="0.325\linewidth"} {width="0.325\linewidth"} {width="0.325\linewidth"} {width="0.325\linewidth"} {width="0.325\linewidth"}
{width="0.325\linewidth"} {width="0.325\linewidth"} {width="0.325\linewidth"} {width="0.325\linewidth"} {width="0.325\linewidth"} {width="0.325\linewidth"}
We note that our results differ somewhat from those previously presented in Ref. [@prior]. Whereas we find approximate agreement with Ref. [@prior] for the cases of protons or iron nuclei, we disagree in the case of helium. In particular, we obtain a photon fraction in the case of helium nuclei that is between the values found in the proton and iron cases, whereas Ref. [@prior] quotes values below those found for iron nuclei. This is puzzling as the photon fraction should predominantly depend upon the fraction of fragmented protons produced locally ([*i.e.*]{} within $\sim 100$ Mpc) with energies above the threshold for pion production. Since a rigidity-dependent cutoff leads to a maximum fragmented proton energy proportional to $Z/A$, the photon fraction for heavier nuclei should decrease monotonically with increasing $A$. Furthermore, pair production losses further reduce the contribution from heavy nuclei relative to lighter nuclei, and should thus decrease the photon fraction below that for lighter nuclei. It appears that although the authors of Ref. [@prior] did consider photopion production by secondary nucleons, they neglected pair production by protons and nuclei and photopion production by secondary nuclei [@gelmini].
[*Summary*]{}: We find that if ultra-high cosmic rays consist largely of heavy or intermediate mass nuclei, then the cosmogenic photon flux will be suppressed by about a factor of 10 relative to that expected for proton primaries. This provides a means of potentially discriminating between composition scenarios that is not subject to the uncertainties associated with hadronic interaction models. As the Pierre Auger Observatory continues to collect data, it is projected to reach the sensitivity required to use this distinction to constrain the chemical composition of the UHECRs. This would be complementary to the information potentially provided by future measurements of the cosmogenic neutrino flux which depends significantly on the cosmological evolution of UHECR sources – greater or fewer sources at high redshifts would lead to a higher or lower neutrino flux, respectively [@evolution]. In contrast, since any observed ultra-high energy photons must have originated within $\sim$100 Mpc, cosmological source evolution cannot affect their flux.
[*Acknowledgements:*]{} We are grateful to Graciela Gelmini for clarifications of earlier work and to Markus Risse for helpful correspondence. DH is supported by the US Department of Energy, including grant DE-FG02-95ER40896, and by NASA grant NAG5-10842. SS acknowledges support by the EU Marie Curie Network “UniverseNet” (HPRN-CT-2006-035863).
[99]{}
J. Abraham [*et al.*]{} \[Pierre Auger Collaboration\], Phys. Rev. Lett. [**104**]{}, 091101 (2010); arXiv:0906.2319; D. Hooper and A. M. Taylor, Astropart. Phys. [**33**]{}, 151 (2010). J. Engel, T. K. Gaisser, T. Stanev and P. Lipari, Phys. Rev. D [**46**]{}, 5013 (1992); R. S. Fletcher, T. K. Gaisser, P. Lipari and T. Stanev, Phys. Rev. D [**50**]{}, 5710 (1994); N. N. Kalmykov, S. S. Ostapchenko and A. I. Pavlov, Nucl. Phys. Proc. Suppl. [**52B**]{}, 17 (1997); K. Werner and T. Pierog, arXiv:0707.3330 \[astro-ph\]. T. Wibig, arXiv:0810.5281 \[hep-ph\]. R. Ulrich, R. Engel, S. Muller, F. Schussler and M. Unger, Nucl. Phys. Proc. Suppl. [**196**]{}, 335 (2009); K. Greisen, Phys. Rev. Lett. [**16**]{}, 748 (1966); G. T. Zatsepin and V. A. Kuzmin, JETP Lett. [**4**]{}, 78 (1966). V. S. Beresinsky and G. T. Zatsepin, Phys. Lett. B [**28**]{} 423 (1969); F. W. Stecker, Astrophys. J. [**228**]{}, 919 (1979); C. T. Hill and D. N. Schramm, Phys. Lett. B [**131**]{}, 247 (1983); R. Engel, D. Seckel and T. Stanev, Phys. Rev. D [**64**]{}, 093010 (2001); Z. Fodor, S. D. Katz, A. Ringwald and H. Tu, JCAP [**0311**]{}, 015 (2003). P. W. Gorham [*et al.*]{} \[ANITA collaboration\], arXiv:1003.2961 \[astro-ph.HE\]; Phys. Rev. Lett. [**103**]{}, 051103 (2009); J. Abraham [*et al.*]{} \[Pierre Auger Collaboration\], Phys. Rev. D [**79**]{}, 102001 (2009); Phys. Rev. Lett. [**100**]{}, 211101 (2008); K. Martens \[HiRes Collaboration\], arXiv:0707.4417 \[astro-ph\]; F. Halzen and D. Hooper, Phys. Rev. Lett. [**97**]{}, 099901 (2006); I. Kravchenko [*et al.*]{}, Phys. Rev. D [**73**]{}, 082002 (2006). P. Allison [*et al.*]{}, Nucl. Instrum. Meth. A [**604**]{}, S64 (2009); H. Landsman, L. Ruckman and G. S. Varner \[for the IceCube Collaboration\], Nucl. Instrum. Meth. A [**604**]{}, S70 (2009); F. Halzen and D. Hooper, JCAP [**0401**]{}, 002 (2004). J. Abraham [*et al.*]{} \[The Pierre Auger Collaboration\], Astropart. Phys. [**31**]{}, 399 (2009); Astropart. Phys. [**29**]{}, 243 (2008). M. Risse and P. Homola, Mod. Phys. Lett. A [**22**]{}, 749 (2007). D. Hooper, A. Taylor and S. Sarkar, Astropart. Phys. [**23**]{}, 11 (2005); M. Ave, N. Busca, A. V. Olinto, A. A. Watson and T. Yamamoto, Astropart. Phys. [**23**]{}, 19 (2005); D. Allard [*et al.*]{}, JCAP [**0609**]{}, 005 (2006); L. A. Anchordoqui, H. Goldberg, D. Hooper, S. Sarkar and A. M. Taylor, Phys. Rev. D [**76**]{}, 123008 (2007). D. Hooper, S. Sarkar and A. M. Taylor, Astropart. Phys. [**27**]{}, 199 (2007); Phys. Rev. D [**77**]{}, 103007 (2008). L. N. Epele and E. Roulet, JHEP [**9810**]{}, 009 (1998); F. W. Stecker and M. H. Salamon, Astrophys. J. [**512**]{}, 521 (1999); G. Bertone, C. Isola, M. Lemoine and G. Sigl, Phys. Rev. D [**66**]{}, 103003 (2002); T. Yamamoto, K. Mase, M. Takeda, N. Sakaki and M. Teshima, Astropart. Phys. [**20**]{}, 405 (2004); E. Khan [*et al.*]{}, Astropart. Phys. [**23**]{}, 191 (2005); E. Armengaud, G. Sigl and F. Miniati, Phys. Rev. D [**72**]{}, 043009 (2005); D. Allard, E. Parizot, E. Khan, S. Goriely and A. V. Olinto, Astron. Astrophys. [**443**]{}, L29 (2005); G. Sigl and E. Armengaud, JCAP [**0510**]{}, 016 (2005); J. Abraham [*et al.*]{} \[The Pierre Auger Collaboration\], Phys. Lett. B [**685**]{}, 239 (2010); Phys. Rev. Lett. [**101**]{}, 061101 (2008). A. M. Taylor, J. A. Hinton, P. Blasi and M. Ave, Phys. Rev. Lett. [**103**]{}, 051102 (2009). A. M. Hillas, Ann. Rev. Astron. Astrophys. [**22**]{}, 425 (1984). A. M. Taylor, A. De Castro and E. Castillo-Ruiz, AIP Conf. Proc. [**1123**]{}, 94 (2009). A. M. Taylor and F. A. Aharonian, Phys. Rev. D [**79**]{}, 083010 (2009). A. Eungwanichayapant and F. A. Aharonian, Int. J. Mod. Phys. D [**18**]{}, 911(2009). T. A. Clark, L. W. Brown and J. K. Alexander, Nature 228, 847 (1970),
U. Keshet, E. Waxman and A. Loeb, Astrophys. J. [**617**]{}, 281 (2004). D. Seckel and T. Stanev, Phys. Rev. Lett. [**95**]{}, 141101 (2005); H. Takami, K. Murase, S. Nagataki and K. Sato, Astropart. Phys. [**31**]{}, 201 (2009); M. Ahlers, L. A. Anchordoqui and S. Sarkar, Phys. Rev. D [**79**]{}, 083009 (2009); V. Berezinsky, A. Gazizov, M. Kachelriess and S. Ostapchenko, arXiv:1003.1496; M. Ahlers, L. A. Anchordoqui, M. C. Gonzalez-Garcia, F. Halzen and S. Sarkar, arXiv:1005.2620 \[astro-ph.HE\]. G. Gelmini, O. Kalashev and D. V. Semikoz, J. Exp. Theor. Phys. [**106**]{}, 1061 (2008); Astropart. Phys. [**28**]{}, 390 (2007); JCAP [**0711**]{}, 002 (2007). G. Gelmini, private communication.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Making use of Murakami’s classification of outer involutions in a Lie algebra and following the Morse-theoretic approach to harmonic two-spheres in Lie groups introduced by Burstall and Guest, we obtain a new classification of harmonic two-spheres in outer symmetric spaces and a Weierstrass-type representation for such maps. Several examples of harmonic maps into classical outer symmetric spaces are given in terms of meromorphic functions on $S^2$.'
address: |
Universidade da Beira Interior\
Rua Marquês d’Ávila e Bolama, 6200-001 Covilhã, Portugal
author:
- 'N. Correia and R. Pacheco'
title: 'Harmonic spheres in outer symmetric spaces, their canonical elements and Weierstrass-type representations'
---
Introduction {#introd}
============
The harmonicity of maps $\varphi$ from a Riemann surface $M$ into a compact Lie group $G$ with identity $e$ amounts to the flatness of one-parameter families of connections. This establishes a correspondence between such maps and certain holomorphic maps $\Phi$ into the based loop group $\Omega G$, the *extended solutions* [@Uh]. Evaluating an extended solution $\Phi$ at $\lambda=-1$ we obtain a harmonic map $\varphi$ into the Lie group. If an extended solution takes values in the group of algebraic loops $\Omega_\mathrm{alg}{G}$, the corresponding harmonic map is said to have *finite uniton number*. It is well known that all harmonic maps from the two-sphere into a compact Lie group have finite uniton number [@Uh].
Burstall and Guest [@BG] have used a method suggested by Morse theory in order to describe harmonic maps with finite uniton number from $M$ into a compact Lie group $G$ with trivial centre. One of the main ingredients in that paper is the Bruhat decomposition of the group of algebraic loops $\Omega_\mathrm{alg}{G}$. Each piece $U_\xi$ of the Bruhat decomposition corresponds to an element $\xi$ in the integer lattice $\mathfrak{I}(G)=(2\pi)^{-1} \exp^{-1}(e)\cap \mathfrak{t}$ and can be described as the unstable manifold of the energy flow on the Kähler manifold $\Omega_\mathrm{alg}{G}$. Each extended solution $\Phi:M\to \Omega_\mathrm{alg}{G}$ takes values, off some discrete subset $D$ of $M$, in one of these unstable manifolds $U_\xi$ and can be deformed, under the gradient flow of the energy, to an extended solution with values in some conjugacy class of a Lie group homomorphism $\gamma_\xi:S^1\to G$. A normalization procedure allows us to choose $\xi$ among the *canonical elements* of $\mathfrak{I}(G)$; there are precisely $2^n$ canonical elements, where $n=\mathrm{rank}(G)$, and consequently $2^n$ classes of harmonic maps. Burstall and Guest [@BG] introduced also a Weierstrass-type representation for such harmonic maps in terms of meromorphic functions on $M$. It is possible to define a similar notion of canonical element for compact Lie groups $G$ with non-trivial centre [@correia_pacheco_4; @correia_pacheco_5]. In the present paper, we will not assume any restriction on the centre of $G$.
Given an involution $\sigma$ of $G$, the compact symmetric $G$-space $N=G/G^\sigma$, where $G^\sigma$ is the subgroup of $G$ fixed by $\sigma$, can be embedded totally geodesically in $G$ via the corresponding Cartan embedding $\iota_\sigma$. Hence harmonic maps into compact symmetric spaces can be interpreted as special harmonic maps into Lie groups. For inner involutions $\sigma=\mathrm{Ad}(s_0)$, where $s_0\in G$ is the geodesic reflection at some base point $x_0\in N$, the composition of the Cartan embedding with left multiplication by $s_0$ gives a totally geodesic embedding of $G/G^\sigma$ in $G$ as a connected component of $\sqrt{e}$. Reciprocally, any connected component of $\sqrt{e}$ is a compact inner symmetric $G$-space. As shown by Burstall and Guest [@BG], any harmonic map into a connected component of $\sqrt{e}$ admits an extended solution $\Phi$ which is invariant under the involution $I(\Phi)(\lambda)=\Phi(-\lambda)\Phi(-1)^{-1}$. Off a discrete set, $\Phi$ takes values in some unstable manifold $U_\xi$ and can be deformed, under the gradient flow of the energy, to an extended solution with values in some conjugacy class of a Lie group homomorphism $\gamma_\xi:S^1\to G^\sigma$. An appropriate normalization procedure which preserves both $I$-invariance and the underlying connected component of $\sqrt{e}$ allows us to choose $\xi$ among the canonical elements of $\mathfrak{I}(G^\sigma)$. As a matter of fact, since $\sigma$ is inner, $\mathrm{rank}(G)=\mathrm{rank}(G^\sigma)$ and we have $\mathfrak{I}(G)=\mathfrak{I}(G^\sigma)$, that is the canonical elements of $\mathfrak{I}(G)$ coincide with the canonical elements of $\mathfrak{I}(G^\sigma)$. Consequently, if $G$ has trivial center, we have $2^n$ classes of harmonic maps with finite uniton number into *all* inner symmetric $G$-spaces.
The theory of Burstall and Guest [@BG] on harmonic two-spheres in compact inner symmetric $G$-spaces was extended by Eschenburg, Mare and Quast [@EMQ] to outer symmetric spaces as follows: each harmonic map from a two-sphere into an outer symmetric space $G/G^\sigma$, with outer involution $\sigma$, corresponds to an extended solution $\Phi$ which is invariant under a certain involution $T_\sigma$ induced by $\sigma$ on $\Omega G$ (see also [@GO]); $\Phi$ takes values in some unstable manifold $U_\xi$, off some discrete set; under the gradient flow of the energy any such invariant extended solution is deformed to an extended solution with values in some conjugacy class of a Lie group homomorphism $\gamma_\xi:S^1\to G^\sigma$; applying the normalization procedure of extended solutions introduced by Burstall and Guest for Lie groups, $\xi$ can be chosen among the canonical elements of $\mathfrak{I}(G^\sigma)\subsetneq \mathfrak{I}(G)$; if $G$ has trivial centre, there are precisely $2^k$ canonical homorphisms, where $k=\mathrm{rank}(G^\sigma)<\mathrm{rank}(G)$; hence there are *at most* $2^k$ classes of harmonic two-spheres in $G/G^\sigma$ if $G$ has trivial centre.
In the present paper, we will establish a more accurate classification of harmonic maps from a two-sphere into compact outer symmetric spaces. This classification takes into consideration the following crucial facts concerning extended solutions associated to harmonic maps into outer symmetric spaces: although any harmonic map from a two-sphere into an outer symmetric space $G/G^\sigma$ admits a $T_\sigma$-invariant extended solution, not all $T_\sigma$-invariant extended solutions correspond to harmonic maps into $G/G^\sigma$; on the other hand, the Burstall and Guest’s normalization procedure does not necessarily preserve $T_\sigma$-invariance.
Our strategy is the following. The existence of outer involutions of a simple Lie algebra $\mathfrak{g}$ depends on the existence of non-trivial involutions of the Dynkin diagram of $\mathfrak{g}^{\mathbb{C}}$ [@BR; @EMQ; @He; @Mu]. More precisely, if $\varrho$ is a non-trivial involution of the Dynkin diagram of $\mathfrak{g}^{\mathbb{C}}$, then it induces an outer involution $\sigma_\varrho$ of $\mathfrak{g}^{\mathbb{C}}$, which we call the *fundamental outer involution*, and, as shown by Murakami [@Mu], all the other outer involutions are, up to conjugation, of the form $\sigma_{\varrho,i}:=\mathrm{Ad}\exp\pi\zeta_i\circ \sigma_\varrho$ where each $\zeta_i$ is a certain element in the integer lattice $\mathfrak{I}(G^{\sigma_\varrho})$. Each connected component of $P^{\sigma_\varrho}=\{g\in G| \,\sigma_\varrho(g)=g^{-1}\}$ is a compact outer symmetric $G$-space associated to some involution $\sigma_\varrho$ or $\sigma_{\varrho,i}$; reciprocally, any outer symmetric space $G/G^\sigma$, with $\sigma$ equal to $\sigma_\varrho$ or $\sigma_{\varrho,i}$, can be totally geodesically embedded in the Lie group $G$ as a connected component of $P^{\sigma_\varrho}$ (see Proposition \[concomp\]). As shown in Section \[basics\], any harmonic map $\varphi$ into a connected component $N$ of $P^{\sigma_\varrho}$ admits a $T_{\sigma_\varrho}$-invariant extended solution $\Phi$; off a discrete set, $\Phi$ takes values in some unstable manifold $U_\xi$. In Section \[normprocedure\] we introduce an appropriate normalization procedure in order to obtain from $\Phi$ a *normalized* extended solution $\tilde{\Phi}$ with values in some unstable manifold $U_\zeta$ such that: $\zeta$ is a canonical element of $\mathfrak{I}(G^{\sigma_\varrho})$; $\tilde{\Phi}$ is $T_\tau$-invariant, where $\tau$ is the outer involution given by $\tau=\mathrm{Ad}\exp\pi(\xi-\zeta)\circ \sigma_{\varrho}$; $\tilde{\Phi}(-1)$ takes values in some connected component of $P^{\sigma_\varrho}$ which is an isometric copy of $N$ completely determined by $\zeta$ and $\tau$; moreover, $\tilde{\Phi}(-1)$ coincides with $\varphi$ up to isometry. Hence, we obtain a classification of harmonic maps of finite uniton number from $M$ into outer symmetric $G$-spaces in terms of the pairs $(\zeta,\tau)$.
Dorfmeister, Pedit and Wu [@DPW] have introduced a general scheme for constructing harmonic maps from a Riemann surface into a compact symmetric space from holomorphic data, in which the harmonic map equation reduces to a linear ODE similar to the classical Weierstrass representation of minimal surfaces. Burstal and Guest [@BG] made this scheme more explicit for the case $M=S^2$ by establishing a “Weierstrass formula" for harmonic maps with finite uniton number into Lie groups and their inner symmetric spaces. In Theorem \[sigmaweirstrass\] we establish a version of this formula to outer symmetric spaces, which allows us to describe the corresponding $T_\sigma$-invariant extended solutions in terms of meromorphic functions on $M$. For normalized extended solutions and “low uniton number", such descriptions are easier to obtain. In Section \[examples\] we give several explicit examples of harmonic maps from the two-sphere into classical outer symmetric spaces: Theorem \[RAs\] interprets old results by Calabi [@calabi_1967] and Eells and Wood [@eells_wood_1983] concerning harmonic spheres in real projective spaces $\mathbb{R}P^{2n-1}$ in view of our classification; harmonic two-spheres into the real Grassmannian $G_3(\mathbb{R}^6)$ are studied in detail; we show that all harmonic two spheres into the *Wu manifold* $SU(3)/SO(3)$ can be obtained explicitly by choosing two meromorphic functions on $S^2$ and then performing a finite number of algebraic operations, in agreement with the explicit constructions established by H. Ma in [@Ma].
Groups of algebraic loops
=========================
For completeness, in this section we recall some fundamental facts concerning the structure of the group of algebraic loops in a compact Lie group. Further details can be found in [@BG; @correia_pacheco_3; @PS].
Let $G$ be a compact matrix semisimple Lie group with Lie algebra $\mathfrak g$ and identity $e$. Denote the *free* and *based* loop groups of $G$ by $\Lambda G$ and $\Omega G$, respectively, whereas $\Lambda_+ G^{\mathbb{C}}$ stands for the subgroup of $\Lambda G^{\mathbb{C}}$ consisting of loops $\gamma:S^1\to G^{\mathbb{C}}$ which extend holomorphically to the unitary disc $|\lambda|<1$.
Fix a maximal torus $T$ of $G$ with Lie algebra $\mathfrak{t}\subset \mathfrak{g}$. Let $\Delta\subset \mathrm{i}\mathfrak{t}^*$ be the corresponding set of roots, where $\mathrm{i}:=\sqrt{-1}$, and, for each $\alpha\in \Delta$, denote by ${\mathfrak{g}}_\alpha$ the corresponding root space. Choose a fundamental Weyl chamber $\mathcal{W}$ in $\mathfrak{t}$, which corresponds to fix a positive root system $\Delta^+$. The intersection $\mathfrak{I}'(G):=\mathfrak{I}(G)\cap\mathcal{W}$ parameterizes the conjugacy classes of homomorphisms $S^1\to G$. More precisely, $\mathrm{Hom}(S^1,G)$ is the disjoint union of $\Omega_\xi(G)$, with $\xi\in \mathfrak{I}'(G)$, where $\Omega_\xi(G)$ is the conjugacy class of homomorphisms which contains $\gamma_\xi(\lambda)=\exp{(-\mathrm{i}\ln(\lambda)\xi)}$.
The *Bruhat decomposition* states that the subgroup of algebraic based loops $\Omega_\mathrm{alg}G$ is the disjoint union of the orbits $\Lambda^+_\mathrm{alg}G^{\mathbb{C}}\cdot\gamma_\xi$, with $\xi\in \mathfrak{I}'(G)$, where $\cdot$ denotes the dressing action of $\Lambda_+G^\mathbb{C}$ on $\Omega G$ induced by the *Iwasawa decomposition* $\Lambda G^{\mathbb{C}}\cong \Omega G\times \Lambda_+G^\mathbb{C}$. According to the Morse theoretic interpretation [@BG; @PS] of the Bruhat decomposition, for each $\xi\in\mathfrak{I}'(G)$, $U_\xi(G):=\Lambda^+_\mathrm{alg}G^{\mathbb{C}}\cdot \gamma_\xi$ is the unstable manifold of $\Omega_\xi(G)$ under the flow induced by the energy gradient vector field $-\nabla E$: each $\gamma\in U_\xi(G)$ flows to some homomorphism $u_\xi(\gamma)$ in $\Omega_\xi(G)$.
In [@BG], the authors proved that, for each $\xi\in\mathfrak{I}'(G)$, the critical manifold $\Omega_\xi(G)$ is a complex homogeneous space of $G^{\mathbb{C}}$ and the unstable manifold $U_\xi(G)$ is a complex homogeneous space of the group $\Lambda^+_{\mathrm{alg}}G^{\mathbb{C}}$. Moreover, $U_\xi(G)$ carries a structure of holomorphic vector bundle over $\Omega_\xi(G)$ and the bundle map $u_\xi:U_\xi(G)\to \Omega_\xi(G)$ is precisely the natural projection given by $[\gamma]\mapsto [\gamma(0)]$.
Define a partial order $\preceq$ over $\mathfrak{I}'(G)$ as follows: $\xi\preceq \xi'$ if $\mathfrak{p}^{\xi}_i\subset \mathfrak{p}^{\xi'}_i$ for all $i\geq 0$, where $\mathfrak{p}_i^\xi=\sum_{j\leq i}{\mathfrak{g}}_j^\xi$ and ${\mathfrak{g}}_j^\xi$ is the $j\mathrm{i}$-eigenspace of $\mathrm{ad}{\xi}$. As shown in [@correia_pacheco_3], one can define a $\Lambda^+_{\mathrm{alg}}G^{\mathbb{C}}$-invariant fibre bundle morphism $\mathcal{U}_{\xi,\xi'}:U_\xi(G)\to U_{\xi'}(G)$ by $$\mathcal{U}_{\xi,\xi'}(\Psi\cdot \gamma_{\xi})=\Psi\cdot\gamma_{\xi'}, \quad \Psi\in\Lambda^+_{\mathrm{alg}}G^{\mathbb{C}},$$ whenever $\xi\preceq \xi'$. Since the holomorphic structures on $U_\xi(G)$ and $U_{\xi'}(G)$ are induced by the holomorphic structure on $\Lambda^+_{\mathrm{alg}}G^{\mathbb{C}}$, the fibre-bundle morphism $\mathcal{U}_{\xi,\xi'}$ is holomorphic.
Harmonic spheres in Lie groups
==============================
Harmonic maps from the two-sphere $S^2$ into a compact matrix Lie group $G$ can be classified in terms of certain pieces of the Bruhat decomposition of $\Omega_\mathrm{alg}G$. Next we recall briefly this theory from [@BG; @correia_pacheco_3; @correia_pacheco_4; @correia_pacheco_5].
Extended Solutions
------------------
Let $M$ be a simply-connected Riemann surface, $\varphi:M\rightarrow G$ be a smooth map and $\rho:G\to \mathrm{End}(V)$ a finite representation of $G$. Equip $G$ with a bi-invariant metric. If $\varphi$ is an harmonic map of *finite uniton number*, it admits an extended solution $\Phi:M\to \Omega G$ with $\Phi(M)\subseteq \Omega_{\mathrm{alg}}G$ and $\varphi=\Phi_{-1}$. In this case, we can write $\rho\circ\Phi=\sum_{i=r}^s\zeta_i\lambda^i$ for some $r\leq s\in\mathbb{Z}$. The number $s-r$ is called the *uniton number* of $\Phi$ with respect to $\rho$, and the minimal value of $s-r$ (with respect to all extended solutions associated to $\varphi$) is called the *uniton number* of $\varphi$ with respect to $\rho$ and it is denoted by $r_\rho(\varphi)$. As explained in [@correia_pacheco_5], this definition of uniton number of an extended solution with respect to the adjoint representation is twice that of Burstall and Guest [@BG]. K. Uhlenbeck [@Uh] proved that all harmonic maps from the two-sphere have finite uniton number. For simplicity of exposition, henceforth we will take $M=S^2$. However, all our results still hold for harmonic maps of finite uniton number from an arbitrary Riemann surface.
[@BG]\[usd\] [Let $\Phi:S^2\to \Omega_{\mathrm{alg}}G$ be an extended solution. Then there exists some $\xi\in \mathfrak{I}'(G)$, and some discrete subset $D$ of $S^2$, such that $\Phi(S^2\setminus D)\subseteq U_\xi(G)$.]{}
Given a smooth map $\Phi:S^2\setminus D\to U_\xi(G)$, consider $\Psi:S^2\setminus D \to \Lambda_{\mathrm{alg}}^+G^{\mathbb{C}}$ such that $\Phi =\Psi\cdot\gamma_\xi $, that is $\Psi\gamma_\xi=\Phi b$ for some $b:S^2\setminus D\to \Lambda^+_{\mathrm{alg}}G^{\mathbb{C}}.$ Write $\Psi^{-1}\Psi_z=\sum_{i\geq 0} X'_i\lambda^i$, and $\Psi^{-1}\Psi_{\bar{z}}=\sum_{i\geq 0} X''_i\lambda^i.$ Proposition 4.4 in [@BG] establishes that $\Phi$ is an extended solution if, and only if, $$\label{im}
\mathrm{Im} X'_i\subset \,\mathfrak{p}^\xi_{i+1},\,\,\,\,\mathrm{Im} X''_i\subset \mathfrak{p}^\xi_{i},$$ where $\mathfrak{p}_i^\xi=\bigoplus_{j\leq i}{\mathfrak{g}}_j^\xi$ and ${\mathfrak{g}}_j^\xi$ is the $j\mathrm{i}$-eigenspace of $\mathrm{ad}{\xi}$. The derivative of the harmonic map $\varphi=\Phi_{-1}$ is given by the following formula.
\[poi\][@correia_pacheco_3] Let $\Phi=\Psi\cdot\gamma_\xi:S^2\to \Omega_{\mathrm{alg}}G$ be an extended solution and $\varphi=\Phi_{-1}:S^2\to G$ the corresponding harmonic map. Then $$\varphi^{-1}\varphi_z=-2\sum_{i\geq 0}b(0){X'_i}^{i+1}b(0)^{-1},$$ where ${X'_i}^{i+1}$ is the component of ${X'_i}$ over ${\mathfrak{g}}^\xi_{i+1}$, with respect to the decomposition ${\mathfrak{g}}^{\mathbb{C}}=\bigoplus {\mathfrak{g}}^\xi_j$.
Both the fiber bundle morphism $\mathcal{U}_{\xi,\xi'}:U_\xi(G)\to U_{\xi'}(G)$ and the bundle map $u_\xi:U_\xi(G)\to \Omega_\xi(G)$ preserve harmonicity.
[@BG; @correia_pacheco_3] \[popo\] Let $\Phi:S^2\setminus D\to U_\xi(G)$ be an extended solution. Then
1. $u_\xi\circ\Phi:S^2\setminus D\to \Omega_\xi(G)$ is an extended solution, with $\xi\in\mathfrak{I}(G)$;
2. for each $\xi'\in \mathfrak{I}'(G)$ such that $\xi\preceq \xi'$, $\mathcal{U}_{\xi,\xi'}(\Phi)=\mathcal{U}_{\xi,\xi'}\circ \Phi:S^2\setminus D\to U_{\xi'}(G)$ is an extended solution.
Weierstrass representation. {#weircond}
---------------------------
Taking a larger discrete subset if necessary, one obtains a more explicit description for harmonic maps of finite uniton number and their extended solutions as follows.
\[BG\][@BG] Let $\Phi:S^2\to \Omega_{\mathrm{alg}}G$ be an extended solution. There exists a discrete set $D'\supseteq D$ of $S^2$ such that $\Phi{\big|_{S^2\setminus D'}}=\exp C\cdot \gamma_\xi$ for some holomorphic vector-valued function $C: S^2\setminus D'\to \mathfrak{u}^0_\xi$, where $\mathfrak{u}^0_\xi$ is the finite dimensional nilpotent subalgebra of $\Lambda^+_{\mathrm{alg}}{\mathfrak{g}}^{\mathbb{C}}$ defined by $$\mathfrak{u}^0_\xi=\bigoplus_{0\leq i<r(\xi)}\lambda^i(\mathfrak{p}^\xi_i)^\perp,\quad (\mathfrak{p}^\xi_i)^\perp=\bigoplus _{i<j\leq r(\xi)}\mathfrak{g}_j^\xi,$$ with $r(\xi)=\mathrm{max}\{j\,|\,\,{\mathfrak{g}}_j^\xi\neq 0\}$. Moreover, $C$ can be extended meromorphically to $S^2$.
Conversely, taking account and the well-known formula for the derivative of the exponential map, we see that if $C: S^2\to \mathfrak{u}^0_\xi$ is meromorphic then $\Phi=\exp{C}\cdot \gamma_{\xi}$ is an extended solution if and only if in the expression $$\label{C_z}
(\exp C)^{-1}(\exp C)_z=C_z-\frac{1}{2!}(\mathrm{ad} C)C_z+\ldots +(-1)^{r(\xi)-1}\frac{1}{r(\xi)!}(\mathrm{ad} C)^{r(\xi)-1}C_z,$$ the coefficient $\lambda^i$ have zero component in each ${\mathfrak{g}}_{i+2}^\xi,\ldots,{\mathfrak{g}}^\xi_{r(\xi)}$.
$S^1$-invariant extended solutions
----------------------------------
Extended solutions with values in some $\Omega_\xi(G)$, off a discrete subset, are said to be *$S^1$-invariant*. If we take a unitary representation $\rho:G \to U(n)$ for some $n$, then for any such extended solution $\Phi$ we have $\rho\circ \Phi_\lambda=\sum_{i=r}^s\lambda^i\pi_{W_i},$ where, for each $i$, $\pi_{W_i}$ is the orthogonal projection onto a complex vector subbundle $W_i$ of $\underline{{\mathbb{C}}}^n:=M\times {\mathbb{C}}^n$ and $\underline{{\mathbb{C}}}^n=\bigoplus_{i=r}^sW_i$ is an orthogonal direct sum decomposition. Set $A_i=\bigoplus_{j\leq i }W_j$ so that $$\{0\}\subset A_r \subset \ldots \subset A_{i-1}\subset A_i\subset A_{i+1}\subset\ldots \subset A_s= \underline{{\mathbb{C}}}^n.$$ The harmonicity condition amounts to the following conditions on this flag: for each $i$, $A_i$ is a holomorphic subbundle of $\underline{{\mathbb{C}}}^n$; the flag is *superhorizontal*, in the sense that, for each $i$, we have $\partial A_i\subseteq A_{i+1}$, that is, given any section $s$ of $A_i$ then $\frac{\partial s}{\partial z}$ is a section of $A_{i+1}$ for any local complex coordinate $z$ of $S^2$.
Normalization of harmonic maps {#BGnorm}
------------------------------
Let $\Delta_0:=\{\alpha_1,\ldots,\alpha_r\}\subset \Delta^+$ be the basis of positive simple roots, with dual basis $\{H_1,\ldots, H_r\}\subset\mathfrak{t}$, that is $\alpha_i(H_j)=\mathrm{i}\,\delta_{ij}$, where $r=\mathrm{rank}(\mathfrak{g})$. Given $\xi=\sum n_iH_i$ and $\xi'=\sum n'_iH_i$ in $\mathfrak{I}'({G})$, we have $n_i,n'_i\geq 0$ and observe that $\xi\preceq\xi'$ if and only if $n'_i\leq n_i$ for all $i$. For each $I\subseteq \{1,\ldots,r\}$, define the cone $\mathfrak{C}_{I}=\Big\{\sum_{i=1}^r n_i H_i|\, n_i\geq 0, \,\mbox{$n_j=0$ iff $j\notin I$}\Big\}.$
[@correia_pacheco_5] Let $\xi\in\mathfrak{I}'({G})\cap \mathfrak{C}_{I}$. We say that $\xi$ is a *$I$-canonical element* of $G$ with respect to $\mathcal{W}$ if it is a maximal element of $(\mathfrak{I}'({G})\cap \mathfrak{C}_{I},\preceq)$, that is: if $\xi\preceq \xi'$ and $\xi'\in \mathfrak{I}'({G})\cap \mathfrak{C}_{I}$ then $\xi=\xi'$.
When $G$ has trivial centre, which is the case considered in [@BG], there exists a unique $I$-canonical element, which is given by $\xi_I=\sum_{i\in I}H_i$, for each $I$. When $G$ has non-trivial centre, the $I$-canonical elements of $G$ were described in [@correia_pacheco_4; @correia_pacheco_5].
Any harmonic map $\varphi:S^2\to G$ admits a *normalized extended solution*, that is, an extended solution $\Phi$ taking values in $U_\xi(G)$, off some discrete set, for some canonical element $\xi$. This is a consequence of the following generalization of Theorem 4.5 in [@BG].
\[nor\][@correia_pacheco_3] Let $\Phi:S^2\setminus D\to U_\xi(G)$ be an extended solution. Take $\xi'\in \mathfrak{I}'({G})$ such that $\xi\preceq {\xi'}$ and ${\mathfrak{g}}_0^\xi={\mathfrak{g}}_0^{\xi'}$. Then $\gamma^{-1}:=\mathcal{U}_{\xi,\xi-\xi'}(\Phi)$ is a constant loop in $\Omega_{\mathrm{alg}}{G}$ and $\gamma\Phi:S^2\setminus D\to U_{\xi'}(G)$.
The uniton number of a normalized extended solution $\Phi:S^2\setminus D\to U_\xi(G)$ can be computed with respect to any irreducible $n$-dimensional representation ${\rho}:G\to\mathrm{End}(V)$ with highest weight $\omega^*$ and lowest weight $\varpi^*$ as follows [@correia_pacheco_5]: $r_\rho(\xi):=\omega^*(\xi)-\varpi^*(\xi)$.
Harmonic spheres in outer symmetric spaces
==========================================
In the following sections we will establish our classification of harmonic maps from $S^2$ into compact outer symmetric spaces and establish a Weierstrass formula for such harmonic maps. These will allow us to produce some explicit examples of harmonic maps from two-spheres into outer symmetric spaces from meromorphic data.
As we have referred in Section \[introd\], although any harmonic map from a two-sphere into an outer symmetric space $G/K$ admits a $T_\sigma$-invariant extended solution, not all $T_\sigma$-invariant extended solutions correspond to harmonic maps into $G/K$; by Proposition \[concomp\] and Theorem \[GO\] below, they correspond to a harmonic map into some possibly different outer symmetric space $G/K'$ (compare Theorem \[RAs\] with Theorem \[36\] for an example where this happens).
Symmetric $G$-spaces and Cartan embeddings
------------------------------------------
Let $N=G/K$ be a symmetric space, where $K$ is the isotropy subgroup at the base point $x_0\in N$, and let $\sigma:G\to G$ be the corresponding involution: we have $G^{\sigma}_0\subseteq K\subseteq G^{\sigma},$ where $G^{\sigma}$ is the subgroup fixed by $\sigma$ and $G_0^{\sigma}$ denotes its connected component of the identity. We assume that $N$ is a *bottom space*, i.e. $K=G^{\sigma}$. Let $\mathfrak{g}=\mathfrak{k}_\sigma\oplus \mathfrak{m}_\sigma$ be the $\pm1$-eigenspace decomposition associated to the involution $\sigma$, where $\mathfrak{k}_\sigma$ is the Lie algebra of $K$. Consider the (totally geodesic) *Cartan embedding* $\iota_\sigma:N\hookrightarrow G$ defined by $\iota_\sigma (g\cdot x_0)=g\sigma(g^{-1})$. The image of the Cartan embedding is precisely the connected component $P^\sigma_e$ of $P^{\sigma}:=\{g\in G|\,\sigma(g)=g^{-1}\}$ containing the identity $e$ of the group $G$. Observe that, given $\xi\in \mathfrak{I}(G)\cap \mathfrak{k}_\sigma$, then $\exp(\pi\xi)\in P^\sigma$. We denote by $P_{\xi}^{\sigma}$ the connected component of $P^{\sigma}$ containing $\exp(\pi\xi)$.
\[concomp\] Given $\xi\in \mathfrak{I}(G)\cap \mathfrak{k}_\sigma$, we have the following.
1. $G$ acts transitively on $P_{\xi}^\sigma$ as follows: for $g\in G$ and $h\in P_{\xi}^{\sigma}$, $$\label{gaction}
g\cdot_\sigma h=gh\sigma(g^{-1}).$$
2. $P^{\sigma}_{\xi}$ is a bottom symmetric $G$-space totally geodesically embedded in $G$ with involution $$\label{tau}
\tau=\mathrm{Ad}(\exp\pi\xi)\circ \sigma.$$
3. For any other $\xi'\in \mathfrak{I}(G)\cap\mathfrak{k}_\sigma$ we have $\exp(\pi\xi')\in P^{\tau}$ and $P^\tau_{\xi'}=\exp(\pi\xi) P_{\xi'-\xi}^\sigma.$
4. The $\pm 1$-eigenspace decomposition $\mathfrak{g}=\mathfrak{k}_\tau\oplus\mathfrak{m}_\tau$ associated to the symmetric $G$-space $P_{\xi}^\sigma$ at the fixed point $\exp(\pi\xi)\in P_{\xi}^\sigma$ is given by $$\begin{aligned}
\label{hc} \mathfrak{k}_\tau^{\mathbb{C}}&= \bigoplus \mathfrak{g}^\xi_{2i}\cap \mathfrak{k}_\sigma^{\mathbb{C}}\oplus \bigoplus\mathfrak{g}^\xi_{2i+1}\cap \mathfrak{m}_\sigma^{\mathbb{C}}\\ \label{pc} \mathfrak{m}_\tau^{\mathbb{C}}&=\bigoplus \mathfrak{g}^\xi_{2i+1}\cap \mathfrak{k}_\sigma^{\mathbb{C}}\oplus \bigoplus\mathfrak{g}^\xi_{2i}\cap \mathfrak{m}_\sigma^{\mathbb{C}}.\end{aligned}$$
Take $h\in P^\sigma$. We have $$\sigma(g\cdot_\sigma h)=\sigma(gh\sigma(g^{-1}))=\sigma(g)h^{-1}g^{-1}=(gh\sigma(g^{-1}))^{-1}=(g\cdot_\sigma h)^{-1}.$$ Then $g\cdot_\sigma h\in P^\sigma$ and we have a continuous action of $G$ on $P^\sigma$. Since $G$ is connected, this action induces an action of $G$ on each connected component of $P^\sigma$. Since $g\cdot_\sigma e=g\sigma(g^{-1})=\iota_\sigma(g\cdot x_0)$ and $\iota_\sigma(N)=P_e^\sigma$, the action $\cdot_\sigma$ of $G$ on $P_e^\sigma$ is transitive.
Take $\xi\in \mathfrak{I}(G)\cap \mathfrak{k}_\sigma$, so that $\sigma(\xi)=\xi$ and $\exp2\pi\xi=e$. Consider the involution $\tau$ defined by . If $g\in P^\sigma$, then $$\tau(\exp(\pi\xi)g)=\exp(\pi\xi)\sigma(\exp(\pi\xi)g)\exp(\pi\xi)=\sigma(g)\exp(\pi\xi)=(\exp(\pi\xi)g)^{-1},$$ which means that $\exp(\pi\xi)g\in P^\tau$. Reciprocally, if $\exp(\pi\xi)g\in P^\tau$, one can check similarly that $g\in P^{\sigma}$. Hence $P^\tau=\exp(\pi\xi)P^\sigma$. In particular, by continuity, $P^\tau_{\xi'}=\exp(\pi\xi) P^\sigma_{\xi'-\xi}$ for any other $\xi'\in \mathfrak{I}(G)$ with $\sigma(\xi')=\xi'$.
Reversing the rules of $\sigma= \mathrm{Ad}(\exp\pi\xi)\circ \tau$ and $\tau$, we also have $P^\sigma_{\xi}=\exp(\pi\xi)P_e^\tau.$ Since $G$ acts transitively on $P_e^\tau$, for each $h\in P^\sigma_{\xi}$ there exists $g\in G$ such that $$h=\exp(\pi\xi) (g\cdot_\tau e)=(\exp(\pi\xi)g)\cdot_\sigma \exp(\pi\xi).$$ This shows that $G$ also acts transitively on $P^\sigma_{\xi}$. The isotropy subgroup at $\exp(\pi\xi)$ consists of those elements $g$ of $G$ satisfying $g\exp(\pi\xi)\sigma(g^{-1})=\exp(\pi\xi)$, that is those elements $g$ of $G$ which are fixed by $\tau$: $$\label{isot}
\exp(\pi\xi)\sigma(g)\exp(\pi\xi)=g.$$ Hence $P^\sigma_{\xi}\cong G/G^{\tau},$ which is a bottom symmetric $G$-space with involution $\tau$. Since $P_e^\tau\subset G$ totally geodesically and $P^\sigma_\xi$ is the image of $P_e^\tau$ under an isometry (left multiplication by $\exp\pi\xi$), then $P_{\xi}^\sigma\subset G$ totally geodesically.
Differentiating at the identity we get $\mathfrak{k}_\tau=\{X\in \mathfrak{g}|\, X=\mathrm{Ad}(\exp\pi\xi)\circ\sigma(X)\}.$ Taking account of the formula $\mathrm{Ad}({\exp (\pi\xi)})=e^{\pi \mathrm{ad} \xi}$ and that $\sigma$ commutes with $\mathrm{ad}\xi$, we obtain ; and follows similarly.
### Outer symmetric spaces.
The existence of outer involutions of a simple Lie algebra $\mathfrak{g}$ depends on the existence of non-trivial involutions of the Dynkin diagram of $\mathfrak{g}^{\mathbb{C}}$ [@BR; @EMQ; @He; @Mu]. Fix a maximal abelian subalgebra $\mathfrak{t}$ of $\mathfrak{g}$ and a Weyl chamber $\mathcal{W}$ in $\mathfrak{t}$, which amounts to fix a system of positive simple roots $\Delta_0=\{\alpha_1,\ldots,\alpha_r\}$, where $r=\mathrm{rank}(\mathfrak{g})$. Let $\varrho$ be a non-trivial involution of the Dynkin diagram and $\sigma_\varrho$ the *fundamental outer involution* associated to $\varrho$ [@BR; @Mu]. The (local isometry classes of) outer symmetric spaces of compact type associated to involutions of the form $\sigma_\varrho$ are precisely
> $SU(2n)/Sp(n)$, $SU(2n+1)/SO(2n+1)$, $E_6/F_4$ and the real projective spaces $\mathbb{R}P^{2n-1}$.
These spaces are called the *fundamental outer symmetric spaces*. The remaining classes of outer involutions are obtained as follows [@BG; @Mu]. Let $\mathfrak{g}=\mathfrak{k}_{\varrho}\oplus \mathfrak{m}_{\varrho}$ be the corresponding $\pm 1$-eigenspace decomposition of $\mathfrak{g}$. As shown in Proposition 3.20 of [@BR], the Lie subalgebra $\mathfrak{k}_{\varrho}$ is simple and the orthogonal projection of $\Delta_0$ onto $\mathfrak{k}_{\varrho}$, $\pi_{\mathfrak{k}_{\varrho}}(\Delta_0)$, is a basis of positive simple roots of $\mathfrak{k}_\varrho$ associated to the maximal abelian subalgebra $\mathfrak{t}_{\mathfrak{k}_{\varrho}}:=\mathfrak{t}\cap \mathfrak{k}_{\varrho}$. Consider the split $\mathfrak{t}=\mathfrak{t}_{{\mathfrak{k}_\varrho}}\oplus \mathfrak{t}_{\mathfrak{m}_\varrho}$ with respect to $\mathfrak{g}={\mathfrak{k}_\varrho}\oplus {\mathfrak{m}_\varrho}$. Set $s=r-k$, where $k=\mathrm{rank}(\mathfrak{k}_\varrho)$. We can label the basis $\Delta_0$ in order to get the following relations: $\varrho (\alpha_j)=\alpha_j$ for $1\leq j\leq k-s$ and $\varrho (\alpha_j)=\alpha_{s+j}$ for $k-s+1\leq j\leq k$. Let $\pi_{\mathfrak{k}_\varrho}$ be the orthogonal projection of $\mathfrak{t}$ onto $\mathfrak{t}_{{\mathfrak{k}_\varrho}}$, that is $\pi_{\mathfrak{k}_\varrho}(H)=\frac12(H+\sigma_\varrho (H))$ for all $H\in \mathfrak{t}$. Set $\pi_{{\mathfrak{k}_\varrho}}(\Delta_0)=\{\beta_1,\ldots,\beta_k\}$, with $$\label{bes}
\beta_j=\left\{\begin{array}{cl} \alpha_j & \mbox{for $1\leq j\leq k-s$}
\\ \frac12(\alpha_j+\alpha_{j+s}) & \mbox{for $k-s+1\leq j\leq k$}
\end{array}\right..$$ This is a basis of $\mathrm{i}\mathfrak{t}_{\mathfrak{k}_\varrho}^*$ with dual basis $\{\zeta_1,\ldots, \zeta_k\}$ given by $$\label{zes}
\zeta_j=\left\{\begin{array}{cl}
H_j & \mbox{for $1\leq j\leq k-s$} \\
H_j+H_{j+s} & \mbox{for $k-s+1\leq j\leq k$}
\end{array}\right..$$
[@Mu]\[murak\] Let $\varrho$ be an involution of the Dynkin diagram of $\mathfrak{g}$. Let $$\omega=\sum_{j=1}^{k-s}n_j\beta_j+\sum_{j=k-s+1}^{k}n'_j\beta_j$$ be the highest root of $\mathfrak{k}_\varrho$ with respect to $\pi_{\mathfrak{k}_{\varrho}}(\Delta_0)=\{\beta_1,\ldots,\beta_k\}$, defined as in . Given $i$ such that $n_i=1$ or $2$, define an involution $\sigma_{\varrho,i}$ by $$\label{sigmasis}
\sigma_{\varrho,i}=\mathrm{Ad}(\exp\pi \zeta_i)\circ \sigma_\varrho.$$ Then any outer involution of $\mathfrak{g}$ is conjugate in $\mathfrak{Aut}(\mathfrak{g})$, the group of automorphisms of $\mathfrak{g}$, to some $\sigma_\varrho$ or $\sigma_{\varrho,i}$. In particular, there are at most $k-s+1$ conjugacy classes of outer involutions.
The list of all (local isometry classes of) irreducible outer symmetric spaces of compact type is shown in Table 1 (cf. [@BR; @EMQ; @He]).
$\mathrm{rank}(G)$ $\mathrm{rank}(K)$ $\mathrm{rank}(G/K)$ $\mathrm{dim}(G/K)$
--------------------------------------------- -------------------- -------------------- ---------------------- ---------------------
$SU(2n)/SO(2n)$ $2n-1$ $n$ $2n-1$ $(2n-1)(n+1)$
$SU(2n+1)/SO(2n+1)$ $2n$ $n$ $2n$ $n(2n+3)$
$SU(2n)/Sp(n)$ $2n-1$ $n$ $ n-1$ $(n-1)(2n+1)$
$G_{p}(\mathbb{R}^{2n})$ ($p$ odd $\leq n$) $n$ $n-1$ $p$ $p(2n-p)$
$E_6/Sp(4)$ $6$ $4$ $6$ $42$
$E_6/F_4$ $6$ $4$ $2$ $26$
: Irreducible outer symmetric spaces.
Given an outer involution $\sigma$ of the form $\sigma_{\varrho,i}$ or $\sigma_\varrho$ and its $\pm 1$-eigenspace decomposition ${\mathfrak{g}}=\mathfrak{k}_\sigma\oplus \mathfrak{m}_\sigma$, set $\mathfrak{t}_{\mathfrak{k}_\sigma}=\mathfrak{t}\cap \mathfrak{k}_\sigma$, which is a maximal abelian subalgebra of $\mathfrak{k}_\sigma$. Following [@EMQ], a non-empty intersection of $\mathfrak{t}_{\mathfrak{k}_\sigma}$ with a Weyl chamber in $\mathfrak{t}$ is called a *compartment*. Each compartment lies in a Weyl chamber in $\mathfrak{t}_{\mathfrak{k}_\sigma}$ and the Weyl chambers in $\mathfrak{t}_{\mathfrak{k}_\sigma}$ can be decomposed into the same number of compartments [@EMQ].
The intersection of the integer lattice $\mathfrak{I}(G)$ with the Weyl chamber $\mathcal{W}$ in $\mathfrak{t}$, which we have denoted by $\mathfrak{I}'(G)$, is described in terms of the dual basis $\{H_1,\ldots,H_r\}\subset \mathfrak{t}$, with $r=\mathrm{rank} (\mathfrak{g})$, by $$\mathfrak{I}'(G)=\big\{\sum_{i=1}^r n_iH_i\in \mathfrak{I}(G) |\,\mbox{$n_i\in \mathbb{N}_0$ for all $i$}\big\}.$$ When $\sigma$ is a fundamental outer involution $\sigma_\varrho$, the compartment $\mathcal{W}\cap \mathfrak{t}_{\mathfrak{k_\varrho}}$ is itself a Weyl chamber in $\mathfrak{t}_{\mathfrak{k_\varrho}}$. Then, the intersection of the integer lattice $\mathfrak{I}(G^{\sigma_\varrho})$ with the Weyl chamber $\mathcal{W}\cap \mathfrak{t}_{\mathfrak{k_\varrho}}$, is given by $$\mathfrak{I}'(G^{\sigma_\varrho})=\big\{\sum_{i=1}^k n_i\zeta_i\in \mathfrak{I}(G) |\,\mbox{$n_i\in \mathbb{N}_0$ for all $i$}\big\}=\mathfrak{I}'(G)\cap \mathfrak{t}_{\mathfrak{k_\varrho}}.$$
### Cartan embeddings of fundamental outer symmetric spaces.
Next we describe those elements $\xi$ of $ \mathfrak{I}'(G^{\sigma_\varrho})$ for which the connected component $P_\xi^{\sigma_\varrho}$ of $P^{\sigma_\varrho}$ containing $\exp(\pi\xi)$ can be identified with the fundamental outer symmetric $G$-space associated to $\varrho$. Start by considering the following $\sigma_\varrho$-invariant subsets of the root system $\Delta\subset \mathrm{i}\mathfrak{t}^*$ of $\mathfrak{g}$: $$\begin{aligned}
\label{deltas}
\Delta(\mathfrak{k}_\varrho)=\{\alpha\in\Delta|\, {\mathfrak{g}}_\alpha\subset \mathfrak{k}_\varrho^{\mathbb{C}}\},\,\,
\Delta(\mathfrak{m}_\varrho)=\{\alpha\in\Delta|\, {\mathfrak{g}}_\alpha\subset \mathfrak{m}_\varrho^{\mathbb{C}}\},\,\, \Delta_\varrho=\Delta\setminus \left(\Delta(\mathfrak{k}_\varrho)\cup \Delta(\mathfrak{m}_\varrho)\right).\end{aligned}$$ Then $$\mathfrak{k}_\varrho^{\mathbb{C}}=\mathfrak{t}^{\mathbb{C}}_{\mathfrak{k}_\varrho}\oplus\pi_{\mathfrak{k}_\varrho}(\mathfrak{r}_\varrho)\oplus\bigoplus_{\alpha\in \Delta(\mathfrak{k}_\varrho)}{\mathfrak{g}}_\alpha,\,\,\, \mathfrak{m}_\varrho^{\mathbb{C}}=\mathfrak{t}^{\mathbb{C}}_{\mathfrak{m}_\varrho}\oplus\pi_{\mathfrak{m}_\varrho}(\mathfrak{r}_\varrho)\oplus\bigoplus_{\alpha\in \Delta(\mathfrak{m}_\varrho)}{\mathfrak{g}}_\alpha,$$ where $\mathfrak{r}_\varrho=\bigoplus_{\alpha\in\Delta_\varrho}{\mathfrak{g}}_\alpha$. Since the involution $\varrho$ acts on $\Delta_\varrho$ as a permutation without fixed points, we can fix some subset $\Delta'_\varrho$ so that $\Delta_\varrho$ is the disjoint union of $\Delta'_\varrho$ with $\varrho(\Delta'_\varrho)$: $$\label{delta1}
\Delta_\varrho= \Delta'_\varrho {\sqcup}\,\, \varrho(\Delta'_\varrho).$$ For each $\alpha \in\Delta'_\varrho$, $\sigma_\varrho$ restricts to an involution in the subspace ${\mathfrak{g}}_\alpha\oplus {\mathfrak{g}}_{\varrho(\alpha)}\subset \mathfrak{r}_\varrho$. Hence we have the following.
\[pmmm\]The orthogonal projections of $\mathfrak{r}_\varrho$ onto $\mathfrak{k}_\varrho^{\mathbb{C}}$ and $\mathfrak{m}_\varrho^{\mathbb{C}}$ are given by $$\pi_{\mathfrak{k}_\varrho}(\mathfrak{r}_\varrho)=\!\bigoplus_{\alpha\in\Delta'_\varrho}\! \mathfrak{k}_\varrho^{\mathbb{C}}\cap \big({\mathfrak{g}}_\alpha\oplus{\mathfrak{g}}_{\varrho(\alpha)}\big),\,\,\pi_{\mathfrak{m}_\varrho}(\mathfrak{r}_\varrho)=\!\!\bigoplus_{\alpha\in\Delta'_\varrho}\! \mathfrak{m}_\varrho^{\mathbb{C}}\cap \big({\mathfrak{g}}_\alpha\oplus{\mathfrak{g}}_{\varrho(\alpha)}\big),$$ and, for each $\alpha\in\Delta'_\varrho$, $$\mathfrak{k}_\varrho^{\mathbb{C}}\cap \big({\mathfrak{g}}_\alpha\oplus{\mathfrak{g}}_{\sigma_\varrho(\alpha)}\big)=\{X_\alpha+\sigma_\varrho(X_\alpha)|\, X_\alpha\in {\mathfrak{g}}_\alpha\},\,\,\, \mathfrak{m}_\varrho^{\mathbb{C}}\cap \big({\mathfrak{g}}_\alpha\oplus{\mathfrak{g}}_{\sigma(\alpha)}\big)=\{X_\alpha-\sigma_\varrho(X_\alpha)|\, X_\alpha\in {\mathfrak{g}}_\alpha\}.$$ In particular, $\dim \mathfrak{r}_\varrho=2\dim \pi_{\mathfrak{k}_\varrho}(\mathfrak{r}_\varrho)= 2\dim \pi_{\mathfrak{m}_\varrho}(\mathfrak{r}_\varrho)$.
\[fundcan\] Consider the dual basis $\{\zeta_1,\ldots,\zeta_k\}$ defined by . Given $\xi \in \mathfrak{I}'(G^{\sigma_\varrho})$ with $\xi=\sum_{i=1}^kn_i\zeta_i$ and $n_i\geq 0$, then $P_\xi^{\sigma_\varrho}$ is a fundamental outer symmetric space with involution (conjugated to) $\sigma_\varrho$ if and only if $n_i$ is even for each $1\leq i\leq k-s$.
There is only one class of outer symmetric $SU(2n+1)$-spaces and, in this case, the involution $\varrho$ does not fix any simple root, that is $k-s=0$. Hence the result trivially holds for $N=SU(2n+1)/SO(2n+1).$
Next we consider the remaining fundamental outer symmetric spaces, which are precisely the symmetric spaces of *rank-split type* [@EMQ], those satisfying $\Delta(\mathfrak{m}_\varrho)=\emptyset$. For such symmetric spaces, the reductive symmetric term $\mathfrak{m}_\varrho$ satisfies $\mathfrak{m}_\varrho=\mathfrak{t}_{\mathfrak{m}_\varrho}\oplus\pi_{\mathfrak{m}_\varrho}(\mathfrak{r}_\varrho)$. On the other hand, in view of , we have, for $\tau=\mathrm{Ad}(\exp\pi\xi)\circ\sigma_\varrho$, $$\begin{aligned}
\nonumber
\mathfrak{m}_\tau^{\mathbb{C}}&=\bigoplus \mathfrak{g}^\xi_{2i+1}\cap \mathfrak{k}^{\mathbb{C}}_\varrho\oplus \bigoplus\mathfrak{g}^\xi_{2i}\cap \mathfrak{m}^{\mathbb{C}}_\varrho\\&=
\mathfrak{t}^{\mathbb{C}}_{\mathfrak{m}_\varrho}\oplus\!\!\! \bigoplus_{{\alpha\in \Delta(\mathfrak{k}_\varrho)\cap \Delta_\xi^-}}\!\!\! {\mathfrak{g}}_\alpha\oplus
\!\!\! \bigoplus_{{\alpha\in \Delta'_\varrho\cap \Delta_\xi^-}} \!\!\!\mathfrak{k}^{\mathbb{C}}_\varrho\cap({\mathfrak{g}}_\alpha\oplus {\mathfrak{g}}_{\varrho(\alpha)})\oplus \!\!\!\bigoplus_{{\alpha\in \Delta'_\varrho\cap\Delta_\xi^+}} \!\!\!\mathfrak{m}^{\mathbb{C}}_\varrho\cap({\mathfrak{g}}_\alpha\oplus {\mathfrak{g}}_{\varrho(\alpha)}),
\end{aligned}$$ where $\Delta_\xi^+:=\{\alpha\in\Delta|\,\mbox{$\alpha(\xi)\mathrm{i}$ is even}\}$ and $\Delta_\xi^-:=\{\alpha\in\Delta|\,\mbox{$\alpha(\xi)\mathrm{i}$ is odd}\}$. Taking into account Lemma \[pmmm\], from this we see that $\dim \mathfrak{m}_\tau=\dim \mathfrak{m}_\varrho$ (which means, by Table 1, that $P_\xi^{\sigma_\varrho}$ is a fundamental outer symmetric space with involution conjugated to $\sigma_\varrho$) if and only if $$\bigoplus_{{\alpha\in \Delta(\mathfrak{k}_\varrho)\cap \Delta_\xi^-}}\!\! {\mathfrak{g}}_\alpha=\{0\},$$ which holds if and only if $\xi=\sum_{i=1}^kn_i\zeta_i$ with $n_i$ even for each $1\leq i\leq k-s$.
Harmonic spheres in symmetric $G$-spaces {#basics}
----------------------------------------
Given an involution $\sigma$ on $G$, define an involution $T_\sigma$ on $\Omega G$ by $T_\sigma(\gamma)(\lambda)=\sigma(\gamma(-\lambda)\gamma(-1)^{-1}).$ Let $\Omega^\sigma G$ be the fixed set of $T_\sigma$.
\[GO\][@EMQ; @GO] Given $\xi\in \mathfrak{I}(G)\cap\mathfrak{k}_\sigma$, any harmonic map $\varphi:S^2\to P^{\sigma}_\xi\subset G$ admits an $T_\sigma$-invariant extended solution $\Phi:S^2\to\Omega^\sigma G$. Conversely, given an $T_\sigma$-invariant extended solution $\Phi$, the smooth map $\varphi=\Phi_{-1}$ from $S^2$ is harmonic and takes values in some connected component of $P^{\sigma}$.
[@EMQ] \[incariance\] Given $\Phi\in U_\xi^\sigma(G):=U_\xi(G)\cap \Omega^{\sigma}G$, with $\xi\in \mathfrak{I}(G)\cap \mathfrak{k}_\sigma$, set $\gamma=u_\xi\circ \Phi$. Then $\gamma$ takes values in $K$. By continuity, $\Phi_{-1}$ and $\gamma(-1)$ take values in the same connected component of $P^\sigma$.
Hence, together with Theorems \[usd\] and \[GO\], this implies the following.
\[tinva\] Any harmonic map $\varphi$ from $S^2$ into a connected component of $P^{\sigma}$ admits an extended solution $\Phi:S^2\setminus D\to U_\xi^{\sigma}(G):=U_\xi(G)\cap \Omega^{\sigma}G$, for some $\xi\in \mathfrak{I}'(G)\cap\mathfrak{k}_\sigma$ and some discrete subset $D$. If $\sigma=\sigma_\varrho$ is the fundamental outer involution, then $\varphi=\Phi_{-1}$ takes values in $P_\xi^{\sigma_\varrho}$.
By Proposition \[incariance\], $\Phi$ and $\gamma:=u_\xi\circ \Phi$ take values in the same connected component of $P^{\sigma}$ when evaluated at $\lambda=-1$. Since $\gamma:S^1\to G^\sigma$ is a homomorphism, $\gamma$ is in the $G^\sigma$-conjugacy class of $\gamma_{\xi'}$ for some $\xi'\in \mathfrak{I}'(G^\sigma)$, where $G^\sigma$ is the subgroup of $G$ fixed by $\sigma$. Consequently, $\gamma(-1)=g\gamma_{\xi'}(-1)g^{-1}=g\cdot_\sigma \gamma_{\xi'}(-1),$ for some $g\in G^{\sigma}$, which means that $\gamma(-1)$ takes values in the connected component $P_{\xi'}^{\sigma}$. On the other hand, $\gamma$ is in the $G$-conjugacy class of $\gamma_\xi$, with $\xi\in \mathfrak{I}'(G)\cap\mathfrak{k}_\sigma$. If $\sigma$ is the fundamental outer involution $\sigma_\varrho$, then $\mathfrak{I}'(G^{\sigma})=\mathfrak{I}'(G)\cap\mathfrak{k}_\sigma$; and we must have $\xi=\xi'$.
If $\sigma$ is not a fundamental outer involution, each Weyl chamber $\mathcal{W}_\sigma$ in $\mathfrak{t}_{\mathfrak{k}_\sigma}$ can be decomposed into more than one compartment: $ \mathcal{W}_\sigma=C_1\sqcup \ldots \sqcup C_l$, where $C_1=\mathcal{W}\cap \mathfrak{t}_{\mathfrak{k}_\sigma}$ and the remaining compartments are conjugate to $C_1$ under $G$ [@EMQ], that is, there exists $g_i\in G$ satisfying $C_i=\mathrm{Ad}(g_i)(C_1)$ for each $i$. Hence, if we have an extended solution $\Phi:S^2\setminus D\to U_\xi^{\sigma}(G)$ with $\xi\in \mathfrak{I}'(G)\cap\mathfrak{k}_\sigma\subset C_1$, the corresponding harmonic map $\Phi_{-1}$ takes values in one of the connected components $P_{g_i\xi g_i^{-1}}^\sigma$.
### $\varrho$-canonical elements.
Let $I$ be a subset of $\{1,\ldots,k\}$, with $k=\mathrm{rank}(\mathfrak{k}_\varrho)$, and set $$\mathfrak{C}^\varrho_{I}=\Big\{\sum_{i=1}^k n_i \zeta_i|\, n_i\geq 0, \,\mbox{$n_j=0$ iff $j\notin I$}\Big\}.$$ Let $\xi\in\mathfrak{I}'({G}^{\sigma_\varrho})\cap \mathfrak{C}^\varrho_{I}$. We say that $\zeta$ is a *$\varrho$-canonical element* of $G$ (with respect to the choice of $\mathcal{W}$) if $\zeta$ is a maximal element of $(\mathfrak{I}'({G}^{\sigma_\varrho})\cap \mathfrak{C}^\varrho_{I},\preceq)$, that is: if $\zeta\preceq \zeta'$ and $\zeta'\in \mathfrak{I}'({G}^{\sigma_\varrho})\cap \mathfrak{C}^\varrho_{I}$ then $\zeta=\zeta'$.
When $G$ has trivial centre, the duals $\zeta_1,\ldots,\zeta_k$ belong to the integer lattice. Then, for each $I$ there exists a unique $\varrho$-canonical element, which is given by $\zeta_I=\sum_{i\in I}\zeta_i$. In this case, our definition of $\varrho$-canonical element coincides with that of $S$-canonical element in [@EMQ].
Now, consider a fundamental outer involution $\sigma_\varrho$ and let $N$ be an associated outer symmetric $G$-space, that is, $N$ corresponds to an involution of $G$ of the form $\sigma_\varrho$ or $\sigma_{\varrho,i}$, with $\zeta_i$ in the conditions of Theorem \[murak\]. If $G$ has trivial centre, we certainly have $\zeta_i\in \mathfrak{I}'({G}^{\sigma_\varrho})$. As a matter of fact, as we will see later, in most cases we have $\zeta_i\in \mathfrak{I}'({G}^{\sigma_\varrho})$, whether $G$ has trivial centre or not, with essentially one exception: for $G=SU(2n)$ and $N=SU(2n)/SO(2n)$. So, we will treat this case separately and assume henceforth that $\zeta_i\in \mathfrak{I}'({G}^{\sigma_\varrho})$.
Consider the Dynkin diagram of $\mathfrak{e}_6$:
(-0.55,-0.5)(4.82,1.30) (0,0)(1,0) (1,0)(2,0) (2,0)(3,0) (3,0)(4,0) (2,0)(2,1) (-0.1,-0.16)[$\alpha_1$]{} (1.5,1.14)[$\alpha_2$]{} (0.92,-0.16)[$\alpha_3$]{} (1.93,-0.16)[$\alpha_4$]{} (2.93,-0.16)[$ \alpha_5 $]{} (3.97,-0.16)[$\alpha_6$]{}
(0,0) (1,0) (2,0) (3,0) (4,0) (2,1)
This admits a unique nontrivial involution $\varrho$. Let $\{H_1,\ldots,H_6\}$ be the dual basis of $\Delta_0=\{\alpha_1,\ldots,\alpha_6\}$. The semi-fundamental basis $\pi_{\mathfrak{k}_\varrho}(\Delta_0)=\{\beta_1,\beta_2,\beta_3,\beta_4\}$ is given by $\beta_1=\alpha_2$, $\beta_2=\alpha_4$, $\beta_3=\frac{\alpha_1+\alpha_6}{2}$ and $\beta_4=\frac{\alpha_3+\alpha_5}{2}$, whereas the dual basis is given by $\zeta_1=H_2$, $\zeta_2=H_4$, $\zeta_3=H_1+H_6$ and $\zeta_4=H_3+H_5$. Taking account that the elements $H_i$ are related with the duals $\eta_i$ of the fundamental weights by $$\left[H_i\right]=\left[\begin{array}{cccccc} 4/3 & 1 & 5/3 & 2 & 4/3 & 2/3\\ 1 & 2 & 2 & 3 & 2 & 1\\ 5/3 & 2 & 10/3 & 4 & 8/3 & 4/3\\ 2 & 3 & 4 & 6 & 4 & 2\\ 4/3 & 2 & 8/3 & 4 & 10/3 & 5/3\\ 2/3 & 1 & 4/3 & 2 & 5/3 & 4/3 \end{array}\right]\left[\eta_i\right],$$ we see that the elements $\zeta_i$ are in the integer lattice $\mathfrak{I}'(\tilde E_6)\subset \mathfrak{I}'( E_6)$, where $\tilde E_6$ is the compact simply connected Lie group with Lie algebra $\mathfrak{e}_6$, which has centre $\mathbb{Z}_3$, and $E_6$ is the adjoint group $\tilde E_6/\mathbb{Z}_3$.
Taking into account Proposition \[concomp\], we can identify $N$ with the connected component $P^{\sigma_\varrho}_{\zeta_i}=\exp(\pi\zeta_i)P_e^{\sigma_{\varrho,i}}$, which is a totally geodesic submanifold of $G$, via $$\begin{aligned}
\label{tttt}
g\cdot x_0\in N\mapsto\exp(\pi\zeta_i)g\sigma_{\varrho,i}(g^{-1})\in P^{\sigma_\varrho}_{\zeta_i}.\end{aligned}$$ By Theorem \[tinva\], each harmonic map $\varphi:S^2\to N\cong P^{{\sigma_\varrho}}_{\zeta_i}$ admits a $T_{\sigma_\varrho}$-invariant extended solution with values, off a discrete set, in some unstable manifold $U_\xi(G)$, with $\xi\in \mathfrak{I}'({G}^{\sigma_\varrho})\cap \mathfrak{C}_I^\varrho$. By Theorem \[nor\], this extended solution can be multiplied on the left by a constant loop in order to get a normalized extended solution with values in some unstable manifold $U_\zeta(G)$ for some ${\varrho}$-canonical element $\zeta$. Hence, if $G$ has trivial centre, the Bruhat decomposition of $\Omega_{\mathrm{alg}}G$ gives rise to $2^k$ classes of harmonic maps into $P^{\sigma_\varrho}$, that is $2^k$ classes of harmonic maps into *all* outer symmetric $G$-spaces.
However, the normalization procedure given by Theorem \[nor\] does not preserve $T_{\sigma_\varrho}$-invariance, and consequently, as we will see next, normalized extended solutions with values in the same unstable manifold $U_\zeta(G)$, for some $\varrho$-canonical element $\zeta$, correspond in general to harmonic maps into different outer symmetric $G$-spaces. Hence the classification of harmonic two-spheres into outer symmetric $G$-spaces in terms of $\varrho$-canonical elements is manifestly unsatisfactory since it does not distinguishes the underlying symmetric space. In the following sections we overcome this weakness by establishing a classification of all such harmonic maps in terms of pairs $(\zeta,\sigma)$, where $\zeta$ is a ${\varrho}$-canonical element and $\sigma$ an outer involution of $G$.
### Normalization of $T_\sigma$-invariant extended solutions {#normprocedure}
Let $\sigma$ be an outer involution of $G$. The fibre bundle morphisms $\mathcal{U}_{\xi,\xi'}$ preserve $T_{\sigma}$-invariance:
\[proposition\] If $\xi\preceq \xi'$ and $\xi,\xi'\in \mathfrak{I}'({G})\cap \mathfrak{k}_\sigma$, then $\mathcal{U}_{\xi,\xi'}(U_\xi^{\sigma}(G))\subset U_{\xi'}^{\sigma}(G)$.
For $\Phi\in U^{\sigma}_\xi(G)$, write $\Phi=\Psi\cdot \gamma_\xi$ for some $\Psi\in \Lambda^+_{\mathrm{alg}}G^{\mathbb{C}}$. If $\Phi$ is $T_\sigma$-invariant we have $\Psi(\lambda)\cdot \gamma_\xi=\sigma(\Psi(-\lambda))\cdot \gamma_\xi.$ Consequently, we also have $\Psi(\lambda)\cdot \gamma_{\xi'}=\sigma(\Psi(-\lambda))\cdot \gamma_{\xi'},$ which means in turn that $\mathcal{U}_{\xi,\xi'}(\Phi)=\Psi\cdot \gamma_\xi'$ is $T_{\sigma}$-invariant.
Hence, if $\Phi:S^2\setminus D\to U^{\sigma}_\xi(G)$ is an extended solution and $\xi\preceq\xi'$, with $\xi,\xi'\in\mathfrak{I}'(G)\cap \mathfrak{k}_{_\varrho}$, by Theorem \[nor\] and Proposition \[proposition\] we know that $\gamma^{-1}:=\mathcal{U}_{\xi,\xi-\xi'}(\Phi)$ is a constant $T_{\sigma}$-invariant loop if ${\mathfrak{g}}_0^\xi={\mathfrak{g}}_0^{\xi'}$. However, in general, the product $\gamma \Phi$ is not $T_{\sigma}$-invariant.
\[poca\] Assume that $\gamma^{-1},\Phi\in \Omega^{\sigma} G$ and $\gamma(-1)\in P_{\xi}^{\sigma}$ for some $\xi\in\mathfrak{I}(G)\cap \mathfrak{k}_\sigma$. Take $h\in G$ such that $\gamma(-1)=h^{-1}\cdot_\sigma \exp(\pi\xi)$. Then $h\gamma\Phi h^{-1}\in\Omega^{\tau} G$, with $\tau=\mathrm{Ad}(\exp \pi\xi)\circ \sigma$.
Since $\gamma^{-1},\Phi\in \Omega^\sigma G$, a simple computation shows that $T_{\sigma}(\gamma \Phi)=\gamma(-1)^{-1}\gamma\Phi\gamma(-1).$ Since $\gamma(-1)\in P_{\xi}^{\sigma}$, there exists $h\in G$ such that $\gamma(-1)=h^{-1}\cdot_\sigma \exp(\pi\xi)=h^{-1}\exp(\pi\xi)\sigma(h)$. One can check now that $T_\tau(h\gamma \Phi h^{-1})=h\gamma \Phi h^{-1}$.
\[norminf\] Take $\xi,\xi '\in \mathfrak{I}'({G})\cap \mathfrak{k}_\sigma$ such that $\xi\preceq \xi'$. Let $\Phi:S^2\setminus D\to U^{\sigma}_\xi(G)$ be a $T_{\sigma}$-invariant extended solution. If $\gamma^{-1}:=\mathcal{U}_{\xi,\xi-\xi'}(\Phi)$ is a constant loop, there exists $h\in G$ such that $\tilde{\Phi}:=h\gamma\Phi h^{-1}$ takes values in $U_{\xi'}^\tau(G)$, with $\tau=\mathrm{Ad}(\exp\pi(\xi-\xi'))\circ {\sigma}.$
Additionally, if $\sigma$ is the fundamental outer involution $\sigma_\varrho$, the harmonic map $\Phi_{-1}$ takes values in $P_\xi^{\sigma}$ and $\tilde{\Phi}_{-1}$ takes values in $P_{\xi'}^{\tau}$, which implies that $\Phi_{-1}$ is given, up to isometry, by $$\exp(\pi(\xi-\xi'))\tilde{\Phi}_{-1}:S^2\to P_\xi^\sigma.$$
Assume that $\gamma^{-1}:=\mathcal{U}_{\xi,\xi-\xi'}(\Phi)=\Psi\cdot \gamma_{\xi-\xi'}$ is a constant loop. We can write $\Psi\gamma_{\xi-\xi'}=\gamma^{-1} b$ for some $b:S^2\setminus D\to \Lambda^+_{\mathrm{alg}}G$. Then $$\Phi =\Psi\cdot\gamma_\xi =\Psi\cdot\gamma_{\xi-\xi'}\gamma_{\xi'}= \gamma^{-1} b\cdot\gamma_{\xi'},$$ which implies that $\gamma\Phi$ takes values in $U_{\xi'}(G)$. On the other hand, since $\gamma^{-1}$ is $T_\sigma$-invariant (by Proposition \[proposition\]), $\gamma(-1)\in P^{\sigma}$.
Take $\eta\in \mathfrak{I}'(G^\sigma)$ and $h\in G$ such that $\gamma(-1)\in P^{\sigma}_{\eta}$ and $\gamma(-1)=h^{-1}\cdot_\sigma\exp\pi\eta$. From Lemma \[poca\], we see that $\tilde\Phi:= h\gamma\Phi h^{-1}$ is $T_\tau$-invariant. Hence $\tilde{\Phi}$ takes values in $U_{\xi'}^\tau(G)$. Since $\gamma$ is constant, $\tilde\Phi$ is an extended solution.
If $\sigma=\sigma_\varrho$, then $\mathfrak{I}'(G^{\sigma_\varrho})=\mathfrak{I}'(G)\cap \mathfrak{k}_{\sigma_\varrho}$, which implies that $\eta=\xi-\xi'$. The element $h\in G$ is such that $$\gamma(-1)=h^{-1}\exp(\pi(\xi-\xi'))\sigma_\varrho(h).$$ On the other hand, since, by Theorem \[tinva\], $\Phi_{-1}$ takes values in $P_\xi^{\sigma_\varrho}$, we also have $\Phi_{-1}=g\exp(\pi\xi)\sigma_\varrho(g^{-1})$ for some lift $g:S^2\to G$. Hence $$\begin{aligned}
\tilde\Phi_{-1}&=h\gamma(-1)\Phi_{-1} h^{-1}=\exp(\pi(\xi-\xi'))\sigma_\varrho(h)g\exp(\pi\xi)\sigma_\varrho(\sigma_\varrho(h)g)^{-1}\\&=\exp(\pi(\xi-\xi'))(\sigma_\varrho(h)g\cdot_{\sigma_\varrho} \exp\pi\xi)
\end{aligned}$$ Hence, in view of Proposition \[concomp\], $\tilde\Phi_{-1}$ takes values in $P_{\xi'}^\tau=\exp(\pi(\xi-\xi'))P_{\xi}^\sigma$.
Under some conditions on $\xi\preceq \xi'$, the morphism $\mathcal{U}_{\xi,\xi-\xi'}(\Phi)$ is always a constant loop.
\[norm2\] Take $\xi,\xi '\in \mathfrak{I}'({G})\cap \mathfrak{k}_\sigma$ such that $\xi\preceq \xi'$. Assume that $$\begin{aligned}
{\mathfrak{g}}_{2i}^\xi\cap \mathfrak{m}_\sigma^{\mathbb{C}}\subset \bigoplus_{0\leq j<2i}{\mathfrak{g}}^{\xi-\xi'}_j,\quad\quad
{\mathfrak{g}}_{2i-1}^\xi\cap \mathfrak{k}_\sigma^{\mathbb{C}}\subset \bigoplus_{0\leq j<2i-1}{\mathfrak{g}}^{\xi-\xi'}_j,\label{toing2}
\end{aligned}$$ for all $i> 0$. Then, $\mathcal{U}_{\xi,\xi-\xi'}:U_\xi^\sigma(G)\to U_{\xi-\xi'}^\sigma(G)$ transforms $T_\sigma$-invariant extended solutions in constant loops.
Given an extended solution $\Phi:S^2\setminus D\to U^\sigma_\xi(G)$, choose $\Psi:S^2\setminus D \to \Lambda_{\mathrm{alg}}^+G^{\mathbb{C}}$ such that $\Phi=\Psi\cdot\gamma_\xi $ and $T_\sigma(\Psi)=\Psi$. Differentiating this we see that $$\label{ondepsi}
\mathrm{Im}\Psi^{-1}\Psi_z\subset \bigoplus_{i\geq 0}\lambda^{2i}\mathfrak{k}_\sigma^{\mathbb{C}}\oplus\bigoplus_{i\geq 0}\lambda^{2i+1}\mathfrak{m}_\sigma^{\mathbb{C}}.$$ Write $\Psi^{-1}\Psi_z=\sum_{r\geq 0}\lambda^rX'_r$. Since $\xi\preceq \xi-\xi'$, by Proposition \[popo\] and Proposition \[proposition\], $\mathcal{U}_{\xi,\xi-\xi'}(\Phi)$ is an extended solution with values in $U^\sigma_{\xi-\xi'}(G)$. Hence, taking into account Lemma \[poi\], in order to prove that $\mathcal{U}_{\xi,\xi-\xi'}(\Phi)$ is constant we only have to check that the component of $X'_r$ over ${\mathfrak{g}}^{\xi-\xi'}_{r+1}$ vanishes for all $r\geq 0$.
From and we see that, for $r=2i$, $X'_{2i}$ takes values in $\bigoplus_{j\leq 2i+1} {\mathfrak{g}}_{j}^\xi\cap \mathfrak{k}_\sigma^{\mathbb{C}}$. But, since $\xi\preceq \xi-\xi'$ and, by hypothesis, holds, we have $$\bigoplus_{j\leq 2i+1} {\mathfrak{g}}_{j}^\xi\cap \mathfrak{k}_\sigma^{\mathbb{C}}=\big(\bigoplus_{j\leq 2i} {\mathfrak{g}}_{j}^\xi\cap \mathfrak{k}_\sigma^{\mathbb{C}}\big)\oplus \big({\mathfrak{g}}_{2i+1}^\xi\cap \mathfrak{k}_\sigma^{\mathbb{C}}\big)\subset
\big(\bigoplus_{j\leq 2i} {\mathfrak{g}}_{j}^{\xi-\xi'}\cap \mathfrak{k}_\sigma^{\mathbb{C}}\big)\oplus \bigoplus_{0\leq j<2i+1}{\mathfrak{g}}^{\xi-\xi'}_j.$$ Hence the component of $X'_{2i}$ over ${\mathfrak{g}}^{\xi-\xi'}_{2i+1}$ vanishes for all $i\geq 0$. Similarly, for $r=2i-1$, $X'_{2i-1}$ takes values in $\bigoplus_{j\leq 2i} {\mathfrak{g}}_{j}^\xi\cap\mathfrak{m}_\sigma^{\mathbb{C}}$, and we can check that the component of $X'_{2i-1}$ over ${\mathfrak{g}}^{\xi-\xi'}_{2i}$ vanishes for all $i>0$.
Hence $\gamma^{-1}:=\mathcal{U}_{\xi,\xi-\xi'}(\Phi)=\Psi\cdot \gamma_{\xi-\xi'}$ is a constant loop.
We say that $\zeta\in \mathfrak{I}'({G}^{\sigma_\varrho})\cap \mathfrak{C}^\varrho_I$ is a $\varrho$-*semi-canonical* element if $\zeta$ is of the form $\zeta=\sum_{i\in I}n_i\zeta_i$ with $1\leq n_i\leq 2m_i$, where $m_i$ is the least positive integer which makes $m_i\zeta_i\in\mathfrak{I}'({G}^{\sigma_\varrho})$.
\[crorol\] Take $\xi\in \mathfrak{I}'({G}^{\sigma_\varrho})\cap \mathfrak{C}^\varrho_I$, with $I\subset\{1,\ldots,k\}$. Let $\Phi:S^2\setminus D\to U^{\sigma_\varrho}_\xi(G)$ be a $T_{\sigma_\varrho}$-invariant extended solution, and let $\varphi:S^2\to P_\xi^{\sigma_\varrho}$ be the corresponding harmonic map. Then there exist $h\in G$, a constant loop $\gamma$, and a $\varrho$-*semi-canonical* $\zeta$ such that $\tilde\Phi:=h\gamma\Phi h^{-1}$ defined on $S^2\setminus D$ takes values in $U^{\sigma_\varrho}_{\zeta}(G)$. The harmonic map $\tilde\Phi_{-1}$ takes values in $P^{\sigma_\varrho}_{\zeta}= P^{\sigma_\varrho}_{\xi}$ and coincides with $\varphi$ up to isometry.
Write $\xi=\sum_{i\in I}r_i\zeta_i$, with $r_i>0$. For each $i\in I$, let $n_i$ be the unique integer number in $\{1,\ldots, 2m_i\}$ such that $n_i=r_i\mod 2m_i.$ Set $\zeta=\sum_{i\in I}n_i\zeta_i$. It is clear that $\xi\preceq \zeta$ and $\zeta \in \mathfrak{I}'({G}^{\sigma_\varrho})\cap \mathfrak{C}^\varrho_I$. Observe also that conditions hold automatically for any $\xi'\in \mathfrak{I}'({G}^{\sigma_\varrho})\cap \mathfrak{C}^\varrho_I$ satisfying $\xi\preceq\xi'$. In particular they hold for $\xi'=\zeta$. Finally, since $\xi-\zeta=2\sum_{i\in I}m_ik_i\zeta_i$ for some nonnegative integer numbers $k_i$, then $\exp{\pi(\xi-\zeta)}=e$, and the result follows from Propositions \[norminf\] and \[norm2\].
### Classification of harmonic two-spheres into outer symmetric spaces {#classs}
To sum up, in order to classify all harmonic two-spheres into outer symmetric spaces we proceed as follows:
1. Start with a fundamental outer involution $\sigma_\varrho$ and let $N$ be an outer symmetric $G$-space corresponding to an involution of the form $\sigma_\varrho$ or $\sigma_{\varrho,i}$ of $G$, according to , where the element $\zeta_i$ is in the conditions of Theorem \[murak\]. We assume that $\exp2\pi\zeta_i=e$, that is $\zeta_i\in \mathfrak{I}'({G}^{\sigma_\varrho})$. Let $\varphi:S^2\to N$ be an harmonic map and identify $N$ with $P_{\zeta_i}^{\sigma_\varrho}=\exp(\pi\zeta_i)P_e^{\sigma_{\varrho,i}}$ via the totally geodesic embedding . If $N$ is the fundamental outer space with involution $\sigma_\varrho$ we simply identify $N$ with $P_e^{\sigma_\varrho}$ via $\iota_{\sigma_\varrho}$.
2. By Theorem \[tinva\], $\varphi:S^2\to N\cong P_{\zeta_i}^{\sigma_\varrho}$ admits a $T_{\sigma_\varrho}$-invariant extended solution $\Phi:S^2\to\Omega^{\sigma_\varrho} G$ which takes values, off some discrete subset $D$, in some unstable manifold $U_{\zeta'}^{\sigma_\varrho}(G)$, with $\zeta'\in\mathfrak{I}'({G}^{\sigma_\varrho})$; moreover, $P_{\zeta'}^{\sigma_\varrho}=P_{\zeta_i}^{\sigma_\varrho}$.
3. By Corollary \[crorol\], we can assume that $\zeta'$ is a $\varrho$-semi-canonical element in $\mathfrak{I}'({G}^{\sigma_\varrho})\cap \mathfrak{C}^\varrho_I$. If $\zeta$ is a $\varrho$-canonical element such that $\zeta'\preceq \zeta$ and $\mathcal{U}_{\zeta',\zeta'-\zeta}(\Phi)$ is constant, then, taking into account Proposition \[norminf\], there exists a $T_\tau$-invariant extended solution $\tilde\Phi: S^2\setminus D\to U^\tau_{\zeta}(G),$ where $$\label{otau}\tau=\mathrm{Ad}(\exp \pi(\zeta'-\zeta))\circ \sigma_\varrho,$$ such that the harmonic map $\varphi$ is given, up to isometry, by $\tilde\Phi_{-1}:S^2\to P_{\zeta}^\tau.$ Here we identify $N$ with $P_{\zeta}^\tau =\exp(\pi(\zeta'-\zeta)) P_{\zeta_i}^{\sigma_\varrho}$ via the composition of with the left multiplication by $\exp(\pi(\zeta'-\zeta))$.
4. By Proposition \[norm2\], there always exists a $\varrho$-canonical element $\zeta$ in such conditions.
Hence, we classify harmonic spheres into outer symmetric $G$-spaces in terms of pairs $(\zeta,\tau)$, where $\zeta$ is a $\varrho$-canonical element and $\tau$ is an outer involution of the form for some $\varrho$-semi-canonical element $\zeta'$ with $\zeta'\preceq \zeta$.
### Weierstrass Representation for $T_\sigma$-invariant Extended Solutions
From and Proposition \[BG\], we obtain the following.
\[sigmaweirstrass\] Let $\Phi:M\to \Omega_{\mathrm{alg}}^\sigma G$ be an extended solution. There exists a discrete set $D'\supseteq D$ of $M$ such that $\Phi{\big|_{M\setminus D'}}=\exp C\cdot \gamma_\xi$ for some holomorphic vector-valued function $C: M\setminus D'\to (\mathfrak{u}^0_\xi)_\sigma$, where $(\mathfrak{u}^0_\xi)_\sigma$ is the finite dimensional nilpotent subalgebra of $\Lambda^+_{\mathrm{alg}}\mathfrak g^{\mathbb{C}}$ defined by $$(\mathfrak{u}^0_\xi)_\sigma=\bigoplus_{0\leq 2i<r(\xi)}\lambda^{2i}(\mathfrak{p}^\xi_{2i})^\perp\cap \mathfrak{k}^{\mathbb{C}}_\sigma\oplus \bigoplus_{0\leq 2i+1<r(\xi)}\lambda^{2i+1}(\mathfrak{p}^\xi_{2i+1})^\perp\cap \mathfrak{m}^{\mathbb{C}}_\sigma,$$ with $(\mathfrak{p}^\xi_i)^\perp=\bigoplus _{i<j\leq r(\xi)}\mathfrak{g}_j^\xi$. Moreover, $C$ can be extended meromorphically to $M$.
Examples
========
Next we will describe explicit examples of harmonic spheres into *classical* outer symmetric spaces.
Outer symmetric $SO(2n)$-spaces
-------------------------------
For details on the structure of $\mathfrak{so}(2n)$ see [@fulton_harris]. Consider on $\mathbb{R}^{2n}$ the standard inner product $\langle \cdot, \cdot \rangle$ and fix a complex basis $\mathbf{u}=\{u_1,\ldots,u_n,\overline{u}_1,\ldots,\overline{u}_n\}$ of $\mathbb{C}^{2n}=(\mathbb{R}^{2n})^\mathbb{C}$ satisfying $$\label{us}
\langle u_i,u_j\rangle=0, \quad \langle u_i,\overline{u}_j\rangle=\delta_{ij},\quad \mbox{for all $1\leq i,j\leq n$}.$$ Throughout this section we will denote by $V_l$ the $l$-dimensional isotropic subspace spanned by $\overline{u}_1,\ldots,\overline{u}_l$.
Set $E_i=E_{i,i}-E_{n+i,n+i}$, where $E_{j,j}$ is a square matrix, with respect to the basis $\mathbf{u}$, whose $(j,j)$-entry is $\mathrm{i}$ and all other entries are $0$. The complexification ${\mathfrak{t}}^{\mathbb{C}}$ of the algebra ${\mathfrak{t}}$ of diagonal matrices $\sum a_iE_i$, with $a_i\in \mathbb{R}$ and $\sum a_i=0$, is a Cartan subalgebra of $\mathfrak{so}(2n)^{\mathbb{C}}$. Let $\{L_1,\ldots,L_n\}$ be the dual basis in $\mathrm{i}{\mathfrak{t}}^*$ of $\{E_1,\ldots,E_n\}$, that is $L_i(E_j)=\mathrm{i}\delta_{ij}$. The roots of $\mathfrak{so}(2n)$ are the vectors $\pm L_i\pm L_j$ and $\pm L_i\mp L_j$, with $i\neq j$ and $1\leq i,j\leq n$.
Consider the endomorphisms $$\label{geradores}
X_{i,j}=E_{i,j}-E_{n+j,n+i},\,\,Y_{i,j}=E_{i,n+j}-E_{j,n+i},\,\, Z_{i,j}=E_{n+i,j}-E_{n+j,i},$$ where $E_{i,j}$, with $i\neq j$, is a square matrix whose $(i,j)$-entry is $1$ and all other entries are $0$. The root spaces of $L_i-L_j$, $L_i+L_j$ and $-L_i-L_j$, respectively, are generated by the endomorphisms $X_{i,j}$, $Y_{i,j}$ and $Z_{i,j}$, respectively.
Fix the positive root system $\Delta^+=\{L_i\pm L_j\}_{i<j}.$ The positive simple roots are $\alpha_i=L_i-L_{i+1}$, for $1\leq i\leq n-1$, and $\alpha_n=L_{n-1}+L_n$. The vectors of the dual basis $\{H_1,\ldots,H_n\}\subset {\mathfrak{t}}$ are given by $H_i =E_1+E_2+\ldots+E_i$, for $1\leq i\leq n-2$, $$\mbox{$H_{n-1} =\frac12(E_1+E_2+\ldots+E_{n-1}-E_n)$, and $H_n =\frac12(E_1+E_2+\ldots+E_{n-1}+E_n)$}.$$
Consider the non-trivial involution $\varrho$ of the corresponding Dynkin diagram,
(-0.9,1.11)(4.85,2.76) (0,0)(1,0) (1,0)(2,0) (2,0)(3,0) (3,0)(4,0) (2,0)(1.95,0.21) (-0.6,1.86)[$\alpha_1$]{} (0.43,1.84)[$\alpha_2$]{} (0.92,-0.13)[$\alpha_3$]{} (1.8,1.86)[$\alpha_{n-3}$]{} (2.7,1.86)[$ \alpha_{n-2}$]{} (3.97,-0.13)[$\alpha_6$]{} (-0.5,2)(0.5,2) (3,2)(3.86,1.5) (3,2)(3.85,2.5) (2,2)(3,2) (0.84,2)[$\ldots\ldots\ldots$]{} (3.96,1.55)[$\alpha_n$]{} (3.96,2.63)[$\alpha_{n-1}$]{}
(0,0) (1,0) (2,0) (3,0) (4,0) (1.95,0.21) (0,4) (-0.5,2) (0.5,2) (3,2) (3.86,1.5) (3.85,2.5) (-2.26,0.91) (2,2)
This involution fixes $\alpha_i$ if $i\leq n-2$ and $\varrho(\alpha_{n-1})= \alpha_n$. The corresponding semi-fundamental basis $\pi_{\mathfrak{k}_\varrho}(\Delta_0)=\{\beta_1,\ldots,\beta_{n-1}\}$ is given by $$\mbox{$\beta_i=\alpha_i=L_i-L_{i+1}$, if $i\leq n-2$, and $\beta_{n-1}=\frac12(\alpha_{n-1}+\alpha_{n})=L_{n-1}$},$$ whereas the dual basis $\{\zeta_1,\ldots,\zeta_{n-1}\} $ is given by $\zeta_i=E_1+\ldots+E_i$, with $i=1,\ldots,n-1$. Since each $\zeta_i$ belongs to the integer lattice $\mathfrak{I}(SO(2n)^{\sigma_\varrho})$, we have:
The $\varrho$-semi-canonical elements of $SO(2n)$ are precisely the elements $\zeta=\sum_{i=1}^{n-1} m_i\zeta_i$ such that $m_i\in\{0,1,2\}$ for $1\leq i\leq n-1$.
The fundamental outer symmetric $SO(2n)$-space is the real projective space $\mathbb{R}P^{2n-1}$, and the associated outer symmetric $SO(2n)$-spaces are the real Grassmannians $G_p(\mathbb{R}^{2n})$ with $p>1$ odd.
### Harmonic maps into real projective spaces $\mathbb{R}P^{2n-1}$.
Consider as base point the one dimensional real vector space $V_0$ spanned by $e_n=(u_n+\overline{u}_n)/\sqrt{2}$ in $\mathbb{R}^{2n}$, which establishes an identification of $\mathbb{R}P^{2n-1}$ with $SO(2n)/S(O(1)O(2n-1)).$ Denote by $\pi_{V_0}$ and $\pi_{V_0}^\perp$ the orthogonal projections onto $V_0$ and $V_0^\perp$, respectively. The fundamental involution is given by $\sigma_\varrho=\mathrm{Ad}(s_0)$, where $s_0=\pi_{V_0}-\pi_{V_0}^\perp$. Following the classification procedure established in Section \[classs\], we start by identifying $\mathbb{R}P^{2n-1}$ with $P_e^{\sigma_\varrho}$.
\[classesproj\] Each harmonic map $\varphi:S^2\to\mathbb{R}P^{2n-1}$ belongs to one of the following classes: $(\zeta_l,\sigma_{\varrho,l})$, with $1\leq l\leq n-1$.
Let $\zeta$ be a $\varrho$-semi-canonical element and write $$\label{ozeta}\zeta=\sum_{i\in I_1}\zeta_i+ \sum_{i\in I_2}2\zeta_i$$ for some disjoint subsets $I_1$ and $I_2$ of $\{1,\ldots,n-1\}$. By Proposition \[fundcan\], $P^{\sigma_\varrho}_\zeta\cong \mathbb{R}P^{2n-1}$ if and only if either $I_1=\emptyset$ or $I_1=\{n-1\}$. Suppose that $I_1=\{n-1\}$. In this case, $\exp\pi\zeta=\exp\pi\zeta_{n-1} \in P_{\zeta_{n-1}}^{\sigma_\varrho}$. We claim that $P_{\zeta_{n-1}}^{\sigma_\varrho}$ is not the connected component of $P^{\sigma_\varrho}$ containing the identity $e$. Write $\exp\pi\zeta_{n-1}=\pi_V-\pi_V^\perp,$ where $V$ is the two-dimensional real space spanned by $e_{n}$ and $e_{2n}$. For each $g\in P_e^{\sigma_\varrho}$, since the $G$-action $\cdot_{\sigma_\varrho}$ defined by is transitive, we have $g=h\cdot_{\sigma_\varrho}e=hs_0h^{-1}s_0$ for some $h\in G$, which means that $gs_0=hs_0h^{-1}$. In particular, the $+1$-eigenspaces of $gs_0$ must be $1$-dimensional. However, a simple computation shows that the $+1$-eigenspace of $\exp(\pi\zeta_{n-1})s_0$ is $3$-dimensional, which establishes our claim.
Then, any harmonic map $\varphi:S^2\to\mathbb{R}P^{2n-1}\cong P_e^{\sigma_\varrho}$ admits a $T_{\sigma_\varrho}$-invariant extended solution $\Phi:S^2\setminus D\to U^{\sigma_\varrho}_\zeta(SO(2n))$ with $\zeta$ a $\varrho$-semi-canonical element of the form $\zeta=\sum_{i\in I_2}2\zeta_i.$ Set $l=\max I_2$. Next we check that $\zeta$ and $\zeta_l$ satisfy the conditions of Proposition \[norm2\], with $\xi=\zeta$ and $\xi'=\zeta_l$. It is clear that $\zeta\preceq \zeta_l$. Now, according to and , we can take $\Delta'_\varrho=\{L_i-L_n, L_n-L_i\}$. Hence, for $i>0$, $$\mathfrak{g}_{2i}^\zeta\cap \mathfrak{m}_\varrho^{\mathbb{C}}=\bigoplus_{\alpha\in \Delta'_\varrho\cap \Delta_\zeta^{2i}}({\mathfrak{g}}_\alpha\oplus {\mathfrak{g}}_{\varrho(\alpha)})\cap\mathfrak{m}^{\mathbb{C}}_\varrho,$$ where $\Delta_\zeta^{2i}=\{\alpha\in \Delta|\, \alpha(\zeta)=2i\mathrm{i}\}$. Since $$(L_j-L_n)(\zeta)=(\alpha_j+\alpha_{j+1}+\ldots+\alpha_{n-1})(\zeta)=2|I_2\cap \{j,\ldots,n-1\}|\mathrm{i},$$ we have $$\Delta'_\varrho\cap \Delta_\zeta^{2i}=\{L_j-L_n|\,\, \mbox{$1\leq j\leq l$, and $|I_2\cap\{j,\ldots,l\}|=i$}\}.$$ Then, given a root $\alpha=L_j-L_n\in \Delta'_\varrho\cap \Delta_\zeta^{2i}$ (in particular, $j\leq l$) we have $\alpha(\zeta-\zeta_l)=(2i-1)\mathrm{i},$ which means that ${\mathfrak{g}}_\alpha\subset {\mathfrak{g}}^{\zeta-\zeta_l}_{2i-1}$. Consequently, $${\mathfrak{g}}_{2i}^\zeta\cap \mathfrak{m}_\varrho^{\mathbb{C}}\subset \bigoplus_{0\leq j<2i}{\mathfrak{g}}^{\zeta-\zeta_l}_j.$$ Since ${\mathfrak{g}}_{2i-1}^\zeta=\{0\}$ for all $i$, we conclude that holds, and the statement follows from Propositions \[norminf\] and \[norm2\].
It is known [@calabi_1967] that there are no full harmonic maps $\varphi:S^2\to\mathbb{R}P^{2n-1}$. The class of harmonic maps associated to $(\zeta_l,\sigma_{\varrho,l})$ consists precisely of those $\varphi$ with $\varphi(S^2)$ contained, up to isometry, in some $\mathbb{R}P^{2l}$, as shown in the next theorem.
\[RAs\] Given $1\leq l\leq n-1$, any harmonic map $\varphi:S^2\to\mathbb{R}P^{2n-1}$ in the class $(\zeta_l,\sigma_{\varrho,l})$ is given by $$\label{projspace}
\varphi={R}\cap(A\oplus \overline{A})^\perp,$$ where ${R}$ is a constant $2l+1$-dimensional subspace of $\mathbb{R}^{2n}$ and $A$ is a holomorphic isotropic subbundle of $S^2\times {R}$ of rank $l$ satisfying $\partial A\subseteq \overline{A}^\perp$. The corresponding extended solutions have uniton number $2$ with respect to the standard representation of $SO(2n)$.
Let $\varphi:S^2\to\mathbb{R}P^{2n-1}$ be a harmonic map in the class $(\zeta_l,\sigma_{\varrho,l})$. This means that $\varphi$ admits an extended solution $\Phi:S^2\setminus D \to U_{\zeta_l}^{\sigma_{\varrho,l}}(SO(2n))$. Up to isometry, $\varphi$ is given by $\Phi_{-1}$, which takes values in $P_{\zeta_l}^{\sigma_{\varrho,l}}=\exp(\pi \zeta_l)P_e^{\sigma_{\varrho}}$. This connected component is identified with $\mathbb{R}P^{2n-1}$ via $$\label{identi}
g\cdot V_0\mapsto \exp(\pi \zeta_l)g\sigma_\varrho(g^{-1}).$$ Write $\gamma_{\zeta_l}(\lambda)=\lambda^{-1}\pi_{V_l}+\pi_{V_l\oplus\overline{V}_l}^\perp+\lambda \pi_{\overline{V}_l},$ where $V_l$ is the $l$-dimensional isotropic subspace spanned by $\overline{u}_1,\ldots,\overline{u}_l$.
We have $r(\zeta_l)=2$ if $l>1$ and $r(\zeta_1)=1$. Consequently, by Theorem \[sigmaweirstrass\], $$(\mathfrak{u}^0_{\zeta_l})_{\sigma_{\varrho,l}}=(\mathfrak{p}^{\zeta_l}_{0})^\perp\cap \mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,l}}\oplus \lambda (\mathfrak{p}^{\zeta_l}_{1})^\perp\cap \mathfrak{m}^{\mathbb{C}}_{\sigma_{\varrho,l}}.$$ Here $(\mathfrak{p}^{\zeta_l}_{1})^\perp={\mathfrak{g}}_2^{\zeta_l}$, which is the null space for $l=1$. For $l>1$, since $\zeta_l=E_1+\ldots+E_l$, we have ${\mathfrak{g}}_2^{\zeta_l}=\{L_i+L_j|\, 1\leq i< j\leq l\}\subset \Delta(\mathfrak{k}_\varrho)$ and, from , $$\mathfrak{m}_{\sigma_{\varrho,l}}^{\mathbb{C}}=\bigoplus \mathfrak{g}^{\zeta_l}_{2i+1}\cap \mathfrak{k}^{\mathbb{C}}_{\varrho}\oplus\bigoplus \mathfrak{g}^{\zeta_l}_{2i}\cap \mathfrak{m}^{\mathbb{C}}_{\varrho}.$$ Hence $(\mathfrak{p}^{\zeta_l}_{1})^\perp\cap \mathfrak{m}^{\mathbb{C}}_{\sigma_{\varrho,l}}= {\mathfrak{g}}_2^{\zeta_l}\cap \mathfrak{m}^{\mathbb{C}}_{\varrho}=\{0\}.$ Then, for any $l\geq 1$, we can write $\Phi=\exp C\cdot \gamma_{\zeta_l}$ for some holomorphic function $$C:S^2\setminus D\to (\mathfrak{p}^{\zeta_l}_{0})^\perp\cap\mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,l}} =({\mathfrak{g}}_1^{\zeta_l}\oplus{\mathfrak{g}}_2^{\zeta_l})\cap \mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,l}},$$ which means that $\Phi$ is a $S^1$-invariant extended solution with uniton number $2$: $$\label{phiw}
\Phi_\lambda=\lambda^{-1}\pi_{W}+\pi_{W\oplus\overline{W}}^\perp+\lambda \pi_{\overline{W}},$$ where $W$ is a holomorphic isotropic subbundle of $S^2\times \mathbb{R}^{2n}$ of rank $l$ satisfying the superhorizontality condition $\partial W\subseteq \overline{W}^\perp$.
Set $\tilde{V}_l= V_l\oplus \overline{V}_l$ and $\tilde{W}= W\oplus \overline{W}$. The $T_{{\sigma_{\varrho,l}}}$-invariance of $\Phi$ implies that $$\label{inva} [\pi_W,\pi_{V_0\oplus \tilde V_l}]=0.$$ Now, write $\varphi=g\cdot V_0$ and consider the identification . We must have $$\label{phirp}
\Phi_{-1}= \exp(\pi \zeta_l)g\sigma_\varrho(g^{-1})=\exp(\pi \zeta_l)(\pi_\varphi-\pi_\varphi^\perp)s_0.$$ From and we obtain $$\label{pias}
\pi_\varphi-\pi_\varphi^\perp=\mathrm{Ad}(s_0)\big(\pi_{V_0\oplus \tilde V_l}\pi_{\tilde W}^\perp+\pi_{V_0\oplus \tilde V_l}^\perp\pi_{\tilde W}
-\pi_{V_0\oplus \tilde V_l}\pi_{\tilde W}-\pi_{V_0\oplus \tilde V_l}^\perp\pi_{\tilde W}^\perp\big).$$ In view of , we see that $\pi_{V_0\oplus \tilde V_l}\pi_{\tilde W}^\perp+\pi_{V_0\oplus \tilde V_l}^\perp\pi_{\tilde W}$ is an orthogonal projection, and implies that this must be an orthogonal projection onto a $1$-dimensional real subspace. Then, one of its two terms vanishes, that is either $\tilde W\subset V_0\oplus \tilde V_l$ or $\tilde W^\perp\subset (V_0\oplus \tilde V_l)^\perp$. For dimensional reasons, we see that the second case can not occur. Hence, we have $$\pi_\varphi=\mathrm{Ad}(s_0)( \pi_{V_0\oplus \tilde V_l}\pi_{\tilde W}^\perp)= \pi_{V_0\oplus \tilde V_l}\mathrm{Ad}(s_0)(\pi_{\tilde W}^\perp),$$ that is holds with ${R}=V_0\oplus V_l\oplus \overline V_l$ and $A=s_0(W)$.
If $\varphi$ is full in ${R}$, then the isotropic subbundle $A$ is the $l$-osculating space of some full totally isotropic holomorphic map $f$ from $S^2$ into the complex projective space of ${R}$, the so called *directrix curve* of $\varphi$. That is, in a local system of coordinates $(U,z)$, we have $A(z)=\mathrm{Span}\big\{g,g',\ldots,g^{(l-1)}\}$, where $g$ is a lift of $f$ over $U$ and $g^{(r)}$ the $r$-th derivative of $g$ with respect to $z$. Hence, formula agrees with the classification given in Corollary 6.11 of [@eells_wood_1983].
Let us consider the case $n=2$. We have only one class of harmonic maps: $(\zeta_1,\sigma_{\varrho,1})$. From Theorem \[RAs\], any such harmonic map $\varphi:S^2\to{\mathbb{R}}P^3$ is given by $\varphi=R\cap(A\oplus\overline{A})^\perp$, where $R$ is a constant 3-dimensional subspace of ${\mathbb{R}}^4$ and $A$ a holomorphic isotropic subbundle of $S^2\times R$ of rank 1 such that $\partial A\subseteq\overline{A}^\perp$. Taking into account Theorem \[sigmaweirstrass\], any such holomorphic subbundles $A$ can be obtained from a meromorphic function $a$ on $S^2$ as follows.
We have $\zeta_1=E_1$ and the corresponding extended solutions have uniton number $r(\zeta_1)=1$ (with respect to the standard representation). Any extended solution $\Phi:S^2\setminus D\to U_{\zeta_1}^{\sigma_{\varrho,1}}(SO(4))$ is given by $\Phi=\exp C\cdot\gamma_{\zeta_1}$, with $\gamma_{\zeta_1}(\lambda)=\lambda^{-1}\pi_{{V}_1}+\pi_{V_1\oplus\overline{V}_1}^\perp+\lambda\pi_{\overline{V}_1}$, for some holomorphic vector-valued function $C:S^2\setminus D\to(\mathfrak{u}_{\zeta_1}^0)_{\sigma_{\varrho,1}}$, where $$(\mathfrak{u}_{\zeta_1}^0)_{\sigma_{\varrho,1}}=(\mathfrak{p}_0^{\zeta_1})^\perp\cap\mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,1}}=\mathfrak{g}_1^{\zeta_1}\cap\mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,1}}
=(\mathfrak{g}_{L_1-L_2}\oplus\mathfrak{g}_{L_1+L_2})\cap\mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,1}}.$$
Considering the root vectors $X_{i,j},Y_{i,j},Z_{i,j}$ as defined in , we have $Y_{1,2}=\sigma_{\varrho,1}(X_{1,2})$. Hence $C=a(z)(X_{1,2}+Y_{1,2})$ where $a(z)$ is a meromorphic function on $S^2$. In this case, from , it follows that $(\exp C)^{-1}(\exp C)_z=C_z$, and it is clear that the extended solution condition for $\Phi$ holds independently of the choice of the meromorphic function $a(z)$. Then, with respect to the complex basis $\mathbf{u}=\{u_1,u_2,\overline{u}_1,\overline{u}_2\}$, $$\label{zeta1rp3}\exp C\cdot\gamma_{\zeta_1}=\left[\begin{array}{cccc} 1& a & -a^2 & a\\ 0 & 1 & -a & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & -a & 1 \end{array}\right]\cdot\gamma_{\zeta_1}$$ and the subbundle $A$ of $R=\mathrm{Span}\{{u}_1,\overline{u}_1, u_2+\overline{u}_2\}$ is given by $A=\exp C\cdot {V}_1=\mathrm{span}\{(a^2,a,-1,a)\}$, which satisfies $\partial A\subseteq\overline{A}^\perp$.
Any harmonic two-sphere into ${\mathbb{R}}P^5$ in the class $(\zeta_1,\sigma_{\varrho,1})$ takes values in some ${\mathbb{R}}P^3$ inside ${\mathbb{R}}P^5$ and so it is essentially of the form (\[zeta1rp3\]). Next we consider the Weierstrass representation of harmonic spheres into ${\mathbb{R}}P^5$ in the class $(\zeta_2,\sigma_{\varrho,2})$, which are given by $\varphi=R\cap(A\oplus\overline{A})^\perp$, where $R$ is a constant 5-dimensional subspace of ${\mathbb{R}}^6$ and $A$ a holomorphic isotropic subbundle of $S^2\times R$ of rank 2 such that $\partial A\subseteq\overline{A}^\perp$. We have $\zeta_2=E_1+E_2$, then $r(\zeta_2)=2$. Any extended solution $\Phi:S^2\setminus D\to U_{\zeta_2}^{\sigma_{\varrho,2}}(SO(6))$ is given by $\Phi=\exp C\cdot\gamma_{\zeta_2}$, with $\gamma_{\zeta_2}(\lambda)=\lambda^{-1}\pi_{{V}_2}+\pi_{V_2\oplus\overline{V}_2}^\perp+\lambda\pi_{\overline{V}_2},$ for some holomorphic vector-valued function $C:S^2\setminus D\to(\mathfrak{u}_{\zeta_2}^0)_{\sigma_{\varrho,2}}$, where $$\begin{aligned}
(\mathfrak{u}_{\zeta_2}^0)_{\sigma_{\varrho,2}} =\left((\mathfrak{g}_{L_1- L_3}\oplus \mathfrak{g}_{L_1+L_3})\cap\mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,2}}\right)\oplus\left((\mathfrak{g}_{L_2- L_3}\oplus \mathfrak{g}_{L_2+L_3})\cap\mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,2}}\right)\oplus \mathfrak{g}_{L_1+L_2}. \end{aligned}$$
We have $Y_{1,3}=\sigma_{\varrho,2}(X_{1,3})$ and $Y_{2,3}=\sigma_{\varrho,2}(X_{2,3})$. Hence we can write $$C=a(z)(X_{1,3}+Y_{1,3})+b(z)(X_{2,3}+Y_{2,3})+c(z)Y_{1,2}$$ where $a(z)$, $b(z)$ and $c(z)$ are meromorphic functions on $S^2$.
Now, $\Phi=\exp{C}\cdot \gamma_{\zeta_2}$ is an extended solution if and only if, in the expression $C_z-\frac{1}{2!}(\mathrm{ad} C)C_z,$ which does not depend on $\lambda$, the component on ${\mathfrak{g}}_2^{\zeta_2}={\mathfrak{g}}_{L_1+L_2}$ vanishes. Since $Y_{1,2}=[Y_{2,3},X_{1,3}]=[X_{2,3},Y_{1,3}]$ and $[X_{1,3},X_{2,3}]=[Y_{1,3},Y_{2,3}]=0$, this holds if and only if $c'=ba'-ab'$, where prime denotes $z$-derivative. Since $A=\exp C\cdot V_2$, we can compute $\exp C$ in order to conclude that the holomorphic subbundle $A$ of $R=\mathrm{Span}\{{u}_1,u_2,\overline{u}_1,\overline{u}_2, u_3+\overline{u}_3\}$ is given by $$A=\mathrm{Span}\{(a^2,ab+c,a,-1,0,a),(ab-c,b^2,b,0,-1,b)\}.$$
### Harmonic maps into Real Grassmanians.
Let $\zeta'$ be a $\varrho$-semi-canonical element of $SO(2n)$ given by , for some disjoint subsets $I_1$ and $I_2$ of $\{1,\ldots,n-1\}$. By Proposition \[fundcan\], we know that $P^{\sigma_\varrho}_{\zeta'}\cong \mathbb{R}P^{2n-1}$ if and only if either $I_1=\emptyset$ or $I_1=\{n-1\}$. More generally we have:
\[grassd\] If $I_1=\{i_1>i_2>\ldots>i_r\}$ and $d=\sum_{j=1}^r(-1)^{j+1}i_j$, then $P^{\sigma_\varrho}_{\zeta'}\cong G_{2d+1}(\mathbb{R}^{2n})$.
For $\zeta'$ of the form , set $\zeta'_{I_1}=\sum_{i\in I_1}\zeta_i$. Clearly, $\exp\pi\zeta'=\exp\pi\zeta'_{I_1}$, and, by Proposition \[concomp\], $P^{\sigma_\varrho}_{\zeta'}$ is a symmetric space with involution $$\tau=\mathrm{Ad}(\exp\pi\zeta'_{I_1})\circ \sigma_\varrho=\mathrm{Ad}(s_0\exp \pi \zeta'_{I_1}).$$ We have $$\zeta'_{I_1}=r(E_1+\ldots + E_{i_r})+(r-1)(E_{i_r+1}+\ldots +E_{i_{r-1}})+\ldots +(E_{i_2+1}+\ldots +E_{i_1}),$$ and consequently, with the convention $V_{i_0}=V_n$ and $V_{i_{r+1}}=\{0\}$, $$\exp\pi\zeta'_{I_1}=\sum_{j=0}^r(-1)^j\pi_{i_j-i_{j+1}}+\sum_{j=0}^r(-1)^j\overline{\pi}_{i_j-i_{j+1}},$$ where $\pi_{i_j-i_{j+1}}$ is the orthogonal projection onto $V_{i_j}\cap V_{i_{j+1}}^\perp$ and $\overline{\pi}_{i_j-i_{j+1}}$ the orthogonal projection onto the corresponding conjugate space. Hence, the $+1$-eigenspace of $s_0\exp\zeta'_{I_1} $ has dimension $2d+1$, with $d=\sum_{j=1}^r(-1)^{j+1}i_j$, which means that $P^{\sigma_\varrho}_{\zeta'}\cong G_{2d+1}(\mathbb{R}^{2n})$.
In particular, we have $P_{\zeta_d}^{\sigma_\varrho}\cong G_{2d+1}(\mathbb{R}^{2n})$ for each $d\in\{1,\ldots,n-1\}$.
\[cangrass\] Each harmonic map from $S^2$ into the real Grassmannian $G_{2d+1}(\mathbb{R}^{2n})$ belongs to one of the following classes: $(\zeta,\mathrm{Ad}\exp\pi(\tilde\zeta-\zeta)\circ \sigma_{\varrho,l}),$ where $\zeta$ and $\tilde \zeta$ are $\varrho$-canonical elements such that $\tilde\zeta\preceq \zeta$ and $\tilde\zeta=\sum_{i\in I_1}\zeta_i+\zeta_l$, where
1. $I_1=\{i_1>i_2>\ldots>i_r\}$ satisfies $d=\sum_{j=1}^r(-1)^{j+1}i_j$;
2. $l\in\{0,1,\ldots,n-1\}$ and $l\notin I_1$ (if $l=0$, we set $\zeta_0=0$).
We consider harmonic maps into $P_{\zeta_d}^{\sigma_\varrho}\cong G_{2d+1}(\mathbb{R}^{2n})$. Let $\zeta'$ be a $\varrho$-semi-canonical element and write $\zeta'=\sum_{i\in I_1}\zeta_i+ \sum_{i\in I_2}2\zeta_i$ for some disjoint subsets $I_1$ and $I_2$ of $\{1,\ldots,n-1\}$. By Proposition \[grassd\], $P^{\sigma_\varrho}_{\zeta'}\cong G_{2d+1}(\mathbb{R}^{2n})$ if and only if either $d=\sum_{j=1}^r(-1)^{j+1}i_j$ or $n-d-1=\sum_{j=1}^r(-1)^{j+1}i_j$, since $G_{2d+1}(\mathbb{R}^{2n})$ and $G_{2d'+1}(\mathbb{R}^{2n}),$ with $d'=n-d-1$, can be identified via $V\mapsto V^\perp$. However, it follows from the same reasoning as in the proof of Theorem \[classesproj\] that, in the second case, $P^{\sigma_\varrho}_{\zeta'}$ does not coincide with the connected component $P_{\zeta_d}^{\sigma_\varrho}$. So we only consider the $\varrho$-semi-canonical elements $\zeta'$ with $d=\sum_{j=1}^r(-1)^{j+1}i_j$.
Set $l=\max I_2$. Next we check that the pair $\zeta'\preceq \tilde\zeta=\sum_{i\in I_1}\zeta_i+\zeta_l$ satisfies the conditions of Proposition \[norm2\]. Considering the same notations we used in the proof of Theorem \[classesproj\], for each $i>0$ we have $$\Delta'_\varrho\cap \Delta_{\zeta'}^{2i}=\{L_j-L_n|\,\,\, 2|I_2\cap\{j,\ldots,l\}|+|I_1\cap\{j,\ldots,n-1\}|=2i\}.$$ In particular, for $i>0$ and $\alpha=L_j-L_n\in \Delta'_\varrho\cap \Delta_{\zeta'}^{2i}$, it is clear that $\alpha(\zeta'-\tilde \zeta)/\mathrm{i}\leq 2i-1,$ and consequently $${\mathfrak{g}}_{2i}^{\zeta'}\cap \mathfrak{m}_\varrho^{\mathbb{C}}\subset \bigoplus_{0\leq j<2i}{\mathfrak{g}}^{\zeta'-\tilde\zeta}_j.$$
For $i>0$, we have the decomposition $${\mathfrak{g}}_{2i-1}^{\zeta'}\cap \mathfrak{k}_\varrho^{\mathbb{C}}=\!\!\! \bigoplus_{{\alpha\in \Delta(\mathfrak{k}_\varrho)\cap \Delta_{\zeta'}^{2i-1}}}\!\!\! {\mathfrak{g}}_\alpha\oplus
\!\!\! \bigoplus_{{\alpha\in \Delta'_\varrho\cap \Delta_{\zeta'}^{2i-1}}} \!\!\!({\mathfrak{g}}_\alpha\oplus {\mathfrak{g}}_{\varrho(\alpha)})\cap \mathfrak{k}^{\mathbb{C}}_\varrho.$$ Given $\alpha\in {\mathfrak{g}}_{2i-1}^{\zeta'}$, since $\alpha(\zeta')/\mathrm{i}$ is odd, we must have $\alpha(\zeta_j)\neq 0$ for some $j\in I_1$. Hence $\alpha(\zeta'-\tilde\zeta)/\mathrm{i}<\alpha(\zeta')/\mathrm{i} $ and we conclude that $${\mathfrak{g}}_{2i-1}^{\zeta'}\cap \mathfrak{k}_\varrho^{\mathbb{C}}\subset \bigoplus_{0\leq j<2i-1}{\mathfrak{g}}^{\zeta'-\zeta}_j.$$
The statement of the theorem follows now from Propositions \[norminf\] and \[norm2\].
Next we will study in detail the case $G_3(\mathbb{R}^6)$. Take as base point of $G_3(\mathbb{R}^6)$ the $3$-dimensional real subspace $V_0\oplus V_1\oplus \overline{V}_1$, where $V_1$ is the one-dimensional isotropic subspace spanned by $\overline{u}_1$. This choice establishes the identification $$G_3(\mathbb{R}^6)\cong SO(6)/S(O(3)\times O(3))$$ and the corresponding involution is $\sigma_{\varrho,1}=\mathrm{Ad}(\exp \pi\zeta_1)\circ \sigma_\varrho$. Following our classification procedure, we also identify $G_3(\mathbb{R}^6)$ with $P_{\zeta_1}^{\sigma_\varrho}$ via the totally geodesic embedding . From Theorem \[cangrass\], we have six classes of harmonic maps into $G_3(\mathbb{R}^6)$: $$\begin{aligned}
&(\zeta_1,\sigma_\varrho),\,\,\,\, (\zeta_1+\zeta_2,\sigma_\varrho),\,\,\,\, (\zeta_2, \sigma_{\varrho,1}),\,\,\,\,(\zeta_1, \sigma_{\varrho,2}), \,\,\,\, (\zeta_1+\zeta_2,\sigma_{\varrho,2}),\,\,\,\, (\zeta_2,\mathrm{Ad}(\exp\pi\zeta_2)\circ \sigma_{\varrho,1}).\end{aligned}$$
\[36\] Let $\varphi:S^2\to G_3(\mathbb{R}^6)$ be an harmonic map.
1. If $\varphi$ is associated to the pair $(\zeta_1,\sigma_\varrho)$ then $\varphi$ is $S^1$-invariant and, up to isometry, is given by $$\label{mixedpair}\varphi=V_0\oplus V\oplus \overline{V},$$ where $V$ is a holomorphic isotropic subbundle of $S^2\times V_0^\perp$ of rank $1$ satisfying $\partial V\subseteq \overline{V}^\perp$.
2. If $\varphi$ is associated to the pair $(\zeta_1+\zeta_2,\sigma_\varrho)$ and is $S^1$-invariant, then, up to isometry, $$\label{varphi1}
\varphi=V_0\oplus (W\cap V^\perp) \oplus (\overline {W\cap V^\perp }),$$ where $V\subset W$ are holomorphic isotropic subbundles of $S^2\times V_0^\perp$ of rank $1$ and $2$, respectively, satisfying $\partial V\subset W$ and $\partial W\subset \overline W^\perp.$
3. If $\varphi$ is associated to the pair $(\zeta_2,\sigma_{\varrho,1})$ and is $S^1$-invariant, then, up to isometry, $$\label{varphi2}\varphi= \{(L_1\oplus \overline{L}_1)^\perp\cap (V_0\oplus V_1\oplus\overline{V}_1)\}\oplus(L_2\oplus \overline{L}_2),$$ where $L_1$ and $L_2$ are holomorphic isotropic bundle lines of $S^2\times (V_0\oplus V_1\oplus\overline{V}_1)$ and $S^2\times (V_0\oplus V_1\oplus\overline{V}_1)^\perp$, respectively.
The corresponding extended solutions have uniton number $2$, $4$, and $2$, respectively, with respect to the standard representation of $SO(6)$. The harmonic maps in the classes $(\zeta_1, \sigma_{\varrho,2})$, $(\zeta_1+\zeta_2,\sigma_{\varrho,2})$, and $(\zeta_2,\mathrm{Ad}(\exp\pi\zeta_2)\circ \sigma_{\varrho,1})$ are precisely the orthogonal complements of the harmonic maps in the classes $(\zeta_1,\sigma_\varrho)$, $(\zeta_1+\zeta_2,\sigma_\varrho)$, and $(\zeta_2, \sigma_{\varrho,1})$, respectively.
For the first two classes, and according to our classification procedure, we identify $G_3(\mathbb{R}^6)$ with $P_{\zeta_1}^{\sigma_\varrho}$ via the totally geodesic embedding $g\cdot (V_0\oplus V_1\oplus \overline{V}_1)\mapsto \exp(\pi\zeta_1)g\sigma_{\varrho,1}(g^{-1}).$ In these two cases, $T_{\sigma_\varrho}$-invariant extended solutions $\Phi$ associated to harmonic maps $\varphi=g\cdot (V_0\oplus V_1\oplus \overline{V}_1)$ satisfy $$\label{phimenos1}
\Phi_{-1}=\exp(\pi\zeta_1)g\sigma_{\varrho,1}(g^{-1})=\exp(\pi\zeta_1)(\pi_\varphi-\pi_{\varphi}^\perp)\exp(\pi\zeta_1)s_0.$$
First we consider the harmonic maps associated to the pair $(\zeta_1,\sigma_\varrho)$. We have $r(\zeta_1)=1$ and $$(\mathfrak{u}^0_{\zeta_1})_{\sigma_{\varrho}}=(\mathfrak{p}^{\zeta_1}_{0})^\perp\cap \mathfrak{k}^{\mathbb{C}}_{{\varrho}}={\mathfrak{g}}_1^{\zeta_1}\cap \mathfrak{k}^{\mathbb{C}}_{{\varrho}}.$$ Consequently any such harmonic map is $S^1$-invariant. Write $\gamma_{\zeta_1}(\lambda)=\lambda^{-1}\pi_{V_1}+\pi^\perp_{V_1\oplus \overline{V}_1}+\lambda \pi_{\overline{V}_1},$ where $V_1$ is the one-dimensional isotropic space spanned by $\overline{u}_1$. Let $\Phi:S^2\setminus D\to U_{\zeta_1}^{\sigma_\varrho}$ be an extended solution associated to the harmonic map $\varphi$. Then, by $S^1$-invariance, we can write $$\label{phicase1}
\Phi_\lambda=\lambda^{-1}\pi_V+ \pi^\perp_{V\oplus \overline{V}}+\lambda\pi_{\overline V},$$ where $V$ is a holomorphic isotropic subbundle of $S^2\times \mathbb{R}^{6}$ of rank $1$ satisfying $\partial V\subseteq \overline{V}^\perp$. The $T_{\sigma_\varrho}$-invariance of $\Phi$ implies that $V_0\subset (V\oplus\overline{V})^\perp$. Equating and , we get, up to isometry, $\varphi=V_0\oplus V\oplus \overline{V}$.
For the case $(\zeta_1+\zeta_2,\sigma_{\varrho})$, since $$\label{zeta1+zeta2}
\gamma_{\zeta_1+\zeta_2}(\lambda)=\lambda^{-2}\pi_{V_1}+\lambda^{-1}\pi_{V_2\cap V_1^\perp}+\pi_{V_2\oplus \overline{V}_2}^\perp+\lambda\pi_{\overline{V}_2\cap \overline{V}_1^\perp}+\lambda^2\pi_{\overline{V}_1},$$ any $S^1$-invariant harmonic map $\varphi$ in this class admits an extended solution of the form $$\label{phicase2}
\Phi_\lambda= \lambda^{-2}\pi_{V}+\lambda^{-1}\pi_{W\cap V^\perp}+\pi_{W\oplus \overline{W}}^\perp+\lambda\pi_{\overline{W}\cap \overline{V}^\perp}+\lambda^2\pi_{\overline{V}},$$ where $V\subset W$ are holomorphic isotropic subbundles of rank $1$ and $2$, respectively, satisfying $\partial V\subset W$ and $\partial W\subset \overline W^\perp.$ By $T_{\sigma_\varrho}$-invariance, we must have $V_0\subset (W\oplus\overline{W})^\perp$, hence $V\subset W$ are subbundles of $S^2\times V_0^\perp$. Equating and , we get .
For the case $(\zeta_2,\sigma_{\varrho,1})$, we identify $G_3(\mathbb{R}^6)$ with $P_{\zeta_2}^{\sigma_{\varrho,1}}=\exp\pi\zeta_1P_{\zeta_2-\zeta_1}^{\sigma_\varrho}$ via the totally geodesic embedding $$\label{oao}
g\cdot (V_0\oplus V_1\oplus \overline{V}_1)\mapsto g\sigma_{\varrho,1}(g^{-1}).$$ Extended solutions $\Phi$ associated to $S^1$-invariant harmonic maps in this class must be of the form $$\label{phiws}
\Phi_\lambda=\lambda^{-1}\pi_W+ \pi_{W\oplus \overline W}^\perp+\lambda\pi_W,$$ where $W$ is a holomorphic isotropic subbundle of rank $2$. By $T_{\sigma_{\varrho,1}}$-invariance, we must have $[\pi_W,\pi_{V_0\oplus V_1\oplus\overline{V}_1}]=0,$ which means that $W$ must be of the form $W=L_1\oplus L_2$, where $L_1$ and $L_2$, respectively, are holomorphic isotropic bundle lines of $S^2\times (V_0\oplus V_1\oplus\overline{V}_1)$ and $S^2\times (V_0\oplus V_1\oplus\overline{V}_1)^\perp$.
On the other hand, in view of , we have $\Phi_{-1}=(\pi_\varphi-\pi_\varphi^\perp)\exp(\pi\zeta_1) s_0.$ Equating this with , we conclude that holds. The remaining cases are treated similarly.
The first two classes of $S^1$-invariant harmonic maps $\varphi:S^2\to G_3(\mathbb{R}^6)$ in Theorem \[36\] factor through $G_2(\mathbb{R}^5)$. That is, for any such harmonic map $\varphi$, there exists $\tilde{\varphi}:S^2\to G_2(\mathbb{R}^5)$, where we identify $\mathbb{R}^5$ with $V_0^\perp$, such that $\varphi=V_0\oplus \tilde{\varphi}$. An explicit construction of all harmonic maps from $S^2$ into $G_2(\mathbb{R}^n)$ can be found in [@woodG2]. In that paper, harmonic maps of the form are called *real mixed pairs*. We emphasise that the harmonic maps into $G_3(\mathbb{R}^6)$ associated to extended solutions in the corresponding unstable manifolds need not to factor through $G_2(\mathbb{R}^5)$ in the same way.
Let us consider the case $(\zeta_1+\zeta_2,\sigma_{\varrho})$. Taking into account the Weierstrass representation of Theorem \[sigmaweirstrass\], any extended solution $\Phi:S^2\setminus D\to U^{\sigma_{\varrho}}_{\zeta}(SO(6))$, with $\zeta=\zeta_1+\zeta_2$, can be written as $\Phi=\exp{C}\cdot \gamma_{\zeta}$, for some meromorphic vector-valued function $C:S^2\to (\mathfrak{u}^0_\zeta)_{\sigma_\varrho}$. We have $r(\zeta)=3$ and $$(\mathfrak{u}^0_{\zeta})_{\sigma_\varrho}=({\mathfrak{g}}_1^\zeta\oplus{\mathfrak{g}}_2^\zeta\oplus {\mathfrak{g}}_3^\zeta)\cap \mathfrak{k}^{\mathbb{C}}_{\varrho}\oplus \lambda ({\mathfrak{g}}_2^\zeta\oplus {\mathfrak{g}}_3^\zeta)\cap \mathfrak{m}^{\mathbb{C}}_{\varrho}\oplus \lambda^2 {\mathfrak{g}}_3^\zeta\cap \mathfrak{k}^{\mathbb{C}}_{\varrho}.$$ Moreover, $$\begin{aligned}
&{\mathfrak{g}}_1^\zeta\cap \mathfrak{k}^{\mathbb{C}}_{\varrho}={\mathfrak{g}}_{L_1-L_2}\oplus \{({\mathfrak{g}}_{L_2- L_3}\oplus{\mathfrak{g}}_{L_2+L_3}) \cap \mathfrak{k}^{\mathbb{C}}_{\varrho}\},\quad
{\mathfrak{g}}_2^\zeta\cap \mathfrak{k}^{\mathbb{C}}_{\varrho}=({\mathfrak{g}}_{L_1+ L_3}\oplus{\mathfrak{g}}_{L_1-L_3} )\cap \mathfrak{k}^{\mathbb{C}}_{\varrho},\\
& {\mathfrak{g}}_3^\zeta\cap \mathfrak{k}^{\mathbb{C}}_{\varrho}={\mathfrak{g}}_{L_1+L_2},\quad
({\mathfrak{g}}_2^\zeta\oplus {\mathfrak{g}}_3^\zeta)\cap \mathfrak{m}^{\mathbb{C}}_{\varrho}={\mathfrak{g}}_2^\zeta\cap \mathfrak{m}^{\mathbb{C}}_{\varrho}=({\mathfrak{g}}_{L_1- L_3}\oplus {\mathfrak{g}}_{L_1+ L_3})\cap \mathfrak{m}^{\mathbb{C}}_{\varrho}.\end{aligned}$$
Write $$\label{CC}
C=C_0+\lambda C_1+\lambda^2 C_2,\quad C_0=c_0^1+c_0^2+c_0^3,\quad C_1=c_1^2+c_1^3,\quad C_2=c_2^3$$ where the functions $c_0^i:S^2\to {\mathfrak{g}}_i^\zeta\cap \mathfrak{k}^{\mathbb{C}}_{\varrho}$, $c_1^i:S^2\to {\mathfrak{g}}_i^\zeta\cap \mathfrak{m}^{\mathbb{C}}_{\varrho}$, and $c_2^3:S^2\to {\mathfrak{g}}_3^\zeta\cap \mathfrak{k}^{\mathbb{C}}_{\varrho}$ are meromorphic functions. Clearly, $c_1^3=0$. Consider the root vectors defined by . Since $\sigma_\varrho(X_{2,3})=-Y_{2,3}$ and $\sigma_\varrho(X_{1,3})=-Y_{1,3}$, we can write $$\begin{aligned}
c_0^1=aX_{1,2}+b(X_{2,3}-Y_{2,3}),\,\,c_0^2=c(X_{1,3}-Y_{1,3}),\,\, c_0^3=dY_{1,2},\,\,
c_1^2=e(X_{1,3}+Y_{1,3}),\,\,c_2^3=fX_{1,2}\end{aligned}$$ in terms of ${\mathbb{C}}$-valued meromorphic functions $a$, $b$, $c$, $d$, $e$, $f$.
Taking into account the results of Section \[weircond\], $\Phi=\exp{C}\cdot \gamma_{\zeta}$ is an extended solution if and only if, in the expression $$(\exp C)^{-1}(\exp C)_z=C_z-\frac{1}{2!}(\mathrm{ad} C)C_z+\frac{1}{3!}(\mathrm{ad} C)^2C_z,$$ we have:
1. the independent coefficient should have zero component in each ${\mathfrak{g}}_{2}^{\zeta}$ and ${\mathfrak{g}}_{3}^{\zeta},$ that is $$\begin{aligned}
\label{o}
{c_0^2}_z-\frac{1}{2}[c_0^1,{c_0^1}_z]=0,\,\,\,\, {c_0^3}_z-\frac{1}{2}[c_0^1,{c_0^2}_z]-\frac{1}{2}[c_0^2,{c_0^1}_z]+\frac16[c_0^1,[c_0^1,{c_0^1}_z]]=0;
\end{aligned}$$
2. the $\lambda$ coefficient should have zero component in ${\mathfrak{g}}_{3}^{\zeta},$ that is $$\label{1}
[c_0^1,{c_1^2}_z]+[c_1^2,{c_0^1}_z]=0.$$
From equations we get the equations (prime denotes $z$-derivative) $$\label{merocond}
2c'=ab'-ba',\,\,\,\,\, 3d'=3 cb'-bc';$$ on the other hand, observe that always holds since $$[c_0^1,{c_1^2}_z]+[c_1^2,{c_0^1}_z]\in [{\mathfrak{g}}_1^\zeta\cap \mathfrak{k}^{\mathbb{C}}_{\varrho},{\mathfrak{g}}_2^\zeta\cap \mathfrak{m}^{\mathbb{C}}_{\varrho}]\subset {\mathfrak{g}}_3^\zeta\cap \mathfrak{m}^{\mathbb{C}}_{\varrho}=\{0\}.$$
Hence we conclude that, any extended solution $\Phi:S^2\setminus D\to U^{\sigma_{\varrho}}_{\zeta}(SO(6))$, with $\zeta=\zeta_1+\zeta_2$, of the form $\Phi=\exp{C}\cdot \gamma_{\zeta}$, can be constructed as follows: choose arbitrary meromorphic functions $a$, $b$, $e$ and $f$; integrate equations to obtain the meromorphic functions $c$ and $d$; $C$ is then given by .
Choose $a(z)=b(z)=z$. From , we can take $c(z)=1$ and $d(z)=z$. This data defines the matrix $C_0$ and the $S^1$-invariant extended solution $\Phi^0=\exp C_0\cdot \gamma_{\zeta}$, where the loop $\gamma_\zeta$, with $\zeta=\zeta_1+\zeta_2$, is given by . The extended solutions $\Phi:S^2\to U_{\zeta}^{\sigma_\varrho}(SO(6))$ satisfying $\Phi^0=u_{\zeta}\circ\Phi$ are of the form $\Phi=\exp C\cdot \gamma_\zeta$, where the matrix $C=C_0+C_1\lambda+C_2\lambda^2$ is given by $$C= \left(
\begin{array}{cccccc}
0 & z & 1 & 0 & z & -1 \\
0 & 0 & z & -z & 0 & -z \\
0 & 0 & 0 & 1 & z & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -z & 0 & 0 \\
0 & 0 & 0 & -1 & -z & 0 \\
\end{array}
\right)+ \left(
\begin{array}{cccccc}
0 & 0 & e\lambda & 0 & f\lambda^2 & -e\lambda \\
0 & 0 & 0 & -f\lambda^2 & 0 & 0 \\
0 & 0 & 0 & e\lambda & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -e\lambda & 0 & 0 \\
\end{array}
\right),$$ with respect to the complex orthonormal basis $\mathbf{u}=\{u_1,u_2,u_3,\overline{u}_1,\overline{u}_2,\overline{u}_3\}$, where $e$ and $f$ are arbitrary meromorphic functions on $S^2$. The holomorphic vector bundles $V$ and $W$ associated to the $S^1$-invariant extended solution $\exp C_0\cdot \gamma_{\zeta}$ are given by $V=\exp{C_0}\cdot {V}_1$ and $W=\exp{C_0}\cdot V_2$, and we have, with respect to the basis $\mathbf{u}$, $$\begin{aligned}
V&=\mathrm{span}\{(12-12z^2-z^4,-4z^3,12-6z^2,12,-12z,-12+6z^2 ) \}\\
W&=\mathrm{span}\{(6z+z^3,3z^2,3z,0,3,-3z ) \}\oplus V.\end{aligned}$$
Outer symmetric $SU(2n+1)$-spaces. {#su(2n+1)-outer}
----------------------------------
Let $E_j$ be the square $(m\times m)$-matrix whose $(j,j)$-entry is $\mathrm{i}$ and all other entries are $0$. The complexification ${\mathfrak{t}}^{\mathbb{C}}$ of the algebra ${\mathfrak{t}}$ of diagonal matrices $\sum a_iE_i$, with $a_i\in \mathbb{C}$ and $\sum a_i=0$, is a Cartan subalgebra of $\mathfrak{su}(m)^{\mathbb{C}}$. Let $\{L_1,\ldots,L_{m}\}$ be the dual basis of $\{E_1,\ldots,E_{m}\}$, that is $L_i(E_j)=\mathrm{i}\delta_{ij}$. The roots of $\mathfrak{su}(m)$ are the vectors $ L_i- L_j$, with $i\neq j$ and $1\leq i,j\leq m-1$ and $\Delta^+=\{L_i-L_j\}_{i<j}$ is a positive root system with positive simple roots $\alpha_i=L_i-L_{i+1}$, for $1\leq i\leq m-1$. For $i\neq j$, the matrix $X_{i,j}$ whose $(i,j)$ entry is $1$ and all other entries are $0$ generate the root space ${\mathfrak{g}}_{L_i-L_j}$. The dual basis of $\Delta_0=\{\alpha_1,\ldots,\alpha_{m-1}\}$ in $\mathrm{i}\mathfrak{t}^*$ is formed by the matrices $$\begin{aligned}
H_i=\frac{m-i}{m}(E_1+\ldots+ E_i)- \frac{i}{m}(E_{i+1}+\ldots+ E_{m}).\end{aligned}$$
### Special Lagrangian spaces
Consider on $\mathbb{R}^{2m}$ the standard inner product $\langle\cdot,\cdot\rangle$ and the canonical orthonormal basis $\mathbf{e}^{2m}=\{e_1,\ldots, e_{2m}\}.$ Define the orthogonal complex structure $I$ by $I(e_i)=e_{2m+1-i}$, for $i\in \{1,\ldots,m\}$. A *Lagrangian subspace* of $\mathbb{R}^{2m}$ (with respect to $I$) is a $m$-dimensional subspace $L$ such that $IL\perp L$. Let $\mathcal{L}_m$ be the space of all Lagrangian subspaces of $\mathbb{R}^{2m}$ and $L_0\in\mathcal{L}_m$ the Lagrangian subspace generated by $\mathbf{e}^{m}=\{e_1,\ldots,e_m\}$. The unitary group $U(m)$ acts transitively on $\mathcal{L}_m$, with isotropy group at $L_0$ equal to $SO(m)$, and $\mathcal{L}_m$ is a reducible symmetric space that can be identified with $U(m)/SO(m)$ (see [@Zi] for details).
The space $\mathcal{L}_m$ can also be interpreted as the set of all orthogonal linear maps $\tau:\mathbb{R}^{2m}\to\mathbb{R}^{2m}$ satisfying $\tau^2=e$ and $I\tau=-\tau I$. Indeed, if $V_\pm$ are the $\pm 1$ eigenspaces of $\tau$, then $IV_+=V_-$ and $IV_+\perp V_+$, that is $V_+$ is Lagrangian. From this point of view, $U(m)$ acts on $\mathcal{L}_m$ by conjugation: $g\cdot \tau=g\tau g^{-1}$. Let $\tau_0\in \mathcal{L}_m$ be the orthogonal linear map corresponding to $L_0$, that is, ${\tau_0}_{|_{L_0}}=e$ and ${\tau_0}_{|_{IL_0}}=-e$. The corresponding involution on $U(m)$ is given by $\sigma(g)=\tau_0g\tau_0$ and the Cartan embedding $\iota:\mathcal{L}_m\hookrightarrow U(m)$ is given by $\iota(\tau)=\tau \tau_0$.
The totally geodesic submanifold $\mathcal{L}^s_m:=SU(m)/SO(m)$ of $U(m)/SO(m)$ is also known as the *space of special Lagrangian subspaces of $\mathbb{R}^{2m}$*. It is an irreducible outer symmetric $SU(m)$-space.
### Harmonic maps into $\mathcal{L}^s_{2n+1}$
Take $m=2n+1$. The non-trivial involution $\varrho$ of the Dynkin diagram of $\mathfrak{su}(2n+1)^{\mathbb{C}}$ is given by $\varrho(\alpha_i)=\alpha_{2n+1-i}$. In particular, $\varrho$ does not fix any root in $\Delta_0$ and there exists only one class of outer symmetric $SU(2n+1)$-spaces. The semi-fundamental basis $\pi_{\mathfrak{k}_\varrho}(\Delta_0)=\{\beta_1,\ldots,\beta_{n}\}$ is given by $\beta_i=\frac12(\alpha_i+\alpha_{2n+1-i})$ whereas the dual basis $\{\zeta_1,\ldots,\zeta_{n}\}$ is given by $$\zeta_i=H_i+H_{2n+1-i}=E_1+\ldots+E_i-(E_{2n+2-i}+\ldots +E_{2n+1}),$$ for $1\leq i\leq n$. Since each $\zeta_i$ belongs to the integer lattice $\mathfrak{I}(SU(2n+1))$, the $\varrho$-semi-canonical elements of $SU(2n+1)$ are precisely the elements $\zeta=\sum_{i=1}^{n}m_i\zeta_i$ with $m_i\in\{0,1,2\}$.
Let $\mathbf{e}^{2n+1}=\{e_1,\ldots,e_{2n+1}\}$ be the canonical orthonormal basis of $\mathbb{R}^{2n+1}$. Identify ${\mathbb{C}}^{2n+1}$ with $(\mathbb{R}^{4n+2},I)$, where $I$ is defined as above. Set $$v_j=\frac{1}{\sqrt2}(e_j+\mathrm{i}e_{2n+2-j}),$$ for $1\leq j\leq n$, $v_{n+1}=e_{n+1}$ and $v_{2n+2-j}=\overline{v}_j$. Take the matrices $E_j$ with respect to the complex basis $\mathbf{v}=\{v_1,\ldots,v_{2n+1}\}$ of ${\mathbb{C}}^{2n+1}$. Hence $\tau_0E_j\tau_0=-E_{2n+2-j}$ and the fundamental involution $\sigma_\varrho$ is given by $\sigma_\varrho(g)=\tau_0g\tau_0$. The fundamental outer symmetric $SU(2n+1)$-space is the space of special Lagrangian subspaces $\mathcal{L}_{2n+1}^s=SU(2n+1)/SO(2n+1)$, and this is the unique outer symmetric $SU(2n+1)$-space.
Next we consider in detail harmonic maps into $\mathcal{L}_{3}^s$. In this case we have two non-zero $\varrho$-semi-canonical elements, $\zeta_1$ and $2\zeta_1$, and consequently two classes of harmonic maps, $(\zeta_1,\sigma_\varrho)$ and $(\zeta_1,\sigma_{\varrho,1})$. Since $\zeta_1=E_1-E_3$, we have $r(\zeta_1)=(L_1-L_3)(\zeta_1)/\mathrm{i}=2$. Let $W_1$, $W_2$ and $W_3$ be the complex one-dimensional images of $E_1$, $E_2$ and $E_3$, respectively. Any extended solution $$\Phi:S^2\setminus D\to U_{\zeta_1}^{\sigma_\varrho}(SU(2n+1))$$ is given by $\Phi=\exp C\cdot \gamma_{\zeta_1}$, with $\gamma_{\zeta_1}(\lambda)=\lambda^{-1}\pi_{W_3}+\pi_{W_2}+\lambda \pi_{W_1}$, for some holomorphic vector-valued function $C:S^2\setminus D\to (\mathfrak{u}^0_{\zeta_1})_{\sigma_\varrho}$, where $$(\mathfrak{u}^0_{\zeta_1})_{\sigma_\varrho}=(\mathfrak{p}_0^{\zeta_1})^\perp\cap\mathfrak{k}^{\mathbb{C}}_{\varrho}+\lambda(\mathfrak{p}_1^{\zeta_1})^\perp\cap
\mathfrak{m}^{\mathbb{C}}_{\varrho}$$ and $$\begin{aligned}
(\mathfrak{p}_0^{\zeta_1})^\perp\cap\mathfrak{k}^{\mathbb{C}}_{\varrho}=({\mathfrak{g}}_{L_1-L_2}\oplus{\mathfrak{g}}_{L_2-L_3}\oplus {\mathfrak{g}}_{L_1-L_3})\cap\mathfrak{k}^{\mathbb{C}}_{\varrho},\quad (\mathfrak{p}_1^{\zeta_1})^\perp\cap\mathfrak{m}^{\mathbb{C}}_{\varrho}= {\mathfrak{g}}_{L_1-L_3}\cap\mathfrak{m}^{\mathbb{C}}_{\varrho}.\end{aligned}$$
Let $X_{i,j}$ be the square matrix whose $(i,j)$ entry is $1$ and all the other entries are $0$, with respect to the basis $\mathbf{v}$. The root space $\mathfrak{g}_{L_i-L_j}$ is spanned by $X_{i,j}$. We have $\sigma_{\varrho}(X_{1,2})=-X_{2,3}$ and $\sigma_\varrho(X_{1,3})= -X_{1,3}$ (consequently, $\mathfrak{g}_{L_1-L_3}\subset \mathfrak{m}_{\varrho}^{\mathbb{C}}$). Hence we can write $C=C_0+C_1\lambda$, with $C_0=a(X_{1,2}-X_{2,3})$ and $C_1=bX_{1,3}$, for some meromorphic functions $a,b$ on $S^2$. The harmonicity equations do not impose any condition on these meromorphic functions, hence any harmonic map $\varphi:S^2\to \mathcal{L}_{3}^s$ in the class $(\zeta_1,\sigma_\varrho)$ admits an extended solution of the form $$\label{su1}
\Phi=\exp\left(
\begin{array}{ccc}
0 & a & b\lambda \\
0 & 0 & -a \\
0 & 0 & 0 \\
\end{array}
\right)\cdot \gamma_{\zeta_1}=\left(
\begin{array}{ccc}
1 & a & \frac12(-a^2+2b\lambda) \\
0 & 1 & -a \\
0 & 0 & 1 \\
\end{array}
\right)\cdot \gamma_{\zeta_1},$$ and $\varphi$ is recovered by setting $\varphi=\Phi_{-1}\tau_0$. Similarly, one can see that the class of harmonic maps in $(\zeta_1,\sigma_{\varrho,1})$ admits an extended solution of the form $$\label{su2}\Phi=\left(
\begin{array}{ccc}
1 & a & \frac12(a^2+2b\lambda) \\
0 & 1 & a \\
0 & 0 & 1 \\
\end{array}
\right)\cdot \gamma_{\zeta_1},$$ with no restrictions on the meromorphic functions $a$ and $b$.
H. Ma established (cf. Theorem 4.1 of [@Ma]) that harmonic maps $\varphi:S^2\to \mathcal{L}_{3}^s$ are essentially of two types: 1) $\iota_\sigma\circ \varphi$ is a *Grassmannian solution* obtained from a full harmonic map $f:S^2\to \mathbb{R}P^2\subset \mathbb{C}P^2$, where $\iota_\sigma$ is the Cartan embedding of $ \mathcal{L}_{3}^s$ in $SU(3)$; 2) up to left multiplication by a constant, $\iota_\sigma\circ \varphi$ is of the form $(\pi_{\beta_1}- \pi_{\beta_1}^\perp) (\pi_{\beta_2}- \pi_{\beta_2}^\perp)$, where $\beta_1$ is a *Frenet pair* associated to a full totally istotropic holomorphic map $g:S^2\to\mathbb{C}P^2$ and $\beta_2$ is a rank $1$ holomorphic subbundle of $G'(g)^\perp$, where $G'(g)$ is the first *Gauss bundle* of $g$. Observe that if, in the second case, $\beta_2$ coincides with $g$, then $\iota_\sigma\circ\varphi$ is a [Grassmannian solution]{} obtained from the full harmonic map $f:=G'(g)$ from $S^2$ to $\mathbb{R}P^2$, that is, $\varphi$ is of type 1). Comparing this with our description, it is not difficult to see that harmonic maps of type 1) are $S^1$-invariant extended solutions (take $b=0$ in and ) and harmonic maps of type 2) are associated to extended solutions with values in the corresponding unstable manifolds (which corresponds to an arbitrary choice of $b$ in and ). H. Ma also established a purely algebraic explicit construction of such harmonic maps in terms of meromorphic data on $S^2$, which is consistent with our results.
Outer symmetric $SU(2n)$-spaces.
--------------------------------
With the same notations of Section \[su(2n+1)-outer\], the non-trivial involution $\varrho$ of the Dynkin diagram of $\mathfrak{su}(2n)$ is given by $\varrho(\alpha_i)=\alpha_{2n-i}$, and $\varrho$ fixes the root $\alpha_n$. The semi-fundamental basis $\pi_{\mathfrak{k}_\varrho}(\Delta_0)=\{\beta_1,\ldots,\beta_{n-1}\}$ is given by $\beta_1=\alpha_n$ and $\beta_i=\frac12(\alpha_i+\alpha_{2n-i})$ if $i\geq 2$; whereas its dual basis $\{\zeta_1,\ldots,\zeta_{n-1}\}$ is given by $$\begin{aligned}
\zeta_1&=H_n=\frac12({E_1}+\ldots+E_n)-\frac12(E_{n+1}+\ldots+ E_{2n})\\
\zeta_i&= H_{i-1}+H_{2n-i+1}=E_1+\ldots+E_{i-1}-(E_{2n+2-i}+\ldots+E_{2n}),\,\,\,\,\mbox{for $2\leq i\leq n-1$}.\end{aligned}$$
By Theorem \[murak\], there exist two conjugacy classes of outer involutions: the fundamental outer involution $\sigma_\varrho$ and $\sigma_{\varrho,1}$. These outer involutions correspond to the symmetric spaces $SU(2n)/Sp(n)$ and $SU(2n)/SO(2n)$, respectively. Observe that $\zeta_1$ does not belong to the integer lattice $\mathfrak{I}'(SU(2n)^{\sigma_\varrho})$ since $\exp 2\pi\zeta_1=-e$.
### Harmonic maps into the space of special unitary quaternionic structures on $\mathbb{C}^{2n}$.
A *unitary quaterninonic structure* on the standard hermitian space $(\mathbb{C}^{2n},\langle\cdot,\cdot\rangle )$ is a conjugate linear map $J:{\mathbb{C}}^{2n}\to {\mathbb{C}}^{2n}$ satisfying $J^2=-Id$ and $\langle v,w\rangle= \langle J\,w,J\,v\rangle$ for all $v,w\in {\mathbb{C}}^{2n}$. Consider as base point the quaternionic structure $J_o$ defined by $J_o(e_i)=e_{2n+1-i}$ for each $1\leq i\leq n$, where $\mathbf{e}^{2n}=\{e_1,\ldots,e_{2n}\}$ is the canonical hermitian basis of $\mathbb{C}^{2n}$. The unitary group $U(2n)$ acts transitively on the space of unitary quaternionic structures on ${\mathbb{C}}^{2n}$ with isotropy group at $J_o$ equal to $Sp(n)$, and thus $M=U(2n)/Sp(n)$. This is a reducible symmetric space with involution $\sigma:U(2n)\to U(2n)$ given by $\sigma(X)=J_oXJ_o^{-1}$, but the totally geodesic submanifold $\mathcal{Q}^s_n:=SU(2n)/Sp(n)$ is an irreducible symmetric space, which we call the space of *special unitary quaternionic structures* on $\mathbb{C}^{2n}$ (see [@Zi] for details). If we consider the matrices $E_i$ with respect to the complex basis $\mathbf{v}=\{v_1,\ldots,v_{2n}\}$ defined by $$\label{ves}
v_j=\frac{1}{\sqrt2}(e_j+\mathrm{i}e_{2n+1-j}),$$ for $1\leq j\leq n$, and $v_{2n+1-j}=\overline{v}_j$, we see that $J_oE_jJ_o^{-1}=-E_{2n+1-j}$, and consequently we have $\sigma=\sigma_\varrho$.
Next we consider with detail harmonic maps into $\mathcal{Q}^s_2$.
Each harmonic map $\varphi:S^2\to\mathcal{Q}^s_2$ belongs to one of the following classes: $(2\zeta_1,\sigma_\varrho)$, and $(\zeta_2,\sigma_{\varrho,2})$.
We start by identifying $\mathcal{Q}^s_2$ with $P_e^{\sigma_\varrho}$.
The $\varrho$-semi-canonical elements of $SU(4)$ are precisely the elements $$2\zeta_1,\,4\zeta_1,\,\zeta_2,\,2\zeta_2,\,2\zeta_1+\zeta_2,\,2\zeta_1+2\zeta_2,\,4\zeta_1+\zeta_2,\,4\zeta_1+2\zeta_2.$$ By Proposition \[fundcan\], all these elements correspond to the symmetric space $\mathcal{Q}^s_2$.
We claim that $\exp\pi \zeta_2$ is not in the connected component $$P_e^{\sigma_\varrho}=\{gJ_og^{-1}J_o^{-1}|\,g\in SU(4)\}.$$ In fact, $\exp(\pi \zeta_2)J_o=gJ_og^{-1}\cong gSp(n)$ for the unitary transformation $g$ defined by $g(e_1)=e_4$, $g(e_4)=e_1$, $g(e_2)=e_3$ and $g(e_3)=-e_2$. Since $\det g\neq 1$ we conclude that $\exp\pi \zeta_2$ does not belong to $P_e^{\sigma_\varrho}$. Similarly, one can check that $\exp\pi (2\zeta_1+\zeta_2)$ is not in $P_e^{\sigma_\varrho}$.
Hence, since $\exp\pi2\zeta_1$ belongs to the centre of $SU(4)$, any harmonic map $\varphi:S^2\to\mathcal{Q}^s_2\cong P_e^{\sigma_\varrho} $ belongs to one of the following classes: $(2\zeta_1,\sigma_\varrho)$, $(\zeta_2,\sigma_{\varrho,2})$, and $(2\zeta_1+\zeta_2,\sigma_{\varrho,2})$. It remains to check that, in view of Proposition \[norm2\], harmonic maps in the class $(2\zeta_1+\zeta_2,\sigma_{\varrho,2})$ can be normalized to harmonic maps in the class $(\zeta_2,\sigma_{\varrho,2})$.
It is clear that $2\zeta_1+\zeta_2\preceq \zeta_2$. On the other hand, for any positive root $L_i-L_j\in \Delta^+$, with $i<j$, we have $(L_i-L_j)(2\zeta_1)/\mathrm{i}\leq (L_i-L_j)({2\zeta_1+\zeta_2})/\mathrm{i},$ where the equality holds in just one case: $(L_2-L_3)(2\zeta_1)= (L_2-L_3)(2\zeta_1+\zeta_2)=2\mathrm{i}.$ However, $\mathfrak{g}_{L_2-L_3}\subset \mathfrak{k}_{\sigma_{\varrho,2}}$, which means that the conditions of Proposition \[norm2\] hold for $\zeta=2\zeta_1+\zeta_2$ and $\zeta'=\zeta_2$, and consequently harmonic maps in the class $(2\zeta_1+\zeta_2,\sigma_{\varrho,2})$ can be normalized to harmonic maps in the class $(\zeta_2,\sigma_{\varrho,2})$.
Following the same procedure as before, one can see that any harmonic map $\varphi\to\mathcal{Q}^s_2$ in the class $(2\zeta_1,\sigma_\varrho)$ admits an extended solution of the form $$\Phi=\left(
\begin{array}{cccc}
1 & 0 & c_1+a\lambda & c_2 \\
0 & 1 & c_3 & c_1-a\lambda \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)\cdot \gamma_{2\zeta_1},$$ where $c_1,c_2,c_3\in{\mathbb{C}}$ are constants, $a$ is a meromorphic function on $S^2$. The harmonic map is recovered by setting $\varphi=\Phi_{-1}J_o$. Reciprocally, given arbitrary complex constants $c_1,c_2,c_3$ and a meromorphic function $a:S^2\to {\mathbb{C}}$, such $\Phi$ is an extended solution associated to some harmonic map in the class $(2\zeta_1,\sigma_\varrho)$ (the harmonicity equations do not impose any restriction to $a$).
Similarly, any harmonic map $\varphi\to\mathcal{Q}^s_2$ in the class $(\zeta_2,\sigma_{\varrho,2})$ admits an extended solution of the form $$\Phi=\left(
\begin{array}{cccc}
1 & b & a & c \\
0 & 1 & 0 & a \\
0 & 0 & 1 & -b \\
0 & 0 & 0 & 1 \\
\end{array}
\right)\cdot \gamma_{\zeta_2},$$ where $a$, $b$ and $c$ are meromorphic functions satisfying $c'=ba'-b'a$. Since $P_{\zeta_2}^{\sigma_{\varrho,2}}=\exp(\pi\zeta_2)P_e^{\sigma_\varrho}$, the harmonic map is recovered by setting $\varphi=\exp(\pi\zeta_2)\Phi_{-1}J_o$.
### Harmonic maps into $\mathcal{L}^s_{2n}$.
The outer symmetric $SU(2n)$-space that corresponds to the involution $\sigma_{\varrho,1}$ is the space of special Lagrangian subspaces $\mathcal{L}^s_{2n}\cong SU(2n)/SO(2n)$. Take as base point the Lagrangian space $L_o=\mathrm{Span}\{e_1,\ldots,e_{2n}\}$ of $\mathbb{R}^{4n}$ and let $\tau_0$ be the corresponding conjugation, so that the Cartan embedding of $\mathcal{L}^s_{2n}$ into $SU(2n)$ is given by $\tau=g\tau_og^{-1}\mapsto g\tau_0 g^{-1} \tau\in P_e^{\sigma_{\varrho,1}}$.
\[lemafixe\] For each $\zeta\in \mathfrak{I}(SU(2n)^{\sigma_{\varrho,1}})$ we have $\exp\pi\zeta\in P_e^{\sigma_{\varrho,1}}$.
Each $\zeta\in \mathfrak{I}(SU(2n)^{\sigma_{\varrho,1}})$ can be written as $\zeta=\sum_{i=1}^{n}n_i(E_i-E_{2n+1-i}).$ Hence, $\exp\pi\zeta=\pi_V-\pi_{V}^\perp$, where $V=\bigoplus_{{n_i\, \mathrm{even}}}\mathrm{Span}\{e_i,e_{2n+1-i}\}$. Define $g\in SU(2n)$ as follows: if $n_i$ is even, then $g(e_i)=e_i$ and $g(e_{2n+1-i})=e_{2n+1-i}$; if $n_i$ is odd, then $g(e_i)=\mathrm{i} e_{i}$ and $g(e_{2n+1-i})=-\mathrm{i}e_{2n+1-i}$. We have $\exp\pi\zeta=g\tau_0g^{-1}\tau_0$, that is $\exp\pi\zeta\in P_e^{\sigma_{\varrho,1}}$.
Now, identify $\mathcal{L}^s_{2n}$ with $P_e^{\sigma_{\varrho,1}}$ via its Cartan embedding. By Theorem \[tinva\], any harmonic map $\varphi:S^2\to P_e^{\sigma_{\varrho,1}}$ admits an extended solution $\Phi:S^2\setminus D\to U_{\zeta'}^{\sigma_{\varrho,1}}(SU(2n))$, for some $\zeta'\in \mathfrak{I}'(SU(2n))\cap\mathfrak{k}_{\sigma_{\varrho,1}}$ and some discrete subset $D$. We can assume that $\zeta'$ is a $\varrho$-semi-canonical element. The corresponding $S^1$-invariant solution $u_\zeta\circ \Phi$ takes values in $\Omega_{\xi}(SU(2n)^{\sigma_{\varrho,1}})$, with $\xi\in \mathfrak{I}'(SU(2n)^{\sigma_{\varrho,1}})$; and both $\Phi_{-1}$ and $(u_\zeta\circ \Phi)_{-1}$ take values in $P_\xi^{\sigma_{\varrho,1}}$. A priori, $\xi$ can be different from $\zeta$ since $\sigma_{\varrho,1}$ is not a fundamental outer involution. However, by Lemma \[lemafixe\] we have $P_\xi^{\sigma_{\varrho,1}}=P_e^{\sigma_{\varrho,1}}=P_{\zeta'}^{\sigma_{\varrho,1}}.$
If $\zeta$ is a $\varrho$-canonical element such that $\zeta'\preceq \zeta$ and $\mathcal{U}_{\zeta',\zeta'-\zeta}(\Phi)$ is constant, then, taking into account Proposition \[norminf\], there exists a $T_\tau$-invariant extended solution $\tilde\Phi: S^2\setminus D\to U^\tau_{\zeta}(SU(2n)),$ where $$\label{invo1}\tau=\mathrm{Ad}(\exp \pi(\zeta'-\zeta))\circ \sigma_{\varrho,1}.$$ such that $\tilde\Phi_{-1}$ take values in $P_\zeta^\tau$ and $\varphi$ is given up to isometry by $$\label{varphil}\varphi=\exp(\zeta'-\zeta)\tilde{\Phi}_{-1}\tau_0.$$ We conclude that, given a pair $(\zeta,\tau)$, where $\zeta\in \mathfrak{I}(SU(2n)^{\sigma_\varrho})$ is a $\varrho$-canonical element and $\tau$ is an outer involution of the form (\[invo1\]), any extended solution $\tilde{\Phi}:S^2\setminus D\to U_\zeta^\tau(SU(2n)))$ gives rise via (\[varphil\]) to an harmonic map $\varphi$ from the two-sphere into $\mathcal{L}^s_{2n}$ and, conversely, all harmonic two-spheres into $\mathcal{L}^s_{2n}$ arise in this way.
For $\mathcal{L}^s_{4}$, since $\exp\pi2\zeta_1$ belongs to the centre of $SU(4)$, we have five classes of harmonic maps into $\mathcal{L}^s_{4}$: $$(2\zeta_1,\sigma_{\varrho,1}),\, (\zeta_2,\sigma_{\varrho,1}),\,(2\zeta_1+\zeta_2,\sigma_{\varrho,1})\, (\zeta_2,\mathrm{Ad}\exp\pi\zeta_2\circ \sigma_{\varrho,1}),\,(2\zeta_1+\zeta_2,\mathrm{Ad}\exp\pi\zeta_2\circ\sigma_{\varrho,1}).$$
Let us consider in detail the class $(\zeta_2, \sigma_{\varrho,1})$. Clearly $r(\zeta_2)=2$. Let $W_1$, $W_2$, $W_3$ and $W_4$ be the complex one-dimensional images of $E_1$, $E_2$, $E_3$ and $E_4$, respectively. That is, $W_i=\mathrm{Span}\{v_i\}$, where $v_i$ are defined by (\[ves\]). Any extended solution $\Phi:S^2\setminus D\to U_{\zeta_2}^{\sigma_{\varrho,1}}$ is given by $\Phi=\exp C\cdot \gamma_{\zeta_2}$, with $\gamma_{\zeta_2}(\lambda)=\lambda^{-1}\pi_{W_4}+\pi_{W_3\oplus W_2 }+\lambda \pi_{W_1}$, for some holomorphic vector-valued function $C:S^2\setminus D\to (\mathfrak{u}^0_{\zeta_2})_{\sigma_{\varrho,1}}$, where $$(\mathfrak{u}^0_{\zeta_2})_{\sigma_{\varrho,1}}=(\mathfrak{p}_0^{\zeta_2})^\perp\cap\mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,1}}+\lambda(\mathfrak{p}_1^{\zeta_2})^\perp\cap
\mathfrak{m}^{\mathbb{C}}_{\sigma_{\varrho,1}}$$ and $$\begin{aligned}
(\mathfrak{p}_0^{\zeta_2})^\perp\cap\mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,1}}&=({\mathfrak{g}}_{L_1-L_2}\oplus{\mathfrak{g}}_{L_3-L_4} \oplus {\mathfrak{g}}_{L_1-L_3} \oplus {\mathfrak{g}}_{L_2-L_4})\cap\mathfrak{k}^{\mathbb{C}}_{\sigma_{\varrho,1}}, \\ (\mathfrak{p}_1^{\zeta_2})^\perp\cap\mathfrak{m}^{\mathbb{C}}_{\sigma_{\varrho,1}}&= {\mathfrak{g}}_{L_1-L_4}\cap\mathfrak{m}^{\mathbb{C}}_{\sigma_{\varrho,1}}= {\mathfrak{g}}_{L_1-L_4}.\end{aligned}$$
We have $\sigma_{\varrho,1}(X_{1,2})=-X_{3,4}$ and $\sigma_{\varrho,1}(X_{1,3})= X_{2,4}$. Hence we can write $C=C_0+C_1\lambda$, with $$C_0=a(X_{1,2}-X_{3,4})+b(X_{1,3}+X_{2,4}),\,\,\,\,\, C_1=cX_{1,4}$$ for some meromorphic functions $a,b,c$ on $S^2$. The harmonicity equations impose that $ab'-ba'=0$, which means that $b=\alpha a$ for some constant $\alpha\in {\mathbb{C}}.$ Hence given arbitrary meromorphic functions $a,c$ on $S^2$ and a complex constant $\alpha$, $$\Phi=\left(
\begin{array}{cccc}
1 & a & \alpha a & c\lambda \\
0 & 1 & 0 & -\alpha a \\
0 & 0 & 1 & a \\ 0 & 0 & 0 & 1 \\
\end{array}
\right)\cdot \gamma_{\zeta_1},$$ is an extended solution associated to some harmonic map in the class $(\zeta_2, \sigma_{\varrho,1})$. Reciprocally, any harmonic map in such class arises in this way.
[10]{}
F. E. Burstall and M. A. Guest, *Harmonic two-spheres in compact symmetric spaces, revisited*, Math. Ann. **309** (1997), no. 4, 541–572.
F. E. Burstall, J. H. Rawnsley, [[T]{}wistor Theory for Riemannian Symmetric Spaces]{}, Lectures Notes in Math. 1424 Berlin, Heidelberg: 1990.
E. Calabi, *Minimal immersions of surfaces in Euclidean spheres*, J. Diff. Geom., **1** (1967), 111–125.
N. Correia and R. Pacheco, *Harmonic maps of finite uniton number into [$G_2$]{}*, Math. Z. **271** (2012), no. 1-2, 13–32.
N. Correia and R. Pacheco, *[Extended Solutions of the Harmonic Map Equation in the Special Unitary Group]{}*, Q. J. Math. **65** (2014), no. 2, 637–654.
N. Correia and R. Pacheco, *[Harmonic maps of finite uniton number and their canonical elements]{}*, Ann. Global Anal. Geom. **47** (2015), no. 4, 335–358.
J. Dorfmeister, F. Pedit and H. Wu, *Weiestrass type representation of harmonic maps into symmetric spaces*, Comm. Anal. Geom. **6** (1998), 633–668.
J.-H. Eschenburg, A.-L. Mare, and P. Quast, *Pluriharmonic maps into outer symmetric spaces and a subdivision of Weyl chambers*, Bull. London Math. Soc. **42** (2010), no. 6, 1121–1133.
J. Eells and J. C. Wood, *Harmonic maps from surfaces to complex projective spaces*, Adv. in Math. **49** (1983), no. 3, 217–263.
W. Fulton and J. Harris, [Representation theory (a first course)]{}, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991.
M. A. Guest and Y. Ohnita, *Loop group actions on harmonic maps and their applications*, Harmonic maps and integrable systems, 273–292, Aspects Math., E23, Vieweg, Braunschweig, 1994.
S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York 1978.
H. Ma, *Explicit construction of harmonic two-spheres in $SU(3)/SO(3)$*, Kyushu J. Math. **55** (2001), 237–247.
S. Murakami, *Sur la classification des algèbres de Lie réelles et simples*, Osaka J. Math. **2** (1965), 291–307 . A.N. Pressley and G.B. Segal, [[L]{}oop Groups]{}, Oxford University Press, 1986.
A. Bahy-El-Dien and J.C. Wood, *The explicit construction of all harmonic two-spheres in $G_2(\mathbb{R}^n)$,* J. Reine Angew. Math. **398** (1989), 36–66.
K. Uhlenbeck, *[H]{}armonic maps into Lie groups (classical solutions of the chiral model)*, J. Diff. Geom. **30** (1989), 1–50.
W. Ziller, W. Lie groups, representation theory and symmetric spaces. Notes for a course given in the fall of 2010 at the University of Pennsylvania and 2012 at IMPA.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we give a simple proof of the functional relation for the Lerch type Tornheim double zeta function. By using it, we obtain simple proofs of some explicit evaluation formulas for double $L$-values.'
address: 'Department of Mathematics Faculty of Science and Technology Tokyo University of Science Noda, CHIBA 278-8510 JAPAN'
author:
- Takashi Nakamura
title: Simple proof of the functional relation for the Lerch type Tornheim double zeta function
---
Introduction and main results
=============================
We define the Lerch type Tornheim double zeta function by $$T (s,t,u \,; x,y) := \lim_{R \to \infty} \sum_{m,n=1}^{m+n=R} \frac{e^{2 \pi i mx} e^{ 2 \pi i ny}}{m^s n^t (m+n)^u},
\label{eq:defT}$$ where $0 \le x,y \le 1$, $\Re (s+t) >1$, $\Re (t+u) > 1$ and $\Re (s+t+u) > 2$. This function is continued meromorphically by [@NakamuraA Theorem 2.1]. Let $k \in {\mathbb{N}} \cup \{ 0 \}$. The function $T (s, t, u \,; x,y)$ can be continued meromorphically to ${\mathbb{C}}^3$, and all of its singularities are located on the subsets of ${\mathbb{C}}^3$ defined by the following equations; $$\begin{split}
t = 1 - k \qquad &{\rm{if}} \quad x \not \equiv 1, \,\, y \equiv 1 \mod 1, \\
s = 1 - k \qquad &{\rm{if}} \quad x \equiv 1, \,\, y \not \equiv 1 \mod 1, \\
{\mbox{no singularity}} \qquad &{\rm{if}} \quad x \not \equiv 1, \,\, y \not \equiv 1 \mod 1.
\end{split}$$
We write $T(s, t, u) := T (s, t, u \,; 1,1)$ and call this function the Tornheim double zeta function. The values $T(a,b,c)$ for $a,b,c \in{\mathbb{N}}$ were investigated by Tornheim in 1950 and later by Mordell in 1958, and some explicit formulas for them were obtained. Subbarao and Sitaramachandrarao, Huard, Williams and Zhang, and Tsumura researched the explicit formulas for $T(a,b,c)$ for $a,b,c \in{\mathbb{N}}$. The value $T (0, a, b \,; x,y)$ and their multiple sum versions have been already defined in Arakawa and Kaneko [@AraKa] for the case $x,y \in {\mathbb{Q}}$ as special cases of their multiple $L$-values.
As a three-variable function, Matsumoto continued $T(s, t, u)$ meromorphically to the whole ${\mathbb{C}}^3$ plane in [@Ma Theorem 1]. Tsumura [@Tsumurap2 Theorem 4.5], afterwards Nakamura [@Na Theorem 1.2] found functional relations for the Tornheim double zeta function. Moreover, generalizations of the functional relations are proved by Matsumoto and Tsumura [@MaTsu], and Nakamura [@NakamuraA].
In this paper, we show the following functional relation. This functional relation is essentially the same as [@NakamuraT Theorem 3.1]. Therefore we can obtain the all results in [@NakamuraT] by this formula. Zhou gave a simple proof of [@Na Theorem 1.2] in [@Zhou2]. Recently, Li gave the proof similar to Zhou’s one in [@Li], independently. By modifying their methods, we can prove the following theorem.
For $0 < x \ne y < 1$, $a,b \in {\mathbb{N}}$ and $s \in {\mathbb{C}}$, we have $$\begin{split}
&T (a,b,s \,; x,y) + (-1)^b T(b,s,a \,; x-y,x) + (-1)^a T(s,a,b \,; y, y-x) \\
&=\sum_{j=1}^{a} \binom{a+b-j-1}{a-j} \zeta (a+b+s-j \, ; y ) \bigl( \zeta (j \, ; x-y) + (-1)^j \zeta (j \, ; y-x) \bigr) \\
&\quad + \sum_{j=1}^{b} \binom{a+b-j-1}{b-j} \zeta (a+b+s-j \, ; x ) \bigl( \zeta (j \, ; y-x) + (-1)^j \zeta (j \, ; x-y )\bigr) \\
&\quad - \binom{a+b-1}{a} \zeta (a+b+s \, ; y) - \binom{a+b-1}{b} \zeta (a+b+s \, ; x) .
\label{eq:th1}
\end{split}$$ \[th:1\]
Taking $x \to y$ in the above formula, we have [@NakamuraT (3.1)] since $$\lim_{x \to y} \bigl( \zeta (a+b+s-1 \, ; y ) - \zeta (a+b+s-1 \, ; x ) \bigr)
\bigl( \zeta (1 \, ; x-y) - \zeta (1 \, ; y-x) \bigr) = 0 .$$ Define $K (a,b \,; x,y)$ by the right-hand side of (\[eq:th1\]) with $s=0$. By using Theorem \[th:1\], we obtain the following propositions. It should be noted that $T(0,a,b \,; -y,x-y) = T(0,a,b \,; y, y-x)$ when $(x,y) = (1,1)$, $(1,1/2)$, $(1/2,1)$ or $(1/2, 1/2)$. In this case, the next proposition coincides with [@NakamuraT Proposition 2.8] or [@Zhou Proposition 1].
For any admissible index, we have $$\begin{split}
&T(0,a,b \,; -y,x-y) -(-1)^{a+b} T(0,a,b \,; y, y-x)
= \\ &(-1)^b \zeta (a \,; x) \zeta (b \,; y) - (-1)^b K (a,b \,; x,y)
+ \zeta (a \,; -y) \zeta (b \,; x) - \zeta (a+b \,; x-y)
\label{eq:tasite}
\end{split}$$ \[pro:1\]
Let $\phi$, $\chi$ and $\psi$ are Dirichlet characters of conductor $h$, $k$, and $q$, respectively. For any admissible index, we define $L (0,a,b \,; \phi, \chi ,\psi)$ by $$L (0,a,b \,; \phi, \chi ,\psi) := \lim_{R \to \infty} \sum_{m,n=1}^{m+n=R}
\frac{\phi (m) \chi (n) \psi (m+n)}{n^a (m+n)^b}.
\label{eq:defL}$$
Terhune [@Terhune] showed that if and $\chi \psi (-1) = (-1)^{a+b+1}$ then $L (0,a,b \,; 1, \chi ,\psi)$ can be expressed as a polynomial in the values of polylogarithms at certain roots of unity, with coefficients in a cyclotomic field. In [@NakamuraT Proposition 4.5], the author obtained explicit evaluation formulas for $L (0,a,b \,; 1, \chi ,\psi)$ when $\chi \psi (-1) = (-1)^{a+b+1}$. Proposition \[pro:1\] and the following proposition give simpler ones. Denote the Gauss sum by $\tau (\chi) := \sum_{l=1}^{k-1} \chi (l) e^{2 \pi i l/k}$.
Define $2U(a,b \,; x,y):= T(0,a,b \,; x,y) -(-1)^{a+b} T(0,a,b \,; -x, -y)$. Let $\phi \chi \psi(-1) = (-1)^{a+b+1}$. For any admissible index, we have $$\begin{split}
&\tau (\overline{\phi}) \tau (\overline{\chi}) \tau (\overline{\psi}) L(0,a,b \,; \phi, \chi ,\psi) = \\
&\sum_{j=1}^{h-1} \sum_{l=1}^{k-1} \sum_{r=1}^{q-1} \overline{\phi}(j) \overline{\chi}(l) \overline{\psi}(r) U (a,b \,; j/h+r/q, l/k+r/q).
\label{eq:dl1}
\end{split}$$ \[pro:2\]
Proof of results
================
First, suppose $\Re (s) >1$. We define $S (a,b,s \,; x,y)$ by $$S (a,b,s \,; x,y) := T (a,b,s \,; x,y) + (-1)^b T(b,s,a \,; x-y,x) + (-1)^a T(s,a,b \,; y, y-x).$$ It is easy to see that $S (a,b,s \,; x,y) = S (a-1,b,s+1 \,; x,y) + S (a,b-1,s+1 \,; x,y)$ because of $T (a,b,s \,; x,y) = T (a-1,b,s+1 \,; x,y) + T (a,b-1,s+1 \,; x,y)$. Hence we have $$\begin{split}
S (a,b,s \,; x,y) = &\sum_{j=1}^{a} \binom{a+b-j-1}{a-j} S (j,0,a+b+s-j \,; x,y) \\ &+ \sum_{j=1}^{b} \binom{a+b-j-1}{b-j} S (0,j,a+b+s-j \,; x,y).
\end{split}$$ Now we consider the function $S (j,0,a+b+s-j \,; x,y)$ in the above formula. By the definition of $S (a,b,s \,; x,y)$ and the harmonic product formula, we have $$\begin{split}
&S (j,0,a+b+s-j \,; x,y) =
T (j,0,a+b+s-j \,; x,y) \\ &+ T(0,a+b+s-j,j \,; x-y,x) + (-1)^j T(a+b+s-j,j,0 \,; y, y-x) \\= \,
&\zeta (a+b+s-j \, ; y ) \bigl( \zeta (j \, ; x-y) + (-1)^j \zeta (j \, ; y-x) \bigr) - \zeta (a+b+s \, ; x ) .
\end{split}$$ By exchanging the parameters $x$ and $y$, we can repeat the same manner for $S (0,j,a+b+s-j \,; x,y)$. Therefore we obtain this theorem when $\Re (s) >1$ by the well-known formula $\sum_{j=1}^{a} \binom{a+b-j-1}{a-j} = \binom{a+b-1}{b}$. By the analytic continuation, we obtain Theorem \[th:1\].
By putting $s=0$ in (\[eq:th1\]), we have $$\zeta (a \,; x) \zeta (b \,; y) + (-1)^b T(b,0,a \,; x-y,x) + (-1)^a T(0,a,b \,; y, y-x) = K (a,b \,; x,y)$$ On the other hand, one has $$T(0,a,b \,; -y,x-y) + T(b,0,a \,; x-y,x) + \zeta (a+b \,; x-y) = \zeta (a \,; -y) \zeta (b \,; x)$$ by the harmonic product formula. Therefore we have Proposition \[pro:1\] by removing the term $T(b,0,a \,; x-y,x)$.
Recall the well-known formula $$\chi (n) = \frac{1}{\tau (\overline{\chi})} \sum_{l=1}^{k-1} \overline{\chi} (l) e^{2 \pi i ln/k} =
\frac{\chi (-1)}{\tau (\overline{\chi})} \sum_{l=1}^{k-1} \overline{\chi} (l) e^{-2 \pi i ln/k}.$$ By using above formula and $\phi \chi \psi(-1) = (-1)^{a+b+1}$, we have $$\begin{split}
&\tau (\overline{\phi}) \tau (\overline{\chi}) \tau (\overline{\psi}) L(0,a,b \,; \phi, \chi ,\psi) \\
&=\sum_{j=1}^{h-1} \sum_{l=1}^{k-1} \sum_{r=1}^{q-1} \overline{\phi}(j) \overline{\chi}(l) \overline{\psi}(r) T (0,a,b \,; j/h+r/q, l/k+r/q)\\
&= (-1)^{a+b+1} \sum_{j=1}^{h-1} \sum_{l=1}^{k-1} \sum_{r=1}^{q-1} \overline{\phi}(j) \overline{\chi}(l) \overline{\psi}(r) T (0,a,b \,; -j/h-r/q, -l/k-r/q) .
\end{split}$$ Hence we obtain Proposition \[pro:2\].
[1]{} T. Arakawa and M. Kaneko, On multiple $L$-values. [*[J. Math. Soc. Japan]{}*]{} [**[56]{}**]{} (2004), no. 4, 967–991. Z. Li, On functional relations for the alternating analogues of Tornheim’s double zeta function. [*[arXiv:1011.2897v1]{}*]{}. K. Matsumoto, On Mordell-Tornheim and other multiple zeta-functions. Proceedings of the Session in Analytic Number Theory and Diophantine Equations, 17 pp., Bonner Math. Schriften, 360, Univ. Bonn, Bonn, 2003. K. Matsumoto and H. Tsumura, Functional relations among certain double polylogarithms and their character analogues. Šiauliai Math. Semin. [**[11]{}**]{} (2008), 189–205. T. Nakamura, A functional relation for the Tornheim double zeta function. [*[Acta Arith.]{}*]{} [**[125]{}**]{} (2006), no. 3, 257–263. T. Nakamura, Double Lerch series and their functional relations. [*[Aequationes Mathematicae]{}*]{} [**[75]{}**]{}, (2008), no. 3, 251–259. T. Nakamura, Double Lerch value relations and functional relations for Witten zeta functions. Tokyo J. Math. [**[31]{}**]{} (2008), no. 2, 551–574. D. Terhune, Evaluation of double $L$-values. J. Number Theory [**[105]{}**]{} (2004), no. 2, 275–301. H. Tsumura, On functional relations between the Mordell-Tornheim double zeta functions and the Riemann zeta function. [*[Math. Proc. Camb. Phil. Soc.]{}*]{} [**[142]{}**]{} (2007), 395–405. X. Zhou, T. Cai and D. M. Bradley, Signed $q$-analogs of Tornheim’s double series. [*[Proc. Amer. Math. Soc.]{}*]{} [**[136]{}**]{} (2008), no. 8, 2689–2698. X. Zhou, A functional relation for the Tornheim double zeta function. [*[preprint]{}*]{}.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'C. López-Sanjuan[^1]'
- 'A. J. Cenarro'
- 'C. Hernández-Monteagudo'
- 'J. Varela'
- 'A. Molino'
- 'P. Arnalte-Mur'
- 'B. Ascaso'
- 'F. J. Castander'
- 'A. Fernández-Soto'
- 'M. Huertas-Company'
- 'I. Márquez'
- 'V. J. Martínez'
- 'J. Masegosa'
- 'M. Moles'
- 'M. Pović'
- 'J. A. L. Aguerri'
- 'E. Alfaro'
- 'N. Benítez'
- 'T. Broadhurst'
- 'J. Cabrera-Caño'
- 'J. Cepa'
- 'M. Cerviño'
- 'D. Cristóbal-Hornillos'
- 'A. Del Olmo'
- 'R. M. González Delgado'
- 'C. Husillos'
- 'L. Infante'
- 'J. Perea'
- 'F. Prada'
- 'J. M. Quintana'
bibliography:
- 'biblio.bib'
date: 'Received 12 August 2013 – Accepted 17 January 2014'
title: 'The ALHAMBRA survey[^2]: an empirical estimation of the cosmic variance for merger fraction studies based on close pairs'
---
Introduction {#intro}
============
Our understanding of the formation and evolution of galaxies across cosmic time have been greatly improved in the last decade thanks to deep photometric and spectroscopic surveys. Some examples of these successful deep surveys are SDSS [Sloan Digital Sky Survey, @sdssdr7], GOODS [Great Observatories Origins Deep Survey, @goods], AEGIS [All-Wavelength Extended Groth Strip International Survey, @aegis], ELAIS [European Large-Area ISO Survey, @elais], COSMOS [Cosmological Evolution Survey, @cosmos], MGC [Millennium Galaxy Catalogue, @mgc], VVDS [VIMOS VLT Deep Survey, @lefevre05; @vvdsud], DEEP [Deep Extragalactic Evolutionary Probe, @deep2], zCOSMOS [@zcosmos10k], GNS [GOODS NICMOS Survey, @gns], SXDS [Subaru/XMM-Newton Deep Survey, @sxds], or CANDELS [Cosmic Assembly NIR Deep Extragalactic Legacy Survey, @candels; @candels2].
One fundamental uncertainty in any observational measurement derived from galaxy surveys is the [*cosmic variance*]{} ($\sigma_v$), arising from the underlying large-scale density fluctuations and leading to variances larger than those expected from simple Poisson statistics. The most efficient way to tackle with cosmic variance is split the survey in several independent areas in the sky. This minimises the sampling problem better than increase the volume in a wide contiguous field [e.g., @driver10]. However, observational constraints (depth vs area) lead to many existing surveys to have observational uncertainties dominated by the cosmic variance. Thus, a proper estimation of $\sigma_v$ is needed to fully describe the error budget in deep cosmological surveys.
The impact of the cosmic variance in a given survey and redshift range can be estimated using two basic methods: theoretically by analysing cosmological simulations [e.g., @somerville04; @trenti08; @stringer09; @moster11], or empirically by sampling a larger survey [e.g., @driver10]. Unfortunately, previous studies estimate only the cosmic variance affecting number density measurements, and do not tackle the impact of $\sigma_v$ in other important quantities as the merger fraction. Merger fraction studies based on close pair statistics measure the correlation of two galaxy populations at small scales ($\leq 100h^{-1}$ kpc), so the amplitude of the cosmic variance and its dependence on galaxy properties, probed volume, etc. should be different than those in number density studies. In the present paper we take advantage of the unique design, depth, and photometric redshift accuracy of the ALHAMBRA[^3] (Advanced, Large, Homogeneous Area, Medium-Band Redshift Astronomical) survey [@alhambra] to estimate empirically, for the first time, the cosmic variance that affect close pair studies. The ALHAMBRA survey has observed 8 separate regions of the northern sky, comprising 48 sub-fields of $\sim180$ arcmin$^{2}$ each that can be assumed as independent for our purposes. Thus, ALHAMBRA provides 48 measurements of the merger fraction across the sky. The intrinsic dispersion in the distribution of these merger fractions, that we characterise in the present paper, is an observational estimation of the cosmic variance $\sigma_v$.
The paper is organised as follows. In Sect. \[data\] we present the ALHAMBRA survey and its photometric redshifts, and in Sect. \[metodo\] we review the methodology to measure close pair merger fractions when photometric redshifts are used. We present our estimation and characterisation of the cosmic variance for close pair studies in Sect. \[analysis\]. In Sect. \[conclusions\] we summarise our work and present our conclusions. Throughout this paper we use a standard cosmology with $\Omega_{\rm m} = 0.3$, $\Omega_{\Lambda} = 0.7$, $H_{0}= 100h$ km s$^{-1}$ Mpc$^{-1}$, and $h = 0.7$. Magnitudes are given in the AB system.
The ALHAMBRA survey {#data}
===================
The ALHAMBRA survey provides a photometric data set over 20 contiguous, equal-width ($\sim$300Å), non-overlapping, medium-band optical filters (3500Å– 9700Å) plus 3 standard broad-band near-infrared (NIR) filters ($J$, $H$, and $K_{\rm s}$) over 8 different regions of the northern sky [@alhambra]. The survey has the aim of understanding the evolution of galaxies throughout cosmic time by sampling a large enough cosmological fraction of the universe, for which reliable spectral energy distributions (SEDs) and precise photometric redshifts ($z_{\rm p}$’s) are needed. The simulations of @benitez09, relating the image depth and $z_{\rm p}$’s accuracy to the number of filters, have demonstrated that the filter set chosen for ALHAMBRA can achieve a photometric redshift precision that is three times better than a classical $4 - 5$ optical broad-band filter set. The final survey parameters and scientific goals, as well as the technical properties of the filter set, were described by @alhambra. The survey has collected its data for the 20+3 optical-NIR filters in the 3.5m telescope at the Calar Alto observatory, using the wide-field camera LAICA (Large Area Imager for Calar Alto) in the optical and the OMEGA–2000 camera in the NIR. The full characterisation, description, and performance of the ALHAMBRA optical photometric system was presented in @aparicio10. A summary of the optical reduction can be found in Cristóbal-Hornillos et al. (in prep.), while of the NIR reduction in @cristobal09.
The ALHAMBRA survey has observed 8 well-separated regions of the northern sky. The wide-field camera LAICA has four chips with a $15\arcmin \times 15\arcmin$ field-of-view each (0.22 arcsec/pixel). The separation between chips is also $15\arcmin$. Thus, each LAICA pointing provides four separated areas in the sky (black or red squares in Fig. \[alfield\]). Six ALHAMBRA regions comprise two LAICA pointings. In these cases, the pointings define two separate strips in the sky (Fig. \[alfield\]). In our study we assumed the four chips in each strip as independent sub-fields. The photometric calibration of the field ALHAMBRA-1 is currently on-ongoing, and the fields ALHAMBRA-4 and ALHAMBRA-5 comprise one pointing each [see @molino13 for details]. We summarise the properties of the 7 ALHAMBRA fields used in the present paper in Table \[alhambra\_fields\_tab\]. At the end, ALHAMBRA comprises 48 sub-fields of $\sim180$ arcmin$^2$, that we assumed independent, in which we measured the merger fraction following the methodology described in Sect. \[metodo\]. When we searched for close companions near the sub-field boundaries we did not consider the observed sources in the adjacent fields to keep the measurements independent. We prove the independence of the 48 ALHAMBRA sub-fields in Sect \[sec7f\].
[lcccc]{} Field & Overlapping & RA & DEC & sub-fields / area\
name & survey & (J2000) & (J2000) & (\# / deg$^2$)\
ALHAMBRA-2 & DEEP2 & 01 30 16.0 & +04 15 40 & 8 / 0.377\
ALHAMBRA-3 & SDSS & 09 16 20.0 & +46 02 20 & 8 / 0.404\
ALHAMBRA-4 & COSMOS & 10 00 00.0 & +02 05 11 & 4 / 0.203\
ALHAMBRA-5 & GOODS-N & 12 35 00.0 & +61 57 00 & 4 / 0.216\
ALHAMBRA-6 & AEGIS & 14 16 38.0 & +52 24 50 & 8 / 0.400\
ALHAMBRA-7 & ELAIS-N1 & 16 12 10.0 & +54 30 15 & 8 / 0.406\
ALHAMBRA-8 & SDSS & 23 45 50.0 & +15 35 05 & 8 / 0.375\
Total & & & & 48 / 2.381\
Bayesian photometric redshifts in ALHAMBRA
------------------------------------------
We rely on the ALHAMBRA photometric redshifts to compute the merger fraction (Sect. \[metodo\]). The photometric redshifts used all over present paper are fully presented and tested in @molino13, and we summarise their principal characteristics below.
The ALHAMBRA $z_{\rm p}$’s were estimated with BPZ2.0, a new version of BPZ [@benitez00]. BPZ is a SED-fitting method based in a Bayesian inference where a maximum likelihood is weighted by a prior probability. The library of 11 SEDs (4 ellipticals, 1 lenticular, 2 spirals, and 4 starbursts) and the prior probabilities used by BPZ2.0 in ALHAMBRA are detailed in Benítez (in prep.). The ALHAMBRA photometry used to compute the photometric redshifts is PSF-matched aperture-corrected and based on isophotal magnitudes. In addition, a recalibration of the zero point of the images was performed to enhance the accuracy of the $z_{\rm p}$’s. Sources were detected in a synthetic $F814W$ filter image, noted $i$ in the following, defined to resemble the HST/$F814W$ filter. The areas of the images affected by bright stars, as well as those with lower exposure times (e.g., the edges of the images), were masked following @arnaltemur13. The total area covered by the ALHAMBRA survey after masking is 2.38 deg$^{2}$. Finally, a statistical star/galaxy separation is encoded in the variable `Stellar_Flag` of the ALHAMBRA catalogues, and throughout present paper we keep as galaxies those ALHAMBRA sources with $\texttt{Stellar\_Flag} \leq 0.5$.
The photometric redshift accuracy, estimated by comparison with spectroscopic redshifts ($z_{\rm s}$’s), is $\delta_z = 0.0108$ at $i \leq 22.5$ with a fraction of catastrophic outliers of $\eta = 2.1$%. The variable $\delta_z$ is the normalized median absolute deviation of the photometric versus spectroscopic redshift distribution [@ilbert06; @eazy], $$\delta_z = 1.48 \times {\rm median}\,\bigg( \frac{|z_{\rm p} - z_{\rm s}|}{1 + z_{\rm s}} \bigg).$$ The variable $\eta$ is defined as the fraction of galaxies with $|z_{\rm p} - z_{\rm s}|/(1 + z_{\rm s}) > 0.2$. We illustrate the high quality of the ALHAMBRA photometric redshifts in Fig. \[zpvszs\]. We refer to @molino13 for a more detailed discussion.
The `odds` quality parameter, noted $\mathcal{O}$, is a proxy for the photometric redshift accuracy of the sources and is also provided by BPZ2.0. The `odds` is defined as the redshift probability enclosed on a $\pm K(1+z)$ region around the main peak in the probability distribution function (PDF) of the source, where the constant $K$ is specific for each photometric survey. @molino13 find that $K = 0.0125$ is the optimal value for the ALHAMBRA survey. The parameter $\mathcal{O} \in [0,1]$ is related with the confidence of the $z_{\rm p}$, making possible to derive high quality samples with better accuracy and lower rate of catastrophic outliers. For example, a $\mathcal{O} \geq 0.5$ selection for $i \leq 22.5$ galaxies yields $\delta_z = 0.0094$ and $\eta = 1$%, while $\delta_z = 0.0061$ and $\eta = 0.8$% for $\mathcal{O} \geq 0.9$ [see @molino13 for further details]. We explore the optimal `odds` selection in ALHAMBRA for close pair studies in Sect. \[optimal\].
Reliable photometric redshift errors ($\sigma_{z_{\rm p}}$) are needed to compute the merger fraction in photometric samples (Sect. \[metodo\]). In addition to the $z_{\rm p}$, we have the $z_{\sigma}^{+}$ and $z_{\sigma}^{-}$ of each source, defined as the redshifts that enclose 68% of the PDF of the source. We estimated the photometric redshift error of each individual source as $\sigma_{z_{\rm p}} = C \times (z_{\sigma}^{+} - z_{\sigma}^{-})$. The constant $C$ is estimated from the distribution of the variable $$\Delta_z = \frac{z_{\rm p} - z_{\rm s}}{\sigma_{z_{\rm p}}} = \frac{z_{\rm p} - z_{\rm s}}{C \times (z_{\sigma}^{+} - z_{\sigma}^{-})}.$$ The variable $\Delta_z$ should be normally distributed with zero mean and unit variance if the $\sigma_{z_{\rm p}}$’s from ALHAMBRA are a good descriptor of the $z_{\rm p}$’s accuracy [e.g., @ilbert09; @carrasco13]. We find that $\Delta_z$ is described well by a normal function when $C = 0.49$ (Fig. \[deltaz\_dist\], see also @molino13). Note that, with the definition of $z_{\sigma}^{+}$ and $z_{\sigma}^{-}$, $C = 0.5$ was expected. This result also implies that the Gaussian approximation of the PDF assumed in the estimation of the merger fraction (Setc. \[metodo\]) is statistically valid, even if the actual PDF of the individual sources could be multimodal and/or asymmetric at faint magnitudes. We estimated $C$ for different $i$-band magnitudes and `odds` selections, finding that the $C$ values are consistent with the global one within $\pm 0.1$. Thus, we conclude that $\sigma_{z_{\rm p}}$ provides a reliable photometric redshift error for every ALHAMBRA source.
Sample selection
----------------
Throughout present paper we focus our analysis in the galaxies of the ALHAMBRA first data release[^4]. This catalogue comprises $\sim500$k sources and is complete ($5\sigma$, $3\arcsec$ aperture) for $i \leq 24.5$ galaxies [@molino13]. We explored different apparent luminosity sub-samples from $i \leq 23$ to $i \leq 20$. That ensures excellent photometric redshifts and provides reliable merger fraction measurements (Sect. \[optimal\]), because the PDFs of $i \leq 23$ sources are defined well by a single Gaussian peak [@molino13]. In Sect. \[secpop\] we also study the cosmic variance in luminosity- and stellar mass-selected samples. The $B-$band luminosities and the stellar masses of the ALHAMBRA sources were also provided by BPZ2.0 and are included in the ALHAMBRA catalogue [see @molino13 for further details]. The mass-to-light ratios from @taylor11 and a @chabrier03 initial mass function were assumed in the estimation of the stellar masses.
Measuring of the merger fraction in photometric samples {#metodo}
=======================================================
The linear distance between two sources can be obtained from their projected separation, $r_{\rm p} = \phi\,d_A(z_1)$, and their rest-frame relative velocity along the line of sight, $\Delta v = {c\, |z_2 - z_1|}/(1+z_1)$, where $z_1$ and $z_2$ are the redshift of the principal (more luminous/massive galaxy in the pair) and the companion galaxy, respectively; $\phi$ is the angular separation, in arcsec, of the two galaxies on the sky plane; and $d_A(z)$ is the angular diameter distance, in kpc arcsec$^{-1}$, at redshift $z$. Two galaxies are defined as a close pair if $r_{\rm p}^{\rm min} \leq r_{\rm p} \leq r_{\rm p}^{\rm max}$ and $\Delta v \leq \Delta v^{\rm max}$. The PSF of the ALHAMBRA ground-based images is $\lesssim 1.4\arcsec$ (median seeing of $\sim1\arcsec$), which corresponds to $7.6h^{-1}$ kpc in our cosmology at $z = 0.9$. To ensure well de-blended sources and to minimise colour contamination, we fixed $r_{\rm p}^{\rm min}$ to $10h^{-1}$ kpc ($\phi > 1.8\arcsec$ at $z < 0.9$). We left $r_{\rm p}^{\rm max} \leq 50h^{-1}$ kpc as a free parameter and estimate its optimal value in Sect. \[optimal\]. Finally, we set $\Delta v^{\rm max} = 500$ km s$^{-1}$ following spectroscopic studies [e.g., @patton00; @lin08]. With the previous constraints 50%-70% of the selected close pairs will finally merge [@patton08; @bell06; @jian12].
To compute close pairs we defined a principal and a companion sample. The principal sample comprises the more luminous or massive galaxy of the pair, and we looked for those galaxies in the companion sample that fulfil the close pair criterion for each galaxy of the principal sample. If one principal galaxy has more than one close companion, we took each possible pair separately (i.e., if the companion galaxies B and C are close to the principal galaxy A, we study the pairs A-B and A-C as independent). In addition, through present paper we do not impose any luminosity or mass difference between the galaxies in the close pair unless noted otherwise.
With the previous definitions the merger fraction is $$f_{\rm m}\ = \frac{N_{\rm p}}{N_{1}},\label{ncspec}$$ where $N_1$ is the number of sources in the principal sample and $N_{\rm p}$ the number of close pairs. This definition applies to spectroscopic volume-limited samples, but we rely on photometric redshifts to compute $f_{\rm m}$ in ALHAMBRA. In a previous work, @clsj10pargoods develop a statistical method to obtain reliable merger fractions from photometric redshift catalogues as those from the ALHAMBRA survey. This methodology has been tested with the MGC [@clsj10pargoods] and the VVDS [@clsj12sizecos] spectroscopic surveys, and successfully applied in the GOODS-South [@clsj10pargoods] and the COSMOS fields [@clsj12sizecos]. We recall the main points of this methodology below and we explore how to apply it optimally over the ALHAMBRA data in Sect. \[optimal\].
We used the following procedure to define a close pair system in our photometric catalogue [see @clsj10pargoods for details]: first we search for close spatial companions of a principal galaxy, with redshift $z_1$ and uncertainty $\sigma_{z_1}$, assuming that the galaxy is located at $z_1 - 2\sigma_{z_1}$. This defines the maximum $\phi$ possible for a given $r_{\rm p}^{\rm max}$ in the first instance. If we find a companion galaxy with redshift $z_2$ and uncertainty $\sigma_{z_2}$ at $r_{\rm p} \leq r_{\rm p}^{\rm max}$, we study both galaxies in redshift space. For convenience, we assume below that every principal galaxy has, at most, one close companion. In this case, our two galaxies could be a close pair in the redshift range $$[z^{-},z^{+}] = [z_1 - 2\sigma_{z_1}, z_1 + 2\sigma_{z_1}] \cap [z_2- 2\sigma_{z_2}, z_2 + 2\sigma_{z_2}].$$ Because of variation in the range $[z^{-},z^{+}]$ of the function $d_A(z)$, a sky pair at $z_1 - 2\sigma_{z_1}$ might not be a pair at $z_1 + 2\sigma_{z_1}$. We thus impose the condition $r_{\rm p}^{\rm min} \leq r_{\rm p} \leq r_{\rm p}^{\rm max}$ at all $z \in [z^{-},z^{+}]$, and redefine this redshift interval if the sky pair condition is not satisfied at every redshift. After this, our two galaxies define the close pair system $k$ in the redshift interval $[z^{-}_k,z^{+}_k]$, where the index $k$ covers all the close pair systems in the sample.
The next step is to define the number of pairs associated to each close pair system $k$. For this, and because all our sources have a photometric redshift, we suppose in the following that a galaxy $i$ in whatever sample is described in redshift space by a Gaussian probability distribution, $$P_i\,(z_i\,|\,z_{{\rm p},i},\sigma_{z_{{\rm p},i}}) = \frac{1}{\sqrt{2\pi}\sigma_{z_{{\rm p},i}}}\exp\bigg[-\frac{(z_i-z_{{\rm p},i})^2}{2\sigma_{z_{{\rm p},i}}^2}\bigg]\label{zgauss}.$$
With the previous distribution we are able to treat statistically all the available information in redshift space and define the number of pairs at redshift $z_1$ in system $k$ as $$\nu_{k}\,(z_1) = {\rm C}_k\, P_1 (z_1\, |\, z_{{\rm p},1},\sigma_{z_{{\rm p},1}}) \int_{z_{\rm m}^{-}}^{z_{\rm m}^{+}} P_2 (z_2\, |\, z_{{\rm p},2},\sigma_{z_{{\rm p},2}})\, {\rm d}z_2,\label{nuj}$$ where $z_1 \in [z^{-}_k,z^{+}_k]$, the integration limits are $$\begin{aligned}
z_{\rm m}^{-} = z_1(1-\Delta v^{\rm max}/c) - \Delta v^{\rm max}/c,\\
z_{\rm m}^{+} = z_1(1+\Delta v^{\rm max}/c) + \Delta v^{\rm max}/c,\end{aligned}$$ the subindex 1 \[2\] refers to the principal \[companion\] galaxy in the system $k$, and the constant ${\rm C}_k$ normalises the function to the total number of pairs in the interest range, $$2 N_{\rm p}^k = \int_{z_k^{-}}^{z_k^{+}} P_1 (z_1\, |\, z_{{\rm p},1},\sigma_{z_{{\rm p},1}})\, {\rm d}z_1 + \!\! \int_{z_k^{-}}^{z_k^{+}} P_2 (z_2\, |\, z_{{\rm p},2},\sigma_{z_{{\rm p},2}})\, {\rm d}z_2.$$ Note that $\nu_k = 0$ if $z_1 < z_k^-$ or $z_1 > z_k^+$. The function $\nu_k$ tells us how the number of pairs in the system $k$, noted $N_{\rm p}^k$, are distributed in redshift space. The integral in Eq. (\[nuj\]) spans those redshifts in which the companion galaxy has $\Delta v \leq \Delta v^{\rm max}$ for a given redshift of the principal galaxy. This translates to $z_{\rm m}^{+} - z_{\rm m}^{-} \sim 0.005$ in our redshift range of interest.
With the previous definitions, the merger fraction in the interval $z_{\rm r} = [z_{\rm min}, z_{\rm max})$ is $$f_{{\rm m}} = \frac{\sum_k \int_{z_{\rm min}}^{z_{\rm max}}{\nu_k\,(z_1)}\, {\rm d}z_1}{\sum_i \int_{z_{\rm min}}^{z_{\rm max}} P_i\, (z_i\,|\,z_{{\rm p},i},\sigma_{z_{{\rm p},i}})\, {\rm d}z_i}.\label{ncphot}$$ If we integrate over the whole redshift space, $z_{\rm r} = [0,\infty)$, Eq. (\[ncphot\]) becomes $$f_{{\rm m}} = \frac{\sum_k N_{\rm p}^k}{N_1},\label{ncphot2}$$ where $\sum_k N_{\rm p}^k$ is analogous to $N_{\rm p}$ in Eq. (\[ncspec\]). In order to estimate the observational error of $f_{{\rm m}}$, noted $\sigma_{f}$, we used the jackknife technique [@efron82]. We computed partial standard deviations, $\delta_k$, for each system $k$ by taking the difference between the measured $f_{{\rm m}}$ and the same quantity with the $k$th pair removed for the sample, $f_{{\rm m}}^k$, such that $\delta_k = f_{{\rm m}} - f_{{\rm m}}^k$. For a redshift range with $N_{\rm p}$ systems, the variance is given by $\sigma_{f}^2 = [(N_{\rm p}-1) \sum_k \delta_k^2]/N_{\rm p}$.
Border effects in redshift and in the sky plane {#border}
-----------------------------------------------
When we search for a primary source’s companion, we define a volume in the sky plane-redshift space. If the primary source is near the boundaries of the survey, a fraction of the search volume lies outside of the effective volume of the survey. @clsj10pargoods find that border effects in the sky plane are representative (i.e., $1\sigma$ discrepancy) only at $r_{\rm p}^{\rm max} \gtrsim 70h^{-1}$ kpc. Thus, we restricted the search radius in our study to $r_{\rm p}^{\rm max} \leq 50h^{-1}$ kpc.
We avoid the incompleteness in redshift space by including in the samples not only the sources inside the redshift range $[z_{\rm min}, z_{\rm max})$ under study, but also those sources with either $z_{{\rm p},i} + 2\sigma_{z_{{\rm p},i}} \geq z_{\rm min}$ or $z_{{\rm p},i} - 2\sigma_{z_{{\rm p},i}} < z_{\rm max}$.
The merger rate {#secmr}
---------------
The final goal of merger studies is the estimation of the merger rate $R_{\rm m}$, defined as the number of mergers per galaxy and Gyr$^{-1}$. The merger rate is computed from the merger fraction by close pairs as $$R_{\rm m} = \frac{C_{\rm m}}{T_{\rm m}}\,f_{\rm m},$$ where $C_{\rm m}$ is the fraction of the observed close pairs than finally merge after a merger time scale $T_{\rm m}$. The merger time scale and the merger probability $C_{\rm m}$ should be estimated from simulations [e.g., @kit08; @lotz10gas; @lotz10t; @lin10; @jian12; @moreno13]. On the one hand, $T_{\rm m}$ depends mainly on the search radius $r_{\rm p}^{\rm max}$, the stellar mass of the principal galaxy, and the mass ratio between the galaxies in the pair, with a mild dependence on redshift and environment [@jian12]. On the other hand, $C_{\rm m}$ depends mainly on $r_{\rm p}^{\rm max}$ and environment, with a mild dependence on both redshift and the mass ratio between the galaxies in the pair [@jian12]. Despite of the efforts in the literature to estimate both $T_{\rm m}$ and $C_{\rm m}$, different cosmological and galaxy formation models provide different values within a factor of two–three [e.g., @hopkins10mer]. To avoid model-dependent results, in the present paper we focus therefore in the cosmic variance of the observational merger fraction $f_{\rm m}$.
Estimation of the cosmic variance for merger fraction studies {#analysis}
=============================================================
Theoretical background {#theory}
----------------------
In this section we recall the theoretical background and define the basic variables involved in the cosmic variance definition and characterisation. The [*relative cosmic variance*]{} ($\sigma_v$) arises from the underlying large-scale density fluctuations and lead to variances larger than those expected from simple Poisson statistics. Following @somerville04 and @moster11, the mean $\langle N \rangle$ and the variance $\langle N^2 \rangle - \langle N \rangle^2$ in the distribution of galaxies are given by the first and second moments of the probability distribution $P_N(V_c)$, which describes the probability of counting $N$ objects within a volume $V_c$. The relative cosmic variance is defined as $$\sigma_v^2 = \frac{\langle N^2 \rangle - \langle N \rangle^2}{\langle N \rangle^2} - \frac{1}{\langle N \rangle}.\label{cosvarteo}$$ The second term represents the correction for the Poisson shot noise. The second moment of the object counts is $$\langle N^2 \rangle = \langle N \rangle^2 + \langle N \rangle + \frac{\langle N \rangle^2}{V_c^2} \int_{V_c} \xi(|{r}_{\rm a} - r_{\rm b}|)\,{\rm d}V_{c,{\rm a}}\,{\rm d}V_{c,{\rm b}},$$ where $\xi$ is the two-point correlation function of the sample under study [@peebles80]. Combining this with Eq. (\[cosvarteo\]), the relative cosmic variance can be written as $$\sigma_v^2 = \frac{1}{V_c^2}\,\int_{V_c} \xi(|{\bf r}_{\rm a} - {\bf r}_{\rm b}|)\,{\rm d}V_{c,{\rm a}}\,{\rm d}V_{c,{\rm b}}.\label{cosvarteoxi}$$ Thus, the cosmic variance of a given sample depends on the correlation function of that population. We can approximate the galaxy correlation function in Eq. (\[cosvarteoxi\]) by the linear theory correlation function for dark matter $\xi_{\rm dm}$, $\xi = b^2\,\xi_{\rm dm}$, where $b$ is the galaxy bias. The bias at a fixed scale depends mainly on both redshift and the selection of the sample under study. With this definition of the correlation function we find that $$\sigma_v \propto \frac{b}{V_c^{1 - \alpha}},$$ where the power law index $\alpha$ takes into account the extra volume dependence from the integral of the correlation function $\xi_{\rm dm}$ in Eq. (\[cosvarteoxi\]).
The bias of a particular population is usually measured from the analysis of the correlation function and is well established that the bias increases with luminosity and stellar mass [see @zehavi11; @coupon12; @marulli13; @arnaltemur13 and references therein]. The estimation of the bias is a laborious task, so we decided to use the redshift and the number density $n$ of the population under study instead of the bias to characterise the cosmic variance. The number density is an observational quantity that decreases with the increase of the luminosity and the mass selection, so a $b \propto n^{-\beta}$ relation is expected. This inverse dependence is indeed suggested by @nuza13 results.
In summary, we expect $$\sigma_v \propto \frac{b}{V_c^{1 - \alpha}} \propto \frac{z^{\gamma}}{n^{\beta}\,V_c^{1 - \alpha}}.\label{cosvarteofin}$$ This equation shows that the number density of galaxies, the redshift, and the cosmic volume can be assumed as independent variables in the cosmic variance parametrisation. Equation (\[cosvarteofin\]) and the deduction above apply to the cosmic variance in the number of galaxies. We are interested on the cosmic variance of the merger fraction by close pairs instead, so a dependence on $V_c$, redshift, and the number density of the two populations under study, noted $n_1$ for principal galaxies and $n_2$ for the companion galaxies, is expected. We used therefore this four variables ($n_1$, $n_2$, $z$, and $V_c$) to characterise the cosmic variance in close pair studies (Sect. \[sigv\]).
The power-law indices in Eq. (\[cosvarteofin\]) could be different for luminosity- and mass-selected samples, as well as for flux-limited samples. In the present paper we use flux-limited samples selected in the $i$ band to characterise the cosmic variance. This choice has several benefits, since we have a well controlled selection function, a better understanding of the photometric redshifts and their errors, and we have access to larger samples at lower redshift that in the luminosity and the stellar mass cases. That improves the statistics and increases the useful redshift range. At the end, future studies will be interested on the cosmic variance in physically selected samples (i.e., luminosity or stellar mass). Thus, in Sect. \[secpop\] we compare the results from the flux-limited $i-$band samples with the actual cosmic variance measured in physically selected samples.
\[C3\] Finally, we set the definition of the number density $n$. In the present paper the number density of a given population is the [*cosmic average number density*]{} of that population. For example, if we are studying the merger fraction in a volume dominated by a cluster, we should not use the number density in that volume, but the number density derived from a general luminosity or mass function work instead. Thanks to the 48 sub-fields in ALHAMBRA we have direct access to the average number densities of the populations under study (Sect. \[secn1\]).
Distribution of the merger fraction and $\sigma_{v}$ estimation {#ffdist}
---------------------------------------------------------------
In this section we explore which statistical distribution reproduces better the observed merger fractions and how to measure reliably the cosmic variance $\sigma_{v}$. As representative examples, we show in Fig. \[ff\_lognormal\] the distributions of the merger fraction $f_{\rm m}$ in the 48 ALHAMBRA sub-fields for $i \leq 22$ and $i \leq 21$ galaxies. The merger fraction was measured from close pairs with $10h^{-1}\ {\rm kpc} \leq r_{\rm p} \leq 30h^{-1}$ kpc. Unless noted otherwise, in the following the principal and the companion samples comprise the same galaxies. We find that the observed distributions are not Gaussian, but follow a log-normal distribution instead, $$P_{LN}\,(f_{\rm m}\,|\,\mu, \sigma) = \frac{1}{\sqrt{2 \pi}\,\sigma f_{\rm m}}\,{\rm exp}\,\bigg[-\frac{(\ln f_{\rm m} - \mu)^2}{2 \sigma^2}\bigg]\,,\label{Plog}$$ where $\mu$ and $\sigma$ are the median and the dispersion of a Gaussian function in log-space $f'_{\rm m} = \ln f_{\rm m}$. This is, $$P_{G}\,(f'_{\rm m}\,|\,\mu, \sigma) = \frac{1}{\sqrt{2 \pi}\,\sigma}\,{\rm exp}\,\bigg[-\frac{(f'_{\rm m} - \mu)^2}{2 \sigma^2}\bigg]\,.\label{Pg}$$ The 68% confidence interval of the log-normal distribution is $[{\rm e}^{\mu}{\rm e}^{-\sigma}, {\rm e}^{\mu}{\rm e}^{\sigma}]$. This functional distribution was expected for two reasons. First, the merger fraction can not be negative, implying an asymmetric distribution [@cameron11]. Second, the distribution of overdense structures in the universe is log-normal [e.g., @coles91; @delatorre10; @kovac10] and the merger fraction increases with density [@lin10; @deravel11; @pawel12]. We checked that the merger fraction follows a log-normal distribution in all the samples explored in the present paper.
The variable $\sigma$ encodes the relevant information about the dispersion in the merger fraction distribution, including the dispersion due to the cosmic variance. The study of the median value of the merger fraction in ALHAMBRA, estimated as ${\rm e}^{\mu}$, and its dependence on $z$, stellar mass, or colour, is beyond the scope of the present paper and we will address this issue in a future work.
A best least-squares fit with a log-normal function to the distributions in Fig. \[ff\_lognormal\] shows that $\sigma$ increases with the apparent brightness, from $\sigma = 0.33$ for $i \leq 22$ galaxies to $\sigma = 0.62$ for $i \leq 21$ galaxies. However, the origin of the observed $\sigma$ is twofold: (i) the intrinsic dispersion due to the cosmic variance $\sigma_v$ (i.e., the field-to-field variation in the merger fraction because of the clustering of the galaxies), and (ii) the dispersion due to the observational errors $\sigma_{\rm o}$ (i.e., the uncertainty in the measurement of the merger fraction in a given field, including the Poisson shot noise term). Thus, the dispersion $\sigma$ reported in Fig. \[ff\_lognormal\] is an upper limit for the actual cosmic variance $\sigma_v$. We deal with this limitation applying a maximum likelihood estimator (MLE) to the observed distributions. In Appendix \[mlmethod\] we develop a MLE that estimates the more probable values of $\mu$ and $\sigma_v$, assuming that the merger fraction follows a Gaussian distribution in log-space (Eq. \[\[Pg\]\]) that is affected by known observational errors $\sigma_{\rm o}$. We prove that the MLE provides an unbiased estimation of $\mu$ and $\sigma_v$, as well as reliable uncertainties of these parameters. Applying the MLE to the distributions in Fig. \[ff\_lognormal\], we find than $\sigma_v$ is lower than $\sigma$, as anticipated, and that the cosmic variance increases with the apparent brightness from $\sigma_v = 0.25 \pm 0.04$ for $i \leq 22$ galaxies to $\sigma_v = 0.44 \pm 0.08$ for $i \leq 21$ galaxies.
We constraint the dependence of $\sigma_v$ on the number density of the populations under study in Sects. \[secn1\] and \[secn2\], on the probed cosmic volume in Sect. \[secvc\], and on redshift on Sect. \[secz\]. That provides a complete description of the cosmic variance for merger fraction studies. We stress that our definition of $\sigma_v$ differs from the classical definition of the relative cosmic variance presented in Sect. \[theory\], which is equivalent to ${\rm e}^{\sigma_v}$. However, $\sigma_v$ encodes the relevant information needed to estimate the intrinsic dispersion in the measurement of the merger fraction due to the clustering of galaxies.
[lcccc]{} $r_{\rm p}^{\rm max}$ & $\sigma_v$ & $\sigma_v$ & $\sigma_v$\
($h^{-1}$ kpc) & ($i \leq 22.5$) & ($i \leq 21.5$) & ($i \leq 21.0$)\
30 & $0.181 \pm 0.030$ & $0.235 \pm 0.053$ & $0.447 \pm 0.091$\
35 & $0.184 \pm 0.027$ & $0.246 \pm 0.045$ & $0.433 \pm 0.079$\
40 & $0.199 \pm 0.026$ & $0.284 \pm 0.041$ & $0.460 \pm 0.073$\
45 & $0.195 \pm 0.024$ & $0.289 \pm 0.040$ & $0.447 \pm 0.067$\
50 & $0.190 \pm 0.023$ & $0.284 \pm 0.038$ & $0.451 \pm 0.066$\
Average & $0.190 \pm 0.011$ & $0.272 \pm 0.019$ & $0.448 \pm 0.033$\
Optimal estimation of $\sigma_{v}$ in the ALHAMBRA survey {#optimal}
---------------------------------------------------------
In the previous section we have defined the methodology to compute the cosmic variance from the observed distribution of the merger fraction. However, as shown by @clsj10pargoods, to avoid projection effects we need a galaxy sample with either small photometric redshift errors or a large fraction of spectroscopic redshifts. In the present study we did not use information from spectroscopic redshifts, so we should check that the photometric redshifts in ALHAMBRA are good enough for our purposes. A natural way to select excellent $z_{\rm p}$’s in ALHAMBRA is by a selection in the `odds` parameter. On the one hand, this selection increases the accuracy of the photometric redshifts of the sample and minimises the fraction of catastrophic outliers [@molino13], improving the merger fraction estimation. On the other hand, our sample becomes incomplete and could be biased toward a population of either bright galaxies or galaxies with marked features in the SED (i.e., emission line galaxies or old populations with a strong $4000\AA$ break). In this section we study how the merger fraction in ALHAMBRA depends on the $\mathcal{O}$ selection and derive the optimal one to estimate the cosmic variance.
Following the methodology from spectroscopic surveys [e.g., @lin04; @deravel09; @clsj11mmvvds; @clsj13ffmassiv], if we have a population with a total number of galaxies $N_{\rm tot}$ in a given volume and we observe a random fraction $f_{\rm obs}$ of these galaxies, the merger fraction of the total population is $$f_{\rm m} = f_{\rm m, obs} \times f_{\rm obs}^{-1},\label{fmp}$$ where $f_{\rm m, obs}$ is the merger fraction of the observed sample. In ALHAMBRA we applied a selection in the parameter $\mathcal{O}$, so Eq. (\[fmp\]) becomes $$f_{\rm m} = f_{\rm m}\,(\geq \mathcal{O}_{\rm sel}) \times \frac{N_{\rm tot}}{N\,(\geq \mathcal{O}_{\rm sel})},\label{fmal}$$ where $N\,(\geq \mathcal{O}_{\rm sel})$ is the number of galaxies with `odds` higher than $\mathcal{O}_{\rm sel}$ (i.e., galaxies with $\mathcal{O} \geq \mathcal{O}_{\rm sel}$), $N_{\rm tot}$ is the total number of galaxies (i.e., galaxies with $\mathcal{O} \geq 0$), and $f_{\rm m}\,(\geq \mathcal{O}_{\rm sel})$ is the merger faction of those galaxies with $\mathcal{O} \geq \mathcal{O}_{\rm sel}$. Because $f_{\rm m}$ must be independent of the $\mathcal{O}$ selection, the study of $f_{\rm m}$ as a function of $\mathcal{O}_{\rm sel}$ provides the clues about the optimal `odds` selection for merger fraction studies in ALHAMBRA. We show $f_{\rm m}$ as a function of $\mathcal{O}_{\rm sel}$ for galaxies with $i \leq 22.5$ at $0.3 \leq z < 0.9$ in Fig. \[fmvsodds\]. We find that
- the merger fraction is roughly constant for $0.2 \leq \mathcal{O}_{\rm sel} \leq 0.6$. This is the expected result if the merger fraction is reliable and measured in a non biased sample. In this particular case, the $\mathcal{O}_{\rm sel} = 0.2$ (0.6) sample comprises 98% (66%) of the total number of galaxies with $i \leq 22.5$;
- the merger fraction is overestimated for $\mathcal{O}_{\rm sel} \leq 0.1$. Even if only a small fraction of galaxies with poor constrains in their $z_{\rm p}$’s are included in the sample, the projection effects become important;
- the merger fraction is overestimated for $\mathcal{O}_{\rm sel} \geq 0.7$. This behaviour at high `odds` (i.e., in samples with high quality photometric redshifts) suggests that the retained galaxies are a biased sub-sample of the general population under study.
In the analysis above we only accounted for close companions of $i \leq 22.5$ galaxies with $10h^{-1}\ {\rm kpc} \leq r_{\rm p} \leq 30h^{-1}$ kpc, but we can use other values of $r_{\rm p}^{\rm max}$ or searching over different samples. On the one hand, we repeated the study for $r_{\rm p}^{\rm max} = 40$ and 50$h^{-1}$ kpc, finding the same behaviour than for $r_{\rm p}^{\rm max} = 30 h^{-1}$ kpc (Fig. \[fmvsodds\]). The only differences are that the merger fraction increases with the search radius and that the $\mathcal{O}_{\rm sel} = 0.2$ point starts to deviate from the expected value (the search area increases with $r_{\rm p}^{\rm max}$ and more accurate $z_{\rm p}$’s are needed to avoid projection effects). On the other hand, we explored a wide range of $i-$band magnitude selections, from $i \leq 23$ to 20, in the three previous $r_{\rm p}^{\rm max}$ cases. We find again the same behaviour. That reinforces our arguments above and suggests $0.3 \leq \mathcal{O}_{\rm sel} \leq 0.6$ as acceptable `odds` limits to select samples for merger fraction studies in ALHAMBRA.
The merger fraction increases with the search radius (Fig. \[fmvsodds\]). However, the merger rate $R_{\rm m}$ (Sect. \[secmr\]) is a physical property of any population and it can not depend on $r_{\rm p}^{\rm max}$. Thus, the increase in the merger fraction with the search radius is compensated with the increase in the merger time scale [e.g., @deravel09; @clsj11mmvvds]. This is, $R_{\rm m} \propto f_{\rm m}(r_{\rm p}^{\rm max})/T_{\rm m}(r_{\rm p}^{\rm max})$. For the same reason, the cosmic variance of the merger rate can not depend on $r_{\rm p}^{\rm max}$. In other words, the 68% confidence interval of the merger rate, $[R_{\rm m}{\rm e}^{-\sigma_v}, R_{\rm m}{\rm e}^{\sigma_v}]$, should be independent of the search radius. Expanding the previous confidence interval we find that $$\begin{aligned}
[R_{\rm m}{\rm e}^{-\sigma_v}, R_{\rm m}{\rm e}^{\sigma_v}] \propto \label{mrconf} \\\nonumber
[f_{\rm m}(r_{\rm p}^{\rm max})\,T^{-1}_{\rm m}(r_{\rm p}^{\rm max})\,{\rm e}^{-\sigma_v}, f_{\rm m}(r_{\rm p}^{\rm max})\,T^{-1}_{\rm m}(r_{\rm p}^{\rm max})\,{\rm e}^{\sigma_v}] = \\\nonumber
[f_{\rm m}(r_{\rm p}^{\rm max})\,{\rm e}^{-\sigma_v}, f_{\rm m}(r_{\rm p}^{\rm max})\,{\rm e}^{\sigma_v}]\,T^{-1}_{\rm m}(r_{\rm p}^{\rm max}).\end{aligned}$$ Note that the dependence on $r_{\rm p}^{\rm max}$ is encoded in the median merger fraction and in the merger time scale. Thus, [*the cosmic variance $\sigma_v$ of the merger fraction should not depend on the search radius*]{}. We checked this prediction by studying the cosmic variance as a function of the search radius for $i \leq 22.5, 21.5$, and 21 galaxies with $\mathcal{O} \geq \mathcal{O}_{\rm sel} = 0.3$ at $0.3 \leq z < 0.9$. We find that $\sigma_v$ is consistent with a constant value irrespective of $r_{\rm p}^{\rm max}$ in the three populations probed, as desired (Table \[sigv\_rp\_tab\] and Fig. \[sigv\_vs\_rp\]). This supports $\sigma_v$ as a good descriptor of the cosmic variance and our methodology to measure it. In the previous analysis we have omitted the merger probability $C_{\rm m}$, which mainly depends on $r_{\rm p}^{\rm max}$ and environment (Sect. \[secmr\]). The merger fraction correlates with environment, so the merger probability could modify the factor ${\rm e}^{\sigma_v}$ in Eq. (\[mrconf\]). Because a constant $\sigma_v$ with $r_{\rm p}^{\rm max}$ is observed, the impact of $C_{\rm m}$ in the $f_{\rm m}$ to $R_{\rm m}$ translation should be similar in the range of $r_{\rm p}^{\rm max}$ explored. Detailed cosmological simulations are needed to clarify this issue.
Finally, we studied the dependence of $\sigma_v$ on the `odds` selection for $i \leq 22.5$ galaxies at $0.3 \leq z < 0.9$. Following the same arguments than before, [*the cosmic variance should not depend on the `odds` selection*]{}. We find that (i) $\sigma_v$ is consistent with a constant value as a function of $r_{\rm p}^{\rm max}$ for any $\mathcal{O}_{\rm sel}$, reinforcing our results above, and (ii) $\sigma_v$ is independent of the `odds` selection at $0.1 \leq \mathcal{O}_{\rm sel} \leq 0.5$ (Fig.\[sigvvsodds\]). As for the merger fraction, we checked that different populations follow the same behaviour. We set therefore $\mathcal{O} \geq \mathcal{O}_{\rm sel} = 0.3$ as the optimal `odds` selection to measure the cosmic variance in ALHAMBRA. This selection provides excellent photometric redshifts and ensures representative samples.
In summary, in the following we estimate the cosmic variance $\sigma_v$ from the merger fractions measured in the 48 ALHAMBRA sub-fields with $10h^{-1}$ kpc $\leq r_{\rm p} \leq 50 h^{-1}$ kpc close pairs (the $\sigma_v$ uncertainty is lower for larger search radii) and in samples with $\mathcal{O} \geq \mathcal{O}_{\rm sel} = 0.3$. That ensures reliable results in representative (i.e., non biased) samples.
Characterisation of $\sigma_{v}$ {#sigv}
--------------------------------
At this stage we have set both the methodology to compute a robust cosmic variance from the observed merger fraction distribution (Sect. \[ffdist\]) and the optimal search radius and `odds` selection to estimate $\sigma_v$ in ALHAMBRA (Sect. \[optimal\]). Now we can characterise the cosmic variance as a function of the populations under study (Sects. \[secn1\] and \[secn2\]), the probed cosmic volume (Sect. \[secvc\]), and the redshift (Sect. \[secz\]).
### Dependence on the number density of the principal sample {#secn1}
In this section we explore how the cosmic variance depends on the number density $n_1$ of the principal population under study. For that, we took the same population as principal and companion sample. We study the dependence on the companion sample in Sect. \[secn2\]. To avoid any dependence of $\sigma_v$ on either the probed cosmic volume and $z$, and to minimise the observational errors, in this section we focus in the redshift range $0.3 \leq z < 0.9$. This range probes a cosmic volume of $V_c \sim 1.4 \times 10^{5}$ Mpc$^{3}$ in each ALHAMBRA sub-field. To explore different number densities, we measured the cosmic variance for different $i-$band selected samples, from $i \leq 20$ to $i \leq 23$ in 0.5 magnitude steps. We estimated the average number density $n_1$ in the redshift range $z_{\rm r}$ as the median number density in the 48 ALHAMBRA sub-fields, with $$n_1^j\,(z_{\rm r}) = \frac{\sum_i \int_{z_{\rm min}}^{z_{\rm max}} P_i^j\, (z_i\,|\,z_{{\rm p},i},\sigma_{z_{{\rm p},i}})\, {\rm d}z_i}{V_c^j (z_{\rm r})}$$ being the number density in the sub-field $j$ and $V_c^j$ the cosmic volume probed by it at $z_{\rm r}$. In the measurement of the number density all the galaxies were taking into account, i.e., any `odds` selection was applied ($\mathcal{O} \geq 0$). We stress that our measured number densities are unaffected by cosmic variance, and they can be used therefore to characterise $\sigma_{v}$. We report our measurements in Table \[sigv\_n1\_tab\].
We find that the cosmic variance increases as the number density decreases (Fig. \[sigv\_vs\_n1\]), as expected by Eq. (\[cosvarteofin\]). The error-weighted least-squares fit of a power-law to the data is $$\sigma_v\,(n_1) = (0.45 \pm 0.04) \times \bigg( \frac{n_1}{10^{-3}\ {\rm Mpc}^{-3}} \bigg)^{-0.54 \pm 0.06}.\label{sigv_n1}$$
In this section and in the following ones we used $i-$band selected samples to characterise $\sigma_v$. We show that the results obtained with these $i-$band samples can be applied to luminosity- and stellar mass-selected samples in Sect. \[secpop\].
[lcc]{} Principal & $n_1$ & $\sigma_v$\
sample & ($10^{-3}$ Mpc$^{-3}$) &\
$i \leq 23.0$ & $6.88 \pm 0.16$ & $0.158 \pm 0.019$\
$i \leq 22.5$ & $4.79 \pm 0.14$ & $0.190 \pm 0.023$\
$i \leq 22.0$ & $3.30 \pm 0.11$ & $0.245 \pm 0.030$\
$i \leq 21.5$ & $2.12 \pm 0.07$ & $0.284 \pm 0.038$\
$i \leq 21.0$ & $1.28 \pm 0.05$ & $0.451 \pm 0.066$\
$i \leq 20.5$ & $0.73 \pm 0.03$ & $0.587 \pm 0.100$\
$i \leq 20.0$ & $0.35 \pm 0.01$ & $0.695 \pm 0.154$\
### Dependence on the cosmological volume {#secvc}
In this section we explore the dependence of the cosmic variance on the cosmic volume probed by the survey. We defined $\sigma_v^{*}$ as $\sigma_v^{*} = \sigma_v / \sigma_v\,(n_1)$. This erased the dependence on the number density of the population and only volume effects were measured. We explored smaller cosmic volumes than in the previous section by studying (i) different redshift ranges over the full ALHAMBRA area (avoiding redshift ranges smaller than 0.1), and (ii) smaller areas, centred in the ALHAMBRA sub-fields, at $0.3 \leq z < 0.9$. All the cases, summarised in Table \[sigv\_vc\_tab\], are for $i \leq 23$ galaxies. At the end, we explored an order of magnitude in volume, from $V_c \sim 0.1\times10^5$ Mpc$^{3}$ to $V_c \sim 1.4\times10^5$ Mpc$^{3}$. The power-law function that better describes the observations (Fig. \[sigv\_vs\_vc\]) is $$\sigma_v^{*}\,(V_c) = (1.05 \pm 0.05) \times \bigg( \frac{V_c}{10^{5}\ {\rm Mpc}^{3}} \bigg)^{-0.48 \pm 0.05}.\label{sigv_vc}$$
We tested the robustness of our result by fitting the two sets of data (variation in redshift and area) separately. We find $\sigma_v^{*} \propto V_c^{-0.43 \pm 0.08}$ for the redshift data, while $\sigma_v^{*} \propto V_c^{-0.48 \pm 0.05}$ for the area data.
[lccccc]{} Redshift & Effective area & $V_c$ & $n_1$ & $\sigma_v$ & $\sigma^*_v$\
range & (deg$^2$) & ($10^{4}$ Mpc$^{3}$) & ($10^{-3}$ Mpc$^{-3}$) & & $\sigma_v/\sigma_v(n_1)$\
$[0.30,0.69)$ & $2.38$ & $6.98 \pm 0.06$ & $9.21 \pm 0.25$ & $0.169 \pm 0.025$ & $1.24 \pm 0.15$\
$[0.69,0.90)$ & $2.38$ & $6.87 \pm 0.06$ & $4.69 \pm 0.16$ & $0.273 \pm 0.040$ & $1.39 \pm 0.18$\
$[0.30,0.60)$ & $2.38$ & $4.68 \pm 0.04$ & $10.32 \pm 0.32$ & $0.205 \pm 0.030$ & $1.60 \pm 0.19$\
$[0.60,0.77)$ & $2.38$ & $4.68 \pm 0.04$ & $5.82 \pm 0.18$ & $0.274 \pm 0.042$ & $1.57 \pm 0.19$\
$[0.77,0.90)$ & $2.38$ & $4.49 \pm 0.04$ & $4.17 \pm 0.19$ & $0.323 \pm 0.051$ & $1.54 \pm 0.21$\
$[0.30,0.55)$ & $2.38$ & $3.60 \pm 0.03$ & $11.23 \pm 0.38$ & $0.230 \pm 0.032$ & $1.88 \pm 0.22$\
$[0.55,0.70)$ & $2.38$ & $3.68 \pm 0.03$ & $6.14 \pm 0.21$ & $0.252 \pm 0.041$ & $1.48 \pm 0.20$\
$[0.70,0.82)$ & $2.38$ & $3.74 \pm 0.03$ & $5.26 \pm 0.21$ & $0.311 \pm 0.056$ & $1.68 \pm 0.23$\
$[0.45,0.60)$ & $2.38$ & $2.91 \pm 0.02$ & $7.64 \pm 0.31$ & $0.268 \pm 0.043$ & $1.78 \pm 0.24$\
$[0.30,0.45)$ & $2.38$ & $1.76 \pm 0.01$ & $14.04 \pm 0.50$ & $0.276 \pm 0.051$ & $2.55 \pm 0.30$\
$[0.30,0.90)$ & $2.38$ & $13.85 \pm 0.11$ & $6.88 \pm 0.16$ & $0.158 \pm 0.019$ & $0.99 \pm 0.12$\
$[0.30,0.90)$ & $1.92$ & $11.15 \pm 0.11$ & $6.91 \pm 0.17$ & $0.158 \pm 0.020$ & $0.99 \pm 0.13$\
$[0.30,0.90)$ & $1.59$ & $9.26 \pm 0.10$ & $7.00 \pm 0.17$ & $0.150 \pm 0.020$ & $0.95 \pm 0.13$\
$[0.30,0.90)$ & $1.19$ & $6.95 \pm 0.07$ & $6.79 \pm 0.18$ & $0.179 \pm 0.024$ & $1.11 \pm 0.15$\
$[0.30,0.90)$ & $0.79$ & $4.61 \pm 0.05$ & $7.06 \pm 0.20$ & $0.259 \pm 0.033$ & $1.64 \pm 0.21$\
$[0.30,0.90)$ & $0.59$ & $3.44 \pm 0.04$ & $6.85 \pm 0.22$ & $0.264 \pm 0.036$ & $1.65 \pm 0.22$\
$[0.30,0.90)$ & $0.48$ & $2.77 \pm 0.03$ & $6.74 \pm 0.21$ & $0.325 \pm 0.045$ & $2.01 \pm 0.28$\
$[0.30,0.90)$ & $0.39$ & $2.29 \pm 0.03$ & $6.73 \pm 0.21$ & $0.354 \pm 0.050$ & $2.19 \pm 0.31$\
$[0.30,0.90)$ & $0.34$ & $1.97 \pm 0.04$ & $6.72 \pm 0.24$ & $0.340 \pm 0.050$ & $2.10 \pm 0.31$\
$[0.30,0.90)$ & $0.30$ & $1.74 \pm 0.03$ & $6.82 \pm 0.26$ & $0.391 \pm 0.055$ & $2.44 \pm 0.34$\
$[0.30,0.90)$ & $0.24$ & $1.40 \pm 0.02$ & $6.99 \pm 0.26$ & $0.411 \pm 0.059$ & $2.60 \pm 0.37$\
### Dependence on redshift {#secz}
The redshift is an expected parameter in the parametrisation the cosmic variance. However, Fig. \[sigv\_vs\_vc\] shows that the results at different redshifts are consistent with those from the wide redshift range $0.3 \leq z < 0.9$. As a consequence, the redshift dependence of the cosmic variance should be smaller than the typical error in our measurements. We tested this hypothesis by measuring $\sigma_v$ in different, non-overlapping, redshift bins. We summarise our measurements, performed for $i \leq 23$ galaxies, in Table \[sigv\_z\_tab\]. We defined $\sigma_v^{**} = \sigma_v / \sigma_v\,(n_1, V_c)$ to isolate the redshift dependence of the cosmic variance. We find that $\sigma_v^{**}$ is compatible with unity, $\sigma_v^{**} = 1.02 \pm 0.07$, and that no redshift dependence remains after accounting for the variation in $n_1$ and $V_c$ (Fig. \[sigv\_vs\_z\]). This confirms our initial hypothesis and we assume therefore $\gamma = 0$ in the following.
[lcccccc]{} Principal & Redshift & $\overline{z}$ & $n_1$ & $V_c$ & $\sigma_v$ & $\sigma^{**}_v$\
sample & range & & ($10^{-3}$ Mpc$^{-3}$) & ($10^{4}$ Mpc$^{3}$) & & $\sigma_v/\sigma_v(n_1,V_c)$\
$i \leq 23$ & $[0.30,0.45)$ & $0.374$ & $14.04 \pm 0.50$ & $ 1.76 \pm 0.01$ & $0.276 \pm 0.033$ & $1.06 \pm 0.13$\
$i \leq 23$ & $[0.45,0.60)$ & $0.524$ & $ 7.64 \pm 0.31$ & $ 2.29 \pm 0.02$ & $0.268 \pm 0.036$ & $0.94 \pm 0.13$\
$i \leq 23$ & $[0.60,0.75)$ & $0.679$ & $ 5.83 \pm 0.19$ & $ 4.06 \pm 0.03$ & $0.286 \pm 0.037$ & $1.02 \pm 0.13$\
$i \leq 23$ & $[0.75,0.90)$ & $0.820$ & $ 4.40 \pm 0.18$ & $ 5.11 \pm 0.04$ & $0.309 \pm 0.043$ & $1.05 \pm 0.15$\
### Dependence on the number density of the companion sample {#secn2}
As we show in Sect. \[metodo\], two different populations are involved in the measurement of the merger fraction: the principal sample and the sample of companions around principal galaxies. In the previous sections the principal and the companion sample were the same, and here we explore how the number density $n_2$ of the companion sample impacts the cosmic variance. We set $i \leq 20.5$ galaxies at $0.3 \leq z < 0.9$ as principals, and varied the $i-$band selection of the companion galaxies from $i \leq 20.5$ to $i \leq 23$ in 0.5 steps. As in Sect. \[secvc\], the variable $\sigma_v^{*} = \sigma_v / \sigma_v\,(n_1)$ was used.
We find that the cosmic variance decreases as the number density of the companion sample increases (Table \[sigv\_n2\_tab\] and Fig. \[sigv\_vs\_n2\]). We fit the dependence with a power-law, forcing it to pass for the point $\sigma_v^{*}\,(n_1,n_1) = 1$. We find that $$\sigma_v^{*}\,(n_1,n_2) = \bigg( \frac{n_2}{n_1} \bigg)^{-0.37 \pm 0.04}.\label{sigv_n2}$$ We checked that if we leave free the intercept, it is consistent with unity, as we assumed: $\sigma_v^{*}\,(n_1,n_1) = 1.04\pm0.12$. In addition, the power-law index changes slightly, $\sigma_v^{*} \propto (n_2/n_1)^{-0.39 \pm 0.08}$.
### The cosmic variance in merger fraction studies bases on close pairs
In the previous sections we have characterised the dependence of the cosmic variance $\sigma_v$ on the basic parameters involved in close pair studies (Sect. \[theory\]): the number density of the principal ($n_1$, Sect. \[secn1\]) and the companion sample ($n_2$, Sect. \[secn2\]), the cosmic volume under study ($V_c$, Sect. \[secvc\]), and the redshift (Sect. \[secz\]). We find that $$\begin{aligned}
\lefteqn{\sigma_v\,(n_1,n_2,V_c) =} \nonumber\\
&& 0.48 \times \bigg( \frac{n_1}{10^{-3}\ {\rm Mpc}^{-3}} \bigg)^{-0.54} \!\!\!\! \times \bigg( \frac{V_c}{10^{5}\ {\rm Mpc}^{3}} \bigg)^{-0.48} \!\!\!\! \times \bigg( \frac{n_2}{n_1} \bigg)^{-0.37}.\label{sigv_final}\end{aligned}$$
This is the main result of the present paper. We estimated through Monte Carlo sampling than the typical uncertainty in $\sigma_v$ from this relation is $\sim15$%. The dependence of $\sigma_v$ on redshift should be lower than this uncertainty. In addition, $\sigma_v$ is independent of the search radius used to compute the merger fraction as we demonstrated in Sect. \[optimal\].
[lccc]{} Companion & $n_2/n_1$ & $\sigma_v$ & $\sigma^{*}_v$\
sample & & & $\sigma_v/\sigma_v(n_1)$\
$i \leq 20.5$ & $1$ & $0.587 \pm 0.100$ & $1.09 \pm 0.18$\
$i \leq 21.0$ & $1.75 \pm 0.10$ & $0.459 \pm 0.067$ & $0.85 \pm 0.12$\
$i \leq 21.5$ & $2.90 \pm 0.15$ & $0.343 \pm 0.045$ & $0.64 \pm 0.08$\
$i \leq 22.0$ & $4.52 \pm 0.24$ & $0.306 \pm 0.038$ & $0.57 \pm 0.07$\
$i \leq 22.5$ & $6.56 \pm 0.33$ & $0.258 \pm 0.031$ & $0.48 \pm 0.06$\
$i \leq 23.0$ & $9.42 \pm 0.44$ & $0.244 \pm 0.029$ & $0.45 \pm 0.05$\
Cosmic variance in spatially random samples {#randsigv}
-------------------------------------------
In this section we further test the significance of our results by measuring both the merger fraction and the cosmic variance in samples randomly distributed in the sky plane. For this we created a set of 100 random samples, with each random sample comprising 48 random sub-samples (one per ALHAMBRA sub-field). We generated each random sub-sample by assigning a random RA and Dec to each source in the original catalogue, but retaining the original redshift of the sources. This erases the clustering signal inside each ALHAMBRA sub-field (i.e., at $\lesssim 15\arcmin$ scales), but the number density variations between sub-fields because of the clustering at scales larger than $\sim 15\arcmin$ remains. We estimated the merger fraction and the cosmic variance for each random sample at $0.3 \leq z < 0.9$ as in Sect. \[secn1\], and computed the median merger fraction, $\langle f_{\rm m} \rangle$, and the median cosmic variance, $\langle \sigma_{v} \rangle$, in the set of 100 random samples to compare them with the values measured in the real samples. To facilitate this comparison, we defined the variables $F_{\rm m} = f_{\rm m}/\langle f_{\rm m} \rangle$ and $\Sigma_{v} = \sigma_{v}/\langle \sigma_{v} \rangle$. We estimated $F_{\rm m}$ and $\Sigma_{v}$ for different selections in $n_1$ following Sect. \[secn1\], and we show our findings in Fig. \[randfig\].
On the one hand, the merger fraction in the real samples is higher than in the random samples by a factor of three–four, $F_{\rm m} = 4.25 - 0.27 \times n_1$ (Fig. \[randfig\], [*top panel*]{}). This reflects the clustering present in the real samples that we erased when randomised the positions of the sources in the sky, as well as the higher clustering of more luminous galaxies. This result is consistent with previous close pair studies comparing real and random samples [e.g., @kar07]. On the other hand, the cosmic variance measured in the random samples is higher than the cosmic variance in the real ones, $\langle \Sigma_{v} \rangle = 0.81 \pm 0.04$ (Fig. \[randfig\], [*bottom panel*]{}). This implies that most of the variance between sub-fields is unrelated with the clustering inside these sub-fields, and that the $\sigma_v$ measured in the present paper is a real signature of the relative field-to-field variation of the merger fraction.
Testing the independence of the 48 ALHAMBRA sub-fields {#sec7f}
------------------------------------------------------
Hitherto we have assumed that the 48 ALHAMBRA sub-fields are independent. However, only the 7 ALHAMBRA fields are really independent and correlations between adjacent sub-fields should exists. This correlations could impact our $\sigma_v$ measurements, and in this section we test the independence assumption.
[lccc]{} Principal & $n_1$ & $\sigma_v$ & $\sigma^{**}_v$\
sample & ($10^{-3}$ Mpc$^{-3}$) & & $\sigma_v/\sigma_v(n_1,V_c)$\
$i \leq 23.0$ & $7.32 \pm 0.19$ & $0.055 \pm 0.019$ & $0.76 \pm 0.26$\
$i \leq 22.5$ & $5.20 \pm 0.18$ & $0.088 \pm 0.028$ & $1.01 \pm 0.32$\
$i \leq 22.0$ & $3.57 \pm 0.14$ & $0.132 \pm 0.040$ & $1.23 \pm 0.37$\
$i \leq 21.5$ & $2.24 \pm 0.09$ & $0.175 \pm 0.054$ & $1.27 \pm 0.39$\
$i \leq 21.0$ & $1.40 \pm 0.06$ & $0.290 \pm 0.089$ & $1.63 \pm 0.50$\
$i \leq 20.5$ & $0.82 \pm 0.05$ & $0.280 \pm 0.107$ & $1.18 \pm 0.45$\
$i \leq 23.0$ & $7.05 \pm 0.14$ & $0.080 \pm 0.024$ & $1.09 \pm 0.32$\
$i \leq 22.5$ & $4.99 \pm 0.09$ & $0.084 \pm 0.027$ & $0.95 \pm 0.30$\
$i \leq 22.0$ & $3.41 \pm 0.11$ & $0.094 \pm 0.034$ & $0.86 \pm 0.31$\
$i \leq 21.5$ & $2.22 \pm 0.05$ & $0.106 \pm 0.047$ & $0.77 \pm 0.34$\
$i \leq 21.0$ & $1.33 \pm 0.05$ & $0.172 \pm 0.069$ & $0.95 \pm 0.38$\
$i \leq 20.5$ & $0.75 \pm 0.02$ & $0.293 \pm 0.112$ & $1.19 \pm 0.45$\
We defined two groups of seven independent pointings, one per ALHAMBRA field. The first group comprises the pointings f02p01, f03p02, f04p01, f05p01, f06p01, f07p03, and f08p02; where f0? refers to the ALHAMBRA field and p0? to the pointing in the field. The second group comprises the pointings f02p02, f03p01, f04p01, f05p01, f06p02, f07p04, f08p01. Note that fields f04 and f05 have only one pointing in the current ALHAMBRA release. Each of the previous pointings probe a cosmic volume four times higher than our fiducial sub-fields, with a median $V_c = (54.49 \pm 0.59)\times10^{4}$ Mpc$^{3}$ for the first group and $V_c = (55.24 \pm 0.50)\times10^{4}$ Mpc$^{3}$ for the second one at $0.3 \leq z < 0.9$. Then, we measured the merger fraction in the seven independent pointings of each group and we obtained $\sigma_v$ applying the MLE. We repeated this procedure for different selections, from $i \leq 23$ to $i \leq 20.5$ in 0.5 magnitude steps. Finally, we defined $\sigma_v^{**} = \sigma_v / \sigma_v\,(n_1, V_c)$, so the values of $\sigma_v^{**}$ would be dispersed around unity if the cosmic variance measured from the 7 independent areas is described well by the cosmic variance measured from the 48 sub-fields. We summarise our results in Table \[sigv\_7f\_tab\] and in Fig. \[sigv7f\].
We find that the cosmic variance from the 7 independent fields nicely agree with our expectations from Eq. (\[sigv\_final\]), with an error-weighted average of $\sigma_v^{**} = 1.01 \pm 0.10$. Thus, assume the 48 ALHAMBRA sub-fields as independent is an acceptable approximation to study $\sigma_v$. In addition, the uncertainties in $\sigma_v$ are lower by a factor of two when we use the 48 sub-fields, improving the statistical significance of our results.
[lccccccc]{} Principal & Companion & Redshift & $n_1$ & $n_2/n_1$ & $V_c$ & $\sigma_v$ & $\sigma^{***}_v$\
sample & sample & range & ($10^{-3}$ Mpc$^{-3}$) & & ($10^{4}$ Mpc$^{3}$) & & $\sigma_v/\sigma_v(n_1,n_2,V_c)$\
$M_B \leq -20.5$ & $M_B \leq -20.5$ & $[0.30, 0.90)$ & $1.63 \pm 0.06$ & $1$ & $13.85 \pm 0.11$ & $0.305 \pm 0.050$ & $0.97 \pm 0.16$\
$M_B \leq -20.0$ & $M_B \leq -20.0$ & $[0.30, 0.90)$ & $2.95 \pm 0.08$ & $1$ & $13.85 \pm 0.11$ & $0.250 \pm 0.034$ & $1.09 \pm 0.15$\
$M_B \leq -20.0$ & $M_B \leq -20.0$ & $[0.30, 0.69)$ & $2.67 \pm 0.08$ & $1$ & $ 6.98 \pm 0.06$ & $0.309 \pm 0.050$ & $0.92 \pm 0.15$\
$M_B \leq -20.0$ & $M_B \leq -20.0$ & $[0.69, 0.90)$ & $3.37 \pm 0.11$ & $1$ & $ 6.87 \pm 0.06$ & $0.276 \pm 0.041$ & $0.92 \pm 0.14$\
$M_B \leq -19.5$ & $M_B \leq -19.5$ & $[0.30, 0.90)$ & $4.63 \pm 0.11$ & $1$ & $13.85 \pm 0.11$ & $0.213 \pm 0.026$ & $1.19 \pm 0.14$\
$M_B \leq -19.5$ & $M_B \leq -19.5$ & $[0.30, 0.60)$ & $4.12 \pm 0.15$ & $1$ & $ 4.68 \pm 0.04$ & $0.284 \pm 0.042$ & $0.88 \pm 0.13$\
$M_B \leq -19.5$ & $M_B \leq -19.5$ & $[0.60, 0.77)$ & $4.72 \pm 0.14$ & $1$ & $ 4.68 \pm 0.04$ & $0.288 \pm 0.038$ & $0.96 \pm 0.13$\
$M_B \leq -19.5$ & $M_B \leq -19.5$ & $[0.77, 0.90)$ & $5.16 \pm 0.19$ & $1$ & $ 4.49 \pm 0.04$ & $0.302 \pm 0.040$ & $1.04 \pm 0.14$\
$M_B \leq -19.0$ & $M_B \leq -19.0$ & $[0.30, 0.90)$ & $6.76 \pm 0.14$ & $1$ & $13.85 \pm 0.11$ & $0.165 \pm 0.020$ & $1.13 \pm 0.14$\
$M_B \leq -19.0$ & $M_B \leq -19.0$ & $[0.30, 0.60)$ & $6.10 \pm 0.18$ & $1$ & $ 4.68 \pm 0.04$ & $0.223 \pm 0.034$ & $0.86 \pm 0.13$\
$M_B \leq -19.0$ & $M_B \leq -19.0$ & $[0.60, 0.77)$ & $6.79 \pm 0.17$ & $1$ & $ 4.68 \pm 0.04$ & $0.251 \pm 0.031$ & $1.02 \pm 0.13$\
$M_B \leq -19.0$ & $M_B \leq -19.0$ & $[0.77, 0.90)$ & $7.25 \pm 0.28$ & $1$ & $ 4.49 \pm 0.04$ & $0.227 \pm 0.029$ & $0.94 \pm 0.12$\
$M_B \leq -20.5$ & $M_B \leq -20.0$ & $[0.30, 0.90)$ & $1.63 \pm 0.06$ & $1.81 \pm 0.08$ & $13.85 \pm 0.11$ & $0.262 \pm 0.035$ & $1.03 \pm 0.14$\
$M_B \leq -20.5$ & $M_B \leq -19.5$ & $[0.30, 0.90)$ & $1.63 \pm 0.06$ & $2.84 \pm 0.12$ & $13.85 \pm 0.11$ & $0.222 \pm 0.027$ & $1.04 \pm 0.13$\
$M_B \leq -20.5$ & $M_B \leq -19.0$ & $[0.30, 0.90)$ & $1.63 \pm 0.06$ & $4.12 \pm 0.12$ & $13.85 \pm 0.11$ & $0.184 \pm 0.022$ & $0.98 \pm 0.12$\
$M_B \leq -20.0$ & $M_B \leq -19.5$ & $[0.30, 0.90)$ & $2.95 \pm 0.06$ & $1.57 \pm 0.06$ & $13.85 \pm 0.11$ & $0.207 \pm 0.026$ & $1.07 \pm 0.13$\
$M_B \leq -20.0$ & $M_B \leq -19.0$ & $[0.30, 0.90)$ & $2.95 \pm 0.06$ & $2.29 \pm 0.08$ & $13.85 \pm 0.11$ & $0.171 \pm 0.020$ & $1.01 \pm 0.12$\
$M_B \leq -19.5$ & $M_B \leq -19.0$ & $[0.30, 0.90)$ & $4.63 \pm 0.11$ & $1.46 \pm 0.05$ & $13.85 \pm 0.11$ & $0.163 \pm 0.019$ & $1.04 \pm 0.12$\
$M_B \leq -20.5$ & $\mathcal{R} = 1/2$ & $[0.30, 0.90)$ & $1.63 \pm 0.06$ & $2.31 \pm 0.10$ & $13.85 \pm 0.11$ & $0.268 \pm 0.040$ & $1.16 \pm 0.17$\
$M_B \leq -20.5$ & $\mathcal{R} = 1/4$ & $[0.30, 0.90)$ & $1.63 \pm 0.06$ & $4.15 \pm 0.17$ & $13.85 \pm 0.11$ & $0.216 \pm 0.026$ & $1.16 \pm 0.14$\
$M_B \leq -20.5$ & $\mathcal{R} = 1/10$ & $[0.30, 0.60)$ & $1.41 \pm 0.05$ & $8.24 \pm 0.37$ & $ 4.68 \pm 0.04$ & $0.296 \pm 0.039$ & $1.12 \pm 0.15$\
$M_B \leq -20.0$ & $\mathcal{R} = 1/2$ & $[0.30, 0.90)$ & $2.95 \pm 0.08$ & $1.92 \pm 0.07$ & $13.85 \pm 0.11$ & $0.210 \pm 0.031$ & $1.16 \pm 0.17$\
$M_B \leq -20.0$ & $\mathcal{R} = 1/4$ & $[0.30, 0.75)$ & $2.74 \pm 0.09$ & $3.18 \pm 0.12$ & $ 8.74 \pm 0.07$ & $0.195 \pm 0.026$ & $1.01 \pm 0.13$\
$M_B \leq -20.0$ & $\mathcal{R} = 1/10$ & $[0.30, 0.45)$ & $2.78 \pm 0.13$ & $5.95 \pm 0.33$ & $ 1.76 \pm 0.01$ & $0.329 \pm 0.045$ & $1.00 \pm 0.14$\
$M_B \leq -19.5$ & $\mathcal{R} = 1/2$ & $[0.30, 0.75)$ & $4.32 \pm 0.10$ & $1.75 \pm 0.05$ & $ 8.74 \pm 0.05$ & $0.194 \pm 0.031$ & $1.03 \pm 0.16$\
$M_B \leq -19.5$ & $\mathcal{R} = 1/4$ & $[0.30, 0.60)$ & $4.12 \pm 0.15$ & $2.82 \pm 0.13$ & $ 4.68 \pm 0.04$ & $0.173 \pm 0.027$ & $0.79 \pm 0.12$\
Expectations for luminosity- and mass-selected samples {#secpop}
------------------------------------------------------
Throughout present paper we have focused our analysis in (aparent) bright galaxies with $i \leq 23$. This ensures excellent photometric redshifts and provides reliable merger fraction measurements (Sect. \[optimal\]). However, one will be interested on the merger fraction of galaxies selected by their luminosity, stellar mass, colour, etc. Because the bias of the galaxies with respect to the underlying dark-matter distribution depends on the selection of the sample, our prescription to estimate $\sigma_v$ could not be valid for physically selected samples (Sect. \[theory\]). In this section we compare the expected cosmic variance from Eq. (\[sigv\_final\]) with the actual cosmic variance of several luminosity- and stellar mass-selected samples to set the limits and the reliability of our suggested parametrisation.
We defined the variable $\sigma_v^{***}=\sigma_v / \sigma_v\,(n_1, n_2, V_c)$, so the values of $\sigma_v^{***}$ would be dispersed around unity if no extra dependence on the luminosity or the stellar mass exists. Throughout the present paper we imposed neither luminosity nor mass ratio constraint between the galaxies in the close pairs. However, merger fraction studies impose such constraints to study major or minor mergers. This ratio is defined as $\mathcal{R} = M_{\star,2}/M_{\star,1}$, where $M_{\star,1}$ and $M_{\star,2}$ are the stellar masses of the principal and the companion galaxy in the pair, respectively. The definition of $\mathcal{R}$ in the $B$-band luminosity $L_{B}$ case is similar. Major mergers are usually defined with $1/4 \leq \mathcal{R} \leq 1$, while minor mergers with $\mathcal{R} \leq 1/4$. We explored different $\mathcal{R}$ cases and estimated $n_2$ as the number density of the $L_{B} \geq \mathcal{R} L_{B,1}$ or the $M_{\star} \geq \mathcal{R} M_{\star,1}$ population. The properties of all the studied samples are summarised in Tables \[sigvpop\_mb\_tab\] and \[sigvpop\_ms\_tab\]. The redshift range probed in each case was chosen to ensure volume-limited companion samples. We stress that the samples in Tables \[sigvpop\_mb\_tab\] and \[sigvpop\_ms\_tab\] mimic typical observational selections and $\mathcal{R}$ values from the literature.
On the one hand, we find that the error-weighted average of all the luminosity-selected samples is $\sigma_v^{***} = 1.01 \pm 0.03$, compatible with unity as we expected if no (or limited) dependence on the selection exists (Fig. \[sigvpop\_mb\]). We obtained $\sigma_v^{***} = 1.03 \pm 0.05$ from samples with the luminosity ratio $\mathcal{R}$ applied, while $\sigma_v^{***} = 1.00 \pm 0.03$ from samples without it. On the other hand, we find $\sigma_v^{***} = 1.02 \pm 0.03$ for the stellar mass-selected samples (Fig. \[sigvpop\_ms\]). As previously, the value is compatible with unity. We obtained $\sigma_v^{***} = 0.98 \pm 0.05$ from samples with the mass ratio $\mathcal{R}$ applied, while $\sigma_v^{***} = 1.03 \pm 0.03$ from samples without it.
We conclude that our results based on $i-$band selected samples provide a good description of the cosmic variance for physically selected samples, with a limited dependence ($\lesssim 15$%) on both the luminosity and the stellar mass selection. Thus, only $n_1$, $n_2$ and $V_c$ are needed to estimate a reliable $\sigma_v$ for merger fractions studies based on close pairs.
[lccccccc]{} Principal & Companion & Redshift & $n_1$ & $n_2/n_1$ & $V_c$ & $\sigma_v$ & $\sigma^{***}_v$\
sample & sample & range & ($10^{-3}$ Mpc$^{-3}$) & & ($10^{4}$ Mpc$^{3}$) & & $\sigma_v/\sigma_v(n_1,n_2,V_c)$\
$M_{\star} \geq 10^{10.75}\ M_{\odot}$ & $M_{\star} \geq 10^{10.75}\ M_{\odot}$ & $[0.30, 0.90)$ & $0.67 \pm 0.03$ & $1$ & $13.85 \pm 0.11$ & $0.386 \pm 0.082$ & $0.76 \pm 0.16$\
$M_{\star} \geq 10^{10.5}\ M_{\odot}$ & $M_{\star} \geq 10^{10.5}\ M_{\odot}$ & $[0.30, 0.90)$ & $1.35 \pm 0.05$ & $1$ & $13.85 \pm 0.11$ & $0.406 \pm 0.063$ & $1.16 \pm 0.18$\
$M_{\star} \geq 10^{10.25}\ M_{\odot}$ & $M_{\star} \geq 10^{10.25}\ M_{\odot}$ & $[0.30, 0.90)$ & $2.33 \pm 0.07$ & $1$ & $13.85 \pm 0.11$ & $0.309 \pm 0.040$ & $1.19 \pm 0.15$\
$M_{\star} \geq 10^{10.25}\ M_{\odot}$ & $M_{\star} \geq 10^{10.25}\ M_{\odot}$ & $[0.30, 0.69)$ & $2.30 \pm 0.08$ & $1$ & $ 6.98 \pm 0.06$ & $0.348 \pm 0.050$ & $0.96 \pm 0.14$\
$M_{\star} \geq 10^{10.25}\ M_{\odot}$ & $M_{\star} \geq 10^{10.25}\ M_{\odot}$ & $[0.69, 0.90)$ & $2.46 \pm 0.09$ & $1$ & $ 6.87 \pm 0.06$ & $0.410 \pm 0.054$ & $1.16 \pm 0.15$\
$M_{\star} \geq 10^{10.0}\ M_{\odot}$ & $M_{\star} \geq 10^{10.0}\ M_{\odot}$ & $[0.30, 0.90)$ & $3.49 \pm 0.11$ & $1$ & $13.85 \pm 0.11$ & $0.226 \pm 0.028$ & $1.08 \pm 0.13$\
$M_{\star} \geq 10^{10.0}\ M_{\odot}$ & $M_{\star} \geq 10^{10.0}\ M_{\odot}$ & $[0.30, 0.69)$ & $3.32 \pm 0.12$ & $1$ & $ 6.98 \pm 0.06$ & $0.242 \pm 0.036$ & $0.81 \pm 0.12$\
$M_{\star} \geq 10^{10.0}\ M_{\odot}$ & $M_{\star} \geq 10^{10.0}\ M_{\odot}$ & $[0.69, 0.90)$ & $3.62 \pm 0.12$ & $1$ & $ 6.87 \pm 0.06$ & $0.323 \pm 0.040$ & $1.13 \pm 0.14$\
$M_{\star} \geq 10^{11.0}\ M_{\odot}$ & $M_{\star} \geq 10^{10.5}\ M_{\odot}$ & $[0.30, 0.90)$ & $0.20 \pm 0.01$ & $ 6.75 \pm 0.40$ & $13.85 \pm 0.11$ & $0.513 \pm 0.084$ & $1.03 \pm 0.14$\
$M_{\star} \geq 10^{11.0}\ M_{\odot}$ & $M_{\star} \geq 10^{10.25}\ M_{\odot}$ & $[0.30, 0.90)$ & $0.20 \pm 0.01$ & $11.65 \pm 0.68$ & $13.85 \pm 0.11$ & $0.439 \pm 0.065$ & $1.11 \pm 0.16$\
$M_{\star} \geq 10^{11.0}\ M_{\odot}$ & $M_{\star} \geq 10^{10.0}\ M_{\odot}$ & $[0.30, 0.90)$ & $0.20 \pm 0.01$ & $17.45 \pm 1.03$ & $13.85 \pm 0.11$ & $0.350 \pm 0.047$ & $1.06 \pm 0.17$\
$M_{\star} \geq 10^{10.75}\ M_{\odot}$ & $M_{\star} \geq 10^{10.5}\ M_{\odot}$ & $[0.30, 0.90)$ & $0.67 \pm 0.03$ & $ 2.01 \pm 0.12$ & $13.85 \pm 0.11$ & $0.423 \pm 0.063$ & $1.08 \pm 0.16$\
$M_{\star} \geq 10^{10.75}\ M_{\odot}$ & $M_{\star} \geq 10^{10.25}\ M_{\odot}$ & $[0.30, 0.90)$ & $0.67 \pm 0.03$ & $ 3.48 \pm 0.19$ & $13.85 \pm 0.11$ & $0.340 \pm 0.043$ & $1.06 \pm 0.13$\
$M_{\star} \geq 10^{10.75}\ M_{\odot}$ & $M_{\star} \geq 10^{10.0}\ M_{\odot}$ & $[0.30, 0.90)$ & $0.67 \pm 0.03$ & $ 5.21 \pm 0.28$ & $13.85 \pm 0.11$ & $0.265 \pm 0.032$ & $0.96 \pm 0.12$\
$M_{\star} \geq 10^{10.5}\ M_{\odot}$ & $M_{\star} \geq 10^{10.25}\ M_{\odot}$ & $[0.30, 0.90)$ & $1.35 \pm 0.05$ & $ 1.73 \pm 0.08$ & $13.85 \pm 0.11$ & $0.307 \pm 0.039$ & $1.08 \pm 0.14$\
$M_{\star} \geq 10^{10.5}\ M_{\odot}$ & $M_{\star} \geq 10^{10.0}\ M_{\odot}$ & $[0.30, 0.90)$ & $1.35 \pm 0.05$ & $ 5.21 \pm 0.13$ & $13.85 \pm 0.11$ & $0.257 \pm 0.031$ & $1.05 \pm 0.13$\
$M_{\star} \geq 10^{10.25}\ M_{\odot}$ & $M_{\star} \geq 10^{10.0}\ M_{\odot}$ & $[0.30, 0.90)$ & $2.33 \pm 0.07$ & $ 1.50 \pm 0.07$ & $13.85 \pm 0.11$ & $0.233 \pm 0.028$ & $1.04 \pm 0.12$\
$M_{\star} \geq 10^{11.0}\ M_{\odot}$ & $\mathcal{R} = 1/4$ & $[0.30, 0.90)$ & $0.20 \pm 0.01$ & $ 8.75 \pm 0.53$ & $12.63 \pm 0.10$ & $0.562 \pm 0.091$ & $1.28 \pm 0.21$\
$M_{\star} \geq 10^{11.0}\ M_{\odot}$ & $\mathcal{R} = 1/10$ & $[0.30, 0.90)$ & $0.20 \pm 0.01$ & $17.45 \pm 1.03$ & $12.63 \pm 0.10$ & $0.367 \pm 0.049$ & $1.08 \pm 0.14$\
$M_{\star} \geq 10^{10.75}\ M_{\odot}$ & $\mathcal{R} = 1/2$ & $[0.30, 0.90)$ & $0.67 \pm 0.03$ & $ 2.28 \pm 0.13$ & $12.63 \pm 0.10$ & $0.332 \pm 0.068$ & $0.88 \pm 0.18$\
$M_{\star} \geq 10^{10.75}\ M_{\odot}$ & $\mathcal{R} = 1/4$ & $[0.30, 0.90)$ & $0.67 \pm 0.03$ & $ 4.13 \pm 0.22$ & $12.63 \pm 0.10$ & $0.354 \pm 0.049$ & $1.17 \pm 0.16$\
$M_{\star} \geq 10^{10.75}\ M_{\odot}$ & $\mathcal{R} = 1/10$ & $[0.30, 0.60)$ & $0.57 \pm 0.03$ & $ 8.05 \pm 0.52$ & $ 4.68 \pm 0.04$ & $0.375 \pm 0.053$ & $0.87 \pm 0.12$\
$M_{\star} \geq 10^{10.5}\ M_{\odot}$ & $\mathcal{R} = 1/2$ & $[0.30, 0.90)$ & $1.35 \pm 0.05$ & $ 1.86 \pm 0.09$ & $12.63 \pm 0.10$ & $0.290 \pm 0.049$ & $1.04 \pm 0.18$\
$M_{\star} \geq 10^{10.5}\ M_{\odot}$ & $\mathcal{R} = 1/4$ & $[0.30, 0.60)$ & $1.28 \pm 0.06$ & $ 2.95 \pm 0.18$ & $ 4.68 \pm 0.04$ & $0.390 \pm 0.058$ & $0.96 \pm 0.14$\
$M_{\star} \geq 10^{10.25}\ M_{\odot}$ & $\mathcal{R} = 1/2$ & $[0.30, 0.60)$ & $2.33 \pm 0.08$ & $ 1.51 \pm 0.08$ & $ 4.68 \pm 0.04$ & $0.311 \pm 0.059$ & $0.83 \pm 0.16$\
$M_{\star} \geq 10^{10.0}\ M_{\odot}$ & $\mathcal{R} = 1/2$ & $[0.30, 0.60)$ & $3.27 \pm 0.11$ & $ 1.50 \pm 0.07$ & $ 4.68 \pm 0.04$ & $0.287 \pm 0.056$ & $0.91 \pm 0.18$\
Summary and conclusions {#conclusions}
=======================
We use the 48 sub-fields of $\sim$180 arcmin${^2}$ in the ALHAMBRA survey (total effective area of $2.38~$deg$^2$) to estimate empirically, for the first time in the literature, the cosmic variance that affect merger fraction studies based on close pairs. We find that the distribution of the merger fraction is log-normal and we use a maximum likelihood estimator to measure the cosmic variance $\sigma_v$ unaffected by observational errors (including the Poisson shot noise term).
We find that the better parametrisation of the cosmic variance for merger fraction studies based on close pairs is (Eq. \[\[sigv\_final\]\]) $$\begin{aligned}
\lefteqn{\sigma_v\,(n_1,n_2,V_c) =} \nonumber\\
&& 0.48 \times \bigg( \frac{n_1}{10^{-3}\ {\rm Mpc}^{-3}} \bigg)^{-0.54} \!\!\!\! \times \bigg( \frac{V_c}{10^{5}\ {\rm Mpc}^{3}} \bigg)^{-0.48} \!\!\!\! \times \bigg( \frac{n_2}{n_1} \bigg)^{-0.37},\nonumber\end{aligned}$$ where $n_1$ and $n_2$ are the cosmic average number density of the principal and the companion populations under study, respectively, and $V_c$ is the cosmological volume probed by our survey in the redshift range of interest. We stress that $n_1$ and $n_2$ should be estimated from general luminosity or mass function studies and that measurements from volumes dominated by structures (e.g., clusters or voids) should be avoided. In addition, $\sigma_v$ is independent of the search radius used to compute the merger fraction. The typical uncertainty in $\sigma_v$ from our relation is $\sim15$%. The dependence of the cosmic variance on redshift should be lower than this uncertainty. Finally, we checked that our formula provides a good estimation of $\sigma_v$ for luminosity- and mass-selected samples, as well as for close pairs with a given luminosity or mass ratio $\mathcal{R}$ between the galaxies in the pair. In the later case, $n_2$ is the average number density of those galaxies brighter or more massive than $\mathcal{R} L_{1}$ or $\mathcal{R} M_{\star,1}$, respectively.
Equation (\[sigv\_final\]) provides the expected cosmic variance of an individual merger fraction measurement $f_{\rm m}$ at a given field and redshift range. The 68% confidence interval of this merger fraction is $[f_{\rm m}{\rm e}^{-\sigma_{v}}, f_{\rm m}{\rm e}^{\sigma_{v}}]$. This interval is independent of the error in the measurement of $f_{\rm m}$, so both sources of uncertainty should be added to obtain an accurate description of the merger fraction error in pencil-beam surveys. If we have access to several independent fields $j$ for our study, we should combine the cosmic variance $\sigma_v^j$ of each single field with the following formula [see @moster11 for details]: $$\sigma_{v, {\rm tot}}^2 = \frac{\sum_j\,({V_{c}^j}\,\sigma_{v}^j)^2}{(\sum_j V_{c}^j)^2},\label{sigv_tot}$$ where $V_{c}^j$ is the cosmic volume probed by each single field in the redshift range of interest.
Thanks to the Eqs. (\[sigv\_final\]) and (\[sigv\_tot\]) we can estimate the impact of cosmic variance in close pair studies from the literature. For example, @bundy09 measure the major merger fraction in the two GOODS fields. We expect $\sigma_v \sim 0.42$ for massive ($M_{\star} \geq 10^{11}\ M_{\odot}$) galaxies, while $\sigma_v \sim 0.16$ for $M_{\star} \geq 10^{10}\ M_{\odot}$ galaxies. The studies of @deravel09 and @clsj11mmvvds explore the merger fraction in the VVDS-Deep. We expect $\sigma_v \lesssim 0.09$ for major mergers and $\sigma_v \lesssim 0.07$ for minor mergers in this survey. @lin08 explore the merger properties of $M_{B} \leq -19$ galaxies in three DEEP2 fields. We estimate $\sigma_v \sim 0.03$ for their results. Several major close pair studies have been conducted in the COSMOS field [e.g., @deravel11; @xu12]. Focussing in mass-selected samples, we expect $\sigma_v \sim 0.17$ for massive galaxies, while $\sigma_v \sim 0.07$ for $M_{\star} \geq 10^{10}\ M_{\odot}$ galaxies. In addition, we estimate $\sigma_v \sim 0.13$ for the minor merger fractions reported by @clsj12sizecos in the COSMOS field. Regarding local merger fractions ($z \lesssim 0.1$), the expected cosmic variance in the study of @depropris05 in the MGC is $\sigma_v \sim 0.03$, while $\sigma_v < 0.03$ in the study of @patton00. Finally, studies based in the full SDSS area are barely affected by cosmic variance, with $\sigma_v \lesssim 0.005$ [e.g., @patton08].
Extended samples over larger sky areas are needed to constraint the subtle redshift evolution of the comic variance, as well as its dependence on the selection of the samples. Future large photometric surveys such as J-PAS[^5] (Javalambre – Physics of the accelerating universe Astrophysical Survey), that will provide excellent photometric redshifts with $\delta_z \sim 0.003$ over 8500 deg$^2$ in the northern sky, are fundamental to progress on this topic.
In the present paper we have studied in detail the intrinsic dispersion of the merger fraction measured in the 48 ALHAMBRA sub-fields. In future papers we will explore the dependence of the [*median*]{} merger fraction, estimated as ${\rm e}^{\mu}$, on stellar mass, colour, or morphology [see @povic13 for details about the morphological classification in ALHAMBRA], and we will compare the ALHAMBRA measurements (both the median and the dispersion) with the expectations from cosmological simulations.
We dedicate this paper to the memory of our six IAC colleagues and friends who met with a fatal accident in Piedra de los Cochinos, Tenerife, in February 2007, with a special thanks to Maurizio Panniello, whose teachings of `python` were so important for this paper.
We thank the comments and suggestions of the anonymous referee, that improved the clarity of the manuscript.
This work has mainly been funding by the FITE (Fondo de Inversiones de Teruel) and the projects AYA2006-14056 and CSD2007-00060. We also acknowledge the financial support from the Spanish grants AYA2010-15169, AYA2010-22111-C03-01 and AYA2010-22111-C03-02, from the Junta de Andalucia through TIC-114 and the Excellence Project P08-TIC-03531, and from the Generalitat Valenciana through the project Prometeo/2009/064.
A. J. C. (RyC-2011-08529) and C. H. (RyC-2011-08262) are [*Ramón y Cajal*]{} fellows of the Spanish government.
Maximum likelihood estimation of the cosmic variance $\sigma_{v}$ {#mlmethod}
=================================================================
Maximum likelihood estimators (MLEs) have been used in a wide range of topics in astrophysics. For example, @naylor06 use a MLE to fit colour-magnitude diagrams, @arzner07 to improve the determination of faint X-ray spectra, @makarov06 to improve distance estimates using red giant branch stars, and @clsj08ml [@clsj09ffgs; @clsj09ffgoods; @clsj10megoods] to estimate reliable merger fractions from morphological criteria. MLEs are based on the estimation of the most probable values of a set of parameters which define the probability distribution that describes an observational sample.
The general MLE operates as follows. Throughout this Appendix we denote as ${\it P}\,({\bf a}\,|\,{\bf b})$ the probability to obtain the values ${\bf a}$, given the parameters ${\bf b}$. Being ${\bf{x}}_j$ the measured values in the ALHAMBRA field $j$ and $\theta$ the parameters that we want to estimate, we may express the joined likelihood function as $$L({\bf x}_j\,|\,\theta ) \equiv -\ln \big[ \prod_j {\it P}\,( {\bf x}_j\,|\,\theta) \big] = - \sum_j \ln \big[{\it P}\,({\bf x}_j\,|\,{\bf \theta})\big].\label{MLdef}$$ If we are able to express ${\it P}\,({\bf x}_j\,|\,\theta)$ analytically, we can minimise Eq. (\[MLdef\]) to obtain the best estimation of the parameters $\theta$, denote as $\theta_{\rm ML}$. In our case, ${\bf x}_j$ is the observed value of the merger fraction in log-space for the ALHAMBRA sub-field $j$, ${\bf x}_j \equiv f'_{{\rm m},j} = \ln f_{{\rm m},j}$. We decided to work in log-space because that makes the problem analytic and simplifies the implementation of the method without losing mathematical rigour.
ALHAMBRA sub-fields are assumed to have a real merger fraction (not affected by observational errors) that define a Gaussian distribution in log-space, $$P_{G}\,(f'_{{\rm real},j}\,|\,\mu, \sigma_v) = \frac{1}{\sqrt{2 \pi}\,\sigma_v}\,{\rm exp}\,\bigg[-\frac{(f'_{{\rm real},j} - \mu)^2}{2 \sigma_v^2}\bigg]\,.$$ Observational errors cause the observed $f'_{{\rm m},j}$ differ from their respective real values $f'_{{\rm real},j}$. The observed $f'_{{\rm m},j}$ are assumed to be extracted for a Gaussian distribution with mean $f'_{{\rm real},j}$ and standard deviation $\sigma_{{\rm o},j}$ (the observational errors), $$P_{G}\,(f'_{{\rm m},j}\,|\,f'_{{\rm real},j}, \sigma_{{\rm o},j}) = \frac{1}{\sqrt{2 \pi}\,\sigma_{{\rm o},j}}\,{\rm exp}\,\bigg[-\frac{(f'_{{\rm m},j} - f'_{{\rm real},j})^2}{2 \sigma_{{\rm o},j}^2}\bigg]\,.$$ We assumed that the observational errors are Gaussian in log-space, i.e., that they are log-normal in observational space. This is a good approximation of the reality because we are dealing with fractions that cannot be negative and that have asymmetric confidence intervals, as shown by @cameron11. In our case, we estimated the observational errors in log-space as $\sigma_{\rm o} = \sigma_{f}/f_{\rm m}$. We checked that the values of $\sigma_{\rm o}$ derived from our jackknife errors are similar to that estimated from the Bayesian approach in @cameron11, with a difference between them $\lesssim 15\%$.
We obtained the probability ${\it P}\,({\bf x}_j\,|\,\theta)$ of each ALHAMBRA sub-field by the total probability theorem: $$\begin{aligned}
\label{tprob}
\lefteqn{P\,(f'_{{\rm m}, j}\,|\,\mu, \sigma_v, \sigma_{{\rm o},j})} \nonumber\\
&& = \int_{-\infty}^{\infty} P_G\,(f'_{{\rm real},j}\,|\,\mu, \sigma_v) \times P_G\,(f'_{{\rm m},j}\,|\,f'_{{\rm real},j}, \sigma_{{\rm o},j})\,{\rm d}f'_{{\rm real},j},\label{Pint}\end{aligned}$$ where $f'_{{\rm m},j} = {{\bf x}_j}$ and $(\mu, \sigma_v, \sigma_{{\rm o},j}) = \theta$ in Eq. (\[MLdef\]). Note that the values of $\sigma_{{\rm o},j}$ are the measured uncertainties for each ALHAMBRA sub-field, so the only unknowns are the variables $\mu$ and $\sigma_v$, that we want to estimate. Note also that we integrate over the variable $f'_{{\rm real},j}$, so we are not be able to estimate the real merger fractions individually, but only the underlying Gaussian distribution that describes the sample.
The final joined likelihood function, Eq. (\[MLdef\]), after integrating Eq. (\[Pint\]), is $$L\,(f'_{{\rm m},j}\,|\,\mu, \sigma_{v}, \sigma_{{\rm o},j}) = -\frac{1}{2} \sum_j \ln\,(\sigma_{v}^2 + \sigma_{{\rm o},j}^2) + \frac{(f'_{{\rm m},j} - \mu)^2}{\sigma_{v}^2 + \sigma_{{\rm o},j}^2}.$$ With the minimisation of this function we obtain the best estimation of both $\mu$ and the cosmic variance $\sigma_{v}$, unaffected by observational errors.
In addition, we can estimate analytically the errors in the parameters above. We can obtain those via an expansion of the function $L\,(f'_{{\rm m},j}\,|\,\mu, \sigma_{v}, \sigma_{{\rm o},j})$ in Taylor’s series of its variables $\theta = (\mu, \sigma_v, \sigma_{{\rm o},j})$ around the minimisation point $\theta_{\rm ML}$. The previous minimisation process made the first $L$ derivative null and we obtain $$L = L(\theta_{\rm ML}) + \frac{1}{2}(\theta - \theta_{\rm ML})^{T} H_{xy} (\theta - \theta_{\rm ML}),$$ where $H_{xy}$ is the Hessian matrix and $T$ denotes the transpose matrix. The inverse of the Hessian matrix provides an estimate of the 68% confidence intervals of $\mu_{\rm ML}$ and $\sigma_{\rm ML}$, as well as the covariance between them. The Hessian matrix of the joined likelihood function $L$ is defined as $$\label{hessian}
H_{xy} = \left(
\begin{array}{cc}
\frac{\partial^2 L}{\partial^2 \mu} & \frac{\partial^2 L}{\partial \mu \, \partial \sigma_{v}}\\
\frac{\partial^2 L}{\partial \sigma_{v} \, \partial \mu} & \frac{\partial^2 L}{\partial^2 \sigma_{v}}
\end{array}\right),$$ with $$\frac{\partial^2 L}{\partial^2 \mu} = - \sum_i \frac{1}{\sigma_{v}^2 + \sigma_{{\rm o}, j}^2},$$ $$\frac{\partial^2 L}{\partial \mu \, \partial \sigma_{v}} = \frac{\partial^2 L}{\partial \sigma_{v} \, \partial \mu} = -2\sum_i \frac{\sigma_{v} (f'_{{\rm m},j} - \mu)}{(\sigma_{v}^2 + \sigma_{{\rm o}, j}^2)^2},$$ and $$\frac{\partial^2 L}{\partial^2 \sigma_{v}} = \sum_i \frac{(\sigma_{{\rm o}, j}^2 - 3\sigma_{v}^2)\times(f'_{{\rm m},j} - \mu)^2}{(\sigma_{v}^2 + \sigma_{{\rm o}, j}^2)^3} - \frac{(\sigma_{{\rm o}, j}^2 - \sigma_{v}^2)}{(\sigma_{v}^2 + \sigma_{{\rm o}, j}^2)^2}.$$
Then, we computed the inverse of the minus Hessian, $h_{xy} = (-H_{xy})^{-1}$. Finally, and because maximum likelihood theory states that $\sigma_{\theta_{x}}^2 \leq h_{xx}$, we estimated the variances of our inferred parameters as $\sigma_{\mu}^2 = h_{11}$ and $\sigma_{\sigma_{v}}^2 = h_{22}$.
We tested the performance and the limitations of our MLE through synthetic catalogues of merger fractions. We created several sets of 1000 synthetic catalogues, each of them composed by a number $n$ of merger fractions randomly drawn from a log-normal distribution with $\mu_{\rm in} = \log 0.05$ and $\sigma_{v, {\rm in}} = 0.2$, and affected by observational errors $\sigma_{\rm o}$. We explored the $n = 50, 250$ and $1000$ cases for the number of merger fractions, and varied the observational errors from $\sigma_{\rm o} = 0.1$ to 0.5 in 0.1 steps. That is, we explored observational errors in the measurement of the merger fraction from $\Delta \sigma \equiv \sigma_{\rm o}/\sigma_{v} = 0.5$ to 2.5 times the cosmic variance that we want to measure. We checked that the results below are similar for any value of $\sigma_{v, {\rm in}}$. We find that
1. The median value of the recovered $\mu$, noted $\overline{\mu}_{\rm ML}$, in each set of synthetic catalogues is similar to $\mu_{\rm in}$, with deviations lower than 0.5% in all cases under study. However, we find that $\overline{\sigma_{v}}_{,{\rm ML}}$ for $n = 50$ catalogues overestimates $\sigma_{v, {\rm in}}$ more than 5% at $\Delta \sigma \gtrsim 2.0$, while for $n = 1000$ we recover $\sigma_{v, {\rm in}}$ well even with $\Delta \sigma = 2.5$ (Fig. \[ml\_sigv\], [*top panel*]{}). This means that larger data sets are needed to recover the underlying distribution as the observational errors increase.
2. We also study the values recovered by a best least-squares (BLS) fit of Eq. (\[Plog\]) to the synthetic catalogues. We find that (i) the BLS fit recovers the right values of $\mu_{\rm in}$ as well as the MLE. That was expected, since the applied observational errors preserve the median of the initial distribution. And (ii) the BLS fit overestimates $\sigma_{v, {\rm in}}$ in all cases. The recovered values depart from the initial one as expected from a convolution of two Gaussians with variance $\sigma_{v, {\rm in}}$ and $\sigma_{\rm o}$, $\sigma_{v, {\rm BLS}}/\sigma_{v, {\rm in}} = \sqrt{1 + (\Delta \sigma)^{2}}$. The MLE performs a de-convolution of the observational errors, recovering accurately the initial cosmic variance (Fig. \[ml\_sigv\], [*top panel*]{}).
3. The estimated variances of $\mu$ and $\sigma_{v}$ are reliable. That is, the median variances $\overline{\sigma}_{\mu}$ and $\overline{\sigma}_{\sigma_v}$ estimated by the MLE are similar to the dispersion of the recovered values, noted $s_{\mu}$ and $s_{\sigma_v}$, in each set of synthetic catalogues. The difference between both variances for $\mu$ is lower than $5$% in all the probed cases. However, we find that $\overline{\sigma}_{\sigma_v}$ for $n = 50$ catalogues overestimates $s_{\sigma_v}$ more than 5% at $\Delta \sigma \gtrsim 1.5$: this is the limit of the MLE to estimate reliable uncertainties with this number of data (Fig. \[ml\_sigv\], [*bottom panel*]{}). Because the estimated variance tends asymptotically to $s_{\sigma_v}$ for a large number of data, $\overline{\sigma}_{\sigma_v}$ for $n = 1000$ catalogues deviates less from the expected value than for $n = 50$ synthetic catalogues. Note that even when the estimated variance $\sigma_{\sigma_v}$ deviates from the expectations at large $\Delta \sigma$, the value of $\sigma_{v}$ is still unbiased as such large observational errors (Fig. \[ml\_sigv\], [*top panel*]{}) and we can roughly estimate $\sigma_{\sigma_v}$ through realistic synthetic catalogues as those in this Appendix.
4. The variances of the recovered parameters decreases with $n$ and increases with $\sigma_{\rm o}$. That reflects the loss of information due to the observational errors. Remark that the MLE takes these observational errors into account to estimate the parameters and their variance.
We conclude that the MLE developed in this Appendix is not biased, provides accurate variances, and we can recover reliable uncertainties of the cosmic variance $\sigma_{v}$ in ALHAMBRA ($n = 48$) for $\Delta \sigma \lesssim 1.5$. Note that reliable values of $\sigma_{v}$ in ALHAMBRA are recovered at $\Delta \sigma \lesssim 2.0$. We checked that the average $\Delta \sigma$ in our study is 0.60 (the average observational error is $\overline{\sigma}_{\rm o} = 0.18$), and the maximum value is $\Delta \sigma = 0.85$. Thus, the results in the present paper are robust against the effect of observational errors.
[^1]:
[^2]: Based on observations collected at the German-Spanish Astronomical Center, Calar Alto, jointly operated by the Max-Planck-Institut für Astronomie (MPIA) at Heidelberg and the Instituto de Astrofísica de Andalucía (IAA-CSIC).
[^3]: http://alhambrasurvey.com
[^4]: http://cloud.iaa.es/alhambra/
[^5]: http://j-pas.org/
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study the hydrodynamics of relativistic fluids with several conserved global charges (i.e., several species of particles) by performing a Kaluza-Klein dimensional reduction of a neutral fluid on a $N$-torus. Via fluid/gravity correspondence, this allows us to describe the long-wavelength dynamics of black branes with several Kaluza-Klein charges. We obtain the equation of state and transport coefficients of the charged fluid directly from those of the higher-dimensional neutral fluid. We specialize these results for the fluids dual to Kaluza-Klein black branes.'
author:
- Adriana Di Dato
title: '**Kaluza-Klein reduction of relativistic fluids and their gravity duals**'
---
*Departament de Física Fonamental and\
Institut de Ciències del Cosmos, Universitat de Barcelona,\
Martí i Franquès 1, ES-08028, Barcelona, Spain.*\
Introduction {#1}
============
Kaluza-Klein dimensional reduction is a well known method to obtain solutions to a gravitational theory coupled to a Maxwell field, plus a scalar (dilaton) field. Velocities (or momenta) along the compactified direction result in electric charges in the reduced theory [@Pope]. Thus, if we take a neutral black string solution of the vacuum Einstein theory, perform a boost along the direction of the string and then dimensionally reduce in this direction, we obtain an electrically charged black hole of the Einstein-Maxwell-dilaton theory, for a particular value of the dilaton coupling [@Horowitz].
It should be clear that this method is not exclusive to gravitational theories. The identification between momenta along the internal direction and conserved charges in the reduced theory is in fact generic. Note, however, that in a non-gravitational theory, one obtains charges of a global symmetry group — e.g., a global $U(1)$ for reduction in a circle — while in the gravitational case, since the relevant spacetime symmetries are gauged, they are charges of a gauge symmetry group.
In this article we are interested in applying the Kaluza-Klein procedure to relativistic hydrodynamics. That is, we begin with a relativistic fluid without any conserved particle number in $p$ spatial dimensions, where $p-N$ of these are non-compact directions and $N$ of them form an $N$-torus. We assume that none of the fluid variables depend on the internal directions, but the fluid can have non-trivial velocity along them. These velocities give internal momenta that in the reduced theory appear as conserved global charges, i.e., particle numbers for $N$ different species. For a perfect fluid, this reduction is a straightforward one. Of more interest is the reduction of the first-order dissipative terms. Viscosity of the higher-dimensional fluid in the internal directions gives rise not only to viscosities but also conductivities in the reduced theory.
We shall do our analysis for a generic relativistic fluid in $p$ spatial dimensions with no conserved particle number, without assuming any specific equation of state nor constituent relation for its first-order transport coefficients. When applying our results to particular fluids, we will consider a class of recent interest in the context of dual relations between fluid dynamics and black brane dynamics. These are the fluids that correspond to neutral black $p$-branes of the vacuum Einstein theory, and which feature in the blackfold approach to black brane dynamics [@Emparan; @Joan]. Ref. [@Joan] developed the dictionary between the spacetime fluctuations of these black branes and the fluctuations of specific fluids. Using this mapping, our results yield a mapping between the dynamics of charged black branes in Kaluza-Klein theory and the hydrodynamics of certain charged fluids. The map includes their fluid equation of state and first-order transport coefficients. Note that the Kaluza-Klein black brane solutions differ from other charged black branes in their coupling to the dilaton. While the dilaton plays no direct role in the dual fluid description, since it is not associated to any conserved quantity of the black brane, the value of its coupling affects the equation of state and constituent relations.
There have been some previous studies of Kaluza-Klein reduction in the context of fluid/gravity correspondences [@Blaise; @Blaise2; @Milena; @Kanitscheider; @Kanitscheider2]. However, these have been restricted to the fluids that are dual to AdS black branes, and moreover they only work out explicitly the cases of circle [@Blaise] and 2-torus reduction [@Milena]. The results of [@Blaise; @Milena] can be mapped, via the AdS/Ricci-flat connection of [@Joan2], to our results for circle reductions of a neutral vacuum black brane. Vice versa, our results can be readily translated into results for AdS black branes with $N$ different charges using this mapping. The perfect fluid dynamics of asymptotically flat charged branes (with arbitrary dilaton coupling) was studied in [@Caldarelli; @Emparan:2011]. Dissipative effects of non-dilatonic asymptotically flat charged branes have been analyzed in [@Gath]. The first-order hydrodynamics of asymptotically flat black D3-branes has been studied in [@Emparan:2013], but the charge in this case can not be redistributed along the worldvolume and therefore the dynamics is qualitatively different. Moreover, in [@Armas; @Armas2] the Kaluza-Klein approach has been applied, in a slightly different manner, to obtain first-derivative corrections of charged black brane (extrinsic) dynamics.
In our opinion it is useful to treat the Kaluza-Klein reduction of fluids separately from any specific fluid/gravity dualities. First, this makes clear how the procedure stands on its own within the context of hydrodynamics without any reference to General Relativity. Second, by not tying the reduction to any particular fluid, we achieve a large degree of generality. Clearly, the method can be extended to the case in which the higher-dimensional fluid carries a particle number or some other property, but we will not pursue this in the present article.
Hydrodynamic Kaluza-Klein ansatz and\
reduction of the perfect fluid
=====================================
Let us consider a neutral relativistic fluid in flat space-time in $p+1$ dimensions. The hydrodynamical behaviour of this fluid is governed by the stress energy tensor conservation equations $\partial_A T^{AB}=0$. In general we split the stress energy tensor into a perfect fluid and a dissipative part, T\_[AB]{}=T\_[AB]{}\^[pf]{}+T\_[AB]{}\^[diss]{} \[eqn:generT\]. The perfect fluid part is given in terms of the energy density $\epsilon$, the pressure $P$ and the normalized velocity field $u^A$ by \[pf\] T\_[AB]{}\^[pf]{}=(+P) u\_A u\_B + P g\_[AB]{} while, to first derivative order, the dissipative part is \[eqn:diss\] T\_[AB]{}\^[diss]{}=-2\_[AB]{}-P\_[AB]{} where $\eta$ and $\zeta$ are the shear and bulk viscosities and $\theta$, $ \sigma_{AB}$ and $P_{AB}$ are defined as &=&\_A u\^A\[espa\],\
\_[AB]{}&=& P\_A\^C P\_B\^D\_[(C]{} u\_[D)]{}-P\_[AB]{}\[shear\],\
P\_[AB]{}&=&g\_[AB]{}+u\_A u\_B\[orto\]. A complete description of the fluid requires the specification of the equation of state, namely the relation between $P$ and $\epsilon$, and of the viscosities. For the most part we will keep them general, and will only specify them in sec. \[6\]. Furthermore, we naturally assume that this uncharged fluid is described in the Landau frame where $u^A T_{AB}^{diss}=0$.
We assume that the spacetime in which the fluid moves contains $N$ compact directions that form an $N$-torus $$\label{ansatz}
d\hat s^2= \sum_{j=1}^{N} dy_j^2 +\eta_{ab} d\sigma^a d\sigma^b \,,$$ where the metric $\eta_{ab}$ is the Minkowski metric in $p-N+1$ spacetime dimensions and the coordinates $y_j$ are identified with periodicity $2\pi R_j$. We take the fluid to move with non-zero velocity along the $N$ compactified dimensions. On Kaluza-Klein reduction this will give rise to charges in the reduced fluid.
The Kaluza-Klein ansatz for the field assumes that none of the fluid variables depend on the internal directions $y_j$. The velocity profile is \[boost\] u\_a=\_a\_[i=1]{}\^[N]{} \_i,u\_[y\_j]{}= \_j \_[k=1]{}\^[j-1]{} \_k ,j=1,…, N where $\alpha_i$ are boost parameters characterizing the velocity along the compact directions and $\hat{u}_a$ is the velocity in the reduced spacetime, which is unit-normalized with respect to $\eta_{ab}$, \^a\^b \_[ab]{}=-1. Note that it should be possible to formulate an ansatz for the velocity field where the different boosts enter in a manner that preserves the local symmetry $SO(N)$ that rotates them (this is broken globally by the compact size of the torus). The above ansatz does not show this, but it is a convenient one for our calculations.[^1]
Let us apply this Kaluza-Klein reduction ansatz to the perfect fluid stress energy tensor. Substituting in we obtain \[eqn:perfectgen\] T\_[ab]{}\^[pf]{}&=&V \[(+ P) u\_a u\_b \_[i=1]{}\^[N]{} \^2\_i + P\_[ab]{}\]\[eqn:perfectgen1\],\
T\_[ay\_j]{}\^[pf]{}&=&V (+ P)\_ju\_a\_[i=1]{}\^[N]{} \_i \_[k=1]{}\^[j-1]{} \_k \[eqn:perfectgen2\],\
T\_[y\_j y\_[j’]{}]{}\^[pf]{}&=&V \[(+ P) \_[j’]{} \_j\_[k=1]{}\^[j-1]{} \_k\_[i=1]{}\^[j’-1]{} \_i +\_[jj’]{}P\]\[eqn:perfectgen3\], where $V=\prod_{j=1}^{N} (2\pi R_j)$ is volume of the torus. The factor $V$ appears because $T_{AB}$ refers to densities so we have to include the internal volume we are going to integrate out. The form of the stress energy tensor in the reduced theory is T\_[ab]{}\^[pf]{}&=&(+P) u\_a u\_b +P \_[ab]{}\[eqn:perfectgenbb1\],\
T\_[ay\_j]{}\^[pf]{}&=& u\_a q\_j\[eqn:perfectgenbb2\], where P &=&P V\[pres\],\
&=&P (-1+\_[i=1]{}\^[N]{} \^2\_i)+ V \_[m=1]{}\^[N]{} \^2\_[m]{}\[en\],\
q\_j&=&(P+) \[car\]. This yields not only the energy density and pressure in the reduced theory, but also a set of $N$ charge densities $\hat q_j$, one for each boost parameter $\alpha_j$.
The temperature of the reduced fluid is given by \[tempe\] T= due to the fact that we have changed the timelike Killing vector. From the conservation of the entropy current for the initial fluid, we can read off the reduced entropy density given by \[redentropy\] s=s V \_[i=1]{}\^[N]{}\_i. From the Euler relation P+= T s+\_[j=1]{}\^Nq\_j \_j the chemical potential for each charge takes the form \[mu\] \_j= .
Since we assume that the first law is satisfied for the initial neutral fluid, it is possible to verify that the same is true for the reduced fluid. The neutral fluid obeys the law d = T d S from which it follows d = T d S+ \_[i=1]{}\^N \_i dq\_i using + P&=&V(+ P)\_[i=1]{}\^[N]{} \^2\_i ,\
d\_[j+1]{}&=&d\_[j]{} and that $\epsilon+ P=TS$.
Reduction of dissipative terms
==============================
The Kaluza-Klein reduction has given us a charged fluid. When including dissipative terms we expect the presence of another set of transport coefficients, namely a heat conductivity matrix. These coefficients measure the response of the charge current to changes in temperature and in chemical potential.
In order to reduce the first-derivative terms in the stress energy tensor, we need to express the expansion $\theta$ defined in Eq.(\[espa\]) in terms of the new velocities. We find that \[eqn:thetagen\] = \_[i=1]{}\^[N]{}\_i(+\_[k=1]{}\^[N]{} \_ku\_a\^a \_k), where $\hat\theta=\partial^a \hat u_a$.
The equation of conservation of the stress energy tensor relates the gradients of the rapidities to $\hat \theta$ as u\_a \^a \_j =, where ’=c\_s\^[-2]{}=, where $c_s$ is the speed of sound. The explicit calculation can be found in the Appendix \[eqmotion\]. Substituting this result in Eq. the expansion becomes \[eqn:thetagen1\] =. The orthogonal projectors tensor in Eq. is given by P\_[ab]{}&=&\_[i=1]{}\^[N]{}\^2\_i u\_a u\_b + \_[ab]{},\
P\_[ay\_j]{}&=& u\_a\_j \_[i=1]{}\^[N]{} \_i \_[k=1]{}\^[j-1]{} \_k ,\
P\_[y\_j y\_[j’]{}]{}&=& \_[j’]{} \_j\_[k=1]{}\^[j-1]{} \_k \_[i=1]{}\^[j’-1]{} \_i +\_[jj’]{}. The shear viscosity tensor in Eq.(\[shear\]) takes the form \_[ab]{}&=&P\^C\_a P\^D\_b\_[(C]{} u\_[D)]{}-\
&=&\_[i=1]{}\^N P\^c\_[(a]{} P\^[y\_i]{}\_[b)]{}\_c u\_[y\_i]{}+P\^c\_a P\^d\_b\_[(c]{} u\_[d)]{}-\[eqn:ab\], \_[ay\_j]{}&=&P\^C\_a P\^D\_[y\_j]{}\_[(C]{} u\_[D)]{}-\
&=&P\^c\_a P\_[y\_j]{}\^d\_[(c]{} u\_[d)]{}+\_[i=1]{}\^N P\^[y\_i]{}\_[(a]{} P\_[y\_[j]{})]{}\^d\_d u\_[y\_i]{} -\[eqn:ay\], \_[y\_j y\_[j’]{}]{}&=&P\^C\_[y\_j]{} P\^D\_[y\_[j’]{}]{}\_[(C]{}u\_[D)]{}-\
&=&P\^c\_[y\_j]{} P\^d\_[y\_[j’]{}]{}\_[(c]{} u\_[d)]{} +\_[i=1]{}\^N P\^[y\_i]{}\_[(y\_j]{} P\^d\_[y\_[j’]{})]{}\_[d]{} u\_[[y\_i]{}]{}-\[eqn:yy\]. We have all the ingredients to compute the new transport coefficients. However, our reduced fluid is not in the Landau frame. Indeed, we find that $u^A T_{AB}=0$ implies u\^a T\_[ab]{}\^[diss]{}&+&\_[j=1]{}\^[N]{} u\^[y\_j]{}T\_[[y\_j]{}b]{}\^[diss]{}=0,\
u\^a T\_[a y\_j]{}\^[diss]{}&+&\_[j’=1]{}\^[N]{} u\^[y\_[j’]{}]{}T\_[[y\_[j’]{}]{} y\_j]{}\^[diss]{} =0, or using Eq.(\[boost\]) u\^a T\_[ab]{}\^[diss]{}&=&- T\_[[y\_j]{}b]{}\^[diss]{}\[eqn:Laundau1\],\
u\^a T\_[a y\_j]{}\^[diss]{}&=&-T\_[[y\_[j’]{}]{} y\_j]{}\^[diss]{}\[eqn:Laundau2\]. This means that we cannot directly extract the coefficients from the reduced stress energy tensor but we need to introduce some frame-invariant formulae. In [@Bhattacharya2] was proposed an efficient way to extract those coefficients based on a general dissipative correction to the stress energy tensor and the charge currents. In order to avoid unphysical solutions we require the semi-positivity of the divergence of local entropy current. Following the same procedure as in [@Bhattacharya2] and generalizing the result for $N$ charges we construct frame invariant formulae P\^a\_c P\^b\_d T\_[ab]{}\^[diss]{} - P\_[cd]{} P\^[ab]{} T\_[ab]{}\^[diss]{}=-2 \_[cd]{}\[eqn:etagen\],\
P\^b\_a( T\_[by\_j]{}\^[diss]{} + u\^c T\^[diss]{}\_[cb]{}) =-\_[j’=1]{}\^[N]{}\_[jj’]{} P\^b\_a\_b()\[eqn:kappagen\],\
-u\^a u\^b T\_[ab]{}\^[diss]{}+\_[j=1]{}\^[N]{}u\^a T\_[ay\_j]{}\^[diss]{} =-\[eqn:zetagen\]. Using these we can extract the viscosities $\hat\eta$, $\hat\zeta$ and the matrix of conductivities $\hat\kappa_{jj'}$. The derivative $\partial \hat P/ \partial \hat\epsilon$ is evaluated at constant charges, while $\partial \hat P/\partial \hat q_j$ are evaluated keeping fixed the energy density and the other charges $q_{k\neq j}$.\
We obtain =V \_[i=1]{}\^[N]{} \_i\[eqn:etaflui\], \_[jj]{}&=& VT 1-\_[i=1]{}\^[N]{} \_i\[eqn:kappafluidjj\],\
\_[jk]{}&=&\_[kj]{}=-VT\_[l=1]{}\^[k-1]{} \_l\[eqn:kappafluidjk\], kj, &=&2V\_[i=1]{}\^[N]{} \_i \[eqn:zetafluid\]\
&+&\
&-&\
&+&V.These are the main results of this article.
The transport coefficients can be rewritten in terms of the independent thermodynamic variables of the reduced theory, the temperature and the chemical potentials, using Eq. and Eq. functions of the rapidities.
Observe that the viscosity to entropy density ratio remains constant under the reduction, \[KSS\] =.
Furthermore, since the entropy current for our charged fluid in a canonical form is J\_s\^a=su\^a- T\^[ab]{}\_[diss]{}-\_[j=1]{}\^N \_j T\^[ay\_j]{}\_[diss]{} using the relations in Eq. and substituting the values of the chemical potentials Eq.(\[mu\]), it is easy to see that J\_s\^a=su\^a. Comparing this result with the entropy density of our neutral initial fluid J\_s\^A=s u\^A multiplied by the volume factor $V$, we recover the result obtain from the Euler relation in Eq.(\[redentropy\]).
Finally, the speed of sound is given by \[sos\] c\_s\^2==, where the derivative is considered at fixed $\hat s/\hat q_j$ for every $ \hat q_j$.\
Charged black brane/fluid duals {#6}
===============================
The previous analysis can be applied to the case of the fluid dual to a black p-brane. Let us consider a black p-brane in $D = p + n + 3$ dimensions with $p+1$ worldvolume coordinates of the p-brane and $n+2$ coordinates in directions transverse to that. Since we perform a Kaluza Klein reduction exclusively on the wordvolume directions, we focus only on the $p+1$ coordinates. This means that the $p$ dimensions of the black p-brane can be seen as the $p$ spatial dimensions of the previous fluid.
In [@Joan] was shown that the long-wavelength dynamics of a neutral black brane in $D = p+ n + 3$ dimensions can be described in terms of a fluid with equation of state \[eqn:p\] =-(n+1)P, and viscosities &=& ,\
&=&2 (+)\[eqn:etazeta\].
If we substitute these values in Eqs., , , , and we obtain the reduced thermodynamic quantities and the transport coefficients of the charged fluid. These are
P&=&P V, =-P (1+n\_[i=1]{}\^[N]{} \^2\_i ),\
q\_j&=&-P n \_j \_[i=1]{}\^[N]{} \_i \_[k=1]{}\^[j-1]{} \_k,\
&=&V \_[i=1]{}\^[N]{} \_i\[eqn:etargen\]\
&=&\^[-n-1]{}(1-\_[i=1]{}\^N \_i\^2)\^,\
\_[jj]{}&=& VT 1-\_[i=1]{}\^[N]{} \_i\[eqn:coefkappagen\]\
&=&\^[-n-1]{}T\^[-n]{}(1-\_j\^2)(1-\_[i=1]{}\^N \_i\^2)\^,\
\_[jk]{}&=&-VT\_[l=1]{}\^[k-1]{} \_l\
&=&-\^[-n-1]{}T\^[-n]{}(\_k\_j)(1-\_[i=1]{}\^N \_i\^2)\^,\
&=&2V\_[i=1]{}\^[N]{} \_i \[eqn:zetared\]\
&-&\
&=&2 - , using the explicit values of the temperature and the shear viscosity \[identconst\] T=,=r\_0\^[n+1]{} where $r_0$ is the horizon radius of the black p-brane and $\Omega_{n+1}$ is the volume of the unit (n+1)-sphere.
We can compare our results with those found in Eq.(3.4.17) and Eqs. (3.4.38)-(3.4.40) in [@Blaise] for $N=1$ in AdS. This can be done using the AdS/Ricci flat map in [@Joan2] which relates the dynamics of Ricci-flat black $p$-branes in $n+p+3$ dimensions to that of black $d$-branes in AdS$_{2\sigma+1}$, by identifying \[map\] -n2, pd.
If we apply this map to the equation of state and the transport coefficients that we obtain for $N=1$, we find the same results as in [@Blaise] with \_i\_i-L. where $L$ is defined after Eq.(3.1.3) in \[5\].
For $N=2$ the map to black $d$-branes in AdS$_{2\sigma+1}$ is \[map2\] -n2, pd+1. In this case we recover the results of [@Milena] in Eq.(3.1.18) and Eqs. (3.2.42)-(3.2.51) for the Kaluza-Klein Einstein-Maxwell-Dilaton theory containing two Maxwell fields, three neutral scalars and an axion in AdS, again using Eqs.. Note that the speed of sound and the other thermodynamic quantities as entropy, temperature and chemical potential agree too.
Finally, let us study whether the bound \[bound\] 2(-c\^2\_s) proposed in [@Buchel] is satisfied. The bulk viscosity for a charged black $(p-N)$-brane takes the form &=&2V\_[i=1]{}\^[N]{} \_i\
&-&. In terms of the speed of sound c\_s\^2=, the bulk to shear viscosity ratio can be written as &=&2 (-c\^2\_s)\
&-&2c\^4\_s . The relation (\[bound\]) requires that n-2+, which is satisfied for all $n\geq 0$. Thus the bound is always satisfied. In contrast, the bound is always violated in [@Blaise; @Milena] for the black $d$-branes with $\sigma> 1$, where $2\sigma+1$ are the initial spacetime dimensions.
An alternative bound was proposed in [@Blaise], \[bound2\] 2(-c\^2\_q) in terms of the ‘speed of sound at constant charge density’ c\_q=\_[q\_j]{}=. It is straightforward to show that in our case this bound is always violated. We obtain &=&2 (-c\^2\_q)\
&-&2c\^4\_s . In order to satisfy the bound in (\[bound2\]) we would need $n\le -2$. The inversion of the results regarding both bounds and as compared to [@Blaise] and [@Milena] is expected from the mappings and . On the other hand, for electrically charged, non-dilatonic asymptotically flat black brane solutions, ref. [@Gath] finds that the bound is satisfied only for small enough charge density, while the bound is always violated.[^2]
Note Eq.(\[KSS\]) implies that the KSS bound [@Kovtun] is saturated.
Acknowledgements
================
I am grateful to Roberto Emparan for discussions that led to this project, for many conversations, and for carefully reading the draft. It is also a pleasure to thank Harvey Reall and Joan Camps for useful discussions and comments, and the warm hospitality provided by the Department of Applied Mathematics and Theoretical Physics of Cambridge University where part of this work was done. This work was supported by FPI scholarship BES-2011-045401 and grants MEC FPA2010-20807-C02-02, AGAUR 2009-SGR-168 and CPAN CSD2007-00042 Consolider-Ingenio 2010.
Equation of motion {#eqmotion}
==================
In order to extract the transport coefficient we need to compute the equations of motion derived from energy momentum conservation relations. These read as \^a T\^[pf]{}\_[ab]{}&=&0=\^a\[(V+P) u\_a u\_b \_[i=1]{}\^[N]{} \^2\_i + P\_[ab]{}\]\[eqn:tuttogen\]\
&=&\[ (1+’)u\_au\_b \_[i=1]{}\^[N]{} \^2\_i\] \^a P+\_b P\
&+&(V+P ) \_[i=1]{}\^[N]{} \^2\_i,\
\^a T\^[pf]{}\_[ay\_j]{}&=&\^a(V + P)\_ju\_a\_[i=1]{}\^[N]{} \_i \_[k=1]{}\^[j-1]{} \_k\[eqn:loggen21\]\
&=&{(1+’)u\_a \^a P+(V+P )}\_j\
&&\_[i=1]{}\^[N]{} \_i \_[k=1]{}\^[j-1]{} \_k=0 where ’==’ .If we contract the Eq.(\[eqn:tuttogen\]) with $\hat u^b$ we find \[eqn:loggen\] u\_a \^a P=\_[i=1]{}\^[N]{}\^2\_i. In addition, Eq. can be rewritten as u\_a \^a P=-.\[eqn:loggen2\] If we take the latter relation for two different indices, $j$ and $k$ with $j\neq k$ (corresponding to the conservation of two different components of the stress energy tensor) and we subtract them, we obtain \[eqn:relazvarjdif\] \_ [i=k]{}\^[j-1]{}\_[i]{}u\_a \^a \_i =\_[k]{}u\_a \^a \_[k]{}-\_[j]{}u\_a \^a \_[j]{},j>k or equivalently u\_a \^a \_[i+1]{}=u\_a \^a \_i . Now, let us compare Eq. and Eq.. This gives \[sumconserv\] &&\_[i=1]{}\^[N]{}\^2\_i=\
&&- \[\_[i=1]{}\^[N]{} \_iu\_a\^a \_i+\_[l=1]{}\^[j-1]{}\_lu\_a\^a \_l +\_ju\_a\^a \_j+\]\
Since we want the relation between the reduced expansion and a derivatives of specific rapidity $\alpha_j$, we replace in Eq. the other derivatives of the remaining rapidities in term of the one chosen using Eq.. This gives u\_a \^a \_j= that can be rewritten as \[eqn:alfathetagen\] u\_a \^a \_j =.
The Eq.(\[eqn:loggen\]) in terms of $\hat u_a \partial^a \alpha_j$ only is given by \[eqn:lopPrapidgen\] u\_a \^a P &=&-\_[j]{} \_[i=1]{}\^[j-1]{}\^2\_[i]{}u\_a \^a \_j using the result in Eq..
The reduced acceleration is obtain from Eq. u\_a\^a u\_b&=&--u\_bu\_a\^a P\
&-&2u\_au\_b\_[k=1]{}\^[N]{} \_k\^a \_k-u\_b.Using Eqs., , and , this becomes \[accel\] u\_a \^au\_b=-+ .
Transport coefficients
======================
Shear viscosity
---------------
The first term in Eq.(\[eqn:etagen\]) is given by \[firstter\] &&P\^a\_c P\^b\_d T\_[ab]{}\^[diss]{}=- V P\^a\_c P\^b\_d(2\_[ab]{}+P\_[ab]{})\
&=&- V P\^a\_c P\^b\_d.Taking into account that P\_c\^aP\_d\^b P\_a\^l P\_b\^m= P\_c\^lP\_d\^m ,P\_c\^l P\_d\^m\_[(l]{} u\_[m)]{}=0,P\_c\^a P\_a\^[yj]{}=0 the Eq.(\[firstter\]) becomes \[shear1\] P\^a\_c P\^b\_d T\_[ab]{}\^[diss]{}=- V. For what concerns the second term in Eq.(\[eqn:etagen\]) we get && P\_[cd]{} P\^[ab]{} T\_[ab]{}\^[diss]{}\[shear2\]\
&=&-Vconsidering that \[abab\] P\^[ab]{} P\_[ab]{}=p-N,P\^[cd]{}\_[(c]{}\_i u\_[d)]{}=0,P\^[ab]{} P\^c\_[(a]{} P\_[b)]{}\^[y\_i]{}= 0. If now we subtract (\[shear2\]) with (\[shear1\]) we are able to extract the shear viscosity. In fact &-&2 V\_[i=1]{}\^N \_i\
&=&-2 V\_[i=1]{}\^N \_i=-2\_[cd]{}, where $\hat P^{ab} \partial_{(a}\hat u_{b)}=\hat \theta$ and $\hat\sigma_{cd}$ is defined as \[eqn:sigmared\] \_[cd]{}= P\_c\^a P\_d\^b\_[(a]{} u\_[b)]{}-P\_[cd]{}. We recover the result anticipated in Eq.(\[eqn:etaflui\]), that is =V \_[i=1]{}\^[N]{} \_i.
Heat conductivity matrix
------------------------
Let us turn to the heat conductivity matrix elements. The Eq.(\[eqn:kappagen\]) can be simplified substituting the Eqs.(\[eqn:Laundau1\]) that leads to \[eqn:kappagen1\] &&P\_a\^b( T\_[by\_j]{}\^[diss]{}-\_[j’=1]{}\^[N]{} T\_[by\_[j’]{}]{}\^[diss]{}) where we have also replaced the values of the reduced density charge, pressure and energy density from Eqs.(\[pres\]), (\[en\]) and (\[car\]). It is convenient to split the above expression into two parts: one with index $j'=j$ and the other with different indeces $k\neq j$ as P\_a\^b T\_[by\_j]{}\^[diss]{}(1-)-P\_a\^b\_[k=1]{}\^[N]{} T\_[by\_[k]{}]{}\^[diss]{}. So, we extract the heat conductivity coefficients from the relations P\_a\^b T\_[by\_j]{}\^[diss]{}(1-)&=&-\_[jj]{} P\^b\_a\_b()\[kappajj\],\
-P\_a\^b\_[k=1]{}\^[N]{} T\_[by\_[k]{}]{}\^[diss]{}&=&-\_[k=1]{}\^[N]{}\_[jk]{} P\^b\_a\_b().\[kappajk\] The $\hat P_a^b T_{by_j}^{diss}$ term is given by P\_a\^b T\_[by\_j]{}\^[diss]{}&=&-2V P\_a\^[c]{}(P\^d\_[y\_j]{}\_[(c]{} u\_[d)]{}+\_[j’=1]{}\^[N]{} P\_[y\_j]{}\^[y\_[j’]{}]{}\_c u\_[y\_[j’]{}]{})\
&=&-V P\_a\^[c]{}.Using the expression (\[accel\]) for the acceleration, and the fact that $\hat P^{ab}u_b=0$, this becomes &-&V P\_a\^[c]{}Making explicit the value of $P_{y_j}^{y_i}$ and simplifying we obtain &-&V P\_a\^[c]{}\_j \_[l=1]{}\^[j-1]{}\_l(\_[l’=1]{}\^[j-1]{}\_[l’]{}\_c \_[l’]{}+ \_j\_c \_[j]{}\
&-&)\[PabTbyfin\],
For what concerns the right part of the formula (\[kappajj\]), if we substitute the values (\[tempe\]) and (\[mu\]) we find &&P\^b\_a \_b ()= P\^b\_a \_b ()\[derivmuT\]\
&&= \[eqn:condu2gen\].
Now we are able to extract the elements of the heat conductivity metric from Eq.(\[kappajj\]) only replacing the result obtained in Eq.(\[PabTbyfin\]) and (\[eqn:condu2gen\]). This leads to &-&V P\_a\^[c]{}\_j \_[l=1]{}\^[j-1]{}\_l(1-)\
&=&-\_[jj]{} .So, the diagonal elements are \_[jj]{}= VT\_[i=1]{}\^[N]{} \_i(1-), as already shown in Eq.(\[eqn:kappafluidjj\]).
The same procedure is performed for the Eq.(\[kappajk\]). We find that &-&P\_a\^b T\_[by\_k]{}\^[diss]{}\_[k=1]{}\^[N]{}\
&=&VP\_a\^[c]{}\_[k=1]{}\^[N]{}\
&=&-\_[k=1]{}\^[N]{}\_[jk]{} . Comparing the last two equations, term by term, in the sum it is easy to find that \_[jk]{}=-VT\_[l=1]{}\^[k-1]{} \_l.
Bulk viscosity
--------------
Let complete the hydrodynamic analysis computing the bulk viscosity.\
First of all, we need to compute the derivatives of the pressure with respect to energy density and the charges. Due to the fact that $\partial\hat P/\partial\hat\epsilon$ is calculated with constant charges $\hat q_j$ and $\partial\hat P/\partial\hat q_j$ keeping fixed the energy density and the remaining $q_{k\neq j}$ charges, we need to use d =0 && d P=-2d\_l\
\[enetgconst\]\
d q\_j=0 && d P=-.\[carconst\]
The considered derivatives are given by \[derivatives3\] &&=\
&&,\
&&=,\[derivatives4\] with A&=&\_j\_[i=1]{}\^[N]{} \_i \_[k=1]{}\^[j-1]{} \_k\
B&=&\_[l=1]{}\^N\_l+\_[k=1]{}\^[j-1]{}\_k+\_j\
Combining conveniently Eqs. and in Eqs. and we obtain that
\[derivatives\] &=&,\[derivatives1\]\
&=&.\[derivatives2\]
Let now evaluate the first term in Eq.(\[eqn:zetagen\]). We find that \[firstterm\] &&\
&&=-V \_[l=1]{}\^[N]{} \_l.
The remaining terms can be simplified considering the Eq.(\[eqn:Laundau1\]) and Eq.(\[eqn:Laundau2\]). These become -\_[j’=1]{}\^[N]{} (\_[j=1]{}\^[N]{} +\_[j=1]{}\^[N]{})T\_[y\_jy\_[j’]{}]{}\^[diss]{} Using the Eqs.(\[eqn:alfathetagen\]), (\[eqn:relazvarjdif\]), , and replacing the components of the stress energy tensor $T_{y_jy_{j'}}^{diss}$ we find &&-=-2V\_[i=1]{}\^[N]{} \_i\
&&+\
&&--V. in terms of $\hat\theta$ only. Remember that for the initial expansion we use the values found in the Eq.(\[eqn:thetagen1\]) . It is straightforward to read now the bulk viscosity as already presented in Eq.(\[eqn:zetafluid\]).
[99]{} C. N. Pope, [*Lectures on Kaluza-Klein Theory.*]{} \[http://faculty.physics.tamu.edu/pope\].
G. T. Horowitz and T. Wiseman, “General black holes in Kaluza-Klein theory,” arXiv:1107.5563 \[gr-qc\]. R. Emparan, T. Harmark, V. Niarchos and N. A. Obers, “Essentials of Blackfold Dynamics,” JHEP [**1003**]{} (2010) 063 \[arXiv:0910.1601 \[hep-th\]\]. J. Camps, R. Emparan and N. Haddad, “Black Brane Viscosity and the Gregory-Laflamme Instability,” JHEP [**1005**]{} (2010) 042 \[arXiv:1003.3636 \[hep-th\]\]. B. Gouteraux, J. Smolic, M. Smolic, K. Skenderis and M. Taylor, “Holography for Einstein-Maxwell-dilaton theories from generalized dimensional reduction,” JHEP [**1201**]{} (2012) 089 \[arXiv:1110.2320 \[hep-th\]\]. B. Gouteraux and E. Kiritsis, “Generalized Holographic Quantum Criticality at Finite Density,” JHEP [**1112**]{} (2011) 036 \[arXiv:1107.2116 \[hep-th\]\]. M. Smolic, “Holography and hydrodynamics for EMD theory with two Maxwell fields,” JHEP [**1303**]{} (2013) 124 \[arXiv:1301.6020 \[hep-th\]\].
I. Kanitscheider and K. Skenderis, “Universal hydrodynamics of non-conformal branes,” JHEP [**0904**]{} (2009) 062 \[arXiv:0901.1487 \[hep-th\]\].
I. Kanitscheider, K. Skenderis and M. Taylor, “Precision holography for non-conformal branes,” JHEP [**0809**]{} (2008) 094 \[arXiv:0807.3324 \[hep-th\]\].
M. M. Caldarelli, J. Camps, B. Gouteraux and K. Skenderis, “AdS/Ricci-flat correspondence and the Gregory-Laflamme instability,” Phys. Rev. D [**87**]{} (2013) 061502 \[arXiv:1211.2815 \[hep-th\]\].
M. M. Caldarelli, R. Emparan and B. Van Pol, “Higher-dimensional Rotating Charged Black Holes,” JHEP [**1104**]{} (2011) 013 \[arXiv:1012.4517 \[hep-th\]\]. R. Emparan, T. Harmark, V. Niarchos and N. A. Obers, “Blackfolds in Supergravity and String Theory,” JHEP [**1108**]{} (2011) 154 \[arXiv:1106.4428 \[hep-th\]\].
J. Gath and A. V. Pedersen, “Viscous Asymptotically Flat Reissner-Nordström Black Branes,” arXiv:1302.5480 \[hep-th\]. R. Emparan, V. E. Hubeny and M. Rangamani, “Effective hydrodynamics of black D3-branes,” JHEP [**1306**]{} (2013) 035 \[arXiv:1303.3563 \[hep-th\]\]. J. Armas, J. Gath and N. A. Obers, “Black Branes as Piezoelectrics,” Phys. Rev. Lett. [**109**]{} (2012) 241101 \[arXiv:1209.2127 \[hep-th\]\]. J. Armas, J. Gath and N. A. Obers, “Electroelasticity of Charged Black Branes,” JHEP [**1310**]{} (2013) 035 \[arXiv:1307.0504 \[hep-th\]\].
S. Bhattacharyya, V. E. Hubeny, S. Minwalla, and M. Rangamani, [*Nonlinear Fluid Dynamics from Gravity*]{}, JHEP [**0802**]{} (2008) 045, \[arXiv:0712.2456\].
J. Bhattacharya, S. Bhattacharyya, S. Minwalla and A. Yarom, “A Theory of first order dissipative superfluid dynamics,” arXiv:1105.3733 \[hep-th\]. P. Kovtun, D. T. Son and A. O. Starinets, “Viscosity in strongly interacting quantum field theories from black hole physics,” Phys. Rev. Lett. [**94**]{} (2005) 111601 \[hep-th/0405231\]; “Holography and hydrodynamics: Diffusion on stretched horizons,” JHEP [**0310**]{} (2003) 064 \[hep-th/0309213\].
A. Buchel, “Bulk viscosity of gauge theory plasma at strong coupling,” Phys. Lett. B [**663**]{} (2008) 286 \[arXiv:0708.3459 \[hep-th\]\].
[^1]: Our choice is in this sense analogous to choosing polar coordinates for a sphere, which allows easy explicit calculation but obscures the rotational symmetry.
[^2]: Jakob Gath informs us (private communication) that for sufficiently large values of the dilaton coupling these bounds are satisfied/violated in the same manner as we have found: is always satisfied, and is always violated, for all values of the charge density.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Based on a new approach on modeling the magnetically dominated outflows from AGNs (Li et al. 2006), we study the propagation of magnetic tower jets in gravitationally stratified atmospheres (such as a galaxy cluster environment) in large scales ($>$ tens of kpc) by performing three-dimensional magnetohydrodynamic (MHD) simulations. We present the detailed analysis of the MHD waves, the cylindrical radial force balance, and the collimation of magnetic tower jets. As magnetic energy is injected into a small central volume over a finite amount of time, the magnetic fields expand down the background density gradient, forming a collimated jet and an expanded “lobe” due to the gradually decreasing background density and pressure. Both the jet and lobes are magnetically dominated. In addition, the injection and expansion produce a hydrodynamic shock wave that is moving ahead of and enclosing the magnetic tower jet. This shock can eventually break the hydrostatic equilibrium in the ambient medium and cause a global gravitational contraction. This contraction produces a strong compression at the head of the magnetic tower front and helps to collimate radially to produce a slender-shaped jet. At the outer edge of the jet, the magnetic pressure is balanced by the background (modified) gas pressure, without any significant contribution from the hoop stress. On the other hand, along the central axis of the jet, hoop stress is the dominant force in shaping the central collimation of the poloidal current. The system, which possesses a highly wound helical magnetic configuration, never quite reaches a force-free equilibrium state though the evolution becomes much slower at late stages. The simulations were performed without any initial perturbations so the overall structures of the jet remain mostly axisymmetric.'
author:
- 'Masanori Nakamura, Hui Li, and Shengtai Li'
title: Structure of Magnetic Tower Jets in Stratified Atmospheres
---
INTRODUCTION
============
A number of astronomical systems have been discovered to eject tightly collimated and hyper-sonic plasma beams and large amounts of magnetic fields into the interstellar, intracluster and intergalactic medium from the central objects during their initial/final (often violent) stages. Magnetohydrodynamic (MHD) mechanisms are frequently invoked to model the launching, acceleration and collimation of jets from Young Stellar Objects (YSOs), X-ray binaries (XRBs), Active Galactic Nuclei (AGNs), Microquasars, and Quasars (QSOs) [see, [*e.g.*]{}, @M01 and references therein].
Theory of magnetically driven outflows in the electromagnetic regime has been proposed by @B76 and @L76 and subsequently applied to rotating black holes [@BZ77] and to magnetized accretion disks [@BP82]. By definition, these outflows initially are dominated by electromagnetic forces close to the central engine. In these and subsequent models of magnetically driven outflows (jets/winds), the plasma velocity passes successively through the hydrodynamic (HD) sonic, slow-magnetosonic, Alfvénic, and fast-magnetosonic critical surfaces.
The first attempt to investigate the nonlinear (time-dependent) behavior of magnetically driven outflows from accretion disks was performed by @US85. The differential rotation in the system (central star/black hole and the accretion disk) creates a magnetic coil that simultaneously expels and pinches some of the infalling material. The buildup of the azimuthal (toroidal) field component in the accretion disk is released along the poloidal field lines as large-amplitude torsional Alfvén waves (“sweeping magnetic twist”). After their pioneering work, a number of numerical simulations to study the MHD jets have been done [see, [*e.g.*]{}, @F98 and references therein]. An underlying large-scale poloidal field for producing the magnetically driven jets is almost universally assumed in many theoretical/numerical models. However, the origin and existence of such a galactic magnetic field are still poorly understood.
In contrast with the large-scale field models, Lynden-Bell [@LB94; @L96; @L03; @L06] examined the expansion of the local force-free magnetic loops anchored to the star and the accretion disk by using the semi-analytic approach. Twisted magnetic fluxes due to the disk rotation make the magnetic loops unstable and splay out at a semi-angle 60 from the rotational axis of the disk. Global magnetostatic solutions of magnetic towers with external thermal pressure were also computed by @Li01 using the Grad-Shafranov equation in axisymmetry [see also, @L02; @LR03; @UM06]. Full MHD numerical simulations of magnetic towers have been performed in two-dimension (axisymmetric) [@R98; @T99; @U00; @K02; @K04a] and three-dimension [@K04b]. Magnetic towers are also observed in the laboratory experiments [@HB02; @L05].
This paper describes the nonlinear dynamics of propagating magnetic tower jets in large scales ($>$ tens of kpc) based on three-dimensional MHD simulations. We follow closely the approach described in Li et al. (2006; hereafter Paper I). Different from Paper I, which studied the dynamics of magnetic field evolution in a uniform background medium, we present results on the injection and the subsequent expansion of magnetic fields in a stably stratified background medium that is described by an iso-thermal King model [@K62]. Since the simulated magnetic structures traverse several scale heights of the background medium, we regard that our simulations can be compared with the radio sources inside the galaxy cluster core regions. Due to limited numerical dynamic range, however, the injection region (see Paper I) assumed in this paper will be large (a few kpc). Our goal here is to provide the detailed analysis of the magnetic tower jets, in terms of its MHD wave structures, its cylindrical radial force balance, and collimation. The paper is organized as follows: In §2, we outline the model and numerical methods. In §3, we describe the simulation results. Discussions and conclusions are given in §4 and §5.
MODEL ASSUMPTIONS AND NUMERICAL METHODS
=======================================
The basic model assumptions and numerical treatments we adopt here are essentially the same as those in Paper I. Magnetic fluxes and energy are injected into a characteristic central volume over a finite duration. The injected fluxes are not force-free so that Lorentz forces cause them to expand, interacting with the background medium. For the sake of completeness, we show the basic equations and other essential numerical setup here again, and refer readers to Paper I for more details.
Basic Equations
---------------
We solve the nonlinear system of time-dependent ideal MHD equations numerically in a 3-D Cartesian coordinate system ($x,\,y,\,z$): $$\begin{aligned}
\label{eq:mass}
\frac{\partial\rho}{\partial t} + \nabla\cdot(\rho\mbox{\bf V}) &=&
\dot{\rho}_{\rm inj}
\\
\label{eq:momentum}
\frac{\partial (\rho {\bf V})}{\partial t} +
\nabla\cdot \left( \rho {\bf V V} + p + B^2/2 -
{\bf B B}\right) &=& - \rho \nabla\psi
\\
\label{eq:energy}
\frac{\partial E}{\partial t} + \nabla\cdot\left[\left(E
+p + B^2/2 \right)
{\bf v}-{\bf B}({\bf v}\cdot{\bf B})\right] &=& -\rho {{\bm V}}\cdot
\nabla \psi \nonumber \\
&+& \dot{E}_{\rm inj}\\
\label{eq:induction}
\frac{\partial {\bf B}}{\partial t} - \nabla\times( {\bf V}
\times {\bf B}) &=& {\dot {\bf B}}_{\rm inj},\end{aligned}$$ Here $\rho$, $p$, ${{\bm V}}$, ${{\bm B}}$, and $E$ denote the mass density, hydrodynamic (gas) pressure, fluid velocity, magnetic field, and total energy, respectively. The total energy $E$ is defined as $E=p/(\gamma-1)+\rho V^2/2+B^2/2$, where $\gamma$ is the ratio of the specific heats (a value of $5/3$ is used). The Newtonian gravity is $-\nabla \psi$. Quantities $\dot{\rho}_{\rm inj}$, $\dot{{{\bm B}}}_{\rm
inj}$, and $ \dot{E}_{\rm inj}$ represent the time-dependent injections of mass, magnetic flux, and energy, whose expressions are given in Paper I.
We normalize physical quantities with the unit length scale $R_{0}$, density $\rho_{0}$, velocity $V_{0}$ in the system, and other quantities derived from their combinations, [*e.g.*]{}, time as $R_0/V_0$, etc. These normalizing factors are summarized in Table \[tbl:unit\]. Hereafter, we will use the normalized variables throughout the paper. Note that a factor of $4 \pi$ has been absorbed into the scaling for both the magnetic field ${{{\bm B}}}$ and the corresponding current density ${{{\bm J}}}$. To put our normalized physical quantities in an astrophysical context, we use the parameters derived from the X-ray observations of the Perseus cluster as an example [@C03]. These values are also given in Table \[tbl:unit\].
The system of dimensionless equations is integrated in time by using an upwind scheme [@LL03]. Computations were performed on the parallel Linux clusters at LANL.
Initial and Boundary Conditions
-------------------------------
One key difference from Paper I is that we now introduce a non-uniform background medium. An iso-thermal model [@K62] has been adopted to model a gravitationally stratified ambient medium. This is applicable, for example, to modeling the magnetic towers from AGNs in galaxy clusters.
The initial distributions of the background density $\rho$ and gas pressure $p$ are assumed to be $$\label{eq:king}
\rho = p = \left[1 + \left(\frac{R}{R_{\rm c}}\right)^2\right]^{-\kappa},$$ where $R=(x^2+y^2+z^2)^{1/2}$ is the spherical radius and $R_{\rm c}$ the cluster core radius. (In the following discussion, both “transverse” and “radial” have the same meaning, referring to the cylindrical radial direction.) The parameter $\kappa$ controls the gradient of the ambient medium. Furthermore, we assume that the ambient gas is initially in a hydrostatic equilibrium under a fixed (in time and space) yet distributed gravitational field $-\nabla
\psi(R)$ (such as that generated by a dark matter potential). From the initial equilibrium, we get $$-\nabla \psi = \frac{\nabla
p|_{t=0}}{\rho|_{t=0}} = -\frac{2 \kappa R}{R_{\rm
c}^2}\left[1+\left(\frac{R}{R_{\rm c}^2} \right)^2\right]^{-1}.$$ In the present paper, we choose $R_{\rm c}=4.0$ and $\kappa=1.0$. The initial sound speed in the system is constant, $C_{\rm
s}|_{t=0}={\gamma}^{1/2} \approx 1.29$, throughout the computational domain. An important time scale is the sound crossing time $\tau
(\equiv R/C_{\rm s}) \approx 0.78$, normalized with $\tau_{\rm s0}
(\equiv R_{0} / C_{\rm s0}) \approx 10.0$ Myrs. Therefore, a unit time scale $t=1$ corresponds to $12.8$ Myrs.
The total computational domain is taken to be $|x| \leq 16$, $|y| \leq
16$, and $|z| \leq 16$ corresponding to a $(80\ {\rm kpc})^3$ box in the actual length scales. The numbers of grid points in the simulations reported here are $N_{x}\times N_{y}\times N_{z}=240
\times 240 \times 240$, where the grid points are assigned uniformly in the $x$, $y$, and $z$ directions. A cell $\Delta x\ (=\Delta
y=\Delta z \sim 0.13)$ corresponds to $\sim 0.65\ {\rm kpc}$. We use the outflow boundary conditions at all outer boundaries. Note that for most of the simulation duration, the waves and magnetic fields stay within the simulation box, and all magnetic fields are self-sustained by their internal currents.
Injections of Magnetic Flux, Mass, and Energy
---------------------------------------------
The injections of magnetic flux, mass and its associated energies are the same as those described in Paper I. The ratio between the toroidal to poloidal fluxes of the injected fields is characterized by a parameter $\alpha = 15$, which corresponds to $\sim 6$ times more toroidal flux than poloidal flux. The magnetic field injection rate is described by $\gamma_b$ and is set to be $\gamma_b=3$. The mass is injected at a rate of $\gamma_\rho =0.1$ over a central volume with a characteristic radius $r_\rho = 0.5$. Magnetic fields and mass are continuously injected for $t_{\rm inj} = 3.1$, after which the injection is turned off. These parameters correspond to an magnetic energy injection rate of $\sim 10^{43}$ ergs/s, a mass injection rate of $\sim 0.046 M_\odot$/yr, and an injection time $\sim 40$ Myrs.
In summary, we have set up an initial stratified cluster medium which is in a hydrostatic equilibrium. The magnetic flux and the mass are steadily injected in a central small volume with a radius of 1. Since these magnetic fields are not in a force-free equilibrium, they will evolve, forming a magnetic tower and interacting with the ambient medium.
RESULTS
=======
In this section we examine the nonlinear evolution and the properties of magnetic tower jets in the gravitationally stratified atmosphere.
Overview of Formation and Propagation of a Magnetic Tower Jet
-------------------------------------------------------------
Before considering our numerical results in detail, it is instructive to give a brief overview of the time development of the magnetic tower jet system. We achieve this by presenting the selected physical quantities using two-dimensional $x-z$ slices at $y=0$. The distributions of density at various times $(t=2.5,\,5.0,\,7.5,\,{\rm
and}\,10.0)$ are shown in Fig. \[fig:de\_evo\]. At the final stage ($t=10.0$), we see the formation of a quasi-axisymmetric (around the jet axis) magnetic tower jet with low density cavities (a factor of $\sim 30$ smaller than the peak density). Inside these cavities, the Alfvén speed is large $V_{\rm A} \gtrsim 5.0$, while plasma $\beta$ is small ($\beta = 2p/B^2 \lesssim 0.1$). However, the temperature $T$ ($\propto C_{\rm s}^2$) becomes large $T \gtrsim 2.5 $ ($\sim$ twice the initial constant value); that is, the hotter gas is confined in the low-$\beta$ magnetic tower. The jet possesses a slender hourglass-shaped structure with a radially confined “body” for $|z|
\leq 3$, which is likely due to the background pressure profile having a core radius $R_c = 4$. As the magnetic tower moves into an increasingly lower pressure background, the expanded “lobes” are formed. A quasi-spherical hydrodynamic (HD) shock wave front moves ahead of the magnetic tower.
[]{}
A snapshot of the gas pressure change ratio $\Delta p/p_{\rm i} =
p/p_{\rm i} -1$ (where $p_{\rm i}$ represents $p|_{t=0}$) at $t=10.0$ is shown in Fig. \[fig:final\_prdiff\]. Positive $\Delta p$ can be seen at both the post-shock region of the propagating HD shock wave and just ahead of the magnetic tower ($|z| \sim 8-10$). The distribution of $\Delta p$ forms a “[U]{}”-shaped bow-shock-like structure around the head of the magnetic tower. This structure, however, does not appear until $t \sim 7.5$. This is apparently caused by the local compression between the head of the magnetic tower and the reverse MHD slow-mode wave (see discussions in the next section). The gas pressure inside the magnetic tower becomes small ($|\Delta p/p_{\rm i}| \lesssim 0.5$) due to the magnetic flux expansion. Note that the light-blue region between the HD shock and the magnetic tower shows a small pressure decrease ($\Delta p/p_{\rm
i} \approx - 0.1$). In later sections, we will discuss the origin of this depression and what role it plays in the dynamics of magnetic tower jets.
The magnetic tower jet has a well-ordered helical magnetic configuration. The 3-D view of the selected magnetic lines of force, as illustrated in Fig. \[fig:3Dlines\], indicates that a tightly wound central helix goes up along the central axis and a loosely wound helix comes back at the outer edge of the magnetic tower jet. The magnetic pitch $B_{\phi}/\sqrt{B_r^2+B_z^2}$ has a broad distribution with a maximum of $\sim 15$. Figure \[fig:final\_Jz\] shows a snapshot of the axial current density $J_z$ at $t=10.0$. Clearly, the axial current flow displays a closed circulating current system in which it flows along the central axis (the “forward” current ${{\bm J}}^{\rm F}$) and returns on the conically shaped path that is on the outside (the “return” current ${{\bm J}}^{\rm R}$). It is well known that an axial current-carrying cylindrical plasma column with a helical magnetic field is subject to current-driven instabilities, such as sausage ($m=0$), kink ($m=1$), and the other higher order $m$ modes ($m$ is the azimuthal mode number). We however do not see any visible evolution of the non-axisymmetric features in this magnetic tower jet.
![\[fig:3Dlines\] Three-dimensional configuration view of the selected magnetic lines at $t=10.0$. ](f3.eps)
From this overview, we see that the magnetic tower jet can propagate through the stratified background medium while keeping well-ordered structures throughout the time evolution. The magnetic fields push away the background gas, forming magnetically dominated, low-density cavities. This action also drives a HD shock wave which is ahead of and eventually separated from the magnetic structures. The magnetic tower has a slender ”body” from the confinement of the background pressure and an expanded “lobe” when the fields expand into a background with the decreasing pressure.
We will now turn to the discussions on the detailed properties of the tower jet, including the HD shock wave and its impact in the axial ($z$) and radial ($x$) directions in §\[sec:A\] and \[sec:B\]. The radial force balance of the jet is examined in §\[sec:C\].
Structure of a Magnetic Tower Jet in the Axial ($z$) Direction {#sec:A}
--------------------------------------------------------------
Figure \[fig:t7.5\_linez\_1\] displays several physical quantities along a line with $(x,y)=(1,0)$ in the axial direction at $t=0.75$. Several features can be identified. First, the HD shock wave front can be seen around $z \sim 13.5$ in the profiles of $\rho$ ([*top*]{} panel), $V_{z}$, and $C_{\rm s}$ ([*bottom*]{} panel). $C_{\rm s}$ is higher than the initial background value $1.29$ in the post-shock region due to shock heating and becomes smaller than $1.29$ at $z \sim
10.7$ due to axial expansion. $V_{z}$ has the similar behavior as $C_{\rm s}$.
![\[fig:t7.5\_linez\_1\] Axial profiles of physical quantities, parallel to the $z$-axis with $(x,\,y)=(1,\,0)$ at $t=7.5$. [*Top*]{}: Density $\rho$ and magnetic field components $(B_r,\,B_\phi,\,B_z)$. [*Bottom*]{}: Sound speed $C_{\rm s}$ and velocity components $(V_r,\,V_\phi,\,V_z)$. The positions of the expanding hydrodynamic shock wave front and the magnetic tower front, which is identified as a tangential discontinuity, are shown in both panels. A horizontal [*solid line*]{} in the [*bottom*]{} panel denotes the initial sound speed (constant throughout the computational domain). ](f5.eps)
Second, a magnetic tower front (“tower front” in the following discussions) is located at $z \sim 8.0$, beyond which the magnetic field goes to zero, as seen in the [*top*]{} panel. This indicates that the gas within the magnetic tower jet is separated from the non-magnetized ambient gas beyond the tower front. We regard this front as a tangential discontinuity as the magnetic fields are tangential to the front without the normal component. This is consistent with the fact that the radial and azimuthal field components ($B_{r}$ and $B_{\phi}$) are dominant near the tower front ($z \lesssim 8.0$) but the axial field component $B_{z}$ becomes dominant only for $z<6.0$. The density and pressure show smooth transition through this front though the gradients of $\rho$ and $C_{\rm s}$ are slightly changed there.
![\[fig:t7.5\_linez\_2\] Similar to Fig. \[fig:t7.5\_linez\_1\]. [*Top*]{}: Shown are the gas pressure $p$ ([*dashed line*]{}), magnetic pressure ([*solid line*]{}) $p_{\rm
m}$, and total pressure $p_{\rm tot}\,(=p+p_{\rm m})$ ([*light gray thick solid line*]{}). [*Bottom*]{}: Shown are the forces in the axial ($z$) direction: the Lorentz force $F_{{{\bm J}}\times
{{\bm B}}}=-\partial/\partial z \left[(B_r^2+B_{\phi}^2)/2 \right]$, ([*solid line*]{}), gas pressure gradient force $F_p = -\partial p/\partial
z$ ([*dashed line*]{}), gravitational force $F_{\rm g}=-\rho \partial
\psi / \partial R \times |z|/R$ ([*dotted line*]{}), and total force $F_{\rm tot}\,(=F_{{{\bm J}}\times {{\bm B}}}+F_p+F_{\rm g})$ ([*light gray thick solid line*]{}). The position of the reverse slow-mode MHD wave front is also shown in the [*top*]{} panel. ](f6.eps)
Third, there is another MHD wave front at $z \sim 7.0$ where $B_{r}$, $C_{\rm s}$, and every velocity component have their local maxima ($\rho$ instead has its local minimum), as seen in both panels. To better understand the nature of this MHD wave front, we plot the axial profiles of pressures and various forces along the line $(x,y) = (1,0)$ at $t=7.5$ in Fig. \[fig:t7.5\_linez\_2\]. The total pressure $p_{\rm
tot}$ consists of only the gas pressure $p$ beyond the tower front ($z
\gtrsim 8.0$) but is dominated by the magnetic pressure $p_{\rm m}$ behind the MHD wave front ($z \lesssim 7.0$), as seen in the [*top*]{} panel. A transition occurs around $7.0 \lesssim z \lesssim 8.0$, where an increase in $p$ is accompanied by a decrease in $p_{\rm m}$. We therefore identify this as a reverse slow-mode compressional MHD wave front. In magnetic towers, the transition region between gas and magnetic pressures can be identified as a reverse slow wave front in the context of MHD wave structures. It does not depend on the resolution and parameters. So, the reverse slow mode wave (sometimes, it can be steepen into a shock) will always be there. In addition, in Fig. \[fig:t7.5\_linez\_3\], we show several snapshots of $V_z$ and $p$ during $t=7.5 \sim 10.0$ (along the same offset axial path with Figs. \[fig:t7.5\_linez\_1\] and \[fig:t7.5\_linez\_2\]). The axial flow is decelerated by the gravitational force in the post-shock region beyond the tower front as seen in the [*top*]{} panel. On the other hand, the narrow region between the tower front and the reverse slow-mode wave front is accelerated by the magnetic pressure gradient (the “magnetic piston” effect). Note that the reverse slow-mode MHD compressional wave could eventually steepen into the reverse slow-mode MHD shock wave via this nonlinear evolution. Consequently, in the frame co-moving with the reverse slow-mode shock, a strong compression occurs behind the shock wave front and causes a local heating, as seen in the [*bottom*]{} panel and also Fig. \[fig:final\_prdiff\]. This heating could have interesting implications for the enhancement of radiation from radio to X-rays at the terminal part of Fanaroff-Riley type II AGN jets, such as lobes and hot spots, which are generally interpreted as heating caused by the jet terminal shock wave [@BR74; @S74].
![\[fig:t7.5\_linez\_3\] Similar to Fig. \[fig:t7.5\_linez\_1\], but with selected snapshots during $t=7.5
\sim 10.0$ (each time-interval is equal to 0.5). [*Top*]{}: The axial velocity $V_z$. [*Bottom*]{}: The gas pressure $p$. ](f7.eps)
Fourth, the HD shock wave breaks the initial background hydrostatic equilibrium. The passage of the shock wave heats the gas and alters its pressure gradient. As shown in the [*bottom*]{} panel of Fig. \[fig:t7.5\_linez\_2\], the gas pressure gradient force $F_{\rm
p}$ stays uupositive at the shock front (which pushes the shock forward), but the total (gravity plus pressure gradient) force $F_{\rm
tot}$ becomes negative behind the shock, implying a deceleration of the gas in the axial direction in the post-shock region. This is consistent with Fig. \[fig:t7.5\_linez\_3\].
Deformation of the Jet “Body” in the Radial Direction {#sec:B}
-----------------------------------------------------
We next examine the structure and dynamics of the magnetic tower jet along the radial direction in the equatorial plane with $(y,z)=(0,0)$. Figure \[fig:t6.0\_liner\] shows the radial profiles of physical quantities along the $x$-axis at $t=6.0$. The boundary of the magnetic tower jet (“tower edge” in the following discussions) is located at $x \sim 3.0$ where $F_{{{\bm J}}\times {{\bm B}}}$ becomes zero. Two distinct peaks of $C_{\rm s}$ and $V_{x}$ around $x \sim$ 7.8 and 9.5 are visible in the [*top*]{} panel. The first front ($x \sim
9.5$) is the propagating HD shock wave front as we showed in the previous section. The second front ($x \sim 7.8$) also indicates another expanding HD shock wave front generated by a bounce when the magnetic flux pinches in the radial direction caused by the “hoop stress”. This secondary shock front appears only in the radial direction (see also Fig. \[fig:t7.5\_linez\_1\]). $C_{\rm s}$ decreases gradually towards the jet axis in the post-shock region of the secondary shock and becomes smaller than its initial value $1.29$ at $x \sim 5.7$. We can confirm that these shock fronts are purely powered by the gas pressure gradients $F_{p}$, as seen in the [*bottom*]{} panel (twin peaks of $F_{\rm p}$ are shifted a bit behind that of $C_{\rm s}$ in the [*top*]{} panel).
![\[fig:t6.0\_liner\] Radial profiles of physical quantities along the $x$-axis in the equatorial plane with $(y,\,z)=(0,\,0)$ at $t=6.0$. [*Top*]{}: The sound speed $C_{\rm s}$ ([*light gray thick solid line*]{}) and the radial velocity $V_x$ ([*solid line*]{}). [*Bottom*]{}: The forces in the radial ($x$) direction: the Lorentz force $F_{{{\bm J}}\times {{\bm B}}}=-\partial/\partial r
\left[(B_{\phi}^2+B_{z}^2)/2\right]-B_{\phi}^2/r$ ([*solid line*]{}), gas pressure gradient force $F_p = -\partial p / \partial r$ ([*dashed line*]{}), gravitational force $F_{\rm g}=-\rho \partial \psi /
\partial R \times |x|/R$ ([*dotted line*]{}), and total force $F_{\rm
tot}\,(=F_{{{\bm J}}\times {{\bm B}}}+F_p+F_{\rm g})$ ([*light gray thick solid line*]{}). The positions of two expanding hydrodynamic shock wave fronts and the edge of the magnetic tower (tangential discontinuity) are shown in the [*top*]{} panel. A horizontal [*solid line*]{} in the [*top*]{} panel denotes the initial sound speed (constant throughout the computational domain) and a horizontal [*dashed line*]{} in the [*top*]{} panel represents a level “0” for the transverse velocity. ](f8.eps)
Part of the region between the secondary HD shock front and the tower edge has $F_{\rm tot}$ being negative (the [bottom]{} panel), meaning that the gas is undergoing gravitational contraction. This is indicated by $V_x < 0$ in the [*top*]{} panel for $x < 5.1$. This behavior is similar to what we have discussed earlier, i.e., the HD shock waves break the background hydrostatic equilibrium, causing a global contraction. Note that this contraction is occurring along the whole jet “body”, e.g., for $|z| \lesssim 4.0$ when $t=7.5$.
The [*bottom*]{} panel also helps to address the question on what forces are confining the magnetic fields in the equatorial plane. Since the total magnetic field $F_{{{\bm J}}\times {{\bm B}}}$ stays positive, this means that the inward hoop stress is not strong enough to confine the magnetic fields. Instead, at the tower edge, it is the background gravity that counters the combined effects of magnetic field pressure and a positive pressure gradient (pushing outward). A bit further into the tower edge (at $x \sim 2.1$), however, the pressure force changes from outwardly directed to inwardly directed. Then, both the pressure gradient and the gravitational forces act to counter the outward ${{{\bm J}}\times {{\bm B}}}$ force. This behavior, which is mostly caused by the magnetic tower expanding in a background gravitational field, is different from the usual MHD models for jets where the inwardly directed hoop stress is balanced by the outwardly directed magnetic pressure gradient [@BP82].
![\[fig:liner\_de\] The radial profiles of density $\rho$ in the equatorial plane with selected snapshots during $t=6.0 \sim 10.0$ (each time-interval is equal to 0.5). The initial profile ($t=0.0$) is also shown ([*dashed line*]{}). ](f9.eps)
To investigate the radius evolution of the magnetic tower jet, we show the radial profile of density at the equatorial plane from $t=6.0$ to $ 10.0$ in Fig. \[fig:liner\_de\]. It shows that the radius has an approximately constant contraction speed for $t=6.5 \sim 9.5$. The time scale for contraction is $\tau_{\rm contr} \sim 6$, which is about 7.5 times longer than the local sound-crossing time scale.
Force Balance in the Radial Direction {#sec:C}
-------------------------------------
We now discuss the jet properties along the radial direction away from the equatorial plane. Figure \[fig:t7.5\_liner\] shows the radial profiles of physical quantities along the $x$-axis with $(y,z)=(0,4)$ at $t=7.5$. The tower edge is now located at $x \sim 3.0$. The plasma $\beta$ in the core of the tower is $\beta \lesssim 0.1$. The [*top*]{} panel shows that the central total pressure (which is dominated by the magnetic pressure) is much bigger than the background “confining” pressure. This illustrates the original suggestion by Lynden-Bell [@L96; @L03] that the hoop stress of the toroidal field component $B_{\phi}$ can act as a pressure amplifier in the central region of the magnetic tower: the pinch effect amplifies $p_{m}$ near the axis of the tower. At the tower edge, however, a finite (albeit small) gas pressure is required [see also @Li01].
![\[fig:t7.5\_liner\] The radial profiles of physical quantities along the $x$-axis with $(y,\,z)=(0,\,4)$ at $t=7.5$. [*Top*]{}: Similar to the [*top*]{} panel in Fig. \[fig:t7.5\_linez\_2\]. [*Bottom*]{}: Similar to the [*bottom*]{} panel in Fig. \[fig:t6.0\_liner\], but with the centrifugal force $F_{\rm
c}=\rho V_{\phi}^2/r$ ([*dash-dotted line*]{}) and the Lorentz force $F_{{{\bm J}}\times {{\bm B}}}$, which can be separated into the magnetic pressure gradient force $-\partial/\partial r
\left[(B_{\phi}^2+B_{z}^2)/2\right]$ ([*light gray thick solid line*]{}) and the hoop stress (magnetic tension force) $-B_{\phi}^2/r$ ([*dark gray thick solid line*]{}). The position of the magnetic tower edge (tangential discontinuity) is shown in both panels. ](f10.eps)
![\[fig:t10.0\_liner\_B\] The radial profiles of the magnetic field components $(B_r,\,B_\phi,\,B_z)$, the poloidal magnetic field $B_{\rm p}=\sqrt{B_r^2+B_{z}^2}$, and the quantity $r B_\phi$ along the $x$-axis with $(y,\,z)=(0,\,7)$ at $t=10.0$. The position of the magnetic tower edge (tangential discontinuity) is shown. ](f11.eps)
The [*bottom*]{} panel shows the detailed distributions of forces along the radial direction. They show that:
- [(Region: $x \gtrsim 3.0$) Beyond the tower edge, the gravitational force $F_{\rm g}$ is slightly stronger than the outwardly directed gas pressure gradient force $F_{p}$, indicating that this edge is subject to the gravitational contraction, as discussed in the previous section.]{}
- [(Region: $2.0 \lesssim x \lesssim 3.0$) Interior to the tower edge, the Lorentz force $F_{{{\bm J}}\times {{\bm B}}}$]{} is dominated by the outwardly directed magnetic pressure gradient force $F_{p_{m}}$, and it is also larger than the inwardly directed $F_{p}$, indicating that the outer shell of the magnetic shell should be expanding at the relatively higher $z$, in contrast to the equatorial region ($z=0$) where the tower radius is contracting. The hoop stress $F_{\rm hp}$ plays a minor role in the force balance around this region. Note that ${{\bm J}}^{\rm R}$ flows inside this region.
- [(Region: $x \lesssim 2.0$) Inside the jet “body”, Contributions from both $F_{p_{m}}$ and $F_{\rm hp}$ become comparable and nearly cancel each other. The residual $F_{{{\bm J}}\times {{\bm B}}}$ is balanced by $F_{p}$ everywhere in this region. $F_{\rm hp}$ becomes dominant in the Lorentz force at the core part. Note that ${{\bm J}}^{\rm F}$ flows within $x \lesssim 1.0$.]{}
In addition, both the $F_{\rm g}$ and the centrifugal force $F_{\rm
c}$ play a minor role in terms of the force balance inside the magnetic tower jet. Thus, the interior of the magnetic tower is magnetically dominated but not exactly force-free, i.e., $- \nabla p +
{{\bm J}}\times {{\bm B}}\simeq 0~~$. This small but finite pressure gradient force could potentially provide some stabilizing effects on the traditionally kink-unstable twisted magnetic configurations. The detailed examinations of stability properties in magnetic tower jets will be discussed in a forthcoming paper.
DISCUSSIONS
===========
On scales of $\sim$ tens of kpc to hundreds of kpc, the background density and pressure profiles have a strong influence on the overall morphology of the magnetic tower jet. Most notably, the transverse size of the magnetic tower grows as the jet propagates into a decreasing pressure environment, showing a similar morphology with the jet/lobe configuration of radio galaxies.
The radial size of the lobe $r_{\rm lobe}$ can be estimated from the following consideration: Figure \[fig:t10.0\_liner\_B\] shows the radial profiles of magnetic field components parallel to the $x$-axis with $z=7.0$ at $t=10.0$. Together with Fig. \[fig:final\_Jz\], we see that both the poloidal field and especially the poloidal current $I_z$ maintain well collimated around the central axis even at the late stage of the evolution. This implies that the toroidal magnetic fields in the lobe region are distributed roughly as $B_\phi \sim
I_z/r$. This is consistent with the results shown in Fig. \[fig:t10.0\_liner\_B\] where $rB_\phi$ have a plateau from $x\approx 1-2.5$. As indicated in Fig. \[fig:t7.5\_liner\], we see that the magnetic pressure and the background pressure try to balance each other at the tower edge. Thus, we expect that $$\label{eq:balance_edge}
\frac{B_{\rm P}^{2}+B_{\phi}^{2}}{8 \pi} \sim p_{\rm e}~,$$ where $p_{\rm e}$ is the external gas pressure at the tower edge. When the lobe experiences sufficient expansion, we expect the poloidal field strength to drop much faster than $1/r_{\rm lobe}$. So we have $$\label{eq:balance_edge2}
\frac{B_{\phi}^{2}}{8 \pi} \sim \frac{(I_z/r_{\rm
lobe})^2}{8\pi}\sim p_{\rm e}~,$$ which gives $$r_{\rm lobe} \sim I_z~p_e^{-1/2}~~.$$ This is generally consistent with our numerical result shown in Fig. \[fig:de\_evo\] though it is difficult to make a firm quantitative determination since the lobes have not expanded sufficiently.
To make direct comparison between our simulations and observations, further analysis is clearly needed. Note that for the magnetic tower model, the Alfvén surface is located at the outer edge of the magnetic tower and flow within the magnetic tower is always sub-Alfvénic. This is quite different from the hydromagnetic models where the MHD flow is accelerated and has passed through several critical velocity surfaces, including the Alfvén surface.
CONCLUSION
==========
By performing 3-D MHD simulations we have investigated in detail the nonlinear dynamics of magnetic tower jets, which propagate through the stably stratified background that is initially in hydrostatic equilibrium. Our simulations, based on the approach developed in @Li06, confirm a number of the global characteristics developed in Lynden-Bell (1996, 2003) and Li et al. (2001). The results presented here give additional details for a dynamically evolving jet in a stratified background. The magnetic tower is made of helical magnetic fields, with poloidal flux and poloidal current concentrated around the central axis. The “returning” portion of the poloidal flux and current is distributed on the outer shell of the magnetic tower. Together they form a self-contained system with magnetic fields and associated currents, being confined by the ambient pressure and gravity. The overall morphology exhibits a confined magnetic jet “body” with an expanded “lobe”. The confinement of the “body” comes jointly from the external pressure and the gravity. The formation of the lobe is due to the expansion of magnetic fields in a decreasing background pressure.
A hydrodynamic shock wave initiated by the injection of magnetic energy/flux propagates ahead of the magnetic tower and can break the hydrostatic equilibrium of the ambient medium, causing a global gravitational contraction. As a result, a strong compression occurs in the axial direction between the magnetic tower front and the reverse slow-mode MHD shock wave front that follows. Furthermore, the magnetic tower jets are deformed radially into a slender-shaped body due to the inward-directed flow of the ambient (non-magnetized) gas.
The lobe is magnetically dominated and is likely filled with the toroidal magnetic fields generated by the central poloidal current. At the edge of the magnetic tower jet, the outward-directed magnetic pressure gradient force is balanced by the inward-directed gas pressure gradient force, so the radial width of the magnetic tower can be determined jointly by the magnitude of the poloidal current and the magnitude of the external gas pressure. The highly wound helical magnetic field in the magnetic tower never reaches the force-free equilibrium precisely, but obtains radial force-balance, including the gas pressure gradient inside the magnetic tower.
The stability of the magnetic tower jets will be considered in our forthcoming papers.
Useful discussions with John Finn, Stirling Colgate and Ken Fowler are gratefully acknowledged. This work was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. It was supported by the Laboratory Directed Research and Development Program at LANL and by IGPP at LANL.
Blandford, R. D., & Rees, M. J. 1974, , 169, 395 Blandford, R. D. 1976, , 199, 883 Blandford, R. D., & Znajek, R. L. 1977, , 179, 433 Blandford, R. D., & Payne, D. G. 1982, , 199, 883 Churazov, E., Forman, W., Jones, C., & Böhringer, H. 2003, , 590, 225 Ferrari, A. 1998, , 36, 539 Hsu, S. C., & Bellan, P. M. 2002, , 334, 257 Kato, Y., Hayashi, M. R., & Matsumoto, R. 2004, , 600, 338 Kato, Y., Mineshige, S., & Shibata, K. 2004, , 605, 307 King, I. 1962, , 67, 471 Kudoh, T., Matsumoto, R., & Shibata, K. 2002, , 54, 26 Lebedev, S. V. et al. 2005, , 361, 97 Li, H., Lovelace, R. V. E., Finn, J. M., & Colgate, S. A. 2001, , 561, 915 Li, H., Lapenta, G., Finn, J. M., Li, S., & Colgate, S. A. 2006, in press (astro-ph/0604469) Li, S., & Li, H. 2003, Technical Report, LA-UR-03-8935, Los Alamos National Laboratory Lovelace, R. V. E. 1976, , 262, 649 Lovelace, R. V. E., Li, H., Koldoba, A. V., Ustyugova, G. V., & Romanova, M. M. 2002, , 572, 445 Lovelace, R. V. E., & Romanova, M. M. 2003, , 596, L159 Lynden-Bell, D. 1996, , 279, 389 Lynden-Bell, D. 2003, , 341, 1360 Lynden-Bell, D. 2006, in press (astro-ph/0604424) Lynden-Bell, D., & Boily, C. 1994, , 267, 146 Meier, D. L., Koide, S., & Uchida, Y. 2001, Science, 291, 84 Romanova, M. M., Ustyugova, G. V., Koldoba, A. V., Chechetkin, V. M., & Lovelace, R. V. E. 1998, , 500, 703 Scheuer, P. A. G. 1974, , 166, 513 Turner, N. J., Bodenheimer, P., & Różyczka M. 1999, , 524, 12 Uchida, Y., & Shibata, K. 1985, , 37, 515 Ustyugova, G. V., Lovelace, R. V. E., Romanova, M. M., Li, H., Colgate, S. A. (2000), , 541, L21 Uzdensky, D. A., & MacFadyen, A. I. 2006, preprint (astro-ph/0602419)
[rlll]{} $R\,(= \sqrt{x^2+y^2+z^2})$& Length & $R_{0} $ & 5 Kpc\
${{{\bm V}}}$& Velocity field & $C_{\rm s0} $ & $4.6 \times 10^7$ cm/s\
$t$& Time & $R_{\rm 0}/C_{\rm s0} $ & $1.0 \times 10^7$ yrs\
$\rho$& Density & $\rho_{0} $ & $5.0 \times 10^{-27}$ g/cm$^3$\
$p$& Pressure & $\rho_{0} C_{\rm s0}^{2} $ & $1.4 \times 10^{-11}$ dyn/cm$^2$\
${{{\bm B}}}$& Magnetic field & $\sqrt{4 \pi \rho_{0} C^{2}_{\rm s0}} $ & 17.1 $\mu$G\
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Dark matter remains one of the most puzzling mysteries in Fundamental Physics of our times. Experiments at high-energy physics colliders are expected to shed light to its nature and determine its properties. This review talk focuses on recent searches for dark-matter signatures at the Large Hadron Collider, either within specific theoretical scenarios, such as supersymmetry, or in a model-independent scheme looking for mono-$X$ events arising in WIMP-pair production.'
address: |
Instituto de Física Corpuscular (IFIC), CSIC – Universitat de València,\
Parc Científic de la U.V., C/ Catedrático José Beltrán 2,\
E-46980 Paterna (Valencia), Spain
author:
- Vasiliki A Mitsou
title: Overview of searches for dark matter at the LHC
---
Introduction {#sc:intro}
============
Both Astroparticle and Particle Physics pursue the exploration of the nature of dark matter (DM) [@dm-review]. Among the long list of well-motivated candidates, the most popular particles are *cold* and weakly interacting, typically predicting missing-energy signals at particle colliders. Supersymmetry [@susy-dm] and models with extra dimensions [@ued-dm] are theoretical ideas that inherently provide such a dark matter candidate. High-energy colliders, such as the Large Hadron Collider [@lhc] at CERN, are ideal machines for producing and eventually detecting DM [@ijmpa].
The structure of this paper is as follows. Section \[sc:intro\] provides a brief introduction to the relevance of colliders, and in particular the LHC experiments ATLAS and CMS, for the production of dark matter. In Section \[sc:monox\], the strategy, methods, and results of the LHC experiments as far as model-independent DM-production is concerned are discussed. In Section \[sc:susy\], the latest results in searches for supersymmetry at the LHC are presented. The paper concludes with a summary and an outlook in Section \[sc:summary\].
Dark matter and colliders {#sc:dm}
-------------------------
The nature of the dark sector of the Universe constitutes one of the major mysteries in fundamental physics. According to recent observations from anisotropies of the cosmic microwave background made by the Planck mission team [@planck2], most of our Universe energy budget consists of unknown entities: $\sim\!26.8\%$ is dark matter and $\sim\!68.3\%$ is dark energy, a form of ground-state energy. Dark energy is believed to be responsible for the current-era acceleration of the Universe. Dark matter, on the other hand, is matter inferred to exist from gravitational effects on visible matter, being undetectable by emitted or scattered electromagnetic radiation.
Evidence from the formation of large-scale structure (galaxies and their clusters) strongly favour cosmologies where non-baryonic DM is entirely composed of cold dark matter (CDM), i.e. non-relativistic particles. CDM particles, in turn, may be weakly interacting massive particles (WIMPs), a class of DM candidates that arise naturally in models which attempt to explain the origin of electroweak symmetry breaking. Furthermore, the typical (weak-scale) cross sections characterising these models are of the same order of magnitude as the WIMP annihilation cross section, thus establishing the so-called *WIMP miracle*; this is precisely where the connection between Cosmology and Particle Physics lies [@mavromatos].
WIMP dark matter candidates include the lightest neutralino in models with weak-scale supersymmetry [@susy-dm], Kaluza-Klein photons arise in scenarios with universal extra dimensions (UED) [@ued-dm], while lightest $T$-odd particles are predicted in Little Higgs models [@little] with a conserved $T$-parity. The common denominator in these theories is that they all predict the existence of an electrically neutral, colorless and *stable* particle, whose decay is prevented by a kind of symmetry: -parity, connected to baryon and lepton number conservation in SUSY models; KK-parity, the four-dimensional remnant of momentum conservation in extra dimension scenarios; and a $Z_2$ discrete symmetry called $T$-parity in Little Higgs models.
Weakly interacting massive particles do not interact neither electromagnetically nor strongly with matter and thus, once produced, they traverse the various detectors layers without leaving a trace, just like neutrinos. However by exploiting the hermeticity of the experiments, we can get a hint of the WIMP presence through the balance of the energy/momentum measured in the various detector components, the so-called *missing energy*. In hadron colliders, in particular, since the longitudinal momenta of the colliding partons are unknown, only the *transverse missing energy*, , can be reliably used to ‘detect’ DM particles.
The ATLAS and CMS experiments at the LHC {#sc:lhc}
----------------------------------------
The Large Hadron Collider (LHC) [@lhc], situated at CERN, the European Laboratory for Particle Physics, outside Geneva, Switzerland, started its physics program in 2010 colliding two counter-rotating beams of protons or heavy ions. Before the scheduled 2013–2014 long shutdown, the LHC succeeded in delivering $\sim5~\ifb$ of integrated luminosity at centre-of-mass energy of $7~\tev$ during 2010–2011 and another $\sim23~\ifb$ at $\sqrt{s}=8~\tev$ in 2012. The LHC has already extended considerably the reach of its predecessor hadron machine, the Fermilab Tevatron, both in terms of instantaneous luminosity and energy, despite the fact that it has not arrived yet to its design capabilities.
The two general-purpose experiments, ATLAS (A Toroidal LHC ApparatuS) [@atlas-det] and CMS (Compact Muon Solenoid) [@cms-det], have been constructed and operate with the aim of exploring a wide range of possible signals of New Physics that LHC renders accessible, on one hand, and performing precision measurements of Standard Model (SM) parameters, on the other. It is worth mentioning that the MoEDAL [@moedal] experiment is specifically designed to explore high-ionisation signatures that may also arise in some theoretical scenarios of dark matter [@moedal-review].
The ATLAS [@atlas-det] and CMS [@cms-det] detectors are designed to overcome difficult experimental challenges: high radiation levels, large interaction rate and extremely small production cross sections of New Physics signals with respect to known SM processes. To this end, both experiments feature separate subsystems to measure charged particle momentum, energy deposited by electromagnetic showers from photons and electrons, energy from hadronic showers of strongly-interacting particles and muon-track momentum.
The most remarkable highlight of ATLAS and CMS operation so far is undoubtedly the discovery of a new particle [@higgs-disc] that so far seems to have all the features pinpointing to a SM(-like) Higgs boson [@higgs-prop]. The observation of this new boson has strong impact not only on our understanding of the fundamental interactions of Nature, as encoded in the SM, but on the proposed theoretical scenarios of Physics beyond the SM (BSM).
Model-independent DM production at the LHC {#sc:monox}
==========================================
Collider searches for dark matter are highly complementary to direct [@col-direct; @col-di-in] and indirect [@col-di-in; @col-indirect] DM detection methods. The main advantage of collider searches is that they do not suffer from astrophysical uncertainties and that there is no lower limit to their sensitivity on DM masses. The leading generic diagrams responsible for DM production [@tevatron1; @tevatron2] at hadron colliders involve the pair-production of WIMPs plus the initial- or final-state radiation (ISR/FSR) of a gluon, photon or a weak gauge boson $Z,\,W$. The ISR/FSR particle is necessary to balance the two WIMPs’ momentum, so that they are not produced back-to-back resulting in negligible . Therefore the search is based on selecting events high- events, due to the WIMPs, and a single jet, photon or boson candidate. The search results are interpreted in terms of a largely model-independent effective-field-theory framework, in which the interactions between a DM Dirac fermion and SM fermions $f$ are described by contact operators. Some of the possible operators are listed in Table \[tb:operators\]. In this framework, the interaction between SM and DM particles is determined by only two parameters, namely the DM-particle mass, , and the suppression scale, $M_*$, which is related to the mediator mass and to its coupling to SM and DM particles. The derived limits are independent of the theory behind the WIMP (SUSY, extra dimensions, etc), yet it *has* been assumed that other hypothetical particles are too heavy to be produced directly in $pp$ collisions. Henceforth, Dirac DM fermions are considered, however conclusions for Majorana fermions can also be drawn, since their production cross section only differs by a factor of two.
[@cccc@]{} Name & Initial state & Type & Operator\
D1 & $qq$ & scalar & $\frac{m_q}{M_*^3}\bar{\x}\x\bar{q}q$\
D5 & $qq$ & vector & $\frac{1}{M_*^2}\bar{\x}\gamma^{\mu}\x\bar{q}\gamma_{\mu}q$\
D8 & $qq$ & axial-vector & $\frac{1}{M_*^2}\bar{\x}\gamma^{\mu}\gamma^5\x\bar{q}\gamma_{\mu}\gamma^{\mu}q$\
D9 & $qq$ & tensor & $\frac{1}{M_*^2}\bar{\x}\sigma^{\mu\nu}\x\bar{q}\sigma_{\mu\nu}q$\
D11 & $gg$ & scalar & $\frac{1}{4M_*^3}\bar{\x}\x\alpha_s(G^s_{\mu\nu})^2$\
Monojet searches {#sc:monojet}
----------------
Event topologies with a single high- jet and large , hereafter referred to as *monojets*, constitute valuable probes of physics beyond the SM at the LHC. Both ATLAS [@atlas-monojet] and CMS [@cms-monojet] experiments have performed searches for an excess of monojet events over SM expectations in a wide range of signatures. The analyses outlined here use the full 2011 $pp$ LHC dataset at a centre-of-mass energy of $\sqrt{s} = 7~\tev$. The primary SM process yielding a true monojet final state is $Z$-boson production in association with a jet, where the $Z$ decays to two neutrinos. Other known processes acting as background in this search are $Z(\to\ell\ell)$+jets —with $\ell=e,\,\mu$—, +jets, $t\bar{t}$ as well as single-top events. All electroweak backgrounds and multijet events passing the selections criteria, as well as non-collision backgrounds, are in most cases determined by data-driven methods. Top and diboson backgrounds are estimated solely from Monte Carlo (MC) simulation.
The monojet analyses for ATLAS and CMS are based on some general requirements: large , with thresholds typically ranging from 120 to 500 and a energetic jet with a variable threshold higher than 110 that fulfils high jet-reconstruction quality criteria. Moreover, events with at least one electron or muon or a third jet are vetoed. Back-to-back dijet events are suppressed by requiring the subleading jet not to be collinear with $\bf p_{\bf T}^{\bf miss}$. The selected data are required to pass a trigger based on high (ATLAS) or large plus one high- jet (CMS).
The data, amounting to $\sim5~\ifb$, are found to be in agreement with the SM expectations. The results are interpreted in a framework of WIMP production with the simulated WIMP-signal MC samples corresponding to various assumptions of the effective field theory, as discussed previously. In this framework, the interaction between SM and DM particles are defined by only two parameters, namely the DM-particle mass, , and the suppression scale, $M_*$, which is related to the mediator mass and to its coupling to SM and DM particles.
Experimental and theoretical systematic uncertainties are considered when setting limits on the model parameters $M_*$ and . The experimental uncertainties on jet energy scale and resolution and on range from $1-20\%$ of the WIMP event yield, depending on the and thresholds and the considered interaction operator. Other experimental uncertainties include the ones associated with the trigger efficiency and the luminosity measurement. On the other hand, the parton-distribution-function set, the amount of ISR/FSR, and the factorisation and renormalisation scales assumed lead to theoretical uncertainties on the simulated WIMP signal.
![ATLAS lower limits at 90% CL on the suppression scale, $M_*$, for different masses of obtained with the monojet analysis for the operators D1 (top left), D5 (top right) and D8 (bottom). The region below the limit lines is excluded. All shown curves and areas are explained in the text. From Ref. [@atlas-monojet].\[fg:atlas-monojet-d\]](atlas-monojet-d1 "fig:"){width="48.00000%"} ![ATLAS lower limits at 90% CL on the suppression scale, $M_*$, for different masses of obtained with the monojet analysis for the operators D1 (top left), D5 (top right) and D8 (bottom). The region below the limit lines is excluded. All shown curves and areas are explained in the text. From Ref. [@atlas-monojet].\[fg:atlas-monojet-d\]](atlas-monojet-d5 "fig:"){width="48.00000%"}
![ATLAS lower limits at 90% CL on the suppression scale, $M_*$, for different masses of obtained with the monojet analysis for the operators D1 (top left), D5 (top right) and D8 (bottom). The region below the limit lines is excluded. All shown curves and areas are explained in the text. From Ref. [@atlas-monojet].\[fg:atlas-monojet-d\]](atlas-monojet-d8){width="48.00000%"}
From the limit on the visible cross section of new BSM physics processes, lower limits on the suppression scale as a function of the WIMP mass have been derived by the ATLAS Collaboration [@atlas-monojet]. The 90% confidence level (CL) lower limits for the D1, D5 and D8 operators are shown in Fig. \[fg:atlas-monojet-d\]. The observed limit on $M_*$ includes experimental uncertainties; the effect of theoretical uncertainties is indicated by dotted $\pm1\sigma$ lines above and below it. Around the expected limit, $\pm1\sigma$ variations due to statistical and systematic uncertainties are shown as a grey band. The lower limits are flat up to $\mx\lesssim100~\gev$ and become weaker at higher mass due to the collision energy. In the bottom-right corner of the $(\mx,\,M_*)$ plane (light-grey shaded area), the effective field theory approach is no longer valid. The rising lines correspond to couplings consistent with the measured thermal relic density [@tevatron1], assuming annihilation in the early universe proceeded exclusively via the given operator. Similar exclusion limits for all operators listed in Table \[tb:operators\] are given in Ref. [@atlas-monojet]. For the operator D1, the limits are much weaker ($\sim30~\gev$) than for other operators. Nevertheless, if heavy-quark loops are included in the analysis, much stronger bounds on $M_*$ can be obtained [@monojet-loop].
The observed limit on the dark matter-nucleon scattering cross section depends on the mass of the dark matter particle and the nature of its interaction with the SM particles. The limits on the suppression scale as a function of can be translated into a limit on the cross section using the reduced mass of -nucleon system [@tevatron2], which can be compared with the constraints from direct and indirect detection experiments, as we shall see at the end of this Section in conjunction with bounds acquired with other mono-$X$ analyses.
![\[fg:atlas-monojet-annih\]ATLAS 95% CL limits on WIMP annihilation rates $\langle\sigma v\rangle$ versus mass , inferred from the monojet analysis. Explanation of the shown curves is given in the text. From Ref. [@atlas-monojet].](atlas-monojet-annih){width="50.00000%"}
The ATLAS collider limits on vector (D5) and axial-vector (D8) interactions are also interpreted in terms of the relic abundance of WIMPs, using the same effective theory approach [@tevatron1]. The upper limits on the annihilation rate of WIMPs into light quarks, defined as the product of the annihilation cross section $\sigma$ and the relative WIMP velocity $v$ averaged over the WIMP velocity distribution $\langle\sigma v\rangle$, are shown in Fig. \[fg:atlas-monojet-annih\]. The results are compared to limits on WIMP annihilation to $b\bar{b}$, obtained from galactic high-energy gamma-ray observations, measured by the Fermi-LAT telescope [@fermilat]. Gamma-ray spectra and yields from WIMPs annihilating to $b\bar{b}$, where photons are produced in the hadronisation of the quarks, are expected to be very similar to those from WIMPs annihilating to light quarks [@annihilation]. Under this assumption, the ATLAS and Fermi-LAT limits can be compared, after scaling up the Fermi-LAT values by a factor of two to account for the Majorana-to-Dirac fermion adaptation. Again, the ATLAS bounds are especially important for small WIMP masses: below 10 for vector couplings and below about $100~\gev$ for axial-vector ones. In this region, the ATLAS limits are below the annihilation cross section needed to be consistent with the thermic relic value, keeping the assumption that WIMPs have annihilated to SM quarks only via the particular operator in question. For masses of $\mx\gtrsim 200~\gev$ the ATLAS sensitivity becomes worse than the Fermi-LAT one. In this region, improvements can be expected when going to larger centre-of-mass energies at the LHC.
The case in which the mediator is light enough to be accessible to the LHC has been considered too by the CMS experiment in a monojet search performed with $\sim20~\ifb$ at 8 [@cms-monojet-lm]. Figure \[fg:cms-light-mediator\] shows the observed limits on the contact interaction scale $\Lambda$ as a function of the mass of the mediator $M$, assuming vector interactions and a dark matter mass of 50 and 500 . The width $\Gamma$ of the mediator is varied between $M/3$ and $M/8\pi$ [@fox]. It shows the resonant enhancement in the production cross section once the mass of the mediator is within the kinematic range and can be produced on-shell. For $m_{\x}\gtrsim 100~\gev$, this approach is adequate and quite conservative in the bounds on $\Lambda$. For $m_{\x}\lesssim 100~\gev$, the collider bounds are considerably weaker. At large mediator masses, i.e. $M\gtrsim 5~\tev$, the limits on $\Lambda$ approximate to those obtained in the effective theory framework.
![\[fg:cms-light-mediator\]Observed limits on the scale $\Lambda$ as a function of the mass $M$, assuming vector interactions and a dark matter mass of $50~\gev$ (blue) and $500~\gev$ (red) in a CMS monojet analysis. The width of the mediator was varied between $M/3$, $M/10$ and $M/8\pi$. From Ref. [@cms-monojet-lm].](cms-light-mediator){width="50.00000%"}
Monophoton-based probes {#sc:monophoton}
-----------------------
In the same fashion as in the monojet searches, the *monophoton* analyses aim at probing dark matter requiring large —from the -pair production— and at least one ISR/FSR photon. Searches in monophoton events by ATLAS [@atlas-monophoton] and CMS [@cms-monophoton] also show an agreement with the SM expectations. The limits are derived in a similar fashion as for the monojet search, however the monophoton search is found to be somewhat less sensitive with respect to the monojet topology.
The primary (irreducible) background for a $\gamma+\met$ signal comes from $Z\gamma\to\nu\bar{\nu}\gamma$ production. This together with other SM backgrounds, including $W\gamma$, $W\to e\nu$, $\gamma+\text{jet}$ multijet, diphoton and diboson events, as well as backgrounds from beam halo and cosmic-ray muons, are taken into account in the analyses. The CMS analysis is based on singe-photon triggers, whilst ATLAS relies on high- triggered events. The photon candidate is required to pass tight quality and isolation criteria, in particular in order to reject events with electrons faking photons. The missing transverse momentum of the selected events should be as high as 150 (130 ) in the ATLAS (CMS) search. In CMS, events with a reconstructed jet are vetoed, while the ATLAS analysis rejects events with an electron, a muon or a second jet.
Both analyses, observe no significant excess of events over the expected background when applied on $\sim5~\ifb$ of $pp$ collision data at $\sqrt{s}=7~\tev$. Hence they set lower limits on the suppression scale, $M_*$ versus the DM fermion mass, , which in turn they are translated into upper limits on the nucleon-WIMP interaction cross section applying the prescription in Ref. [@tevatron1]. Figure \[fg:atlas-monophoton\] shows the 90% CL upper limits on the nucleon-WIMP cross section as a function of derived from the ATLAS search [@atlas-monophoton]. The results are compared with previous CDF [@cdf2], CMS [@cms-monojet; @cms-monophoton] and direct WIMP detection experiments [@xenon100a; @cdmsii; @cogent; @simple; @picasso2] results. The CMS limit curve generally overlaps the ATLAS curve.
![ATLAS 90% CL upper limits on the nucleon-WIMP cross section as a function of for spin-dependent (left) and spin-independent (right) interactions, corresponding to D8, D9, D1, and D5 operators, derived from the monophoton analysis. Explanation of the shown curves is given in the text. From Ref. [@atlas-monophoton].\[fg:atlas-monophoton\]](atlas-monophoton){width="85.00000%"}
The observed limits on $M_*$ typically decrease by 2% to 10% if the $-1\sigma$ theoretical uncertainty, resulting from the same sources as the one cited in the monojet analysis, is considered. This translates into a 10% to 50% increase of the quoted nucleon-WIMP cross section limits. To recapitulate, the exclusion in the region $1~\gev<\mx<1~\tev$ ($1~\gev<\mx<3.5~\gev$) for spin-dependent (spin-independent) nucleon-WIMP interactions is driven by the results from collider experiments, always under the assumption of the validity of the effective theory, and is still dominated by the monojet results.
Mono-$W$ and mono-$Z$ final states {#sc:monoWZ}
----------------------------------
As demonstrated in the previous sections, searches for monojet or monophoton signatures have yielded powerful constraints on dark matter interactions with SM particles. Other studies propose probing DM at LHC through a $pp\to\x\bar{\x}+W/Z$, with a leptonically decaying $W$ [@monow] or $Z$ [@monoz]. The final state in this case would be large and a single charged lepton (electron or muon) for the *mono-W* signature (*monolepton*) or a pair of charged leptons that reconstruct to the $Z$ mass for the *mono-Z* signature. In either case, the gauge boson radiations off a $q\bar{q}$ initial state and an effective field theory is deployed to describe the contact interactions that couple the SM particle with the WIMP.
In Ref. [@cms-monow], the existing $W'$ searches from CMS [@cms-wprime8] —which share a similar final state with mono-$W$ searches— are used to place a bound on mono-$W$ production at LHC, which for some choices of couplings are better than monojet bounds. This is illustrated in the left (right) panel of Fig. \[fg:cms-monolep\], where the spin-dependent (spin-independent) WIMP-proton cross section limits are drawn. The parameter $\xi$ parametrises the relative strength of the coupling to down-quarks with respect to up-quarks: $\xi=+1$ for equal couplings; $\xi=-1$ for opposite-sign ones; and $\xi=0$ when there is no coupling to down-quarks. Even in cases where the monoleptons do not provide the most stringent constraints, they provide an interesting mechanism to disentangle WIMP couplings to up-type versus down-type quarks. Such an interpretation has also been performed in Ref. [@monow] yielding similar limits.
![CMS monolepton search with 20 at 8 . Excluded proton-dark matter cross section for axial-vector-like, i.e. spin dependent (left), and vector-like, i.e. spin independent (right), for the combination of electron and muon channels. The CMS monojet result is for 20 of 2012 data [@cms-monojet-lm]. From Ref. [@cms-monow].\[fg:cms-monolep\]](cms-monolep-sd "fig:"){width="48.00000%"} ![CMS monolepton search with 20 at 8 . Excluded proton-dark matter cross section for axial-vector-like, i.e. spin dependent (left), and vector-like, i.e. spin independent (right), for the combination of electron and muon channels. The CMS monojet result is for 20 of 2012 data [@cms-monojet-lm]. From Ref. [@cms-monow].\[fg:cms-monolep\]](cms-monolep-si "fig:"){width="48.00000%"}
The ATLAS Collaboration has extended the range of possible mono-$X$ probes by looking for $pp\to\x\bar{\x}+W/Z$, when the gauge boson decays to two quarks [@atlas-monowz], as opposed to the leptonic signatures discussed so far. The analysis searches for the production of $W$ or $Z$ bosons decaying hadronically and reconstructed as a single massive jet in association with large from the undetected $\x\bar{\x}$ particles. For this analysis, the jet candidates are reconstructed using a filtering procedure referred to as *large-radius jets* [@fat-jets]. This search, the first of its kind, is sensitive to WIMP pair production, as well as to other DM-related models, such as invisible Higgs boson decays ($WH$ or $ZH$ production with $H\to\x\bar{\x}$).
![ATLAS-derived limits on -nucleon cross sections as a function of at 90% CL for spin-independent (left) and spin-dependent (right) cases, obtained with the mono-$W/Z$ analysis and compared to previous limits. From Ref. [@atlas-monowz].\[fg:atlas-monowz\]](atlas-monowz){width="80.00000%"}
Figure \[fg:atlas-monowz\] shows the 90% CL upper limits on the dark matter-nucleon scattering cross section as a function of the mass of DM particle for the spin-independent (left) and spin-dependent (right) models obtained by the ATLAS mono-$W/Z$ analysis [@atlas-monowz]. The new limits are also compared to the limits set by ATLAS in the $7~\tev$ monojet analysis [@atlas-monojet]. Limits from XENON100 [@xenon100b], CoGent [@cogent], CDMS II [@cdmsii], SIMPLE [@simple], COUPP [@coupp2], Picasso [@picasso2], IceCube [@icecube2] are superimposed for comparison. For the spin-independent case with the opposite-sign up-type and down-type couplings, the limits are improved by about three orders of magnitude. For other cases, the bounds are similar. Comparable limits have been obtained by the CMS experiment.
It is worth noting that the spin-dependent limits derived from the operator D9, give a smaller, hence better, bound on the WIMP-nucleon cross section throughout the range of , compared to direct DM experiments. In the spin-independent case the bounds from direct detection experiments are stronger for $\mx\gtrsim10~\gev$, whereas the collider bounds, acquired with the operator D5, get important for the region of low DM masses.
Searches for supersymmetry {#sc:susy}
==========================
Supersymmetry (SUSY) [@susy] is an extension of the Standard Model which assigns to each SM field a superpartner field with a spin differing by a half unit. SUSY provides elegant solutions to several open issues in the SM, such as the hierarchy problem and the grand unification. In particular, SUSY predicts the existence of a stable weakly interacting particle —the lightest supersymmetric particle (LSP)— that has the pertinent properties to be a dark matter particle, thus providing a very compelling argument in favour of SUSY [@lisboa].
SUSY searches in the ATLAS [@atlas-det] and CMS [@cms-det] experiments typically focus on events with high transverse missing energy, which can arise from (weakly interacting) LSPs, in the case of -parity conserving SUSY, or from neutrinos produced in LSP decays, if -parity is broken (c.f. Section \[sc:rpv\]). Hence, the event selection criteria of inclusive channels are based on large , no or few leptons ($e$, $\mu$), many jets and/or $b$-jets, $\tau$-leptons and photons. In addition, kinematical variables such as the transverse mass, , and the effective mass, , assist in distinguishing further SUSY from SM events, whilst the *effective transverse energy* [@alberto-ete] can be useful to cross-check results, allowing a better and more robust identification of the SUSY mass scale, should a positive signal is found. Although the majority of the analysis simply look for an excess of events over the SM background, there is an increasing application of distribution shape fitting techniques [@shape].
Typical SM backgrounds are top-quark production —including single-top—, $W$/$Z$ in association with jets, dibosons and QCD multijet events. These are estimated using semi- or fully data-driven techniques. Although the various analyses are optimised for a specific SUSY scenario, the interpretation of the results are extended to various SUSY models or topologies.
Analyses exploring $R$-parity conserving SUSY models at LHC are roughly divided into inclusive searches for squarks and gluinos, for third-generation fermions, and for electroweak production (pairs of $\tilde{\chi}^0$, $\tilde{\chi}^{\pm}$, $\tilde{\ell}$). Although these searches are designed and optimised to look for $R$-parity conserving SUSY, interpretation in terms of $R$-parity violating (RPV) models is also possible. Other analyses are purely motivated by oriented by RPV scenarios and/or by the expectation of long-lived sparticles. Recent summary results from each category of ATLAS and CMS searches are presented in this section.
Gluinos and first two generations of quarks {#sc:strong}
-------------------------------------------
At the LHC, supersymmetric particles are expected to be predominantly produced hadronically, i.e. through gluino-pair, squark-pair and squark-gluino production. Each of these (heavy) sparticles is going to decay into lighter ones in a cascade decay that finally leads to an LSP, which in most of the scenarios considered is the lightest neutralino . The two LSPs would escape detection giving rise to high transverse missing energy, hence the search strategy followed is based on the detection of high , many jets and possibly energetic leptons. The analyses make extensive use of data-driven Standard Model background measurements.
The most powerful of the existing searches are based on all-hadronic final states with large missing transverse momentum [@atlas-0l; @cms-0l]. In the 0-lepton search, events are selected based on a jet+ trigger, applying a lepton veto, requiring a minimum number of jets, high , and large azimuthal separation between the and reconstructed jets, in order to reject multijet background. In addition, searches for squark and gluino production in a final state with one or two leptons have been performed [@atlas-lep; @cms-lep]. The events are categorised by whether the leptons have higher or lower momentum and are referred to as the *hard* and *soft* lepton channels respectively. The soft-lepton analysis which enhances the sensitivity of the search in the difficult kinematic region where the neutralino and gluino masses are close to each other forming the so-called *compressed spectrum.* [@compressed] Leptons in the soft category are characterised by low lepton- thresholds ($6-10~\gev$) and such events are triggered by sufficient . Hard leptons pass a threshold of $\sim25~\gev$ and are seeded with both lepton and triggers. Analyses based on the *razor* [@razor] variable have also been carried out by both experiments [@atlas-razor; @cms-razor].
The major backgrounds ($t\bar{t}$, $W$+jets, $Z$+jets) are estimated by isolating each of them in a dedicated control region, normalising the simulation to data in that control region, and then using the simulation to extrapolate the background expectations into the signal region. The multijet background is determined from the data by a matrix method. All other (smaller) backgrounds are estimated entirely from the simulation, using the most accurate theoretical cross sections available. To account for the cross-contamination of physics processes across control regions, the final estimate of the background is obtained with a simultaneous, combined fit to all control regions.
![\[fg:msugra\]Exclusion limits at 95% CL for 8 ATLAS analyses [@atlas-0l; @atlas-msugra] in the $(m_0,\,m_{1/2})$ plane for the mSUGRA model. From Ref. [@atlas-susy-results].](atlas-msugra){width="80.00000%"}
In the absence of deviations from SM predictions, limits for squark and gluino production are set. Figure \[fg:msugra\] illustrates the 95% CL limits set by ATLAS under the minimal Supergravity (mSUGRA) model in the $(m_0,\,m_{1/2})$ plane [@atlas-0l; @atlas-msugra]. The remaining parameters are set to $\tan\beta = 30$, $A_0 = -2\,m_0$, $\mu > 0$, so as to acquire parameter-space points where the predicted mass of the lightest Higgs boson, $h^0$, is near $125~\gev$, i.e. compatible with the recently observed Higgs-like boson [@higgs-disc; @higgs-prop]. Exclusion limits are obtained by using the signal region with the best expected sensitivity at each point. By assumption, the mSUGRA model avoids both flavour-changing neutral currents and extra sources of $CP$ violation. For masses in the TeV range, it typically predicts too much cold dark matter, however these predictions depend of the presence of stringy effects that may dilute [@mavro] or enhance [@vergou] the predicted relic dark matter density. In the mSUGRA case, the limit on squark mass reaches 1750 and on gluino mass is 1400 .
Third-generation squarks {#sc:third}
------------------------
The previously presented limits from inclusive channels indicate that the masses of gluinos and first/second generation squarks are expected to be above 1 . Nevertheless, in order to solve the hierarchy problem in a *natural* way, the masses of the stops, sbottoms, higgsinos and gluinos have to be below the TeV-scale to properly cancel the divergences in the Higgs mass radiative corrections. Despite their production cross sections being smaller than for the first and second generation squarks, stop and sbottom may well be directly produced at the LHC and could provide the only direct observation of SUSY at the LHC in case the other sparticles are outside of the LHC energy reach. The lightest mass eigenstates of the sbottom and stop particles, $\t{b}_1$ and $\t{t}_1$, could hence be produced either directly in pairs or through $\t{g}$ pair production followed by $\t{g}\to\t{b}_1b$ or $\t{g}\to\t{t}_1t$ decays. Both cases will be discussed in the following.
For the aforementioned reasons, direct searches for third generation squarks have become a priority in both ATLAS and CMS. Such events are characterised by several energetic jets (some of them $b$-jets), possibly accompanied by light leptons, as well as high . A suite of channels have been considered, depending on the topologies allowed and the exclusions generally come with some assumptions driven by the shortcomings of the techniques and variables used, such as the requirement of 100% branching ratios into particular decay modes.
In the case of the gluino-mediated production of stops, a simplified scenario (“Gtt model”), where $\tilde{t}_1$ is the lightest squark but $m_{\tilde{g}} < m_{\tilde{t}_1}$, has been considered. Pair production of gluinos is the only process taken into account since the mass of all other sparticles apart from the $\tilde{\chi}_1^0$ are above the scale. A three-body decay via off-shell stop is assumed for the gluino, yielding a 100% branching ratio for the decay $\tilde{g}\rightarrow t\bar{t}\tilde{\chi}_1^0$. The stop mass has no impact on the kinematics of the decay and the exclusion limits [@cms-razor; @cms-gluino; @cms-ss2l] set by the CMS experiment are presented in the $(m_{\tilde{g}},m_{\tilde{\chi}_1^0})$ plane in Fig. \[fg:gl-ttlsp\]. For a massless LSP, gluinos with masses from 560 to 1320 are excluded. Similar results are obtained if the decay $\tilde{g}\rightarrow b\bar{b}\tilde{\chi}_1^0$ is considered instead, as shown in Fig. \[fg:gl-bblsp\].
![\[fg:gl-ttlsp\]Summary of observed and expected limits [@cms-razor; @cms-gluino; @cms-ss2l] for gluino pair production with gluino decaying via a 3-body decay to a top, an anti-top and a neutralino. From Ref. [@cms-susy-results].](cms-gl-ttlsp){width="50.00000%"}
![\[fg:gl-bblsp\]Summary of observed and expected limits [@cms-razor; @cms-gluino; @cms-ss2l] for gluino pair production with gluino decaying via a 3-body decay to a bottom, an anti-bopttom and a neutralino. From Ref. [@cms-susy-results].](cms-gl-bblsp){width="50.00000%"}
If the gluino is too heavy to be produced at the LHC, the only remaining possibility is the direct $\t{t}_1\t{t}_1$ and $\t{b}_1\t{b}_1$ production. If stop pairs are considered, two decay channels can be distinguished depending on the mass of the stop: $\t{t}_{1}\to b\t{\chi}_1^{\pm}$ and $\t{t}_{1}\to t\t{\chi}_1^0$. CMS and ATLAS carried out a wide range of different analyses in each of these modes at both 7 and 8 centre-of-mass energy. In all these searches, the number of observed events has been found to be consistent with the SM expectation. Limits have been set by ATLAS on the mass of the scalar top for different assumptions on the mass hierarchy scalar top-chargino-lightest neutralino [@atlas-tt], as shown in the left panel of Fig. \[fg:directstop\]. A scalar top quark of mass of up to 480 is excluded at 95% CL for a massless neutralino and a 150 chargino. For a 300 scalar top quark and a 290 chargino, models with a neutralino with mass lower than 175 are excluded at 95% CL.
![Summary of the dedicated ATLAS searches [@atlas-tt] for stop pair production based on $20-21~\ifb$ of $pp$ collision data taken at $\sqrt{s} = 8~\tev$, and $4.7~\ifb$ of $pp$ collision data taken at $\sqrt{s} = 7~\tev$. Exclusion limits at 95% CL are shown in the $(\t{t}_{1},\,\t{\chi}_1^0)$ mass plane for channels targeting $\t{t}_{1}\to b\t{\chi}_1^{\pm},\:\t{\chi}_1^{\pm}\to W^{\pm}\t{\chi}_1^0$ decays (left) and $\t{t}_{1}$ decays to $t\t{\chi}_1^0$ or $W b \t{\chi}_1^0$ or $c \t{\chi}_1^0$ (right). The dashed and solid lines show the expected and observed limits, respectively, including all uncertainties except the theoretical signal cross section uncertainty. From Ref. [@atlas-susy-results].\[fg:directstop\]](atlas-directstop-bchargino "fig:"){width="50.00000%"} ![Summary of the dedicated ATLAS searches [@atlas-tt] for stop pair production based on $20-21~\ifb$ of $pp$ collision data taken at $\sqrt{s} = 8~\tev$, and $4.7~\ifb$ of $pp$ collision data taken at $\sqrt{s} = 7~\tev$. Exclusion limits at 95% CL are shown in the $(\t{t}_{1},\,\t{\chi}_1^0)$ mass plane for channels targeting $\t{t}_{1}\to b\t{\chi}_1^{\pm},\:\t{\chi}_1^{\pm}\to W^{\pm}\t{\chi}_1^0$ decays (left) and $\t{t}_{1}$ decays to $t\t{\chi}_1^0$ or $W b \t{\chi}_1^0$ or $c \t{\chi}_1^0$ (right). The dashed and solid lines show the expected and observed limits, respectively, including all uncertainties except the theoretical signal cross section uncertainty. From Ref. [@atlas-susy-results].\[fg:directstop\]](atlas-directstop-tlsp "fig:"){width="50.00000%"}
For the case of a high-mass stop decaying to a top and neutralino ($\t{t}_{1}\to t\t{\chi}_1^0$), analyses requiring one, two or three isolated leptons, jets and large have been carried out. No significant excess of events above the rate predicted by the SM is observed and 95% CL upper limits are set on the stop mass in the stop-neutralino mass plane. The region of excluded stop and neutralino masses is shown on the right panel of Fig. \[fg:directstop\] for the ATLAS analyses. Stop masses are excluded between 200 and 680 for massless neutralinos, and stop masses around 500 are excluded along a line which approximately corresponds to neutralino masses up to 250 . It is worth noting that a monojet analysis with $c$-tagging is deployed to cover part of the low-$m_{\t{t}_1}$, low-$m_{\t{\chi}_1^0}$ region through the $\t{t}_1\to c \t{\chi}_1^0$ channel.
Electroweak gaugino production {#sc:gaugino}
------------------------------
If all squarks and gluinos are above the TeV scale, weak gauginos with masses of few hundred gigaelectronvolts may be the only sparticles accessible at the LHC. As an example, at $\sqrt{s}Ê= 7~\tev$, the cross-section of the associated production $\t{\chi}_1^{\pm}\t{\chi}_2^0$ with degenerate masses of 200 is above the 1- gluino-gluino production cross section by one order of magnitude. Chargino pair production is searched for in events with two opposite-sign leptons and using a jet veto, through the decay $\t{\chi}_1^{\pm} \to \ell^{\pm}\nu\t{\chi}_1^0$. A summary of related analyses [@cms-ewkino] performed by CMS is shown in Fig. \[fg:ewkino\]. Charginos with masses between 140 and 560 are excluded for a massless LSP in the chargino-pair production with an intermediate slepton/sneutrino between the $\t{\chi}_1^{\pm}$ and the $\t{\chi}_1^0$. If $\t{\chi}_1^{\pm}\t{\chi}_2^0$ production is assumed instead, the limits range from 11 to 760 . The corresponding limits involving intermediate $W$, $Z$ and/or $H$ are significantly weaker.
![\[fg:ewkino\]Summary of observed limits for electroweak-gaugino production from CMS [@cms-ewkino]. From Ref. [@cms-susy-results].](cms-ewkino){width="60.00000%"}
$R$-parity violating SUSY and meta-stable sparticles {#sc:rpv}
----------------------------------------------------
-parity is defined as: $R = (-1)^{3(B-L)+2S}$, where $B$, $L$ and $S$ are the baryon number, lepton number and spin, respectively. Hence $R=+1$ for all Standard Model particles and $R=-1$ for all SUSY particles. It is stressed that the conservation of -parity is an *ad-hoc* assumption. The only firm restriction comes from the proton lifetime: non-conservation of both $B$ and $L$ leads to rapid proton decay. -parity conservation has serious consequences in SUSY phenomenology in colliders: the SUSY particles are produced in pairs and the lightest SUSY particle is absolutely stable, thus providing a WIMP candidate. Here we highlight the status of RPV supersymmetry [@rpv] searches at the LHC.
Both ATLAS and CMS experiments have probed RPV SUSY through various channels, either by exclusively searching for specific decay chains, or by inclusively searching for multilepton events. ATLAS has looked for resonant production of $e\mu$, $e\tau$ and $\mu\tau$ [@atlas-rpv-emu], for multijets [@atlas-rpv-multijets], for events with at least four leptons [@atlas-rpv-4l] and for excesses in the $e\mu$ continuum [@atlas-rpv-emu-cont]. Null inclusive searches in the one-lepton channel [@atlas-brpv] have also been interpreted in the context of a model where RPV is induced through bilinear terms [@brpv].
Recent CMS analyses are focused on studying the lepton number violating terms $\lambda_{ijk}L_iL_j\bar{e}_{k}$ and $\lambda'_{ijk}L_iQ_j\bar{d}_{k}$, which result in specific signatures involving leptons in events produced in $pp$ collisions at LHC. A search for resonant production and the following decay of $\t{\mu}$ which is caused by $\lambda'_{211}\neq0$ has been conducted [@cms-rpv-ssmu]. Multilepton signatures caused by LSP decays due to various $\lambda$ and $\lambda'$ terms in stop production have been probed [@cms-rpv-stop]. Reference [@cms-rpv-4l] discusses the possibility of the generic model independent search for RPV SUSY in 4-lepton events. A summary of the limits set by several CMS analyses [@cms-ss2l; @cms-rpv-stop; @cms-rpv] are listed in Fig. \[fg:rpv\].
![Best exclusion limits for the masses of the mother particles, for RPV scenarios, for each topology, for all CMS results [@cms-ss2l; @cms-rpv-stop; @cms-rpv]. In this plot, the lowest mass range is $m_{\text{mother}}=0$, but results are available starting from a certain mass depending on the analyses and topologies. Branching ratios of 100% are assumed, values shown in plot are to be interpreted as upper bounds on the mass limits. From Ref. [@cms-susy-results].\[fg:rpv\]](cms-rpv){width="95.00000%"}
In view of the null results in other SUSY searches, it became mandatory to fully explore the SUSY scenario predicting meta-stable or long-lived particles. These particles, not present in the Standard Model, would provide striking signatures in the detector and rely heavily on a detailed understanding of its performance. In SUSY, non-prompt particle decay can be caused by (i) very weak RPV [@atlas-dv], (ii) low mass difference between a SUSY particle and the LSP [@atlas-kinked], or (iii) very weak coupling to the gravitino in GMSB models [@rhadrons; @nonp-phot]. A small part of these possibilities have been explored by the ATLAS [@atlas-susy-results] and CMS [@cms-susy-results] experiments covering specific cases, difficult to summarise here. There is still a wide panorama of signatures to be explored, in view of various proposed SUSY scenarios pointing towards this direction.
As a last remark, we address the issue of (not necessarily cold) dark matter in RPV SUSY models. These seemingly incompatible concepts *can* be reconciled in models with a gravitino [@rpv-grav; @trpv-gravitino] or an axino [@rpv-axino; @brpv-axino] LSP with a lifetime exceeding the age of the Universe. In both cases, RPV is induced by bilinear terms in the superpotential that can also explain current data on neutrino masses and mixings without invoking any GUT-scale physics [@brpv]. Decays of the next-to-lightest superparticle occur rapidly via RPV interaction, and thus they do not upset the Big-Bang nucleosynthesis, unlike the -parity conserving case. Such gravitino DM is proposed in the context of $\mu\nu$SSM [@munussm] with profound prospects for detecting $\gamma$ rays from their decay [@munussm-dm].
Recent evidence on the four-year Fermi data that have found excess of a 130 gamma-ray line from the Galactic Center [@fermi130] have been studied in the framework of $R$-parity breaking SUSY. A decaying axino DM scenario based on the SUSY KSVZ axion model with the bilinear $R$-parity violation explains the Fermi 130 gamma-ray line excess from the GC while satisfying other cosmological constraints [@brpv-axino]. On the other hand, gravitino dark matter with trilinear RPV —in particular models with the $LLE$ RPV coupling— can account for the gamma-ray line, since there is no overproduction of anti-proton flux, while being consistent with big-bang nucleosynthesis and thermal leptogenesis [@trpv-gravitino].
Summary and outlook {#sc:summary}
===================
The nature of dark matter remains one of the mysteries of Particle Physics and Cosmology. Mono-$X$ searches at the LHC provide strong constraints on dark matter properties in an effective field theory formalism. Colliders are superior to direct searches if dark matter is very light ($< 10~\gev$) or if interactions are spin-dependent. Extensive efforts are currently in progress on the validity of the effective field-theory approach, the proper comparison with the results from direct detection experiments and the use of simplified models with light mediators. Analyses looking for specific models providing DM candidates, such as Supersymmetry, are ongoing. Searches continue with the full 2012 dataset but a new discovery might eventually require more energy and more data coming up in 2015.
The author is grateful to the XIV MWPF organisers for the kind invitation and support that gave her the opportunity to present this plenary talk. She acknowledges support by the Spanish Ministry of Economy and Competitiveness (MINECO) under the projects FPA2009-13234-C04-01 and FPA2012-39055-C02-01, by the Generalitat Valenciana through the project PROMETEO II/2013-017 and by the Spanish National Research Council (CSIC) under the JAE-Doc program co-funded by the European Social Fund (ESF).
References {#references .unnumbered}
==========
[99]{}
For a pedagogical introduction, see e.g.: Profumo S 2013 TASI 2012 Lectures on Astrophysical Probes of Dark Matter [*Preprint*]{} arXiv:1301.0952 \[hep-ph\] Ellis J R, Falk T, Ganis G and Olive K A 2000 Supersymmetric dark matter in the light of LEP and the Tevatron collider [*Phys. Rev.*]{} D [**62**]{} 075010 ([*Preprint*]{} hep-ph/0004169)\
Drees M, Kim Y G, Nojiri M M, Toya D, Hasuko K and Kobayashi T 2001 Scrutinizing LSP dark matter at the CERN LHC [*Phys. Rev.*]{} D [**63**]{} 035008 ([*Preprint*]{} hep-ph/0007202)\
Chattopadhyay U and Nath P 2001 Upper limits on sparticle masses from $g-2$ and the possibility for discovery of SUSY at colliders and in dark matter searches [*Phys. Rev. Lett.*]{} [**86**]{} 5854 ([*Preprint*]{} hep-ph/0102157)\
Lahanas A B, Mavromatos N E and Nanopoulos D V 2003 WMAPing the universe: Supersymmetry, dark matter, dark energy, proton decay and collider physics [*Int. J. Mod. Phys.*]{} D [**12**]{} 1529 ([*Preprint*]{} hep-ph/0308251)\
Baer H, Belyaev A, Krupovnickas T and O’Farrill J 2004 Indirect, direct and collider detection of neutralino dark matter [*JCAP*]{} [**0408**]{} 005 ([*Preprint*]{} hep-ph/0405210)\
Allanach B C, Belanger G, Boudjema F and Pukhov A 2004 Requirements on collider data to match the precision of WMAP on supersymmetric dark matter [*JHEP*]{} [**0412**]{} 020 ([*Preprint*]{} hep-ph/0410091)\
Roszkowski L, Ruiz de Austri R and Choi K-Y 2005 Gravitino dark matter in the CMSSM and implications for leptogenesis and the LHC [*JHEP*]{} [**0508**]{} 080 ([*Preprint*]{} hep-ph/0408227\])\
Baer H, Mustafayev A, Profumo S, Belyaev A and Tata X 2005 Direct, indirect and collider detection of neutralino dark matter in SUSY models with non-universal Higgs masses [*JHEP*]{} [**0507**]{} 065 ([*Preprint*]{} hep-ph/0504001)\
Arbey A, Battaglia M and Mahmoudi F 2012 Light Neutralino Dark Matter in the pMSSM: Implications of LEP, LHC and Dark Matter Searches on SUSY Particle Spectra [*Eur. Phys. J.*]{} C [**72**]{} 2169 ([*Preprint*]{} arXiv:1205.2557 \[hep-ph\]) Hooper D and Profumo S 2007 Dark matter and collider phenomenology of universal extra dimensions [*Phys. Rept.*]{} [**453**]{} 29 ([*Preprint*]{} hep-ph/0701197) Evans L and Bryant P 2008 LHC Machine [*JINST*]{} [**3**]{} S08001 Mitsou V A 2013 Shedding Light on Dark Matter at Colliders [*Int. J. Mod. Phys.*]{} A [**28**]{} 1330052 ([*Preprint*]{} arXiv:1310.1072 \[hep-ex\]) Ade P A R [*et al.*]{} \[Planck Collab.\] 2013 Planck 2013 results. XVI. Cosmological parameters [*Preprint*]{} arXiv:1303.5076 \[astro-ph.CO\] Mavromatos N E 2012 Some Aspects of String Cosmology and the LHC [*Preprint*]{} arXiv:1210.0211 \[hep-ph\] in [*Proc. 1$^{\rm st}$ Int. Conf. on New Frontiers in Physics*]{}, to appear in [*Eur. Phys. J. Web Conf.*]{} Birkedal A, Noble A, Perelstein M and Spray A 2006 Little Higgs dark matter [*Phys. Rev.*]{} D [**74**]{} 035002 ([*Preprint*]{} arXiv:hep-ph/0603077) Aad G [*et al*]{}. \[ATLAS Collab.\] 2008 The ATLAS Experiment at the CERN Large Hadron Collider [*JINST*]{} [**3**]{} S08003 Adolphi R [*et al*]{}. \[CMS Collab.\] 2008 The CMS experiment at the CERN LHC [*JINST*]{} [**3**]{} S08004 MoEDAL Collaboration 2009 Technical Design Report of the MoEDAL Experiment [*CERN Preprint*]{} CERN-LHC-2009-006, MoEDAL-TDR-1.1
Acharya B [*et al.*]{} 2014 The Physics Program of the MoEDAL Experiment at the LHC, to appear in [*Int. J. Mod. Phys.*]{} A
Aad G [*et al.*]{} \[ATLAS Collab.\] 2012 Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC [*Phys. Lett.*]{} B [**716**]{} 1 ([*Preprint*]{} arXiv:1207.7214 \[hep-ex\])\
Chatrchyan S [*et al.*]{} \[CMS Collab.\] 2012 Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC [*Phys. Lett.*]{} B [**716**]{} 30 ([*Preprint*]{} arXiv:1207.7235 \[hep-ex\]) Chatrchyan S [*et al.*]{} \[CMS Collab.\] 2013 Study of the Mass and Spin-Parity of the Higgs Boson Candidate Via Its Decays to Z Boson Pairs [*Phys. Rev. Lett.*]{} [**110**]{} 081803 ([*Preprint*]{} arXiv:1212.6639 \[hep-ex\])\
Aad G [*et al.*]{} \[ATLAS Collab.\] 2013 Measurements of Higgs boson production and couplings in diboson final states with the ATLAS detector at the LHC [*Phys. Lett.*]{} B [**726**]{} 88 ([*Preprint*]{} arXiv:1307.1427 \[hep-ex\])\
Aad G [*et al.*]{} \[ATLAS Collab.\] 2013 Evidence for the spin-0 nature of the Higgs boson using ATLAS data [*Phys. Lett.*]{} B [**726**]{} 120 ([*Preprint*]{} arXiv:1307.1432 \[hep-ex\]) Baer H and Brhlik M 1998 Neutralino dark matter in minimal supergravity: Direct detection versus collider searches [*Phys. Rev.*]{} D [**57**]{} 567 ([*Preprint*]{} hep-ph/9706509)\
Pato M, Baudis L, Bertone G, Ruiz de Austri R, Strigari L E and Trotta R 2011 Complementarity of Dark Matter Direct Detection Targets [*Phys. Rev.*]{} D [**83**]{} 083505 ([*Preprint*]{} arXiv:1012.3458 \[astro-ph.CO\]) Rajaraman A, Shepherd W, Tait T M P and Wijangco A M 2011 LHC Bounds on Interactions of Dark Matter [*Phys. Rev.*]{} D [**84**]{} 095013 ([*Preprint*]{} arXiv:1108.1196 \[hep-ph\])\
Cheung K, Tseng P-Y, Tsai Y-L S and Yuan T-C 2012 Global Constraints on Effective Dark Matter Interactions: Relic Density, Direct Detection, Indirect Detection, and Collider [*JCAP*]{} [**1205**]{} 001 ([*Preprint*]{} arXiv:1201.3402 \[hep-ph\])\
Bauer D [*et al.*]{} 2013 Dark Matter in the Coming Decade: Complementary Paths to Discovery and Beyond [*Preprint*]{} arXiv:1305.1605 \[hep-ph\] Bertone G, Cerdeno D G, Fornasa M, Pieri L, Ruiz de Austri R and Trotta R 2012 Complementarity of Indirect and Accelerator Dark Matter Searches [*Phys. Rev.*]{} D [**85**]{} 055014 ([*Preprint*]{} arXiv:1111.2607 \[astro-ph.HE\]) Goodman J, Ibe M, Rajaraman A, Shepherd W, Tait T M P and Yu H-B 2010 Constraints on Dark Matter from Colliders [*Phys. Rev.*]{} D [**82**]{} 116010 ([*Preprint*]{} arXiv:1008.1783 \[hep-ph\]) Bai Y, Fox P J and Harnik R 2010 The Tevatron at the Frontier of Dark Matter Direct Detection [*JHEP*]{} [**1012**]{} 048 ([*Preprint*]{} arXiv:1005.3797 \[hep-ph\])\
Beltran M, Hooper D, Kolb E W, Krusberg Z A C and Tait T M P 2010 Maverick dark matter at colliders [*JHEP*]{} [**1009**]{} 037 ([*Preprint*]{} arXiv:1002.4137 \[hep-ph\]) Aad G [*et al.*]{} \[ATLAS Collab.\] 2013 Search for dark matter candidates and large extra dimensions in events with a jet and missing transverse momentum with the ATLAS detector [*JHEP*]{} [**1304**]{} 075 ([*Preprint*]{} arXiv:1210.4491 \[hep-ex\]) Chatrchyan S [*et al.*]{} \[CMS Collab.\] 2012 Search for dark matter and large extra dimensions in monojet events in $pp$ collisions at $\sqrt{s}=7~\tev$ [*JHEP*]{} [**1209**]{} 094 ([*Preprint*]{} arXiv:1206.5663 \[hep-ex\]) Haisch U, Kahlhoefer F and Unwin J 2013 The impact of heavy-quark loops on LHC dark matter searches [*JHEP*]{} [**1307**]{} 125 ([*Preprint*]{} arXiv:1208.4605 \[hep-ph\]) Ackermann M [*et al.*]{} \[Fermi-LAT Collab.\] 2011 Constraining Dark Matter Models from a Combined Analysis of Milky Way Satellites with the Fermi Large Area Telescope [*Phys. Rev. Lett.*]{} [**107**]{} 241302 ([*Preprint*]{} arXiv:1108.3546 \[astro-ph.HE\]) Bergstrom L, Ullio P and Buckley J H 1998 Observability of gamma-rays from dark matter neutralino annihilations in the Milky Way halo [*Astropart. Phys.*]{} [**9**]{} 137 ([*Preprint*]{} astro-ph/9712318)\
Cirelli M [*et al.*]{} 2011 PPPC 4 DM ID: A Poor Particle Physicist Cookbook for Dark Matter Indirect Detection [*JCAP*]{} [**1103**]{} 051 \[Erratum-ibid. 2012 [**1210**]{} E01\] ([*Preprint*]{} arXiv:1012.4515 \[hep-ph\]) CMS Collaboration 2012 Search for new physics in monojet events in $pp$ collisions at $\sqrt{s}=8~\tev$ [*CMS Note*]{} CMS-PAS-EXO-12-048 Fox P J, Harnik R, Kopp J and Tsai Y 2012 Missing Energy Signatures of Dark Matter at the LHC [*Phys. Rev.*]{} D [**85**]{} 056011 ([*Preprint*]{} arXiv:1109.4398 \[hep-ph\]) Aad G [*et al.*]{} \[ATLAS Collab.\] 2013 Search for dark matter candidates and large extra dimensions in events with a photon and missing transverse momentum in $pp$ collision data at $\sqrt{s}=7~\tev$ with the ATLAS detector [*Phys. Rev. Lett.*]{} [**110**]{} 011802 ([*Preprint*]{} arXiv:1209.4625 \[hep-ex\]) Chatrchyan S [*et al.*]{} \[CMS Collab.\] 2012 Search for Dark Matter and Large Extra Dimensions in pp Collisions Yielding a Photon and Missing Transverse Energy [*Phys. Rev. Lett.*]{} [**108**]{} 261803 ([*Preprint*]{} arXiv:1204.0821 \[hep-ex\]) Aaltonen T [*et al.*]{} \[CDF Collab.\] 2012 A Search for dark matter in events with one jet and missing transverse energy in $p\bar{p}$ collisions at $\sqrt{s} = 1.96~\tev$ [*Phys. Rev. Lett.*]{} [**108**]{} 211804 ([*Preprint*]{} arXiv:1203.0742 \[hep-ex\]) Aprile E [*et al.*]{} \[XENON100 Collab.\] 2011 Dark Matter Results from 100 Live Days of XENON100 Data [*Phys. Rev. Lett.*]{} [**107**]{} 131302 ([*Preprint*]{} arXiv:1104.2549 \[astro-ph.CO\]) Aalseth C E [*et al.*]{} \[CoGeNT Collab.\] 2011 Results from a Search for Light-Mass Dark Matter with a P-type Point Contact Germanium Detector [*Phys. Rev. Lett.*]{} [**106**]{} 131301 ([*Preprint*]{} arXiv:1002.4703 \[astro-ph.CO\]) Ahmed Z [*et al.*]{} \[CDMS-II Collab.\] 2010 Dark Matter Search Results from the CDMS II Experiment [*Science*]{} [**327**]{} 1619 ([*Preprint*]{} arXiv:0912.3592 \[astro-ph.CO\])\
Ahmed Z [*et al.*]{} \[CDMS-II Collab.\] 2011 Results from a Low-Energy Analysis of the CDMS II Germanium Data [*Phys. Rev. Lett.*]{} [**106**]{} 131302 ([*Preprint*]{} arXiv:1011.2482 \[astro-ph.CO\]) Felizardo M [*et al.*]{} 2012 Final Analysis and Results of the Phase II SIMPLE Dark Matter Search [*Phys. Rev. Lett.*]{} [**108**]{} 201302 ([*Preprint*]{} arXiv:1106.3014 \[astro-ph.CO\]) Archambault S [*et al.*]{} \[PICASSO Collab.\] 2012 Constraints on Low-Mass WIMP Interactions on $^{19}F$ from PICASSO [*Phys. Lett.*]{} B [**711**]{} 153 ([*Preprint*]{} arXiv:1202.1240 \[hep-ex\]) Bai Y and Tait T M P 2013 Searches with Mono-Leptons [*Phys. Lett.*]{} B [**723**]{} 384 ([*Preprint*]{} arXiv:1208.4361 \[hep-ph\]) Bell N F, Dent J B, Galea A J, Jacques T D, Krauss L M and Weiler T J 2012 Searching for Dark Matter at the LHC with a Mono-Z [*Phys. Rev.*]{} D [**86**]{} 096011 ([*Preprint*]{} arXiv:1209.0231 \[hep-ph\])\
Carpenter L M, Nelson A, Shimmin C, Tait T M P and Whiteson D 2013 Collider searches for dark matter in events with a $Z$ boson and missing energy [*Phys. Rev.*]{} D [**87**]{} 074005 ([*Preprint*]{} arXiv:1212.3352 \[hep-ex\]) CMS Collaboration 2013 Search for dark matter in the mono-lepton channel with $pp$ collision events at $\sqrt{s}=8~\tev$ [*CMS Note*]{} CMS-PAS-EXO-13-004 CMS Collaboration 2012 Search for new physics in the final states with a lepton and missing transverse energy at $\sqrt{s}=8~\tev$ [*CMS Note*]{} CMS-PAS-EXO-12-060 Aad G [*et al.*]{} \[ATLAS Collab.\] 2014 Search for dark matter in events with a hadronically decaying $W$ or $Z$ boson and missing transverse momentum in $pp$ collisions at $\sqrt{s}=8~\tev$ with the ATLAS detector [*Phys. Rev. Lett.*]{} [**112**]{} 041802 ([*Preprint*]{} arXiv:1309.4017 \[hep-ex\]) Aad G [*et al.*]{} \[ATLAS Collab.\] 2013 Performance of jet substructure techniques for large-$R$ jets in proton-proton collisions at $\sqrt{s} = 7~\tev$ using the ATLAS detector [*JHEP*]{} [**1309**]{} 076 ([*Preprint*]{} arXiv:1306.4945 \[hep-ex\]) Aprile E [*et al.*]{} \[XENON100 Collab.\] 2012 Dark Matter Results from 225 Live Days of XENON100 Data [*Phys. Rev. Lett.*]{} [**109**]{} 181301 ([*Preprint*]{} arXiv:1207.5988 \[astro-ph.CO\]) Behnke E [*et al.*]{} \[COUPP Collab.\] 2012 First Dark Matter Search Results from a 4-kg CF$_3$I Bubble Chamber Operated in a Deep Underground Site [*Phys. Rev.*]{} D [**86**]{} 052001 ([*Preprint*]{} arXiv:1204.3094 \[astro-ph.CO\]) Aartsen M G [*et al.*]{} \[IceCube Collab.\] 2013 Search for dark matter annihilations in the Sun with the 79-string IceCube detector [*Phys. Rev. Lett.*]{} [**110**]{} 131302 ([*Preprint*]{} arXiv:1212.4097 \[astro-ph.HE\]) Nilles H P 1984 Supersymmetry, Supergravity and Particle Physics [*Phys. Rept.*]{} [**110**]{} 1 Mitsou V A 2013 Experimental status of particle and astroparticle searches for supersymmetry [*J. Phys. Conf. Series*]{} [**447**]{} 012019 ([*Preprint*]{} arXiv:1304.1414 \[hep-ph\]) Cabrera M E and Casas J A 2012 Understanding and improving the Effective Mass for LHC searches [*Preprint*]{} arXiv:1207.0435 \[hep-ph\] Cabrera M E, Casas J A, Mitsou V A, Ruiz de Austri R and Terron J 2012 Histogram comparison as a powerful tool for the search of new physics at LHC. Application to CMSSM [*JHEP*]{} [**1204**]{} 133 ([*Preprint*]{} arXiv:1109.3759 \[hep-ph\]) Aad G [*et al.*]{} \[ATLAS Collab.\] 2013 Search for new phenomena in final states with large jet multiplicities and missing transverse momentum at $\sqrt{s}=8~\tev$ proton-proton collisions using the ATLAS experiment [*JHEP*]{} [**1310**]{} 130 ([*Preprint*]{} arXiv:1308.1841 \[hep-ex\]) Chatrchyan S [*et al.*]{} \[CMS Collab.\] 2012 Search for supersymmetry in hadronic final states using MT2 in $pp$ collisions at $\sqrt{s} = 7~\tev$ [*JHEP*]{} [**1210**]{} 018 ([*Preprint*]{} arXiv:1207.1798 \[hep-ex\])\
Chatrchyan S [*et al.*]{} \[CMS Collab.\] 2012 Search for new physics in the multijet and missing transverse momentum final state in proton-proton collisions at $\sqrt{s} = 7~\tev$ [*Phys. Rev. Lett.*]{} [**109**]{} 171803 ([*Preprint*]{} arXiv:1207.1898 \[hep-ex\]) Aad G [*et al.*]{} \[ATLAS Collab.\] 2012 Further search for supersymmetry at $\sqrt{s}=7~\tev$ in final states with jets, missing transverse momentum and isolated leptons with the ATLAS detector [*Phys. Rev.*]{} D [**86**]{} 092002 ([*Preprint*]{} arXiv:1208.4688 \[hep-ex\]) Chatrchyan S [*et al.*]{} \[CMS Collab.\] 2013 Search for supersymmetry in $pp$ collisions at $\sqrt{s}=7~\tev$ in events with a single lepton, jets, and missing transverse momentum [*Eur. Phys. J.*]{} C [**73**]{} 2404 ([*Preprint*]{} arXiv:1212.6428 \[hep-ex\])\
Chatrchyan S [*et al.*]{} \[CMS Collab.\] 2013 Search for supersymmetry in events with opposite-sign dileptons and missing transverse energy using an artificial neural network [*Phys. Rev.*]{} D [**87**]{} 072001 ([*Preprint*]{} arXiv:1301.0916 \[hep-ex\]) Baer H, Box A, Park E-K and Tata X 2007 Implications of compressed supersymmetry for collider and dark matter searches [*JHEP*]{} [**0708**]{} 060 ([*Preprint*]{} arXiv:0707.0618 \[hep-ph\])\
LeCompte T J and Martin S P 2012 Compressed supersymmetry after 1 at the Large Hadron Collider [*Phys. Rev.*]{} D [**85**]{} 035023 ([*Preprint*]{} arXiv:1111.6897 \[hep-ph\]) Rogan C 2010 Kinematical variables towards new dynamics at the LHC [*Preprint*]{} arXiv:1006.2727 \[hep-ph\] Chatrchyan S [*et al.*]{} \[CMS Collab.\] 2013 Inclusive search for supersymmetry using the razor variables in $pp$ collisions at $\sqrt{s}=7~\tev$ [*Phys. Rev. Lett.*]{} [**111**]{} 081802 ([*Preprint*]{} arXiv:1212.6961 \[hep-ex\]) Aad G [*et al.*]{} \[ATLAS Collab.\] 2013 Multi-channel search for squarks and gluinos in $\sqrt{s}=7~\tev$ $pp$ collisions with the ATLAS detector [*Eur. Phys. J.*]{} C [**73**]{} 2362 ([*Preprint*]{} arXiv:1212.6149 \[hep-ex\]) ATLAS Collaboration 2013 Search for strongly produced superpartners in final states with two same sign leptons with the ATLAS detector using 21 of proton-proton collisions at $\sqrt{s}=8~\tev$ [*ATLAS Note*]{} ATLAS-CONF-2013-007\
ATLAS Collaboration 2013 Search for Supersymmetry in Events with Large Missing Transverse Momentum, Jets, and at Least One Tau Lepton in 21 of $\sqrt{s} = 8~\tev$ Proton-Proton Collision Data with the ATLAS Detector [*ATLAS Note*]{} ATLAS-CONF-2013-026\
ATLAS Collaboration 2013 Measurement of multi-jet cross-section ratios and determination of the strong coupling constant in proton-proton collisions at $\sqrt{s}=7~\tev$ with the ATLAS detector [*ATLAS Note*]{} ATLAS-CONF-2013-041\
ATLAS Collaboration 2013 Search for strong production of supersymmetric particles in final states with missing transverse momentum and at least three $b$-jets using 20.1 of $pp$ collisions at $\sqrt{s} = 8~\tev$ with the ATLAS Detector [*ATLAS Note*]{} ATLAS-CONF-2013-061\
ATLAS Collaboration 2013 Search for squarks and gluinos in events with isolated leptons, jets and missing transverse momentum at $\sqrt{s} = 8~\tev$ with the ATLAS detector [*ATLAS Note*]{} ATLAS-CONF-2013-062 ATLAS Supersymmetry Searches Public Results:\
<https://twiki.cern.ch/twiki/bin/view/AtlasPublic/SupersymmetryPublicResults>
Lahanas A B, Mavromatos N E and Nanopoulos D V 2007 Smoothly evolving supercritical-string dark energy relaxes supersymmetric-dark-matter constraints [*Phys. Lett.*]{} B [**649**]{} 83 ([*Preprint*]{} hep-ph/0612152) Mavromatos N E, Sarkar S and Vergou A 2011 Stringy Space-Time Foam, Finsler-like Metrics and Dark Matter Relics [*Phys. Lett.*]{} B [**696**]{} 300 ([*Preprint*]{} arXiv:1009.2880 \[hep-th\])\
Mavromatos N E, Mitsou V A, Sarkar S and Vergou A 2012 Implications of a Stochastic Microscopic Finsler Cosmology [*Eur. Phys. J.*]{} C [**72**]{} 1956 ([*Preprint*]{} arXiv:1012.4094 \[hep-ph\]) Chatrchyan S [*et al.*]{} \[CMS Collab.\] 2013 Search for gluino mediated bottom- and top-squark production in multijet final states in $pp$ collisions at 8 [*Phys. Lett.*]{} B [**725**]{} 243 ([*Preprint*]{} arXiv:1305.2390 \[hep-ex\])\
CMS Collaboration 2013 Search for supersymmetry in $pp$ collisions at a center-of-mass energy of 8 in events with a single lepton, multiple jets and $b$-tags [*CMS Note*]{} CMS-PAS-SUS-13-007\
CMS Collaboration 2013 Search for supersymmetry in $pp$ collisions at $\sqrt{s} = 8~\tev$ in events with three leptons and at least one $b$-tagged jet [*CMS Note*]{} CMS-PAS-SUS-13-008 CMS Collaboration 2013 Search for new physics in events with same-sign dileptons and jets in $pp$ collisions at $\sqrt{s} = 8~\tev$ [*CMS Note*]{} CMS-PAS-SUS-13-013 CMS Supersymmetry Physics Results:\
<https://twiki.cern.ch/twiki/bin/view/CMSPublic/PhysicsResultsSUS>
Aad G [*et al.*]{} \[ATLAS Collab.\] 2012 Search for a supersymmetric partner to the top quark in final states with jets and missing transverse momentum at $\sqrt{s}=7~\tev$ with the ATLAS detector [*Phys. Rev. Lett.*]{} [**109**]{} 211802 ([*Preprint*]{} arXiv:1208.1447 \[hep-ex\])\
Aad G [*et al.*]{} \[ATLAS Collab.\] 2012 Search for direct top squark pair production in final states with one isolated lepton, jets, and missing transverse momentum in $\sqrt{s}=7~\tev$ $pp$ collisions using 4.7 of ATLAS data [*Phys. Rev. Lett.*]{} [**109**]{} 211803 ([*Preprint*]{} arXiv:1208.2590 \[hep-ex\])\
Aad G [*et al.*]{} \[ATLAS Collab.\] 2012 Search for light scalar top quark pair production in final states with two leptons with the ATLAS detector in $\sqrt{s}=7~\tev$ proton-proton collisions [*Eur. Phys. J.*]{} C [**72**]{} 2237 ([*Preprint*]{} arXiv:1208.4305 \[hep-ex\])\
Aad G [*et al.*]{} \[ATLAS Collab.\] 2012 Search for a heavy top-quark partner in final states with two leptons with the ATLAS detector at the LHC [*JHEP*]{} [**1211**]{} 094 ([*Preprint*]{} arXiv:1209.4186 \[hep-ex\])\
Aad G [*et al.*]{} \[ATLAS Collab.\] 2013 Search for light top squark pair production in final states with leptons and $b$-jets with the ATLAS detector in $\sqrt{s}=7~\tev$ proton-proton collisions [*Phys. Lett.*]{} B [**720**]{} 13 ([*Preprint*]{} arXiv:1209.2102 \[hep-ex\])\
Aad G [*et al.*]{} \[ATLAS Collab.\] 2013 Search for direct third-generation squark pair production in final states with missing transverse momentum and two $b$-jets in $\sqrt{s} =8~\tev$ $pp$ collisions with the ATLAS detector JHEP [**1310**]{} 189 ([*Preprint*]{} arXiv:1308.2631 \[hep-ex\])\
ATLAS Collaboration 2013 Search for direct production of the top squark in the all-hadronic $t\bar{t}+\met$ final state in 21 of $pp$ collisions at $\sqrt{s}=8~\tev$ with the ATLAS detector [*ATLAS Note*]{} ATLAS-CONF-2013-024\
ATLAS Collaboration 2013 Search for direct top squark pair production in final states with one isolated lepton, jets, and missing transverse momentum in $\sqrt{s}=8~\tev$ $pp$ collisions using 21 of ATLAS data [*ATLAS Note*]{} ATLAS-CONF-2013-037\
ATLAS Collaboration 2013 Search for direct top squark pair production in final states with two leptons in $\sqrt{s}=8~\tev$ $pp$ collisions using 20 of ATLAS data [*ATLAS Note*]{} ATLAS-CONF-2013-048\
ATLAS Collaboration 2013 Searches for direct scalar top pair production in final states with two leptons using the stransverse mass variable and a multivariate analysis technique in $\sqrt{s}=8~\tev$ $pp$ collisions using 20.3 of ATLAS data [*ATLAS Note*]{} ATLAS-CONF-2013-065\
ATLAS Collaboration 2013 Search for pair-produced top squarks decaying into charm quarks and the lightest neutralinos using 20.3 of $pp$ collisions at $\sqrt{s}=8~\tev$ with the ATLAS detector at the LHC [*ATLAS Note*]{} ATLAS-CONF-2013-068 CMS Collaboration 2013 Search for electroweak production of charginos, neutralinos, and sleptons using leptonic final states in $pp$ collisions at $\sqrt{s} = 8~\tev$ [*CMS Note*]{} CMS-PAS-SUS-13-006\
CMS Collaboration 2013 Search for electroweak production of charginos and neutralinos in final states with a Higgs boson in $pp$ collisions at $\sqrt{s} = 8~\tev$ [*CMS Note*]{} CMS-PAS-SUS-13-017 Barbier R [*et al.*]{} 2005 $R$-parity violating supersymmetry [*Phys. Rept.*]{} [**420**]{} 1 ([*Preprint*]{} hep-ph/0406039) Aad G [*et al.*]{} \[ATLAS Collab.\] 2011 Search for a heavy particle decaying into an electron and a muon with the ATLAS detector in $\sqrt{s}=7~\tev$ $pp$ collisions at the LHC [*Phys. Rev. Lett.*]{} [**106**]{} 251801 ([*Preprint*]{} arXiv:1103.5559 \[hep-ex\])\
Aad G [*et al.*]{} \[ATLAS Collab.\] 2011 Search for a heavy neutral particle decaying into an electron and a muon using 1 of ATLAS data [*Eur. Phys. J.*]{} C [**71**]{} 1809 ([*Preprint*]{} arXiv:1109.3089 \[hep-ex\])\
Aad G [*et al.*]{} \[ATLAS Collab.\] 2013 Search for a heavy narrow resonance decaying to $e\mu$, $e\tau$, or $\mu\tau$ with the ATLAS detector in $\sqrt{s} = 7~\tev$ $pp$ collisions at the LHC [*Phys. Lett.*]{} B [**723**]{} 15 ([*Preprint*]{} arXiv:1212.1272 \[hep-ex\]) Aad G [*et al.*]{} \[ATLAS Collab.\] 2012 Search for pair production of massive particles decaying into three quarks with the ATLAS detector in $\sqrt{s}=7~\tev$ $pp$ collisions at the LHC [*JHEP*]{} [**1212**]{} 086 ([*Preprint*]{} arXiv:1210.4813 \[hep-ex\]) Aad G [*et al.*]{} \[ATLAS Collab.\] 2012 Search for $R$-parity-violating supersymmetry in events with four or more leptons in $\sqrt{s}=7~\tev$ $pp$ collisions with the ATLAS detector [*JHEP*]{} [**1212**]{} 124 ([*Preprint*]{} arXiv:1210.4457 \[hep-ex\]) Aad G [*et al.*]{} \[ATLAS Collab.\] 2012 Search for lepton flavour violation in the $e\mu$ continuum with the ATLAS detector in $\sqrt{s}=7~\tev$ $pp$ collisions at the LHC [*Eur. Phys. J.*]{} C [**72**]{} 2040 ([*Preprint*]{} arXiv:1205.0725 \[hep-ex\]) Aad G [*et al.*]{} \[ATLAS Collab.\] 2012 Search for supersymmetry in final states with jets, missing transverse momentum and one isolated lepton in $\sqrt{s} = 7~\tev$ $pp$ collisions using 1 of ATLAS data [*Phys. Rev.*]{} D [**85**]{} 012006 ([*Preprint*]{} arXiv:1109.6606 \[hep-ex\])\
ATLAS Collaboration 2012 Search for supersymmetry at $\sqrt{s} = 7~\tev$ in final states with large jet multiplicity, missing transverse momentum and one isolated lepton with the ATLAS detector [*ATLAS Note*]{} ATLAS-CONF-2012-140 Romao J C, Diaz M A, Hirsch M, Porod W and Valle J W F 2000 A Supersymmetric solution to the solar and atmospheric neutrino problems [*Phys. Rev.*]{} D [**61**]{} 071703 ([*Preprint*]{} hep-ph/9907499)\
Hirsch M, Diaz M A, Porod W, Romao J C and Valle J W F 2000 Neutrino masses and mixings from supersymmetry with bilinear $R$ parity violation: A Theory for solar and atmospheric neutrino oscillations [*Phys. Rev.*]{} D [**62**]{} 113008 \[Erratum-ibid. 2002 D [**65**]{} 119901\] ([*Preprint*]{} hep-ph/0004115)\
Porod W, Hirsch M, Romao J and Valle J W F 2001 Testing neutrino mixing at future collider experiments [*Phys. Rev.*]{} D [**63**]{} 115004 ([*Preprint*]{} hep-ph/0011248)\
Diaz M A, Hirsch M, Porod W, Romao J C and Valle J W F 2003 Solar neutrino masses and mixing from bilinear $R$ parity broken supersymmetry: Analytical versus numerical results [*Phys. Rev.*]{} D [**68**]{} 013009 \[Erratum-ibid. 2005 D [**71**]{} 059904\] ([*Preprint*]{} hep-ph/0302021)\
Hirsch M and Valle J W F 2004 Supersymmetric origin of neutrino mass [*New J. Phys.*]{} [**6**]{} 76 ([*Preprint*]{} hep-ph/0405015) CMS Collaboration 2013 Search for RPV SUSY resonant second generation slepton production in same-sign dimuon events at $\sqrt{s}=7~\tev$ [*CMS Note*]{} CMS-PAS-SUS-13-005 Chatrchyan S [*et al.*]{} \[CMS Collab.\] 2013 Search for top squarks in $R$-parity-violating supersymmetry using three or more leptons and $b$-tagged jets [*Phys. Rev. Lett.*]{} [**111**]{} 221801 ([*Preprint*]{} arXiv:1306.6643 \[hep-ex\]) CMS Collaboration 2013 Search for RPV SUSY in the four-lepton final state [*CMS Note*]{} CMS-PAS-SUS-13-010 Chatrchyan S [*et al.*]{} \[CMS Collab.\] 2013 Search for pair production of third-generation leptoquarks and top squarks in $pp$ collisions at $\sqrt{s}=7~\tev$ [*Phys. Rev. Lett.*]{} [**110**]{} 081801 ([*Preprint*]{} arXiv:1210.5629 \[hep-ex\])\
CMS Collaboration 2012 Search for RPV supersymmetry with three or more leptons and $b$-tags [*CMS Note*]{} CMS-PAS-SUS-12-027\
CMS Collaboration 2012 Search for Light- and Heavy-flavour Three-Jet Resonances In Multijet Final States at 8 CMS-PAS-EXO-12-049 Aad G [*et al.*]{} \[ATLAS Collab.\] 2013 Search for long-lived, heavy particles in final states with a muon and multi-track displaced vertex in proton-proton collisions at $\sqrt{s}=7~\tev$ with the ATLAS detector [*Phys. Lett.*]{} B [**719**]{} 280 ([*Preprint*]{} arXiv:1210.7451 \[hep-ex\]) Aad G [*et al.*]{} \[ATLAS Collab.\] 2013 Search for direct chargino production in anomaly-mediated supersymmetry breaking models based on a disappearing-track signature in $pp$ collisions at $\sqrt{s}=7~\tev$ with the ATLAS detector [*JHEP*]{} [**1301**]{} 131 ([*Preprint*]{} arXiv:1210.2852 \[hep-ex\]) Aad G [*et al.*]{} \[ATLAS Collab.\] 2013 Searches for heavy long-lived sleptons and R-Hadrons with the ATLAS detector in $pp$ collisions at $\sqrt{s}=7~\tev$ [*Phys. Lett.*]{} B [**720**]{} 277 ([*Preprint*]{} arXiv:1211.1597 \[hep-ex\])\
Chatrchyan S [*et al.*]{} \[CMS Collab.\] 2013 Searches for long-lived charged particles in $pp$ collisions at $\sqrt{s}=7$ and 8 122 ([*Preprint*]{} arXiv:1305.0491 \[hep-ex\]) Aad G [*et al.*]{} \[ATLAS Collab.\] 2013 Search for non-pointing photons in the diphoton and final state in $\sqrt{s} = 7~\tev$ proton-proton collisions using the ATLAS detector [*Phys. Rev.*]{} D [**88**]{} 012001 ([*Preprint*]{} arXiv:1304.6310 \[hep-ex\])\
Chatrchyan S [*et al.*]{} \[CMS Collab.\] 2013 Search for long-lived particles decaying to photons and missing energy in proton-proton collisions at $\sqrt{s} = 7~\tev$ [*Phys. Lett.*]{} B [**722**]{} 273 ([*Preprint*]{} arXiv:1212.1838 \[hep-ex\]) Takayama F and Yamaguchi M 2000 Gravitino dark matter without $R$-parity [*Phys. Lett.*]{} B [**485**]{} 388 ([*Preprint*]{} hep-ph/0005214)\
Hirsch M, Porod W and Restrepo D 2005 Collider signals of gravitino dark matter in bilinearly broken $R$-parity [*JHEP*]{} [**0503**]{} 062 ([*Preprint*]{} hep-ph/0503059)\
Buchmuller W, Covi L, Hamaguchi K, Ibarra A and Yanagida T 2007 Gravitino Dark Matter in $R$-Parity Breaking Vacua [*JHEP*]{} [**0703**]{} 037 ([*Preprint*]{} hep-ph/0702184) Liew S P 2013 Gamma-ray line from radiative decay of gravitino dark matter [*Phys. Lett.*]{} B [**724**]{} 88 ([*Preprint*]{} arXiv:1304.1992 \[hep-ph\]) Chun E J and Kim H B 2006 Axino Light Dark Matter and Neutrino Masses with $R$-parity Violation [*JHEP*]{} [**0610**]{} 082 ([*Preprint*]{} hep-ph/0607076) Endo M, Hamaguchi K, Liew S P, Mukaida K and Nakayama K 2013 Axino dark matter with $R$-parity violation and 130 gamma-ray line [*Phys. Lett.*]{} B [**721**]{} 111 ([*Preprint*]{} arXiv:1301.7536 \[hep-ph\]) Lopez-Fogliani D E and Munoz C 2006 Proposal for a new minimal supersymmetric standard model [*Phys. Rev. Lett.*]{} [**97**]{} 041801 ([*Preprint*]{} hep-ph/0508297)\
Escudero N, Lopez-Fogliani D E, Munoz C and Ruiz de Austri R 2008 Analysis of the parameter space and spectrum of the $\mu\nu$SSM [*JHEP*]{} [**0812**]{} 099 ([*Preprint*]{} arXiv:0810.1507 \[hep-ph\])\
Ghosh P, Lopez-Fogliani D E, Mitsou V A, Munoz C and Ruiz de Austri R 2013 Probing the $\mu$ from $\nu$ supersymmetric standard model with displaced multileptons from the decay of a Higgs boson at the LHC [*Phys. Rev.*]{} D [**88**]{} 015009 ([*Preprint*]{} arXiv:1211.3177 \[hep-ph\]) Choi K-Y, Lopez-Fogliani D E, Munoz C and Ruiz de Austri R 2010 Gamma-ray detection from gravitino dark matter decay in the $\mu\nu$SSM [*JCAP*]{} [**1003**]{} 028 ([*Preprint*]{} arXiv:0906.3681 \[hep-ph\]) Bringmann T, Huang X, Ibarra A, Vogl S and Weniger C 2012 Fermi LAT Search for Internal Bremsstrahlung Signatures from Dark Matter Annihilation [*JCAP*]{} [**1207**]{} 054 ([*Preprint*]{} arXiv:1203.1312 \[hep-ph\])\
Weniger C 2012 A Tentative Gamma-Ray Line from Dark Matter Annihilation at the Fermi Large Area Telescope [*JCAP*]{} [**1208**]{} 007 ([*Preprint*]{} arXiv:1204.2797 \[hep-ph\])\
Tempel E, Hektor A and Raidal M 2012 Fermi 130 gamma-ray excess and dark matter annihilation in sub-haloes and in the Galactic centre [*JCAP*]{} [**1209**]{} 032 \[Addendum-ibid. 2012 [**1211**]{} A01\] ([*Preprint*]{} arXiv:1205.1045 \[hep-ph\])
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The Monge-Ampère equation arises in the theory of optimal transport. When more complicated cost functions are involved in the optimal transportation problem, which are motivated e.g. from economics, the corresponding equation for the optimal transportation map becomes a Monge-Ampère type equation. Such Monge-Ampère type equations are a topic of current research from the viewpoint of mathematical analysis. From the numerical point of view there is a lot of current research for the Monge-Ampère equation itself and rarely for the more general Monge-Ampère type equation. Introducing the notion of discrete $Q$-convexity as well as specifically designed barrier functions this purely theoretical paper extends the very recently studied two-scale method approximation of the Monge-Ampère itself [@NochettoNtogkasZhang2018] to the more general Monge-Ampère type equation as it arises e.g. in [@PhillippisFigalli2013] in the context of Sobolev regularity.'
address: 'Universität Duisburg-Essen, Fakultät für Mathematik, Mathematikcarrée, Thea-Leymann-Straße 9, 45127 Essen, Germany'
author:
- Heiko Kröner
title: 'Definition and certain convergence properties of a two-scale method for Monge-Ampère type equations'
---
Introduction {#intro}
============
The starting point and motivation on the very basic level for our paper is Monge’s transportation problem which is formulated in [@Monge1781]. Here we recall it by using its formulation from the introduction of [@FigalliKimMcCann2013]. Let $0\le f^{-}, f^{+}\in L^1(\mathbb{R}^n)$ be probability densities with respect to the Lebesgue measure $L^n$ on $\mathbb{R}^n$ and $c:\mathbb{R}^n\times \mathbb{R}^n\rightarrow [0, +\infty]$ a cost function. Then Monge’s optimal transport problem consists in finding a mapping $G:\mathbb{R}^n\rightarrow \mathbb{R}^n$ which pushes $d\mu^{+}=f^{+}dL^n$ forward to $d\mu^{-}=f^{-}dL^n$ and which minimizes the expected transportation cost $$\inf_{G_{\#}\mu^{+}=\mu^{-}} \int_{\mathbb{R}^n}c(x, G(x)) d \mu^{+}(x)$$ where $G_{\#}\mu^{+}=\mu^{-}$ means $\mu^{-}[Y]=\mu^{+}[G^{-1}(Y)]$ for each Borel set $Y \subset \mathbb{R}^n$. It is of interest under which conditions such a map $G$ exists and, furthermore, under which conditions such a map has a certain classical or Sobolev regularity, for details concerning this we refer to [@FigalliKimMcCann2013] and [@PhillippisFigalli2013] and the references to the literature therein. Under appropriate assumptions which are not stated here explicitly it turns out that the optimal transportation map $u$ satisfies the following Monge-Ampère type equation $$\label{1}
\begin{aligned}
\det \left(D^2u-A(x,Du)\right) = &f \quad \text{in }\Omega \\
u=& g, \quad \text{on } \partial \Omega
\end{aligned}$$ where $\Omega \subset \mathbb{R}^n$ is a bounded open set, $f> c_0$ where $c_0>0$ is a constant, $$\label{positivity}
D^2u-A(x, Du)>0$$ in $\Omega$ and $$\label{A_smooth}
\bar \Omega \times \mathbb{R}^n \ni (x,p) \mapsto A(x,p) \in \mathbb{R}^{n\times n}$$ is a $C^{\infty}$-smooth matrix valued function and $f\in C^0(\bar \Omega)$, $g\in C^0(\partial \Omega)$. Note that the assumed regularity for $A,f$ and $g$ is not minimal from the point of view of mathematical analysis but still quite low as well as challenging and interesting for a first approach with numerical analysis. In the next section we will present the most important examples of cost functions and derive in these special cases some properties for $A$ which result from these special cases. While basically working with a general $A$ in our paper we will for the sake of simplicity assume that $A$ satisfies these latter assumptions, see Section \[section2\].
Note that reduces to the classical Monge-Ampère equation when $A=0$.
As a survey and without claiming completeness we give the following list of references concerning approximation schemes for the Monge-Ampère equation [@1; @2; @3; @4; @5; @6; @7; @8; @9; @11; @12; @13]. We are not aware of any works about the finite element approximation error analysis for the Monge-Ampère type equation (\[1\]) with $A\neq 0$.
The first purpose of our paper is to adapt the two-scale method definition from [@NochettoNtogkasZhang2018] to a modified two-scale method for an approximation of the Monge-Ampère type equation . This is not completely straightforward and unique and we make an appropriate choice for the definition. Second and mainly we show convergence of the discrete solutions defined by our two-scale method to the solution of when the two discrete parameters go to zero as well as their quotients satisfy certain bounds. For it we make certain (regularity) assumptions for the solution of , cf. Remark \[regularity\_assumption\]. Basically the convergence proof is achieved following the strategy from [@NochettoNtogkasZhang2018] by using suitable barriers and comparison principles. The main advance and crucial difference of our paper from [@NochettoNtogkasZhang2018] is that we design completely new and much more complicated barrier functions. Apart from the barriers themselves the arguments are much more involved since we have to handle the terms arising from $A$. This becomes especially obvious from the fact that we have only a so called $\mod O(h)$ uniqueness in the comparison principle on the discrete level which is still non-trivial. Furthermore, our convergence result for the convergence of the discrete solutions to the solution of the original problem is different since we require certain regularity assumptions for the solution (Remark \[regularity\_assumption\]). We are aware that this assumption is strong from the view point of viscosity solutions and a kind of artificial. We are also aware that there are basically a large variety of numerical methods which are candidates which can be tested for our equation. This includes also methods working with regularization methods in order to achieve better regularity. The perspective and challenge of our paper is to test the beautiful theory developed in [@NochettoNtogkasZhang2018] in this more general case and to see how far that is possible. Hereby we include, of course, as main tool the barriers modeled on the exponential function instead of quadratic functions as in [@NochettoNtogkasZhang2018]. The exponential function as helping function is a very common tool in maths in general but here it should be analyzed what can be achieved by leaving the quite restricted class of quadratic functions.
Our paper is organized as follows. In section \[section2\] we introduce the assumptions on the cost function and discuss the two most important cases. In section \[section3\] we define our two-scale method. In section \[section4\] we derive a discrete comparison principle and uniqueness for the discrete equation $\mod O(h)$. In section \[section5\] we show existence of a discrete solution. In sections \[section6\] and \[section7\] we present some auxiliary facts. In section \[section8\] we study the convergence properties of the discrete model.
Assumptions on the cost function and the setting in general {#section2}
===========================================================
Here we first recall the setting from [@PhillippisFigalli2013] and [@FigalliKimMcCann2013] concerning the general setup for the optimal transport problem. Then we specify the cost function and derive further properties for the matrix function $A$ in these specific cases. These motivate further assumptions for the matrix function $A$ (in addition to those from the above mentioned and below described general setup) which we will assume throughout the paper.
The general setting in [@PhillippisFigalli2013] and [@FigalliKimMcCann2013] is motivated from the applications and the purpose to achieve certain regularity properties. We will present these assumptions in the following and will afterwards discuss the two most important examples of cost functions, especially it turns out that the corresponding matrices $A$ for these examples are smooth. Let $X\subset \mathbb{R}^n$ be an open set and $u: X\mapsto \mathbb{R}$ be a $c$-convex function, i.e., $u$ can be written as $$u(x) = \max_{y\in \bar Y}\{-c(x,y)+\lambda_y\}$$ for some open set $Y\subset \mathbb{R}^n$ and $\lambda_y\in \mathbb{R}$ for all $y\in \bar Y$. We are going to assume that $u$ is an Alexandrov solution of (\[1\]) inside some open set $\Omega \subset X$, i.e., $$|\partial^cu(E)| = \int_E f \quad \forall E\subset \Omega \quad \text{Borel},$$ where $$\partial^cu(E) := \bigcup_{x\in E}\partial^cu(x), \quad \partial^cu(x):=
\{y\in \bar Y:u(x)=-c(x,y)+\lambda_y\}$$ and $|F|$ denotes the Lebesgue measure of a set $F$. For $y\in \bar Y$ we define the contact set $$\Lambda_y:= \{x\in X: u(x) = -c(x,y)+\lambda_y\}.$$ Let $O\subset \subset Y$ be an open neighborhood of $\partial^cu(\Omega)$. We define $$|||c|||:= \|c\|_{C^3(\bar \Omega \times \bar O)}+\|D_{xxyy}c\|_{L^{\infty}(\bar \Omega\times \bar O)}$$ and assume
1. $|||c|||< \infty $
2. For every $x\in \Omega$ and $p:= -D_xc(x,y)$ with $y\in O$ it holds that $$D_{p_lp_k}A_{ij}(x,p)\xi_i\xi_j\eta_k\eta_l\ge 0 \quad \forall \xi, \eta \in \mathbb{R}^n, \quad \xi \cdot \eta =0$$ where $A$ is defined through $c$ by $$A_{ij}(x,p):= -D_{x_ix_j}c(x,y).$$
3. For every $(x,y)\in \Omega \times O$ the maps $x \in \Omega\mapsto -d_yc(x,y)$ and $y \in O \mapsto -D_xc(x,y)$ are diffeomorphisms on their respective ranges.
Special choices for the cost function arise from the applications, for a motivation of such choices in an economical context we refer to [@FigalliKimYoung-HeonMcCann2011]. Nevertheless, the two most relevant special cases for the cost function $c$ are the following functions $c=c_1$ and $c=c_2$, cf. [@FigalliKimMcCann2013], for which we will derive the mapping $A$ explicitly, namely $$c_1(x,y)=\frac{1}{2}|x-y|^2 \quad \text{and} \quad c_2(x,y)= -\log |x-y|.$$ For $c=c_1$ we have $$D_{x}c= x-y, \quad p=y-x$$ and hence $$A_{ij}(x,y-x)=-I,$$ or, equivalently, $$A_{ij}(x, \xi)=-I\quad \forall \xi.$$ For $c=c_2$ we have $$\begin{aligned}
D_xc=& -\frac{x-y}{|x-y|^2}, \\
D_{x_ix_j}c=& 2\frac{(x_i-y_i)(x_j-y_j)}{|x-y|^4}-\frac{\delta_{ij}}{|x-y|^2} \\
p=& -D_xc = \frac{x-y}{|x-y|^2}
\end{aligned}$$ and hence $$\begin{aligned}
A_{ij}\left(x, \frac{x-y}{|x-y|^2}\right) =& \frac{\delta_{ij}}{|x-y|^2}-2\frac{(x_i-y_i)(x_j-y_j)}{|x-y|^4},
\end{aligned}$$ or, equivalently, $$A_{ij}(x, \xi) = |\xi|^2\delta_{ij}-2\xi_i\xi_j \quad \forall \xi.$$ In these special cases the following assumption is valid.
\[ass1\] $A$ is $C^{\infty}$-smooth and in addition there holds $$\label{general_assumptions}
A(x,0) =0 \quad \forall x \in \bar \Omega \quad \text{or} \quad A=-I.$$
Motivated by these two special cases and since we need such properties for technical reasons we will assume throughout the paper that Assumption \[ass1\] holds.
Definition of the two-scale method for Monge-Ampère type equations {#section3}
==================================================================
In this section we adapt the definition of the two-scale method from [@LiNochetto2018] to the more general equation . Let $T_h=\{T_1, ..., T_N\}$, $h>0$, be a shape-regular and quasi-uniform mesh consisting of closed simplices $T_i$, $i=1, ..., N$, of diameter $ch$ where here and in the following $c$ denotes a generic constant which may vary from line to line. We furthermore denote $$\Omega_h = \operatorname{int}\left(\bigcup_{i=1}^{N}T_i\right),$$ let $N_h$ be the nodes of $T_h$ and write $N_h^b=\{x_i\in N_h: x_i \in \partial \Omega_h\}$ for the boundary nodes and $N_h^0=N_h\setminus N_h^b$ for the interior nodes. We furthermore assume that $\Omega$ is convex, that $N_h^b\subset \partial \Omega$ and denote the space of continuous functions on $\Omega_h$, which are linear on $T_i$ for every $i=1, ..., N$, by $V_h$. We denote the set of $n\times n$ matrices of real numbers by $\mathbb{R}^{n\times n}$ and the subset of orthogonal matrices by $O(n)$, furthermore, we write elements $V \in \mathbb{R}^{n\times n}$ by $V=(v_j)_{j=1}^d$ where $v_j$ are the columns of $V$ with respect to the standard basis in $\mathbb{R}^n$.
We denote the unit sphere in $\mathbb{R}^n$ by $S$ and for $\theta>0$ we let $S_{\theta}$ be a finite subset of $S$ with the property that $$\forall\ v \in S� \quad \exists\ v_{\theta}\in S_{\theta} : \quad |v-v_{\theta}| \le \theta.$$ Especially, we may assign to an element $V=(v_j)\in O(n)$ a matrix $V=(v_j^{\theta})$ where $v^{\theta}_j=(v_j)_{\theta}$ and denote the set of all such matrices by $O^{\theta}(n)$. Note that $O^{\theta}(n) \not \subset O(n)$ in general.
In addition to the meshsize $h$ which will serve as the fine scale in the remaining part of the paper we introduce in the following a coarse scale $\delta>h$ as a second discrete parameter which will serve as step size in difference quotients defining discrete derivatives. For $x_i \in N_h^0$ let $$\delta_i = \min\{\delta, \operatorname{dist}(x_i, \partial \Omega_h)\}$$ and note that $\delta_i \ge ch$ where $c$ does not depend on $h$ and that $B(x_i, \delta_i)\subset \Omega_h$. Here, $B(x_i, \delta_i)$ denotes the open ball of radius $\delta_i$ around $x_i$. For $w \in C^0(\overline{\Omega_h})$ we define the one-sided first order difference operator $$\nabla_{\delta}w(x_i, v_j)=\frac{w(x_i+\delta_iv_j)-w(x_i)}{\delta_i}$$ and the centered second order difference operator $$\label{2.1}
\nabla^2_{\delta}w(x_i; v_j)=\frac{w(x_i+\delta_iv_j)-2w(x_i)+w(x_i-\delta_iv_j)}{\delta_i^2}$$ for $x_i\in N_h^0$ and $v_j \in S_{\theta}$. Here we choose one-sidedness in the definition for the first order difference operator but remark that centered differences might also work. Altogether we have three discrete parameters which we will summarize as $$\varepsilon=(h, \delta, \theta)$$ where $h, \delta, \theta>0$ and $\delta >h$. To the two latter inequalities we will sometimes refer to by writing $\varepsilon>0$. For the following we will fix $\varepsilon$ for a while and will analyze the corresponding discrete model. Then later in a second step we will discuss the limit $\varepsilon \rightarrow 0$ and a necessary coupling between the parameters in order to achieve convergence of the solutions of the discrete equations to the solution of the original equation.
In the following definition we generalize the two-scale operator from [@LiNochetto2018].
\[defi\_1\] For $x_i \in N_h^0$ we define for any $w_h\in V_h$ $$\label{equ_1}
\begin{aligned}
T_{\varepsilon}[w_h](x_i):=&\min_{v^{\theta}\in O^{\theta}(n) }
\Big(\prod_{j=1}^d\left(\nabla_{\delta}^2w(x_i, v^{\theta}_j)-(v^{\theta}_j)^TA(x_i, \nabla_{\delta}w(x_i, e_k))v^{\theta}_j\right)^{+}\\
&-\sum_{j=1}^d\left(\nabla_{\delta}^2w(x_i, v^{\theta}_j)-(v^{\theta}_j)^TA(x_i, \nabla_{\delta}w(x_i, e_k))v^{\theta}_j\right)^{-}
\Big)
\end{aligned}$$ where two remarks are in order concerning our notation. Firstly, we write $(\cdot)^{+}=\max(\cdot, 0)$ and $(\cdot)^{-}=-\min(\cdot, 0)$ to denote the non-negative and non-positive part of $(\cdot)$, respectively. Secondly, we abbreviate $$\nabla_{\delta}w(x_i, e_k):=\left(\nabla_{\delta}w(x_i, e_k)\right)_{k=1}^d$$ where $w\in V_h$, $\delta>h$, $x_i\in N_h^0$ and $(e_k)_{k=1}^d$ denotes the canonical basis in $\mathbb{R}^d$ to simplify the notation in expression (\[equ\_1\]).
By using the discrete two-scale operator from Definition \[defi\_1\] we obtain the following discrete version of the Monge-Ampère type problem (\[1\]).
For a given (triple) $\varepsilon>0$ a two-scale method solution of (\[1\]) is a function $u_{\varepsilon}\in V_h$ such that $u_{\varepsilon}(x_i)=g(x_i)$ for all $x_i \in N^b_h$ and $$\label{discreteMongeAmpere}
T_{\varepsilon}[u_{\varepsilon}](x_i):= f(x_i)$$ for all $x_i \in N^0_h$.
In view of the widely used convention in numerical analysis to denote discrete solutions with the subscript $h$, i.e. $u_h$, we will write in the following ocassionally $u_h$ instead of $u_{\varepsilon}$.
Discrete $Q$-convexity, monotonicity and discrete comparison principle $\mod O(h)$ {#section4}
==================================================================================
To simplify the notation we use the following conventions. Firstly, in the setting from Definition \[defi\_1\] we will abbreviate in the following $$\label{def_Q}
Q(x_i, v_j)=Q_w(x_i, v_j)=\nabla_{\delta}^2w(x_i, v_j)-(v_j)^TA(x_i, \nabla_{\delta}w(x_i, e_k))v_j$$ so that (\[equ\_1\]) takes the form $$\label{equ_2}
\begin{aligned}
T_{\varepsilon}[w_h](x_i)=&\min_{v^{\theta}\in O^{\theta}(n) }
\Big(\prod_{j=1}^d\left(Q(x_i, v^{\theta}_j)\right)^{+}
-\sum_{j=1}^d\left(Q(x_i, v^{\theta}_j)\right)^{-}
\Big).
\end{aligned}$$ Secondly, when a variable ranges in a discrete set we sometimes emphasize this fact by adding a superscript to this variable which is linked to this discrete set, e.g. we write $v^{\theta} \in O^{\theta}(n)$ and $w_h \in V_h$ but equally $v\in O^{\theta}(n)$ and $w \in V_h$, respectively. Throughout this section we assume that (the triple) $\varepsilon>0$ is fixed.
\[definition\_1\] We say that $w_h \in V_h$ is discretely $Q$-convex if $$\label{equ_2_}
Q(x_i; v_j) \ge 0 \quad \forall x_i \in N^0_h, \quad \forall v_j \in O^{\theta}(n).$$
Note, that discrete $Q$-convexity of $w_h$ does not imply convexity of $w_h$ in general.
\[lemma\_1\] If $w_h \in V_h$ satisfies $$T_{\varepsilon}[w_h](x_i)\ge 0 \quad \forall x_i \in N_h^0,$$ then $w_h$ is discretely $Q$-convex and as a consequence $$T_{\varepsilon}[w_h](x_i) = \min_{v\in O^{\theta}(n)}\prod_{j=1}^dQ(x_i, v_j).$$
We distinguish two cases depending on whether $T_{\varepsilon}[w_h](x_i)>0$ or not. Let $v=(v_j)_{j=1}^d\in O^{\theta}(n)$ be a $d$-tuple that realizes the minimum in the definition of $T_{\varepsilon}[w_h](x_i)$ and note that $$\prod_{j=1}^dQ(x_i; v_j)^{+}\ge 0, \quad \sum_{j=1}^dQ(x_i; v_j)^{-}\ge 0.$$
\(i) Assume that $T_{\varepsilon}[w_h](x_i)>0$. The expression $T_{\varepsilon}[w_h](x_i)$ is defined as a product, cf. (\[equ\_2\]), so that its positivity implies the positivity of all its factors. These positive factors are differences of type $a-b$ of non-negative numbers $a$ and $b$ so that we also always necessarily have $a>0$. This implies that each quantity $Q(x_i; v_j)^{+}$ is also positive and hence the sum-term in (\[equ\_2\]) vanishes.
\(ii) Assume that $T_{\varepsilon}[w_h](x_i)=0$. Using again the representation from (\[equ\_2\]) of this expression as a difference of a product and a sum we make the following conclusion. If this product is positive then by (i) the sum vanishes and hence the product and the sum vanish so that $Q(x_i; v_j)= 0$ and the claim follows as well.
Note that Lemma \[lemma\_1\] and Definition \[definition\_1\] make formally sense when in (\[equ\_1\]) and (\[equ\_2\]) the superscript $\theta$ is omitted and Lemma \[lemma\_1\] is then even also true.
We need a definition in which we introduce a family of subspaces of $V_h$.
\[Definition\_V\_h\] For $\Lambda, h>0$ we define $$\label{def_V_h}
\begin{aligned}
V_h^{\Lambda} = \left\{
v_h \in V_h: \exists \eta \in C^{\infty}(\bar \Omega) , I_h\eta = v_h,
\|\eta\|_{C^2(\bar \Omega)}\le \Lambda\right\}
\end{aligned}$$ where $I_h$ denotes the usual Lagrange interpolation operator.
In the course of the paper it will turn out that when fixing a sufficiently large $\Lambda>0$ all considerations can be done (and will be done) for the sequence of discrete spaces $(V_h^{\Lambda})_{h>0}$. Here, $\Lambda$ will be chosen depending on the data of the problem, i.e. depending on $A$, $g$, $f$, $\Omega$ and the uniform (with respect to $h$) parameters of the triangulation. Interestingly, we only bound derivatives up to the [*second*]{} order in the definition of $V_h^{\Lambda}$. So arguments solely based on interpolation do not work hence they require at least bounds for the third derivative of the interpolating function.
\[remark\_1\] For the sake of a simplier notation we will write in the following again $V_h$ instead of $V_h^{\Lambda}$ with $\Lambda$ large and fixed. We will comment on $\Lambda$ where necessary.
As a consequence of Remark \[remark\_1\] we have the following discrete version of the fact that the derivative of a differentiable function vanishes in interior extremal points. Let $v_h\in V_h$, $v_j\in S_{\theta}$ and $z\in N_h^0$ be a maximum (or a minimum) of $v_h$ then $$\label{rolle}
\nabla_{\delta}v_h(z, v_j) = O(h).$$
In the next lemma we show that $T_{\varepsilon}$ is monotone$\mod\ O(h)$.
\[lemma10\] Let $u_h, w_h\in V_h$ be discretely $Q$-convex. If $u_h-w_h$ attains a maximum at an interior node $z \in N_h^0$ then $$T_{\varepsilon}[w_h]\ge T_{\varepsilon}[u_h]+O(h)$$ in $\Omega_h$. Here, the constant hidden in the $O(h)$-notation depends on $\Lambda$.
If $u_h-w_h$ attains a maximum at $z \in N_h^0$ then $$\label{equ1}
u_h(z)-w_h(z) \ge u_h(x_i)-w_h(x_i) \quad \forall x_i \in N_h.$$ Since $u_h$ and $w_h$ are piecewise linear this inequality can be generalized to $$u_h(z)-w_h(z) \ge u_h(x)-w_h(x) \quad \forall x \in \Omega_h.$$ Especially, evaluating this for difference quotients gives in view of (\[2.1\]) that $$\label{equ10}
\nabla_{\delta}^2u_h(z, v_j)\le \nabla_{\delta}^2w_h(z, v_j) \quad \forall v_j \in S_{\Theta}.$$ It remains to show that $$Q_{u_h}(z, v_j) \le Q_{w_h}(z, v_j)+O(h)$$ which can be reduced by (\[equ10\]) to $$\label{equ11}
(v_j)^TA(z, \nabla_{\delta}w_h(z, e_k))v_j \le (v_j)^TA(z, \nabla_{\delta}u_h(z, e_k))v_j+O(h).$$ But this follows since $$\nabla_{\delta}w_h(z, e_k) -\nabla_{\delta}u_h(z, e_k)=O(h),$$ cf. (\[rolle\]), and $$|\nabla_{\delta}w_h(x_i, e_k)|\le C,$$ cf. Definition \[Definition\_V\_h\], from the continuity of $A$.
We will use the following notation.
Given two functions $f_1=f_1(x,h)$ and $f_2=f_2(x,h)$ where $x\in S$ ranges in a certain parameter set $S$ as well as the discretization parameter $h>0$ we write $$f_1 \le f_2 \text{ for all } x\in S \mod O(h)$$ if there exists a constant $C>0$ which does not depend on $h$, $f_1$ or $f_2$ such that $$f_1(x,h) \le f_2(x,h) + Ch \text{ for all }x\in S \text{ and all } h>0.$$ When the parameter set $S$ is clear from the context we will not mention it explicitly.
We show the following discrete comparison principle, cf. Lemma \[comparison\]. Note that the inequality in the lemma includes the error term $O(h)$. This linear error order is not trivial in the context of a nonlinear second order operator and the use of first order finite elements for the following reason. If one derives the inequality in the following lemma firstly via a well-known comparison principle on the continuous level and later on transfers this by using interpolation estimates to the discrete level then usually third derivatives appear. But according to Definition \[Definition\_V\_h\] third derivatives of the interpolating functions may be arbitrary large and hence there appears an error term which is of the size of the product of the third derivative of an artificial interpolating function and $h$ and hence possibly large and especially larger than $O(h)$.
\[comparison\] Let $u_h, w_h \in V_h$ with $u_h\le w_h$ on the boundary $\partial \Omega_h$ be such that $$T_{\varepsilon}[u_h](x_i) \ge T_{\varepsilon}[w_h](x_i) > 0 \quad \forall x_i \in N_h^0. \label{equ12}$$ Then we have $u_h \le w_h$ in $\Omega_h$$\mod O(h)$.
Since $u_h, w_h \in V_h$, it suffices to prove $u_h(x_i)\le w_h(x_i)$ for all $x_i \in N_h^0$. In view of Lemma \[lemma\_1\] we may write inequality (\[equ12\]) as $$\label{equ13}
\min_{v\in O^{\theta}(n)}\prod_{j=1}^dQ_{u_h}(x_i, v_j) \ge \min_{v\in O^{\theta}(n)}\prod_{j=1}^dQ_{w_h}(x_i, v_j)> 0 \quad \forall x_i \in N_h^0.$$ Now we distinguish cases. For it we fix constants $C_1, C_2>0$ which depend only on the data of the problem, i.e. on $A$, $f$, $\Omega$, and which will be specified later.
\(i) Let us assume $$\label{equ13_}
\min_{v\in O^{\theta}(n)}\prod_{j=1}^dQ_{u_h}(x_i, v_j) -C_1 h> \min_{v\in O^{\theta}(n)}\prod_{j=1}^dQ_{w_h}(x_i, v_j) \quad \forall x_i \in N_h^0.$$ We argue by contradiction and assume that there is $x_k \in N_h^0$ such that $$\label{equ20}
u_h(x_k)-w_h(x_k)=\max_{x_i\in N_h^0}u_h(x_i)-w_h(x_i)> 0.$$ Similarly, as in Lemma \[lemma10\] we conclude that $$Q_{u_h}(x_k, v_j) \le Q_{w_h}(x_k, v_j)+C_3 h \quad \forall v_j \in S_{\theta}$$ where $C_3>0$ is a suitable constant which depends only on the data of the problem (and especially not on $h$). Taking the product on both sides and after this the infimum on the left-hand side of the equation yields $$\min_{v\in O^{\theta}(n)}\prod_{j=1}^dQ_{u_h}(x_k, v_j)\le \prod_{j=1}^dQ_{w_h}(x_k, \tilde v_j) +C_2h \quad \forall \tilde v \in O^{\theta}(n)$$ where $C_2$ is a suitable constant which depends only on $C_3$ and the data of the problem. W.l.o.g. we may also take the infimum over all $\tilde v\in O^{\theta}(n)$ on the right-hand side of the inequality.
Combining this with nequality (\[equ13\_\]) we obtain a contradiction provided $C_1$ is sufficiently large compared to $C_2$. This finishes case (i).
\(ii) Let us assume the other case, i.e. we have $$\label{equ14_}
\min_{v\in O^{\theta}(n)}\prod_{j=1}^dQ_{u_h}(x_i, v_j) -C_1 h \le \min_{v\in O^{\theta}(n)}\prod_{j=1}^dQ_{w_h}(x_i, v_j) \quad \forall x_i \in N_h^0$$ where now $C_1$ is fixed as it turned out to be necessary in case (i).
The strategy of the proof will now be as follows. We show that there are constants $h_0, C_4, C_5>0$ and an auxiliary function $q_h \in V_h$ which depend on the data of the problem such that $$\label{consequence_barrier}
\begin{aligned}
T_{\varepsilon}[u_h+\alpha q_h](x) > T_{\varepsilon}[w_h](x)+C_1h \\
\forall 0<h<h_0 \quad \quad \forall \alpha \ge C_4h \text{ sufficiently small}
\end{aligned}$$ and $$\|q_h\|_{L^{\infty}(\Omega)}\le C_5.$$ From this we conclude by using (i) that $$u_h \le w_h + C_4C_5h$$ and hence the claim.
We choose $\tilde x \in \mathbb{R}^n$ such that $\operatorname{dist}(\tilde x, \bar \Omega)\ge 1$, $\lambda>0$ large and define the strictly convex function $$\label{ansatz_function1}
q(x) = e^{\lambda |x-\tilde x|^2}-R$$ where $R=R(\lambda, \Omega)$ is so that $q\le 0$ in $\Omega$ and especially in $\bar \Omega_h$. We now define $$q_h=I_hq,$$ perform some relevant calculations on the level of $q$ instead of $q_h$ and translate them to $q_h$ afterwards by using the standard interpolation estimate $$\label{interpolation}
\|q_h-q\|_{C^m(\Omega_h)} \le c_{m,r} h^r \|q\|_{C^{m+r}(\bar \Omega)}, \quad m,r \in \mathbb{N},$$ where $c_{m,r}$ are suitable constants. We have $$\begin{aligned}
D_iq(x) =& 2\lambda (x_i-\tilde x_i) e^{\lambda |x-\tilde x|^2} \\
D_iD_jq(x) =& 2\lambda e^{\lambda|x-\tilde x|^2}\delta_{ij}+4 \lambda^2e^{\lambda |x-\tilde x|^2}(x_i-\tilde x_i)(x_j-\tilde x_j).
\end{aligned}$$ For $x \in N_h^0$ and $v \in O^{\theta}(n)$ we calculate $$\label{representation_Q}
\begin{aligned}
Q_{u_h+\alpha q}(x, v_r) =&
\nabla_{\delta}^2u_h(x, v_r)+\alpha \nabla_{\delta}^2q(x, v_r) \\
& - v_r^TA(x, \nabla_{\delta}u_h(x, e_k)+\alpha \nabla_{\delta}q(x, e_k))v_r \\
=& \nabla_{\delta}^2u_h(x, v_r)+\alpha\left(O(\delta)+D_{v_r}D_{v_r}q(x)\right)\\
&-v_r^TA(x, \nabla_{\delta}u_h(x, e_k)+\alpha D_kq(x)+\alpha O(\delta))v_r \\
=& \nabla_{\delta}^2u_h(x, v_r)+\alpha O(\delta) + 2 \alpha \lambda e^{\lambda |x-\tilde x|^2}\delta_{ij}v_{ri}v_{rj} \\
&+4 \alpha \lambda^2e^{\lambda |x-\tilde x|^2}(x_i-\tilde x_i)(x_j-\tilde x_j)v_{ri}v_{rj} \\
&-v_r^TA(x, \nabla_{\delta}u_h(x, e_k)+ 2 \alpha\lambda (x_k-\tilde x_k)e^{\lambda|x-\tilde x|^2}+\alpha O(\delta))v_r
\end{aligned}$$ Here, the constant hidden in the $O(\delta)$-notation depends on $q$, more precisely, on its higher order derivatives. Since $T_{\varepsilon}[u_h]=:f>0$ there is $\alpha_0>0$ such that $$\label{infimum_attained}
T_{\varepsilon}[u_h+\alpha q_h]>0$$ for all $\alpha \in (0, \alpha_0)$. Hence for these $\alpha$ we have $$T_{\varepsilon}[u_h+\alpha q_h](x) = \min_{v \in O^{\theta}(d)}\prod_{j=1}^dQ_{u_h+\alpha q_h}(x, v_j)$$ and therefore by abbreviating $Q(\alpha, j)=Q_{u_h+\alpha q_h}(x, v_j)$ (where the arguments $x$ and $v_j$ are assumed to be implicitly clear from the context) for $\alpha \in (0, \alpha_0)$ and $j\in \{1, ..., d\}$ we write $$\label{deriv_of_prod}
\begin{aligned}
\frac{d}{d\alpha}T_{\varepsilon}[u_h&+\alpha q_h](x)_{|\alpha=0} \\
=& \sum_{j=1}^dQ(0, 1)...
Q(0, j-1)\frac{d}{d\alpha}Q(\alpha, j)_{|\alpha=0}Q(0, j+1)...Q(0, d).
\end{aligned}$$ Note that the arguments $x$ and $v$ which appear here implicitly on the right-hand side are chosen obviously - namely as on the left-hand side of the equation as far as $x$ is concerned; the matrix $v$ is chosen so that in the point $x$ the infimum is attained in the definition (\[infimum\_attained\]). In order to evaluate (\[deriv\_of\_prod\]) we first calculate the derivative of the expression in (\[representation\_Q\]). Observe that there holds for all $k \in \{1, ..., d\}$ that $$\begin{aligned}
\frac{d}{d\alpha}Q(\alpha, k)
=& O(\delta)+2 \lambda e^{\lambda |x-\tilde x|^2}\delta_{ij}v_{ik}v_{jk} \\
&+ 4 \lambda^2e^{\lambda |x-\tilde x|^2}(x_i-\tilde x_i)(x_j-\tilde x_j)v_{ki}v_{kj}
\\
&-v_k^T \left(\frac{\partial A}{\partial p_l}(x, \nabla_{\delta}u_h(x, e_k))2 \lambda(x_l-\tilde x_l)e^{\lambda |x-\tilde x|^2}+O(\delta)\right)v_k.
\end{aligned}$$ Assuming that $\theta$ is sufficiently small we have $$(x_i-\tilde x_i)(x_j-\tilde x_j)v_{ki}v_{kj} \ge \frac{1}{2}|x-\tilde x|^2$$ for all $k \in \{1, ..., d\} \setminus \{k_0\} $ and all $v \in O^{\theta}(d)$ where $k_0=k_0(v)\in \{1, ..., d\}$ is suitable. Now having $\Lambda$ fixed in the definition of $V^{\Lambda}_h$ and choosing $0<h_0\le 1$ (at the moment not further specified) we may assume that $$\frac{d}{d\alpha}Q(\alpha, k)_{|\alpha=0} \ge \lambda^2e^{\lambda |x-\tilde x|^2}|x-\tilde x|^2>0$$ provided $\lambda>0$ is sufficiently large. Furthermore, for $\alpha_0=\alpha_0(\lambda)>0$ sufficiently small the quantities $Q( \alpha, k)$ are uniformly with respect to $k$ and with respect to $\alpha \in (-\alpha_0, \alpha_0)$ bounded by a positive constant from below. Hence we arrive at $$\frac{d}{d\alpha}T_{\varepsilon}[u_h+\alpha q_h](x)_{|\alpha=0}\ge \mu_0>0$$ with a suitable fixed $\mu_0>0$. An expansion of $T_{\varepsilon}[u_h+\alpha q_h](x)$ around 0 yields the existence of $\alpha \in (0, \alpha_0)$ such that $$T_{\varepsilon}[u_h+\alpha q_h](x)>T_{\varepsilon}[u_h](x)+ \frac{\alpha}{2}\mu_0.$$ Clearly, by assuming that $h_0$ is sufficiently small the above construction shows that we can realize property (\[consequence\_barrier\]) with this specific $\alpha$. Actually, we have in addition to choose $C_4>0$ but as long it is not too large it does not matter how we choose it. This finishes the proof.
Existence of discrete solutions {#section5}
===============================
We now prove uniqueness$\mod\ O(h)$ and existence of a discrete solution $u_{\varepsilon}\in V_h$ of (\[discreteMongeAmpere\]). Here, the uniqueness $\mod O(h)$ means, that given two discrete solutions $u^1_{\varepsilon}, u^2_{\varepsilon}$ of (\[discreteMongeAmpere\]) there holds $u^{i}_{\varepsilon}\le u^{j}_{\varepsilon}$ $\mod O(h)$ for all $i,j\in \{1,2\}$.
There exists $u_{\varepsilon}\in V_h$ which satisfies the discrete Monge-Ampère type equation (\[discreteMongeAmpere\]) and which is unique$\mod O(h)$. Furthermore, $\|u_{\varepsilon}\|_{L^{\infty}(\Omega)}$ does not depend on the parameter $\varepsilon=(h, \delta, \theta)$.
Let us fix $\varepsilon>0$. The uniqueness$\mod O(h)$ of a solution of (\[discreteMongeAmpere\]) follows from Lemma \[comparison\]. Hence it remains to show existence. For it we construct a special monotone sequence of discretely $Q$-convex subsolutions $\{u_{\varepsilon}^k\}_{k=0}^{\infty}$ of (\[discreteMongeAmpere\]) from which we will select a subsequence which converges to the desired discrete solution of (\[discreteMongeAmpere\]). The construction is by induction and works as follows.
\(i) [*Claim: There is $u_h^0 \in V_h$ such that $u_h^0=I_hg$ on $\partial \Omega_h$ and $$T_{\varepsilon}[u_h^0](x_i)\ge f(x_i) \quad \forall x_i \in N_h^0.$$* ]{}
[*Proof of the claim:*]{}
\(a) We give a short proof in the case that there is $C^{2,\alpha}$-regularity of the solution available. Let us assume that there is $0<\alpha<1$ such that the problem (\[1\]) with right-hand side $f$ replaced by $f+1$ has a solution $u \in C^{2,\alpha}(\bar \Omega)$. Setting $u_h^0=I_hu$ and using the interpolation estimates from [@NochettoNtogkasZhang2018 Lemma 4.1] as well as the continuity of $A$ yields the claim.
\(b) In the general case we proceed without using (regularity of) the solution of (\[1\]). Let $q$ denote the auxiliary function from with arbitrary choice of $R$, e.g. $R=0$. Let $w$ be a smooth function in a ball $B_{L}(0)$, $L>0$ large, let us say $2\bar \Omega \subset B_L(0)$, with $$w(x_i)=g(x_i)-q(x_i), \quad x_i \in N_h^b.$$ Such a function can easily be obtained by fixing it in $N_h^b$ and then extending it as smooth function to $B_L(0)$. But we would like to have that the size of $|Dw(x)|$ and $|D^2w(x)|$ is of order $ O(\lambda e^{\lambda |x-\tilde x|^2})$ and hence small compared to the order $O(\lambda^2 e^{\lambda |x-\tilde x|^2})$ which is the size of $D^2q(x)$. For it we define an artificial domain $\tilde \Omega \subset \mathbb{R}^d$ with smooth boundary $\partial \tilde \Omega$ passing through all elements of $N_h^b$, i.e. $N_h^b \subset \partial \tilde \Omega$. We extend $g-q$ from $N_h^b$ to a function $b\in C^{1, \alpha}(\partial \tilde \Omega)$ with $$\label{representation}
\|b\|_{C^{1, \alpha}_x(\partial \tilde \Omega)} \le \mu_x\lambda^2 e^{\lambda |x-\tilde x|^2}$$ for all $x \in \partial \tilde \Omega$ where we may and will choose here the constant $$0<\mu_x<\mu_0$$ with $\mu_0>0$ small. Here, we denote $$\|b\|_{C^{1, \alpha}_x(\partial \tilde \Omega)} = |b(x)| + \sum_{i=1}^d\|D_ib\|_{C^{0,\alpha}_x(\partial \tilde \Omega)}$$ where $$\|D_ib\|_{C^{0,\alpha}_x(\partial \tilde \Omega)} = |D_ib(x)|+\sup_{y \in \partial \tilde \Omega, y \neq x}\frac{|D_ib(x)-D_ib(y)|}{|x-y|^{\alpha}}, \quad x \in \partial \tilde \Omega.$$ To give derivatives (and their norms) of a function being defined on the hypersurface $\partial \tilde \Omega$ a sense we either consider these with respect to a fixed finite selection of local coordinate systems covering $\partial \tilde \Omega$ or with respect to an arbitrary but fixed extension to an open neighborhood of $\partial \tilde \Omega$ of the corresponding functions. In order to understand how the representation (\[representation\]) is possible, we explain this for the most non-trivial case, i.e. on the level of the Hölder norm of the derivative. Given a small choice for $\mu_x>0$, we estimate for $i\in \{1, ..., d\}$ and $x,y \in \partial \tilde \Omega$ that $$\begin{aligned}
\frac{|D_ib(x)-D_ib(y)|}{|x-y|^{\alpha}} =& \frac{|D_ib(x)-D_ib(y)|}{|x-y|^{\alpha}|x-y|^{1-\alpha}}|x-y|^{1-\alpha} \\
\approx& D^2 q(x) |x-y|^{1-\alpha} \\
\le& D^2q(x) \mu_x^{1-\alpha}
\end{aligned}$$ if $|x-y|\le \mu_x$. In the other case, i.e. when $|x-y|>\mu_x$, we estimate $$\begin{aligned}
\frac{|D_ib(x)-D_ib(y)|}{|x-y|^{\alpha}} \le&
\frac{|D_ib(x)|+|D_ib(y)|}{\mu_x^{\alpha}} \\
\le& O\left(\frac{\lambda}{\mu_x}e^{\lambda |x-\tilde x|^2}\right).
\end{aligned}$$ Then we solve the Dirichlet problem $$\begin{aligned}
\Delta w =&0 \quad \text{in } \tilde \Omega \\
w =&b \quad \text{on } \partial \tilde \Omega
\end{aligned}$$ and obtain by classical PDE-theory a solution $w \in C^{3, \alpha}\left(\overline{\tilde \Omega}\right)$ which satisfies the Schauder-estimate $$\label{schauder}
\|w\|_{C^{3,\alpha}(\overline{\tilde \Omega})} \le c \left(
\|w\|_{C^0\left(\overline{\tilde \Omega}\right)}+\|b\|_{C^{1, \alpha}(\partial \tilde \Omega)}\right).$$ Noting that $$\|w\|_{C^0_x(\tilde \Omega)} = O(q(x))$$ we conclude that $w$ satisfies the desired properties. Now we set $$u^0=w+q$$ and then $$u^0_{\varepsilon}:= I_hu^0.$$ By construction $u^0_{\varepsilon}$ has the correct boundary values, i.e. $u^0_{\varepsilon}(x_i)=g(x_i)$ when $x_i \in N_h^b$ and it satisfies $$T_{\varepsilon}[u^0_{\varepsilon}](x_i) \ge f(x_i)$$ for all $x_i \in N_h^0$ provided $\lambda$ is large in view of the interpolation error estimates in [@NochettoNtogkasZhang2018 Lemma 4.1]. Hence we have constructed $u_{\varepsilon}^0$ as desired. Note that we proved here a little bit more than needed. In order to apply [@NochettoNtogkasZhang2018 Lemma 4.1] it suffices to have only the Schauder estimate (\[schauder\]) on the level of $C^{2,\alpha}$ available. Furthermore, we remark that the construction can be done so that the $L^{\infty}$-norm of $u^0_{\varepsilon}$ can be estimated uniformly in $h$.
\(ii) We follow a Perron construction from [@NochettoNtogkasZhang2018] and use induction. First we label all interior nodes, let us say, $N_h^0=\{x_1, ..., x_m\}$, $m\in \mathbb{N}$. The induction begins with $u_h^0 \in V_h$ from (i). Let us assume we already have constructed $u_h^k\in V_h$ for some $k\in \mathbb{N}$ such that $$\label{properties}
\begin{aligned}
u_h^k \ge& u_h^0 \\
u_h^k(x_i) =& I_hg(x_i), \quad x_i \in N_h^b, \\
T_{\varepsilon}[u_h^k](x_i)\ge& f(x_i), \quad x_i \in N_h^0.
\end{aligned}$$ In order to construct $u_h^{k+1}\in V_h$ which satisfies $$\label{property_additional}
u_h^{k+1}\ge u_h^k$$ as well as the properties with $k$ replaced by $k+1$ we first define auxiliary functions $u_h^{k,i}\in V_h$, $i=0, ..., m$. We set $$u_h^{k,0}:=u_h^k.$$ Assume that $u_h^{k, i-1}\in V_h$ is already defined, $i\ge 1$. In order to define $u_h^{k,i}\in V_h$ we increase (only) the value of $u_h^{k,i-1}(x_i)$ (eventually) until $$\label{increasement}
T_{\varepsilon}[u_h^{k,i}](x_i)=f(x_i).$$ This defines $u_h^{k,i}$. The equality in can indeed be achieved under this process which becomes clear when we look at Lemma \[lemma\_1\] and (\[def\_Q\]). Noting that the centered second differences appearing in this definition of $Q$ are decreasing with slope $\frac{c}{h^2}$, $c$ a generic constant, with respect to the central value for all directions and that all other expressions therein change under this process at most by a rate of $\frac{c}{h}$ the equality in can clearly be achieved for $h$ sufficiently small. This process potentially increases the second centered differences at all the other nodes $x_j$, $j \neq i$ at a rate $\frac{c}{h^2}$ and changes lower order terms at most at a rate $\frac{c}{h}$. Hence $$T_{\varepsilon}[u_h^{k,i}](x_j)\ge T_{\varepsilon}[u_h^{k,i-1}](x_j) \ge f(x_j) \quad \forall j \neq i.$$ We repeat this process with the remaining nodes $x_j$ for $i<j\le m$ and set $$u_h^{k+1}:=u_h^{k,m}.$$ Note that the ’sufficient smallness’ of $h$ can be chosen here uniformly. Clearly, $u_h^{k+1}$ satisfies and .
\(iii) We derive an a priori $L^{\infty}$-bound for the sequence $(u_h^k)_{k\in \mathbb{N}}$. The lower bound for this sequence follows from the remarks at the end of steps (i) and (ii). Recall that by (\[general\_assumptions\]) we have $A(x, \cdot)=0$ or $A=-I$. The upper bound is chosen as follows. We set $\tilde b_h=\max_{x_i \in N_h^b}g(x_i) \in V_h$. In the case $A(x, \cdot)=0$ we set $b_h=\tilde b_h$ and in the case $A=-I$ we set $b_h=\tilde b_h +c(\Omega)- (1- \frac{1}{4} \min f)I_h|x|^2 $ where $c(\Omega)$ is a positive constant which depends on $\Omega$. Clearly, by the comparison principle $b_h$ is an upper barrier$\mod O(h)$ for the sequence $(u_h^k)_{k\in \mathbb{N}}$ and we are finished, note that we assume here that $h$ is small.
\(iv) Since $(u_h^k(x_i))_{k=1}^{\infty}$ is monotone and bounded from above for all $x_i \in N_h^0$ it converges and we set $$u_{\varepsilon}(x_i)=\lim_{k\rightarrow \infty}u_h^k(x_i)\quad \forall x_i \in N_h^0$$ and extend $u_{\varepsilon}$ without relabeling to $u_{\varepsilon}\in V_h$. Then we have $u_{\varepsilon}=I_hg$ on $\partial \Omega_h$ and $$\label{91}
T_{\varepsilon}[u_{\varepsilon}](x_i)\ge f(x_i)\quad \forall x_i \in N_h^0.$$ We show that even equality holds in (\[91\]) and assume for it that the inequality in (\[91\]) is strict for a certain $x_i \in N_h^0$. Then we find arbitrary large $k$ such that $$T_{\varepsilon}[u_h^k](x_i)>f(x_i).$$ But then in the construction of $u_h^{k+1}$ in step (ii) there was a certain ’space’ for increasement which contradicts that $(u_h^k(x_i))_k$ is especially pointwisely a Cauchy sequence. Hence we have shown existence of $u_{\varepsilon}$ as desired and we also have obtained an a priori $L^{\infty}$-bound which is independent from the discretization parameters.
Hereby, our analysis of the discrete model is finished. The remaining sections are concerned with the convergence analysis of the discrete solutions $u_{\varepsilon}$, i.e. they show that the discrete solutions $u_{\varepsilon}$ of (\[discreteMongeAmpere\]) converge to the solution $u$ of (\[1\]) when $\varepsilon \rightarrow 0$ and the relative size of $h$ and $\delta$ satisfies a certain relation provided the original equation satisfies appropriate properties. Recall that $\varepsilon = (h, \delta, \theta)$.
A special auxiliary function {#section6}
============================
We construct in the following lemma a special auxiliary function.
\[lemma\_5\_1\] Let $\Omega$ be uniformly convex, $h_0>0$ and $1<E\le c(\Omega, h_0)$ be sufficiently large within this range, $c(\Omega, h_0)$ a suitable constant which depends only on $\Omega$ and $h_0$ with $$c(\Omega, h_0) \rightarrow \infty$$ as $h_0 \rightarrow 0$ (this relation becomes more explicit in the proof). There exists a $h_0=h_0(\Omega)$ such that for all $0<h<h_0$ the following holds. For each node $z\in N_h^0$ and $\delta>0$ with $\operatorname{dist}(z, \partial \Omega_h)\le \delta$ there exists a function $p_h \in V_h$ and $E'>E$ such that $T_{\varepsilon}[p_h](x_i)\ge E'$ for all $x_i\in N_h^0$, $p_h\le 0$ on $\partial \Omega_h$ and $$|p_h(z)|\le CE'\delta$$ with $C$ depending on $\Omega$.
Let $z \in N_h^0$ and $\delta>0$ be arbitrary. Let $\tilde z \in \partial \Omega$ be a nearest boundary point, i.e. $$|z-\tilde z| = \operatorname{dist}(z, \partial \Omega).$$ Let $$0< \kappa_1(x) \le ... \le \kappa_n(x)$$ be the ordered-by-size $n$ principal curvatures of $\partial \Omega$ in $x\in \partial \Omega$ with respect to the outer unit normal of $\partial \Omega$ in $x$ (the convention is here as usual so that e.g. a unit sphere has principal curvatures equal to 1). In view of the uniform convexity of $\partial \Omega$ we have $$\kappa := \min_{\partial \Omega} \kappa_1 >0.$$ For the moment we fix a point $\tilde x \in \mathbb{R}^n$ and a large $\lambda>0$ and we will adjust them later appropriately. We define $$f(x) = e^{\lambda |x-\tilde x|^2}-e^{\lambda |\tilde z-\tilde x|^2}, \quad x \in \mathbb{R}^n.$$ The function $f$ looks roughly spoken like a bowl, attains a global minimum in $\tilde x$, is rotationally symmetric around $\tilde x$. Furthermore, it is strictly monotone increasing and strictly convex along rays starting from $\tilde x$ where in addition this convexity in radial direction at a point $x\in \mathbb{R}^n$ can be quantified as being of size $O(\lambda^2e^{\lambda |x-\tilde x|^2})$. Here, the constant hidden in the $O$-notation depends on $\Omega$ and $\tilde x$.
Let us now adjust $\tilde x \in \mathbb{R}^n$ where we assume w.l.o.g. that $\tilde x \notin \bar \Omega$ and choose $R>\frac{1}{\kappa}$ such that $$\partial \Omega \cap \partial B_R(\tilde x) = \{\tilde z\} \quad \text{and} \quad
\Omega \subset B_R(\tilde x).$$ Let $$c_1 = \max_{\bar \Omega}|x-\tilde x|, \quad c_2 = \min_{\bar \Omega}|x-\tilde x|�$$ then the second derivatives of $f$ in $\bar \Omega$ are of size at least $O(\lambda^2c_2^2e^{\lambda c_2^2})$ and the first derivatives are of size at most $O(c_1\lambda e^{\lambda c_1})$. We increase $\lambda$ until $O(\lambda^2c_2^2e^{\lambda c_2^2})$ is large compared to $E$ and $O(c_1\lambda e^{\lambda c_1})$. Then we set $$E' = \max\{O(\lambda^2c_2^2e^{\lambda c_2^2}), O(c_1\lambda e^{\lambda c_1})\}$$ as well as $$p_h=I_h(f-f(z)).$$ Now we may assume that $E$ and $E'$ are bounded by a constant which may become arbitrary large provided $h_0(\Omega)$ is correspondingly small so that for $0<h<h_0$ the previous interpolation does not produce relevant errors. This finishes the proof of the lemma. Note that we tacitly introduced concrete but generic constants in order to replace the $O$-notation in the context of inequalities.
An approximating problem {#section7}
========================
In the following remark we formulate some rather weak assumptions for equation and its solution which allow us to construct an approximating smooth problem with a smooth solution (which is also the main purpose of this section).
\[regularity\_assumption\]
1. (Regularity) $u\in C^{1}(\bar \Omega)$ is a viscosity solution of .
2. (Comparison principle one sided around the solution) There is $\varepsilon_1>0$ such that the following holds. Given continuous $\tilde f_1, \tilde f_2, \tilde g_1, \tilde g_2$ with $f\le \tilde f_2\le \tilde f_1 \le f+\varepsilon_1$ in $\Omega$ and $g-\varepsilon_1\le \tilde g_1\le \tilde g_2 \le g$ on $\partial \Omega$ and continuous viscosity solutions $\tilde u_1$ and $\tilde u_2$ of with respect to the data $\tilde f_1, \tilde g_1$ and $\tilde f_2, \tilde g_2$, respectively, then there holds a comparison principle in the usual sense, i.e. $\tilde u_1 \le \tilde u_2$ in $\Omega$.
Let $\Omega_n \supset \Omega$, $n\in \mathbb{N}$, be an approximation of $\Omega$ by smooth convex sets with respect to the Hausdorff distance $d_H$, i.e. $$0< \operatorname{dist}_H(\Omega, \Omega_n) \le \delta_n \rightarrow 0.$$ Let $p\in \Omega$ be arbitrary and fixed. The family of rays $$\{R_p=\{p+te: t\ge 0\}:e\in \mathbb{R}^n, \|e\|=1\}$$ clearly defines a bijection $$b_n: \partial \Omega \rightarrow \partial \Omega_n$$ by mapping $R_p \cap \partial \Omega$ to $R_p \cap \partial \Omega_n$. Let $f_n$ and $g_n$ be smooth functions in $\mathbb{R}^n$ approximating $f$ and $g$, respectively, such that $$\label{inequality}
f<f_n, \quad g_n<g,$$ $$|g(x)-g_n(b_n(x))|\le \delta_n, \quad |f(x)-f_n(y)|\le \delta_n$$ for all $$x \in \partial \Omega, y \in [x, b_n(x)]=\{tx+(1-t)b_n(x): 0\le t \le 1\}$$ and $$|f(x)-f_n(x)|\le \delta_n, \quad x \in \bar \Omega.$$ We note that the above approximations can be obtained in a standard fashion and indicate that especially inequalites (\[inequality\]) can be achieved by first replacing $f$ by $f+\frac{\delta_n}{2}$ and $g$ by $g-\frac{\delta_n}{2}$ and then extending and mollifying these modified functions.
Let $u_n\in C^{\infty}(\bar \Omega)$ be classical solutions of $$\label{Approx_1}
\det \left(D^2u_n-A(x, Du_n)\right) = f_n \quad \text{in }\Omega_n$$ and $$\label{Approx_2}
u_n=g_n \quad \text{on }\partial \Omega_n,$$ cf. the useful exposition in the introduction of [@FigalliKimYoung-HeonMcCann2011] for an overview of different assumptions and corresponding references leading to different regularities. Here, we mention especially the reference [@LiuTrudingerWang2010] mentioned on page 2 of [@FigalliKimYoung-HeonMcCann2011] for the smooth case: smooth data imply smoothness of the solution. In the next lemma we estimate $u_n$ on the boundary $\partial \Omega$ by constructing suitable barriers. We have the following plausible lemma which we will also prove rigorously in the following without using any a priori estimates for the solution. Our proof without using the last named type of estimates has the advantage that the lemma also holds when only the sufficient regularity without a priori estimates is available.
There holds $$|u_n-g|\rightarrow 0$$ uniformly on $\partial \Omega$ as $n \rightarrow \infty$.
Let us fix $z \in \partial \Omega$ and evaluate $g$ and $u_n$ at $z$ and compare them. Let $y$ be the closest point to $z$ in $\partial \Omega_n$, then $|z-y|\le \delta_n$ and given $\delta >0$ we have $$|g(z)-g_m(y)|\le |g(z)-g_m(z)|+|g_m(z)-g_m(y)|\le \delta$$ provided $m$ is sufficiently large and also $n=n(m)$ is sufficiently large. Let $p$ be the (not discrete) barrier function from the proof of Lemma \[lemma\_5\_1\] associated with $\Omega_n$, $z\in \Omega_n$, i.e. $$p(x) = e^{\lambda |x-\tilde x|^2}-e^{\lambda |\tilde z-\tilde x|^2}$$ where $\tilde x, \tilde z$ are chosen accordingly to the proof of Lemma \[lemma\_5\_1\] and we may arrange it so that $\tilde z$ equals the above specified $y$, i.e. $y=\tilde z$. We define the function $$b_m^{-}:= p(x)+g_m(y)-C_0|x-y|$$ where $C_0\ge \|g_m\|_{C^1(\bar \Omega)}$. Clearly, $b_m^{-}\le g_m$ in $\bar \Omega$ in view of $p\le 0$ in $\bar \Omega$ and we also have that $$T_{\varepsilon}[b_m^{-}]\ge f_m\quad \text{ in } \Omega_n$$ provided $\lambda$ is sufficiently large. Hence by the comparison principle we conclude that $$b_m^{-}\le u_m \quad \text{ in } \Omega_n.$$ Evaluating this inequality in $z$ and retranslation by using the definition of $b_m^{-}$ leads to $$g_m(y)-C_m\delta_n \le u_m(z)$$ where $C_m>0$ is a constant which may depend on $m$ (but not on $n$). Similarly, using $$b_m^{+}(x):=-p(x)+g_m(y)+C_0|x-y|$$ as upper barrier function for $u_m$ which is $Q$-convex on the one-dimensional line $$\bar \Omega \cap \{x_1=0\}$$ we conclude from the maximum principle in one variable that $b_m^{+}\ge u_m$ in $\bar \Omega$. Similarly as before we then get $$u_m(z) \le g_m(y)+C_m\delta_z.$$ Putting this by using the triangle inequality together we conclude that $$|g(z)-u_m(z)| \le |g(z)-g_m(y)|+|g_m(y)-u_m(z)| \le C_m\delta_n+\delta.$$
The following lemma gives the desired arbitrary good approximation of by smooth problems (i.e. with smooth data) with smooth solutions.
\[lemma7\] Let $f_n$, $g_n$, $\Omega_n$ and $u_n$ as before. Let $\varepsilon>0$ then there is $n \in \mathbb{N}$ such that $$|u_n-u|\le \varepsilon$$ in $\Omega$.
Let $q\le 0$ be the function from (\[ansatz\_function1\]) and $\alpha, \beta>0$ suitable constants which will be specified later. We consider the auxiliary function $$u^{-}:=u+\alpha q-\beta.$$ We observe that $$u^{-}\le u-\beta = g -\|g-g_n\|_{L^{\infty}(\partial \Omega)} \le g_n$$ on $\partial \Omega$ for $\beta= \|g-g_n\|_{L^{\infty}(\partial \Omega)}$. Let $\phi \in C^2(\Omega)$ and $x_0 \in \Omega$ be a point where $$u^{-}-\phi=u-(\phi-\alpha q+\beta)$$ attains a maximum. Abbreviating $$w=\phi - \alpha q + \beta \in C^2(\Omega)$$ and using that $u$ is a viscosity subsolution of we conclude that $$T[w]\ge f.$$ We would like to show that $$\label{goal_to_show}
T[\phi]\ge f_n$$ from which we deduce that $u^{-}$ is a viscosity subsolution of the problem , . For it we evaluate $T[\phi]$ more explicitly. As a tool we use the following straightforward and general relation. For positive numbers $a_1, ..., a_n, \varepsilon$ holds when setting $$\prod_{i=1}^na_i = z>0$$ that $$\label{deliberation}
\prod_{i=1}^n(a_i+\varepsilon)\ge \prod_{i=1}^na_i + \varepsilon^{n-1} \sum_{i=1}^na_i\ge z+\varepsilon^{n-1} z^{\frac{1}{n}}.$$ For fixed $x\in \bar \Omega$ we let $a_1, ..., a_n$ be the eigenvalues of $$D^2w(x)-A(x, Dw(x)).$$ From the min-max characterization of eigenvalues (given by the Courant-Fisher-Weyl maximum principle) we conclude that the ordered by size eigenvalues $\lambda_1 \le ... \le \lambda_n$ of $$D^2\phi(x)-A(x, D\phi(x))$$ satisfy $$\lambda_i \ge a_i+\varepsilon$$ for some $\varepsilon>0$ provided $\lambda$ in the definition of $p$ is sufficiently large. Hence we have $$T[\phi] \ge f+\varepsilon^{n-1} \min f^{\frac{1}{n}}$$ in view of our previous deliberation (\[deliberation\]). Clearly, we can achieve that holds. Since $u\in C^1(\bar \Omega)$ we may assume w.l.o.g. in the previous argumentation that $\|\phi\|_{C^1(\bar \Omega)}\le c(\|u\|_{C^1(\bar \Omega)})$. Hence the previous mechanism works for $\lambda$ sufficiently large depending only on $f$, $g$ and $\|u\|_{C^1(\bar \Omega)}$ and independently from the choice of $\alpha$. Hence we see that for $n$ sufficiently large we may choose $\alpha$ sufficiently small and the claim follows since $$u^{-} \le u_n \le u$$ and $$u-u^{-} = -\alpha q+\beta$$ can be made small for large $n$.
Convergence properties of the discrete solutions when the scales go to zero {#section8}
===========================================================================
Since $u_{\varepsilon}$ is defined in the computational domain $\Omega_h$ and $\Omega_h\subset \Omega$, we extend $u_{\varepsilon}$ to $\Omega$ as follows. Given $x\in \Omega\setminus \Omega_h$ we choose $z \in \partial \Omega_h$ as the nearest point in $\Omega_h$ to $x$ which is unique because $\Omega_h$ is convex and let $$\label{boundary_definition}
u_{\varepsilon}(x):= u_{\varepsilon}(z)=I_hg(z) \quad \forall x \in \Omega\setminus \Omega_h.$$
In the following theorem we prove convergence of the discrete solutions to the solution of the original problem.
Let $\Omega$ be uniformly convex, $f, g \in C(\bar \Omega)$ and $f> 0$ in $\bar \Omega$. Let $u$ be a solution of satisfying the assumptions in Remark \[regularity\_assumption\]. The discrete solutions $u_{\varepsilon}$ of and converge uniformly to $u$ as $\varepsilon=(h, \delta, \theta)\rightarrow 0$ and $\frac{h}{\delta}\rightarrow 0$. Here, the constant $\Lambda$ in the definition of the finite element space, cf. (\[def\_V\_h\]), depends on $\varepsilon$ in the general case. If in addition the sequence of solutions $u_n$ of the approximating problems as constructed in the previous section is uniformly bounded in $C^3$ then $\Lambda$ can be chosen uniformly in $\varepsilon$.
We first split the domain $$\begin{aligned}
\|u-u_{\varepsilon}\|_{L^{\infty}(\Omega)}
\le& \| u-u_{\varepsilon}\|_{L^{\infty}(\Omega_h)}+\|u-u_{\varepsilon}\|_{L^{\infty}(\Omega\setminus \Omega_h)} \\
=& I_1 + I_2.
\end{aligned}$$ Estimating the first term with the triangle inequality gives $$I_1 \le \|u-u_{n}\|_{L^{\infty}(\Omega_h)}+\|u_n-I_hu_n\|_{L^{\infty}(\Omega_h)}
+\|I_h u_n-u_{\varepsilon}\|_{L^{\infty}(\Omega_h)}$$ where $u_n$ is the solution of the approximating problem from the previous section and $n$ is assumed to be sufficiently large, and hence $$\|u-u_n\|_{L^{\infty}(\Omega_h)}$$ can be assumed to be arbitrarily small. In view of the standard interpolation estimate $$\|u_n-I_hu_n\|_{L^{\infty}(\Omega_h)}\le c \|u_n\|_{W^{2, \infty}(\Omega)}h^2$$ we may assume that $h=h(n)$ is so small that the norm on the left-hand side is as small as desired as well as that $\Lambda=\Lambda(\varepsilon)$ is sufficiently large. From (\[boundary\_definition\]) we conclude that for all $x\in \Omega \setminus \Omega_h$ and corresponding $z=z(x)\in \partial \Omega_h$ we have $$\begin{aligned}
|u(x)-u_{\varepsilon}(x)| =& |u(x)-u_{\varepsilon}(z)|\\
\le& |u(x)-u(z)|+|u(z)-u_{\varepsilon}(z)|.
\end{aligned}$$ Denoting the modulus of continuity of $u \in C(\bar \Omega)$ by $\tau$ we have $$I_2= \|u-u_{\varepsilon}\|_{L^{\infty}(\Omega\setminus \Omega_h)}
\le \tau(\operatorname{dist}_H(\Omega, \Omega_h))+\|u-u_{\varepsilon}\|_{L^{\infty}(\Omega_h)}.$$ Since $\operatorname{dist}_H(\Omega, \Omega_h)\rightarrow 0$ as $h\rightarrow 0$ the proof reduces to showing that $$\|I_h u_n-u_{\varepsilon}\|_{L^{\infty}(\Omega_h)}$$ can be made arbitrarily small which will be shown in the remaining part of the proof. Note that instead of arguing with the modulus of continuity of $u$ we could have also used the $C^{0}$-estimates which we derived in the proof of Lemma \[lemma7\] and the modulus of continuity of the corresponding approximating $u_n$.
Recall that we have chosen and will choose for the following $n$ sufficiently large. Furthermore, we will assume that $h=h(n)$ is chosen sufficiently small and $\Lambda=\Lambda(\varepsilon)$ sufficiently large.
We use the function $q_h=I_hq$ where $$q(x) = e^{\lambda |x-\tilde x|^2}-R$$ with $\tilde x$ outside $\bar \Omega$ and $R>0$ so that $q < 0$ in $\bar \Omega$. We define the discrete lower barrier as $$b_{\varepsilon}^{-}=u_{\varepsilon}+\rho q_h$$ where $\rho>0$ so that $$b_{\varepsilon}^{-} \le g_n$$ on $\partial \Omega_h$. W.l.o.g. let us assume that $$T_{\varepsilon}[I_hu_n] \le f_n+\|f-f_n\|_{L^{\infty}(\bar \Omega)}+\frac{1}{n}.$$ Choosing $\lambda>0$ sufficiently large we achieve that $$T_{\varepsilon}[b_{\varepsilon}^{-}] \ge T_{\varepsilon}[I_hu_n]$$ and hence $$b_{\varepsilon}^{-} \le I_hu_n + O(h).$$ A similar argument with $b_{\varepsilon}^{+}:= u_{\varepsilon}-\rho q_h$, $\rho>0$ suitable, results in $b_{\varepsilon}^{+}\ge I_hu_n-O(h)$.
Clearly, this leads summarized to $$|I_hu_n-u_{\varepsilon}| \le -2\rho q_h+O(h).$$ Now, choosing $\varepsilon$ (resp. $h$) small, $\Lambda$ sufficiently large, and $\rho$, $\lambda$ suitable (not depending on $h$ or $\Lambda$) we get the desired convergence. This completes the proof.
G. Awanou, Convergence rate of a stable, monotone and consistent scheme for the Monge-Ampère equation, Symmetry, 8 (18), pp. 1-7, (2016).
J.-D. Benàmou, F. Collino, J.-M. Mirebeau, Monotone and consistent discretization of the Monge-Ampère operator, arXiv:1409.6694, (2014).
S. C. Brenner, T. Gudi, M. Neilan, L.-Y. Sung, $C^0$ penalty methods for the fully nonlinear Monge-Ampère equation, Math. Comp., 80 (276), pp. 1979-1995, (2011).
H. Chen, G. Huang, X.-J. Wang, Convergence rate estimates for Aleksandrov’s solution to the Monge-Ampère equation, SIAM J. Numer. Anal., 57, No. 1, 173-191 (2019).
M. G. Crandall, H. Ishii, P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27(1), pp. 1-67, 1992.
E. J. Dean, R. Glowinski, An augmented Lagrangian approach to the numerical solution of the Dirichlet problem for the elliptic Monge-Ampère equation in two dimensions, Electron. Trans. Numer. Anal., 22, pp. 71-96, (2006).
E. J. Dean, R. Glowinski, On the numerical solution of the elliptic Monge-Ampère equation in dimension two: A least-squares approach, Partial Differential Equations, Comput. Meths. Appl. Sci. 16, Springer, Dordrecht, pp. 43-63, (2008).
X. Feng, M. Jensen, Convergent semi-Lagrangian methods for the Monge-Ampère equation on unstructured grids, arXiv:1602.04758v2, (2016).
X. Feng, M. Neilan, Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations, J. Sci. Comput., 38 (1), pp. 74-98, 2009.
X. Feng, M. Neilan, Mixed finite element methods for the fully nonlinear Monge-Ampère euquation based on the vanishing moment method, SIAM J. Numer. Anal., 47(2), pp. 1226-1250, (2009).
A. Figalli, Y.-H. Kim and R. J. McCann, When is multidimensional screening a convex program?, J. Econ. Theory., 146, No. 2, 454-478 (2011)
A. Figalli, Y.-H. Kim and R. J. McCann, Hölder Continuity and Injectivity of Optimal Maps, Arch. Ration. Mech, Anal., 209, pp. 747-795, (2013)
B. Froese, A. Oberman, Convergent finite difference solvers for viscosity solutions of the elliptic Monge-Ampère equation in dimensions two and higher, SIAM J. Numer. Anal., 49(4), pp. 1692-1714, (2012).
D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order, volume 224 of Grundlehren der Mathematischen Wissenschaften \[Fundamental Principles of Mathematical Sciences\]. Springer-Verlag, Berlin, second edition, 1983.
R. Glowinski, Numerical methods for fully nonlinear elliptic equations, Proceedings of the 6th International Congress on Industrial and Applied Mathematics, R. Jeltsch and G. Wanner, eds., ICIAM 07, Invited Lectures, pp. 155-192, (2009).
H. Ishii, P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Diff. Eqs. 83 (1), pp. 26-78, 1990.
J. Jost, Partielle Differentialgleichungen, Springer-Verlag Berlin Heidelberg, 1998.
W. Li and R. H. Nochetto, Optimal pointwise error estimates for two-scale methods for the Monge-Ampère equation, SIAM J. Numer. Anal., Vol 56, No. 3, pp. 915-1941 (2018) J. Liu, N. S. Trudinger and X.-J. Wang, Interior $C^{2,\alpha}$ regularity for potential functions in optimal transportation, Comm. Partial Differential Equations, 35(1), pp. 165-184, (2010)
X. N. Ma, N. S. Trudinger and X.-J. Wang, Regularity of potential functions of the optimal transportation problem, Arch. Ration. Mech, Anal., 177, pp. 151-183, (2005)
J.-M- Mirebeau, Discretization of the 3D Monge-Ampère operator, between wide stencils and power diagrams, arXiv:1503.00947, 2014.
G. Monge, Mémoire sur la théorie des déblais et de remblais, Histoire de l’Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année, pp. 666-704, (1781).
R. H. Nochetto, D. Ntogkas and W. Zhang, Two-scale method for the Monge-Ampère equation: Convergence to the viscosity solution, Math. Comp., (2018)
V. I. Oliker, L. D. Prussner, On the numerical solution of the equation $(\partial^2z/\partial z^2)(\partial^2z/\partial y^2)-
(\partial^2z/\partial x\partial y)^2=f$ and its discretizations, I. Numer. Math., 54(3), pp. 271-293, (1988).
G. D. Phillippis and A. Figalli, Sobolev regularity for Monge-Ampère type equations, SIAM J. Math. Anal., Vol. 45, No. 3, pp. 1812-1824, (2013)
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We solve some noncommutative analogue of the Noether’s problem for the reflection groups by showing that the skew field of fractions of the invariant subalgebra of the Weyl algebra under the action of any finite complex reflection group is a Weyl field, that is isomorphic to the skew field of fractions of some Weyl algebra. We also extend this result to the invariants of the ring of differential operators on any finite dimensional torus. The results are applied to obtain analogs of the Gelfand-Kirillov Conjecture for Cherednik algebras and Galois algebras.'
address:
- 'Department of Mathematics, University of Sichuan, Chengdu, China'
- 'Instituto de Matemática e Estatística, Universidade de S˜\~ ao Paulo São Paulo, Brasil'
- 'Instituto de Matemática e Estatística, Universidade de S˜\~ ao Paulo São Paulo, Brasil'
author:
- Farkhod Eshmatov
- Vyacheslav Futorny
- Sergiy Ovsienko
- Joao Fernando Schwarz
title: 'Noncommutative Noether’s problem for complex reflection groups'
---
Introduction
============
Let $k$ be a field of characteristic $0$. Let $G$ be a finite subgroup of $\GL_n(k)$ acting naturally by linear automorphisms on $S:=k[x_1, \ldots, x_n]$, and hence on the field of fractions $F:={{\rm{Frac}}}(S)= k(x_1, \ldots, x_n)$. It is easy to show $F^G={{\rm{Frac}}}(S^G)$. Then a well-known result due to E.Artin claims that the transcendence degree of $F^G$ over $k$ is equal to $n$. Now one can ask the following natural question: $\textit{Noether's problem. }$ Is $F^G$ a purely transcendental extension of $k$ or equivalently is $F^G$ isomorphic to $F$? This problem has been studied by many authors. Let us briefly recall some of the results. For more detailed discussion see [@D].
By a classical theorem of E.Fisher, the answer to Noether’s problem is positive for all $n\ge 1$ when $G$ is abelian and $k$ algebraically closed. For a general $k$ this is no longer true. One of the first counterexamples were produced in [@Le] for $k=\mathbb{Q}$ and when $G$ is a cyclic group of order eight. The answer is also positive for all $n\ge 1$ when $G$ is a complex reflection group, since it is a consequence of the Chevalley-Shephard-Todd theorem: $ S^G$ is isomorphic to $S$.
In general, for $n=1$ the positive answer is a straightforward consequence of the classical theorem of Lüroth, while for $n=2$ it is a simple consequence of Castelnuovo’s theorem. For $n=3$ the positive answer was proved by Burnside using Miyata’s theorem. Now, we will discuss a noncommutative version of the Noether problem. Let $A_n=A_n(k)$ be the $n$-th Weyl algebra with usual generators $x_1, \ldots, x_n$ and $\partial_1, \ldots, \partial_n$, and let $F_n$ be its skew field of fractions. Then the action of $G$ on $S$ naturally extends to $A_n$ and to $F_n$. The following question was originally posed by J.Alev and F.Dumas (see [@AD Section $1.2.2$]) $ \textit{Noether's problem for} \, \, A_n $: Is $F_n^G$ isomorphic to $F_n \, ?$\
One of the motivations to study this problem comes from [@L Theorem $5$], where it has been shown that the the subalgebra of the invariant differential operators on the affine space under the action of a finite unitary reflection group $G$ is not isomorphic to the whole algebra of differential operators, that is $A_n^G$ is not isomorphic to $A_n$.
Let $V$ be a finite dimensional vector space of dimension $n$ over $k$. By fixing a basis in $V$ one can identify $S(V^*)$ with $k[x_1, \ldots, x_n]$, where $x_1, \ldots, x_n$ is the dual basis in $V^*$. If $G$ be a finite subgroup of $GL(V)$ (more generally $V$ is a $G$-module) then it acts on $S(V^*)$ by linear automorphisms: $g.f(v)=f(g^{-1}v),$ $g \in G$, $f \in S(V^*)$, $v \in V$. This action can be naturally extended to the ring of differential operators $\D(k[x_1, \ldots, x_n])$ on $S(V^*)$. This induces a group of linear automorphisms of the Weyl algebra $A_n$. The following was proved in [@AD]
$(a)$ Let $V$ be a representation of $G$ which is a direct sum of $n$ representations of dimension one. Then $F_n^G \cong F_n$.\
$(b)$ For any $2$-dimensional representation of $G$, we have $F_2^G \cong F_2$.
It follows from part $(a)$ that we have the positive answer to Noether’s problem for $A_n$ for all $n \ge 1$ when $G$ is abelian and $k$ is algebraically closed. We should also point out that in [@AD], this problem was discussed for the case when $G$ is not necessarily a finite group. In this case the above question should be slightly modified.
From now on we assume that all algebras and varieties are defined over $\c$.
To our knowledge, the only other case for which Noether’s problem for $A_n$ has been considered is when $G \cong S_n$, a symmetric group of degree $n$, acting on $S(V^*)$ by permuting variables $x_i$. In [@FMO], it was shown among other things that $F_n^{S_n} \cong F_n$.
Our first main result in this paper is
Let $V$ be an $n$-dimensional vector space over $\c$ and $W$ be a finite complex reflection subgroup of $\GL(V)$. Then $F_n^W \cong F_n$.
Perhaps, this result is known to specialists but we could not find any proof. We are aware of an unpublished manuscript by I.Gordon where a similar statement is claimed without a proof.
Next we extend this technique to the study of the skew field of fractions of the invariants for classical reflection groups in the case of any finite dimensional torus.
Our second main result is
Let $X=T^n$ be an $n$-dimensional torus, $\D(X)$ the ring of differential operators on $X$, $F(\D(X))$ the skew filed of fractions of $\D(X)$, $W$ a classical complex reflection group. Then there exists a natural action of $W$ on $\D(X)$ which extends to $F(\D(X))$ and $F(\D(X))^W \cong F_n$.
As one of the applications of Theorem \[main\], we will show that an analogue of the Gelfand-Kirillov conjecture for Lie algebras holds for spherical subalgebras of rational Cherednik algebras $H_k:=H_k(W)$ associated to $W$. We also discuss the Gelfand-Kirillov conjecture for a class of linear Galois algebras which includes the universal enveloping algebras of $gl_n$ and $sl_n$.
[**Acknowledgements.**]{} V.F. would like to thank the Mittag-Leffler Institute for its hospitality during his stay where part of this work was done. F.E. is supported in part by Fapesp ( 2013/22068-6), V.F. is supported in part by CNPq (301320/2013-6) and by Fapesp (2014/09310-5).
Invariant differential operators
================================
Let $B$ be a commutative algebra. The ring of differential operators $\D(B)$ is defined to be $\D(B)=\cup_{n=0}^{\infty}\D(B)_n$, where $\D(B)_0=B$ and $$\D(B)_n \, = \, \{ \, d \in \End_{\c} (B) \, :\, d\, b - b\, d \in \D(B)_{n-1}\, \mbox{ for all }\, b \in B \} \, .$$ Let $B$ be a reduced, finitely generated algebra and let $G$ be a group acting on $B$ by algebra automorphisms. Then $G$ acts on $\D(B)$, via $ (g \ast \partial) \cdot f = g \circ \partial \circ g^{-1} \cdot f$. By [@L Theorem 5] restricting differential operators gives an injective homomorphism $ \D(B)^{G} \to \D(B^G)$. It is interesting to know when this map is an isomorphism. The following is a special case of [@CH Theorem 3.7]
Let $X$ be a normal, irreducible, affine algebraic variety, and let $G$ be a finite group acting freely on $X$. Then $\D(X)^G \cong \D(X/G)$.
Proof of Theorem \[main\].
==========================
Let $V$ be a finite-dimensional vector space over $\c$. An element $s \in \GL(V)$ is a *complex reflection* if it acts as identity on some hyperplane $H_s$ in $V$. A finite subgroup $W$ of $\GL(V)$ is called a complex reflection group if it is generated by its complex reflections. Let $(\cdot ,\cdot)$ be a positive definite Hermitian form on $V$, which is invariant under the action of $W$. We may assume that $(\cdot, \cdot)$ is antilinear in the first argument and linear on its second argument: if $x\in V $, we write $x^*$ for the linear form $V \to \c, \, v \mapsto (x,v)$.
Let $\mathcal{A}=\{H_s \}$ denote the set of reflection hyperplanes of $W$, corresponding to $s\in W$. The group $W$ acts on $\mathcal{A}$ by permutations. If $H \in \mathcal{A}$, the (pointwise) stabilizer of $H$ in $W$ is a cyclic subgroup $W_H$ of order $n_H$. Let $\alpha_{H}$ be a linear form for which $H$ is the zero set. It is defined up to a constant. We set $$\delta \, := \, \prod_{H\in \mathcal{A}} \alpha_H \quad , \quad J\, := \, \prod_{H\in \mathcal{A}} \alpha_{H}^{n_H-1}\, .$$ It is easy to show that $w.J = \det(w) J$ for any $w\in W$ (see [@Sp Exercise $4.3.5$]). Let $N$ be the order of $W$. Then $\triangle :=J^N$ is an invariant polynomial.
We fix a basis $\{v_1,...,v_n\}$ of $V$ and let $\{x_1,...,x_n\}$ be the corresponding dual basis of $V^*$. Then $S:=\c[V]= \c[x_1,...,x_n]$. Let $S_\delta$ be the localization of $S$ by $\{1,\delta, \delta^2, ... \}$. Then $S_\delta=S_J=S_\triangle$ since they are just localizations by $\{ \alpha_H^k\}_{ k\ge 0, H \in \mathcal{A}}.$ In fact, $S_\delta \cong \c[V^{\rm{reg}}]$, where $ V^{\rm{reg}}:=V \setminus \bigcup_{H\in \mathcal{A}} H$.
\[lem-free\] The action of $W$ restricts to a free action on $V^{\rm reg}$ and on $\c[V^{\rm reg}]$.
Assume that for some $w\in W$ and $v\in V^{\rm reg}$, $wv$ belongs to a hyperplane fixed by some reflection $s$. Then $w^{-1}sw$ belongs to the isotropy group of $v$ which is also a reflection group by the Steinberg’s theorem. Since $v\in V^{\rm reg}$, we conclude that $s=id$, which is a contradiction. Hence, $wv\in V^{\rm reg}$ and this action is clearly free.
If we let $X:={{\rm{Spec}}}(S_\triangle)$ and $Y:={{\rm{Spec}}}((S_\triangle)^W)$, then both $X$ and $Y$ are normal, irreducible, affine algebraic varieties. Since the action of $W$ on $X$ is free, the dominant morphism $\phi : X \to Y$ is unramified in codimension $1$. So we can use Theorem \[cor1\] to get $$\la{DiffIden}
\D(S_\triangle)^W \, \cong \D((S_\triangle)^W) \, .$$
\[prop1\] $(i)$ If $A$ is a domain and $M$ is an Ore subset then ${{\rm{Frac}}}(A_M)\cong {{\rm{Frac}}}(A)$.\
$(ii)$ $(S_\triangle)^W \cong (S^W)_\triangle$.\
$(iii)$ $\D(S_\triangle)^W \cong (\D(S)^W)_\triangle$.\
$(iv)$ ${{\rm{Frac}}}(A_n)^W\simeq {{\rm{Frac}}}(A_n^W)$.\
$(i)$ This statement is clear.\
$(ii)$ Since $\triangle$ is an invariant polynomial then $f\in (S_\triangle)^W$ iff $\triangle^k f \in S^W$ for some $k\ge 0$ iff $f \in (S^W)_\triangle$.\
$(iii)$ Note that $\D(S_M)\cong D(S)_M$ for a multiplicative set $M$, [@MR Theorem 15.1.25]. If $d \in \D(S_\triangle)^W$ then $\triangle ^k d \in \D(S)^W$ for some $k\ge 0$. Finally, (iv) follows from [@Fa], Theorem 1, see also [@D].
Now $(S_\triangle)^W \cong (S^W)_\triangle \cong S_{\triangle}$ , where the first identity holds by part $(ii)$ while the second one follows from the Chevalley-Shephard-Todd theorem. Therefore, the right hand side of is isomorphic to $\D(S_\triangle)\cong \D(S)_\triangle $. Thus, using part $(ii)$ we have $$( \D(S)^W)_\triangle \, \cong \, \D(S)_\triangle \, .$$ Finally, taking the skew field of fractions on both sides, we obtain $$F_n^W \cong F_n \, .$$
Gelfand-Kirillov conjecture for rational Cherednik algebras
===========================================================
Let us first recall the definition of rational Cherednik algebras. As before $W$ is a finite complex reflection subgroup of $\GL(V)$ and $(\cdot, \cdot)$ is a $W$-invariant positive definite Hermitian form. For $H\in \mathcal{A}$, let $v_H \in V $ be such that $\alpha_H=v_H^*$. Next for each $H$ from $\mathcal{A}$, we set $$e_{H,i}\, := \, \frac{1}{n_H} \, \sum_{w\in W_H} \,(\det w )^{-i} w \, .$$ Since $W_H$ is a cyclic group of order $n_H$, this is a complete set of orthogonal idempotents in $\c W_H$. Now, for $H\in \mathcal{A}$, we fix a sequence of non-negative integers $k_H=\{k_{H,i}\}_{i=0}^{n_H-1}$ so that $k_H=k_{H'}$ if $H$ and $H$ are on same orbit of $W$ on $\mathcal{A}$. The *rational Cherednik algebra* $H_k=H_k(W)$ is generated by elements $x\in V^*, \xi \in V $ and $w \in W$ subject to the following relations $$\begin{aligned}
&& [x,x'] = 0 \, , \quad [ \xi,\xi']=0\, , \quad w\, x \, w^{-1} =w(x)\, , \quad w\, \xi \, w^{-1} = w(\xi)\, , \nonumber \\
&& [\xi,x] = \langle \xi , x \rangle + \sum_{H\in \mathcal{A}} \frac{\langle \alpha_H, \xi \rangle\,
\langle x, v_H \rangle }{\langle \alpha_H , v_H \rangle } \sum_{i=0}^{n_H-1} n_H ( k_{H,i}-k_{H,i+1}) e_{H,i}\, . \nonumber \end{aligned}$$ Next, we introduce the *spherical subalgebra* $U_k(W)$ of $H_k$: by definition, $U_k(W):=eH_ke$, where $e:=|W|^{-1} \sum_{w \in W} w$ is the symmetrizing idempotent in $\c W \subset H_k$. The skew field of fractions of the spherical subalgebra was studied by Etingof and Ginzburg [@EG Theorem $17.7^*$]. Combining this result with Theorem \[main\], we get the following analogue of the Gelfand-Kirillov conjecture for rational Cherednik algebras:
For a complex reflection group $W$ we have ${{\rm{Frac}}}(U_k (W)) \cong F_n$.
Noether’s problem for $n$-dimensional torus
===========================================
Let $X =\mathbb{T}_n= \mathrm{Spec} \,( k[x_1^{\pm 1}, \ldots, x_n^{\pm 1}])$ be the $n$-dimensional torus. Then $\D(X)\simeq \Tilde{A_{n}}$ where $\tilde{A_{n}}$ is the localization of $A_{n}$ by the multiplicative set generated by $\{x_i | i=1, \ldots, n\}$. We consider the action of classical reflection groups on $\D(X)$.
For each $j=1,\ldots, n$ consider the involutions $\tau_j^{\pm}$ on $\c[x_1^{\pm1}, \ldots, x_n^{\pm1}]$ such that $$\tau_j^{\pm}(x_j) = \pm x_j^{-1} \, \mbox{ and } \, \tau_j^{\pm}(x_i)=x_i \, \mbox { for } \, i\neq j \, .$$ They induce the involutions $\varepsilon_{n,j}^{\pm}$ on $\tilde{A_{n}}$ such that $\varepsilon_{n,j}^{\pm}(x_i)=\tau_j^{\pm}(x_i)$, $$\varepsilon_{n,j}^{\pm}(\partial_j)=\mp x_j^{2}\partial_j \, \mbox{ and }
\, \varepsilon_{n,j}(\partial_i)=\partial_i \, \mbox{ if } i\neq j \, ,$$ $i,j=1, \ldots, n$. We show it for $n=1$. Suppose we have a group action $G\times \mathbb{T}^1\rightarrow \mathbb{T}^1$ on one dimensional torus. Then the induced action on $\tilde{A}_{1}$ is given as $$x \stackrel{g}\longmapsto x^{g}, \partial\stackrel{g}\longmapsto \partial^{g}= g\partial g^{-1}.$$ Hence, if $\tau^{-}$ sends $x$ to $-x^{-1}$ then we have
$$\varepsilon^{-}(\partial)(x^r)=\partial^{\tau^{-}}(x^r)= \tau^{-}\partial \tau^{-}(x^r)=(-1)^r \tau^{-}\partial(x^{-r})=(-1)^{r+1} r \tau^{-}(x^{-(r+1)})=$$ $=rx^{r+1},$ and $\varepsilon^{-}(x)(x^r)=(-1)^r \tau^{-}(x^{-r+1})=-x^{r-1}.$\
We obtain $\varepsilon^{-}(\partial)=x^{2}\partial$, $\varepsilon^{-}(x)=-x^{-1}$. This is easily generalized to $n$-dimensional torus $\mathbb{T}^n$ and $\tau_i^{\pm}$, $i=1, \ldots, n$.
We will consider the action of the reflection group of type $B_n$ ($n\geq 2$) and the reflection group of type $D_n$ ($n\geq 4$) on $X$. We recall, the group $B_n$ is the semi-direct product of the symmetric group $S_{n}$ and $(\mathbb{Z}/2\mathbb{Z})^n$. There is a natural action of $B_n$ on $\D(X)=\tilde{A_{1}}^{\otimes n}$, where $S_n$ acts by permutations and $(\mathbb{Z}/2\mathbb{Z})^n$ acts by $\varepsilon_{n,i}^{-}$, $i=1, \ldots, n$.
We have
\[proposition-action-in-odd-case\] (i) The subalgebra of $B_n$-invariants of $\D(X)$ is a polynomial algebra in $$s_{i}=e_{i}(x_{1}-x_{1}^{-1},\dots,x_{n}-x_{n}^{-1}), \, i=1, \dots, n,$$ where $e_i$ is the $i$-th elementary symmetric polynomial. In particular, $X/B_n$ is $n$-dimensional affine space.
\(ii) Let $Z\subset \mathbb{T}^{n}$ be the subvariety defined by the following equation $$\prod_{1\leq i\leq j\leq n} (x_{i}^{2}-\frac{1}{x_{j}^{2}})\prod_{1\leq i<j\leq n}(x_{i}^{2}-x_{j}^{2})=0$$ and $U=\mathbb{T}^{n}\setminus Z$. Then $U$ is an affine $B_n$-invariant subvariety of $X$ and the action of $B_n$ on $U$ is free. In particular, the projection $\pi:U\mapsto U/B_n$ is etale.
\[proof-of-the-proposition-action-in-odd-case\] For $(i)$, consider the lexicographical order on Laurent monomials. Let $\pi=(k_{1},\dots,k_{n})$ be a sequence of integers with the property $k_{1}\geq k_{2}\geq\dots\geq k_{n}\geq 0$ and $x_{1}^{k_{1}}\dots x_{n}^{k_{n}}$ the corresponding monomial. Denote by $m_{\pi}=x_{1}^{k_{1}}\dots x_{n}^{k_{n}}+\dots$ a $B_n$-invariant polynomial with a minimal number of monomials. We will call $\pi$ the degree of $m_{\pi}$. The polynomials $m_{\pi}$ form a basis of the subalgebra of $B_n$-invariants. The leading monomial $x_{1}^{k_{1}}\dots x_{n}^{k_{n}}$ of $m_{\pi}$ coincides with the leading monomial of $M_{\pi}:=s_{1}^{k_{1}-k_{2}}\dots s_{n-1}^{k_{n-1}-k_{n}}s_{n}^{k_{n}}$. Then $m_{\pi}-M_{\pi}$ has a smaller leading monomial and we can proceed by induction on the degree.
For $(ii)$ we denote $$\Delta=\prod_{1\leq i, j\leq n} \bigg(x_{i}^{2}-\frac{1}{x_{j}^{2}}\bigg)\, \prod_{1\leq i<j\leq n}(x_{i}^{2}-x_{j}^{2})\,
\bigg(\frac{1}{x_{i}^{2}}-\frac{1}{x_{j}^{2}}\bigg)\, \prod_{i=1}^{ n}
\bigg(x_{i}^{2}-\frac{1}{x_{i}^{2}}\bigg).$$ Then one can easily that $\Delta$ is $B_n$-invariant and $U=X\setminus V(\Delta)$, where $V(\Delta)$ is the algebraic subset of $\mathbb{T}^n$ corresponding to $\Delta$.
The group $D_n$ is generated by $S_{n}$ and $(\mathbb{Z}/2\mathbb{Z})^{n-1}$ which consists of the transformations $(\varepsilon_{1}^{d_{1}},\dots, \varepsilon_{n}^{d_{n}})\in
(\mathbb{Z}/2\mathbb{Z})^{n}$, $d_{i}=0,1,i=1,\dots, n$, such that $d_{1}+\dots+d_{n}$ is even. Consider now the action of $D_n$ on $\D(X)=\tilde{A_{1}}^{\otimes n}$ where $S_n$ acts by natural permutations and $\varepsilon_i$ acts as $${{\rm{Id}}}_{\tA_{1}}^{i-1}\otimes \varepsilon_{n,i}^{+}\otimes \varepsilon_{n,i+1}^{+}\otimes {{\rm{Id}}}_{\tA_{1}}^{n-i-1}, i=1,\dots, n-1.$$
\[proposition-action-in-even-case\] (i) The subalgebra of $D_n$-invariants of $\D(X)$ is generated by $$s_{i}=e_{i}(x_{1}+x_{1}^{-1},\dots,x_{n}+x_{n}^{-1}),\, i=1,\dots,n-1$$ and $$\Delta_{n}^{\pm}=\frac{1}{2}\bigg(\, \prod_{i=1}^{n}(x_{i}+\dfrac{1}{x_{i}})
\pm
\prod_{i=1}^{n}(x_{i}-\frac{1}{x_{i}})\,\bigg).$$ Moreover, $\Delta_{n}^{-}\in \c[s_{1},\dots,s_{p-1},\Delta^{+}]_P$, where $P$ is some polynomial in\
$\c[s_{1},\dots,s_{n-1},\Delta_{n}^{+}]$. In particular, $X/D_n$ is isomorphic to a principal open subset of $n$-dimensional affine space.
\(ii) Let $Z\subset \mathbb{T}^n$ be the variety defined by equation $\Delta=0$ and $U=X\setminus Z$. Then $U$ is an affine $D_n$-invariant subvariety of $X$ and the action of $D_n$ on $U$ is free. In particular, the projection $\pi:U \rightarrow U/D_n$ is etale.
\[proof-of-the-proposition-action-in-even-case\] Proof of $(i)$ is similar to the proof of $(i)$ in Proposition \[proposition-action-in-odd-case\]. Order the Laurent monomials lexicographically. Let $\pi=(k_{1}, \dots, k_{n})$ be a sequence of integers such that $k_{1}\geq k_{2}\geq\dots\geq |k_{n}|\geq 0$. Note that $k_{n}$ can be negative. Set $$\lambda_{\pi}=|\{g\in B_n\mid g\cdot (x_{1}^{k_{1}}\dots x_{n}^{k_{n}})
=x_{1}^{k_{1}}\dots x_{n}^{k_{n}}\}|, \, m_{\pi}=\lambda_{k}^{-1}
\sum_{g\in B_n}{}g\cdot (x_{1}^{k_{1}}\dots x_{n}^{k_{n}}).$$ Then polynomials $m_{\pi}$ form a basis of the space of $D_n$-invariant Laurent polynomials. The leading monomial in $m_{\pi}$ is $x_{1}^{k_{1}}\dots x_{n}^{k_{n}}$ and the same leading monomial has the element $$M_{k}:=s_{1}^{k_{1}-k_{2}}\dots s_{n-1}^{k_{n-1}-k_{n}}(\Delta_{n}^{sign(k_{n})})^{|k_{n}|}.$$ Then $ m_{k}-M_{k}$ has a smaller leading monomial and we can proceed by induction. Next we show how to choose $P$. Note that both $s_n=\Delta^{+}_{n}+\Delta^{-}_{n}$ and $D=\Delta^{+}_{n}\Delta^{-}_{n}$ are $D_n$-invariant and $D$ can be expressed as a polynomial in $s_{1},\dots, s_{n}$. The leading monomial in $D$ has the degree $(2,2,\dots,2,0)$ and, hence, $s_{n}$ can not enter in the expression for $D$ in the degree greater than $1$, as the degree of the leading monomial in $s_{n}$ is $(1,1,\dots,1,1)$. It is easy to see that the polynomial part of $D$ consists of the squares and hence $D\not\in \c[s_{1},\dots,s_{n-1}]$ since it has the same leading monomial as $s_{n-1}^{2}$ and the second in lexicographical order monomial in $s_{n-1}^{2}$ has degree $(2,2,\dots,2,1,1)$. We conclude that $$\Delta_{n}^{+}\Delta_{n}^{-}=p_{1}(s_{1},\dots,s_{n-1})
+s_{n}p_{0}(s_{1},\dots,s_{n-1}), \text{ i.e }
\Delta_{n}^{-}=\frac{\Delta_{n}^{+}p_{0}+p_{1}}{\Delta_{n}^{+}-p_{0}}.$$ Set $P=\Delta_{n}^{+}-p_{0}.$ Then $(i)$ follows.
To show $(ii)$, we take the same polynomial $\Delta$ as in the proof of Proposition \[proposition-action-in-odd-case\]. Then $\Delta$ is $D_n$-invariant and $U=X\setminus V(\Delta)$.
Proof of Theorem 3
------------------
Let $X = \mathbb{T}^n = \mathrm{Spec} \,( k[x_1^{\pm 1}, \ldots, x_n^{\pm 1}])$, $\Lambda = k[x_1,\ldots,x_n]$ and $\Gamma = \Lambda_f$, where $f=x_1\ldots x_n$. Then $X = \mathrm{Spec} \, \Gamma$ is an affine, regular, normal irreducible variety. Then the statement for the symmetric group $S_n$ is analogous to Theorem 2.
We consider first the action of the group $B_n$. By Proposition \[proposition-action-in-odd-case\], the action of $B_n$ restricts to a free action on $U= {{\rm{Spec}}}\, \Gamma_{\Delta}$ which is an affine, irreducible, regular and normal variety. By applying Theorem \[cor1\], we have $\D(U)^{B_n} \cong \D(U/B_n)$ and $D(\Gamma_\Delta)^{B_n} \cong \D(\Gamma_\Delta^{B_n})$. By Proposition \[prop1\], we may conclude $\D(\Gamma)^{B_n}_\Delta \cong \D(\Gamma^{B_n}_\Delta)$. Since $\Gamma^{B_n} \cong \Lambda$ we have $\D(\Gamma)^{B_n}_\Delta \cong \D(\Lambda_\Delta) \cong \D(\Lambda)_\Delta$. Forming the skew fields of fractions we conclude ${{\rm{Frac}}}\, \D(X)^{B_n} \cong {{\rm{Frac}}}\, \D(X)$.
Consider now the case of the group $D_n$. Repeating the same steps as above we have $\D(\Gamma)^{D_n}_\Delta \cong \D(\Gamma^{D_n}_\Delta)$. By Proposition \[proposition-action-in-even-case\], $\Gamma^{D_n} \simeq \Lambda_P$ for some polynomial $P$. Therefore, $$\D(\Gamma)^{D_n}_{\Delta} \cong \D((\Gamma_P)_{\tilde{\Delta}})=\D(\Gamma_{P\Delta}) = \D(\Gamma)_{P\Delta}.$$ Forming the skew fields of fractions we conclude ${{\rm{Frac}}}\, \D(X)^{D_n} \cong {{\rm{Frac}}}\, \D(X)\cong F_n$, which completes the proof of Theorem 3.
Galois algebras
===============
Let $\Gamma$ be an integral domain, $K$ the field of fractions of $\Gamma$, $K\subset L$ is a finite Galois extension with the Galois group $G$. Let $\mathcal M\subset {{\rm{Aut}_{\c}}}L$ be a monoid on which $G$ acts by conjugations, Recall that an associative $\c$-algebra $U$ containing $\Gamma$ is called a *Galois $\Gamma$-algebra* with respect to $\Gamma$ if it is finitely generated $\Gamma$-subalgebra in $(L*\mathcal M)^{G}$ and $KU=(L*\mathcal M)^{G}, UK=(L*\mathcal M)^{G}$ [@FO].
If $U$ is such algebra then $S=\Gamma\setminus
\{0\}$ satisfies both left and right Ore condition and the canonical embedding $U\hookrightarrow (L*\mathcal M)^{G}$ induces the isomorphisms of rings of fractions $[S^{-1}]U\simeq (L*\mathcal M)^{G}$, $
U[S^{-1}]\simeq (L*\mathcal M)^{G}$.
The following is standard.
\[corol-skew-field-invariants\] If $L*\mathcal M$ is an Ore domain, then $(L*\mathcal M)^G$ is an Ore domain. If $\mathcal L$ is the skew field of fractions of $L*\mathcal M$, then the skew field of fractions of $(L*\mathcal M)^G$ coincides with $\mathcal L^G$, where the action of $G$ on $\mathcal L$ is induced by the action of $G$ on $L*\mathcal M$.
We immediately have
\[corol-skew-field-Galois\] Let $U$ be a Galois $\Gamma$-algebra and the skew group algebra $L*\mathcal M$ is the left and the right Ore domain with the skew field of fractions $\mathcal L$. Then $U$ is the left and right Ore domain and for its skew field of fractions $\mathcal U$ holds $\mathcal U=\mathcal L^G$. In particular, all Galois subalgebras with respect to $\Gamma$ in $(L*\mathcal M)^G$ have the same skew field of fractions.
Due to Proposition \[corol-skew-field-invariants\], $U[S^{-1}]\simeq
(L*\mathcal M)^{G}$ is an Ore domain, $S=\Gamma\setminus \{0\}$. Hence for any $u_{1},u_{2}\in U$ there exist $v_{1},v_{2}\in U,s_{1},s_{2}\in S$, such that $u_{1}v_{1}s_{1}^{-1}=u_{2}v_{2}s_{2}^{-1}$. Since $S$ is commutative we get $u_{1}v_{1}s_{2}=u_{2}v_{2}s_{1}$. Other conditions of Ore rings are proved analogously.
Let $V$ be a $\c$-vector space, $\dim_{\c}V=N$, $L$ the field of fractions of the symmetric algebra $S(V)$ and $G\subset {{\rm{Aut}_{\c}}}(L)$ is the classical reflection group whose action on $L$ is induced by its action by reflections on $V$. Then $L=\c(t_{1},\dots, t_{N})$ and $K=L^G$ is a purely transcendental extension of $\c$ by the Chevalley-Shephard-Todd Theorem. If $\mathcal M\subset {{\rm{Aut}_{\c}}}(L)$ is a subgroup such that $G$ normalizes $\mathcal M$ and $U$ is a Galois algebra in $(L*\mathcal M)^G$ with respect to a polynomial subalgebra $\Gamma$ such that the field of fractions $F(\Gamma)$ is isomorphic to $ K$, then such $U$ will be called a *linear Galois algebra*. In particular, $\mathcal K=(L*\mathcal M)^G$ is itself a linear Galois algebra with respect to $\Gamma$ if $L$, $G$, $\mathcal M$, $K$ and $\Gamma$ are as above. Other examples of linear Galois algebras are the universal enveloping algebras of $gl_{n}$ and $sl_{n}$ with respect to their Gelfand-Tsetlin subalgebras.
We will need the following action of $\mathcal M$ on $L$. Suppose $\mathcal M = \mathbb Z^{N}$ is a free abelian group of rank $N$ generated by $\sigma_{i}, i=1, \ldots, N$. We say that $\mathbb Z^{N}$ acts by shifts on $L$ if $\sigma_i$, $i=1, \ldots, N$ act on $t_j$ as follows: $\sigma_i(t_j)=t_j-\delta_{ij}$. Thus we can construct a skew group algebra $L*\mathcal M$. As a subgroup of automorphisms of $L$, $G$ normalizes $\mathcal M$ $$\sigma_{k}^{\pi}=\delta_{\pi(k)}, \pi\in S_{p}, \sigma_{k}^{\epsilon_{i}}=\left\{\begin{array}
{ll}\sigma_{k},&\text{ if } i\neq k\\
\sigma_{k}^{-1}&\text{ otherwise, }
\end{array}\right.$$
Hence $G$ acts on $L*\mathcal M$. We will call this action *natural*.
Let $R_N=\c[t_1, \ldots, t_N]*\mathbb Z^N$, where the group $\mathbb Z^N$ is generated by the elements $\sigma_i$, $i=1, \ldots, N$ as above. For each $i=1, \ldots, N$ and any $c\in k$ consider the involutions $\epsilon_{R_N,c,i}^{\pm}$ on $R_N$ defined by $\epsilon_{R_N,c,i}^{\pm}(\sigma_i)=\pm\,\sigma_i^{-1}$, $\epsilon_{R_N,c,i}^{\pm}(\sigma_j)=\sigma_j$ if $i\neq j$, $\epsilon_{R_N,c,i}^{\pm}(t_i)=c-t_i$ and $\epsilon_{R_N,c,i}^{\pm}(t_j)=t_j$ if $j\neq i$.
\[lemma-on-transformation-inversion-mult-inversion-add\] Let $\Tilde{A_{N}}$ be the localisation of the Weyl algebra $A_N$ by the multiplicative set generaled by $\{x_1, \ldots, x_N\}$. The homomorphism $\phi^{\pm}_{c}:\Tilde{A_{N}}\rightarrow R_N$, given by $$\phi^{\pm}_{c}(x_i)=\sigma_i,\ \phi^{\pm}_{c}(\partial_i)=\big(t_i+1-\dfrac{c}{2}\big)\sigma_i^{-1}+ (1\mp\sigma_i^{-2}),$$ $i=1, \ldots, N$ is an isomorphism of algebras with involutions.
It is sufficient to consider the case $N=1$. Set $R=R_1$. Since $\phi^{\pm}_{c}(\partial x - x \partial)=t-\sigma t\sigma^{-1}=1$, $\phi^{\pm}_{c}$ are homomorphisms, $(\phi^{\pm}_{c})^{-1}(\sigma)=x$ and $(\phi^{\pm}_{c})^{-1}(t)=\big(\partial x+\dfrac{1}{2}\big)+(\dfrac{1}{x}\mp x).$ We also have $$\epsilon_{R,c}^{\pm}\phi^{\pm}_{c}(\partial)=\mp\big(t-1-\dfrac{c}{2}\big)\sigma+ (1\mp\sigma^{2})$$ and $$\phi^{\pm}_{c}\epsilon_{\tilde{A}_{1},c}^{\pm}(\partial)= \mp\sigma^{2}
\bigg(\big(t+1-\dfrac{c}{2}\big)\sigma^{-1}+ (1\mp\sigma^{-2}) \bigg)=\mp\big(t-1-\dfrac{c}{2}\big)\sigma+ (1\mp\sigma^{2}).$$
For integers $n\geq 1$, $m\geq 0$ denote $A_{n,m}$ the $n$-th Weyl algebra over the field of rational functions $\c(z_1, \ldots, z_m)$. Then $A_{n,m}$ admits the skew field of fractions $F_{n, m}\simeq F_n\otimes \c(z_1, \ldots, z_m)$. We have
\[lem-skew-main\] Let $\mathcal K=(L*\mathcal M)^G$ be a linear Galois algebra where $G=G_N$ is a classical Weyl group and
- $L=\c(t_{1},\dots, t_{N})$;
- $G$ acts naturally on $\mathcal K$;
- $\mathcal M\simeq \mathbb Z^{n}$ acts by shifts on $t_{1},\dots, t_{n}$, $n\leq N$.
Then $\mathcal K$ admits the skew field of fractions $F(\mathcal K)$ and $F(\mathcal K)\simeq F_{n,N-n}.$
Since the action of $\mathcal M$ is trivial on $L(t_{n+1}, \ldots, t_N)$ then we have the $G$-equivariant embedding $$(L(t_1, \ldots, t_n)*\mathcal M)\otimes L(t_{n+1}, \ldots, t_N) \hookrightarrow L*\mathcal M$$ and $$((L(t_1, \ldots, t_n)\otimes L(t_{n+1}, \ldots, t_N))*\mathcal M)^G\hookrightarrow (L*\mathcal M)^G.$$ Moreover, both algebras have the same skew fields. But $$((L(t_1, \ldots, t_n)*\mathcal M)\otimes L(t_{n+1}, \ldots, t_N))^G\simeq ((L(t_1, \ldots, t_n)*\mathcal M)^G\otimes L(t_{n+1}, \ldots, t_N)^G.$$ Since $(L(t_1, \ldots, t_n)*\mathcal M\simeq R_n$, $R_n^G=R_n^{G_n}$, $F(R_n)\simeq F(A_n)$ and $L(t_{n+1}, \ldots, t_N)^G\simeq
L(z_1, \ldots z_{N-n})$ we obtain $$F(\mathcal K)\simeq F(A_n)^G\otimes L(z_1, \ldots z_{N-n}).$$ The result follows from Theorem \[main\].
\[theorem-main-theorem\] Let $U$ be a linear Galois algebra in $(L*\mathcal M)^G$ such that
- $L=\c(t_{ij}, i=1, \ldots, N; j=1, \ldots, n_i; z_1, \ldots z_m)$, for some integers $m$, $n_1$, $\ldots$, $n_N$;
- $G=G_1\times \ldots \times G_N$, where $G_s$ acts normally only on variables $t_{s1}$, $\ldots$, $t_{s, n_s}$, $s=1, \ldots, N$;
- $\mathcal M\simeq \mathbb Z^{n}$ acts by shifts on $t_{11},\dots, t_{N,n_N}$, $n=n_1+\ldots + n_N$.
Then $U$ admits the skew field of fractions $F(U)$ and $F(U)\simeq F_{n, m}.$
Follows from Theorem 3 and Lemma \[lem-skew-main\].
\[remark-case-of-s-n\] The action of the symmetric group $S_{N}$ on $\c^{N}$ by permutations of the coordinates is obviously linear and it normalizes the action of $\mathcal M=\mathbb Z^{N}$ on $\c^{N}$ by shifts. Recall that $U(gl_{n})$ and $U(sl_{n})$ are linear Galois algebras with respect to their Gelfand-Tsetlin subalgebras [@FO]. Then Theorem \[theorem-main-theorem\] implies immediately the Gelfand-Kirillov conjecture for $gl_{n}$ and $sl_{n}$. In a similar manner one obtains the Gelfand-Kirillov conjecture for restricted Yangians of type $A$ which was shown in [@FMO].
[100]{}
J. Alev, F. Dumas, [*Invariants d’operateurs differentiels et probleme de Nother*]{}, Studies in Lie Theory : A.Joseph Festschrift (eds. J.Bernstein, V.Hinich and A.Melnikov) Birkhauser, 2006.
R. Cannings, M.P. Holland,[*Differential operators on varieties with a quotient subvariety.* ]{} J. Algebra 170 (1994), no. 3, 735-753.
F. Dumas, [*An Introduction to Non commutative polynomial invariants*]{}, Lecture Notes, Homological methods and representations of non commutative algebras, Mar del Plata, Argentina, March 6-16, 2006.
P.Etingof, V.Ginzburg, [*Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism*]{}, Invent. Math. 147 (2002), no 2, 243-348.
Faith C., [*Galois subrings of Ore domains are Ore domains*]{}, Bull AMS, 78 (1972), no.6, 1077-1080.
V.Futorny, S.Ovsienko, [*Galois orders in skew monoid rings*]{}, J. Algebra 324 (2010), 598-630.
V.Futorny, A.Molev, S.Ovsienko, [*Gelfand-Kirillov conjecture and Gelfand-Tsetlin modules for finite $W$-algebras*]{}, Adv. Math., Advances in Mathematics 223 (2010), 773-796.
H.W. Lenstra Jr., [*Rational functions invariant under a finite abelian group*]{}, Invent. Math. 25 (1974), 299-325.
T. Levausseur, [*Anneaux d’operateurs differentiels*]{}, Lecture Notes in Mahematics, 867 (1981), 157-173.
F. Knop, [*Graded cofinite rings of differential operators*]{}, Michigan Math. J. 54 (2006), 3-23.
J. C. McConnell, J. C. Robson, [*Noncommutative Noetherian Rings*]{}, J. Wiley $\&$ Sons, New York, 1987.
T.A. Springer, [*Invariant theory*]{}, Lecture Notes in Mahematics, 585, Springer, 1977, 120 pp.
| {
"pile_set_name": "ArXiv"
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---
abstract: 'We have studied the partition function of a free compact boson on a $n$-sheeted covering of torus gluing along $m$ branch cuts. It is interesting because when the branched cuts are chosen to be real, the partition function is related to the $n$-th Rényi entanglement entropy of $m$ disjoint intervals in a finite system at finite temperature. After proposing a canonical homology basis and its dual basis of the covering surface, we find that the partition function can be written in terms of theta functions.'
author:
- Feihu Liu
title: Free compact boson on branched covering of the torus
---
Introduction {#sec:intro}
============
Conformal field theory on higher genus Riemann surfaces is a fruitful field for both physics and mathematics. It has many important applications in physics, for example, string perturbation theory and statistical physics. Recently, entanglement entropy of two dimension field theory has attracted much attention due to its calculability. In Ref.[@ee-and-cft; @Calabrese:2009ez], the Rényi entanglement entropy of a free complex boson compactified on a torus was proved to be the same as the partition function for the $n$-sheeted covering surfaces of $\mathbb{CP}^1$. Indeed, there are three different cases: 1). The subsystem is in a infinite system at zero temperature; 2). The subsystem is in a finite system at zero temperature; 3). The subsystem is in a infinite system at finite temperature. The Rényi entanglement entropy for these three cases are related by a conformal transformation and all can be derived from the partition function on the $n$-sheeted covering of $\mathbb{CP}^1$. More specifically, for $n$-th Rényi entanglement entropy of $m$ disjoint intervals, the corresponding surface should be represented by a singular $Z_n$ curve $$\label{eq:zncurve}
{y}^n\equiv \prod_{i=0}^{m-1}(z-z_{2i-1})\prod_{j=1}^{m}(z-z_{2j})^{n-1}.$$ This curve has been well studied in Ref.[@enolski:2006; @Bobenko:2011], where the period matrix and the Thomae formula are given explicitly. As a matter of fact, in Ref.[@Coser:2013qda], the partition function of free compact boson on surface has been derived by using the results of Ref.[@Bobenko:2011].
For $n$-th Rényi entanglement entropy of $m$ disjoint intervals in a *finite* system at *finite* temperature, the corresponding Riemann surface should be the branched covering of torus, which is the main studying object of this paper. These surfaces are constructed by gluing $n$ replica torus along $m$ branch cuts denoted by $\mathcal{T}_{n,m}$. In order to find the partition function, one can introduce the so-called twist fields, which first appear in the orbifold theory [@Dixon:1986qv; @Dijkgraaf:1989hb; @Bershadsky:1986fv; @Knizhnik:1987xp]. By inserting a pairs of twist-antitwist fields at the ends of each interval, the total partition function on $\mathcal{T}_{n,m}$ is just the correlation function of twist fields on the torus [@ee-and-cft] $$\label{eq:totalcorrelationfunction}
\begin{split}
Z&=\prod_{k=0}^{n-1}\braket{\sigma_{k}(z_1,\bar{z}_1)\sigma_{n-k}(z_2,\bar{z}_2)\cdots\sigma_{k}(z_{2m-1},\bar{z}_{2m-1})\sigma_{n-k}(z_{2m},\bar{z}_{2m})}\\
&=\sum_{windings}\prod_{k=0}^{n-1}Z_{qu}(k)e^{-S_{cl}(k)}.
\end{split}$$ Here $\sigma_{k}(z_i,\bar{z}_i)$ is the twist field which introduces the local monodromy around point $z_i$: $$\begin{split}
\partial\phi\rightarrow e^{2\pi {\mathrm{i}}k/n}\partial\phi,\quad
\partial\bar{\phi}\rightarrow e^{-2\pi {\mathrm{i}}k/n}\partial\bar{ \phi},
\end{split}$$ and $\sigma_{n-k}(z_i,\bar{z}_i)$ denotes the antitwist field. Note that the quantum part and the classical part of the correlation function have been separated for each $k$-mode. However the summation must be performed after the product, because the compactification condition introduces non-trivial couplings between the winding numbers for different $k$-modes, which is actually an obstacle to generalizing the results to higher genus cases.
By far, the most well studied example is the Rényi entropy for a single interval at finite temperature, which is corresponding to the surface $\mathcal{T}_{n,1}$. One can find the calculation for free boson theory in Ref.[@Datta:2013hba; @Chen:2015cna]. While in Ref.[@Schnitzer:2015ira; @Schnitzer:2016xaj], the Rényi entropy was obtained by the similar way.
The main purpose of this paper is to evaluate for free compact boson defined on $\mathcal{T}_{n,m}$. Unlike the singular $Z_n$ curve , the basis of holomorphic differentials and the homology basis of $\mathcal{T}_{n,m}$ are not known. However, by using the cut abelian differentials defined in Ref.[@Atick:1987kd], we are able to construct the basis explicitly. In this way, we do *not* need to separate the classical part into different $k$ modes, therefore the winding numbers are all independent and there will be no annoying infinity that appears in the summation.
Quantum partition function on $\mathcal{T}_{n,m}$
=================================================
Since the quantum part of the partition function doesn’t depend on the windings, so the correlation function can be derived following Ref.[@Atick:1987kd]. Let’s first define the two Greens functions $$\label{eq:greensfunctions}
\begin{split}
g(z,\omega)&=\frac{\braket{\partial \phi(z)\partial \bar{\phi}(\omega)\prod_{i=1}^{2m}\sigma_{k_i}(z_i,\bar{z}_i)}}{\braket{\prod_{i=1}^{2m}\sigma_{k_i}(z_i,\bar{z}_i)}},\\
h(\bar z,\omega)&=\frac{\braket{\bar{\partial} \phi(\bar{z})\partial \bar{\phi}(\omega)\prod_{i=1}^{2m}\sigma_{k_i}(z_i,\bar{z}_i)}}{\braket{\prod_{i=1}^{2m}\sigma_{k_i}(z_i,\bar{z}_i)}}.
\end{split}$$ The Greens function $g(z,\omega)$ is a doubly periodic holomorphic function for both $z$ and $\omega$. Based on the OPE $$\label{eq:ope}
\partial \phi(z) \partial \phi(\bar{\omega})\sim \frac{1}{(z-\omega)^2}+T(\omega),,$$ we know that $g(z,\omega)$ should have a double pole with coefficient $1$ as $z\rightarrow \omega$ but no simple pole. Additionally, $g(z,\omega)$ should satisfy the correct local monodromy for $z$ and $\omega$ around the branch points. It is similar for $h(\bar{z},\omega)$, the only difference is that there are no poles as $\bar{z}\rightarrow \omega$. It turns out that these conditions are quite restrict. The two Greens functions can be fixed up to an undetermined function which is irrelevant to the partition function. As proposed in Ref.[@Atick:1987kd], one should introduce the so-called cut abelian differentials, which are defined as doubly-periodic holomorphic funtions on the torus yet have the appropriate monodromy around the branch points. By using the theta function $\vartheta_1(z)\equiv \vartheta_1(z|\tau)$, there are two sets of cut abelian differentials: $$\label{eq:abeliandifferentials}
\begin{split}
w_{n-k}^{\alpha_i}(z)&=\prod_{i=1}^{2m}\vartheta_1(z-z_i)^{-(1-\frac{k_i}{n})}\vartheta_1(z-z_{\alpha_i}-p_1)\prod_{j\ne i}\vartheta_1(z-z_{\alpha_j}),\\
w_k^{\beta_i}(z)&=\prod_{i=1}^{2m}\vartheta_1(z-z_i)^{-\frac{k_i}{n}}\vartheta_1(z-z_{\beta_i}-p_2)\prod_{j\ne i}\vartheta_1(z-z_{\beta_j}),
\end{split}$$ where $\{z_{\alpha_i}\}$ is an arbitrary chosen set of $m$ branch points and $\{z_{\beta_i}\}$ are the rest ones. The two shifts $p_1$ and $p_2$ are given by $$\label{eq:p1p2}
p_1=\sum_{i=1}^{2m}(1-\frac{k_i}{n})z_i-\sum_{i=1}^{m}z_{\alpha_i}, \quad p_2=\sum_{i=1}^{2m}\frac{k_i}{n}z_i-\sum_{i=1}^{m}z_{\beta_i}.$$ Knowing the property of theta functions, one can easily check that the cut abelian differentials are doubly-periodic. Since $\theta_1(z-\omega)\sim (z-\omega)$ as $z\rightarrow \omega$, one can see that the cut abelian differentials also satisfy the appropriate local monodromy around each branch points. It is also worth to mention that, within each set, the cut abelian differentials are independent. However, there is no requirement that $w_{n-k}^{\alpha_i}(z)$ should be independent of $w_{k}^{\beta_i}(z)$.
From the pole structure of $z \rightarrow \omega $, one can fix the Green’s function up to the most general form $$\label{eq:greenfunctionsolution_undetermined}
\begin{split}
g(z,\omega)&= g_s(z,\omega)-\sum_{i=1}^{m}\sum_{j=1}^{m}A_{ij}w_k^{\beta_j}(\omega) w_{n-k}^{\alpha_i}(z),\\
h(\bar z,\omega)&= \sum_{i=1}^{m}\sum_{j=1}^{m} B_{ij}w_{k}^i(\omega) \bar{w}_{k}^j(\bar z),
\end{split}$$ where $g_s(z,w)$ is singular part of the Greens function $$\label{eq:gs}
g_s(z,\omega)=\prod_{i=1}^{2m}\vartheta_1(z-z_i)^{-(1-\frac{k_i}{n})}\prod_{i=1}^{2m}\vartheta_1(w-z_i)^{-\frac{k_i}{n}}\left[\frac{\vartheta_1^\prime (0)}{\vartheta_1(z-\omega)}\right]^2 P(z,\omega).$$ It can be shown that the exact form of $P(z,\omega)$ turns out to be irrelevant in the end [@Atick:1987kd].
To determine the coefficients $A_{ij}$ and $B_{ji}$ in , one can impose the global monodromy conditions: $$\label{eq:globalqu}
\oint_{C_a} dz g(z,\omega)+\oint_{C_a} d \bar z h(\bar z,\omega)=0,$$ where $C_a$ are the independent net twist zero loops on the torus. The global monodromy conditions just mean that $\phi$ is single valued around the closed loops $C_a$. There are two cycles inherit from the original torus, while the remaining loops are chosen as shown in figure \[fig:torus2\].
![We can always put the twist fields at the left ends and the antitwist fields at the right ends. The branched cuts can be chosen as connecting branch points $z_{2i-1}$ to $z_{2i}$, so that the number of branch cuts is $m$. Besides the two cycles of the original torus, the remaining loops $\gamma_a,a=1,\cdots,2m-2$, which encloses branch points $z_1$ and $z_{a+1}$ respectively.[]{data-label="fig:torus2"}](torus2.pdf){width="80.00000%"}
These equations can be solved by introducing the cut period matrix $\mathbf{W}_a^i$ defined by $$\label{eq:cutperiodmatrix}
\begin{split}
\mathbf{W}_a^i &\equiv \oint _{C_a} dz w_{n-k}^{\alpha_i}(z), \,i=1,\cdots,m,\\
\mathbf{W}_a^{m+j} &\equiv \oint_{C_a} d\bar z \bar{w}_{k}^{\beta_j}( \bar z),\,j=1,\cdots, m.
\end{split}$$ However we have to say, the loops chosed in figure \[fig:torus2\] are not suitable for doing numerical integration. As an alternative, we can use the contours $\alpha_a$ described in figure \[fig:torus\]. The relations between $\gamma_a$ and $\alpha_a$ are simple: $$\gamma_1=\alpha_1, \, \gamma_a=\alpha_a-\gamma_{a-1} \text{ for } 1<a\le 2m-2.$$ Further, by using the OPEs one can get $$\label{eq:Tz}
\begin{split}
\braket{\braket{T(z;z_i)}}&\equiv \frac{\braket{T(z)\prod_{i=1}^{2m}\sigma_{k_i}(z_i,\bar{z}_i)}}{\braket{\prod_{i=1}^{2m}\sigma_{k_i}(z_i,\bar{z}_i)}}= \lim_{\omega\rightarrow z}\left [ g(z,\omega;z_i)-\frac{1}{(z-\omega)^2} \right ]\\
\partial_{z_i}\ln Z_{qu}&=\lim_{z\rightarrow z_i}\left[ (z-z_i)\braket{\braket{T(z;z_i)}}-\frac{h_i}{z-z_i} \right],
\end{split}$$ where the second equation of is essentially the conformal Ward identity. By integrating the conformal Ward identity, one can obtain the quantum part up to an unfixed normalization function $N$. Since the middle calculation is rather straightforward but lengthy, we will not repeat them here, one can find all the missing steps in Ref.[@Atick:1987kd]. Finally, multiplying the quantum part for different $k$-modes together, we have $$\label{eq:quantumpart}
\begin{split}
Z_{qu}=&\frac{N}{(\det \Im \tau) |\eta(\tau)|^4}\left(\prod_{k=1}^{n-1}\vartheta_1(p_1)^{(m-1)(n-1)} \bar{\vartheta}_1(p_2)^{(n-1)(m-1)}\right) \prod_{i<j}^{m}\vartheta^{n-1}_{\alpha_i\alpha_j}\\
&\quad \prod_{i<j}^{m}\bar{\vartheta}^{n-1}_{\beta_i\beta_j} \left(\prod_{k=1}^{n-1}|\det{\mathbf{W}}|^{-1}\prod_{i<j}^{2m}\vartheta_{ij}^{-(1-\frac{k_i}{n})(1-\frac{k_j}{n})}(\bar{\vartheta}_{ij})^{-\frac{k_i}{n}\frac{k_j}{n}}\right),
\end{split}$$ where we have denoted $\vartheta_1(z_i-z_j)$ by $\vartheta_{ij}$. Note that the factor $\frac{1}{(\det \Im \tau) |\eta(\tau)|^4}$ is coming from the $k=0$ mode, which means that there are no twist field insertions, i.e., it is just the torus partition function of a free complex boson [@cft]. For each $k$-mode, since there are pairs of twist-antitwist insertions, one can set $k_i=k$ for $i$ odd and $k_i=n-k$ for $i$ even. Here $N$ is the normalization function came from the integration of conformal Ward identity, which can be fixed by factorizing the correlation function on to lower genus ones.
![The solid line is on the first sheet, while the broken line is on the $n$th sheet []{data-label="fig:torus"}](torus.pdf){width="80.00000%"}
Classical partition function on $\mathcal{T}_{n,m}$
===================================================
As we mentioned earlier, in , the winding numbers for different $k$-modes are correlated due to the compactification condition. Hence if one tries to compute the classical solution for different $k$-modes separately, then one has to introduce the redundant winding numbers. As a result, in the final summation, there will be some unpleasant infinity need to be regularized. In general, the regularization is quite involved. Roughly speaking, one has to find the zero eigenvectors of some complicated matrix. A concrete example of regularization can be found in [@Calabrese:2009ez]. However, for more general cases, this regularization process could be very complicated. Therefore, we are not going to calculate the classical solution for different $k$-mode separately as in [@Calabrese:2009ez]. Instead, without using the trick of replicated target space, we construct the independent classical solutions for the $n$-sheeted covering surface directly. Such that there are no redundant summation. Actually, this similar more direct strategy has been implemented to calculate the higher-genus correlation functions of WZW models long time ago[@Naculich:1989ii].
The classical action is given by $$\label{eq:classicalaction}
S=\frac{1}{8\pi}\int_{\mathcal{T}_{n,m}}\left(\partial \phi(z){\mathrm{d}}z\wedge \bar{ \partial} \bar{ \phi}(\bar{z}) {\mathrm{d}}\bar{z}+ \partial \bar{ \phi}(z){\mathrm{d}}z\wedge \bar{ \partial} \phi(\bar{z}) {\mathrm{d}}\bar{z}\right).$$ From the classical equation of motion we know that $\partial \phi(z){\mathrm{d}}z$ and $\partial \bar{ \phi}(z){\mathrm{d}}z$ are the holomorphic one forms defined on $\mathcal{T}_{n,m}$, while $\bar{ \partial} \bar{ \phi}(\bar{z}) {\mathrm{d}}\bar{z}$ and $\bar{ \partial} \phi(\bar{z}) {\mathrm{d}}\bar{z}$ are anti-holomorphic. Thus, if we denote the basis of holomorphic one forms by $w_{i}(z)$, then the classical solutions can be written as a linear summation of the basis $$\partial \phi(z){\mathrm{d}}z=\alpha^i w_i(z), \quad \partial \bar{\phi}(z){\mathrm{d}}z=\beta^i w_j(z).$$ To fix the coefficients $\alpha^i$ and $\beta^i$, we need to use the global monodromy conditions similar as , only for this time there will be a shift on the RHS due to the windings. For simplicity, we will assume both the real and imaginary part of the compactified radii are $R$, then the global monodromy condition is $$\label{eq:globalmonodromy}
\oint_{C_a}\partial \phi(z) {\mathrm{d}}z+\oint_{C_a}\bar{\partial}\phi(\bar{z}){\mathrm{d}}\bar{z}= 2\pi R(m_a+{\mathrm{i}}n_a)\equiv v_a,\quad m_a,n_a \in \mathbb{Z}.$$ It is important to notice that the number of independent closed loops is the same as the number of independent holomorphic one forms [@Atick:1987kd]. Plugging the solution of $\alpha^i$ and $\beta^i$ back into , the classical action can always be written as $$\label{eq:classical}
\begin{split}
S_{cl} = \frac{1}{8\pi} v_a M^{ab}\bar v_b.
\end{split}$$ Here $M^{ab}$ is dependent on the choice of homology basis, although $S_{cl}$ doesn’t. Thus, one can always find a convenient basis to simplify $M^{ab}$. The procedure is explained as follows.
We first recall some basic definitions. Let $a_i, b_i, i = 1,\cdots,g$, be a canonical homology basis of genus $g$ Riemann surface $\mathcal{R}$ and let $w_k , k = 1,\cdots,g$, be the dual basis of $H^1(\mathcal{R},\mathbb{C})$. Then the period matrix of $\mathcal{R}$ is defined by $$\mathbf{P}=(\mathbf{A},\mathbf{B}),$$ where $$\mathbf{A}_{ij}=\oint_{a_i}w_j, \quad \mathbf{B}_{ij}=\oint_{b_i}w_j.$$ For many cases, it is more convenient to use the canonical basis of $H^1(\mathcal{R},\mathbb{C})$ defined by $$\tilde{w}_k=\sum_i(\mathbf{A}^{-1})_{ki}w_i.$$ Then the Riemann matrix of the surface $\mathcal{R}$ can be defined by $$\mathbf{\Omega}_{ij}=\oint_{b_i}\tilde{w}_j=(\mathbf{B} \cdot \mathbf{A}^{-1} )_{ij}.$$ By using the Riemann bilinear relation [@Bobenko:2011] $$\begin{split}
(w_i,w_j)&\equiv {\mathrm{i}}\int_{\mathcal{R}}w_i \wedge \bar{w_j}={\mathrm{i}}\sum_{k=1}^g \oint_{a_k}w_i \oint_{b_k}\bar{w_j}-\oint_{b_k}w_i \oint_{a_k}\bar{w_j}
\end{split}$$ and arranging the loops $\{C_1,\cdots,C_{2g}\}$ to be $\{a_1,\cdots,a_g,b_1,\cdots,b_g\}$, we have $$M^{ab}=\left[(\mathbf{G}^{-1})^{\mathrm{T}}\cdot \mathbf{H} \cdot (\bar{\mathbf{G}}^{-1}) \right]^{ab},$$ where $$\mathbf{G}=\begin{bmatrix}
\mathbf{1}_g & \mathbf{1}_g\\
\mathbf{\Omega} & \bar{\mathbf{\Omega}}
\end{bmatrix},
\mathbf{H}=
\begin{bmatrix}
2{\mathrm{i}}\Im \mathbf{\Omega}& 0\\
0 & 2{\mathrm{i}}\Im \mathbf{\Omega}
\end{bmatrix}.$$ A straightforward block-matrix calculation shows that $$\label{eq:m}
M=
\left(\begin{array}{c|c}
\Im{\Omega}+\Re{\Omega}\cdot \Im{\Omega}^{-1} \cdot \Re{\Omega}\quad &\quad\Re{\Omega}\cdot \Im{\Omega}^{-1}\\
\hline
\Im{\Omega}^{-1}\cdot \Re{\Omega} & \quad \Im{\Omega}^{-1}
\end{array}\right).$$ Putting all things together, the classical part of the partition function can be written as $$\label{eq:zcl}
\begin{split}
\sum_{\mathrm{windings}}\exp(-S_{cl})&=\sum_{\mathrm{\{m_a,n_a\}}}\exp\left[\frac{-1}{8\pi}v_a M^{ab}\bar{v}_b\right]=\left | \Theta(0|{\mathrm{i}}\frac{R^2}{2}M)\right |^2.
\end{split}$$ Here function $\Theta(0|\Omega)$ is the multi-dimensional theta function [@nist]. We have noticed that the form of the classical summation is the same as [@Coser:2013qda]. Thus the classical partition function should only depend on the Riemann matrix of the surface. However, it is very hard to evaluate the Riemann matrix for a generic Riemann surface. The most well studied examples are the plane algebraic curves, i.e., the branched covering of $\mathbb{CP}^1$, for which one can find the Riemann matrix numerically [@Bobenko:2011; @Coser:2013qda]. For our case $\mathcal{T}_{n,m}$, which is defined as the $n$-sheeted covering of the torus gluing along $m$ branch cuts, the problem is quite different. Luckily, by using the powerful theta functions, we are able to propose a basis of $H_1(\mathcal{T}_{n,m},\mathbb{Z})$ and its dual basis of $H^1(\mathcal{T}_{n,m},\mathbb{C})$, such that the explicit form of the classical partition function can be found in terms of theta functions.
In the following we will construct the basis of $H^1(\mathcal{T}_{n,m},\mathbb{C})$. Notice that the genus of $\mathcal{T}_{n,m}$ is given by $
g=n+(m-1)(n-1)=m(n-1)+1,
$ which also gives the number of independent holomorphic one-forms. Remember that there is only one nontrivial holomorphic one-form for the torus, i.e., ${\mathrm{d}}z$. So all we need to do is to construct the rest $m(n-1)$ differentials, which should encode the information of the gluing procedure. Let’s now look more closely into the cut abelian differentials defined in , one should notice that they are nothing but a subsets of $H^1(\mathcal{T}_{n,m},\mathbb{C})$. Actually, there is another example: For free boson on the singular $Z_n$ curve , the collection of cut abelian differentials for $k=1,\cdots,n-1$ are exactly the basis of the holomorphic one-forms of curve [@Dixon:1986qv; @Bershadsky:1986fv]. Hence, one can similarly construct the basis of $H^1(\mathcal{T}_{n,m},\mathbb{C})$ out of the cut abelian differentials.
Inspired by the work of [@Atick:1987kd], we denote the branch points by $\{z_1,z_2,\cdots, z_{2m}\}$, so that there are $m$ branch cuts connecting $z_{2i-1}$ to $z_{2i}$ for $i=\{1,\cdots,m\}$. We propose the basis of holomorphic one-forms to be $$\label{eq:holomorphicdifferentials}
\begin{split}
&w_{s,i}(z){\mathrm{d}}z=w_{i+(s-1)m}(z){\mathrm{d}}z=\gamma_s(z)\vartheta_1(z-z_{2i-1}-Y_s)\prod_{j\ne i}^m\vartheta_1(z-z_{2j-1}){\mathrm{d}}z,\\
& s\in \{1,\cdots,n-1\}, i \in \{1,\cdots,m\}, \quad w_{n,1}{\mathrm{d}}z=w_{g}{\mathrm{d}}z={\mathrm{d}}z
\end{split}$$ where $$\begin{split}
\gamma_s(z)&\equiv \prod_{i=1}^{m}\vartheta_1(z-z_{2i-1})^{-(1-\frac{s}{n})}\vartheta_1(z-z_{2i})^{-\frac{s}{n}},\\
Y_s&=\left( \{ \frac{s(n-1)}{n}\}-1\right)\sum_{i=1}^{m}z_{2i-1}+\frac{s}{n}\sum_{i=1}^{m}z_{2i}.
\end{split}$$ Here $\{\bullet\}$ is a convention representing the fractional part of $\bullet$. Since $w_{s,i}(z){\mathrm{d}}z$ are just the independent cut abelian differentials for the $k=s$ mode defined in [@Atick:1987kd], as a consequence, the basis in are all independent by construction. One can also see that, the number of independent holomorphic one-forms is exactly the genus $g$.
Following [@enolski:2006], we are going to choose a basis of the homology a-cycles $\alpha_a$ and b-cycles $\beta_a$ similar as curve , except that there are additional loops inherited from the torus. As shown in figure \[fig:branchtorus\], they are denoted by $$\label{eq:canonicalbasis}
\begin{split}
&\alpha_{s,i}=\alpha_{(s-1)(m-1)+i}, \,\beta_{s,i}=\alpha_{(s-1)(m-1)+i},\\
&s\in \{1,\cdots,n-1\}, \, i \in \{1,\cdots,m-1\},\\
&a_{l}=\alpha_{(n-1)(m-1)+l}, \, b_{l}=\beta_{(n-1)(m-1)+l},\quad l\in \{1,\cdots, n\}.
\end{split}$$ We think the subscript $\{s,i\}$ is useful for the bookkeeping: $s$ label the sheet, counting from top to bottom, while $i$ label the branch points of odd order, counting from left to right.
![The canonical homology basis of $\mathcal{T}_{n,m}$, where $a_s$, $b_s$ are the canonical cycles of the $s$-th sheet torus. The $\alpha$ and $\beta$ cycles are chosen similarly as the algebraic curve in [@enolski:2006]. The solid line is on the first sheet, the broken line is on the $s$-th sheet, while the dotted line is on the $n$-th sheet. In each sheet, the number of $\alpha$ and $\beta$ cycles are both $m-1$.[]{data-label="fig:branchtorus"}](branchtorus.pdf){width="70.00000%"}
In order to do numerical calculation, the integrals in the period matrix should be performed on the first sheet. Let’s define the $\mathbf{A}$ and $\mathbf{B}$ periods as $$\label{eq:4integral}
\begin{split}
&\mathbf{A}_{s,r}^{i,j}=\oint_{\alpha_{s,i}}w_{r,j}, \, \mathbf{B}_{s,r}^{i,j}=\oint_{\beta_{s,i}}w_{r,j} \\
&\mathbf{A}_{l,r}^{j}=\oint_{a_{l}}w_{r,j},\, \mathbf{B}_{l,r}^{j}=\oint_{b_{l}}w_{r,j}\\
&i=1,\cdots,m-1,\,j=1,\cdots,m\\
&l=1,\cdots,n, \, r=1,\cdots,n-1
\end{split}$$ The contour integral of ${\mathrm{d}}z$ is trivial, the only non-zero integrals are $$\oint_{a_l}{\mathrm{d}}z=1, \,\oint_{b_l}{\mathrm{d}}z=\tau,$$ where $\tau$ is the moduli of the torus. The integrals in are related to the integral on the first sheet by a phase $$\begin{split} \label{eq:first3}
\mathbf{A}_{s,r}^{i,j}=\rho^{r(s-1)}\mathbf{A}_{1,r}^{i,j},\,\mathbf{A}_{l,r}^{j}=\rho^{l-1}\mathbf{A}_{1,r}^{j}, \,\mathbf{B}_{l,r}^{j}=\rho^{l-1}\mathbf{B}_{1,r}^{j},
\end{split}$$ where $\rho\equiv e^{2\pi {\mathrm{i}}/n}$. For $\mathbf{B}_{s,r}^{i,j}$, things get a little bit tricky, but we notice that $$\mathbf{B}_{s,r}^{i,j}=(\mathbf{B}_{s,r}^{i,j}-\mathbf{B}_{s+1,r}^{i,j})+(\mathbf{B}_{s+1,r}^{i,j}-\mathbf{B}_{s+2,r}^{i,j})+\cdots (\mathbf{B}_{n-2,r}^{i,j}-\mathbf{B}_{n-1,r}^{i,j})+\mathbf{B}_{n-1,r}^{i,j}.$$ Further, the action of the cyclical automorphism $J$ on the homology basis tells us that [@enolski:2006] $$\begin{split}
&J(\beta_{s,j})=\beta_{s+1,j}-\beta_{1,j}, \, s=1,\cdots,n-2,\\
&J(\beta_{s+1,j}-\beta_{s,j})=\beta_{s+2,j}-\beta_{s+1,j}, \quad J(\beta_{n-1,j})=-\beta_{1,j}
\end{split}$$ from which one can see that $$\rho^r(\mathbf{B}_{s,r}^{i,j}-\mathbf{B}_{s+1,r}^{i,j})=(\mathbf{B}_{s+1,r}^{i,j}-\mathbf{B}_{s+2,r}^{i,j}).$$ Then we have $$\label{eq:last1}
\begin{split}
\mathbf{B}_{s,r}^{i,j}&=\rho^{r(s-1)}(\mathbf{B}_{1,r}^{i,j}-\mathbf{B}_{2,r}^{i,j})+ \rho^{rs}(\mathbf{B}_{1,r}^{i,j}-\mathbf{B}_{2,r}^{i,j})+\cdots+\rho^{r(n-2)}(\mathbf{B}_{1,r}^{i,j}-\mathbf{B}_{2,r}^{i,j})\\
&=\frac{\rho^{r(s-1)}(1-\rho^{r(n-s)})}{1-\rho^r}(\mathbf{B}_{1,r}^{i,j}-\mathbf{B}_{2,r}^{i,j})\equiv \frac{\rho^{r(s-1)}(1-\rho^{r(n-s)})}{1-\rho^r}\mathbf{C}_{1,r}^{i,j},
\end{split}$$ where we have defined $\mathbf{C}_{1,r}^{i,j}=\mathbf{B}_{1,r}^{i,j}-\mathbf{B}_{2,r}^{i,j}$, which is an integral along a contour just enclosing the branch points $z_{2i}$ and $z_{2i+1}$.
Now, from and , one can see that all the integrals in the period matrix can written as the integrals on the first sheet, which can be calculated by contour integrals: $$\label{eq:firstsheetintegral}
\begin{split}
\mathbf{A}_{1,r}^{i,j}&=(\rho^{-r}-1)\sum_{k=1}^{i}\int_{z_{2k-1}}^{z_{2k}} w_{r,j},\\
\mathbf{C}_{1,r}^{i,j}&=\rho^{r/2}(\rho^{-r}-1)(-1)^{1/n}\int_{z_{2i}}^{z_{2i+1}} w_{r,j},\\
\mathbf{A}_{1,r}^j&=\int_{0}^{1}w_{r,j},\quad \mathbf{B}_{1,r}^j=\int_{0}^{{\mathrm{i}}\beta} w_{r,j}.
\end{split}$$ By now, all the elements in the period matrix $(\mathbf{A},\mathbf{B})$ can be numerically computed, so is the Riemann matrix $\mathbf{\Omega}$. To be more concrete, we give some examples of calculation in the appendix.
Finally, we are able to give the most general form of the partition function on $\mathcal{T}_{n,m}$: $$\begin{split} \label{eq:torus-general}
Z_{\mathcal{T}_{m,n}}=&\frac{N}{(\det \Im \tau) |\eta(\tau)|^4}\left(\prod_{k=1}^{n-1}|\vartheta_1(p)|^{2(m-1)(n-1)} \right) \prod_{i<j}^{m}\vartheta^{n-1}_{\alpha_i\alpha_j}\prod_{i<j}^{m}\bar{\vartheta}^{n-1}_{\beta_i\beta_j}\\
& \left(\prod_{k=1}^{n-1}|\det{\mathbf{W}}|^{-1}\prod_{i<j}^{2m}\vartheta_{ij}^{-(1-\frac{k_i}{n})(1-\frac{k_j}{n})}(\bar{\vartheta}_{ij})^{-\frac{k_i}{n}\frac{k_j}{n}}\right)
\left | \Theta(0|{\mathrm{i}}\frac{R^2}{2}M)\right |^2,
\end{split}$$ where $k_i=k$ for $i$ odd and $k_i=n-k$ for $i$ even and $
p=\frac{k}{n}\sum_{i=1}^{m}(z_{2i}-z_{2i-1}).
$ If one set $\tau={\mathrm{i}}\beta$ to be pure imaginary and all the branch cuts lie on the real interval: $0<z_1<z_2\cdots<z_{2m-1}<z_{2m}<1$, then the partition function corresponds to the Rényi entanglement entropy for $m$ disjoint intervals in a finite system at finite temperature $1/\beta$.
Conclusions
===========
In conclusion, we obtain the partition function of free compact boson on the $n$-sheeted covering of a torus gluing along $m$ branch cuts. In order to achieve that goal, we proposed a canonical homology basis and construct the holomorphic differentials in terms of theta functions, such that the period matrix can be directly constructed, as importantly, it is numerically computable. Given the period matrix, we are able to generalized the earlier results of $g=3$ results [@Liu:2015iia] to higher genus, which means that we can calculate the $n$-th Rényi entanglement entropy for arbitrary disjoint intervals in a finite system at finite temperature.
Mathematically, it is also very interesting to know if there are some Thomae type formulas for branched covering of torus, which may relate the positions of the branch points on the torus to the period matrix by theta functions. The basis and we proposed here may be useful for that purpose as well. Also, we should mention that in Ref.[@Matone:2001uy], there is a theorem (Theorem 3.) which states that the period matrix of the branch covering of the torus should satisfy the conditon: $$m_j^\prime -\sum_{k=1}^{g}\Omega_{ij}n_k^\prime=\bar{c}(m_j^\prime,n_k^\prime;m_j,n_k) \left(m_j-\sum_{k=1}^{g}\Omega_{ij}n_k \right), \quad m_j^\prime,n_k^\prime,m_j,n_k \in \mathbb{Z}.$$ One may use the method we proposed to calculate the coefficient $\bar{c}(m_j^\prime,n_k^\prime;m_j,n_k)$.
Acknowledgement
===============
We would like to thank Tianjun Li for carefully reading the manuscript. We also want to thank Wei Fu, Lina Wu for useful discussing.
Example: $\mathcal{T}_{2,2}$
============================
We are going to discuss $2$-sheeted covering of torus with four branch points, which was studied in [@Liu:2015iia]. The basis of holomorphic differentials are given by $$\begin{split}
&w_{1,1}=\gamma(z)\vartheta_1(z-z_1-Y_1)\vartheta_1(z-z_3){\mathrm{d}}z\\
&w_{1,2}=\gamma(z)\vartheta_1(z-z_3-Y_1)\vartheta_1(z-z_1){\mathrm{d}}z, \quad w_3={\mathrm{d}}z,
\end{split}$$ where $$Y_1=\frac{1}{2}(z_4-z_3+z_2-z_1),\, \gamma(z)=\prod_{i=1}^{4}\vartheta_1(z-z_i)^{\frac{1}{2}}.$$ The $\mathbf{A}$ and $\mathbf{B}$ periods are given in terms of the cut period matrix $$\mathbf{A}=
\begin{pmatrix}
{W_4}^1 & {W_4}^2 & 0 \\
{W_1}^1 & {W_1}^2 & 1 \\
-{W_1}^1 & -{W_1}^2 & 1
\end{pmatrix},
\,\mathbf{B}=
\begin{pmatrix}
{W_2}^1 & {W_2}^1 & 0 \\
-{W_3}^1 & -{W_3}^1 & \tau \\
{W_3}^1 & {W_3}^2 & \tau
\end{pmatrix}.$$ They are the contour integrals on the first sheet defined by : $$\begin{split}
W_2^{1(2)}& =\oint_{\gamma_2} {\mathrm{d}}z \omega^1(z)=(e^{2 \pi \mathrm{i}\frac{1}{2}}-1)\int_{z_2}^{z_3}{\mathrm{d}}z w^{1(2)}(z)\\
{W_3}^{1(2)} & =\oint_{\gamma_3} {\mathrm{d}}z w^{1(2)}(z)=\int_{0}^{-\tau }w^{1(2)}{\mathrm{d}}z \\
{W_4}^1 &=e^{-\frac{\mathrm{i} \pi}{2}}2 \mathrm{i} \sin(\frac{3}{2}\pi)(-1)^{-1/2}\int_{z_1}^{z_2}{\mathrm{d}}z \vartheta_1(z_1-z)^{-1/2}\vartheta_1(z-z_2)^{1/2} \vartheta_1(z-z_3)^{1/2}\vartheta_1(z-z_4)^{-1/2}\\
{W_4}^2 &=e^{-\frac{\mathrm{i} \pi}{2}}2 \mathrm{i} \sin(\frac{3}{2}\pi)(-1)^{1/2}\int_{z_1}^{z_2}{\mathrm{d}}z \vartheta_1(z_1-z)^{1/2}\vartheta_1(z-z_2)^{-1/2} \vartheta_1(z-z_3)^{-1/2}\vartheta_1(z-z_4)^{1/2}.
\end{split}$$ In order to simplify the formula, we define $$\begin{split}
x&= \mathrm{i} \frac{-{W_2}^2{W_3}^1+{W_2}^1{W_3}^2}{{W_1}^2{W_2}^1-{W_1}^1{W_2}^2}\\
y&= \mathrm{i} \frac{-2{W_1}^2{W_3}^1+2{W_1}^1{W_3}^2+{W_2}^2{W_4}^1-{W_2}^1{W_4}^2}{-4{W_1}^2{W_2}^1+4{W_1}^1{W_2}^2}\\
z&= \mathrm{i} \frac{-{W_1}^2{W_4}^1+{W_1}^1{W_4}^2}{-2{W_1}^2{W_2}^1+2{W_1}^1{W_2}^2},
\end{split}$$ and also assuming that $\tau$ is pure imaginary $\tau={\mathrm{i}}\beta$, then the period matrix $\Omega=\mathbf{B}\cdot\mathbf{A}^{-1}$ have a simpler form: $$\label{eq:omega22}
\Omega=
\left(
\begin{array}{ccc}
\frac{{\mathrm{i}}}{2 z} & \,-\frac{y}{2 z} &\, \frac{y}{2 z} \\
-\frac{y}{2 z} & \,{\mathrm{i}}\left(\frac{x z-y^2}{2 z}+\frac{\beta }{2}\right) &\, {\mathrm{i}}\left(\frac{y^2-x z}{2 z}+\frac{\beta }{2}\right) \\
\frac{y}{2 z} & \, {\mathrm{i}}\left(\frac{y^2-x z}{2 z}+\frac{\beta }{2}\right) & \,{\mathrm{i}}\left(\frac{x z-y^2}{2 z}+\frac{\beta }{2}\right) \\
\end{array}
\right).$$ where we have used a rather nontrivial results in [@Liu:2015iia] $$\label{eq:mir}
y^2-xz=\frac{1}{2}\frac{{W_3}^1{W_4}^2-{W_4}^1{W_3}^2}{{W_1}^1{W_2}^2-{W_2}^1{W_1}^2}.$$ The equality actually ensures the $T$-duality of the classical action [@Liu:2015iia]. One can also numerically check that is indeed a Riemann matrix, i.e., $\Im{\Omega}$ is symmetric and positive definite.
[99]{}
P. Calabrese and J. Cardy, *Entanglement entropy and conformal field theory*, J. Phys.A [**42**]{} (2009), 504005, arXiv:0905.4013
P. Calabrese, J. Cardy and E. Tonni, *“Entanglement entropy of two disjoint intervals in conformal field theory,”* J. Stat. Mech. [**0911**]{}, P11001 (2009) \[arXiv:0905.2069 \[hep-th\]\].
V.Z. Enolski and T. Grava, *“Thomae Type Formulae For Singular $Z_N$ Curves”*, Letters in Mathematical Physics (2006) 76:187-214.
Alexander I. Bobenko, Christian Klein, *“Computational Approach to Riemann Surfaces”*, Lecture Notes in Mathematics Volume 2013 (2011), Springer.
A. Coser, L. Tagliacozzo and E. Tonni, *“On Rényi entropies of disjoint intervals in conformal field theory,”* J. Stat. Mech. [**1401**]{} (2014) P01008 \[arXiv:1309.2189 \[hep-th\]\].
L. J. Dixon, D. Friedan, E. J. Martinec and S. H. Shenker, *“The Conformal Field Theory of Orbifolds,”* Nucl. Phys. B [**282**]{}, 13 (1987). R. Dijkgraaf, C. Vafa, E. P. Verlinde and H. L. Verlinde, *“The Operator Algebra of Orbifold Models,”* Commun. Math. Phys. [**123**]{} (1989) 485. M. Bershadsky and A. Radul, *“Conformal Field Theories with Additional Z(N) Symmetry,”* Int. J. Mod. Phys. A [**2**]{} (1987) 165. V. G. Knizhnik, *“Analytic Fields on Riemann Surfaces. 2,”* Commun. Math. Phys. [**112**]{}, 567 (1987). S. Datta and J. R. David, JHEP [**1404**]{}, 081 (2014) doi:10.1007/JHEP04(2014)081 \[arXiv:1311.1218 \[hep-th\]\]. B. Chen and J. q. Wu, Phys. Rev. D [**91**]{}, no. 10, 105013 (2015) doi:10.1103/PhysRevD.91.105013 \[arXiv:1501.00373 \[hep-th\]\]. H. J. Schnitzer, arXiv:1510.05993 \[hep-th\]. H. J. Schnitzer, arXiv:1611.03116 \[hep-th\].
J. J. Atick, L. J. Dixon, P. A. Griffin and D. Nemeschansky, *“Multiloop Twist Field Correlation Functions for $Z(N$) Orbifolds,”* Nucl. Phys. B [**298**]{}, 1 (1988).
S. G. Naculich and H. J. Schnitzer, Nucl. Phys. B [**332**]{}, 583 (1990). doi:10.1016/0550-3213(90)90003-V F. Liu and X. Liu, *“Two intervals Rényi entanglement entropy of compact free boson on torus,”* JHEP [**1601**]{} (2016) 058 \[arXiv:1509.08986 \[hep-th\]\].
P. Di Francesco, P. Mathieu and D. Senechal, *“Conformal Field Theory”*, Springer. *“NIST Handbook of Mathematical Functions”*, Cambridge University Press.
M. Matone, Trans. Am. Math. Soc. [**356**]{} (2004) 2989 doi:10.1090/S0002-9947-04-03587-1 \[math/0105051 \[math-ag\]\].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'While deep neural networks take loose inspiration from neuroscience, it is an open question how seriously to take the analogies between artificial deep networks and biological neuronal systems. Interestingly, recent work has shown that deep convolutional neural networks (CNNs) trained on large-scale image recognition tasks can serve as strikingly good models for predicting the responses of neurons in visual cortex to visual stimuli, suggesting that analogies between artificial and biological neural networks may be more than superficial. However, while CNNs capture key properties of the average responses of cortical neurons, they fail to explain other properties of these neurons. For one, CNNs typically require large quantities of labeled input data for training. Our own brains, in contrast, rarely have access to this kind of supervision, so to the extent that representations are similar between CNNs and brains, this similarity must arise via different training paths. In addition, neurons in visual cortex produce complex time-varying responses even to static inputs, and they dynamically tune themselves to temporal regularities in the visual environment. We argue that these differences are clues to fundamental differences between the computations performed in the brain and in deep networks. To begin to close the gap, here we study the emergent properties of a previously-described recurrent generative network that is trained to predict future video frames in a self-supervised manner. Remarkably, the model is able to capture a wide variety of seemingly disparate phenomena observed in visual cortex, ranging from single unit response dynamics to complex perceptual motion illusions. These results suggest potentially deep connections between recurrent predictive neural network models and the brain, providing new leads that can enrich both fields.'
author:
- |
William Lotter$^1$, Gabriel Kreiman$^1$, David Cox$^{1,2,3}$\
$^1$Harvard University, $^2$MIT-IBM Watson AI Lab, $^3$IBM Research\
[[email protected], [email protected], [email protected]]{}
bibliography:
- 'main.bib'
title: A neural network trained to predict future video frames mimics critical properties of biological neuronal responses and perception
---
Introduction
============
The fields of neuroscience and machine learning have long enjoyed productive dialogue, with neuroscience offering inspiration for how artificial systems can be constructed, and machine learning providing tools for modeling and understanding biological neural systems. Recently, as deep convolutional neural networks (CNNs) have emerged as leading systems for visual recognition tasks, these same models have emerged—without any modification or tailoring to purpose—as leading models for explaining the population responses of neurons in primate visual cortex [@Yamins_2014; @Yamins_2016; @Kriegeskorte_2014]. These results suggest that the connections between artificial deep networks and brains may be more than skin deep.
However, while deep CNNs capture some important details of the responses of visual cortical neurons, they fail to explain other key properties of the brain. Notably, the level of strong supervision used to train state-of-the-art CNNs is much greater than that available to our brain. To the extent that representations in the brain are similar to those in CNNs trained on e.g. ImageNet, the brain must be arriving at these representations by different, largely unsupervised routes. Another key difference is that CNNs are fundamentally static and lack a notion of time, whereas neuronal systems are highly dynamic, producing responses that vary dramatically in time, even in response to static inputs. Figure \[on\_off\]a shows a typical response profile of a visual cortical neuron to a static input [@Schmolesky_1998]. The neuron produces a brief transient response to the onset of the visual stimulus, followed by near total suppression of that response. When the stimulus is removed, the neuron responds again with a transient burst of activity (known as an “off” response). Neurons throughout visual cortex show a variety of dynamic response profiles, and the computational purpose of these dynamics is currently not well understood.
To further complicate matters, the responses of neurons in the primate visual cortex are also sensitive to long range temporal structure in the visual world. For instance, Meyer and Olson [@Meyer_2011] showed that neurons in inferior temporal cortex (IT) could be strongly modulated by prior experience with sequences of presented images. After repeated presentations of arbitrary images with predictable transition statistics (e.g. “image B always follows image A”), neurons appeared to learn the sequence statistics, responding robustly only to sequence transitions that were unexpected. The importance of temporal context in perception is further illustrated in various motion illusions, such as the flash-lag effect [@Nijhawan_1994; @Mackay_1958; @Eagleman_2000] and static motion illusions [@Watanabe_2018], where the motion of objects is incorrectly perceived by humans in predictable ways. Again, standard feedforward CNNs are insufficient to explain these temporal phenomena.
Here, inspired by past success in using “out-of-the-box” artificial deep neural networks as models of visual cortex, we explore whether modern predictive recurrent neural networks built for unsupervised learning can also explain dynamic phenomena in the brain. In particular, we consider a deep predictive coding network (“PredNet”; [@Lotter_2017]), a network that learns to perform next-frame prediction in video sequences [@Ranzato_2014; @Brabandere_2016; @Finn_2016; @Mathieu_2015; @SVVP; @VPN; @Vondrick_2017; @Villegas_2017v2; @Villegas_2017]. The PredNet is motivated by the principle of “predictive coding” [@Rao_1999; @Friston_2005; @Spratling_2012; @Chalasani_2013; @Wen_2018]; the network continually generates predictions of future sensory data via a top-down path, and it sends prediction errors in its feedforward path (Fig. \[architecture\]). At its lowest layer, the network predicts the input pixels at the next time-step, and it has been shown to make successful predictions in real-world settings (e.g. the KITTI car-mounted camera dataset [@Geiger2013IJRR]). The internal representations learned from video prediction also proved to be useful for subsequent decoding of underlying latent parameters of the video sequence, consistent with the suggestion of prediction as a good loss function for unsupervised learning [@Softky_1996; @Palm_2012; @Lotter_2015; @Wang_2015; @Mathieu_2015; @Srivastava_2015; @OReilly_2014; @Dosovitskiy_2017; @Finn_2016].
Predictive coding has a rich history in neuroscience literature [@Rao_2000; @Summerfield_2006; @Bastos_2012; @Kanai_2015; @Srinivasan_1982; @Atick_1992]. Rao and Ballard helped popularize the notion of predictive coding in neuroscience in 1999, proposing that spatial predictive coding could explain a key property of neurons in primary visual cortex (V1) known as end-stopping ([@Rao_1999]; see Section \[end\_stopping\]). Predictive coding has also been proposed as an explanatory framework for a variety of sensory systems in neuroscience [@Sukhbinder_2011; @Zelano_2011; @Mumford_1992]. The PredNet formulates predictive coding principles in a deep learning framework to work on natural sequences, providing an opportunity to test a wide range of neuroscience phenomena using a single model. Below, we show that despite being trained only to predict next frames in video sequences, the PredNet naturally captures a wide array of seemingly unrelated fundamental properties of neuronal responses and perception, including on/off dynamics, length suppression, sequence learning effects in visual cortex, norm-based coding of faces, illusory contours, and the flash-lag illusion.
Deep Predictive Coding Networks
===============================
The deep predictive coding network proposed in [@Lotter_2017] (“PredNet”) consists of repeated, stacked modules where each module generates a prediction of its own feedforward inputs, computes errors between these predictions and the observed inputs, and then forwards these error signals to subsequent layers. The model consists of four components: targets to be predicted ($A_l$), predictions ($\hat{A}_l$), errors between predictions and targets ($E_l$), and a recurrent representation from which predictions are made ($R_l$). On an initial time step, the feedforward pass can be viewed as a standard CNN, consisting of alternating convolutional and pooling layers. Predictions are made in a top-down pass via convolutions over the representational units, which are first updated using the representational units from the layer above and errors from the previous time step as inputs. The $R_l$ units are implemented as convolutional LSTMs [@Hochreiter_1997; @Shi_2015]. Here, for the sake of biological interpretability, we replace the $tanh$ output activation function for the LSTMs with a $relu$ activation, enforcing positive “firing rates”. On the KITTI dataset this leads to a marginally ($8$%) worse prediction mean-squared error (MSE) than the standard formulation, but it is still $2.6$ times better than the MSE that would be obtained by simply copying the last frame seen (compared to $2.8$ for $tanh$).
The error layers in the model, $E_l$, are calculated as a simple difference between the targets and predictions, followed by splitting into positive and negative error populations with $relu$ rectification. The loss function for the network is set as the (weighted) sum of the error activations across each layer. We utilize the $L_{all}$ formulation presented in the original paper, which places a non-zero loss on the error unit activity in every level in the network. Except where stated otherwise, results presented here use a model trained on the KITTI car-mounted camera dataset [@Geiger2013IJRR]. The same model hyperparameters were used as presented in the paper (besides the $relu$ activation in the LSTM units). Particularly, the model consists of four layers. With $0$-indexing used here, Layer $1$ would be analogous to V1 in visual cortex.
![Deep Predictive Coding Networks (PredNets) [@Lotter_2017]. Left: Each layer consists of representation neurons ($R_l$), which output a layer-specific prediction at each time step ($\hat{A}_l$), which is compared against a target ($A_l$) to produce an error term ($E_l$), which is then propagated laterally and vertically. Right: Module operations for case of video sequences. The target at the lowest layer of the network, $\hat{A}_0$, is set to the current input image.[]{data-label="architecture"}](prednet.pdf){width="62.00000%"}
Single Neuron Response Properties
=================================
We begin by comparing the response properties of units in the PredNet to established single unit response properties of neurons in the primate visual system, which have been studied extensively using microelectrode recordings. Here, we primarily compare response properties in the PredNet’s error (“E”) units, the output units of each layer, to neuronal recordings in the superficial layers of cortex. Response properties of other units in the PredNet (e.g. the “R” units) are included in the Supplemental Materials, and would likely map onto other parts of the cortical circuit.
#### On/Off Temporal Dynamics
As mentioned in the introduction, a conspicuous feature of visual cortical neuron responses is that they are highly dynamic, even when a static, unchanging image is presented to the subject. As an example of the commonly seen pattern of image on/off dynamics, Fig. \[on\_off\]a shows a raster plot and peri-stimulus-time histogram of a recorded neuron in the secondary visual cortex (V2) of a macaque monkey [@Schmolesky_1998]. Peaks in firing rate shortly after image display and removal are prominent. Fig. \[on\_off\]b shows the average response of PredNet $E$ units in different layers over a set of $25$ naturalistic objects appearing on a gray background. The on/off dynamics are apparent on the population average level, for all four layers of the network. These dynamics are also evident at the individual unit level, as illustrated in the Supplement, though there is variability. While on/off dynamics have an an “error”-like quality—an object unpredictably appears and disappears—these dynamics manifest in the $A$ and $R$ layers as well (see Supplement).
![On/off temporal dynamics. Left: Exemplar macaque V2 neuron responding to a static image. Reproduced with permission from Schmolesky et al. [@Schmolesky_1998]. Right: PredNet response to a set of naturalistic objects appearing on a gray background, after training on KITTI. Responses are grouped by layer for the $E$ units, and averaged across all units and all stimuli, per layer. ](on_off_kitti_relu_E.pdf){width="90.00000%"}
\[on\_off\]
#### End-Stopping and Length Suppression {#end_stopping}
Prediction in time and prediction in space are inextricably intertwined. As Rao and Ballard [@Rao_1999] illustrated, end-stopping in V1 can be explained by spatial predictive coding. End-stopping, or length suppression, is the phenomenon where a neuron tuned for a particular orientation becomes less responsive to a bar at this orientation, when the bar extends beyond its classical receptive field [@Hubel_1968]. The predictive coding explanation is that lines/edges tend to be continuous in nature, and thus the center of a long bar can be predicted from its flanks. A short, discontinuous bar, however, deviates from natural statistics, and responding neurons signal the deviation. One potential source for conveying the long range predictions in the case of an extended bar could be feedback from higher visual areas with larger receptive fields. This hypothesis was elegantly tested in Nassi et al. [@Nassi_2013] using reversible inactivation of V2 paired with V1 recordings in the macaque. As illustrated in the left side of Fig. \[length\_suppression\], cryoloop cooling of V2 led to a significant reduction in length suppression, indicating that feedback from V2 to V1 is essential for the effect.
![Length suppression. Left: Responses of example macaque V1 units to bars of different lengths before (red), during (blue), and after (green) inactivation of V2 via cryoloop cooling. Reproduced with permission from [@Nassi_2013]. Right: PredNet after training on the KITTI dataset – average $E_1$ and $A_1$, and examples. Red: Original network. Blue: Feedback weights from $R_2$ to $R_1$ set to zero.[]{data-label="length_suppression"}](length_suppression_relu.pdf){width="100.00000%"}
The right side of Fig. \[length\_suppression\] demonstrates that length suppression, and its mediation through top-down feedback, are also present in the PredNet. The upper left and right panels contain the mean normalized response for units in the $E_1$ and $A_1$ layers, respectively, to bars of different lengths. The red curves correspond to the original network (trained on KITTI) and the blue curves correspond to zero-ing the feedback from $R_2$ to $R_1$. For each filter channel, a set of 2D Gabor wavelet stimuli was first used to determine the optimal orientation. Responses to bars of different length at this orientation were then measured, as a sum of the activations over stimulus duration ($10$ time steps). The bottom row contains exemplar $E_1$ and $A_1$ units. Quantifying percent length suppression (%LS) as $100*\frac{R_{max} - R_{longest \: bar}}{R_{max}}$, the median decrease in %LS upon removing top-down signaling was $16$% for $E_1$ units ($p < 0.05$, Wilcoxon signed rank test) and $33$% for $A_1$ units ($p < 0.0005$). For $R_1$ units, the median %LS decrease was $2$% ($p=0.18$). Indeed, the average $R_1$ response did not exhibit much length suppression (see Supplement), though, there were particular examples with a strong effect.
#### Sequence Learning Effects in Visual Cortex
Predictions are often informed by recent experience, and violations of these predictions can be highly salient. Meyer and Olson [@Meyer_2011] provided a striking example of this in visual cortex via image sequence learning. The authors exposed monkeys to image pairs in a fixed order for over $800$ trials for each pair. The left panel of Fig. \[image\_pairing\] shows the mean response of $81$ IT neurons in a subsequent testing period, for predicted and unpredicted pairs. When the second image differs from expectations, the response is much stronger than when the expected image is presented.
![Predicted vs. unpredicted image transitions. Left: Mean of $81$ neurons recorded in macaque (IT). Reproduced with permission from [@Meyer_2011]. Right: Mean ($\pm$ SE) across PredNet $E_3$ units.[]{data-label="image_pairing"}](image_pairing-relu.pdf){width="90.00000%"}
![Learned image transitions. Top: Predictions of a KITTI-trained PredNet model on an example sequence. Middle: PredNet predictions after repeated “training” on the sequence. Bottom: PredNet predictions for an unpredicted image transition.[]{data-label="image_pairing_examples"}](image_pairing_examples-relu_v2.pdf){width="90.00000%"}
The right panel of Fig. \[image\_pairing\] demonstrates a similar effect in the PredNet after an analogous experiment. The model was trained on five image pairs for $800$ epochs. Fig. \[image\_pairing\_examples\] contains an example sequence and the corresponding next-frame predictions before and after the training. The model, prior to exposure to the images in this experiment (trained only on KITTI), settles into a noisy, copy-last-frame prediction mode. After exposure, the model is able to successfully make predictions for the expected image pair (row $2$). Since the chosen image pair is unknown *a priori* and the model is fully convolutional, the initial prediction is the constant gray background when the first image appears. The model then rapidly copies this image for the ensuing three frames. Next, the model successfully predicts the transition to the second object (a stack of tomatoes in this case). In row $3$, a sequence that differs from the training pair is presented. The model still makes the prediction of a transition to tomatoes, even though a chair is presented, but then copies the chair into subsequent predictions. Fig. \[image\_pairing\] shows that the unexpected transitions result in a larger average response in the final $E$ layer of the network. In fact, in all levels and all unit types ($E$, $A$, $R$), there is a larger response to the unpredicted images (Supp. Table 1). The overall magnitude of the difference is similar for $A$ and $E$, and is lower for $R$.
#### Norm-Based Coding of Faces
The representation of deviations from expectations can also explain observed neural embeddings of familiar stimuli such as faces. The norm-based coding theory suggests that faces are encoded with respect to a mean face [@Rhodes_2006]. Leopold et al. [@Leopold_2006] demonstrated that face-responsive neurons in macaque anterior IT are frequently tuned monotonically (often positively) to directions away from an average face (Fig. \[norm\_faces\]). This was tested by using synthetic faces with continuously varying levels of caricature. Training for next-frame prediction of rotating, computer-generated faces [@Lotter_2017], we see similar effects in the PredNet. The faces were created using software implementing a principal component analysis of a corpus of human faces (FaceGen [@facegen]). For training, $16K$ sequences were generated with a random face, initial orientation, and rotation velocity. The blue curve in Fig. \[norm\_faces\]b illustrates the post-training response of $E$ units in the network as a function of caricaturization level for $200$ previously unseen faces. The response is calculated by first averaging the response of all units in a given layer, then averaging the layer responses. The PredNet $E$ units become significantly more responsive to increasing levels of caricature after predictive training compared to the random initialization (red curve). This effect is diminished when the same network is trained on the same set of images, but in a static, autoencoder fashion (yellow curve). Note that the randomly initialized network already responds more highly to caricature, likely because the caricature faces tend to have higher contrast, sharper edges, etc., which even random CNNs can be tuned for [@Saxe_2010]. When training on an unrelated dataset (e.g. the KITTI dataset), this effect is reduced (purple curve). Training the network on rotating faces that had been generated using half the standard deviation for each principal component results in an even larger caricature response (green curve). All of these effects are consistent in the $A$ units as well, though the results are mixed for $R$ (Supplement).
![Norm-based coding of faces. a): Population response of $37$ neurons recorded in macaque IT for four different faces, as a function of caricaturization. Reproduced with permission from [@Leopold_2006]. b): Mean PredNet $E$ unit responses for models trained on different stimuli. Faces - Rotating synthetic faces [@Lotter_2017]. RandInit - Random initial weights. Static Faces - Same collection of images used in Rotating Faces except presented statically. Faces 0.5 SD - Rotating faces except each face generated from a distribution with half of the original standard deviation. c): Comparing PredNet with VGG16. VGG Faces - VGG trained in a siamese manner on a same/different task using the static face images.](norm_faces_summary-relu_v4.pdf){width="90.00000%"}
\[norm\_faces\]
Fig. \[norm\_faces\]c compares the PredNet responses to a popular CNN, VGG16 [@Simonyan_2015]. VGG responses were quantified by averaging over the outputs of its five convolutional blocks (after the max-pooling). Similar to the PredNet, the randomly initialized VGG16 displayed higher activity for more caricatured faces. ImageNet training decreased this effect, akin to the PredNet KITTI training. To train VGG on the synthetic faces, a same/different task was performed using the network in a siamese fashion. Using the same images as the PredNet, a training example consisted of a pair of images at different orientations with a binary cross-entropy objective for same/different identity classification. This training procedure resulted in an increased response to higher caricature levels (at least on the original dataset – yellow curve in Fig. \[norm\_faces\]c), although somewhat less than the PredNet $E$ units. While analogous norm-based face encoding effects can be seen with discriminatively-trained (VGG) models, the PredNet architecture is able to capture these same effects (even more strongly) in a fully unsupervised way.
Visual Illusions
================
Visual illusions can provide powerful insight into the underpinnings of perception. Here we demonstrate that the PredNet exhibits correlates of two illusions: illusory contours and the flash-lag effect, both of which have aspects of spatial and temporal prediction. The PredNet has also recently been shown to predict the illusory motion perceived in the rotating snakes illusion [@Watanabe_2018].
#### Illusory Contours
Illusory contours, as in the Kanizsa figures [@Kaniza], elicit perceptions of edges and shapes, despite the lack of enclosing lines. Lee et al. [@Lee_2001] found that neurons in monkey V1 can be responsive to illusory contours, albeit at a reduced response and increased latency to physical contours. Fig. \[illusory\_contours\]a contains an example of such a neuron. The stimuli in the experiment consisted of sequences starting with an image of four circles, which then abruptly transitioned to one of numerous test images, including the illusion. Illustrated in Fig. \[illusory\_contours\]b, the population average of $49$ superficial V1 neurons responded more strongly to the illusion than similar, but non-illusory stimuli. This preference was also apparent in V2, with a response that was, interestingly, of a shorter latency compared to V1.
![Illusory contours. Top: Reproduced with permission from Lee et al. [@Lee_2001]. A given trial consisted of the four circles image abruptly transitioning to one of the displayed test images.[]{data-label="illusory_contours"}](illusory_contours-kitti_relu_v2.pdf){width="100.00000%"}
Fig. \[illusory\_contours\]c-e demonstrate that the effects discovered by [@Lee_2001] are also present in the PredNet. In the population average of $E_1$ units, there is indeed a response to the illusory contour, and at an increased latency compared to a physical line square (Fig. \[illusory\_contours\]c). The response was calculated separately for each filter channel, by first finding the optimal orientation using short bar segments. The responses were then normalized (division by max response over all stimuli) and averaged. Fig. \[illusory\_contours\]d illustrates that the average $E_1$ response was higher for the illusory contour than the similar control images. This was also the case for $E_2$ units, with a peak response one time step before $E_1$. Indeed, the size of the stimuli was such that it was larger than the feedforward receptive field of the layer $1$ neurons, but smaller than that of the layer $2$ neurons (matching the protocol of [@Lee_2001]). Quantifying the preference of the illusion to the amodal and rotated “J” images for each individual unit [@Lee_2001]), we find that the average is positive (more preference to the illusion) for all tested layers ($E$, $A$, $R$, layers 1,2).
#### The Flash-Lag Effect
Another illusion for which prediction has been proposed as having a role is the flash-lag effect. Fundamentally, the flash-lag effect describes illusions where an unpredictably appearing stimulus (e.g. a line or dot) is perceived as “lagging” a predictably moving stimulus nearby, even when the stimuli are, in fact, precisely aligned in space. These illusions are sometimes interpreted as evidence that the brain is performing inference to predict the likely true current position of a stimulus, even in spite of substantial latency (up to hundreds of milliseconds) in the visual system [@Khoei_2017]. The version of the illusion tested here consists of an inner, continuously rotating bar and an outer bar that periodically flashes on. Fig. \[flash\_lag\_example\] contains an example prediction by the PredNet on a sample sequence within a flash-lag stimulus. The rotation speed of the inner bar in the clip was set to $6$ degrees per time step. The first feature of note is that the PredNet is indeed able to make reasonable predictions for the inner rotating bar. Quantifying this, the average angle of the bar in the outputted predictions is 1.41.2(s.d.) behind the actual bar (see Supp. Methods), which is significantly less than a $6$difference, which would result from simply copying the last seen frame. Again, the model was trained on real-world videos, so the generalization to this impoverished stimulus is non-trivial. Secondly, the post-flash predictions made by the model tend to resemble the perceived illusion. The average angular difference between the predicted outer bar and inner bar is 6.82.0. Considering that the model was trained for next frame prediction on a corpus of natural videos, this suggests that our percept matches the statistically predicted next frame (as estimated by the PredNet) more than the actual next frame. This natural statistics interpretation of the flash-lag illusion has, in fact, been similarly suggested by Wojtach et al. [@Wojtach_2008].
![Flash-lag effect. Top: Stimulus clip. Bottom: PredNet predictions after KITTI training.[]{data-label="flash_lag_example"}](flash_lag_example_relu-v3.pdf){width="80.00000%"}
Discussion
==========
We have shown that an off-the-shelf recurrent neural network trained to predict future video frames can explain a wide variety of seemingly unrelated phenomena observed in visual cortex and visual perception. These phenomena range from the details of responses of individual neurons, to complex visual illusions. Importantly, throughout, we used a base model trained on natural videos. Our work adds to a growing body of literature showing that deep neural networks trained to perform relevant tasks can serve as surprisingly good models of biological neural networks, often even outperforming models designed to explain neuroscience phenomena.
While we have shown that the PredNet architecture demonstrates a wide range of phenomena reminiscent of biology, we do not claim that the PredNet architecture *per se* is required to explain these phenomena. Rather, we argue that the network is *sufficient* to produce these phenomena, and we note that explicit representation of prediction errors in units within the feedforward path of the PredNet provides a straightforward explanation for the transient nature of responses in visual cortex in response to static images. That a single, simple objective—prediction—can produce such a wide variety of observed neural phenomena underscores the idea that prediction may be a central organizing principle in the brain [@Rao_1999], and points toward fruitful directions for future study in both neuroscience and machine learning.
### Acknowledgments {#acknowledgments .unnumbered}
This work was supported by IARPA (contract D16PC00002), the National Science Foundation (NSF IIS 1409097), and the Center for Brains, Minds and Machines (CBMM, NSF STC award CCF-1231216).
Supplementary Material
======================
On/Off Temporal Dynamics
------------------------
Temporal dynamics were tested with a set of $25$ objects. Examples of the objects can be seen in Fig. \[image\_pairing\_examples\]. Testing sequences consisted of a gray background for $7$ time steps, followed by an object on the background for $6$ time steps. As a general theme, we see some diversity in the response profiles of all units, but especially those in the “R” layers. We have focused in the main text on the “E” units, which most naturally map onto Layer 2/3 cortical pyramidal neurons (which are the output units in a putative cortical microcircuit). Diversity of responses is also observed throughout the neuroscience literature, and we hypothesize that to the extent that units in the PredNet map in a direct way onto cortical circuits [@Bastos_2012], the less-often experimentally sampled deep neurons might be reasonable analogs to the “R” units. We present representative units from all parts of the PredNet here for completeness.
Summary responses for the $A$ and $R$ units are contained in Fig. \[on\_off\_AR\]. The responses are grouped per layer and consist of an average across all the units (all filters and spatial locations) in a layer. The mean responses were then normalized between $0$ and $1$. Responses for layer $0$, the pixel layer, are omitted in Fig. \[on\_off\_AR\] because of their heavy dependence on the input pixels for the $A$ and $R$ layers. Note that, by notation in the network’s update rules, the input image reaches the $R$ layers at a time step after the $E$ and $A$ layers.
As illustrated in Fig. \[on\_off\_AR\], the $A$ and $R$ layers seem to generally exhibit on/off dynamics, similar to the $E$ layers. $R_1$ also seems to have another mode in its response, specifically a ramp up between time steps $3$ and $5$ post image onset. As will be illustrated below, this results from a few strongly firing neurons that exhibit this pattern.
![Average temporal dynamics in $A$ and $R$ units in response to set of naturalistic objects on a gray background, after training on the KITTI dataset.[]{data-label="on_off_AR"}](on_off_AR_relu.pdf){width="100.00000%"}
Fig. \[E\_raster\]-\[R\_raster\] illustrate the variety of responses present among the units in the model. Each plot for a given layer shows all the active units in the layer at the central receptive field. The average response for each unit over the $25$ images is shown, and each row is normalized to have a max of one. In each of the plots, it is apparent that a large proportion of the neurons have a peak response closely following image onset and/or offset. However, there are a good number of neurons that have peaks at different times. Overall, the on and off responses for individual neurons tend to be asymmetric – some neurons have stronger “on” responses, some have stronger “off” responses.
![Mean response of each active unit in the $E$ layer at the central receptive field. Each row (unit) is normalized by its max response.[]{data-label="E_raster"}](rasters_E_kitti_relu.pdf){width="100.00000%"}
![Mean response of each active unit in the $A$ layer at the central receptive field. Each row (unit) is normalized by its max response.[]{data-label="A_raster"}](rasters_A_kitti_relu.pdf){width="100.00000%"}
![Mean response of each active unit in the $R$ layer at the central receptive field. Each row (unit) is normalized by its max response.[]{data-label="R_raster"}](rasters_R_kitti_relu_v2.pdf){width="90.00000%"}
Figure \[Layer1\_raster\] is similar to the previous plots, except a global normalization is used instead of row normalization. There are subsets of neurons that are particularly more active than others. For $R_1$, rows $23$-$25$ contain examples of units that contribute to the ramping behavior from time-steps $3$ to $5$ in Fig. \[on\_off\_AR\].
![Mean response of each active unit with a central receptive field for all three unit types at Layer $1$. Responses are normalized globally per each unit type.[]{data-label="Layer1_raster"}](rasters_layer1_kitti_relu-not_normed.pdf){width="100.00000%"}
End-Stopping and Length Suppression {#end-stopping-and-length-suppression}
-----------------------------------
Responses to bars of different length were quantified for the length suppression experiment as a sum over the stimulus duration of $10$ time steps. The bars appeared on a gray background, which was first presented to the network for $5$ time steps, to allow the network to settle to a steady state before stimulus presentation. For each filter channel, the response at the central receptive field was quantified and normalized to unit maximum before averaging. The average $R_1$ response and two exemplar units are displayed in Fig. \[end\_stopping\_R\]. As mentioned in the main text, $R_1$ did not have a significant length suppression effect, with some neurons showing length suppression (right panel) and others showing an opposite effect (middle panel).
![Length suppression analysis for $R_1$ units.[]{data-label="end_stopping_R"}](end_stopping-kitti_keras2_relu_R.pdf){width="100.00000%"}
Sequence Learning Effects in Visual Cortex
------------------------------------------
For the exposure training phase in the learned sequence experiment, the Adam [@Kingma_2014] optimizer was used with default parameters. Table \[sequence\_table\] contains the percent increase in response between predicted and unpredicted sequences for each layer.
Unit Type Layer $0$ Layer $1$ Layer $2$ Layer $3$
----------- ----------- ----------- ----------- -----------
E $308$ $90$ $109$ $108$
A N/A $78$ $109$ $108$
R N/A $18$ $19$ $30$
: Percent increase of response between predicted and unpredicted sequences[]{data-label="sequence_table"}
Norm-Based Coding of Faces
--------------------------
For the faces generated for the norm-based coding experiment, a caricature level of, say $2$, corresponds to having all principal components with a of magnitude $2$ (either positive or negative). The hyperparameters of the tested PredNet model were chosen to match those of the rotating faces model in the original paper [@Lotter_2017]. Fig. \[norm\_faces\_AR\] shows the responses of the $A$ and $R$ units to the caricature faces. Responses are calculated as an average per each layer, and then averaged across layers. Training on rotating faces led to a much higher caricature response in the $A$ units, especially for training on faces generated with half of the original principal component standard deviation. The lower standard deviation had a similar effect in the $R$ units, although training on the original rotating faces actually led to a smaller caricature response than the initial weights. The siamese VGG network used for the same/diff face identification task was constructed by taking the squared element-wise difference between the flattened features at the last convolutional layer for the two inputs, followed by a fully-connected, softmax classification layer.
![Responses of PredNet $A$ and $R$ units to varying levels of caricaturized faces, trained in different settings. Faces - Rotating synthetic faces. RandInit - Random initial weights. Static Faces - Same collection of images used in Rotating Faces except presented statically. Faces 0.5 SD - Rotating faces except each face generated from a distribution with half of the original standard deviation. []{data-label="norm_faces_AR"}](norm_faces_summary-relu_AR_v2.pdf){width="80.00000%"}
Illusory Contours
-----------------
Fig. \[illusory\_contours\_AR\] contains the illusory contour response plots for the $A$ and $R$ layers. The stimuli sequences consisted of $10$ time steps of the “four circles” image (see main text) followed by a test image for $10$ time steps. The response to the illusory stimuli begins one time step after the response to the line square for all unit types in the first layer.
![Illusory contours responses for $A$ and $R$ units.[]{data-label="illusory_contours_AR"}](illusory_contours-kitti_relu-AR.pdf){width="100.00000%"}
To quantify illusory responsiveness, we follow Lee et al. [@Lee_2001] in calculating the following two measures: $IC_a = \frac{R_i - R_a}{R_i + R_a}$ and $IC_r = \frac{R_i - R_r}{R_i + R_r}$, where $R_i$ is the response to the illusory contour (sum over stimulus duration), $R_a$ is the response to amodal stimuli, and $R_r$ is the response to the rotated J image. These indices were calculated separately for each unit with a non-uniform response. For both measures and all examined layers, the average across the layer was positive (significant in half of the calculations (Table \[illusory\_table\])).
Source Layer $IC_A$ $IC_R$ Layer $IC_A$ $IC_R$ Layer $IC_A$ $IC_R$ Layer $IC_A$ $IC_R$
---------- ------- -------- ---------- ------- ---------- -------- ------- ---------- -------- ------- ---------- --------
Monkey A V1S $0.19$ $0.31$ V2S $0.21$ $0.11$ V1D $0.09$ $0.11$ V2D $0.28$ $0.24$
Monkey B V1S $0.10$ $0.16$ V2S $0.08$ $0.12$ V1D $0.04$ $0.13$ V2D $0.07$ $0.20$
PredNet $E_1$ $0.09$ $0.14$\* $E_2$ $0.15$\* $0.09$ $R_1$ $0.11$\* $0.04$ $R_2$ $0.12$\* $0.03$
PredNet $A_1$ $0.03$ $0.10$\* $A_2$ $0.15$\* $0.09$
: Illusory responsiveness measures for the units in Lee et al. [@Lee_2001] and the PredNet. $IC_a$ and $IC_r$ compare the response of the illusion to the amodal and rotated stimuli, respectively. Positive measures indicate a preference to the illusion. \*$p<0.05$ (T-test)[]{data-label="illusory_table"}
Flash-Lag Effect
----------------
The flash-lag stimulus was created with a rotation speed of $6$per time step, with a flash every $6$ time steps for $3$ full rotations. Angles of the predictions were quantified over the last two rotations. The angles of the predicted bars were estimated by calculating the mean-squared error (MSE) between the prediction and a probe bar generated at $0.1$increments and a range of centers, and taking the angle with the minimum mean-squared error. Fig. \[flash\_lag\_post\_flashes\] contains additional predictions by the model after four consecutive flashes.
![Four consecutive post-flash predictions by the PredNet model following training on the KITTI dataset.[]{data-label="flash_lag_post_flashes"}](flash_lag_post_flashes-4nt_kitti_keras2_relu_v3.pdf){width="100.00000%"}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
The present works deals with gravitational collapse of cylindrical viscous heat conducting anisotropic fluid following the work of Misner and Sharp. Using Darmois matching conditions, the dynamical equations are derived and the effect of charge and dissipative quantities over the cylindrical collapse are analyzed. Finally, using the Miller-Israel-Steward causal thermodynamic theory, the transport equation for heat flux are derived and its influence on collapsing system has been studied.\
Keywords : Cylindrical collapse, Dissipation, heat flux, Junction conditions, Dynamical equations .
---
0.2cm 0.2cm
\
Introduction
============
A challenging but curious issue in gravitational physics as well as in relativistic astrophysics is to know the final fate of a continual gravitational collapse. The stable configuration of a massive star persists as long as the inward pull of gravity is neutralized by the outward pressure of the nuclear fuel at the core of the star. Subsequently, when the star has exhausted its nuclear fuel there is no longer any thermonuclear burning and there will be endless gravitational collapse. However, depending on the mass of the collapsing star, the compact objects such as white dwarfs, neutron stars and black holes are formed. In white dwarf and neutron star gravity is counter balanced by electron and neutron degeneracy pressure respectively while black hole is an example of the end state of collapse.\
The study of gravitational collapse was initiated long back in 1939 by Oppenheimer and Snyder \[1\]. They have studied the collapse of a homogeneous spherical dust cloud in the frame work of general relativity. Then after a quarter century, a more realistic investigation was done by Misner and Sharp\[2\] with perfect fluid in the interior of a collapsing star. In both the studies, the exterior of the collapsing star was chosen as vacuum. Vaidya\[3\] formulated the non-vacuum exterior of a star having radiating fluid in the interior. An inhomogeneous spherically symmetric dust cloud was analytically studied by Joshi and Singh \[4\] and they have shown that the final fate of the collapsing star depend crucially on the initial density profile and the radius of the star. Debnath etal\[5\] investigated collapse dynamics of the non-adiabatic fluid, considering quasi-spherical Szekeres space-time in the interior and plane symmetric Vaidya solution in the exterior region.\
Although most of the works on collapse dynamics are related to spherical objects, still there are interesting information about self-gravitating fluids for collapsing object with different symmetries. The natural choice for non-spherical symmetry is axis symmetric objects. The vacuum solution for Einstein field equations in cylindrically symmetric space-time was obtained first by Levi-Civita \[6\] but still it is a challenging issue of interpreting two independent parameters in the solution. Herrera etal \[7\] studied cylindrical collapse of non-dissipative fluid with exterior Einstein -Rosen space-time and showed wrongly a non-vanishing radial pressure on the boundary surface and subsequently in collaboration with M.A.H. Maccallum \[8\] they corrected the result. Then Herrera and collaborators investigated cylindrical collapse of matter with\[9\] or without shear \[10\].\
Further, the junction conditions due to Darmois \[11\] has a very active role in dealing collapsing problems. Sharif etal \[12-14\] showed the effect of positive cosmological constant on the collapsing process by using junction conditions between static exterior and non-static interior with a cosmological constant. Also Herrera etal \[15\], using junction conditions were able to prove that any conformally flat cylindrically symmetric static source cannot be matched to the Levi-Civita space-time. Then Kurita and Nakao \[16\] formulated naked singularity along the axis of symmetry, considering cylindrical collapse with null dust.\
Moreover from realistic point of view it is desirable to consider dissipative matter in the context of collapse dynamics \[17-19\]. Considering collapse of a radiating star with dissipation in the form of radial heat flow and shear viscosity, Chan\[20\] has showed that shear viscosity plays a significant role in the collapsing process. Collapse dynamics with dissipation of energy as heat flow and radiation has been studied by Herrera and Santos\[18\]. Subsequently, by Considering of causal transport equations related to different dissipative components (heat flow, radiation, shear and bulk viscosity) Herrera etal \[15,21,22\] investigated the collapse dynamics. The same collapsing process with plane symmetric geometry or others has been examined by Sharif etal \[23,24\] .\
On the other-hand, in the context of gravitational waves, the sources must have non spherical symmetry. Further, cylindrical collapse of non-dissipative fluid with exterior containing gravitational waves shows non-vanishing pressure on the boundary surface by using Darmois matching conditions. Recently, it has been verified \[25\] in studying cylindrical collapse of anisotropic dissipative fluid with formation of gravitational waves outside the collapsing matter.\
In the present work, following Misner and Sharp collapse dynamics of viscous, heat conducting charged anisotropic fluid in cylindrically symmetric background will be studied. The paper is organized as follows. Section 2 deals with basic equations related to interior and exterior space-time. The junction conditions are evaluated and discussed in Section 3. The dynamical equations are derived and studied in Section 4. Finally, the process of mass, heat and momentum transfer through transport equation is discussed in section 5.\
Interior and exterior space-time: Basic equations.
==================================================
Mathematically, the whole four dimensional space-time manifold having a cylindrical collapsing process can be written as $M=M^+U \Sigma U M^-$ with $M^{-}\bigcap M^{+}=\phi$. Here, $\Sigma$, the collapsing cylindrical surface is a time-like three surface and is the boundary of the two four dimensional sub-manifolds $M^-$ (interior) and $M^+$ (exterior).\
In $M^-$ choosing co-moving coordinates the line element can be written as \[25\] $$d{s_-^2}=-{A^2}d{t^2} +{B^2 }d{r^2} +{C^2}d{\phi^2} +{D^2}d{z^2}$$\
where the metric coefficients are functions of t and r i.e. A=A(t,r) and so on. Also due to cylindrical symmetry, the coordinates are restricted as:\
$-\infty\leq t\leq +\infty,~~~r\geq 0,~~~-\infty<z<+\infty,~~~0\leq \phi \leq 2\pi$\
For compact notation we write $\lbrace x^{-\mu} \rbrace \equiv[t,r,\phi,z]~~~,~~~ (\mu=0,1,2,3) $.\
The anisotropic fluid having dissipation in the form of shear viscosity and heat flow has the energy- momentum tensor of the form \[7,9\] $$T_{\mu\nu}=(\rho+p_t){v_\mu}{v_\nu}+{p_t}g_{\mu\nu}+({p_r}-{p_t}){\chi_\mu}{\chi_\nu}-2\eta\sigma_{\mu\nu}+2q_{(\mu}v_{\nu)}$$\
Here $\rho,~p_r,~p_t~ ~~\eta~~~ and~~q_{\mu}$ stands for energy density ,the radial pressure, the tangential pressure, coefficient of shear viscosity and radial heat flux vector respectively. Also $v_\mu$ and $\chi_\mu$ are unit time-like and space-like vectors satisfying the following relations $$v_\mu v^\mu =-\chi_\mu\chi^\mu=-1~~~,~~~~\chi^\mu v_\mu=0~~~,~~~q_{\mu}v^{\mu}=0~~~~$$\
Moreover, the shear tensor $\sigma_{\mu\nu}$ has the expression $$\sigma_{\mu\nu}= v_{(\mu;\nu)}+a_{(\mu}v_{\nu)}-\frac{1}{3}\Theta(g_{\mu\nu}+v_\mu v_\nu)$$\
where $a_\mu=v_{\mu;\nu}v^\nu $ is the acceleration vector and $\Theta=v^\mu;_\mu$ is the expansion scalar.\
For the above metric one may choose the unit time-like vector, space-like vector and heat flux vector in a simple form as $$v^\mu=A^{-1}\delta_0^\mu~~~,~~~~~~~~~~~~~~\chi^\mu=B^{-1}\delta_1^\mu~~~,~~~~~q^{\mu}=q\delta^{\mu}_{1}~~~~~$$\
The shear tensor has only non-zero diagonal components as $$\sigma_{11}=\frac{B^2}{3A}[\Sigma_1-\Sigma_3]~~,~~~~~~\sigma_{22}=\frac{C^2}{3A}[\Sigma_2-\Sigma_1]~~~and~~~\sigma_{33}=\frac{D^2}{3A}[\Sigma_3-\Sigma_2]~~~with~~~~~\sigma^2=\frac{1}{6A^2}[\Sigma_1^2+\Sigma_2^2+\Sigma_3^2]$$\
where $ \Sigma_1=\frac{\dot{B}}{B}-\frac{\dot{C}}{C}~~,~~~\Sigma_2=\frac{\dot{C}}{C}-\frac{\dot{D}}{D}~~,~~~\Sigma_3=\frac{\dot{D}}{D}-\frac{\dot{B}}{B}
$\
Also the acceleration vector and the expansion scalar have the explicit expressions $$a_1=\frac{A^\prime}{A}~~,\Theta=\frac{1}{A}(\frac{\dot{B}}{B}+\frac{\dot{C}}{C}+\frac{\dot{D}}{D})$$\
In the above, by notation we have used $\cdot$ $\equiv\frac{\partial}{\partial t}$ and $^\prime$ $\equiv\frac{\partial}{\partial r}.$\
If in addition we assume the above fluid distribution to be charged then the energy-momentum tensor for the electromagnetic field has the form $$E_{\alpha\beta}=\frac{1}{4\pi}(F^{\alpha}_{\mu}F_{\nu\alpha}-\frac{1}{4}F^{\alpha\beta}F_{\alpha\beta}g_{\mu\nu})~~~~~$$
where the Maxwell field tensor $F_{\alpha\beta}$ is related to the four potential $\phi_{\alpha}$ as $$F_{\alpha\beta}=\phi_{\beta , \alpha}-\phi_{\alpha , \beta}$$ and the evolution of the field tensor corresponds to Maxwell equations $$F^{\alpha\beta}_{;\beta}=4\pi J^{\alpha}$$ where $J^{\alpha}$ the four current vector.\
As the charge per unit length of the cylinder is at rest with respect to comoving co-ordinates so the magnetic field will be zero in this local coordinate system \[26,27\]. Hence the four potential and the four current takes the simple form $$\phi_{\alpha}=\phi\delta^{0}_{\alpha} ,~~~~J^{\alpha}=\epsilon v^{\alpha}$$ where $\phi=\phi(t,r)$ is the scalar potential and $\epsilon=\epsilon(t,r)$ is the charge density.\
From the law of conservation of charge : $J^{\alpha}_{;\alpha}=0$, one obtains the total charge distribution interior to radius $r$ and per unit length of the cylinder as $$s(r)=2\pi\int^{r}_{0}\epsilon BCD dr$$ Now the explicit form of the Maxwell’s equations (10) for the interior space-time $M^{-}$ are given by $$\phi^{''}-(\frac{A^{'}}{A}+\frac{B^{'}}{B}-\frac{C^{'}}{C}-\frac{D^{'}}{D})\phi^{'}=4\pi\epsilon AB^{2}$$ and $$\dot{\phi}^{'}-(\frac{\dot{A}}{A}+\frac{\dot{B}}{B}-\frac{\dot{C}}{C}-\frac{\dot{D}}{D})\phi^{'}=0$$ A first integral of equation (13) gives $$\phi^{'}=\frac{2sAB}{CD}$$ which satisfies identically the other Maxwell’s equation(14). Hence one obtains the electric field intensity as $E(t,r)=\frac{s(r)}{2\pi C}$
Further, in the interior space-time $M^{-}$ the Einstein field equations $G_{\alpha\beta}=8\pi(T_{\alpha\beta}+E_{\alpha\beta})$ have the explicit form
$$\frac{A^{2}}{B^{2}}(-\frac{C^{''}}{C}-\frac{D^{''}}{D}+\frac{B^{'}}{B}(\frac{C^{'}}{C}+\frac{D^{'}}{D})-\frac{C^{'}D^{'}}{CD})+(\frac{\dot{B}\dot{C}}{BC}+\frac{\dot{B}\dot{D}}{BD}+\frac{\dot{C}\dot{D} }{CD})=8\pi(\rho A^{2}-2\eta\sigma_{00})+4\frac{s^{2}A^{2}}{C^{2}D^{2}}$$
\
$$-\frac{B^{2}}{A^{2}}(\frac{\ddot{C}}{C}+\frac{\ddot{D}}{D}+\frac{\dot{C}\dot{D}}{CD}-\frac{\dot{A}\dot{C}}{AC}-\frac{\dot{A}\dot{D}}{AD})+(\frac{C^{'}D^{'}}{CD}+\frac{A^{'}C^{'}}{AC}+\frac{A^{'}D^{'}}{AD})=8\pi(p_{r}B^{2}-2\eta\sigma_{11})-4\frac{s^{2}B^{2}}{C^{2}D^{2}}$$
$$-\frac{C^2}{A^2}[\frac{\ddot{B}}{B}+\frac{\ddot{D}}{D}-\frac{\dot{A}}{A}(\frac{\dot{B}}{B}+\frac{\dot{D}}{D})+\frac{\dot{B}\dot{D}}{BD}]+\frac{C^2}{B^{2}}[\frac{A^{''}}{A}+\frac{D^{''}}{D}-\frac{A^{'}}{A}(\frac{B^{'}}{B}-\frac{D^{'}}{D})-\frac{D^{'}}{D}\frac{B^{'}}{B}]=8\pi(p_{t}C^2-2\eta\sigma_{22})+4\frac{s^{2}}{D^{2}}$$
$$-\frac{D^{2}}{A^{2}}[\frac{\ddot{B}}{B}+\frac{\ddot{C}}{C}-\frac{\dot{A}}{A}(\frac{\dot{B}}{B}+\frac{\dot{C}}{C})+\frac{\dot{B}\dot{C}}{BC}]+\frac{D^{2}}{B^{2}}[\frac{A^{''}}{A}+\frac{C^{''}}{C}-\frac{A^{'}}{A}(\frac{B^{'}}{B}-\frac{C^{'}}{C})-\frac{C^{'}}{C}\frac{B^{'}}{B}]=8\pi(p_{t}D^{2}-2\eta\sigma_{33})+4\frac{s^{2}}{C^{2}}$$
and $$\frac{1}{AB}(\frac{\dot{C^{'}}}{C}+\frac{\dot{D^{'}}}{D}-\frac{C^{'}}{C}\frac{\dot{B}}{B}-\frac{\dot{B}}{B}\frac{D^{'}}{D}-\frac{A^{'}}{A}\frac{\dot{C}}{C}-\frac{A^{'}}{A}\frac{\dot{D}}{D})=8\pi q$$
The gravitational energy per specific length in cylindrically symmetric space-time is defined as \[28-30\] $$E=\frac{(1-l^{-2}\nabla^{a}r \nabla_{a}r)}{8}$$ In principle, $E$ is the charge associated with a general current which combines the energy-momentum of the matter and gravitational waves. It is usually referred in the literature as $C$-energy for the cylindrical symmetric space-time. For cylindrically symmetric model with killing vectors the circumference radius $\rho$ and specific length $l$ are defined as \[28-30\]\
$\rho^{2}=\xi_{(1)a}\xi_{(1)}^{a}$ , $l^{2}=\xi_{(2)a}\xi_{(2)}^{a}$, so that $r=\rho l$ is termed as areal radius.\
For the present model with the contribution of electromagnetic field in the interior region the $C-$energy takes the form
$$E'=\frac{l}{8}+\frac{1}{8D}[\frac{1}{A^{2}}(C\dot{D}+\dot{C}D)^{2}-\frac{1}{B^{2}}(CD^{'}+C^{'}D)^{2}]+\frac{s^{2}}{2C}$$
\
It should be noted that the above energy is also very similar to Tabu’s mass function in the plane symmetric space-time\[31\]\
The exterior space-time manifold $(M^{+})$ of the cylindrical surface $\Sigma$ is described by the metric in the retarded time co-ordinate as \[32,33\]
$$ds_{+}^{2}=-(\frac{-2M(v)}{R}+\frac{Q^{2}(v)}{R^{2}})dv^{2}-2dRdv+R^{2}(d\phi^{2}+\lambda^{2}dz^{2})$$
where $v$ is the usual retarded time, $M(v)$ is the total mass inside $\Sigma$, $Q(v)$ is the total charge bounded by $\Sigma$ and $\lambda$ is an arbitrary constant. Further, from the point of view of the interior manifold $(M^{-})$ the bounding three surface $\Sigma$ (comoving surface) is described as $$f_{-}(t,r)=r-r_{\Sigma}=0$$ and hence the interior metric on $\Sigma$ takes the form $$ds_-^2\stackrel{\Sigma}{=}-d\tau^2 +C^2dz^2+D^2d\phi^2$$ where $$d\tau\stackrel{\Sigma}{=}Adt,$$\
defines the time co-ordinate on $\Sigma$ and $\stackrel{\Sigma}{=}$ by notation implies the equality of both sides on the surface $\Sigma$.\
Similarly, from the perspective of the exterior manifold the boundary three surface $\Sigma$ is characterized by
$$f_{+}(v,R)\equiv R-R_{\Sigma}(v)=0$$
so that the exterior metric on $\Sigma$ takes the form
$$ds_{+}^{2}\stackrel{\Sigma}{=}-(\frac{-2M(v)}{R}+\frac{Q^{2}(v)}{R^{2}}+\frac{2dR_{\Sigma}(v)}{dv})dv^{2}+R^{2}(d\phi^{2}+\lambda^{2}dz^{2})$$
Here by notation we write $[x^{+\mu}]=[v,R,\phi, z]$
Junction conditions
===================
In order to have a smooth matching of the interior and exterior manifolds over the bounding three surface (not a surface layer), the following conditions due to Darmois \[11\] are to be satisfied:\
(i)The continuity of the first fundamental form i.e.
$$(ds^2)_{\Sigma}=(ds^2_{-})_{\Sigma}=(ds^{2}_{+})_{\Sigma}$$
(ii)The continuity of the second fundamental form i.e $K_{ij}d\xi^{i}d\xi^{j}$. This implies the continuity of the extrinsic curvature $K_{ij}$ over the hypersurface \[11\] i.e. $$[K_{ij}]\equiv K_{ij}^+ -K_{ij}^-=0$$
where $K_{ij}^\pm$ is given by
$$K_{ij}^\pm=-n_\sigma^\pm[\frac{\partial^2x_\pm ^\sigma}{\partial\xi^i \partial\xi^j}+\Gamma_{\mu\nu}^\sigma \frac{\partial x_\pm^\mu}{\partial\xi^i}\frac{\partial x_\pm^\nu}{\partial\xi^j}],~~~(\sigma,~\mu,~\nu~=0,1,2,3)$$
In the above expression for extrinsic curvature, $n_{\sigma}^{\pm}$ are the components of the outward unit normal to the hyper-surface with respect to the manifolds $M^{\pm}$ (i.e. in the co-ordinates $x^{\pm \mu}$) and have explicit expressions\
$n_\sigma^- \stackrel{\Sigma}{=}(0,B,0,0)~~and~~~~n_\sigma^+ \stackrel{\Sigma}{=}\mu(\frac{-dR}{dv},1,0,0)$ with $\mu=[\frac{-2M(v)}{R}+\frac{Q^{2}(v)}{R^{2}}+2\frac{dR}{dv}]^{\frac{-1}{2}}$\
Also in the above the christoffel symbols are evaluated for the metric in $M^{-}$ or $M^{+}$ accordingly and we choose $\xi^{0}=\tau$, $\xi^{2}=z$, $\xi^{3}=\phi$ as the intrinsic co-ordinates on $\Sigma$ for convenience.\
The continuity of the 1st fundamental form gives
$$C(t,r_{\Sigma})\stackrel{\Sigma}{=}R_{\Sigma}(v)~~~,~~~D(t,r_{\Sigma})\stackrel{\Sigma}{=}\lambda R_{\Sigma}(v)$$
$$\frac{dt}{d\tau}=1/A~~~~\frac{dv}{d\tau}=\mu$$
Now the non vanishing components of extrinsic curvature $K^{\pm}_{ij}$ are\
$$K_{00}^-{=}-(\frac{A^\prime}{AB})_{\Sigma}$$ $$K_{00}^+{=}[(\frac{d^{2}v}{d\tau^{2}})(\frac{dv}{d\tau})^{-1}-(\frac{dv}{d\tau})(\frac{M}{R^{2}}-\frac{Q^{2}}{R^{3}})]_{\Sigma}$$\
$$K_{22}^-{=}(\frac{C{C}^\prime}{B})_{\Sigma}$$ $$K_{33}^-{=}(\frac{D{D}^\prime}{B})_{\Sigma}$$
$$K_{22}^+{=}[R(\frac{dR}{d\tau})-\frac{(dv)}{d\tau}(2M-\frac{Q^{2}}{R})]_{\Sigma}=\lambda ^{-2}K^{+}_{33}$$
\
and\
Hence continuity of the extrinsic curvature together with equations (32) and (33) gives the following relations over $\Sigma$ \[33,34\] $$M(v)\stackrel{\Sigma}{=}\frac{R}{2}[(\frac{\dot{R}}{A})^{2}-(\frac{R^{'}}{B})^{2}]+\frac{Q^{2}}{2R}$$\
$$E\stackrel{\Sigma}{=}\frac{l}{8}+\lambda M$$\
and $$q\stackrel{\Sigma}{=}p_{r}-\frac{2\eta \sigma_{11}}{B^{2}}-\frac{s^{2}}{2\pi c^{4}}(\frac{1}{\lambda^{2}}-1/4
)$$\
Thus equations (39) gives the total mass inside the boundary surface $\Sigma$ while equation (40) shows the linear relationship between the $C$ energy for the cylindrically symmetric space-time with the bounding mass over $\Sigma$. Further, equation (41) shows a linear relationship among the fluid parameters $(p_{r}, \eta, q)$ on the bounding surface $\Sigma$. Hence radial pressure is in general non zero on the bounding surface due to dissipative nature of the fluid and the charge on the bounding surface. But when dissipative components of the fluid are switch off then the above result (uncharged) agrees with the results of Herrera etal \[8\]. Also it should be noted that the radial pressure on the boundary does not depend on the charge bounded by $\Sigma$, it depends only on the charge on the surface $\Sigma$.\
Analysis of Dynamical equations:
================================
From the conservation of energy-momentum i.e. $(T^{\alpha\beta}+E^{\alpha\beta})_{;\beta}=0$ we can have two zero scalars namely $(T^{\alpha\beta}+E^{\alpha\beta});_{\beta}v_{\alpha}~~~and~~~(T^{\alpha\beta}+E^{\alpha\beta});_{\beta}\chi_{\alpha}$\
Using equations (2) and(8) the explicit expressions for these two scalars are $$\frac{\dot{\rho}}{A}+\frac{\dot{B}}{A}(\frac{\rho}{B}+\frac{p_{r}}{B}-2\eta\sigma^{11})+\frac{\dot{C}}{A}(\frac{p_{\bot}}{C}+\frac{\rho}{C}-2\eta\sigma^{22})+\frac{\dot{D}}{A}(\frac{\rho}{D}+\frac{p_{\bot}}{D}-2\eta\sigma^{33})+\frac{q^{'}}{B}+\frac{q}{B}(2\frac{A^{'}}{A}+\frac{C^{'}}{C}+\frac{D^{'}}{D})=0$$ and $$\begin{aligned}
(\frac{p_{r}}{B^{2}}-2\eta\sigma^{11})^{'}+\frac{\dot{q}}{AB}+\frac{q}{AB}(\frac{\dot{C}}{C}+\frac{\dot{D}}{D})+\frac{A^{'}}{A}(\frac{\rho}{B^{2}}+\frac{p_{r}}{B^{2}}-2\eta\sigma^{11})+\frac{B^{'}}{B}(\frac{p_{r}}{B^{2}}-2\eta\sigma^{11})+\frac{C^{'}}{C}(\frac{p_{r}}{B^{2}}
\nonumber
\\
-\frac{p_{\bot}}{B^{2}}-2\eta\sigma^{11}-2\eta\sigma^{22}\frac{C^{2}}{B^{2}})+\frac{D^{'}}{D}(\frac{p_{r}}{B^{2}}-\frac{p_{\bot}}{B^{2}}-2\eta\sigma^{11}+2\eta\sigma^{33}\frac{D^{2}}{B^{2}})-\frac{ss^{'}}{\pi C^{2}D^{2}B}=0 \end{aligned}$$\
Now following the formulation of Misner and Sharp \[2\], we introduce the proper time derivative and proper radial derivative as $$D_{T}=\frac{1}{A}\frac{\delta}{\delta t}~~~~~~and~~~~ D_{R}=\frac{1}{R^{'}}\frac{\delta}{\delta r}$$
so that the fluid velocity in the collapsing situation, can be defined as \[35\] $$U=D_{T}(R)=D_{T}(C) <0
~~~and~~~ V=D_{T}(Rr)=D_{T}(D) <0$$ Using equations (17)-(22) and(45), we can obtain the acceleration of a collapsing matter inside $\Sigma$ as
$$D_{T}(U)=-4\pi R(p_{r}-\frac{4\eta \sigma}{\sqrt{3}})+\tilde{E}\frac{A^{'}}{AB}+\frac{s^{2}}{R^{3}}(2+\frac{1}{2\lambda})-\frac{1}{R^{2}\lambda}(E^{'}-\frac{l}{8})$$
Now combining (43) and (46) we obtain $$\begin{aligned}
(\rho+p_{r}-\frac{4\eta\sigma}{\sqrt{3}})D_{T}(U)&=&(\rho+p_{r}-\frac{4\eta\sigma}{\sqrt{3}})[\frac{1}{R^{2}\lambda}(E^{'}-\frac{l}{8})+4\pi R(p_{r}-\frac{4\eta \sigma}{\sqrt{3}})-\frac{s^{2}}{R^{3}}(2+\frac{1}{2\lambda})]-\tilde{E}^{2}[D_{R}(p_{r}
\nonumber
\\
&-&\frac{4\eta\sigma}{\sqrt{3}})+\frac{2}{R}(p_{r}-p_{\bot}-2\sqrt{3}\eta\sigma)-\frac{s}{\pi R^{4}}D_{R}(s)]-\frac{2q\tilde{E}}{A}(\frac{\dot{B}}{B}+\frac{\dot{C}}{C})-\frac{\dot{q}\tilde{E}}{A}\end{aligned}$$\
Using (22) and the junction condition $D\stackrel{\Sigma}{=}\lambda C$, we write \[24\] $$\tilde{E} =\frac{C^{'}}{B}=[U^{2}+\frac{s^{2}}{\lambda c^{2}}-\frac{2}{\lambda c}(E^{'}-\frac{l}{8})]^{\frac{1}{2}}$$
Hence using the field equations for the interior manifold we obtain the time rate of change of C-energy as $$D_{T}E^{'}=-4\pi R^{2}\lambda[(p_{r}-\frac{4\eta\sigma}{\sqrt{3}})U +q\tilde{E}]+\frac{s^{2}\dot{C}}{R^{2}A}(2\lambda-\frac{1}{2})$$ Also the above equation can be interpreted as the variation of the total energy inside the collapsing cylinder. Note that due to negativity, of the fluid velocity $v$ the first term on the r.h.s will contribute to the energy of the system provided the radial pressure is restricted as $p_{r}>\frac{4\eta\sigma}{\sqrt{3}}$. Due to negativity, the second term indicates an outflow of energy in the form of radiation during the collapsing process. The third term is coulomb-like force term and it will increase the energy of the system provided $\lambda~>~\frac{1}{4}$.\
Further, using the Einstein field equations (16), (20) and the expression for C- energy in equation (22), the radial derivative of the C energy takes the form $$D_{R}E'=4\pi\rho R^{2}\lambda+\frac{s^{2}}{R^{2}}(2\lambda-\frac{1}{2})+\frac{s}{R}D_{R}(s)+\frac{4\pi q BR^{2}\lambda}{R'}D_{T}(C)+\frac{1}{8\rho R'}$$
This radial derivative can be interpreted as the energy variation between the adjacent cylindrical surfaces within the matter distribution. The first term on the r.h.s. is the usual energy density of the fluid element while the second term and third terms are the conditions due to the electromagnetic field. The fourth term represents contribution due to the dissipative heat flux and the last term will increase or decrease the energy of the system during the collapse of the cylinder provided $R^{'}> ~or~<~0$\
Finally, the collapse dynamics is completely characterized by the equation of motion in equation (47). Normally, for collapsing situation $D_{T}U$ should be negative, i.e, indicating an inward radial flow of the system. Consequently, terms on the r.h.s (of eq.(47)) contributing negatively favours the collapse and positive terms oppose the collapsing process. In an extreme situation the system will be in hydrostatic equilibrium if terms of both signs balance each other. Further, from dimensional analysis the factor $(\rho+p_{r}-\frac{4\eta\sigma}{\sqrt{3}})$ can be considered as an inertial mass density, independent of heat flux contribution. The first term on the r.h.s. of eq. (47) can be identified as the gravitational force, indicating the effects of specific length and electric charge in the gravitational contribution. The second term has three contributing components- the pressure gradient (which is negative), local anisotropy of the fluid and electromagnetic field term. The remaining terms represent the heat flux contribution and due to negativity they seem to leave the system along the radial outward streamlines.\
Causal Thermodynamics: The Transport equation
=============================================
In causal thermodynamics due to Miller-Israel-Stewart, the transport equation for heat flow is given by \[21\]
$$\tau h^{ab}V^{c}q_{b;c}+q^{a}=-\kappa h^{ab}(T_{,b}+a_{b}T)-\frac{1}{2}\kappa T^{2}(\frac{\tau V^{b}}{\kappa T^{2}})_{;b}q^{a}$$
where $h^{ab}=g^{ab}+V^{a}V^{b}$ is the projection tensor of the $3-$surface orthogonal to the unit time-like vector $V^{a}$, $\kappa$ represents the thermal conductivity, $T$ is the temperature, $\tau$ denotes the relaxation time and $a_{b}T$ is the inertial term due to Tolman. Now due to cylindrical symmetry, the above transport equation (51)simplifies to
$$\tau\dot{q}=-\frac{1}{2A}\kappa \frac{qT^{2}}{\tau}(\frac{\tau}{\kappa T^{2}})-q[\frac{3U}{2R}+G+\frac{1}{\tau}]-\frac{\kappa \tilde{E}D_{R}T}{\tau}-\frac{\kappa TD_{T}U}{\tau\tilde{E}}-\frac{\kappa T}{\tau\tilde{E}R^{2}}[\frac{1}{\lambda}(E^{'}-\frac{l}{8})+4\pi R^{3}(p_{r}-\frac{4\eta\sigma}{\sqrt{3}})-\frac{S^{2}}{R}(2+\frac{1}{2\lambda})]$$
with $G=\frac{1}{A}(\frac{\dot{B}}{B}-\frac{\dot{C}}{C})$\
Now considering proper derivatives in equation(44) of the above equation and using the field velocity (in eq.(45)), and equation of motion (i.e. eq. (47)) one obtains the effects of heat flux or dissipation in the collapsing process as\
$$\begin{aligned}
(1-\alpha)(\rho+ p_{r}-\frac{4\eta\sigma}{\sqrt{3}})D_{T}U =(1-\alpha)F_{grav}+F_{hyd}+ \alpha \tilde{E}^{2}[D_{R}p_{r}+2(p_{r}-p_{\bot}-2\sqrt{3}\eta\sigma)\frac{1}{R}
\nonumber
\\
-\frac{SD_{R}(S)}{\pi R^{4}\lambda^{2}}]-\tilde{E}[D_{T}q+2qG+\frac{4qU}{R}]+\alpha \tilde{E}[D_{T}q+\frac{4qU}{R}+2qG]\end{aligned}$$\
with $$\alpha=\frac{\kappa T}{\tau}(\rho+p_{r}-\frac{4\eta\sigma}{\sqrt{3}})^{-1}$$
$$F_{grav}=-(\rho+p_{r}-\frac{4\eta\sigma}{\sqrt{3}})[(E^{'}-\frac{l}{8})\frac{1}{\lambda}+4\pi p_{r}R^{3}-(2+\frac{1}{2\lambda})\frac{S^{2}}{R}](\frac{1}{R^{2}})$$
$$F_{hyd}=\tilde{E}^{2}[D_{R}(p_{r}-\frac{4\eta\sigma}{\sqrt{3}})+\frac{2}{R}(p_{r}-p_{\bot}-2\sqrt{3}\eta\sigma)-\frac{S}{\pi R^{4}}D_{R}(S)]$$
The l.h.s. of equation (53) can be interpreted as Newtonian force $F$ with $(\rho+p_{r})(1-\alpha)$ as the inertial mass density. So as $\alpha\rightarrow 1$, $F\rightarrow 0$ i.e. there is no inertial force and collapse will be inevitable due to gravitational attraction. Further, the inertial mass density decreases as long as $0~<~\alpha~<1$ and it increases for $\alpha~>~1$. Moreover, due to equivalence principle the gravitational mass also decrease or increase according as $\alpha~<~or~>~1$ and gives a clear distinction between the expanding and collapsing process due to dynamics of dissipative system. Note that although the gravitational force is affected by the same factor $(1-\alpha)$ but the hydrodynamical force is free from it. Further, combination of all these terms on the r.h.s of equation (53) results the l.h.s i.e. $(1-\alpha)(\rho+p_{r}-\frac{4\pi\sigma}{\sqrt{3}})D_{T}U~<0$, there will be gravitational collapse while there will be expansion if the l.h.s to be positive. Interestingly, if $\alpha$ continuously decreases from a value larger than unity to one less than unity, then there will be a phase transition (collapse to expansion) and bounce will occur. As a result, there will be loss of energy of the system and collapsing cylinder with non-adiabatic source causes emission of gravitational radiation. Therefore, there will be radiation outside the collapsing cylinder and hence the choice of the exterior metric (in eq.(23)) is justified.\
J.R. Oppenheimer and H. Snyder , [*Phys.Rev.*]{} [**56,**]{} (1939)455 ; C.W. Misner and D. Sharp, [*Phys.Rev.*]{} [**136**]{} (1964) B571; P.C.Vaidya, [*Proc.Indian Acad.Sci.A*]{} [**33**]{} (1951) 264; P.S. Joshi and T.P. Singh, [*Phys.Rev.D*]{} [**51**]{} (1995) 6778; U. Debnath, S. Nath and S. Chakraborty [*Gen. Relt. Grav.*]{} [**37**]{} (2005) 215. T. Levi-Civita, [*Rend. Accad. Lincei*]{} [**28**]{} (1919) 101. L. Herrara and N.O. Santos, [*Class. Quant. Grav*]{} [**22**]{} (2005) 2407. L. Herrera, N.O. Santos and M.A.H. Maccallum, [*Class. Quant. Grav*]{} [**24**]{} (2007) 1033. A. Di Prisco, L. Herrera, M. Maccallum and N.O. Santos, [*Phys. Rev. D.*]{} [**80**]{} (2009) 064031. L. Herrera, A. Di Prisco, J. Ospino, [*Gen.Relt. Grav.*]{} [**44**]{} (2012) 2645. G. Darmois, Memorial des Sciences Mathematiques (Gautheir-Viuars, Paris,) Fasc. 25. M. Sharif and Z. Ahmed [*Mod.Phys.Lett.A*]{} [**22**]{} (2007) 1493 M. Sharif and Z. Ahmed [*Mod.Phys.Lett.A*]{} [**22**]{} (2007) 2947. M. Sharif and Z. Ahmed [*Gen.Relt. Grav.*]{} [**39**]{} (2007) 1331. L. Herrera , G. Le Denmat, G. Marcilhacy, N.O. Santos [*Int. J. Mod. Phys.D*]{} [**14**]{} (2005) 657. Y. Kurita and K. Nakao, [*Phys.Rev. D*]{} [**73**]{} (2006) 064022. L. Herrera, A. DiPrisco, J.Martin, J.Ospino, N.O. Santos and O.Troconis [*Phys.Rev.D*]{} [**69**]{} (2004) 084026. L. Herrera, N.O. Santos, [*Phys. Rev. D.*]{} [**70**]{} (2004) 084004. A. Mitra, [*Phys.Rev.D.*]{} [**74**]{} (2006) 024010. R. Chan, [*Astron.Astrophys.*]{} [**368**]{} (2001) 325. L. Herrera, [*Int. J. Mod Phys.D*]{} [**15**]{} (2006) 2197. L. Herrera, A. Di Prisco, E.Fuenmayor and O. Troconics, [*Int. J. Mod. Phys. D*]{} [**18**]{} (2009) 129. M. Sharif and Z. Rehmat, [*Gen.Relt.Grav.*]{} [**42**]{} (2010) 1795. M. Sharif and G. Abbas, [*Astrophys.Space Sci*]{} [**335**]{} (2011) 515. S. Chakraborty and S. Chakraborty [*Gen.Relt. Grav.*]{} [**46**]{}, (2014) 1784; [*Annals of Physics*]{} [**364**]{}, (2016) 110 M. Sharif and G. Abbas, [*J.Phys.Soc.Jpn*]{} [**80**]{} (2011) 104002. A.Di Prisco etal, [*Phys.Rev.D*]{} [**76**]{} (2007) 064017. K.S Throne, [*Phys.Rev.*]{} [**138**]{} (1965) B251. T. Chiba, [*Prog.Theo.Phys.*]{} [**95**]{} (1996) 321. S.A. Hayward, [*Class.Quant.Grav.*]{} [**17**]{} (2000) 1749. M. Sharif and Z. Rehamat [*Gen.Relt.Grav*]{} [**42**]{} (2010) 1795. H. Chao-Guang [*Acta Phys.Sin.*]{} [**4**]{} (1995) 617. M. Sharif and M. Azam [*Jour.of cosmology and Astro.Physics*]{} [**02**]{} (2012) 043 S.Guha and R.Banerji [*Int J Theorn Phys.*]{} [**53**]{} (2014)2332 M. Sharif and S. Fatima [*Gen.Relt.Grav.*]{} [**43**]{} (2011) 127
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We show that one can decide if a rational equivalence relation can be given as the equivalence kernel of a sequential letter-to-letter transduction. This problem comes from the setting of games with imperfect information. In [@BerwangerD18 p. 6] the authors propose to model imperfect information by a rational equivalence relation and leave open the problem of deciding if one can synthesize a sequential letter-to-letter transducer (Mealy machine) which maps equivalent histories to the same sequence of observations. We also show that knowing if an equivalence relation can be given as the equivalence kernel of a sequential transducer is *undecidable*, even if the relation is given as a letter-to-letter transducer.'
author:
- Paulin Fournier
- Nathan Lhote
bibliography:
- 'biblio.bib'
title: Equivalence kernels of sequential functions and sequential observation synthesis
---
Introduction {#introduction .unnumbered}
============
##### Motivation: games with imperfect information
The motivation for the present article comes from the paper: *Observation and Distinction. Representing Information in Infinite Games* by Dietmar Berwanger and Laurent Doyen, submitted to the arXiv in 2018 [@BerwangerD18]. The authors propose an alternative way of representing imperfect information in games. The standard way to model imperfect information for a player is through a Mealy machine which transforms a sequence of game locations (a history) into a sequence of observations, which we call in the following an *observation function*. The proposed model of [@BerwangerD18 p. 6] is to give instead a transducer recognizing an *indistinguishability relation*, *i.e.* an equivalence relation over game histories which recognizes those pairs of histories that are indistinguishable from the player’s perspective.
This new model is actually more expressive than the standard one (composing a Mealy machine with its inverse yields a transducer recognizing the indistinguishability relation), and one of the problems left open in [@BerwangerD18 p. 22] is to decide when an indistinguishability relation can be transformed into an observation function, given as a Mealy machine.
Given a class **R** of equivalence relations and a class **F** of functions we define the **R**,**F**-*observation synthesis problem* as the problem of deciding if an equivalence relation in **R** can be expressed as the equivalence kernel[^1] of a function in **F**, and if possible computing such a function.
The main goal of this article is to solve this problem for rational relations and functions given by Mealy machines. Moreover, we also consider the problem of constructing an observation function given, not as a Mealy machine but, as a sequential transducer, *i.e.* the outputs are not restricted to single letters but can be arbitrary words. In terms of observations, Mealy machines characterize the fact that each game move produces exactly one piece of observation (in some finite alphabet), while for sequential transducers, a move might produce several observations, or even none, in which case this step is *invisible* to the player.
##### Contributions
We don’t use the vocabulary of games, but that of transducers, which is actually more suited to this problem: most of the proof techniques that we use stem from the theory of transducers. We consider several subclasses of [[**RatEq**]{}]{}, the set of rational equivalence relations, that is relations realized by transducers. The *equivalence kernel* of a total function $f$, is the equivalence relation defined by having the same image under $f$. The class ${{\bf KerSeq }\xspace}$ contains the equivalence relations that are the equivalence kernels of *sequential* transductions (a transducer is *sequential* if it is deterministic with respect to the input). The subclass ${{\bf KerSeq }\xspace}^{{ll}}$ is the set of equivalence relations that are the equivalence kernel of a transduction given as a *sequential letter-to-letter* transducer (also known as a Mealy machine).
We start by studying the simpler class of ${{\bf KerSeq }\xspace}^{{ll}}$ in Sec. \[sec:ltl\] and then consider the class ${{\bf KerSeq }\xspace}$ in Sec. \[sec:seq\]. Our main contribution is to give explicit characterizations for both classes ${{\bf KerSeq }\xspace}$ and ${{\bf KerSeq }\xspace}^{{ll}}$. For relations satisfying these properties, we exhibit a construction of a sequential, resp. letter-to-letter sequential, transducer whose kernel is the original relation. Finally we show that for rational equivalence relations, membership in ${{\bf KerSeq }\xspace}^{{ll}}$ is *decidable*. In contrast, membership in ${{\bf KerSeq }\xspace}$ is *undecidable* even for letter-to-letter rational relations (also known as automatic, synchronous or regular relations).
Note that while the characterization of ${{\bf KerSeq }\xspace}^{{ll}}$, as well as the construction were already given in [@BerwangerD18 Thm. 29, p. 19], the decidability status was left open. We reprove these results in our framework. Moreover, while extending the construction from ${{\bf KerSeq }\xspace}^{{ll}}$ to ${{\bf KerSeq }\xspace}^{{lp}}$ is rather straightforward, obtaining the characterization for this class is difficult and actually the *most* challenging part of this article.
Words, relations, automata and transducers
==========================================
##### Words, languages and relations
An *alphabet* $A$ is a set of symbols called *letter*. A word is a finite sequence of letters and we denote by $A^*$ the set of finite words with $\epsilon$ denoting the *empty word*. The length of a word $w$ is denoted by $|w|$ with $|\epsilon|=0$. Given a non-empty word $w$ and an integer $1\leq i\leq |w|$ we denote by $w(i)$ the $i$th letter of $w$, by $w{{({:}i)}}$ the prefix of $w$ up to position $i$ included, and by $w{{(i{:})}}$ the suffix of $w$ from position $i$ included. Given two words $u,v$ we write $u\preceq v$ (resp. $u\prec v$) to denote that $u$ is a (resp. strict) prefix of $v$, and we write $u^{-1}v$ the unique word $w$ such that $uw=v$. A *language* over an alphabet $A$ is a subset of $A^*$. A *word relation* $R$ (or *transduction*) over alphabets $A,B$ is a subset of $A^*\times B^*$ and we often write $uRv$ to denote $(u,v)\in R$. Let $R(u)={\left\{v\mid\ uRv \right\}}$, and if $R$ is a partial function from $A^*$ to $B^*$, we rather write $R(u)=v$ instead of $R(u)={\left\{v \right\}}$. The *composition* of two relations $R$ and $S$ is $R\circ S={\left\{(u,w)|\ \exists v,\ uSv\text{ and } vRw \right\}}$. The *inverse* of a relation $R$ is $R^{-1}={\left\{(v,u)|\ uRv \right\}}$. The *identity relation* over an alphabet $A$ is ${Id}={\left\{(u,u)|\ u\in A^* \right\}}$. The *domain* and *range* of a relation $R$ are respectively: ${\mathrm{dom}}(R)={\left\{u|\ \exists v,\ uRv \right\}}$ and ${\mathrm{ran}}(R)={\left\{v|\ \exists u,\ uRv \right\}}$.
We say that a relation $S$ is *finer* than $R$ (or that $R$ is *coarser* than $S$) if for any words $u,v$, $uSv \Rightarrow uRv$, which we denote by $S\subseteq R$.
An equivalence relation $R$ over alphabet $A$ is a relation over alphabets $A,A$ such that it is reflexive (${Id}\subseteq R$), symmetric ($R^{-1}\subseteq R$) and transitive ($R\circ R \subseteq R$). Taking the terminology of [@Johnson86 Sec. 2], the *(equivalence) kernel* of a total function $f:A^*\rightarrow B^*$ is the equivalence relation $\ker(f)={\left\{(u,v)|\ f(u)=f(v) \right\}}=f^{-1}\circ f$. A *canonical function* for an equivalence relation $R$ is a function $f$ such that $\ker (f)=R$. The *transitive closure* of a relation $R$, denoted by $R^+$, is the finest transitive relation coarser than $R$. Given two equivalence relations $S \subseteq R$ then any equivalence class of $R$ is a union of equivalence classes of $S$ and the *index* of $S$ with respect to $R$ is the supremum of the number of equivalence classes of $S$ included in a unique equivalence class of $R$. We extend the notion of index to arbitrary relations $S\subseteq R$: the index of $S$ with respect to $R$ is the value $\sup_{\begin{smallmatrix}
{u,T\subseteq R(u)}\\
\forall v\neq w\in T,\ v{\cancel S}w
\end{smallmatrix}} |T|$. We denote by $S\subseteq_k R$ that the index of $S$ with respect to $R$ is at most $k$, by $S\subseteq_{\mathit{ fin}}R$ that the index of $S$ with respect to $R$ is finite, and by $S\subseteq_\infty R$ that the index of $S$ with respect to $R$ is infinite.
The *valuedness* of a relation $R$ is the supremum of the cardinal of the image set of a word, [*i.e.* ]{}$\sup_{u}|R(u)|$.
##### Automata and transducers
A *finite automaton* (or just automaton) over an alphabet $A$ is a tuple ${\mathcal A}={\left(Q,\Delta, I, F \right)}$ where $Q$ is a finite set of *states*, $\Delta\subseteq Q\times A\times Q$ is a finite *transition relation* and $I,F\subseteq Q$ are the sets of *initial states* and *final states*, respectively. A *run* of ${\mathcal A}$ over a word $w\in A^*$ is a word $r\in Q^*$ of length $|w|+1$ such that for $1\leq i\leq |w|$, ${\left(r(i),w(i),r(i+1) \right)}\in \Delta$. We use the notation $p\xrightarrow{w}_{\mathcal A}q$ (or just $p\xrightarrow{w} q$ when ${\mathcal A}$ is clear from context) to denote that there exists a run $r$ of ${\mathcal A}$ over $w$ such that $r(1)=p$ and $r(|r|)=q$. Let $r$ be a run of ${\mathcal A}$, if $r(1)\in I$ then $r$ is called *initial*, if $r(|r|)\in F$ then $r$ is called *final* and a run which is both initial and final is called *accepting*. A word $w$ is *accepted* by ${\mathcal A}$ if there is an accepting run over it and the set of words accepted by ${\mathcal A}$ is called the *language recognized* by ${\mathcal A}$ and denoted by ${ \llbracket {\mathcal A}\rrbracket}$. A language is called *rational* if it is recognized by some automaton.
An automaton is called *deterministic* if it has a unique initial state, and for any pair of transitions $(p,a,q_1),(p,a,q_2)\in \Delta$ we have $q_1=q_2$.
A *finite transducer* over alphabets $A,B$ is an automaton over $A^*\times B^*$. We define the natural projections $\pi_A:(A^*\times B^*)^*\rightarrow A^*$ and $\pi_B:(A^*\times B^*)^*\rightarrow B^*$. We say that a pair of words $(u,v)\in A^*\times B^*$ is *realized* by a transducer ${\mathcal T}$ if there exists a word $w$ such that ${\mathcal T}$ has an accepting run $r$ over $w$, $\pi_A(w)=u$ and $\pi_B(w)=v$, and we write $(u,v)\in { \llbracket {\mathcal T}\rrbracket}$ with ${ \llbracket {\mathcal T}\rrbracket}$ denoting the *relation realized* by ${\mathcal T}$. A relation realized by a transducer is called *rational*. Given a transducer ${\mathcal T}={\left(Q,\Delta,I,F \right)}$ we define $\pi_A({\mathcal T})$ the *input automaton* of ${\mathcal T}$ by ${\left(Q,\pi_A(\Delta),I,F \right)}$, where $\pi_A(\Delta)={\left\{(p,a,q)|\ \exists b\in B^*\ (p,a,b,q)\in \Delta \right\}}$. A transducer is called *real-time* if its transitions are over the alphabet $A\times B^*$ and *letter-to-letter* if its transitions are over $A\times B$. A real-time transducer whose input automaton is deterministic is called *sequential* and the function it realizes is also called sequential. We say that a relation $R$ is *length-preserving* if for any words $u,v$, $uRv \Rightarrow |u|=|v|$. A letter-to-letter transducer realizes a length-preserving relation and it is known that any length-preserving rational relation can be given as a letter-to-letter transducer. However, one can easily see that a sequential length-preserving function cannot in general be given as a letter-to-letter sequential transducer. For instance the function mapping $aa$ to $aa$ and $ab$ to $bb$ is sequential and length-preserving yet cannot be given as a sequential letter-to-letter transducer.
##### Classes of rational equivalence relations
We define classes of equivalence relations: [[**RatEq**]{}]{}the class of all rational equivalence relations, [[**KerRat**]{}]{}the class of relations which are kernels of rational functions and [[**KerSeq** ]{}]{}the class of relations which are kernels of sequential functions. For each of the previous classes $\mathbf C$, we define $\mathbf C^{{lp}}$ as the class of *length-preserving* relations of $\mathbf C$. Similarly we define $\mathbf C^{{ll}}$ by restricting to letter-to letter transducers, and we have obviously that $\mathbf C^{{ll}}\subseteq \mathbf C^{{lp}}$. For instance ${{\bf RatEq}\xspace}^{{ll}}$ is the class of equivalence relations which are given by letter-to-letter transducers while ${{\bf KerSeq }\xspace}^{{ll}}$ is the class of relations which are kernels of letter-to-letter sequential transducers. Fig. \[fig:classes\] gives the relative inclusions of the classes considered in this article, and a similar one can be found in [@Johnson86 Fig. 1].
at (0,1) ; at (0,0) ; at (0,-.5) ; at (0,-1) ; at (0,-1.5) ; at (0,-2) ;
(1.5,1) -> (1.5,-2.5); (-1,.7) -> (6,.7);
at (4,1) ; at (4,0) ; at (4,-.5) ; at (4,-1) ; at (4,-1.5) ; at (4,-2) ;
It is not known whether the classes [[**RatEq**]{}]{}and [[**KerRat**]{}]{}are equal or not. The generic problem we want to study is: given a rational equivalence relation, can we effectively decide if it is in [[**KerSeq** ]{}]{}? Let $R$ be a length-preserving equivalence relation given by a transducer ${\mathcal T}$, we know (*e.g.* from [@Johnson85 Thm. 5.1]) that there is a canonical function given by a transducer which maps any word to the minimum, for the lexicographic order, of its equivalence class. Hence we have that ${{\bf RatEq}\xspace}^{{ll}}={{\bf RatEq}\xspace}^{{lp}}={{\bf KerRat}\xspace}^{{ll}}={{\bf KerRat}\xspace}^{{lp}}$ and ${{\bf KerSeq }\xspace}^{{ll}}\varsubsetneq {{\bf KerSeq }\xspace}^{{lp}}$, as we have seen above.
\(0) at (0,0) ; (1) at (2,0) ;
\(0) edge node\[above\] (1); (0) edge\[loop above\] node\[above\] (0); (1) edge\[loop above\] node\[above\] (1); (1) edge\[bend left\] node\[below\] (0); (0) edge +(-.5,0); (0) edge +(0,-.5);
\(0) at (0,0) ; (1) at (2,-.9) ; (2) at (2,.9) ; (0) edge node\[above,sloped\] (1); (1) edge\[bend left\] node\[below\] (0); (0) edge\[\] node\[above,sloped\] (2);
\(0) edge\[loop above\] node\[above\] (0); (1) edge\[loop above\] node\[above\] (1); (2) edge\[loop above\] node\[above\] (2);
\(0) edge +(-.5,0); (2) edge +(-.5,0); (2) edge +(.5,0);
\(0) at (0,0) ; (1) at (2,0) ;
\(0) edge\[\] node\[above\] (1); (0) edge\[loop above\] node\[above\] (0); (1) edge\[loop above\] node\[above\] (1); (1) edge\[bend left\] node\[below\] (0); (0) edge +(-.5,0); (0) edge +(0,-.5);
\(0) at (0,0) ; (1) at (2,0) ; (0) edge\[bend left\] node\[above\] (1); (1) edge\[bend left\] node\[below\] (0);
\(0) edge\[loop above\] node\[above\] (0); (1) edge\[loop above\] node\[above\] (1);
\(0) edge +(-.5,0); (0) edge +(.5,0); (1) edge +(.5,0);
We give in Fig. \[fig:lasta\] an example of length-preserving rational equivalence relation $R$, and we exhibit a rational canonical function for it. This equivalence relation is not in ${{\bf KerSeq }\xspace}$ and this can be shown using the characterization we prove in Sec. \[sec:seq\]. Intuitively, one has to *guess* when reading an $a$ if it is the last one or not, which cannot be done sequentially. In Fig. \[fig:evena\], we exhibit an equivalence relation which is length-preserving and is the kernel of a sequential function. However it is not the kernel of a *letter-to-letter* sequential function, which we will be able to show using the characterization from Sec. \[sec:ltl\].
Kernels of sequential letter-to-letter functions {#sec:ltl}
================================================
The goal of this section is to characterize relations which are kernels of sequential letter-to-letter functions. First, in Sections \[subsec:synt\] and \[subsec:pref-close\] we give two necessary conditions for a relation to be in ${{\bf KerSeq }\xspace}^{{ll}}$. Then in Sec. \[subsec:cons-seq-ltl\] we provide an algorithm to construct a sequential letter-to-letter canonical function when the two aforementioned conditions are satisfied, showing that they are indeed sufficient and thus characterize ${{\bf KerSeq }\xspace}^{{ll}}$. Finally in Sec. \[subsec:dec-seq-ltl\], we state the characterization established before and show that it is decidable.
Syntactic congruence {#subsec:synt}
--------------------
We start by introducing a notion of syntactic congruence associated with an equivalence relation, which will prove crucial throughout the paper. Given a relation $R$, we define $S_R$ the *syntactic congruence* of $R$ by $uS_Rv$ if for any word $w$, we have $uwRvw$. In particular $S_R$ is finer than $R$ and $S_R$ is a (right) congruence meaning that if $uSv$ then for any letter $a$ we have $uaS_Rva$. Furthermore, if $R$ is an equivalence relation then so is $S_R$.
We now exhibit a first necessary condition to be in ${{\bf KerSeq }\xspace}$, and *a fortiori* in ${{\bf KerSeq }\xspace}^{{ll}}$.
\[prop:finite-index\] Let $R$ be an equivalence relation. If $R\in {{\bf KerSeq }\xspace}$ then $S_R$ has finite index with respect to $R$.
Let ${\mathcal T}$ be a sequential transducer realizing a function $f$ whose kernel is $R$, and let $n$ be the number of states of ${\mathcal T}$. Let $uRv$, then we have $f(u)=f(v)$. Furthermore, if $u,v$ reach the same state in ${\mathcal T}$, since ${\mathcal T}$ is sequential, $f(uw)=f(vw)$ for any word $w$ which means that $uS_Rv$. Let $u_1Ru_2R\ldots R u_{n+1}$. By a pigeon-hole argument, there must be two indices $1\leq i<j\leq n+1$, such that $u_{i}$ and $u_{j}$ reach the same state in ${\mathcal T}$, hence $u_{i}S_Ru_{j}$. Thus we have shown that the index of $S_R$ with respect to $R$ is less than $ n$, and is thus finite.
\(0) at (0,0) ; (1) at (2,-.9) ; (2) at (2,.9) ;
\(0) edge node\[below\] (1); (0) edge\[loop above\] node\[above\] (0); (1) edge\[loop above\] node\[above\] (1); (0) edge node\[above,sloped\] (2); (2) edge\[loop above\] node\[right\] (2); (0) edge +(-.5,0); (0) edge +(0,-.5); (1) edge +(.5,0); (2) edge +(.5,0);
We give in Fig. \[fig:indexinf\] an example of a length-preserving equivalence relation such that its syntactic congruence does not have a finite index with respect to it. Two different words are never syntactically equivalent, however two words of same length without any $c$s are equivalent. Thus by Prop. \[prop:finite-index\], this relation is not in [[**KerSeq** ]{}]{}.
In the next two propositions, we show that 1) the syntactic congruence can be computed for a relation in ${{\bf RatEq}\xspace}^{{lp}}$ and 2) that the finiteness of its index can also be decided.
\[prop:comp-synt\] Let $R$ be an equivalence relation given as a pair-deterministic letter-to-letter transducer. One can compute a transducer recognizing its syntactic congruence in [<span style="font-variant:small-caps;">PTime</span>]{}.
Let $R$ be given by a letter-to-letter pair-deterministic transducer ${\mathcal R}$, and let $S_R$ denote its syntactic congruence. Let $(u,v)$ be a pair of words of equal length, and let us denote by $p$ the state reached in ${\mathcal R}$ after reading $(u,v)$. Then $uS_Rv$ if and only if the automaton ${\mathcal R}_p$ (obtained by taking $p$ as initial state) recognizes a reflexive relation. This property can be easily checked and thus $S_R$ is obtained by taking ${\mathcal R}$ and restricting the final states to states $p$ such that ${\mathcal R}_p$ recognizes a reflexive relation.
\[prop:dec-fin-index\]
Let $R$ be a rational relation given as a transducer ${\mathcal R}$, and let $f$ be a rational function given by a transducer ${\mathcal F}$ such that $S=\ker(f)$ is finer than $R$. Then one can decide if $S$ has finite index with respect to $R$ in [<span style="font-variant:small-caps;">PTime</span>]{}.
Let $f$ be a rational function such that $\ker(f)=S$. We show that the index of $S$ with respect to $R$ is equal to the valuedness of $T=f \circ R$. We want to show that for any $u$, $|T(u)|=\max_{\begin{smallmatrix}
{u,X\subseteq R(u)}\\
\forall v\neq w\in X,\ v{\cancel S}w
\end{smallmatrix}} |X|$.
Let $X\subseteq R(u)$ be such that $\forall v\neq w\in X,\ v{{\cancel S}}w$. Then $f$ is injective over $X$ since $f$ maps words to the same value if and only if they are $S$ equivalent, thus $|X|=|f(X)|$. Moreover $f(X)\subseteq T(u)$, which means that $|T(u)|\geq\max_{\begin{smallmatrix}
{u,X\subseteq R(u)}\\
\forall v\neq w\in X,\ v{\cancel S}w
\end{smallmatrix}} |X|$.
For each $v\in T(u)$, we can find a word $v'\in f^{-1}(v)$ (for instance the minimum word in the lexicographic order). Let $X$ be the set of these words, we have by construction $|X|=|T(u)|$. Moreover, for each pair of distinct words $v',w'\in X$, we have $f(v')\neq f(w')$ and thus in particular $v' {{\cancel S}} w'$. Thus we have shown $|T(u)|\leq\max_{\begin{smallmatrix}
{u,X\subseteq R(u)}\\
\forall v\neq w\in X,\ v{\cancel S}w
\end{smallmatrix}} |X|$.
Hence the index of $S$ with respect to $R$ is equal to the valuedness of $T$. Since finite valuedness can be decided in [<span style="font-variant:small-caps;">PTime</span>]{}[@Weber89 Thm. 3.1], then one can decide if $S$ has finite index with respect to $R$, also in [<span style="font-variant:small-caps;">PTime</span>]{}.
\[cor:dec-finite-index\] Let $R\in {{\bf RatEq}\xspace}^{{lp}}$, one can decide if its syntactic congruence $S_R$ has finite index with respect to it.
From Prop. \[prop:comp-synt\] we can compute a transducer realizing $S_R$. According to [@Johnson85 Thm. 5.1], we can even compute a transducer realizing a function whose kernel is $S_R$. Hence from Prop. \[prop:dec-fin-index\] we can decide the finiteness of the index of $S_R$ with respect to $R$.
Prefix closure {#subsec:pref-close}
--------------
Here we consider a second necessary condition of relations in ${{\bf KerSeq }\xspace}^{{ll}}$, namely that they are prefix-closed.
The *prefix closure* of a relation $R$ is the relation $P_R$ defined by $uP_Rv$ if there exists $u',v'$, with $|u'|=|v'|$, such that $uu'Rvv'$. A relation is called *prefix-closed* if it is equal to its prefix closure. We often say that $u,v$ are *equivalent in the future* when $uP_R v$.
\[prop:prefix-closed\] Let $R$ be an equivalence relation. If $R\in {{\bf KerSeq }\xspace}^{{ll}}$ then $R$ is prefix-closed.
Let $R$ be an equivalence relation, let $f$ be realized by a transducer in ${{\bf KerSeq }\xspace}^{{ll}}$ such that $\ker f=R$ and let $P_R$ denote the prefix closure of $R$. Let $uP_Rv$, then there exist $u',v'$ with $|u'|=|v'|$ such that $f(uu')=f(vv')$. Since $f$ is letter-to-letter sequential, we have $f(u)\preceq f(uu')$, $f(v)\preceq f(vv')$ and $|f(u)|=|f(v)|$ which means that $f(u)=f(v)$. Hence $uRv$, $R=P_R$ and $R$ is prefix-closed.
The equivalence relations given in Figures \[fig:lasta\] and \[fig:evena\] are *not* prefix-closed, which explains why they are not in ${{\bf KerSeq }\xspace}^{{ll}}$, according to Prop. \[prop:prefix-closed\].
Construction of a canonical function {#subsec:cons-seq-ltl}
------------------------------------
The main technical lemma of this section says that the two necessary conditions given above are sufficient:
\[lem:cons-seq\] Let $R\in {{\bf RatEq}\xspace}^{{lp}}$ be prefix-closed with a finite index syntactic congruence with respect to it. Then we can construct a sequential letter-to-letter transducer whose kernel is $R$.
Let $R\in {{\bf RatEq}\xspace}^{{lp}}$ be given by a transducer ${\mathcal R}={\left(Q,\Delta_R, I, F_R \right)}$ which is letter to letter, over the alphabet $A\times A$, and such that $S_R\subseteq_k R = P_R$ for some $k\in{\mathbb N}$. Without loss of generality, we assume that ${\mathcal R}$ is deterministic. A state of ${\mathcal R}$ will be called *diagonal* if the identity is accepted from that state, and let $D\subseteq F_R$ be the set of diagonal states. According to Prop. \[prop:comp-synt\], we can obtain a letter-to-letter transducer ${\mathcal S}$ realizing $S$ (just by setting $D$ as the set of final states).
Our goal is to define a sequential letter-to-letter transducer ${\mathcal T}$ whose kernel is the relation $R$. The main idea to obtain this construction is to distinguish three kinds of relationships between two words: 1) $uSv$ 2) $u{{\cancel S}}v$ and $uRv$ and 3) $u{\cancel R} v$. Then the key idea, as seen in the proof of Prop. \[prop:finite-index\], is that two words in case number 2) *cannot* end up in the same state in ${\mathcal T}$. Two words in situation number 1) might as well reach the same state in ${\mathcal T}$ since they have the exact same behavior. Then two words in situation 3) may or may not reach the same state, it does not matter since their image by ${\mathcal T}$ should be different.
For each equivalence class of $R$ containing $l\leq k$ different $S$-equivalence classes we define $l$ distinct states. The states will be pairs ${\left(M,i \right)}$ where $M\in {\mathcal M}_l(Q)$ is an $l\times l$ square matrix with values in $Q$, the state space of ${\mathcal R}$, and $i\in {\left\{1,\ldots,l \right\}}$. Let $u_1,\ldots, u_l$ be the least lexicographic representatives of the $l$ $S$-equivalence classes, in lexicographic order. Then $M(i,j)=p$ if $p$ is the state reached in ${\mathcal R}$ after reading $(u_i,u_j)$. Then the state ${\left(M,i \right)}$ is supposed to be the state reached after reading $u_i$, or any other $S$-equivalent word. Let us remark that the reachable states will only contain matrices where all states are accepting, [*i.e.* ]{}with values in $F_R$. Moreover, all values on the diagonal are in $D$.
Let us define a sequential transducer ${\mathcal T}={\left(Q_{\mathcal M},\Delta,{\left\{(M_0,1) \right\}} \right)}$ whose kernel will be the relation $R$ (we don’t specify the final states since all states are final). As we have seen, we define $Q_{\mathcal M}=\bigcup_{l\in{\left\{1,\ldots,k \right\}}}{\mathcal M}_l(Q)\times {\left\{1,\ldots,l \right\}}$. Since the word $\epsilon$ is the only word of length $0$, it is alone in its $R$ and $S$-equivalence classes, hence $M_0$ is the $1\times 1$ matrix with value $q_0$ the initial state of ${\mathcal R}$. We have left to define $\Delta$ and then show that the construction is correct. This will be done by induction on the length of the words. More precisely, let us state the induction hypothesis for words of length $n$:
- Let $u_1,\ldots,u_l$ be the minimal representatives of the $S$-equivalence classes of some $R$-equivalence class, of words of length $\leq n$. Then any word $uSu_i$ with $i\in {\left\{1,\ldots,l \right\}}$, reaches the state $(M,i)$ where $M(j,j')$ is the state reached in ${\mathcal R}$ by reading the pair $(u_j,u_{j'})$.
- Two words, of length $\leq n$, are $R$-equivalent if and only if their outputs in ${\mathcal T}$ are equal.
This trivially holds for the word of length $0$, and let us assume that it holds for words of length $\leq n$. Let $u_1,\ldots,u_l$ be the minimal representatives of the $S$-equivalence classes of some $R$-equivalence class, of words of length $n$. Let us consider the corresponding matrix $M\in {\mathcal M}_l(Q)$.
Let us define an equivalence relation $\sim_R$ over ${\left\{1,\ldots,l \right\}}\times A$ which will separate word which are no longer $R$-equivalent. Let $q_{i,j,a,b}$ be the state reached in ${\mathcal R}$ from $M(i,j)$ by reading $(a,b)$. Two pairs $(i,a),(j,b)$ are $\sim_R$-equivalent if $q_{i,j,a,b} \in F_R$. By $Hn.1$ we know that this is indeed an equivalence relation. We define a second equivalence relation $\sim_S$. Two pairs $(i,a),(j,b)$ are equivalent if $q_{i,j,a,b} \in D$. Finally, we consider a linear order on ${\left\{1,\ldots,l \right\}}\times A$ which is just the lexicographic order (with some fixed order over $A$).
Let $(i,a)\in {\left\{1,\ldots,l \right\}}\times A$, let us consider the set of minimal $\sim_S$-representatives of the $\sim_R$-equivalence class of $(i,a)$: $$I_R={\left\{(j,b)|\ (j,b)\sim_R (i,a)\text{ and }\forall (j',b')<(j,b),\ (j,b)\mathrel{{\cancel \sim}_S}(j',b') \right\}}$$ Let $l'$ denote the cardinal of $I_R$, [*i.e.* ]{}the number of $\sim_S$ equivalence classes in the $\sim_R$-equivalence class of $i$. We define the state $(N,j)$ and the output $b\in B$ such that $((M,i),(a,b),(N,j))\in \Delta$. The output $b$ is defined by $\min I_R$. The matrix $N$ has dimension $l'$ and let $(i_1,a_1),\ldots, (i_{l'},a_{l'})$ be the elements of $I_R$ in increasing order. The matrix $N$ is defined by $N(j,j')=p$ where $p$ is the state reached from $M(i_j,i_{j'})$ by reading $(a_j,a_{j'})$. Let $j$ be the index such that $(i,a)\sim_S (i_j,a_j)$, then we have $((M,i),(a,b),(N,j))\in \Delta$.
Let us show $Hn+1.1$. Let $uSu_ja$, we need to show that $u$ reaches the state $(N,j)$. Let $vc=u$, with $c\in A$. Since $vcSu_ja$, we have $vcSu_ja$, which means that $vRu_j$, since $R$ is prefix closed. hence there exists $u_{j'}$ such that $vS u_{j'}$. This means that we have $u_{j'}cSu_ja$ and $u_{j'}c\geq u_ja$ in the lexicographic order. By induction hypothesis, $v$ reaches the state $(M,j')$, and by construction we have $((M,j'),(c,b),(N,j))\in \Delta$.
We now show $Hn+1.2$. Let $v_1=w_1a_1,v_2=w_2a_2$ be two words of length $n+1$, with $a_1,a_2\in A$. If $v_1Rv_2$, then $w_1Rw_2$ since $R$ is prefix closed. By induction hypothesis, the outputs over $w_1$ and $w_2$ are the same. Moreover, by construction of $\Delta$, the final outputs reading $a_1$ and $a_2$, respectively, are the same. If $w_1{{\cancel R}}w_2$, then by induction, their outputs are different, and so are the outputs over $v_1,v_2$. The only remaining case is when $w_1Rw_2$ and $v_1{{\cancel R}}v_2$. By induction, we have that the outputs over $w_1,w_2$ are the same, hence we need to show that the outputs from the letters $a_1$, $a_2$ are different. By the construction of $\Delta$, the outputs are linked with $\sim_R$ equivalence classes, which means that the outputs corresponding to $w_1,a_1$ and $w_2,a_2$ are different.
Characterization of ${{\bf KerSeq }\xspace}^{{ll}}$ and decidability {#subsec:dec-seq-ltl}
--------------------------------------------------------------------
As a corollary we obtain a characterization of ${{\bf KerSeq }\xspace}^{{ll}}$.
\[thm:char-seq\] Let $R\in {{\bf RatEq}\xspace}$. The following are equivalent:
1. $R\in {{\bf KerSeq }\xspace}^{{ll}}$
2. $R$ is length-preserving and $S_R\subseteq_{\mathit{ fin}}R = P_R$
1.$\Rightarrow$2. comes from the results of Prop. \[prop:finite-index\] and Prop. \[prop:prefix-closed\]. To obtain 2.$\Rightarrow$1. we use the construction of Lem. \[lem:cons-seq\].
From the previous result we get an algorithm deciding if an equivalence relation is in ${{\bf KerSeq }\xspace}^{{ll}}$.
\[thm:dec-ltl\] The following problem is decidable.
1. **Input:** ${\mathcal R}$ a transducer realizing an equivalence relation $R$.
2. **Question:** Does $R$ belong to ${{\bf KerSeq }\xspace}^{{ll}}$?
Without loss of generality, we can assume that ${\mathcal R}$ is a letter-to-letter pair-deterministic transducer. From Cor. \[cor:dec-finite-index\] we can decide if $S_R$ has finite index with respect to $R$. Deciding if $R$ is prefix-closed, is easy: just check if a reachable state is not final.
According to Thm. \[thm:char-seq\], we thus have an algorithm to decide the problem.
Kernels of sequential functions {#sec:seq}
===============================
We turn to the problem of deciding membership in ${{\bf KerSeq }\xspace}^{{lp}}$. To tackle this we introduce another kind of transducers called *subsequential*, which are transducers allowed to produce a final output at the end of a computation. A *subsequential transducer* over alphabets $A,B$ is a pair $({\mathcal T}, {\mathit{t}})$, where $ {\mathit{t}}:F\rightarrow B$ is called the *final output function* ($F$ being the set of final states of ${\mathcal T}$). We denote by [[**KerSub**]{}]{}the class of equivalence relations which are kernels of subsequential functions.
Our results are obtained in two steps. First we exhibit sufficient conditions for being in ${{\bf KerSub}\xspace}^{{ll}}$ very similar to the characterization of ${{\bf KerSeq }\xspace}^{{ll}}$. Second we show that ${{\bf KerSub}\xspace}^{{ll}}={{\bf KerSeq }\xspace}^{{lp}}={{\bf KerSub}\xspace}^{{lp}}$.
Construction for ${{\bf KerSub}\xspace}^{{ll}}$
-----------------------------------------------
When studying relations in ${{\bf KerSub}\xspace}^{{ll}}$, we lose the property of being prefix-closed. We have to consider instead the transitive closure of the prefix closure.
\[thm:cons-kersseq\] Let $R\in {{\bf RatEq}\xspace}^{{lp}}$, let $P_R$ be the prefix closure of $R$ such that $S_R$ has finite index with respect to $P_R^+$. Then we can construct a subsequential letter-to-letter transducer whose kernel is $R$.
From Prop. \[prop:comp-synt\], we can obtain a transducer ${\mathcal S}$ realizing $S_R$. Let ${\mathcal P}$ be a transducer realizing $P_R^+$. Without loss of generality, we assume that ${\mathcal R},{\mathcal S},{\mathcal P}$ are letter-to-letter and deterministic. Let us assume that $S_R\subseteq_k P_R^+$. We use the algorithm defined in the proof of Lem. \[lem:cons-seq\] to obtain a transducer which realizes $P_R^+$, with state space $\bigcup_{l\leq k}{\mathcal M}_l(Q)$, where $Q=Q_{\mathcal R}\times Q_{\mathcal P}$, the product of the state spaces of ${\mathcal R}$ and ${\mathcal P}$. Using the same construction we can obtain a sequential transducer realising $P_R^+$ with the following properties:
- Let $u_1,\ldots,u_l$ be the minimal representatives of the $S_R$-equivalence classes of some $P_R^+$-equivalence class. Then any word $uS_Ru_i$ with $i\in {\left\{1,\ldots,l \right\}}$, reaches the state $(M,i)$ where $M(j,j')$ is the state reached in ${\mathcal R}\times{\mathcal P}$ by reading the pair $(u_j,u_{j'})$.
- Two words are $P_R^+$ equivalent if and only if their outputs in ${\mathcal T}$ are equal.
We only need to define a final output function $t:Q_{\mathcal R}\times Q_{\mathcal P}\rightarrow B$ which will differentiate words that are $P_R^+$ equivalent but not $R$ equivalent. Let $u_1,\ldots,u_l$ be the minimal representatives of the $S_R$-equivalence classes of some $P_R^+$-equivalence class, and let $M\in {\mathcal M}_l(Q)$ be the corresponding matrix such that $M(i,j)$ is the state reached by reading $(u_u,u_{j})$ in ${\mathcal R}\times{\mathcal P}$. Then let us consider the equivalence relation $\sim_R$ over ${\left\{1,\ldots,l \right\}}$ defined by $i\sim_Rj$ if and only if $u_iRu_j$. Then we define $t(M,i)=\min_{j\sim_R i}j$.
According to $H.1$ we only need to show that this construction is correct for minimal lexicographic representatives of $S$ classes. Let $u,v$ be two words of same length, and let us assume that $u {\cancel {P_R^+}}v$. Then the images of $u$ and $v$ are already different, even without taking the final output into account. Let us assume that $u P_R^+ v$, then $u,v$ have the same image by ${\mathcal T}$. If $u {\cancel R}v$ considering that $u, v$ are representatives of their respective $S$ class, we have by definition that $t(M,i_u)\neq t(M,i_v)$, where $(M,i_u)$ and $(M,i_v)$ are the states reached by reading $u$ and $v$, respectively. Similarly, we show that if $uRv$, then final outputs are the same which means that the image of $u,v$ by $({\mathcal T},t)$ is the same.
Equality of classes
-------------------
Let us start by stating the obvious inclusions which are just obtained by syntactic restrictions: ${{\bf KerSub}\xspace}^{{ll}}\subseteq {{\bf KerSub}\xspace}^{{lp}}$ and ${{\bf KerSeq }\xspace}^{{lp}}\subseteq {{\bf KerSub}\xspace}^{{lp}}$.
We now show that one can remove the final outputs by adding modulo counting.
\[lem:sseq-seq\] ${{\bf KerSub}\xspace}^{{ll}}\subseteq {{\bf KerSeq }\xspace}$
Let $({\mathcal T}, {\mathit{t}})$ with ${\mathcal T}={\left(Q,\Delta, {\left\{q_0 \right\}}, F \right)}$ be a subsequential letter-to-letter transducer over $A,B$ realizing a function $f$, and let $g$ be the function realized by ${\mathcal T}$. Let $\sim_{\mathit{t}}$ be an equivalence relation defined over $F$ by $p\sim_{\mathit{t}}q$ if ${\mathit{t}}(p)={\mathit{t}}(q)$. Let $u,v$ be two words that reach states $p,q$ respectively from $q_0$. Then, $f(u)=f(v)$ if and only if $g(u)=g(v)$ and $p\sim_{\mathit{t}}q$. We know that the number of equivalence classes of $\sim_{\mathit{t}}$ is less than $n=|B|$, so we number the equivalence classes from $1$ to $n$. The main idea is to consider $g^n$ which multiplies in $g$ every occurrence of each letter by $n$, except for the last letter. Then, the number of occurrences of the last letter encodes, modulo $n$, the equivalence class of the state. Hence for any words $u,v$ we have $g^n(u)=g^n(v)$ if and only if $g(u)=g(v)$ and $p\sim_{\mathit{t}}q$ if and only if $f(u)=f(v)$, which means that the equivalence kernel of $f$ is equal to that of $g^ n$.
Let us now show that $g^n$ is sequential. We extend the equivalence relation $\sim_{\mathit{t}}$ arbitrarily to non final states, and to simplify things, we assume that the equivalence class of the initial state is $n$. Let us define a transducer ${\mathcal T}^n={\left( Q\times B,\Delta^n, {\left\{(q_0,b_0) \right\}}, F\times B \right)}$ realizing $g^n$ (where $b_0$ is some fixed letter in $B$). Let $p,q\in Q$ with respective equivalence classes $i,j\in {\left\{1,\ldots, n \right\}}$ such that $(p,(a,b),q)\in \Delta$. For any $c\in B$ we have $((p,c),(a,c^{n-i}b^{j}),(q,b))\in \Delta^n$.
We only have left to show that relations in ${{\bf KerSeq }\xspace}^{{lp}}$ satisfy the sufficient conditions to be in ${{\bf KerSub}\xspace}^{{ll}}$.
We need a few technical results before showing the main lemma. The next claim is quite simple and just says that if two words can be equivalent in the future, then they can be equivalent in a near future.
\[lem:fut-close\] Let $R\in {{\bf RatEq}\xspace}^{{ll}}$ and let $P_R$ be the prefix closure of $R$. There exists $D\geq 0$ such that for all $u,v$ with $uP_Rv$ there exists $w_1,w_2$ with $|w_1|=|w_2|\leq D$ and $uw_1Ruw_2$.
Let ${\mathcal R}$ be a letter-to-letter transducer recognizing $R$. We assume without loss of generality that ${\mathcal R}$ is pair-deterministic. Then the relation $P_R$ is recognized by ${\mathcal R}'$ which is just ${\mathcal R}$ where all states that can reach a final state become final. Let $D$ be the number of states of ${\mathcal R}$. If $uP_Rv$ there exists $w_1,w_2$ with $|w_1|=|w_2|\leq D$ and $uw_1Ruw_2$.
This next statement is a quite simple consequence of the previous one. If two words are equivalent in the future, then their images by a subsequential kernel function have to be close too.
\[lem:fut-out-close\] $R\in {{\bf KerSub}\xspace}^{{lp}}$, let $f$ be a subsequential function such that $\ker(f)=R$ and let $P_R$ be the prefix closure of $R$. There exists $\delta\geq 0$ such that for all $u,v$ with $uP_Rv$, $||f(u)|-|f(v)||\leq \delta$.
$R\in {{\bf KerSub}\xspace}^{{lp}}$, let $f$ be a subsequential function such that $\ker(f)=R$ and let $P_R$ be the prefix closure of $R$. Let $({\mathcal T},{\mathit t})$ be a subsequential transducer realizing $f$ and let $K$ be the maximal size of an output of $({\mathcal T},{\mathit t})$. According to Lem. \[lem:fut-close\], we know that there exists $D$ such that, if $uP_Rv$, then there exists $w_1,w_2$ with $|w_1|=|w_2|\leq D$ and $uw_1Ruw_2$. This means that $f(uw_1)=f(vw_2)$, and thus $||f(u)|-|f(v)||\leq 2KD$.
The next lemma is the most technical part of this section, and its proof is given in App. \[app:fplus-close\] due to a lack of space. It says that if two words are *transitively* future equivalent, then their images by a subsequential canonical function have to be close.
\[lem:fplus-close\] $R\in {{\bf KerSub}\xspace}^{{lp}}$, let $f$ be a subsequential function such that $\ker(f)=R$ and let $P_R$ be the prefix closure of $R$. There exists $D\geq 0$ such that for all $u,v$ with $uP_R^+v$, $||f(u)|-|f(v)||\leq D$.
The previous lemma shows that two words that are *transitively* future equivalent must have close output from a subsequential canonical function. By a pigeon-hole argument we obtain in the next corollary that a relation in ${{\bf KerSub}\xspace}^{{lp}}$ must have finite index with respect to the transitive closure of the future equivalence.
\[cor:char-kerseq\] Let $R\in {{\bf KerSub}\xspace}^{{lp}}$, and let $P_R$ denote the prefix closure of $R$. Then $S_R$ has finite index with respect to $P_R^+$.
Let $({\mathcal T},{\mathit{t}})$ be a subsequential transducer realizing $f$ such that $\ker f=R$, and let $n$ be the number of states of ${\mathcal T}$. According to Lem. \[lem:fplus-close\], there exists $D$ such that for all $u,v$ with $uP_R^+v$, $||f(u)|-|f(v)||\leq D$. Let $N=|B|^{D+1}$ and let $u_1P_R^+u_2P_R^+\ldots P_R^+ u_{(n+1)N}$. For all $i,j\in{\left\{1,\ldots,(n+1)N \right\}}$, $||f(u_i)|-|f(u_j)||\leq D$. This means that the set ${\left\{f(u_i)|\ 1\leq i\leq (n+1)N \right\}}$ has cardinality less than $N$. Thus there exists $i_1<\ldots<i_{n+1}$ such that $f(u_{i_1})=\ldots=f(u_{i_{n+1}})$. One can see that there must be two indices $1\leq j<k\leq n+1$, such that $u_{i_{j}}$ and $u_{i_{k}}$ reach the same state in ${\mathcal T}$, hence $u_{i}Ru_{j}$, and even $u_{i}S_Ru_{j}$. Thus we have shown that the index of $S_R$ with respect to $P_R^+$ is less than $ (n+1)N$, and is thus finite.
\[prop:eq-classes\] The following classes of equivalence relations are identical:
1. ${{\bf KerSeq }\xspace}^{{lp}}$
2. ${{\bf KerSub}\xspace}^{{ll}}$
3. ${{\bf KerSub}\xspace}^{{lp}}$
The proof is given in Fig. \[fig:proof\]. The arrows represent class inclusion. Black arrows are trivial syntactic restrictions.
(sl) at (0,0) ; (p) at (4,0) ; (sp) at (2,2) ;
(sl) edge\[\] (sp); (p) edge\[\] (sp);
(sp) edge\[bend right\] node\[above left\] (sl); (sl) edge\[\] node\[above\] (p);
Deciding membership in [[**KerSeq** ]{}]{}
==========================================
We show here that knowing if a rational equivalence relation is in ${{\bf KerSeq }\xspace}$ is an undecidable problem, and this even if the relation is length-preserving. The trouble lies with computing the equivalence relation $P_R^+$. Indeed, transitive closures of even very simple relations are known not to be computable (the next configuration of a Turing machine can be computed by a simple transduction).
Let us first state a characterization of ${{\bf KerSeq }\xspace}^{{lp}}$, by combining the results of the previous subsections.
\[thm:char-seq2\] Let $R\in {{\bf RatEq}\xspace}^{{ll}}$. The following are equivalent:
1. $R\in {{\bf KerSeq }\xspace}$
2. $S_R\subseteq_{\mathit{ fin}}R\subseteq_{\mathit{ fin}}P_R^+$
3. $S_R\subseteq_{\mathit{ fin}}R\subseteq_{\mathit{ fin}}P_R$ and $\exists k\ P_R^k=P_R^{k+1}$
$1\rightarrow 2$. Let $R\in {{\bf RatEq}\xspace}^{{ll}}$. Let us first assume that $R\in {{\bf KerSeq }\xspace}$. Then according to Cor. \[cor:char-kerseq\], we have $S_R\subseteq_{\mathit{ fin}}R\subseteq_{\mathit{ fin}}P_R^+$.
$2\rightarrow 1$. Conversely, let us assume that $S_R\subseteq_{\mathit{ fin}}R\subseteq_{\mathit{ fin}}P_R^+$. According to Theorem \[thm:cons-kersseq\], we can construct a subsequential letter-to-letter transducer whose kernel is $R$. From Lem. \[lem:sseq-seq\], we have $R\in {{\bf KerSeq }\xspace}$.
$2\rightarrow 3$. Let us assume $S_R\subseteq_{\mathit{ fin}}R\subseteq_{\mathit{ fin}}P_R^+$. In particular $S_R\subseteq_{\mathit{ fin}}R\subseteq_{\mathit{ fin}}P_R$. Let us assume that $S_R\subseteq_N P_R^+$. Let $uP_R^+ v$ and let $u=u_0P_Ru_1\ldots P_R u_m=v$ be a chain of minimal length $m$. If we assume $m>N$, then there must exist $i,j\leq m$ such that $u_iS_Ru_j$. Since $u_i$ and $u_j$ are *syntactically* equivalent, this means that $u_iP_Rw \Leftrightarrow u_jP_Rw$. Thus we can obtain a strictly smaller chain, which contradicts the assumption, thus $P_R^N=P_R^{N+1}$.
$3\rightarrow 2$. Finally, let us assume $S_R\subseteq_{\mathit{ fin}}R\subseteq_{\mathit{ fin}}P_R$ and $\exists k\ P_R^k=P_R^{k+1}$. Let us assume that $S\subseteq_N P_R$. We only have to show $S_R\subseteq_{N^k}P_R^k$ to conclude the proof. Let us assume that for some $i$ we have $S_R\subseteq_{N^i}P_R^i$. We want to show that $S_R\subseteq_{N^i}P_R^{i+1}$. Let $u\in A^+$, let $T= P_R^{i}(u)$. Let $T'\subseteq T$ be such that $\forall v\in T,\ \exists! w\in T',\ vS_Rw$. Thus we have $P_R(T)=P_R(T')$ since $S_R$ is the syntactic equivalence relation of $R$. Moreover, we have $|T'|\leq N^i$ by assumption, since for all words $v,w\in T'$, $v{{\cancel S_R}}w$. For each $v\in T'$, for each $X\subseteq P_R(v)$ verifying $\forall x,y\ x{\cancel{S}_R}y$, we know by assumption that $|X|\leq N$. Thus for any $Y\subseteq P_R(T)=P_R(T')$ verifying $\forall x,y\ x{\cancel{S}_R}y$, we know that $|Y|\leq |T'|\cdot N\leq N^{i+1}$, which concludes the proof.
From this characterization we obtain two decidability results, one negative and one positive.
\[thm:undec-seq\] The following problem is undecidable:
**Input:** ${\mathcal R}$ a letter-to-letter transducer realizing an equivalence relation $R$.
**Question:** Does $R$ belong to ${{\bf KerSeq }\xspace}$?
The proof of this theorem relies on a reduction of the *mortality problem*, see [@Hooper66 p. 226] and is given in App. \[app:undec-seq\]. The next theorem shows that we are able to identify exactly where the undecidability comes from: computing the transitive closure of the relation $P_R$.
The following problem is decidable:
**Input:** ${\mathcal R},{\mathcal P}$ two transducers realizing equivalence relations $R,P$, respectively, such that $P$ is the transitive closure of the prefix closure of $R$.
**Question:** Does $R$ belong to ${{\bf KerSeq }\xspace}^{{lp}}$?
To show this we rely on the characterization from Thm .\[thm:char-seq2\]. We proceed as in the proof of Thm. \[thm:dec-ltl\], except that we want to check whether $S_R$ has finite index with respect to $P=P_R^+$ instead of $R$. First we can compute a transducer realizing $S_R$, according to Prop. \[prop:comp-synt\]. Then from [@Johnson85 Thm. 5.1], we know we can obtain a transducer realizing a function $f$ whose kernel is $S_R$. Then, using Prop. \[prop:dec-fin-index\], we can decide if $S_R$ has finite index with respect to $P$.
We sum up the decidability of the problem for different classes of equivalence relations in the table of Fig. \[fig:table\]. New results are shown in red.
$\quad$Relations $\backslash$ Kernels$\quad$ $\quad {{\bf KerSeq }\xspace}^{{ll}}\quad$ $\quad{{\bf KerSeq }\xspace}\quad$ $\quad {{\bf KerRat}\xspace}\quad$
------------------------------------------------- -------------------------------------------- ------------------------------------ ------------------------------------
${{\bf RatEq}\xspace}^{{ll}}$ (Thm. \[thm:undec-seq\])
${{\bf RatEq}\xspace}^{{\color{white} {{ll}}}}$ (Thm. \[thm:dec-ltl\])
Conclusion {#conclusion .unnumbered}
==========
We have studied the observation synthesis problem for two classes of observation functions: ${{\bf KerSeq }\xspace}$ and ${{\bf KerSeq }\xspace}^{{ll}}$. A natural question would be to consider the same problem for different classes of functions. However, the term *observation function* is only justified (and related to games with imperfect information) if the functions considered are *monotone* meaning that if $h_1\prec h_2$ denotes that history $h_1$ is a prefix of history $h_2$, then any reasonable class of observation function should ensure that $f(h_1)\prec f(h_2)$, for any function $f$.
Since bounded memory and monotonicity somehow characterize the sequential functions, this means that such a class of observation functions would have to use unbounded memory, for instance the class of regular function, *i.e.* functions realized by two-way transducers. In terms of observations, this would mean that a single game step could give an arbitrary long (actually linear in the size of the history) sequence of observations.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Bruno Guillon for his help in obtaining the undecidability result.
Proof of Lem. \[lem:fplus-close\] {#app:fplus-close}
=================================
$R\in {{\bf KerSub}\xspace}^{{lp}}$, let $f$ be a subsequential function such that $\ker(f)=R$ and let $P$ be the prefix closure of $R$. Let $({\mathcal T},{\mathit t})$ be a subsequential transducer realizing $f$ and let $K$ be the maximal size of an output of $({\mathcal T},{\mathit t})$ . Let us remark that there are no loops in ${\mathcal T}$ that produce nothing. Indeed, if we assume otherwise, then we can find a loop in ${\mathcal T}$ producing nothing, contradicting the fact that $R$ is length-preserving. Hence let $k$ be the smallest ratio of output length over input length for a simple loop in ${\mathcal T}$. Thus we have for any words $u, v$, $k|v|-b\leq ||f(uv)|-|f(u)||\leq K|v|$. Let us assume towards a contradiction that the statement does not hold. This means that for any $D$, we can find a sequence $u_0Pu_1P\ldots Pu_N$, such that $||f(u_0)|-|f(u_N)||\geq D$. Without loss of generality, let us assume that $|f(u_0)|$ is minimal among ${\left\{|f(u_i)|\ \mid\ 0\leq i\leq N \right\}}$. According to Lem. \[lem:fut-out-close\] there exists $\delta$ such that $||f(u_{i-1})|-|f(u_{i})||\leq \delta$, for any $i\in{\left\{1,\ldots,N \right\}}$. Thus, for any integer $M\in {\left\{|f(u_0)|,|f(u_0)|+1,\ldots, |f(u_N)| \right\}}$, there exists $i\in{\left\{0,\ldots,N \right\}} $ and $d\in {\left\{0,\ldots, \delta \right\}}$ such that $|f(u_i)|+d=M$.
Let $C> \max(3,K,\frac {2K} k,\delta,b)$ be a large enough integer. We extract a subsequence of the $u_i$s defined in the following way. Let $i\geq 0$ be such that $|f(u_0)|+C^i+\delta \leq |f(u_N)|$, then there exists $u\in {\left\{u_0,\ldots,u_N \right\}}$ such that $|f(u)|=C^i+d_i$, with $d_i\in {\left\{0,\ldots, \delta \right\}}$, and we set $v_i=u$. Let $v_i'$ be the smallest prefix of $v_i$ such that $|f(v_{i}')|= |f(u_0)|+d_i'$ with $d_i'\in {\left\{0,\ldots, \delta \right\}}$. Then we obtain:
$$\begin{array}{rcccl}
k{ \lVert v_i,v_i' \rVert}-b &\leq& { \lVert f(v_{i}),f(v_{i}') \rVert} &\leq & K{ \lVert v_i,v_i' \rVert}\\
k{ \lVert v_i,v_i' \rVert}-b &\leq & C^i+d_i-d_i' &\leq& K{ \lVert v_i,v_i' \rVert}\\
\end{array}$$ Using these inequalities for $i+1$ and $i$ we have:
$$\begin{array}{rllllll}
{ \lVert v_{i+1},v_{i+1}' \rVert}
&\geq & \frac{C}{K}C^i+\frac{d_{i+1}-d_{i+1}'}{K} \\
&\geq & \frac{C}{K}(k{ \lVert v_i,v_i' \rVert}-b+d_i'-d_i)+\frac{d_{i+1}-d_{i+1}'}{K} \\
&\geq & C\frac{k}{K}{ \lVert v_i,v_i' \rVert}+\frac{C(-b+d_i'-d_i)+d_{i+1}-d_{i+1}'}{K} \\
&\geq & C\frac{k}{K}{ \lVert v_i,v_i' \rVert}-\frac{C(b+\delta)+\delta}{K} \\
&> & C\frac{k}{K}{ \lVert v_i,v_i' \rVert}-\frac{2C^2+C}{K} \\
\end{array}$$ It suffices to show $C\frac{k}{K}{ \lVert v_i,v_i' \rVert}-\frac{2C^2+C}{K}\geq { \lVert v_i,v_i' \rVert}$ in order to obtain ${ \lVert v_{i+1},v_{i+1}' \rVert}> { \lVert v_i,v_i' \rVert}$.
$$\begin{array}{rrrllll}
&C\frac{k}{K}{ \lVert v_i,v_i' \rVert}-\frac{2C^2+C)}{K} &\geq &{ \lVert v_i,v_i' \rVert}\\
\Longleftrightarrow\quad &Ck{ \lVert v_i,v_i' \rVert}-(2C^2+C) &\geq &K{ \lVert v_i,v_i' \rVert}\\
\Longleftrightarrow\quad & { \lVert v_i,v_i' \rVert} &\geq &\frac{2C^2+C}{Ck-K}
\end{array}$$ Since $C>\frac{2K}{k}$, we only have to show ${ \lVert v_i,v_i' \rVert} \geq 2C^2+C$. Moreover, we know that ${ \lVert v_i,v_i' \rVert} \geq \frac{C^i-\delta}{K} $. Thus it suffices to show that $\frac{C^i-C}{C}\geq 2C^2+C$, since $C$ is larger than both $K$ and $\delta$. The inequality holds, as long as $i\geq 4$, since $C$ is larger than 3.
This means that ${ \lVert v_{i+1},v_{i+1}' \rVert}> { \lVert v_i,v_i' \rVert}$ for any $i>3$. Since all $v_i$s have the same length, this means that all $v_i'$s have different length, for $i>3$. For $D$ large enough, we can assume that there are more than $B^{\delta+1}$ $v_i'$s of different lengths. Thus there must exist two with the same image, which contradicts the assumption that $R$ is length-preserving.
Proof of Thm. \[thm:undec-seq\] {#app:undec-seq}
===============================
We use a reduction from the following problem, which well call the *bounded configuration problem*:
**Input:** $M$ a reversible Turing machine
**Question:** Is there a computation $c_1\rightarrow c_2 \rightarrow \ldots$ which visits an infinite number of configurations.
We first give the reduction and then show that the problem is actually undecidable.
Let $M$ be a Turing machine with alphabet $\Sigma$, a state space $Q$ and a transition function $\delta:Q\times\Sigma\rightarrow Q\times \Sigma\times {\left\{\mathrm{left}, \mathrm{right} \right\}}$. A *configuration* is a word over $\Sigma\cup Q$, with exactly one occurrence of a letter in $Q$.
We define a letter-to-letter transducer ${\mathcal R}$ recognizing an equivalence relation $R$, with a prefix closure $P$. Let $c_1,c_2$ be a pair of consecutive configurations, then ${\mathcal R}$ recognizes the pairs $(c_1\sharp1,c_2\sharp2)$, and $(c_2\sharp2,c_1\sharp1)$ by symmetry. Note that these equivalence classes of $R$ have size $2$. Words of the shape $c\sharp$, with $c$ a configuration are only equivalent to themselves. Words that are strict prefixes of words of the shape $c\sharp$ are all equivalent, if they have the same size. All other words are only equivalent to themselves.
On can easily see that there exists $k$ such that $P^k=P^{k+1}$ if and only if computations of $M$ visit at most $k+1$ different configurations. We only have left to check that there are computations of unbounded size if and only if there is an infinite computation. Let $c_1,c_2,\ldots$ be configurations sur that from $c_n$, the machine $M$ visits at least $n$ distinct configurations. Then we can extract a subsequence $d_1,d_2,\ldots$ such that all configurations start in the same state. Extracting a subsequence we can assume that all cells of the tape at distance $1$ from the reading head agree. Repeating the operation, we end up with a configuration which visits more than $n$ configurations for any $n$, *i.e.* is infinite.
The bounded configuration problem is undecidable.
This is shown by a reduction from the *mortality problem* which amounts to deciding if a Turing machine has an infinite computation. Note that this is different from the halting problem, because we ask if the machine halts on *all* possible configurations. This problem was shown to be undecidable in [@Hooper66 p. 226] for Turing machines and in [@KariO08 Thm. 7] for *reversible* Turing machines.
Assume that the bounded configuration problem is decidable. Given a reversible machine $M$, if is has a computation visiting an infinite number of configurations, then it has an infinite computation. If there is no computation visiting an infinite number of distinct configurations then there is a uniform bound on the number of configurations that a computation can visit. This bound $k$ can be computed, just by simulating the machine on larger and larger configurations. Then, on can easily see if one of the computations loops, and thus decide if the machine has an infinite computation.
[^1]: The equivalence kernel of a total function $f$ is defined by $x\sim y \Leftrightarrow f(x)=f(y)$
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Within the Glauber formalism and a BUU transport model we analyze the $\eta$-photoproduction data from nuclei and evaluate the in-medium $\eta N$ cross section. Our results indicate that the $\eta N$ cross section is almost independent of the $\eta$ energy up to 200 MeV.'
author:
- |
M. Effenberger and A. Sibirtsev\
Institut für Theoretische Physik, Universität Giessen\
D-35392 Giessen, Germany
date:
title: 'The energy dependence of the in-medium $\eta N$ cross section evaluated from $\eta$-photoproduction [^1]'
---
=-10mm
Introduction
============
For a long time $\eta$-meson production in nuclei has been of interest as a source of information about the $\eta$-nucleus final state interaction. The present knowledge about the $\eta N$ interaction even in the vacuum comes from either simple analysis of the inverse $\pi N \to \eta N$ reaction or as a free parameter fitted to experimental data by theoretical calculations [@Liu; @Wilkin; @Chiavassa; @Sibirtsev1].
Note that the value of the $\eta N$ scattering length is still an open problem and there is not actual agreement between a bulk of theoretical investigations.
The analysis of $\eta$-production from $pA$ collisions indicated strong sensitivity of the calculations to the prescription of the $\eta$-meson final state interaction [@Chiavassa; @Golubeva; @Sibirtsev2]. It was found that the $\eta$-energy spectrum [@Chiavassa2] is mostly influenced by the variation of the $\eta N$ cross section [@Golubeva; @Sibirtsev2]. However the experimental data [@Chiavassa2] had large uncertainties and there was no continuation of the systematical studies.
Recent measurements on $\eta$-meson photoproduction in nuclei [@Krusche] are more detailed and accurate. Among the theoretical investigations [@Lee; @Carrasco; @Effenberger] only the calculations within the Distorted Wave Impulse Approximation (DWIA) from Lee et al.[@Lee] are able to reproduce the experimental data by incorporating the $\eta$-nucleus potential proposed by Bennhold and Tanabe [@Ben].
The present paper is organized as follows. In section \[glauber\] we use the Glauber formalism to extract the in-medium $\eta N$ cross section from the experimental data. In section \[buu\] these results are compared to Boltzmann-Uehling-Uhlenbeck transport model calculations. The sensitivity of the theoretical results to the prescription of $\eta N$ scattering is investigated.
Analysis within the Glauber Model {#glauber}
=================================
In an incoherent approximation the cross section of $\eta$-meson photoproduction off nuclei is given by $$\label{app1}
{\sigma}_{\gamma \eta}^A =
{\sigma}_{\gamma \eta }^p \times
\left[ Z + \zeta (A-Z) \right]$$ with $A$, $Z$ being the mass and charge of the target,respectively, while the factor $\zeta=2/3$ [@Krusche2] stands for the ratio of the elementary $\eta$-photoproduction cross sections from $\gamma n$ and $\gamma p$ reactions.
In nuclei the cross section differs from the approximation (\[app1\]) due to nuclear effects. ([*i*]{}) The Fermi motion of nucleons as well as ([*ii*]{}) Pauli blocking are important at energies below and close to the reaction threshold in free space [@Cassing1; @Salcedo]. We should also take into account [*(iii)*]{} the modification of the $N^*$-resonance by the nuclear medium [@Carrasco; @Effenberger].
However the most important effect is [*(iv)*]{} the strong final state interaction of $\eta $-mesons in nuclear matter. The deviation of the $A$-dependence of the ${\sigma}_{\gamma A \rightarrow \eta X}$ from $A^1$ mostly reflects the strength of the final state interaction.
Here we present an analysis of the experimental data on $\gamma A \rightarrow \eta X$ reactions in order to extract the in-medium cross section ${\sigma}_{\eta N}$. Our approach is based on the Glauber model [@Glauber] and first was developed by Margolis [@Margolis1] for evaluation of the $\rho N$ cross section from both incoherent and coherent $\rho$-meson photoproduction off nuclei. The most detailed description and application of the Glauber model to photoproduction reactions may be found from review of Bauer, Spital and Yennie [@Bauer]. A similar formalism is adopted for studying color transparency [@Bertch; @Kopeliovich1], where the in-medium cross sections is treated as a function of a transverse separation of the hadronic wave function.
In the Glauber model the cross section of the incoherent $\eta$-meson photoproduction reads $$\label{fact}
{\sigma}_{\gamma \eta }^A=
{\sigma}_{\gamma \eta }^p
\frac
{Z + \zeta (A-Z)} { A} \times A_{eff}$$ where $$\label{aef}
A_{eff} = \frac {1} {2 \pi}
\int_0^{+\infty} d{\bf b} \int_{-\infty}^{+\infty} dz \ {\rho}({\bf b},z)
\int_0^{2\pi} d\phi \ exp \left[-{\sigma}_{\eta N}
\oint d\xi \ {\rho}({\bf r}_{\xi}) \right]$$ Here ${\rho}(r)$ is the single particle density function, which was taken of Fermi type with parameters for each nucleus from [@Jager]. The last integration in Eq. (\[aef\]) being over the path of the produced $\eta$-meson $$r_{\xi}^2 = (b+ \xi cos\phi sin\theta)^2+(\xi sin\phi sin\theta )^2
+(z + \xi cos\theta )^2$$ Here $\theta$ is the emission angle of the $\eta$-meson relative to $\gamma$-momentum.
Eq. (\[aef\]) is similar to those from [@Vercellin; @Hufner] and in the low energy limit, i.e. by integration over the $\eta$-emission angle $\theta$ becomes as [@Benhar] $$\label{ave}
A_{eff} =
\int_0^{+\infty} d{\bf b} \int_{-\infty}^{+\infty} dz \ {\rho}({\bf b},z)
exp \left( -{\sigma}_{\eta N}
\int_z^{\infty} d\xi \ {\rho}({\bf b}, \xi) \right)$$ In the high energy limit, i.e. with the small angle scattering approximation $\theta =0$, Eq. (\[aef\]) reduces to simple formula from [@Margolis2] $$\label{eq2}
A_{eff}=
\frac {1} {{\sigma}_{\eta N}}
\int_0^{\infty} d{\bf b}
\ \left( 1- exp \left[ -{\sigma}_{\eta N} \
\int_{-\infty}^{+\infty} dz \ {\rho}({\bf b},z)
\right] \right)$$
The nuclear transparency is defined now as $$T^A = \frac {{\sigma}_{\gamma \eta }^A }
{{\sigma}_{\gamma \eta }^p \times \left[ Z + \zeta (A-Z) \right]}$$ and in the Glauber model it is simply given by $$T^A = A_{eff} / A$$ being the function of the target mass $A$, emission angle $\theta$ and in-medium $\eta N$ cross section ${\sigma}_{\eta N}$. Note that this model neglect the in-medium effects [*(i-iii)*]{}, and takes only the final state interactions into account.
We analyze now the recent MAMI data [@Krusche] on $\eta$-photoproduction from $^{12}C$, $^{40}Ca$, $^{93}Nb$ and $^{207}Pb$ at $E_{\gamma}<$800 MeV in order to resolve the dependence (\[aef\]) with respect to the target mass. The $\eta$-production threshold on a free nucleon lies at $\simeq$706 MeV, thus our analysis is expected to be valid for $E_{\gamma} \geq 750$ MeV, in order to minimize effects [*(i-ii)*]{}. Moreover, to minimize the uncertainties related to [*(iii)*]{}, which also are valid at high $E_{\gamma}$, we analyze the ratios of the differential cross sections integrated over the $\eta$-meson emission angle as $$\label{rat}
R(A/^{12}C) = \frac{d{\sigma}_{\gamma \eta }^A} {dT}
\left( \frac {d{\sigma}_{\gamma \eta }^{^C}}{dT} \right)^{-1}$$ We thus assume that the medium modifications of the $N^{\star}$-resonance are almost the same for all nuclear targets.
The ratios (\[rat\]) are shown in Fig. \[fi1\] for several kinetic energies of $\eta$-mesons and as a function of the target mass. The lines indicate our calculations performed for different $\eta N$ cross sections. The model results are integrated over the $\theta$. Note that for ${\sigma}_{\eta N}$=0 the ratio (\[rat\]) saturates at $R=A/C$ as was expected neglecting the final state interaction.
We now fit the experimental ratios for each $T_{\eta}$ by minimizing the ${\chi}^2$ in order to evaluate ${\sigma}_{\eta N}$. A similar analysis was perfomed recently by Kharzeev et al. [@Kharzeev] for the evaluation of the $J/\Psi$-nucleon cross section. Fig.\[fi2\] illustrates the minimization procedure and shows sensitivity of the data to the variation of $\eta N$ cross section. We fixed the confidence level that gives the value of the reduced ${\chi}^2/n >2$ can be expected no more than 10% of the time. With respect to the statistical errors of the experimental data the minimization produces three types of results. Namely, 1) with extraction of ${\sigma}_{\eta N}$ and indication its uncertainty, 2) with evaluation only the lower limit for ${\sigma}_{\eta N}$ or 3) with obtaining the minima behind the confidence level.
Fig. \[fi3\] shows our final results in comparison with the experimental data and illustrate excellent agreement for wide range of the $\eta$-energies. Nevertheless we keep in mind the uncertainties in evaluating of $\eta N$ cross section and collect the ${\sigma}_{\eta N}$ in Fig. \[fi4\] as function of the $\eta$ energy and indication of confidence level. Note that within present analysis we evaluate the inelastic (or absorption) $\eta$-nucleon cross section, because the elastic scattering does not remove the $\eta$-meson from the total flux, which was detected experimentally.
Our results indicate almost constant in-medium $\eta N$ cross section as function of the $\eta$-energy in strong contradiction with the ${\sigma}_{\eta N}$ from the scattering in vacuum.
To make a more definite conclusion about the suppression of the $\eta N$ cross section in nuclear matter we need an accurate data on the coherent $\eta$-photoproduction off nuclei. The coherent reactions are more sensitive to the nuclear transparency ($\propto A_{eff}^2$ [@Bauer; @Margolis2; @Bochman]) and might solve the uncertanties of the present analysis performed with the Glauber model.
Results from BUU calculations {#buu}
=============================
In order to verify the results from the previous section we use a BUU transport model [@Cassing1; @Bertsch1; @Teis1] to calculate energy differential $\eta$-photoproduction cross sections in nuclei. This allows to drop several assumptions needed for the Glauber calculations. Fermi motion and Pauli blocking are taken into account as well for the primary $\eta$-production process as for the final state interaction of the produced particles.
In Ref. [@Effenberger] the BUU model was used to calculate $\eta$-photoproduction in nuclei with the resonance model for the $\eta$ final state interaction from Ref. [@Teis1]. Here the $\eta$-rescattering was described by intermediate excitations of N(1535) resonances. The elastic and inelastic $\eta N$ cross sections calculated with this model are shown in Fig. \[abs\] with the solid lines. It turned out that this model was able to describe the total $\eta$-photoproduction cross sections reasonably well but failed in the description of angular and energy differential cross sections. Compared to the experimental data the calculated cross sections were shifted to smaller angles and larger $\eta$-energies for all considered target nuclei and all photon energies up to 800 MeV. In Fig. \[new\] the solid line shows the calculation of an energy differential $\eta$-photoproduction cross section on $^{40}$Ca with this model.
It was already reported in Ref. [@Effenberger] that the discrepancy to the experimental data is cured by using an energy independent $\eta N$ cross section. The corresponding result is shown in Fig. \[new\] by the line labelled ’constant cross sections (1)’ where an inelastic cross section $\sigma^\eta_{in}=30\,$mb and an elastic cross section $\sigma^\eta_{el}=20\,$mb was used.
Now we want to study the influence of different prescriptions for $\eta N$ scattering. The dashed line in Fig. \[new\] indicates the results calculated with a modified $\eta$-rescattering model: $$\begin{aligned}
\label{modi}
{\sigma}_{\eta N \to \pi N} &=& \frac{q_{\pi}}{q_{\eta}s} \
\frac{c^2 }{c^2+q_{\eta}^2} \ 40 \, {\rm mb \,GeV^2}
\, ,\, c=0.3\,{\rm GeV} \\
{\sigma}_{\eta N \to \eta N} &=& \frac{45 \, {\rm mb\,GeV^2}}{s}
\nonumber\end{aligned}$$ where $q_{\pi}$, $q_{\eta}$ are the cms momenta of the $\pi$- and $\eta$-meson, respectively, and $s$ stands for the squared invariant energy. The inelastic cross section was obtained by fitting the experimental data for the reaction $\pi N \to \eta N$, while the elastic cross section was assumed to be equal to the one in the resonance model at an $\eta$-energy of 125 MeV. The resulting elastic and inelastic cross sections are shown in Fig. \[abs\] with the dashed line. As can be seen from Fig. \[new\] (dashed line) this model improves the description of the energy differential cross section for small $\eta$-energies but still overestimates the cross section for higher energies.
We have also used the $\eta N$ cross sections from Green and Wycech [@Green] and Lee et al. [@Lee]. The calculation of Green and Wycech is based on the K-matrix method and includes the $S_{11}(1535)$ and $S_{11}(1650)$ resonances. Following the authors this model is valid up to an invariant energy of about 100 MeV from $\eta N$ threshold which corresponds to an $\eta$ kinetic energy in the nucleon rest frame of 160 MeV. Lee et al. use a parameterization of the $\eta N$ scattering amplitude that is based on the calculation of Bennhold and Tanabe [@Ben]. This model contains the $P_{11}(1440)$, $D_{13}(1520)$ and $S_{11}(1535)$ resonances and might therefore be limited to an $\eta$ kinetic energy of about 100 MeV.
The total cross sections within these models are shown in Fig. \[abs\]. Both models give about the same inelastic cross section which is basically due to the fact that in both models the dominating inelastic channel is given by the process $\eta N \to N \pi$. The experimental data [@Landolt] for this reaction, obtained by detailed balance from $\pi^- p \to \eta n$, are also shown. But one should note that the theoretical curves contain additional, even though small, contributions from $\eta N \to N \pi \pi$ and therefore can not directly be compared to these data points. The elastic $\eta N$ cross section in both models is very different which is an indication for the large theoretical uncertainties in the models for $\eta N$ scattering even in the vacuum.
The corresponding results of the BUU calculations for photoproduction are given in Fig. \[new\]. As in the calculations within the resonance model and the model from Eq. (\[modi\]) we again fail to describe the shape of the energy differential cross section. The same holds for all target nuclei and photon energies as well as for the angular differential cross sections. Apart from the resonance model [@Teis1] all models give a satisfactory description of the cross section for $\eta$-energies below 50 MeV. The failure of the resonance model is due to the fact that this model was fitted to a larger class of elementary processes and a wider kinematical range and overestimated the cross section for $\eta N \to N \pi$ in the considered energy range.
In Fig. \[new\] we also show the result of a model calculation with a constant inelastic cross section $\sigma_{in}^\eta=30\,$mb where we neglected elastic $\eta N$ scattering (curve labelled ’constant cross sections (2)’). Compared to the previous calculation with constant cross sections that included an elastic cross section the energy differential cross section is shifted to larger energies and fails to describe the data. Moreover the integrated cross section is slightly larger because the elastic cross section increases, in average, the length of the path of the produced etas through the nucleus and therefore reduces the number of etas that escape from the nucleus. One sees that in our model a constant inelastic $\eta N$ cross section alone is not sufficient to describe the data but an elastic cross section is also needed.
Since the models for $\eta N$ scattering in the vacuum show a rather strong decrease of the total $\eta N$ cross section with $\eta$-energy we are not able to reproduce the data for $\eta$-photoproduction with any of these models within our transport model approach. For $\eta$-energies larger than 100 MeV we need an $\eta N$ cross section that is significantly larger than the one from the vacuum models while for lower energies our calculations are not very sensitive to the size of the cross section. A possible explanation is that the vacuum models [@Green; @Ben] are simply not applicable to the considered energy range. After all, due to the Fermi motion of the nucleons, the $\eta N$ cross section up to an invariant energy of $\sqrt{s}=1.74\,$GeV ($T_\eta=446\,$MeV in the nucleon rest frame) enters the calculations for an eta with a kinetic energy of 250 MeV in the rest frame of the nucleus.
Our findings are in line with the Glauber analysis from section \[glauber\] and Ref. [@Krusche] which need a constant inelastic cross section of 30 mb in order to describe the mass dependence of the energy differential cross sections. However, Lee et al. [@Lee] were able to reproduce energy differential cross sections within the DWIA framework by using vacuum $\eta N$ cross sections. One crucial difference to our calculation is that in their calculation the outgoing nucleon in the elementary photoproduction process $\gamma N \to N \eta$ is set on-shell while in our semi-classical treatment the elementary process takes place instantaneously with following propagation of the produced particles through the nucleus. The potential energy which is needed to set the nucleon on-shell clearly shifts the $\eta$-spectrum to lower energies. A priori it is not obvious which of the two prescriptions is better suited to model the physical reality. Only a DWIA calculation along the line of Ref. [@Li] without the local approximation of Ref. [@Lee] could clarify this question. An indication for a larger $\eta N$ cross section at higher energies is the fact that we are able to describe energy and angular differential cross section for $\eta$-photoproduction simultaneously by using a constant cross section [@Effenberger] while in the calculations of Lee et al. the angular differential cross sections are shifted to smaller angles compared to the data.
Summary
=======
We have analyzed the $\eta$-photoproduction in nuclei within the framework of the Glauber model and a BUU transport model approach.
Using the standard Glauber theory we investigate the $A$-dependence of the reaction $\gamma A \to \eta X$ in order to extract the data on $\eta$-meson final state interaction in nuclei. It was found that the in-medium $\eta N$ cross section is almost energy independent from $\eta N \to \pi N$ threshold up to $\eta$-kinetic energy of 130 MeV.
Within a BUU transport model calculation we are able to reproduce energy and angular differential data for $\eta$ photoproduction only by using an energy independent $\eta N$ cross section but not with any available $\eta N$ vacuum cross section. However, the effect of the nucleon potential that can not be treated in our semi-classical calculation in a correct way might have an impact on that conclusion.
Acknowledgement
===============
The authors are grateful to B. Krusche and H. Stroeher for productive discussions. They especially like to thank U. Mosel for valuable suggestions and a careful reading of the manuscript.
[99]{} R.S. Bhalerao and L.C. Liu, Phys. Rev. Lett. 54 (1985) 865. C. Wilkin, Phys. Rev. C 47 (1993) R938. E. Chiavassa et al. Z. Phys. A 344 (1993) 345. A.A. Sibirtsev, Phys. Scr. RS 21 (1993) 167. Ye.S. Golubeva, A.S. Ilijnov, E.Y. Paryev and I.A. Pshenichnov, Z. Phys. A 345 (1993) 223. A.A. Sibirtsev, Phys. At. Nucl. 56 (1993) 608. E. Chiavassa et al., Nuovo Cim. A 107 (1994) 1195. M. Roebig-Landau, J. Ahrens, G. Anton et al., Phys. Lett. B 373 (1996) 45. F.X. Lee, L.E. Wright, C. Bennhold and L. Tiator, Nucl. Phys. A 603 (1996) 345. R.C. Carrasco, Phys. Rev. C 48 (1993) 2333. M. Effenberger, A. Hombach, S. Teis and U. Mosel, Nucl. Phys. A 614 (1997) 501. C. Bennhold and H. Tanabe, Nucl. Phys. A 530 (1991) 625. B. Krusche et al., Phys. Lett. B 358 (1995) 40. W. Cassing, V. Metag, U. Mosel and K. Niita, Phys. Rep. 188 (1990) 363. L.L. Salcedo, E. Oset, M.J. Vicente-Vacas and C. Garcia-Recio, Nucl. Phys. A 484 (1988) 557. R.J. Glauber and J. Mathiae, Nucl. Phys. B 21 (1970) 135. B. Margolis, Phys. Lett. B 26 (1968) 254. T.H. Bauer, R.D. Spital and D.R. Yennie, Rev. Mod. Phys. 50 (1978) 261. J. Bertch, S.J. Brodsky, A.S. Golhaber and J.G. Gunion, Phys. Rev. Lett. 47 (1981) 297. B.Z. Kopeliovich and B.G. Zakharov, Phys. Lett. 264 (1991) 434. C.W. Jager, C. Vries and H. Vries, At. Data Nucl. Data Tables 14 (1974) 480. E. Vercellin, E. Chiavassa, G. Dellacasa et al., Nuovo Cim. A 106 (1993) 861. J. Hüfner, B. Kopeliovich and J. Nemchik, nucl-th/9605007 (1996) O. Benhar, B.Z. Kopeliovich, C. Mariotti, N.N. Nikolaev and B.G. Zaharov, Phys. Rev. Lett. 69 (1992) 1156. K.S. Kölbig and B. Margolis, Nucl. Phys. B 6 (1968) 85. D. Kharzeev, C. Lourenco, M. Nardi and H. Satz, Z. Phys. C 74 (1997) 307. G. Bochman, B. Margolis and C.L. Tang, Phys. Lett. B 30 (1969) 254. G.F. Bertsch, H. Kruse and S. Das Gupta, Phys. Rev. C 29 (1984) 673. S. Teis, W. Cassing, M. Effenberger, A. Hombach, U. Mosel and Gy. Wolf, Z. Phys. A 356 (1997) 421. A.M. Green and S. Wycech, Phys. Rev. C 55 (1977) 2167. Baldini et al., Landolt-Börnstein, Band 12 (Springer, Berlin, 1987). X. Li et al., Phys. Rev. C 48 (1993) 816.
[^1]: Supported by Forschungszentrum Jülich, GSI, BMBF and DFG
| {
"pile_set_name": "ArXiv"
} |
---
author:
-
title: Making Use of Affective Features from Media Content Metadata for Better Movie Recommendation Making
---
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'This article describes a quantum bit commitment protocol, QBC1, based on entanglement destruction via forced measurements and proves its unconditional security.'
author:
- |
\
\
Horace P. Yuen\
Department of Electrical Engineering and Computer Science\
Department of Physics and Astronomy\
Northwestern University, Evanston Il. 60208\
[email protected]
title: An Unconditionally Secure Quantum Bit Commitment Protocol
---
**Note:** This paper is an elaboration of my 2006 QCMC paper, arXiv: 0702074v4(2007), published in its Proceedings volume. It was submitted in Dec 2009 as an “invited paper” to a journal, which was withdrawn half a year later because the editors found it incomprehensible. I hope it may make better sense to some other readers.
My reasons for the impossibility of QBC “impossibility proofs” are described in ref \[1\]. Over the years I have produced several QBC protocols that I thought were secure, but when concealing they are not binding due to the scope of entanglement attack that works even across teleportation. I did not and do not see such scope spelled out anywhere, though before putting such papers on the arXiv I should have tried harder to find out whether entanglement attack works in my cases, which I eventually did. Since 2003 I have not received any substantial negative comment on my QBC arXiv papers, only getting a few questions and agreements, and thus the arXiv papers have not served the purpose of soliciting technical disagreements I sought in this controversial subject.
I have been as sure that the present protocol is secure as most results I ever published, but I knew the environment of disagreement and did not submit any QBC paper to any journal until Dec 2009. If this present paper is indeed incomprehensible, it would have to be expanded before submission to a journal. In the meantime a QBC possibility paper by G. P. He, J. Phys. A: Math. Theor. 44, 445305 (2011) has appeared in a reputable journal. That protocol is based on an entirely different mechanism from that of this paper, and gives a weaker form of security. Generally, the best a QBC impossibility proof can do is to show a certain type of QBC protocols cannot be unconditionally secure. It cannot show general impossibility for the simple reason that not all QBC protocols can be captured in any mathematical formulation just within nonrelativistic quantum mechanics \[1\].
My view is that QBC can actually be practically developed and it could perform cryptographic functions with security that is impossible to achieve classically. However, it would not be through the impractical protocol of this paper and the security would not be “unconditional” which is never needed in practice. It appears such QBC development is only possible after the entrenched contrary view on unconditionally secure QBC is sufficiently softened up. I hope this paper would contribute to such end.
Introduction {#sec:intro}
============
It is nearly universally accepted that unconditionally secure quantum bit commitment (QBC) is impossible. This is taken to be a consequence of the Einstein-Podolsky-Rosen(EPR) type entanglment cheating. For detailed discussion with historical remarks on the impossibility of secure QBC and the various impossibility proofs, see ref [@yuen1]-[@dariano07]. In the following, a new approach is described that lies outside the formulation of these impossible proofs. A secure QBC protocol, to be called QBC1, is presented together with a full proof of its unconditional security. This paper is completely self-contained other than background knowledge of quantum mechanics.
QBC Formulation and the Impossibility Proof {#sec:form}
===========================================
In a [*bit commitment*]{} scheme, one party, Alice, provides another party, Bob, with a piece of evidence that she has chosen a bit (0 or 1) which is committed to him. Later, Alice would [*open*]{} the commitment by revealing the bit to Bob and convincing him that it is indeed the committed bit with the evidence in his possession and whatever further evidence Alice then provides, which he can [*verify*]{}. The usual concrete example is for Alice to write down the bit on a piece of paper, which is then locked in a safe to be given to Bob, while keeping for herself the safe key that can be presented later to open the commitment. The scheme should be [*binding*]{}, i.e., after Bob receives his evidence corresponding to a given bit value, Alice should not be able to open a different one and convince Bob to accept it. It should also be [*concealing*]{}, i.e., Bob should not be able to tell from his evidence what the bit is. Otherwise, either Alice or Bob would be able to cheat successfully.
In standard cryptography, secure bit commitment is to be achieved either through a trusted third party, or by invoking an unproved assumption concerning the complexity of certain computational problems. By utilizing quantum effects, specifically the intrinsic uncertainty of a quantum state, various QBC schemes not involving a third party have been proposed to be unconditionally secure, in the sense that neither Alice nor Bob could cheat with any significant probability of success as a matter of physical laws. In 1995-1996, a supposedly general proof on the impossibility of unconditionally secure QBC and the insecurity of previously proposed protocols were presented [@may1]-[@lc1]. Henceforth it has been generally accepted that secure QBC and related objectives are impossible as a matter of principle [@lc2]-[@sr].
There is basically just one impossibility proof, which gives the EPR attacks for the cases of equal and nearly equal density operators that Bob has for the two different bit values. The proof purports to show that if Bob’s successful cheating probability $P^B_c$ is close to the value $\frac{1}{2}$, which is obtainable from pure guessing of the bit value, then Alice’s successful cheating probability $P^A_c$ is close to the perfect value 1. The impossibility proof describes the EPR attack on a specific type of protocols, and then argues that all possible QBC protocols are of this type.
The formulation of the standard impossibility proof can be cast as follows. Alice and Bob have available to them two-way quantum communications that terminate in a finite number of exchanges, during which either party can perform any operation allowed by the laws of quantum physics, all processes ideally accomplished with no imperfection of any kind. During these exchanges, Alice would have committed a bit with associated evidence to Bob. It is argued that, at the end of the commitment phase, there is an openly known entangled pure state $\ket{\Phi_\sb}$, $\sb \in \{0,1\}$, shared between Alice who possesses state space $\cH^A$, and Bob who possesses $\cH^B$. For example, if Alice sends Bob one of $M$ possible states $\{
\ket{\phi_{\sb i}} \}$ for bit with probability $p_{\sb i}$, then $$\ket{\Phi_{\sb }} = \sum_i \sqrt{p_{\sb i}}\ket{e_i}\ket{\phi_{\sb i}}
\label{eq:1}$$ with orthonormal $\ket{e_i} \in \cH^A$ and known $\ket{\phi_{\sb i}}
\in \cH^B$. Alice would open by making a measurement on $\cH^A$, say $\{ \ket{e_i} \}$, communicating to Bob her result $i_0$, then Bob would verify by measuring the corresponding projector ${|\phi_{\sb i_0}\rangle\langle \phi_{\sb i_0}|}$ on $\cH^B$.
When classical random numbers known only to one party are used in the commitment, they are to be replaced by corresponding quantum entanglement purification. The commitment of $\ket{\phi_{\sb i}}$ with probability $p_{\sb i}$ in (\[eq:1\]) is, in fact, an example of such purification. Generally, for any random $k$ used by Bob, it is argued from the doctrine of the “Church of the Larger Hilbert Space” that it is to be replaced by the purification $\ket{\Psi}$ in $\cH^{B_1} \otimes \cH^{B_2}$, $$\ket{\Psi} = \sum_k \sqrt{\lambda_k} \ket{f_k}\ket{\psi_k},
\label{eq:2}$$ where $\ket{\psi_k} \in \cH^{B_2}$. The {$\ket{f_k}$} are complete orthonormal in $\cH^{B_1}$ kept by Bob while $\cH^{B_2}$ would be sent to Alice.
For unconditional, rather than perfect, security, one demands that both cheating probabilities $P^B_c - \frac{1}{2}$ and $P^A_c$ can be made arbitarily small when a security parameter $n$ is increased [@may1]. Thus, [*unconditional security*]{} is quantitatively expressed as $$\qquad \lim_n P^B_c = \frac{1}{2},\quad \lim_n P^A_c = 0.
\label{eq:3}$$ The condition (\[eq:3\]) says that, for any $\epsilon > 0$, there exists an $n_0$ such that for all $n > n_0$, $P^B_c - \frac{1}{2} <
\epsilon$ and $P^A_c < \epsilon$, to which we may refer as $\epsilon$-[*concealing*]{} and $\epsilon$-[*binding*]{}. These cheating probabilities are to be computed purely on the basis of logical and physical laws, and thus would survive any change in technology, including an increase in computational power. In general, one can write down explicitly the optimal $P^B_c$, $$P^B_c = \frac{1}{4}\left(2 + {\| \rho^B_0 - \rho^B_1 \|_1}\right),
\label{eq:4}$$ where ${\| \cdot \|_1}$ is the trace norm, ${\| \tau \|_1} \equiv \tr
(\tau^\dag \tau)^{1/2}$ for a trace-class operator $\tau$.
The entanglement cheating mechanism is explicitly spelled out in the impossibility proof. Under perfect concealing $P^B_c=\frac{1}{2}$, it follows from (\[eq:4\]) that the state $\rho^B_b$ at Bob’s possession obeys $\rho^B_0=\rho^B_1$. Hence by the Schmidt decomposition Alice can turn $\ket{\phi_{0i}}$ into $\ket{\phi_{1i}}$ by a unitary transformation on $\cH^A$ in her possession, thus succeeds in cheating perfectly. Under approximate concealing, an explicit transformation on $\cH^A$ can be similarly identified [@yuen2]-[@yuec] which leads to $$4(1-P^B_c)^2 \le P^A_c \le 2 \sqrt{P^B_c
(1-P^B_c)}.
\label{eq:5}$$ The lower bound in (\[eq:5\]) yields the following impossibility result, $$\lim_n P^B_c = \frac{1}{2} \,\, \Rightarrow
\,\, \lim_n P^A_c = 1
\label{eq:6}$$ Note that the impossibility proof makes a stronger statement than the mere impossibility of unconditional security, i.e., (\[eq:6\]) is stronger than (\[eq:3\]) not being possible.
The assumption in the impossibility formulation that $\ket{\Phi_{\sb }}$ are openly known has been challenged. In a multi-pass protocol where Alice and Bob exchange states, each $\ket{\phi_{\sb i}}$ becomes of the form $\ket{\phi_{\sb i k}}$ [@sr] $$\ket{\phi_{\sb ik}} = U^A_{\sb i_n} \ldots U^A_{\sb i_2} U^B_{k_1}
U^A_{\sb i_1} \ket{\phi_0}.
\label{eq:7}$$ where $U^A_{\sb il}$ are unitaries that Alice applies and $U^B_{\sb kj}$ are applied by Bob. The ancilla state $\ket{e_i}$ also separates into $\ket{e^A_i}\ket{e^B_k}$ with $\ket{e^A_i}$ in Alice’s possession and $\ket{e^B_k}$ in Bob’s. It is clear that the exact $\ket{e^B_k}$ may be kept secret by Bob, in an unnormalized form that would include both the entanglement basis and the probability of each state in it. The question is why secure QBC is impossible under such added randomness, whose quantum purification is either unknown to anyone as in the case of classical random number generation from a piece of macroscopic equipment, or at least known only to the party who preforms the entanglement purification.
For some discussion of this point of employing unknown randomness, see [@yuec]-[@yueb2] and references cited therein. It turnes out it appears impossible to get a secure protocol with this approach. For the case of perfect concealing, a general proof of this impossibility was given in [@yuen2] for a two-pass protocol. A different argument applicable to multi-pass protocol was given by Ozawa[@oz] and later independently by Cheung[@chau1]. Simple as well as more complicated proofs concerning all natural protocols of this kind in the case of approximate concealing are also available. See [@yuen1]-[@dariano07], [@chau2].
In the above formulation one may consider, *more generally*, the whole $\ket{\Phi_{\sb }}$ of (1) as the state corresponding to the bit with Alice sending $\cH^A$ to Bob at opening who verifies by measuring on the total $\ket{\Phi_{\sb }}$. Similarly in the multi-pass case, (7) is generalized from $\ket{\phi_{\sb ik}}$ to $\ket{\Phi_{\sb ik}}$ with different subspaces of $\cH^A$ and $\cH^B$ being exchanged during each pass. The above quantitative conclusion is not affected. Note, however, that either Alice or Bob has to provide the initial state $\ket{\phi_0}$. Indeed, $\ket{\phi_0}$ must be on a large enough dimension state space and openly known to both parties if either can perform random number purification. It is more convenient to just let each party supply its own state space at each turn when needed, and let $\cH^A$ and $\cH^B$ be their individual total spaces as just indicated. In contrast to one-pass protocols in [@lc1], there is then always the question of “*honesty*” in multi-pass protocols. It is clear that some form of state checking may be necessary to execute these protocols.
New Approach {#sec:newappr}
============
In the impossibility proof formulation the probability of interactive checking between Alice and Bob, similar to there in QKD protocol such as BB84, is not explicitly accounted for. Even if Bob’s check on Alice can be postponed to just before opening, Alice’s check on Bob must be carried out during the commitment phase to maintain $\epsilon$-concealing, The implicit assumption must be, therefore, that such checking could be satisfied perfectly without affecting the protocol. In this section, a new approach to QBC protocol would be described that shows such implicit assumption cannot be true. This approach would be utilized in the next Section \[sec:qbc1\] to show how a specific secure protocol can be obtained.
Consider the following situation or “protocol": Bob sends Alice a sequence of $n$ qubits, each randomly in one of the two orthogonal states $\ket{l_j}, j \in \{1,2\}$, which are themselves chosen randomly on a fixed great circle $C$ of the qubit Bloch sphere. The index $l$ indicates the position in the $n$-qubit sequence. We assume for convenience that Bob entangled each $l$th qubit to a qubit ancilla he keeps. Alice randomly picks one $\ket{\bar{l}}$, modulates it by $U_{0}=R(\frac{\pi}{2})$ or $U_{1}=R(-\frac{\pi}{2})$, rotation by two different angles on $C$, depending on $\sb\ \in \{0,1\}$, and sends it back to Bob as commitment. Alice opens by sending back the rest and revealing everything. Let $|k\rangle \in
\cH^{A}$ be the orthogonal entanglement ancilla states, $P$ the cyclic shift unitary operator on $n$ qubits, $P^n=I$. Suppose Alice entangles in a minimal way, $$\label{eq:8}
|\Psi_{\sb}\rangle=U_{\sb}\frac{1}{\sqrt{n}}\sum_{k=1}^n|k\rangle\otimes P^{k}|1_{j}\rangle ...|n_{j}\rangle$$ where $|1_{j}\rangle$ is acted on by $U_{\sb}$. This “protocol” can be shown to be $\epsilon$-concealing, and Alice can locally turn $|\Psi_{0}\rangle$ to $|\Psi_{1}\rangle$ near perfectly in a standard entanglement cheating.
Consider a protocol with the following added checking to the above. Before opening, Bob asks Alice to send back a fraction $\lambda$, say $\lambda=\frac{1}{2}$, of the $n$ qubits chosen randomly by Bob for checking. If Alice replies that fraction contains the committed one, Bob would ask to check the remaining $1-\lambda$ fraction instead. Assuming Alice has to answer correctly, she must measure on $\cH^A$ to get a specific $\ket{k}$. After Bob’s checking, he still has a uniform distribution on exactly what the original committed qubit is according to his own positions. Thus the protocol *remains* $\epsilon$-concealing if $n$ is sufficiently large, while Alice has lost her entanglement cheating capability. This is what was referred to as “the destruction of entanglement for cheating" in several of my previous protocols, beginning with a first one at the 2000 QCMC meeting in Capri, Italy.
Such ploy did not lead to a secure protocol because the entanglement (\[eq:8\]) or a similar sparsely entangled one was not insisted upon as part of the protocol prescription. Before it will be discussed in the following how the entanglement (\[eq:8\]) can be enforced, note that Alice can retain her entanglement cheating capability by other entanglements, in particular by the full $n$-permutation group. She could name her entanglement basis vectors $\ket{k}$ by the original positions of the qubits Bob sent, $$\label{eq:9}
\ket{k}\rightarrow\ket{1(k_1),\ldots,n(k_n)}$$ where $l(k_l)$ indicated that original qubit $l$ is at position $k_l$ corresponding to $\ket{k}$. When she is asked to return a fraction $\lambda$ that has positions $\lambda(m)$, $m\in\overline{1-n}$, she would perform a Lüders measurement, that is, a projection $P^{\prime}$ into the subspace in $\cH^A$ that fixes the {$\lambda(m)$} position. If the entanglement is sufficient dense, the remaining $1-\lambda$ fraction is still entangled in the remaining ancilla space $(1-P^{\prime})\cH^A$, and entanglement cheating remains possible. With the entanglement (\[eq:8\]) there is no such degeneracy. In fact, $P^{\prime}\cH^A=\cH^A$. Thus, fixing the position of just one qubit already fixes the positions of all the others.
Note that the checking of ancilla is naturally included in the generalized formulation discussed in the last paragraph of section \[sec:form\]—there is no system or subsystem that cannot be exchanged. The question now is why Alice should entangle as in (\[eq:8\]) rather than one which allows her to cheat later. In the QBC literature, with the possible excepting of (\[eq:2\]), the claim has always been that even under *honest* following of the protocol prescription, no protocol can be secure [@lc2]-[@sr]. In the presence of interactive checking as above, we here conclusively shown that such claim is incorrect.
The “honesty" assumption is widely used in the literature to describe multi-pass protocol including those for quantum coin tossing [@dk]. It may or may not make sense depending on whether the“honest" action can in principle be checked by the other party without rendering the protocol ineffective. For example, in the simple one-pass protocol of [@lc1], it makes no sense to require Alice to be honest and does *not* entangle. It is clear that an actual physical entanglement is needed for the EPR cheating even when the protocol is perfectly concealing. Note that this is in fact the *basis* of the success of checking for preventing entanglement cheating with (\[eq:8\]), that only classical randomness is left after checking. Thus, Alice would entangle anyway in the situation of [@lc1] and the simple protocol that requires such “honesty” is not secure.
In a multi-pass protocol, there is always the question whether “honest entanglement” or any other prescription of the protocol is followed. Even with just a two-pass QBC protocol in which Bob first sends Alice some qubits in prescribed states, including the above “protocol” involving (\[eq:8\]), he can easily cheat by sending in other qubits instead. For example, he could send in identical fixed qubit states and so he would know how to measure to distinguish $\sb=0,1$ from the committed qubit with considerable $P^B_c>\frac{1}{2}$ for any $\{U_0, U_1\}$ pair. He is prevented from such cheating via checking of one form or another.
A crucial question is: what happens when one party is found cheating during protocol execution. Clearly the party cannot be allowed to keep cheating indefinitely, if only because of “intent” [@yuec] since the party does not need to participate to begin with. I have previously described [@app] several approaches to deal with this problem which has *not* yet received an adequate discussion in the literature, but which can be solved in one stroke by an honesty assumption that requires all the parties to be perfectly honest in their prescribed actions and thus no cheating would even be found before opening. This is a perfectly reasonable working assumption for the ideal protocol under discussion *as long as* the action can be checked, in view of the discussion just given above. It is equivalent to the assignment of infinite penalty in a game type formulation [@app], and it allows us to bring forth our new point without the burden of technicalities. It is also exactly what has been implicitly assumed in the literature as we mentioned.
Note that the whole protocol may need to be started all over again after a checking. It is easy to see that in the absence of resource constraint as in the case of all QBC impossibility proof formulations thus far, one party can check the same state an arbitrarily large number of times before proceeding. The total number of checks may grow multiplicatively, not just linearly, with the number of state checking. It is reasonable to count cheating detection probability as the party’s failure probability in $P^A_c$ and $P^B_c$. Thus, whenever a bound is imposed on the allowable total number of cheatings getting caught, an unconditionally secure protocol would be obtained which is equivalent to the honest assumption. This is because both $P^A_c$ and $P^B_c$ can be brought arbitrarily close to their prescribed $\epsilon$-level with a large enough number of checkings on each state.
The point that was made in this section in connection with (\[eq:8\]) has the following *general implication* independently of whether a secure protocol can be made on that basis: There is no general impossibility proof that shows the entanglement formed by one party as prescribed by a QBC protocl would have effective remaining entanglement after checking. In the next section, however, we do exhibit such a specific secure protocol.
Secure Protocol QBC1 {#sec:qbc1}
====================
We consider the following protocol QBC1 [@qbc3] in which Bob sends Alice a sequence of $n$ qubits as described in the last section, requiring Alice to entangle as in (\[eq:8\]). We will show later in appropriate places how that as well as any other prescribed states for Alice and Bob can be checked. That the protocol is $\epsilon$-concealing is intuitively obvious, and can be proved as follow. For simplicity we let the protocol prescribe that each of the qubit state Bob sends is entangled with an ancilla in his possession. Alice can check this before proceeding by asking Bob to send her the qubit ancilla and measuring to verify.\
Concealing Proof for QBC1:
First we assume that Bob does not permutation entangle the $n$ qubit. It is technically messy to show concealing if she does, but the absence of such permutation entanglement can be assured by requiring Bob to permutation entangle as in (\[eq:8\]), and destroyed by Alice asking to check one or more of the qubits. That Bob did entangle in such manner in the first place can be checked by asking him to send in the ancilla for Alice to check.
For simplicity we do not distinguish here a qubit state from the qubit which is clear from context. Let $a_l$ be the ancilla part of Bob’s states entangled to the $l$th qubit. Then $a_l=\frac{I}{2}$ without the $i$th qubit and $$\label{eq:10}
\rho_\sb=\frac{1}{n}\sum_{l=1}^n a_1\otimes\ldots\otimes(\sigma_\sb a_l)\otimes\cdots\otimes a_n$$
In (10), $(\sigma_\sb a_ l)$ denotes the state obtained by pairing of the $l$th ancilla state to the committed qubit, $\sigma_\sb$ is the committed part of the committed qubit-ancilla entangled pair. We have $(\sigma_\sb a_l)=\sigma_\sb\otimes a_l$ when the pairing is incorrect but is a properly qubit-ancilla state when they match. From (\[eq:10\]), $$\label{eq:11}
n(\rho_0-\rho_1)=[(\sigma_0 a_{\bar{l}})-(\sigma_1 a_{\bar{l}})]\otimes_{l\neq \bar{l}} a_l+(\sigma_0-\sigma_1)\otimes_{l}a_l$$
where $\bar{l}$ is the actual position of the committed qubit. Since $\sigma_0=\sigma_1=\frac{I}{2}$ for incorrect matching and $\|(\sigma_0-\sigma_1)\otimes a\|_1=\|\sigma_0-\sigma_1\|_1$ for all density operators $\sigma_0$, $\sigma_1$ and $a$, from (\[eq:11\]) $$\label{eq:12}
\|\rho_0-\rho_1\|_1=\frac{1}{n}\|(\sigma_0 a_{\bar{l}})-(\sigma_1 a_{\bar{l}})\|_1=\frac{2}{n}$$ which can be made arbitrarily small with large n. Equation (\[eq:12\]) expresses exactly the intuitively obvious fact that Bob succeeds in cheating when and only when he guesses correctly which original qubit the committed qubit is.\
Binding Proof for QBC1:
Since $\sigma_\sb=\frac{I}{2}$ without Bob’s ancilla, the probability that Alice can determine the state of the committed qubit she chooses for commitment is arbitrarily small, given by $\frac{1}{M}$ for $M$ possible states on $C$ and is zero asymptotically. Without the possibility of entanglement cheating, Alice can simply declare the bit she wants to open. In that situation $P^A_c$ is given by the inner product square of the two possible committed state. With our choice $P^A_c=0$ since the two states for the two different $\sb$ values are orthogonal.\
Note that as in the discussion of QBC since the beginning, an $\epsilon$-concealing protocol can be made $\epsilon$-binding in a sequence of committed qubits to obtain a single secure bit whenever $P^A_c$ is not too close to 1 for each original qubit. In the above QBC1, such a sequence has also been indicated in our previous version in [@qbc3] for such purpose. It is not needed if the two bit states are orthogonal or nearly orthogonal and if completely random qubit states on $C$ are supplied by Bob.
It remains to show that (\[eq:8\]) can be checked and no security leak could occur during the checking process. In contrast to the states sent in by Bob, it is more complicated to check (\[eq:8\]) since Alice already committed by then, but it can be done as follow.\
Checking of Entanglement (\[eq:8\]):
Alice would first send her ancillas of (8) to Bob with an entanglement basis unknown to him. Then Bob sends back Alice’s committed qubit to her who would turn it back to the original state by reversing her $U_b$. Then she sends back all qubits to Bob who can thus (\[eq:8\]).
We now show there can be no security compromise in the checking and each party must follow the prescription as all relevant states can be checked. First, Bob can derive no information on which qubit he sent is committed without first knowing what qubit positions are indicated by what ancilla state. The ancilla he so receives back from Alice is a totally random state to him. Secondly, Alice must send in her entanglement ancilla and tell Bob later exactly what the total state is, as prescribed in the checking. Third, that Bob then sends back the correct qubit can be checked by Alice via asking Bob to send back all the relevant states in his possession which include his ancilla, Alice’s committed qubit, and her ancilla that was sent him. Alice can then check similar to the beginning check on Bob’s $n$ qubit ancilla state. Finally, Alice must send back the proper states or else Bob cannot verify (\[eq:8\]).
We have completed the security proof with proper operation procedure for QBC1. Assuming honest operation that we have shown can all be checked, the protocol can be simply summarized in the following:
0.1in
0.11in
Scope of QBC Possibility {#sec:scope}
========================
It has long been known that a trusted third party or special relativistic effects can be used to establish secure bit commitment protocol both classically and quantum mechanically. Furthermore, D’Ariano has suggested [@dariano09] that casuality or time order cannot be purified and is built into quantum mechanics already in a way that would imply special relativity. If true this would imply quantum mechanics by itself would ensure the possibility of secure QBC similar to Kent’s relativistic protocol [@kent]. Cheung [@chau3] has recently proposed a secure protocol on the basis of timing effect. In this paper, we show that quantum mechanics allows secure QBC without invoking causality or timing, in a way that was first described in [@qbc3].
The exact mechanism of how our QBC1 falls outside the standard impossibility proof is made clear in section \[sec:newappr\] above. There seem to be some vague claims of universal QBC impossibility in ref [@dariano07] and [@chis]. Both papers are presented in unfamiliar mathematical formulation of $C^\ast$-algebra or “quantum comb” with no translation into the usual formulation. In both of these new formulations, there is no clear indication on exactly what would happen when one party is found cheating during protocol execution. Just aborting the protocol is not enough as one party can keep on cheating as discussed in section \[sec:newappr\]. While the number of allowable protocol abortions may be bounded in [@chis], cheating detection entails no penalty in any form. More significantly, it appears there is no restriction put on the parties’ entanglement purification and a private ancilla not to be checked is allowed, thus excluding QBC1 in these formulations.
A most important point that is not addressed before in all the impossible proofs that claim universality is *what the proof is* that all possible QBC protocols have been included. A general discussion of this issue can be found in [@yuen1]. A main point that has *not* even been made clear in [@yuen1] is that a ‘machine’ formulation cannot capture all the possible protocols, classical or quantum, that can be clearly formulated with ordinary natural language due to the ‘meaning’ problem. Specific intended meaning can be captured by a mechanical process, but not all possible meaning in a general context. This is the situation of human knowledge that, I believe, would not be changed in the future. In the present QBC issue, one manifestation of this situation is that there is no general mathematical definition which captures all possible QBC protocols.
As a concluding remark, practical QBC protocols can be developed that can be proved secure within technological limits that are unlikely to be removed in the foreseeable future. Entanglement across many qubits already by itself falls under these limits. Such implementable protocols could be practically significant even if they are not unconditionally secure under the impractical assumption of ideal system devices and components.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank C.Y. Cheung, G.M. D’Ariano, and M. Ozawa for very useful discussions.
[99]{}
H.P. Yuen, quant-ph/0808.2040v1, (2008).G.M. D’Ariano, Kretschmann D, Schlingemann D and Werner R F, Phys. Rev. A [**76**]{}, 032328 (2007).D. Mayers, Phys. Rev. Lett. [**78**]{}, 3414 (1997)H.K. Lo and H.F. Chau, Phys. Rev. Lett. [**78**]{}, 3410 (1997).H.K. Lo and H.F. Chau, Fortschr. Phys. [**46**]{}, 907 (1998).H.K. Lo and H.F. Chau, Physica D [**120**]{}, 177 (1998).G. Brassard, C. Crépeau, D. Mayers, and L. Salvail, preprint quant-ph/9712023.G. Brassard, C. Crépeau, D. Mayers, and L. Salvail, preprint quant-ph/9806031.R.W. Spekkens and T. Rudolph, Phys. Rev. A [**65**]{}, 012310 (2001).,H.P. Yuen, quant-ph/0109055.H.P. Yuen, in [*Quantum Communication, Measurement, and Computing*]{}, ed. by J.H. Shapiro and O. Hirota, Rinton Press, 2003; p. 371; also preprint quant-ph/0210206.H.P. Yuen, quant-ph/0207089v3 (2002).H.P. Yuen, quant-ph/0305144v3 (2003).M. Ozawa, private communication, Sep 2001.C.Y. Cheung, quant-ph/0508180 (2005).C.Y. Cheung, quant-ph/0601206 (2006).C. Döscher and M. Keyl, Fluct. Noise Lett. [**4**]{}, R125 (2002); also quant-ph/0206088.See [@yuen1] and the appendices of H.P. Yuen, quant-ph/0305142 (2003) and quant-ph/0305143 (2003).Essentially the same protocol is called QBC3 in H.P. Yuen, in [*Quantum Communication, Measurement, and Computing*]{}, ed. by O. Hirota, J.H. Shapiro and M. Sasaki, NICT Press, 249 (2007), Japan; also quant-ph/0702074v4 (2007).G.M. D’Ariano, private communication, Nov 2009.A. Kent, Phys. Rev. Lett. [**83**]{}, 1447 (1999).C.Y. Cheung, quant-ph/0910.2645 (2009).G. Chisibella, G.M. D’Ariano, P. Pesinotti, D. Schlingemann, and R.F. Werner, quant-ph/0905.3801 (2009).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this article, I present the questions that I seek to answer in my PhD research. I posit to analyze natural language text with the help of semantic annotations and mine important events for navigating large text corpora. Semantic annotations such as named entities, geographic locations, and temporal expressions can help us mine events from the given corpora. These events thus provide us with useful means to discover the locked knowledge in them. I pose three problems that can help unlock this knowledge vault in semantically annotated text corpora: *i.* identifying important events; *ii.* semantic search; and *iii.* event analytics.'
author:
- |
Dhruv Gupta\
\
\
subtitle: 'Detecting Events in Semantically Annotated Corpora for Search & Analytics'
title: Event Search and Analytics
---
Introduction
============
Information retrieval systems have largely relied on word statistics in text corpora to satisfy information needs of users by retrieving documents with high relevance for a given keyword query. In my PhD research I hypothesize that information needs of users can be satisfied to a greater extent by using *events* as a means of navigating text corpora. Events in our context would be an act performed by certain actor(s) at a specific location during a specific time interval. An example would be : *Usain Bolt won the gold medal at the 2008 summer Olympics in Bejing*. With the availability of annotators that can provide us with accurate semantic annotations in form of named entities, geographic locations, and temporal expressions; we can leverage the growing number of knowledge resources such as Wikipedia [@wiki] and ontologies such as Freebase [@freebase] to understand natural language text and mine important events. Formally the central hypothesis can be stated as follows:
As a toy example consider the following text snippet [^1] with demonstrative semantic annotations in Figure \[fig:text\] :
In the text snippet (Figure \[fig:text\]), we obtain the named entity whose mention has been identified and disambiguated to point to an external knowledge source. Also identified is a geographical location - , which is disambiguated and resolved to its geographical coordinates. Likewise the temporal expression has also been resolved to time range. Having these semantic annotations we can now devise algorithms that can deduce that the event is that of Usain Bolt winning Olympic competition in Beijing, China.
The goal of the proposed research is to leverage the semantic annotations for mining important events and use them to navigate text corpora. The research will find application in many domains of research such as *digital humanities*, in which social scientists are interested in computational history in large digital-born text collections. Anthropologists are interested in cultural and linguistic shifts that occur in such collections. Collectively we can allow *computational culturomics* [@culturomics] on corpora to study cultural trends. Events can also be used to link information in multiple and diverse text collections. In short, important events provide a way to create a gist from semantically annotated corpora, which otherwise is not possible through manual human effort.
**Outline**. The article consists of:
- a literature survey (Section \[sec:background\]);
- an overview of the research problems (Section \[sec:problem\]);
- available corpora, test sources and evaluation measures for research (Section \[sec:evaluation\]);
- discussion of few open technical problems (Section \[sec:discussion\]).
Related Work
============
\[sec:background\]
In this section I discuss the progress already made in the area of analyzing different semantic annotations in isolation as well as in conjunction for some of the problems proposed.
**Temporal Information Retrieval and Extraction**. Researchers have considered only temporal annotations in text corpora to improve retrieval effectiveness by analyzing the time sensitivity of keyword queries and incorporating the time dimension in retrieval models. Some methods of analysis of time-sensitive queries rely on publication dates of documents [@diaz_profile; @nattiya_2010], while others also look at the temporal expressions in document contents [@dhruv_2014]. Several works also take into account the time dimension for re-ranking documents [@klaus_2010] and diversifying them along time [@klaus_2013; @nattiya_2014]. One of the seminal works in extracting temporal events was by Ling and Weld [@ling_tie]. They outline a probabilistic model to solve the problem of extracting relations from text with temporal constraints.
**Important Events in Annotated Corpora**. One of the most important seminal works in identifying existing and emerging events were the various tasks in *Topic Detection and Tracking* (TDT) [@tdt_book]. The TDT program aimed to “search, organize and structure” broadcast news media from multiple sources. The five tasks laid within the ambit of TDT where topic tracking, link detection, topic detection, first story detection, and story segmentation. The task of topic tracking required to build a system to detect *on topic stories* from an evaluation corpus after being trained on a set *on topic* stories. The link detection task involved answering a boolean query to whether two given *stories* are related by a common topic. The topic detection task comprised of declaring new *topics* from incoming *stories* which had not been presented to the system. First story detection was another boolean decision task of determining whether the given *story* is a seed story (first-story) to create a new *topic* cluster. Story segmentation task required segmentation of an incoming stream of text into *stories*.
Focusing specifically on extracting and summarizing events in the future, Jatowt and Yeung [@adam_cikm11] present a model-based clustering algorithm. The clustering considers both textual and temporal similarities. For computing temporal similarity, the authors model time as a probability distribution by utilizing different family of distributions based on whether its is singular time point, starting date or and ending date. The similarity is then computed using KL-divergence.
Radinsky et al. [@kira_jair] present an algorithm *Pundit*, which based on the past events in text is able to predict a future event given a query to the system. The events are represented as multidimensional attributes such as time, geographic location and participating entities. The algorithm derives these events from external text collection and builds an *abstraction tree*, which is the result from hierarchical agglomerative clustering. In order to predict the future *Pundit* is trained to select the most similar cluster from the *abstraction tree* and produce an event representation.
The work by Yeung and Jatowt [@yeung_cikm11] tackles the problem of analysis of historical events in multiple large document collections. They utilize *latent Dirichlet allocation* to identify *topic* distributions along time. Thereafter they perform analytics to answer questions such as *i.* significant years and topics, *ii.* triggers that caused remembrance of the past and *iii.* historical similarity of countries.
Most recently, Abujabal and Berberich [@mppf] present a system which identifies important events in text collections by counting frequent itemsets of sentences containing named entities and temporal expressions. For evaluation they resort to *Wikipedia’s* event directory as a ground truth.
**Semantic Search**. Summarizing text collections in a timeline visualization is a natural choice. Swan and Allan [@swan_timeline] present an approach for producing a timeline that depicts most important topics and events closely modeled on the *Topic Detection and Tracking* task. The algorithms analyzes features based on named entities and noun phrases. The analysis involves construction of $2 \times 2$ contingency table on presence or absence of features, and subsequent measurement of $\chi^2$ statistic for measuring significance of co-occurrence of a pair of features.
The seminal work by Baeza-Yates [@yates_future] proposed a *future retrieval* (FR) system. The FR system considers both text and temporal expressions to identify future events that might be relevant to an input query. Baeza-Yates outlined the components of a FR system to be composed of an *information extraction* (IE) module, *information retrieval* (IR) module, and a *text mining* (TM) module. The IE module would act as a temporal annotator; identify temporal expressions and normalize them. The IR system is designed to incorporate the time dimension in an index; thus retrieving documents with text and time similarity. The TM module would identify the most relevant topics given a time period. He presented a retrieval model, in which each document consists of a multiple temporal events. A temporal event consists of a time segment and its associated likelihood of occurring. The score of the document is thus obtained by its textual similarity and the maximum likelihood of all the temporal events in that document.
Bast and Buchhold [@bast_index] outline a joint index structure over ontologies and text. Which allows for fast semantic search and provide context sensitive auto-complete suggestions.
Events as a means of search document collections has also been explored by Strötgen and Gertz [@jannik_event]. Events were modeled by the geographic location and time of their occurrence. For temporal queries expressed in simple natural language they outline an extended Backus-Naur form (EBNF) language that incorporates time intervals with standard boolean operations. Geographical queries are also modeled as EBNF language, however the input for them is a minimum bounding rectangle (MBR). Using this multidimensional querying model the user is able to visualize search results in form of events; which are additionally represented on a map.
Giving special attention to geographical information retrieval, Samet et al. [@samet_news] present a system *NewsStand*, that is able to resolve and pinpoint a news article based on the geographic information present in its content. They discuss various methods for toponymn resolution, which is in essence disambiguating the geographic location based on its surface form in the news content. The system involves a streaming clustering algorithm that can keep track of emerging news in new locations and present them in a map-based interface.
[l|p[4cm]{}p[4cm]{}p[4cm]{}]{}
\
**Event** & $c_1$ & $c_2$ & $c_3$\
**Words** & micheal, phelps, bejing, china, tibet & london, usain, bolt, england, badminton & rio, brazil,copacabana, deodoro, maracanã\
**Time** &$[08-08-2008, 24-08-2008]$ & $[27-07-2012, 12-08-2012]$ & $[05-08-2016, 21-08-2016]$\
**Location** & $\langle Beijing, China \rangle$ & $\langle London, England \rangle$ & $\langle Rio de Janeiro, Brazil \rangle$\
**Entities** & $\langle China \rangle$, $\langle Micheal\_Phelps \rangle$ & $\langle England \rangle$, $\langle Badminton \rangle$ & $\langle Brazil \rangle$, $\langle Copacabana \rangle$\
**Event Analytics**. By disambiguating and linking named entities to ontologies, Hoffart et al. [@aesthetics; @stics] provide a framework for semantic search and performing analytics on them. They provide features for giving auto-complete suggestions in the form of similar entities for the input named-entity. In [@aesthetics] they provide analytics that leverage accurate entity counts and provide entity co-occurrence statistics which is helpful in analyzing semantically similar named-entities.
Research Objectives
===================
\[sec:problem\] Given the text corpora with semantic annotations, I describe three important research problems in this section: *i.* identifying important events; *ii.* using identified events for improving retrieval effectiveness; and *iii.* using identified events for analytics.
Notation
--------
Let us consider multiple corpora for the purpose of analysis. This allows us to capture frequently occurring events as well as link similar events across corpus. Given corpora $$D = \bigcup_{k=1}^{N} D_k,$$ where each document $d \in D$ consists of word sequence $x$ at appropriate granularity (e.g. paragraph or sentence): $$d = \bigcup_{i=1}^{n} x_i.$$ Further each $x \in d$ contains semantic annotations in form of *i.* named entities ($e$), *ii.* geographical location ($g$), and *iii.* temporal expressions ($t$). Additionally $x$ also consists of the a bag of words $\mathcal{W}$ drawn from a vocabulary $\mathcal{V}$. Formally represented as: $$x = \langle \mathcal{E}, g, t, \mathcal{W} \rangle$$
Problem Definition
------------------
The objective is to design a family of algorithms: $$\textsc{Event*}(X,Q,\alpha)$$
where $X = \bigcup x$, $Q$ represents an input query and $\alpha \in \mathds{R}^m$, where $\alpha$ is set of parameters.
The input query $Q$ can be a combination of following input components: *i.* keyword query $q$, *ii.* time $q_{time}$, *iii.* geographical location $q_{geo}$, and *iv.* named entity $q_{entity}$.
Given the input, we need to design the algorithms $\textsc{Event*}$ according to the different problems. We discuss the design objectives for the three different purposes in this section.
**Identifying Important Events**. *Events* are the proposed building blocks for further text analysis. An *event* in our context is defined to be an activity or an act involving named entities that happens in a specific geographical location anchored to a specific time interval. Mathematically, given a multidimensional query : $$Q = \langle q, q_{time}, q_{geo}, g_{entity} \rangle,$$ and a subset of highly relevant documents $R \subseteq D$, the algorithm for this purpose $\textsc{EventDetect}$ should produce a set of ordered events : $$\mathcal{C} = \{ c_1, c_2, \ldots, c_k \},$$
where, $c = \langle \mathcal{E}, g, t, \mathcal{W} \rangle$. The event $c$ is hence described by the participating named entities $\mathcal{E}$, its location $g$, its time of occurrence $t$, and frequently occurring contextual terms around these semantic annotations $\mathcal{W}$. This requires proposing a probability mass function, $P(\mathcal{C}, R)$, using which we can impose a total order on $\mathcal{C}$.
As an example consider the keyword-only query `summer olympics` to the processed corpora of news articles. The designed algorithm shall then identify the important events as in Figure \[fig:event\].
**Diversifying and Summarizing Search Results** are retrieval tasks that try to address the information need underlying an ambiguous query at different levels of textual granularity. Each task tries to maximize the coverage of different information needs underlying the given ambiguous query. As information intents, we propose to use the mined set of *events*. Accomplishing these tasks would allow for automatic creation of *event timelines* or *entity biographies*. We briefly discuss an intuition of achieving the same.
When diversifying search results we would like to present users with *documents* such that the user finds *at least* one document that satisfies her information intent. For this we need to devise an algorithm $\textsc{EventDiverse}$ which considers as an input $Q$ and $R \subseteq D$. As an output it returns a set of documents $S \subset R$ which cover all events in $\mathcal{C}$.
Summarizing search results would require us to construct an algorithm $\textsc{EventSummary}$ to piece together, *sentences* $\hat{S} = \bigcup x$, such that the text summary covers all events in $\mathcal{C}$.
**Semantic Search and Analytics**. The mined set of *events* can further be utilized for search and analytics. For this purpose we can utilize inherent hierarchy in the semantic annotations. For example a given year can be broken down to different months and subsequently days in those months. Similarly, we can utilize the *type hierarchies* in named entities. Such as and are subtypes of . This can jointly be modeled by using the concept of a *data cube* [@han_dm] as shown in Figure \[fig:data\_cube\].
Formally, given a query $Q$, the objective would to first model the mined set of events as a *data cube* and subsequently provide *data cube operations* [@han_dm]:
- roll ($\bigcirc$),
- slice ($\ominus$),
- dice ($\oplus$),
- drill up ($\bigtriangleup$),
- drill down ($\bigtriangledown$).
![Example data cube based on set of events $\mathcal{C}$[]{data-label="fig:data_cube"}](cube.pdf)
![Example data cube operations for the query `all races won during 2008 by usain bolt in china` []{data-label="fig:cube_opr"}](cubeopr.pdf)
As a concrete example consider the query `all races won during 2008 by usain bolt in china`. To produce an appropriate result the sequence of operations would be: first a slice on the entity ; second dice on ; and finally drill up to year (see Figure \[fig:cube\_opr\]).
Data
====
**Corpora**. There are several readily available massive data sets. They are available from news corporation such as the *New York Times* [@nyt], *English Gigaword* [@gigaword]. These corpora have the benefit of being available with reliable publication dates and grammatically well-formed text. On larger scale are Web collections such as *ClueWeb’09* [@clueweb09]/’12 [@clueweb12], which are not always accompanied by reliable creation dates and many are ill-formed documents.
**Semantic Annotations**. The text corpora next need to be annotated for text mining. I explain how to obtain the different semantic annotations in the following paragraphs.
**Named Entities**. For disambiguating and linking named entities in text to an external knowledge source such as *Wikipedia* [@wiki] or an ontology such as YAGO [@yago] or Freebase [@freebase]; I use the AIDA system [@aida]. The AIDA system does named entity disambiguation and linking by leveraging contexts extracted from ontologies such as YAGO. For Web collections such ClueWeb’09/’12 the entity disambiguation and linking has been released as *facc1 : Freebase annotation of ClueWeb Corpora* [@facc].
**Geographical Locations** can be obtained by utilizing *geographic* named entities such as those known to be cities, countries, or continents. Geographical relations stored in an ontology can be used to resolve these locations to its geographical coordinates. Having obtained a set of coordinates, we can subsequently construct a geographical representation such as a *minimum bounding rectangle* over the coordinates.
**Temporal Expressions**, both implicit and explicit, can be extracted and normalized from text by using *temporal taggers* such as HeidelTime [@heideltime] or SUTime [@sutime].
Evaluation
==========
\[sec:evaluation\]
To test our approach we need to construct query sets that contain an event description associated with the query; along with participating named entities, geographical locations where the event took place and relevant time interval associated with it. I describe a tentative approach to achieve this here.
**Test Data**. To evaluate the correctness of the various algorithms, I plan to use reliable encyclopedic resources on the Web such as *Wikipedia* [@wiki] or other curated knowledge sources. For an objective evaluation, I propose the following different sources depending on the algorithm under evaluation.
- Identify important events
- Events in a particular year/decade etc. pages available on *Wikipedia* [@wiki_year].
- Testing of past events can be done by extracting important topics from *Category* pages on various historical topics on *Wikipedia* [@wiki_past].
- Events in the future can be evaluated by using important infrastructure projects, engineering projects etc. These can be extracted from *Wikipedia* and other sources on the Internet.
- Current events extracted from *Wikipedia* [@wiki_current].
- Alternatively, we can manually construct a list of prominent events and extract relevant information such as named entities, geographical location, and time from ontologies such as: YAGO [@yago], Freebase [@freebase], etc.
- Diversifying and summarizing search events
- Biographies of eminent personalities, for example United States presidents [@wiki_potus].
- Historical timelines of various countries, for example for India [@wiki_timeline].
**Structure**. Each event in our test bed is then composed of a fact with an accompanying query. Formally, a *fact* in our testbed is a 4-tuple extracted from one of the aforementioned sources: $$\langle q,\mathcal{E}, g, t, \mathcal{W} \rangle$$
where $q$ consists of keyword query describing the event, $\mathcal{E}$ is a bag of participating entities, $g$ is the geographic location, $t$ is the time of its occurrence, and $\mathcal{W}$ are important terms describing the event.
**Metrics**. Based on the structure of the testbed of events, metrics such as *precision*, *recall* and *F$_1$* can be utilized to measure the effectiveness of the algorithms for detecting important events in semantically annotated corpora. How effectively the algorithm diversifies documents along multiple dimensions can be evaluated by metrics such as $\alpha$-nDCG [@diverse]. Quality of summaries can be measured by an automatic evaluation metric called *Rouge* [@rouge].
Discussion
==========
\[sec:discussion\]
I briefly present some open technical challenges that I will address along with the research objectives in my PhD dissertation.
**Mathematical Models**. One key aspect that occurs in the design of the algorithms is that of computational models for named entities, geographical locations and temporal expressions. What would be the most descriptive mathematical models for each of these semantic annotations?
**Similarity Functions**. Given a pair of named entities, geographical locations or temporal expressions; how can we efficiently compute the similarity between the same type of annotations?
**Efficiency & Scalability**. Identifying data structures for indexing corpora along with their semantic annotations, such that their asymptotic run times scale linearly with the size of the corpora.
**Evaluation**. Since evaluation of the solutions outlined are very subjective in nature; what are other reliable sources of objective ground truth ? What other metrics can be employed to test the effectiveness of our methods ?
Conclusion
==========
\[sec:conclusion\]
In this article I laid out an outline of the research work that I envisage to carry out for my PhD dissertation. The research would in its culmination provide us methods to computationally extract world history as sequence of temporally ordered events and portray future events to take place from semantically annotated corpora. The research would also provide ways to perform semantic search and large scale event analytics on these annotated corpora. I further described already available resources that can be utilized for carrying out the research; test cases that can be built from encyclopedic resources on the Internet; and the metrics that can be utilized for evaluation.
[10]{} Clueweb’09. <http://www.lemurproject.org/clueweb09.php/>.
Clueweb’12. <http://www.lemurproject.org/clueweb12.php/>.
English gigaword. <https://catalog.ldc.upenn.edu/LDC2003T05>.
New york times anotated corpus. <https://catalog.ldc.upenn.edu/LDC2008T19>.
Abujabal A. and Berberich K. Important events in the past, present, and future. WWW’15 Companion Volume.
Agrawal R. et al. Diversifying search results. WSDM’09.
Allan J., editor. . Kluwer Academic Publishers, Norwell, MA, USA, 2002.
Baeza-Yates R. Searching the future. SIGIR’05 Workshop MF/IR.
Bast H. and Buchhold B. An index for efficient semantic full-text search. CIKM’13.
Berberich K. et al. A language modeling approach for temporal information needs. ECIR’10.
Berberich K. and Bedathur S. Temporal diversification of search results. TAIA’13.
Bollacker K. et al. Freebase: A collaboratively created graph database for structuring human knowledge. SIGMOD’08.
Chang A. X. and Manning C. D. : A library for recognizing and normalizing time expressions. LREC’12.
Ringgaard M. et al. Facc1: Freebase annotation of clueweb corpora, version 1 (release date 2013-06-26, format version 1, correction level 0), June 2013. <http://lemurproject.org/clueweb12/>.
Gupta D. and Berberich K. Temporal query classification at different granularities. SPIRE’15.
Han J. et al. . Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 3rd edition, 2011.
Hoffart J. et al. analytics with strings, things, and cats. CIKM’14.
Hoffart J. et al. searching with strings, things, and cats. SIGIR’14.
Hoffart J. et al. Robust disambiguation of named entities in text. EMNLP’11.
Jatowt A. and Yeung C. A. Extracting collective expectations about the future from large text collections. CIKM’11.
Jones R. and Diaz F. Temporal profiles of queries. , 25(3), July 2007.
Kanhabua N. and N[ø]{}rv[å]{}g K. Determining time of queries for re-ranking search results. ECDL’10.
Ling X. and Weld D. S. Temporal information extraction. AAAI’10.
Michel J. B. et al. Quantitative analysis of culture using millions of digitized books. Science’10.
Nguyen T. N. and Kanhabua N. Leveraging dynamic query subtopics for time-aware search result diversification. ECIR’14.
Radinsky K. et al. Learning to predict from textual data. , 45:641–684, 2012.
Samet H. et al. Reading news with maps by exploiting spatial synonyms. , 57(10):64–77, 2014.
Str[ö]{}tgen J. and Gertz M. Event-centric search and exploration in document collections. JCDL’12.
Strötgen J. and Gertz M. Multilingual and cross-domain temporal tagging. , 47(2):269–298, 2013.
Suchanek F. M. et al. Yago: A large ontology from wikipedia and wordnet. , 6(3):203–217, September 2008.
Swan R. C. and Allan J. Automatic generation of overview timelines. SIGIR’00.
Wikipedia. List of presidents of the united states — [W]{}ikipedia[,]{} the free encyclopedia, 2015. <https://en.wikipedia.org/wiki/List_of_Presidents_of_the_United_States>.
Wikipedia. List of years — [W]{}ikipedia[,]{} the free encyclopedia, 2015. <https://en.wikipedia.org/wiki/List_of_years>.
Wikipedia. List of years in india — [W]{}ikipedia[,]{} the free encyclopedia, 2015. <https://en.wikipedia.org/wiki/List_of_years_in_India>.
Wikipedia. Portal:contents/history and events — [W]{}ikipedia[,]{} the free encyclopedia, 2015. <https://en.wikipedia.org/wiki/Portal:Contents/History_and_events>.
Wikipedia. Portal:current\_events — [W]{}ikipedia[,]{} the free encyclopedia, 2015. <https://en.wikipedia.org/wiki/Portal:Contents/History_and_events>.
Wikipedia. ikipedia[,]{} the free encyclopedia, 2015. <http://en.wikipedia.org/>.
Yeung C. A. and Jatowt A. Studying how the past is remembered: towards computational history through large scale text mining. CIKM’11.
Lin C. Y. Rouge: a package for automatic evaluation of summaries. ACL-04 workshop.
[^1]: <http://www.bbc.com/sport/0/athletics/34032366>
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Generating nonclassical states of photons such as polarization entangled states on a monolithic chip is a crucial step towards practical applications of optical quantum information processing such as quantum computing and quantum key distribution. Here we demonstrate two polarization entangled photon sources in a single monolithic semiconductor waveguide. The first source is achieved through the concurrent utilization of two spontaneous parametric down-conversion (SPDC) processes, namely type-0 and type-I SPDC processes. The chip can also generate polarization entangled photons via the type-II SPDC process, enabling the generation of both co-polarized and cross-polarized entangled photons in the same device. In both cases, polarization entanglement is generated directly on the chip without the use of any off-chip compensation, interferometry or bandpass filtering. This enables direct, chip-based generation of both Bell states $(|H,H\rangle+|V,V\rangle)/\sqrt{2}$ and $(|H,V\rangle+|V,H\rangle)/\sqrt{2}$ simultaneously utilizing the same pump source. In addition, based on compound semiconductors, this chip can be directly integrated with it own pump laser. This technique ushers an era of self-contained, integrated, electrically pumped, room-temperature polarization entangled photon sources.'
author:
- Dongpeng Kang
- Minseok Kim
- Haoyu He
- 'Amr S. Helmy'
title: Two Polarization Entangled Sources from the Same Semiconductor Chip
---
Introduction
============
Entangled photons are essential building blocks for optical quantum information processing, such as quantum computing (QC) [@Ladd_Nature_2010] and quantum key distribution (QKD) [@Gisin_RMP_2002]. Conventionally, entangled photons have been generated using a myriad of techniques, most notably by using the process of spontaneous parametric down-conversion (SPDC) utilizing second order nonlinearities in crystals [@Christ_2013]. Properties such as brightness, scalability, compact form-factor and room temperature operation play key roles in enabling us to fully profit from entangled photon sources in applications such as QC and QKD. As such, the physics and technology of generating and manipulating entangled photons in monolithic settings have recently been topics of immense interest. Harnessing such effects in a monolithic form-factor also enables further incorporation of other photonic components that may be necessary for the aforementioned applications [@O'Brien_NP_2009; @Gaggero_APL_2010; @Silverstone_NP_2014; @Jin_PRL_2014]. This provided the drive that motivated the early work on implementing entangled sources in waveguides of crystals with appreciable second order nonlinearities such as Lithium Niobate [@Suhara_LPR_2009].
Realizing entangled photon sources in monolithic settings enables much more than the inclusion of numerous necessary components simultaneously: It can enable the direct generation of novel and useful photonic quantum states with specified properties, without moving parts, while benefiting from the accurate alignment of nano-lithography, precision of epitaxial growth and thin film deposition techniques. For example, monolithic platforms offer opportunities to provide photons that are entangled in one or several degrees of freedom simultaneously without the need for any extra component on the chip [@Zhukovsky_OL_2011; @Kang_PRA_2014]. In addition, monolithic sources can offer significant control over the spectral-temporal properties of the entangled photons with relative ease and high precision [@Abolghasem_OL_2010]. This in turn provides a powerful tool for tailoring the temporal correlation or the spectral bandwidth of the photon states. Such states can be of extremely short correlation times, which can enhance the accuracy of protocols for quantum positioning and timing [@valencia1] and the sensitivity offered by quantum illumination [@lloyd1]. The same integrated sources can generate states with extremely large temporal correlation times. This in turn leads to narrow spectral bandwidth, which can provide a more efficient atom-photon interface and improved sources for long-haul QKD [@Narrowband_Sauge].
The vast majority of the aforementioned applications use polarization entangled photon sources. Entanglement in the polarization degree of freedom has been the most widely utilized to implement entangled sources for experiments and applications that probe or exploit quantum effects. Photon pairs in polarization entangled sources need to be indistinguishable in every degree of freedom, except for polarization, which is challenging to achieve for states produced directly in waveguides [@Suhara_LPR_2009; @Kaiser_NJP_2012; @Arahira_OE_2013]. For photon pairs generated in a type-II process, in which the down-converted photons are cross-polarized, the birefringence in the group velocities of the modes, where the photons propagate, will cause a temporal walk-off between the pair, allowing polarization to be inferred from the photon arrival time. On the other hand, for photon pairs generated in a type-0 or type-I process, where the photons in a pair are co-polarized, there is a lack of two orthogonal polarizations necessary for polarization entanglement. As a result, most waveguide sources of photon pairs require an off-chip compensation setup [@Kaiser_NJP_2012] or an interferometer [@Arahira_OE_2013] to generate polarization entanglement, which increases the source complexity and decreases the system stability significantly.
Recently, several techniques have been demonstrated to generate polarization entangled photons from a monolithic chip [@Matsuda_SR_2012; @Olislager_OL_2013; @Orieux_PRL_2013; @Horn_SR_2013]. The approaches which use spontaneous four-wave mixing (SFWM) in Si-based chips utilize integrated photonic components such as on-chip polarization rotators [@Matsuda_SR_2012] or 2D grating couplers [@Olislager_OL_2013], and benefit from mature fabrication technologies. However, the indirect bandgap of Si presents significant challenges for further integration with the pump lasers. To this end, III-V semiconductor material systems offer an optimal solution in terms of functionality to tailor the dispersion and birefringence as well as monolithic integration with the pump lasers [@Bijlani_APL_2013; @Boitier_PRL_2014]. Techniques using the counterpropagating phase-matching (PM) scheme [@Orieux_PRL_2013] and modal PM in Bragg reflection waveguides (BRWs) [@Horn_SR_2013] based on AlGaAs have been demonstrated. In the former case, however, the requirement of two pump beams with strictly controlled incidence angles and beam shapes imposes significant challenge for further integration, while in the latter case, the spectral distinguishability and walk-off due to modal birefringence compromises the quality of entanglement.
In this work, we demonstrate how the waveguiding physics associated with BRWs can be used to simultaneously produce two polarization entangled photon sources using alternative approaches in a single self-contained, room-temperature semiconductor chip. The waveguide structure utilized is schematically shown in Fig. \[Fig:structure\_SPDC\_SEM\](a). The chip, based on a single monolithic semiconductor BRW, is straightforward to design and implement and has no moving parts. The technique allows direct polarization entanglement generation using an extremely simple setup without any off-chip walk-off compensation, interferometer, or even bandpass filtering. The first source is achieved through the concurrent utilization of two second order processes, namely type-0 and type-I SPDC processes, pumped by a single waveguide mode [@Kang_OL_2012] as opposed to two modes of different polarizations [@Matsuda_SR_2012] or modes propagating in opposite directions [@Orieux_PRL_2013]. This approach permits the integration of the pump with the source in a monolithic form. Within the same waveguide, there exists a second source based on type-II process due to the lack of material birefringence [@Horn_SR_2013]. The virtual energy diagrams of the two sources are also schematically shown in Fig. \[Fig:structure\_SPDC\_SEM\](a). As such, in this approach, by varying the pump polarization and wavelength, one can select between both polarization entangled sources or use them concomitantly. The direct generation of both Bell states $(|H,H\rangle+|V,V\rangle)/\sqrt{2}$ and $(|H,V\rangle+|V,H\rangle)/\sqrt{2}$ on a single chip can be envisaged. In addition, by lithographically tuning the waveguide ridge width, one can tune the degree of entanglement of the first source.
Methods
=======
For concurrent type-0 and type-I processes with a shared TM polarized pump, paired photons can be either generated in TM polarizations via the type-0 process, or in TE polarizations via type-I process. In the ideal case, photon pairs can be produced from the two processes with the same efficiency and identical spectrum, which renders them in a maximally entangled state $(|H,H\rangle+\exp{i\phi}|V,V\rangle)/\sqrt{2}$. This is the approach which we shall pursue to obtain a chip-based entangled source using the type-0 and type-I interactions.
The AlGaAs structure used to demonstrate these sources was grown on a \[001\] GaAs substrate and the waveguide direction was oriented along \[110\]. Due to the zinc-blende crystal symmetry, the nonlinear tensor $\chi^{(2)}_{ijk}$ is non-zero only when $i\neq j\neq k$, with $i,j,k=x,y,x$ being the crystal coordinates. As a result, three SPDC processes, namely type-0, type-I, and type-II could coexist provided PM is satisfied. Among them, type-0 process depends on the electric field components of the interacting modes along the propagation direction, which are usually negligible in weakly guided waveguides. In BRWs, however, due to the index variations between different layers, the efficiency of type-0 process can be significant and can even be markedly tuned by engineering the epitaxial structure [@Kang_OL_2012]. In order to achieve concurrent PM of type-0 and type-I processes, the effective indices of the pump $n_{\text{TM}}(2\omega)$ should be equal to those of the down-converted photons $n_{\text{TE}}(\omega)$ and $n_{\text{TM}}(\omega)$ simultaneously, i.e., $n_{\text{TM}}(2\omega)=n_{\text{TE}}(\omega)=n_{\text{TM}}(\omega)$, with $\omega$ indicating the degenerate PM frequency of the down-converted photons, and TE, TM indicating the polarizations. This requirement can be satisfied lithographically by tuning the ridge width.
The two photon state generated via the two concurrent SPDC processes is given by [@Kang_PRA_2014; @Zhukovsky_JOSAB_2012] $$\begin{aligned}
\left|\psi\right\rangle=&\frac{1}{\sqrt{\eta_{\text{I}}+\eta_0}}\iint d\omega_1 d\omega_2[\sqrt{\eta_{\text{I}}}\Phi_{HH}(\omega_1,\omega_2)|H\omega_1,H\omega_2\rangle \nonumber\\
&+\sqrt{\eta_0}\Phi_{VV}(\omega_1,\omega_2)|V\omega_1,V\omega_2\rangle],
\label{Eq:state}\end{aligned}$$ where $\eta_{\text{I}}$, $\eta_{0}$ are the generation rates (pairs per pump photon) of the two processes after taking into account the losses. $\Phi_{HH}(\omega_1,\omega_2)$ and $\Phi_{VV}(\omega_1,\omega_2)$ are the biphoton wavefunctions, with the subscripts indicating the photon polarizations, and satisfy the normalization condition $\iint{d\omega_1 d\omega_2|\Phi_{HH(VV)}(\omega_1,\omega_2)|^2}=1$. When spatially separated, the paired photons are polarization entangled. The two spectra are found to be almost identical, as shown in Fig. \[Fig:spectra\](a). As a result, Eq. (\[Eq:state\]) is maximally entangled when the generation rates are the same, i.e., $\eta_{\text{I}}=\eta_0$. In this case, there is no way, even in principle, to tell in which process a pair of photons are generated unless polarizations are measured. Therefore, polarization entangled photons can be generated inherently on the chip without the need of any extra component.
Polarization entanglement can also be produced by the type-II process on the same chip. Following the same formalism, the two-photon state of the type-II process can be explicitly written as $$\begin{aligned}
\left\vert\psi^{\prime}\right\rangle=&\frac{1}{\sqrt{2}}\iint{d\omega_{1}d\omega_{2}}[\Phi_{HV}(\omega_1,\omega_2)|H\omega_1,V\omega_2\rangle \nonumber\\
&+\Phi_{VH}(\omega_1,\omega_2)|V\omega_1,H\omega_2\rangle].
\label{Eq:state_type-II}\end{aligned}$$ For maximal entanglement, it requires $\Phi_{HV}(\omega_1,\omega_2)=\Phi_{VH}(\omega_1,\omega_2)$. This is not satisfied for the waveguide under test. However, due to the lack of material birefringence, and thus very small temporal walk-off, there exists a significant amount of overlap between $\Phi_{HV}(\omega_1,\omega_2)$ and $\Phi_{VH}(\omega_1,\omega_2)$. The corresponding spectra are shown in Fig. \[Fig:spectra\](b). As a result, entanglement exists even without any compensation and bandpass filtering.
![(Color online) (a) The simulated spectral intensities of $H$ and $V$ polarized photons generated via the type-I and type-0 process, respectively; and (b) those of photons generated via the type-II process. The waveguide length is assumed to be 1.05 mm, the same as the waveguide tested in the experiment.[]{data-label="Fig:spectra"}](spectra_all_2){width="0.99\columnwidth"}
Photons in a pair need to be spatially separated. In this work, we used a 50:50 beamsplitter to separate photons non-deterministically followed by post-selection. However, paired photons can also be separated deterministically by a dichroic mirror or integrated dichroic splitter as was done in [@Horn_SR_2013]. For an ideal dichroic mirror which has a splitting wavelength at the degenerate point, the degree of entanglement is identical to that using a 50:50 beamsplitter.
Sample Description and Experimental Details
===========================================
As a proof of principle demonstration, a wafer designed for type-I PM around 1550 nm [@Abolghasem_OE_2010] is used to demonstrate this entangled photon source. Waveguides of this structure are lithographically tuned in order to satisfy concurrent PM of type-0 and type-I processes, with an etch depth of 6.5 $\mu$m and multiple ridge widths centered at 1.5 $\mu$m with a step size of 20 nm. An SEM image of a waveguide is shown in Fig. \[Fig:structure\_SPDC\_SEM\](b). Numerical simulations predict that concurrent PM of the two types can be achieved at a wavelength around 1.63 $\mu$m with a ridge width of $\sim$1.5 $\mu$m. Note that redesigning the epitaxial structure can shift the center wavelength to 1550 nm [@Kang_OL_2012]. The sample under test had a length of 1.05 mm.
In order to select the waveguide that has the best alignment of their PM wavelengths, second harmonic generation (SHG) for both type-0 and type-I processes were tested. The experiment was carried out on a standard end-fire coupling setup by pumping the waveguides with an optical parametric oscillator (OPO) pumped by a femtosecond pulsed Ti:Sapphire laser. The normalized SHG tuning curves of the waveguide under test are shown in Fig. \[Fig:PM wavelength\](a). According to Fig. \[Fig:PM wavelength\](a), PM wavelengths of both types are near 1640 nm, the longest achievable wavelength of the OPO used. Due to the large bandwidth of the pump pulse, the exact PM wavelengths could not be accurately identified. SPDC was then carried out by pumping the waveguides with a CW Ti:Sapphire laser where the dependence of single photon count rate on the pump wavelength was measured. This allowed us to locate the waveguide which has the best overlap in PM wavelengths among all tested waveguides. The results of the waveguide selected, shown in Fig. \[Fig:PM wavelength\](b), indicates that the PM wavelengths of both types are $816.7\pm0.3$ nm. The uncertainty in the PM wavelength measurement was due to the pump power fluctuation in the waveguide because of Fabry-–Pérot resonances, which could not be resolved with instrumentation available.
To generate polarization entangled photons via concurrent type-0 and type-I processes, the TM polarized pump beam from a CW Ti:sapphire laser was coupled into the waveguide using a 100$\times$ (N.A.=0.90) objective lens, with a power of 1.13 mW before the lens. Photon pairs generated were collected by a 40$\times$ objective lens at the output facet and passed through long pass filters to eliminate the pump. After their separation using a 50:50 beamsplitter, the signal and idler photons were collected into multimode fibers and detected by two InGaAs single photon detectors. The signal arm detector (ID220, ID Quantique) operates in a free-running mode and provided 20% efficiency at 1550 nm. The idler detector (ID210, ID Quantique) was operating in a gated mode and provided 25% at 1550 nm. It was externally triggered by the detection events of the first detector. An optical delay was added before the second detector to compensate for the electronic delay between the two detectors. Both detectors had a deadtime of 20 $\mu$s. The coincidence counts were recorded with the help of a time-to-digital converter (TDC) circuitry. At the degenerate wavelength of $\sim$1635 nm, the detection efficiencies are around 4% and 5%, respectively. A pair of quarter-wave plates (QWPs) and polarizers were used to measure the polarizations of the photon pairs. The schematic of the experimental setup is illustrated in Fig. \[Fig:setup\].
Considering the transmission coefficients of the output objective lens (90%), long pass filters (70%), beamsplitter (43%), QWPs and polarizers (75%) and fiber collection efficiencies in each path (53% and 34%), the overall collection efficiency of photon pairs with respect to the position right after the waveguide output facet was found to be $\sim$1.5%.
Results and Discussion
======================
Typical coincidence histograms are given in Fig. \[Fig:histograph\](a) for two $H$ polarized photons and Fig. \[Fig:histograph\](b) for two $V$ polarized photons for a pump wavelength of 816.76 nm and an integration time of 3 minutes. The coincidence peaks indicate that photon pairs were generated via both type-I and type-0 processes. The high level of accidental counts is due to detector dark counts and broken photon pairs due to waveguide losses as well as limited collection (1.5%) and detection (0.2%) efficiencies. By blocking the idler arm, we found the dark counts of the second detector consists of 83% of the total accidental counts. Thus we can expect a much higher coincidence-to-accidental (CAR) ratio by redesigning a sample which generates photon pairs in region where the detectors are more efficient (e.g. at 1550 nm).
![(Color online) Coincidence histograms of photon pairs in the (a) $HH$ and (b) $VV$ basis for an integration time of 3 minutes. The red bars around the peaks represent the counts in the coincidence window of $\sim$0.5 ns.[]{data-label="Fig:histograph"}](histograph){width="0.95\columnwidth"}
The net coincidence count rates for both type-0 and type-I processes are around 0.7 Hz, after subtracting the accidental counts. Taking into account the signal arm detector’s dead time, single count rate (16 kHz), as well as the overall collection and detection efficiencies, we estimate the photon pair generation rates after the waveguide are $3.4\times10^4$ pairs/s. The input objective lens has a transmission of 70% and the coupling efficiency into the pump Bragg mode is estimated to be 6%, resulting in an internal pump power of 47.3 $\mu$W right after the input facet. Therefore, the photon pair production rates are both around $7.3\times10^5$ pairs/s/mW with respect to the internal pump power and external photon pairs, or equivalently, $1.8\times10^{-10}$ pairs/pump photon. The fact that both processes have roughly the same generation rates, as opposed to type-I being more efficient than type-0 according to theoretical calculation, could be because that the TM mode has a smaller loss than that of the TE mode in this type of deeply etched waveguides. We confirmed this by measuring the losses using Fabry-–Pérot method and found the losses are 4.3 cm$^{-1}$ and 2.5 cm$^{-1}$ for TE and TM modes, respectively. It could also be because the pump wavelength is slightly detuned from the degenerate PM wavelength of the more efficient process, as the two processes may not have exactly identical PM wavelengths.
Quantum state tomography measurements were then subsequently performed by projecting the paired photons into 16 polarization combinations and the density matrix was reconstructed using the maximal likelihood method [@James_PRA_2001; @note2]. The net density matrix $\rho$ of the output two-photon state, given by Fig. \[Fig:rho\], is found to have a concurrence [@Wootters_PRL_1998] of $0.85\pm 0.07$. The maximum fidelity with a maximally entangled state $|\Phi\rangle=(|H,H\rangle+\exp{(i\phi)}|V,V\rangle)/\sqrt{2}$, defined by $F=\max_{\phi}{\langle\Phi|\rho|\Phi\rangle}$, is 0.89 with a corresponding phase angle $\phi=40^{\circ}$. The non-zero phase $\phi$ is because of the slightly dissimilar degenerate PM wavelengths of the two processes. Theoretical calculation according to Eq. (\[Eq:state\]) shows that this phase value is due to the type-I PM wavelength being $\sim$0.02 nm shorter than that of the type-0 process. The imperfection of the entanglement could be mainly because the pump wavelength is not optimal, causing extra spectral distinguishability between the two processes. This can be improved by using a tunable diode laser which has a fine spectral tunability. In addition, the mechanical drift of the characterization setup could result in an increase of mixture and a decrease of entanglement for measurements longer than a few minutes.
![(Color online) (a) The turning curves of the type-I (top) and type-0 (bottom) processes weighted by the corresponding efficiencies. The white dashed lines represent the pump wavelength, and (b) the corresponding spectra of the down-converted photons.[]{data-label="Fig:tuning_curves_and_spectra"}](tuning_curves_and_spectra_2){width="0.95\columnwidth"}
The fact that a slightly different PM wavelengths causes one of the processes to take place below its maximal generation rate can be used to balance the generation rates of the two processes and therefore increase the degree of entanglement. Although engineering efforts can be made to achieve almost identical efficiencies, in reality, there may still be differences especially with the existence of polarization dependent losses. In such a case, the waveguide ridge width can be lithographically tuned such that the stronger process has a shorter PM wavelength, while the pump wavelength should be tuned to the degenerate PM wavelength of the weaker process.
To illustrate this point, we consider an example where the type-I process is twice efficient as the type-0 process, i.e., $\eta_{\text{I}}=2\eta_{0}$. In the case where the two PM wavelengths are identical, the theoretical fidelity to a maximally entangled state is 0.96, and the concurrence is 0.93. The degree of entanglement can be increased to near maximum by tuning the ridge width such that the PM wavelength of type-I process is 0.55 nm below that of type-0 process. The corresponding tuning curves of the two processes weighted by their efficiencies are shown in Fig. \[Fig:tuning\_curves\_and\_spectra\](a). The pump wavelength, marked by the white line, is fixed at the degenerate PM wavelength of type-0 process. In this case, the spectra, given by Fig. \[Fig:tuning\_curves\_and\_spectra\](b), show almost the same amplitude, indicating the two processes having the same spectral brightness near the degenerate wavelength. By utilizing a weak spectral filtering with a bandwidth of 80 nm, the calculated fidelity and concurrence are increased to 0.997 and 0.995, respectively, with $\phi=34^{\circ}$, indicating the generation of maximally entangled photons. Without bandpass filtering, the maximum fidelity and concurrence can still reach 0.99 and 0.97, respectively, with a phase $\phi=14^{\circ}$, when the type-I PM wavelength is 0.36 nm below that of type-0 process.
In addition, this tuning approach could be used to generate non-maximally entangled states $(|H,H\rangle+r\exp{(i\phi)}|V,V\rangle)/\sqrt{1+r^2}$ with a tunable value of $r$, which offers significant advantages over maximally entangled states in some applications such as closing the detection loophole in quantum nonlocality tests [@Christensen_PRL_2013].
Lastly, we show the generation of cross-polarized entangled photons on the same chip via the type-II process. For a TE polarized pump at 812.92 nm, which is only less than 4 nm below those of type-0 and type-I, the output state, again without any compensation and bandpass filtering, shows a concurrence of $0.55\pm 0.08$ and a fidelity of 0.74 to the maximally entangled state $(|H,V\rangle+\exp{i\phi}|V,H\rangle)/\sqrt{2}$. The degree of entanglement is comparable with that obtained in [@Horn_SR_2013]. The reconstructed density matrix is given by Fig. \[Fig:rho\_type-II\]. The utility of this device becomes most apparent when one considers its capability to have all three types of PM simultaneously achieved at the same wavelength via the tuning of the epitaxial structure combined with ridge width control [@Zhukovsky_OC_2013]. The unique ability to achieve this monolithically allows the generation of both co-polarized and cross-polarized polarization entangled photons on the same chip at the same pump wavelength, by simply selecting the pump polarization. Nevertheless, a few nanometres’ difference in the PM wavelength can be covered by a femtosecond pump laser.
Conclusion
==========
In conclusion, we have demonstrated how the waveguiding physics associated with BRWs can be used to simultaneously produce two polarization entangled photon sources using alternative approaches in a single self-contained, room-temperature semiconductor chip. Direct generation of polarization entangled photons from a monolithic compound semiconductor chip via concurrent type-0 and type-I SPDC processes has been characterized. Simultaneous PM of the two processes was achieved using simple lithographic control on the ridge width of BRWs. Without the need of off-chip compensation, interferometer, and bandpass filter, the degree of entanglement is among the highest in previous demonstrations from monolithic III-V and Si chips. In addition, the same device can also directly generate polarization entanglement via the type-II process, with a pump wavelength of only 4 nm shorter.
Further improving the device performance relies largely on improved fabrications. By reducing the waveguide losses, the generation rates can potentially be increased by more than two orders of magnitudes, as predicted in [@Zhukovsky_JOSAB_2012] for lossless waveguides. In addition, the degree of entanglement can be increased by fine-tune the ridge width via more precise fabrications, as we have shown that the entanglement can be nearly maximal in the ideal case [@Kang_OL_2012]. The degree of entanglement can also be easily increased by bandpass filtering [@note].
Note that previous work on ferro-electric waveguides also studied the diversity of multiple PM processes to generate quantum states of particular properties, such as using two quasi-phase matching (QPM) gratings to generate polarization entangled photons [@Herrmann_OE_2013; @Gong_PRA_2011] and using different spatial modes to achieve mode entanglement [@Mosley_PRL_2009]. While they are fundamentally important, realizing such effects in monolithic fashions, especially on III-V semiconductor platforms as in this work, ushers a new era of practical optical quantum information processing.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to acknowledge R. Marchildon, G. Egorov, F. Xu, E. Zhu, and Z. Tang for helpful discussions. This work was supported by Natural Sciences and Engineering Research Council of Canada (NSERC).
[99]{} T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, Nature **464**, 45 (2010).
N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys. **74**, 145 (2002).
A. Christ, A. Fedrizzi, H. Hübel, T. Jennewein, and C. Silberhorn, Exper. Methods Phys. Sci. **45**, 351 (2013).
J. L. O’Brien, A. Furusawa, and J. Vučković, Nature Photon. **3**, 687 (2009).
A. Gaggero, S. Jahanmiri Nejad, F. Marsili, F. Mattioli, R. Leoni, D. Bitauld, D. Sahin, G. J. Hamhuis, R. Nötzel, R. Sanjines, and A. Fiore, Appl. Phys. Lett. **97**, 151108 (2010).
J. W. Silverstone, D. Bonneau, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, C. M. Natarajan, M. G. Tanner, R. H. Hadfield, V. Zwiller, G. D. Marshall, J. G. Rarity, J. L. O’Brien, and M. G. Thompson, Nature Photon. **8**, 104 (2014)
H. Jin, F. M. Liu, P. Xu, J. L. Xia, M. L. Zhong, Y. Yuan, J. W. Zhou, Y. X. Gong, W. Wang, and S. N. Zhu, Phys. Rev. Lett. **113**, 103601 (2014).
T. Suhara, Laser Photon. Rev. **3**, 370 (2009).
S. V. Zhukovsky, D. Kang, P. Abolghasem, L. G. Helt, J. E. Sipe, and A. S. Helmy, Opt. Lett. **36**, 3548 (2011).
D. Kang, L. G. Helt, S. V. Zhukovsky, J. P. Torres, J. E. Sipe, and A. S. Helmy, Phys. Rev. A **89**, 023833 (2014).
P. Abolghasem, M. Hendrych, X. Shi, J. P. Torres, and A. S. Helmy, Opt. Lett. **34**, 2000 (2009).
A. Valencia, G. Scarcelli, and Y. H. Shih, Appl. Phys. Lett. **85**, 2655 (2004).
S. Lloyd, Science [**321**]{}, 1463 (2008).
S. Sauge, M. Swillo, S. Alber-Seifried, G. B. Xavier, J. Waldeback, M. Tengner, D. Ljunggren and A. Karlsson, Opt. Express **15**, 6926 (2007). F. Kaiser, A. Issautier, L. A. Ngah, O. Dănilă, H. Herrmann, W. Sohler, A. Martin, and S. Tanzilli, New J. Phys. **14**, 085015 (2012).
S. Arahira and H. Murai, Opt. Express **21**, 7841 (2013).
N. Matsuda., H. L. Jeannic, H. Fukuda, T. Tsuchizawa, W. J. Munro, K. Shimizu, K. Yamada, Y. Tokura, and H. Takesue, Sci. Rep. **2**, 817 (2012).
L. Olislager, J. Safioui, S. Clemmen, K. P. Huy, W. Bogaerts, R. Baets, P. Emplit, and S. Massar, Opt. Lett. **38**, 1960 (2013).
A. Orieux, A. Eckstein, A. Lemaître, P. Filloux, I. Favero, G. Leo, T. Coudreau, A. Keller, P. Milman, and S. Ducci, Phys. Rev. Lett. **110**, 160502 (2013).
R. T. Horn, P. Kolenderski, D. Kang, P Abolghasem, C. Scarcella, A. D. Frera, A. Tosi, L. G. Helt, S. V. Zhukovsky, J. E. Sipe, G. Weihs, A. S. Helmy, and T. Jennewein, Sci. Rep. **3**, 2314 (2013).
B. J. Bijlani, P. Abolghasem, and A. S. Helmy, Appl. Phys. Lett. **103**, 091103 (2013).
F. Boitier, A. Orieux, C. Autebert, A. Lemaitre, E. Galopin, C. Manquest, C. Sirtori, I. Favero, G. Leo, and S. Ducci, Phys. Rev. Lett. **112**, 183901 (2014).
D. Kang and A. S. Helmy, Opt. Lett., **37**, 1481 (2012).
S. V. Zhukovsky, L. G. Helt, P. Abolghasem, D. Kang, J. E. Sipe, and A. S. Helmy, J. Opt. Soc. Am. B **29**, 2516 (2012).
P. Abolghasem, J. B. Han, B. J. Bijlani, and A. S. Helmy, Opt. Express **18**, 12861 (2010).
D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, Phys. Rev. A **64**, 052312 (2001).
The density matrix reconstruction was performed using the program provided on [P. Kwiat group’s website](http://webusers.physics.illinois.edu/~tomowww/TomographyDemo.php).
W. K. Wootters, Phys. Rev. Lett. **80**, 2245 (1998).
B. G. Christensen, K. T. McCusker, J. B. Altepeter, B. Calkins, T. Gerrits, A. E. Lita, A. Miller, L. K. Shalm, Y. Zhang, S. W. Nam, N. Brunner, C. C. W. Lim, N. Gisin, and P. G. Kwiat, Phys. Rev. Lett. **111**, 130406 (2013).
S. V. Zhukovsky, L.G. Helt, D. Kang, P. Abolghasem, A. S. Helmy, and J. E. Sipe, Opt. Comm. **301-302**, 127 (2013).
In a separate experiment, we have demonstrated a concurrence of 0.98 via the type-II process in a similar waveguide with bandpass filtering only. The results will be published somewhere else.
H. Herrmann, X. Yang, A. Thomas, A. Poppe, W. Sohler, and C. Silberhorn, Opt. Express **21**, 27981 (2013).
Y. X. Gong, Z. D. Xie, P. Xu, X. Q. Yu, P. Xue, and S. N. Zhu, Phy. Rev. A **84**, 053825, (2011).
P. J. Mosley, A. Christ, A. Eckstein, C. Silberhorn, Phys. Rev. Lett. **103**, 233901, (2009).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We represent a theory of polymer gelation as an analogue of liquid-glass transition in which elastic fields of stress and strain shear components appear spontaneously as a consequence of the cross-linking of macromolecules. This circumstance is explained on the basis of obvious combinatoric arguments as well as a synergetic Lorenz system, where the strain acts as an order parameter, a conjugate field is reduced to the elastic stress, and the number of cross-links is a control parameter. Both the combinatoric and synergetic approaches show that an anomalous slow dependence of the shear modulus on the number of cross-links is obtained.
[*PACS:*]{} 05.70Ln, 47.17+e, 61.43Fs
[*Keywords:*]{} [Polymer gelation; sol-gel transition; Shear stress/strain; Shear modulus]{}
address:
- |
Sumy State University\
Rimskii-Korsakov St. 2, Sumy, 40007 Ukraine\
- |
Department of Macromolecular Physics, Faculty of Mathematics & Physics\
Charles University\
V Holešovičkách 2, 180 00 Prague 8, Czech Republic\
author:
- 'Alexander I. Olemskoi'
- Ivan Krakovský
title: 'Two Simple Approaches to Sol-Gel Transition'
---
INTRODUCTION
============
Within the phenomenological framework the basic distinction between a liquid and a glass consists in character of the relaxation law of shear components of the elastic stress: if in an ideal glass they are kept infinitely long, in a liquid such relaxation proceeds in the finite time $\tau =
\eta /G$, where $\eta$ is the dynamical shear viscosity and $G$ is the shear modulus [@1]. In naive manner it is possible to assume, that a glass transition is caused purely by kinetic effect of a liquid freezing, for which viscosity $\eta$ gets infinite value for a finite value of the shear modulus $G$ [@2]. However in the course of usual second-order phase transition, where infinite increase of the $\tau$ at critical point is also observed the situation is reverse. Really, proceeding from the viscoelastic liquid to a general case, one has $\tau = \chi / \gamma$, where $\chi$ is the generalized susceptibility and $\gamma$ is the kinetic coefficient (in the case under consideration they are reduced to the quantities $G^{-1},~\eta^ {-1}$, respectively) [@3]. The infinite increase of a susceptibility $\chi$ occurs and a kinetic coefficient $\gamma$ does not manifest any peculiarity at the phase transition. In our case, this is equivalent to the situation when the shear modulus $G$ is approaching zero value at a fixed viscosity $\eta$. This situation corresponds to a viscoelastic transition [@4].
Usually under a glass transition, the transition into a glassy state which occurs during fast cooling of a liquid is understood. However, an analogous type of transition occurs during polymer network formation ([*polymer gelation*]{}) from monomers or linear macromolecules ([*sol*]{}) by means of a chemical reaction at constant temperature. This transition is referred to as the [*sol-gel transition*]{}. In such a case, thermodynamic peculiarities are observed as, e.g., appearance of the shear modulus when the system reaches a critical point.[^1] The first theoretical representations of such type transitions have been elaborated in classical works by Flory [@flory1] – [@flory3]. Modern description of the glassy state of the macromolecule networks (see [@G] – [@E]) is based on pioneering contributions to the theory of soft condensed matter: the Deam-Edwards theory [@DE] of a cross-linked tangled macromolecule, and the Edwards-Anderson theory [@EA] of spin glass.
The aim of this paper is to elaborate the simplest theoretical scheme so that the transition occurring during polymer gelation can be explained on the base of obvious combinatoric arguments (Section 2) as well as within framework of a synergetic theory (Section 3). In the former case, the critical point and behaviour of the system behind the critical point are obtained using information about the cross-linking properties of the constituent parts of the system ([*monomers*]{}) and extent of the cross-linking process, only. In Section 3, the starting point is that a sol-gel transition is ensured by self-organization of the elastic fields of stress and strain shear components and the number of cross-links. Both the combinatoric approach and the synergetic one allow to obtain anomalous dependence of the shear modulus on the number of cross-links.
Combinatoric approach
=====================
The gelation process has been described first by Flory[@flory1] and Stockmayer[@stock]. They used a laborious approach based on combinatoric considerations of the most probable composition of the system. A much more effective variant of this approach, exploiting theory of branching processes, has been developed later by Dobson and Gordon [@dg1; @dg2]. This approach allows also an analysis of structural details of the system behind the critical point, see, e.g. [@dusek].
As the simplest case let us consider the gelation process in the system which at the beginning consists of a large number of monomers $\cal N$, wearing $f$ functional terminal groups of independent reactivity. Monomers react mutually and irreversibly via their terminal groups and if $f>2$, branched molecules of increasing size and complexity are formed progressively in the system. Extent of the reaction is described by the conversion of terminal groups $\alpha$, which is defined as the ratio of the number of groups consumed by the reaction at given time and the starting number of groups. Eventually, when a critical conversion $\alpha_{\rm c}$ is achieved a molecule of macroscopic dimensions ([*gel*]{}) appears in the system [[@flory1], [@flory2]]{}. Gel contains cycles of long sequences of linked monomers and, consequently, attains elastic properties. Flory [@flory3] has shown that shear modulus of gel $G$ is proportional to its “cycle rank” $\xi$ defined as the number of “superfluous” links formed in gel which can be cut without breaking integrity of gel. In other words, cycle rank is the number of cuts needed for elimination of all cycles from gel.
The key role in the approach is played by the extinction probability $v$, which is a probability that a link formed in the process has just a finite continuation. It can be shown easily [@dg1] that for the system considered here the extinction probability can be obtained as a root of the equation $$v=(1-\alpha+\alpha v)^{f-1}
\label{v1}$$ satisfying the condition $0\le v\le 1$. Eq.(\[v1\]) expresses fact that a given link has a finite continuation in a direction only if $f-1$ functional groups remaining on the monomer connected by the link are either unreacted (with probability $1-\alpha$) or reacted giving rise to links with finite continuation only (with probability $\alpha v$). Below the critical conversion, the only solution of Eq.(\[v1\]) is $v=1$, i.e., only molecules of finite size are formed in the system. However, behind the critical conversion monomers can be found either in sol or in gel: $v<1$. As a measure of the “distance” from the critical conversion let us introduce the parameter $\epsilon$ defined as $$\epsilon\equiv\alpha-\alpha_{\rm c}
\label{eps}$$ and expand the extinction probability in a series: $$v=1+A_{1}\epsilon+A_{2}\epsilon^{2}+\ldots
\label{v3}$$ Substituting Eqs.(\[eps\]) and (\[v3\]) into Eq.(\[v1\]), expressions for the critical conversion and the parameters $A_{i}$ are obtained as $$\begin{aligned}
\alpha_{\rm c}&=&\frac{1}{f-1}\label{alpha},\\
A_{1}&=&-2~\frac{(f-1)^{2}}{f-2}\label{a1},\\
A_{2}&=&\frac{4}{3}~\frac{(2f-3)(f-1)^{3}}{(f-2)^{2}}.
\label{a2}\end{aligned}$$
The expression for the cycle rank can be found in the following way. Obviously, to join $\cal N$ monomers into a cycle-free structure, ${\cal N}-1$ links are necessary. By definition, the cycle rank $\xi$ is the number of “superfluous” links formed in gel, i.e., the difference between the number ${\rm N}_{\rm G}$ of all links formed in gel and the number ${\cal N}_{\rm G}-1$ of links sufficient to join together ${\cal N}_{\rm G}$ monomers of gel: $$\xi\equiv {\rm N}_{\rm G}-({\cal N}_{\rm G}-1)\simeq
{\rm N}_{\rm G}-{\cal N}_{\rm G} \label{xi1}$$ as ${\rm
N}_{\rm G}$, ${\cal N}\gg 1$. Correspondingly, the number of links formed in gel is the difference between the numbers of links formed in the total system and in sol, i.e., $${\rm N}_{\rm
G}=\frac{1}{2}{\cal N}\alpha f(1-v^2) \label{ng}$$ as any link in sol has to have finite continuations in two directions. On the other hand, the number ${\cal N}_{\rm G}$ of monomers incorporated in gel is the difference between the total number of monomers and the number of monomers in sol which is made of monomers with links of only finite continuation: $${\cal
N}_{\rm G}={\cal N}-{\cal N}(1-\alpha+\alpha v)^{f}. \label{Ng}$$ By virtue of Eqs.(\[xi1\]) – (\[Ng\]), one gets for cycle rank of the system considered $$\frac{\xi}{\cal N} = (1-\alpha+\alpha
v)^{f}+\frac{1}{2}\alpha f(1-v^{2})-1. \label{xi2}$$ Finally, substituting Eqs.(\[v3\]),(\[a1\]) and (\[a2\]) into the formula (\[xi2\]), one gets the necessary expansion: $$\frac{\xi}{\cal N} =
\frac{2}{3}~\frac{(f-1)^{4}}{(f-2)^2}f~\epsilon^3+O(\epsilon^{4}).
\label{xi3}$$ Respectively, the weight fraction of gel $w_{\rm G}\equiv{\cal N}_{\rm G}/{\cal N}$ is determined by Eq.(\[Ng\]) to read $$w_{\rm G}=2~\frac{(f-1)^{2}}{f-2}~\epsilon+O(\epsilon^{2}).
\label{wg2}$$ So, if the critical exponent of the gel weight fraction is equal 1 as usual, this for the cycle rank is anomalous large being equal 3. It is worthwhile to note that combinatoric approach is based exclusively on the information about the functionality of the monomers and extent of the chemical reaction between the monomers in the system considered.
Synergetic approach
===================
Now, let us consider the polymer network as a viscoelastic continuum matter that is characterized by the shear modulus $G$ and the shear viscosity $\eta$. Process of polymer gelation is determined by the number of the cross-links $N$, which value is different from a stationary magnitude $N_0$ at a time $t$. Therein, an elastic state of the polymer is defined by the shear component of the proper (internal) values of deformation $\varepsilon(t)$ and stress $\sigma(t)$. The keypoint is that these values are not reduced to the external elastic deformation $e\ll 1$ and stress $\sigma_e\ll G$, in particular they can get large magnitudes $\varepsilon\sim 1$, $\sigma\sim G$.
Our consideration of evolution of the elastic continuum with the internal structure is stated on the phenomenological equations by Maxwell-Kelvin [@1] $${d\varepsilon\over dt}=-{\varepsilon\over \tau}+{\sigma\over\eta},
\label{a}$$ $${d\sigma\over dt}=-{\sigma\over \tau_{\sigma}}+
g_{\sigma}\varepsilon N.
\label{b}$$ Here we introduce a macroscopic relaxation time $\tau$ for the strain and a microscopic one $\tau_{\sigma}$ for the stress, as well as a constant $g_{\sigma} > 0$ of the positive feedback between the deformation $\varepsilon$ and the number of cross-links $N$. Within the microscopic interval $t\gg\tau_{\sigma}$, steady-state condition $d\sigma/dt=0$ in Eq.(\[b\]) leads to the Hooke law with the microscopic shear modulus $$G_{\sigma}\equiv\tau_{\sigma} g_{\sigma}N
\label{c}$$ being determined by the number of cross-links $N$. Respectively, within a macroscopic interval $t\gg\tau$ Eq.(\[a\]) gives the magnitude $G\equiv\eta/\tau$ that is characteristic for the usual modulus of the viscoelastic matter. Lastly, a variation rate $dN/dt$ of the internal degree of freedom is supposed to be determined by the equation $${dN\over dt}={N_0-N\over
\tau_N}-g_N\sigma\varepsilon \label{e}$$ where $\tau_N$ is a mesoscopic relaxation time, $g_N > 0$ is constant of negative feedback between the deformation $\varepsilon$ and the stress $\sigma$. Within a mesoscopic interval $\tau_N\ll t\ll\tau$, Eq.(\[e\]) determines a steady-state value $$N=N_0-\tau_N g_N\sigma\varepsilon
\label{f}$$ that is smaller than the magnitude $N_0$ fixed by external conditions due to the fact that the elastic energy is proportional to the product $\sigma\varepsilon$.
System of Eqs.(\[a\]), (\[b\]) and (\[e\]) is known in synergetics [@Haken] as the Lorenz system where the deformation $\varepsilon$, the stress $\sigma$ and the number of cross-links $N$ play roles of an order parameter, a conjugate field and a control parameter, respectively. It is very important for following considerations that the relation between micro-, meso- and macroscopic values of the relaxation times $$\tau_{\sigma}, \tau_N \ll
\tau \label{g}$$ is satisfied. Due to this condition the evolution of the quantities $\sigma$, $N$ turns out to be subordinated to the long-time variation of $\varepsilon$. A peculiarity of the Lorenz system consists in linear character of the equation (\[a\]) for the order parameter $\varepsilon$ and in non-linearity of equations (\[b\]), (\[e\]) for the conjugate field $\sigma$ and the control parameter $N$. The negative nature of non-linearity in Eq.(\[e\]) means a decrease of the number $N$ of cross-links. Evidently, this fact reflects Le Chatelier principle. A non-linear term in Eq.(\[b\]) for a field $\sigma$ describes the positive feedback causing the system self-organization.
Expressions (\[a\]), (\[b\]) and (\[e\]) form the complete system of equations determining the polymer cross-linking behaviour. Because of a slow evolution, the order parameter $\varepsilon(t)$ subordinates variations of quantities $\sigma (t)$, $N(t)$, so that one can take $d\sigma/dt =dN/dt =0$ within the framework of the adiabatic approximation [@Haken]. Then $N$, $\sigma$ are expressed in terms of $\varepsilon$ by the equations: $$N={N_0\over 1+
\varepsilon^2/
\varepsilon^2_m},\quad
\varepsilon_m^{-2}\equiv \tau_{\sigma} \tau_N g_{\sigma} g_N;
\label{h}$$ $$\sigma=G_0{\varepsilon \over 1+
\varepsilon^2/
\varepsilon^2_m},\quad
G_0 \equiv \tau_{\sigma} g_{\sigma} N_0.
\label{i}$$ In accordance with Eq.(\[h\]) the number of cross-links $N$ decreases monotoneously with increase of the strain $\varepsilon$ from the value $N_0$ at $\varepsilon = 0$ to $N_0/2$ at $\varepsilon
=\varepsilon_m$.[^2] In Eq.(\[i\]) the elastic stress in terms of the strain has the linear form of the Hooke law at $\varepsilon\ll
\varepsilon_m$ with the effective shear modulus $G_0$. Then, at $\varepsilon = \varepsilon_m$ the function $\sigma (\varepsilon) $ has a maximum and at $\varepsilon > \varepsilon_m$ it decreases which is physically meaningless. Thus, the constant $\varepsilon_m$, defined by the second equation (\[h\]), has the meaning of the maximum achievable strain.
Substituting Eq.(\[i\]) in Eq.(\[a\]) we find the equation describing evolution of a system in the course of the sol-gel transition: $${d\varepsilon\over dt}=-\gamma
{\partial E \over \partial\varepsilon},\qquad
\gamma\equiv {\varepsilon_m^2\over\tau T~N_0}
\label{j}$$ where constant $\gamma$ plays a role of the kinetic coefficient. Behaviour of the system under consideration is determined by the dependence $E (\varepsilon) $ of the elastic energy on the strain: $$E \equiv {T~N_0\over 2}
\left [{\varepsilon^2 \over \varepsilon^2_m} - {N_0\over N_c}
\ln\left(1+{\varepsilon^2\over
\varepsilon^2_m}\right)\right]
\label{k}$$ where the characteristic value of the number of cross-links is introduced $$N_c \equiv {\eta\over\tau_{\sigma} g_{\sigma}}.
\label{l}$$ At $N_0\le N_c$ dependence (\[k\]) is monotoneously increasing with a minimum at the point $\varepsilon = 0$. It means that in the stationary state ($\dot\varepsilon {=} 0$) the elastic strain is not spontaneous. Thus, a liquid state is realized, in which the strain caused by the external stress relaxes during the time $$\tau_ {ef} = \tau \left (1 -
N_0/N_c\right)^{-1}.
\label{m}$$ The relaxation time increases infinitely when the number of cross-links $N_0$ reaches the critical value $N_c$ and at $N_0> N_c$ the system undergoes a sol-gel transition. In the gel state the multiplier 1/2 appears in the dependence (\[m\]), and the minimum of the elastic energy corresponds to the elastic strain $${\varepsilon^2 \over \varepsilon^2_m} = {N_c
\over N_0}~{N_0-N_c\over N_c}\equiv {\epsilon \over 1 +
\epsilon}
\label{n}$$ where we introduce the distance from the critical value $N_c$ $$\epsilon\equiv
{N_0-N_c\over N_c} \label{o}$$ being equivalent to the definition (\[eps\]). Inserting Eq.(\[m\]) into the dependence (\[k\]), we obtain the elastic energy of the steady-state: $$E_0\equiv E(\varepsilon_0)=-~{TN_0\over
4}~\epsilon^2 + O(\epsilon^3).
\label{p}$$ As would be expected, this energy is proportional to the second power of the parameter (\[o\]) and is negative in nature (the latter means the energy benefit of the gel state in comparison with the liquid state).
Taking into account that the glassy state is determined by density of the localized monomers, let us find now the shear modulus of the appeared gel state. It is principally important in our considerations that the gel state is determined by the value of the Deam-Edwards parameter of localization $\omega$ [@DE] which is supposed to be proportional to the square of the proper strain $\varepsilon_0$ of the matter. Under the condition of the appearance of the elastic strains $e$, a generalized Deam-Edwards parameter $\omega(e)$ has to be considered which is related to the condition $\omega(e=0)\equiv \omega$. Then, expanding the function $\omega(e)$ into a series and keeping the first two terms only, one obtains $$\omega(e)\simeq
\omega(1+e^2), \quad \omega\propto\varepsilon^2_0,\quad e\ll 1.
\label{r1}$$ By virtue of the parity condition $\omega(e)=\omega(-e)$, this expansion does not contain a linear term.[^3] Because of that the total strain $\varepsilon\propto\sqrt{\omega(e)}$, internal one $\varepsilon_0\propto\sqrt{\omega}$ and elastic one $e$ are connected by the following relation: $$\varepsilon^2 =
\varepsilon_0^2 (1+e^2), \qquad e\ll 1. \label{r}$$ The key point is that Eq.(\[r\]) supposes the additivity rule holds not for quantities $\varepsilon$, $\varepsilon_0$, $e$ themselves, but for their squares. The physical reason for such a situation is that the system under consideration is random in character and described by a symmetrical distribution function. Therefore, all odd-power moments vanish identically and making use of the additivity rule (\[r\]) for variances follows.
It is worthwhile to note the seeming contradiction between above relations for the localization parameter $\omega$ and the Deam-Edwards results [@DE]. Obvious reason consists in that the formers are obtained within framework of the mean-field theory, whereas the latters suppose fluctuation effects. According to [@DE] the cross-linking process is not sensible to strain $\varepsilon$ and corresponding localization parameter $\omega$ is proportional to the cross-link number $N$ but not to the difference $N-N_c$, as in Eqs.(\[n\]), (\[o\]). In our opinion, this is caused by non-self-consistency of the approach [@DE] in sense that the stress field $\sigma$ is switched off. On the other hand, making use of the statistical scheme [@DE] arrives at the strain-dependence for the localization parameter $\omega$ due to appearance of the polymer network entanglement, whereas within the above phenomenological approach this dependence has to be postulated.
A contribution to the elastic energy caused by the external strain is determined by equality $$\Delta
E(e)\equiv | E\left(\varepsilon(e)\right)- E(\varepsilon_0)|.
\label{s}$$ With use of Eqs. (\[k\]), (\[p\]), (\[r\]) and expansion $\ln(1+x)\approx x-x^2/2+x^3/3$, $ x\ll 1$ the expression $$\Delta E(e)\approx N_0~{T\epsilon^3\over 2} e^2
\label{u}$$ is easily obtained where only the first non-zero term is kept. Comparing this relation with the usual expression for the elastic energy [@1] $$\Delta E(e)\equiv V~{G\over
2} e^2 \label{v}$$ where $V$ is volume, we arrive at the final expression for the shear modulus of the gel state of the polymer network: $$G={T\epsilon^3\over\Omega},
\quad \epsilon\equiv {N_0-N_c\over N_c}\ll 1,
\quad \Omega\equiv
{V\over N_0}.
\label{w}$$ The notable peculiarity of this result consists in that, in accordance with previous result (\[xi3\]), the shear modulus is proportional to the third power of the distance $\epsilon$. It is worthwhile to note that an expression of this kind can be obtained only within framework of the synergetic approach used, but not on the basis of the phase transition theory by Landau.
Finally, comparing Eqs. (\[xi3\]) and (\[w\]), we obtain the relation between the micro- and macroscopic parameters of the gel under consideration in the proximity of the critical point ($\epsilon\ll 1$): $$G=g~\frac{\xi}{\cal N}~\frac{T}{\Omega},\quad
g\equiv\frac{3}{2}~f^{-1}~\frac{(f-2)^2}{(f-1)^{4}}.
\label{xy}$$
[00]{}
L. D. Landau, E. M. Lifshitz, [*Elasticity Theory*]{} (Nauka, Moscow, 1987).
A. Havranek, M. Marvan, Ferroelectrics [**176**]{}, 25 (1996).
A. I. Olemskoi, [*Theory of Structure Transformations in Non-equilibrium Condensed Matter*]{} (NOVA Science, N.-Y., 1999).
V. G. Bar’yakhtar, A. I. Olemskoi, Sov. Phys. Solid State [**33**]{}, 2705 (1991).
P. J. Flory, J.Am.Chem.Soc. [**63**]{}, 3083 (1941).
P. J. Flory, [*Principles of Polymer Chemistry*]{} (Cornell University Press, 1953).
P. J. Flory, Proc.Roy.Soc.London A[**351**]{}, 351 (1976).
P. M. Goldbart, H. E. Castillo, A. Zippelius, Adv. Phys. [**45**]{}, 393 (1996).
S. Panyukov, Y. Rabin, Phys. Rep. [**269**]{}, 1 (1996).
S. F. Edwards, H. Takano, E. M. Terentjev, cond-mat/0007270.
R. T. Deam, S. F. Edwards, Phil.Trans.R.Soc. A[**280**]{}, 317 (1976).
S. F. Edwards, P. W. Anderson, J.Phys. F[**5**]{}, 965 (1975).
W. H. Stockmayer, J.Chem.Phys. [**11**]{}, 45 (1943).
G. R. Dobson, M. Gordon, J.Chem.Phys. [**41**]{}, 2389 (1964).
G. R. Dobson, M. Gordon, J.Chem.Phys. [**42**]{}, 705 (1965).
K. Dušek, Adv.Polym.Sci. [**78**]{}, 1 (1986).
G. Haken, [*Synergetics*]{} (Springer-Verlag, Berlin, Heidelberg, N.Y., 1983).
[^1]: Of course, when the temperature of the polymer network formed is decreased the kinetic glass transition in above sense occurs, as well.
[^2]: Obviously this decrease is caused by the negative feedback in Eq.(\[e\]), that is reflection of Le Chatelier principle for analyzes problem. Actually, a liquid self-organization, resulting in a sol-gel transition, is caused by the positive feedback between the strain $\varepsilon$ and the number of cross-links $N$ in Eq.(\[b\]). Hence, the increase of the number of cross-links should intensify the self-organization effect. However, according to Eq.(\[h\]) system behaves in such way that the consequence of self-organization, i.e., growth of the elastic strain, leads to decrease of its origin – the number of cross-links.
[^3]: Otherwise, the usual squared dependence (\[u\]) will not be obtained.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'C. Horellou'
- 'H.T. Intema'
- 'V. Smol[č]{}i[ć]{}'
- 'A. Nilsson'
- 'F. Karlsson'
- 'C. Krook'
- 'L. Tolliner'
- 'C. Adami'
- 'C. Benoist'
- 'M. Birkinshaw'
- 'C. Caretta'
- 'L. Chiappetti'
- 'J. Delhaize'
- 'C. Ferrari'
- 'S. Fotopoulou'
- 'V. Guglielmo'
- 'K. Kolokythas'
- 'F. Pacaud'
- 'M. Pierre'
- 'B.M. Poggianti'
- 'M.E. Ramos-Ceja'
- 'S. Raychaudhury'
- 'H.J.A. Röttgering'
- 'C. Vignali'
bibliography:
- 'Horellou\_final\_20180723.bib'
date: 'Received 6 March 2018 / Accepted 12 July 2018'
subtitle: 'A tale of two radio galaxies in a supercluster at z = 0.14'
title: 'The XXL Survey: [XXXIV]{}. Double irony in XXL-North'
---
[We show how the XXL multiwavelength survey can be used to shed light on radio galaxies and their environment.]{} [Two prominent radio galaxies were identified in a visual examination of the mosaic of XXL-North obtained with the Giant Metrewave Radio Telescope at 610 MHz. Counterparts were searched for in other bands. Spectroscopic redshifts from the GAMA database were used to identify clusters and/or groups of galaxies, estimate their masses with the caustic method, and quantify anisotropies in the surrounding galaxy distribution via a Fourier analysis.]{} [Both radio galaxies are of FR [i]{} type and are hosted by early-type galaxies at a redshift of 0.138. The first radio source, named the Exemplar, has a physical extent of $\sim 400$ kpc; it is located in the cluster XLSSC 112, which has a temperature of $\sim 2$ keV, a total mass of $\sim10^{14} M_\odot$, and resides in an XXL supercluster with eight known members. The second source, named the Double Irony, is a giant radio galaxy with a total length of about 1.1 Mpc. Its core coincides with a cataloged point-like X-ray source, but no extended X-ray emission from a surrounding galaxy cluster was detected. However, from the optical data we determined that the host is the brightest galaxy in a group that is younger, less virialized, and less massive than the Exemplar’s cluster. A friends-of-friends analysis showed that the Double Irony’s group is a member of the same supercluster as the Exemplar. There are indications that the jets and plumes of the Double Irony have been deflected by gas associated with the surrounding galaxy distribution. Another overdensity of galaxies (the tenth) containing a radio galaxy was found to be associated with the supercluster.]{} [Radio galaxies can be used to find galaxy clusters/groups that are below the current sensitivity of X-ray surveys.]{}
Introduction
============
{width="8.8cm"} {width="8.8cm"}
The study of radio galaxies benefits enormously from a multiwavelength approach, from the X-ray to the radio (e.g., @2006LNP...693...39W). X-ray emission can be produced both by the active galactic nucleus (AGN) and the diffuse surrounding hot gas. Optical and infrared imaging and spectroscopy provide valuable information on the host galaxy, which most of the time is an early-type galaxy with very little or no star formation (some host galaxies, however, do contain dust and gas in a disk; e.g., ). In the radio, synchrotron radiation directly shows the extent and structure of the emitting plasma that contains relativistic electrons and magnetic fields. The jets may cover tens to hundreds of kiloparsecs before terminating in extended lobes. These jets and lobes may heat the intergalactic gas, displace material, and create cavities and shocks that are seen in the X-ray surface brightness distribution of galaxy clusters (e.g., @2005Natur.433...45M). This process, called radio-mode AGN feedback, is believed to play an important role in preventing gas infall on the host galaxy and regulating its evolution (e.g., @croton06).
The XXL survey (@pierre16, hereafter ) is particularly well suited to the study of AGN and galaxy clusters. With a total observing time of 6.9 Ms, it is the largest X-ray survey carried out by XMM-[*Newton*]{}. It covers two fields of 25 square degrees each, one in the southern hemisphere (XXL-South) and the other at an equatorial declination (XXL-North). The X-ray observations have been complemented by an extensive multiwavelength program.[^1]
In this paper we discuss two double-lobed radio galaxies (see Fig. \[figCFHTcolGMRTcont\]) that we have identified in the mosaic of XXL-North obtained with the Giant Metrewave Radio Telescope (GMRT). The GMRT-XXL-N 610 MHz survey is presented in this issue (@smolcic2018, hereafter ). It is of interest to compare the properties of the two sources as they are both at the same redshift ($z\simeq 0.14$) and the exact same data set is available for both. Their redshift is near the peak of the redshift distribution of the most securely detected clusters in XXL (, hereafter ). As our analysis shows, the radio galaxies are both part of the same supercluster, but their immediate surroundings are different.
The first radio galaxy appears regular, with symmetric jets leading to two radio lobes. It is located at the center of a cluster and hosted by the brightest cluster galaxy (BCG), both of which have been detected and characterized in studies of the sample of the brightest 100 XXL clusters (, hereafter ; ; , hereafter ; , hereafter ; @2016MNRAS.462.4141L, hereafter ). Because of its appearance with twin jets and plumes and its association with an early-type galaxy in a cluster, we refer to this source in the remainder of this paper as [*the Exemplar*]{}. The other radio galaxy stands out for its very large extent (a total length of $7\farcm56$, or 1100 kpc in our adopted cosmology[^2]) and its peculiar shape reminiscent of two reversed question marks, the reason why we named it [*the Double Irony.*]{}[^3]
------------ ----------- ----------------- ------------- -----------------------
Survey Frequency Resolution Pixel size $\sigma$
(MHz) ($\arcsec$) ($\arcsec$) (mJy beam$^{-1}$)
TGSS ADR1 150 25 6.4 5.0
GMRT-XXL-N 610 6.5 1.9 0.060
FIRST 1400 $6.8\times 5.4$ 1.8 0.15
NVSS 1400 45 15 0.45 (I), 0.29 (Q, U)
------------ ----------- ----------------- ------------- -----------------------
\[tabRadioData\]
This paper is organized as follows. The available data sets are presented in Sect. \[SectData\]. In Sect. \[SectExemplar\] we discuss the Exemplar radio galaxy, its host galaxy, and host cluster; the radio images are presented and analyzed. Section \[SectDbleIrony\] is about the Double Irony and follows a similar structure; an overdensity of galaxies is found. In Sect. \[SectCompa\] the two radio galaxies are compared and discussed. In particular, the distributions of surrounding galaxies are examined in relation to the orientation of the radio galaxies to search for environmental effects. In Sect. \[SectSuperclu\] it is shown that the group of galaxies associated with the Double Irony radio galaxy is part of the same supercluster as the Exemplar’s cluster and this large-scale structure is discussed. In Sect. \[SectDiscussionRadiogalsSuperclu\] we discuss the occurrence of radio galaxies in superclusters and their potential use as tracers of the large-scale structure. We conclude in Sect. \[SectSummary\].
Data and data analysis {#SectData}
======================
Radio {#SectDataRadio}
-----
As detailed below, we used images at three different frequencies obtained from four different surveys carried out with the Very Large Array (VLA) and the GMRT. These surveys have different sensitivities and angular resolutions, and are also sensitive to different angular scales. The characteristics of the data are summarized in Table \[tabRadioData\].
The most important data set is the GMRT-XXL-N 610 MHz radio continuum survey described in . XXL-North contains the XMM-LSS area that was observed earlier with the GMRT (@tasse06, [-@tasse07]). The rest of the XXL-North field was recently observed at higher sensitivity, and the data were combined. The final mosaic has an angular resolution of $6\farcs5$. The two radio galaxies discussed here are located outside the XMM-LSS region, in the newly covered region of 12.66 square degrees that has an average sensitivity of 45 $\mu$Jy beam$^{-1}$.
The other radio continuum data were taken from the following, publicly available surveys:
- the Tata Institute of Fundamental Research (TIFR) GMRT Sky Survey (TGSS) first alternative data release (TGSS ADR1) at 150 MHz and 25$\arcsec$ resolution [@TGSS_ADR1];
- the Faint Images of the Radio Sky at Twenty Centimeters (FIRST) at 1.4 GHz, $6\farcs8\times 5\farcs4$ resolution (@first95; @first15);
- the NRAO VLA Sky Survey (NVSS) at 1.4 GHz, 45$\arcsec$ resolution (@NVSS). NVSS also contains polarization information. We downloaded Stokes $Q$ and $U$ images from the NVSS website[^4] and produced images of the polarized intensity and of the polarization angle using the task [POLC]{} in the Astronomical Image Processing System[^5] ([AIPS]{}) that corrects for the Ricean bias in polarized intensity [@1974ApJ...194..249W].
Total-intensity images do not always do justice to the details of the surface brightness distributions in astronomical sources. Images of the norm of the intensity gradient, $|\nabla I|$, have been used to reveal the paths of jets and substructures in radio galaxies (e.g., @2011MNRAS.417.2789L). We computed images of $|\nabla I|$ at 610 MHz using a Sobel filter (Sobel & Feldman 1968, unpublished, reported in Wikipedia[^6]). Flux densities and spectral indices were measured within defined regions. The statistical uncertainties on the flux density measurements were calculated as $\sigma \sqrt{N_{\rm beam}}$, where $\sigma$ is the standard deviation of the noise and $N_{\rm beam}$ is the number of beams within the region. The survey images were used and no attempt was made to re-image the data to match the $(u,v)$ ranges or the imaging weighting scheme. Only the FIRST image of the Double Irony may have extended emission that has been filtered out (the largest angular scale of FIRST is $2'$), and this is discussed in Sect. \[SectDbleIronySpectralIndex\]. The sources are barely resolved in TGSS and in NVSS, and some of the features of the Double Irony are only marginally detected in TGSS, leading to large uncertainties in the flux measurements that would not be improved much by re-imaging.
Throughout the paper we use the following convention for the radio spectral index, $\alpha$: $S_\nu \propto \nu^\alpha$, where $S_\nu$ is the flux density at frequency $\nu$. The uncertainties on the spectral indices measured between two frequencies were calculated using standard error propagation, $$\Delta\alpha =
\sqrt{
\left(
\frac{\Delta S_1}{S_1}
\right)^2
+
\left(
\frac{\Delta S_2}{S_2}
\right)^2
}
\Bigg/
\ln\left(\frac{\nu_2}{\nu_1}\right) \, ,
\label{equncalpha}$$ where $\Delta S_i$ is the uncertainty on the flux density $S_{i}$ measured at frequency $\nu_i$ and $\nu_2 > \nu_1$.
A systematic uncertainty of 10% on each flux density measurement (due to uncertainties on the absolute flux calibration and from the imaging process) results in the following systematic uncertainties on the spectral indices (Eq. \[equncalpha\]): $$\Delta\alpha_{\rm sys}(\nu_1,\nu_2) =
\left \{
\begin{tabular}{l}
0.10\\
0.17\\
0.06\\
\end{tabular}
\right.
{\rm if\, } \frac{\nu_1}{\nu_2} =
\left \{
\begin{tabular}{l}
150 MHz/610 MHz\\
610 MHz/1400 MHz\\
150 MHz/1400 MHz\\
\end{tabular}
.
\right.
\label{eqDeltaAlphaSys}$$
Multiwavelength
---------------
We used data from the following surveys:
- The X-ray measurements with XMM-[*Newton*]{} and the XXL project are described in . The cluster catalogs are presented in and , and the AGN catalogs by (; ) and by @lucio2018 ([-@lucio2018], hereafter ).
- There is a large overlap between XXL-North and the CFHT-LS W1 region covered in several optical bands by the Canada-France-Hawaii Telescope (CFHT). All XXL-North clusters but five have $ugriz$ photometry from MegaCam [@2012AJ....143...38G].
- The Sloan Digital Sky Survey (SDSS) Data Release 14 (SDSS DR14[^7]) contains optical and near-infrared images of our fields and spectra of some of the sources.
- The Galaxy and Mass Assembly (GAMA) database[^8] contains spectra, spectroscopic redshifts, and information on the detected spectral lines for the host galaxies of our two radio sources and other galaxies in the field [@2018MNRAS.474.3875B].
- The Wide-field Infrared Survey Explorer (WISE; @2010AJ....140.1868W) has detected the host galaxies of our two radio sources in several bands.
The Exemplar {#SectExemplar}
============
[lll]{} Quantity & Value & Notes Cluster name & XLSSC 112 &(1)\
RA$_{\rm cluster}$(J2000) & $32\fdg514$ &(1)\
Dec$_{\rm cluster}$(J2000) & $-5\fdg462$ &(1)\
$z_{\rm cluster}$ & 0.139 &(1)\
$N_{\rm galaxies}$ & 14 &(1)\
$L_{\rm 500,MT}^{\rm XXL}$ &$(0.61\pm0.08) \cdot 10^{43}$ erg s$^{-1}$ &(1)\
$F_{60}$ & $(5.89\pm0.62) \cdot 10^{-17}$ Wm$^{-2}$ &(2)\
$T_{\rm 300~kpc}$ & $1.76^{+0.25}_{-0.15}$ keV &(2)\
$M_{\rm 500,MT}$ & $(9.0 \pm4.1) \cdot 10^{13} M_\odot$ &(2)\
$r_{\rm 500,MT}$ & 0.653 Mpc &(2)\
$M_{\rm gas,500}$ & $(0.42\pm0.12)\cdot 10^{13} M_\odot$ &(3)\
$r_{\rm 500,WL}$ & $0.6^{+0.1}_{-0.2}$ Mpc &(4)\
$M_{\rm 500,WL}$ & $0.8^{+0.6}_{-0.5} \cdot 10^{14} M_\odot$ &(4)\
$M_{\rm 200,WL}$ & $1.2^{+0.9}_{-0.8} \cdot 10^{14} M_\odot$ &(4)\
RA$_{\rm BCG}$ (J2000) &$32\fdg5093$ &(5)\
Dec$_{\rm BCG}$ (J2000) &$-5\fdg4678$ &(5)\
Offset from X-ray center &$24\farcs5$ &(5)\
$z_{\rm BCG}$ & 0.138 &(5)\
$M_{\rm BCG}$ &$5.35^{+0.41}_{-0.29} \cdot 10^{11} M_\odot$ &(5)\
$M_{\rm stellar}$ & $2.13^{+0.30}_{-0.56}\cdot10^{11}~M_\odot$ &(6)\
SFR1 (2–20 Myr ago) & $0~M_\odot$ yr$^{-1}$ &(6)\
SFR2 (20–600 Myr ago) & $5.6~M_\odot$ yr$^{-1}$ &(6)\
SFR3 (0.6–5.6 Gyr ago) & $46.2~M_\odot$ yr$^{-1}$ &(6)\
SFR4 ($> 5.6$ Gyr ago) & $14.9~M_\odot$ yr$^{-1}$ &(6)\
LW-age & $3.6\times10^9$ yr &(6)\
MW-age & $5.8\times10^9$ yr &(6)\
$\sigma_V$ &$282.0\pm 8.6$ km s$^{-1}$ &(7)\
$z_{\rm SDSS\, DR14}$ &$0.13818\pm 0.00002$ &(7)\
$z_{\rm GAMA}$ &0.13814 &(8)\
In Fig. \[figExemplarOptXrayradio\] we show a multiband false-color image of the field of the Exemplar: the optical image is shown in green, the X-ray emission in blue and in contours, and the 610 MHz radio emission from the GMRT in red. Several X-ray point sources are seen, in addition to diffuse X-ray emission associated with a cluster. The two brightest X-ray point-like sources (X3 and X4) are galaxies with optical counterparts and photometric redshifts in SDSS DR14; they are not associated with the cluster (see caption of Fig. \[figExemplarOptXrayradio\]). The fainter X-ray source X1 coincides with the radio source S1 (see discussion in Sect. \[subSectExemplarRadio\]). It has an optical galaxy counterpart, but no redshift is available.
Host galaxy in the cluster XLSSC 112 in the supercluster XLSSsC N03
-------------------------------------------------------------------
### Cluster XLSSC 112
The Exemplar is located at the center of a galaxy cluster listed among the 100 brightest clusters in XXL (). Cataloged as XLSSC 112, the cluster has a mean redshift of $z = 0.139$ calculated from the spectroscopic redshifts of 14 member galaxies (Table \[tabTheOtherOne\]). The intracluster gas has a temperature of $\sim1.8$ keV measured within the central radius of 300 kpc . The total cluster mass was estimated from a mass–temperature scaling relation and from weak gravitational lensing , yielding consistent values of about 10$^{14} M_\odot$ within a radius $r_{500} = 0.6$ Mpc (the parameter $r_{500}$ is the radius within which the mean density is 500 times the critical density of the universe at the cluster’s redshift).
![Multiband false-color image of the Exemplar: X-ray emission in the \[0.5–2\] keV band in blue, optical emission ($i$-band CFHT image) in green, radio emission (GMRT 610 MHz) in red. The ten contours showing the X-ray surface brightness increase logarithmically between $3\times 10^{-17}$ and $6\times 10^{-16}$ erg s$^{-1}$ cm$^{-2}$. Four X-ray point sources are visible in the image, labeled X1 to X4 above the yellow circles that indicate their positions. The point sources were subtracted before calculating the flux density of the cluster given in and quoted in Table \[tabTheOtherOne\], but they are shown in this image. X3 corresponds to 3XLSS J021002.1-052655 and X4 to 3XLSS J020958.2-052853 in the XXL point source catalog . There are optical counterparts to X3 (a galaxy at photometric redshift of $0.028\pm 0.014$, SDSS J021002.25-052655.5) and to X4 (a galaxy at a photometric redshift of $0.351\pm0.118$, SDSS J020958.27-052853.5). The other two X-ray sources did not match the criteria to be included in the X-ray catalog. The size of the image is about $4'\times4'$. Diffuse X-ray emission is seen, and is associated with the galaxy cluster XLSSC 112. []{data-label="figExemplarOptXrayradio"}](fig2ExemplarRGBradioOptXrays.png){width="8.8cm"}
### Supercluster XLSSsC N03
The cluster was found to be part of a larger structure of seven clusters, all at a redshift of $z \simeq 0.14$, with a total mass of about $10^{15} M_\odot$ (XLSSC-b; ). Five such superstructures were identified in the sample of the 100 brightest clusters in XXL using a friends-of-friends algorithm in 3D space (). New supercluster candidates were found in the larger sample of 365 clusters cataloged by @adami2018 ([-@adami2018], hereafter ). The five superclusters identified in were confirmed. However, the supercluster hosting the Exemplar, which was described as a double structure in , with an eastern component of four clusters and a western component of three clusters (including XLSSC 112), has been split into two superclusters in the new study: the four clusters in the eastern part form XLSSsC N08, at a mean redshift of $z = 0.141$, and the three clusters in the western part joined five other clusters to form a larger structure, the supercluster XLSSsC N03, at a mean redshift of $z = 0.139$ . XLSSC 112 is therefore affiliated to a supercluster with eight cluster members. This is discussed in Sect. \[SectSuperclu\] (see also Fig. \[figSuperclu\]).
{width="8.8cm"} {width="8.8cm"} {width="8.8cm"} {width="8.8cm"}
### Brightest cluster galaxies and the nearest neighbor {#subSectExemplarBCG}
The properties of the BCG of the 100 brightest XXL clusters have been studied in . @guglielmo2018a ([-@guglielmo2018a]; ) have assembled a catalog of optically detected galaxies in the X-ray–detected clusters and groups of XXL-North. In Fig. \[figGAMAspectra\] (top panel) we show the GAMA spectrum of the BCG, typical of an old stellar population. We fitted the GAMA spectrum using the full spectral fitting code [SINOPSIS]{} () and obtained an estimate of its stellar mass of about $2\times10^{11}~M_\odot$ (using a @2003PASP..115..763C initial mass function). This is lower than had been estimated from broadband photometry (). This analysis also provided an estimate of the star formation history of the galaxy. The characteristics of the cluster and of its BCG are summarized in Table \[tabTheOtherOne\].
There is a second, smaller galaxy located 7$\arcsec$ to the NE of the BCG (about 17 kpc on the sky) at a similar redshift ($z = 0.13623$, galaxy GAMA J021002.58-052759.8). Companions like this are very common for radio sources of this type (e.g., 3C296).
Radio {#subSectExemplarRadio}
-----
### Radio morphology
![Grayscale image of the norm of the intensity gradient of the Exemplar at 610 MHz. It reveals the path of the jets and substructures in the brightness distribution of the radio lobes. The beam is shown in orange in the bottom left corner. []{data-label="figExemplarGradI"}](fig4ExemplarGMRTgrad.png){width="8.8cm"}
Figure \[figSarcasmAllradioimages\] shows the four radio continuum images of the Exemplar used in this work. The Exemplar has two jets and lobes that appear clearly in the higher resolution images (GMRT 610 MHz and the FIRST 1.4 GHz images, Figs. \[figSarcasmAllradioimages\]b and d), but are not resolved in the lower resolution images (TGSS and NVSS, Figs. \[figSarcasmAllradioimages\]a and c). The TGSS ADR1 catalog of [@TGSS_ADR1] lists three individual Gaussian sources.[^9] The NVSS catalog lists two sources[^10] [@NVSS]. The XXL-GMRT-610 MHz catalog[^11] lists three resolved sources within a radius of 4$'$ from the core of the Exemplar . From end to end the radio source stretches over 160$\arcsec$ (396 kpc) in the GMRT 610 MHz image.
Zooming into the central region, we see that the radio surface brightness peaks about $3\farcs5$ (8.7 kpc) away from the optical center of the BCG on either side of the jets, with a flux density of about 8.8 mJy in the NW and 9.8 mJy to the SE at 610 MHz. There are two corresponding sources in the FIRST catalog[^12]. We note that these brighter radio spots are located just outside the optical extent of the host galaxy. No cataloged FIRST source is found at the central location of the BCG. The radio core is unresolved and is likely embedded in the jet and/or is synchrotron self-absorbed.
The jets have position angles of $+130^\circ$ and $-50^\circ$ (measured counterclockwise from the north). They are straight out to a distance of 32$\arcsec$ to the NW and 40$\arcsec$ to the SE from the center of the BCG, at which point they change direction and turn right (that is, the NW jet turns toward the W and the SE jet turns toward the E). The NW jet moves into two brighter regions embedded in a diffuse lobe. The SE jet continues to grow fainter but remains collimated for another 15$\arcsec$ before reaching a diffuse lobe. This morphology suggests an interaction with the surrounding medium at a distance of about 80 kpc (30$\arcsec$) to the NW and about 90 kpc (35$\arcsec$) to the SE.
Two compact radio sources are seen near the radio galaxy. They are named S1 and S2 in Tables \[tabExemplarFlux\] and \[tabExemplarAlpha\] and their flux densities and their spectral index were estimated between 610 MHz and 1.4 GHz (FIRST image). The spectral indices are used to estimate the flux at 150 MHz. The contributions of those two sources are subtracted from the total flux densities measured within the regions outlined in Fig. \[figSarcasmAllradioimages\]. The radio photometry of the Exemplar is presented in the coming section.
In Fig. \[figExemplarGradI\] we show an image of the gradient of the intensity in the GMRT image at 610 MHz, $|\nabla I|$. The path of the jet is clearly delineated in white where the intensity has a local maximum. Strong gradients are seen near the inner jets where the jet bends as it enters the NW lobe, and at the extremity of the SE lobe. Similar features have been seen in deep images of nearby low-luminosity radio galaxies (@2011MNRAS.417.2789L).
---------------------------------------------------- -------------------- ------------------- --------------------------- --------------------------
Region $S_{\rm 150~MHz}$ $S_{\rm 610~MHz}$ $S_{\rm 1.4~GHz}$ (FIRST) $S_{\rm 1.4~GHz}$ (NVSS)
(mJy) (mJy) (mJy) (mJy)
S1 25.73 $\pm$ 3.11 3.27 $\pm$ 0.14 0.97 $\pm$ 0.40 –
S2 6.74 $\pm$ 3.30 2.48 $\pm$ 0.14 1.38 $\pm$ 0.40 –
A (SE lobe) 61.14 $\pm$ 9.46 32.25 $\pm$ 0.43 8.21 $\pm$ 1.19 12.98 $\pm$ 0.37
B (NW lobe) 71.98 $\pm$ 7.85 38.76 $\pm$ 0.36 13.11 $\pm$ 1.00 10.91 $\pm$ 0.28
C 97.07 $\pm$ 10.94 56.64 $\pm$ 0.50 29.89 $\pm$ 1.40 30.60 $\pm$ 0.41
C $\setminus$ S2 90.33 $\pm$ 11.42 54.16 $\pm$ 0.52 28.52 $\pm$ 1.45 29.23 $\pm$ 0.44
T 284.79 $\pm$ 26.63 138.16 $\pm$ 1.21 44.81 $\pm$ 3.38 74.95 $\pm$ 0.99
T$_{\rm A \cup B \cup C}$ 230.20 $\pm$ 16.45 127.66 $\pm$ 0.75 51.21 $\pm$ 2.09 54.50 $\pm$ 0.63
T$_{\rm (A \cup B \cup C) \setminus (S1 \cup S2)}$ 197.72 $\pm$ 28.60 121.90 $\pm$ 1.30 48.87 $\pm$ 3.63 52.16 $\pm$ 1.09
T $\setminus$ (S1 $\cup$ S2) 252.32 $\pm$ 27.01 132.40 $\pm$ 1.23 42.47 $\pm$ 3.43 72.61 $\pm$ 1.01
---------------------------------------------------- -------------------- ------------------- --------------------------- --------------------------
------------------------------ -------------------------------------- --------------------------------------- ---------------------------------------
Region $\alpha_{\rm 150~MHz}^{\rm 610~MHz}$ $\alpha_{\rm 610~MHz}^{\rm 1400~MHz}$ $\alpha_{\rm 150~MHz}^{\rm 1400~MHz}$
(1) (2) (3)
$\Delta\alpha_{\rm sys}$ 0.1 0.17 0.06
A (SE lobe) $-0.46 \pm 0.11$ $-1.65 \pm 0.18$ $-0.69 \pm 0.07$
B (NW lobe) $-0.44 \pm 0.00$ $-1.30 \pm 0.09$ $-0.84 \pm 0.05$
C $-0.38 \pm 0.08$ $-0.77 \pm 0.06$ $-0.52 \pm 0.05$
C $\setminus$ S2 $-0.36 \pm 0.09$ $-0.77 \pm 0.04$ $-0.51 \pm 0.06$
T $-0.52 \pm 0.07$ $-1.36 \pm 0.09$ $-0.60 \pm 0.04$
T $\setminus$ (S1 $\cup$ S2) $-0.46 \pm 0.08$ $-1.37 \pm 0.06$ $-0.56 \pm 0.05$
------------------------------ -------------------------------------- --------------------------------------- ---------------------------------------
: Spectral index measurements in the different regions of the Exemplar shown in Fig. \[figSarcasmAllradioimages\].
\[tabExemplarAlpha\]
### Luminosity, flux density, spectral index {#SectExemplarSpectralIndex}
Flux densities were measured at the three different frequencies in the regions indicated in Fig. \[figSarcasmAllradioimages\]; the values are listed in Table \[tabExemplarFlux\].
The spectral luminosity, $L_\nu$, can be calculated following $$\left(
\frac{L_{\nu}}
{{\rm W~ Hz}^{-1}}
\right) =
5.257\times 10^{22} \,
\left(
\frac{S_{\nu}}
{{\rm mJy}}
\right)
\left(
\frac{D_L}
{{\rm 662.8~Mpc}}
\right)^2 \, .
\label{eqLnu}$$
In the commonly used classification of [@fanaroff74], FR [i]{} radio galaxies have been observed to have a spectral luminosity lower than $5\times 10^{25}$ W Hz$^{-1}$ at 178 MHz, while the FR [ii]{} radio galaxies are more luminous. The Exemplar has a total flux density of about 250 mJy at 150 MHz (taking an average of the value obtained by summing up the different regions and by measuring in a larger ellipse). For a spectral index of $-0.7$, this gives $L_{\rm 178~MHz}\simeq 1.5\times 10^{25}$ W Hz$^{-1}$, making it an FR [i]{} radio galaxy. Its morphology is also that of an FR [i]{}.
Spectral indices were calculated in the regions overlaid on Fig. \[figSarcasmAllradioimages\] (Table \[tabExemplarAlpha\]). The central region has flatter spectral indices than the lobes. The spectral indices calculated at high frequencies are systematically steeper than those at low frequencies[^13] Using the NVSS 1.4 GHz image and the TGSS 150 MHz image gives intermediate values.
Unfortunately, the quality of the data does not allow a quantitative analysis of the evolution of the cosmic-ray electrons, and deeper images with matching angular resolutions are required to measure the curvature of the radio spectrum.
### Polarization
No polarization is seen in the NVSS images of the Exemplar. The lobes have a flux density at 1.4 GHz ranging between 8 and 13 mJy, and the flux density of the core/central region is about 30 mJy (Table \[tabExemplarFlux\]). The noise level in polarized intensity is about 0.4 mJy in NVSS. So we place a $3\sigma$ upper limit on the fractional polarization of 15% in the lobes and 4% in the central region.
The Double Irony as a giant radio galaxy {#SectDbleIrony}
========================================
[llc]{} Quantity & Value & Notes\
RA$_{\rm galaxy}$(J2000) &$30\fdg97696$ & (1)\
Dec$_{\rm galaxy}$(2000) &$-4\fdg23247$ & (1)\
$z_{\rm spec}$ & 0.13751 & (1)\
Central point source:\
RA$_{\rm X-ray}$(J2000) & $30\fdg977942$ & (2)\
Dec$_{\rm X-ray}$(J2000) & $-4\fdg232405$ & (2)\
$F_{\rm X,[0.5-2]~keV}$ & $2.4\times 10^{-15}$ erg s$^{-1}$ cm$^{-2}$ & (2)\
$F_{\rm X,[2-10]~keV}$ & undetected &(2)\
$L_{\rm X,[0.5-2]~keV}$ & $1.26\times10^{41}$ erg s$^{-1}$ &(3)\
Limit on cluster:\
Count rate & $< 0.032$ ct s$^{-1}$ &(4)\
$T_{\rm 300~kpc}$ & $< 1.26$ keV &(4)\
$M_{500}$ & $< 5.13\times 10^{13} M_\odot$ &(4)\
$L_{500}$ & $< 1.76\times 10^{42}$ erg s$^{-1}$ &(4)\
$F_{\rm X,[0.5-2]~keV, < 300~kpc}$ &$< 2.82\times 10^{14}$ erg s$^{-1}$ cm$^{-2}$ &(4)\
$M_{\rm stellar}$ & $(2.97\pm0.25)\times10^{11}~M_\odot$ &(5)\
SFR1 (2–20 Myr ago) & $0~M_\odot$ yr$^{-1}$ &(5)\
SFR2 (20–600 Myr ago) & $1.48~M_\odot$ yr$^{-1}$ &(5)\
SFR3 (0.6–5.6 Gyr ago) & $33.08~M_\odot$ yr$^{-1}$ &(5)\
SFR4 ($> 5.6$ Gyr ago) & $42.36~M_\odot$ yr$^{-1}$ &(5)\
LW-age & $5.7\times10^9$ yr &(5)\
MW-age & $7.9\times10^9$ yr &(5)\
The Double Irony stands out in XXL-North because of its large angular size and peculiar shape. As we show in Sect. \[sectDbleIronyRadio\], it has a total physical length of about 1.1 Mpc, which makes it a member of the rare class of giant radio galaxies (GRGs), i.e., Mpc-sized radio galaxies. Samples of GRGs have been built based on searches in large radio surveys at GHz frequencies such as the Westerbork Northern Sky Survey[^14] (e.g., ); the NVSS[^15] and the FIRST survey[^16] (e.g., , ; ; @2016ASSP...42..231S; @2016ApJS..224...18P), or the Sydney University Molonglo Sky Survey[^17] (e.g., @2005AJ....130..896S). The total number of known GRGs is about 300 [@2017MNRAS.469.2886D], but this number is likely to increase rapidly with the advent of radio surveys at lower frequencies where the flux densities are stronger because most of the emission comes from the more numerous lower energy relativistic electrons that have a longer lifetime. Surveys carried out for example with the LOw Frequency ARray (LOFAR, ), MSSS[^18] [*Herschel*]{}-ATLAS, and LoTSS[^19] have begun to reveal even larger GRGs than those discovered at higher frequencies (e.g., @2016MNRAS.462.1910H; ). GRGs are interesting in their own right, but also as probes of their environment. As radio lobes expand beyond the halos of their host galaxies, they may interact with the so-called Warm-Hot Intergalactic Medium (WHIM), a diffuse plasma at a temperature of $10^5$ to $10^7$ K that is associated with galaxies in the large-scale structure (e.g., @2008ApJ...677...63S; @2009MNRAS.393....2S). They are therefore useful probes of the intergalactic medium and possibly even of its evolution if they can be found at high redshift [@2007AcA....57..227M]. The large sizes of GRGs do not seem to be due to stronger nuclear activity (@1999MNRAS.309..100I) and GRGs do not seem to form a class that is fundamentally different from normal-sized radio galaxies (; @2012MNRAS.426..851K). The general picture is that GRGs are able to grow to larger sizes because of their lower density environments and/or simply because they are older (e.g., @2008MNRAS.385.1286J). In this section we begin by examining the host galaxy of the Double Irony; then we look at the X-ray information and the galaxy distribution in the optical before presenting the radio maps; finally the possible nature of this unusual radio galaxy is discussed. In the next section the two radio galaxies will be compared and more quantities will be derived.
Host galaxy
------------
![Multiband false-color image of the Double Irony: X-ray emission in blue displayed at the noise level, optical emission ($i$-band image from the CFHT) in green, radio emission from the GMRT at 610 MHz as yellow contours. []{data-label="figDbleIronyOptXrayradio"}](fig5DbleIronyRGB.png){width="8.8cm"}
{width="8.8cm"} {width="8.8cm"} {width="8.8cm"} {width="8.8cm"}
We identify a galaxy with a spectroscopic redshift of $z= 0.13751$ (GAMA J020354.47-041356.8) as the host galaxy of the Double Irony based on its position with respect to the radio contours. The GAMA spectrum is displayed in Fig. \[figGAMAspectra\] (bottom panel). It is very similar to the spectrum of the host galaxy of the Exemplar, and reflects an old stellar population. We derived an estimate of its stellar mass of about $3\times10^{11}~M_\odot$. This analysis provides some insight into the star formation history of the galaxy: it has no ongoing star formation, but had a star formation rate (SFR) of about 1.5 $M_\odot$ yr$^{-1}$ $\sim20$–600 Myr ago, and an even higher SFR of $\sim30$–45 $M_\odot$ yr$^{-1}$ in the more distant past (see Table \[tabDbleIrony\]).
There is also an X-ray point-like counterpart to the optical/near-IR host (see Table \[tabDbleIrony\]). The source is detected only in the \[0.5 – 2\] keV band, and the detection is not of sufficient quality to say anything about the nature of the X-ray spectrum (thermal or non-thermal). The corresponding X-ray luminosity is $1.26\times10^{41}$ erg s$^{-1}$. found a relation between the X-ray luminosity and the SFR of nearby star-forming galaxies. Using their relation (their Eq. 14) gives a SFR estimate of about 28 $M_\odot$ yr$^{-1}$, which is significantly higher than that inferred from the analysis of the optical spectrum. It is very unlikely that the X-ray emission is related to star formation; since the X-ray emission is point-like and the optical major axis of the galaxy is about 20$\arcsec$ (more than double the full width at half maximum (FWHM) of the point response function of XMM-[*Newton*]{}), it indicates that the X-ray emission does not come from hot gas surrounding the elliptical galaxy. We conclude that it is likely to come from the AGN. The signal is, however, very weak, with only about 12 counts, and a detection likelihood of 15.6, just above the threshold of 15 below which sources are considered spurious and not included in the catalog (@lucio2018).
Host cluster/group {#SectDbleIronyHostCluster}
-------------------
In the following we review the information that can be gathered from X-ray and optical observations on the possible existence of a cluster or group around the Double Irony.
– X-ray. No cluster is listed at the location of the Double Irony radio galaxy in the new XXL catalog of X-ray detected clusters . From the 1$\sigma$ upper limit on the count rate within a radius of 300 kpc, we were able to place an upper limit on the X-ray flux, temperature, and luminosity within a radius $r_{500}$ using scaling relations established for XXL clusters (see Table \[tabDbleIrony\]): a cluster/group around the Double Irony would have a temperature lower than 1.3 keV and a mass within $r_{500}$ lower than $5.1\times10^{13}~M_\odot$. This is significantly lower than the corresponding values measured for the Exemplar’s cluster.
A closer inspection of the X-ray image (Fig. \[figDbleIronyOptXrayradio\]) does not reveal any clear extended emission. The noise varies across the image and the quality of the data does not allow us to say anything about the presence or absence of gas interaction with the radio lobes.
– Optical.
A search in the GAMA database led to the identification of 32 galaxies with spectroscopic redshifts in the $0.13 \leq z \leq 0.15$ range and within a radius of $6.3'$ (0.92 Mpc) of the core of the Double Irony. This radius was chosen as it corresponds to $r_{200}$ (which is approximately the virial radius) of the Exemplar galaxy cluster. The actual virial radius of the Double Irony galaxy might be different, but this shows that the radio galaxy is surrounded by a large number of galaxies. In Fig. \[figDbleIronyGAMAgalaxies\] we show the GMRT 610 MHz image of the Double Irony radio galaxy in red, superimposed on an optical $i$-band image from the CFHT in blue. The yellow circles indicate the positions of some of the galaxies in the field cataloged in the GAMA survey; their spectroscopic redshifts are also indicated. We defer the analysis of the spectroscopic redshifts to Sect. \[sectDiscusssionClusterMass\] (cluster mass estimates) and Sect. \[sectCompaEnvironment\] (study of the anisotropy of the surrounding galaxy distribution).
Radio {#radio}
-----
### Radio morphology {#sectDbleIronyRadio}
![Grayscale image of the norm of the intensity gradient of the Double Irony at 610 MHz. The path of the jets is clearly visible, in particular the abrupt turn in the region of the Elbow. The beam is shown in the bottom left corner. []{data-label="figDbleIronyGradI"}](fig7DbleIronyGMRTgrad.png){width="8.8cm"}
In Fig. \[figDbleIronyAllradioimages\] we show the radio images of the Double Irony. The structure of the source is best seen in the GMRT 610 MHz image that combines high angular resolution and good sensitivity.[^20] The yellow cross indicates the location of the center of the host galaxy. Several features appear clearly in this image: the central east–west jets, a bright lobe to the southwest, a diffuse plume farther out (SW2), a bright region to the northeast (known as the “Elbow”), and a second diffuse plume-like feature to the north that breaks into several lumps. These five main features are also seen in the lower resolution (45$\arcsec$) 1.4 GHz image from the NVSS. The higher resolution FIRST image at the same frequency is badly affected by stripes that prevent us from seeing the two plumes in the north and the southwest. The TGSS ADR1 image is noisy as well. Only the lobe in the SW1 region is clearly visible. While the lumps of emission in the north and in the Elbow are discernible, it is clear that flux measurements in these regions are affected by the artifacts from the imaging.
Most striking perhaps are the multiple bends: on the eastern side the nearly horizontal jet turns abruptly, but remains collimated until the bright Elbow; the continued structure to the north of the Elbow is fainter and at about 90 degrees from its original direction. On the SW side, the jet shows a small deviation before entering the SW1 lobe, and the SW2 plume is at another angle. The entire structure covers about 4$'$ and shows a remarkable symmetry by rotation of 180 degrees, which suggests that it is not a superposition of several sources, but forms one system. We examined optical images and could not find any counterparts to the individual features described above.
The XXL-GMRT-610 MHz catalog[^21] lists seven resolved sources within 4$'$ of the core of the Double Irony . Figure \[figDbleIronyGradI\] shows the GMRT 610 MHz intensity gradient map. The white line (which corresponds to the peaks in the intensity map where the gradient is zero) extends far out and traces the path of the jets. Areas of strong gradients in the inner region and in the Elbow region have a flatter spectral index (Table \[tabDbleIronyAlpha\]). This has been observed in more nearby low-luminosity FR [i]{} radio galaxies by [@2011MNRAS.417.2789L]. Following the jet’s path out to the 5$\sigma$ contour of the GMRT 610 MHz map, we measure a total length of $\sim 7.56'$, which corresponds to a projected physical size of 1100 kpc. [*This means that the Double Irony is a GRG.*]{}
{width="8.8cm"} {width="8.8cm"}
### Luminosity, flux density, spectral index {#SectDbleIronySpectralIndex}
-------- -------------------- ------------------- --------------------------- --------------------------
Region $S_{\rm 150~MHz}$ $S_{\rm 610~MHz}$ $S_{\rm 1.4~GHz}$ (FIRST) $S_{\rm 1.4~GHz}$ (NVSS)
(mJy) (mJy) (mJy) (mJy)
C 31.60 $\pm$ 10.06 23.75 $\pm$ 0.45 10.95 $\pm$ 1.23 11.54 $\pm$ 0.84
Elbow 17.09 $\pm$ 9.14 20.83 $\pm$ 0.40 7.59 $\pm 1.10$ 8.67 $\pm$ 0.68
N 136.25 $\pm$ 23.15 77.76 $\pm$ 1.03 7.06 $\pm 2.79^*$ 29.29 $\pm$ 1.82
SW1 98.17 $\pm$ 12.54 46.29 $\pm$ 0.56 14.43 $\pm 1.52^*$ 23.98 $\pm$ 0.99
SW2 99.51 $\pm$ 22.84 63.97 $\pm$ 1.02 22.93 $\pm 2.77^*$ 27.88 $\pm$ 1.79
T 434.95 $\pm$ 79.70 205.76 $\pm$ 3.54 21.57 $\pm 9.65^*$ 131.23 $\pm$ 6.27
-------- -------------------- ------------------- --------------------------- --------------------------
--------------------------- -------------------------------------- ---------------------------------------
Region $\alpha_{\rm 150~MHz}^{\rm 610~MHz}$ $\alpha_{\rm 150~MHz}^{\rm 1400~MHz}$
$\Delta \alpha_{\rm sys}$ 0.1 0.17
C $-0.20 \pm 0.23$ $-0.45 \pm 0.15$
Elbow $+0.14 \pm 0.38$ $-0.30 \pm 0.24$
N $-0.40 \pm 0.12$ $-0.69 \pm 0.08$
SW1 $-0.54 \pm 0.09$ $-0.63 \pm 0.06$
SW2 $-0.31 \pm 0.16$ $-0.57 \pm 0.11$
T $-0.53 \pm 0.13$ $-0.54 \pm 0.08$
--------------------------- -------------------------------------- ---------------------------------------
: Spectral index measurements in the different regions of the Double Irony shown in Fig. \[figDbleIronyAllradioimages\].[]{data-label="tabDbleIronyAlpha"}
From the NVSS flux density measurement (Table \[tabDbleIrony\]) we derive a spectral luminosity at 1.4 GHz of $6.9\times10^{24}$ W Hz$^{-1}$ (Eq. \[eqLnu\]), or of $2.9\times10^{25}$ W Hz$^{-1}$ at 178 MHz assuming a spectral index of $-0.7$. This is about twice the spectral luminosity of the Exemplar, but still in the range of luminosities of FR [i]{} radio galaxies. Some GRGs are powerful radio galaxies with FR [ii]{} structures (e.g., @1996MNRAS.279..257S; @1999MNRAS.309..100I; ); as the sensitivity of observations increases, more FR [i]{} GRGs (or hybrid FR [i]{}/FR [ii]{}) with lobes of lower surface brightness are found (e.g., ; @2005AJ....130..896S). In Tables \[tabDbleIronyFlux\] and \[tabDbleIronyAlpha\] we list the values for the flux densities and the spectral indices measured in different regions displayed in Fig. \[figDbleIronyAllradioimages\]. The spectral indices measured between 150 MHz and 1.4 GHz are systematically steeper than the low-frequency spectral indices measured between 150 MHz and 610 MHz, an indication of spectral aging of the cosmic-ray electrons.
The Elbow and the central region both have a relatively flat spectral index, but no systematic pattern is seen. We intend to model the evolution of cosmic-ray electrons in the Double Irony and in the Exemplar and to derive spectral age estimates by analyzing more sensitive low-frequency data (pending GMRT observations below 500 MHz).
### Polarization
Figure \[figDbleIronyPolar\] shows images of the polarization detected by NVSS[^22] at 1.4 GHz. The left panel shows the polarized intensity image overlaid with vectors showing the direction of the electric field of the polarized wave rotated by 90$^\circ$. In the absence of Faraday rotation, this would correspond to the direction of the magnetic field component on the plane of the sky, $B_\perp$, in the source. The length of the vectors is proportional to the fractional polarization.
The vectors follow the large-scale distribution of the total intensity rather well, except in the very center where they are almost perpendicular to the direction of the inner jets and may trace a poloidal magnetic field. We note, however, that one vector per pixel is shown, so the vectors (like the pixels) are correlated, and variations on scales smaller than the synthesized beam (45$\arcsec$, shown in the bottom left corner) are smoothed out.
In the right panel of Fig. \[figDbleIronyPolar\] we show an image of the fractional polarization. The position of the core is indicated by a cross. It appears that the fractional polarization peaks on the right side of the core, while the left side is less polarized. This could be a signature of the Laing–Garrington effect (@1988Natur.331..149L; @1988Natur.331..147G): stronger external Faraday depolarization [@1966MNRAS.133...67B] occurs along the line of sight of the counter-jet due to the longer path length through a fluctuating magnetic field component. This would mean that [the eastern part of jet points away from us, and the more strongly polarized western jet points toward us.]{}
Dedicated observations of the polarization at several radio frequencies are required to map the magnetic field pattern in this radio galaxy and measure the Faraday rotation.
Nature of the Double Irony {#SectDiscussionNatureDbleIrony}
--------------------------
The nature of the Double Irony radio galaxy remains somewhat of a mystery. Given the very large extent on the sky, we suspect that the inclination is not very large, perhaps 30$^\circ$ to 45$^\circ$ relative to the plane of the sky, but the structure is probably not entirely coplanar. The radio galaxy displays a remarkable symmetry by rotation of 180$^\circ$, except for the strong bend near the Elbow in the northeast, while the western part is more regular.
Similar features have been seen in other FR [i]{} radio galaxies. For example 3C 31 has two 90-degree elbows, and the overall radio structure could be nicely reproduced in the dynamical model of an unbound gravitational encounter with a neighboring galaxy (@1978MNRAS.185..527B). The morphology of the Double Irony is similar to that of NGC 326, a radio galaxy that has been the subject of different interpretations, such as jet precession (@1978Natur.276..588E), jet realignment due to gravitational interaction between two galaxies (@1982AJ.....87..602W), or buoyancy forces that redirect the lobes into regions of lower pressure in the surrounding medium (@1995ApJ...449...93W). In the following we discuss some physical effects that may have shaped the Double Irony.
### A restarted radio galaxy?
Some other GRGs have a morphology that is indicative of a recurrent nuclear activity. For example [@2002ApJ...565..256S] found an inner double radio structure within larger edge-brightened lobes. The two bright regions of the Double Irony (the SW lobe and the Elbow) and the two diffuse outer plumes beyond the SW lobe and the Elbow may be due to two separate events, where the two brighter inner features would be more recent. However, it would be a coincidence for the restarting lobes to have just reached the bends. It seems more likely that it is a quasi-steady structure influenced by the intergalactic medium.
### Gravitational interaction with the second brightest galaxy {#sectDiscussion2ndGal}
A second galaxy is clearly visible to the NE of the host of the Double Irony. It is the next brightest galaxy within a radius of 1 Mpc and is listed in SDSS (SDSS J020357.62-041321.5). With a spectroscopic redshift (from BOSS) of 0.13433, the line-of-sight velocity difference with the host of the Double Irony is 840 km s$^{-1}$. It is at a distance on the sky of 59$\arcsec$, or a projected distance of 146 kpc. An analysis of the spectrum in the GAMA database (GAMA J20357.62-041321.5) yields a stellar mass of about $0.9\times10^{11}~M_\odot$, which is about one-third of the stellar mass of the host of the Double Irony (Table \[tabDbleIrony\]). There is no sign of tidal perturbations in the optical surface brightness distribution of this galaxy, which is an argument against a prograde encounter with the host of the Double Irony. Nevertheless, given the relative line-of-sight velocity difference and their projected distance, the two galaxies could have been interacting gravitationally in the past $\sim100$ Myr, causing a motion of the host of the Double Irony in the general direction of the second galaxy during the lifetime of the radio source. The drop-like morphology of the SW1 radio lobe is consistent with [a global motion of the AGN of the Double Irony toward the east, in the general direction of the companion]{}.
An observation of the distribution of atomic hydrogen via the H[i]{} 21 cm line in the companion galaxy may reveal a possible interaction between the two galaxies. The galaxy must contain some atomic hydrogen gas since it is disk-like and star forming: from the GAMA spectrum, we estimated a SFR of about 0.8 $M_\odot$ yr$^{-1}$ in the last 2–20 Myr and a higher SFR earlier (8.8 $M_\odot$ yr$^{-1}$ 20–600 Myr ago and about 19 $M_\odot$ yr$^{-1}$ 0.6–5.6 Gyr ago). The detection of H[i]{} tidal tails would constrain the dynamical history of this galaxy. Unfortunately, its redshift places the H[i]{} line near 1250 MHz, in a region that is usually affected by radio-frequency interference.
### Interaction with the gaseous environment
In the absence of a deeper X-ray image not much can be said about the surrounding medium of the Double Irony. The upper limit on the X-ray emission provides an upper limit on the temperature of about 1.3 keV and a limit on the mass within $r_{500}$ of about $5\times 10^{13} M_\odot$ (Table \[tabDbleIrony\]). The ejected radio-emitting plasma may have been able to travel farther out because of the low-density of the surrounding medium. The absence of significant X-ray emission may also indicate that the atmosphere of the cluster/group has been so spread out by the radio galaxy’s heating and pushing that the $n^2 L$ (density squared times pathlength) through the cluster has dropped from what it was when the radio source turned on. The change in direction of the outer plumes may also be a sign of backflow, as seen for instance in the FR [i]{} radio galaxies 3C 296 [@2006MNRAS.372..510L], 3C 270 [@2015MNRAS.450.1732K], or NGC 326 (@2012ApJ...746..167H). The circumgalactic medium (the gas located outside the optical body of a galaxy but inside the galaxy’s virial radius) extends in some cases on scales larger than 100 kpc () and the Double Irony may have interacted with the circumgalactic medium of its bright neighbor located beyond the bend and mentioned in the previous section.
Comparison of the two radio galaxies {#SectCompa}
====================================
{width="17.7cm"}
In Sect. \[SectSuperclu\] we show that the Double Irony and the Exemplar are part of the same large-scale structure, a supercluster at $z\simeq 0.14$. In this context it is of interest to compare the two radio galaxies and their environments.
In Sect. \[SectDiscussionMBH\] we estimate their black hole masses from scaling relations. In Sect. \[SectDiscussionLERGs\] we discuss the properties of the optical spectra of the host galaxies in relation to what is known from large samples of radio galaxies. In Sect. \[sectDiscusssionClusterMass\] we analyze the distribution of galaxies surrounding the radio galaxies and make estimates of the cluster/group masses. In Sect. \[SectDiscussionClusterAges\] we derive an age estimate from the properties of the two brightest galaxies in each cluster/group. Finally, in Sect. \[sectCompaEnvironment\] we examine the distributions of galaxies that surround each radio galaxy and quantify their degrees of anisotropy in relation to the orientation of the radio jets and lobes.
Estimation of the black hole masses {#SectDiscussionMBH}
-----------------------------------
There is a well-known correlation between the mass of the black hole and the stellar mass (bulge mass) of the host galaxy (e.g., @2000ApJ...539L...9F). This correlation is expressed as $$\log_{10}\left(
\frac{M_{\rm BH}}{M_\odot}
\right) =
a + b \log_{10}(X) \, ,$$ where $X$ is the stellar mass itself or an indicator of it (e.g., the velocity dispersion, the $K$-band absolute magnitude).
We have stellar mass estimates for both galaxies (Tables \[tabTheOtherOne\] and \[tabDbleIrony\]). Using the parameters given by [@2013ApJ...764..184M] in their fits to the stellar mass–black hole mass relation ($a = 8.56\pm 0.10$; $b = 1.34\pm0.15$), we obtain $$\begin{array}{lcl}
M_{\rm BH}^{\rm Exemplar} &\simeq &1.0\times10^9 M_\odot\\
\medskip
M_{\rm BH}^{\rm Double~Irony} &\simeq &1.6\times10^9 M_\odot \, .\\
\end{array}$$
For the Exemplar we can also use the value of the velocity dispersion from SDSS DR 14 (Table \[tabTheOtherOne\]). The fits of [@2013ApJ...764..184M] for elliptical galaxies when $X$ is the velocity dispersion in units of 200 km s$^{-1}$ have the following parameters: $a = 8.39 \pm 0.06$ and $b = 5.20 \pm 0.36$. This gives a black hole mass of about $1.5\times10^9$ $M_\odot$.
We see that [the black holes of the Double Irony and Exemplar both have a mass on the order of $10^9$ $M_\odot$]{}.
Brightest cluster galaxies {#SectDiscussionLERGs}
--------------------------
The host galaxy of the Double Irony is somewhat brighter than that of the Exemplar in all bands except the WISE W1 and W2 bands (the difference is largest in the $u$ band, Tables \[tabOptNIRphot\] and \[tabWISEphot\]).
The two host galaxies have very similar optical spectra (Fig. \[figGAMAspectra\]) characteristic of old stellar populations. Detection of H$\alpha$, H$\beta$, and \[O[ii]{}\] is reported in the GAMA database for both galaxies. However, there is no detection of high-excitation emission lines such as \[O[iii]{}\] that is seen in the host galaxies of high-excitation radio galaxies (HERGs). It has been shown that the most luminous radio galaxies (FR [ii]{}) are predominantly HERGs, while the hosts of the less powerful FR [i]{} radio galaxies do not exhibit such spectral lines. Low-excitation radio galaxies (LERGs) are in general hosted by galaxies that have a lower star formation rate, are located in clusters, and are more massive than the host galaxies of HERGs. They are believed to have different modes of accreting gas: the LERGs accrete mostly hot gas, whereas the HERGs accrete cold gas and radiate more efficiently (e.g., @hardcastle07; ; @smolcic09; @best12; @best14). [The Exemplar and the Double Irony can both be classified as LERGs.]{}
Both radio galaxies can be classified as FR [i]{} from their morphologies. By examining the mid-infrared photometry and colors of the host galaxies of radio galaxies, [@2014MNRAS.438..796S] was able to show that FR [i]{} and FR [ii]{} fall in different parts of the (W1-W2)-(W2-W3) color-color diagram ((\[3.4 $\mu$m\] - \[4.6 $\mu$m\])-(\[4.6 $\mu$m\]-\[12 $\mu$m\]), their Fig. 13). The WISE colors of the hosts of our radio galaxies are consistent with those of FR [i]{} radio galaxies (Table \[tabWISEphot\]).
Mass of the clusters {#sectDiscusssionClusterMass}
--------------------
The spectroscopic redshifts obtained from the GAMA database allow us to obtain a new mass estimate of the galaxy cluster that hosts the Exemplar, and to search for a cluster/group of galaxies around the Double Irony. The velocities $v_i$ in the rest frame of a cluster at redshift $z_{\rm cluster}$ are simply $$v_i = c \frac{z_i - z_{\rm cluster}}{1 + z_{\rm cluster}}\, ,$$ where $z_i$ are the redshifts of the galaxies.
The velocity dispersion within the virial radius can be used to obtain an estimate of the virial mass. However, this mass estimate relies on a single number and does not take into account the full distribution of galaxies on the sky and in redshift space. A more sophisticated method is the caustic method developed by [@1997ApJ...481..633D] and [@1999MNRAS.309..610D]. The idea is to estimate the group’s escape velocity from the characteristic trumpet shape of the distribution of galaxies in the plane defined by the group/cluster-centric distance and the line-of-sight velocities with respect to the median recession velocity of the group. We used the modified caustic mass estimation algorithm written by [@2012MNRAS.426.2832A] to calculate the total masses of galaxy groups in the GAMA group catalog (G$^3$Cv1; @2011MNRAS.416.2640R).
### Cluster/group membership
The first step is to identify the member galaxies of our two clusters/groups. We simply selected all the galaxies located within an estimated radius (which corresponds to $r_{200}$ for the Exemplar’s cluster) and within a certain spectroscopic redshift range: $$\left \{
\begin{tabular}{l}
$R < r_{200} = r_{500}/0.7 = 6.3' = 934$~kpc\\
$0.13 < z < 0.15\, .$\\
\end{tabular}
\right.$$ To estimate the radius $r_{200}$, we used the relation of : $r_{200} = r_{500}/0.7$, which gives $r_{200} = 934$ kpc $= 6.3'$ for the Exemplar’s cluster, using the value of $r_{500}$ given in Table \[tabTheOtherOne\]. In the absence of other information on the Double Irony, we took the same value of $r_{200}$ as for the Exemplar. The number of galaxies selected in this manner is given in Table \[tabClusterMasses\]. It should be noted that mass estimates depend on how accurately the cluster members have been selected. However, as we see below, the caustic mass method provides a way to assess cluster membership that is complete to a very large degree for massive clusters (@2013ApJ...768..116S).
### Caustic mass method
The caustic mass method requires the cluster’s redshift and velocity dispersion as inputs in addition to the catalog of positions and redshifts of the individual member galaxies. Rather than using a simple median value and standard deviation, we used the “gapper estimator” introduced by [@1990AJ....100...32B]. This estimator has been shown to be unbiased, even for groups with few member galaxies and robust to weak variations in group memberships. The estimated “scale” in velocity space (an estimator of the “velocity dispersion”) is calculated as $$\sigma_{\rm gap} = \frac{\sqrt{\pi}}{N(N-1)} \sum_{i = 1}^{N-1} w_i g_i \, ,
\label{eqBeersGap}$$ where $g_i = v_{i+1} - v_{i}$ is the velocity difference between each velocity pair computed after having sorted the galaxies’ recession velocities in the catalog in increasing order; the weights are $w_i = i(N-i)$, where $N$ is the number of galaxy redshifts and $i$ varies between 1 and $N-1$.
Following [@2011MNRAS.416.2640R], we increased the velocity dispersion by a factor $\sqrt{N/(N-1)}$ to take into account the fact that the central galaxy is moving with the center of mass of the halo (associated cluster); we also corrected for the measurement uncertainties on the recession velocities of the individual galaxies (50 km s$^{-1}$ for GAMA, @2011MNRAS.413..971D) by subtracting the contribution for the $N$ galaxies in the cluster ($\sigma_{\rm err} = 50 \sqrt{N}$ km s$^{-1}$): $$\sigma = \left(
\frac{N}{N-1} \sigma_{\rm gap}^2 - \sigma_{\rm err}^2
\right)^{1/2} \, .
\label{eqBeersGapcorr}$$For the Exemplar we used the position of the BCG as the central position. For the Double Irony we used the position of the host galaxy, which is also the brightest galaxy within a radius of about 1 Mpc.
In Fig. \[figCaustic\] we show the results of the caustic mass analysis. The dots show the distribution of the galaxies in phase space, and the magenta lines show the caustic lines fitted to the black parabola. The dots outside the caustics are, by definition, beyond the turn-around radius of the cluster and have a velocity that is greater than the escape velocity. The results are summarized in Table \[tabClusterMasses\].
While there are more galaxies in the area around the Double Irony, the estimated mass of the Double Irony cluster is about three times smaller than that of the Exemplar. The caustic mass analysis shows that many galaxies fall outside the caustic line (see Fig. \[figCaustic\]), which indicates that they are not cluster members.
![Result of the caustic mass analysis for the Exemplar (top panel) and the Double Irony (bottom panel). The dots shows the positions of the galaxies in phase space ($R$ is the distance on the sky to the brightest galaxy, and $v_{\rm los}$ is the line-of-sight velocity relative to the mean line-of-sight velocity of the galaxies in the region. The magenta lines show the caustic lines. Galaxies outside the caustic lines have a $v_{\rm los}$ greater than the escape velocity of the cluster/group. []{data-label="figCaustic"}](fig10CausticExemplar.png "fig:"){width="8.8cm"} ![Result of the caustic mass analysis for the Exemplar (top panel) and the Double Irony (bottom panel). The dots shows the positions of the galaxies in phase space ($R$ is the distance on the sky to the brightest galaxy, and $v_{\rm los}$ is the line-of-sight velocity relative to the mean line-of-sight velocity of the galaxies in the region. The magenta lines show the caustic lines. Galaxies outside the caustic lines have a $v_{\rm los}$ greater than the escape velocity of the cluster/group. []{data-label="figCaustic"}](fig10CausticDbleIrony.png "fig:"){width="8.8cm"}
### Scaling relations
Now let us estimate the mass within $r_{200}$ using a simple scaling relation with the velocity dispersion within the same radius, $\sigma_{200}$. [@2004cgpc.symp....1E] showed that the following relation is an excellent fit to a wide range of simulated clusters over a wide range of redshifts, $$M_{200} = \frac{10^{15} h^{-1} M_\odot}{H(z)/H_0}
\left(
\frac{\sigma_{200}}{1080~{\rm km~s}^{-1}}
\right)^3 \, ,
\label{eqM200}$$ where $H(z)$ is the value of the Hubble parameter at the redshift $z$ of the cluster. This relation is very close to the mass–velocity dispersion relation for a singular isothermal sphere truncated at the virial radius. For such a model in hydrostatic equilibrium, there is a simple relation between mass and temperature (e.g., @2005RvMP...77..207V) $$k_B T_{200} = ({\rm 8.2~keV})
\left(
\frac{M_{200}}
{10^{15} h^{-1} M_\odot}
\right)^{2/3}
\left(
\frac{H(z)}{H_0}
\right)^{2/3} \, ,
\label{eqIsoSphereT200M200}$$ where $k_B$ is the Boltzmann constant and $T_{200}$ is the temperature within $r_{200}$.
$M_{200}$ and $M_{500}$ are related by the factor $$M_{500} = M_{200}/1.35
\label{eqM500}$$ for an appropriate value of the concentration parameter.
The mass estimates are given in Table \[tabClusterMasses\]. The caustic mass estimates are somewhat larger than the estimates from a single scaling relation with the velocity dispersion, and from the X-ray derived masses (an upper limit in the case of the Double Irony). Several galaxies in the field of the Double Irony fall outside the caustic line (e.g., the brightest neighbor on the sky has a relatively high velocity relative to the mean velocity of the cluster/group). Given the relatively small number of galaxies, the caustic method may not be reliable to assess cluster membership for individual galaxies. It is also difficult to attach a realistic uncertainty to the caustic mass estimate when the number of galaxies is small. For the GAMA group catalog (G3Cv1), [@2012MNRAS.426.2832A] showed that “on average, the caustic mass estimates agree with dynamical mass estimates within a factor of 2 in about 90% of the groups and compares equally well to velocity dispersion based mass estimates for both high and low multiplicity groups over the full range of masses probed by the G3Cv1.” [According to all indicators, the cluster/group of the Double Irony is less massive than that of the Exemplar.]{}
The temperature estimates derived from Eq. \[eqIsoSphereT200M200\] are 2.1 keV for the Exemplar and 0.7 keV for the Double Irony. These values are consistent with the X-ray estimates ($\approx 1.8$ keV for the Exemplar, Table \[tabTheOtherOne\], and $\lessapprox 1.3$ keV for the Double Irony, Table \[tabDbleIrony\]).
From the estimates of $M_{500}$ in Table \[tabClusterMasses\], we can estimate that the corresponding radius, $r_{500}$, for the Double Irony’s group is about 0.6 times that of the Exemplar’s cluster, or about 380 kpc, which corresponds to $2\farcm6$. A circle of that radius roughly encompasses the entire radio structure.
Exemplar Double Irony Note
----------------------------------- ---------------------------- --------------------- ------
$N (R < 6\farcm3)$ 46 55 (1)
$N (< 6\farcm3, 0.13 < z < 0.15)$ 23 32 (2)
$z_{\rm cluster}$ 0.1387 0.1368 (3)
$\sigma_v [{\rm km~s}^{-1}]$ 475 275 (4)
$M_{\rm Caustic} [M_\odot]$ $3.2\times10^{14}$ $9.1\times10^{13}$ (5)
$M_{\rm 200} [M_\odot]$ $1.7\times10^{14}$ $3.3\times10^{13}$ (6)
$M_{\rm 500} [M_\odot]$ $1.3\times10^{14}$ $2.5\times10^{13}$ (7)
$M_{\rm 500}^{MT} [M_\odot]$ $(0.9\pm0.4)\times10^{14}$ $<5.1\times10^{13}$ (8)
Cluster ages and dynamical states {#SectDiscussionClusterAges}
---------------------------------
The luminosity difference between the BCG and the second brightest galaxy in a cluster is used as an indicator of the evolutionary state of a cluster (e.g., ). In the hierarchical model of galaxy formation, BCGs located at the center of a cluster will grow by accretion and mergers faster than other galaxies. The difference in mass (and luminosity) is expected to increase with time, and galaxy groups/clusters with a large magnitude difference $\Delta m_{12}$ between their two brightest galaxies are expected to have assembled early. [@2014MNRAS.442.1578R] used galaxies drawn from semi-analytic models of [@2011MNRAS.413..101G] based on the Millennium simulations, and investigated how some measurable parameters are related to the age of the groups. They defined “young” galaxy groups as those that have assembled up to 30% of their mass by redshift $z = 1$, and “old” groups as the ones that had assembled more than 50% of their mass at redshift $z = 1$. Plotting the distributions of the magnitude gap $\Delta m_{12}$ versus the absolute magnitude in the $r$ band ($M_r$) of the BCG, they were able to quantify the fraction of old and young groups in different parts of the diagram.
For the Exemplar’s cluster (XLSSC 112), the difference in the $r$-band magnitude is 0.97, while for the BCG of the Double Irony group it is much lower, 0.35. The second brightest galaxy in the Double Irony environment is a late-type galaxy located near the Elbow radio structure (see Fig. \[figDbleIronyOptXrayradio\] and Sect. \[sectDiscussion2ndGal\]). The Exemplar has $M_r = -22.73$ and the Double Irony has $M_r = -22.85$. Placing them in the diagram of @2014MNRAS.442.1578R, we see that the Double Irony group has a higher probability of being a young galaxy group (box (3) with 63% probability of being young and 4% probability of being old). The Exemplar group, XLSSC 112, falls in box (7) with a 30% probability of being young and a 22% probability of being old. This means that the probability that the Exemplar’s cluster is older that the Double Irony’s group/cluster is 51%, while the probability that the Double Irony is older than the Exemplar is 13%; there is a 36% probability that they are in the same age group (young, old, or in between). While this is not conclusive in itself, it is consistent with the idea that the Double Irony’s group/cluster is not as evolved as XLSSC 112.
Comparing the surrounding galaxy distributions {#sectCompaEnvironment}
----------------------------------------------
{height="6.5cm"} {height="6.5cm"}
[lccccrcccc]{} Name & $R_{\rm max}$ & $N$ & $n$ & $\bar{N}_{\rm env}$ & $N/\bar{N}_{\rm env}$ & $A_2$ &$A_3$ &$A_4$ &$A_5$& (Mpc) & & (Mpc$^{-3}$) & Exemplar & 0.5 & 15 & 0.794 & 1.12 & 13.3 & ${\bf -0.337 \pm 0.014}$ & ${\bf -0.086 \pm 0.012}$ & $+0.048 \pm 0.012$ & ${\bf -0.250 \pm 0.015}$ Exemplar & 1.0 & 24 & 0.318 & 3.62 & 6.6 & ${\bf -0.183 \pm 0.006}$ & ${\bf -0.188 \pm 0.006}$ & $-0.017 \pm 0.006$ & $+0.041 \pm 0.007$ Double Irony & 0.5 & 19 & 1.006 & 0.75 & 25.3 & ${\bf +0.119 \pm 0.009}$ & ${\bf -0.178 \pm 0.009}$ & ${\bf +0.161 \pm 0.008}$ & $-0.007 \pm 0.010$ Double Irony & 1.0 & 38 & 0.503 & 3.38 & 11.3 & ${\bf +0.187 \pm 0.003}$ & ${\bf -0.173 \pm 0.003}$ & ${\bf +0.097 \pm 0.003}$ & ${\bf +0.261 \pm 0.003}$
{width="17.7cm"}
A number of studies have been carried out to search for a relation between the large-scale distribution of galaxies surrounding GRGs and the orientation of the radio lobes and plumes (e.g., ; @2008ApJ...677...63S; @2009MNRAS.393....2S; @2013MNRAS.434.2877T; @2015MNRAS.449..955M). There is increased evidence that overdensities of galaxies are found on the side of the shorter radio lobes, and that jets are oriented in directions perpendicular to the overall galaxy distribution surrounding the host galaxy; also, non-colinear radio jets and lobes seem to have directions perpendicular to the surrounding galaxy distribution, suggesting that they have been deflected [@2015MNRAS.449..955M]. XXL-North has been covered by the GAMA spectroscopic survey (G02 region in the GAMA Data Release 3; @2018MNRAS.474.3875B) that is 95.5% redshift complete to a magnitude $r < 19.8$ mag. In this section, we begin with a qualitative description of the distribution of galaxies located within a radius of 1 Mpc of the centers of the two radio galaxies and with a spectroscopic redshift from the GAMA database close to that of the radio galaxy (see Fig. \[figRadiogalsEnv\]). Then we carry out a more quantitative analysis in terms of Fourier components of the distribution of the galaxies surrounding the radio sources.
In Fig. \[figRadiogalsEnv\] we show the distribution of the galaxies with a spectroscopic redshift $z_{\rm RG} -\Delta z < z < z_{\rm RG} +\Delta z$, where $z_{\rm RG} = 0.138$ is the redshift of the radio galaxy and $\Delta z = 0.003$. The galaxies are color-coded in redshift bins of 0.001, increasing from blue to red. The overlaid contour corresponds to the 5$\sigma$ level (0.3 mJy beam$^{-1}$) in the 610 MHz images of the radio galaxies. The arrows point in the directions of the brightest regions in the lobes. The lines were extended to separate the regions above and below the radio axes.
In the Exemplar, the jets and lobes point in opposite directions, but the SW lobe is farther away from the core than the NE lobe. The red arrow points to the brightest region in the longer lobe and the blue arrow to the shorter one. We note the following:
– more galaxies are clearly situated above the Exemplar’s radio axis;
– an overdensity of galaxies are in the direction of the shorter lobe relative to the opposite direction;
– in the central region, the galaxy distribution is roughly perpendicular to the radio axis.
The Double Irony radio galaxy is not a linear structure. The red arrow points to the brightest region in the SW lobe and the blue arrow to the Elbow; these two regions are at about the same distance from the center of the host galaxy ($90\arcsec$), but have different position angles. We see:
– an overdensity of galaxies above the radio axis (at positive angles, measured counterclockwise).
– in the direction of the blue arrow, an overdensity of galaxies beyond the Elbow and to the east of the northern plume;
– an overdensity to the northwest of the SW plume.
To quantify the distribution of the galaxies around our two radio galaxies and those apparent features, let us use the Fourier component method proposed by [@2013MNRAS.434.2877T] and applied to a sample of GRGs by [@2015MNRAS.449..955M]. The method consists of estimating five components defined as $$A_1 = \sum\limits_{i=1}^{N} f_k (\theta_i) \, , \\$$ and $$A_k = \frac{1}{N} \sum\limits_{i=1}^{N} f_k (\theta_i) \,\,\,\,\, {\rm for}\,\, k = 2, ..., 5,$$ where the summation is over the number of galaxies, $N$, within a radius $R$ (taken to 0.5 or 1 Mpc) and a redshift $\Delta z = \pm 0.003$ of the position and redshift of the radio galaxy; the values of $\theta_i$ are the angles measured counterclockwise on the sky between the direction of galaxy $i$ and that of the longer radio lobe; the $f_k$ functions are $f_1 = 1$, $f_2 = \sin(\theta_i)$, $f_3 = \cos(\theta_i)$, $f_4 = \sin(2 \theta_i)$, $f_5 = \cos(2 \theta_i)$.
We use a slightly different notation from that of [@2015MNRAS.449..955M] ($A_k$ rather than $a_k$) to make it clear that our $A_k$ are normalized by dividing by the number of galaxies $N$ in the corresponding volume surrounding the radio galaxy, while [@2015MNRAS.449..955M] divided by the number of galaxies located farther out in the environment of the radio sources. The absolute values of their $a_k$ are therefore a mixture of information related to the actual local anisotropy of the galaxy distribution and the overdensity with respect to the more distant environment. The difference is therefore simply a normalization factor (which is different for every radio galaxy) and we find the use of $A_k$ more intuitive because they are actual mean quantities.
The $A_k$ components are illustrated in Fig. \[figSketchAnisotropies\] and can be understood as follows:
1. $A_1 = N$ is simply the number of galaxies located within the pre-defined radius from the core of the radio galaxy.
2. $A_2$ and $A_3$ are sensitive to a dipolar distribution of galaxies. The $A_3$ parameter (average of the cosines) is sensitive to asymmetries in the density distribution of galaxies along the radio axis (e.g., a positive $A_3$ indicates an overdensity of galaxies along the longer radio lobe compared to the opposite direction; a negative $A_3$ indicates an overdensity along the shorter radio lobe). The $A_2$ parameter (average of the sines) expresses asymmetries between the two sides of the radio axis (i.e., in the direction perpendicular to that of the radio axis, above or below the radio axis);
3. $A_4$ and $A_5$ are sensitive to a quadrupolar distribution. A positive $A_5$ (average of $\cos(2\theta_i)$) corresponds to overdensities of galaxies along the radio axis compared to the direction perpendicular to the radio axis; a negative $A_5$ will indicate an overdensity above and below the radio axis compared to along the radio axis. $A_4$ (average of $\sin(2\theta_i)$ also indicates a quadrupolar anisotropy, but at a $45^\circ$ angle compared to the previous distribution (see Fig. \[figSketchAnisotropies\]).
In the case of a non-uniform distribution of surrounding galaxies, all $A_k$ components will differ significantly from zero.
The quoted uncertainties on the $A_k$ Fourier coefficients ($k\geq 2)$ are standard errors and were calculated using jackknives: the $A_k$ coefficients were recalculated by removing one galaxy at a time; then the standard error on the $A_k$ values was calculated.
To estimate the overdensity of galaxies around the radio galaxy, we defined a $3\times3$ grid centered on the radio galaxy host at (RA$_{\rm RG}$,DEC$_{\rm RG}$) with eight points placed at RA = RA$_{\rm RG}\pm \Delta\theta$ and DEC = DEC$_{\rm RG}\pm\Delta\theta$, with $\theta = 0.5^\circ$. Then we calculated the mean number of galaxies in eight 24 Mpc$^3 $ volumes (radius of 1 Mpc and $z = z_{RG}\pm 0.003$), $\bar{N}_{\rm env}$. The overdensity is $N/N_{\rm env}$, where $N$ is the number of galaxies within the same volume centered on the radio galaxy (in the middle of the grid).
The distribution of galaxies with spectroscopic redshifts from the GAMA database close to that of the two radio galaxies is displayed in Fig. \[figRadiogalsEnv\]. The Fourier analysis was applied to this data set and the resulting $A_k$ components are listed in Table \[tabFourierComps\]. The main results can be summarized as follows:
{width="9.0cm"} {width="8.6cm"}
– $N$ ($= A_1$) and $N/\bar{N}_{env}$: There are significantly more galaxies within a radius of 1 Mpc of the host of the Double Irony than around the Exemplar. The overdensity compared to the local environment is almost twice as large for the Double Irony as for the Exemplar. Given such a large number of galaxies in the Double Irony, the non-detection of X-ray emitting gas may seem surprising. Could it be that the galaxies around the Double Irony are less massive than in and around the Exemplar’s cluster? To answer this question we retrieved the $ugriz$ magnitudes (model magnitudes from SDSS) of the galaxies with spectroscopic redshifts from GAMA. Figure \[figColorDiagSDSS\] (right panel) is a color-magnitude diagram for both regions. Error bars are shown only for the two BCGs so as not to clutter the figure (the error bars on the $z$-magnitudes are smaller than the widths of the points). The distributions of galaxies in the radio galaxies’ environment are similar: most galaxies are red, with a $(g-r)$ color around 1.05. On the $x$-axis we show the $z$-band magnitude which can be used as a proxy for the mass. The two BCGs stand out in the upper left corner of the plot as the brightest (and likely most massive galaxies), with a $z$-band magnitude difference of at least one compared to the other galaxies, except for the second brightest galaxy (marked as a red star) discussed in Sect. \[sectDiscussion2ndGal\] and located about 1$'$ to the NE of the Double Irony host; we note that this galaxy was not included in the Fourier component analysis because its redshift falls just below the selected redshift interval of $0.138\pm0.003$. [The distribution of galaxies is more concentrated around the Exemplar than around the Double Irony]{} (Fig. \[figRadiogalsEnv\]; left panel of Fig. \[figColorDiagSDSS\]).
– Dipole ($A_2$ and $A_3$): $A_2 >0$ for the Exemplar and $<0$ for the Double Irony. In both cases this means that there are more galaxies above the radio axes (see Fig. \[figSketchAnisotropies\]; we note that for the Examplar the origin of the angles is taken as the longer lobe (red arrow), directed toward the SE, so the positive $A_2$ are under the radio axis; this is the opposite for the Double Irony where the red arrow points toward the SW). There is also an anisotropy in the other direction ($A_3$).
– Quadrupole ($A_4$ and $A_5$): Most interesting are the negative $A_5$ values for the Exemplar that reflect overdensities in the direction perpendicular to the radio axis, as noted through visual examination. This anisotropy is considerably stronger within a radius of 0.5 Mpc. For the Double Irony, the effect is the opposite: the anisotropy corresponds to overdensities in the direction of the radio axis that corresponds to the orange arrow). Interestingly, the jets seems to be deflected in both directions and redirected into regions of lower galaxy densities (see right panel of Fig. \[figRadiogalsEnv\]).
It seems that the Double Irony radio galaxy has been able to expand and become a GRG in an environment that may not be significantly less dense in galaxies than that of the Exemplar, and has been strongly affected by this environment: the jets deviated from their paths and the lobes expanded away from regions of higher galaxy density into sparser regions.
{width="8.8cm"} {width="8.8cm"}
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Large-scale structure at [*z*]{} = 0.14 {#SectSuperclu}
=======================================
In Fig. \[figSuperclu\] we show the distribution of the galaxies with a spectroscopic redshift from the GAMA database between 0.135 and 0.141 (top panel) and the GMRT 610 MHz image (bottom panel) of the region of XXL-North that encompasses the supercluster XLSSsC N03 identified in . The supercluster contains eight cluster members, five of which are securely detected clusters in X-rays (C1 class). The cluster hosting the Exemplar, XLSSC 112, is a secure X-ray detection and its position is circled in red. There is an interesting alignment on the sky of four clusters, with a decrease in size and mass from the more massive Abell cluster (XLSSC 60) to the other clusters on the right. This large structure might lead to the formation of a massive cluster if infall and merger occur.
The Double Irony radio galaxy is seen in the top right corner of Fig. \[figSuperclu\] (left panel). The redshift of its surrounding cluster/group is very similar to the redshifts of the members of the supercluster ($z \simeq 0.14$). We performed a new friends-of-friends analysis with exactly the same parameters as those used in the study presented in , which used 326 clusters with $0.03 < z < 1.0$, but adding the Double Irony’s cluster to the list of clusters. Very interestingly, [*the Double Irony’s cluster was identified as a new member (the ninth) of the supercluster XLSSsC N0*]{}.
We note an overdensity of galaxies at RA$\simeq 32\fdg445$, DEC$\simeq -4\fdg38$ (upper panel) where there is no reported XXL cluster. When this structure is added to the list of identified clusters/groups, [the friends-of-friends algorithm identifies it as a new member (the tenth) of the same supercluster.]{} [Even at this location, a radio source is found in the GMRT 610 MHz mosaic;]{} we call it “the Unhinged” because of its bizarre appearance (see Fig. \[figMoreRadiogals\]).
Radio galaxies in superclusters {#SectDiscussionRadiogalsSuperclu}
===============================
We note that XLSSC 201, another cluster member of the supercluster, also hosts an extended radio source, shown in Fig. \[figMoreRadiogals\] and denoted the “Eyebrow.” It is remarkable that four out of ten clusters/groups in this structure contain well-developed radio galaxies. The Eyebrow and the Unhinged may be the superposition of radio sources at different redshifts and deserve a separate detailed study.
More work is clearly needed to quantify the incidence of radio galaxies in clusters residing inside superclusters. A number of superclusters have been identified in XXL (21 in XXL-North and 14 in XXL-South; ). [@guglielmo2018b] characterized the stellar populations of the galaxies in the richest XXL supercluster (at $z\simeq 0.3$) and found evidence for an active role of the environment on the star formation rates and stellar masses. detected a number of X-ray AGN in three XXL superclusters, but a larger study is needed to reach firm conclusions on the frequency of AGN in superclusters. The first supercluster detected in XXL is at $z = 0.43$ (; ). Its galaxy and AGN populations have been studied in the optical and in the radio, but no large radio galaxies were found within the supercluster’s overdensities (@baran16; ). The GMRT-XXL-N 610 MHz survey () can be used to search for radio galaxies in superclusters in XXL-North; XXL-South has been covered with the Australia Telescope Compact Array (@butler2018a; , @butler2018b; ).
In the hierarchical model of structure formation, structures grow by accretion and mergers, small structures forming first. Superclusters of X-ray-detected clusters/groups are large structures that may still be collapsing. Both the AGN of a radio galaxy and its host galaxy (usually a massive elliptical) have accreted significant amounts of material, and the jets and lobes trace the relatively recent (a few $10^7$ years) ejection of plasma. Because of their sizes, radio galaxies are easily seen and might be a powerful way to find new clusters/groups that have escaped detection in X-ray surveys (e.g., @2016MNRAS.460.2376B, from the Radio Galaxy Zoo).
Summary and conclusion {#SectSummary}
======================
We have presented an analysis of two radio galaxies at $z\sim 0.14$ identified in the GMRT-XXL-N 610 MHz survey of XXL-North. We have made use of the extensive multiwavelength coverage of XXL to gain a better understanding of those sources. We were able to identify their host galaxies, and in the case of the Exemplar the host cluster that had been detected in the X-ray and cataloged as an XXL cluster. The second and more spectacular source, the Double Irony, is a giant radio galaxy with a linear size of about 1100 kpc on the sky. We were able to show that it is hosted by the BCG of a lower mass cluster with member galaxies detected in the optical and listed in the GAMA database. Using a friends-of-friends algorithm, we were able to show that both clusters are part of the same supercluster. We identified another overdensity of galaxies in the supercluster that is associated with a radio galaxy of peculiar shape. This brings the total number of members of this supercluster to ten. Four of these clusters/group host a radio galaxy. Anisotropies were found in the distribution of the surrounding galaxies, possibly indicating that the jets and lobes/plumes reside in lower density regions. This shows the potential of the XXL survey to lead to the discovery of new structures, and the use of radio galaxies as tracers of large-scale structure.
Future work on the two clusters presented here will require more sensitive, higher angular resolution radio and X-ray data. In particular, the JVLA (including polarization) at GHz frequencies and the GMRT below 500 MHz will allow a detailed modeling of the aging of the cosmic-ray electrons in the radio jets and lobes and will constrain timescales. In the X-ray, a deeper XMM-[*Newton*]{} observation of the Double Irony is necessary to detect the hot gas that is likely to be associated with the optically detected group; higher resolution images with [*Chandra*]{} may reveal the structure of the gaseous atmosphere around the radio galaxies and possibly shocks.
The XXL fields contain other remarkable radio galaxies that will be analyzed following an approach similar to the one presented here. Special attention will be given to their potential relation to clusters and superclusters.
XXL is an international project based on an XMM Very Large Programme surveying two 25 deg$^{2}$ extragalactic fields at a depth of $\sim5\times10^{-15}$ erg cm$^{-2}$ s$^{-1}$ in the [\[]{}0.5–2[\]]{} keV band for point-like sources. The XXL website is [<http://irfu.cea.fr/xxl>]{}. Multiband information and spectroscopic follow-up of the X-ray sources are obtained through a number of survey programs, summarized at [<http://xxlmultiwave.pbworks.com/>]{}.
We thank the staff of the GMRT who made these observations possible. The GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research.
This work is based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA.
It is also based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/IRFU, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at Terapix available at the Canadian Astronomy Data Centre as part of the Canada-France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS.
This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
This research has made use of the VizieR catalog access tool, CDS, Strasbourg, France. The original description of the VizieR service was published by . We have also made use of the table analysis software [topcat]{} [@topcat] and of the caustic mass estimation algorithm written by [@2012MNRAS.426.2832A]. This research made use of Astropy, a community-developed core Python package for Astronomy [@astropy].
This research also made use of the Matplotlib plotting library [@matplotlib].
GAMA is a joint European-Australasian project based around a spectroscopic campaign using the Anglo-Australian Telescope. The GAMA input catalog is based on data taken from the Sloan Digital Sky Survey and the UKIRT Infrared Deep Sky Survey. Complementary imaging of the GAMA regions is being obtained by a number of independent survey programs including GALEX MIS, VST KiDS, VISTA VIKING, WISE, Herschel-ATLAS, GMRT, and ASKAP providing UV to radio coverage. GAMA is funded by the STFC (UK), the ARC (Australia), the AAO, and the participating institutions. The GAMA website is [<http://www.gama-survey.org/>]{}.
This research has made use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
VS acknowledges support from the European Union’s Seventh Frame-work program under grant agreement 337595 (ERC Starting Grant, “CoSMass”). MERC acknowledges support from the German Aerospace Agency (DLR) with funds from the Ministry of Economy and Technology (BMWi) through grant 50 OR 1514. The Saclay group acknowledges long-term support from the Centre National d’Etudes Spatiales (CNES). SF acknowledges financial support from the Swiss National Science Foundation. CH thanks Michael Olberg for his help with the $R$ software and John H. Black for useful comments and moral support. We thank the referee for pointing out relevant references and for other constructive comments.
Mid-infrared photometry and optical spectra of the host galaxies
================================================================
Table \[tabWISEphot\] contains the WISE photometry and Table \[tabOptNIRphot\] the optical and near-IR photometry. The optical spectra of the host galaxies are shown in Fig. \[figGAMAspectra\].
[lcccccc]{} Source &W1 &W2 &W3 &W1-W2 &W2-W3 Exemplar &12.93 &12.79 &12.04 &0.14 &0.75Double Irony &13.10 &12.95 &11.56 &0.15 &1.39Difference &$-0.17$ &$-0.16$ & 0.48 &$-0.01$ &$-0.64$
[ccccccccc]{} Source &$u$ &$g$ &$r$ &$i$ &$z$ &$J$ &$H$ &$K_s$ Exemplar & $19.59\pm0.08$ &$17.43\pm0.01$ &$16.37\pm0.00$ &$15.90\pm0.00$ &$15.56\pm0.01$ &15.5 &14.742 &14.176Double Irony &$19.26\pm0.06$ &$17.35\pm0.91$ &16.26 &15.82 &$15.47\pm0.01$ &15.223 &14.634 &13.987Mag. difference & 0.33 &0.08 & 0.11 &0.08 &0.09 &0.28 &0.11 &0.189
{width="17.7cm"} {width="17.7cm"}
[^1]: <http://xxlmultiwave.pbworks.com>
[^2]: Throughout this paper and for consistency with the first XXL series of papers, we use the WMAP9 cosmology ($\Omega_{\rm m} = 0.28$, $\Omega_\Lambda = 0.72$, $H_0 = 70$ km s$^{-1}$ Mpc$^{-1}$; @2013ApJS..208...19H). At the redshift of the radio galaxies discussed here ($z \simeq 0.138$), this gives a scale of 2.443 kpc/$\arcsec$ and a luminosity distance of 652.5 Mpc.
[^3]: The percontation point (a reversed question mark, ?) was invented by the English printer Henry Denham at the end of the sixteenth century to mark the end of a rhetorical question and was later used to denote irony.
[^4]: <http://www.cv.nrao.edu/nvss>
[^5]: <http://www.aips.nrao.edu>
[^6]: [ https://en.wikipedia.org/wiki/Sobel\_operator]( https://en.wikipedia.org/wiki/Sobel_operator), page last edited on 22 February 2018, at 14:58
[^7]: <http://skyserver.sdss.org/dr14>
[^8]: <http://www.gama-survey.org>
[^9]: TGSS J021006-052841, which corresponds to the SE lobe, is listed as an isolated single-Gaussian source with a flux density of $57.2\pm7.0$ mJy; TGSS J021002-052804, which corresponds to the core and the inner jets, has $90.5\pm9.9$ mJy; TGSS J020959-052738, which corresponds to the NW lobe, has $73.3\pm 3.3$ mJy. The second and third sources overlap.
[^10]: NVSS J021007-052842 has a total flux density of $15.3\pm1.0$ mJy. NVSS J021001-052758 has a total flux density $58.8\pm 2.2$ mJy,
[^11]: The three sources listed as resolved within a radius of 4$'$ from the core of the Exemplar are XXL-GMRT J021006.7-052840, XXL-GMRT J021004.3-052826, and XXL-GMRT J021000.9-052753. The two compact sources that we named S1 and S2 are XXL-GMRT J021009.1-052834 and XXL-GMRT J021003.7-052751, respectively.
[^12]: FIRST J021001.9-052800 with a peak flux density of 4.15 mJy beam$^{-1}$ (integrated flux density $S_{\rm T} = 5.53$ mJy) and FIRST J021002.4-052806 with a peak flux density of 6.32 mJy beam$^{-1}$ ($S_{\rm T} = 10.19$ mJy). From the peak values this gives a spectral index of $-0.90$ for the fainter NW peak and $-0.53$ for the SE peak between 610 MHz and 1.4 GHz.
[^13]: Flux loss in the FIRST 1.4 GHz image would produce a similar effect. The largest angular scale to which the FIRST observations are sensitive is 2$'$ and the radio galaxy stretches over $\sim$$2\farcm5$. On the other hand, the lobes themselves are smaller than 1$'$ across. For the SE lobe, the flux density measured in the FIRST image within the elliptical region displayed in Fig. \[figSarcasmAllradioimages\] is lower than that measured in the NVSS image (about 8.2 vs. 13 mJy, see Table \[tabExemplarFlux\]), an indication that some of the extended emission might have been filtered out. For the NW lobe on the other hand, the FIRST flux is higher than the NVSS flux, perhaps because the region (within a circle of 50$\arcsec$ diameter) does not include all the flux of the 45$\arcsec$ resolution NVSS image.
[^14]: WENSS,
[^15]: NVSS, [@NVSS]
[^16]: FIRST, [@first95], [@first15]
[^17]: SUMSS, [@1999AJ....117.1578B]
[^18]: The LOFAR Multifrequency Snapshot Sky Survey,
[^19]: The LOFAR Two-metre Sky Survey,
[^20]: The Double Irony lies in the northwestern corner of the GMRT 610 MHz mosaic at the intersection of two pointings where the noise is slightly higher than average (see Fig. 5 of ).
[^21]: Seven sources are listed as resolved in the XXL-GMRT-610 MHz catalog (@smolcic2018) within 4$'$ of the core of the Double Irony: XXL-GMRT J020354.8-041356, XXL-GMRT J020353.0-041358, XXL-GMRT J020357.6-041244, XXL-GMRT J020359.1-041303, XXL-GMRT J020348.5-041416, XXL-GMRT J020356.1-041216, and XXL-GMRT J020344.7-041505.
[^22]: Four continuum sources associated with the Double Irony are listed in the NVSS catalog [@NVSS]; all of them are listed as polarized, with polarized flux densities ranging from 1.71 to 5.95 mJy. None of these sources appears in the NVSS rotation measure (RM) catalog of [@2009ApJ...702.1230T], which had a selection threshold of 5 mJy on the Stokes $I$ intensity and of 8$\sigma$ on the polarized intensity.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
The addition of the two Forward TPCs to the STAR detector allows one to measure anisotropic flow at forward pseudorapidities. This made possible the first measurement of directed flow at collision energies of $\sqrtsNN =
200$GeV. PHOBOS’ results on elliptic flow at forward rapidities were confirmed, and the sign of $v_2$ was determined to be positive for the first time at RHIC energies. The higher harmonic, $v_4$, is consistent with the recently suggested $v_2^2$ scaling behavior.
This write-up contains results presented as a poster [@poster] at the Quark Matter conference in Oakland, California in January 2004.
author:
- 'Markus D. Oldenburg'
date: 'March 17, 2004'
title: 'Anisotropic flow in the forward directions at $\sqrt{s_\mathrm{NN}} = 200$GeV'
---
\[intro\]Introduction
=====================
In non-central heavy-ion collisions the initial spatial anisotropy of the collision region translates into a final state anisotropy in momentum space. In a hydrodynamical picture this is believed to be due to pressure gradients in the dense medium which lead to collective motion — so called transverse flow — of the generated particles.
The simplest way of characterizing these final state anisotropies is to perform a Fourier decomposition on the particle’s emission angles $\phi$ with respect to the reaction plane $\Psi_{RP}$ [@vol]. The reaction plane is given by the incident beam direction and the impact parameter and it is experimentally not known *a priori*. It has to be estimated for every event by looking at the anisotropy of particle emission itself [@fourier]. This leads to a finite resolution of the measured event plane which one has to correct for.
Spurious contributions to the measured transverse flow signal are particle correlations due to non-flow effects (e.g. resonance decays). To cope with these, several new methods of the anisotopic flow analysis, based either on cumulants [@BDO1; @BDO2] or on Lee-Yang zeros [@LeeYangZeros], have been proposed.
\[expsetup\]Experimental setup
==============================
The two Forward TPCs (FTPCs [@Ftpc]) of the STAR experiment [@STAR] extend the pseudorapidity coverage of STAR into the region $2.5<|\eta|<4.0$. The pseudorapidity resolution of these radial drift chambers is better than 5% for their full acceptance. During RHIC run 2 about 70 thousand Au+Au collisions at a center of mass energy of $\sqrtsNN = 200$GeV were taken with both FTPCs and the STAR TPC [@Tpc].
Measurements
============
Directed Flow $v_1$
-------------------
The first measurement of directed flow at RHIC energies was recently published [@v1v4Paper] (see Fig. \[v1\]). It showed that while $v_1(\eta)$ is close to zero at mid-rapidities, the signal rises to a couple of percent near pseudorapidity $|\eta| \approx 4$.
It was noted that our measurement greatly differs from the NA49 results [@NA49Paper] at lower beam energies of $158A$GeV. But if the NA49 data are shifted and both measurements are seen in the projectile frame, they look similar.
Elliptic flow $v_2$
-------------------
The comparison of our new measurement on elliptic flow $v_2(\eta)$ at forward pseudorapidities confirms the published result [@Phobosv2] obtained by the PHOBOS collaboration (see Fig. \[v2phobos\]) at high $\eta$. We observe a similar fall-off by a factor of 1.8 comparing $v_2(\eta = 0)$ with $v_2(\eta
=3)$. Both measurements were done using the event plane method.
If we compare our results for $v_2$ obtained with the method of two-particle cumulants, $v_2\{2\}$, to the four-particle cumulants, $v_2\{4\}$, we observe almost no difference in the FTPC region, while the two-particle cumulant measurement gives a 15% higher signal in the TPC. Since four-particle cumulants are much less prone to non-flow contributions we conclude that non-flow effects are less strong in the forward regions.
A new method to measure directed flow
-------------------------------------
From the above measurements it became clear that the STAR TPC sitting at mid-rapidity has very good capabilities to measure elliptic flow, while the Forward TPCs allow to measure directed flow (which appears to be close to zero at mid-rapidities).
### $v_1\left\{\mathrm{EP}1,\mathrm{EP}2\right\}$
In order to utilize the method of Fourier decomposition but to reduce non-flow contributions at the same time, we measured $v_1$ with respect to the first and second order reaction plane $\Psi_1$ and $\Psi_2$, where $\Psi_1$ was determined in the FTPCs while $\Psi_2$ was measured in the TPC. Within the recently proposed notation (see [@v1v4Paper]) we denote this measurement as $v_1\left\{\mathrm{EP}1,\mathrm{EP}2\right\}$. $$\begin{aligned}
v_1\{\mathrm{EP}1,\mathrm{EP}2\} =\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\
\frac{\left\langle\cos\left(\phi+\Psi_1^{\mathrm{FTPC}}-2\Psi_2^{\mathrm{TPC}}\right)\right\rangle}{\sqrt{\left\langle\cos\left(\Psi_1^{\mathrm{FTPC}_1}+\Psi_1^{\mathrm{FTPC}_2}-2\Psi_2^{\mathrm{TPC}}\right)\right\rangle\cdot \mathrm{Res}(\Psi_2^{\mathrm{TPC}})}}\nonumber\;.\end{aligned}$$
As shown in Fig. \[v1Ep1Ep2\], the results are in reasonable agreement with the published measurement obtained by the three-particle cumulant method $v_1\{3\}$.
### The sign of $v_2$
This new method provides an elegant tool to measure the sign of $v_2$, which was assumed to be positive but had not yet been determined at RHIC energies. One of the quantities involved in the measurement of $v_1\{\mathrm{EP}1,\mathrm{EP}2\}$ is approximately equal to the product of integrated values of $v_1^2$ and $v_2$: $$\begin{aligned}
v_1^2\cdot v_2 \approx
\frac{\left\langle\cos\left(\Psi_1^{\mathrm{FTPC}_1}+\Psi_1^{\mathrm{FTPC}_2}-2\Psi_2^{\mathrm{TPC}}\right)\right\rangle}{\sqrt{M_{\mathrm{FTPC}_1}\cdot
M_{\mathrm{FTPC}_2}\cdot M_{\mathrm{TPC}}}}\;,\end{aligned}$$ where $M_{\mathrm{FTPC}_1}$, $M_{\mathrm{FTPC}_2}$, and $M_{\mathrm{TPC}}$ denote the multiplicities for a given centrality bin in the two FTPCs and the TPC, respectively. Since $v_1^2$ is always positive, the sign of $v_1^2\cdot
v_2$ determines the sign of $v_2$.
Averaged over centralities 20–60% we measure $v_1^2\cdot v_2$ in Fig. \[v1v1v2\] to be $(1.08\pm0.46) \cdot10^{-5}$. In this region the expected non-flow contributions are much smaller than for the most central and peripheral centrality bins. Therefore the sign of $v_2$ is determined to be positive: [*In-plane*]{} elliptic flow is confirmed. (This stated value for $v_1^2\cdot v_2$ and its uncertainty is based on an approximation that does not affect the statistical significance of the conclusion that $v_2$ is [*in-plane*]{}.)
The fourth harmonic $v_4$
-------------------------
Since elliptic flow $v_2$ is strong, the second order reaction plane $\Psi_2$ can be estimated with high precision at RHIC energies. This makes the study of higher order flow feasible [@v1v4Paper].
The fourth harmonic $v_4$ shows an average value of $(0.4\pm0.1)\%$ in pseudorapidity coverage of the TPC ($|\eta| < 1.2$), see Fig. \[v2v4\]. In contrast, its value of $(0.06\pm0.07)\%$ in the forward regions is consistent with zero and we place a $2\sigma$ upper limit of 0.2%. Therefore the fall-off of $v_4$ from mid-rapidities to forward rapidities appears to be stronger than for $v_2$. This behavior is consistent with scaling like $v_4\sim v_2^2$.
Future developments
===================
First attempts to make use of the newly proposed method [@LeeYangZeros] utilizing Lee-Yang zeros are encouraging. This method eliminates higher order non-flow contributions by construction. It is mathematically equivalent to the *all*-particle cumulant method $v\{\infty\}$ which takes into account all higher order non-flow effects. The great advantage of the new method is its simplicity and speed compared to the evaluation of the cumulants.
The upcoming RHIC run 4 will greatly enhance our data sample. With it we will reduce our statistical uncertainties in the forward pseudorapiditiy region significantly.
[99]{}
see http://www.star.bnl.gov/STAR/central/\
presentations/2004/qm2004/Oldenburg\_Markus.pdf
S.A. Voloshin and Y. Zhang, Z. Phys. C [**70**]{}, 665 (1996).
A.M. Poskanzer and S.A. Voloshin, , 1671 (1998).
N. Borghini, P.M. Dinh, and J.-Y. Ollitrault, , 054901 (2001).
N. Borghini, P.M. Dinh, and J.-Y. Ollitrault, , 014905 (2002).
R.S. Bhalerao, N. Borghini, J.-Y. Ollitrault, Nucl. Phys. A [**727**]{}, 373 (2003).
K.H. Ackermann , Nucl. Instrum. Meth. A [**499**]{}, 713 (2003).
K.H. Ackermann , Nucl. Instrum. Meth. A [**499**]{}, 624 (2003).
M. Anderson , Nucl. Instrum. Meth. A [**499**]{}, 659 (2003).
J. Adams , , 062301 (2004).
NA49 Collaboration, C. Alt , , 034903 (2003).
S. Manly for the PHOBOS Collaboration, Nucl. Phys. A [**715**]{}, 611c (2003).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We prove the Gromov non-hyperbolicity with respect to the Kobayashi distance for $\mathcal{C}^{1,1}$-smooth convex domains in $\mathbb{C}^{2}$ which contain an analytic disc in the boundary or have a point of infinite type with rotation symmetry. The same is shown for “generic” product spaces, as well as for the symmetrized polydisc and the tetrablock. On the other hand, examples of smooth, non-pseudoconvex, Gromov hyperbolic domains in $\Bbb C^n$ are given.'
address:
- |
Institute of Mathematics and Informatics\
Bulgarian Academy of Sciences\
Acad. G. Bonchev 8, 1113 Sofia, BulgariaFaculty of Information Sciences\
State University of Library Studies and Information Technologies\
Shipchenski prohod 69A, 1574 Sofia, Bulgaria
- |
Université de Toulouse\
UPS, INSA, UT1, UTM\
Institut de Mathématiques de Toulouse\
F-31062 Toulouse, France
- |
Institute of Mathematics, Faculty of Mathematics and Computer Science\
Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
author:
- Nikolai Nikolov
- 'Pascal J. Thomas'
- Maria Trybuła
title: 'Gromov (non-)hyperbolicity of certain domains in $\mathbb{C}^{n}$'
---
Introduction and statements
===========================
In [@Gromov], Gromov introduced the notion of almost hyperbolic space. He discovered that “negatively curved” space equipped with some distance share many properties with the prototype, even though the distance does not come from a Riemannian metric. This gave the impulse to intensive research to find new interesting classes of spaces which are hyperbolic in that sense. In this paper we are mainly interested in investigating this concept with respect to the Kobayashi distance of convex domains. One may suspect that it is a restriction to consider only the Kobayashi metric. Actually, because the Kobayashi distance of a ($\Bbb C$-)convex domain containing no complex lines, as well as of a bounded strictly pseudoconvex domain, is bilipschitz equivalent to the (inner) Carathéodory and Bergman distances (see [@NPZ Theorem 12] and [@Nikolov Proposition 4]), it does not matter which one we choose (see below). Recall that a set $E$ in $\Bbb C^n$ is called *$\Bbb C$-convex* if any intersection of $E$ with a complex line $l$ and its complement in $l$ are both connected in $l$ (cf. [@APS]).
The notion of a bilipschitz equivalence has the following generalization.
Let $(X_1,d_1)$ and $(X_2,d_2)$ be two metric spaces. Then a map $\varphi:X_1\to X_2$ is said to be a *quasi-isometry* if there are constants $c_1,c_2 >0$ such that for any $x,y\in X_1$, $$c_1^{-1} d_1(x,y) - c_2 \le d_2\left( \varphi(x), \varphi(y) \right) \le c_1 d_1(x,y) + c_2.$$ Two distances $d_1,d_2$ on a set $X$ are said to be quasi-isometrically equivalent if the identity map is a quasi-isometry from $(X,d_1)$ to $(X,d_2)$.
Gromov hyperbolicity is well-known to be invariant under bijective quasi-isometries of path metric spaces (cf. [@Jesus Theorems 3.18, 3.20]).
\[defghyp\] Let $(D,d)$ be a metric space. Given points $x,y,z\in D,$ the *Gromov product* is $$(x,y)_{z}=d(x,z)+d(z,y)-d(x,y).$$ Let $$S_d(p,q,x,w)= \min\{(p,x)_{w},(x,q)_{w}\}-(p,q)_{w}.$$ $(D,d)$ is *Gromov hyperbolic* if $$\sup_{p,q,x,w\in D} S_d(p,q,x,w) <\infty.$$ If $S_d(p,q,x,w)\le 2\delta$, then $(D,d)$ is called *$\delta$-hyperbolic*.
We refer to [@Jesus] for other characterizations of Gromov hyperbolicity, especially for path metric spaces. We chose this one because it does not use geodesics explicitly.
$(D,d)$ is a *path metric space* if, for any two points $x,y\in D$ and any number $\varepsilon >0,$ there exists a rectifiable path joining $x$ and $y$ with length at most $d(x,y)+\varepsilon.$ Then the distance $d$ is called *intrinsic*.
From now on, let $D$ be a domain in $\mathbb{C}^{n}$.
Denote by $c_D$ and $l_{D}$ the Carathéodory distance and the Lempert function of $D$: $$c_{D}(z,w)=\sup\{\tanh^{-1}|f(w)|:f\in\mathcal{O}(D,\Bbb D), f(z)=0\},$$ $$l_{D}(z,w)=\inf\{\tanh^{-1}|\alpha|:\exists\varphi\in\mathcal{O}(\mathbb{D},D)
\hbox{ with }\varphi(0)=z,\varphi(\alpha)=w\},$$ where $\mathbb{D}$ is the unit disc. The Kobayashi distance $k_{D}$ is the largest pseudodistance not exceeding $l_{D}.$ The inner Carathéodory distance $c_D^i$ is the inner pseudodistance associated to $c_D.$ So, $c_D\le c_D^i\le k_D\le l_D.$ By Lempert’s seminal paper [@Lempert], we have equalities above if $D$ is convex (or bounded, $\mathcal{C}^2$-smooth and $\Bbb C$-convex).
An important property of $k_{D}$ is that it is the integrated form of the Kobayashi metric $\kappa_{D}$ of $D,$ i.e. $$\begin{gathered}
k_{D}(z,w)=\inf\{\int_0^1\kappa_{D}(\gamma (t);\gamma^{\prime}(t))dt:\\
\gamma:[0,1]\to D\textup{ is a smooth curve with }\gamma(0)=z\textup{ and }\gamma(1)=w\},\end{gathered}$$ where $$\kappa_{D}(z;X)=\inf\{|\alpha|:\exists\varphi\in\mathcal{O}(\mathbb{D},D)
\hbox{ with }\varphi(0)=z,\ \alpha\varphi'(0)=X\},$$ $z,w\in D,\ X\in\mathbb{C}^{n}.$
We refer to [@Jarnicki] for basic properties of the invariants defined here and of the Bergman distance $b_D.$
We shall say that $D$ is Gromov *$s$-hyperbolic* if $(D,s_D)$ is Gromov hyperbolic with respect to the distance $s$ (this should not be confused with $\delta$-hyperbolicity for some constant $\delta>0$).
The first result concerning Gromov $k$-hyperbolicity for domains in ${\Bbb C}^n$ was given by Balogh and Bonk [@Bonk] who gave both positive and negative examples. They proved that any bounded strictly pseudoconvex domain is Gromov $k$-hyperbolic [@Bonk Theorem 1.4]. They also showed that the Cartesian product of bounded strictly pseudoconvex domains is not Gromov $k$-hyperbolic [@Bonk Proposition 5.6] which is a special case of a general situation mentioned in many places, but without proof (cf. [@Gaussier]).
\[I\] Assume that $(X_1,d_1)$ is a path metric space with $d_1$ unbounded and $(X_2,d_2)$ a metric space with unbounded $d_2$. Let $d=\max\{d_1,d_2\}$. Then $(X_1\times X_2,d)$ is not Gromov hyperbolic.
\[I\]
The next proposition is more general than the previous one. However its proof uses Proposition \[I\].
\[II\] Let $(X_1,d_1)$ and $(X_2,d_2)$ be metric spaces, such that one of them is a path metric space. Let $d=\max\{d_1,d_2\}$. Then $(X_1\times X_2,d)$ is Gromov hyperbolic if and only if one of the factors is Gromov hyperbolic and the metric of the second one is bounded (in particular, it is also Gromov hyperbolic).
Moreover, the proof of Proposition \[I\] and Remark 1 (following this proof) show that the path property in Proposition \[II\] can be replaced the following.
A metric space $(Y,d)$ admits the *weak midpoints property* if either $d$ is bounded or there exist sequences $(x_k),(y_k),(z_k)\subset Y$ such that $d(x_k,y_k)\to\infty$ and $$\label{wmp}
\frac{d(x_k,z_k)}{d(x_k,y_k)}\to\frac{1}{2},\ \frac{d(y_k,z_k)}{d(x_k,y_k)}\to\frac{1}{2}.$$
\[III\] Let $D_1$ and $D_2$ be Kobayashi hyperbolic domains (i.e. $k_{D_1}$ and $k_{D_2}$ are distances) admitting non-constant bounded holomorphic functions (for example, bounded domains). Then $D_1\times D_2$ is not Gromov $k$-hyperbolic.
To see this, it is enough to observe that if a domain $G$ in $\Bbb C^n$ admits a non-constant bounded holomorphic function $f$ and $|f(z_j)|\to\sup_G|f|,$ then $k_G(z,z_j)\ge c_G(z,z_j)\to\infty.$
Note also that Proposition \[I\] implies that if $D_1$ and $D_2$ are planar domains with complements containing more than one point (i.e. they are Kobayashi hyperbolic), then $D_1\times D_2$ is not Gromov $k$-hyperbolic (use that $k_{D_k}(z,z_j)\to\infty$ as $z_j\to\partial
D_k,$ $k=1,2$).
As an immediate consequence we obtain that the polydisc is not Gromov $k$-hyperbolic. Moreover, even its “symmetrized” counterpart is not.
\[G\_n\] $\mathbb{G}_{n}$ is not Gromov $c$- nor $k$-hyperbolic for $n\geq 2$.
For the convenience of the reader, recall that the [*symmetrized polydisc*]{} $\mathbb{G}_{n},$ which is of great relevance due to its properties and role (cf. [@AY], [@Costara]), is the image of the holomorphic map $$\pi :\mathbb{D}^{n}\rightarrow\mathbb{C}^{n},\ \pi=(\pi_1,\ldots,\pi_n),$$ $$\pi_k(z_1,\ldots,z_n)=\sum_{1\leq j_1<\ldots<j_k\leq n}z_{j_1}\ldots z_{j_k},\ z_1,\ldots,z_n\in\mathbb{D},\ 1\leq k\leq n,$$ which is proper from $\mathbb{D}^{n}$ to $\mathbb{G}_{n}$.
Another interesting domain, the [*tetrablock*]{} (cf. [@AWY]), fails to be Gromov $k$-hyperbolic, too. Let $$\varphi:\mathcal{R}_{II}\rightarrow \mathbb{C}^{3},\ \varphi(z_{11},z_{22},z)=(z_{11},z_{22},z_{11}z_{22}-z^{2}),$$ where $\mathcal{R}_{II}$ denotes the classical Cartan domain of the second type (in $\mathbb{C}^{3}$), i.e. $$\mathcal{R}_{II} =\{\widetilde{z}\in\mathcal{M}_{2\times 2}(\mathbb{C}): \widetilde{z}=\widetilde{z}^{t},\ \lVert \widetilde{z}\rVert<1\},$$ where $\lVert \cdot \rVert$ is the operator norm and $\mathcal{M}_{2\times 2}(\mathbb{C})$ denotes the space of $2 \times 2$ complex matrices (we identify a point $(z_{11},z_{22},z)\in\mathbb{C}^{3}$ with a $2\times 2$ symmetric matrix $ \left( \begin{array}{ll}
z_{11} & z \\
z & z_{22}
\end{array} \right) ). $ Then $\varphi$ is a proper holomorphic map and $\varphi(\mathcal{R}_{II})=\mathbb{E}$ is a domain, called the tetrablock.
\[tetrablock\] $\mathbb{E}$ is not Gromov $k$-hyperbolic.
Since $\Bbb G_2$ and $\mathbb{E}$ are bounded $\mathbb{C}$-convex domains (see [@G2 Theorem 1 (i)] and [@Zwonek Corollary 4.2]), it follows that they are not Gromov $c^i$- nor $b$-hyperbolic either.
Buckley in [@Buckley], claimed that it is because of the flatness of the boundary rather than the lack of smoothness that Gromov hyperbolicity fails. Recently, Gaussier and Seshadri have provided a proof of that conjecture. More precisely, their main result in [@Gaussier Theorem 1.1] states that any bounded convex domain in $\mathbb{C}^{n}$ whose boundary is $\mathcal{C}^{\infty}$-smooth and contains an analytic disc, is not Gromov $k$-hyperbolic. Lemma 5.4 in their proof used the $\mathcal{C}^{\infty}$ assumption in an essential way. Our aim is to prove this result in a shorter way in $\mathbb{C}^{2}$, assuming only $\mathcal{C}^{1,1}$-smoothness. Moreover, the proofs of the facts we use are more elementary.
\[Gaussier\] Let $D$ be a convex domain in $\mathbb{C}^{2}$ containing no complex lines.[^1] Assume that $\partial D$ is $\mathcal{C}^{1,1}$-smooth and contains an analytic disc. Then $D$ is not Gromov $k$-hyperbolic.
Besides, we give a partial answer to the question raised in [@Bonk].
\[Pascal\] Let $D$ be a $\mathcal{C}^{1,1}$-smooth convex bounded domain in $\mathbb{C}^{2}$ admitting a defining function of the form $\varrho (z)=-\Re z_{1}+\psi (|z_{2}|)$ near the origin, where $\psi$ is a $\mathcal{C}^{1,1}$-smooth nonnegative convex function near $0$ satisfying $\psi (0)=0,$ and $$\label{infinite type}
\limsup _{x\rightarrow 0}\frac{\log \psi (|x|)}{\log |x|}=\infty.\footnote{If $\psi$ is $\mathcal{C}^{\infty},$ then $0$ is of infinite type if and only if condition (\ref{infinite type}) holds.}$$ Then $D$ is not Gromov $k$-hyperbolic.
Finally, note that there is no connection between Gromov hyperbolicity and pseudoconvexity. Indeed, take any strictly pseudoconvex domain $G.$ As we have already mentioned, $G$ is Gromov $k$-hyperbolic, and $k_G$ and $c_G$ are bilipschitz equivalent. Hence $G$ is Gromov $c$-hyperbolic, too. Assume that, respectively, $A\Subset G$ and $B$ is a relatively closed subset of $G$ such that $G\setminus A$ is a domain and that $B$ is negligible with respect to the $(2n - 2)$-dimensional Hausdorff measure. Then $G\setminus A$ is Gromov $c$-hyperbolic and $G\setminus B$ is Gromov $k$-hyperbolic, since $$c_{G\setminus A} = c_{G}|_{(G\setminus A)\times (G\setminus A)}$$ (by the Hartogs extension theorem) and $$k_{G\setminus B} = k_{G}|_{(G\setminus B)\times (G\setminus B)}$$ (cf. [@Jarnicki Theorem 3.4.2]).
However, the example with $G\setminus B$ does not have a smooth boundary. The next proposition yields, in particular, a family of non-pseudoconvex domains with smooth boundaries which are Gromov $k$-hyperbolic.
\[ball\] Let $G$ be a bounded domain in $\Bbb C^n\ (n\geq 2).$ Assume that $D\Subset G$ is a $\mathcal{C}^{2}$-smooth domain in $\Bbb C^n$ and its Levi form has at least one positive eigenvalue at each boundary point. Then $G\setminus D$ is a domain such that $k_{G\setminus D}$ is quasi-isometrically equivalent to $k_{G}|_{(G\setminus D)\times (G\setminus D)}.$[^2]
In particular, if $G$ is Gromov $k$-hyperbolic, then so is $G\setminus\overline{D}.$
\[str\_psc\] If $D\Subset G$ are strictly pseudoconvex domains in $\Bbb C^n,$ then $G\setminus\overline{D}$ is a Gromov $k$-hyperbolic domain.
The estimates that we use in the proof of Proposition 5 do not hold for the planar annulus $\mathbb{A}_r=\{z\in\mathbb{C}: r^{-1}<|z|<r\}$ ($r>1$). However, any finitely connected proper planar domain is Gromov $k$-hyperbolic (cf. [@RT Proposition 3.2]).
\[compact\] Let $G$ be a bounded domain in $\Bbb C^n\ (n\geq 2).$ Assume that $K$ is compact subset of $G$ such that through any point $z\in\Bbb C^n\setminus K$ passes a complex line disjoint from $K.$ Then $G\setminus D$ is a domain such that $k_{G\setminus D}$ is quasi-isometrically equivalent to $k_{G}|_{(G\setminus D)\times (G\setminus D)}$.[^3]
In particular, if $G$ is Gromov $k$-hyperbolic, then so is $G\setminus K.$
Note that we may take $K$ to be any compact (${\Bbb C}$-)convex set, since any compact or open ${\Bbb C}$-convex set $E$ in $\Bbb C^n$ is *linearly convex*, i.e. through any point in $\Bbb C^n\setminus E$ passes a complex line disjoint from $E$ (cf. [@APS Theorem 2.3.9]).
Throughout the paper $d_{D}$ denotes the (Euclidean) distance to $\partial D.$ A point $z\in\mathbb{C}^{n}$ we write as $(z_{1},\ldots,z_{n}),\ z_{j}\in\mathbb{C}.$
An appendix at the end of the paper includes some of the estimates for the Kobayashi distance and metric used in the proofs.
Proofs
======
[*Proof of Proposition \[I\].*]{} Assume that $(X,d)$ is $\frac{\delta}{2}$-hyperbolic. Put $k=3+\delta$. Then there are points $y_1,y_2\in X_2$ such that $d_2(y_1,y_2)=2s\geq 2k$. Choose points $x_1,x_2^\ast\in X_1$ with $d_1(x_1,x_2^\ast)\geq 2s$. By the path property of $X_1$, there is a $d_1$-continuous curve $\gamma:[0,1]\to X_1$ joining the points $x_1$ and $x_2^\ast$ such that $L_{d_1}(\gamma)<d_1(x_1,x_2^\ast)+1$. Note that $t\to
d_1(x_1,\gamma(t))$ is continuous. Hence there is a smallest number $t_0$ such that $d_1(x_1,\gamma(t_0))=2s$. Set $x_2=\gamma(t_0)$.
Now $L(\gamma|_{[0,t_0]}) \ge d_1(x_1,x_2)=2s$, and $$L(\gamma|_{[0,t_0]}) = L(\gamma) - L(\gamma|_{[t_0,1]})
\le d_1(x_1,x_2^\ast)+1 - d_1(x_2,x_2^\ast) \le d_1(x_1,x_2) +1.$$ Let $t_1$ be the smallest number in $[0,t_0]$ such that $d_1(x_1,\gamma(t_1))=s$. Set $x_3=\gamma(t_1)$. Then $$d_1(x_2,x_3) \ge d_1(x_1,x_2) - d_1(x_1,x_3) = s, \mbox{ and }$$ $$d_1(x_2,x_3) = L(\gamma|_{[0,t_1]}) = L(\gamma|_{[0,t_0]}) - L(\gamma|_{[t_1,t_0]})
\le 2s+1 - d_1(x_1,x_2) = s+1.$$ Hence, $s= d_1(x_1,x_3)\leq
d_1(x_3,x_2)<s+1$.
Now define the following points in $X_1 \times X_2$: $x=(x_1,y_1)$, $y=(x_2,y_1)$, $w=(x_3,y_1)$, and $z=(x_3,y_2)$. Then $d(z,w)=d(z,x)=d(z,y)=2s$ and $(x,y)_w\leq 1$, $(x,z)_w= d(x,w)=s$, $(y,z)_w=d(y,w)\ge s-1$. By the assumption of $\frac{\delta}{2}$-hyperbolicity we reach the inequalities $$1\geq (x,y)_w\geq \min\{(y,z)_w,(x,z)_w\}-\delta\geq
s-1-\delta\ge 2$$ which is a contradiction.
An essential ingredient in the proof of Proposition \[I\] is the existence of points $x_{1},x_{2},x_{3}$ such the triangle inequality is a near-equality, namely $(x_1, x_2)_{x_3} \le 1$. The condition is equivalent to $(x_1, x_2)_{x_3} = o(d(x_1,x_2))$, and $\left| d(x_1,x_3) - d(x_2,x_3)\right| = o(d(x_1,x_2))$.
Using this weaker hypothesis and following the steps of the above proof, setting $2s =d(x_1,x_2)$ as before, we find $$o(s)\geq (x,y)_w\geq s - o(s) -\delta,$$ leading to a contradiction when $s\to\infty$. Similar changes can be made in the proof below.
[*Proof of Proposition \[II\].*]{} Let first $(X_1,d_1)$ be $2\delta$-hyperbolic and $d_2\le 2c.$ Since $d\le d_1+2c,$ it follows that $$(x_1,y_1)_{w_1}-2c\le (x,y)_w\le(x_1,y_1)_{w_1}+4c$$ and then $(X,d)$ is $(\delta+3c)$-hyperbolic.
Assume now that $(X,d)$ is $\delta$-hyperbolic. Following the proof of Proposition \[I\], we deduce that one of the distances is bounded, say $d_2\le 2c.$ Then we get as above that $(X_1,d_1)$ is $(\delta+3c)$-hyperbolic.
[*Proof of Proposition \[G\_n\].*]{} Let $a\in\Bbb D,$ $p_a=\pi (a,\ldots,a),$ $q_a=\pi (a,\ldots,a,-a)$ and $m_a=\pi (a,\ldots,a,0).$ We shall show that $$S_{c_{\Bbb G_n}}(p_a,q_a,m_a,0)\to\infty\mbox{ as }|a|\to 1.$$ It follows exactly in the same way that $S_{k_{\Bbb G_n}}(p_a,q_a,m_a,0)\to\infty\mbox{ as }|a|\to 1.$ So, $\Bbb G_n$ is not Gromov $c$- nor $k$-hyperbolic for $n\geq 2.$
The holomorphic contractibility implies that $$c_{\Bbb G_n}(p_a,q_a) \ge c_{\Bbb D}(a^n,-a^n)=2c_{\Bbb D}(a^n,0),$$ $$\max\{c_{\Bbb G_n}(p_a,0), c_{\Bbb G_n}(q_a,0),c_{\Bbb G_n}(p_a,m_a),c_{\Bbb G_n}(q_a,m_a)\}\le c_{\Bbb D}(a^n,0).$$ Thus $$S_{c_{\Bbb G_n}}(p_a,q_a,m_a,0)\ge c_{\Bbb G_n}(m_a,0)+2c_{\Bbb D}(a^n,0)-2c_{\Bbb D}(a,0).$$ Since $$2c_{\Bbb D}(a,0)-2c_{\Bbb D}(a,0)\to\log n\mbox{ as }|a|\to 1,$$ it remains to see that $c_{\Bbb G_n}(m_a,0)\to\infty$ as $|a|\to 1.$ This follows by the fact that any point $b\in\Bbb G_n$ is a weak peak point, i.e. there exists $f_b\in\mathcal O(\Bbb G_n,\Bbb D)$ such that $|f_b(z)|\to 1$ as $z\to b$ (a consequence of [@Costara Corollary 3.2]).
[*Proof of Proposition \[tetrablock\].*]{} Let $a\in (0,1),$ and put $P_a=\varphi(\textup{diag}(a,a)),\ Q_a=\varphi(\textup{diag}(a,-a)).$ Recall that $\Phi_a(Z)=(Z-a\textup{I})(\textup{I}-aZ)^{-1}$ is an automorphism of $\mathcal{R}_{II}.$ Direct computations show that $$\varphi\circ\Phi_a (\left( \begin{array}{ll}
z_{11} & z \\
z & z_{22}
\end{array} \right))=\varphi\circ\Phi_a (\left( \begin{array}{ll}
z_{11} & -z \\
-z & z_{22}
\end{array} \right)),$$ whenever $\left( \begin{array}{ll}
z_{11} & z \\
z & z_{22}
\end{array} \right)\in\mathcal{R}_{II}.$ Thus, $\Phi_a$ induces an automorphism $\widetilde{\Phi}_{a}$ of $\mathbb{E}.$ It follows from this and [@AWY Corollary 3.7] that $$\begin{gathered}
k_{\mathbb{E}}(0,(a,b,p))=\tanh^{-1}\max\Big{\{}\frac{|a-\overline{b}p|+|ab-p|}{1-|b|^{2}},\frac{|b-\overline{a}p|+|ab-p|}{1-|a|^{2}}\Big{\},}\\
(a,b,p)\in\mathbb{E},\end{gathered}$$ $$2k_{\mathbb{E}}(P_a,0),\ 2k_{\mathbb{E}}(Q_a,0),\ k_{\mathbb{E}}(P_a,Q_a)
=-\log d_{\mathbb{D}}(a) +\textup{O}(1).$$ Observe that if $f(\lambda)=(0,\lambda,0),$, then $g_a=\widetilde{\Phi}_{-a}\circ f$ is a complex geodesic for $k_{\Bbb E}$ with $P_a=g_a(0)$, $Q_a=g\left(-\frac{2a}{1+a^2}\right)$. Note that the Kobayashi middle point $R_{a}$ of $g_{a}|_{[-\frac{2a}{1+a^2},0]}$ tends to the boundary; more precisely, $$R_{a}=g_{a}(-a)\rightarrow \textup{diag}(1,0) \textup{ as }a\rightarrow 1.$$ Consequently, $S_{k_{\Bbb E}}(P_a,Q_a,R_a,0)$ is comparable with $k_{\mathbb{E}}(R_a,0).$ By Proposition A1(b) (see Appendix), $k_{\mathbb{E}}(R_a,0)\to\infty$ as $a\to 1,$ which finishes the proof.
[*Proof of Theorem \[Gaussier\].*]{} Since $\partial D$ contains an analytic disc, it is well known that it contains an affine disc (cf. [@NPZ Proposition 7]). We assume that this disc has center $0$ and lies in $\{z_{1}=0\},$ and that $D\subset\{\Re z_{1}>0\}.$
We can find an $r>0$ such that for any $\delta >0$ small enough there exist two discs $\Delta (\tilde{p}_{\delta},r)$ and $\Delta (\tilde{q}_{\delta},r)$ in $D_{\delta}=D\cap\{z_{1}=\delta\}$ which touch $\partial{D}$ at two points $\hat{p}_{\delta}$ and $\hat{q}_{\delta}$ with $\lVert \hat{p}_{\delta}-\hat{q}_{\delta}\rVert >5r.$
We identify $\partial D\cap \{ z_{1}=0\}$ with a closed, bounded, convex subset of $\mathbb C$, which is the closure of its interior. Call this interior $D_0.$
There exists $\zeta_0 \in D_0$ such that $d_{D_0}(\zeta_0)=\max_{\zeta\in D_0}d_{D_0}(\zeta).$ Then $$M= \left\{ p \in \partial D_0 : |p- \zeta_0 | = \min_{\xi \in \partial D_0} |\xi- \zeta_0 |
\right\}$$ is a not empty set which cannot be contained in any half plane $$H_\theta = \{ \zeta :\\ \Re [(\zeta-\zeta_0)e^{-i\theta}] <0\}:$$ if it were, one could find $\varepsilon>0$ such that $d_{D_0}( \zeta_0 + \varepsilon e^{i\theta})
> d_{D_0}(\zeta_0)$. So there are $\hat p \neq \hat q
\in M$ such that $\arg( (\hat p -\zeta_0) (\hat q-\zeta_0)^{-1} ) \ge 2\pi/3$. Take $r\in (0, \frac{\sqrt 3}{5+\sqrt 3}|\hat p -\zeta_0|)$, $\tilde{p} = \zeta_0 + ( 1 - r|\hat p -\zeta_0|^{-1} ) (\hat p -\zeta_0)$, and $\tilde{q}$ chosen likewise. Then the discs $\Delta (\tilde{p},r)\subset D_{0}$ and $\Delta (\tilde{q},r)\subset D_{0}$ are tangent to $\partial D_{0}$ at $\hat{p}$ and $\hat{q}$.
Now we want to move these discs inside $D.$ By $\mathcal{C}^{1,1}$-smoothness of $D,$ we can move them (in $\mathbb{C}^{2}$) along the vector $(1,0)$ inside $D,$ i.e. $\Delta (\tilde{p},r),\,\Delta (\tilde{q},r)\subset D\cap\{z_{1}=\delta\}=D_{\delta},$ for $0<\delta <\delta_{0}.$ If they do not touch $\partial D_{\delta},$ then shift them (separately at every sublevel set) to the boundary but leaving their centers on the real line passing through $\tilde{p}+(\delta,0)$ and $\tilde{q}+(\delta,0).$ Denote new discs by $\Delta (\tilde{p}_{\delta},r),\,\Delta (\tilde{q}_{\delta},r),$ and by $\hat{p}_{\delta},\,\hat{q}_{\delta}$ points of contact of those discs with $\partial D_{\delta}.$
Choose now a point $a=(\delta_{0},0)\in D\textup{ (}\delta_{0}>0\textup{)}$ and consider the cone with vertex at $a$ and base $\partial D\cap\{z_{1}=0\}.$ Denote by $G_{\delta}$ the intersection of this cone and $\{z_{1}=\delta\}.$ For any $\delta>0$ small enough the line segment with ends at $\tilde{p}_{\delta}$ and $\hat{p}_{\delta}$ intersects $\partial G_{\delta},$ say at $p_{\delta}.$ Define $q_{\delta}$ in a similar way.
Set $\tilde{s}_{\delta}=\frac{ \tilde{p}_{\delta}+\tilde{q}_{\delta}}{2}.$ We shall show that $S_{k_D}(p_{\delta},q_{\delta}, \tilde{s}_{\delta}, a) \rightarrow \infty$ as $\delta\rightarrow 0.$ For this we will see that $(p_{\delta},\tilde{s}_{\delta})_{a}-(p_{\delta},q_{\delta})_{a}\rightarrow \infty$ as $\delta\rightarrow 0.$ It will follow in the same way that $(q_{\delta},\tilde{s}_{\delta})_{a}-(p_{\delta},q_{\delta})_{a}\rightarrow \infty.$
It is enough to prove that $$\label{1}
k_{D}(q_{\delta},a)-k_{D}(\tilde{s}_{\delta},a)<c_{1}$$ and $$\label{2}
k_{D}(p_{\delta},q_{\delta})-k_{D}(p_{\delta},\tilde{s}_{\delta})\rightarrow \infty.$$ Here and below $c_{1},c_{2},\ldots$ denote some positive constants which are independent of $\delta.$
For (\[1\]), observe that, by Propositions A1(a) and A2 (see Appendix), $$\label{Kobayashi estimates}
k_{D}(\tilde{s}_{\delta},a)\geq\frac{1}{2}\log\frac{d_{D}(a)}{d_{D}(\tilde{s}_{\delta})}\textup{ and }2k_{D}(q_{\delta},a)\leq -\log d_{D}(q_{\delta})+c_{2}.$$ It remains to use that $d_{D}(\tilde{s}_{\delta})=d_{D}(q_{\delta})$ for any $\delta >0$ small enough.
To prove (\[2\]), denote by $F_{\delta}$ the convex hull of $\Delta (\tilde{p}_{\delta},r)$ and $\Delta (\tilde{s}_{\delta},r).$ Then by inclusion $k_{D}(p_{\delta},\tilde{s}_{\delta})\leq k_{F_{\delta}}(p_{\delta},\tilde{s}_{\delta})$.
$k_{F_{\delta}}(p_{\delta},\tilde{s}_{\delta}) < -\frac{1}{2}\log d^{\prime}_{D}(p_{\delta})+c_{3},$ where $d^{\prime}_{D}$ is the distance to $\partial D$ in the $z_{2}$-direction.
[*Proof.*]{} For $\delta>0$ small enough we have that $$d^{\prime}_{D}(p_{\delta}) = d_{D_\delta}(p_{\delta}) =
d_{F_\delta}(p_{\delta}) = d_{\Delta (\tilde{p}_{\delta},r)}(p_{\delta})$$ because the closest point on $\partial D_\delta$ belongs to $\partial \Delta (\tilde{p}_{\delta},r)$. Now $k_{F_{\delta}}(p_{\delta},\tilde{s}_{\delta}) \le
k_{F_{\delta}}(p_{\delta},\tilde{p}_{\delta}) + k_{F_{\delta}}(\tilde p_{\delta},\tilde{s}_{\delta}) $.
Since $\Delta (\tilde{p}_{\delta},r)\subset F_{\delta}$, $$\begin{gathered}
k_{F_{\delta}}(p_{\delta},\tilde{p}_{\delta})
\le k_{\Delta (\tilde{p}_{\delta},r)}(p_{\delta},\tilde{p}_{\delta})
= \frac{1}{2}\log \frac{1+\frac{|p_{\delta}-\tilde{p}_{\delta}|}{r} }{1-\frac{|p_{\delta}-\tilde{p}_{\delta}|}{r}}
\\
\le - \frac{1}{2} \log d_{\Delta (\tilde{p}_{\delta},r)}(p_{\delta}) +\frac12 \log (2r)
= -\frac{1}{2}\log d^{\prime}_{D}(p_{\delta})+ \frac12 \log (2r) .\end{gathered}$$ On the other hand, by using a finite chain of discs of radius $r$ with centers on the line segment from $\tilde{p}_{\delta}$ to $\tilde{s}_{\delta}$, we obtain that $$k_{F_{\delta}}(\tilde p_{\delta},\tilde{s}_{\delta}) \le
4 \frac{|\tilde p_{\delta}-\tilde{s}_{\delta}|}{r} \le C(r).\qed$$
Now, we shall show that $$\label{3}
2k_{D}(p_{\delta},q_{\delta})>-\log d^{\prime}_{D}(p_{\delta})-\log d^{\prime}_{D}(q_{\delta})-c_{4},$$ which implies (\[2\]), because $d^{\prime}_{D}(q_{\delta})\rightarrow 0$ as $\delta\rightarrow 0.$
Since the Kobayashi distance is intrinsic, we may find a point $m_{\delta}\in D$ such that $$\lVert p_{\delta} - m_{\delta}\rVert=\lVert q_{\delta}-m_{\delta}\rVert\geq\frac{\lVert p_{\delta}-q_{\delta}\rVert}{2},$$ $$k_{D}(p_{\delta},q_{\delta})>
k_{D}(p_{\delta},m_{\delta})+k_{D}(m_{\delta},q_{\delta})-1.$$
Let $\check{p}_{\delta}\in\partial D$ be the closest point to $p_{\delta}$ in the direction of the complex line through $p_{\delta}$ and $m_{\delta}.$
Recall that $d_D'$ is the distance to $\partial D$ in the $z_{2}$-direction and $d_D(p_\delta)$ is attained in $z_{1}$-direction for any $\delta>0$ small enough. This means that the standard basis is adapted to the local geometry of $\partial D$ near $p_\delta$, and more precisely, if $X=(X_1,X_2)\in\Bbb C^2$ is a unit vector, [@NPZ (4)] states in this case that there exists a constant $C$ such that $$\frac{1}{d_D(p_\delta,X)}\le \frac{|X_1|}{d_{D}(p_\delta)} +
\frac{|X_2|}{d'_{D}(p_\delta)}\le\frac{C}{d_D(p_\delta,X)},$$ where $d_D(\cdot;X)$ is the distance to $\partial D$ in direction $X.$ Since $d'_{D}\ge d_D$, we obtain $$d_D(p_\delta;X)\le c_5d_D'(p_\delta).$$
Let $X= \frac{m_\delta-p_{\delta}}{\|m_\delta-p_{\delta}\|}.$ Then $\| p_{\delta}-\check{p}_{\delta}\| = d_X (p_{\delta})$ and thus $$\label{4}
\lVert p_{\delta}-\check{p}_{\delta}\rVert< c_{5}d^{\prime}_{D}(p_{\delta}).$$
By convexity, $D$ is on the one of the sides, say $H_{\delta},$ of the real tangent plane to $\partial D$ at $\check{p} _{\delta}.$ Since $\frac{\lVert m_{\delta}-\check{p}_{\delta}\rVert}{d_{H_\delta}(m_{\delta})}
=\frac{\lVert p_{\delta}-\check{p}_{\delta}\rVert}{d_{H_\delta}(p_{\delta})}$, it follows by that $$2k_{D}(p_{\delta},m_{\delta})
\geq 2k_{H_{\delta}}(p_{\delta},m_{\delta})
\geq \log \frac{d_{H_\delta}(m_{\delta})}{d_{H_\delta}(p_{\delta})}
=
\log\frac{\lVert m_{\delta}-\check{p}_{\delta}\rVert}{\lVert p_{\delta}-\check{p}_{\delta}\rVert}.$$ Applying the triangle inequality and , we get that $$\begin{gathered}
\log\frac{\lVert m_{\delta}-\check{p}_{\delta}\rVert}{\lVert p_{\delta}-\check{p}_{\delta}\rVert}\ge
\log\frac{\lVert m_\delta -p_\delta\rVert -\lVert p_\delta -\check{p}_\delta\rVert}{\lVert p_{\delta}-\check{p}_{\delta}\rVert}\ge\\
\log\left(\frac{r}{2\lVert p_{\delta}-\check{p}_{\delta}\rVert}-1\right)\ge\log\frac{r}{2c_5 d^{\prime}(p_\delta)}\,-\,1,\end{gathered}$$ for any $\delta >0$ small enough. So $2k_{D}(p_{\delta},m_{\delta})>-\log d^{\prime}_{D}(p_{\delta})-c_{6}.$ Similarly, $2k_{D}(q_{\delta},m_{\delta})>-\log d^{\prime}_{D}(q_{\delta})-c_{6},$ which implies (\[3\]), and completes the proof.
All the above arguments hold in $\mathbb{C}^{n}$, $n\ge 3$, except (\[4\]).
[*Proof of Theorem \[Pascal\].*]{} Since the case when $\psi(z_{0}) =0$ for some $z_{0}\not= 0,$ is covered by Proposition \[Gaussier\], we may assume that $\psi^{-1}\{0\}=\{0\}$. Also assume $p=(1,0)\in D.$
Let $\alpha (x),$ small enough, an increasing function such that for any $x>0,\ \psi^{\prime}(x)\geq\psi^{\prime}((1-\alpha (x))x)\geq\frac{1}{2}\psi' (x)$. We choose, for $x>0,\ q(x)=(\psi(x),0),\ r(x)=(\psi (x),-(1-\alpha (x))x),\ s(x)=(\psi (x),(1-\alpha (x))x).$
We claim that for $x$ small enough:
1. $d_{D}(q)=\psi (x),$\[A\]
2. $\frac{\alpha (x)}{4}x\psi^{\prime}(x)\leq d_{D}(s),d_{D}(r)\leq \alpha (x)x\psi^{\prime}(x),$\[B\]
3. the functions $k_{D}(s,q)+\frac{1}{2}\log\alpha (x)$ and $k_{D}(r,q)+\frac{1}{2}\log\alpha (x)$ are bounded,\[C\]
4. the function $k_{D}(r,s)+\log\alpha (x)$ is bounded.\[D\]
Before we proceed to prove the claims we make some general observation about infinite order of vanishing.
\[psipsi\] For any $\varepsilon>0$ and $A>0$, there exists $x \in (0,\varepsilon)$ such that $\frac{x \psi'(x) }{\psi(x)} >A$.
Suppose instead that there exist $\varepsilon>0$ and $A>0$ such that $ \frac{x \psi'(x) }{\psi(x)} \le A$ for $0<x\le \varepsilon$. Then $$\frac{d}{dx} \left( \log \psi(x)\right) \le \frac{A}x, \quad 0<x\le \varepsilon,$$ so $\log(\psi(\varepsilon)) - \log (\psi(x)) \le A \left( \log \varepsilon - \log x \right)$, i.e. $$\psi(x) \ge \frac{\psi(\varepsilon)}{\varepsilon^{A}} x^{ A}, \quad 0<x\le \varepsilon,$$ which means that at the point $0$ there is finite order of contact with the tangent hyperplane, a contradiction.
Assume the claims for a while, and observe that for any $x$ verifying the conclusion of Lemma \[psipsi\] we have $$(r,p)_{q}-(r,s)_{q}, \quad (p,s)_{q}-(r,s)_{q}\geq -\frac{1}{2}\log\frac{\psi (x)}{x\psi^{\prime}(x)}+C_{1}.$$ Since the above quantity can be made arbitrarily large, it finishes the proof.
It remains to prove (\[A\])-(\[D\]).
(\[A\]) is clear. Next, since $(\psi((1-\alpha (x))x),(1-\alpha (x))x)\in D,\ d_{D}(s)\leq \psi (x)-\psi ((1-\alpha (x))x))\leq \alpha (x)x\psi^{\prime}(x)$ by convexity. Let $L$ be the real line through $(\psi((1-\alpha (x))x),(1-\alpha (x))x)$ and $(\psi (x),x).$ Its slope is less than $\psi^{\prime}(x),$ so $d_{D}(s)\geq \textup{dist}\,(s,L^{\prime}),$ where $L^{\prime}$ is the line through $(\psi((1-\alpha (x))x),(1-\alpha (x))x)$ with slope $\psi^{\prime}(x),$ so $$\begin{gathered}
d_{D}(s) \geq \frac{\psi (x) - \psi ((1-\alpha(x))x)}{\sqrt{1+\psi^{\prime}(x)^{2}}} \\
\geq \frac{1}{2} {\alpha (x) \times \psi^{\prime}((1-\alpha (x))x)} \geq \frac{1}{4} \alpha (x) \times \psi^{\prime}(x).\end{gathered}$$ Thus, $\frac{\alpha (x)}{4}x\psi^{\prime}(x)\leq d_{D}(s)\leq \alpha (x)x\psi^{\prime}(x).$ Analogous estimates hold for $r,$ which gives (\[B\]).
The analytic disc $\zeta \mapsto (\psi (x),x\zeta)$ provides immediate upper estimates in (\[C\]) and (\[D\]).
To get lower estimate for $k_{D}(s,q),$ we map $D$ to a domain in $\mathbb{C}$ by the complex affine projection $\pi_{s}$ to $\{z_{1}=\psi (x)\},$ parallel to the complex tangent space to $\partial D$ at the point $(\psi(x),x).$ Then $\pi_{s}(D)=\{\psi (x)\}\times D_{s},$ where $D_{s}$ is a convex domain in $\mathbb{C},$ containing the disc $\{|z_{2}|<x\},$ and its tangent line at the point $x$ is the real line $\{\Re z_{2}=x\}.$ The projection is given by the explicit formula $$\pi_{s}(z_{1},z_{2})=\Big{(}\psi (x),z_{2}+\frac{\psi (x)-z_{1}}{\psi^{\prime}(x)}\Big{)}.$$ We renormalize by setting $f_{+}(z)=1-\frac{1}{x}[\pi_{s}(z)]_{2}.$ Therefore $f_{+}(D) \subset H=\{z\in\mathbb{C}:\Re z>0\},$ so $$\label{E}
k_{D}(s,q)\geq
k_{H}(f_{+}(s),f_{+}(q)) = k_H (\alpha(x),0)
\geq -\frac{1}{2}\log \alpha (x) +C_{2},$$ where $C_{2}>0$ does not depend on $x.$
The estimate for $k_{D}(r,q)$ proceeds along the same lines, but we use the projection $\pi_{r}$ to $\{z_{1}=\psi (x)\}$ along the complex tangent space to $\partial D$ at $(\psi (x),x),$ given by $$\pi_{r}=\Big{(}\psi (x),z_{2}-\frac{\psi (x)-z_{1}}{\psi^{\prime}(x)}\Big{)}.$$ Note that choosing $f_{-}(z)=1+\frac{1}{x}[\pi_{r}(z)]_{2},$ we have $f_{-}(D)\subset\{\Re z>0\}.$
Now we tackle the lower estimates for $k_{D}(r,s).$ Let $\gamma$ be any piecewise $\mathcal{C}^{1}$ curve such that $\gamma (0)=s,\,\gamma (1)=r.$ Let $c_{0}<\frac{1}{2}.$ We claim that there exists $t_{0}\in (0,1)$ such that if we set $u=\gamma (t_{0}),$ then $|f_{+}(u)|,|f_{-}(u)|\geq c_{0}.$
For this write $\gamma=(\gamma_{1},\gamma_{2}).$ Set $\zeta_{1}=1-\frac{\psi (x)\,-\,\gamma_1(t_0)}{x\psi^{\prime}(x)}.$ By the explicit form of $\pi_{s},$ the condition $|f_{+}(u)|\geq c_{0}$ reads $|\zeta_{1}-\frac{\gamma_{2}(t_0)}{x}|\geq c_{0},$ and the condition $|f_{-}(u)|\geq c_{0}$ reads $|\zeta_{1}+\frac{\gamma_{2}(t_0)}{x}|\geq c_{0}.$ We claim that the discs $\overline{\mathbb{D}}(\zeta_{1},c_{0})$ and $\overline{\mathbb{D}}(-\zeta_{1},c_{0})$ are disjoint for any $t.$ Indeed, they would intersect if and only if $0\in\overline{\mathbb{D}}(\zeta_{1},c_{0}),$ which implies $$\Re\Big{(}\frac{\gamma_{1}(t_0)}{x \psi^{\prime}(x)}\Big{)}
\leq -1+c_{0}+\frac{\psi (x)}{x\psi^{\prime}(x)}\leq -\frac{1}{3}$$ for any $x$ such that $\frac{\psi (x)}{x\psi^{\prime}(x)}\leq \frac{1}{6}$, which we may assume by Lemma \[psipsi\]. In particular $\Re\gamma_{1}(t_0)<0,$ which is excluded for any $\gamma (t)\in D.$ Now let $t_{1}=\max\{t:\frac{\gamma_{2}(t)}{x}\in\overline{\mathbb{D}}(\zeta_{1},c_{0})\}.$ Then $\frac{\gamma_{2}(t_{1})}{x}\notin\overline{\mathbb{D}}(-\zeta_{1},c_{0}),$ and by continuity there is $\eta >0$ such that $\frac{\gamma_{2}(t_{1}+\eta)}{x}\notin\overline{\mathbb{D}}(-\zeta_{1},c_{0}),$ and of course $\frac{\gamma_{2}(t_{1}+\eta)}{x}\notin\overline{\mathbb{D}}(\zeta_{1},c_{0})$ by maximality of $t_{1},$ so $t_{0}=t_{1}+\eta$ will provide a point satisfying the claim.
Consequently, taking a curve $\gamma$ such that $$k_D(r,s)+1 > \int_{0}^{1}\kappa_{D}(\gamma (t);\gamma^{\prime}(t))dt,$$ $$\begin{gathered}
\int_{0}^{1}\kappa_{D}(\gamma (t);\gamma^{\prime}(t))dt
\geq\int_{0}^{t_{0}}\kappa_{D}(\gamma (t);\gamma^{\prime}(t))dt
+\int_{t_{0}}^{1}\kappa_{D}(\gamma (t);\gamma^{\prime}(t))dt \\
\geq k_{D}(r,u)+k_{D}(u,s).\end{gathered}$$ We end the proof by estimating $k_{D}(r,u)$ in the same way as we did $k_D(r,q)$ above, and $k_{D}(u,s)$ as as we did $k_D(s,q)$ above, using the maps $f_+,f_-$ and estimates about the Kobayashi distance in a half plane.
[*Proof of Proposition \[ball\].*]{} Set $G'=G\setminus\overline{D}.$
Assume first that $G'$ is not a domain. Let $G''$ be a bounded connected component of $G'.$ Consider a farthest point $a\in\partial G''$ from the origin. Then $a$ is a concave boundary point of $D$ which a contradiction.
Choose now a smooth domain $E$ such that $D\Subset E\Subset G.$ By smoothness and compactness, there is a constant $C>0$ such that any two points in $G'\cap\overline{E}$ may be jointed by a path in $G'\cap\overline{E}$ of length (at most) $C.$ By Propositions A3 (after integration) and A4, we may find a constant $c>0$ such that $$k_{G'}(z,w)\le c||z-w||^{1/4} \le c^2,\ z,w\in G'\cap\overline{E},$$ $$k_{G'}\le c k_G,\ z,w\in G\setminus E.$$ So there are constants $c_1, c_2$ such that $$k_G\le k_{G'}\le c_1 k_G + c_2,\ z,w\in G\setminus E\mbox{ or }z,w\in G'\cap\overline{E}.$$
Finally, let $z\in G\setminus E$ and $w\in G'\cap\overline{E}.$ Since the Kobayashi distance is intrinsic and $\partial E$ is compact, we may find a point $u\in\partial E$ such that $$k_G(z,w)=k_G(z,u)+k_G(u,w).$$ It follows that $$k_G(z,w)\ge ck_{G'}(z,u)+ck_{G'}(u,w)\ge ck_{G'}(z,w).$$
[*Proof of Proposition \[compact\].*]{} It clear that $G'=G\setminus K$ is a domain. Following the previous proof, let $E$ be a domain such that $K\subset E\Subset G.$ All the arguments in the previous proof work except the estimate on $G'\cap\overline{E}.$ We need to find a constant $c>0$ such that $$k_{G'}(z,w)\le c,\ z,w\in G'\cap\overline{E}.$$
Take a complex line $L$ through $z$ which is disjoint from $K.$ Then the disc in $L$ with center of $z$ and radius $d_{G\cap L}(z)$ lies in $G'.$ Choose a common point $z'$ of this disc and $\partial E$ such that $||z-z'||=d_{E\cap L}(z).$ Then $$\tanh l_{G'}(z,z')\le\frac{||z-z'||}{d_{G\cap L}(z)}\le
1-\frac{r}{d_{G\cap L}(z)}\le 1-\frac{2r}{s},$$ where $r=\mbox{dist}(E,\partial G)$ and $s=\mbox{diam }G.$
Choosing $w'$ for $w$ in the same way, it follows that $$k_{G'}(z,w)\le k_{G'}(z,z')+k_{G'}(z',w')+k_{G'}(w',w)\le 2c'+c''$$ where $c'=\tanh^{-1}(1-2r/s)$ and $c''=\max k_{G'}|_{\partial E\times\partial E}.$
Appendix
========
\(a) Let $D$ be proper convex domain in $\Bbb C^n.$ Then $$c_{D}(z,w)\geq\frac{1}{2}\left|\log\frac{d_{D}(z)}{d_{D}(w)}\right|,\quad z,w\in D.$$
\(b) Let $D$ be proper $\Bbb C$-convex domain in $\Bbb C^n.$ Then $$c_{D}(z,w)\geq\frac{1}{4}\left|\log\frac{d_{D}(z)}{d_{D}(w)}\right|,\quad z,w\in D.$$
Let $b$ be a $\mathcal{C}^{1,1}$-smooth boundary point of a domain $D$ is $\Bbb C^n$ and let $K\Subset D.$ Then there exist a neighborhood $U$ of $b$ and a constant $C>0$ such that $$2k_D(z,w)\le-\log d_D(z)+C,\quad z\in D\cap U,\ w\in K.$$
Let $b$ is a $\mathcal{C}^2$-smooth non-pseudoconvex boundary point of a domain $D$ in $\Bbb C^2.$ Then there exist a neighborhood $U$ of $b$ and a constant $c>0$ such that $$c\kappa_D(z;X)\le\frac{|\langle\nabla d_D(z),X\rangle|}{(d_D(z))^{3/4}}+|X|,\quad z\in D\cap U,\ X\in\Bbb C^n.$$
Let $D$ be a bounded domain in $\Bbb C^n.$ Let $U$ and $V$ be neighborhoods of $\partial D$ with $V\Subset U.$ Then there exists a constant $c>0$ such that for any connected component $D'$ of $D\cap U$ one has that $$ck_{D'}(z,w)\le k_D(z,w),\quad z,w\in D'\cap V.$$
[*Proof.*]{} Let $\varepsilon>0.$ Take a smooth curve $\gamma:[0,1]\to D$ such that $\gamma(0)=z,$ $\gamma(1)=w$ and $$k_D(z,w,\varepsilon):=k_D(z,w)+\varepsilon>\int_0^1\kappa_{D}(\gamma (t);\gamma^{\prime}(t))dt.$$ Let $s=\sup\{t\in(0,1):\gamma(0,t)\subset D'\cap V\}$ and $r=\inf\{t\ge s:\gamma([t,1])\subset D'\cap V\}.$ Set $z'=\gamma(s)$ and $w'=\gamma(r).$ The localization property of the Kobayashi metric (cf. ) provides a constant $c'>0$ such that $$c'\kappa_{D'}(u;X)\le\kappa_D(u;X),\quad z\in {D'}\cap V,\ X\in\Bbb C^n.$$ It follows that $$\begin{gathered}
k_D(z,w,\varepsilon)>c'k_{D'}(z,z')+k_D(z',w')+c'k_{D'}(w',w)\\
\ge c'k_{D'}(z,w)+k_D(z',w')-c'k_{D'}(z',w').\end{gathered}$$
If $z'\neq w',$ then $z',w'\in {D'}\cap\partial V\Subset {D'}.$ Then there exists a constant $c_1>0$ such that $$k_{D'}(u,v)\le c_1||u-v||,\quad u,v\in {D'}\cap\partial V.$$ On the other hand, since $D$ is bounded, we may find a constant $c_2>0$ such that $$k_D(u,v)\ge c_2||u-v||,\quad u,v\in {D'}\cap\partial V.$$
Then $$k_D(z,w,\varepsilon)>c'k_{D'}(z,w)+(c_2-c'c_1)||z'-w'||.$$ Since $$k_D(z,w,\varepsilon)>k_D(z',w')\ge c_2||z'-w'||,$$ we get that $$k_D(z,w,\varepsilon)>c'k_{D'}(z,w)-(c'c_1/c_2-1)^+k_D(z,w,\varepsilon).$$
The last inequality also holds if $z'=w'.$ Letting $\varepsilon\to 0,$ we obtain that $$k_D(z,w)\ge\min\{c',c_2/c_1\}k_{D'}(z,w).$$
[M]{} Abouhajar, A., White, M., Young, N.: A Schwarz lemma for a domain related to $\mu$-synthesis, J. Geom. Anal. **17** (2007), 717–750. Andersson, M., Passare, M., Sigurdsson, R., Complex convexity and analytic functionals, Birkhäuser, Basel-Boston-Berlin, 2004. Agler, J., Young, N.J.: A commutant lifting theorem for a domain in $\Bbb C^2$ and spectral interpolation, J. Funct. Anal. **161** (1999), 452–477. Balogh, Z.M., Bonk, M.: Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains, Comment. Math. Helv. **75** (2000), 504–533. Buckley, S.: Gromov hyperbolicity of invariant metrics, preprint (2008); www.uma.es/investigadores/grupos/cfunspot/research/0806pBuckley.pdf. Costara, C.: On the spectral Nevanlinna-Pick problem, Studia Math. **170** (2005), 23–55. Dieu, N.Q., Nikolov N., Thomas P.J.: Estimates for invariant metrics near non-semipositive boundary points , J. Geom. Anal. **23** (2013), 598–610. Forstneric, F., Rosay, J-P.: Localization of the Kobayashi metric and the boundary continuity of proper holomorphic mappings, Math. Ann. **279** (1987), 239–252. Gaussier, H., Seshadri, H.: On the Gromov hyperbolicity of convex domains in $\mathbb{C}^{n}$, arXiv:1312.0368. Gromov, M.: Hyperbolic groups. Essays in group theory, 75–263, Math. Sci. Res. Ins. Publ., 8, Springer, New York, 1987. Jarnicki, M., Pflug, P.: Invariant distances and metrics in complex analysis, de Gruyter Exp. Math. 9, de Gruyter, Berlin-New York, 1993. Lempert, L.: La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France **109** (1981), 427–474. Nikolov, N.: Comparison of invariant functions on strongly pseudoconvex domains, J. Math. Anal. Appl. **421** (2015), 180–185. Nikolov, N., Pflug, P., Zwonek, W.: An example of a bounded $\Bbb C$-convex domain which is not biholomorphic to a convex domain, Math. Scan. **102** (2008), 149–155. Nikolov, N., Pflug, P., Zwonek, W.: Estimates for invariant metrics on $\mathbb{C}$-convex domains, Trans. Amer. Math. Soc. **363** (2011), 6245–6256. Nikolov, N., Trybuła, M.: The Kobayashi balls of ($\Bbb C$-)convex domains, Monatsh. Math. DOI 10.1007/s00605-015-0746-3. Rodriguez J.M., Touris, E.: Gromov hyperbolicity through decomposition of metrics spaces, Acta Math. Hung. **103** (2004), 107–138. Väisälä, J.: Gromov hyperbolic spaces, Expo. Math. **23** (2005), 187–231. Zwonek, W.: Geometric properties of the tetrablock, Arch. Math. **100** (2013), 159–165.
[^1]: Then D is biholomorphic to a bounded domain (cf. [@Jarnicki Theorem 7.1.8]).
[^2]: One can show that these distances are not bilipschitz equivalent.
[^3]: One can show that these distances are not bilipschitz equivalent if, for example, $K$ is a closed polydisc.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'One-dimensional structures of non-Hermitian plasmonic metallic nanospheres are studied in this paper. For a single sphere, solving Maxwell’s equations results in quasi-stationary eigenmodes with complex quantized frequencies. Coupled mode theory is employed in order to study more complex structures. The similarity between the coupled mode equations and the effective non-Hermitian Hamiltonians governing open quantum systems allows us to translate a series of collective phenomenon emerging in condensed matter and nuclear physics to the system of plasmonic spheres. A nontrivial physics emerges as a result of strong non-radiative near field coupling between adjacent spheres. For a system of two identical spheres, this occurs when the width of the plasmonic resonance of the uncoupled spheres is twice the imaginary component of the coupling constant. The two spheres then become coupled through a single continuum channel and the effect of coherent interaction between the spheres becomes noticeable. The eigenmodes of the system fall into two distinct categories: superradiant states with enhanced radiation and dark states with no radiation. The transmission through one-dimensional chains with an arbitrary number of spheres is also considered within the effective Hamiltonian framework which allows us to calculate observables such as the scattering and transmission amplitudes. This nano-scale waveguide can undergo an additional superradiance phase transition through its coupling to the external world. It is shown that perfect transmission takes place when the superradiance condition is satisfied.'
author:
- Amin Tayebi
- Scott Rice
bibliography:
- 'Refs.bib'
title: 'Superradiant and Dark States in Non-Hermitian Plasmonic Antennas and Waveguides'
---
[^1]
Introduction
============
Manipulation of light in nanometer scales via surface plasmonic resonances of metallic structures has attracted a great deal of attention over the past two decades [@intro1]. Optical antennas capable of localizing light in sub-wavelength regions have resulted in a new generation of photonic devices with applications ranging from imaging [@intro2; @intro2p5; @intro3] to biosensing [@intro4; @intro5] and emission enhancement of photon sources [@intro6; @intro7; @intro8]. In addition, plasmonics ought to play an important role in the efficient reception and transport of optical energy in light harvesting devices [@intro15]. Therefore, various waveguide structures are being investigated in order to control and further improve the propagation of light in micrometer length scales [@intro13; @intro14; @Plasmonics.9.925; @PhysRevB.82.035434].
More recently, it has been shown that surface plasmons can exhibit quantum interference [@intro9; @intro10]. This has sparked a great interest in studying the quantum properties of surface plasmons and exploiting plasmonic devices as potential building blocks of quantum computers and quantum circuits [@intro11; @intro12].
It was previously suggested that plasmonic structures could be mapped to quantum systems governed by non-Hermitian Hamiltonians [@NonhemitianPlasmonic1; @NonhemitianPlasmonic2]. In [@NonhemitianPlasmonic2], the radiation properties of an array of optical dipole antennas are manipulated by altering the anti-Hermitian coupling strength between the elements of the array. However, due to the complexity of nano-dipole antennas and the lack of closed form expressions for the fields, the mapping to a non-Hermitian Hamiltonian was achieved via numerical simulation and curve fitting. In this paper we consider systems of plasmonic nano-spheres using the effective non-Hermitian Hamiltonian framework. This framework provides a general platform for studying different physical systems; it has been previously utilized in various problems ranging from quantum signal transmission in nano-structures [@celardo09; @Greenberg_transport] to solid state quantum computing [@Tayebi_qc; @PhysRevA.78.062116] and nuclear reactions [@VZ777; @AZ07; @PhysRevLett.115.052501; @DrZ_New; @PhysRevC.86.044602]. The description of plasmonic structures via the effective Hamiltonian is achieved due to the correspondence between the Feshbach formalism and the coupled mode theory of optical resonators. The effective non-Hermitian Hamiltonian approach allows us to translate phenomena already known in condensed matter and nuclear systems to the plasmonic system under study. One such example is the existence of superradiant and subradiant, or dark, states. In addition, the effective Hamiltonian framework can be readily used in order to calculate observables, such as the transmission coefficient through a plasmonic waveguide.
In Section \[secII\] we discuss the effective non-Hermitian Hamiltonian formalism used for studying *open* system in condensed matter [@Greenberg_transport; @Tayebi_qTrans; @Amin_Scott_Conf; @YakovG3], nuclear physics [@Volya_NucP; @Volya2014_new; @Volya_new2; @AUERBACH200445] and quantum optics [@YakovG1; @YakovG2; @YakovG4]. We then discuss the coupled mode formalism used for studying optical and plasmonic systems. The connection between the effective Hamiltonian and coupled mode theory is illustrated through simple two-level examples. We then consider a single plasmonic metallic nanosphere.
Section \[secIII\] considers a single plasmonic metallic nanosphere. The wave equation is solved in order to find the natural resonant frequencies of the single sphere, and discuss the intrinsic radiative nature of nanospheres. This provides a basis for the consideration of two spheres via coupled mode theory.
The signature of superradiance emerges when the interaction between adjacent optical nano antennas occurs through a single continuum channel, resulting in states with enhanced radiation and confined dark modes. This is discussed in Section \[secIV\]. The effect of these states on energy transmission through a one-dimensional chain of spheres is considered in Section \[secV\], with applications to optical frequency nanoscale antennas and waveguide-like structures.
Section \[secVI\] includes the summary, concluding remarks and future work.
The effective non-hermitian hamiltonian and coupled mode theory {#secII}
===============================================================
In this section we briefly discuss the effective non-Hermitian Hamiltonian approach and the coupled mode theory and show how both theories result in matrices of similar structures.
The Effective Hamiltonian
-------------------------
Consider a quantum system described by the Hamiltonian $H_0$ and its discrete set of eigenvectors $\ket{i}$ that interacts with its surrounding environment. The environment is thermodynamically large and is characterized by infinitely many channels with continuous energy spectrum $\ket{c;E}$. The Hilbert space of this problem can be divided into two subspaces with the help of projection operators $\mathcal{Q}$ and $\mathcal{P}$. The operator $\mathcal{Q}$ only acts on the subspace of the closed system and the operator $\mathcal{P}$ acts on the environment only. Without loss of generality, one can further assume that the projections are orthogonal, therefore $\mathcal{P}+\mathcal{Q}=1$ and $\mathcal{P}\mathcal{Q}=\mathcal{Q}\mathcal{P}=0$. The total Hamiltonian $H$ i.e. the Hamiltonian of the system, the environment and their interactions, is then decomposed into $H=H_{\mathcal{Q}\mathcal{Q}}+H_{\mathcal{Q}\mathcal{P}}+H_{\mathcal{P}\mathcal{Q}}+H_{\mathcal{P}\mathcal{P}}$, where $H_{\mathcal{Q}\mathcal{Q}}=\mathcal{Q}H\mathcal{Q}$, $H_{\mathcal{Q}\mathcal{P}}=\mathcal{Q}H\mathcal{P}$, $H_{\mathcal{P}\mathcal{Q}}=\mathcal{P}H\mathcal{Q}$ and $H_{\mathcal{P}\mathcal{P}}=\mathcal{P}H\mathcal{P}$. The effective Hamiltonian is achieved by projecting the stationary wave function in the Schrödinger equation $H\Psi=E\Psi$ into the subspace of the closed system $$\label{effectiveHamiltonian1}
\mathscr{H}_{\text{eff}} (E) \mathcal{Q} \Psi = E\mathcal{Q} \Psi,$$ where $$\label{effectiveHamiltonian2}
\mathscr{H}_{\text{eff}} (E) = H_{\mathcal{Q}\mathcal{Q}} + H_{\mathcal{Q}\mathcal{P}} \frac{1}{E-H_{\mathcal{P}\mathcal{P}} } H_{\mathcal{P}\mathcal{Q}} .$$ The first term in the left handside of (\[effectiveHamiltonian2\]) is the Hamiltonian of the closed system $\mathcal{Q} H \mathcal{Q}=H_0$. The second term in (\[effectiveHamiltonian2\]), can be further simplified by calculating the matrix element between two intrinsic states of the closed system $\ket{i}$ and $\ket{j}$ $$\label{effectiveHamiltonian3}
\langle i| H_{\mathcal{Q}\mathcal{P}} \frac{1}{E-H_{\mathcal{P}\mathcal{P}} } H_{\mathcal{P}\mathcal{Q}} \ket{j}= \sum_{c}\int{dE'\frac{A_{i}^{c}(E'){A_{j}^{c}}^{*}(E')}{E-E'}},$$ where $A_{i}^{c}(E)$ are transition amplitudes from continuum channel $\ket{c;E}$ to internal state $\ket{i}$; $A_{i}^{c}(E)=\langle i|H_{\mathcal{Q}\mathcal{P}}\ket{c;E}$. Usually continuum channels couple to the system only if the energy is above a certain threshold in which case the channel is open, otherwise the channel is closed and the transition amplitude vanishes. Using the Sokhotski-Plemelj theorem, the integral in (\[effectiveHamiltonian3\]) can be decomposed into Hermitian and anti-Hermitian parts $$\label{effectiveHamiltonian4}
\vspace*{-16mm} \sum_{c} \int{dE'}\frac{A_{i}^{c}(E'){A_{j}^{c}}^{*}(E')}{E-E'} = \\$$ $$\sum_{c}\mathcal{P}.\mathcal{V}.\int{dE'\frac{A_{i}^{c}(E'){A_{j}^{c}}^{*}(E')}{E-E'}}
- i \pi \sum_{c_{open}} A_{i}^{c}(E){A_{j}^{c}}^{*}(E), \nonumber$$ where P.V. denotes the Cauchy principal value. Accordingly, two operators are defined; $\Delta(E)$, corresponding to the Hermitian component in (\[effectiveHamiltonian4\]) with matrix elements $$\label{realpartHeff}
\Delta_{i,j}(E) = \sum_{c}\mathcal{P}.\mathcal{V}.\int{dE'\frac{A_{i}^{c}(E'){A_{j}^{c}}^{*}(E')}{E-E'}},$$ and $W(E)$ corresponding to the anti-Hermitian component in (\[effectiveHamiltonian4\]), with matrix elements $$\label{imagpartofHeff}
W_{ij}(E)= 2\pi \sum_{c_{open}} A_{i}^{c}(E){A_{j}^{c}}^{*}(E).$$ Thus, in operator form, the *energy-dependent* effective Hamiltonian is $$\label{effectiveHamiltonian_complete}
\mathscr{H}_{\text{eff}} (E)= H_0+\Delta(E)-\frac{i}{2}\,W(E).$$ The two terms, $\Delta(E)$ and $W(E)$, also known as the *self energy*, completely take the effect of the interaction with the environment into account. The Hermitian part, $\Delta(E)$, renormalizes the energies of the closed system. Notice that the summation in (\[realpartHeff\]) runs over all continuum channels, open and close. This is because even when the running energy, $E$, is below the threshold such that $A_{i}^{c}(E)$ vanishes, the integral is still non-vanishing and thus takes the *virtual* coupling to the continuum into account. On the other hand, $W(E)$ which is responsible for the decay width of the energy states of the effective Hamiltonian (\[effectiveHamiltonian\_complete\]), arises only due to real interaction processes with the environment.
In many situations, including the case we are concerned with in this paper, the energy window of interest is relatively narrow and the transition amplitudes $A_{i}^{c}(E)$ are smooth functions of energies. These amplitudes can therefore be considered as energy-independent quantities. Consequently, the integral in the Hermitian component of the self energy (\[realpartHeff\]) vanishes and the effective Hamiltonian reduces to $$\label{ReducedHeff}
\mathscr{H}_{\text{eff}}=H_{0}-\frac{i}{2}\,W.$$
It is interesting to look at the statistics of the *quasi-stationary* eigenenergies of the effective Hamiltonian $$\label{Heff_energies}
E_{n}=\mathrm{E}_{n}-\frac{i}{2}\Gamma_n,$$ where $\mathrm{E}_{n}$ is the real component of the energy and $\Gamma_n$ is the width of the state and related to the lifetime by $\tau_n=\hbar/\Gamma_n$. The positiveness of $\Gamma_n$ is guaranteed by the Cholesky factorized form of $W$ in (\[imagpartofHeff\]) which makes $W$ a positive definite matrix. We define the quantity $\xi$ that parameterizes the strength of interaction with the external world, thus $$\label{ReducedHeff_xi}
\mathscr{H}_{\text{eff}}=H_{0}-\frac{i}{2}\xi\,W.$$ Because the framework is exact and no approximation was used, $\xi$ can take arbitrarily small values representing weak interactions or extremely large values representing strong interactions with the external world. For weak interactions, when $\xi$ is small, the anti-Hermitian component, $W$ is a perturbation to $H_0$. The complex eigenenergies are then narrow resonances with almost uniform width distribution [@SOKOLOV1; @Volya2016]. In the opposite limit of strong interactions, the anti-Hermitian component becomes the dominant term and $H_0$ is the perturbation. It is clear from (\[imagpartofHeff\]) that the rank of $W$ is equal to the number of open channels which is normally much smaller than the number of intrinsic states of the closed system. Therefore a few states become dominant resonances that consume the entire width and the remaining states become long-lived states that decouple from the environment. Due to the resemblance of this phenomenon to the Dicke superradiance in quantum optics, we term the broad short-lived resonances as superradiant states and the narrow short-live resonances as subradiant states. It was shown [@Tayebi_qc; @Amin_Thesis] that if the system is connected to the environment through its physical boundaries then the superradiant states are localized to the boundaries of the system, and the subradiant states are pushed away from the boundaries and trapped within the interior of the system. The superradiance *phase transition* is discussed more rigorously in [@SOKOLOV1; @Auerbach]. It was shown that the important parameter to consider is the ratio of $\langle \Gamma \rangle$, the average widths of the energies in (\[Heff\_energies\]), to $D$, the mean level spacing of the close system. The transition occurs when $\langle \Gamma \rangle / D \approx 1$, this is when the resonances are maximally overlapped and the superradiant states emerge.
The effective non-Hermitian Hamiltonian framework also provides us with useful expressions corresponding to observables such as the scattering and transmission amplitudes [@SOKOLOV1; @Auerbach; @SOKOLOV2; @Amin_Thesis]. Here we are particularly interested in transmission through a one dimensional chain of metallic nano-spheres. The transmission amplitude from a continuum channel $a$ to channel $b$ through the system is given by $$\label{Transmission_amp}
Z^{ab}(E)=\sum_{i,j} A_{i}^{a} \bigg( \frac{1}{E-\mathscr{H}_{\text{eff}}} \bigg)_{i,j} {A_{j}^{b}}^*.$$ The transmission probability, $T^{ab}(E)$ is therefore $$\label{ProcessAmplitude}
T^{ab}(E)=|Z^{ab}(E)|^2.$$ The amplitude in (\[Transmission\_amp\]) has a simple interpretation: the particle enters into state $\ket{j}$ through channel $b$. Next the propagator takes the particle from state $\ket{j}$ to $\ket{i}$ considering all possible *paths*. Finally the particle escapes the system from site $\ket{i}$ through continuum channel $a$.
Coupled Mode Theory
-------------------
In this paper, we employ coupled mode theory in order to study systems of interacting plasmonic spheres. This method is reminiscent of the time-dependent perturbation theory in quantum mechanics; it has been used for the investigation of coupled resonators in optical systems [@CMT_optic1; @CMT_optic2; @CMT_optic3], wireless energy transfer loop antennas [@CMT_antennas1], and plasmonic structures including antennas [@NonhemitianPlasmonic2; @CMT_antennas1] and waveguides [@CMT_waveguide1; @CMT_waveguide2; @CMT_waveguide3]. The coupled mode approach significantly simplifies the complexity of the problem: instead of solving the wave equation one needs to solve a system of linear algebraic equations. In addition, it provides a clear and intuitive picture of how interactions between the constituents of the system can dramatically change the dynamics.
The formulation provided in this paper, similar to [@cmt_qnm_new], is rather general. No specific boundary conditions are assumed and hence it is applicable to the system of coupled plasmonic nanoantennas discussed in the future sections.
Consider two non-magnetic dielectric resonators with relative dielectric constants $\epsilon_1(\vec{r})$ and $\epsilon_2(\vec{r})$. The resonators occupy a volume in space, $V_1$ and $V_2$, respectively. In addition, consider that the relative dielectric constants $\epsilon_1(\vec{r})$ and $\epsilon_2(\vec{r})$ are equal to unity for points outside of the resonators. In a time harmonic scenario each resonator, when isolated, satisfies the wave equation $$\label{wave_eqn_cmt}
\vec{\nabla} \times \vec{\nabla} \times \vec{\mathcal{E}}^{\alpha}_n(\vec{r})-\bigg(\frac{ \omega_{\alpha , n}}{c}\bigg)^2 \epsilon_{\alpha} (\vec{r}) \vec{\mathcal{E}}^{\alpha}_n(\vec{r})=0,$$ where $\alpha=1, 2$ denotes the resonator number and $c$ is the speed of light in the background medium which is assumed to be the free space for simplicity. Due to the sharp discontinuity between the resonator and the background at the resonator boundaries, the modes are quantized and characterized by the integer number $n=1, 2, 3, ...$ and their eigenfrequency $\omega_{\alpha,n}$. The modes of isolated resonators are normalized according to $$\label{normalization_isolated_mode_gen}
\int_{V} \epsilon_{\alpha}(\vec{r}) {\vec{\mathcal{E}}^{\alpha^*}_m}(\vec{r}). \vec{\mathcal{E}}^{\alpha}_n(\vec{r}) d^3r = \delta_{mn},$$ where $V$ is the total volume in which fields are present and $m$ and $n$ are the mode indices and $\delta_{mn}$ is the Kronecker delta. The normalization expression is of crucial importance and its form is dictated by the boundary conditions of the problem. For instance in [@cmt_qnm_new] the normalization is similar to (\[normalization\_isolated\_mode\_gen\]), however with no complex conjugation. As we will see, the normalization expression has to be modified when discussing spherical plasmonic particles in the following sections, in order for the normalization expression to remain finite when integrated over all space. For now, we assume that the normalization rule is given by the general dot product definition provided in (\[normalization\_isolated\_mode\_gen\]) with the integration volume being all space, as this is usually the case in electromagnetic textbooks. We will revisit the normalization definition later in this paper.
Next we assume that, for a system of two coupled resonators, the total electric field, $\vec{\mathcal{E}}(\vec{r})$, can be written as a superposition of a finite number of individual modes of the two resonators: $$\label{ansatz_cmt}
\vec{\mathcal{E}}(\vec{r})=\sum_{n=1}^{N} \Big[ a_1(n) \vec{\mathcal{E}}^{1}_n(\vec{r}) + a_2(n) \vec{\mathcal{E}}^{2}_n(\vec{r})\Big],$$ where $N$ is the total number of modes. The total electric field satisfies the wave equation $$\label{wave_eqn_cmt_total_field}
\vec{\nabla} \times \vec{\nabla} \times \vec{\mathcal{E}}_n(\vec{r})-\Big(\frac{\omega_n}{c}\Big)^2 \epsilon(\vec{r}) \ \vec{\mathcal{E}}_n(\vec{r})=0,$$ where $\omega_n$ are the eigenfrequencies of the coupled system and $\epsilon(\vec{r})$ is the dielectric constant at a given point, $\vec{r}$, when both resonators are simultaneously present. The function $\epsilon(\vec{r})$ is equal to $\epsilon_1(\vec{r})$ and $\epsilon_2(\vec{r})$ for points inside the first and second resonator, respectively, and is equal to unity otherwise. Plugging the ansatz (\[ansatz\_cmt\]) into the wave equation (\[wave\_eqn\_cmt\_total\_field\]) and using its linearity and (\[wave\_eqn\_cmt\]) we arrive at $$\begin{aligned}
\label{series_eqn}
\sum_{n=1}^N \Big[ & a_1(n) \big(\omega_{1,n}\big)^2 \epsilon_{1} (\vec{r}) \vec{\mathcal{E}}^{1}_n(\vec{r}) + a_2(n) \big(\omega_{2,n}\big)^2 \epsilon_{2} (\vec{r}) \vec{\mathcal{E}}^{2}_n(\vec{r}) \Big] \nonumber \\
& =\omega_n^2 \epsilon (\vec{r}) \sum_{n=1}^N \Big[ a_1(n) \vec{\mathcal{E}}^{1}_n(\vec{r}) + a_2(n) \vec{\mathcal{E}}^{2}_n(\vec{r}) \Big].\end{aligned}$$ Using the normalization rule (\[normalization\_isolated\_mode\_gen\]) to project (\[series\_eqn\]) onto $\vec{\mathcal{E}}^{1}_m(\vec{r})$ and $ \vec{\mathcal{E}}^{2}_m(\vec{r})$ for all values of $m$: $m=1,2,..,N$, we obtain a system of $2N$ linear equations. In the matrix form $$\label{cmt_equation_complete_form}
\begin{pmatrix}
\bm{T^{11}} & \bm{T^{12}} \\
\bm{T^{21}} & \bm{T^{22}}
\end{pmatrix}
\begin{pmatrix}
\bm{\Omega_{1}}^2 & \bm{0} \\
\bm{0} & \bm{\Omega_{2}}^2
\end{pmatrix}
\begin{bmatrix}
\vec{A}_1 \\
\vec{A}_2
\end{bmatrix}=
\omega^2
\begin{pmatrix}
\bm{L^{11}} & \bm{L^{12}} \\
\bm{L^{21}} & \bm{L^{22}}
\end{pmatrix}
\begin{bmatrix}
\vec{A}_1 \\
\vec{A}_2
\end{bmatrix},$$ where $\vec{A}_1$ and $\vec{A}_2$ are $N \times 1$ vectors of the coefficients $a_1(n)$ and $a_2(n)$ in the ansatz (\[ansatz\_cmt\]), respectively. $\bm{\Omega_{1}}$ and $\bm{\Omega_{2}}$ are $N \times N$ diagonal matrices containing the eigenfrequencies of the isolated resonators with matrix elements $$\big(\Omega_{\alpha}\big)_{mn}= \omega_{\alpha,n} \ \delta_{mn}$$ where as previously $\alpha=1, 2$. The matrix elements of the four square $N \times N$ matrices $\bm{T^{\alpha \beta}}$, where $\alpha, \beta=1, 2$, are given by $$T^{\alpha\beta}_{mn}= \int_{V} \epsilon_{\beta}(\vec{r}) {\vec{\mathcal{E}}^{\alpha^*}_m}(\vec{r}). \vec{\mathcal{E}}^{\beta}_n(\vec{r}) d^3r.$$ According to (\[normalization\_isolated\_mode\_gen\]), $\bm{T^{11}}$ and $\bm{T^{22}}$ are equal to the identity matrix $\bm{1}$. Finally the elements of the matrices $\bm{L^{\alpha, \beta}}$ are given by $$\label{L_matrix_cmt}
L^{\alpha\beta}_{mn}=\int_{V} \epsilon(\vec{r}) {\vec{\mathcal{E}}^{\alpha^*}_m}(\vec{r}). \vec{\mathcal{E}}^{\beta}_n(\vec{r}) d^3r.$$ Because the dielectric function $\epsilon(\vec{r})$ is the sum of the two dielectric function, the matrix elements $L^{12}_{m,n}$ and $T^{12}_{m,n}$ are related via $$\label{T_matrix_cmt}
L^{12}_{mn}=T^{12}_{mn}+\int_{V_1} \big(\epsilon_1(\vec{r})-1 \big) \vec{\mathcal{E}}^{1^{*}}_m(\vec{r}).\vec{\mathcal{E}}^{2}_n(\vec{r}) d^3r,$$ where the integration is carried out over the volume of the first resonator, $V_1$ only. Accordingly, we define the matrix $\bm{K^{12}}$ with matrix elements $$\label{coupling_coeff_K1}
K^{12}_{mn}=\int_{V_1} \big(\epsilon_1(\vec{r})-1 \big) \vec{\mathcal{E}}^{1^{*}}_m(\vec{r}).\vec{\mathcal{E}}^{2}_n(\vec{r}) d^3r.$$ Therefore $$\label{kappa_matrix1}
\bm{L^{12}}=\bm{T^{12}}+\bm{K^{12}}.$$ Similarly $L^{21}_{mn}$ is related to $T^{21}_{mn}$ as $$L^{21}_{mn}=T^{21}_{mn}+\int_{V_2} \big(\epsilon_2(\vec{r})-1 \big) \vec{\mathcal{E}}^{2^{*}}_m(\vec{r}).\vec{\mathcal{E}}^{1}_n(\vec{r}) d^3r.$$ Correspondingly $\bm{K^{21}}$ is defined with matrix elements $$\label{coupling_coeff_K2}
K^{21}_{mn}=\int_{V_2} \big(\epsilon_2(\vec{r})-1 \big) \vec{\mathcal{E}}^{2^{*}}_m(\vec{r}).\vec{\mathcal{E}}^{1}_n(\vec{r}) d^3r.$$ Hence $$\label{kappa_matrix2}
\bm{L^{21}}=\bm{T^{21}}+\bm{K^{21}}.$$
In order to simplify (\[cmt\_equation\_complete\_form\]), we accept a number of approximations that are commonly used in studying systems of weakly coupled resonators [@Elnaggar_app; @Elnaggar_CMT1; @Elnaggar_CMT2]. We assume that the diagonal matrix elements in (\[L\_matrix\_cmt\]) are approximately equal to unity, i.e. $L^{11}_{m,n}=L^{22}_{m,n} \approx 1$ and therefore $\bm{L^{11}}=\bm{L^{22}}\approx \bm{1}$. This is justified due to the strong field confinement within the dielectric regions. Furthermore, we assume that the coupling is weak and therefore the coupling elements in (\[T\_matrix\_cmt\]) satisfy the condition $T^{12}_{mn} T^{21}_{m'n'} \ll 1$. Using these approximations along with (\[kappa\_matrix1\]) and (\[kappa\_matrix2\]), the coupled mode equation (\[cmt\_equation\_complete\_form\]) reduces to $$\label{cmt_equation_reduced_form1}
\begin{pmatrix}
\bm{1} & -\bm{K^{12}} \\
-\bm{K^{21}} & \bm{1}
\end{pmatrix}
\begin{pmatrix}
\bm{\Omega_{11}^2} & \bm{0} \\
\bm{0} & \bm{\Omega_{22}^2}
\end{pmatrix}
\begin{bmatrix}
\vec{A}_1 \\
\vec{A}_2
\end{bmatrix}=
\omega^2
\begin{bmatrix}
\vec{A}_1 \\
\vec{A}_2
\end{bmatrix}.$$ It is also helpful to linearize the system of equations (\[cmt\_equation\_reduced\_form1\]). This can be done by noting that the eigenmodes of the isolated resonators are not far apart and are clustered around their mean value [@Haus_CMT], i.e. $\omega \approx \omega_{\alpha,n}$. Under this approximation $\omega$ and $\omega_{\alpha,n}$ satisfy the following $$\omega^2-\big(\omega_{\alpha,n} \big)^2 \approx 2\omega_{\alpha,n}(\omega-\omega_{\alpha,n} ).$$ This brings us to the final form of the coupled mode equations $$\label{cmt_equation_reduced_form2}
\begin{pmatrix}
\bm{1} & -\frac{1}{2}\bm{K^{12}} \\
-\frac{1}{2}\bm{K^{21}} & \bm{1}
\end{pmatrix}
\begin{pmatrix}
\bm{\Omega_{11}} & \bm{0} \\
\bm{0} & \bm{\Omega_{22}}
\end{pmatrix}
\begin{bmatrix}
\vec{A}_1 \\
\vec{A}_2
\end{bmatrix}=
\omega
\begin{bmatrix}
\vec{A}_1 \\
\vec{A}_2
\end{bmatrix}.$$
In the simplest situation when the two resonators are identical and only one mode of an isolated resonator is considered, the electric field of the coupled system can be expressed as $\vec{\mathcal{E}}(\vec{r})= a_1 \vec{\mathcal{E}}^1(\vec{r}) + a_2 \vec{\mathcal{E}}^1(\vec{r})$. According to (\[cmt\_equation\_reduced\_form2\]) the coupled mode equations are then given by $$\begin{aligned}
\label{CMT2res_timeDomain}
\omega_0 a_1 + \kappa \ a_2=\omega a_1, \nonumber \\
\omega_0 a_2 + \kappa^*a_1=\omega a_2, \end{aligned}$$ where $\omega_0$ is the eigenfrequency of the isolated resonators. Using (\[coupling\_coeff\_K1\]) and (\[cmt\_equation\_reduced\_form2\]), $\kappa$ is given by $$\label{coupling_coefficient}
\kappa=-\frac{1}{2}\omega_0 \int_{\text{V}_1} \big(\epsilon_1(\vec{r})-1 \big) \vec{\mathcal{E}}^{1^*}(\vec{r}). \vec{\mathcal{E}}^2(\vec{r}) d^3r.$$ The coupling coefficient $\kappa$ has a simple interpretation: it is the interaction energy between the field generated by the second resonator and the dipole moment of the first resonator averaged over one period. The complex conjugation of the coupling coefficient in (\[CMT2res\_timeDomain\]) is dictated by the energy conservation, assuming there is no loss or gain in the system [@Haus_CMT]. The eigenfrequencies of the coupled system, which are guaranteed to be real due to the Hermitian form of the equations in (\[CMT2res\_timeDomain\]), are $$\omega_{\pm}=\omega_0 \pm |\kappa|,$$ where the frequencies of the coupled system, $\omega_{+}$ and $\omega_{-}$, correspond to the symmetric eigenstate with $a_1=a_2=1/\sqrt{2}$ and the anti-symmetric eigenstate with $a_1=-a_2=1/\sqrt{2}$, respectively.
One can readily see the similarity between the coupled mode theory and the quantum theory as both are a theory of waves. Equation (\[CMT2res\_timeDomain\]) is the Schrödinger equation for a two-level system (a qubit). In quantum mechanical language, $\omega_0$ is the energy of the *unperturbed* states and the off-diagonal matrix element $\kappa$ represents the interaction strength between the two states which is responsible for the level repulsion and avoided crossing of the final *mixed* states.
An interesting dynamic of the two-level system is the so-called Rabi oscillation. Let the system at time $t=0$ be prepared in the unperturbed state with energy $\omega_1$. Then the probability $P(t)$ to find the system in the same state at time $t$ is [@zelevinskybook] $$P(t)=1-\text{sin}^2\Big(\frac{\omega_R t}{2} \Big),$$ where $\omega_R=\omega_{+}-\omega_{-}=2|\kappa|$ is the Rabi frequency of the excitation oscillating back and forth between the two levels. Because the unperturbed energies of the two states are equal, the probability goes through the minimum, $P=0$, which indicates that the excitation can be completely transferred, leaving no residue in the initial state. The Rabi oscillation was predicted in systems of optical waveguides [@Rabi_waveguide] and in coupled ring resonators [@Rabi_ringRes].
We now focus on a more realistic case: two coupled identical dielectric resonators where in general, due to damping and leakage of the resonators, the energy is no longer conserved. Therefore the governing equations need not be Hermitian. In the case of open systems one has to modify the normalization expression (\[normalization\_isolated\_mode\_gen\]) which leads to an altered expression for $\kappa$. The coupling coefficient in this case is similar to (\[coupling\_coefficient\]) but with no complex conjugation (see [@cmt_qnm_new] for detail). The coupled equations (\[CMT2res\_timeDomain\]) are modified to a more general form $$\begin{aligned}
\label{CMT2res_freqDomian}
\omega_0 a_1 + \kappa a_2 &=\omega a_1 \nonumber \\
\omega_0 a_2 + \kappa a_1 &=\omega a_2, \end{aligned}$$ where $\omega_0$ can now be complex: $\omega_0=\eta_0-i\gamma_0/2$ representing loss and radiation. Similarly, $\kappa$ is in general complex as well: $\kappa=\kappa'-i\kappa''$ where $\kappa'$ and $\kappa''$ are real numbers. The left hand side of the coupled equations (\[CMT2res\_freqDomian\]) can then be written as the summation of two matrices, a Hermitian matrix, $H'_0$, and an anti-Hermitian matrix $W'$: $$\label{CMT_to_eff_Hamiltonian}
\begin{pmatrix}
\omega_0 & \kappa \\
\kappa & \omega_0
\end{pmatrix}=
H'_0-\frac{i}{2}W',$$ where $$\label{CMT_Matrix_hermitian}
H'_0=
\begin{pmatrix}
\eta_0 & \kappa' \\
\kappa' & \eta_0
\end{pmatrix},$$ and $$\label{CMT_Matrix_nonhermitian}
W'=
\begin{pmatrix}
\gamma_0 & 2\kappa'' \\
2\kappa'' & \gamma_0
\end{pmatrix}.$$ The similarity between the coupled mode theory and the effective non-Hermitian Hamiltonian formalism becomes apparent by comparing (\[ReducedHeff\]) and (\[CMT\_to\_eff\_Hamiltonian\]). The problem of two coupled dielectric resonators is mapped to a two level quantum system where in general each level is coupled to an independent continuum channel. This is because in general the rank of $W'$ is 2, therefore, according to (\[imagpartofHeff\]), one requires two independent open channels to construct the anti-Hermitian matrix $W'$. In the particular case when $\gamma_0=\pm 2\kappa''$, the rank of $W'$ is equal to unity, therefore the superradiance condition is fulfilled and only one open channel is required to construct the matrix $W'$. In this case, the effective Hamiltonian has two distinct eigenvalues, a purely real eigenvalue or the subradiant state , reminiscent of dark modes in open quantum systems, with eigenfrequency $\eta_0-\kappa'$, and a complex eigenvalue with enhanced radiation properties and eigenfrequency $\eta_0+\kappa'-i\gamma_0$, which is the superradiant state.
A Single Metallic Sphere {#secIII}
========================
In this section we consider a single isolated metallic sphere embedded in a homogeneous background dielectric material. The problem is treated classically by solving Maxwell equations. In the absence of external sources and assuming a harmonic time dependence of the form $$\label{Phase_convention}
\vec{\mathbb{E}}(\vec{r},t)=e^{i\omega t} \vec{\mathcal{E}}(\vec{r}),$$ the governing equation is the well known Helmholtz equation $$\label{Helmholtz_eqn}
\nabla^2 \vec{\mathcal{E}}(\vec{r})-k^2\vec{\mathcal{E}}(\vec{r})=0.$$ The wave number $k$ is defined for the interior and exterior regions of the plasmonic sphere according to $$\begin{aligned}
k = \begin{cases}
k_{\text{in}}=\frac{\omega}{c} \sqrt{\epsilon_{\text{in}}} & r\leq a,\\
k_{\text{out}}=\frac{\omega}{c} \sqrt{\epsilon_{\text{out}}} & r> a,
\end{cases}\end{aligned}$$ where $c$ is the speed of light in vacuum and $a$ is the radius of the sphere which is located at the origin of the coordinate system. The relative dielectric constants of the metallic sphere and the background medium are denoted by $\epsilon_{\text{in}}$ and $\epsilon_{\text{out}}$, respectively. Plasmonic structures are usually made of noble metals, such as gold and silver, with face-centered cubic lattice, or alkali metals, such as sodium and potassium, with body-centered cubic lattice. Due to their symmetric crystal lattice types, they are isotropic to light and their relative permittivity is characterized by a scalar. This dielectric constant is well described by the Drude-Sommerfeld model $$\begin{aligned}
\label{Drude_dielectric_func}
\epsilon_{\text{in}}(\omega)=\epsilon_{\infty}-\frac{\omega_p^2}{\omega^2-i\omega\gamma_s},\end{aligned}$$ where $\omega_p$ is the plasma frequency which is defined by the electron effective mass $m^*$, the vacuum permittivity $\epsilon_0$, electron charge $e$, and electron density $n$; $\omega_p^{2}= ne^2/\epsilon_0m^*$. The loss within the dielectric material due to various processes, such as electron-phonon interaction, impurities and scattering, is incorporated into the relaxation rate $\gamma_s$. The negative sign of this term in the denominator is dictated by the phase convention adopted in (\[Phase\_convention\]). The phenomenological parameter $\epsilon_{\infty}$ accounts for the contribution of the bound electrons to the polarization of the dielectric material. For typical metals the plasma frequency, $\omega_p$, ranges from 3 to 15 eV (700-3600 THz) which mainly falls into the ultraviolet spectrum [@Drude_model_plasma_1; @Drude_model_plasma_2; @Drude_model_plasma_3; @Drude_model_plasma_4; @Drude_model_plasma_5]. The damping rate $\gamma_s$ is much smaller than the plasma frequency, $\gamma_s \ll \omega_p$, being of the order $10^{-2}-10^{-1}$ eV (2.4-24 THz). Finally, the correction term $\epsilon_{\infty}$ typically ranges from 1 to 10 [@Silver_drude]. In an ideal electron gas, $\epsilon_{\infty}=1$ and $\gamma_s=0$, therefore the dielectric function (\[Drude\_dielectric\_func\]) reduces to $\epsilon_{\text{in}}(\omega)=1-\omega_p^2/\omega^2$. Below the plasma frequency the dielectric function is negative and the field can not penetrate inside the metal. For frequencies larger than the plasma frequency however, the dielectric constant becomes positive and the fields can penetrate the metal i.e. the metal becomes transparent.
Using the spherical coordinate system, the solutions of the Helmholtz equation (\[Helmholtz\_eqn\]) for the plasmonic sphere can be divided into two categories: transverse magnetic (TM) modes with no radial magnetic field and transverse electric (TE) modes with no radial electric field component. In this work we consider the TM modes only. The components of the electric field are given by [@harringtonbook]: $$\begin{aligned}
\label{field_exprs}
\mathcal{E}_r&=\zeta \ C(r;a) \ell(\ell+1) \frac{f_{\ell}(kr)}{\epsilon(r)kr} Y_\ell^m(\theta,\phi), \nonumber \\
\mathcal{E}_\theta&=\zeta \ C(r;a)\frac{1}{\epsilon(r)kr} \frac{\partial}{\partial(kr)}\Big(krf_\ell(kr)\Big)\frac{\partial}{\partial \theta} Y_\ell^m(\theta,\phi), \\
\mathcal{E}_\phi&=\zeta \ C(r;a)\frac{1}{\epsilon(r)kr} \frac{\partial}{\partial(kr)}\Big(krf_\ell(kr)\Big)\frac{1}{\sin\theta}\frac{\partial}{\partial \phi}Y_\ell^m(\theta,\phi), \nonumber\end{aligned}$$ where $\zeta$ is the normalization constant discussed in detail in the next section. The dielectric constant is equal to $\epsilon_{\text{in}}$ and $\epsilon_{\text{out}}$ for $r\leq a$ and $r>a$, respectively. $Y_\ell^m(\theta,\phi)$ are the spherical harmonics with $\ell=0,1,2,...$ and $m=0,\pm 1,\pm 2,...,\pm \ell$. The case of $\ell=0$ results in the trivial solution. The first non-trivial solution corresponds to the dipole mode, $\ell=1$. The function $f_{\ell}(kr)$ is equal to the spherical Bessel function of the first kind and the spherical Hankel function of the second kind, for $r\leq a$ and $r>a$, respectively. $$\begin{aligned}
f_{\ell}(kr) = \begin{cases}
j_{\ell}(k_{\text{in}}r) & r\leq a,\\
h_{\ell}^{(2)}(k_{\text{out}}r) & r> a.
\end{cases}\end{aligned}$$ The Bessel function $j_{\ell}(k_{\text{in}}r)$ represent standing waves within the plasmonic sphere while, noting the phase convention (\[Phase\_convention\]), the Hankel function $h_{\ell}^{(2)}(k_{\text{out}}r)$ describes radially outward traveling waves which satisfy the Sommerfeld radiation boundary condition. The coefficient $C(r;a)$ guarantees that the boundary conditions are satisfied at the boundary of the sphere (see [@SingleSph1] for details) $$\begin{aligned}
\label{general_constant}
C(r;a) = \begin{cases}
\big[ \ j_{\ell}(k_{\text{in}}a)\big]^{-1} & r\leq a,\\
\big[ \ h_{\ell}^{(2)}(k_{\text{out}}a)\big]^{-1} & r> a.
\end{cases}\end{aligned}$$ Matching the interior and the exterior fields leads to the characteristic equation of the discrete eigenfrequencies of the system: $$\label{charac_eqn_single_sphere}
\epsilon_{\text{in}}\bigg[1+k_{\text{out}}a \ \frac{h_{\ell}^{(2)'}(k_{\text{out}}a)}{h_{\ell}^{(2)}(k_{\text{out}}a)} \bigg]=\epsilon_{\text{out}}\bigg[1+k_{\text{in}}a \ \frac{j'_{\ell}(k_{\text{in}}a)}{j_{\ell}(k_{\text{in}}a)} \bigg].$$ Here, the prime denotes differentiation with respect to the argument of the function, i.e. $j'_{\ell}(k_{\text{in}}r)= \partial j_{\ell}(k_{\text{in}}r)/ \partial (k_{\text{in}}r )$. For a given radius, different modes can be labeled by $\ell$: $\omega_{0,\ell}$, where the subscript $0$ denotes isolated single spheres. In case of small spherical particles, when $ka \ll 1$, considering the dipole mode $\ell=1$, the spherical Bessel and Hankel functions can be approximated by their leading order terms: $j_{1}(k_{\text{in}}a) \sim k_{\text{in}}a/3$ and $h_{1}^{(2)}(k_{\text{out}}a) \sim i(k_{\text{out}}a)^{-2}$. Therefore the characteristic equation (\[charac\_eqn\_single\_sphere\]) reduces to $\epsilon_{\text{in}}=-2\epsilon_{\text{out}}$ which leads to the well known resonance frequency of $\omega=\omega_{p}/\sqrt{3}$ for an ideal electron gas with $\epsilon_{\infty}=1$ and $\gamma_s=0$.
Next, we numerically solve eq. (\[charac\_eqn\_single\_sphere\]) for a silver sphere with a free space background. The parameters of the Drude-Sommerfeld model for silver are [@Silver_drude]: the plasma frequency $\omega_p=8.9$ eV, the damping rate $\gamma_s=0.1$ eV, and $\epsilon_{\infty}=5$. The eigenfrequencies are always complex, which indicates the radiative nature of the nanospheres [@SingleSphDAMPING]. The real and imaginary components of the eigenfrequencies as a function of radius and for various values of $\ell=1, 2, 3, 4$, are shown in Fig. \[SilverResonance\]. The real part is the frequency required to excite a mode, for instance with a laser, and the imaginary component is the associated width of the mode. For all the modes, as expected, the real component decreases monotonically as the radius increases. To see the capability of the plasmonic sphere to manipulate light in sub-wavelength dimensions consider the dipole resonance ($\ell=1$) for a 50 nm sphere. The resonant frequency is about 3 eV corresponding to a free space wavelength of approximately 413 nm which is an order of magnitude larger than the radius of the sphere. The imaginary part of the eigenfrequencies consists of both non-radiative and radiative components, $\text{Im}(\omega_{0,\ell})=\gamma^{\text{nrad}}+\gamma_{\ell}^{\text{rad}}$. The non-radiative damping is associated with the loss within the plasmonic sphere. It was discussed in [@SingleSphDAMPING] that the non-radiative component can approximately be considered size-independent and is of equal value for all different modes, $\gamma^{\text{nrad}}=(1/2) \gamma_{s}$. It is therefore clear from Fig. \[SilverResonance\](b) that the dipole is the most radiative mode. For $\ell=1$, initially the sphere becomes more radiative as the radius increases. However, larger spheres have less pronounced radiation properties. For all higher order modes, the imaginary part grows as the radius increases.
The electric field patterns of a silver sphere with a radius of 40 nm and various values of $\ell$ are shown in Fig. \[Fieldplots\]. The field values are displayed logarithmically, and normalized to the maximum of the electric field. As the $\ell$ value increases the fields become more tightly bound to the surface of the sphere. This becomes important when we consider the coupling between two spheres in the next section.
It is also instructive to look at the electric field pattern for the dipole mode. Fig. \[Field2Dplots\] represents the electric field pattern in polar coordinates for various radial distances from the center of the sphere. The black arrow represents the dipole orientation. In all six figures, $\rho$ is the radial distance, perpendicular to the dipole axis and $z$ is the direction along the dipole. The blue line represents the relative field strength at a given polar angle. Furthermore, the field strength is normalized to the maximum value of the electric field. Fig. \[Field2Dplots\](a) shows the pattern at the surface of the plasmonic sphere, $r/a=1$. At locations close to the surface of the sphere, the fields reach a maximum along the direction of the dipole. At $r/a=3$ and $r/a=5$ the patterns are almost omnidirectional (see Figs. \[Field2Dplots\](b) and (c)). The well known torus shape radiation pattern of the dipole only emerges in the far field. This is shown in Figs. \[Field2Dplots\](d), (e) and (f).
The eigenmodes and plasmonic resonances of single metallic nanospheres of various sizes and the effect of different background dielectrics have been considered in detail in [@SingleSph3] for gold nanoparticles and in [@SingleSph2] for alkali metals such as sodium, lithium and Cesium; see [@SingleSphDAMPING] for a more detail description of the size dependency properties of nanospheres. Below we consider the case of two identical plasmonic spheres and discuss the coupling between them and the eigenmodes of the system.
Superradiant and Dark States in The System of Two Coupled Spheres {#secIV}
=================================================================
Given the importance of two level systems in physics [@Amin_Holstein], we now consider the case of two coupled metallic spheres within the coupled mode theory framework discussed earlier. The interaction between the spheres can greatly alter the radiation properties of the system and result in resonance frequencies profoundly different from those for the isolated spheres. Here, we limit our consideration to the dipole modes only. This is justified by the earlier discussion that the dipole mode is the most radiative mode and also has the longest range compared to higher multipolarities. The isolated frequencies discussed in the previous section serve as the diagonal elements in the coupled mode matrix (\[CMT\_to\_eff\_Hamiltonian\]). The real and imaginary components of the dipole eigenmode are the diagonal elements of the Hermitian (\[CMT\_Matrix\_hermitian\]) and the anti-Hermitian (\[CMT\_Matrix\_nonhermitian\]) matrices, respectively. The coupling between two modes, $\kappa$, makes up the off-diagonal matrix elements of the final matrix (\[CMT\_to\_eff\_Hamiltonian\]).
However, a difficulty arises due to the normalization of the single sphere modes. In the far field region, $k_{\text{out}}r \gg 1$, the spherical Hankel functions behave asymptotically as $$\label{asymptotic_fields}
h_{\ell}^{(2)}(k_{\text{out}}r) \approx (i)^{\ell+1} \frac{e^{-ik_{\text{out}}r}}{k_{\text{out}}r}.$$ Because of the complex nature of the eigenfrequencies, the asymptotic form of the fields given by (\[asymptotic\_fields\]) grows exponentially in space as $r \rightarrow \infty$. This growth is however compensated by the exponential decay in time in the complete expression of the field (\[Phase\_convention\]) when the time dependency is considered. As a result the amplitude of the wave front of the total field (\[Phase\_convention\]) reaching any point in the asymptotic region is proportional to $1/r$, as it is expected. Nevertheless, the modes (\[field\_exprs\]) should be properly normalized. The correct normalization of such modes was discussed in [@normalization_1D2; @normalization_1D1; @normalization_1D3; @Amin_Thesis] for one-dimensional problems. The generalization to three dimensions by three different methods is discussed in [@normalization_3D1], [@normalization_3D2], and [@normalization_3D3]. However, in [@normalization_comparison], it was shown that all three expressions are compatible.
The normalization condition is given by [@normalization_comparison]: $$\label{normalization_int}
\int_V \sigma(\vec{r},\omega) \vec{\mathcal{E}}(\vec{r}).\vec{\mathcal{E}}(\vec{r}) d^3r + \frac{i\epsilon_{\text{out}}}{2k} \int_{\partial V} \vec{\mathcal{E}}(\vec{r}).\vec{\mathcal{E}}(\vec{r}) d^2r=1$$ where $V$ is the integration volume and $\partial V$ is its surface. The integration volume is assumed to be sufficiently large, so the fields at its surface are accurately approximated by asymptotic expressions of the spherical Hankel function provided in (\[asymptotic\_fields\]). The modified dielectric function $\sigma(\vec{r},\omega)$ which incorporates the dispersiveness of the medium is given, according to [@DispersiveMedia] as $$\label{modified_dielectric}
\sigma(\vec{r},\omega)=\frac{1}{2\omega} \frac{\partial}{\partial \omega} \Big(\omega^2 \epsilon(\vec{r},\omega)\Big).$$ Contrary to the normalization discussed earlier, the dot-product in (\[normalization\_int\]) does not require any complex conjugation of the fields. It is therefore easier to use the so-called tesseral harmonics instead of the conventional spherical harmonics in the field expression (\[field\_exprs\]). The tesseral harmonics (sometimes also called real spherical harmonics) are nothing but even and odd superpositions of the traditional spherical harmonics, see Appendix \[THandIs\]. The normalization condition (\[normalization\_int\]) defines the constant $\zeta$ in the field expressions (\[field\_exprs\]) up to a phase. It is shown in Appendix \[QNMNAp\] that assuming the volume of integration as a sphere, the volume and surface terms in the normalization expression can be evaluated explicitly. It is furthermore proved in the same appendix that the condition (\[normalization\_int\]) reduces to: $$I \big[ j_{\ell}(k_{\text{in}}a) \big] - I \big[h_{\ell}^{(2)}(k_{\text{out}}a)\big]=1,$$ where the functional $I \big[f_{\ell}(kr) \big]$ is given by (\[volumeterm4\])
$$\begin{aligned}
I \big[f_{\ell}(kr)\big]=\sigma(\vec{r},\omega) \ \zeta^2 \ C^2(r;a)\frac{\ell(\ell+1)}{k^2} \bigg[ r f_{\ell}^2(kr)+kr^2 f_{\ell}(kr) f_{\ell}^{'}(kr)
+\frac{k^2r^3}{2}\Big(f_{\ell}^2(kr)-f_{\ell-1}(kr)f_{\ell+1}(kr) \Big) \bigg].\end{aligned}$$
Once the normalization constant $\zeta$ is found the coupling between two modes can be calculated according to $$\label{coupling_coefficient_modified}
\kappa=-\frac{1}{2}\omega_0 \int_{\text{V}_1} \big(\epsilon_1(\vec{r})-1 \big) \vec{\mathcal{E}}^{1}(\vec{r}). \vec{\mathcal{E}}^2(\vec{r}) d^3r,$$ where $\text{V}_1$ is again the volume of sphere 1. Contrary to (\[coupling\_coefficient\]), the fields from both spheres are treated on equal footing and no complex conjugation is required due to the normalization definition (\[normalization\_int\]).
In what follows we show the eigenfrequencies of a system of two coupled spheres with different sizes and different separation distances. The coupling $\kappa$ is calculated numerically and the coupled mode matrix (\[CMT\_to\_eff\_Hamiltonian\]) is constructed and diagonalized. In the first case two identical silver spheres with radii of 10 nm are considered and the eigenfrequencies of the system are calculated for two different dipole orientations and as a function of the separation distance between the spheres. It is important to note that due to the symmetry of the dipole modes, dipoles with perpendicular orientations do not couple. Therefore we consider two parallel orientations only.
As a check of the coupled mode theory approximation, we have also computed the resonant frequencies of two coupled metallic spheres via a modal solution. We refer to this modal solution as exact even though we only retain a finite number of modes that adequately resolves the resonant frequencies. The modal solutions are rigorously based upon solving Maxwell’s equations, and as such we treat them as exact, in contrast to the approximations used to generate the coupled mode theory results.
The modal solution is derived by considering the discrete modes for an isolated sphere, which have electric fields as given by (\[field\_exprs\]), and corresponding magnetic fields determined via Maxwell’s equations. If we numerically truncate the modes at a maximum $\ell$ value, then this results in a finite set of possible $\ell$ and $m$ values. Each of these ($\ell$,$m$) combinations can be considered as defining a basis function, with distinct basis functions defined for the region interior and exterior to the sphere. The weighted summation of these basis functions can then approximate the fields supported everywhere by the metallic sphere.
We now consider two spheres with these allowed modal solutions, and our goal is to find which particular combination of modal coefficients satisfies the boundary condition of the tangential electric and magnetic fields being continuous at the surfaces of both spheres. This is accomplished by choosing $N$ points spaced approximately equally over the surface of each sphere, where $N$ exceeds the number $M$ of modal coefficients we wish to determine.
Each point on the surface of a sphere defines a constraint equation, with $4N$ total constraint equations due to there being four field components to match at each surface point: two tangential electric field components, and two magnetic field components. If we were able to exactly represent the fields, then at resonance the sum of the mode contributions at each point would equal zero, resulting in a matrix equation $Ax=0$ involving a $4N$-by-$M$ matrix $A$ and a length-$M$ vector $x$ representing the mode coefficients.
However, because we are working with a truncated set of modes, we cannot solve the boundary condition exactly. We instead seek complex frequencies at which the smallest singular value of $A$ achieves a local minimum. These frequencies are approximately equal to the true resonant frequencies, and for the results plotted in this paper, frequency convergence was observed for maximum $\ell$ values ranging from 2 to 8, with a larger $\ell$ value being required as the sphere separation decreases.
Fig. \[coupling\_10nm\_sphs\](a) shows the case where the two dipoles are parallel (vertical orientation). Both the real and imaginary components of the two eigenfrequencies of the system are plotted as a function of the separation $d$ between the two spheres normalized by the sphere radius $a$. The red solid lines and the black diamonds correspond to calculations via coupled mode theory (CMT) and modal expansion (Exact), respectively. There is good agreement between the two methods across the entire separation distances, even for small separations or strong interaction regime. This indicates that the dipole mode has the largest contribution in the coupling strength between the spheres and therefore coupled mode theory provides an acceptable approximation of the solution. The maximum coupling occurs when the two spheres are in contact with one another which in turn results in the largest deviation from the unperturbed frequency of a single sphere (dotted black lines). As the separation increases, the coupling decreases and the two eigenfrequencies approach the unperturbed eigenmode. A similar phenomenon is seen for the case of two dipoles in Fig. \[coupling\_10nm\_sphs\](b) (horizontal orientation). However, for a small separation distance between the spheres, the splitting of the two eigenmodes is larger for horizontal (colinear) dipole orientation than for vertical (parallel) dipole orientation. This is because, as shown in the previous section, the electric field is maximum along the dipole direction in the near field (see Fig. \[Field2Dplots\]). Consequently, this orientation results in larger coupling coefficients between the spheres.
According to our findings in the previous section, silver spheres with the larger 40 nm radii are more radiative than the 10 nm radii spheres just considered. It is therefore desirable to look at the case of coupling between larger spheres since the coupling is stronger. Fig. \[coupling\_40nm\] shows the eigenfrequencies of a system of two silver spheres with radii of 40 nm. Because the coupling coefficient between two horizontal dipoles is greater than that of two vertical dipoles, we only consider the horizontal case. The difference between the superradiant and subradiant states is more pronounced in this case. At $d/a \approx 3$, coupled mode theory predicts a maximal difference between the imaginary components of the two eigenmodes. Similar to the previous case, the two eigenfrequencies approach the unperturbed resonance as the separation distance increases. At lower separations however, when the spheres are strongly interacting, coupled mode theory solution deviates more from the exact solution compared to the 10 nm spheres shown in Fig. \[coupling\_10nm\_sphs\](b). This indicates the importance of higher order modes in the interaction strength of larger spheres, and that the coupled mode results can improve if these modes are included in the calculations.
![Real and imaginary components of the eigenfrequencies of a system of two coupled identical silver spheres with horizontal dipole orientation calculated via coupled mode theory shown in red (solid lines) and modal expansion shown in black (diamonds). The radii of the spheres are 40 nm. $d$ is the center-to-center separation and $a$ is the radius of the spheres. The black dotted lines represent the unperturbed eigenfrequency of a single sphere.[]{data-label="coupling_40nm"}](coupling40nmh_silver_NEW2.pdf)
According to our calculations, an exact dark mode does not exist for a system of two silver plasmonics dipoles. Therefore a numerical search over the parameters of the Drude-Sommerfeld dielectric function (\[Drude\_dielectric\_func\]) was performed in order to find material properties for which two plasmonic spheres can support a dark mode. Fig. \[coupling\_darkmode\] shows the eigenfrequencies of a system of two spheres with Drude-Sommerfeld parameters of $\epsilon_{\infty}=1$, $\omega_p=10.918$ eV and $\gamma_s=0$. At $d/a \approx 3.5$, the rank of the anti-Hermitian part of the coupled mode matrix is almost unity indicating that the interaction between the two spheres occurs through a single continuum channel. Consequently, the imaginary component of the subradiant mode is extremely small, in the order of 10$^{-3}$ eV. This is indicated with a black circle in the figure.
![Real and imaginary components of the eigenfrequencies of a system of two coupled identical spheres with horizontal dipole orientation. The material properties of the spheres are: $\epsilon_{\infty}=1$, $\omega_p=10.918$ eV and $\gamma_s=0$. The radii of the spheres are 20 nm. $d$ is the center to center separation and $a$ is the radii of the spheres. The black dotted lines represent the unperturbed eigenfrequency of a single sphere.[]{data-label="coupling_darkmode"}](couplingDARKMODE_NEW2.pdf)
Plasmonic Waveguide {#secV}
===================
We now consider the signal transmission through a plasmonic waveguide, namely a one-dimensional chain of identical spheres. The idea that such a structure acts as a waveguide due to interparticle coupling was proposed in [@Quinten98] and experimentally verified in [@waveguide_exp]. Here, it is assumed that the two edges of the waveguide are connected to an instrument capable of exciting the system of spheres with frequency $\omega_e$ and measuring the electric field intensity. Fig. \[plasmonic\_waveguide\_schematics\] depicts the schematic of the plasmonic waveguide and the two probes symmetrically coupled to the edges of the chain with coupling constants $\gamma_e$. Similar to the tight binding model of crystals in condensed matter physics, it is further assumed that each sphere in the chain only interacts with its nearest neighbor. This system can be modeled with the effective non-Hermitian Hamiltonian (\[ReducedHeff\]) $$\label{effective_hamiltonian_plasmonic_waveguide}
\mathscr{H}_{\text{eff}}=
\begin{bmatrix}
\frac{i}{2}\gamma_e+\omega_0 & \kappa & 0 & \dots & 0 & 0 \\
\kappa & \omega_0 & \kappa & \dots & 0 & 0 \\
0 & \kappa & \omega_0 & \kappa & \dots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & \dots & \kappa & \frac{i}{2}\gamma_e+\omega_0
\end{bmatrix}
,$$ where $\omega_0$ is the unperturbed dipole frequency of an isolated sphere, and $\kappa$ is the coupling coefficient (\[coupling\_coefficient\_modified\]) between adjacent dipoles. It is important to mention that the addition of the anti-Hermitian matrix elements, $\frac{i}{2}\gamma_e$, with a positive sign is due to the phase convention adopted earlier in (\[Phase\_convention\]).
![Schematics of a plasmonic waveguide; a one-dimensional chain of five silver spheres with nearest neighbor coupling $\kappa$. The two edges are symmetrically coupled to continuum, the excitation source with frequency $\omega_e$, with coupling coefficient $\gamma_e$.[]{data-label="plasmonic_waveguide_schematics"}](TransportSchematic.pdf)
Through its coupling to the two probes, the system can undergo an additional superradiance phase transition, other than that discussed in the previous section. This is illustrated by considering two different plasmonic waveguides. In both cases, according to our findings in the last section, in order to maximize the coupling between the neighboring sites the spheres are in contact with one another and the dipole orientation of the spheres is considered to be along the waveguide (horizontal orientation). In the first case, a chain of five silver spheres with radii of 10 nm is considered (Fig. \[plasmonic\_waveguide\_schematics\]). The resulting effective Hamiltonian (\[effective\_hamiltonian\_plasmonic\_waveguide\]) describing the system is a 5x5 square matrix with diagonal elements, $\omega_0=3.3468 + i0.0519$ eV, and off-diagonal matrix elements $\kappa=-0.2459 + i0.0029$ eV. The continuum coupling coefficient $\gamma_e$ is treated as a variable that changes from small, $\gamma_e=0.01$ eV, to extreme values $\gamma_e=10$ eV. The evolution of the complex eigenvalues of the effective Hamiltonian as the coupling to the continuum varies is shown in Fig. \[complex\_eigenvals\_sphrs\](a). At small values of $\gamma_e$ all the eigenvalues acquire a small width through the coupling to the continuum. The widths of the complex eigenmodes almost uniformly increase as the system is more strongly coupled to the continuum, up until $\gamma_e \approx 1$. At this point, the eigenvalues have reached their maximum width and, with further increasing $\gamma_e$, the system undergoes a phase transition (superradiance transition) when the eigenmodes become segregated into two distinct categories: superradiant and subradiant states. At strong coupling, the two superradiant states, their number being equal to the number of continuum channels (two probes), steal the entire available width of the system and leave the remaining states as narrow resonances.
The second waveguide differs only in that the size of the spheres now have radii of 40 nm. In this case, the diagonal unperturbed frequencies are $\omega_0=3.1172 + i0.1910$ eV and the off-diagonal coupling coefficients are $\kappa=-0.2606 + i0.0475$ eV. The continuum coupling $\gamma_e$ is again varied from $\gamma_e=0.01$ eV to $\gamma=10$ eV, and the complex eigenvalues are plotted in Fig. \[complex\_eigenvals\_sphrs\](b). In general the picture is similar to the previous case. The superradiant transition can be clearly seen as the coupling $\gamma_e$ increases to extreme values.
We now study the propagation of a signal through the two waveguides by calculating the transmission coefficient. Using (\[ProcessAmplitude\]) and (\[Transmission\_amp\]) we arrive at the following expression for the transmission coefficient $$T(\hbar \omega_e) = \Bigg| \frac{\gamma_e/ \kappa}{ \prod_{r=1}^{N}\big[(\hbar \omega_e - \hbar \omega_r)/ \kappa \big]} \Bigg|^2,$$ where $\omega_r$ are the complex frequencies of the effective Hamiltonian (\[effective\_hamiltonian\_plasmonic\_waveguide\]) and $N$ is its dimension.
Transmission as a function of the excitation frequency, $\omega_e$, is shown in Fig. \[PlasmonicTransport10nm\] for the waveguide with 10 nm spheres. At weak coupling to the continuum, $\gamma_e=0.03$ eV, Fig. \[PlasmonicTransport10nm\](a), the five resonances are distinguishable. However, the resonances are not well separated due to the complex coupling coefficient between spheres, $\kappa$, which provide the eigenvalues of the effective Hamiltonian an initial width even for the closed system ($\gamma_e=0$). The case of intermediate coupling, when $\gamma_e=0.55$ eV, is shown in Fig. \[PlasmonicTransport10nm\](b). This is when the system is on the road to superradiance transition and all the eigenvalues of the Hamiltonian have large widths. Consequently, the resonances overlap and the transmission is dramatically enhanced. The case of strong couplings, Fig. \[PlasmonicTransport10nm\](c), has a picture similar to that of the weak coupling case. However, only three resonances remain. The two giant superradiant states do not participate in signal transmission and transmission is greatly suppressed due to the small width of the remaining subradiant states.
We follow the same steps of weak, intermediate and strong coupling to continuum in order to study transmission through the waveguide with 40 nm spheres. Due to larger coupling, $\kappa$, between adjacent spheres the eigenvalues of the effective Hamiltonian poses a relatively large initial width even for small coupling to the continuum. Therefore, contrary to the previous case, the resonances overlap and are not separated even at weak coupling, Fig. \[PlasmonicTransport40nm\](a). Similar to before, the transmission is greatly enhanced at the superradiance transition, and at extreme couplings, we are back to suppressed transmission.
Conclusion {#secVI}
==========
We studied the resonant frequencies of plasmonic spherical nanoantennas by solving the full wave equation. These eigenfrequencies are always complex due to radiation and damping. Utilizing the effective non-Hermitian Hamiltonian framework, it was shown that a system of coupled two spheres can have modes with distinct properties; a superradiant mode with enhanced radiation and a dark mode with extremely damped radiation.
Signal transmission through one dimensional chains was also considered. The coupling of the edge spheres to the continuum can drastically change transport properties of the system. A different superradiant transition arises through this interaction. Transmission is greatly enhanced at this transition.
A possible direction to improve the accuracy of the results is to modify the Drude-Sommerfeld model by including terms that take into account surface scattering effects. It would be interesting to study the contribution of surface scattering to the total damping and radiation of the nanospheres. Another possibility is to consider higher order modes and their effect on the eigenfrequencies of the coupled system. These are left for future work.
A. T. is grateful to his PhD advisor, Vladimir Zelevinsky for his guidance and many helpful discussions and thanks A. Stain for her support and assistance.
S. R. thanks his doctoral advisor J. Verboncoeur for his mentorship.
Quasi Normal Modes Normalization {#QNMNAp}
================================
In this appendix we explicitly normalize the fields of a plasmonic sphere by evaluating the normalization expression (\[normalization\_int\]) $$\label{normalization_int2}
\int_V \sigma(\vec{r},\omega) \vec{\mathcal{E}}(\vec{r}).\vec{\mathcal{E}}(\vec{r}) d^3r + \frac{i\epsilon_{\text{out}}}{2k} \int_{\partial V} \vec{\mathcal{E}}(\vec{r}).\vec{\mathcal{E}}(\vec{r}) d^2r=1.$$ Due to the homogeneity of the sphere dielectric function $\epsilon_{\text{in}}$, and the surrounding background $\epsilon_{\text{out}}$, the modified dielectric function $\sigma(\omega)$ given in (\[modified\_dielectric\]) is only a function of frequency and can be taken out of the integral. In what follows we first evaluate the volume term in (\[normalization\_int2\]) assuming the normalization volume itself is a sphere with a radius $R$ where $a \ll R$. We first consider the volume term. Using the field expressions given in (\[field\_exprs\]) the volume term of the normalization (\[normalization\_int\]) is expressed as $$\begin{aligned}
\label{volumeterm1}
& \sigma(\omega) \int_V \ \vec{\mathcal{E}}(\vec{r}).\vec{\mathcal{E}}(\vec{r}) d^3r =\sigma(\omega) \ \zeta^2 \ \int_0^{R} dr r^2 \ C^2(r;a) \Bigg\{ \big(\ell(\ell+1)\big)^2 \bigg( \frac{f_{\ell}(kr)}{kr} \bigg)^2 \int d\Omega \bigg(Y^m_\ell(\theta,\phi)\bigg)^2 \nonumber \\
&+ \bigg( \frac{1}{kr} \frac{\partial}{\partial(kr)} \big( krf_{\ell}(kr) \big) \bigg)^2 \int d\Omega \bigg[ \bigg( \frac{\partial}{\partial \theta} Y_\ell^m(\theta,\phi) \bigg)^2 +\frac{1}{\text{sin}^2\theta} \bigg( \frac{\partial}{\partial \phi} Y_\ell^m(\theta,\phi) \bigg)^2 \bigg] \Bigg\},\end{aligned}$$ where $d\Omega=\sin \theta d\theta d \phi$ is the solid angle differential in spherical coordinates. The integrals involving spherical harmonics can be evaluated by using the orthogonality relation of the tesseral harmonics and identity (\[SphericalHarmonic\_angular\_integral\_identity\]) in Appendix \[THandIs\]. Thus, (\[volumeterm1\]) reduces to $$\begin{aligned}
\label{volumeterm2}
& \sigma(\omega) \int_V \ \vec{\mathcal{E}}(\vec{r}).\vec{\mathcal{E}}(\vec{r}) d^3r= \nonumber \\
& \sigma(\omega) \ \zeta^2 \ \ell(\ell+1) \int_0^R dr r^2 \ C^2(r;a) \Bigg\{ \ell(\ell+1) \bigg( \frac{f_{\ell}(kr)}{kr} \bigg)^2
+ \bigg( \frac{1}{kr} \frac{\partial}{\partial(kr)} \big( krf_{\ell}(kr) \big) \bigg)^2 \Bigg\}.\end{aligned}$$ Due to the discontinuity of the function $f_{\ell}(kr)$ and the coefficients $C(r;a)$ at the surface of the plasmonic sphere \[see eqn. (\[general\_constant\])\], the radial integral in (\[volumeterm2\]) has to be divided into two terms: $\int_0^R=\int_0^a + \int_a^R$. Each term can be evaluated with the help of (\[SphericalHarmonic\_integral\_identity2\]) and (\[SphericalHarmonic\_integral\_identity3\]). This brings us to the final expression for the volume term of the normalization $$\label{volumeterm3}
\sigma(\omega) \int_V \ \vec{\mathcal{E}}(\vec{r}).\vec{\mathcal{E}}(\vec{r}) d^3r = I \big[ j_{\ell}(k_{\text{in}}a) \big] - I\big[h_{\ell}^{(2)}(k_{\text{out}}a)\big] + I\big[h_{\ell}^{(2)}(k_{\text{out}}R)\big],$$ where the functional $I\big[f_{\ell}(kr)\big]$ is defined as $$\begin{aligned}
\label{volumeterm4}
& I\big[f_{\ell}(kr)\big]= \sigma(\omega) \ \zeta^2 \ C^2(r;a)\frac{\ell(\ell+1)}{k^2} \bigg[ r f_{\ell}^2(kr)+kr^2 f_{\ell}(kr) f_{\ell}^{'}(kr)+\frac{k^2r^3}{2}\Big(f_{\ell}^2(kr)-f_{\ell-1}(kr)f_{\ell+1}(kr) \Big) \bigg].\end{aligned}$$ As before, $f_{\ell}^{'}(kr)$ implies differentiation with respect to the argument i.e. $\frac{\partial}{\partial (kr)} f_{\ell}(kr)$. Note that in evaluating the first term of the right hand side of (\[volumeterm3\]), $I \big[ j_{\ell}(k_{\text{in}}a) \big] $, one has to use all the parameters corresponding to the region interior to the plasmonic sphere. i.e. $\epsilon_{\text{in}}(\omega)$, $k_{\text{in}}$ and $C(r;a)$ for $r \leq a$ as it is defined in (\[general\_constant\]). Accordingly, the same applies to the second and third terms, $I\big[h_{\ell}^{(2)}(k_{\text{out}}a)\big]$ and $I\big[h_{\ell}^{(2)}(k_{\text{out}}R)\big]$, for which one has to use the parameters corresponding to the background material.
Next we show that the last term in the right hand side of (\[volumeterm3\]), $I\big[h_{\ell}^{(2)}(k_{\text{out}}R)\big]$, exactly cancels out with the surface term in (\[normalization\_int\]). Therefore, as expected, the normalization condition becomes independent of the integration volume. Because the integration sphere was assumed to be sufficiently large, asymptotic expressions can be used to evaluate both these terms. The spherical Hankel functions at large radial distances have the following asymptotic form $$\label{Asymptotic_hankel}
h_{\ell}^{(2)}(kr)\sim i^{(\ell+1)} \frac{e^{-ikr}}{kr} \Bigg(1-i \frac{\ell(\ell+1)}{2kr} \Bigg).$$ The last term in the right hand side of (\[volumeterm4\]) can therefore be approximated as $$\label{VolumeLargeContr}
I\big[h_{\ell}^{(2)}(k_{\text{out}}R)\big] \approx \frac{-i \epsilon_{\text{out}}}{2k} C^2(r;a) (-)^{\ell+1}e^{-2ik_{\text{out}}R}.$$ We now consider the surface term of the normalization condition (\[normalization\_int\]). Considering that in the far field the dominant terms are $\mathcal{E}_\theta$ and $\mathcal{E}_\phi$ and using the asymptotic form of the spherical Hankel function (\[Asymptotic\_hankel\]), the leading order of the surface term is $$\label{Surfaceterm_final}
\frac{i \epsilon_{\text{out}}}{2k} \int_{\partial V} \vec{\mathcal{E}}(\vec{r}).\vec{\mathcal{E}}(\vec{r}) d^2r \approx \frac{i \epsilon_{\text{out}}}{2k^3} C^2(r;a) (-)^{\ell+1} e^{-2ik_{\text{out}}R}$$
It is clear that the surface term (\[Surfaceterm\_final\]) and (\[VolumeLargeContr\]) of the volume term cancel out. With this, the normalization condition (\[normalization\_int2\]) reduces to $$\label{normalization_reduced}
I \big[ j_{\ell}(k_{\text{in}}a) \big] - I\big[h_{\ell}^{(2)}(k_{\text{out}}a)\big]=1,$$ which provide us the coefficient $\zeta$ in (\[field\_exprs\]).
Tesseral Harmonics and Spherical Functions Identities {#THandIs}
=====================================================
This appendix contains the definition of tesseral harmonics and a list of identities that are used throughout the paper.
The tesseral harmonic are linear superpositions of the complex spherical harmonics with same $\ell$ and opposite sign $m$ values. Therefore the azimuthal dependency of the functions are in the form of $\text{sin}(m\phi)$ and $\text{cos}(m\phi)$ instead of the usual exponential form $e^{im\phi}$. They are defined as [@THarmonics]:
$$\begin{aligned}
Y_{\ell}^m(\theta,\phi) = \begin{cases}
\sqrt{\frac{2\ell+1}{2 \pi}\frac{(\ell-|m|)!}{(\ell+|m|)!}} P_{\ell}^{|m|}(\text{cos} \theta) \text{sin}(|m|\phi) & m<0 \\
\sqrt{\frac{2\ell+1}{4 \pi}} P_{\ell}^{m} & m=0\\
\sqrt{\frac{2\ell+1}{2 \pi}\frac{(\ell-m)!}{(\ell+m)!}} P_{\ell}^{m}(\text{cos} \theta) \text{cos}(m\phi) & m>0
\end{cases}\end{aligned}$$
where $P_{\ell}^{m}$ are the associated Legendre polynomials. The tesseral harmonics satisfy the same orthogonality relation as the complex spherical harmonics.
The tesseral harmonics also satisfy the following identity
$$\begin{aligned}
\label{SphericalHarmonic_angular_integral_identity}
& \int d\Omega \Bigg\{ \bigg( \frac{\partial}{\partial \theta} Y_\ell^m(\theta,\phi) \bigg)^2 +\frac{1}{\sin^2\theta} \bigg( \frac{\partial}{\partial \phi} Y_\ell^m(\theta,\phi) \bigg)^2 \Bigg\} = \nonumber \\
& \ell (\ell+1).\end{aligned}$$
This can be proven by starting with the fact that the harmonics fulfill the identity $r^2 \nabla^2 Y_{\ell}^m(\theta,\phi)=-\ell(\ell+1)Y_{\ell}^m(\theta,\phi)$. One can then get to (\[SphericalHarmonic\_angular\_integral\_identity\]), by calculating the matrix element of the operator $r^2 \nabla^2$ and using integration by parts.
A useful identity of the spherical Bessel and Hankel functions is the following
$$\begin{aligned}
\label{SphericalHarmonic_integral_identity2}
\int dr \Bigg\{ \ell (\ell+1) f_{\ell}^2(kr) + \bigg( \frac{\partial}{\partial (kr)} krf_{\ell}(kr) \bigg)^2 \Bigg\}= \nonumber \\
r f_{\ell}^2(kr)+kr^2 f_{\ell}(kr) f_{\ell}^{'}(kr)+k^2 \int dr r^2 f_{\ell}^2(kr).\end{aligned}$$
where as previously $f_{\ell}^{'}(kr)$ implies differentiation with respect to the argument. The equality can be proven by using integration by parts and the spherical Bessel differential equation to get $$\begin{aligned}
& \ell (\ell+1) f_{\ell}^2(kr) = \nonumber \\
& k^2r^2 f_{\ell}(kr) f_{\ell}^{''}(kr)+2krf_{\ell}(kr)f_{\ell}^{'}(kr)+k^2r^2f_{\ell}^2(kr),\end{aligned}$$ and $$\label{SphericalHarmonic_integral_identity3}
\int dr r^2 f_{\ell}^2(kr) = \frac{r^3}{2} \bigg( f_{\ell}^2(kr)- f_{\ell-1}(kr) f_{\ell+1}(kr) \bigg).$$
[^1]: [email protected]
| {
"pile_set_name": "ArXiv"
} |
=1
Introduction
============
Associated with any system of linear differential equations is a linear representation of the fundamental group of a sphere punctured at the poles of the system, called the monodromy representation. An isomonodromic deformation is the way in which the system’s coefficients change while preserving this monodromy representation [@Jimbo:Monodromy1]. It is known that all the Painlevé equations arise as isomonodromic deformations of second-order differential equations [@Jimbo:Monodromy2]. The Garnier system arises as an isomonodromic deformation of a second-order Fuchsian scalar differential equation with $m$ apparent singularities and $m+3$ poles [@Garnier]. When we fix three poles, we have $m$ remaining poles that are considered time variables [@Jimbo:Monodromy2]. The simplest nontrivial case, where $m=1$, corresponds to the sixth Painlevé equation [@Fuchs2; @Fuchs1].
The focus of this study is a collection of systems that may be regarded as discrete analogues of the Garnier system. We regard these to be nonlinear integrable systems arising as discrete isomonodromic deformations [@Gramani:Isomonodromic]. Our starting point is a regular system of difference equations of the form $$\begin{gathered}
\label{linearsigma}
\sigma Y(x) = A(x) Y(x),\end{gathered}$$ where $A(x)$ is a $2\times 2$ matrix polynomial whose determinant is of degree $N$ in $x$, which is called a spectral variable, and where $\sigma = \sigma_h \colon f(x) \to f(x+h)$ or $\sigma = \sigma_q \colon f(x) \to f(qx)$. These operators are defined in terms of two constants $h, q \in \mathbb{C}$ subject to the constraints $\Re h > 0$ and $0 < | q | < 1$. The goal of this work is to specify a parameterization of these matrices by giving a factorization, $$\begin{gathered}
\label{prodform}
A(x) = L_1(x) \cdots L_N(x),\end{gathered}$$ which will be conducive to finding the discrete isomonodromic deformations of .
A discrete isomonodromic deformation is a transformation induced by an auxiliary system of difference equations, which may be written in matrix form as $$\begin{gathered}
\label{Atildeev}
\tilde{Y}(x) = R(x) Y(x).\end{gathered}$$ The transformed matrix, $\tilde{Y}(x)$, satisifes a new equation of the form , given by $$\begin{gathered}
\label{transformedlinearsigma}
\sigma \tilde{Y}(x) = \tilde{A}(x)\tilde{Y}(x),\end{gathered}$$ where consistency in the calculation of $\sigma \tilde{Y}(x)$ imposes the relation $$\begin{gathered}
\label{comp}
\tilde{A}(x)R(x) = \sigma R(x) A(x),\end{gathered}$$ which is compatible with . Comparing the left and right-hand sides of defines a rational map between the entries of $A(x)$ and $\tilde{A}(x)$. The two operators appearing in and define a Lax pair for the resulting map. Compatibility conditions of the form give rise to discrete isomonodromic deformations in the sense of Papageorgiou et al. [@Gramani:Isomonodromic]. It was shown later by Jimbo and Sakai that compatibility relations of the form , as a map between linear systems, also preserves a connection matrix [@Sakai:qP6]. This connection matrix, introduced by Birkhoff [@Birkhoff; @Birkhoffallied], is considered to be a discrete analogue of a monodromy matrix. It is known that various discrete Painlevé equations, QRT maps and general classes of integrable mappings that characterize reductions of partial difference equations arise in this way [@Ormerod2014a; @OvdKQ:reductions; @Gramani:Isomonodromic].
Discrete isomonodromic deformations share more in common with Schlesinger transformations than isomonodromic deformations. That is, given an $A(x)$ we have a collection of transformations of the form . In a similar manner to Schlesinger transformations, the system of transformations governed by and has the structure of a finitely generated lattice [@Ormerodlattice; @Ormerod2011b]. Our discrete Garnier systems are systems of elementary transformations generating an action of $\mathbb{Z}^d$ for some dimension, $d$. One of the consequences of , for the particular choice of $L_i$ we propose, is that the resulting analogues of elementary Schlesinger transformations are simply expressed in terms of commutation relations between factors. This factorization, and their commutation relations, are also features of the work of Kajiwara et al. [@KajiwaraSym].
An additional novel feature of our work is the presence of [*symmetric*]{} Lax pairs, in which solutions of satisfy an extra symmetry constraint. We may take this into consideration by letting $Y(x)$ satisfy relations involving two operators
\[lineartau\] $$\begin{gathered}
\label{tau1}\tau_1 Y(x) = A(x) Y(x),\\
\label{tau2}\tau_2 Y(x) = Y(x),\end{gathered}$$
where $\tau_1 \colon f(x) \to f(-h-x)$ or $\tau_1 \colon f(x) \to f(1/qx)$ while $\tau_2 \colon f(x) \to f(-x)$ or $\tau_2 \colon f(x) \to f(1/x)$. The composition of $\tau_1$ and $\tau_2$ recovers while gives a constraint on the entries of $A(x)$. The operators $\tau_1$ and $\tau_2$ generate a copy of the infinite dihedral group.
The presence of this additional symmetry is a structure that plays an important role in hypergeometric and basic hypergeometric orthogonal polynomials, biorthogonal functions and related special functions. This constraint naturally manifests itself in the known Lax pairs for the elliptic Painlevé equation [@rains:isomonodromy; @Yamada:E8Lax] via their parameterization in terms of theta functions, but this property has not manifested itself in any obvious way in the known Lax pairs for more degenerate Painlevé equations.
There are a number of technical issues in presenting the discrete isomonodromic deformations of such systems: Firstly, the classical theory of Birkhoff (see [@Birkhoff; @Birkhoffallied]) is no longer sufficient to guarantee the existence of solutions. For this we appeal to the work of Praagman [@Praagman:Solutions]. Secondly, the theorems that prescribe discrete isomonodromic deformations do not necessarily preserve the required symmetry. What makes finding the isomonodromic deformations of the symmetric cases tractable is that $A(x)$ can be shown to admit a factorization $$\begin{gathered}
\label{AviaB}
A(x) = ( \tau_1 B(x))^{-1} B(x),\end{gathered}$$ for some rational matrix $B(x)$. By insisting that $B(x)$ takes the same factored form, namely $$\begin{gathered}
\label{prodform2}
B(x) = L_1(x) \cdots L_{N'}(x),\end{gathered}$$ we =-1 are able to describe the discrete isomonodromic deformations of these systems in terms of the same commutation relations as the non-symmetric case. Thirdly, since the classical fundamental solutions of Birkhoff do not necessarily exist, it is not clear that the analogue of monodromy involving Birkhoff’s connection matrix (see [@Sakai:qP6]) is appropriate. To address this, we give a short account of how discrete isomonodromic deformations preserve the associated Galois group of .
This gives us four classes of system; two difference Garnier systems whose associated linear problems are of the form of , and two symmetric difference Garnier systems, whose associated linear problems are of the form of . We reparameterize these systems in terms of the image and kernel vectors at the singular points. This provides a correspondence between one of our systems and Sakai’s $q$-Garnier system [@Sakai:Garnier]. We also consider specializations whose evolution coincides with discrete Painlevé equations of type $q$-$\mathrm{P}\big(A_k^{(1)}\big)$ for $k = 0,1,2,3$ and $d$-$\mathrm{P}\big(A_k^{(1)}\big)$ for $k=0,1,2$. The convention we use is that we list the type of the Affine root system associated with the surface of initial conditions [@Sakai:Rational]. This means that the systems we treat appear as the top cases of the disrete Painlevé equations and $q$-Painlevé equations.
It should be recognised that the phrase symmetric and asymmetric discrete Painlevé equations has been applied to equations arising as deautonomized symmetric and asymmetric QRT maps respectively [@Kruskal:AsymmetricdPs]. The way in which the word symmetric is used in the context of this article is that the associated linear problem possesses an additional symmetry. The ideas of having a symmetric system of difference equations and having a symmetric QRT mapping or its deautonomization are very different and should not be confused.
The product form for $A(x)$ in arises naturally in recent work on reductions of partial difference equations [@Ormerod:qP6; @Ormerod2014a]. We present a way in which these systems characterize certain periodic and twisted reductions of the lattice Korteweg–de Vries (KdV) equation and the lattice Schwarzian KdV equation [@Ormerod2014a]. A corollary of this work is that Sakai’s $q$-Garnier system arises as a twisted reduction of the lattice Schwarzian KdV equation, as do any specializations. This work also gives an explicit expression for the evolution in terms of known Yang–Baxter maps.
The plan of the paper is as follows. In Section \[sec:isomonodromy\] we give an overview of the theory of linear systems of difference equations where we formalize the way in which we consider our systems to be isomonodromic. In Section \[sec:Garnier\] we provide evolution equations for the discrete Garnier systems in terms of viables naturally associated with and , whereas Section \[sec:reparam\] gives the same evolution equations in terms of variables associated with the image and kernel at each value of $x$ in which $A(x)$ is singular. Section \[sec:special\] gives a number of cases in which the evolution of the discrete Garnier systems coincide with known case of discrete Painlevé equations. Section \[sec:reductions\] shows how both cases of the non-symmetric Garnier systems, and their special cases, arise as reductions of discrete potential KdV equation and the discrete Schwarzian KdV equation.
Linear systems of difference equations {#sec:isomonodromy}
======================================
This section aims to provide the relevant theorems concerning systems of linear difference equations. This includes a recapitulation of the classical results of Birkhoff on linear systems of difference and $q$-difference equations [@Birkhoff; @Birkhoffallied]. While the work of Birkhoff gives fundamental solutions to systems of difference equations of the form , they are not sufficient to ensure solutions of systems of the form of .
Secondly, given a system of the form of or , we wish to specify the type of transformations we expect. The set of transformations has the structure of a finite-dimensional lattice. Characterizating these transformations follows the work of Borodin [@Borodin:connection], who developed this theory in application to gap probabilities of random matrices [@Borodin2003]. We extend this to $q$-difference equations [@Ormerodlattice].
A secondary issue concerns what structures are being preserved by discrete isomonodromic deformations. The celebrated work of Jimbo and Sakai [@Sakai:qP6] argues that preserves the connection matrix, when it exists. A fundamental object that is preserved under transformations of the form of is the structure of the difference module [@VanderPut2003]. This provides a more robust definition of what it means to be a discrete isomonodromic deformation. In particular, this holds for discrete isomonodromic deformations of , or any system in which the existence of a connection matrix may not be assumed.
Systems of linear $\boldsymbol{h}$-difference equations
-------------------------------------------------------
We start with where $\sigma= \sigma_h$, which we write as $$\begin{gathered}
\label{lineardiff}
Y(x+h) = A(x)Y(x),\end{gathered}$$ where $A(x)$ is a rational $M \times M$ matrix that is invertible almost everywhere and $\Re h > 0$ as above. We may reduce to the case in which $A(x)$ is polynomial by multiplying $Y(x)$ by gamma functions. This means that the form of $A(x)$ can generally be taken to be $$\begin{gathered}
\label{polydiff}
A(x) = A_0 + A_1 x + \dots + A_nx^n,\end{gathered}$$ where $A_n \neq 0$. Furthermore, if $A_n$ is invertible and semisimple then, by applying constant gauge transformations, we can assume that $A_n$ is diagonal. By the same argument, if $A_n = I$ and $A_{n-1}$ is semisimple, we may assume that $A_{n-1}$ is diagonal. Under these assumptions, it is useful to describe an asymptotic form of formal solutions which is the subject of the following theorem due to Birkhoff [@Birkhoff].
Let $A_n = \operatorname{diag}(\rho_1, \ldots, \rho_M)$, where the $\rho_i$ are pairwise distinct, or if $A_n = I$ and $A_{n-1} = \operatorname{diag}(r_1, \ldots, r_M)$ subject to the non resonnancy constraint $$\begin{gathered}
r_i - r_j \notin \mathbb{Z}\setminus\{0\},\end{gathered}$$ then there exists a unique formal matrix solution of the form $$\begin{gathered}
\label{seriessolhdiff}
\hat{Y}(x) = x^{\frac{nx}{h}} e^{-{nx}{h}}\left(I + \frac{\mathcal{Y}_1}{x}+ \frac{\mathcal{Y}_2}{x^2} +\cdots \right)\operatorname{diag}\left( \rho_1^{x/h} \left(\frac{x}{h}\right)^{d_1}, \ldots, \rho_M^{x/h} \left(\frac{x}{h}\right)^{d_n}\right),\end{gathered}$$ where $\{d_i\}$ is some set of constants.
Given a solution of that is convergent when $\Re x \gg 0$ or $\Re x \ll 0$, and since $\Re h > 0$, we may use to extend the solution by $$\begin{gathered}
\hat{Y}(x) = A(x-h)A(x-2h) \cdots A(x-kh) \hat{Y}(x-kh),\\
\hat{Y}(x) = A(x)^{-1}A(x+h)^{-1} \cdots A(x+(k-1)h)^{-1} \hat{Y}(x+kh).\end{gathered}$$ This extension introduces possible singularities at translates by integer multiples of $h$ of the points where $\det A(x) = 0$. The values of $x$ in which $\det A(x) = 0$ play an important role in the theory of discrete isomonodromy, hence, it is useful to parameterize the determinant by $$\begin{gathered}
\det A(x) = \rho_1 \cdots \rho_M(x-a_1)(x-a_2) \cdots (x-a_{Mn}).\end{gathered}$$
\[thm:diffexistence\] Assume that $A_0 = \operatorname{diag}(\rho_1, \ldots, \rho_M)$, with $$\begin{gathered}
\rho_i \neq 0, \qquad \rho_i/\rho_j \notin \mathbb{R} \qquad \textrm{for all} \ \ i \neq j,\end{gathered}$$ then there exists unique solutions of , $Y_l(x)$ and $Y_r(x)$, such that
1. The functions $Y_l(x)$ and $Y_r(x)$ are analytic throughout the complex plane except at translates to the left and right by integer multiples of $h$ of the poles of $A(x)$ and $A(x-h)^{-1}$ respectively.
2. In any left or right half-plane, $Y_l(x)$ and $Y_r(x)$ are asymptotically represented by .
Both $Y_l(x)$ and $Y_r(x)$ form a basis for the solutions of , which are both non-degenerate in the sense that in the limit as $x \to \pm \infty$, they possess a non-zero determinant, hence, are invertible almost everywhere. The notion that any two non-degenerate solutions of the same difference equation should be related leads us to the concept of a connection matrix. It should be clear as $Y_l(x)$ and $Y_r(x)$ are both solutions of , the connection matrix, defined by $$\begin{gathered}
P(x) = (Y_{l}(x))^{-1}Y_{r}(x),\end{gathered}$$ is periodic in $x$ with period $h$. The connection matrix and the monodromy matrices for systems of linear differential equations play very similar roles [@Borodin:connection].
Given a linear system of difference equations, it is useful to talk about the Riemann–Hilbert or monodromy map, which sends the Fuchsian system of differential equations to a set of monodromy matrices [@Boalch2005]. The monodromy matrices depend on the coefficients of a given Fuchsian system. The collection of variables that specify the monodromy matrices are called the characteristic constants. The next theorem defines the characteristic constants for systems of difference equations.
Under the general assumptions of Theorem [\[thm:diffexistence\]]{} the entries of connection matrix take the general form $$\begin{gathered}
(P(x))_{i,j} = \begin{cases}
p_{i,i}\big( e^{2\pi ix/h} \big) +e^{2\pi i (d_k + x/h)} & \text{if} \ i = j,\\
e^{2\pi i \lambda_{i,j}x/h} p_{i,j}\big(e^{2\pi ix/h}\big) , &
\end{cases}\end{gathered}$$ where each $p_{i,j}(x)$ is a polynomial of degree $n-1$ with $p_{i,i}(0) = 1$ and $\lambda_{i,j}$ denotes the least integer as great as the real part of $(\log(\rho_i) - \log(\rho_j))/2\pi i$.
Perhaps the simplest nontrivial example of such a connection matrix arises from solutions of the one dimension case, which can be broken down into linear factors of the form $$\begin{gathered}
y(x+h) = \left(1+\frac{d}{x}\right)y(x).\end{gathered}$$
In this way we associate a set of constants to each system of linear difference equations, by giving a map $$\begin{gathered}
(A_0, \ldots, A_n) \mapsto (\{d_k\}; \{p_{i,j}(x)\} ).\end{gathered}$$ This gives us $M(Mn+1)$ constants in total, which is also the number of entries in the coefficient matrices.
Assume there exists two polynomials $A(x) = A_0 + A_1 x + \cdots + A_nx^n$ and $\tilde{A}(x)=\tilde{A}_0 + \tilde{A}_1 x + \cdots + \tilde{A}_nx^n$ such that $A_n = \tilde{A}_n = \operatorname{diag}(\rho_1, \ldots, \rho_M)$ with the same sets of characteristic constants, then there exists a rational matrix $R(x)$, such that $\tilde{A}(x)$ and $A(x)$ are related by $($where $\sigma = \sigma_h)$. The fundamental solutions of , $Y_l(x)$ and $Y_r(x)$, are related to the fundamental solutions of , denoted $\tilde{Y}_l(x)$ and $\tilde{Y}_r(x)$, by respectively.
If we fix $A_n = \operatorname{diag}(\rho_1, \ldots, \rho_M)$, we may denote the algebraic variety of all $n$-tuples, $(A_0, \ldots$, $A_{n-1})$ such that $$\begin{gathered}
\det\big( A_n x^n + A_{n-1} x^{n-1} + \cdots + A_0\big) = \prod_{j=1}^{M} \rho_j \prod_{k = 1}^{Mn}(x-a_k),\end{gathered}$$ by $\mathcal{M}_h(a_1, \ldots, a_{Mn}; d_1, \ldots, d_M;\rho_1, \ldots, \rho_M)$. The space of discrete isomonodromic deformations is characterized by the following theorem of Borodin [@Borodin:connection].
\[hisolattice\] For any $\epsilon_1, \ldots, \epsilon_{Mn} \in \mathbb{Z}$, $\delta_1, \ldots, \delta_M \in \mathbb{Z}$ such that $$\begin{gathered}
\sum_{i=1}^{Mn} \epsilon_i + \sum_{j=1}^{M} \delta_j = 0,\end{gathered}$$ there exists a non-empty Zariski open subset, $\mathcal{A} \subset \mathcal{M}_h(a_1, \ldots, a_{Mn}; d_1, \ldots, d_M;\rho_1, \ldots, \rho_M)$, such that for any $(A_0,\ldots, A_{n-1}) \in \mathcal{A}$ there exists a unique rational matrix, $R(x)$ and a matrix, $\tilde{A}(x)$ related by such that $$\begin{gathered}
\tilde{A} = \tilde{A}_0 + \tilde{A}_1x + \cdots + \tilde{A}_{n-1}x^{n-1} + A_nx^n, \\
(\tilde{A}_0, \ldots, \tilde{A}_{n-1}) \in \mathcal{M}_h(a_1 + \epsilon_1, \ldots,a_{Mn} + \epsilon_{Mn}; d_1 + \delta_1, \ldots, d_M + \delta_M;\rho_1, \ldots, \rho_M),\end{gathered}$$ and where $\tilde{Y}_{\pm \infty}(x)$ are related to $Y_{\pm \infty}(x)$ by .
Fixing some translation, we obtain that and are a Lax pair for a birational map of algebraic varieties $$\begin{gathered}
\phi \colon \ \mathcal{M}_h (a_1, \ldots, a_{Mn}; d_1, \ldots, d_M;\rho_1, \ldots, \rho_M) \\
\hphantom{\phi \colon}{} \ {}\to \mathcal{M}_h(a_1 + \epsilon_1, \ldots,a_{Mn} + \epsilon_{Mn}; d_1 + \delta_1, \ldots, d_M + \delta_M;\rho_1, \ldots, \rho_M),\end{gathered}$$ which we wish to identify as some integrable system. For reasons of simplification, if $A_n = I$ we will also assume that $A_{n-1}$ is semisimple, in which case we we may apply a constant gauge transformation so that $A_{n-1}$ is also diagonal. Hence, if $A_n = I$, we will impose the condition that $A_{n-1}$ is diagonal for any tuple in $\mathcal{M}_h(a_1, \ldots, a_{Mn}; d_1, \ldots, d_M;1, \ldots, 1)$.
Classical $\boldsymbol{q}$-difference results
---------------------------------------------
We write where $\sigma= \sigma_q$ as $$\begin{gathered}
\label{linearqdiff}
Y(qx) = A(x)Y(x),\end{gathered}$$ where $A(x)$ is a rational $M\times M$ matrix that is invertible almost everywhere and $0 < |q| < 1$ as above.
The functions required to express the solution of any scalar linear first-order system of $q$-difference equations are not as commonly used as the gamma function. Hence, before discussing some of the particular existence theorems, let us introduce some standard functions, all of which may be found in [@GasperRahman]. We define the $q$-Pochhammer symbol by $$\begin{gathered}
( a;q )_{\infty} = \prod_{n=0}^{\infty} \big(1-aq^{n}\big),\qquad
( a;q )_m = \frac{(a;q)_{\infty}}{(aq^{m};q)_{\infty}}.\end{gathered}$$ The important property of $(x;q)_{\infty}$ is that $$\begin{gathered}
(qx;q)_{\infty} = \frac{(x;q)_{\infty}}{1-x}.\end{gathered}$$ We also have the Jacobi theta function, $$\begin{gathered}
\theta_q(x) = \sum_{n=-\infty}^{\infty} q^{\frac{n(n-1)}{2}} x^{n},\end{gathered}$$ which is analytic over $\mathbb{C}^{*}$ and satisfies $$\begin{gathered}
\theta_q(qx) = qx\theta_q(x),\qquad
\theta_q(x) = (q;q)_{\infty}(-xq;q)_{\infty}(-q/x;q)_{\infty}.\end{gathered}$$ The last expression is known as the Jacobi triple product identity [@GasperRahman]. The function $\theta_q(x)$ has simple roots on $-q^{\mathbb{Z}}$. We define the $q$-character to be $$\begin{gathered}
e_{q,c}(x) = \frac{\theta_q(x)}{\theta_q (x/c)},\end{gathered}$$ which satisfies $e_{q,c}(qx) = ce_{q,c}(x)$ and has simple zeroes at $x=q^{\mathbb{Z}}$ and simple poles at $x=cq^{\mathbb{Z}}$. In the special case in which $c = q^n$ then $e_{q,q^n}(x)$ is proportional to $x^n$. Lastly, we have the $q$-logarithm $$\begin{gathered}
l_q(x) = x\frac{\theta_q'(x)}{\theta_q(x)},\end{gathered}$$ which satisfies $$\begin{gathered}
\sigma_q l_q(x) = l_q(x)+1,\end{gathered}$$ and is meromorphic over $\mathbb{C}^{*}$ with simple poles on $q^{\mathbb{Z}}$.
We may use the functions above to solve any scalar $q$-difference equation, hence transform in which $A(x)$ is rational to a case in which $A(x)$ is polynomial, given by . If $A_0$ and $A_n$ are semisimple and invertible, then by using constant gauge transformations we can assume that one of them, say $A_n$, is diagonal. Under these circumstances, we may specify two formal solutions.
\[lemq\] Suppose $A_n = \operatorname{diag}(\kappa_1, \ldots, \kappa_M)$ and $A_0$ is semisimple with non-zero eigenvalues, $\theta_1, \ldots, \theta_M$, such that the non resonnancy conditions $$\begin{gathered}
\frac{\kappa_i}{\kappa_j}, \ \frac{\theta_i}{\theta_j} \neq q, \ q^2, \ \ldots\end{gathered}$$ are satisfied, then there exists two formal matrix solutions
$$\begin{gathered}
\label{basicqdif}
Y_0(x) = \left(\mathcal{Y}_0 + \mathcal{Y}_1x + \cdots \right) \operatorname{diag}(e_{q,\theta_1}, \ldots, e_{q,\theta_M}),\\
Y_{\infty}(x) = \left(I + \frac{\mathcal{Y}_{-1}}{x} + \cdots \right) \operatorname{diag}(e_{q,\kappa_1}, \ldots, e_{q,\kappa_M}) \theta_q(x/q)^{-n},\end{gathered}$$
where $\mathcal{Y}_0$ diagonalizes $A_0$.
By using $\theta_q$ as our building block for the multiplicative factors appearing on the right in , this formulation is slightly different from the original formulation of Birkhoff [@Birkhoffallied]. These functions have nicer properties with respect to the Galois theory of difference equations [@Sauloy; @VanderPut2003]. We should mention that the above form can be generalized to the case in which some of the eigenvalues are $0$ by using the so-called Birkhoff–Guenther form [@BirkhoddAdamsSum]. Formal solutions defined in terms of the Birkhoff–Guenther form do not necessarily define convergent solutions. This issue of convergence gives rise to a $q$-analogue of the Stokes phenomenon for systems of linear differential equations [@FlaschkaNewell]. Regardless of the convergence, these solutions may be used to derive deformations of the form , as shown in [@Ormerodlattice].
We are interested in solutions defined in open neighborhoods of $x= 0$ and $x= \infty$, which may be extended by $$\begin{gathered}
Y_{\infty}(x) = A(x/q)A\big(x/q^2\big) \cdots A\big(x/q^k\big) Y_{\infty}\big(x/q^k\big),\\
Y_0(x) = A(x)^{-1}A(qx)^{-1} \cdots A\big(q^{k-1} x\big)^{-1} Y_0\big(q^kx\big).\end{gathered}$$ The resulting solutions are singular at $q$-power multiples of the values of $x$ where $\det A(x) = 0$. For this reason, it is once again convenient to fix where $A(x)$ is not invertible. If $A_n$ is semisimple with eigenvalues $\kappa_1, \ldots, \kappa_M$ ($\kappa_i \neq 0$) then we parameterize the determinant as $$\begin{gathered}
\label{qdetA}
\det A(x) = \kappa_1 \cdots \kappa_M (x-a_1) \cdots (x-a_{Mn}).\end{gathered}$$ The series part of the solution around $x= 0$, which we denote $\hat{Y}_0(x)$, satisfies $$\begin{gathered}
\hat{Y}_0(qx) A_0 = A(x)\hat{Y}_0(x),\end{gathered}$$ whereas the series part of the solution around $x=\infty$, denoted $\hat{Y}_{\infty}(x)$, satisfies a similar equation. By a succinct argument featured in van der Put and Singer [@VanderPut2003 Section 12.2.1] we have solutions, $\hat{Y}_0(x)$ and $\hat{Y}_{\infty}(x)$, that are convergent in neighborhoods of $x=0$ and $x=\infty$ respectively.
The series part of the solutions, $\hat{Y}_0(x)$ and $\hat{Y}_{\infty}(x)$, specified by are holomorphic and invertible at $x=0$ and $x = \infty$ respectively. Moreover, $\hat{Y}_{\infty}(x)$ and $\hat{Y}_0(x)^{-1}$ have no poles, while $Y_{\infty}(x)^{-1}$ and $\hat{Y}_0(x)$ have possible poles at $q^{k+1}a_i$ and $q^{-k}a_i$ respectively for $i = 1,\ldots, mn$ and $k \in \mathbb{N}$.
While we do not have an explicit presentation of the connection matrix, it is generally known to be expressible in terms of elliptic theta functions. In particular, the entries of the connection matrix lie in the field of meromorphic functions on an elliptic curve, i.e., $\mathbb{C}^{*}/\langle q \rangle$.
Let us specify the required lattice actions in a similar way to the $h$-difference case. We denote the algebraic variety of all $n$-tuples of $M\times M$ matrices, $(A_1, \ldots, A_n)$ such that $A_n$ has eigenvalues $\kappa_1,\ldots, \kappa_M$ and $A_0 = \operatorname{diag}(\theta_1, \ldots, \theta_M)$ with determinant specified by by $\mathcal{M}_q(a_1, \ldots, a_{Mn}; \kappa_1, \ldots, \kappa_M; \theta_1, \ldots, \theta_M)$. The natural constraint obtained by evaluating at $x=0$, is that $$\begin{gathered}
\label{qgenprodcon}
\prod_{j=1}^{m} \kappa_j \prod_{k = 1}^{Mn} a_k (-1)^{Mn} =\prod_{j=1}^{M} \theta_j.\end{gathered}$$
\[qlattice\] For any $\epsilon_1, \ldots, \epsilon_{Mn} \in \mathbb{Z}$ and $\delta_1, \ldots, \delta_M \in \mathbb{Z}$ such that $$\begin{gathered}
\sum_{j=1}^{Mn} \epsilon_{j} + \sum_{i=1}^{M} \delta_i = 0,\end{gathered}$$ there exists a non-empty Zariski open subset, $\mathcal{A} \subset \mathcal{M}_q(a_1, \ldots, a_{Mn}; \kappa_1, \ldots, \kappa_M; \theta_1, \ldots, \theta_M)$, such that for any $(A_1,\ldots, A_{n-1}) \in \mathcal{A}$ there exists a rational matrix, $R(x)$, and a matrix, $\tilde{A}(x)$ related by with $$\begin{gathered}
\tilde{A}(x) = A_0 + \tilde{A}_1x + \cdots + \tilde{A}_{n-1}x^{n-1} + \tilde{A}_nx^n, \\
(\tilde{A}_1, \ldots, \tilde{A}_{n-1}) \in \mathcal{M}_q\big(a_1q^{\epsilon_1}, \ldots,a_{mn} q^{\epsilon_{mn}}; \kappa_1q^{\delta_1}, \ldots, \kappa_Mq^{\delta_M}; \theta_1, \ldots, \theta_M\big),\end{gathered}$$ and where $\tilde{Y}_{0}(x)$ and $\tilde{Y}_{\infty}(x)$ are related to $Y_{0}(x)$ and $Y_{\infty}(x)$ by .
It is sufficient to specify an atomic operation that performs the following invertible operation $$\begin{gathered}
e_{1,1} \colon \ \kappa_1 \to \kappa_1/q, \qquad a_1 \to qa_1,\end{gathered}$$ which when composed with actions that permutes $a_1,\ldots, a_{Mn}$ and $\kappa_1, \ldots, \kappa_n$ give us all transformations we require. A matrix that does this is found by using a constant gauge transformation to change the basis so that the vectors $$\begin{gathered}
\ker(A(a_1)), \ \ker(A_m - \kappa_2), \ \ldots, \ \ker(A_m - \kappa_M)\end{gathered}$$ are the new coordinate vectors. We then perform a gauge transformation of the form whose effect is dividing the first column by $(1-x/a_1)$ and multiplying the first row by $(1- x/a_1q)$. Reverting back to a basis in which $A_0$ is the constant coefficient matrix using another constant matrix gives the required matrix. It should be clear from the determinant that $a_1 \to q a_1$, while looking at $\tilde{A}(x)$ asymptotically around $x=\infty$ it is clear $\kappa_1 \to \kappa_1/q$. Since all these steps were invertible, the inverse atomic operation is also rational, hence, we obtain all possible transformations this way.
Systems of linear $q$-difference equations can also be treated as discrete connections, where the matrix presentations of these systems of linear $q$-difference equations arise as trivializations of linear maps between the fibres of a vector bundle. In this framework, the theorem above may also be deduced by purely geometric means, as was done in the $h$-difference case in [@Arinkin2006]. The $q$-difference version of this framework was the subject of a recent paper by Kinzel [@Knizel2015].
The elementary translations are those that multiply any collection of up to $m$ of the $a_i$’s by $q$ and multiply the same number of $\kappa_j$’s by $q^{-1}$. For the applications that follow, this formulation will be sufficient, however, this is a slightly less general result than possible. One may generally find a rational matrix in which the $\theta_i$ values are shifted by $q$-powers in a way that preserves .
Difference equations and vector bundles
---------------------------------------
The aim of this section is to present the theorems required for the existence of meromorphic solutions to , which we write as two cases: $$\begin{gathered}
Y(-x-h) = A(x)Y(x)\qquad \text{and} \qquad Y(x) = Y(-x),\end{gathered}$$ or $$\begin{gathered}
Y(1/qx) = A(x)Y(x)\qquad \text{and} \qquad Y(x) = Y(1/x).\end{gathered}$$ To prove the general existence of solutions with these symmetry properties, we turn to some general results concerning sheaves on compact Riemann surfaces (see [@Forster1980] for example). For a connected Riemann surface, $\Sigma$, we may denote the sheaves of holomorphic and meromorphic functions on $\Sigma$ by $\mathcal{O}_{\Sigma}$ and $\mathcal{M}_{\Sigma}$ respectively. A holomorphic or meromorphic vector bundle of rank $n$ is a sheaf of $\mathcal{O}_{\Sigma}$-modules or $\mathcal{M}_{\Sigma}$-modules which is locally isomorphic to $\mathcal{O}_{\Sigma}^n$ or $\mathcal{M}_{\Sigma}^n$ respectively.
\[Praagman\] Let $G$ be a group of automorphisms of $\mathbb{P}_1$, $L$ is the limit set of $G$ and $U$ a component of $\mathbb{P}_1\setminus L$ such that $G(U) = U$. If there is a map, $G \to \mathrm{GL}_M(\mathcal{M}_U)$, $g \to A_g(x)$ satisfying $$\begin{gathered}
A_{gh}(x) = A_g(h(z)) A_h(z),\end{gathered}$$ then the system of equations $$\begin{gathered}
Y(\gamma(z)) = A_{\gamma}(z) Y(z) ,\qquad \gamma \in G,\end{gathered}$$ possesses a meromorphic solution.
The two important examples in the context pertain to the case in which $G$ is a group of automorphisms of $\mathbb{P}_1$ admitting the presentation $$\begin{gathered}
G = \big\langle \tau_1,\tau_2 \,|\, \tau_1^2 = \tau_2^2 = 1 \big\rangle,\end{gathered}$$ which is often called the infinite dihedral group. In particular, we are interested in the case in which the groups of automorphisms are $$\begin{gathered}
G_h = \langle \tau_1,\tau_2 \,|\, \tau_1(x) = -h-x, \, \tau_2(x) = -x \rangle,\\
G_q = \left\langle \tau_1, \tau_2 \,|\, \tau_1(x) = \frac{1}{qx}, \, \tau_2(x) = \frac{1}{x} \right\rangle.\end{gathered}$$ If we let $A_{\tau_2} = I$ in each case and $A_{\tau_1}(x)$ be some rational matrix, $A(x)^{-1}$, the commutation relation on $\tau_1$ and $\tau_2$ requires $$\begin{gathered}
A(x) = A(-h-x)^{-1} \qquad \text{or}\qquad A(x) = A\left(\frac{1}{qx}\right)^{-1},\end{gathered}$$ respectively.
\[lem:existencegenatu\] Let $\mathbb{L}/\mathbb{K}$ be a quadratic field extension and $A \in \mathrm{GL}_n(\mathbb{L})$ be a matrix such that $\bar{A}=A^{-1}$, where $\bar{A}$ is the conjugation of $A$ in $\mathbb{L}$ over $\mathbb{K}$. Then there exists a matrix $B \in \mathrm{GL}_n(\mathbb{L})$ such that $A = \bar{B}B^{-1}$ and $B$ is unique up to right-multiplication by $\mathrm{GL}_n(\mathbb{K})$.
Given a vector $w \in \mathbb{L}^n$, it is easy to see that if $v = \bar{w} + A^{-1} w$ then $\bar{v} = w + A\bar{w} = A v$. Applying to any basis for $\mathbb{L}^n$ over $\mathbb{K}$ gives at least $n$ vectors satisfying $\bar{v} = Av$ that are linearly independent over $\mathbb{K}$, whose columns give a matrix $B$ such that $$\begin{gathered}
\label{hil90proof}
\bar{B} = AB.\end{gathered}$$ For uniqueness we suppose two such matrices, $B_1$ and $B_2$, satisfy , then $C = B_1 B_2^{-1}$ satisfies $C = \bar{C}$, in which case $C \in \mathrm{GL}_n(\mathbb{K})$.
This lemma is a special case of what is often called “Hilbert’s theorem 90", which states that any $1$-cocycle of a Galois group with values in $\mathrm{GL}_n$ is trivial. Hilbert dealt with the case in which $\mathrm{Gal}(\mathbb{L}/\mathbb{K})$ is cyclic, and $n=1$.
Specializing to the function fields $\mathbb{L}= \mathbb{C}(x)$ and $\mathbb{K}$ is the subfield of rational functions invariant under $x \to -x-h$ or $x \to 1/x$ allows us to write $A(x)$ as one of two cases;
$$\begin{gathered}
\label{symmdiff}A(x) = B(-h-x) B(x)^{-1},\\
\label{symmqdiff}A(x) = B\left( \frac{1}{qx} \right) B(x)^{-1},\end{gathered}$$
where $B(x)$ is rational. This reduces the problem of determining the algebraic variety of all $n$-tuples of matrices with a symmetry condition to determining $n$-tuples of matrices with prescribed properties. In particular, it makes sense to let $$\begin{gathered}
B(x) = B_0 + B_1 x + \cdots + B_nx^n,\end{gathered}$$ where either $$\begin{gathered}
(B_0,\ldots, B_{n-1}) \in \mathcal{M}_h(a_1, \ldots, a_{Mn}; d_1, \ldots, d_M;\rho_1, \ldots, \rho_M),\\
(B_0,\ldots, B_{n-1}) \in \mathcal{M}_q(a_1, \ldots, a_{Mn}; \kappa_1, \ldots, \kappa_M;\theta_1,\ldots, \theta_M).\end{gathered}$$ In discussing the isomonodromic deformations, we specify two different types of discrete isomonodromic deformations; those that act on the left and those that act on the right, which are given as follows
\[symlax\] $$\begin{gathered}
\label{leftRB}\tilde{B}(x) = \lambda(x) R_l(x) B(x),\\
\label{rightRB}\tilde{B}(x) = \lambda(x) B(x) R_r(x),\end{gathered}$$
where $\lambda(x)$ is some rational scalar factor. This scalar factor only swaps poles and roots of the determinant and should be considered trivial from the perspective of integrability. These two equations should be thought of as the symmetric equivalent of . We may rigidify the definitions of $R_l(x)$ or $R_r(x)$ by insisting that these matrices are proportional to identity matrices around $x=\infty$.
If we insist that $R_l(x)$ is invariant under $\tau_2$, i.e., we have the symmetry $R_l(x) = \tau_2 R_l(x)$, then it is clear that a transformation of the form coincides with a transformation of the form , hence, will be considered a discrete isomonodromic deformation. Furthermore, if we may find such a matrix, Theorem \[hisolattice\] or Theorem \[qlattice\], depending on the case, tells us that this matrix and resulting transformation are unique, hence, the discrete isomonodromic deformation does preserve the required symmetry.
Preserving the Galois group {#sec:Galois}
---------------------------
The main reason for passing from connection preserving deformations to the Galois theory of difference equations is that we have not shown that systems of the form possess connection matrices. While mechanically, we still have Lax pairs using or , the implications of possessing a discrete Lax pair of any form are not generally known. We wish to show that and preserve the associated Galois group.
This is an issue that is not confined to symmetric Lax pairs. Various Painlevé equations are known to arise as relations of the form of where the series part of the formal solutions at $x=\pm \infty$ or $x= 0$ are not convergent [@OrmerodqPV; @Ormerodlattice]. From an integrable systems perspective, it is useful to know precisely what is preserved, and it turns out the associated difference module is always preserved under transformations of the form . We require some of the formalism described in [@VanderPut2003] to demonstrate this.
A difference ring is a commutative ring/field, $R$, with $1$, together with an automorphism $\sigma \colon R \to R$. The constants, denoted $C_R$ are the elements satisfying $\sigma(f) = f$. An ideal of a difference ring is an ideal, $I$, such that $\phi(I) \subset I$. If the only difference ideals are $0$ and $R$ then the difference ring is called simple.
This is a natural discrete analogue of a differential field. In Picard–Vessiot theory, a Picard–Vessiot extension is formed by extending the field of constants by the solutions of a homogenous linear ordinary differential equation [@VanderPut2003]. The analogue of this for difference equations is the following construction.
Let $\mathbb{K}$ be a difference field and be a first-order system with $A(x) \in \mathrm{GL}_n(\mathbb{K})$. We call a $\mathbb{K}$-algebra, $R$, a Picard–Vessiot ring for if:
1. an extension of $\sigma$ to $R$ is given,
2. $R$ is a simple difference ring,
3. there exists a solution of with coefficients in $R$,
4. $R$ is minimal in the sense that no proper subalgebra satisfies $1$, $2$ and $3$.
We are treating $\mathbb{C}(x)$ as a difference field where $\sigma_h$ and $\sigma_q$ are the relevant automorphisms. The field of constants contain $\mathbb{C}$ extended by the $\sigma$-periodic functions (e.g., $e^{2 i \pi x/h}$ and $\phi_{c,d} = e_{q,c}e_{q,d}/e_{q,cd}$). We may formally construct a Picard–Vessiot ring for by considering a matrix of inderminants, $Y(x) = (y_{i,j}(x))$. We extend $\sigma$ to $\mathbb{K}(Y)$ via the entries of . If $I$ is a maximal difference ideal, then we obtain a Picard–Vessiot ring for by considering the quotient $\mathbb{K}(Y)/ I$. This quotient by a maximal difference ideal ensures the resulting construction is a simple difference ring.
This formal construction may be replaced by a fundamental system of meromorphic solutions of either or specified by Theorem \[lem:existencegenatu\]. For $q$-difference equations, in general (see [@vanderPut]) the entries of any solution are elements of the field $\mathcal{M}(\mathbb{C})(l_q, (e_{q,c})_{c \in \mathbb{C}^*})$.
If $R$ is a Picard–Vessiot ring for , the Galois group, $G = \mathrm{Gal}(R/C_R)$ is the group of automorphisms of $R$ commuting with $\sigma$.
Let us briefly describe the role of the connection matrix in this context. We have given conditions for there to exist two fundamental solutions, which we will call $Y_1(x)$ and $Y_2(x)$, which are distinguished by the regions of the complex plane in which they define meromorphic functions. If we adjoin the entries of $Y_1(x)$ or $Y_2(x)$ we describe two Picard–Vessiot extensions, denoted $R_1$ and $R_2$. We expect $R_1$ and $R_2$ to be isomorphic to the formal construction above, in particular, there exists an isomorphism between $R_1$ and $R_2$. The connection matrix, $P(x)$, relates solutions via $$\begin{gathered}
Y_1(x) = P(x)Y_2(x),\end{gathered}$$ which defines such an isomorphism between $R_1$ and $R_2$. For any generic value of $x$ for which $P(x)$ is defined, $P(x)$ describes a connection map, which is an isomorphism of Picard–Vessiot extensions, hence, for generic values of $u$ and $v$ for which the connection matrix is defined, the matrix $P(u)P(v)^{-1}$ defines an automorphism of $R_1$. In the case of regular systems of $q$-difference equations, it is a result of Etingof that the Galois group is a linear algebraic group over $\mathbb{C}$ that is generated by matrices of the form $P(u)P(v)^{-1}$ for $u,v \in C$ where defined [@Etingof1995]. This mirrors differential Galois theory where it is generally known that the differential Galois group is generated by the monodromy matrices, the Stokes matrices and the exponential torus [@Martinet1989]. More generally, this relation between values of the connection matrix and the Galois group has been the subject of works of a number of authors [@Sauloy; @vanderPut].
We may generalize the definition of the Galois group from a category theoretic perspective. Given a difference field, $\mathbb{K}$ (e.g., $\mathbb{C}(x)$), with a difference operator $\sigma$, we can consider the ring of finite sums of difference operators in a new operator, $\phi$, $$\begin{gathered}
k[\phi, \phi^{-1}] = \left\{ \sum_{n \in \mathbb{Z}} a_n \phi^n \right\},\end{gathered}$$ where $\phi$ is defined by the relation $\phi(\lambda) = \sigma(\lambda) \phi$ for $\lambda \in \mathbb{K}$. We can consider the category of left modules, $M$, over $\mathbb{K}$. Under a suitable basis, we may identify $M$ with $\mathbb{K}^m$. In this basis, the action of $\phi$ is identified with a matrix by $$\begin{gathered}
\label{moduledef}
\phi Y = A \sigma Y.\end{gathered}$$ Conversely, given a difference equation of the form $\sigma Y = AY$, we may endow $\mathbb{K}^m$ with the structure of a difference module via .
\[diffmods\] Two systems, $\sigma Y(x) = A(x)Y(x)$ and $\sigma \tilde{Y}(x) = \tilde{A}(x)\tilde{Y}(x)$ define isomorphic difference modules if and only if the matrices $A(x)$ and $\tilde{A}(x)$ are related by .
The object that is being preserved under these deformations is the local system/sheaf of solutions. We could also call these transformations isomodular since the difference module is preserved.
The advantage of this definition is that the category of difference modules over a difference field is a rigid abelian tensor category. We may use the definitions of [@Deligne1981] to define the Galois group from a category theoretic perspective. While it is difficult to see a priori that a transformation of the form necessarily preserves the Galois group, from the perspective of the category theory, isomorphic difference modules resulting from Theorem \[diffmods\] yield isomorphic Galois groups.
\[corintegrability\] Two systems, $\sigma Y(x) = A(x)Y(x)$ and $\sigma \tilde{Y}(x) = \tilde{A}(x)\tilde{Y}(x)$, related by defines a transformation that preserves the Galois group.
These structures can be defined without reference to a connection matrix, it only requires the existence of a linearly independent set of solutions specified by Theorem \[Praagman\]. In particular, it specifies that the birational maps of Theorems \[hisolattice\] and \[qlattice\] are integrable with respect to the preservation of a Galois group. What may be interesting from an integrable systems perspective is to consider the combinatorial data that specifies the difference module. Such data would be the analogue of the characteristic constants involved in isomonodromic deformations, and the map from the given difference module to this data would constitute a discrete analogue of the Riemann–Hilbert map [@Boalch2005].
Discrete Garnier systems {#sec:Garnier}
========================
We now turn to the parameterization of our discrete Garnier systems, which has drawn inspiration from a series of results concerning the description of various integrable autonomous mappings and discrete Painlevé equations in terms of reductions of partial difference equations [@Ormerod2014a]. We have denoted the various cases of discrete Garnier systems by a value $m$ in a way that the case $m=1$ coincides with a discrete Garnier system that possesses the sixth Painlevé equation as a limit. With respect to the Garnier systems increasing $m$ increases the number of poles of the matrix of the associated linear problem whereas increasing $m$ by one in what we are calling the discrete Garnier systems increases the number of roots of the determinant of the matrix for the associated linear problem by two.
The asymmetric $\boldsymbol{h}$-difference Garnier system
---------------------------------------------------------
We start with where $A(x)$ is specified by for $N = 2m+4$ with each factor of is taken to be of the form $L_i(x) = L(x,u_i,a_i)$ where $$\begin{gathered}
\label{difffactor}
L(x,u,a) = \begin{pmatrix} u & 1 \\ x-a+u^2 & u \end{pmatrix}.\end{gathered}$$ The variable $a$ parameterizes the value of the spectral parameter, $x$, in which $L$ is singular. Some of the useful properties of these matrices are
\[detL\] $$\begin{gathered}
\det L(x,u,a) = a-x, \label{detrel}\\
L(x+\delta,u,a+\delta) = L(x,u,a),\label{hinvar} \\
L(x,u,a)^{-1} = \frac{1}{x-a} L(x,-u,a),\label{hinv}\\
\operatorname{Ker} L(a,u,a) = \left\langle \begin{pmatrix} 1 \\ - u \end{pmatrix} \right\rangle, \label{hker}\\
\operatorname{Im} L(a,u,a) = \left\langle \begin{pmatrix} 1 \\ u \end{pmatrix} \right\rangle, \label{hIm}\end{gathered}$$
hence we think of $u$ as the variable parameterizing the image and kernel vectors. The resulting matrix, $A(x)$, takes the general form $$\begin{gathered}
\label{mtupleh}
A(x) = A_0 + A_1 x + \dots + A_{m+1} x^{m+1} + A_{m+2}x^{m+2}.\end{gathered}$$
Let $A(x)$ be the matrix specified by where each factor is given by subject to the constraints $$\begin{gathered}
\label{constrainth}\sum_{i = 1}^{2m+4} u_i = 0,\\
\label{constrainth2} \sum_{k \textrm{ even}} \big(u_{k}^2-a_k\big) \neq \sum_{k \textrm{ odd}} (u_{k}^2-a_k),\end{gathered}$$ then $A(x)$ defines an $(m+2)$-tuple $(A_0,\ldots, A_{m+1})$ via where $A_{m+2} = I$ with $$\begin{gathered}
(A_0, \ldots, A_{m+1}) \in \mathcal{M}_h(a_1,\ldots, a_{2m+4};d_1, d_2;1,1),\end{gathered}$$ where the values of $d_1$ and $d_2$ are
\[dvals\] $$\begin{gathered}
\label{d1}d_1 = \sum_{i=1}^{N}\sum_{j=1}^{i-1} u_i u_j + \sum_{k \textrm{ even}} \big(u_{k}^2-a_k\big), \\
\label{d2}d_2 = \sum_{i=1}^{N}\sum_{j=1}^{i-1} u_i u_j + \sum_{k \textrm{ odd}} \big(u_{k}^2-a_k\big).\end{gathered}$$
The determinant of $A(x)$ is given by $$\begin{gathered}
\label{deth}
\det A(x) = (x-a_1) \cdots (x-a_{N}),\end{gathered}$$ which follows from . The first two terms in the asymptotic expansion around $x= \infty$ are $$\begin{gathered}
A(x) = x^{m+2} \begin{pmatrix}
1 & 0 \\
r_{1,2} & 1
\end{pmatrix}
+ x^{m+1}
\begin{pmatrix} d_1 & r_{1,2} \\
r_{2,1} & d_2
\end{pmatrix}
+ O\big(x^{m-1}\big),\end{gathered}$$ where $d_1$ and $d_2$ are given by (\[dvals\]) and $r_{1,2}$ is given by the left-hand side of , hence, $A_{m+2} = I$ when assuming the constraints. The value of $r_{2,1}$ is $$\begin{gathered}
\label{r21h}
r_{2,1} = \sum _{i=1}^{m+2} \left(\big(a_{2 i-1}+u_{2 i-1}^2\big) \sum _{k=2 i}^{N} u_k+\big(a_{2 i}+u_{2 i}^2\big) \sum _{k=1}^{2 i-1} u_k\right) + \sum_{1\leq k < j < i \leq N} u_i u_j u_k,\end{gathered}$$ which =-1 may be used in a constant lower triangular gauge transformation that diagonalizes $A_n\!$. This naturally preserves $d_1$ and $d_2$, hence, defines an element of $\mathcal{M}_h(a_1,\ldots, a_{2m+4};d_1, d_2;1,1)$.
While it is a consequence of , , and , it should be noted that $d_1$ and $d_2$ satisfy $$\begin{gathered}
\label{drel}
d_1 + d_2 + \sum_{i=1}^{N} a_i = 0,\end{gathered}$$ which is a constraint that is necessarily satisfied by any element of $\mathcal{M}_h(a_1,\ldots, a_{2m+4};d_1, d_2;1,1)$.
Suppose we are given $A_{m+2} = I$, and an $(m+2)$-tuple $$\begin{gathered}
(A_0,\ldots, A_{m+1}) \in \mathcal{M}_h(a_1,\ldots, a_N;d_1,d_2;1,1),\end{gathered}$$ where $A_{m+1}$ has been diagonalized, we wish to know whether there is a corresponding matrix of the form . We claim that the subvariety of $(m+2)$-tuples arising from is of the same dimension. If we fix $A_{m+2} = I$ and $A_{m+1} = \operatorname{diag}(d_1,d_2)$ then each of the $4(m+1)$ entries of the $A_i$’s, for $i = 0, \ldots, m$, are considered free. We have $2m+3$ coefficients of the determinant not automatically satisfied. Conjugating by diagonal matrices may also be used to fix one additional off-diagonal entry, which also removes any gauge freedom, making a algebraic variety of dimension $2m$ (or $2m+1$ with a gauge freedom).
Similarly, a product of the form is specified by $2m+4$ values, $u_i$ for $i = 1,\ldots, 2m+4$ subject to two constraints, namely and and , one gauge freedom and two constants related by , giving a total of $2m+2$ free variables. Fixing $r_{2,1}$ removes another variable, as does conjugating by diagonal matrices, which gives an algebraic variety dimension $2m$ (or $2m+1$ with a gauge freedom), as above.
We may also describe maps between $\mathcal{M}_h(a_1,\ldots, a_N;d_1,d_2;1,1)$ and matrices given by . To obtain an element of $\mathcal{M}_h(a_1,\ldots, a_N;d_1,d_2;1,1)$, we expand the product and diagonalize. To obtain we obtain left (or right) factors of $A(x)$ by observing the corresponding image (kernel) vectors at the points $x = a_1$ (or $x=a_n$).
The property we will use to parameterize the system of discrete isomonodromic deformations is given by the following observation.
\[com:diff\] The matrices of the form of satisfy the commutation relation $$\begin{gathered}
L(x,u_i,a_i)L(x,u_j,a_j) = L(x,\tilde{u}_j,a_j)L(x,\tilde{u}_i,a_i),\end{gathered}$$ where the map $(u_i,u_j ) \to \left(\tilde{u}_i,\tilde{u}_j\right)$ is given by $$\begin{gathered}
\label{FV}
\tilde{u}_i = u_j + \frac{a_i - a_j}{u_i+u_j}, \qquad \tilde{u}_j = u_i - \frac{a_i - a_j}{u_i+u_j}.\end{gathered}$$
This is a well known relation for these matrices [@Adler1993; @KajiwaraSym; @Suris2003]. This map is related to the discrete potential Korteweg–de Vries equation [@Papageorgiou2006]. If we let $R_{i,j}$ be the map $$\begin{gathered}
\label{RijYB}
R_{i,j} \colon \ ( u_1, \ldots, u_i, \ldots, u_j, \ldots, u_n ) \to (u_1, \ldots, \tilde{u}_i, \ldots, \tilde{u}_j, \ldots, u_n),\end{gathered}$$ then this map satisfies the relation $$\begin{gathered}
\label{YangBaxter}
R_{23}R_{13}R_{12}(u,v,w) = R_{12}R_{13}R_{23}(u,v,w),\end{gathered}$$ which is known as the Yang–Baxter property for maps. This map appears as $\mathrm{F}_{\mathrm{V}}$ in the classification of quadrirational Yang–Baxter maps [@Adler2003]. A common pictorial representation of this property appears in Fig. \[fig:YangBaxter\]. More generally, it has been remarked upon in [@Borodin:connection] that the set of commuting transformations obtained by discrete isomonodromic deformations define solutions to the set-theoretic Yang–Baxter maps [@Veselov2003].
(0,0) – (1,.5) – (1,1.5) – (0,2) – (-1,1.5) – (-1,.5)–cycle; (-1,.5) –(0,1) – (0,2); (0,1) – (1,.5); (1,1) arc (60:120:2cm); (-.5,.25) arc (-61:-7:2cm); (.5,.25) arc (-120:-173:2cm); (.5,.25) circle (.08cm); (1,1) circle (.08cm); (.5,1.75) circle (.08cm); (-.5,.25) circle (.08cm); (-1,1) circle (.08cm); (-.5,1.75) circle (.08cm); at (-.6,0) [$w$]{}; at (-.6,2) [$u$]{}; at (-1.3,1) [$v$]{}; at (.6,2) [$\tilde{w}$]{}; at (.6,0) [$\tilde{u}$]{}; at (1.3,1) [$\tilde{v}$]{}; at (-1.1,1.7) [$R_{12}$]{}; at (0,-.2) [$R_{13}$]{}; at (1.1,1.7) [$R_{23}$]{};
(0,0) – (1,.5) – (1,1.5) – (0,2) – (-1,1.5) – (-1,.5)–cycle; (0,0) –(0,1) – (1,1.5); (0,1) – (-1,1.5); (1,1) arc (-60:-120:2cm); (-.5,.25) arc (-187:-237:2cm); (.5,.25) arc (7:60:2cm); (.5,.25) circle (.08cm); (1,1) circle (.08cm); (.5,1.75) circle (.08cm); (-.5,.25) circle (.08cm); (-1,1) circle (.08cm); (-.5,1.75) circle (.08cm); at (-.6,0) [$w$]{}; at (-.6,2) [$u$]{}; at (-1.3,1) [$v$]{}; at (.6,2) [$\tilde{w}$]{}; at (.6,0) [$\tilde{u}$]{}; at (1.3,1) [$\tilde{v}$]{}; at (-1.1,.3) [$R_{23}$]{}; at (0,2.2) [$R_{13}$]{}; at (1.1,.3) [$R_{12}$]{};
We may use Lemma \[com:diff\] to define an action of $S_N$ on $A(x)$. Given a permutation, $\sigma \in S_N$, we denote the corresponding rational transformation of the $u_i$ and $a_i$ by $s_{\sigma} u_i$ and $s_{\sigma} a_i$ respectively. The group $S_N$ is generated by 2-cycles of the form $(i,i+1)$, whose action we denote by $s_i = s_{(i,i+1)}$ for $i = 1, \ldots, N-1$. Using Lemma \[com:diff\] these are given by
\[permute\] $$\begin{gathered}
s_i \colon \ u_i \to u_{i+1} + \frac{a_i - a_{i+1}}{u_i+u_{i+1}}, \qquad s_i \colon \ a_i = a_{i+1},\\
s_i \colon \ u_{i+1} \to u_i - \frac{a_i - a_{i+1}}{u_i+u_{i+1}}, \qquad s_i \colon \ a_{i+1} = a_i.\end{gathered}$$
By construction for any $\sigma \in S_N$, the effect of $s_{\sigma}$ on $A(x)$ is trivial. We may use the action of $S_n$ to determine the image or kernel of $A(x)$ at $x=a_i$ by acting on $A(x)$ by a permutation that sends the factor that is singular at $x=a_i$ to either the first or last term of respectively.
We are now in a position to define an elementary collection of translations, $T_i$, whose effect on the parameters, $a_i$, is given by $$\begin{gathered}
T_i \colon \ a_j \to \begin{cases}
a_i + h & \text{if $i = j$},\\
a_j & \text{if $i\neq j$},
\end{cases}\end{gathered}$$ and whose action on the $u_i$ variables is the subject of the following proposition.
The matrix $R(x) = L(x-h,u_1,a_1)^{-1}$ in defines a birational map between linear algebraic varieties $$\begin{gathered}
T_1 \colon \ \mathcal{M}_h(a_1, a_2, a_3, \ldots, a_N; d_1, d_2;1,1) \to \mathcal{M}_h(a_1+h, a_2+h, a_3, \ldots, a_N;d_2-h, d_1;1,1).\end{gathered}$$ The effect of $T_1$ on the $u_i$ variables is given by $$\begin{gathered}
T_1 u_k = \begin{cases}
u_{1,k} + \dfrac{a_1+h-a_k}{u_{1,k}+u_k} & \text{for $k = 2, \ldots, N$},\\
u_{1,1} & \text{for $k = 1$},
\end{cases}\end{gathered}$$ where $$\begin{gathered}
u_{1,k-1} = \begin{cases}
u_{k-1} - \dfrac{a_1+h-a_k}{u_{1,k}+u_k} & \text{for $k = 2, \ldots, N-1$}, \\
u_1 & \text{for $k= N+1$}.
\end{cases}\end{gathered}$$
To ascertain the how this transformation acts on $A(x)$, we observe that a rearrangement of is that $$\begin{gathered}
\label{tildeprod}
\tilde{A}(x) = L(x,u_2,a_2) \cdots L(x,u_{2m+4},a_{2m+4}) L( x, u_1, a_1 + h),\end{gathered}$$ where we have used . It is convenient to leave it in this form and read off the transformed values of $d_1$ and $d_2$ in the expansion of to be given by $$\begin{gathered}
T_1 d_1 = \sum_{i=1}^{N}\sum_{j=1}^{i-1} u_i u_j + \sum_{k \textrm{ odd}} \big(u_{k}^2-a_k\big) - h = d_2 - h,\\
T_1 d_2 =\sum_{i=1}^{N}\sum_{j=1}^{i-1} u_i u_j + \sum_{k \textrm{ even}} \big(u_{k}^2-a_k\big) = d_1,\end{gathered}$$ which determines that $\tilde{A}(x)$ is an element of $\mathcal{M}_h(a_1+h, a_2+h, a_3, \ldots, a_{2m+4};d_2-h, d_1;1,1)$. We may inductively determine $T_1 u_k$ by observing that the kernel of $\tilde{A}(a_{2m+2})$, giving us $$\begin{gathered}
T_1 \colon \ u_{2m+4} = u_1 +\frac{a_1+h-a_{2m+4}}{u_{1}+u_{2m+4}},\end{gathered}$$ by applying $s_{2m+3}$ and . Any subsequent kernels may be found inductively by examining the kernel of $$\begin{gathered}
L(a_k,u_2,a_2) \cdots L(x,u_{k},a_{k}) L( a_k, u_{1,k}, a_1 + h),\end{gathered}$$ for $k > 1$ and where $u_{1,2m+2} = u_1$.
Rather than computing compatibility relations explicitly, we have simply exploited the commutation relations between the $L_i$ factors. All the other elementary transformations may be obtained by conjugating by elements of $S_N$. One of the issues with this type of transformation is that it is singular at $x =\infty$, which manifests itself in the way it has swapped the roles of $d_1$ and $d_2$. If we conjugate by the matrix with $1$’s on the off diagonal, we can also swap the roles of $d_1$ and $d_2$, however, the effect this has on the $u_i$ variables is not so clear, as it requires a nontrivial refactorization into a product of the appropriate form. We may now present the generators for the discrete Garnier systems, which are compositions of the form $T_{i,j} = T_i \circ T_j$ where $i \neq j$.
\[tranha1a2\] The matrix $R(x) = L(x-h,u_2,a_2)^{-1}L(x-h,u_1,a_1)^{-1}$ in defines a birational map between linear algebraic varieties $$\begin{gathered}
T_{1,2} \colon \ \mathcal{M}_h (a_1, a_2, a_3, \ldots, a_N;d_1,d_2;1,1) \\
\hphantom{T_{1,2} \colon}{} \ {}\to \mathcal{M}(a_1+h, a_2+h, a_3, \ldots, a_N;d_1-h, d_2-h;1,1).\end{gathered}$$ The effect of $T_{1,2}$ on the $u_i$ variables is given by $$\begin{gathered}
\label{conprevuh}
T_{1,2} \colon \ u_i = \begin{cases}
u_{1,2} & \text{for $i = 1$},\\
u_{2,2} & \text{for $i = 2$},\\
u_k + (u_{k,1} -u_{k-1,1})+ (u_{2,k}- u_{2,k-1}) & \text{for $k = 3,\ldots, N$},
\end{cases}\end{gathered}$$ where $$\begin{gathered}
u_{1,k-1}= u_k + \frac{a_1+h-a_k}{u_{1,k}+u_k},\\
u_{2,k-1}= u_k - u_{1,k-1} + u_{1,k}+ \frac{a_2+h-a_k}{u_k + u_{2,k} + u_{1,k} - u_{1,k-1}},\end{gathered}$$ for $k = 2, \ldots, N$, $u_{1,N} = u_1$ and $u_{2,N} = u_2$.
As was the case in the previous proposition, using the identification of $\tilde{A}(x)$ with the action of $T_{1,2}$ we find that $$\begin{gathered}
\tilde{A}(x) = L(x,u_3,a_3) \cdots L(x,u_{N}, a_{N}) L(x,u_1,a_1+ h)L(x,u_2,a_2+h),\end{gathered}$$ whose expansion around $x=\infty$ reveals that $T_{1,2} d_i = d_i - h$ for $i = 1,2$, showing that the image of $T_{1,2}$ is indeed in $\mathcal{M}_h(a_1+h, a_2+h, a_3, \ldots, a_N;d_1-h, d_2-h)$. To compute the action on the $u_i$ variables, we inductively compute the kernel of $$\begin{gathered}
L(a_k,u_3,a_3) \cdots L(a_k,u_{k},a_{k}) L( a_k, u_{1,k}, a_1 + h)L( a_k, u_{2,k}, a_2 + h),\end{gathered}$$ using the action of $S_N$, which gives with an initial step where $u_{1,N} = u_1$ and $u_{2,N} = u_2$ as above.
We may construct a generic element $T_{i,j}$, whose action on the space of parameters is $$\begin{gathered}
T_{i,j} \colon \ \mathcal{M}_h (a_1, \ldots, a_i, \ldots, a_j, \ldots, a_N;d_1,d_2;1,1) \nonumber\\
\hphantom{T_{i,j} \colon}{} \ {} \to \mathcal{M}_h(a_1, \ldots, a_i+h, \ldots, a_j+h, \ldots, a_N;d_1-h,d_2-h;1,1),\label{hGarnierAction}\end{gathered}$$ by conjugating by the element $\sigma_{(1i)(2j)}$. That is to say $$\begin{gathered}
T_{i,j} = \sigma_{(1i)(2j)} \circ T_{1,2} \circ \sigma_{(1i)(2j)}.\end{gathered}$$ The system of transformations of the form $T_{i,j}$ constitutes what we call the $h$-Garnier system. The simplest case, when $m=1$, is shown to coincide with the difference analogue of the sixth Painlevé equation in Section \[dP6sec\].
As a consequence of Theorem \[hisolattice\], we have the following.
The set of transformations of the form $T_{i,j}$ satisfy the following
1. The action is symmetric in $i$ and $j$, i.e., $$\begin{gathered}
T_{i,j} = T_{j,i}.\end{gathered}$$
2. These actions commute, i.e., $$\begin{gathered}
T_{i_1, j_1} \circ T_{i_2, j_2} = T_{i_2, j_2} \circ T_{i_1, j_1}.\end{gathered}$$
The symmetric $\boldsymbol{h}$-difference Garnier system
--------------------------------------------------------
Let us consider difference equations whose solutions satisfy $Y(x) = Y(-x)$. The consistency of requires that $$\begin{gathered}
A(x)A(-h-x) =I.\end{gathered}$$ Under these conditions we express $A(x)$ by , in which $$\begin{gathered}
B(x) = L_1(x) \cdots L_{N'}(x),\end{gathered}$$ where $L_i(x) = L(x,u_i, a_i)$ given by and $N' = 2m+4$ as before. In this case, by using we may write $$\begin{gathered}
A(x) = B(-h-x)^{-1}B(x)= \!\left[\prod_{k=1}^{N'} \frac{1}{x+a_k+h}\right]\! L(-x,-u_{N'}, a_{N'}+h){\cdots} L(-x,-u_1, a_1+h)\\
\hphantom{A(x) = B(-h-x)^{-1}B(x)=}{} \times L(x,u_1, a_1+h) \cdots L(x,-u_{N'}, a_{N'}+h).\end{gathered}$$ This could be transformed via $\Gamma$ functions to a matrix of the form of for $N = 2N'$ and where the last $N$ factors take a slightly different form. If we were to apply Theorem \[hisolattice\], it is not clear at this point that the solutions would preserve the symmetry.
Due to and the invariance of under changes to the spectral variable, it is easy to see that one may simultaneously act on $B(x)$ and $B(-h-x)^{-1}$ by $S_n$ in the same way as . As discussed previously, we expect to find transformations induced by multiplication on the left and the right. The left multiplication is expected to define a trivial transformation of $A(x)$, but what is not expected is that the transformation is similar to the transformation specified by Lemma \[com:diff\].
The rational matrix $$\begin{gathered}
R_l(x) = \begin{pmatrix}(x-a_1)(x+a_1) & 0 \\ 0& (x-a_2)(x+a_2) \end{pmatrix}
+ (u_1+u_2)(a_1+ a_2) \begin{pmatrix} u_1 & -1 \\ u_1 E_{1,2} u_1 & -u_1 \end{pmatrix}\end{gathered}$$ defines a birational transformation $$\begin{gathered}
E_{1,2} \colon \ \mathcal{M}_h (a_1, \ldots, a_{N'}; d_1,d_2;1,1) \\
\hphantom{E_{1,2} \colon}{} \ {} \to \mathcal{M}_h(-a_1,-a_2, a_3, \ldots, a_{N'}; d_1+a_1+a_2,d_2+a_1+a_2;1,1),\end{gathered}$$ via with $\lambda = (x-a_1)^{-1}(x-a_2)^{-1}$. The effect on the $u_i$ variables is given by $$\begin{gathered}
E_{1,2} u_1 = u_1 - \frac{a_1-a_2}{u_1+ u_2},\qquad E_{1,2} u_2 = u_2 + \frac{a_1-a_2}{u_1+ u_2}.\end{gathered}$$
This is an elementary calculation that is easily verified. It is also seen that $R_l(x) = R_l(-x)$, as required, and that $$\begin{gathered}
\det R_l(x) = (x-a_1)(x+a_1)(x-a_2)(x+a_2).\end{gathered}$$ This defines an involution on the parameter space.
The rational matrix $$\begin{gathered}
R_r(x) = \begin{pmatrix}(x-a_{N'-1})(x+h+a_{N'-1}) & 0 \\ 0& (x-a_{N'})(x+h+a_{N'}) \end{pmatrix} \\
\hphantom{R_r(x) = }{} + (u_{N'}+u_{N'-1})(a_{N'}+ a_{N'-1}) \begin{pmatrix} u_{N'} & -1 \\ u_{N'} F_{N',N'-1}u_{N'} & -u_{N'} \end{pmatrix}\end{gathered}$$ defines a birational transformation $$\begin{gathered}
F_{N',N'-1}\colon \ \mathcal{M}_h(a_1,\ldots, a_{N'}; d_1,d_2;1,1) \\
\hphantom{F_{N',N'-1}\colon}{} \ {}\to \mathcal{M}_h(a_1, \ldots, a_{N'-2}, -a_{N'-1}-h,-a_{N'}-h;\\
\hphantom{F_{N',N'-1}\colon \ {}\to \mathcal{M}_h(}{} d_1+a_{N'}+a_{N'-1}+h,d_2+a_{N'}+a_{N'-1}+h;1,1),\end{gathered}$$ via where $\lambda(x) = (x-a_{N'-1})(x-a_{N'})$ whose effect on the $u_i$ variables is given by $$\begin{gathered}
F_{N',N'-1}u_{N'} = u_{N'} - \frac{a_{N'}-a_{N'-1}}{u_{N'}+ u_{N'-1}},\\
F_{N',N'-1}u_{N'-1} = u_{N'-1} + \frac{a_{N'}-a_{N'-1}}{u_{N'}+ u_{N'-1}}.\end{gathered}$$
It is easy to see $R_r(x) = R_r(-x-h)$ and $$\begin{gathered}
\det R_r(x) = (x-a_1)(x+a_1+h)(x-a_2)(x+a_2+h).\end{gathered}$$ These matrices are not of the same form as $L_i(x)$, yet the resulting transformation takes the form specified in Lemma \[com:diff\] where the roles of $u_i$ and $u_j$ have been swapped.
It is fitting that we define the generators of the symmetric difference Garnier system to be the maps $E_{i,j}$ and $F_{i,j}$, which may be expressed as
\[symgenh\] $$\begin{gathered}
\label{heij}E_{i,j} = s_{(1i)(2j)} \circ E_{1,2} \circ s_{(1i)(2j)},\\
\label{hfij}F_{i,j} = s_{(1i)(2j)} \circ F_{1,2} \circ s_{(1i)(2j)}.\end{gathered}$$
The translations, $T_{i,j}$, are specified in terms of these generators as $$\begin{gathered}
T_{i,j} = F_{i,j} \circ E_{i,j},\end{gathered}$$ which form the generators for the system of translations in the $h$-difference Garnier system. While this bears some similarity with , the difference is that given an $A(x)$ with the appropriate symmetry, the resulting action is inequivalent since the resulting transformations of do not necessarily preserve the symmetry, whereas by acting upon $B(x)$, the resulting matrix $A(x)$ necessarily possesses the required symmetry. The key difference is not the moduli space itself, but the actions being considered on them.
$\boldsymbol{q}$-difference Garnier systems
-------------------------------------------
As we did with the $h$-difference systems, we start with where $\sigma = \sigma_q$ and where $A(x)$ is specified by . Before defining $L_i(x)$, we specify two matrices, $L(x,u,a)$ and a diagonal matrix which we call $D$ given by $$\begin{gathered}
\label{qdifffactor} L(x,u,a) = \begin{pmatrix}
1 & u \\
\dfrac{x}{au} & 1
\end{pmatrix},\qquad D = \begin{pmatrix}
\theta_1 & 0 \\
0& \theta_2
\end{pmatrix},\end{gathered}$$ which satisfy the commutation relation $$\begin{gathered}
\label{commDL}
L(x,u,a)D = D L\left(x, \frac{\theta_2 u}{\theta_1},a\right) .\end{gathered}$$ Due to , rather than letting each factor take the form $D L(x,u_i,a_i)$ it is sufficient to letting only the first first factor take the form $DL(x,u_1,a_1)$ while all other factors are of the form $L(x,u_i,a_i)$, i.e., we let $A(x)$ take the general form where $$\begin{gathered}
\label{qfactor}
L_i(x) = \begin{cases} D L(x,u_1,a_1) & \text{for $i = 1$}, \\
L(x,u_i,a_i) & \text{for $i \neq 1$}.
\end{cases}\end{gathered}$$ As in the previous section, some of the desirable properties of $L(x,u,a)$ are
$$\begin{gathered}
\label{detLq}\det L(x,u,a) = 1- \frac{x}{a},\\
\label{qident2}L(qx,u,qa) = L(x,u,a),\\
\label{qinv}L(x,u,a)^{-1} = \frac{a}{x-a} L(x,-u,a),\\
\label{kerq} \operatorname{Ker} L(a,u,a) = \left\langle \begin{pmatrix} -u \\ 1 \end{pmatrix} \right\rangle, \\
\label{Imq} \operatorname{Im} L(a,u,a) = \left\langle \begin{pmatrix} u \\ 1 \end{pmatrix} \right\rangle.\end{gathered}$$
By expanding we have that $A(x)$ takes the general form $$\begin{gathered}
A_0 + A_1 x + \cdots + A_{m+1} x^{m+1}.\end{gathered}$$ As we did with the previous section, we find that the properties of $A(x)$ are given by the following proposition.
\[qalgvariety\] Given $A(x)$ specified by where each factor is given by , with the constraints $\theta_1 \neq \theta_2$ and $$\begin{gathered}
\label{con2q}
\theta_1\prod_{i \ \textrm{odd}} \big(a_iu_i^2\big) \neq \theta_2\prod_{j \ \textrm{odd}} \big(a_ju_j^2\big),\end{gathered}$$ defines an element, $$\begin{gathered}
(A_0, \ldots, A_{m+1}) \in \mathcal{M}_q(a_1,\ldots, a_N; \kappa_1, \kappa_2; \theta_1, \theta_2),\end{gathered}$$ where
\[kappavals\] $$\begin{gathered}
\kappa_1 = \theta_1\prod_{i \ \textrm{odd}} u_i \prod_{j \ \textrm{even}} (a_ju_j)^{-1},\\
\kappa_2 = \theta_2 \prod_{i \ \textrm{even}} u_i \prod_{j \ \textrm{odd}} (a_ju_j)^{-1},\end{gathered}$$
and $\theta_1$ and $\theta_2$ appear as they do in .
The property is sufficient to tell us $$\begin{gathered}
\label{detq} \det A(x) = \theta_1\theta_2 \left( 1- \frac{x}{a_1} \right) \cdots \left( 1- \frac{x}{a_N} \right),\end{gathered}$$ where expansions around $x = \infty$ and $x = 0$ are $$\begin{gathered}
A(x) = x^{m+1} \begin{pmatrix} \kappa_1 & 0 \\ r_{1,2} & \kappa_2 \end{pmatrix} + O\big(x^m\big),\\
A(x) = \begin{pmatrix} \theta_1 & \theta_1 \left( \sum\limits_{i = 0}^N u_i \right) \\
0 & \theta_2 \end{pmatrix} + O(x),\end{gathered}$$ where the values of $\kappa_1$ and $\kappa_2$ are as above and $$\begin{gathered}
\label{r12q}
r_{1,2} = \frac{\theta_2 \kappa_1}{\theta_1 u_1} \sum_{j = 1}^{N-1} \prod_{i=1}^{j} \left(\frac{a_{i+1}}{a_i}\right)^{\frac{1 - (-1)^i}{2}}\left( \frac{u_i}{u_{i+1}}\right)^{(-1)^i}.\end{gathered}$$ The constraints that both $\theta_1$ and $\theta_2$ and are sufficient (but not necessary) to ensure that $A_0$ and $A_{m+1}$ are semisimple.
By a similar counting argument to the $h$-difference case, we may show that matrices taking the form given by and $\mathcal{M}_q(a_1,\ldots, a_N; \kappa_1, \kappa_2; \theta_1, \theta_2)$ both describe algebraic varieties of dimension $2m$ with birational maps between the two. This justifies that we may parameterize our discrete isomonodromic deformations in terms of actions on matrices of the form .
As we mentioned above, the conditions that $\theta_1= \theta_2$ or equality holds in are not necessary for $A_0$ and $A_{m+1}$ to be semisimple, as requires that the matrix $A_0$ is diagonalizable, which amounts to requiring that $A_0$ is diagonal, which imposes the constraint $$\begin{gathered}
\sum_{i=0}^N u_i = 0.\end{gathered}$$ On the other hand, if equality holds in , we require that $r_{1,2} = 0$. If $\theta_1 = \theta_2$ then the case of $m = 1$ has too many constraints to be interesting, hence, it is more natural to consider $m=2$ to be the first interesting case. Similarly, if both $\theta_1 = \theta_2$ and equality holds in , then we have an additional constraint, making $m=3$ the first interesting case for similar reasons.
\[lem:qcom\] Matrices of the form of satisfy the commutation relation $$\begin{gathered}
L(x,u_i,a_i)L(x,u_j,a_j) = L(x,\tilde{u}_j,a_j)L(x,\tilde{u}_i,a_i),\end{gathered}$$ where the map $\left(u_i,u_j \right) \to \left(\tilde{u}_i,\tilde{u}_j\right)$ is given by $$\begin{gathered}
\label{commD}
\tilde{u}_i = \frac{a_i u_i(u_i + u_j)}{a_i u_i + a_j u_j},\qquad \tilde{u}_j = \frac{a_j u_j(u_i + u_j)}{a_i u_i + a_j u_j} .\end{gathered}$$
Once again, this map satisfies the Yang–Baxter property in that if we define $R_{i,j}$ in the same manner as then holds. In the classification of quadrirational Yang–Baxter maps appears as $F_{\rm III}$ [@Adler2003].
In the same manner as the previous section, it is useful to utilize Lemma \[lem:qcom\] to define the action of $S_N$ on $A(x)$. Given a permutation, $\sigma \in S_N$, we denote the action of $\sigma$ on the $u_i$ and $a_i$ by $s_{\sigma} u_i$ and $s_{\sigma} a_i$. Following the notation from the previous section, we specify the action of the generators is computed using to be $$\begin{aligned}
{3}
& s_i \colon \ u_i \to \frac{a_{i+1}u_{i+1}(u_i + u_{i+1})}{a_i u_i + a_{i+1} u_{i+1}}, \qquad && s_i \colon \ a_i = a_{i+1},&\\
& s_i \colon \ u_{i+1} \to \frac{a_{i+1}u_{i+1}(u_i + u_{i+1})}{a_i u_i + a_{i+1} u_{i+1}}, \qquad && s_i \colon \ a_{i+1} = a_i.&\end{aligned}$$ Once again, the effect of $s_\sigma$ is trivial on $A(x)$. This action and will be sufficient to express the discrete isomonodromic deformations. We wish to specify the transformation whose action on the parameters is $$\begin{gathered}
T_i \colon \ a_j \to \begin{cases}
qa_i & \text{if $i = j$},\\
a_j & \text{if $i\neq j$},
\end{cases}\end{gathered}$$ and whose action on the $u_i$ is to be specified. Once again, the matrices that define the elementary Schlesinger transformations are of the form of $L(x,u,a)$. The most basic transformation is specified in terms of the left-most factor.
The matrix $R(x) = L(x/q,u_1,a_1)^{-1} D^{-1}$ in defines a birational map between algebraic varieties $$\begin{gathered}
T_1 \colon \ \mathcal{M}_q(a_1,\ldots, a_N; \kappa_1, \kappa_2 ; \theta_1, \theta_2) \to \mathcal{M}_q(a_1,\ldots, a_N; \kappa_2, \kappa_1/q; \theta_1, \theta_2).\end{gathered}$$ The effect of $T_1$ on the $u_i$ variables is given by $$\begin{gathered}
\label{qtrans1}
T_1 u_k = \begin{cases}
\dfrac{qa_1u_{1,k}(u_k \theta_2 + \theta_1 u_{1,k})}{a_k u_k\theta_2 + q a_1 \theta_1 u_{1,k}} & \text{for $k = 2, \ldots, N$},\\
u_{1,1} & \text{for $k = 1$},
\end{cases}\end{gathered}$$ where $$\begin{gathered}
u_{1,k-1} =
\frac{a_ku_k\theta_2(u_k \theta_2 + \theta_1 u_{1,k})}{\theta_1(a_k u_k\theta_2 + q a_1 \theta_1 u_{1,k})}
u_1 \qquad \text{for $k= N+1$}.\end{gathered}$$
This proposition follows in a similar manner, in that we identify $\tilde{A}(x)$ with $$\begin{gathered}
\tilde{A}(x) = L(x,u_2,a_2) \cdots L(x,u_{N},a_{N})DL(x,u_1,q a_1),\end{gathered}$$ using and , which allows us to compute the determinant. Secondly, we note that we may use to show $$\begin{gathered}
\tilde{A}(x) = DL\left(x,\frac{\theta_2u_2}{\theta_1},a_2\right) \cdots L\left(x,\frac{\theta_2u_N}{\theta_1},a_{N}\right) L(x,u_1,q a_1),\end{gathered}$$ which is of the form in Proposition \[qalgvariety\], which shows $$\begin{gathered}
T_1 \kappa_1 = \theta_1 (qa_1 u_1)^{-1} \prod_{i\ \mathrm{ even}} \left( \frac{\theta_2u_i}{\theta_1} \right)\prod_{i\ \mathrm{ odd}, \; i \neq 1} \left(a \frac{\theta_2u_i}{\theta_1} \right)^{-1}
= \frac{\theta_2}{q} \prod_{i\ \mathrm{ even}} u_i\prod_{i\ \mathrm{ odd}} \left(a u_i \right)^{-1} = \frac{\kappa_2}{q},\\
T_1 \kappa_2 = \kappa_1.\end{gathered}$$ This shows that the image of $T_1$ is indeed in $\mathcal{M}_q(a_1,\ldots, a_N; \kappa_2, \kappa_1/q; \theta_1, \theta_2)$. To determine the effect on the $u_i$ variables, the only difference in the inductive step is that we need to use in combination with Lemma \[lem:qcom\]. We compute the kernel of $$\begin{gathered}
L(x,u_2,a_2) \cdots L(x,u_{k},a_{k}) D L(x,u_{1,k},q a_1),\end{gathered}$$ using the action of $S_N$ and , which inductively provides us with with $u_{1,N} = u_1$.
The $T_i$ transformations may be obtain through conjugation by the action of $S_N$. This transformation is also not a transformation of the form specified in Theorem \[qlattice\], since it swaps the role of $\kappa_1$ and $\kappa_2$. To define the generators of the $q$-Garnier system, we compute $T_1 \circ T_2$, which may be used to compute $T_{i,j}$.
The matrix $R(x) = L(x/q,u_2,a_2)^{-1} L(x/q,u_1,a_1)^{-1}D^{-1}$ in defines a birational map between algebraic varieties $$\begin{gathered}
T_{1,2} \colon \ \mathcal{M}_q(a_1, \ldots, a_N; \kappa_1, \kappa_2; \theta_1, \theta_2) \to \mathcal{M}_q(qa_1, qa_2, \ldots, a_N; \kappa_1/q, \kappa_2/q; \theta_1, \theta_2).\end{gathered}$$ The effect of $T_{1,2}$ on the $u_k$ coordinates is given by $$\begin{gathered}
\label{qT12} T_{1,2} u_k = \begin{cases}
u_{1,2} & \text{for $k = 1$},\\
u_{2,2} & \text{for $k = 2$},\\
\dfrac{a_2u_ku_{2,k-1}u_{2,k}\theta_2}{a_1u_{1,k}u_{1,k-1}\theta_1} & \text{for $k = 3, \ldots, N$},
\end{cases}\end{gathered}$$ where $$\begin{gathered}
u_{1,k-1} = \frac{\theta _2 a_k u_k \left(\theta _1 u_{1,k}+\theta _2 u_k\right)}{\theta _1 \left(qa_1 \theta _1 u_{1,k}+\theta _2 a_k u_k\right)},\\
u_{2,k-1} =\frac{a_1 u_{1,k-1} u_{1,k} \left(\theta _1 \left(u_{1,k}-u_{1,k-1}+u_{2,k}\right)+\theta _2
u_k\right)}{a_1 \theta _1 u_{1,k-1} u_{1,k}+a_2 \theta _2 u_k u_{2,k}}.\end{gathered}$$
The induction follows in the same way as it did for Proposition \[tranha1a2\]. In a similar way, we specify that that what we call the $q$-Garnier system is the system of transformations whose action on the parameters is specified by $$\begin{gathered}
T_{i,j} \colon \ \mathcal{M}_q (a_1, \ldots, a_i, \ldots, a_j, \ldots, a_N;\kappa_1,\kappa_2;\theta_1,\theta_2)\nonumber\\
\hphantom{T_{i,j} \colon}{} \ {}\to \mathcal{M}_q(a_1, \ldots, qa_i, \ldots, qa_j, \ldots, a_N;\kappa_1/q,\kappa_2/q;\theta_1,\theta_2),\label{qGarnierAction}\end{gathered}$$ which is given by $$\begin{gathered}
T_{i,j} = \sigma_{(1i)(2j)} \circ T_{1,2} \sigma_{(1i)(2j)}.\end{gathered}$$ The simplest case, when $m=1$, is the $q$-analogue of the sixth Painlevé equation.
The following arises as a consequence of Theorem \[qlattice\].
The set of transformations of the form $T_{i,j}$ satisfy the following
1. The action is symmetric in $i$ and $j$, i.e., $$\begin{gathered}
T_{i,j} = T_{j,i}.\end{gathered}$$
2. These actions commute, i.e., $$\begin{gathered}
T_{i_1, j_1} \circ T_{i_2, j_2} = T_{i_2, j_2} \circ T_{i_1, j_1}.\end{gathered}$$
Symmetric $\boldsymbol{q}$-Garnier system
-----------------------------------------
We now impose the symmetry constraint that the solutions satisfy $Y(x) = Y(1/x)$. The consistency of requires that $$\begin{gathered}
A(x)A(1/(qx))=I.\end{gathered}$$ We assume that $A(x)$ takes the form of where $B(x)$ is given by the product of $L$-matrices as $$\begin{gathered}
B(x) = L_1(x) \cdots L_{N'}(x),\end{gathered}$$ where $L_i$ is given by where the diagonal entry cancels, hence, without loss of generality, we may choose $D = I$. Using , we may write $A(x)$ as $$\begin{gathered}
A(x) = \left[\prod_{i=1}^{N'} \frac{1}{1-x/a_i}\right] L\left(\frac{1}{x},-u_{N'},qa_{N'}\right) \cdots L\left(\frac{1}{x},-u_{1},qa_{1}\right) \\
\hphantom{A(x) =}{} \times
L(x,u_{1},a_{1}) \cdots L(x,u_{N'},a_{N'}) ,\end{gathered}$$ which defines an matrix in terms of a product of $L$-matrices.
The rational matrix $$\begin{gathered}
R_l(x) = \begin{pmatrix} x+\dfrac{1}{x} - \dfrac{1}{a_1} - \dfrac{1}{a_2}\!\! & 0 \\ 0 & x+\dfrac{1}{x} -a_1 - a_2 \end{pmatrix}
+ \frac{(1-a_1a_2)}{a_2u_2}\! \begin{pmatrix} -u_1 & u_1(u_1 + u_2) \vspace{1mm}\\ -1- \dfrac{a_2u_2}{a_1 u_1}\!\! & u_1 \end{pmatrix},\end{gathered}$$ defines a birational transformation $$\begin{gathered}
E_{1,2}\colon \ \mathcal{M}_q(a_1,a_2, a_3, \ldots, a_{N'}; \kappa_1, \kappa_2; 1, 1) \\
\hphantom{E_{1,2}\colon}{} \ {} \to \mathcal{M}_q\left(\frac{1}{a_1},\frac{1}{a_2}, a_3, \ldots, a_{N'}; a_1a_2\kappa_1, a_1a_2 \kappa_2; 1, 1\right) ,\end{gathered}$$ via where $\lambda(x) = (x-a_1)^{-1}(x-a_2)^{-1}$, whose effect on the $u_i$ variables is given by $$\begin{gathered}
E_{1,2} u_1 = \frac{a_1 u_1 (u_1 + u_2)}{a_1 u_1 + a_2 u_2},\qquad
E_{1,2} u_2 = \frac{a_2 u_2 (u_1 + u_2)}{a_1 u_1 + a_2 u_2}.\end{gathered}$$
This is easy to verify directly. Furthermore, we have that $R_l(x)= R_l(1/x)$ and that $$\begin{gathered}
\det R_l(x) = (x-a_1)(x-a_2)\big(x-a_1^{-1}\big)\big(x-a_2^{-1}\big).\end{gathered}$$ The transformation induced by right multiplication is given by the following proposition.
The rational matrix $$\begin{gathered}
R_r(x) = \begin{pmatrix} x+\dfrac{1}{qx} - \dfrac{1}{qa_{N'-1}} - \dfrac{1}{qa_{N'}} & 0 \\ 0 & x+\dfrac{1}{x} -a_{N'-1} - a_{N'} \end{pmatrix} \\
\hphantom{R_r(x) =}{} + \frac{(1-qa_{N'-1}a_{N'})}{qa_{N'-1}u_{N'-1}} \begin{pmatrix} -u_{N'} & -u_{N'}(u_{N'-1} + u_{N'}) \vspace{1mm}\\ 1 + \dfrac{a_{N'-1}u_{N'-1}}{a_{N'} u_{N'}} & u_{N'} \end{pmatrix}\end{gathered}$$ defines a birational transformation $$\begin{gathered}
F_{N',N'-1}\colon \ \mathcal{M}_q(a_1, \ldots,a_{N'-2},a_{n-2}, a_n; \kappa_1, \kappa_2; 1, 1) \\
\hphantom{F_{N',N'-1}\colon}{} \ {} \to \mathcal{M}_q\left(a_1, \ldots, a_{n-2}, 1/(qa_{n-1}), 1/q a_n; \kappa_1, \kappa_2; 1, 1\right),\end{gathered}$$ via where $\lambda = x(1-x/(qa_{N'-1}))^{-1}(1-x/(qa_{N'}))^{-1}$, whose effect on the $u_i$ variables is given by $$\begin{gathered}
F_{N',N'-1} u_{N'-1} = \frac{a_{N'-1} u_{N'-1} (u_{N'} + u_{N'-1})}{a_{N'-1} u_{N'-1} + a_{N'} u_{N'}},\\
F_{N',N'-1} u_{N'} = \frac{a_{N'} u_{N'} (u_{N'} + u_{N'-1})}{a_{N'-1} u_{N'-1} + a_{N'} u_{N'}}.\end{gathered}$$
This is also easy to verify, as is the property that $R_r(x)= R_r(1/qx)$ and $$\begin{gathered}
\det R_r(x) = (x-a_1)(x-a_2)\big(x-(qa_1)^{-1}\big)\big(x-(qa_2)^{-1}\big).\end{gathered}$$ In the same way as the $h$-difference case, we define the $q$-difference Garnier system to be generated by maps $E_{i,j}$ and $F_{i,j}$, which may be expressed as
\[symgenh2\] $$\begin{gathered}
\label{qeij}E_{i,j} = s_{(1i)(2j)} \circ E_{1,2} \circ s_{(1i)(2j)},\\
\label{qfij}F_{i,j} = s_{(1i)(2j)} \circ F_{1,2} \circ s_{(1i)(2j)}.\end{gathered}$$
The translations, $T_{i,j}$, also specified by $$\begin{gathered}
T_{i,j} = F_{i,j} \circ E_{i,j},\end{gathered}$$ generate the translational portion of the symmetric $q$-Garnier system.
Reparameterization {#sec:reparam}
==================
The aim of this section is to express the above systems in terms of variables that have been chosen to make a correspondence between our $q$-Garnier systems and the $q$-Garnier system specified in the work of Sakai [@Sakai:Garnier]. This choice makes sense in both the $h$-difference and $q$-difference setting.
$\boldsymbol{h}$-difference Garnier systems
-------------------------------------------
Let us consider the $h$-difference Garnier system defined by where $\sigma = \sigma_h$, where $A(x)$ is specified by for $N = 2m+2$ and $L_i$ is given by subject to the constraint . As implies that the leading coefficient of $A(x)$ is proportional to the identity matrix, provided $d_1 \neq d_2$, which is specified and , we may gauge by a constant lower triangular matrix so that the next leading coefficient is diagonal. Under these conditions, we specify a new set of variables, $y_i$, $z_i$ and $w_i$, related to $A(x)$ by $$\begin{gathered}
A(a_i) = y_i \begin{pmatrix} 1 \vspace{1mm}\\ \dfrac{z_i}{w} \end{pmatrix} \begin{pmatrix} w_i & w \end{pmatrix} = \begin{pmatrix} w_i y_i & w y_i \vspace{1mm}\\
\dfrac{w_i y_i z_i}{w} & y_i z_i \end{pmatrix},\end{gathered}$$ for $i=1, \ldots, N$. This choice is inspired by many works on the matter, such as the work on the $q$-Garnier systems [@Sakai:Garnier], and various works on the Lagrangian approaches to difference equations [@Dzhamay2007; @Dzhamay2013]. This defines $3N$ parameters, many of which are redundant. After diagonalizing, with $A(x) = (a_{i,j}(x))$, we have that each $a_{i,j}(x)$ is a polynomial with the following properties specifying their coefficients:
- $a_{i,i}(x) = x^{m+2} + d_i x^{m+1} + O(x^m)$ with $a_{1,1}(a_k) = y_kw_k$ and $a_{2,2}(a_k) = y_kz_k$ for $k=1,\ldots, m+1$,
- $a_{1,2}(a_k) = wy_k$ and $a_{2,1} = w_ky_kz_k/w$ for $k=1, \ldots, m+1$.
We use a form of Lagrangian interpolation in the following way: If we let $$\begin{gathered}
D(x) = \prod_{i=1}^{m+1} (x-a_i),\end{gathered}$$ then the polynomial $D(x)/(x-a_k)$ satisfies $$\begin{gathered}
\label{deltapoly}
\frac{D(x)}{(x-a_k) D'(a_k)} =
\begin{cases}
0 & \text{if $x = a_j$ for $j \neq k$},\\
1 & \text{if $x = a_k$}.
\end{cases}\end{gathered}$$ This allows us to express the entries of $A(x)$ as
\[largrangeinth\] $$\begin{gathered}
a_{1,1}(x) = D(x)\left( x + d_1 + \sum_{i=1}^{m+1} a_i+ \sum_{i=1}^{m+1} \frac{y_iw_i}{(x-a_i)D'(a_i)}\right),\\
a_{2,2}(x) = D(x)\left( x + d_2 + \sum_{i=1}^{m+1} a_i+ \sum_{i=1}^{m+1} \frac{y_iz_i}{(x-a_i)D'(a_i)}\right),\\
a_{1,2}(x) = w D(x) \sum_{i=1}^{m+1} \frac{y_i}{D'(a_i)(x-a_i)},\\
a_{2,1}(x) = \frac{D(x)}{w} \sum_{i=1}^{m+1} \frac{y_iz_iw_i}{D'(a_i)(x-a_i)}.\end{gathered}$$
It is convenient to write the expressions for each of the $y_k$, $z_k$ and $w_k$ as $$\begin{gathered}
y_k := \frac{a_{1,2}(a_k)}{w}, \qquad w_k = \frac{a_{1,1}(a_k)}{y_i}, \qquad z_k = \frac{a_{2,2}(a_k)}{y_i},\end{gathered}$$ which is trivially true for $k = 1,\ldots, m+1$ and defines an expression for $y_k$, $z_k$ and $w_k$ in terms of the first $m+1$ values for $k = m+2, \ldots, N$. This also produces expressions for each of the new variables in terms of the $u_i$. Naturally, this does not take into account any constants with respect to $T_{i,j}$. After diagonalizing the leading coefficient in the polynomial expansion in $x$, it is easy to see that the matrix inducing $T_{i,j}$ takes the form $$\begin{gathered}
\label{Rhform}
R(x) = \frac{xI + R_0}{(x-a_i-h)(x-a_j-h)},\end{gathered}$$ hence, we may calculate the equivalent of on the $y_k$, $z_k$ and $w_k$ variables.
The system is equivalent to the following action on the variables $y_k$, $z_k$ and $w_k$
\[hGarnierBirrational\] $$\begin{gathered}
\label{tijrhy} \frac{(T_{i,j} wy_k)}{w}= y_k \frac{z_i (a_k-a_j-h) - z_j(a_k-a_i-h) + w_k(a_i-a_j)}{z_i(a_k-a_i)-z_j(a_k-a_j)+ w (T_{i,j}z_k)(a_i-a_j)},\\
T_{i,j}z_k = \left( \frac{T_{i,j}w}{w} \right)\frac{z_iz_j(a_i-a_j) + z_iz_k(a_k-a_i) + z_jz_k(a_j-a_k)}{z_i(a_k-a_j) + z_j(a_i-a_k) + z_k(a_j-a_i)},\\
\label{whrew}T_{i,j}w_k = \left( \frac{T_{i,j}w}{w} \right)\frac{z_iz_j(a_i-a_j) + z_iw_k(a_i+h-a_k) - z_jw_k(a_j+h-a_k)}{z_i(a_j+h-a_k) - z_j(a_i+h-a_k) - w_k(a_i-a_j)},\end{gathered}$$ for $k \neq i, j$ $$\begin{gathered}
\allowdisplaybreaks
\frac{T_{i,j}y_i}{a_i - a_j} = \frac{w^2 D(a_i+h)-(w a_{1,1}(a_i+h)+z_i
a_{1,2}(a_i+h)) (w a_{2,2}(a_i+h)-z_i
a_{1,2}(a_i+h))}{h a_{1,2}(a_i+h) (T_{i,j}w)
(z_i-z_j){}^2}\nonumber\\
{} +\frac{(w a_{1,1}(a_i+h)+z_i
a_{1,2} (a_i+h ) ) (w a_{2,2} (a_i+h )-z_j
a_{1,2} (a_i+h ) )-w^2 P (a_i+h )}{a_{1,2} (a_i+h ) (T_{i,j}w)
(z_i-z_j ){}^2 (a_i-a_j+h )},\!\!\!\\
\frac{T_{i,j} z_i}{a_i-a_j} = \frac{w^2 a_{2,1} (a_i+h )+w z_i a_{2,2} (a_i+h )+z_i z_j T_{i,j}(w y_i)}{w
(T_{i,j}y_i) (z_i (a_i-a_j+h )-h z_j )},\\
w (T_{i,j}w_i) = -z_j T_{i,j}w, \qquad \frac{T_{i,j}w}{w} = 1 + \frac{(a_i-a_j)(d_1 -d_2+h)}{z_i-z_j},\end{gathered}$$
whereas for $k = j$ we swap the roles of $i$ and $j$ above.
We temporarily use the notation $\tilde{u} = T_{i,j}u$. Given , we multiply the left and right-hand sides of by $(x-a_i)(x-a_j)(x-a_i-h)(x-a_j-h)$ whereby evaluating the resulting expression at $x = a_i$ gives us $$\begin{gathered}
((a_i+h)I + R_0)\tilde{y}_i \begin{pmatrix} 1 \vspace{1mm}\\ \dfrac{z_i}{w} \end{pmatrix} \begin{pmatrix} w_i & w \end{pmatrix},\end{gathered}$$ which specifies that the rows of $((a_i+h)I + R_0)$ are annihilated by the image of $A(a_i)$. Imposing the same condition for $x=a_j$ uniquely specifies $R_0$ by $$\begin{gathered}
\label{R0h1}
R_0 = \frac{1}{z_i-z_j} \begin{pmatrix} (a_i+h)z_j - (a_j+h)z_i & (a_j-a_i)w \vspace{1mm}\\ \dfrac{(a_i-a_j)z_iz_j}{w} & (a_j+h)z_j - (a_i+h)z_i \end{pmatrix}.\end{gathered}$$ Using the values $x= a_i + h$ and $x=a_j+h$ gives us $$\begin{gathered}
R_0 = \frac{1}{\tilde{w}_i - \tilde{w}_j} \begin{pmatrix} (a_j+h)\tilde{w}_j - (a_i+h)\tilde{w} & (a_j-a_i)\tilde{w} \vspace{1mm}\\
\dfrac{(a_i-a_k)\tilde{w}_i\tilde{w}_j}{\tilde{w}} & (a_i+h)\tilde{w} \end{pmatrix},\end{gathered}$$ whose equivalence with gives the first part of . Using with at $x= a_k$ gives us $$\begin{gathered}
y_kR(a_k+h) \begin{pmatrix} 1 \\ \frac{z_k}{w} \end{pmatrix} \begin{pmatrix} w_k & w \end{pmatrix} = \tilde{y}_k \begin{pmatrix} 1 \\ \frac{\tilde{z}_k}{\tilde{w}} \end{pmatrix} \begin{pmatrix} \tilde{w}_k & \tilde{w} \end{pmatrix} R(a_k),\end{gathered}$$ which is equivalent to (\[tijrhy\])–(\[whrew\]). The remaining parts may be calculated from evaluating $$\begin{gathered}
\tilde{A}(x) = R(x+h)A(x)R(x)^{-1},\end{gathered}$$ which is equivalent to using at $x = a_i +h$. The symmetry and uniqueness of $R(x)$ determines that the corresponding formula for $k=j$ may be obtained by swapping the roles of $i$ and $j$.
While we have chosen to express the system in this way, this is not to be considered a $3(m+1)$-dimensional map since it has enough constants with respect to $T_{i,j}$ to be considered a $(N - 2)$-dimensional system in terms of the $u_i$.
The symmetric version may be treated in the same way by considering transformations of $B(x)$ instead of $A(x)$. We take $A(x)$ to be given by where $B(x)$ is given by , in which case we may parameterize $B(x)$ in the same way by introducing the variables $y_i$, $z_i$ and $w_i$ by $$\begin{gathered}
B(a_i) = y_i \begin{pmatrix} 1 \vspace{1mm}\\ \dfrac{z_i}{w} \end{pmatrix} \begin{pmatrix} w_i & w \end{pmatrix} = \begin{pmatrix} w_i y_i & w y_i \vspace{1mm}\\
\dfrac{w_i y_i z_i}{w} & y_i z_i \end{pmatrix},\end{gathered}$$ for $i = 1,\ldots, N'$. The Lagrangian interpolation is the same as it was for $A(x)$ above, hence the entries of $B(x) = (b_{i,j}(x))$ are also given by . We may calculate the effect of $E_{i,j}$ and $F_{i,j}$ on these new variables.
\[eh\] The system is equivalent to the following action on the variables $y_k$, $z_k$ and $w_k$
$$\begin{gathered}
\label{hEij1}E_{i,j}y_k = \frac{w y_k \big(a_i^2 (z_k-z_j )+a_k^2(z_j-z_i)+a_j^2(z_i-z_k)\big)}{(a_i-a_k) (a_k-a_j)
w(h+t) (z_i-z_j)},\\
E_{i,j} z_k = \frac{(E_{i,j}w) \big(a_i^2 z_i (z_k-z_j)+a_j^2 z_j
(z_i-z_k)+a_k^2 z_k (z_j-z_i)\big)}{w \big(a_i^2 (z_k-z_j )+a_k^2 (z_j-z_i )+a_j^2 (z_i-z_k )\big)},\\
\label{hEij3} wE_{i,j} w_k = w_k E_{i,j}w, \qquad \frac{E_{i,j} w}{w} = 1 - \frac{a_i^2 - a_j^2}{z_i - z_j},\end{gathered}$$
for $k \neq i,j$ and for $k = i$ $$\begin{gathered}
(E_{i,j}y_iw) = \frac{(a_i-a_j)(wa_{2,2}(-a_i) - a_{1,2}(-a_i)z_i)}{2a_i(z_j-z_i)},\\
wE_{i,j}z_i = z_j E_{i,j}w, \qquad E_{i,j} w_i = E_{i,j}w \frac{z_i a_{1,1}(-a_i) - w a_{2,1}(-a_i)}{z_i a_{1,2}(-a_i) - w a_{2,2}(-a_i)},\end{gathered}$$
whereas for $k=j$ we swap the roles of $i$ and $j$ above.
\[fh\] The system is equivalent to the following action on the variables $y_k$, $z_k$ and $w_k$
$$\begin{gathered}
\label{hFij1}\frac{F_{i,j}wy_k}{w y_k} =\frac{w_i(a_k-a_j(a_j+h))- w_j(a_k-a_i(a_i+h)) - w_k(a_i-a_j)(a_i+a_j+h) }{(a_i-a_k) (a_k-a_j)
(w_i-w_j) (a_i-a_k+h)(-a_j+a_k-h)}, \\
\frac{wF_{i,j} w_k}{F_{i,j}w} = -\frac{w_iw_j(a_j\!-\!a_i)(a_i\!+\!a_j\!+h)\! +\! w_i w_k (a_i(a_i\!+\!h)\! -\! a_k)\! -\! w_j w_k(a_j(a_j\!+\!h)\! - \! a_k)}{w_k(a_j\!-\!a_i)(a_i\!+\!a_j\! +\! h)\! +\! w_j(a_i(a_i\!+\!h) \!-\! a_k)\! -\! w_i(a_j(a_j\!+h)\!-a_k)},\!\!\!\!\\
\label{hFij3}wF_{i,j}z_k =z_k F_{i,j} w, \qquad F_{i,j} w = 1- \frac{(a_i-a_j)(a_i+a_j+h)}{w_i-w_j},\end{gathered}$$
for $k \neq i,j$ and for $k = i$ $$\begin{gathered}
(F_{i,j}y_iw) = \frac{(a_i-a_j)(wa_{1,1}(-a_i-h) - a_{1,2}(-a_i-h)w_i)}{(2a_i+h)(w_j-w_i)},\\
w F_{i,j} w_i = w_j F_{i,j} w, \qquad \frac{F_{i,j}}{F_{i,j} w} = \frac{w_i a_{2,2}(-h-a_i) - w a_{2,1}(-h-a_i)}{a_{1,2}(-h-a_i)- w a_{1,1}(-h-a_i)},\end{gathered}$$
whereas for $k=j$ we swap the roles of $i$ and $j$ above.
We note that for $B(x)$ to be of the same form we require that $R_l(x)$ and $R_r(x)$ from and take the forms $$\begin{aligned}
{3}
& R_l(x) = x^2 I + R_0, \qquad && \det R_l(x) = (x-a_i)(x-a_j)(x+a_i)(x+a_j),& \\
& R_r(x) = x(x+h) + R_1, \qquad && \det R_r(x) = (x-a_i)(x-a_j)(x+a_i+h)(x+a_j+h),&\end{aligned}$$ with $\lambda = (x-a_i)^{-1}(x-a_j)^{-1}$. We may multiply the left and right-hand sides of and by $(x-a_i)(x-a_j)$ to see that $R_0$ and $R_1$ satisfy $$\begin{gathered}
\big(x^2+R_0\big)A(x) = (x-a_i)(x-a_j)\tilde{A}(x), \qquad \big(x^2+R_1\big)A(x) = (x-a_i)(x-a_j)\hat{A}(x),\end{gathered}$$ where we use the notation $E_{i,j}u = \tilde{u}$ and $F_{i,j}u = \hat{u}$ for the parameters of $A(x)$ and $A(x)$ itself. Evaluating at $x = a_i$ and $x = a_j$ gives us the two matrices $$\begin{gathered}
R_l(x) = \begin{pmatrix} (x-a_i)(x+a_i) & 0 \\ 0 & (x-a_j)(x+a_j) \end{pmatrix}
+ \frac{\big(a_i^2-a_j^2\big)}{z_i-z_j} \begin{pmatrix} z_i & -w \vspace{1mm}\\ \dfrac{z_iz_j}{w} & -z_i \end{pmatrix}, \\
R_r(x) = \begin{pmatrix} (x-a_i)(x+a_i+h) & 0 \\ 0 & (x-a_j)(x+a_j+h) \end{pmatrix}\\
\hphantom{R_r(x) =}{}
+ \frac{(a_i-a_j)(h+a_i+a_j)}{w_i-w_j} \begin{pmatrix}- w_j & w \vspace{1mm}\\ \dfrac{w_iw_j}{w} & w_j \end{pmatrix},\end{gathered}$$ from which using these values in and evaluated at $x= a_k$ give (\[hEij1\])–(\[hEij3\]) and (\[hFij1\])–(\[hFij3\]) easily follow.
$\boldsymbol{q}$-difference Garnier systems
-------------------------------------------
Let us consider the $q$-difference Garnier system, defined by where $\sigma = \sigma_q$, where $A(x)$ is specified by for $N = 2m+2$ and $L_i$ is given by . We may diagonalize the leading coefficient matrices around $x = 0$ and $x=\infty$ provided $\theta_1 \neq \theta_2$ and $\kappa_1 \neq \kappa_2$ using a lower diagonal constant matrix. From this matrix, we define a new set of variables, $y_i$, $z_i$ and $w_i$, for $$\begin{gathered}
A(a_i) = y_i \begin{pmatrix} 1 \vspace{1mm}\\ \dfrac{z_i}{w} \end{pmatrix} \begin{pmatrix} w_i & w \end{pmatrix} = \begin{pmatrix} w_i y_i & w y_i \vspace{1mm}\\
\dfrac{w_i y_i z_i}{w} & y_i z_i \end{pmatrix},\end{gathered}$$ for $i=1, \ldots, N$. This specification in terms of the image and kernel of $A(a_i)$ means that we may use and/or and the action of $S_n$ to determine the values of $z_i/w$ and $w_i/w$. This defines $3N$ parameters, many of which are redundant. However, if we choose the first $N$ (or any collection), we may reconstruct $A(x)$ using Lagrangian interpolation using any collection of $m+1$ values with the following data:
- $a_{i,i}(x) = \kappa_i x^{m+1} + O(x^m)$ with $a_{1,1}(a_k) = y_kw_k$ and $a_{2,2}(a_k) = y_kz_k$ for $k=1,\ldots, m+1$,
- $a_{1,2}(a_k) = wy_k$ and $a_{2,1} = w_ky_kz_k/w$ for $k=1, \ldots, m+1$.
If we let this collection be the first $m$ values, and let $D(x)$ satisfy . We use this to express the entries of $A(x)$ as
\[LagrangianIntq\]$$\begin{gathered}
a_{1,1}(x) = \kappa_1 D(x) \left[1+ \sum_{i=1}^{m} \frac{w_iy_i}{D'(a_i)(x-a_i)} \right], \\
a_{1,2}(x) = \kappa_2 w D(x) \left[\sum_{i=1}^{m} \frac{y_i}{D'(a_i)(x-a_i)} \right],\\
a_{2,1}(x) = \frac{\kappa_1 D(x)}{w} \left[\sum_{i=1}^{m} \frac{w_iz_iy_i}{D'(a_i)(x-a_i)} \right],\\
a_{2,2}(x) = \kappa_2 D(x) \left[1+\sum_{i=1}^{m} \frac{z_iy_i}{D'(a_i)(x-a_i)} \right].\end{gathered}$$
After diagonalizing the leading coefficient in the polynomial expansion in $x$, it is easy to see that the matrix inducing $T_{i,j}$ takes the form $$\begin{gathered}
\label{Rqformywz}
R(x) = \frac{xI + R_0}{(x-qa_i)(x-qa_j)},\end{gathered}$$ from which we may calculate the equivalent action on the variables $y_i$, $z_i$, $w_i$ and $w$.
The action of $T_{i,j}$ specified by the action of $S_n$ and is equivalent to the following action on the variables $y_k$, $z_k$ and $w_k$:
\[qGarnier\] $$\begin{gathered}
\frac{(T_{i,j} wy_k)}{w}= y_k\frac{(qa_j - a_k)(qa_i-a_k)}{(z_i-z_j)^2} \left(\frac{w_k+z_j}{qa_j-a_k} - \frac{w_k+z_i}{qa_i-a_k} \right)\left( \frac{z_k-z_j}{a_k-a_j} - \frac{z_k-z_i}{a_k-a_i}\right),\!\!\!\!\\
\frac{T_{i,j}z_k}{z_iz_j} \left( \frac{z_k-z_j}{a_k-a_j} - \frac{z_k-z_i}{a_k-a_i} \right) = \frac{T_{i,j}w}{w}\left(\frac{1}{z_j} \frac{z_k-z_j}{a_k-a_j} - \frac{1}{z_i}\frac{z_k-z_i}{a_k-a_i}\right),\\
\label{qwk} (T_{i,j}w_k)\left(\frac{w_k+z_i}{a_k-qa_i}-\frac{w_k+z_j}{a_k-qa_j}\right) = z_iz_j \frac{T_{i,j}w}{w} \left(\frac{1}{z_j} \frac{z_k-z_j}{a_k-a_j} - \frac{1}{z_i}\frac{z_k-z_i}{a_k-a_i}\right),\\
\label{qw} \frac{T_{i,j}w}{w} = \frac{T_{i,j}w_i - T_{i,j}w_j}{z_i - z_j} = 1+\frac{(\kappa _1 q/\kappa_2-1) (a_i-a_j)}{z_i-z_j},\end{gathered}$$ for $k \neq i,j$ where for $k=i$ we have $wT_{i,j} w_i = z_iT_{i,j}w$ and $$\begin{gathered}
\frac{T_{i,j} y_i}{a_i-a_j} = \frac{\big(w^2 \kappa_1\kappa_2D(qa_i)- (w a_{1,1} (q a_i )+z_i a_{1,2} (q a_i ) ) (w a_{2,2} (q a_i )-z_i a_{1,2} (q a_i ) )\big)}{(q-1) a_i a_{1,2} (q a_i ) (z_i-z_j ){}^2 (T_{i,j}w)}\nonumber\\
\hphantom{\frac{T_{i,j} y_i}{a_i-a_j} =}{} -\frac{\big(\kappa_1\kappa_2w^2 D(qa_i)- (w a_{1,1} (q a_i )+z_i a_{1,2} (q a_i ) ) (w a_{2,2} (q a_i )-z_j a_{1,2} (q a_i ) )\big)}{a_{12}(q a_i) (z_i-z_j){}^2 (q a_i-a_j)(T_{i,j}w)},\!\!\!\!\label{tijyi}\\
\label{tijzi}\frac{T_{i,j} z_i}{a_i-a_j}= \frac{\big(w a_{2,2} (q a_i ) z_i+w^2 a_{2,1} (q a_i )+z_i z_j (T_{i,j}w y_i)\big)}{w (T_{i,j}y_i) (z_i (q a_i-a_j )-(q-1) a_i z_j )}.\end{gathered}$$
Swapping $i$ and $j$ gives the case for $k = j$.
For a parameter or matrix, $u$, we use the notation $\tilde{u} = T_{i,j}u$. After establishing , we may multiply by $(x-a_i)(x-a_j)(x-qa_i)(x-qa_j)$, whereby cancelling the denominators and evaluating at $x = qa_i$ shows that $$\begin{gathered}
\tilde{y}_i \begin{pmatrix} 1 \vspace{1mm}\\ \dfrac{\tilde{z}_i}{\tilde{w}} \end{pmatrix} \begin{pmatrix} \tilde{w}_i & \tilde{w} \end{pmatrix}(qa_i I + R_0) = 0,\end{gathered}$$ which specifies that the columns are in the kernel of $\tilde{A}(qa_i)$, whereas evaluating at $x=qa_j$ gives a similar equation which is enough to uniquely specifies $R_0$, which can be written explicitly as $$\begin{gathered}
\frac{1}{\tilde{w}_i-\tilde{w}_j} \begin{pmatrix} qa_j\tilde{w}_j - qa_i \tilde{w}_i & q(a_j-a_i)\tilde{w} \\ q(a_i-a_j)\tilde{w}_i\tilde{w}_j/\tilde{w} & qa_i\tilde{w}_j - qa_j \tilde{w}_i \end{pmatrix}.\end{gathered}$$ Evaluating at $x= a_i$ gives that $$\begin{gathered}
(qa_iI + R_0) y_i \begin{pmatrix} 1 \vspace{1mm}\\ \dfrac{z_i}{w} \end{pmatrix} \begin{pmatrix} w_i & w \end{pmatrix},\end{gathered}$$ which specifies that the rows are in the kernel, which means $R_0$ may be computing in terms of $z_i$ and $z_j$, which is explicitly given by $$\begin{gathered}
R_0 =\frac{1}{z_i-z_j} \begin{pmatrix} q a_i z_j-q a_j z_i & q w (a_j-a_i )\vspace{1mm} \\
\dfrac{q \left(a_i-a_j\right) z_i z_j}{w} & q a_j z_j-q a_i z_i
\end{pmatrix}.\end{gathered}$$ The comparison of these values specifies $wT_{i,j} w_i = z_iT_{i,j}w$ which implies . The second part of is specified by looking at the leading order expansion of in the top right-hand entry. The remaining values of are easily and uniquely determined by evaluating at $x= a_k$.
We need only determine the action on $y_i$ and $z_i$, which can be achieved by evaluating $$\begin{gathered}
\tilde{A}(x) = R(qx)A(x)R(x)^{-1},\end{gathered}$$ at $x = qa_i$, whereby using the value of $R_0$ above gives and . By Proposition \[qlattice\], the uniqueness of $R(x)$ shows $T_{i,j} = T_{j,i}$, and the symmetry of $A(x)$ with respect to swapping $i$ and $j$ implies the action on $y_j$ and $z_j$ are obtained by swapping $i$ and $j$ in and .
The resulting form of the evolution was called the birational form of the $q$-Garnier system in [@Sakai:Garnier].
The author of [@Sakai:Garnier] also produces another parameterization in which every root of the polynomial $a_{1,2}(x)$ is a parameter say $y_1, \ldots, y_m$, while the other parameter are the values of $z_i = a_{1,1}(y_i)$ for $i=1,\ldots, n$. This may be considered a natural extension of known parameterizations of Lax pairs for Painlevé equations and discrete Painlevé equations. The issue in defining a collection of variables in this way is that we can only formally distinguish the roots of $a_{1,2}(x)$. A discrete isomonodromic will produce $\tilde{a}_{1,2}(x)$, whose roots are $\tilde{y}_1, \ldots, \tilde{y}_n$, yet there is no way of ordering the $y_i$ and $\tilde{y}_i$ in a way that makes the mapping $y_i \to \tilde{y}_i$. The space formed by considering set of roots of monic polynomials of degree $n$ is a construction for the $n$-th symmetric power of $\mathbb{C}$, which may be consider the correct setting for such a parameterization. In the continuous setting, this parameterization makes more sense as the variables change continuously.
Let us now start with a matrix satisfying $A(x)A(1/qx) = I$, then we take $A(x)$ to be given by where $B(x)$ is given by . We define variables $y_i$, $z_i$ and $w_i$ by $$\begin{gathered}
B(a_i) = y_i \begin{pmatrix} 1 \vspace{1mm}\\ \dfrac{z_i}{w} \end{pmatrix} \begin{pmatrix} w_i & w \end{pmatrix} = \begin{pmatrix} w_i y_i & w y_i\vspace{1mm}\\
\dfrac{w_i y_i z_i}{w} & y_i z_i \end{pmatrix},\end{gathered}$$ for $i = 1,\ldots, N'$. The Lagrangian interpolation is equivalent to the formulation for $A(x)$ above, hence the entries of $B(x) = (b_{i,j}(x))$ are also given by . We may calculate the effect of $E_{i,j}$ and $F_{i,j}$ on these new variables.
\[eq\] The system is equivalent to the following action on the variables $y_k$, $z_k$ and $w_k$
$$\begin{gathered}
\label{qEij1}\frac{(E_{i,j}y_k) \tilde{w}}{wy_k} = \frac{a_j (a_i a_k-1)(z_j-z_k)}{(a_j-a_k) (z_i-z_j)}-\frac{a_i (a_j a_k-1)(z_i-z_k)}{(a_i-a_k) (z_i-z_j)},\\
\frac{E_{i,j}z_ky_k}{w_ky_k} - \frac{a_ia_jz_k}{w} =\frac{a_i z_j (a_i a_j-1) (z_i-z_k)}{w(a_k-a_i) (z_i-z_j)}+\frac{a_j z_i (a_i a_j-1) (z_j-z_k)}{w
(a_j-a_k) (z_i-z_j)},\\
\label{qEij3}wE_{i,j}w_k = w_kE_{i,j}w, \qquad
\frac{E_{i,j} w}{w} = 1 + \frac{(a_j-a_i)(a_ia_j-1)}{a_ia_j(z_i-z_j)},\end{gathered}$$ for $k \neq i,j$ and for $k=i$ with $$\begin{gathered}
\label{qEij4}\frac{(E_{i,j}wy_i)}{a_i -a_j} = \frac{b_{1,2}\left(\frac{1}{a_i}\right) z_i-b_{2,2}\left(\frac{1}{a_i}\right) w}{\left(a_i-\frac{1}{a_i}\right) (z_i-z_j)},\\
\label{qEij5} w E_{i,j} z_i = z_j E_{i,j}w, \qquad E_{i,j}\frac{w_i}{w} = \frac{wb_{2,1}\left(\frac{1}{a_i}\right) - b_{1,1}\left(\frac{1}{a_i}\right)z_i}{wb_{2,2}\left(\frac{1}{a_i}\right) - b_{1,2}\left(\frac{1}{a_i}\right)z_i},\end{gathered}$$
with the equivalent form for $k=j$ obtained by interchanging $i$ and $j$.
\[fq\] The system is equivalent to the following action on the variables $y_k$, $z_k$ and $w_k$
$$\begin{gathered}
\label{qFij1}\frac{(F_{i,j}y_kw)-qa_ia_jwy_k}{(1-qa_ia_j)wy_k} = \frac{a_i (w_i-w_k)}{(a_i-a_k) (w_i-w_j)}+\frac{a_j (w_j-w_k)}{(a_k-a_j) (w_i-w_j)},\\
F_{i,j} z_k - \frac{w_jz_kF_{i,j}w}{w^2} = \frac{a_j y_k z_k (w_j-w_k) (q a_i a_k-1)}{w (a_j-a_k) (F_{i,j}y_k)},\\
\label{qFij3} z_k F_{i,j}w_k = w F_{i,j} z_k, \qquad \frac{F_{i,j} w}{w} = 1 + \frac{(a_i-a_j)(1-qa_ia_j)\kappa_1}{qa_ia_j(w_i-w_j)\kappa_2},\end{gathered}$$
for $k \neq i,j$ whereas for $k =i$ we have $$\begin{gathered}
(F_{i,j} wy_i) = \frac{(1-q)qa_ia_ja_{1,2}\left( \frac{1}{qa_i} \right)}{(1-qa_i^2)(1-qa_ia_j)}\nonumber \\
\hphantom{(F_{i,j} wy_i) =}{} - \frac{q^2 a_i \left(a_i-a_j\right) (a_i a_j-1) \left(z_j a_{12}\left(\frac{1}{q a_i}\right)+w a_{1,1}\left(\frac{1}{q a_i}\right)\right)}{\big(q a_i^2-1\big)(z_i-z_j) (q a_i a_j-1)},\label{qFij4}\\
\label{qFij5}\frac{F_{i,j} w_i y_i}{z_i} + \frac{F_{i,j}w y_i}{w} = \frac{(1-q)qa_ia_j a_{1,2}\left(\frac{1}{qa_i}\right)}{\big(1-qa_i^2\big)(1-qa_ia_j)w} +\frac{(1-q)qa_ia_j a_{1,1}\left(\frac{1}{qa_i}\right)}{\big(1-qa_i^2\big)(1-qa_ia_j)z_i},\end{gathered}$$
with the equivalent form for $k=j$ obtained by interchanging $i$ and $j$.
We wish to take a different approach from the proofs of Propositions \[eh\] and \[fh\] by deducing $R_l(x)$ and $R_r(x)$ in terms of $\tilde{A}(x)$ and $\hat{A}(x)$ where $\tilde{u} = E_{i,j} u$ and $\hat{u} = F_{i,j}u$ respectively. Since we know the determinant of $R_l(x)$ must include and factor of $(x-a_i)(x-a_i)$ and is symmetric with respect to the action $x \to 1/x$, we have that $R_l(x)$ takes the form $$\begin{gathered}
R_l(x) = I\left(x+ \frac{1}{x}\right) + R_0, \qquad \det R_l(x) = (x-a_i)(x-a_j)(xa_i-1)(xa_j-1)/x^2,\end{gathered}$$ and $\lambda(x) = (x-a_i)^{-1}(x-a_j)^{-1}$, whereas $R_r$ is symmetric with respect to $x \to 1/qx$, hence $R_r(x)$ takes the form $$\begin{gathered}
R_r(x) = I\left(x+ \frac{1}{qx}\right) + R_1, \qquad \det R_r(x) = (x-a_i)(x-a_j)(qxa_i-1)(qxa_j-1)/q^2x^2,\end{gathered}$$ with the same $\lambda(x)$. Due to the involutive nature of the transformation, it is natural that the $R_0$ and $R_1$ satisfy $$\begin{gathered}
(xa_i-1)(xa_j-1) A(x) = \left( \left( x + \frac{1}{x} \right)I + R_0^*\right) \tilde{A}(x),\\
(qxa_i-1)(qxa_j-1) A(x) =\hat{A}(x) \left( \left( x + \frac{1}{x} \right)I + R_1^*\right) ,\end{gathered}$$ where $R_i^*$ is the cofactor matrix for $R_i$ for $i = 0,1$. Since $\tilde{\tilde{u}} = \hat{\hat{u}} = u$, these two equations are equivalent to and applied to the transformed values of $A(x)$. This gives $$\begin{gathered}
R_l(x) = \begin{pmatrix} x + \dfrac{1}{x} - a_j - \dfrac{1}{a_j} & 0 \\ 0 & x + \dfrac{1}{x} - a_i - \dfrac{1}{a_i} \end{pmatrix}
+ \frac{(a_i-a_j)(1-a_ia_j)}{a_ia_j(\tilde{z}_i-\tilde{z}_j)} \begin{pmatrix} \tilde{z}_j & -\tilde{w} \vspace{1mm}\\ \dfrac{\tilde{z}_i\tilde{z}_j}{\tilde{w}} & -\tilde{z}_i \end{pmatrix},\nonumber\\
R_r(x) = \begin{pmatrix} x + \dfrac{1}{qx} - a_j - \dfrac{1}{qa_j} & 0 \\ 0 & x + \dfrac{1}{qx} - a_i - \dfrac{1}{qa_i} \end{pmatrix}\\
\hphantom{R_r(x) =}{}
+ \frac{(a_i-a_j)(1-qa_ia_j)}{qa_ia_j(\hat{w}_i-\hat{w}_j)} \begin{pmatrix} \hat{w}_j & \hat{w} \vspace{1mm}\\ -\dfrac{\hat{w}_i\hat{w}_j}{\tilde{w}} & -\hat{w}_j \end{pmatrix},\end{gathered}$$ from which we may calculate and equivalent form of (\[qEij1\])–(\[qEij3\]) and (\[qFij1\])–(\[qFij3\]) in terms of $\tilde{w}_i$’s and $\hat{w}_i$ respectively. Comparing entries of and using these values at $x = 1/a_i$ and $x= 1/qa_i$ gives the remaining values and brings gives (\[qEij4\])–(\[qEij5\]) and (\[qFij4\])–(\[qFij5\]). The first parts of and bring (\[qEij1\])–(\[qEij3\]) and (\[qFij1\])–(\[qFij3\]) into their presented form, similarly with $x= 1/a_j$ and $x=1/qa_j$.
Special cases {#sec:special}
=============
We wish to demonstrate that the simplest cases of the $h$-difference and $q$-difference Garnier systems are known to coincide with discrete versions of the sixth Painlevé equation. Specializing the higher cases coincide with discrete Painlevé equations that appear higher in Sakai’s hierarchy. We summarize the results in Table \[sumtable\]. To avoid confusion, we have used the value of $N$ and since we have used the notation $r_{1,2}$ and $r_{2,1}$ in both sections we state that the value of $r_{2,1}$ in Table \[sumtable\] is specified by and the value of $r_{1,2}$ is given by .
$N$ conditions Painlevé equation
----------------------- ----- ------------------------------------------------------------- -------------------------------------
$h$-Garnier $6$ $d_1 \neq d_2$ $d$-$\mathrm{P}\big(A_2^{(1)}\big)$
$8$ $d_1 = d_2$, $r_{2,1} = 0$ $d$-$\mathrm{P}\big(A_1^{(1)}\big)$
symmetric $h$-Garnier $8$ $d_1 = d_2$, $r_{2,1} = 0$ $d$-$\mathrm{P}\big(A_1^{(1)}\big)$
$q$-Garnier $4$ $\kappa_1\neq \kappa_2$, $\theta_1\neq \theta_2$ $q$-$\mathrm{P}\big(A_3^{(1)}\big)$
$6$ $\kappa_1 = \kappa_2$, $\theta_1=\theta_2$ $q$-$\mathrm{P}\big(A_2^{(1)}\big)$
$8$ $\kappa_1 = \kappa_2$, $\theta_1=\theta_2$, $r_{1,2} = 0$ $q$-$\mathrm{P}\big(A_1^{(1)}\big)$
symmetric $q$-Garnier $8$ $\kappa_1 = \kappa_2$, $\theta_1 = \theta_2$, $r_{1,2} = 0$ $q$-$\mathrm{P}\big(A_0^{(1)}\big)$
: A summary of the special cases of discrete Garnier systems whose evolution coincides with discrete Painlevé equations.\[sumtable\]
We remark that scalar Lax pairs for the $q$-difference cases of discrete Painlevé equations we present have also been presented in [@Yamada:LaxqEs] and more recently scalar Lax pairs for the $h$-difference cases appeared in [@Kajiwara2015]. A correspondence between the scalar Lax pairs and matrix Lax pairs for the $q$-$\mathrm{P}\big(A_2^{(1)}\big)$ case that appears here was constructed in [@Ormerod:qE6]. Such correspondences are almost sure to exist for the other cases, however, we do not pursue these lengthy correspondences here. We do however remark that the characteristic properties of the Lax pairs presented in [@Yamada:LaxqEs] and [@Kajiwara2015] and scalar versions of the Lax pairs we present here seem to coincide up to some nontrivial transformations.
The twisted $\boldsymbol{m=1}$ asymmetric $\boldsymbol{q}$-difference Garnier system
------------------------------------------------------------------------------------
The first system we present as a special case is the $q$-analogue of the sixth Painlevé equation, which we write as
\[qP6\] $$\begin{gathered}
z(q t)z(t) = \frac{b_3 b_4(y(t) - a_1t)(y(t)-a_2t)}{(y-a_3)(y-a_4)},\\
y(q t)y(t) = \frac{a_3a_4(z(q t) - b_1 t)(z(q t) - b_2 t)}{(z(q t) - b_3)(z(q t) - b_4)},\end{gathered}$$
where $$\begin{gathered}
q = \frac{a_1a_2b_3b_4}{b_1 b_2 a_3a_4},\end{gathered}$$ which was first presented by Jimbo and Sakai [@Sakai:qP6]. We consider an associated linear problem of the from , where $$\begin{gathered}
\label{ALqP6}
A(x) = \begin{pmatrix} \theta_1 & 0 \\ 0 & \theta_2 \end{pmatrix} \begin{pmatrix}
1 & u_1 \vspace{1mm}\\
\dfrac{x}{a_1u_1} & 1
\end{pmatrix}\begin{pmatrix}
1 & u_2 \vspace{1mm}\\
\dfrac{x}{a_2u_2} & 1
\end{pmatrix} \begin{pmatrix}
1 & u_3 \vspace{1mm}\\
\dfrac{x}{a_3u_3} & 1
\end{pmatrix} \begin{pmatrix}
1 & u_4 \vspace{1mm}\\
\dfrac{x}{a_4u_4} & 1
\end{pmatrix}.\end{gathered}$$ This matrix is of the form $$\begin{gathered}
A(x) = A_0 + A_1 x + A_2 x^2,\end{gathered}$$ where $A_0$ is upper triangular with diagonal entries $\theta_1$ and $\theta_2$, while $A_2$ is lower triangular with diagonal entries $$\begin{gathered}
\kappa_1 = \frac{\theta_1u_1 u_3}{a_2a_4u_2u_4}, \qquad \kappa_1 = \frac{\theta_2u_2 u_4}{a_1a_3u_1u_3}.\end{gathered}$$ The two natural consequences that $$\begin{gathered}
\label{detqP6}\det A(x) = \kappa_1\kappa_2 (x-a_1)(x-a_2)(x-a_3) (x-a_4),\\
\theta_1 \theta_2 = \kappa_1 \kappa_2 a_1 a_2 a_3 a_4,\nonumber\end{gathered}$$ which means that by diagonalizing the constant coefficient, we may let $A_0 = \operatorname{diag}(\theta_1,\theta_2)$ and have a pair $$\begin{gathered}
(A_1,A_2) \in \mathcal{M}(a_1, \ldots, a_4; \kappa_1, \kappa_2;\theta_1, \theta_2).\end{gathered}$$ We may diagonalize $A_2$ in order to bring this Lax pair into the form of Jimbo and Sakai [@Sakai:qP6]. We propose a slightly different form in which $A_0$ and $A_2$ are upper and lower triangular respectively. This gives us a simple alternative parameterization, which takes the general form $$\begin{gathered}
\label{quasisakaiqP6}
A(x,t) = \begin{pmatrix}
\kappa_2 x^2 + \alpha x + \theta_1 & w (x-y) \vspace{1mm}\\
\dfrac{x^2 \gamma + \delta x}{w} & \kappa_2 x^2 + \beta x + \theta_2
\end{pmatrix}.\end{gathered}$$ We satisfy when $x=y$ by letting $$\begin{aligned}
{3}
&a_{1,1}(x) = \kappa_1z_1, \qquad &&a_{11}(x) = \kappa_2z_2,& \\
&z_1 = \frac{(y-a_1)(y-a_2)}{z}, \qquad &&z_2 = (y-a_3)(y-a_4)z.&\end{aligned}$$ We may solve for $\alpha$, $\beta$, $\gamma$ and $\delta$ in terms of $y$, $z$ and $w$ to show $$\begin{gathered}
\allowdisplaybreaks
\alpha = \frac{z_1 - \kappa_1 y^2 - \theta_1}{y}, \qquad \beta = \frac{z_1 - \kappa_1 y^2 - \theta_1}{y},\\
\gamma = \kappa_1\kappa_2(a_1 + a_2 + a_3 + a_4) + \frac{\alpha}{\kappa_1} + \frac{\beta}{\kappa_2}, \\
\delta= \frac{\kappa_1\kappa_2(a_1a_2a_3 + a_1a_2a_4 + a_1a_3a_4+ a_2a_3a_4)}{y}-\frac{\theta_1\alpha + \theta_2 \beta}{y}.\end{gathered}$$ The only minor difference in the theory presented above is that the constant coefficient in the series part of the solution, $Y_{\infty}(x)$, is lower triangular, rather than the identity, as is the leading term in the discrete isomonodromic deformation.
As above, we wish to we have four variables, $u_1, \ldots,u_4$, with one constant with respect to $T_{1,2}$, which we wish to identify with the variables $y$, $z$ and $w$. Equating the various coefficients of with the corresponding expressions in gives the following expressions for $y$ and $z$
\[qP6yz\] $$\begin{gathered}
y = -\frac{a_2 a_3 u_2 u_3 (u_1+u_2+u_3+u_4)}{a_2 u_2 (u_1+u_2) u_4+a_3 u_1 u_3 (u_3+u_4)},\\
z = -\frac{a_3a_4(y-a_1)(y-a_2)(u_3+u_4)}{(y-a_3)(y-a_4)(u_1+u_2)\theta_1}.\end{gathered}$$
Conversely, we notice that since the right-most factor of $A(a_4)$, $L(a_4,u_4,a_4)$, has a $0$ eigenvector of the form $(u_4,-1)$, we may iteratively define $u_i$ by determining the $0$-eigenvector at $x=a_i$ for $i = 1,\ldots, 4$. For example, using $x= a_4$ we see $$\begin{gathered}
u_4 = -a_{1,2}(a_4)/a_{1,1}(a_4),\end{gathered}$$ which is given in terms of $y$, $w$ and $z$ above. This gives a right factor which we may remove to iteratively proceed for $x= a_3$ and so on and so forth. This gives us a one-to-one correspondence between $u_1, \ldots, u_4$ and $y$, $z$ and $w$ with $\kappa_1$ and $\kappa_2$ specified, with constraint, in terms of $u_1, \ldots, u_4$.
\[qP6ident\] The birational transformation of algebraic varieties $$\begin{gathered}
T_{1,2} \colon \ \mathcal{M}_q(a_1,a_2,a_3, a_4; \kappa_1, \kappa_2;\theta_1, \theta_2) \to \mathcal{M}_q(qa_1,qa_2,a_3, a_4; \kappa_1/q, \kappa_2/q;\theta_1, \theta_2)\end{gathered}$$ is equivalent to the mapping $t \to qt$ in where the values of $b_i$ are given by $$\begin{gathered}
\label{qP6bvals}
b_1 = \frac{q^2 a_1a_2}{\theta_1}, \qquad b_2 = \frac{q^2 a_1 a_2}{\theta_2}, \qquad b_3 = \frac{q}{\kappa_1}, \qquad b_4 = \frac{q^2}{\kappa_2}.\end{gathered}$$
Using , we see that the image of $A(a_1)$ and $A(a_2)$ gives $u_1$ and $u_2$, which are explicitly given by $$\begin{gathered}
u_1 = \frac{\theta _2 w (a_1-y)}{\theta _1 \big(a_1 \beta +a_1^2 \kappa _2+\theta _2\big)}, \\
u_2 = -\frac{a_1 \theta _2 w (a_1-y) (\kappa _2 (a_1 (y-a_2)+a_2 y)+\theta _2+\beta y)}{\theta _1 (a_1
(a_1 \kappa _2+\beta )+\theta _2) (a_1 (a_2 (\beta +\kappa _2 y)+\theta _2)+\theta _2
(a_2-y))}.\end{gathered}$$ This determines the an $R(x)$ inducing $T_{1,2}$ in terms of $y$, $w$ and $z$. This is used into be used in . If we temporarily introduce the notation $T_{1,2} f = \tilde{f}$ then these calculations reveal $$\begin{gathered}
\tilde{w}= w\frac{q^2-\tilde{z}\kappa_1}{q-\tilde{z}\kappa_2},\qquad
\tilde{z}z = \frac{q^2}{\kappa_1\kappa_2} \frac{(y-a_1)(y-a_2)}{(y-a_3)(y-a_4)},\\
\tilde{y}y = \frac{(\theta_1\tilde{z}-q^2a_1a_2)(\kappa_1\kappa_2a_3a_4 \tilde{z}-q^2 \theta_1)}{(\kappa\tilde{z}-q)(\kappa_2 \tilde{z}-q^2)},\end{gathered}$$ which coincides with when the $b_i$ are specified by .
Alternatively, we may simply use and and the expressions for the $u_i$ in terms of $y$ and $z$.
A special case of the $\boldsymbol{m=1}$ $\boldsymbol{h}$-difference Garnier system {#dP6sec}
-----------------------------------------------------------------------------------
The second system we present is the case of the difference analogue of the sixth Painlevé equation, which we present as
\[hP6\] $$\begin{gathered}
(y(t) + z(t))(y(t+h)+z(t)) = \frac{(z(t)+a_3)(z(t)+a_4)(z(t)+a_5)(z(t)+a_6)}{(z(t)+a_7+t)(z(t)+a_8+t)},\\
(y(t+h) + z(t))(y(t+h)+z(t+h))\nonumber\\
\qquad{}= \frac{(y(t+h)-a_3)(y(t+h)-a_4)(z(t+h)-a_5)(y(t+h)-a_6)}{(y(t+h)-a_1-t-h)(y(t+h)-a_2-t-h)},\end{gathered}$$
where $$\begin{gathered}
h = a_3 + a_4 + a_5 + a_6 - a_1 - a_2 - a_7 - a_8.\end{gathered}$$ We consider an associated linear problem of the from , where $$\begin{gathered}
A(x) = \begin{pmatrix} u_1 & 1 \\ x-a_1+u_1^2 & u_1 \end{pmatrix}\begin{pmatrix} u_2 & 1 \\ x-a_2+u_2^2 & u_2 \end{pmatrix}\begin{pmatrix} u_3 & 1 \\ x-a_3+u_3^2 & u_3 \end{pmatrix}\nonumber\\
\hphantom{A(x) =}{} \times\begin{pmatrix} u_4 & 1 \\ x-a_4+u_4^2 & u_4 \end{pmatrix}\begin{pmatrix} u_5 & 1 \\ x-a_5+u_5^2 & u_5 \end{pmatrix}\begin{pmatrix} u_6 & 1 \\ x-a_6+u_6^2 & u_6 \end{pmatrix},\label{ALhP6}\end{gathered}$$ where we impose the constraint $$\begin{gathered}
\sum_{i=1}^{8} u_i = 0.\end{gathered}$$ This product takes the general form $$\begin{gathered}
A(x) = A_0 + A_1 x + A_2 x^2 + A_3x^3,\end{gathered}$$ and may be expressed in the general form $$\begin{gathered}
A(x) = x^3I + \begin{pmatrix} d_1 ((x-\alpha)(x-y) + z_1) & w(x-y) \\ a_{2,1}(x) & d_2((x-\beta)(x-y)+z_2) \end{pmatrix},\end{gathered}$$ where $a_{2,1}(x)$ is a polynomial of degree $2$ before we diagonalize $A_2$. After diagonalizing $A_2$ it becomes a linear function in $x$, which we write as $$\begin{gathered}
a_{2,1}(x) = \frac{\gamma x + \delta}{w}.\end{gathered}$$ The values of $\alpha$, $\beta$, $\gamma$ and $\delta$ are uniquely determined by . The values of $z_1$ and $z_2$ are satisfy $$\begin{gathered}
\big( y^3 + d_1 z_1 \big)\big( y^3 + d_2 z_2 \big) = (y-a_1)(y-a_2)(y-a_3)(y-a_4)(y-a_5)(y-a_6).\end{gathered}$$ This relation is solved by introducing a variable, $z$, via $$\begin{gathered}
y^3 + d_1 z_1 = \frac{(y-a_3)(y-a_4)(y-a_5)(y-a_6)}{y+z},\\
y^3 + d_2 z_2 = (y-a_1)(y-a_2)(y+z).\end{gathered}$$ We also have that the variables $d_1$ and $d_2$ are specified by $$\begin{gathered}
d_1 = a_1 + a_3 + a_5 + u_1^2 + u_3^2 + u_5^2 + \sum_{i=1}^6 \sum_{j=1}^{i-1} u_iu_j,\\
d_2 = a_2 + a_4 + a_5 + u_2^2 + u_4^2 + u_6^2 + \sum_{i=1}^6 \sum_{j=1}^{i-1} u_iu_j,\end{gathered}$$ which are known to be constant with repect to $T_{i,j}$. Using the determinantal relations, and the correspondence between $r_i$ and $d_i$, we have by setting $A_3 = I$, we have the $3$-tuple $$\begin{gathered}
(A_0, A_1, A_2) \in \mathcal{M}(a_1, \ldots, a_6; d_1,d_2;1,1).\end{gathered}$$
\[hP6ident\] The action of the translation $T_{1,2}$ is equivalent to where $a_7$ and $a_8$ are given by $$\begin{gathered}
\label{hP6achoice}
a_7 = -h - a_1 - a_2 - d_1,\qquad a_8= a_3 + a_4 + a_5 + a_6 + d_1.\end{gathered}$$
This follows much the same way as Proposition \[qP6ident\], however, there is an added difficulty in that the diagonalization of $A_2$ introduces a non-trivial correspondence between the matrix and its corresponding $R(x)$, denoted $R'(x)$, to be used in . The resulting matrix, $R'(x)$, can be shown to be of the form $$\begin{gathered}
R'(x) = \frac{x I + R_0}{(x- a_1-h)(x- a_2-h)},\end{gathered}$$ for some constant matrix $R_0$, which can be calculated using . The uniqueness of $R(x)$ ensures this calculation coincides with $T_{1,2}$, as defined by . Using this same over determined relation, namely , we may determine that the mapping in terms of the variables $y$ and $z$ are specified by $$\begin{gathered}
(\tilde{y}+z)(\tilde{z}+\tilde{y}) = \frac{(\tilde{y}-a_3)(\tilde{y}-a_4)(\tilde{y}-a_5)(\tilde{y}-a_6)}{(\tilde{y}-a_1-h)(\tilde{y}-a_2-h)},\\
(\tilde{y}+z)(y+z) = \frac{(z+a_3)(z+a_4)(z+a_5)(z+a_6)}{(z+a_3+a_4+a_5+a_6+d_1)(z-d_1-a_1-a_2-h)},\end{gathered}$$ where $\tilde{y}$ and $\tilde{z}$ are identified with $T_{1,2} y$ and $T_{1,2}z$ respectively. This coincides with with $a_7$ and $a_8$ specified by .
An extra special case of the $\boldsymbol{m=3}$ asymmetric\
and symmetric $\boldsymbol{h}$-difference Garnier system
-----------------------------------------------------------
We have one more special case to consider in the symmetric case, when we allow $A(x)$ to be given by the product $$\begin{gathered}
A(x) =\begin{pmatrix} u_1 & 1 \\ x-a_1+u_1^2 & u_1 \end{pmatrix}
\begin{pmatrix} u_2 & 1 \\ x-a_2+u_2^2 & u_2 \end{pmatrix}
\begin{pmatrix} u_3 & 1 \\ x-a_3+u_3^2 & u_3 \end{pmatrix}\nonumber\\
\hphantom{A(x) =}{} \times \begin{pmatrix} u_4 & 1 \\ x-a_4+u_4^2 & u_4 \end{pmatrix}
\begin{pmatrix} u_5 & 1 \\ x-a_5+u_5^2 & u_5 \end{pmatrix}
\begin{pmatrix} u_6 & 1 \\ x-a_6+u_6^2 & u_6 \end{pmatrix}\nonumber\\
\hphantom{A(x) =}{} \times \begin{pmatrix} u_7 & 1 \\ x-a_7+u_7^2 & u_7 \end{pmatrix}
\begin{pmatrix} u_8 & 1 \\ x-a_8+u_8^2 & u_8 \end{pmatrix},\label{hAprodform8}\end{gathered}$$ where we impose the constraint that $$\begin{gathered}
u_1 + u_2 + u_3 + u_4 + u_5 +u_6 + u_7 + u_8 = 0.\end{gathered}$$ Under this constraint, the coefficient of $x^3$, denoted $A_3$, takes the form $$\begin{gathered}
A_3 = \begin{pmatrix}
d_1 & 0 \\
d_{2,1} & d_2
\end{pmatrix}.\end{gathered}$$ We introduce one more constraint that $d_1 = d_2$, where $d_1$ and $d_2$ are defined by . It is clear from above that $T_{1,2} d_i = d_i + h$, however, it is also easy to show that $$\begin{gathered}
(T_{1,2} - I) d_{2,1} = (d_1-d_2)(u_1 + u_2),\end{gathered}$$ hence, $d_{1,2}$ is constant with respect to $T_{1,2}$ if $d_1 = d_2$. We impose the constraint that the expression for $d_{1,2}$ is identically $0$ for $A(x)$ to define a regular system, so that $A_3 = d_1 I = d_2I$, where the equality implies that $$\begin{gathered}
d_1 = d_2 = \frac{1}{2}\sum_{i=1}^{8} a_i.\end{gathered}$$ As $d_1$ and $d_2$ are defined in terms of the $u_i$ by $$\begin{gathered}
d_1 = a_1 + a_3 + a_5 +a_7+ u_1^2 + u_3^2 + u_5^2 +u_7^2+ \sum_{i=1}^8 \sum_{j=1}^{i-1} u_iu_j,\\
d_2 = a_2 + a_4 + a_5 +a_8+ u_2^2 + u_4^2 + u_6^2 + u_8^2+ \sum_{i=1}^8 \sum_{j=1}^{i-1} u_iu_j,\end{gathered}$$ this is considered an extra constraint on the $u_i$. The map resulting from $T_{1,2}$ is two-dimensional, which an additional difference equations satisfied by one additional gauge freedom. The result is a matrix of the general form $$\begin{gathered}
A(x) = (x-a_1)(x-a_2) \begin{pmatrix} x^2 + \alpha_1 x + \alpha_2 & w \vspace{1mm}\\ \dfrac{\gamma}{w} & x^2 + \beta_1 x + \beta_2 \end{pmatrix}\nonumber\\
\hphantom{A(x) =}{} + \frac{x-a_2}{a_1-a_2} y_1 \begin{pmatrix} 1 \vspace{1mm}\\ \dfrac{z_1}{w} \end{pmatrix} \begin{pmatrix} w_1 & w \end{pmatrix} + \frac{x-a_1}{a_2-a_1} y_2 \begin{pmatrix} 1 \vspace{1mm}\\ \dfrac{z_2}{w} \end{pmatrix} \begin{pmatrix} w_2 & w \end{pmatrix},\label{A0form}\end{gathered}$$ where $$\begin{gathered}
\alpha_1 = \beta_1 = \frac{a_1}{2} + \frac{a_2}{2}-\sum_{i=3}^{8} \frac{a_i}{2}.\end{gathered}$$ The determinant at $x = a_1$ and $x=a_2$ are automatically $0$ by construction. This is also a polynomial of degree six with six nontrivial conditions to satisfy, which are sufficient to write down expressions for $\alpha_2$, $\beta_2$ and $\gamma$.
The map $T_{1,2}$ on the variables $(z_1,z_2,w_1,w_2,y_1,y_2)$ is given by $$\begin{gathered}
\allowdisplaybreaks
T_{1,2} z_1 =
\big\{\tilde{w} \big(w^2 \bar{a}_{21} (z_2 (a_2-a_1-h)+h z_1)+w z_1 (h z_1 \bar{a}_{22}+z_2 (\bar{a}_{22} (a_2-a_1-h)\\
\hphantom{T_{1,2} z_1 = }{} -(a_2-a_1)
\bar{a}_{11}))+(a_1-a_2) z_1^2 z_2 \bar{a}_{12}\big)\big\}\big/\big\{w (z_1 (w (\bar{a}_{11} (a_1-a_2+h)+(a_2-a_1) \bar{a}_{22})\\
\hphantom{T_{1,2} z_1 = }{}
-h z_2 \bar{a}_{12})+w ((a_2-a_1) w \bar{a}_{21}-h z_2 \bar{a}_{11})+z_1^2 \bar{a}_{12} (a_1-a_2+h))\big\}, \\
T_{1,2} z_2 =
\big\{\tilde{w} \big(\hat{a}_{21} w^2 (z_1 (a_1-a_2-h)+h z_2)+w z_2 (\hat{a}_{22} h z_2+z_1 (\hat{a}_{22} (a_1-a_2-h)\\
\hphantom{T_{1,2} z_2 =}{} -(a_1-a_2)
\hat{a}_{11}))+(a_2-a_1) \hat{a}_{12} z_1 z_2^2\big)\big\}\big/\big\{w (z_2 (w (\hat{a}_{11} (a_2-a_1+h)+(a_1-a_2) \hat{a}_{22})\\
\hphantom{T_{1,2} z_2 =}{}
-\hat{a}_{12} h z_1)+w ((a_1-a_2) \hat{a}_{21} w-\hat{a}_{11} h z_1)+\hat{a}_{12} z_2^2 (-a_1+a_2+h))\big\}, \\
T_{1,2} y_1 = \frac{(a_1-a_2) (w \bar{a}_{11}+z_1 \bar{a}_{12})}{(z_1-z_2) \tilde{w} (a_1-a_2+h)}-\frac{(a_1-a_2){}^2 (w^2 \bar{a}_{21}+w z_1
(\bar{a}_{22}-\bar{a}_{11})-z_1^2 \bar{a}_{12})}{h (z_1-z_2){}^2 \tilde{w} (a_1-a_2+h)},\\
T_{1,2} y_2 = \frac{(a_2-a_1) (\hat{a}_{11} w+\hat{a}_{12} z_2)}{(z_2-z_1) \tilde{w} (-a_1+a_2+h)}-\frac{(a_2-a_1){}^2 (\hat{a}_{21} w^2+(\hat{a}_{22}-\hat{a}_{11}) w
z_2-\hat{a}_{12} z_2^2)}{h (z_2-z_1){}^2 \tilde{w} (-a_1+a_2+h)},\\
T_{1,2} w_1 = -z_2 \frac{T_{1,2} w}{w}, \qquad T_{1,2} w_2 = -z_1 \frac{T_{1,2} w}{w}, \qquad T_{1,2}w = w+ \frac{(a_1-a_2) h w}{z_1-z_2},\end{gathered}$$ where we use the notation $\bar{a}_{ij} = a_{ij} (a_1 +h)$ and $\hat{a}_{ij} = a_{ij}(a_2+h)$.
While this has been written as a 6-dimensional map, the action and constraints in the $u$ variables tells us there are $4$ invariants. For example, we could determine expressions for $A(x)$ in terms of the two values $z_1$ and $z_2$ and $w$, specified by $$\begin{gathered}
z_1 = w u_1, \qquad z_2 = w \left( u_1 + \frac{a_1-a_2}{u_1 + u_2} \right),\end{gathered}$$ where $w$ is determined by the coefficient of $x^2$ in the top right entry of $A(x)$. The resulting mapping is a difference equation that sits above the $d$-$\mathrm{P}\big(A_2^{(1)}\big)$ and under some rigidification, it is clearer that the compactifying the moduli space of linear difference equations is indeed $\mathbb{P}_2$ blown up at $9$ points, which has been the subject of one of the authors work [@Rains2013].
Another two-dimensional mapping may be obtained by allowing $A(x)$ to symmetric with respect to the change $x \to -x$, in which case we may allow $A(x)$ to be given by and $B(x)$ to be given by subject to the same constraints. It is easy to show that, under the conditions, that $$\begin{gathered}
(E_{1,2} - I) d_{2,1} = 0, \qquad (F_{1,2} - I) d_{2,1} = 0,\end{gathered}$$ indicating that is a valid parameterization of $B(x)$ that is invariant under the actions $E_{i,j}$ and $F_{i,j}$.
The maps $E_{1,2}$ and $F_{1,2}$ on the variables $(z_1,z_2,w_1,w_2,y_1,y_2)$ are given by $$\begin{gathered}
\allowdisplaybreaks
E_{1,2} z_1 =z_2 \frac{E_{1,2} w}{w},\qquad E_{1,2} w_1 = E_{1,2} w \frac{z_1a_{11}(-h-a_1) - wa_{21}(-h-a_1)}{z_1a_{12}(-h-a_1) - wa_{22}(-h-a_1)},\\
E_{1,2} z_2 =z_1 \frac{E_{1,2} w}{w},\qquad E_{1,2} w_2 = E_{1,2} w \frac{z_2a_{11}(-h-a_2) - wa_{21}(-h-a_2)}{z_2a_{12}(-h-a_2) - wa_{22}(-h-a_2)},\\
E_{1,2} y_1 = \frac{(a_1-a_2) (w a_{22}(-a_1-h)-z_1 a_{12}(-a_1-h))}{(z_2-z_1) \tilde{w} (2 a_1+h)},\\
E_{1,2} y_2 = \frac{(a_2-a_1)(w a_{22}(-a_2-h)-z_2 a_{12}(-a_2-h))}{(z_1-z_2) \tilde{w} (2 a_2+h)},\end{gathered}$$ and $$\begin{gathered}
\allowdisplaybreaks
F_{1,2} z_1 = \frac{F_{1,2}w (a_{21}(-a_1) w-a_{22}(-a_1) w_1)}{a_{11}(-a_1) w-a_{12}(-a_1) w_1},\qquad F_{1,2} w_1 = w_2 \frac{F_{1,2} w}{w},\\
F_{1,2} z_2 = \frac{F_{1,2}w (a_{21}(-a_2) w-a_{22}(-a_2) w_2)}{a_{11}(-a_2) w-a_{12}(-a_2) w_2}, \qquad F_{1,2} w_2 = w_1 \frac{E_{1,2} w}{w},\\
F_{1,2} y_1 = \frac{(a_1-a_2) (w a_{22}(-a_1-h)-z_1 a_{12}(-a_1-h))}{(z_2-z_1) \tilde{w} (2 a_1+h)},\\
F_{1,2} y_2 = \frac{(a_2-a_1) (w a_{22}(-a_2-h)-z_1 a_{12}(-a_1-h))}{(z_2-z_1) \tilde{w} (2 a_1+h)}.\end{gathered}$$
The map $E_{1,2} \circ F_{1,2}$ is once again 2-dimensional and specializes to $T_{1,2}$. This map is also acting on a surface obtained by blowing up $\mathbb{P}_2$ at $9$ points, hence, this map coincides with $d$-$\mathrm{P}\big(A_0^{(1)}\big)$. We seek to establish a more explicit correspondence with well established versions of $d$-$\mathrm{P}\big(A_0^{(1)}\big)$ the future.
An extra special case of the $\boldsymbol{m=3}$ asymmetric\
and symmetric $\boldsymbol{q}$-difference Garnier system
-----------------------------------------------------------
Let us consider the multiplicative version of the previous section, where $A(x)$ is given by the product $$\begin{gathered}
A(x) = \begin{pmatrix}
1 & u_1 \vspace{1mm}\\
\dfrac{x}{a_1^2u_1} & 1
\end{pmatrix}\begin{pmatrix}
1 & u_2 \vspace{1mm}\\
\dfrac{x}{a_2^2u_2} & 1
\end{pmatrix} \begin{pmatrix}
1 & u_3 \vspace{1mm}\\
\dfrac{x}{a_3^2u_3} & 1
\end{pmatrix} \begin{pmatrix}
1 & u_4 \vspace{1mm}\\
\dfrac{x}{a_4^2u_4} & 1
\end{pmatrix}\nonumber\\
\hphantom{A(x) =}{} \times \begin{pmatrix}
1 & u_1 \vspace{1mm}\\
\dfrac{x}{a_5^2u_5} & 1
\end{pmatrix}\begin{pmatrix}
1 & u_6 \vspace{1mm}\\
\dfrac{x}{a_6^2u_6} & 1
\end{pmatrix} \begin{pmatrix}
1 & u_7 \vspace{1mm}\\
\dfrac{x}{a_7^2u_7} & 1
\end{pmatrix} \begin{pmatrix}
1 & u_8 \vspace{1mm}\\
\dfrac{x}{a_8^2u_8} & 1
\end{pmatrix}.\label{Aprod8}\end{gathered}$$ As discussed above, if the $u_i$ variables satisfy $$\begin{gathered}
u_1 + u_2 + u_3 + u_4 + u_5 + u_6 + u_7 + u_8,\end{gathered}$$ then $A_0 = I$. With $\kappa_1$ and $\kappa_2$ specified by , then $$\begin{gathered}
A_4 = \begin{pmatrix} \kappa_1 & 0 \\ \kappa_{2,1} & \kappa_2 \end{pmatrix}.\end{gathered}$$ If $\kappa_1 = \kappa_2$ then we find that $$\begin{gathered}
T_{1,2} \kappa_{2,1} = \frac{a_1 u_1^2}{qa_2u_2^2} \kappa_{2,1},\end{gathered}$$ which means that if $\kappa_{2,1} = 0$ then $T_{1,2} \kappa_{2,1} = 0$. For similar reasons as the previous section, this defines a two-dimensional mapping. Using a similar approach as the previous section, we may specify that the matrix $A(x)$ takes the general form $$\begin{gathered}
A(x) = \frac{(x-a_1)(x-a_2)}{a_1a_2} \begin{pmatrix} \alpha_2 x^2 + \alpha_1 x + 1 & x w\vspace{1mm}\\
\dfrac{\gamma}{w} & \beta_2x^2 + \beta_1 x + 1 \end{pmatrix}\nonumber\\
\hphantom{A(x) =}{} + \frac{x(x-a_2)}{a_1(a_1 - a_2)} y_1 \begin{pmatrix} 1 \vspace{1mm}\\ \dfrac{z_1}{w} \end{pmatrix} \begin{pmatrix} w_1 & w \end{pmatrix} + \frac{x(x-a_1)}{a_2(a_2 - a_1)} y_2 \begin{pmatrix} 1 \vspace{1mm}\\ \dfrac{z_2}{w} \end{pmatrix} \begin{pmatrix} w_2 & w \end{pmatrix}, \label{qBformA0A1}\end{gathered}$$ with $\kappa_1 = \kappa_2$ and the implying that $$\begin{gathered}
\alpha_2 = \beta_2 = \sqrt{\frac{a_1a_2}{a_3a_4a_5a_6a_7}},\end{gathered}$$ in which the mapping $T_{1,2}$ is may be computed accordingly.
The map $T_{1,2}$ on the variables $(z_1,z_2,w_1,w_2,y_1,y_2)$ is given by $$\begin{gathered}
\allowdisplaybreaks
T_{1,2} z_1 =\frac{z_2 T_{1,2}w}{w}+
\big\{a_1 (q-1) (z_2-z_1) T_{1,2}w (w (\bar{a}_{21} w+\bar{a}_{22} z_1)\\
\hphantom{T_{1,2} z_1 =}{} -z_2 (\bar{a}_{11} w+\bar{a}_{12} z_1))\big\}\big/\big\{w \big(a_1 \big(w z_1 (\bar{a}_{22}-\bar{a}_{11}
q)+(q-1) z_2 (\bar{a}_{11} w+\bar{a}_{12} z_1)\\
\hphantom{T_{1,2} z_1 =}{}
-\bar{a}_{12} q z_1^2+\bar{a}_{21} w^2\big)+a_2 \big((\bar{a}_{11}-\bar{a}_{22}) w z_1+\bar{a}_{12}
z_1^2-\bar{a}_{21} w^2\big)\big)\big\}, \\
T_{1,2} z_2 =\frac{z_1 T_{1,2}w}{w}+
\big\{a_2 (q-1) (z_1-z_2) T_{1,2}w (w (\hat{a}_{21} w+\hat{a}_{22} z_2)\\
\hphantom{T_{1,2} z_2 =}{}
-z_1 (\hat{a}_{11} w+\hat{a}_{12} z_2))\big\}\big/\big\{w \big(a_2 \big(w z_2 (\hat{a}_{22}-\hat{a}_{11}
q)+(q-1) z_1 (\hat{a}_{11} w+\hat{a}_{12} z_2)\\
\hphantom{T_{1,2} z_2 =}{}
-\hat{a}_{12} q z_2^2+\hat{a}_{21} w^2\big)+a_1 \big((\hat{a}_{11}-\hat{a}_{22}) w z_2+\hat{a}_{12}
z_2^2-\hat{a}_{21} w^2\big)\big)\big\},\\
T_{1,2}w_1 = z_2 \frac{T_{1,2}w}{w}, \qquad\! T_{1,2}w_2 = z_1 \frac{T_{1,2}w}{w},\qquad\! T_{1,2} w =\frac{q w ((a_1-a_2) (\alpha _2 q-\alpha _2)+z_1-z_2)}{z_1-z_2},\\
T_{1,2}y_1 = \frac{(a_2-a_1){}^2 ((\bar{a_{11}}-\bar{a_{22}}) w z_1+\bar{a_{12}} z_1^2-\bar{a_{21}} w^2)}{a_1 (q-1) (z_2-z_1){}^2 \tilde{w} (a_1
q-a_2)}-\frac{(a_2-a_1) (\bar{a_{11}} w+\bar{a_{12}} z_1)}{(z_2-z_1) \tilde{w} (a_2-a_1 q)}, \\
T_{1,2}y_2 = \frac{(a_1-a_2){}^2 ((\hat{a_{11}}-\hat{a_{22}}) w z_2+\hat{a_{12}} z_2^2-\hat{a_{21}} w^2)}{a_2 (q-1) (z_1-z_2){}^2 \tilde{w} (a_2
q-a_1)}-\frac{(a_1-a_2) (\hat{a_{11}} w+\hat{a_{12}} z_2)}{(z_1-z_2) \tilde{w} (a_1-a_2 q)},\end{gathered}$$ where $\bar{a}_{ij} = a_{ij}(q a_1)$ and $\hat{a}_{ij} = a_{ij}(q a_2)$.
This map induces a map on the two-dimensional moduli space of linear systems of $q$-difference equations and also sits above the $q$-$\mathrm{P}\big(A_2^{(1)}\big)$ case, hence, we naturally expect this to coincide with the $q$-$\mathrm{P}\big(A_1^{(1)}\big)$ case.
We obtain a distinct two-dimensional mapping when we allow the matrix $Y(x)$ to be symmetric with respect to the change $x \to 1/x$, in which case we have $A(x)$ given by and $B(x)$ given by . The first thing to check is that if $\kappa_1 = \kappa_2$, then $E_{1,2} \kappa_1 = E_{1,2} \kappa_2$ and $F_{1,2} \kappa_1 = F_{1,2} \kappa_2$, which is easily done. It is also easy to check that if $\kappa_{2,1} = 0$ then $E_{1,2} \kappa_{2,1} = 0$ and $F_{1,2} \kappa_{2,1} = 0$. This ensures that the mappings $E_{1,2}$ and $F_{1,2}$ may be applied to a matrix $B(x)$ that takes the form .
The maps $E_{1,2}$ and $F_{1,2}$ on the variables $(z_1,z_2,w_1,w_2,y_1,y_2)$ are given by $$\begin{gathered}
\allowdisplaybreaks
E_{1,2} z_1 = z_2 \frac{E_{1,2}w}{w}, \qquad E_{1,2}w_1 = \frac{w \left(a_{21}\left(\frac{1}{a_1}\right) w-a_{11}\left(\frac{1}{a_1}\right) z_1\right)}{a_{22}\left(\frac{1}{a_1}\right) w-a_{12}\left(\frac{1}{a_1}\right) z_1},\\
E_{1,2} z_2 = z_1 \frac{E_{1,2}w}{w}, \qquad E_{1,2}w_1 = \frac{w \left(a_{21}\left(\frac{1}{a_1}\right) w-a_{11}\left(\frac{1}{a_1}\right) z_1\right)}{a_{22}\left(\frac{1}{a_1}\right) w-a_{12}\left(\frac{1}{a_1}\right) z_1},\\
E_{1,2}wy_1 = \frac{a_1 a_2 a_{22}\left(\frac{1}{a_1}\right) w \left(1-a_1^2 (q-1)\right)}{\big(a_1^2-1\big) (a_1 a_2-1) z_2 }
-\frac{a_1^2 \left(a_1-a_2\right) a_2 q \left(a_{22}\left(\frac{1}{a_1}\right)
w-a_{12}\left(\frac{1}{a_1}\right) z_1\right)}{\big(a_1^2-1\big) (a_1 a_2-1 ) (z_1-z_2 ) }, \\
E_{1,2}wy_2 = \frac{a_1 a_2 a_{22}\left(\frac{1}{a_2}\right) w \big(1-a_2^2 (q-1)\big)}{\big(a_2^2-1\big) (a_1 a_2-1) z_1 }
-\frac{a_2^2 (a_2-a_1) a_1 q \left(a_{22}\left(\frac{1}{a_2}\right)
w-a_{12}\left(\frac{1}{a_2}\right) z_2\right)}{\big(a_2^2-1\big) (a_1 a_2-1) (z_2-z_1) },\end{gathered}$$ and $$\begin{gathered}
\allowdisplaybreaks
F_{1,2} w_1 = w_2 \frac{F_{1,2}w}{w}, \qquad F_{1,2} z_1 = F_{1,2}w \frac{w a_{21}\left(\frac{1}{a_1 q}\right)-w_1 a_{22}\left(\frac{1}{a_1 q}\right)}{w a_{11}\left(\frac{1}{a_1 q}\right)-w_1 a_{12}\left(\frac{1}{a_1 q}\right)},\\
F_{1,2} w_2 = w_1 \frac{F_{1,2}w}{w}, \qquad F_{1,2} z_2 = F_{1,2}w \frac{w a_{21}\left(\frac{1}{a_2 q}\right)-w_2 a_{22}\left(\frac{1}{a_2 q}\right)}{w a_{11}\left(\frac{1}{a_2 q}\right)-w_2 a_{12}\left(\frac{1}{a_2 q}\right)},\\
F_{1,2}wy_1 = \frac{a_1 a_2 q \big(1-a_1^2 (q-1) q\big) a_{12}\left(\frac{1}{a_1 q}\right)}{\big(a_1^2 q-1\big) (a_1 a_2 q-1)} \\
\hphantom{F_{1,2}wy_1 =}{}
-\frac{a_1^2 (a_1-a_2) a_2 q^3 \left(w a_{11}\left(\frac{1}{a_1
q}\right)+z_2 a_{12}\left(\frac{1}{a_1 q}\right)\right)}{(z_1-z_2) \big(a_1^2 q-1\big) (a_1 a_2 q-1)},\\
F_{1,2}wy_2 = \frac{a_1 a_2 q \left(1-a_2^2 (q-1) q\right) a_{12}\left(\frac{1}{a_2 q}\right)}{\big(a_2^2 q-1\big) (a_1 a_2 q-1)} \\
\hphantom{F_{1,2}wy_2 =}{}
-\frac{a_2^2 (a_2-a_1) a_1 q^3 \left(w a_{11}\left(\frac{1}{a_2
q}\right)+z_1 a_{12}\left(\frac{1}{a_2 q}\right)\right)}{(z_2-z_1) \big(a_2^2 q-1\big) (a_1 a_2 q-1)}.\end{gathered}$$
By construction, this is a map that sits above the case of $q$-$\mathrm{P}\big(A_1^{(1)}\big)$, hence, it should be $q$-$\mathrm{P}\big(A_0^{(1)}\big)$. We seek to establish a more explicit correspondence with well established versions of $q$-$\mathrm{P}\big(A_0^{(1)}\big)$ in future works.
Reductions of partial difference equations {#sec:reductions}
==========================================
One of the consequences of this work is that we will able to show that the $q$-Garnier system of [@Sakai:Garnier] arises as a set of reduction the lattice Schwarzian Korteweg–de Vries equation. By specializing the $h$-Garnier systems, this also means that $q$-$\mathrm{P}\big(A_1^{(1)}\big)$ and $d$-$\mathrm{P}\big(A_1^{(1)}\big)$ also arise as reductions of the lattice Schwarzian Korteweg–de Vries equation and lattice potential Korteweg–de Vries equation.
The general setting for reductions of partial difference equations on the quad(rilateral) is that we take a function $w \colon \mathbb{Z}^2 \to \mathbb{C}$, whose values are denoted $w_{l,m}$, in which for every $(l,m)\in \mathbb{Z}^2$ we impose the constraint $$\begin{gathered}
\label{quad}
Q(w_{l,m}, w_{l+1,m}, w_{l,m+1}, w_{l+1,m+1}; \alpha_l, \beta_m )=0,\end{gathered}$$ where $Q$ is linear in each of the variables. Given a staircase of initial conditions, we are able to determine each value on $\mathbb{Z}^2$, hence, we require an infinite number of initial conditions to specify a solution [@VanderKamp:IVPs]. For this reason, these systems are commonly referred to as infinite-dimensional systems, and are considered to be the discrete analogues of partial differential equations.
The twisted $(n_1,n_2)$-reduction is the system of solutions that satisfy an additional relation of the form $$\begin{gathered}
\label{twist}
w_{l+n_1, m+n_2} = T(w_{l,m}),\end{gathered}$$ where the function, $T$, is called the twist [@Ormerod2014b]. We require that is invariant under $T$, i.e., we require that $$\begin{gathered}
Q(w_{l,m}, w_{l+1,m}, w_{l,m+1}, w_{l+1,m+1}; \alpha_l, \beta_m )=0 \\
\qquad {} \iff \ Q(Tw_{l,m}, Tw_{l+1,m}, Tw_{l,m+1}, Tw_{l+1,m+1}; \alpha_l, \beta_m )=0.\end{gathered}$$ The staircase of initial conditions for a twisted reduction consists of the $n_1 + n_2$ initial conditions in some finite staircase extended infinitely in both directions using . Secondly, we require that the parameters change in a way that if two points are related by , then the points calculated using those $n_1 + n_2$ initial conditions are also related by . This means we require $$\begin{gathered}
Q(w_{l,m}, w_{l+1,m}, w_{l,m+1}, w_{l+1,m+1}; \alpha_l, \beta_m )=0\nonumber\\
\qquad \iff \ Q(w_{l,m}, w_{l+1,m}, w_{l,m+1}, w_{l+1,m+1}; \alpha_{l+n_1}, \beta_{m+n_2})=0. \label{parchange}\end{gathered}$$ The resulting system may be described by a $(n_1 + n_2)$-dimensional map, which we call a twisted reduction of [@Ormerod2014a]. If $\alpha_l = \alpha_{l+n_1}$ and $\beta_m = \beta_{m+n_2}$ then the resulting ordinary difference equation is necessarily autonomous, otherwise, the system is non-autonomous. In the special case that $T$ is the identity, we call the reduction a periodic reduction.
(-.3,-.3) grid (8.3,4.4); (1,1) – (4,1) – (4,2)–(6,2) –(6,3)– (7,3); (1,1) – (0,1)–(0,0) – (-.3,0); (7,3) – (8.3,3); (1,1) circle (.08cm); (7,3) circle (.08cm); (2,1) circle (.08cm); (3,1) circle (.08cm); (4,1) circle (.08cm); (4,2) circle (.08cm); (5,2) circle (.08cm); (6,2) circle (.08cm); (6,3) circle (.08cm); (0,1) circle (.08cm); (0,0) circle (.08cm); (8,3) circle (.08cm);
One definition of integrability for systems of the form is 3-dimensional consistency. If we impose a constraint of the form on each of the faces of a cube, then 3-dimensional consistency requires that each way of determining the values on the vertices of the cube agree [@Nijhoff:CAC]. A classification of 3-dimensionally consistent multilinear equations of the form of was the subject of the classification of Adler et al. [@ABS:ListI; @ABS:ListII].
We consider two equations of the form ; the lattice potential Korteweg–de Vries equation [@Nijhoff:lkdvreview], $$\begin{gathered}
\label{lpkdv}
(w_{l,m} - w_{l+1,m+1})(w_{l+1,m} - w_{l,m+1}) = \alpha_l - \beta_m,\end{gathered}$$ which is also known as H1 in [@ABS:ListI; @ABS:ListII] and the lattice Schwarzian Korteweg–de Vries equation [@Nijhoff:dSKdV], $$\begin{gathered}
\alpha _l \left(\frac{1}{w_{l,m+1}-w_{l+1,m+1}}+\frac{1}{w_{l+1,m}-w_{l,m}}\right)\nonumber\\
\qquad {} = \beta _m \left(\frac{1}{w_{l+1,m}-w_{l+1,m+1}}+\frac{1}{w_{l,m+1}-w_{l,m}}\right), \label{dSKdV}\end{gathered}$$ which is also known as $\mathrm{Q1}_{\delta= 0}$ in [@ABS:ListI; @ABS:ListII].
It is easy to see that is invariant under translational twists, i.e., those of the form $T(u) = u + \lambda$ whereas is invariant under any twist in the full group of invertible Möbius transformations. Secondly, we see that holds for if $$\begin{gathered}
\label{parchangeh}
\alpha_{l + n_1} = \alpha_l + h, \qquad \beta_{m+n_2} = \beta_m + h,\end{gathered}$$ and holds for if $$\begin{gathered}
\alpha_{l + n_1} = q\alpha_l, \qquad \beta_{m+n_2} = q\beta_m,\end{gathered}$$
A Lax pair in the context of equations of the form of is a pair of equations of the form $$\begin{gathered}
\Phi(l+1,m) = L_{l,m} \Phi(l,m),\qquad \Phi(l,m+1) = M_{l,m} \Phi(l,m),\end{gathered}$$ whose compatibility reads $$\begin{gathered}
L_{l,m+1}M_{l,m} - M_{l+1,m} L_{l,m} = 0.\end{gathered}$$ The matrices, $L_{l,m}$ and $M_{l,m}$, for the lattice potential Korteweg–de Vries equation are
\[LaxdpKdV\] $$\begin{gathered}
L_{l,m} = \begin{pmatrix} w_{l+1,m} - w_{l,m} & 1 \\ \gamma -\alpha_l + (w_{l+1,m} - w_{l,m})^2 & w_{l+1,m} - w_{l,m} \end{pmatrix},\\
M_{l,m} = \begin{pmatrix} w_{l,m+1} - w_{l,m} & 1 \\ \gamma -\beta_m + (w_{l,m+1} - w_{l,m})^2 & w_{l,m+1} - w_{l,m} \end{pmatrix},\end{gathered}$$
which are both of the form for $u = w_{l+1,m} - w_{l,m}$ and $u = w_{l,m+1} - w_{l,m}$. The matrices, $L_{l,m}$ and $M_{l,m}$, for the lattice Schwarzian Korteweg–de Vries equation are
\[LaxdSKdV\] $$\begin{gathered}
L_{l,m} = \begin{pmatrix} 1&w_{l+1,m} - w_{l,m} \vspace{1mm}\\ \dfrac{\gamma}{\alpha_l(w_{l+1,m} - w_{l,m})}& 1 \end{pmatrix},\\
M_{l,m} = \begin{pmatrix} 1&w_{l,m+1} - w_{l,m} \vspace{1mm}\\ \dfrac{\gamma}{\beta_m(w_{l,m+1} - w_{l,m})} & 1 \end{pmatrix},\end{gathered}$$
which are also both of the form for $u = w_{l+1,m} - w_{l,m}$ and $u = w_{l,m+1} - w_{l,m}$. This determines a well known relation between integrable equations of the form and Yang–Baxter maps [@Papageorgiou2006]. The general framework for determining Lax pairs for ordinary difference equations arising as twisted reductions of partial difference was recently outlined in [@Ormerod2014b].
From the point of view of symmetries of reductions [@Ormerod2014b], it is slightly more conducive to regard a twisted $(n_1,n_2)$-reduction as a reduction on an $(n_1 + n_2)$-dimensional hypercube [@Joshi2014]. The symmetries of the reductions arise from different paths on this hypercube from the points connected via .
(0,0) – (1,0) node\[below, midway\] [$u_1$]{} – (1.4,.2) node\[below right, midway\] [$u_2$]{} – (1.4,1.2) node\[below right, midway\] [$u_3$]{} – (1.1,1.6) node\[above right, midway\] [$u_4$]{} ; (1,0)–(1.4,.2) – (1.4,1.2) – (1,1) – cycle; (0.,0.)–(-0.3,0.4); (0.,0.)–(0.4,0.2); (0.,0.)–(1.,0.); (0.,0.)–(0.,1.); (-0.3,0.4)–(0.1,0.6); (-0.3,0.4)–(0.7,0.4); (-0.3,0.4)–(-0.3,1.4); (0.4,0.2)–(0.1,0.6); (0.4,0.2)–(1.4,0.2); (0.4,0.2)–(0.4,1.2); (0.1,0.6)–(1.1,0.6); (0.1,0.6)–(0.1,1.6); (1.,0.)–(0.7,0.4); (1.,0.)–(1.4,0.2); (1.,0.)–(1.,1.); (0.7,0.4)–(1.1,0.6); (0.7,0.4)–(0.7,1.4); (1.4,0.2)–(1.1,0.6); (1.4,0.2)–(1.4,1.2); (1.1,0.6)–(1.1,1.6); (0.,1.)–(-0.3,1.4); (0.,1.)–(0.4,1.2); (0.,1.)–(1.,1.); (-0.3,1.4)–(0.1,1.6); (-0.3,1.4)–(0.7,1.4); (0.4,1.2)–(0.1,1.6); (0.4,1.2)–(1.4,1.2); (0.1,1.6)–(1.1,1.6); (1.,1.)–(0.7,1.4); (1.,1.)–(1.4,1.2); (0.7,1.4)–(1.1,1.6); (1.4,1.2)–(1.1,1.6); at (.8,.7) [$s_{2,3} u_3$]{}; at (1,1.12) [$s_{2,3} u_2$]{};
The key to constructing Lax pairs for periodic reductions is that we have two parameters, $\alpha_l$ and $\beta_m$, whereas the reductions of and depend upon a single variable $t= \alpha_l - \beta_m$ and $t = \alpha_l/\beta_m$ respectively, which is constant with respect to shifts $(l,m) \to (l+n_1,m + n_2)$. We simply need to choose a spectral variable that is not constant with respect to the shift $(l,m) \to (l+n_1,m + n_2)$.
Let us first treat the $h$-difference case. The correspondence between the discrete Garnier systems is made simple by taking periodic reductions of with $$\begin{gathered}
\label{parchoiceh}
x = \alpha_l + a_l, \qquad t = \alpha_l - \beta_m + b_m,\end{gathered}$$ where $a_l$ and $b_m$ are $n_1$-periodic and $n_2$-periodic functions of $l$ and $m$ respectively. Note that an operator that shifts $(l,m) \to (l+n_1,m + n_2)$ by has the effect of fixing $t$, and has the effect of shifting $x \to x+h$. The operator that shifts $(l,m) \to (l,m - n_2)$ fixes $x$ and shifts $t \to t+h$. A matrix inducing the shift in $x$ may be written as $$\begin{gathered}
A(x,t) = M_{l+n_1,m+n_2-1}\cdots M_{l+n_1,m} L_{l,n_1-1} \cdots L_{l,m},\end{gathered}$$ where $L_{l,m}$ and $M_{l,m}$ are given by and we have assumed the correspondence between $\alpha_l$ and $\beta_m$ and $x$ and $t$ is given by . By writing $A(x,t)$ in this way, we have chosen a path of initial conditions represents an L-shaped path. For the other operator, we have $$\begin{gathered}
R(x,t) = M_{l,m-n_2}^{-1} \cdots M_{l,m-1}^{-1},\end{gathered}$$ which brings us to the following result.
The $h$-Garnier system, as defined by the mapping , arises as a periodic $(N-2,2)$-reduction of .
We start by showing that $T_{1,2}$ arises as a periodic reduction. Up to relabeling for some fixed $l$, $m$, we let $$\begin{gathered}
b_{m+1} = a_1,\qquad b_m = a_2,\qquad a_{l+N-3} = a_3,\qquad \ldots, \qquad a_l = a_N,\end{gathered}$$ which is extended periodically with periods $2$ and $N-2$. For that same fixed $l$, $m$ we let $$\begin{gathered}
u_1 = w_{l+ N-2, m+2} - w_{l+ N-2, m+ 1}, \qquad u_2 = w_{l+ N-2,m+ 2} - w_{l+ N-2, m+ 1},\\
u_3 = w_{l+ N-2, m} - w_{l+ N-1, m}, \qquad \ldots, \qquad u_N = w_{l+ 1, m} - w_{l, m},\end{gathered}$$ which, due to in the case that $T$ is the identity (i.e., the periodic case), then these values are also extended periodically in the lattice. Furthermore, the constraint $$\begin{gathered}
u_1 + \dots + u_N = w_{l+N-2,m+2} - w_{l,m} = 0,\end{gathered}$$ by . Under this labelling of the initial conditions the operator $A(x,t)$ is of the form where each factor is of the form . Furthermore, due to the periodicity, we have that $$\begin{gathered}
R(x,t) = L(x,u_2,a_2+t+h)^{-1} L(x,u_1,a_1+t+h)^{-1} = L_2(x-h)^{-1} L_1(x-h)^{-1}.\end{gathered}$$ The compatibility, given by where $\tilde{A}(x,t) = A(x,t+h)$ coincides with the computations in Proposition \[tranha1a2\]. To complete the correspondence, one notices that the action of $S_n$ defined by is equivalent to using to define a different path of initial conditions.
Due to the equivalence between and the birational form, namely , also arises as a reduction, as do any of the special cases that have arisen in Section \[sec:special\].
The correspondence between the $q$-Garnier systems is made simple by taking twisted reductions of with $$\begin{gathered}
\label{parchoiceq}
x = \alpha_l, \qquad t = \alpha_l/\beta_m.\end{gathered}$$ We may construct the matrix connecting points connected via , which is given by $$\begin{gathered}
A'(x,t) = M_{l+n_1,m+n_2-1}\cdots M_{l+n_1,m} L_{l,n_1-1} \cdots L_{l,m},\end{gathered}$$ where the matrices, $L_{l,m}$ and $M_{l,m}$ are given by and we have assumed the correspondence between $x$ and $t$ and $\alpha_l$ and $\beta_m$ are given by . As noted in [@Ormerod2014a], this has a non-trivial effect on the solutions of the linear problem. This introduces a twist matrix whose effect is given by $$\begin{gathered}
T Y(x,t) = S Y(x,t) ,\end{gathered}$$ where $S$ is independent of the spectral parameter, $x$. It follows that the corresponding associated linear problem takes the form $$\begin{gathered}
Y(qx,t) = A(x,t) Y(x,t),\end{gathered}$$ where $$\begin{gathered}
A(x,t) = S^{-1} A'(x,t),\end{gathered}$$ where $A'(x,t)$ is as above. This is the same path, but for different matrices. Also, the matrix inducing the transformation $T_{1,2}$ is also changed by the twist to $$\begin{gathered}
R(x,t) = S M_{l,m-n_2}^{-1} \cdots M_{l,m-1}^{-1}\end{gathered}$$ and is also in terms of the corresponding $L_{l,m}$ and $M_{l,m}$. As in [@Ormerod2014a], we may calculate the twist via $$\begin{gathered}
\label{twistmatcalc}
T(R(x,t)) S = (T_{1,2} S) R(x,t),\end{gathered}$$ which only requires $R(x,t)$ to obtain.
The $q$-Garnier system, as defined by the mapping , arises as a twisted $(N-2,2)$-reduction of where the twist is an affine linear transformation, $$\begin{gathered}
\label{affine}
T(u) = \theta_2/\theta_1 u + b.\end{gathered}$$
The twist matrix in the case of is given by the diagonal matrix $$\begin{gathered}
S = \begin{pmatrix} \theta_1^{-1} & 0 \\ 0 & \theta_2^{-1} \end{pmatrix},\end{gathered}$$ which is a solution to , as guaranteed by the relation . The remainder of this proof follows in a similar manner to before; up to relabeling for some fixed $l$, $m$, we let $$\begin{gathered}
b_{m+1}= a_1,\qquad b_m = a_2,\qquad a_{l+N-3} = a_3,\qquad \ldots, \qquad a_l = a_N,\end{gathered}$$ which is extended periodically with periods $2$ and $N-2$. For that same fixed $l$, $m$ we let $$\begin{gathered}
u_1 = w_{l+ N-2, m+2} - w_{l+ N-2, m+ 1}, \qquad u_2 = w_{l+ N-2,m+ 2} - w_{l+ N-2, m+ 1},\\
u_3 = w_{l+ N-2, m} - w_{l+ N-1, m}, \qquad \ldots, \qquad u_N = w_{l+ 1, m} - w_{l, m}.\end{gathered}$$ in which case the form of $A(x,t)$ is precisely given by in which case the discrete isomonodromic deformations follow.
The discrete Painlevé equations of types $q$-$\mathrm{P}\big(A_3^{(1)}\big)$, $q$-$\mathrm{P}\big(A_2^{(1)}\big)$ and $q$-$\mathrm{P}\big(A_1^{(1)}\big)$ arise as special cases of twisted reductions of . Similarly, $d$-$\mathrm{P}\big(A_2^{(1)}\big)$ and $d$-$\mathrm{P}\big(A_1^{(1)}\big)$ arise as special cases of periodic reductions of .
In the cases of discrete Painlevé equations of types $q$-$\mathrm{P}\big(A_3^{(1)}\big)$, $q$-$\mathrm{P}\big(A_2^{(1)}\big)$ and $d$-$\mathrm{P}\big(A_2^{(1)}\big)$, by replacing $u$ with a difference in $w_{l,m}$-values, we have obtained explicit correspondences between the variables parameterizing the lattice equation and the Painlevé variables. However, in the cases of $d$-$\mathrm{P}\big(A_1^{(1)}\big)$ and $q$-$\mathrm{P}\big(A_1^{(1)}\big)$, we do not have this. We only know that the moduli spaces of the relevant difference equations have anticanonical divisors with two irreducible components [@Rains2013].
At this point, we are unsure as to how the symmetric cases may arise as twisted or periodic reductions. The problem with these cases is that the reductions described require that the spectral parameter enters via in the same way for each factor, whereas in the symmetric cases, the product form requires some involution of the spectral parameter. It may be the case that this is more natural from the perspective of reductions of cases higher than or .
Discussion
==========
By far the most interesting feature of the above Lax pairs is the existence of a symmetry, which manifests itself in the elliptic case quite naturally, which we will present separately. There are two interesting consequences we may derive from this work; the existence of symmetric Lax pairs for the lower discrete Painlevé equations and the existence of discrete symmetric Lax pairs possessing continuous isomonodromic deformations, which we shall present in another paper.
From the above work, we have shown that the $d$-$\mathrm{P}\big(A_1^{(1)}\big)$ and $q$-$\mathrm{P}\big(A_1^{(1)}\big)$ arise as reductions of partial difference equations, however, it is unclear at this point how to make the full correspondence between the $u$-variables and the Painlevé variables. We have also shown that the $q$-Garnier system defined by Sakai in [@Sakai:Garnier] arises as a reduction, but the problem of finding reductions for the symmetric Garnier systems is not clear from this work.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The work of EMR was partially supported by the National Science Foundation under the grant DMS-1500806.
[99]{}
Adler V.E., Cutting of polygons, [*Funct. Anal. Appl.*](http://dx.doi.org/10.1007/BF01085984) **27** (1993), 141–143.
Adler V.E., Bobenko A.I., Suris Yu.B., Classification of integrable equations on quad-graphs. [T]{}he consistency approach, [*Comm. Math. Phys.*](http://dx.doi.org/10.1007/s00220-002-0762-8) **233** (2003), 513–543, [nlin.SI/0202024](http://arxiv.org/abs/nlin.SI/0202024).
Adler V.E., Bobenko A.I., Suris Yu.B., Geometry of [Y]{}ang–[B]{}axter maps: pencils of conics and quadrirational mappings, *Comm. Anal. Geom.* **12** (2004), 967–1007, [math.QA/0307009](http://arxiv.org/abs/math.QA/0307009).
Adler V.E., Bobenko A.I., Suris Yu.B., Discrete nonlinear hyperbolic equations: classification of integrable cases, [*Funct. Anal. Appl.*](http://dx.doi.org/10.1007/s10688-009-0002-5) **43** (2009), 3–17, [arXiv:0705.1663](http://arxiv.org/abs/0705.1663).
Arinkin D., Borodin A., Moduli spaces of [$d$]{}-connections and difference [P]{}ainlevé equations, [*Duke Math. J.*](http://dx.doi.org/10.1215/S0012-7094-06-13433-6) **134** (2006), 515–556, [math.AG/0411584](http://arxiv.org/abs/math.AG/0411584).
Birkhoff G.D., General theory of linear difference equations, [*Trans. Amer. Math. Soc.*](http://dx.doi.org/10.2307/1988577) **12** (1911), 243–284.
Birkhoff G.D., The generalized [R]{}iemann problem for linear differential equations and the allied problems for linear difference and $q$-difference equations, [*Proc. Amer. Acad. Arts Sci.*](http://dx.doi.org/10.2307/20025482) **49** (1913), 512–568.
Birkhoff G.D., Guenther P.E., Note on a canonical form for the linear [$q$]{}-difference system, [*Proc. Nat. Acad. Sci. USA*](http://dx.doi.org/10.1073/pnas.27.4.218) **27** (1941), 218–222.
Boalch P., Some explicit solutions to the [R]{}iemann–[H]{}ilbert problem, in Differential Equations and Quantum Groups, [*IRMA Lect. Math. Theor. Phys.*](http://dx.doi.org/10.4171/020-1/6), Vol. 9, Eur. Math. Soc., Zürich, 2007, 85–112, [math.DG/0501464](http://arxiv.org/abs/math.DG/0501464).
Borodin A., Discrete gap probabilities and discrete [P]{}ainlevé equations, [*Duke Math. J.*](http://dx.doi.org/10.1215/S0012-7094-03-11734-2) **117** (2003), 489–542, [math-ph/0111008](http://arxiv.org/abs/math-ph/0111008).
Borodin A., Isomonodromy transformations of linear systems of difference equations, [*Ann. of Math.*](http://dx.doi.org/10.4007/annals.2004.160.1141) **160** (2004), 1141–1182, [math.CA/0209144](http://arxiv.org/abs/math.CA/0209144).
Deligne P., Milne J.S., Tannakian categories, in Hodge Cycles, Motives, and Shimura Varieties, *Lecture Notes in Math.*, Vol. 900, Springer, 101–228.
Dzhamay A., On the [L]{}agrangian structure of the discrete isospectral and isomonodromic transformations, [*Int. Math. Res. Not.*](http://dx.doi.org/10.1093/imrn/rnn102) **2008** (2008), Art. ID rnn 102, 22 pages, [arXiv:0711.0570](http://arxiv.org/abs/0711.0570).
Dzhamay A., Sakai H., Takenawa T., Discrete Schlesinger transformations, their Hamiltonian Formulation, and difference Painlevé equations, [arXiv:1302.2972](http://arxiv.org/abs/1302.2972).
Etingof P.I., Galois groups and connection matrices of [$q$]{}-difference equations, [*Electron. Res. Announc. Amer. Math. Soc.*](http://dx.doi.org/10.1090/S1079-6762-95-01001-8) **1** (1995), 1–9.
Flaschka H., Newell A.C., Monodromy- and spectrum-preserving deformations. [I]{}, [*Comm. Math. Phys.*](http://dx.doi.org/10.1007/BF01197110) **76** (1980), 65–116.
Forster O., Riemann surfaces, Mir, Moscow, 1980.
Fuchs R., Über lineare homogene [D]{}ifferentialgleichungen zweiter [O]{}rdnung mit drei im [E]{}ndlichen gelegenen wesentlich singulären [S]{}tellen, [*Math. Ann.*](http://dx.doi.org/10.1007/BF01449199) **63** (1907), 301–321.
Fuchs R., Über lineare homogene [D]{}ifferentialgleichungen zweiter [O]{}rdnung mit drei im [E]{}ndlichen gelegenen wesentlich singulären [S]{}tellen, [*Math. Ann.*](http://dx.doi.org/10.1007/BF01564511) **70** (1911), 525–549.
Garnier R., Sur des équations différentielles du troisième ordre dont l’intégrale générale est uniforme et sur une classe d’équations nouvelles d’ordre supérieur dont l’intégrale générale a ses points critiques fixes, *Ann. Sci. École Norm. Sup. (3)* **29** (1912), 1–126.
Gasper G., Rahman M., Basic hypergeometric series, [*Encyclopedia of Mathematics and its Applications*](http://dx.doi.org/10.1017/CBO9780511526251), Vol. 35, 2nd ed., Cambridge University Press, Cambridge, 2004.
Jimbo M., Miwa T., Ueno K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. [I]{}. [G]{}eneral theory and [$\tau $]{}-function, [*Phys. D*](http://dx.doi.org/10.1016/0167-2789(81)90013-0) **2** (1981), 306–352.
Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. [II]{}, [*Phys. D*](http://dx.doi.org/10.1016/0167-2789(81)90021-X) **2** (1981), 407–448.
Jimbo M., Sakai H., A [$q$]{}-analog of the sixth [P]{}ainlevé equation, [*Lett. Math. Phys.*](http://dx.doi.org/10.1007/BF00398316) **38** (1996), 145–154.
Joshi N., Nakazono N., Shi Y., Geometric reductions of [ABS]{} equations on an [$n$]{}-cube to discrete [P]{}ainlevé systems, [*J. Phys. A: Math. Theor.*](http://dx.doi.org/10.1088/1751-8113/47/50/505201) **47** (2014), 505201, 16 pages, [arXiv:1402.6084](http://arxiv.org/abs/1402.6084).
Kajiwara K., Noumi M., Yamada Y., Discrete dynamical systems with [$W\big(A_{m-1}^{(1)}\times A_{n-1}^{(1)}\big)$]{} symmetry, [*Lett. Math. Phys.*](http://dx.doi.org/10.1023/A:1016298925276) **60** (2002), 211–219, [nlin.SI/0106029](http://arxiv.org/abs/nlin.SI/0106029).
Kajiwara K., Noumi M., Yamada Y., Geometric aspects of Painlevé equations, [arXiv:1509.08186](http://arxiv.org/abs/1509.08186).
Knizel A., Moduli spaces of $q$-connections and gap probabilities, [arXiv:1506.06718](http://arxiv.org/abs/1506.06718).
Kruskal M.D., Tamizhmani K.M., Grammaticos B., Ramani A., Asymmetric discrete [P]{}ainlevé equations, [*Regul. Chaotic Dyn.*](http://dx.doi.org/10.1070/rd2000v005n03ABEH000149) **5** (2000), 273–280.
Nijhoff F., Capel H., The discrete [K]{}orteweg–de [V]{}ries equation, [*Acta Appl. Math.*](http://dx.doi.org/10.1007/BF00994631) **39** (1995), 133–158.
Nijhoff F.W., Quispel G.R.W., Capel H.W., Direct linearization of nonlinear difference-difference equations, [*Phys. Lett. A*](http://dx.doi.org/10.1016/0375-9601(83)90192-5) **97** (1983), 125–128.
Nijhoff F.W., Walker A.J., The discrete and continuous [P]{}ainlevé [VI]{} hierarchy and the [G]{}arnier systems, [*Glasg. Math. J.*](http://dx.doi.org/10.1017/S0017089501000106) **43A** (2001), 109–123, [nlin.SI/0001054](http://arxiv.org/abs/nlin.SI/0001054).
Ormerod C.M., A study of the associated linear problem for $q$-${\rm P}_{\rm
V}$, [*J. Phys. A: Math. Theor.*](http://dx.doi.org/10.1088/1751-8113/44/2/025201) **44** (2011), 025201, 26 pages, [arXiv:0911.5552](http://arxiv.org/abs/0911.5552).
Ormerod C.M., The lattice structure of connection preserving deformations for [$q$]{}-[P]{}ainlevé equations [I]{}, [*SIGMA*](http://dx.doi.org/10.3842/SIGMA.2011.045) **7** (2011), 045, 22 pages, [arXiv:1010.3036](http://arxiv.org/abs/1010.3036).
Ormerod C.M., Symmetries in connection preserving deformations, [*SIGMA*](http://dx.doi.org/10.3842/SIGMA.2011.049) **7** (2011), 049, 13 pages, [arXiv:1101.5422](http://arxiv.org/abs/1101.5422).
Ormerod C.M., Reductions of lattice m[K]{}d[V]{} to [$q$]{}-[${\rm P}_{\rm VI}$]{}, [*Phys. Lett. A*](http://dx.doi.org/10.1016/j.physleta.2012.09.008) **376** (2012), 2855–2859, [arXiv:1112.2419](http://arxiv.org/abs/1112.2419).
Ormerod C.M., Symmetries and special solutions of reductions of the lattice potential [K]{}d[V]{} equation, [*SIGMA*](http://dx.doi.org/10.3842/SIGMA.2014.002) **10** (2014), 002, 19 pages, [arXiv:1308.4233](http://arxiv.org/abs/1308.4233).
Ormerod C.M., van der Kamp P.H., Hietarinta J., Quispel G.R.W., Twisted reductions of integrable lattice equations, and their [L]{}ax representations, [*Nonlinearity*](http://dx.doi.org/10.1088/0951-7715/27/6/1367) **27** (2014), 1367–1390, [arXiv:1307.5208](http://arxiv.org/abs/1307.5208).
Ormerod C.M., van der Kamp P.H., Quispel G.R.W., Discrete [P]{}ainlevé equations and their [L]{}ax pairs as reductions of integrable lattice equations, [*J. Phys. A: Math. Theor.*](http://dx.doi.org/10.1088/1751-8113/46/9/095204) **46** (2013), 095204, 22 pages, [arXiv:1209.4721](http://arxiv.org/abs/1209.4721).
Papageorgiou V.G., Nijhoff F.W., Grammaticos B., Ramani A., Isomonodromic deformation problems for discrete analogues of [P]{}ainlevé equations, [*Phys. Lett. A*](http://dx.doi.org/10.1016/0375-9601(92)90905-2) **164** (1992), 57–64.
Papageorgiou V.G., Tongas A.G., Veselov A.P., Yang–[B]{}axter maps and symmetries of integrable equations on quad-graphs, [*J. Math. Phys.*](http://dx.doi.org/10.1063/1.2227641) **47** (2006), 083502, 16 pages, [math.QA/0605206](http://arxiv.org/abs/math.QA/0605206).
Praagman C., Fundamental solutions for meromorphic linear difference equations in the complex plane, and related problems, [*J. Reine Angew. Math.*](http://dx.doi.org/10.1515/crll.1986.369.101) **369** (1986), 101–109.
Rains E.M., An isomonodromy interpretation of the hypergeometric solution of the elliptic [P]{}ainlevé equation (and generalizations), [*SIGMA*](http://dx.doi.org/10.3842/SIGMA.2011.088) **7** (2011), 088, 24 pages, [arXiv:0807.0258](http://arxiv.org/abs/0807.0258).
Rains E.M., Generalized [H]{}itchin systems on rational surfaces, [arXiv:1307.4033](http://arxiv.org/abs/1307.4033).
Ramis J.-P., Martinet J., Théorie de [G]{}alois différentielle et resommation, in Computer Algebra and Differential Equations, *Comput. Math. Appl.*, Academic Press, London, 1990, 117–214.
Sakai H., Rational surfaces associated with affine root systems and geometry of the [P]{}ainlevé equations, [*Comm. Math. Phys.*](http://dx.doi.org/10.1007/s002200100446) **220** (2001), 165–229.
Sakai H., A [$q$]{}-analog of the [G]{}arnier system, [*Funkcial. Ekvac.*](http://dx.doi.org/10.1619/fesi.48.273) **48** (2005), 273–297.
Sauloy J., Galois theory of [F]{}uchsian [$q$]{}-difference equations, [*Ann. Sci. École Norm. Sup. (4)*](http://dx.doi.org/10.1016/j.ansens.2002.10.001) **36** (2003), 925–968, [math.QA/0210221](http://arxiv.org/abs/math.QA/0210221).
Suris Yu.B., Veselov A.P., Lax matrices for [Y]{}ang–[B]{}axter maps, [*J. Nonlinear Math. Phys.*](http://dx.doi.org/10.2991/jnmp.2003.10.s2.18) **10** (2003), suppl. 2, 223–230, [math.QA/0304122](http://arxiv.org/abs/math.QA/0304122).
van der Kamp P.H., Initial value problems for lattice equations, [*J. Phys. A: Math. Theor.*](http://dx.doi.org/10.1088/1751-8113/42/40/404019) **42** (2009), 404019, 16 pages.
van der Put M., Reversat M., Galois theory of [$q$]{}-difference equations, [*Ann. Fac. Sci. Toulouse Math. (6)*](http://dx.doi.org/10.5802/afst.1164) **16** (2007), 665–718, [math.QA/0507098](http://arxiv.org/abs/math.QA/0507098).
van der Put M., Singer M.F., Galois theory of linear differential equations, [*Grundlehren der Mathematischen Wissenschaften*](http://dx.doi.org/10.1007/978-3-642-55750-7), Vol. 328, Springer-Verlag, Berlin, 2003.
Veselov A.P., Yang–[B]{}axter maps and integrable dynamics, [*Phys. Lett. A*](http://dx.doi.org/10.1016/S0375-9601(03)00915-0) **314** (2003), 214–221, [math.QA/0205335](http://arxiv.org/abs/math.QA/0205335).
Witte N.S., Ormerod C.M., Construction of a [L]{}ax pair for the [$E_6^{(1)}$]{} [$q$]{}-[P]{}ainlevé system, [*SIGMA*](http://dx.doi.org/10.3842/SIGMA.2012.097) **8** (2012), 097, 27 pages, [arXiv:1207.0041](http://arxiv.org/abs/1207.0041).
Yamada Y., A [L]{}ax formalism for the elliptic difference [P]{}ainlevé equation, [*SIGMA*](http://dx.doi.org/10.3842/SIGMA.2009.042) **5** (2009), 042, 15 pages, [arXiv:0811.1796](http://arxiv.org/abs/0811.1796).
Yamada Y., Lax formalism for [$q$]{}-[P]{}ainlevé equations with affine [W]{}eyl group symmetry of type [$E^{(1)}_n$]{}, [*Int. Math. Res. Not.*](http://dx.doi.org/10.1093/imrn/rnq232) **2011** (2011), 3823–3838, [arXiv:1004.1687](http://arxiv.org/abs/1004.1687).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Quantum systems of interest are typically coupled to several quantum channels (more generally environments). In this paper, we develop an exact stochastic Schrödinger equation for an open quantum system coupled to a hybrid environment containing both bosonic and fermionic particles. Such a stochastic differential equation may be obtained directly from a microscopic model through employing a classical complex Gaussian noise and a non-commutative fermionic noise to simulate the hybrid bath. As an immediate application of our developed stochastic approach, we show that the evolution of the reduced density matrix can be derived by taking the average over both the bosonic noise and the fermionic noise. Three specific examples are given in this paper to illustrate that the hybrid quantum trajectory is fully consistent with the standard quantum mechanics. Our examples also shed new light on the special features exhibited by the fermionic bath and bosnoic bath.'
address:
- '$^{1}$Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030, USA'
- '$^{2}$Beijing Computational Science Research Center, Beijing 100094, China'
- '$^{3}$Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China'
- '$^{4}$School of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China'
author:
- 'Xinyu Zhao$^{1}$'
- 'Wufu Shi$^{1}$'
- 'J. Q. You$^{2,3}$'
- 'Ting Yu$^{1,2,4}$'
title: 'Non-Markovian Dynamics of Quantum Open Systems Embedded in a Hybrid Environment'
---
Open Quantum System,Non-Markovian,Stochastic
Introduction
============
A quantum system, when it is not isolated, can be in contact with several types of environments. Physically, such open quantum systems like an electron relaxation in a solid may interact with a bosonic system and be coupled to some fermionic systems at the same time [@hybrid1; @hybrid2; @hybrid3]. In a similar manner, one can recognize that an atomic system of interest can be coupled to both classical laser fields and quantized radiation fields [@Gardiner1]. Therefore, a hybrid quantum open system theory is potentially useful since it provides a systematic approach to dealing with the dynamics of an open system coupled to multiple environments in a direct manner. Fundamentally, the dynamics of open quantum systems embedded in one or more environments has attracted the wide-spread interest in recent years [@Unrhu; @Breuer; @Xinyu2011; @Nlevel]. On the one hand, the temporal behaviours of quantum open systems are essential for understanding many fundamental issues of quantum theory such as quantum dissipation and decoherence [@Deco1; @Deco2; @Deco3; @Deco4; @Yu-Eberly04; @Nqubit; @Ncavity]. On the other hand, many novel applications based on quantum devices also require a better understanding on the interaction between the quantum system of interest and its environment in order to manipulate and control the system’s dynamics [@DD; @FBC]. Although a realistic environment can be very complicated, it is typically composed of bosons and fermions. For a bosonic bath, a set of powerful tools have been developed to investigate the open system dynamics, such as path integral approach [@Feynman-Vernon; @ZhangWM-2cav], master equation approach [@H-P-Z; @Hu1; @Hu2; @Leggett], and Markov and non-Markovian quantum trajectory approach [@Gisin-Percival; @Dalibardetal; @QSD; @Yu1999; @YuQBM]. For fermionic bath, similar tools have also been developed, including scattering theory [@scattering], non-equilibrium Green’s function approach [@NEGF], and fermionic path integral [@ZhangDQD; @Zhang2012PRL]. Notably, the fermionic quantum state diffusion equations have been developed recently [@ZhaoFB; @ShiFB; @ChenFB]. Although bosonic and fermionic formalisms share many similarities, a unified description of both types of baths is still useful for the purpose of direct applications.
![(Color online) Schematic diagram of a quantum system coupled to a fermionic bath and a bosonic bath simultaneously.[]{data-label="Fig1"}](Sketch.eps){width="1\columnwidth"}
In this paper, we will consider a hybrid case that the environment is composed of both bosons and fermions as shown in Fig. \[Fig1\]. Of many applications is the primary example of quantum dot model where the quantum dots may interact with two fermionic reservoirs (source and drain) and other agents such as a phonon bath. In this context, the dynamics of the quantum dot system is determined by both the fermionic reservoirs and the bosonic bath. The theoretical approach to be developed in this paper will be capable of taking into account of the environmental effects arising from both types of environments. We shall show that the non-Markovian quantum state diffusion (NMQSD) approach is applicable to this extended case. In this method, the dynamic evolution of the open system is decomposed into an ensemble of quantum trajectories of pure states, and the reduced density matrix is described by the statistical average over these generated trajectories. It is worth noting the fact that this method is well developed for both bosonic bath and fermionic bath, and the method becomes one of the most hopeful candidates to solve the hybrid bath problem. For the case of bosonic bath, several interesting physical systems have been studied in the past fifteen years [@QSD; @Yu1999; @YuQBM; @Yu-FiniteT; @Jing-Yu2010]. As a fundamentally theoretical study, it has been developed from solving single-particle system to solving many-body systems [@Nlevel; @Nqubit; @Ncavity]. Furthermore, as a computing tool in real applications [@QSDHierachy; @Lam], the NMQSD approach also showed its potential value in many interesting problems including precision quantum measurement [@Chen], quantum control dynamics [@JingPQ], and quantum biology [@Eisfeld2011]. In either the bosonic or fermionic NMQSD approach, the central idea is to encode all the influences of the environments on the system into a set of classical random variables forming a stochastic process. Taking the statistic average over the stochastic variables is equivalent to taking the partial trace over the environment to obtain the reduced density matrix. The difference between bosonic and fermionic approaches is that a bosonic bath can be represented by a complex Gaussian noise, while the fermionic bath is simulated by a non-commutative Grassmann noise. For a hybrid bath, an exact dynamical equation will typically contain both the complex Gaussian noise and Grassmann Gaussian noise.
The paper is organized as follows. In Sec. \[sec:II\], we describe a model of hybrid bath and point out some new features arising from the hybrid bath case where the fermionic bath is assumed to commutes or anti-commutes with the system of interest. In Sec. \[sec:III\], we analyze the commutative case with two examples. First, a general NMQSD equation and the corresponding master equation are derived. Then, we examine our general formalism by exactly solving a simple example of the single qubit dissipative model. It is shown that, as expected, the result predicted by the newly developed NMQSD approach is identical to the solution based on the ordinary quantum mechanics. Moreover, the two-qubit dissipative model is also studied for the hybrid bath case. Sec. \[sec:IV\] is devoted to investigating the anti-commutative case. Again, a general NMQSD approach can be developed. As an important example, we show how to use the new approach to study the Anderson model in the hybrid bath context. We identify the different impacts of fermionic bath and bosonic bath on the system dynamics. We conclude in Sec. \[sec:V\].
\[sec:II\] Two Types of Hybrid Bath: Commutative and Anti-commutative
=====================================================================
An open system embedded in a hybrid bath may be described by the following Hamiltonian $$H_{tot}=H_{S}+H_{FB}+H_{BB}+H_{FI}+H_{BI},\label{Hybrid}$$ where $H_{S}$ describes the Hamiltonian of the system, $$H_{BB}=\sum_{r}\Omega_{r}b_{r}^{\dagger}b_{r},\;
H_{FB}=\sum_{k}\epsilon_{k}c_{k}^{\dagger}c_{k},\label{HB}$$ are the bosonic bath and the fermionic bath respectively, where “$b_{r}$” and “$c_{k}$” are the annihilation operators for a single mode of bosonic bath and fermionic bath respectively. The interaction between system and two baths is given by $$H_{BI}=\sum_{r}\lambda_{r}b_{r}^{\dagger}L_{b}+{\rm H.c.},\;
H_{FI}=\sum_{k}\mu_{k}c_{k}^{\dagger}L_{f}+{\rm H.c.},\label{HInt}$$ where $L_{b}$ and $L_{f}$ are the bosonic and fermionic coupling operators. Typically, the bosonic bath commutes with both the fermionic bath and the system no mater the system is composed of fermions or bosons. However, the commutation relation between the system and the fermionic bath could fall into two different categories. Depending on the commutation relation between the system and the fermionic bath (commutative or anti-commutative), the physical model for this Hamiltonian and the technique of solving this model are totally different. Therefore, we need to develop two parallel schemes to deal with these two different cases.
Case 1: System commutes with fermionic bath. {#case-1-system-commutes-with-fermionic-bath. .unnumbered}
--------------------------------------------
The Hamiltonian (\[Hybrid\]) for this case typically describes an effective fermionic bath. For example, a spin-chain bath can be transformed into an effective fermionic bath by using the Jordan-Wigner transformation and the Fourier transform [@ZhaoFB; @Barouch]. After the transformation, t hese effective fermions satisfying fermionic commutation relations. However, since the original spins living in a Hilbert space separated from the system degree of freedom, they all commute with the system. Therefore, in the case of effective fermionic bath, the creation and annihilation operators $c_{k}^{\dagger}$ and $c_{k}$ will commute with any operators living in the Hilbert space of the system $H_{S}$.
Case 2: System anti-commutes with fermionic bath. {#case-2-system-anti-commutes-with-fermionic-bath. .unnumbered}
-------------------------------------------------
The anti-commutative case naturally arises when both the system and the bath are composed of a set of electrons. A well-known example is a quantum dot connected to a source and a drain reservoirs, where the system Hamiltonian of this model is $H_{S}=\omega_{d}d^{\dagger}d$. Obviously, the annihilation operator “$d$” for the system and the operators “$c_{k}$” for the bath satisfy the anti-commutation relation $\{d,c_{k}\}=0$ and $\{d^\dagger,c_{k}=0\}$.
We will develop two different schemes in the following sections for the two cases described above. Several specific examples are provided.
\[sec:III\] Commutative Case
============================
\[sub:IIIA\]General Stochastic Schrödinger Equation
---------------------------------------------------
First, we consider the commutative case. In this case, the total Hamiltonian can be transformed into the interaction picture as $$H_{tot}^{int}=H_{S}+(\sum_{k}\mu_{k}c_{k}^{\dagger}L_{f}e^{i\epsilon_{k}t}+\sum_{r}\lambda_{r}b_{r}^{\dagger}L_{b}e^{i\Omega_{r}t}+\mathrm{H.c.}).$$ By introducing multi-mode bosonic coherent states and fermionic coherent states $$|z\rangle=\prod_{r}\exp\left\{ z_{r}b_{r}^{\dagger}\right\} |0\rangle,$$ $$|\xi\rangle=\prod_{k}\left(1-\xi_{k}c_{k}^{\dagger}\right)|0\rangle,$$ the stochastic state vector can be defined as $$|\psi_{t}(z^{*},\xi^{*})\rangle=\langle z^{*},\xi^{*}|\psi_{tot}(t)\rangle.$$ Throughout the paper, we will use the short-notation $|\psi_{t}\rangle\equiv|\psi_{t}(z^{*},\xi^{*})\rangle$ if no confusion arises. Because of the different properties of bosonic coherent states and fermionic coherent states [@ZhangRMP], the noise variables introduced here are rather different. For bosonic coherent states, $z_{r}$ is an ordinary complex variable, while for fermionic coherent states, $\xi_{k}$ is a Grassmann variable satisfying anti-commutative relations $\{\xi_{i},\xi_{j}\}=0$. Starting with the Schrödinger equation for the total system, one can derive the dynamic equation for the stochastic state vector, $$\begin{aligned}
\frac{\partial}{\partial t}|\psi_{t}\rangle & = & -i\langle z^{*},\xi^{*}|H_{tot}^{int}(t)|\psi_{tot}(t)\rangle\nonumber \\
& = & [-iH_{S}+L_{f}\xi_{t}^{\ast}-L_{f}^{\dagger}\int dsK_{f}(t,s)\frac{\delta_{l}}{\delta\xi_{s}^{\ast}}\nonumber \\
& & +L_{b}z_{t}^{\ast}-L_{b}^{\dagger}\int dsK_{b}(t,s)\frac{\delta}{\delta z_{s}^{\ast}}]|\psi_{t}\rangle,\label{QSDC}\end{aligned}$$ where $K_{b}(t,s)=\sum_{r}\lambda_{r}^{2}e^{-i\Omega_{r}(t-s)}$ and $K_{f}(t,s)=\sum_{k}\mu_{k}^{2}e^{-i\epsilon_{k}(t-s)}$ are the correlation functions for the bosonic bath and the fermionic bath, respectively. The equation (\[QSDC\]) is the fundamental equation governing the dynamics of the stochastic state vector $|\psi_{t}\rangle$. Note that this equation contains two types of noises as $$z_{t}^{*}=-i\sum_{r}z_{r}^{*}e^{i\Omega_{r}t},$$ $$\xi_{t}^{*}=-i\sum_{k}\xi_{k}^{*}e^{i\epsilon_{k}t},$$ where $z_{t}^{*}$ is a complex Gaussian process and $\xi_{t}^{*}$ is a Grassmann stochastic process. They satisfy the following statistical relations $$\langle z_{t}\rangle_{b}=\langle z_{t}^{*}\rangle_{b}=0,\;\langle z_{t}z_{s}^{*}\rangle_{b}=K_{b}(t,s),$$ $$\langle\xi_{t}\rangle_{f}=\langle\xi_{t}^{*}\rangle_{f}=0,\;\langle\xi_{t}\xi_{s}^{*}\rangle_{f}=K_{f}(t,s).$$ The statistical averages over both the complex noise and Grassmann noises are defined as $\langle\cdot\rangle_{b}=\int\prod_{r}\frac{1}{\pi}e^{-|z_{r}|^{2}}dz_{r}^{2}[\cdot]$ and $\langle\cdot\rangle_{f}=\int\prod_{k}d\xi_{k}^{*}d\xi_{k}e^{-\xi_{k}^{*}\xi_{k}}[\cdot]$, respectively.
In order to solve Eq. (\[QSDC\]), one has to deal with the functional derivatives. Similar to the technique used in Refs. [@Yu1999; @ShiFB; @ZhaoFB; @ChenFB], we can always replace these functional derivatives by some time-dependent operators $O$ and $Q$ as $$\frac{\delta}{\delta z_{s}^{*}}|\psi_{t}\rangle=O(t,s,z^{*},\xi^{*})|\psi_{t}\rangle,$$ $$\frac{\delta}{\delta\xi_{s}^{*}}|\psi_{t}\rangle=Q(t,s,\xi^{*},z^{*})|\psi_{t}\rangle.$$ Then, the NMQSD equation (\[QSDC\]) can be written in a more compact form, $$\frac{\partial}{\partial t}|\psi_{t}\rangle=[-iH_{s}+L_{f}\xi_{t}^{\ast}-L_{f}^{\dagger}\bar{Q}+L_{b}z_{t}^{*}-L_{b}^{\dagger}\bar{O}]|\psi_{t}\rangle,\label{QSDC2}$$ where $\bar{O}(t,z^{*},\xi^{*})=\int_{0}^{t}K_{b}(t,s)O(t,s,z^{*},\xi^{*})ds$, $\bar{Q}(t,z^{*},\xi^{*})=\int_{0}^{t}K_{f}(t,s)Q(t,s,z^{*},\xi^{*})ds$. If these time-dependent operators $O$ and $Q$ can be determined, the NMQSD equation will take a time-local form and the equation can be solved numerically in a more straightforward way. The consistency condition may be employed, $$\frac{\partial}{\partial t}\frac{\delta}{\delta z_{s}^{*}}|\psi_{t}\rangle=\frac{\delta}{\delta z_{s}^{*}}\frac{\partial}{\partial t}|\psi_{t}\rangle,$$ $$\frac{\partial}{\partial t}\frac{\delta}{\delta\xi_{s}^{*}}|\psi_{t}\rangle=\frac{\delta}{\delta\xi_{s}^{*}}\frac{\partial}{\partial t}|\psi_{t}\rangle,$$ and gives rise to $$\begin{aligned}
\frac{\partial}{\partial t}O & = & [-iH_{s}+L_{f}\xi_{t}^{\ast}-L_{f}^{\dagger}\bar{Q}+L_{b}z_{t}^{*}-L_{b}^{\dagger}\bar{O},O]\nonumber \\
& & -L_{b}^{\dagger}\frac{\delta}{\delta z_{s}^{*}}\bar{O}-L_{f}^{\dagger}\frac{\delta}{\delta z_{s}^{*}}\bar{Q}, \label{EqO}\end{aligned}$$
$$\begin{aligned}
\frac{\partial}{\partial t}Q & = & [-iH_{s},Q]-\{L_{f}\xi_{t}^{*},Q\}+[L_{b}z_{t}^{*},Q]\nonumber \\
& & -L_{f}^{\dagger}\bar{Q}(-\xi^{*})Q+QL_{f}^{\dagger}\bar{Q}-L_{b}^{\dagger}\bar{O}(-\xi^{*})Q\nonumber \\
& & +QL_{b}^{\dagger}\bar{O}-L_{b}^{\dagger}\frac{\delta_{l}}{\delta\xi_{s}^{\ast}}\bar{O}-L_{f}^{\dagger}\frac{\delta_{l}}{\delta\xi_{s}^{*}}\bar{Q}, \label{EqQ}\end{aligned}$$ with the initial conditions $$O(t,t,z^{*},\xi^{*})=L_{b},$$ $$Q(t,t,z^{*}\xi^{*})=L_{f}.$$ Given these conditions, the $O$ and $Q$ operators can be fully determined, as a result, the NMQSD equation (\[QSDC2\]) becomes more useful for the analytical purpose and numerical simulations. However, A single solution of the NMQSD equation can not fully describe the dynamic evolution of the system. Actually, it only gives one possible realization of many possible solutions of the NMQSD equation corresponding a specific sample path taken by the stochastic process $z_{t}^{*}$ and $\xi_{t}^{*}$. In order to obtain the full picture of the evolution of the system, we need to reproduce the reduced density matrix from the stochastic state vector $|\psi_{t}\rangle$ as $$\rho(t)=\langle\langle P_{t}\rangle_{f}\rangle_{b},\label{ReproduceRho}$$ where $P_{t}\equiv|\psi_{t}(z^{*},\xi^{*})\rangle\langle\psi_{t}(z^{*},-\xi^{*})|$ is the stochastic density operator. Given the relation Eq. (\[ReproduceRho\]), the physical meaning of the NMQSD equation becomes clear. By choosing a random realization of the noises $z_{t}^{*}$ and $\xi_{t}^{*}$ (reflecting the states of the environment), the evolution of the reduced density matrix is decomposed into many pure-state quantum trajectories $|\psi_{t}\rangle$. However, taking the statistical average over all of these trajectories, the reduced density matrix is reproduced. Therefore, the complicated properties of the environment are all encoded into noise functions $z_{t}^{*}$ and $\xi_{t}^{*}$, so that tracing out the environment is equivalent to taking average over all the realizations of the noises. Based on the relation (\[ReproduceRho\]), the master equation can be derived as $$\begin{aligned}
\frac{d}{dt}\rho & = & -i[H_{S},\rho]+[L_{b},\langle\langle P_{t}\bar{O}^{\dagger}\rangle_{f}\rangle_{b}]+[\langle\langle\bar{O}P_{t}\rangle_{f}\rangle_{b},L_{b}^{\dagger}]\nonumber \\
& & +[L_{f},\langle\langle P_{t}\bar{Q}^{\dagger}(-\xi)\rangle_{f}\rangle_{b}]+[\langle\langle\bar{Q}P_{t}\rangle_{f}\rangle_{b},L_{f}^{\dagger}], \label{MEQ1}\end{aligned}$$ where the Novikov theorem for fermionic case [@ZhaoFB] and bosonic case [@Yu1999] are used in the derivation. Although taking the statistical averages $\langle\cdot\rangle_{b}$ and $\langle\cdot\rangle_{f}$ are not simple in the general case, there is still a special case. When the operators $O$ and $Q$ are noise-independent, the master equation can be written in a simpler form as $$\frac{d}{dt}\rho=-i[H_{S},\rho]+\{[\bar{Q}\rho,L_{f}^{\dagger}]+[\bar{O}\rho,L_{b}^{\dagger}]+\mathrm{H.c.}\},\label{eq:MEQ2}$$ Actually, this special case is very common in many interesting models [@QSD; @ZhaoFB; @Yu1999; @Ncavity] in which the exact $O$ or $Q$ operators just contain no noises. Moreover, in general case, we can still expand $O$ and $Q$ into functional series and only taking the first term (with zeroth order of noise variables) of the expansions as $O(t,s,z^{*},\xi^{*})\approx O^{(0)}(t,s)$ and $Q(t,s,z^{*},\xi^{*})\approx Q^{(0)}(t,s)$. This approximation is called the zeroth order approximation [@Yu1999]. The validity and accuracy of this approximation is analyzed in Ref. [@Xu2014]. Actually, the accuracy is proved to be much better than the weak coupling approximation.
Example 1: Two qubits in a hybrid bath
--------------------------------------
In order to illustrate the NMQSD approach for a hybrid bath we discussed above, we will solve some particular examples in details. In the first example, we will consider a two-qubit system interacting with two dissipative baths, one is bosonic and the other fermionic. From this example, we show that dynamic equation for a hybrid bath is not the simple combination of a fermionic bath and a bosonic bath. The cross-terms in $O$ and $Q$ operators reflect the correlation between two baths through interaction with the system of interest. In the general model described by Eqs. (\[Hybrid\]-\[HInt\]), the two qubits example is the special case that $$H_{S}=\frac{\omega}{2}(\sigma_{z}^{A}+\sigma_{z}^{B}),$$ $$L_{b}=L_{f}=\sigma_{-}^{A}+\kappa_{B}\sigma_{-}^{B}.\label{L2qu}$$ where $\kappa_{B}$ is a parameter describing the coupling strength between the second qubit and the baths. We will first investigate the case with $\kappa_{B}=1$ in this subsection. A special case with $\kappa_{B}=0$ will be considered later, which means the second qubit evolves independently from the other part of the total system and the model reduces to a single qubit case. Given this specific model, the NMQSD equation can be formally written as $$\begin{aligned}
\frac{\partial}{\partial t}|\psi_{t}\rangle & = & [-i\frac{\omega}{2}(\sigma_{z}^{A}+\sigma_{z}^{B})+(\sigma_{-}^{A}+\sigma_{-}^{B})(\xi_{t}^{\ast}+z_{t}^{*})\nonumber \\
& & -(\sigma_{+}^{A}+\sigma_{+}^{B})(\bar{Q}+\bar{O})]|\psi_{t}\rangle. \label{QSD2Q}\end{aligned}$$ In fact, it is instructive to compare this dynamic equation for a hybrid bath with the model that two qubits interact with either a single bosonic bath or a single fermionic bath. For a single bath, the NMQSD equation should be $\frac{\partial}{\partial t}|\psi_t\rangle=[-iH_S+L_b z_t^\ast-L_b^\dagger\bar{O}]|\psi_t\rangle$ (bosonic [@Xinyu2011]) or $\frac{\partial}{\partial t}|\psi_t\rangle=[-iH_S+L_f \xi_t^\ast-L_f^\dagger\bar{Q}]|\psi_t\rangle$ (fermionic [@ZhaoFB]). It seems that Eq. (\[QSD2Q\]) is nothing more than a direct summation of a fermionic bath and a bosonic bath. However, more information is encoded in the $O$ and $Q$ operators. According to Eqs. (\[EqO\]-\[EqQ\]), the exact $O$ and $Q$ operators can be determined as $$\begin{aligned}
O & = & f_{1}(t,s)O_{1}+f_{2}(t,s)O_{2}+i\int_{0}^{t}ds^{\prime}f_{3}(t,s,s^{\prime})z_{s^{\prime}}^{*}O_{3}\nonumber \\
& & +i\int_{0}^{t}ds^{\prime}f_{4}(t,s,s^{\prime})\xi_{s^{\prime}}^{*}O_{4},\end{aligned}$$ $$\begin{aligned}
Q & = & g_{1}(t,s)Q_{1}+g_{2}(t,s)Q_{2}+i\int_{0}^{t}ds^{\prime}g_{3}(t,s,s^{\prime})z_{s^{\prime}}^{*}Q_{3}\nonumber \\
& & +i\int_{0}^{t}ds^{\prime}g_{4}(t,s,s^{\prime})\xi_{s^{\prime}}^{*}Q_{4}.\end{aligned}$$ The basis operators are $O_{1}=\sigma_{-}^{A}+\sigma_{-}^{B}$, $O_{2}=(\sigma_{z}^{A}+\sigma_{z}^{B})(\sigma_{-}^{A}+\sigma_{-}^{B})$, $O_{3}=O_{4}=\sigma_{-}^{A}\sigma_{-}^{B}$, $Q_{1}=\sigma_{-}^{A}+\sigma_{-}^{B}$, $Q_{2}=(\sigma_{z}^{A}+\sigma_{z}^{B})(\sigma_{-}^{A}+\sigma_{-}^{B})$, $Q_{3}=Q_{4}=\sigma_{-}^{A}\sigma_{-}^{B}$. The time-dependent coefficients satisfy the following relations $$\frac{\partial}{\partial t}f_{1}(t,s)=i\omega f_{1}+4f_{1}F_{2}+4f_{1}G_{2}+iF_{3}+iG_{3},$$ $$\begin{aligned}
\frac{\partial}{\partial t}f_{2}(t,s) & = & i\omega f_{2}+f_{1}(4F_{2}+4G_{2}-F_{1}-G_{1})-\frac{i}{2}F_{3}\nonumber \\
& & +f_{2}(2F_{1}+2G_{1}-4F_{2}-4G_{2})-\frac{i}{2}G_{3},\end{aligned}$$ $$\begin{aligned}
\frac{\partial}{\partial t}f_{3}(t,s,s^{\prime}) & = & 2i\omega f_{3}+2f_{1}F_{3}+2f_{1}G_{3}-4f_{2}F_{3}\nonumber \\
& & -4f_{2}G_{3}+2f_{3}F_{1}+2f_{3}G_{1},\end{aligned}$$ $$\begin{aligned}
\frac{\partial}{\partial t}f_{4}(t,s,s^{\prime}) & = & 2i\omega f_{4}+2f_{1}F_{4}+2f_{1}G_{4}-4f_{2}F_{4}\nonumber \\
& & -4f_{2}G_{4}+2f_{4}F_{1}+2f_{4}G_{1},\end{aligned}$$ $$\frac{\partial}{\partial t}g_{1}(t,s)=i\omega g_{1}+4g_{1}F_{2}+4g_{1}G_{2}+iF_{3}+iG_{3},$$ $$\begin{aligned}
\frac{\partial}{\partial t}g_{2}(t,s) & = & i\omega g_{2}+g_{1}(4F_{2}+4G_{2}-F_{1}-G_{1})-\frac{i}{2}F_{3}\nonumber \\
& & +g_{2}(2F_{1}+2G_{1}-4F_{2}-4G_{2})-\frac{i}{2}G_{3},\end{aligned}$$ $$\begin{aligned}
\frac{\partial}{\partial t}g_{3}(t,s,s^{\prime}) & = & 2i\omega g_{3}+2g_{1}F_{3}+2g_{1}G_{3}-4g_{2}F_{3}\nonumber \\
& & -4g_{2}G_{3}+2g_{3}F_{1}+2g_{3}G_{1},\end{aligned}$$ $$\begin{aligned}
\frac{\partial}{\partial t}g_{4}(t,s,s^{\prime}) & = & 2i\omega g_{4}+2g_{1}F_{4}+2g_{1}G_{4}-4g_{2}F_{4}\nonumber \\
& & -4g_{2}G_{4}+2g_{4}F_{1}+2g_{4}G_{1},\end{aligned}$$ with the initial conditions $$f_{1}(t,t)=g_{1}(t,t)=1,\quad f_{2}(t,t)=g_{2}(t,t)=0,$$ $$f_{3}(t,t,s^{\prime})=f_{4}(t,t,s^{\prime})=g_{3}(t,t,s^{\prime})=g_{4}(t,t,s^{\prime})=0,$$ $$g_{3}(t,s,t)=-4ig_{2}(t,s),\; g_{4}(t,s,t)=-4ig_{1}(t,s)+4ig_{2}(t,s),$$ $$f_{3}(t,s,t)=f_{4}(t,s,t)=-4if_{2}(t,s).$$ where $$F_{i}(t)=\int_{0}^{t}K_{b}(t,s)f_{i}(t,s)ds,\quad(i=1,2)$$ $$G_{i}(t)=\int_{0}^{t}K_{f}(t,s)g_{i}(t,s)ds,\quad(i=1,2)$$ $$F_{i}(t,s^{\prime})=\int_{0}^{t}K_{b}(t,s)f_{i}(t,s,s^{\prime})ds,\quad(i=3,4)$$ $$G_{i}(t,s^{\prime})=\int_{0}^{t}K_{f}(t,s)g_{i}(t,s,s^{\prime})ds,\quad(i=3,4)$$ In this example, both $O$ and $Q$ operators contain the fermionic noise $\xi^{*}$, but the fermionic $Q$ operator also contains the bosonic noise $z^{*}$. Although the NMQSD equation (\[QSD2Q\]) seems to be a direct summation of two individual baths, the $O$ and $Q$ operators in a hybrid bath are not a simple combination of these operators obtained in the single bath case (the exact $O$ or $Q$ operators for a single bosonic bath or a fermionic bath can be found in Ref. [@Xinyu2011] and Ref. [@ZhaoFB] respectively). Instead, there are many cross terms and they are coupled to each other reflecting the fact that *the effect of a hybrid bath cannot be simply treated as the direct summation of a fermionic bath plus a bosonic bath.* Through the system, two baths are also coupled indirectly. Such kind of indirect coupling can be also considered as interference between two independent baths which has been recently discussed in Ref. [@Lukin]. With the exact solution, it is possible to investigate the interference between fermionic bath and bosonic bath in the future research. Here, we just show one numerical result as an example to demonstrate that the interference between two baths can be dominant under certain conditions. In Fig. \[Coeff\], we compare the time evolutions of the coefficients in $O$ and $Q$ operators. The functions $F_{i}^{\prime}(t)$ ($i=3,4$) are defined as $F_{i}^{\prime}(t)=\int ds^{\prime}K_{b}(t,s^{\prime})F_{i}(t,s^{\prime})$ ($i=3,4$). Similarly, the functions $G_{i}^{\prime}(t)$ ($i=3,4$) are defined as $G_{i}^{\prime}(t)=\int ds^{\prime}K_{f}(t,s^{\prime})G_{i}(t,s^{\prime})$ ($i=3,4$). In the single bath case [@Xinyu2011], $O$ operator should not contain fermionic noise term $i\int_{0}^{t}ds^{\prime}f_{4}(t,s,s^{\prime})\xi_{s^{\prime}}^{*}O_{4}$. However, according to the numerical results for hybrid bath, the coefficient for fermionic noise term, $F_{4}^{\prime}(t)$ can be dominant under certain conditions. Similarly, in the $Q$ operator, the bosonic noise term $G_{3}^{\prime}(t)$ is also larger than $G_{1}(t)$ and $G_{2}(t)$. These results imply that the interference between two baths can be rather complicated and important. It is also worth noting that the results in Fig. \[Coeff\] is obtained in a strongly non-Markovian environment. In a Markov case, $F_1(t)$ and $G_1(t)$ will be dominant. Thus, our exact treatment of the non-Markovian hybrid bath problem could be a valuable tool to study those properties in the future.
 Time evolution for the coefficients in $O$ and $Q$ operators. In the left subplot, the red (solid), green (dashed), blue (dash-dotted) and black (dash-dotted) curves are the absolute values of the coefficients $|F_{1}(t)|$, $|F_{2}(t)|$, $|F_{3}^{\prime}(t)|$, and $|F_{4}^{\prime}(t)|$, respectively. In the right subplot, those curves represent $|G_{1}(t)|$, $|G_{2}(t)|$, $|G_{3}^{\prime}(t)|$, and $|G_{4}^{\prime}(t)|$, respectively.](Coeff){width="1\columnwidth"}
Single qubit case: Consistency with ordinary quantum mechanics
--------------------------------------------------------------
Although the $O$ and $Q$ operators in the two-qubit example are rather complicated, it is still possible to find a simple form when a special case- the single qubit case is considered, namely $\kappa_{B}=0$. In this case, the second qubit evolves independently from all the other parts, so that it can be removed in the interaction picture. Therefore, the NMQSD equation is reduced to $$\frac{\partial}{\partial t}|\psi_{t}\rangle=[-i\frac{\omega}{2}\sigma_{z}^{A}+\sigma_{-}^{A}(\xi_{t}^{\ast}+z_{t}^{*})-\sigma_{+}^{A}(\bar{Q}+\bar{O})]|\psi_{t}\rangle,$$ where the exact $O$ and $Q$ operators can be determined as $$O=Q=f(t,s)\sigma_{-}^{A}.$$ The time-dependent coefficient $f(t,s)$ satisfies the equation $$\frac{\partial}{\partial t}f(t,s)=[i\omega+F(t)]f(t,s),$$ where $F(t)=\int_{0}^{t}[K_{b}(t,s)+K_{f}(t,s)]f(t,s)ds$. Finally, the exact master equation for this model is derived as $$\frac{d}{dt}\rho=-i[H_{S},\rho]+\{F(t)[\sigma_{-}\rho,\sigma_{+}]+\mathrm{H.c.}\}.\label{MEQ1qu}$$ In general, the correlation functions $K_{b}(t,s)$ and $K_{f}(t,s)$ can be very complicated. However, here, we will use a special case to show that the result derived from NMQSD approach is consistent with the ordinary quantum mechanics. Consider the special case that there are only one boson and one fermion in the bosonic bath and fermionic bath respectively, i.e., $H_{FB}=\epsilon c^{\dagger}c$, $H_{BB}=\Omega b^{\dagger}b$. Therefore, the correlation functions is reduced to $$K_{b}(t,s)=\lambda^{2}e^{-i\Omega(t-s)},$$ $$K_{f}(t,s)=\mu^{2}e^{-i\epsilon(t-s)}.$$ In the resonance case, $\omega=\epsilon=\Omega$, $\lambda=\mu$, the equation for $F(t)$ is $$\frac{\partial}{\partial t}F(t)=2\lambda^{2}+F^{2}(t).$$ The solution is $$F(t)=\sqrt{2}\lambda\tan(\sqrt{2}\lambda t).$$ From the master equation (\[MEQ1qu\]), the evolution of the off-diagonal element $\rho_{21}$ is $$\frac{d}{dt}\rho_{21}(t)=i\omega\rho_{21}-F^{*}(t)\rho_{21}.$$ Finally, the solution of $\rho_{21}(t)$ is $$\rho_{21}(t)=\rho_{21}(0)e^{i\omega t}\cos(\sqrt{2}\lambda t).\label{eq:rho21}$$ On the other hand, it is also straightforward to solve the whole system (system plus two “baths”) with the standard Schrödinger equation since the total system only contains three particles. It is easy to confirm that solving the whole system gives the identical result as we obtained in Eq. (\[eq:rho21\]) by using the NMQSD approach. It confirms that our NMQSD approach is indeed consistent with ordinary quantum mechanics as expected.
Example 2: Single qubit with dephasing bosonic bath and dissipative fermionic bath
----------------------------------------------------------------------------------
In the second example, we will investigate the case that the system is coupled to the bosonic bath and fermionic bath in two different ways. The model we considered is given by $$H_{S}=\frac{\omega}{2}\sigma_{z},$$ $$L_{b}=\sigma_{z},\quad L_{f}=\sigma_{-}.$$ According to the general discussion in subsection \[sub:IIIA\], the NMQSD equation for this model can be written as $$\frac{\partial}{\partial t}|\psi_{t}\rangle=[-i\frac{\omega}{2}\sigma_{z}+\sigma_{-}\xi_{t}^{\ast}-\sigma_{+}\bar{Q}+\sigma_{z}z_{t}^{*}-\sigma_{z}\bar{O}]|\psi_{t}\rangle,$$ In this example, the $O$ and $Q$ operators contains infinite order of noise, therefore, for simplicity, we use the zeroth order functional expansion to give the approximate zeroth order operators $O^{(0)}$ and $Q^{(0)}$. By assuming all the terms associated with noises are zero [@Xu2014], the zeroth order $O^{(0)}$ and $Q^{(0)}$ operators can be obtained from Eqs. (\[EqO\]-\[EqQ\]) as $$O^{(0)}=\sigma_{z},$$ $$Q^{(0)}=g(t,s)\sigma_{-},$$ where the function $g(t,s)$ satisfies $$\frac{\partial}{\partial t}g(t,s)=[i\omega+G(t)]g(t,s),$$ where $G(t)=\int_{0}^{t}g(t,s)K_{f}(t,s)ds$. Finally, the corresponding approximate master equation is $$\frac{d}{dt}\rho=-i[\frac{\omega}{2}\sigma_{z},\rho]+\{G(t)[\sigma_{-}\rho,\sigma_{+}]+F(t)[\sigma_{z}\rho,\sigma_{z}]+\mathrm{H.c.}\}, \label{MEQ1q}$$ where $F(t)=\int_{0}^{t}K_{b}(t,s)ds$. Different from the first example where the model can be solved exactly, we show how to use the zeroth order (for higher order expansion, see Ref. [@Xu2014]) approximation to derive an approximate master equation in this second example. It is worth noting that Eq. (\[MEQ1q\]) still contains incomplete non-Markovian information, although it is derived from the zeroth order approximation. In the Markov limit, the correlation functions $K_b(t,s)$ and $K_f(t,s)$ are all $\delta$-functions, and the coefficients are no longer time-dependent but reduced to constants. The zeroth-order approximation can partially capture the non-Markovian features as a way to improve the Markov approximation. Actually, in the real application of the NMQSD approach, this systematic approximation method is shown to be very useful since the exact $O$ and $Q$ are often difficult to find in many realistic models. With this approximation approach, one can still solve these non-Markovian problems with satisfactory accuracy.
\[sec:IV\]Anti-commutative Case
===============================
\[sub:IV-A\]General Stochastic Schrödinger Equation
---------------------------------------------------
After discussing the commutative hybrid bath, we will consider the case that the system is assumed to anti-commutes with the fermionic bath, which often describes an electronic system such as a quantum dot system. Following a similar procedure, we can also derive the NMQSD equation for the anti-commutative hybrid bath as $$\begin{aligned}
\frac{\partial}{\partial t}|\psi_{t}\rangle & = & -i\langle z^{*},\xi^{*}|H_{tot}^{int}(t)|\psi_{tot}(t)\rangle\nonumber \\
& = & [-iH_{S}-L_{f}\xi_{t}^{\ast}-L_{f}^{\dagger}\int dsK_{f}(t,s)\frac{\delta_{l}}{\delta\xi_{s}^{\ast}}\nonumber \\
& & +L_{b}z_{t}^{\ast}-L_{b}^{\dagger}\int dsK_{b}(t,s)\frac{\delta}{\delta z_{s}^{\ast}}]\left\vert \psi_{t}\right\rangle .\label{QSDAC}\end{aligned}$$ Since the operators $L_{f}$ typically anti-commutes with the fermionic bath, the derivation is slightly different from the commutative case [@ChenFB; @ShiFB; @ZhaoFB]. As a result, there is a minor difference between Eq. (\[QSDC\]) and Eq. (\[QSDAC\]). The bosonic noise $z_{t}^{*}$, the fermionic niose $\xi_{t}^{*}$ and the corresponding correlation functions $K_{b}(t,s)$ and $K_{f}(t,s)$ are all defined in the same way. Similarly, we can also define the bosonic $O$ operator and the fermionic $Q$ operator, however, they satisfy different differential equations in the anti-commutative case as $$\begin{aligned}
\frac{\partial}{\partial t}O & = & [-iH_{s}-L_{f}\xi_{t}^{\ast}-L_{f}^{\dagger}\bar{Q}+L_{b}z_{t}^{*}-L_{b}^{\dagger}\bar{O},O]\nonumber \\
& & -L_{b}^{\dagger}\frac{\delta}{\delta z_{s}^{*}}\bar{O}-L_{f}^{\dagger}\frac{\delta}{\delta z_{s}^{*}}\bar{Q},\end{aligned}$$
$$\begin{aligned}
\frac{\partial}{\partial t}Q & = & [-iH_{s},Q]+[L_{f}\xi_{t}^{*},Q]+[L_{b}z_{t}^{*},Q]\nonumber \\
& & -L_{f}^{\dagger}\bar{Q}Q+QL_{f}^{\dagger}\bar{Q}-L_{b}^{\dagger}\bar{O}Q+QL_{f}^{\dagger}\bar{O}\nonumber \\
& & -L_{b}^{\dagger}\frac{\delta_{l}}{\delta\xi_{s}^{\ast}}\bar{O}+L_{f}^{\dagger}\frac{\delta_{l}}{\delta\xi_{s}^{*}}\bar{Q},\end{aligned}$$
with the initial conditions $$O(t,t,z^{*},\xi^{*})=L_{b},$$ $$Q(t,t,z^{*}\xi^{*})=L_{f}.$$ Then, the density matrix can be also reproduced as $$\rho(t)=\langle\langle P_{t}\rangle_{f}\rangle_{b},$$ and the master equation can be derived as $$\begin{aligned}
\frac{d}{dt}\rho & = & -i[H_{S},\rho]+[L_{b},\langle\langle P_{t}\bar{O}^{\dagger}\rangle_{f}\rangle_{b}]+[\langle\langle\bar{Q}P_{t}\rangle_{f}\rangle_{b},L_{b}^{\dagger}]\nonumber \\
& & +[L_{f},\langle\langle P_{t}\bar{Q}^{\dagger}(-\xi)\rangle_{f}\rangle_{b}]+[\langle\langle\bar{Q}P_{t}\rangle_{f}\rangle_{b},L_{f}^{\dagger}].\label{MEQAC}\end{aligned}$$ In the derivation of the master equation, an anti-commutative version of the Novikov theorem [@ShiFB; @ChenFB] has been used. It is different from either the commutative version of Novikov theorem for fermionic bath [@ZhaoFB] or the one for bosonic bath [@Yu1999]. Similarly, when $O$ and $Q$ are noise-independent, the master equation is reduced to $$\frac{d}{dt}\rho=-i[H_{S},\rho]+\{[\bar{Q}\rho,L_{f}^{\dagger}]+[\bar{O}\rho,L_{b}^{\dagger}]+\mathrm{H.c.}\}.$$
Example 3: Quantum dot in a hybrid bath {#sec:IV-B}
---------------------------------------
In order to show the details of solving an anti-commutative hybrid bath problem, we consider a specific example that is the Anderson model in a bosonic environment (see Ref. [@Chung] for example). In this particular example the general Hamiltonian Eq. (\[Hybrid\]) becomes $$H_{S}=\varepsilon d^{\dagger}d,$$ describing the quantum dot, $$H_{B}=\sum_{k,i=L,R}[\epsilon(k)-\mu_{i}]c_{ki}^{\dagger}c_{ki}+\sum_{r}\omega_{r}b_{r}^{\dagger}b_{r},$$ describing the two fermionic baths (“$L$” and “$R$”) and one phonon bath, and $$H_{I}=\sum_{k,i=L,R}t_{k,i}c_{ki}^{\dagger}d+\mathrm{H.c.}+\sum_{r}\lambda_{r}(d^{\dagger}d-\frac{1}{2})(b_{r}+b_{r}^{\dagger}),\label{HI}$$ describing the transport process between two fermionic bath and the dissipation process caused by a bosonic bath.
In the finite temperature case [@Yu-FiniteT], we need to introduce two fictitious bath “$a_{L}$” and “$a_{R}$” with the negative eigen-frequencies as: $$\begin{aligned}
H & = & H_{S}+\sum_{k,i=L,R}[\epsilon(k)-\mu_{i}]c_{ki}^{\dagger}c_{ki}+\{t_{ki}c_{ki}^{\dagger}d+ \mathrm{H.c.}\}\nonumber \\
& & +\sum_{r}\lambda_{r}(d^{\dagger}d-\frac{1}{2})(b_{r}+b_{r}^{\dagger})+\sum_{r}\Omega_{r}b_{r}^{\dagger}b_{r}\nonumber \\
& & +\sum_{k,i=L,R}-[\epsilon(k)-\mu_{i}]a_{ki}^{\dagger}a_{ki}.\end{aligned}$$ Then, performing the Bogoliubov transformation $$c_{ki}=\sqrt{1-\bar{n}_{ki}}c_{ki}^{\prime}+\sqrt{\bar{n}_{ki}}a_{ki}^{\prime\dagger}\quad(i=L,R),$$ $$a_{ki}=\sqrt{1-\bar{n}_{ki}}a_{ki}^{\prime}-\sqrt{\bar{n}_{ki}}c_{ki}^{\prime\dagger}\quad(i=L,R),$$ the Hamiltonian become $$\begin{aligned}
H & = & H_{S}+\sum_{k,i=L,R}[\epsilon(k)-\mu_{i}]c_{ki}^{\prime\dagger}c_{ki}^{\prime}\nonumber \\
& & +\{t_{ki}(\sqrt{1-\bar{n}_{ki}}c_{ki}^{\prime\dagger}+\sqrt{\bar{n}_{ki}}a_{ki}^{\prime})d+ \mathrm{H.c.}\}\nonumber \\
& & +\sum_{r}\lambda_{r}(d^{\dagger}d-\frac{1}{2})(b_{r}+b_{r}^{\dagger})+\sum_{r}\Omega_{r}b_{r}^{\dagger}b_{r}\nonumber \\
& & +\sum_{k,i=L,R}-[\epsilon(k)-\mu_{i}]a_{ki}^{\prime\dagger}a_{ki}^{\prime}.\end{aligned}$$ Redefining $\omega_{ki}=\epsilon(k)-\mu_{i}$, $g_{ki}=t_{ki}\sqrt{1-\bar{n}_{ki}},\; f_{ki}=t_{ki}\sqrt{\bar{n}_{ki}}$, then, in the interaction picture, the Hamiltonian can be written as $$\begin{aligned}
H_{int}(t) & =H_{S}+\sum_{r}\lambda_{r}(d^{\dagger}d-\frac{1}{2})(b_{r}e^{-i\Omega_{r}t}+b_{r}^{\dagger}e^{i\Omega_{r}t})\nonumber \\
& +\{\sum_{k,i=L,R}g_{ki}e^{i\omega_{k}t}c_{ki}^{\prime\dagger}d+f_{ki}e^{i\omega_{k}t}a_{ki}^{\prime}d+ \mathrm{H.c.}\}.\end{aligned}$$ By introducing one bosonic coherent state and two fermionic coherent states as $$|z\rangle=\prod_{r}\exp\{z_{r}b_{r}^{\dagger}\}|0\rangle,$$ $$|\xi_{ia}\rangle=\prod_{k}(1-\xi_{kia}a_{ki}^{\prime\dagger})|0\rangle\;(i=L,R),$$
$$|\xi_{ic}\rangle=\prod_{k}(1-\xi_{kic}c_{ki}^{\prime\dagger})|0\rangle\;(i=L,R).$$
The stochastic state vector can be defined as $$|\psi_{t}(z^{*},\xi_{La}^{*},\xi_{Ra}^{*},\xi_{Lc}^{*},\xi_{Rc}^{*})\rangle=\langle z^{*},\xi_{La}^{*},\xi_{Ra}^{*},\xi_{Lc}^{*},\xi_{Rc}^{*}|\psi_{tot}(t)\rangle.$$ Following the general approach discussed in the last subsection, the NMQSD equation for the stochastic state vector is derived as $$\begin{aligned}
\frac{\partial}{\partial t}|\psi_{t}\rangle & = & H_{eff}|\psi_{t}\rangle,\end{aligned}$$ where $$\begin{aligned}
H_{eff} & = & [-iH_{S}+d^{\dagger}\int_{0}^{t}dsK_{La}(t,s)\frac{\delta}{\delta\xi_{La,s}^{*}}+d\xi_{La,t}^{*}\nonumber \\
& & +d^{\dagger}\int_{0}^{t}dsK_{Ra}(t,s)\frac{\delta}{\delta\xi_{Ra,s}^{*}}+d\xi_{Ra,t}^{*}\nonumber \\
& & -d\int_{0}^{t}dsK_{Lc}(t,s)\frac{\delta}{\delta\xi_{Lc,s}^{*}}-d^{\dagger}\xi_{Lc,t}^{*}\nonumber \\
& & -d\int_{0}^{t}dsK_{Rc}(t,s)\frac{\delta}{\delta\xi_{Rc,s}^{*}}-d^{\dagger}\xi_{Rc,t}^{*}\nonumber \\
& & -b^{\dagger}\int_{0}^{t}ds\alpha(t,s)\frac{\delta}{\delta z_{s}^{*}}+dz_{t}^{*}].\end{aligned}$$ In this equation, we introduced five noises as $$z_{t}^{*}=-i\sum_{r}z_{r}^{*}e^{-i\Omega_{r}t},$$ $$\xi_{ia,t}^{*}=-i\sum_{k}\xi_{kia}^{*}e^{-i\omega_{k}t},\;(i=L,R),$$
$$\xi_{ic,t}^{*}=-i\sum_{k}\xi_{kic}^{*}e^{-i\omega_{k}t},\;(i=L,R),$$
and the corresponding correlation functions are $$\alpha(t,s)=\sum_{r}\lambda_{r}^{2}e^{-i\Omega_{r}(t-s)},$$ $$K_{ia}(t,s)=\sum_{k}g_{ki}^{2}e^{-i\omega_{k}(t-s)}\quad(i=L,R),$$ $$K_{ic}(t,s)=\sum_{k}f_{ki}^{2}e^{i\omega_{k}(t-s)}\quad(i=L,R),$$ Among the noises above, $z_{t}^{*}$ is a complex Gaussian noise, $\xi_{ia,t}^{*}$ and $\xi_{ic,t}^{*}$ are Grassmann Gaussian noises. They satisfy the following statistical relations $$\langle z_{t}\rangle_{b}=\langle z_{t}^{*}\rangle_{b}=0,$$ $$\langle z_{t}^{*}z_{s}\rangle_{b}=\alpha(t,s),$$ $$\langle\xi_{ia,t}^{*}\rangle_{f}=\langle\xi_{ia,t}\rangle_{f}=\langle\xi_{ic,t}^{*}\rangle_{f}=\langle\xi_{ic,t}\rangle_{f}=0,$$ $$\langle\xi_{ia,t}^{*}\xi_{ia,s}\rangle_{f}=K_{ia}(t,s),\;\langle\xi_{ic,t}^{*}\xi_{ic,s}\rangle_{f}=K_{ic}(t,s).$$
Following the technique discussed in subsection \[sub:IV-A\], the time dependent operators $O$ and $Q$ are defined as $$\frac{\delta}{\delta z_{s}^{*}}|\psi_{t}\rangle=O(t,s,z^{*})|\psi_{t}\rangle,$$ $$\frac{\delta}{\delta\xi_{ia,s}^{*}}|\psi_{t}\rangle=Q_{ia}(t,s,\xi_{ia}^{*})|\psi_{t}\rangle\quad(i=L,R),$$ $$\frac{\delta}{\delta\xi_{ic,s}^{*}}|\psi_{t}\rangle=Q_{ic}(t,s,\xi_{ic}^{*})|\psi_{t}\rangle\quad(i=L,R),$$ and the the zeroth order approximation gives the solution of these operators as $$O\approx f_{1}(t,s)d^{\dagger}d,$$ $$Q_{ic}\approx f_{ic}(t,s)d\;(i=L,R),$$ $$Q_{ia}\approx f_{ia}(t,s)d^{\dagger}\;(i=L,R),$$ while the coefficients satisfy $$\frac{\partial}{\partial t}f_{1}(t,s)=0,$$ $$\frac{\partial}{\partial t}f_{Lc}(t,s)=(i\varepsilon+F_{1}+F_{La}+F_{Ra}+F_{Lc}+F_{Rc})f_{Lc},$$ $$\frac{\partial}{\partial t}f_{Rc}(t,s)=(i\varepsilon+F_{1}+F_{La}+F_{Ra}+F_{Lc}+F_{Rc})f_{Rc},$$ $$\frac{\partial}{\partial t}f_{La}(t,s)=(-i\varepsilon-F_{1}-F_{La}-F_{Ra}-F_{Lc}-F_{Rc})f_{La},$$ $$\frac{\partial}{\partial t}f_{Ra}(t,s)=(-i\varepsilon-F_{1}-F_{La}-F_{Ra}-F_{Lc}-F_{Rc})f_{Ra},$$ where $F_{1}=\int_{0}^{t}\alpha(t,s)f_{1}(t,s)ds$, $F_{Lc}=\int_{0}^{t}K_{Lc}(t,s)f_{Lc}(t,s)ds$, $F_{Rc}=\int_{0}^{t}K_{Rc}(t,s)f_{Rc}(t,s)ds$, $F_{La}=\int_{0}^{t}K_{La}(t,s)f_{La}(t,s)ds$, $F_{Ra}=\int_{0}^{t}K_{Ra}(t,s)f_{Ra}(t,s)ds$. Finally, the master equation is derived as $$\begin{aligned}
\frac{\partial}{\partial t}\rho & =-i\varepsilon[d^{\dagger}d,\rho]+\{(F_{Lc}+F_{Rc})[d\rho,d^{\dagger}]\nonumber \\
& +(F_{La}+F_{Ra})[d,d^{\dagger}\rho]+F_{1}[d^{\dagger}d\rho,d^{\dagger}d]+\mathrm{H.c.}\}.\label{MEQdot}\end{aligned}$$ In the third example, the hybrid NMQSD approach is applied to a very important system called Anderson model embedded in a bosonic dephasing environment. First, we show how to map a finite temperature problem into a zero temperature problem to apply the hybrid NMQSD approach in finite temperature case. More important, we show the hybrid NMQSD approach provides us a powerful tool to investigate the dynamics of a quantum system in a non-Markovian regime. With the NMQSD approach, open quantum systems coupled to a hybrid bath such as the example discussed above can be solved systematically in non-Markovian regimes. Typically, in the Markov case, all the coefficients in the master equation, $F_{Lc}$, $F_{Rc}$, $F_{La}$, $F_{Ra}$, and $F_{1}$ are constants. However, in Eq. (\[MEQdot\]), those coefficients are time-dependent, which reflects the non-Markovian behavior even if the zeroth-order $O$ and $Q$ operators are employed. The higher-order non-Markovian approximations can be implemented in a similar way.
Fermionic Bath vs. Bosonic Bath
-------------------------------
The master equation derived in Eq. (\[MEQdot\]) can be used to illustrate the difference between the fermionic bath and bosonic bath. For this purpose, two parameters in the original Hamiltonian are specified to describe the coupling strength to the fermionic bath and bosonic bath. As it is shown in Eq. (\[HI\]), $t_{ki}$ determine how strong the interaction between the system and fermionic bath, and $\lambda_{r}$ determine the strength of the coupling to bosonic bath. In the numerical simulation, we will introduce $c_{f}$ and $c_{b}$ to control the global coupling strengths for fermionic bath and bosonic bath respectively. Namely, we replace $t_{ki}$ by $\sqrt{c_{f}}t_{ki}$ and $\lambda_{r}$ by $\sqrt{c_{b}}\lambda_{r}$. Then, these two parameters reflect the global coupling strength. For example, if we take $c_{b}=0$, then the bosonic bath is switched off, and we can observe the evolution without presence of the bosonic bath. In the numerical simulations, we use four Ornstein-Uhlenbeck noises $K_{mn}(t,s)=\frac{\Gamma_{mn}}{2}\exp[(-\gamma_{mn}+i\phi_{mn})|t-s|]$ ($m=L,R$; $n=a,c$) to model the correlation functions $K_{La}$, $K_{Lc}$, $K_{Ra}$, $K_{Rc}$. The parameters are chosen as $\Gamma_{Lc}=0.017$, $\gamma_{Lc}=0.3$, $\phi_{Lc}=1.1$, $\Gamma_{Rc}=0.034$, $\gamma_{Rc}=0.5$, $\phi_{Rc}=1.65$, $\Gamma_{La}=0.012$, $\gamma_{La}=0.4$, $\phi_{La}=0.75$, $\Gamma_{Ra}=0.044$, $\gamma_{Ra}=0.45$, $\phi_{Ra}=1.2$.
![(Color online) Time evolution for different coupling strengths of fermionic bath. The coupling strength of bosonic bath is fixed as 1. The red (solid), green (dashed), and black (dash-dotted) curves are the elements of density matrix $|\rho_{11}|$, $|\rho_{22}|$, $|\rho_{12}|$ respectively.[]{data-label="FB"}](FB){width="0.95\columnwidth"}
![(Color online) Time evolution for different coupling strength of bosonic bath. The coupling strength of fermionic bath is fixed as 1. The red (solid), green (dashed), and black (dash-dotted) curves are the elements of density matrix $|\rho_{11}|$, $|\rho_{22}|$, $|\rho_{12}|$ respectively.[]{data-label="BB"}](BB){width="0.95\columnwidth"}
Fig. \[FB\] and \[BB\] clearly show the different influences of bosonic bath and fermionic bath on the system. Generally, For the model under consideration, the fermionic bath contributes to both of the energy dissipation and the decoherence, while the bosonic bath mainly contributes to the dephasing process. From Fig. \[FB\], we can see that the dephasing process (off-diagonal elements) remains almost the same while changing the fermionic coupling strength. On the contrary, the dissipative process is significantly modified. From Fig. \[BB\], we can see the dissipative process is not significantly affected by changing the bosonic coupling strength, as a compassion, the dephasing rate is affected. These results can be also predicted by analyzing the master equation or Hamiltonian. Since the coupling form of the bosonic bath is a dephasing type, therefore it will fundamentally affect the dephasing process. In a similar fashion, the coupling form of the fermionic bath is expected to affect the dissipative process.
Conclusion {#sec:V}
==========
In this paper, we have developed a stochastic Schrödingier equation for the open systems interacting with a hybrid environment containing both bosons and fermions. By combining the bosonic and fermionic NMQSD approaches, two types of noises are used simultaneously to derive the NMQSD equation for the hybrid bath case. As a simple application, we show that the corresponding non-Markovian master equation for the hybrid bath case can be recovered from the hybrid NMQSD equation. For more applications, two types of models are discussed including the commutative and anti-commutative cases. In these examples, we have demonstrated the consistency between NMQSD approach and the ordinary quantum mechanics when the system and its environment are relatively simple. Moreover, with these examples, we illustrate the relationship between bosonic bath and fermionic bath, the exact and approximate solutions of $O$ and $Q$ operators, and the different effects of bosonic bath and fermionic bath. The hybrid NMQSD approach established in this paper can be served as a convenient tool in the study of the dynamic evolution of the hybrid open systems. Particularly, it is helpful to investigate some early stage evolution caused by memory effect of the environments since our approach is systematically derived from the microscopic model which goes beyond the standard Born-Markov approximation.
Acknowledgements {#acknowledgements .unnumbered}
================
We acknowledge grant support from the NSF PHY-0925174, The NBRPC No. 2009CB929300,the NSFC Nos. 91121015 and the MOE No. B06011. J.Q.Y is supported by the NSAF (Grant Nos. U1330201 and U1530401) and the National Basic Research Program of China (Grant Nos. 2016YFA0301201 and 2014CB921401).
References {#references .unnumbered}
==========
[10]{}
X. Hu, R. de Sousa, S. Das Sarma, [*Foundations of Quantum Mechanics in the Light of New Technology*]{} (Edited by Y. A. Ono and K. Fujikawa, World Scientific, 2002).
G. Ritschel, J. Roden, W. T. Strunz, A. Aspuru-Guzik, and A. Eisfeld, J. Phys. Chem. Lett. [**2**]{}, 2912 (2011).
N. Lambert and F. Nori Phys. Rev. B [**78**]{}, 214302 (2008).
C. W. Gardiner and P. Zoller, *Quantum Noise* (Springer-Verlag, Berlin, 2004).
W. G. Unruh, Phys. Rev. A [**51**]{}, 992 (1995).
H. P. Breuer and F. Petruccione, *Theory of Open Quantum Systems* (Oxford University, New York, 2002).
X. Zhao, J. Jing, B. Corn, and T. Yu, Phys. Rev. A **84**, 032101 (2011).
J. Jing, X. Zhao, J. Q. You, and T. Yu, Phys. Rev. A **85,** 042106 (2012).
W. H. Zurek, Rev. Mod. Phys. **75**, 715 (2003).
M. Schlosshauer, Rev. Mod. Phys. **76**, 324 (2005).
W. H. Zurek, S. Habib, and J. P. Paz, Phys. Rev. Lett. **70**, 1187 (1993); W. H. Zurek and J. P. Paz, Phys. Rev. Lett. **72**, 2508 (1994).
T. Yu and J. H. Eberly, Quantum Inform. Comp. **7**, 459 (2007).
T. Yu and J. H. Eberly, Phys. Rev. Lett. **93**, 140404 (2004); Phys. Rev. Lett. **97**, 140403 (2006).
J. Jing, X. Zhao, J. Q. You, W. T. Strunz, and T. Yu, Phys. Rev. A **88**, 052122 (2013).
X. Zhao, J. Jing, J. Q. You, and T. Yu, Quantum Inform. Comp. **14**, 0741 (2014).
E. L. Hahn, Phys. Rev. **80**, 580 (1950); W.-K. Rhim, A. Pines, and J. S. Waugh, Phys. Rev. Lett. **25**, 218 (1970); L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. **82**, 2417 (1999).
H. M. Wiseman and G. J. Milburn, Phys. Rev. Lett. **70**, 548 (1993); H. M. Wiseman, Phys. Rev. A **49**, 2133 (1994).
R. P. Feynman, and F. L. Vernon, Ann. Phys. **24**, 118 (1963).
J.-H. An and W.-M. Zhang, Phys. Rev. A **76**, 042127 (2007); C. U. Lei, and W. M. Zhang, Ann. Phys. **327**, 1408 (2012).
B. L. Hu, J. P. Paz, and Y. Zhang, Phys. Rev. D **45**, 2843 (1992); B. L. Hu, J. P. Paz, and Y. Zhang, Phys. Rev. D **47**, 1576 (1993).
B. L. Hu, J. P. Paz, and Y. Zhang, Phys. Rev. D **45**, 2843 (1992); Phys. Rev. D **47**, 1576 (1993).
E. A. Calzetta, and B. L. Hu, *Nonequilibrium Quantum Field Theory* (Cambridge University Press, New York, 2008).
A. O. Caldeira and A. J. Leggett, Physica A **121**, 587 (1983).
J. Dalibard, Y. Castin, and K. Mølmer, Phys. Rev. Lett. **68**, 580 (1992).
N. Gisin and I. C. Percival, J. Phys. A **25**, 5677 (1992); *ibid*. **26**, 2233 (1993).
L. Diósi, N. Gisin, and W. T. Strunz, Phys. Rev. A **58**, 1699 (1998); W. T. Strunz, L. Diósi, and N. Gisin, Phys. Rev. Lett. **82**, 1801 (1999).
T. Yu, L. Diósi, N. Gisin, and W. T. Strunz, Phys. Rev. A **60**, 91 (1999).
W. T. Strunz and T. Yu, Phys. Rev. A **69**, 052115 (2004).
Büttiker, Phys. Rev. B **46**, 12485 (1992).
Y. Meir, N. S. Wingreen, and P. A. Lee, Phys. Rev. Lett. [**66,**]{} 3048 (1991); N. S. Wingreen, A.-P. Jauho, and Y. Meir, Phys. Rev. B **48**, 8487 (1993).
M. W. Y. Tu and W.-M. Zhang, Phys. Rev. B **78**, 235311 (2008).
W.-M. Zhang, P.-Y. Lo, H.-N. Xiong, M.W.-Y. Tu, and F. Nori, Phys. Rev. Lett. **109**, 170402 (2012).
X. Zhao, W. Shi, L.-A. Wu, and T. Yu, Phys. Rev. A **86**, 032116 (2012).
W. Shi, X. Zhao, and T. Yu, Phys. Rev. A **87**, 052127 (2013).
M. Chen and J. Q. You, Phys. Rev. A **87**, 052108 (2013).
T. Yu, Phys. Rev. A **69**, 062107 (2004).
J. Jing and T. Yu, Phys. Rev. Lett. **105**, 240403 (2010).
D. Suess, A. Eisfeld, and W. T. Strunz, Phys. Rev. Lett. **113**, 150403 (2014).
Z.-Z. Li, C.-T. Yip, H.-Y. Deng, M. Chen, T. Yu, J. Q. You, and C.-H. Lam, Phys. Rev. A **90**, 022122 (2014).
H. Yang, H. Miao, and Y. Chen, Phys. Rev. A **85**, 040101(R) (2012).
J. Jing, L.-A. Wu, J. Q. You, and T. Yu, Phys. Rev. A **85**, 032123 (2012).
G. Ritschel *et al*., J. Phys. Chem. Lett. **2**, 2912 (2011).
E. Barouch, B. McCoy, and M. Dresden, Phys. Rev. A **2**, 1075 (1970).
W. -M. Zhang, D. H. Feng, and R. Gilmore, Rev. Mod. Phys. **62**, 867 (1990).
J. Xu, X. Zhao, J. Jing, W. T. Strunz, and T. Yu, J. Phys. A: Math. Theor. **47**, 435301 (2014).
C.-K. Chan, G.-D. Lin, S. F. Yelin, and M. D. Lukin, Phys. Rev. A **89**, 042117 (2014).
C.-H. Chung, K. Le Hur, M. Vojta, and P. Wölfle, Phys. Rev. Lett. **102**, 216803 (2009); C.-H. Chung, Phys. Rev. B **83**, 115308 (2011).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A new tetrad introduced within the framework of geometrodynamics for non-null electromagnetic fields allows for the geometrical analysis of the Lorentz equation and its solutions. This tetrad, through the use of the Frenet-Serret formulae and Fermi-Walker transport, exhibits explicitly the set of solutions to the Lorentz equation in a curved spacetime.'
address: '1. Instituto de Física, Facultad de Ciencias, Iguá 4225, esq. Mataojo, Montevideo, Uruguay.'
author:
- 'Alcides Garat$^{1}$'
date: 'August 15th, 2006'
title: The Lorentz equation in geometrodynamics
---
\#1[\#1]{}
\#1[\#1]{}
Introduction
============
The new tetrads introduced in [@A], yield maximum simplification in the expression of a non-null electromagnetic field in a curved spacetime, for instance. It is through this simplified expression of the electromagnetic field that we will get to the geometry associated with the Lorentz equation and its solutions. It is the purpose of this manuscript two-fold. On one hand we will prove that the Lorentz equation has its origin in the Fermi-Walker transport of the electromagnetic tetrads already mentioned. On the other hand we will study different kinds of solutions to the Lorentz equation from a geometric point of view through the use of the electromagnetic tetrads previously introduced. If $F_{\mu\nu}$ is the electromagnetic field and $f_{\mu\nu}= (G^{1/2} / c^2) \: F_{\mu\nu}$ is the geometrized electromagnetic field, then the Einstein-Maxwell equations can be written,
$$\begin{aligned}
f^{\mu\nu}_{\:\:\:\:\:;\nu} &=& 0 \label{EM1}\\
\ast f^{\mu\nu}_{\:\:\:\:\:;\nu} &=& 0 \label{EM2}\\
R_{\mu\nu} &=& f_{\mu\lambda}\:\:f_{\nu}^{\:\:\:\lambda}
+ \ast f_{\mu\lambda}\:\ast f_{\nu}^{\:\:\:\lambda}\ , \label{EM3}\end{aligned}$$
where $\ast f_{\mu\nu}={1 \over 2}\:\epsilon_{\mu\nu\sigma\tau}\:f^{\sigma\tau}$ is the dual tensor of $f_{\mu\nu}$. The symbol $``;''$ stands for covariant derivative with respect to the metric tensor $g_{\mu\nu}$. At every point in spacetime there is a duality rotation by an angle $-\alpha$ that transforms a non-null electromagnetic field into an extremal field,
$$\xi_{\mu\nu} = e^{-\ast \alpha} f_{\mu\nu}\ = \cos(\alpha)\:f_{\mu\nu} - \sin(\alpha)\:\ast f_{\mu\nu}.\label{dref}$$
The local scalar $\alpha$ is known as the complexion of the electromagnetic field. Extremal fields are essentially electric fields and they satisfy,
$$\xi_{\mu\nu} \ast \xi^{\mu\nu}= 0\ . \label{i0}$$
They also satisfy the identity given by equation (64) in [@MW],
$$\begin{aligned}
\xi_{\alpha\mu}\:\ast \xi^{\mu\nu} &=& 0\ .\label{i1}\end{aligned}$$
As antisymmetric fields the extremal fields also verify the identity,
$$\begin{aligned}
\xi_{\mu\alpha}\:\xi^{\nu\alpha} -
\ast \xi_{\mu\alpha}\: \ast \xi^{\nu\alpha} &=& \frac{1}{2}
\: \delta_{\mu}^{\:\:\:\nu}\ Q \ ,\label{i2}\end{aligned}$$
where $Q=\xi_{\mu\nu}\:\xi^{\mu\nu}=-\sqrt{T_{\mu\nu}T^{\mu\nu}}$ according to equations (39) in [@MW]. $Q$ is assumed not to be zero, because we are dealing with non-null electromagnetic fields. $T_{\mu\nu} = \xi_{\mu\lambda}\:\:\xi_{\nu}^{\:\:\:\lambda}
+ \ast \xi_{\mu\lambda}\:\ast \xi_{\nu}^{\:\:\:\lambda}$ is the electromagnetic stress-energy tensor [@MW]. In geometrodynamics, the Maxwell equations,
$$\begin{aligned}
f^{\mu\nu}_{\:\:\:\:\:;\nu} &=& 0 \label{L1}\nonumber\\
\ast f^{\mu\nu}_{\:\:\:\:\:;\nu} &=& 0 \ , \label{L2}\end{aligned}$$
are telling us that two potential vector fields exist,
$$\begin{aligned}
f_{\mu\nu} &=& A_{\nu ;\mu} - A_{\mu ;\nu}\label{ER}\nonumber\\
\ast f_{\mu\nu} &=& \ast A_{\nu ;\mu} - \ast A_{\mu ;\nu} \ .\label{DER}\end{aligned}$$
We can define then, a tetrad,
$$\begin{aligned}
U^{\alpha} &=& \xi^{\alpha\lambda}\:\xi_{\rho\lambda}\:A^{\rho} \:
/ \: (\: \sqrt{-Q/2} \: \sqrt{A_{\mu} \ \xi^{\mu\sigma} \
\xi_{\nu\sigma} \ A^{\nu}}\:) \label{UO}\\
V^{\alpha} &=& \xi^{\alpha\lambda}\:A_{\lambda} \:
/ \: (\:\sqrt{A_{\mu} \ \xi^{\mu\sigma} \
\xi_{\nu\sigma} \ A^{\nu}}\:) \label{VO}\\
Z^{\alpha} &=& \ast \xi^{\alpha\lambda} \: \ast A_{\lambda} \:
/ \: (\:\sqrt{\ast A_{\mu} \ast \xi^{\mu\sigma}
\ast \xi_{\nu\sigma} \ast A^{\nu}}\:)
\label{ZO}\\
W^{\alpha} &=& \ast \xi^{\alpha\lambda}\: \ast \xi_{\rho\lambda}
\:\ast A^{\rho} \: / \: (\:\sqrt{-Q/2} \: \sqrt{\ast A_{\mu}
\ast \xi^{\mu\sigma} \ast \xi_{\nu\sigma} \ast A^{\nu}}\:) \ .
\label{WO}\end{aligned}$$
The four vectors (\[UO\]-\[WO\]) have the following algebraic properties,
$$-U^{\alpha}\:U_{\alpha}=V^{\alpha}\:V_{\alpha}
=Z^{\alpha}\:Z_{\alpha}=W^{\alpha}\:W_{\alpha}=1 \ .\label{TSPAUX}$$
Using the equations (\[i1\]-\[i2\]) it is simple to prove that (\[UO\]-\[WO\]) are orthogonal. When we make the transformation,
$$\begin{aligned}
A_{\alpha} \rightarrow A_{\alpha} + \Lambda_{,\alpha}\ , \label{G1}\end{aligned}$$
$f_{\mu\nu}$ remains invariant, and the transformation,
$$\begin{aligned}
\ast A_{\alpha} \rightarrow \ast A_{\alpha} +
\ast \Lambda_{,\alpha}\ , \label{G2}\end{aligned}$$
leaves $\ast f_{\mu\nu}$ invariant, as long as the functions $\Lambda$ and $\ast \Lambda$ are scalars. Schouten defined what he called, a two-bladed structure in a spacetime [@SCH]. These blades are the planes determined by the pairs ($U^{\alpha}, V^{\alpha}$) and ($Z^{\alpha}, W^{\alpha}$). It was proved in [@A] that the transformation (\[G1\]) generates a “rotation” of the tetrad vectors ($U^{\alpha}, V^{\alpha}$) into ($\tilde{U}^{\alpha}, \tilde{V}^{\alpha}$) such that these “rotated” vectors ($\tilde{U}^{\alpha}, \tilde{V}^{\alpha}$) remain in the plane or blade one generated by ($U^{\alpha}, V^{\alpha}$). It was also proved in [@A] that the transformation (\[G2\]) generates a “rotation” of the tetrad vectors ($Z^{\alpha}, W^{\alpha}$) into ($\tilde{Z}^{\alpha}, \tilde{W}^{\alpha}$) such that these “rotated” vectors ($\tilde{Z}^{\alpha}, \tilde{W}^{\alpha}$) remain in the plane or blade two generated by ($Z^{\alpha}, W^{\alpha}$). Once we introduced all these elements found in [@A], we state the purpose of this manuscript. A particle of charge $q$ and mass $m$ in the presence of the electromagnetic field $f_{\mu\nu}$ is going to follow a timelike curve determined by the Lorentz equation [@ROH][@JAW][@DeWB][@DeWCM][@MC1]. It will be proved that the set of all curves that satisfy the Lorentz equation can be generated through Fermi-Walker transport.
Fermi-Walker transport in geometrodynamics
==========================================
The Frenet-Serret formulae for a tetrad associated for instance with a timelike curve $x^{\beta}(s)$ in spacetime [@SY][@HS] are,
$$\begin{aligned}
{\delta A^{\alpha} \over \delta s} &=& b \: B^{\alpha} \label{A}\\
{\delta B^{\alpha} \over \delta s} &=& c \: C^{\alpha} + b \: A^{\alpha} \label{B}\\
{\delta C^{\alpha} \over \delta s} &=& d \: D^{\alpha} - c \: B^{\alpha} \label{V}\label{C}\\
{\delta D^{\alpha} \over \delta s} &=& -d \: C^{\alpha} \label{D} \ ,\end{aligned}$$
where $ A^{\alpha}( x^{\beta}(s)) = {dx^{\alpha} \over ds}$ is the unit timelike tangent vector to the curve. The triad $( B^{\alpha}, C^{\alpha}, D^{\alpha})$ is formed by unit spacelike vector fields. We know that when the local scalar functions $b$, $c$ and $d$ are equal to zero, the curve is a geodesic [@MM]. If $b \neq 0$, then we can define the Fermi-Walker transport of a vector $F^{\alpha}$ as,
$${\delta F^{\alpha} \over \delta s} = b \: F_{\rho} \: (A^{\alpha} \: B^{\rho} - A^{\rho} \: B^{\alpha}) \ , \label{FWT}\\$$
where $B^{\alpha}$ and $b$ are the first normal and first curvature to the timelike curve $x^{\beta}_{t}(s)$. The “absolute derivative” ${\delta \over \delta s}$ is defined in [@SY]. Analogously, Fermi-Walker transport of a vector $F^{\alpha}$ can be defined along a spacelike curve as [@SY],
$${\delta F^{\alpha} \over \delta s} = - d \: F_{\rho} \: (C^{\alpha} \: D^{\rho} - C^{\rho} \: D^{\alpha}) \ . \label{FWS}\\$$
Fermi-Walker transport given by (\[FWS\]) is along the spacelike curve $x^{\beta}_{s}(s)$, whose unit spacelike tangent vector is $D^{\alpha}$. In [@A] we defined a new orthonormal tetrad, expressions (\[UO\]-\[WO\]) for spacetimes where non-null electromagnetic fields are present $ (U^{\alpha}, V^{\alpha}, Z^{\alpha}, W^{\alpha})$. Written in terms of these tetrad vectors, the electromagnetic vector field is,
$$f_{\alpha\beta} = -2\:\sqrt{-Q/2}\:\:\cos\alpha\:\:U_{[\alpha}\:V_{\beta]} +
2\:\sqrt{-Q/2}\:\:\sin\alpha\:\:Z_{[\alpha}\:W_{\beta]}\ .\label{EMF}$$
It is precisely the tetrad structure of (\[EMF\]) that allows for a “connection” to the Fermi-Walker transport. It is also the possibility of “rotating” through a scalar angle $\phi$ on blade one, the tetrad vectors $U^{\alpha}$ and $V^{\alpha}$, such that $U_{[\alpha}\:V_{\beta]}$ remains invariant [@A], that we have the freedom to build timelike curve congruences as a function of the “angle” $\phi$ on blade one. Similar on blade two. A “rotation” of the tetrad vectors $Z^{\alpha}$ and $W^{\alpha}$ through an “angle” $\varphi$, such that $Z_{[\alpha}\:W_{\beta]}$ remains invariant [@A], entails the freedom to build spacelike curve congruences as a function of the “angle” $\varphi$ on blade two. We must remember that to each local function $\phi$ there corresponds, through an isomorphic relationship, an electromagnetic gauge transformation local scalar $\Lambda$ as a counterpart to $\phi$ on blade one [@A]. Similar for $\varphi$ on blade two. That is, an electromagnetic local gauge transformation of the vectors $(U^{\alpha}, V^{\alpha})$ generates a “rotation” of these vectors on blade one at every point in spacetime, such that these vectors remain on blade one after the “rotation”. We called the Abelian group of “rotations” on blade one, $LB1$. Similar for $(Z^{\alpha}, W^{\alpha})$ on blade two, under the group $LB2$.
Summarizing the above ideas, we are going to find for each tetrad, solution to the Einstein-Maxwell equations, congruences of timelike curves, solutions to the Lorentz equation.
The Lorentz equation {#Lorentzequation}
====================
We are going to consider a timelike curve in spacetime, such that at each point, the tangent to this curve is given by the unit timelike vector $ u^{\alpha} = u\:U^{\alpha}_{(\phi)} + v\:V^{\alpha}_{(\phi)} + z\:Z^{\alpha}_{(\varphi)} + w\:W^{\alpha}_{(\varphi)}$; $\:\:\:u^{\alpha}\: u_{\alpha} = -1$. The functions $u$, $v$, $z$ and $w$ are local scalars. The “angles” $(\phi,\varphi)$ are also local scalars and we are introducing them in order to express $u^{\alpha}$ in the most general way. The tetrad vectors $ (U^{\alpha}, V^{\alpha}, Z^{\alpha}, W^{\alpha})$ are provided through (\[UO\]-\[WO\]). Simultaneously, we are going to consider the “rotated” tetrad $(\tilde{U}^{\alpha}, \tilde{V}^{\alpha}, \tilde{Z}^{\alpha}, \tilde{W}^{\alpha})$ defined below through expressions (\[uUO\]-\[uWO\]). For the sake of simplicity we are going to assume that the transformation of the two vectors $(U^{\alpha},\:V^{\alpha})$ on blade one, given in (\[UO\]-\[VO\]) by the “angle” $\phi$ is a proper transformation. For improper transformations the result follows the same lines [@A]. Therefore we can write,
$$\begin{aligned}
U^{\alpha}_{(\phi)} &=& \cosh(\phi)\: U^{\alpha} + \sinh(\phi)\: V^{\alpha} \label{UT} \\
V^{\alpha}_{(\phi)} &=& \sinh(\phi)\: U^{\alpha} + \cosh(\phi)\: V^{\alpha} \label{VT} \ .\end{aligned}$$
The transformation of the two tetrad vectors $(Z^{\alpha},\:W^{\alpha})$ on blade two, given in (\[ZO\]-\[WO\]) by the “angle” $\varphi$, can be expressed as,
$$\begin{aligned}
Z^{\alpha}_{(\varphi)} &=& \cos(\varphi)\: Z^{\alpha} - \sin(\varphi)\: W^{\alpha} \label{ZT} \\
W^{\alpha}_{(\varphi)} &=& \sin(\varphi)\: Z^{\alpha} + \cos(\varphi)\: W^{\alpha} \label{WT} \ .\end{aligned}$$
We can define then, a “rotated” orthonormal tetrad with respect to the tetrad (\[UO\]-\[WO\]), along the integral timelike curve as,
$$\begin{aligned}
\tilde{U}^{\alpha} &=& \xi^{\alpha\lambda}\:\xi_{\rho\lambda}\:u^{\rho} \:
/ \: (\: \sqrt{-Q/2} \: \sqrt{u_{\mu} \ \xi^{\mu\sigma} \
\xi_{\nu\sigma} \ u^{\nu}}\:) \label{uUO}\\
\tilde{V}^{\alpha} &=& \xi^{\alpha\lambda}\:u_{\lambda} \:
/ \: (\:\sqrt{u_{\mu} \ \xi^{\mu\sigma} \
\xi_{\nu\sigma} \ u^{\nu}}\:) \label{uVO}\\
\tilde{Z}^{\alpha} &=& \ast \xi^{\alpha\lambda} \: u_{\lambda} \:
/ \: (\:\sqrt{ u_{\mu} \ast \xi^{\mu\sigma}
\ast \xi_{\nu\sigma} u^{\nu}}\:)
\label{uZO}\\
\tilde{W}^{\alpha} &=& \ast \xi^{\alpha\lambda}\: \ast \xi_{\rho\lambda}
\: u^{\rho} \: / \: (\:\sqrt{-Q/2} \: \sqrt{ u_{\mu}
\ast \xi^{\mu\sigma} \ast \xi_{\nu\sigma} u^{\nu}}\:) \ ,
\label{uWO}\end{aligned}$$
where $u^{\alpha}$ is the tangent along the timelike curve. We must be careful about the definition of tetrad vectors (\[uZO\]-\[uWO\]). There is the possibility that the timelike tangent vector $u^{\alpha}$ does not have non-zero components on blade two. We solve this particular problem as an example in the next section. As a comment we can say that if we replace the two tetrad vectors (\[uZO\]-\[uWO\]) for the vectors (\[ZO\]-\[WO\]), we would have an orthonormal tetrad too. We proceed then, to write the Frenet-Serret expressions associated to the timelike curve whose tangent is $u^{\alpha}$.
$$\begin{aligned}
{\delta u^{\alpha} \over \delta s} &=& A\:\left(u_{\beta}\: \xi^{\alpha\beta}\right) + B\:\left(u_{\beta}\: \ast \xi^{\alpha\beta}\right) + C\: \left( (u_{\beta}\:\tilde{W}^{\beta})\:\tilde{U}^{\alpha} - (u_{\beta}\:\tilde{U}^{\beta})\:\tilde{W}^{\alpha} \right)
\label{FSE} \ ,\end{aligned}$$
where $ \tilde{U}^{\alpha}$ and $ \tilde{W}^{\beta}$ are provided by expressions (\[uUO\]) and (\[uWO\]). The functions $A$, $B$ and $C$ are local scalars. It is very simple to prove that $u_{\alpha}$ is orthogonal to $u_{\beta}\: \xi^{\alpha\beta}$, $u_{\beta}\: \ast \xi^{\alpha\beta}$, and $(u_{\beta}\:\tilde{W}^{\beta})\:\tilde{U}^{\alpha} - (u_{\beta}\:\tilde{U}^{\beta})\:\tilde{W}^{\alpha}$. It is also very simple to prove using the tetrad vectors (\[uUO\]-\[uWO\]) and the identities (\[i1\]-\[i2\]) that the three vectors $u_{\beta}\: \xi^{\alpha\beta}$, $u_{\beta}\: \ast \xi^{\alpha\beta}$, and $(u_{\beta}\:\tilde{W}^{\beta})\:\tilde{U}^{\alpha} - (u_{\beta}\:\tilde{U}^{\beta})\:\tilde{W}^{\alpha}$ are orthogonal among them. The extremal field and its dual can be written as [@A],
$$\begin{aligned}
\xi_{\alpha\beta} &=& -2\:\sqrt{-Q/2}\:U_{[\alpha}\:V_{\beta]}\label{ET}\\
\ast \xi_{\alpha\beta} &=& 2\:\sqrt{-Q/2}\:Z_{[\alpha}\:W_{\beta]}\ .\label{DET}\end{aligned}$$
Because of gauge invariance we also know that [@A],
$$\begin{aligned}
U^{[\alpha}_{(\phi)}\:V^{\beta]}_{(\phi)} = U^{[\alpha}\:V^{\beta]} = \tilde{U}^{[\alpha}\:\tilde{V}^{\beta]} \label{GEQ1}\\
Z^{[\alpha}_{(\varphi)}\:W^{\beta]}_{(\varphi)} = Z^{[\alpha}\:W^{\beta]} = \tilde{Z}^{[\alpha}\:\tilde{W}^{\beta]} \label{GEQ2} \ .\end{aligned}$$
We can consider the curvature associated scalars to be, $A = (q/m)\:\cos(\alpha)$, $B = (q/m)\:\sin(\alpha)$ and $C = 0$ where $q$ is the charge and $m$ is the mass of the particle following the integral timelike curve to which $u^{\alpha}$ is tangent. We would like to emphasize that expression $\left( \tilde{W}^{\beta}\:\tilde{U}^{\alpha} - \tilde{W}^{\alpha}\:\tilde{U}^{\beta} \right)$ is not invariant under gauge transformations of the tetrad vectors. Therefore, we have one further justification for considering the case where the curvature scalar $C = 0$. Then, we can rewrite (\[FSE\]) as,
$$\begin{aligned}
{\delta u^{\alpha} \over \delta s} &=& -(q/m)\:\sqrt{-Q/2}\:\cos(\alpha) \:u_{\beta}\: \left(U^{\alpha}\:V^{\beta}- U^{\beta}\:V^{\alpha}\right) + (q/m)\:\sqrt{-Q/2}\:\sin(\alpha)\:u_{\beta}\: \left(Z^{\alpha}\:W^{\beta}- Z^{\beta}\:W^{\alpha}\right) \label{FSESIMPALPHA} \ .\end{aligned}$$
Making use of the expression for the electromagnetic field (\[EMF\]) we can write (\[FSESIMPALPHA\]) as,
$$\begin{aligned}
{\delta u^{\alpha} \over \delta s} = (q/m)\:u_{\beta}\:f^{\alpha\beta} \label{FSEMF} \ .\end{aligned}$$
A particular case {#anotherview}
=================
We are going to consider a curve in spacetime, such that at each point, the projection of its tangent vector on blade one is given by ${dx_{1}^{\alpha}(s) \over ds}$, and on blade two by ${dx_{2}^{\alpha}(s) \over ds} = 0$. It is our objective to find special congruences of solutions to the Lorentz equation through Fermi-Walker transport, if they exist at all. Then, we are going to proceed as follows. For a “generic” scalar “angle” $\phi_{1}$ on blade one, and a “generic” scalar “angle” $\varphi_{1}$ on blade two, we are going to study the Frenet-Serret formulae associated to the timelike curve on blade one, whose tangent at every point is given by the unit timelike vector $U^{\alpha}_{(\phi_{1})} = {dx_{1}^{\alpha}(s) \over ds}= (u\:U^{\alpha}_{(\phi)} + v\:V^{\alpha}_{(\phi)})/\sqrt{u^2-v^2} $. Therefore, we are going to analyze the Frenet-Serret formulae for the tetrad $(U^{\alpha} _{(\phi_{1})},\:V^{\alpha} _{(\phi_{1})},\:Z^{\alpha} _{(\varphi_{1})},\:W^{\alpha} _{(\varphi_{1})})$. For the sake of simplicity we are going to assume that the transformation of the two vectors $(U^{\alpha},\:V^{\alpha})$ on blade one, given in (\[UTAV\]-\[VTAV\]) by the “angle” $\phi_{1}$ is a proper transformation. For improper transformations the result follows the same lines [@A]. Therefore we can write,
$$\begin{aligned}
U^{\alpha}_{(\phi_{1})} &=& \cosh(\phi_{1})\: U^{\alpha} + \sinh(\phi_{1})\: V^{\alpha} \label{UTAV} \\
V^{\alpha}_{(\phi_{1})} &=& \sinh(\phi_{1})\: U^{\alpha} + \cosh(\phi_{1})\: V^{\alpha} \label{VTAV} \ .\end{aligned}$$
The transformation of the two tetrad vectors $(Z^{\alpha},\:W^{\alpha})$ on blade two, given in (\[ZTAV\]-\[WTAV\]) by the “angle” $\varphi_{1}$, can be expressed as,
$$\begin{aligned}
Z^{\alpha}_{(\varphi_{1})} &=& \cos(\varphi_{1})\: Z^{\alpha} -\sin(\varphi_{1})\: W^{\alpha} \label{ZTAV} \\
W^{\alpha}_{(\varphi_{1})} &=& \sin(\varphi_{1})\: Z^{\alpha} + \cos(\varphi_{1})\: W^{\alpha} \label{WTAV} \ .\end{aligned}$$
We proceed then, to write the Frenet-Serret expressions associated to the timelike curve on blade one $x_{1}^{\alpha}(s)$ whose tangent is $U^{\alpha}_{(\phi_{1})}$,
$$\begin{aligned}
{\delta U^{\alpha}_{(\phi_{1})} \over \delta s} &=& a_{1} \: V^{\alpha}_{(\phi_{1})} + b_{1}\: Z^{\alpha}_{(\varphi_{1})} + q_{1}\:W^{\alpha}_{(\varphi_{1})} \label{FSU1}\\
{\delta V^{\alpha}_{(\phi_{1})} \over \delta s} &=& a_{1} \: U^{\alpha}_{(\phi_{1})} + p_{1}\: Z^{\alpha}_{(\varphi_{1})} + d_{1}\: W^{\alpha}_{(\varphi_{1})} \label{FSV1}\\
{\delta Z^{\alpha}_{(\varphi_{1})} \over \delta s} &=& c_{1} \: W^{\alpha}_{(\varphi_{1})} - p_{1} \: V^{\alpha}_{(\phi_{1})} + b_{1} \: U^{\alpha}_{(\phi_{1})} \label{FSZ1}\\
{\delta W^{\alpha}_{(\varphi_{1})} \over \delta s} &=& -c_{1} \: Z^{\alpha}_{(\varphi_{1})} - d_{1} \: V^{\alpha}_{(\phi_{1})} + q_{1}\:U^{\alpha}_{(\phi_{1})} \label{FSW1} \ .\end{aligned}$$
The functions $a_{1}$, $b_{1}$, $c_{1}$, $d_{1}$, $p_{1}$ and $q_{1}$ are local scalars. In addition, we are also going to introduce the auxiliary scalar variables, $$\begin{aligned}
A_{z} &=& Z_{\alpha}\:U^{\alpha}_{\:\:\:;\beta}\:U^{\beta} \label{ASz} \\
B_{z} &=& Z_{\alpha}\:U^{\alpha}_{\:\:\:;\beta}\:V^{\beta} \label{BSz} \\
C_{z} &=& Z_{\alpha}\:V^{\alpha}_{\:\:\:;\beta}\:U^{\beta} \label{CSz} \\
D_{z} &=& Z_{\alpha}\:V^{\alpha}_{\:\:\:;\beta}\:V^{\beta} \label{DSz} \\
A_{w} &=& W_{\alpha}\:U^{\alpha}_{\:\:\:;\beta}\:U^{\beta} \label{ASw} \\
B_{w} &=& W_{\alpha}\:U^{\alpha}_{\:\:\:;\beta}\:V^{\beta} \label{BSw} \\
C_{w} &=& W_{\alpha}\:V^{\alpha}_{\:\:\:;\beta}\:U^{\beta} \label{CSw} \\
D_{w} &=& W_{\alpha}\:V^{\alpha}_{\:\:\:;\beta}\:V^{\beta} \label{DSw} \ .\end{aligned}$$
Now comes the core on the idea to find this special solution to the Lorentz equation. Let us demand or impose in (\[FSU1\]) the following equations,
$$\begin{aligned}
Z_{\alpha(\varphi_{1})} \:{\delta U^{\alpha}_{(\phi_{1})} \over \delta s} &=& 0 \label{eq1b1} \\
W_{\alpha(\varphi_{1})} \:{\delta U^{\alpha}_{(\phi_{1})} \over \delta s} &=& 0 \ .\label{eq2b1} \end{aligned}$$
These equations (\[eq1b1\]-\[eq2b1\]) represent equations for the two local scalar functions $(\phi_{1},\varphi_{1})$. If we replace in (\[eq1b1\]-\[eq2b1\]) expressions (\[UTAV\]-\[VTAV\]) and (\[ZTAV\]-\[WTAV\]), work out the algebraic equations, and make use of (\[ASz\]-\[DSw\]) we find,
$$\begin{aligned}
\tan(\varphi_{1}) &=& {\left(\cosh^{2}(\phi_{1})\:A_{z} + \sinh^{2}(\phi_{1})\:D_{z} + \cosh(\phi_{1})\:\sinh(\phi_{1})\:(B_{z} + C_{z})\right) \over \left(\cosh^{2}(\phi_{1})\:A_{w} + \sinh^{2}(\phi_{1})\:D_{w} + \cosh(\phi_{1})\:\sinh(\phi_{1})\:(B_{w} + C_{w})\right)} \label{eqvarphi1b1} \end{aligned}$$
$$\begin{aligned}
\tan(\varphi_{1}) &=& - { \left(\cosh^{2}(\phi_{1})\:A_{w} + \sinh^{2}(\phi_{1})\:D_{w} + \cosh(\phi_{1})\:\sinh(\phi_{1})\:(B_{w} + C_{w})\right) \over \left(\cosh^{2}(\phi_{1})\:A_{z} + \sinh^{2}(\phi_{1})\:D_{z} + \cosh(\phi_{1})\:\sinh(\phi_{1})\:(B_{z} + C_{z})\right) } \ .\label{eqvarphi2b1}\end{aligned}$$
Now, in order for both equations (\[eqvarphi1b1\]-\[eqvarphi2b1\]) to be compatible, it can be readily noticed that the following equality would have to be true, $\tan(\varphi_{1}) = - 1 / \tan(\varphi_{1})$. Then, there are no solutions to the “pure” blade one Lorentz equation problem.
A more general example {#anothergenview}
======================
It is our objective to find special congruences of solutions to the Lorentz equation through Fermi-Walker transport, as a more general example. We are increasing the level of complexity following a pedagogical scheme. Thus, we are going to proceed as follows. For a “generic” scalar “angle” $\phi_{1}$ on blade one, and a “generic” scalar “angle” $\varphi_{1}$ on blade two, we are going to study the Frenet-Serret formulae associated to the timelike curve, whose projection on blade one at every point is given by the timelike vector $(M / \sqrt{M^2 - N^2}\:)\:U^{\alpha}_{(\phi_{1})} = (M / \sqrt{M^2 - N^2}\:)\: (u\:U^{\alpha}_{(\phi)} + v\:V^{\alpha}_{(\phi)})/\sqrt{u^2-v^2} $ and its projection on blade two is given by $(N / \sqrt{M^2 - N^2}\:)\:W^{\alpha}_{(\varphi_{1})} = (N / \sqrt{M^2 - N^2}\:)\: (z\:Z^{\alpha}_{(\varphi)} + w\:W^{\alpha}_{(\varphi)})/\sqrt{z^2+w^2} $. Both $M$ and $N$ are local scalars. We have already analyzed the Frenet-Serret formulae for the tetrad $(U^{\alpha} _{(\phi_{1})},\:V^{\alpha} _{(\phi_{1})},\:Z^{\alpha} _{(\varphi_{1})},\:W^{\alpha} _{(\varphi_{1})})$. We are simply going to take advantage of this previous analysis to find solutions to the Lorentz equation of a more general nature. For brevity we are going to call $A = M / \sqrt{M^2 - N^2}$ and $B = N / \sqrt{M^2 - N^2}$. We proceed then, to write the Frenet-Serret expressions associated to the projections of the curve both on blade one and two,
$$\begin{aligned}
{\delta \left( A\:U^{\alpha}_{(\phi_{1})}\right) \over \delta s} &=& {\delta \left( A \right) \over \delta s}\: U^{\alpha}_{(\phi_{1})} + A\:{\delta U^{\alpha}_{(\phi_{1})} \over \delta s}\label{GEN1}\\
{\delta U^{\alpha}_{(\phi_{1})} \over \delta s} &=& a_{1} \: V^{\alpha}_{(\phi_{1})} + b_{1}\: Z^{\alpha}_{(\varphi_{1})} + q_{1}\:W^{\alpha}_{(\varphi_{1})} \label{GEN11}\\
{\delta V^{\alpha}_{(\phi_{1})} \over \delta s} &=& a_{1} \: U^{\alpha}_{(\phi_{1})} + p_{1}\: Z^{\alpha}_{(\varphi_{1})} + d_{1}\: W^{\alpha}_{(\varphi_{1})} \label{GEN2}\\
{\delta Z^{\alpha}_{(\varphi_{1})} \over \delta s} &=& c_{1} \: W^{\alpha}_{(\varphi_{1})} - p_{1} \: V^{\alpha}_{(\phi_{1})} + b_{1} \: U^{\alpha}_{(\phi_{1})} \label{GEN3}\\
{\delta W^{\alpha}_{(\varphi_{1})} \over \delta s} &=& -c_{1} \: Z^{\alpha}_{(\varphi_{1})} - d_{1} \: V^{\alpha}_{(\phi_{1})} + q_{1}\:U^{\alpha}_{(\phi_{1})} \label{GEN44}\\
{\delta \left( B\:W^{\alpha}_{(\varphi_{1})}\right) \over \delta s} &=& {\delta \left( B \right) \over \delta s}\: W^{\alpha}_{(\varphi_{1})} + B\:{\delta W^{\alpha}_{(\varphi_{1})} \over \delta s} \label{GEN4} \ .\end{aligned}$$
The functions $a_{1}$, $b_{1}$, $c_{1}$, $d_{1}$, $p_{1}$ and $q_{1}$ are again local scalars.
Once more comes the core on the idea to find these more general solutions to the Lorentz equation. Let us demand or impose in (\[GEN1\]-\[GEN4\]) the following equations,
$$\begin{aligned}
{\delta \left( A \right) \over \delta s} &=& 0 \label{SCALAR1} \\
{\delta \left( B \right) \over \delta s} &=& 0 \label{SCALAR2} \\
W_{\alpha(\varphi_{1})} \:{\delta U^{\alpha}_{(\phi_{1})} \over \delta s} &=& 0 \ .\label{ORTHO2} \end{aligned}$$
These equations (\[SCALAR1\]-\[SCALAR2\]) represent equations for the two local scalar functions $(M, N)$. Expression (\[ORTHO2\]) is just one equation for $(\phi_{1},\varphi_{1})$. In order to find one solution for the pair $(\phi_{1},\varphi_{1})$ we need another equation. It can be supplemented by different additional equations, for instance, $Z_{\alpha(\varphi_{1})} \:{\delta V^{\alpha}_{(\phi_{1})} \over \delta s} = 0$. Each possible choice for an additional equation yields different solutions $(\phi_{1},\varphi_{1})$ to the Lorentz equation.
The problem is posed as follows. First, we are supposed to know the tetrad vectors and their covariant derivatives from solutions to the Einstein-Maxwell equations (\[EM1\]-\[EM3\]). Second, we solve equations (\[SCALAR1\],\[SCALAR2\]) for the scalars $(M, N)$. Then we have available equation (\[ORTHO2\]) for the pair $(\phi_{1},\varphi_{1})$. As mentioned above we can supplement this last equation (\[ORTHO2\]) with another one in order to find $(\phi_{1},\varphi_{1})$. We finally see that Frenet-Serret equations (\[GEN11\]) and (\[GEN44\]) turn out to be,
$$\begin{aligned}
{\delta U^{\alpha}_{(\phi_{1})} \over \delta s} &=& a_{1} \: V^{\alpha}_{(\phi_{1})} + b_{1}\: Z^{\alpha}_{(\varphi_{1})} \label{RED11} \\
{\delta W^{\alpha}_{(\varphi_{1})} \over \delta s} &=& -c_{1} \: Z^{\alpha}_{(\varphi_{1})} - d_{1} \: V^{\alpha}_{(\phi_{1})} \label{RED44}
\ . \end{aligned}$$
Our intention is to Fermi-Walker transport the timelike vector $A\:U^{\alpha}_{(\phi_{1})} + B \: W^{\alpha}_{(\varphi_{1})} $ along the integral timelike curve whose tangent is this same vector that in turn satisfies equations (\[SCALAR1\]-\[SCALAR2\]),
$$\begin{aligned}
{\delta \left( A\: U^{\alpha}_{(\phi_{1})} + B\: W^{\alpha}_{(\varphi_{1})} \right) \over \delta s} &=& A \: {\delta U^{\alpha}_{(\phi_{1})} \over \delta s} + B \: {\delta W^{\alpha}_{(\varphi_{1})} \over \delta s} \ . \label{GENEQ1}\end{aligned}$$
Now, using equations (\[RED11\]-\[RED44\]) we can write,
$$\begin{aligned}
{\delta \left( A\: U^{\alpha}_{(\phi_{1})} + B\: W^{\alpha}_{(\varphi_{1})} \right) \over \delta s} &=& A \: \left( a_{1} \: V^{\alpha}_{(\phi_{1})} + b_{1}\: Z^{\alpha}_{(\varphi_{1})} \right) + B \: \left( -c_{1} \: Z^{\alpha}_{(\varphi_{1})} - d_{1} \: V^{\alpha}_{(\phi_{1})} \right) \ . \label{GENEQ2}\end{aligned}$$
It is simple to see that equation (\[GENEQ2\]) can be rewritten,
$$\begin{aligned}
{\delta \left( A\: U^{\alpha}_{(\phi_{1})} + B\: W^{\alpha}_{(\varphi_{1})} \right) \over \delta s} = (q/m) \: \left( A \: U_{\rho(\phi_{1})} + B\: W_{\rho(\varphi_{1})} \right) \nonumber \\ \nonumber \\ \left( - \sqrt{-Q/2}\:\:\cos\alpha\:\: \left( U^{\alpha}_{(\phi_{1})}\:V^{\rho}_{(\phi_{1})} - U^{\rho}_{(\phi_{1})}\:V^{\alpha}_{(\phi_{1})} \right) +
\sqrt{-Q/2}\:\:\sin\alpha\:\:\left( Z^{\alpha}_{(\varphi_{1})}
\:W^{\rho}_{(\varphi_{1})} - Z^{\rho}_{(\varphi_{1})}
\:W^{\alpha}_{(\varphi_{1})} \right)\right)\ , \label{GENEQ3}\end{aligned}$$
as long as we ask for the scalar functions $a_{1}$, $b_{1}$, $c_{1}$ and $d_{1}$ to satisfy the following algebraic equations,
$$\begin{aligned}
-\:\sqrt{-Q/2}\:\:\cos\alpha\:A\:q/m &=& A\:a_{1} - B\:d_{1} \label{algeb1}\\
\:\sqrt{-Q/2}\:\:\sin\alpha\:B\:q/m &=& A\:b_{1} - B\:c_{1} \label{algeb2}\ .\end{aligned}$$
We can also remind ourselves of the gauge invariant equalities [@A],
$$\begin{aligned}
U^{[\alpha}_{(\phi_{1})}\:V^{\beta]}_{(\phi_{1})} = U^{[\alpha}\:V^{\beta]} = U^{[\alpha}_{(\phi)}\:V^{\beta]}_{(\phi)} \label{GEQphi1}\\
Z^{[\alpha}_{(\varphi_{1})}\:W^{\beta]}_{(\varphi_{1})} = Z^{[\alpha}\:W^{\beta]} = Z^{[\alpha}_{(\varphi)}\:W^{\beta]}_{(\varphi)} \label{GEQvarphi2} \ .\end{aligned}$$
Therefore, equation (\[GENEQ3\]) can be expressed as,
$$\begin{aligned}
{\delta \left( A\: U^{\alpha}_{(\phi_{1})} + B\: W^{\alpha}_{(\varphi_{1})} \right) \over \delta s} &=& \left( A \: U_{\rho(\phi_{1})} + B\: W_{\rho(\varphi_{1})} \right) \: \left( -2\:\sqrt{-Q/2}\:\:\cos\alpha\:\:U^{[\alpha}\:V^{\rho]} +
2\:\sqrt{-Q/2}\:\:\sin\alpha\:\:Z^{[\alpha}\:W^{\rho]}\right)\ . \label{GENEQ4}\end{aligned}$$
Calling $u^{\alpha} = A\: U^{\alpha}_{(\phi_{1})} + B\: W^{\alpha}_{(\varphi_{1})}$, reminding equation (\[EMF\]), and taking into account the orthogonality relations, $U_{\rho}\:Z^{\rho} = U_{\rho}\:W^{\rho} = W_{\rho}\:V^{\rho} = 0$, we can rewrite (\[GENEQ4\]) as,
$${\delta u^{\alpha} \over \delta s} = (q/m)\: u_{\rho}\: f^{\alpha\rho}\ , \label{Lorentz}\\$$
that is, the Lorentz “force” equation for the particle of mass $m$ and charge $q$, through Fermi-Walker transport of the timelike vector $u^{\alpha}$ along the integral timelike curve whose tangent is $u^{\alpha} = A\: U^{\alpha}_{(\phi_{1})} + B\: W^{\alpha}_{(\varphi_{1})}$.
Conclusions
===========
The introduction of new tetrads for non-null electromagnetic fields in [@A] allows for the geometrical analysis of the Lorentz equation. Through Fermi-Walker transport and Frenet-Serret expressions it is possible to identify curvature scalars in terms of the local scalar functions associated to the non-null electromagnetic field present in the curved spacetime. A blade one “pure” solutions are studied through the use of Frenet-Serret expressions, and more general solutions are found for solutions with non-trivial projections on both blades. The new tetrads allow for a better insight regarding the relationship between traditional Abelian gauge theories on one hand, and traditional Riemannian geometry on the other hand. The key lies upon the two theorems proved in [@A] relating through isomorphisms the groups $U(1)$ and LB1, LB2. The local “internal” transformations that leave invariant the electromagnetic field are proved to be isomorphic to the local tetrad transformations either on blade one or two. In other words, the local “internal” group of transformations is isomorphic to a subgroup of the local group of Lorentz tetrad transformations. This was not known before. It is precisely this “gauge geometry” the tool that allows for a new understanding of a traditional equation like the Lorentz equation. Other future applications are possible. For instance, the study of the kinematics in these spacetimes [@RG][@RW][@MC2][@EJKS], that is, the search for a “connection” to the theory of embeddings, time slicings [@KK][@JDBKK][@FM][@FAEP], the initial value formulation [@JWY1][@JWY2][@NOMJWY][@HY][@JY][@LICH][@YCB][@CMDWYCB], the Cauchy evolution [@ADM][@LSJWY], etc. If there is a geometry such that the gradient of the complexion is a timelike vector field, then there would be the possibility of finding a slicing such that the leaves of constant complexion foliate spacetime, and potentially could be a simplification tool.
Simplicity is the most sought after quality in all these geometrical descriptions, and the previously introduced tetrads, blades, isomorphisms, prove to be practical and useful. We quote from [@MW] “The question poses itself insistently to find a point of view which will make Rainich-Riemannian geometry seem a particularly natural kind of geometry to consider”.
I am grateful to R. Gambini and J. Pullin for reading this manuscript and for many fruitful discussions.
A. Garat, J. Math. Phys. [**46**]{}, 102502 (2005).
C. Misner and J. A. Wheeler, Annals of Physics [**2**]{}, 525 (1957).
J. A. Schouten, [*Ricci Calculus: An Introduction to Tensor Calculus and Its Geometrical Applications*]{} (Springer, Berlin, 1954).
F. Rohrlich, [*Classical Charged Particles*]{} (Addison Wesley Publishing Company, 1965).
J. A. Wheeler, Rev. Mod. Phys. [**33**]{}, 63 (1961).
B. S. DeWitt and R. W. Brehme, Annals of Physics [**9**]{}, 220 (1960).
B. S. DeWitt and C. M. DeWitt, Physics [**1**]{}, 3 (1964).
M. Carmeli, Phys. Rev, [**138**]{}, 1003 (1965).
J. L. Synge , [*Relativity: The General Theory*]{} (North Holland Publishing Company, Amsterdam, 1960).
H. Stephani , [*General Relativity*]{} (Cambridge University Press, Cambridge, 2000).
F. K. Manasse and C. W. Misner, J. Math. Phys. [**4**]{}, 735 (1963).
R. Geroch, [*General Relativity: From A to B*]{} [(University of Chicago Press, Chicago, 1978)]{}.
R. Wald, [*General Relativity*]{} (University of Chicago Press, Chicago, 1984).
M. Carmeli, [*Classical Fields: General Relativity and Gauge Theory*]{} (J. Wiley & Sons, New York, 1982).
J. Ehlers, P. Jordan, W. Kundt and R. Sachs, Akad. Wiss. Lit. Mainz Abh. Math.-Natur. Kl, [**11**]{} (Mainz 4), 793 (1961).
K. Kuchař, J. Math. Phys. [**13**]{}, 768 (1972); [**17**]{}, 777 (1976); [**17**]{}, 792 (1976); [**17**]{}, 801 (1976);[**18**]{}, 1589 (1977); Phys. Rev D, [**4**]{}, 955 (1971).
J. D. Brown and K. Kuchař, Phys. Rev D, [**51**]{}, 5600 (1995).
A. E. Fischer and J. E. Marsden, J. Math. Phys. [**13**]{}, 546 (1972).
F. A. E. Pirani, [*Les Theories Relativistes de la Gravitation*]{} (CNRS, Paris, 1962).
J. W. York, J. Math. Phys. [**13**]{}, 125 (1972); [**14**]{}, 456 (1973)
J. W. York, Phys. Rev D, [**10**]{}, 428 (1974).
N. O‘Murchadha and J. W. York, [**14**]{}, 1551 (1973).
H. P. Pfeiffer and J. W. York, Phys. Rev D, [**67**]{}, 044022 (2003).
R. T. Jantzen and J. W. York /gr-qc 0603069 (2006).
A. Lichnerowicz, J. Math. Pure and Appl. [**23**]{}, 37 (1944).
Y. Choquet-Bruhat in [*Gravitation: An Introduction to Current Research*]{} edited by L. Witten (Wiley, New York, 1962).
C. M. DeWitt and Y. Choquet-Bruhat [*Analysis, Manifolds and Physics*]{} edited by (North-holland, The Netherlands , 1982).
R. Arnowitt, S. Deser and C. W. Misner, [*“The Dynamics of General Relativity”*]{} in [*Gravitation: An Introduction to Current Research*]{} edited by L. Witten (Wiley, New York, 1962).
L. Smarr and J. W. York, Phys. Rev. D [**17**]{}, 2529 (1978).
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