diff --git "a/1tE4T4oBgHgl3EQfzg0x/content/tmp_files/2301.05274v1.pdf.txt" "b/1tE4T4oBgHgl3EQfzg0x/content/tmp_files/2301.05274v1.pdf.txt"
new file mode 100644--- /dev/null
+++ "b/1tE4T4oBgHgl3EQfzg0x/content/tmp_files/2301.05274v1.pdf.txt"
@@ -0,0 +1,5182 @@
+arXiv:2301.05274v1  [math.PR]  12 Jan 2023
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE
+CHAOS ON PHASE BOUNDARIES
+HUBERT LACOIN
+Abstract. The complex Gaussian Multiplicative Chaos (or complex GMC) is infor-
+mally defined as a random measure eγXdx where X is a log correlated Gaussian field on
+Rd and γ “ α ` iβ is a complex parameter. The correlation function of X is of the form
+Kpx, yq “ log
+1
+|x ´ y| ` Lpx, yq,
+where L is a continuous function. In the present paper, we consider the cases γ P PI{II
+and γ P P1
+II{III where
+PI{II :“ tα ` iβ : α, β P R ; |α| ą |β| ; |α| ` |β| “
+?
+2du,
+and
+P1
+II{III :“ tα ` iβ : α, β P R ; |α| “
+a
+d{2 ; |β| ą
+?
+2du,
+We prove that if X is replaced by an approximation Xε obtained via mollification, then
+eγXεdx, when properly rescaled, converges when ε Ñ 0. The limit does not depend on
+the mollification kernel. When γ P PI{II, the convergence holds in probability and in Lp
+for some value of p P r1,
+?
+2d{αq. When γ P P1
+II{III the convergence holds only in law.
+In this latter case, the limit can be described a complex Gaussian white noise with a
+random intensity given by a critical real GMC. The regions PI{II and P1
+II{III correspond to
+phase boundary between the three different regions of the complex GMC phase diagram.
+These results complete previous results obtained for the GMC in phase I [18] and III [16]
+and only leave as an open problem the question of convergence in phase II.
+2010 Mathematics Subject Classification: 60F99, 60G15, 82B99.
+Keywords: Random distributions, log-correlated fields, Gaussian Multiplicative Chaos.
+Contents
+1.
+Introduction
+2
+2.
+Main results
+5
+3.
+The martingale approximation for GMC
+8
+4.
+Proof of convergence results on for γ P PI{II
+13
+5.
+Proof of Proposition 3.5
+18
+6.
+Proof of Proposition 2.8
+27
+7.
+Proof of Theorem 3.6
+34
+Appendix A.
+Technical results and their proof
+36
+Appendix B.
+The convergence of Mγ
+ε as a distribution
+39
+Appendix C.
+Beyond star-scale invariance
+43
+Appendix D.
+Proof of Lemma 4.1
+47
+References
+49
+1
+
+2
+HUBERT LACOIN
+1. Introduction
+Let K : Rd ˆ Rd Ñ p´8, 8s be a positive definite kernel on Rd (d ě 1 is fixed) which
+admits a decomposition of the form
+Kpx, yq “ log
+1
+|x ´ y| ` Lpx, yq,
+(1.1)
+(with the convention logp1{0q “ 8) where L is a continuous function on R2d . A kernel
+K is positive definite if for ρ P CcpRdq (ρ continuous with compact support)
+ż
+R2d Kpx, yqρpxqρpyqdxdy ě 0.
+(1.2)
+Given a centered Gaussian field X with covariance K and γ “ α ` iβ a complex number
+(α, β P R) the complex Gaussian Multiplicative Chaos (or complex GMC) with parameter
+γ is the random distribution formally defined by the expression
+Mγpdxq “ eγXpxqdx.
+(1.3)
+A difficulty comes up when trying to give an interpretation to the r.h.s. of (1.3). A field
+X with a covariance given by (1.1) can be defined only as a random distribution. For a
+fixed x P Rd it is not possible to make sense of Xpxq.
+The problem of providing a mathematical construction of Mγ that gives a meaning to
+(1.3) was first considered by Kahane in [15] in the case where γ P R, we refer to [25, 27]
+for reviews on the subject. The case of γ P C was considered only more recently, see for
+instance [11, 12, 13, 16, 18, 19, 20] and references therein. The standard procedure to define
+the GMC is to use a sequence of approximation of the field X, consider the exponential
+of the approximation and then pass to the limit. Mostly two kinds of approximation of X
+have been considered in the literature:
+(A) A mollification of the field, Xε, via convolution with a smooth kernel on scale ε,
+(B) A martingale approximation, Xt, via an integral decomposition of the kernel K.
+In the present paper we present convergence results for the random distribution eγXεpxqdx
+and eγXtpxqdx and in a certain range of parameter γ. Before describing our results in more
+details and provide some motivation, we first rigorously introduce the setup.
+1.1. The mollification of a log-correlated field.
+Log-correlated fields defined as distributions. Since K is infinite on the diagonal, it is not
+possible to define a Gaussian field indexed by Rd with covariance function K. We consider
+instead a process indexed by test functions. We define pK, a bilinear form on CcpRdq (the
+set of compactly supported continuous functions) by
+pKpρ, ρ1q “
+ż
+R2d Kpx, yqρpxqρ1pyqdxdy.
+(1.4)
+Since pK is positive definite (in the usual sense: for any pρiqk
+i“1, the matrix pKpρi, ρjqk
+i,j“1 is
+positive definite), it is possible to define X “ xX, ρyρPCcpRdq a centered Gaussian process
+indexed by CcpRdq with covariance kernel given by pK.
+Remark 1.1. There exists a modification of the process X which take value in a distri-
+bution space (more specifically, such that X takes values in the Sobolev space Hs
+locpRdq for
+every s ă 0 (see the definition (B.3) in the appendix). For this reason (and although we
+will not use this fact) we refer to X as a random distribution.
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES 3
+Approximation of X via mollification. The random distribution X can be approximated
+by a sequence of functional fields - processes indexed by Rd - by the mean of mollification
+by a smooth kernel. Consider θ a nonnegative function in C8
+c pRdq (the set of infinitely
+differentiable functions in CcpRdq) whose compact support is included in Bp0, 1q (for the
+remainder of the paper Bpx, rq denotes the closed Euclidean ball of center x and radius r)
+and which satisfies
+ş
+Bp0,1q θpxqdx “ 1. We define for ε ą 0, θε :“ ε´dθpε´1¨q and consider
+pXεpxqqxPRd, the mollified version of X, that is
+Xεpxq :“ xX, θεpx ´ ¨qy
+(1.5)
+From (1.4), the field Xεp¨q has covariance
+Kεpx, yq :“ ErXεpxqXεpyqs “
+ż
+R2d θεpx ´ z1qθεpy ´ z2qKpz1, z2qdz1dz2.
+(1.6)
+We set Kεpxq :“ Kεpx, xq and extend this convention to other functions of two variables
+in Rd.
+Since Kε is infinitely differentiable - thus in particular is H¨older continuous -
+by Kolmogorov’s Continuity Theorem (e.g. [21, Theorem 2.9]) there exists a continuous
+modification of Xεp¨q. In the remainder of the paper, we always consider the continuous
+modification of a process when it exists.
+This ensures that integrals such as the one
+appearing in (1.7) are well defined. We define the distribution Mγ
+ε by setting for f P CcpRdq
+Mγ
+ε pfq :“
+ż
+Rd fpxqeγXεpxq´ γ2
+2 Kεpxqdx.
+(1.7)
+The question of interest in the present paper is the convergence of Mγ
+ε when ε Ñ 0.
+Remark 1.2. Note that, even if we have chosen to omit this dependence in the notation,
+Xε and Mγ
+ε both depend on the particular convolution kernel θ. An important feature of
+our results is that the limits obtained for Mγ
+ε pfq do not depend on θ.
+1.2. Star-scale invariance and our assumption on K. On top of assuming that K
+admits a decomposition like (1.1), we also assume that it has an almost star-scale invariant
+part (see the definition (1.8)-(1.9)). This assumption might seem at first quite restrictive,
+but it has been shown in [12] that it is locally satisfied as soon as the function L in
+(1.1) is sufficiently regular. In Appendix C we provides details concerning the regularity
+assumption for L and explain how to extend the validity of our results to all sufficiently
+regular log-correlated kernels using the ideas in [12].
+Following a terminology introduced in [12], we say that a the kernel K defined on Rd is
+almost star-scale invariant if it can be written in the form
+@x, y P Rd, Kpx, yq “
+ż 8
+0
+p1 ´ η1e´η2tqκpetpx ´ yqqdt,
+(1.8)
+where η1 P r0, 1s and η2 ą 0 are constants and the function κ P C8
+c pRdq is radial, nonneg-
+ative and definite positive. More precisely we assume the following:
+(i) κ P C8
+c pRdq and there exists rκ : R` Ñ r0, 8q such that κpxq :“ rκp|x|q,
+(ii) rκp0q “ 1 and rκprq “ 0 for r ě 1,
+(iii) The mapping px, yq ÞÑ κpx ´ yq defines a positive definite kernel on Rd ˆ Rd.
+We say furthermore that a kernel K has an almost star-scale invariant part, if
+@x, y P Rd, Kpx, yq “ K0px, yq ` Kpx, yq
+(1.9)
+where Kpx, yq is an almost star-scale invariant kernel and K0 is H¨older continuous on R2d
+and positive definite.
+
+4
+HUBERT LACOIN
+1.3. Phase transitions and phase diagrams for GMC. Our main results concerns
+the asymptotic behavior of Mγ
+ε in the specific range of γ given in the abstract. In order
+to properly motivate and present these results, it is necessary to introduce some context,
+and recall known facts about the phase diagram of the complex GMC.
+Phase transition at |α| “
+?
+2d for the real valued GMC. The question of the existence and
+identification of the limit
+lim
+εÑ0 Mα
+ε p¨q,
+has first been considered in the work of Kahane in the eighties [15], in the case when
+α P R. The obtained limit in that case crucially depends on α: when |α| ă
+?
+2d - referred
+to as the subcritical case - then Mα
+ε converges in probability to a non-trivial limiting
+distribution (see for instance [2, Theorem 1.1] for a short and self contained proof, we
+refer to the introduction in [2] for a detailed chronological account of results obtained for
+the subcritical case).
+When |α| ě
+?
+2d, we have limεÑ0 Mα
+ε pfq “ 0 and a rescaling procedure is needed in
+order to obtain a non-trivial limit. The phenomenology is however different according to
+whether |α| “
+?
+2d (α critical) or |α| ą
+?
+2d (α supercritical).
+In the critical case (α “ ˘
+?
+2d), is has been shown, under fairly mild assumptions (see
+[4, 5, 10, 26] and Theorem A below) that
+a
+log p1{εqMα
+ε converges in probability to a
+non-trivial limit called the critical GMC.
+When |α| ą
+?
+2d, the results are less complete. So far the convergence has not been
+proved for Mα
+ε but only for an approximating martingale sequence Mα
+t (see (3.6)) in [24].
+Besides this technical point, the most important differences with the case |α| ď
+?
+2d
+concerns the type of the convergence and the nature of limiting object. The convergence
+only holds only in law, and the limit is a purely atomic measure (a measure supported by
+a countable set) see [24, Corollary 2.3].
+Phase diagram for complex GMC. When γ is allowed to assume complex value, the phase
+diagram becomes more intricate. The complex plane can be divided in three open regions
+with intersecting boundaries
+PI :“
+␣
+α ` iβ : α2 ` β2 ă d
+(
+Y
+!
+α ` iβ : α P p
+a
+d{2,
+?
+2dq ; |α| ` |β| ă
+?
+2d
+)
+,
+PII :“
+!
+α ` iβ : |α| ` |β| ą
+?
+2d ; |α| ą
+a
+d{2
+)
+,
+PIII :“
+!
+α ` iβ : α2 ` β2 ą d ; |α| ă
+a
+d{2
+)
+.
+(1.10)
+This diagram first appeared in the context of complex Gaussian multiplicative cascade
+[3], and also serves to describe the behavior of other related models such as the complex
+REM [14] or complex branching Brownian Motion [6, 7, 23].
+The region PI corresponds to the subcritical phase. For γ P PI it has been proved
+[12, 18] that Mγ
+ε converges to a limit that does not depend on the mollifier θ.
+The region PII corresponds to the supercritical phase, in which it is believed that Mγ
+ε
+- after proper renormalization - converges only in law to a purely atomic random distri-
+bution. This conjecture is supported by rigorous results obtained in the case of Complex
+Branching Brownian Motion [6, 23].
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES 5
+PSfrag replacements
+β
+α
+?
+d
+a
+d{2
+?
+2d
+´
+?
+2d
+PI
+PII
+PIII
+Figure 1. The phase diagram of the complex GMC in the complex plane. Each region
+correspond to a different limiting behavior for M γ
+ε in terms of renormalization factor,
+type of convergence and properties of the limit. In the present paper, we prove results
+concerning the asymptotic behavior on frontier of PI Y PIII with PII. Results concerning
+convergence in PI Y PIII where proved in [12, 18] (for PI) and [16] (for PIII Y PI{III). The
+convergence in the region PII remains a challenging conjecture.
+Finally the region PIII corresponds to yet another asymptotic behavior for Mγ
+ε . Like in
+PII, Mγ
+ε - properly rescaled - only converges in law. The limit is given by a white noise
+whose intensity is random and is given by the real valued GMC with parameter 2α (which
+is subcritical, according to the definition of PIII). A similar convergence result holds on
+the boundary between PII and PIII that is
+PII{III :“
+!
+α ` iβ : α2 ` β2 “ d ; |α| ă
+a
+d{2
+)
+.
+These convergence statements for γ P PIII Y PI{III are proved in [16].
+The present contribution. The aim of the present paper is to come closer to a completition
+of the phase diagram by stating and proving convergence results for Mγ
+ε on the phase
+transition curves PI{II and PI{III as well as at the triple points PI{II{III.
+In each case,
+the limit obtained does not depend on the regularization kernel θ. We leave as an open
+problem the challenging task of proving a convergence result in the frozen phase PII.
+2. Main results
+For simplicity of notation, we consider, for the remainder of the paper and without loss
+of generality that γ is in the upper-right quarterplane of C, that is α, β ě 0.
+2.1. The boundary between phase I and II. Our first result concerns the case when
+γ lies on the boundary between regions I and II
+PI{II :“ tα ` iβ : α ą β ą 0 ; α ` β “
+?
+2du.
+(2.1)
+
+6
+HUBERT LACOIN
+Note that our definition of PI{II excludes one point of the boundary which correspond to
+Critical Gaussian multiplicative chaos γ “
+?
+2d (see Section 2.2 below).
+Theorem 2.1. If X is a centered Gaussian field whose covariance kernel K has an almost
+star-scale invariant part, γ P PI{II, and f P CcpRdq then there exists a complex valued
+random variable Mγ
+8pfq such that for any choice of mollifier θ the following convergence
+holds in Lp if p P
+”
+1,
+?
+2d{α
+¯
+.
+lim
+εÑ0 Mγ
+ε pfq “ Mγ
+8pfq.
+(2.2)
+The above result extends [18, Theorem 2.2] which established convergence for γ P PI.
+The method which we use to prove it however, completely differs from the one employed in
+[18]. In fact the method of proof that we employ in Section 4 balso provides an alternative
+and much shorter proof of [18, Theorem 2.2], with the additional benefit of establishing
+convergence in Lp for an optimal range of p.
+Remark 2.2. We have chosen to denote the limit by Mγ
+8 rather than Mγ
+0 . While the
+latter may seem a more natural choice, it is already in use for the initial condition of the
+martingale GMC approximation introduced in Section 3 (see for instance (4.2)).
+Remark 2.3. We have chosen to put the emphasis on the proof of the convergence of
+Mγ
+ε pfq for all fixed f, but it is true also that Mγ
+ε p¨q converges as a random distribution.
+More precisely the convergence (in probability) of Mγ
+ε in a local Sobolev space of negative
+index can in fact be deduced from the estimates obtained in the proof of (2.2). We include
+the argument in Appendix B.
+2.2. The boundary between phase II and III, and the triple point. Our second
+result concerns the case when γ P P1
+II{III where
+PII{III :“ t
+a
+d{2 ` iβ : β ą
+a
+d{2u,
+P1
+II{III :“ PII{III Y t
+a
+d{2p1 ` iqu
+(2.3)
+In that case Mγ
+ε needs to be rescaled in order to obtain a non-trivial limit. The convergence
+holds only in law. To describe the limit we need to introduce two notions: Critical Gaussian
+Multiplicative Chaos, and Gaussian White Noise with a random intensity.
+Critical GMC. As explained in the introduction critical Gaussian Multiplicative Chaos
+is obtained as the limit of Mα
+ε when α “
+?
+2d. The value
+?
+2d represent a threshold
+for the convergence of Mα
+ε . The convergence result below follows from a combination of
+[5, Theorem 5] - which establishes the convergence for the martingale sequence Mα
+t (see
+Section 3) and [10, Theorems 1.1 and 4.4] which establish that the limit is the same for
+the exponential of the mollified field Mα
+ε . Alternative concise proofs of these results have
+been recently given in [17].
+Theorem A. Let X be a Gaussian random field with an almost-star scale invariant kernel.
+There exists a locally finite random measure M1 with dense support and no atoms such
+that for every f P CcpRdq the following convergence holds in probability
+lim
+εÑ0
+c
+π log p1{εq
+2
+M
+?
+2d
+ε
+pfq “ M1pfq.
+(2.4)
+Remark 2.4. Note that we have set different conventions and that our M1 differs from
+that in [5, Theorem 5] by a factor
+b
+2
+π.
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES 7
+Complex white noise with random intensity given by a Real GMC. For γ P P1
+II{III we
+define Mγ to be a complex white noise with intensity measure given by M1pe|γ|2L¨q It is a
+random linear form which is constructed jointly with X, on an extended probability space.
+Conditionally on X, for f P C8
+c pDq, Mγpfq is a complex Gaussian random variable, with
+independent real and imaginary parts, both with a variance equal to
+M1pe|γ|2Lf 2q “
+ż
+D
+e|γ|2Lpx,xqfpxq2M1pdxq.
+Formally, letting P and P denote respectively the law of X and the joint law of pX, Mγp¨qq,
+Mγp¨q is the random process indexed by CcpRdq which satisfies for any m, n ě 1, ρ1, . . . , ρm, f1, . . . , fn P
+CcpRdq and any bounded measurable function F on Cn`m
+E
+“
+F
+`
+pxX, ρiyqm
+i“1, pMγpfjqqn
+j“1
+˘‰
+“ E b En
+“
+F
+`
+pxX, ρiyqm
+i“1, Σrγ, X, pfjqn
+j“1s ¨ Nn
+˘‰
+(2.5)
+where under Pn, Nn is an n dimensional vector whose coordinate are IID standard com-
+plex Gaussian variables, and Σrγ, X, pfjqn
+j“1s is the positive definite square root of the
+Hermitian matrix
+´
+M1pe|γ|2Lfif jq
+¯n
+i,j“1 .
+The result. Let us define the function ℓθ on Rd, obtained by convoluting z ÞÑ log 1{|z|
+twice with θ, that is
+ℓθpzq :“
+ż
+Rd log
+ˆ
+1
+|z ` z1 ´ z2|
+˙
+θpz1qθpz2qdz1dz2,
+(2.6)
+and set (recall (2.3))
+vpε, θ, γq :“
+$
+’
+&
+’
+%
+p2π logp1{εqq´1{4ε
+d´|γ|2
+2
+´ş
+Rd e|γ|2ℓθpzqdz
+¯1{2
+if γ P PII{III,
+a
+Σd´1
+´
+2 logp1{εq
+π
+¯1{4
+if γ “
+a
+d{2pi ` 1q
+(2.7)
+where Σd is the volume of the d ´ 1 dimensional sphere. Note that limεÑ0 vpε, θ, γq “ 8
+in all cases.
+Theorem 2.5. Let X be a Gaussian random field with an almost star-scale covariance.
+Then given γ P P1
+II{III, we have the following joint convergence in law
+ˆ
+X,
+Mγ
+ε
+vpε, θ, γq
+˙
+εÑ0
+ñ pX, Mγq,
+(2.8)
+Remark 2.6. The convergence in (2.8) implies that vpε, θ, γq´1Mγ
+ε does not converge
+in probability. On the heuristic level, this can be explained as follows: The white noise
+that appears in the limit is the product of local fluctuations of Xε. These fluctuations are
+produced by high frequencies in the Fourier spectrum of X. The set of frequencies that
+produce the fluctuations diverges to infinity when ε Ñ 0. This means that the randomness
+that produces the white noise become asymptotically independent of X in the limit.
+Remark 2.7. The convergence (2.8) means that for any collection pρiqm
+i“1 and pfjqn
+j“1 we
+have the convergence in law of the Cm`n valued vector
+lim
+εÑ0
+˜
+pxX, ρiyqm
+i“1,
+ˆ Mγ
+ε pfjq
+vpε, θ, γq
+˙n
+j“1
+¸
+“
+´
+pxX, ρiyqm
+i“1, pMγpfjqqn
+j“1
+¯
+.
+(2.9)
+
+8
+HUBERT LACOIN
+The convergence can also be shown to hold in a space of distribution. More precisely, there
+exists a modification of the process Mγ taking values in the local Sobolev space H´u
+loc pRdq
+with u ą d{2 and Mγ
+ε pfjq
+vpε,θ,γq converges in law in that space. See Appendix B.
+Since both X and Mγ
+ε are linear forms, the convergence of finite dimensional marginals
+follows from that of one dimensional marginals (this can simply be checked using Fourier
+transform and L´evy Theorem). More precisely, we only need to prove the convergence for
+every f P CcpRdq and ω P r0, 2πq of the real valued variable (Re denotes the real part)
+Mγ
+ε pf, ωq :“ Re
+`
+e´iωMγ
+ε pfq
+˘
+(2.10)
+Hence Theorem 2.5 can be reduced to the proof of the following statement
+Proposition 2.8. Under the assumption of Theorem 2.5, given ρ, f P CcpRdq, ω P r0, 2πq,
+we have
+lim
+εÑ0 E
+„
+eixX,ρy`i Mγ
+ε pf,ωq
+vpε,θ,γq
+
+“ E
+„
+eixX,ρy´ 1
+2M1pe|γ|2L|f|2q
+
+.
+(2.11)
+The r.h.s. in (2.11) of corresponds to the Fourier transform of pX, Mγq (cf. (2.5))
+E
+„
+eixX,ρy´ 1
+2 M1pe|γ|2L|f|2q
+
+“ E
+”
+eixX,ρy`iMγpfqı
+,
+and the convergence of the Fourier transform implies that of finite dimensional marginals.
+More detailed justifications are exposed in [16, Section 1.2].
+3. The martingale approximation for GMC
+Before getting to the technical core of the paper, we need one more introductory section
+to present an essential tool which is used in the proof of both Theorem 2.1 and Theorem
+2.5: the martingale decomposition of the field X. Under the almost star-scale assumption
+for K, besides mollification, there is another natural way to approximate the log-correlated
+field X by a smooth field. Extending the probability space, one can define a martingale
+sequence of smooth fields pXtqtě0 that converges to X.
+This allows for another approach to the construction of GMC, considering the exponen-
+tial of the martingale approximation of X (see (3.7)) which we call Mγ
+t (see Remark 3.2
+concerning the conflict of notation). Convergence results for Mγ
+t which are a analogous
+to Theorem 2.1 and 2.5 are also presented in this section. In section 3.5 we introduce an
+important technical tool which is used to prove Theorem 2.5. The result (a central limit
+Theorem proving convergence to a Gaussian with random variance) may find applications
+in other context, so it is stated in a rather general setup.
+3.1. The martingale decomposition of X. Given K with an almost star-scale invariant
+part, and using the decomposition (1.8) for K, we set Qtpx, yq :“ κpet1px ´ yqq where t1 is
+defined as the unique positive solution of
+t1 ´ η1
+η2
+p1 ´ e´η2t1q “ t.
+(3.1)
+We set
+Ktpx, yq :“ K0px, yq `
+ż t
+0
+Qspx, yqds
+“ K0px, yq `
+ż t1
+0
+p1 ´ η1e´η2sqκpespx ´ yqqds “: K0px, yq ` Ktpx, yq.
+(3.2)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES 9
+Note that we have limtÑ8 Ktpx, yq “ Kpx, yq. We define pXtpxqqxPRd,tě0 to be a centered
+Gaussian field with covariance given by (using the notation a ^ b :“ minpa, bq, a _ b :“
+maxpa, bq)
+ErXtpxqXspyqs “ Ks^tpx, yq.
+(3.3)
+Since ps, t, x, yq ÞÑ Ks^tpx, yq is H¨older continuous, the field admits a continuous modifi-
+cation. We let Ft :“ σ
+´
+pXspxqqxPRd,sPr0,ts
+¯
+denote the natural filtration associated with
+X¨p¨q. The process X indexed by CcpRdq and defined by xX, ρy “ limtÑ8
+ş
+Rd Xtpxqρpxqdx,
+is a centered Gaussian field with covariance, so that Xt is an approximation sequence for
+a log-correlated field with covariance K. We also define X¨ :“ X¨ ´ X0. Recalling (3.2)
+we have
+ErXtpxqXspyqs “ Ks^tpx, yq.
+(3.4)
+An important observation is that since Ktpxq :“ Ktpx, xq “ t, for any fixed x P Rd, the
+process pXtpxqqtě0 is a standard Brownian Motion. We also introduce the field Xt,ε which
+is the mollification of Xt, that is
+Xt,εpxq :“
+ż
+Rd θεpx ´ zqXtpzqdz “ E rXεpxq | Fts .
+We let Kt,εpx, yq denote the covariance of the field Xt,ε and Kt,ε,0px, yq the cross-covariance
+of Xt,ε and Xt
+Kt,εpx, yq :“ ErXt,εpxqXt,εpyqs “
+ż
+Rd θεpx ´ z1qθεpy ´ z2qKtpz1, z2qdz1dz2,
+Kt,ε,0px, yq :“ ErXt,εpxqXtpyqs “
+ż
+Rd θεpx ´ zqKtpz, yqdz.
+(3.5)
+The quantity Kt,ε is defined similarly and we use the notation Qt,ε and Qt,ε,0 the corre-
+sponding mollified versions of Qt.
+3.2. The martingale approximation for the GMC. We define the distribution Mγ
+t
+by setting for f P CcpRdq
+Mγ
+t pfq :“
+ż
+Rd fpxqeγXtpxq´ γ2
+2 Ktpxqdx.
+(3.6)
+Using the independence of the increments of X, it is elementary to check that Mtpfq is an
+pFtq-martingale. We also define
+Mγ
+t,εpfq :“
+ż
+Rd fpxqeγXt,εpxq´ γ2
+2 Kt,εpxqdx “ E rMγ
+ε pfq | Fts .
+(3.7)
+3.3. A few properties of the covariance kernels. We introduce some technical nota-
+tion and estimates that are going to be of use throughout the article. Let us first not that
+if a kernel K has an almost star-scale invariant part then it can be written in the form
+(1.1). Indeed, if K satisfies (1.8) then the function L defined for x ‰ y by
+Lpx, yq :“ Kpx, yq ` log |x ´ y|,
+(3.8)
+can be extended to a continuous function on R2d. Note that we have
+Lpxq “ lim
+yÑx pKpx, yq ` log |x ´ y|q “ K0pxq ´ j
+(3.9)
+
+10
+HUBERT LACOIN
+where the difference term j does not depend on x and can be computed explicitely
+j :“ lim
+zÑ0
+`
+logp1{|z|q ´ Kp0, zq
+˘
+“ η1
+η2
+`
+ż 8
+0
+`
+1 ´ rκpe´sq
+˘
+ds ă 8.
+(3.10)
+The above comes from the fact that
+logp1{|z|q ´ Kp0, zq “
+ż logp1{|z|q
+0
+p1 ´ Qlogp1{|z|q´up0, zqqdu
+and the fact that the integrand on the r.h.s. converges to 1 ´ rκpe
+η1
+η2 ´sq. Lastly one can
+observe that the following identity holds
+ℓpzq :“ lim
+tÑ8
+`
+Ktp0, e´tzq ´ t
+˘
+“ lim
+tÑ8
+ż t
+0
+pκpes1´tzq ´ 1qds “
+ż 8
+0
+pκpe
+η1
+η2 ´uzq ´ 1qdu, (3.11)
+where in the integral in s, s1 is related to s via (3.1). To obtain the third equality, one
+simply observe that s1 “ s ` η1{η2 ` op1q in the large s limit and make the change of
+variable u “ t ´ s. Note that ℓpzq is a continuous negative function and that for any
+|z| ě e´ η1
+η2 we have ℓpzq “ log 1
+|z| ´ j.
+To conclude this subsection, we gather in a a technical lemma a couple of useful estimates
+concerning Kt, Qt and their variant.
+Lemma 3.1. Given R ą 0, there exists a constant CR such that for any x, y P Bp0, Rq,
+t ą 0 and ε P r0, 1s
+ˇˇˇˇKt,εpx, yq ´ log
+ˆ
+1
+maxpe´t, ε, |x ´ y|q
+˙ˇˇˇˇ ď CR.
+(3.12)
+The bound (3.12) remains valid with Kt,ε replaced by Kt (with ε “ 0), Kt,ε,0, Kt,ε etc...
+We also have ż
+Rd Qtpx, yq “
+ż
+Rd Qt,εpx, yqdy “
+ż
+Rd Qt,ε,0px, yqdy ď Ce´dt,
+(3.13)
+and
+0 ď t ´ Kt,εpx, yq ď C
+`
+etp|x ´ y| ` εq
+˘2
+(3.14)
+The estimates above can be proved rather directly from the definition. A detailed proof
+of (3.12) is provided in [17, Appendix A.3]. The bound (3.13) follows directly from the
+definition of Qt given above (3.1) and the fact that |t´t1| is uniformy bounded. The upper
+bound in (3.14) can be obtained by integrating (in time and space) the inequality
+1 ´ Qtpz1, z2q ď Cret1|z1 ´ z2|s2
+which follows directly from the Taylor expansion at second order of κ.
+Remark 3.2. There is an obvious conflict of notation between Kt introduced above and
+Kε introduced in (1.6) and the same can be said about Xt and Mγ
+t . This should not cause
+any confusion since we keep using the letter ε for quantities related to the mollified field
+Xε and latin letters for quantities related to the martingale approximation Xt.
+3.4. Convergence results for the martingale approximation. An intermediate step
+to prove Theorem 2.1 and Theorem 2.5 is to show that similar results hold for the mar-
+tingale approximation Mγ
+t . These results present of course an interest in their own right.
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES11
+The case of γ P PI{II.
+Proposition 3.3. When γ P PI{II, the martingale Mγ
+t pfq is bounded Lp for p P r1,
+?
+2d{|α|q.
+As a consequence the limit
+lim
+tÑ8 Mγ
+t pfq “: Mγ
+8pfq
+(3.15)
+exists almost surely. The convergence holds in Lp and the limit is non-trivial.
+Remark 3.4. The martingale limit in (3.15) is the same as the limit of Mγ
+ε appearing in
+Theorem 2.1 (this is the reason why we use the same notation), we have
+lim
+tÑ8 Mγ
+t pfq “ lim
+εÑ0 Mγ
+ε pfq.
+(3.16)
+This observation is important, since it establishes that the limit in (2.2) does not depend
+on the choice of the mollifier.
+The case γ P P1
+II{III. In order to state the convergence in law result for Mγ
+t , we need to
+introduce a normalization factor vpt, γq (analogous to (2.7) for the mollified case). Let us
+set
+vpt, γq “
+$
+&
+%
+e
+|γ2|j
+2
+` 1
+2πt
+˘1{4 e
+p|γ|2´dqt
+2
+´ş
+Rd e|γ|2ℓpzqdz
+¯1{2
+,
+if |γ|2 ą d
+a
+Σd´1
+` 2t
+π
+˘1{4
+if |γ|2 “ d.
+(3.17)
+and define
+Mγ
+t pf, ωq :“ Re
+`
+e´iωMγ
+t pfq
+˘
+.
+(3.18)
+The following analogue of Proposition 2.8 holds.
+Proposition 3.5. If X is an almost star-scale invariant field and γ P P1
+II{III we have for
+any ρ, f P CcpRdq
+lim
+tÑ0 E
+„
+eixX,ρy`i
+Mγ
+t pf,ωq
+vpt,γq
+
+“ E
+„
+eixX,ρy´ 1
+2 M1pe|γ|2L|f|2q
+
+.
+(3.19)
+As a consequence we have the following convergence in law (in the sense of finite dimen-
+sional marginals)
+ˆ
+X,
+Mγ
+t
+vpt, γq
+˙
+tÑ8
+ùñ
+pX, Mγq.
+(3.20)
+3.5. CLT towards a Gaussian with random variance. We conclude this section by
+introducing a technical results which is essential to prove the convergence of a sequence of
+variable towards a Gaussian with random intensity in Theorem 2.5. We provide the result
+and its proof in a reasonably high level of generality since it may find application in other
+contexts.
+Consider pFtqtě0 a filtration and pWnqně1 a sequence of real valued random variables
+in L1. We introduce for each n ě 1 the martingale
+Wn,t :“ E rWn | Fts .
+(3.21)
+We assume that the martingale Wn,t admits a modification which is continuous in t for
+every n ě 1 We prove that Wn converges to to a Gaussian with random variance if the
+quadratic variation of pWn,tqtě0 satisfy a law of large number and a couple of additional
+technical assumptions. The result generalizes a similar CLT established for a single mar-
+tingale process (see [9, Theorem 5.50, Chap. VIII-Section 5c] or [16, Theorem 2.5]).
+
+12
+HUBERT LACOIN
+Theorem 3.6. Let us assume that and that there exists a non-negative valued random-
+variable Z which is such that the three following convergences in probability hold
+lim
+nÑ8
+xWny8
+v2pnq “ Z,
+@t ě 0 lim
+nÑ8
+xWnyt
+v2pnq “ 0, and lim
+nÑ8
+Wn,0
+vpnq “ 0.
+(3.22)
+Then Xn{vpnq converges in distribution towards a random Gaussian with variance given
+by Z, that is to say that for any F8 bounded measurable H we have
+lim
+nÑ8 E
+”
+HeiξWn{vpnqı
+“ lim
+nÑ8 E
+„
+He´ ξ2Z
+2
+
+(3.23)
+This is equivalent to saying that for any F8 random variable Y we have the following
+convergence in law
+pY, Wnq ùñ pY,
+?
+ZNq
+where N is a standard Gaussian which is independent of Z and Y .
+Remark 3.7. We believe that with adequate assumption on the size of the jumps, the
+result may extend to the case where pWn,tqtě0 is a c`ad-l`ag martingale, with the quadratic
+variation is replaced by the predictable bracket. Since we have no application in that setup,
+we restricted ourselves to the continuous case where the proof is technically simpler.
+Remark 3.8. In Section 6 we apply Theorem 3.6 for a sequence of variables indexed by
+ε P p0, 1q (namely Mγ
+ε pf, ωq) in the limit when ε Ñ 0 rather than n ě 1 and n Ñ 8.
+These setups are equivalent.
+3.6. Organization of the paper. The remainder of the paper is organized as follows
+‚ In Section 4 we prove all the statements concerning convergence in PI{II. Section
+4.1 is devoted to the proof of Proposition 3.3. The more technical proof of Theorem
+2.1, which uses Proposition 3.3 as in imput is displayed in Section 4.2.
+‚ The statements concerning γ P P1
+II{III, namely Proposition 3.5 and Proposition
+2.8, while relying on relatively simple ideas, require a certain amount of technical
+computations. In Section 5 we prove Proposition 3.5, in Section 6 Proposition 2.8.
+‚ In Section 7, we present the proof of Theorem 3.6.
+A significant amount of material is presented in appendices.
+‚ In Appendix A, we prove a couple of auxilliary results used in Section 5/6.
+‚ In Appendix B, we present and prove an extension of our main results, that is,
+the convergence of Mγ
+ε p¨q as a distribution. After identifying the right topology,
+the proof mostly boils down to repeating the computation made in Section 4 (for
+γ P PI{II) and Section 6 (for γ P P1
+II{III).
+‚ In Appendix C, we explain how our results can be extended to the case of a
+(sufficiently regular) log-correlated Gaussian field defined on an arbitrary open
+domain D Ă Rd.
+‚ In Appendix D, we present a relatively short proof of Lemma 4.1 for the sake
+of completeness. It the same as the one presented in [20, Lemma 3.15], except
+that we include a short proof of Lemma D.2 instead of relying on the branching
+random walk literature where more general results have been shown, albeit with
+much longer proofs (see for instance [8, 22]).
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES13
+A comment on notation. Throughout the paper, we use the letter C for a generic positive
+constant when we need to compare two quantities. It may depend on some parameters
+(for instance on γ or on the kernel K) but never on the variable t or ε. The value of C
+might change from one equation to the other withing the same proof. We use C1 and C2
+if we need several constants in the same display.
+4. Proof of convergence results on for γ P PI{II
+In this section we prove Proposition 3.3 and Theorem 2.1. The first one is easier, recall
+that due to the martingale property of Mγ
+t , it is sufficient to show that the sequence
+is bounded in Lp to prove convergence. This is performed in Section 4.1. We rely on
+the Burkeholder-Davis-Gundy (BDG) inequality, compute the quadratic variation of the
+martingale and studying its moment of order p{2.
+In Section 4.2, we adapt the same method to estimate the Lp norm of Mγ
+ε ´ Mγ
+8.
+More precisely the BDG inequality for the martingale pMγ
+t,ε ´ Mγ
+t qtě0, and show that the
+moment of order p{2 of its quadratic variation is uniformly small in t.
+4.1. Proof of Proposition 3.3. Recalling that
+?
+2d{α ą 1, we are going to prove that
+Mγ
+t pfq is bounded in Lp for
+p P
+´?
+8d{p3αq _ 1,
+?
+2d{α
+¯
+.
+(4.1)
+In the whole paper, when Mt is a complex valued continuous martingale, we use the
+notation xMyt to denote the the bracket between M and M. It is the predictable process
+such that |Mt|2 ´ xMyt is a local martingale. Using Burkeholder-Davis-Gundy (BDG)
+inequality for Mγ
+t pfq, there exists a constant Cp such that for every t ą 0
+E
+“
+|Mγ
+t pfq|p‰
+ď Cp
+´
+ErxMγpfqyp{2
+t
+s ` E r|Mγ
+0 pfq|ps
+¯
+.
+(4.2)
+We have
+E
+“
+|Mγ
+0 pfq|2‰
+“
+ż
+R2d e|γ|2K0px,yqfpxqfpyqdxdy ă 8.
+(4.3)
+Since p ă 2 by assumption, Jensen’s inequality implies that E r|Mγ
+0 pfq|ps ă 8. Using Itˆo
+calculus, we obtain an explicit expression for the quadratic variation
+xMγpfqy8 “ |γ|2
+ż 8
+0
+Atdt
+(4.4)
+where
+At :“
+ż
+R2d fpxqfpyqQtpx, yqeγXtpxq`γXtpyq´ γ2
+2 Ktpxq´ γ2
+2 Ktpyqdxdy.
+(4.5)
+Note that At is real and positive. From (4.2), we deduce that Mγ
+t pfq is bounded in Lp if
+E
+“
+p
+ş8
+0 Atdtqp{2‰
+ă 8. To bound At from above, we take the modulus of the integrand in
+(4.5) and using the assumption that β “
+?
+2d ´ α (γ P PI{II) we obtain that
+At ď
+ż
+R2d |fpxqfpyq|Qtpx, yqeαpXtpxq`Xtpyqq` 2d´2
+?
+2dα
+2
+pKtpxq`Ktpyqqdxdy.
+(4.6)
+Then using the inequality ab ď a2
+2 ` b2
+2 with
+a “ |fpxq|eαXtpxq` 2d´2
+?
+2dα
+2
+Ktpxq
+and
+b “ |fpyq|eαXtpyq` 2d´2
+?
+2dα
+2
+Ktpyq
+
+14
+HUBERT LACOIN
+and symmetry in x and y, we have
+At ď
+ż
+R2d |fpxq|2Qtpx, yqe2αXtpxq`p2d´2
+?
+2dαqKtpxqdxdy.
+(4.7)
+We use (3.13) to integrate over y and (3.12) to replace replace Ktpxq by t (at the cost of
+multiplicative constant) and we have
+At ď Cedt
+ż
+Rd |fpxq|2e2αpXtpxq´
+?
+2dtqdx.
+(4.8)
+Now, as α ą
+a
+d{2, we have by p{2 ă 1 by assumption. We can use thus the following
+inequality (valid for an arbitrary collection of positive real numbers paiqiPI and q P p0, 1q)
+˜ÿ
+iPI
+ai
+¸q
+ď
+ÿ
+iPI
+aq
+i ,
+(4.9)
+with q “ p{2. In the remainder of the paper, we simply say “by subadditivity” when using
+(4.9). Using (4.9) and Jensen’s inequality we have
+E
+«ˆż 8
+0
+Atdt
+˙p{2ff
+ď
+ÿ
+ně0
+E
+«ˆż n`1
+n
+Atdt
+˙p{2ff
+ď
+ÿ
+ně0
+E
+«ˆż n`1
+n
+ErAs | Fnsds
+˙p{2ff
+.
+(4.10)
+Averaging with respect to pXs ´ Xnq we obtain from (4.8)
+ż n`1
+n
+E rAs | Fns ď Cedn
+ż
+R2d |fpxq|2e2αpXnpxq´
+?
+2dnqdx “: CBn.
+(4.11)
+As p ą
+?
+8d{3α by assumption, we can conclude using the estimate in Lemma 4.1 below
+for the fractional moments of Bn (the assumption on p makes the r.h.s. of (4.12) summable
+in n). More precisely, we deduce from (4.10),(4.11) and (4.12) that E
+”`ş8
+0 Atdt
+˘p{2ı
+ă 8.
+Lemma 4.1. For α ą
+a
+d{2 and p ă
+?
+2d{α we have
+E
+”
+Bp{2
+n
+ı
+ď Cn´ 3αp
+?
+8d plog nq6.
+(4.12)
+This result is a weaker version of [20, Lemma 3.15]. We provide, for the commodity of the
+reader a self-contained of Lemma 4.1 in Appendix D.
+4.2. Proof of Theorem 2.1. We prove Theorem 2.1 in the setup where our probability
+space contains a martingale approximation pXtqtě0 of the field X with covariance 3.3.
+More precisely we show that Mγ
+ε pfq converges to the same limit as Mγ
+t pfq. Working in
+an enlarged probability space entails by no mean a loss of generality since the validity of
+the statement “the sequence pMγ
+ε pfqqεPp0,1s is Cauchy in Lp” is entirely determined by the
+distribution of pXεpxqqxPRd,εPp0,1s.
+Proposition 4.2. Given γ P PI{II and p P r1,
+?
+2d{αq we have
+lim
+εÑ0 sup
+tą0
+E
+“
+|pMγ
+t ´ Mγ
+t,εqpfq|p‰
+“ 0.
+(4.13)
+As a consequence the following convergence holds in Lp
+lim
+εÑ0 Mγ
+ε pfq “ Mγ
+8pfq
+(4.14)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES15
+Proof. Let us first show indicate how (4.14) follows from (4.13). We observe that
+E
+“
+|pMγ
+t,ε ´ Mγ
+ε qpfq|2‰
+“
+ż 8
+0
+fpxqfpyq
+´
+e|γ|2Kεpx,yq ´ e|γ|2Kt,εpx,yq¯
+dxdy.
+(4.15)
+Since limtÑ8 Kt,εpx, yq “ Kεpx, yq, using dominated convergence the r.h.s. tends to 0 when
+t Ñ 8 and thus limtÑ8 Mγ
+t,εpfq “ Mγ
+ε pfq in L2, and hence also in Lp. Using Proposition
+3.3 we thus have the following convergence in Lp
+lim
+tÑ8pMγ
+t ´ Mγ
+t,εqpfq “ pMγ
+8 ´ Mγ
+ε qpfq.
+Taking the limit when ε to zero, we obtain that
+lim
+εÑ0 E r|pMγ
+8 ´ Mγ
+ε qpfq|ps “ lim
+εÑ0 lim
+tÑ8 E
+“
+|pMγ
+t ´ Mγ
+t,εqpfq|p‰
+.
+(4.16)
+and we conclude using (4.13).
+To prove (4.13), we assume that (4.1) holds. Then using the BDG inequality (we omit the
+dependence in f for ease of reading). We have for every t ě 0
+Er|Mγ
+t ´ Mγ
+t,ε|ps ď CpErxMγ ´ Mγ
+¨,εyp{2
+8 ` |Mγ
+0 ´ Mγ
+0,ε|ps.
+(4.17)
+The reader can then check by an explicit calculation of the second moment that
+lim
+εÑ0 E
+”
+|Mγ
+0 pfq ´ Mγ
+0,εpfq|pı
+ď lim
+εÑ0 E
+”
+|Mγ
+0 pfq ´ Mγ
+0,εpfq|2ıp{2
+“ 0
+(4.18)
+Hence in view of (4.17)-(4.18), to prove (4.13) we need to show that
+lim
+εÑ0 ErxMγ ´ Mγ
+¨,εyp{2
+8 s “ 0.
+(4.19)
+Expanding the product, using Itˆo calculus (Re denotes the real part) we obtain
+xMγ ´ Mγ
+¨,εy8 “ |γ|2
+ż 8
+0
+´
+At ´ 2Re
+´
+Ap1q
+t,ε
+¯
+` Ap2q
+t,ε
+¯
+dt.
+(4.20)
+where, At is defined in (4.8), and recalling (3.5), Ap1q
+t,ε and Ap2q
+t,ε are defined by
+Ap1q
+t,ε :“
+ż
+R2d fpxqfpyqQt,ε,0px, yqeγXtpxq`γXt,εpyq´ γ2
+2 Ktpxq´ γ2
+2 Kt,εpyqdxdy,
+Ap2q
+t,ε :“
+ż
+R2d fpxqfpyqQt,εpx, yqeγXt,εpxq`γXt,εpyq´ γ2
+2 Kt,εpxq´ γ2
+2 Kt,εpyqdxdy.
+(4.21)
+We are going to reduce the proof of (4.19) to that of two convergence statements concerning
+Apiq
+t,ε for i P t1, 2u (the first being valid for any fixed r ą 0)
+lim
+εÑ0 sup
+tPr0,rs
+E
+”
+|At ´ Apiq
+t,ε|
+ı
+“ 0
+for i P t1, 2u.
+(4.22)
+lim
+rÑ8 sup
+εPp0,1s
+E
+«ˆż 8
+r
+|Apiq
+t,ε|dt
+˙p{2ff
+“ 0
+for i P t1, 2u.
+(4.23)
+Before proving (4.22)-(4.23) let us explain how (4.19) is deduced from it. Note that (4.23)
+is also valid for At (this can be extracted from the proof in Section 4.1). Given δ ą 0,
+
+16
+HUBERT LACOIN
+using subadditivity (4.9), and (4.23) we can find rδ such that for every ε ą 0
+E
+«ˆż 8
+rδ
+´
+At ´ 2Re
+`
+Ap1q
+t,ε
+˘
+` Ap2q
+t,ε
+¯
+dt
+˙p{2ff
+ď E
+«ˆż 8
+rδ
+Atdt
+˙p{2
+`
+ˆż 8
+rδ
+2|Ap1q
+t,ε |dt
+˙p{2
+`
+ˆż 8
+rδ
+Ap2q
+t,ε dt
+˙p{2ff
+ď δ{2.
+(4.24)
+Now using first Jensen’s inequality and then (4.22) (recall that At is real valued) we can
+find εδ such that for every ε P p0, εδq
+E
+«ˆż rδ
+0
+´
+At ´ 2Re
+`
+Ap1q
+t,ε
+˘
+` Ap2q
+t,ε
+¯
+ds
+˙p{2ff
+ď
+ˆż rδ
+0
+E
+”
+At ´ 2Re
+`
+Ap1q
+t,ε
+˘
+` Ap2q
+t,ε
+ı
+ds
+˙p{2
+ď
+ˆż rδ
+0
+E
+”
+2|At ´ Ap1q
+t,ε | ` |Ap2q
+t,ε ´ At|
+ı
+ds
+˙p{2
+ď δ{2.
+(4.25)
+Using subadditivity again we deduce from (4.24)-(4.25) that if ε P p0, εδq we have
+E
+«ˆż rδ
+0
+´
+At ´ 2Re
+`
+Ap1q
+t,ε
+˘
+` Ap2q
+t,ε
+¯
+dt
+˙p{2ff
+ď δ,
+(4.26)
+which (recalling (4.20)) concludes the proof of (4.19). Let us now prove (4.22)-(4.23).
+The proof (4.22) follows from a rather pedestrian but rather cumbersome computation
+of the L2 norm of pAt ´ Apiq
+t,εq. The following lemma summarizes the key points of this
+computation.
+Lemma 4.3. Consider the following:
+‚ Let pX, µq be a measured space and T be a set of indices.
+‚ Let Zt,εp¨q, t P T , ε P p0, 1s be a collection of complex valued Gaussian processes
+defined on X. We set
+Gt,εpx, yq :“ ErZt,εpxqZt,εpyqs
+and
+Ht,εpx, yq :“ ErZt,εpxqZt,εpyqs.
+(4.27)
+‚ Let Zt be defined on the same probability space in such a way that pZt, Zt,εq is
+jointly Gaussian. We let Gt and Ht be defined as in (4.27) and set
+Ht,ε,0px, yq :“ ErZt,εpxqZtpyqs.
+‚ Let gt,ε and gt be deterministic functions X Ñ R.
+We assume that:
+(i) The covariance functions are uniformly bounded, that is
+sup
+tPT
+εPp0,1s
+sup
+x,yPX
+max pHt,εpx, yq, Htpx, yq, Ht,ε,0px, yqq ă 8.
+(ii) There exists a µ-integrable function h such that for every t P T and ε P p0, 1s
+@x P X,
+maxp|gt,εpxq|, |gtpxq|q ď hpxq
+(iii) That for every t P T , we have the following pointwise convergence
+lim
+εÑ0 gt,ε “ gt,
+and
+lim
+εÑ0 Ht,ε “ lim
+εÑ0 Ht,ε,0 “ Ht
+(4.28)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES17
+Then setting
+Wt,ε :“
+ż
+X
+gt,εpxqeZt,εpxq´ 1
+2 Gt,εpxqµpdxq
+and
+Wt :“
+ż
+X
+gtpxqeZtpxq´ 1
+2Gtpx,xqµpdxq.
+We have
+lim
+εÑ0 sup
+tPT
+E
+“
+|Wε ´ Wt,ε|2‰
+“ 0.
+(4.29)
+Proof of Lemma 4.3. The proof is actually much shorter than the statement. We have
+E
+“
+|Wt,ε ´ Wt|2‰
+“
+ż
+X 2
+ˆ
+gt,εpxqgt,εpyqeHt,εpx,yq ´ 2Re
+´
+gt,εpxqgtpyqeHt,ε,0px,yq¯
+` gtpxqgtpyqeHtpx,yq
+˙
+µpdxqµpdyq
+(4.30)
+and using our assumptions we can apply dominated convergence.
+□
+Proof of (4.22). We consider the case i “ 2 but the other one is identical. We set X “ R2d,
+µ is Lebesgue measure, T “ r0, rs, and
+Zt,εpx, yq “ γXt,εpxq ` γXt,εpyq,
+Ztpx, yq “ γXtpxq ` γXtpyq,
+gt,εpx, yq “ Qt,εpx, yqfpxqfpyqe|γ|2Kt,εpx,yq,
+gtpx, yq “ Qtpx, yqfpxqfpyqe|γ|2Ktpxq.
+Then the assumptions of Lemma 4.3 are immediate to check.
+□
+We now provide the details for the proof of (4.23) i “ 2 (the case i “ 1 is similar). Let
+us set n0pεq “ rlogp1{εqs and assume (without loss of generality) that r is an integer and
+is smaller than n0. Using - as in the proof of Proposition 4.2 - subadditivity (4.9) and
+Jensen’s inequality we obtain
+E
+«ˆż 8
+r
+|Ap2q
+t,ε |ds
+˙p{2ff
+ď
+n0
+ÿ
+n“r
+E
+«ˆż n`1
+n
+|Ap2q
+t,ε |dt
+˙p{2ff
+ď
+n0´1
+ÿ
+n“r
+E
+«ˆż n`1
+n
+E
+”
+|Ap2q
+t,ε | | Fn
+ı
+dt
+˙p{2ff
+` E
+«ˆż 8
+n0
+E
+”
+|Ap2q
+t,ε | | Fn0
+ı
+dt
+˙p{2ff
+.
+(4.31)
+Proceeding as in (4.11), we obtain that if t P rn, n ` 1q, n P �r, n0 ´ 1�, or t ě n0, n “ n0,
+we have (using Lemma 3.1 to replace the covariance Kn,εpxq by n)
+E
+”
+|Ap2q
+t,ε | | Fn
+ı
+ď C
+ż
+R2d |fpxq|2Qs,εpx, yqe2αpXn,εpxq´
+?
+2dnq`2dndxdy.
+(4.32)
+Using (3.13) to integrate over y and setting
+Bp2q
+n,ε :“
+ż
+Rd |fpxq|2e2αpXn,εpxq´
+?
+2dnq`dndx
+(4.33)
+we obtain that
+E
+«ˆż 8
+r
+|Ap2q
+t,ε |ds
+˙p{2ff
+ď C
+n0
+ÿ
+n“r
+E
+”
+pBp2q
+n,εqp{2ı
+.
+(4.34)
+
+18
+HUBERT LACOIN
+Using Jensen’s inequality for the probability θεpy ´ xqdy, we can replace the mollification
+acting on Xn in the exponential by one acting of |f|2, we have
+e2αXn,εpxq ď
+ż
+Rd θεpx ´ yqe2αXnpyqdy
+which after multiplying by |fpxq|2 and integrating with respect to x implies that
+Bp2q
+n,ε ď
+ż
+D
+`
+|f|2 ˚ θε
+˘
+pyqe2αpXnpyq´
+?
+2dnq`dndy.
+(4.35)
+Since |f|2˚θε ď }f}2
+81t|x|ďR`1u if f is supported in Bp0, Rq, we can conclude using Lemma
+4.1, that
+E
+”
+pBp2q
+n,εqp{2ı
+ď Cn´ 3αp
+?
+8d
+for a constant which does not depend on ε. Recalling that p ą
+?
+8d{3α (cf. (4.1)) we
+obtain combining(4.32), (4.34) and (4.35) that
+E
+«ˆż 8
+r
+|Ap2q
+t,ε |ds
+˙p{2ff
+ď Cr1´ 3αp
+2
+?
+2d .
+(4.36)
+This concludes the proof of (4.23), and thus of Proposition 4.2.
+□
+5. Proof of Proposition 3.5
+5.1. Reduction to a statement concerning the total variation. Using [16, Theorem
+2.5] (which is a simpler version of Theorem 3.6 displayed above) we can reduce the proof
+of (3.19) to the following convergence statement about the quadratic variation of the
+martingale.
+Proposition 5.1. We have the following
+lim
+tÑ8 vpt, γq´2xMγpf, ωqyt “ M1pe|γ|2L|f|2q.
+(5.1)
+Proof of Proposition 3.5 from Proposition 5.1. We simply apply [16, Theorem 2.5] to the
+martingale Mγ
+t pf, ωq.
+□
+Setting, for notational simplicity Wt :“ Mγ
+t pfq. Recall that for a complex value mar-
+tingale such as Wt we use the notation xWyt for the bracket between W and its conjugate.
+Using bilinearity of the martingale brackets we have
+xMγpf, ωqyt “ 1
+2
+`
+xWyt ` Repe´2iωxW, Wytq
+˘
+(5.2)
+Hence to prove (5.1), it is sufficient to prove that following convergences hold in probability.
+lim
+tÑ8 vpt, γq´2xWyt “ 2M1pe|γ|2L|f|2q,
+lim
+tÑ8 vpt, γq´2xW, Wyt “ 0.
+(5.3)
+The expression for the bracket of Wt can be obtained by using Itˆo calculus (recall (4.4))
+More precisely we have
+xWyt “ |γ|2
+ż t
+0
+Asds
+and
+xW, Wyt “ γ2
+ż t
+0
+Bsds,
+(5.4)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES19
+where At is defined in (4.5) and
+Bt :“
+ż
+R2d fpxqfpyqQtpx, yqeγpXtpxq`Xtpyqq´ γ2
+2 pKtpxq`Ktpyqqdxdy.
+(5.5)
+Now using (5.4) our first idea is to deduce (5.3) from a convergence statement concerning
+At and Bt. A really important point here is that while At, properly rescaled, converges
+to M1pe|γ|2Lfq in probability, this type of convergence is not sufficient to say something
+about the integral
+şt
+0 Asds. A convenient framework to work with integrals is L1 conver-
+gence, but the issue we encounter is that At certainly does not converge in L1 (we have
+E
+”
+|M1pe|γ|2Lfq|
+ı
+“ 8 when f is non trivial).
+To bypass this problem, restrict ourselves to likely family of event and prove L1 convergence
+for the restriction. Recalling the definition of X (3.4), given q ě 0 and R ą 0, t ě 0 and
+x P Rd we introduce the events
+At,qpxq :“
+"
+max
+sPr0,tspXspxq ´
+?
+2dsq ă q
+*
+,
+Aq,R :“
+#
+sup
+sě0,|x|ďR
+pXspxq ´
+?
+2dtq ă q
++
+“
+č
+xPBp0,Rq
+tě0
+At,qpxq.
+(5.6)
+A very important fact, which is a direct consequence of [4, Proposition 19] (see also [17,
+Proposition 2.4] for a concise proof).
+Lemma 5.2. We have for any fixed R ą 0
+lim
+qÑ8 P rAq,Rs “ 1
+(5.7)
+We introduce (we drop the dependence in γ in most displays to make them easier to read)
+φptq “ φpt, γq :“
+d
+2
+πpt _ 1qe|γ2|j
+ˆż
+Rd Qtp0, zqe|γ2|Ktp0,zqdz
+˙
+,
+(5.8)
+which plays the role of a rescaling function for At. Our main technical result in this section
+is the proof that At{φptq converges in L1 towards M1pe|γ|2L|f|2q after restriction to the
+event Aq,R.
+Proposition 5.3. The following convergences hold for any q ě 0 and any R such that
+Supppfq Ă Bp0, Rq
+lim
+tÑ8 E
+”
+|At{φptq ´ M1pe|γ|2L|f|2q|1Aq,R
+ı
+“ 0,
+(5.9)
+lim
+tÑ8 E
+“
+|Bt{φptq| 1Aq,R
+‰
+“ 0,
+(5.10)
+and the above quantities are finite for every t ě 0.
+To show that Proposition 5.3 implies the convergence stated in Proposition 5.1, we need
+to ensure that the rescaling by φptq matches that proposed for xWyt (which is vpt, γq2)
+after integrating with respect to time. This is the purpose of the following lemma.
+Lemma 5.4. We have for any |γ| ě d
+lim
+tÑ8
+|γ|2 şt
+0 φpsqds
+2vpt, γq2
+“ 1.
+(5.11)
+
+20
+HUBERT LACOIN
+The proof of Lemma 5.4 is presented in Appendix A.3.
+Note that the goal of the
+lemma is only to obtain a more presentable expression for vpt, γq since without it, we can
+still prove that Proposition 5.1 and hence Proposition 3.5 are valid with v replaced by
+vpt, γq :“ |γ|
+b
+p
+şt
+0 φpsqdsq{2.
+Proof of Proposition 5.1. As we have seen, it is sufficient to prove (5.3). We provide the
+details concerning the convergence of xWyt (the first line in (5.3)) but that of xW, Wyt can
+be obtained exactly in the same manner. Using (5.4) and Jensen’s inequality we have
+E
+«ˇˇˇˇˇ
+xWyt
+|γ|2 şt
+0 φpsqds
+´ M1pe|γ|2L|f|2q
+ˇˇˇˇˇ 1Aq,R
+ff
+ď
+şt
+0 φpsqE
+”ˇˇˇ As
+φpsq ´ M1pe|γ|2L|f|2q
+ˇˇˇ 1Aq,R
+ı
+ds
+şt
+0 φpsqds
+.
+(5.12)
+Observing that
+ş8
+0 φpsqds “ 8, the r.h.s. of (5.12) is simply a weighted Cesaro mean and
+thus we deduce from Proposition 5.3 and more precisely from (5.9) that
+lim
+tÑ8 E
+«ˇˇˇˇˇ
+xWyt
+|γ|2 şt
+0 φpsqds
+´ M1pe|γ|2L|f|2q
+ˇˇˇˇˇ 1Aq,R
+ff
+“ 0
+(5.13)
+Since this holds for every q ą 0 we obtain that the following convergence holds in proba-
+bility (the replacement of |γ|2 şt
+0 φpsqds by 2vpt, γq2 simply comes from Lemma 5.2) that
+lim
+tÑ0
+ˇˇˇˇ
+xWyt
+2vpt, γq2 ´ M1pe|γ|2L|f|2q
+ˇˇˇˇ 1Ť
+qě1 Aq,R “ 0
+which, since the event in the indicator has probability one (cf. Lemma 5.2) is the desired
+conclusion.
+□
+5.2. Restricted convergence in L2 for the critical GMC. Before starting the proof
+of Proposition 5.3, we recall a result which play a key role in the proof, the L2 convergence
+of M
+?
+2d
+t
+pgq towards M1pgq when considering the restriction to the event Aq,R. This also
+implies convergence in L1 which is what we require for the proof of Proposition 5.3. The
+result can be deduced from the L2 convergence of the truncated version of M
+?
+2d
+t
+pgq which
+is proved in [17].
+Lemma 5.5. We have for any g in CcpRdq such that Supppgq Ă Bp0, Rq and any q ą 0
+lim
+tÑ8 E
+»
+–
+ˇˇˇˇˇ
+c
+πt
+2 M
+?
+2d
+t
+pgq ´ M1pgq
+ˇˇˇˇˇ
+2
+1Aq,R
+fi
+fl “ 0,
+(5.14)
+and E
+“
+|M1pgq|21Aq,R
+‰
+ă 8.
+Proof. The fact that E
+“
+|M1pgq|21Aq,R
+‰
+ă 8 is a simple consequence of the convergence
+since for any fixed t, Er|M
+?
+2d
+t
+pgq|2s ă 8. We set (recall (5.6))
+M
+?
+2d,pqq
+t
+pgq :“
+ż
+gpxqe
+?
+2dXtpxq´dKtpxq1At,qpxqdx,
+(5.15)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES21
+From [17, Proposition 4.1], there exists an L2 variable D
+pqq
+8 pgq such that
+lim
+tÑ8 E
+»
+–
+ˇˇˇˇˇ
+c
+πt
+2 M
+?
+2d,pqq
+t
+pgq ´ D
+pqq
+8 pgq
+ˇˇˇˇˇ
+2fi
+fl “ 0.
+(5.16)
+It satisfies D
+pqq
+8 pgq “ M1pgq on the event Aq,R. More precisely [17, Proposition 4.1] is only
+stated in the special case where g is an indicator function (to keep notation light) but
+the proof for g P CcpRdq is identical. On the event Aq,R we have M
+?
+2d,pqq
+t
+pgq “ M
+?
+2d
+t
+pgq.
+Hence
+lim sup
+tÑ8
+E
+»
+–
+ˇˇˇˇˇ
+c
+πt
+2 M
+?
+2d
+t
+pgq ´ M1pgq
+ˇˇˇˇˇ
+2
+1Aq,R
+fi
+fl
+“ lim sup
+tÑ8
+E
+»
+–
+ˇˇˇˇˇ
+c
+πt
+2 M
+?
+2d,pqq
+t
+pfq ´ D
+pqq
+8 pgq
+ˇˇˇˇˇ
+2
+1Aq,R
+fi
+fl “ 0.
+(5.17)
+where the last equality follows from (5.16).
+□
+5.3. Organizing the proof of Proposition 5.3. The two convergences rely on similar
+ideas, we focus on (5.9) which is the more delicate of the two. The main idea is that since
+the integrand in the definition of At
+At :“
+ż
+R2d fpxqfpyqQtpx, yqeγXtpxq`γXtpyq´ γ2
+2 Ktpxq´ γ2
+2 Ktpyqdxdy,
+vanishes when |x ´ y| ě e´t (due to the presence of the multiplicative Qtpx, yq), the value
+of the integral should not be much affected much if one changes fpyq, Xtpyq and Ktpyq by
+fpxq, Xtpxq and Ktpxq in the expression.
+The quantity obtained after this replacement is, up to a multiplicative factor, of the
+form M
+?
+2d
+t
+pgq (recall that γ ` γ “
+?
+2d) for some function g. Hence we should be able to
+conclude the proof of the convergence statement using Lemma 5.5.
+While this idea is relatively simple, it requires several steps to be implemented. We set
+K˚
+t px, yq :“ K0pxq ` Ktpx, yq
+and
+r “ rptq :“ t ´ log log t
+(5.18)
+(we are assuming that t ą e so that 0 ď r ď t). We introduce the quantity rAt which will
+appear after all our “replacement” steps have been performed, it is defined by
+rAt :“
+ż
+R2d Qtpx, yqe|γ|2K˚
+t px,yq|fpxq|2e
+?
+2dXrpxq´dKrpxqdxdy
+“
+ˆż
+Rd Qtp0, zqe|γ|2Ktp0,zqdz
+˙ ż
+Rd e|γ|2K0ptq|fpxq|2e
+?
+2dXrpxq´dKrpxqdx
+“ φptq
+c
+πt
+2
+ż
+Rd e|γ|2Lpxq|fpxq|2e
+?
+2dXrpxq´dKrpxqdx “ φptq
+c
+πt
+2 M
+?
+2d
+r
+pe|γ|2L|f|2q.
+(5.19)
+As a direct consequence of Lemma 5.5 (since r “ t ´ optq the presence of
+?
+t instead of ?r
+does not affect the convergence), we have
+lim
+tÑ8 E
+«ˇˇˇˇˇ
+rAt
+φptq ´ M1pe|γ|2L|f|2q
+ˇˇˇˇˇ 1Aq,R
+ff
+“ 0.
+(5.20)
+
+22
+HUBERT LACOIN
+With this observation the proof of (5.9) reduces to showing that
+lim
+tÑ0
+1
+φptqE
+”
+|At ´ rAt|1Aq,R
+ı
+“ 0.
+(5.21)
+This requires some care but before going in the depth of the proof, let us explain the
+heuristic behind (5.21). Note that rAt is obtained from At with two simple modifications:
+‚ We have replaced fpxqfpyq by |fpxq|2.
+‚ In the exponential, we have replaced γXtpxq`γXtpyq by
+?
+2dXrpxq “ pγ`γqXrpxq
+and ´ γ2
+2 Ktpxq ´ γ2
+2 Ktpyq by ´dKrpxq ` |γ|2K˚
+t px, yq.
+The first modification is rather straightfoward, we are integrating close to the diagonal so
+that fpyq is close to fpxq. For the second modification, the idea is that replacing Xtpyq
+with Xtpxq (and t with r) should not yield big modifications provided that we change the
+normalization to keep the expectation of the exponential unchanged (or almost so). In
+our case we have
+E
+„
+eγXtpxq`γXtpyq´ γ2
+2 Ktpxq´ γ2
+2 Ktpyq
+
+“ e|γ|2Ktpx,yq,
+E
+”
+e
+?
+2dXrpxq´dKrpxq`|γ|2K˚
+t px,yqı
+“ e|γ|2K˚
+t px,yq.
+(5.22)
+and, on the considered domain of integration, K˚
+t px, yq and Ktpx, yq are very close since
+|x ´ y| ď e´t when Qtpx, yq ‰ 0. The proof of (5.21) requires three distinct steps which
+are detailed in the next subsection.
+5.4. The proof of (5.21).
+Step 1: Changing the deterministic prefactor in the integrand. The integrand of At and
+rAt have different expectations. Our first step aims to fix this by replacing fpyq by fpxq in
+At and doing a small modification in the exponential factor. We set
+Ap1q
+t
+:“
+ż
+R2d |fpxq|2Qtpx, yqeγXtpxq`γXtpyq` γ2
+2 Ktpxq` γ2
+2 Ktpyq`|γ|2pK0pxq´K0px,yqqdxdy (5.23)
+We are going to prove that
+lim
+tÑ8 φptq��1E
+”
+|At ´ Ap1q
+t |1Aq,R
+ı
+“ 0
+(5.24)
+Since f and K0 are uniformly continuous on the support of f and Supppfq Ă Bp0, Rq,
+there exists a positive function δ with limtÑ8 δptq “ 0, such that for |x ´ y| ď e´t setting
+Fpx, yq :“ fpxqfpyq ´ |fpxq|2e|γ|2pK0pxq´K0px,yqq
+we have
+|Fpx, yq| ď δptq1Bp0,Rqpxq1Bp0,Rqpyq
+(5.25)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES23
+Hence we obtain (since α “
+a
+d{2, we have Repγ2q “ d ´ |γ|2)
+|At ´ Ap1q
+t | “
+ˇˇˇˇ
+ż
+R2d Qtpx, yqFpx, yqeγXtpxq`γXtpyq` γ2
+2 Ktpxq` γ2
+2 Ktpyqdxdy
+ˇˇˇˇ
+ď
+ż
+R2d Qtpx, yq|Fpx, yq|e
+b
+d
+2 pXtpxq`Xtpyqq`p|γ|2´dq Ktpxq`Ktpyq
+2
+dxdy
+ď δptq
+ż
+Bp0,Rq2 Qtpx, yqe
+b
+d
+2 pXtpxq`Xtpyqq`p|γ|2´dq Ktpxq`Ktpyq
+2
+dxdy
+ď δptq
+ż
+Bp0,Rq2 Qtpx, yqe
+?
+2dXtpxq`p|γ|2´dqKtpxqdxdy
+(5.26)
+where the first inequality is simply obtained by taking the modulus of the integrand and
+in the third one we simply used
+ZpxqZpyq ď 1
+2pZpxq2 ` Zpyq2q
+with Zpxq “ e
+b
+d
+2 Xtpxq`p|γ|2´dq Ktpxq
+2
+and then symmetry in x and y. Then we observe that
+(λ denotes the Lebesgue measure) since At,qpxq Ă Aq,R we have
+E
+”
+e
+?
+2dXtpxq´dKtpxq1Aq,R
+ı
+ď E
+”
+e
+?
+2dXpxq´dKtpxq1At,qpxq
+ı
+“ Pr@s P r0, ts, Bs ď qs ď
+c
+2
+πtq
+(5.27)
+where in the last line, we used Cameron-Martin formula (see Proposition A.1 in the ap-
+pendix) and the fact that pXtpxqqtě0 is a standard Brownian Motion. The last inequality
+is simply Lemma A.2. Combining (5.26) and (5.27) and the fact that K0 is bounded, we
+have
+E
+”
+|At ´ Ap1q
+t |1Aq,R
+ı
+ď Cδptq
+?
+t
+ż
+Bp0,Rq2 Qtpx, yqe|γ|2Ktpxqdxdy ď C1δptqφptq.
+(5.28)
+□
+Step 2: Taking conditional expectation. Recalling the definition of rptq (5.18) we set
+Ap2q
+t
+:“ ErAp1q
+t
+| Frs
+(5.29)
+For this step of the proof (and only this step), we are going to assume that K0 ” 0 (and
+hence X0 ” 0q. Treating the case where X0 is a non-trivial field does not present any
+extra difficulty besides the challenge of making the equations fit within the margins. This
+assumption allows to replace Ktpxq and Ktpyq by t, and we get the following simplification
+for the expression of Ap1q
+t .
+Ap1q
+t
+:“
+ż
+R2d |fpxq|2Qtpx, yqeγXtpxq`γXtpyq`p|γ|2´dqtdxdy
+(5.30)
+Then we have
+Ap2q
+t
+“
+ż
+R2d |fpxq|2Qtpx, yqeγXrpxq`γXrpyq`p|γ|2´dqr`|γ|2Krr,tspx,yqdxdy,
+(5.31)
+
+24
+HUBERT LACOIN
+where Krr,ts “ Kt ´ Kr (in the remainder of the paper, we use this convention for other
+quantities indexed by t). We are going to show that
+lim
+tÑ8 φptq´1E
+”
+|Ap1q
+t
+´ Ap2q
+t |1Aq,R
+ı
+“ 0
+(5.32)
+Recalling (5.6) we define
+A
+p1q
+t
+:“
+ż
+R2d |fpxq|2Qtpx, yqeγXtpxq`γXtpyq`p|γ|2´dqt1Ar,qpxqdxdy,
+A
+p2q
+t
+:“
+ż
+R2d |fpxq|2Qtpx, yqeγXrpxq`γXrpyq`p|γ|2´dqr`|γ|2Krr,tspx,yq1Ar,qpxqdxdy.
+(5.33)
+Since on Aq,R, Apiq
+t
+and A
+piq
+t
+coincide, We have
+E
+”
+pAp2q
+t
+´ Ap1q
+t q21Aq,R
+ı
+ď E
+”
+pA
+p2q
+t
+´ A
+p1q
+t q2ı
+(5.34)
+and thus we can prove that (5.32) holds by showing that
+lim
+tÑ8 φptq´2E
+”
+pA
+p2q
+t
+´ A
+p1q
+t q2ı
+“ 0.
+(5.35)
+To bound ErpA
+p2q
+t
+´ A
+p1q
+t q2s we expand the square, making it an integral on R4d. We set
+ξpx, yq :“ |fpxq|2Qtpx, yqe´p|γ|2´dqt ´
+eγXspxq`γXspyq ´ E
+”
+eγXspxq`γXspyqq | Fr
+ı¯
+1Ar,qpxq.
+We have
+E
+”
+pA
+p2q
+t
+´ A
+p1q
+t q2ı
+“
+ż
+R4d E
+“
+ξpx1, y1qξpx2, y2q
+‰
+dx1dy1dx2dy2.
+(5.36)
+As the range of correlation of the increment field Xrr,ts :“ Xt ´ Xr is smaller that e´r
+have, whenever |x1 ´ x2| ě 3e´r
+E
+“
+ξpx1, y1qξpx2, y2q | Fr
+‰
+“ 0.
+(5.37)
+Hence we only need to integrate the r.h.s. of (5.36) on the set |x1 ´ x2| ď 3e´r. In that
+case we use
+E
+“
+ξpx1, y1qξpx2, y2q
+‰
+ď E
+“
+|ξpx1, y1q|2‰1{2 E
+“
+|ξpx2, y2q|2‰1{2 .
+(5.38)
+and
+E
+“
+|ξpx, yq|2‰
+“ |fpxq|4Qtpx, yq2e2p|γ|2´dqtE
+”
+e
+?
+2dpXtpxq`Xtpyqq1Ar,qpxq
+ı
+(5.39)
+Using Cameron-Martin formula and the fact that pXtpxqqtě0 is a standard Brownian
+motion we have
+E
+”
+e
+?
+2dpXtpxq`Xtpyqq1Ar,qpxq
+ı
+“ e2dpt`Ktpx,yqqP r@u P r0, rs, Bu ď q ´ Kupx, yqs .
+Using (3.12) (and then Lemma A.2) we obtain for a constant q1 ą q
+e´4dtE
+”
+e
+?
+2dpXtpxq`Xtpyqq1Aq,rpxq
+ı
+ď P
+”
+@u P r0, rs, Bu ď q1 ´
+?
+2du
+ı
+ď Cr´3{2e´dr.
+Altogether , setting hpx, y, tq :“ 1t|x1´x2|ď3e´ru|fpx1qfpx2q|2Qtpx1, y1qQtpx2, y2q (recall
+that r is a function of t) we obtain that for t sufficiently large
+E
+”
+pA
+p2q
+t
+´ A
+p1q
+t q2ı
+ď Cr´3{2e2p|γ|2`dqt´dr
+ż
+R4d hpx, y, tqdxdy
+ď C1t´3{2e2|γ|2t´2dr ď C2t´1{2e2dpt´rqφptq2 ď t´1{4φptq2.
+(5.40)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES25
+To get the second inequality, simply observe that h is smaller than a constant times the
+indicator of the set t|x1| ď R, |x2 ´ x1| ď 3e´r, |yi ´ xi| ď e´t, i “ 1, 2u, which has volume
+of order e´dpr`2tq. The third inequality is a consequence of (A.10) (see the computation
+in the Appendix, while the last inequality follows from the the fact that with our choice
+of parameters (5.18) we have t ´ r “ oplog tq.
+Step 3: Comparing Ap2q
+t
+and rAt. Finally, we show that
+lim
+tÑ8 φptq´1E
+”
+|Ap2q
+t
+´ rAt|1Aq,R
+ı
+“ 0
+(5.41)
+which together with (5.24)-(5.32), concludes the proof of (5.21). We introduce another
+smaller time parameter, namely r “ t{2 and define p
+Xpx, yq “ Xrpxq ` Xrr,rspyq. We want
+to replace Xrpyq by Xrpxq in the exponential with an intermediate steps, so we set
+Z1pxq :“
+?
+2dXrpxq,
+Z2px, yq :“ γXrpxq ` γ p
+Xpx, yq,
+Z3px, yq :“ γXrpxq ` γXrpyq.
+(5.42)
+The reader can check that we have
+rAt :“
+ż
+R2d |fpxq|2Qtpx, yqe|γ|2K˚
+t px,yqeZ1pxq´ 1
+2ErZ1pxqsdxdy,
+Ap2q
+t
+:“
+ż
+R2d |fpxq|2Qtpx, yqe|γ|2K˚
+t px,yqeZ3px.yq´ 1
+2ErZ3px,yqsdxdy.
+(5.43)
+In order to prove (5.41) taking absolute value inside the integrand, we have
+E
+”
+|Ap2q
+t
+´ rAt|1Aq,R
+ı
+ď
+max
+|x|ďR
+|x´y|ďe´t
+E
+„ˇˇˇeZ1pxq´
+ErZ2
+1 s
+2
+´ eZ3´
+ErZ2
+3 s
+2
+ˇˇˇ1Aq,R
+
+ˆ
+ż
+R2d |fpxq|2Qtpx, yqe|γ|2K˚
+t px,yqdxdy.
+(5.44)
+Since the integral is of order ep|γ|2´dqt (cf. (3.12)), which is the same order as φptq
+?
+t (cf.
+(A.10)), the estimate (5.41) boilds down to proving
+lim
+tÑ8
+?
+t
+max
+|x|ďR
+|x´y|ďe´t
+E
+„ˇˇˇeZ1pxq´
+ErZ2
+1 s
+2
+´ eZ3´
+ErZ2
+3 s
+2
+ˇˇˇ1Aq,R
+
+“ 0.
+(5.45)
+To prove (5.45) we start with the decomposition
+E
+„ˇˇˇeZ1pxq´
+ErZ2
+1 s
+2
+´ eZ3´
+ErZ2
+3 s
+2
+ˇˇˇ1Aq,R
+
+ď E
+„ˇˇˇeZ1´
+ErZ2
+1 s
+2
+´ eZ2´ ErZ2s
+2
+ˇˇˇ1Ar,qpxq
+
+` E
+„ˇˇˇeZ3´
+ErZ2
+3 s
+2
+´ eZ2
+2´
+ErZ2
+2 s
+2
+ˇˇˇ
+
+(5.46)
+
+26
+HUBERT LACOIN
+(this is just the triangle inequality and replacing Aq,R with a larger event Ar,qpxq) and
+show that each term is opt´1{2q. We start with the second one. From Lemma A.3, we have
+E
+„
+eZ3´
+ErZ2
+3 s
+2
+´ eZ2
+2´
+ErZ2
+2 s
+2
+|
+
+ď C
+a
+Er|Z3 ´ Z2|2s
+“ C|γ|
+a
+ErpXrpxq ´ Xrpyqq2s ď C1e´ct,
+(5.47)
+where we have used that
+ErpXrpxq ´ Xrpyqq2s “ 2pr ´ Krpx, yqq ` pK0pxq ` K0pyq ´ 2K0px, yqq.
+The second part of the sum is smaller than |x ´ y|c since K0 is H¨older continuous and the
+first part is smaller than |x ´ y|2e2r (from (3.14)), both are exponentially small in t. For
+the first term in (5.46) we factorize the part that is Fr measureable and use independence
+to obtain
+E
+„
+|eZ1´
+ErZ2
+1 s
+2
+´ eZ2´ ErZ2s
+2
+|1Aq,rpxq
+
+“ E
+”
+e
+?
+2dXrpxq´dKrpxq1Aq,rpxq
+ı
+E
+„
+|eZ1
+1´
+ErpZ1
+1q2s
+2
+´ eZ1
+2´
+ErpZ1q2
+2s
+2
+|
+
+,
+(5.48)
+where Z1
+i “ Zi ´
+?
+2dXrpxq. Using Cameron-Martin formula and Lemma A.2, we have
+E
+”
+e
+?
+2dXrpxq´dKrpxq1Ar,qpxq
+ı
+“ P r@s P r0, rs, Bs ď qs ď
+c
+2
+rπq.
+(5.49)
+The factor r´1{2 is sufficient to cancel the
+?
+t in (5.45) and we just have to show that the
+second factor in (5.48) is small. From Lemma A.3 we have
+E
+„
+|eZ1
+1´
+ErpZ1
+1q2s
+2
+´ eZ1
+2´
+ErpZ1
+2q2s
+2
+|
+
+ď
+b
+E r|Z1
+1 ´ Z1
+2|2s
+“ |γ|
+b
+E
+“
+|Xrr,rspxq ´ Xrr,rspyq|2‰
+ď Cer|x ´ y| ď Cer´t,
+(5.50)
+where the penultimate inequality can be deduced from (3.14). The combination of (5.47)-
+(5.49) and (5.50) concludes the proof of (5.45).
+□
+Bonus step: the case of Bt. To conclude let us sketch rapidly the proof of (5.10). We can
+repeat the argument of step 2 to show that
+lim
+tÑ8 φptq´2Er|Bt ´ ErBt | Frs|2s “ 0.
+(5.51)
+Then it is rather direct to check that
+lim
+tÑ8 φptq´1E
+“
+|ErBt | Frs|1Aq,R
+‰
+“ 0.
+(5.52)
+More precisely we have
+ErBt | Frs “
+ż
+R2d fpxqfpyqQtpx, yqeγpXrpxq`Xrpyqq` γ2
+2 p2Krr,tspx,yq´Krpxq´Krpyqqdxdy.
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES27
+Taking the absolute value of the integrand, using (3.12) to evaluate Kt and Krr,ts, then
+the inequality ab ď pa2 ` b2q{2 and symmetry, and finally (3.13)
+|ErBt | Frs| ď Cep|γ2|´dqp2r´tq
+ż
+R2d |fpxqfpyq|Qtpx, yqe
+?
+d{2pXrpxq`Xrpyqqdxdy
+ď Cep|γ2|´dqp2r´tq
+ż
+R2d |fpxq|2Qtpx, yqe
+?
+2dXrpxqdxdy
+ď C1ep|γ2|´dqp2r´tq´dt
+ż
+R2d |fpxq|2e
+?
+2dXrpxqdx
+(5.53)
+Hence we have
+E
+“
+|ErBt | Frs|1Aq,R
+‰
+ď ep|γ2|´dqp2r´tq´dt
+ż
+R2d |fpxq|2E
+”
+e
+?
+2dXrpxq1Ar,qpxq
+ı
+dx.
+(5.54)
+Using Cameron Martin formula, (3.12) and Lemma A.2 (recall that r „ t) we obtain that
+E
+”
+e
+?
+2dXrpxq1Ar,qpxq
+ı
+ď Ct´1{2edr.
+(5.55)
+Overall using (A.10) we have φptq´1E
+“
+|ErBt | Frs|1Aq,R
+‰
+ď Ce´|γ2|pt´rq.
+□
+6. Proof of Proposition 2.8
+6.1. Organization of the proof. Like for the proof of Theorem 2.1, we assume that our
+probability space contains a martingale approximation sequence pXtqtě0 of the field X,
+with covariance given by (3.3). For the same reason as the one exposed at the beginning
+of Section 4.2 this entails no loss of generality.
+The main idea is to apply Theorem 3.6 (for the filtration corresponding to pXtq) to the
+family Mγ
+ε pf, ωq with rate vpε, θ, γq and with the variable Z being equal to M1pe|γ|2L|f|2q.
+Hence need to check that the martingale Mγ
+t,εpf, ωq :“ E rMγ
+ε pf, ωq | Fts satisfy all the
+requirements in (3.22). Setting W pεq
+t
+:“ Mγ
+t,ε (recall (3.7)), and using the bilinearity of the
+martingale bracket like in (5.2) we obtain
+xMγ
+¨,εpf, ωqyt “ 1
+2
+´
+xW pεqyt ` Repe´2iωxW pεq, W pεqytq
+¯
+.
+(6.1)
+The requirements concerning the quadratic variation of Mγ
+t,εpf, ωq can be obtained as
+consequences of the following,
+Proposition 6.1. The following convergences hold
+lim
+εÑ0 E
+«ˇˇˇˇˇ
+xW pεqy8
+2vpε, θ, γq2 ´ M1pe|γ|2L|f|2q
+ˇˇˇˇˇ 1Aq,R
+ff
+“ 0,
+lim
+εÑ0 E
+«ˇˇˇˇˇ
+xW pεq, W pεqy8
+vpε, θ, γq2
+ˇˇˇˇˇ 1Aq,R
+ff
+“ 0.
+(6.2)
+Furthermore we have for any fixed t we have
+sup
+εPp0,1q
+ErxW pεqyts ă 8.
+(6.3)
+Proposition 6.1 is proved in the next subsection, let us first show how our main results
+can be deduced from it.
+
+28
+HUBERT LACOIN
+Proof of Proposition 2.8. We must check that the three requirements in (3.22) are satis-
+fied since the result follows then from Theorem 3.6. Given that limεÑ0 vpε, θ, γq “ 8,
+it is sufficient for the second and third requirements to show that that the sequences
+pMγ
+0,εpf, ωqqεPp0,1q, and pxMγ
+¨,εpf, ωqytqεPp0,1q (for a fixed t) are tight. The sequences are in
+fact uniformly bounded in L1. We have
+sup
+εPp0,1q
+Er|Mγ
+0,εpf, ωq|s ď sup
+εPp0,1q
+Er|Mγ
+0,εpfq|s ă 8.
+(6.4)
+Indeed taking the absolute value of the integrand, we have
+Er|Mγ
+0,εpfq|s ď
+ż
+Rd E
+„
+fpxqe
+?
+d{2X0,εpxq` β2´pd{2q
+2
+K0,εpxq
+
+dx “
+ż
+Rd fpxqeβ2K0,εpxqdx,
+(6.5)
+and the uniform bound follows from (3.12). From (6.1) we have xMγ
+¨,εpf, ωqyt ď xW pεqyt
+and thus the uniform boundedness in L1 is consequence of (6.3). Let us now turn to
+the first and main requirement in (3.22). The convergences in (6.2) imply the following
+convergence in probability
+lim
+εÑ0
+xW pεqy8
+2vpε, θ, γq2 1Ť
+qě1 Aq,R “ M1pe|γ|2L|f|2q,
+lim
+εÑ0
+xW pεq, W pεqy8
+vpε, θ, γq2
+1Ť
+qě1 Aq,R “ 0.
+(6.6)
+Using Lemma 5.2 and (6.1), we conclude that
+lim
+εÑ0 vpε, θ, γq´2xMγ
+¨,εpf, ωqy8 “ M1pe|γ|2L|f|2q
+in probability.
+□
+As another preliminary step to our proof, we reduce the convergence statement in
+Proposition 6.1 to a convergence of the derivative of the martingale brackets. Using Itˆo
+calculus we obtain that for T P r0, 8s,
+xW pεqyT “
+ż T
+0
+At,εdt and xW pεqyT “
+ż T
+0
+Bt,εdt
+where
+At,ε :“
+ż
+R2d fpxqfpyqQt,εpx, yqeγXt,εpxq`γXt,εpyq´ γ2
+2 Kt,εpxq´ γ2
+2 Kt,εpyqdxdy,
+Bt,ε :“
+ż
+R2d fpxqfpyqQt,εpx, yqeγXt,εpxq`γXt,εpyq´ γ2
+2 pKt,εpxq`Kt,εpyqqdxdy.
+(6.7)
+Similarly to what has been done in Proposition 5.3, we are going to show that, with ap-
+propriate renormalizations and restrictions, At,ε and Bt,ε converge in L1 to M1pe|γ|2L|f|2q
+and 0 respectively. To this end we introduce a couple of parameters (recall (3.5))
+tpt, εq :“ t ^ logp1{εq
+φpt, εq :“
+d
+2
+πpt _ 1qe|γ|2j
+ˆż
+Rd e|γ|2Kt,εp0,zqQt,εp0, zqdz
+˙
+.
+(6.8)
+The quantity r will on Our aim is to prove the following
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES29
+Proposition 6.2.
+lim
+εÑ0
+tÑ8
+E
+„ˇˇˇˇ
+At,ε
+φpt, εq ´ M1pe|γ|2L|f|2q
+ˇˇˇˇ 1Aq,R
+
+“ 0,
+lim
+εÑ0
+tÑ8
+E
+„ˇˇˇˇ
+Bt,ε
+φpt, εq
+ˇˇˇˇ 1Aq,R
+
+“ 0,
+(6.9)
+and for any T ă 8
+sup
+tPr0,Ts
+εPp0,1q
+E r|At,ε|s ă 8.
+(6.10)
+Remark 6.3. Let us underline that lim εÑ0
+tÑ8 Fpt, εq “ 0 means that that there exists t0pδq
+and ε0pδq such that |Fpt, εq| ď δ when t ě t0 AND ε P p0, ε0q. This is a stronger statement
+than both limεÑ0 limtÑ8 Fpt, εq “ 0 or limtÑ8 limεÑ0 Fpt, εq “ 0
+Clearly (6.10) implies (6.3). To deduce (6.2) from (6.9), we need to check that renormal-
+izing factor 2vpε, θ, γq2 corresponds to the integral of φpt, εq. This is the content of the
+following lemma whose proof is presented in Appendix A.4.
+Lemma 6.4. We have for any |γ| ě d
+lim
+εÑ0
+|γ|2 ş8
+0 φpt, εqdt
+2vpε, θ, γq2
+“ 1
+(6.11)
+We can now complete the proof of Proposition 6.1 using Proposition 6.2
+Proof of Proposition 6.1. From Lemma 6.4 it is sufficient to prove the convergence of
+E
+«ˇˇˇˇˇ
+xW pεqy8
+ş8
+0 |γ|2φpt, εq ´ M1pe|γ|2L|f|2q
+ˇˇˇˇˇ 1Aq,R
+ff
+ď
+1
+ş8
+0 φpt, εqdt
+ż 8
+0
+φpt, εqE
+„ˇˇˇˇ
+At,ε
+φpt, εq ´ M1pe|γ|2L|f|2q
+ˇˇˇˇ 1Aq,R
+
+dt.
+(6.12)
+Let us fix δ ą 0, and let T and ε0 be such that for all t ą T and ε ă ε0 we have
+E
+„ˇˇˇˇ
+At,ε
+φpt, εq ´ M1pe|γ|2L|f|2q
+ˇˇˇˇ 1Aq,R
+
+ď δ
+2
+(6.13)
+In the integral we can distinguish the contribution from r0, Ts from the rest. We have
+from (6.10) and the fact that φpt, εq is bounded from below
+sup
+tPr0,Ts
+εPp0,ε0q
+E
+„ˇˇˇˇ
+At,ε
+φpt, εq ´ M1pe|γ|2L|f|2q
+ˇˇˇˇ 1Aq,R
+
+ă 8.
+(6.14)
+As a consequence, since
+ş8
+0 φpt, εqdt diverges when ε Ñ 0, taking ε1 sufficiently small we
+have forall ε P p0, ε1q
+1
+ş8
+0 φpt, εq
+ż T
+0
+φpt, εqE
+„ At,ε
+φpt, εq ´ M1pe|γ|2L|f|2q
+
+dt ď δ
+2,
+(6.15)
+
+30
+HUBERT LACOIN
+which implies that for ε ď ε0 ^ ε1 we have
+E
+«ˇˇˇˇˇ
+xW pεqy8
+ş8
+0 φpt, εqdt ´ M1pe|γ|2L|f|2q
+ˇˇˇˇˇ 1Aq,R
+ff
+ď δ
+(6.16)
+and thus conclude the proof.
+□
+6.2. Proof of Proposition 6.2. Let us start with the proof of (6.10). Noting that At,ε
+is positive, we have
+E rAt,εs “
+ż
+R2d fpxqfpyqQt,εpx, yqe|γ|2Kt,εpx,yqdxdy.
+(6.17)
+We can just use (3.12) and bound Qt,εpx, yq by 1 and Kt,εpx, yq by T `C to conclude. For
+the proof of the convergence of At,ε we proceed exactly as for the proof of Proposition 5.3.
+We assume that tpt, εq ą e (recall (6.8)) and set
+r “ rpt, εq :“ t ´ log log t,
+(6.18)
+Setting K˚
+t,εpx, yq :“ K0pxq ` Kt,εpx, yq, we define
+rAt,ε :“
+ż
+R2d |fpxq|2Qt,εpx, yqe|γ|2K˚
+t,εpx,yqe
+?
+2dXrpxq´2dKrpxqdxdy
+“ φpt, εqM
+?
+2d
+r
+pe|γ|2L|f|2q
+(6.19)
+Since lim εÑ0
+tÑ8 rpt, εq “ 8, we obtain, as a direct consequence of Lemma 5.5 that
+lim
+εÑ0
+tÑ8
+E
+«ˇˇˇˇˇ
+rAt,ε
+φpt, εq ´ M1pe|γ|2L|f|2q
+ˇˇˇˇˇ 1Aq,R
+ff
+“ 0.
+(6.20)
+Our task is thus to prove that
+lim
+εÑ0
+tÑ8
+φpt, εq´1E
+”
+| rAt,ε ´ At,ε|1Aq,R
+ı
+“ 0.
+(6.21)
+Like for the proof of (5.21) in the previous section, we proceed in three steps.
+Step 1: Changing the deterministic prefactor in the integrand. Set
+Ap1q
+t,ε :“
+ż
+R2d |fpxq|2Qt,εpx, yqeγXt,εpxq`γXt,εpyq´ γ2
+2 Kt,εpxq´ γ2
+2 Kt,εpyq`|γ|2pK0pxq´K0,εpx,yqqdxdy
+Let us prove that
+lim
+εÑ0
+tÑ8
+φpt, εq´1E
+”
+|Ap1q
+t,ε ´ At,ε|1Aq,R
+ı
+“ 0.
+(6.22)
+Let As a direct consequence of the continuity of f and of K0, if one sets
+sup
+|x|,|y|ďR
+|x´y|ďet`2ε
+ˇˇˇfpxqfpyq ´ |fpxq|2e|γ|2pK0pxq´K0,εpx,yqqˇˇˇ “: δpε, tq,
+(6.23)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES31
+we have lim εÑ0
+tÑ8 δpε, tq “ 0. Since Qt,εpx, yq “ 0 when |x ´ y| ě et ` 2ε repeating the
+computation (5.26) and using (3.13) we get
+|Ap1q
+t,ε ´ At,ε| ď δpε, tq
+ż
+Bp0,Rq2 Qt,εpx, yqe
+?
+2dXt,εpxq`p|γ|2´dqKt,εpxqdxdy
+ď Ce´dtδpε, tq
+ż
+Bp0,Rq
+e
+?
+2dXt,εpxq`p|γ|2´dqKt,εpxqdx
+(6.24)
+Using Cameron-Martin formula, we obtain (assuming |x|, |y| ď R and |x ´ y| ď et ` 2ε)
+for a constant q1 ą q
+E
+”
+e
+?
+2dXt,εpxq´dKt,εpxq1Aq,R
+ı
+ď E
+”
+e
+?
+2dXt,εpxq`p|γ|2´dqKt,εpxq1At,qpxq
+ı
+“ P
+”
+@s P r0, ts, Bs ď
+?
+2dps ´ Ks,ε,0pxqq ` q
+ı
+ď Pp@s P r0, ts, Bs ď q1q ď Cpt _ 1q´1{2et|γ|2
+(6.25)
+where in the second inequality we have used (3.12) to estimate covariances. Hence, using
+(A.20) we deduce from (6.24) that
+E
+”
+|Ap1q
+t,ε ´ At,ε|1Aq,R
+ı
+ď Cδpε, tqt´1{2et|γ|2´dt ď C1δpε, tqφpt, εq.
+This concludes the proof of (6.22).
+□
+Step 2: Taking conditional expectation. We set
+Ap2q
+t,ε :“ E
+”
+Ap1q
+t,ε | Fr
+ı
+(6.26)
+and we are going to prove
+lim
+εÑ0
+tÑ8
+φpt, εq´2E
+”
+|Ap2q
+t,ε ´ Ap1q
+t,ε |21Aq,R
+ı
+“ 0.
+(6.27)
+Like what we did in the previous section, we assume here that K0 ” 0 to simplify the
+writing (but this does not affect the proof).
+In that case note that since Kt,εpxq “
+Kt,εpyq “ Kt,εpxq, we can factorize the term. We have (recall that Krr,ts,ε “ Kt,ε ´ Kr,ε)
+Ap2q
+t,ε :“
+ż
+R2d |fpxq|2Qt,εpx, yqeγXr,εpxq`γXr,εpyq`p|γ2|´dqKr,εpxq`|γ|2Krr,ts,εpx,yqdxdy
+(6.28)
+Setting
+ζpx, yq :“ eγXt,εpxq`γXt,εpyq`p|γ2|´dqKt,εpxq,
+A
+p1q
+t,ε :“
+ż
+R2d |fpxq|2Qt,εpx, yqζpx, yq1Ar,qpxqdxdy,
+A
+p2q
+t,ε :“
+ż
+R2d |fpxq|2Qt,εpx, yqErζpx, yq |Frs1Ar,qpxqdxdy
+(6.29)
+we realize that
+E
+”
+|Ap2q
+t,ε ´ Ap1q
+t,ε |21Aq,R
+ı
+ď E
+”
+|A
+p2q
+t,ε ´ A
+p1q
+t,ε |2ı
+(6.30)
+Now we set
+ξt,εpx, yq :“ |fpxq|2Qt,εpx, yq pζpx, yq ´ Erζpx, yq |Frsq 1Ar,qpxq.
+(6.31)
+
+32
+HUBERT LACOIN
+We have
+E
+”
+|A
+p2q
+t,ε ´ A
+p1q
+t,ε |2ı
+ď
+ż
+R4d E
+“
+ξpx1, y1qξpx2, y2q
+‰
+dx1dx2dy1dy2
+(6.32)
+The range of the convariance of Xrr,ts,ε is smaller than e´r ` 2ε, and Qt,εpx, yq vanishes
+when |x ´ y| ě e´t ` 2ε. All of this implies that if if |x1 ´ x2| ě 2e´r (if ε is sufficiently
+small, then e´r is much larger than both ε and e´t cf. (6.8)) then
+E
+“
+ξpx1, y1qξpx2, y2q | Fr
+‰
+“ 0.
+(6.33)
+When |x1 ´ x2| ď 2e´r we can use Cauchy-Schwarz to bound the covariance. We have
+E
+“
+|ξpx, yq|2‰
+ď |fpxq|4 pQt,εpx, yqq2 Er|ζpx, yq|21Ar,qpxqs
+(6.34)
+and from Cameron-Martin formula, we have, for |x ´ y| ď e´t ` 2ε
+Er|ζpx, yq|21Ar,qpxqs
+“ e2|γ|2Kt,εpxq`2dKt,εpx,yqP
+´
+@s P r0, rs, Bs ď
+?
+2dps ´ Ks,ε,0pxq ´ Ks,ε,0py, xqq ` q
+¯
+ď Cep|γ|2`dqtP
+´
+@s P r0, rs, Bs ď ´
+?
+2ds ` q1¯
+ď C1etp2|γ|2`dqr´3{2.
+where in the first inequality we used (3.12) which replace the Kt,ε and Ks,ε by t and
+s respectively at the cost of an additive constant and in the second inequality we used
+Lemma A.2. Altogether we obtain that
+E
+”
+|A
+p2q
+t,ε ´ A
+p1q
+t,ε |2ı
+ď Cetp2|γ|2`dqr´3{2
+ˆ
+ż
+R4d 1t|x1´x2|ď2e´ru|fpx1q|2|fpx2q|2Qt,εpx1, y1qQt,εpx2, y2qdxdy
+ď C1e2|γ|2t`dpt´rqr´3{2
+ˆż
+Rd Qt,εp0, zqdz
+˙2
+ď C2edpt´rqr´1{2φpt, εq2.
+(6.35)
+We conclude the proof of (6.27) by observing (recall (6.8)) that
+lim
+εÑ0
+tÑ8
+edpt´rqr´1{2 “ 0.
+(6.36)
+□
+Step 3: Final comparison. Finally to conclude we need to show that
+lim
+εÑ0
+tÑ8
+φpt, εq´2E
+”
+|Ap2q
+t,ε ´ rAt,ε|21Aq,R
+ı
+“ 0.
+(6.37)
+We set (recall that Xrs,ts,ε “ Xt,ε ´ Xs,ε´)
+Z1pxq :“
+?
+2dXrpxq,
+Z2px, yq :“
+?
+2dXrpxq ` γXrr,rs,εpxq ` γXrr,rs,εpyq,
+Z3px, yq :“ γXr,εpxq ` γXr,εpyq.
+(6.38)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES33
+The reader can check (using (6.26) rather than (6.28) since the latter assumes K0 ” 0)
+that
+rAt,ε :“
+ż
+R2d |fpxq|2Qt,εpx, yqe|γ|2K˚
+t,εpx,yqeZ1pxq´ 1
+2 ErZ1pxqsdxdy,
+Ap2q
+t,ε :“
+ż
+R2d |fpxq|2Qt,εpx, yqe|γ|2K˚
+t,εpx,yqeZ3px,yq´ 1
+2 ErZ3px,yqsdxdy.
+(6.39)
+In order to prove (6.37) taking absolute value inside the integrand using Jensen’s inequality,
+we just have to obtain a uniform bound on the integrand, that is, to show that
+lim
+tÑ8
+εÑ0
+t1{2
+max
+|x|ďR
+|x´y|ďe´t`2ε
+E
+„ˇˇˇeZ1pxq´
+ErZ2
+1 pxqs
+2
+´ eZ3px,yq´
+ErZ2
+3 px,yqs
+2
+ˇˇˇ1Aq,R
+
+“ 0.
+(6.40)
+The restriction for x and y comes from the support of f and Qt,ε respectively. To prove
+(5.45) we start with the decomposition
+E
+„ˇˇˇeZ1pxq´
+ErZ2
+1 s
+2
+´ eZ3´
+ErZ2
+3 s
+2
+ˇˇˇ1Aq,R
+
+ď E
+„ˇˇˇeZ1´
+ErZ2
+1 s
+2
+´ eZ2´ ErZ2s
+2
+ˇˇˇ1Ar,qpxq
+
+` E
+„ˇˇˇeZ3´
+ErZ2
+3 s
+2
+´ eZ2
+2´
+ErZ2
+2 s
+2
+ˇˇˇ
+
+(6.41)
+(the inequality is just the triangle inequality and replacing Aq,R with a larger event), and
+show that each term is opt´1{2q. Let us start with the second one. Using Lemma A.3, we
+have
+E
+„
+eZ3´
+ErZ2
+3 s
+2
+´ eZ2
+2´
+ErZ2
+2 s
+2
+|
+
+ď C
+a
+Er|Z3 ´ Z2|2s
+“ C
+b
+Er|
+?
+2dXrpxq ´ γXr,εpxq ´ γXr,εpyq|2s
+ď C1
+ˆb
+E r|Xrpxq ´ Xr,εpxq|2s `
+b
+E r|Xrpxq ´ Xr,εpyq|2s
+˙
+“ C1 ´
+pKrpxq ` Kr,εpxq ´ 2Kr,ε,0pxqq1{2 ` pKrpxq ` Kr,εpxq ´ 2Kr,ε,0py, xqq1{2¯
+ď C2 pε ` |x ´ y|qc ď C1e´ct
+(6.42)
+where to obtain the last line we have used (3.14) and the H¨older continuity of K0. For the
+first term in (6.41) we factorize the Fr-measureable part and use independence to obtain
+E
+„
+|eZ1´
+ErZ2
+1 s
+2
+´ eZ2´ ErZ2s
+2
+|1Ar,qpxq
+
+“ E
+”
+e
+?
+2dXrpxq´dKrpxq1Ar,qpxq
+ı
+E
+„
+|eZ1
+1´
+ErpZ1
+1q2s
+2
+´ eZ1
+2´
+ErpZ1q2
+2s
+2
+|
+
+,
+(6.43)
+where Z1
+i “ Zi ´
+?
+2dXrpxq. We have from Cameron-Martin Formula and Lemma A.2
+E
+”
+e
+?
+2dXrpxq´dKrpxq1Ar,qpxq
+ı
+“ P r@s P r0, rs, Bs ď qs ď Cr´1{2.
+(6.44)
+
+34
+HUBERT LACOIN
+This is obviously Opt´1{2q so to conclude we only need to show that the other factor in
+(6.43) goes to zero. We also have from Lemma A.3 and (3.14)
+E
+„
+|eZ1
+1´
+ErpZ1
+1q2s
+2
+´ eZ1
+2´
+ErpZ1q2
+2s
+2
+|
+
+ď C
+b
+E r|Z1
+1 ´ Z1
+2|2s
+ď C
+`
+Krr,rspxq ` Krr,rs,εpxq ` Krr,rs,εpyq ´ 2Krr,rs,ε,0pxq ´ 2Krr,rs,ε,0py, xq
+˘1{2
+ď C1erp|x ´ y| ` εq ď Cepr´tq.
+(6.45)
+This concludes the proof.
+□
+The convergence of Bt,ε. To prove the second convergence in (6.9), it is sufficient again to
+show first that
+lim
+tÑ8
+εÑ0
+ϕpt, εq´2E
+“
+|Bt,ε ´ ErBt,ε | Frs|21Aq,R
+‰
+“ 0
+(6.46)
+repeating the computation of step two, and then prove that
+lim
+tÑ8
+εÑ0
+Er|ErBt,ε | Frs|1Aq,Rs “ 0.
+We leave this part to the reader, since this is very similar to the computation performed
+at the end of Section 5.
+□
+7. Proof of Theorem 3.6
+We need to show that for any H bounded and F8-measurable and ξ P R we have
+lim
+nÑ8 E
+„
+H
+ˆ
+eiξ Wn
+vpnq ´ e´ ξ2Z
+2
+˙
+“ 0.
+(7.1)
+We first assume that the collection of variables vpnq´2xWny8 is uniformly essentially
+bounded, that is, that there exists M such that for every n ě 1
+P
+“
+vpnq´2xWny8 ě M
+‰
+“ 0
+(7.2)
+Note that this implies also that P rZ ě Ms “ 0. We assume, to simplify notation that
+ξ “ 1 (this entails no loss of generality). We set Ht :“ E rH | Fts and Zt :“ E rZ | Fts we
+have
+E
+„
+H
+ˆ
+ei ξWn
+vpnq ´ e´ ξ2Z
+2
+˙
+“ E
+”
+Hpe´ Z
+2 ´ e´ Zt
+2 q
+ı
+` E
+”
+pH ´ Htq
+´
+ei Wn
+vpnq ´ e´ Zt
+2
+¯ı
+` E
+”
+Ht
+´
+ei Wn
+vpnq ´ e´ Zt
+2
+¯ı
+“: E1pt, nq ` E2pt, nq ` E3pt, nq.
+(7.3)
+We prove the convergence (7.1) by showing that for i “ 1, 2, 3
+lim
+tÑ8 lim sup
+nÑ8 |Eipt, nq| “ 0.
+(7.4)
+Using the fact that z ÞÑ ez is 1-Lipshitz (first line) and has modulus bounded by 1 (second
+line) in tz P C : Repzq ď 0u we have
+|E1pt, nq| ď E
+”
+|H|
+ˇˇˇe´ Z
+2 ´ e´ Zt
+2
+ˇˇˇ
+ı
+ď }H}8
+2
+E r|Z ´ Zt|s ,
+|E2pt, nq| ď E
+”
+|H ´ Ht|
+ˇˇˇei Wn
+vpnq ´ e´ Z
+2
+ˇˇˇ
+ı
+ď 2E r|H ´ Ht|s .
+(7.5)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES35
+Since Ht and Zt converge respectively to H and Z in L1, (7.4) holds for i “ 1, 2. For
+i “ 3, we observe that for fixed t the process
+Mpnq
+u
+:“ e
+iWn,t`u´Wn,t
+vpnq
+`
+xWnyt`u´xWnyt
+2vpnq2
+´ Zt
+2
+is a martingale for the filtration pGuq :“ pFt`uq, which converges in L1 when u Ñ 8. In
+particular we have
+E
+„
+e
+iWn´Wn,t
+vpnq
+` xWny8´xWnyt
+2vpnq2
+| Ft
+
+“ 1.
+(7.6)
+Multiplying by Hte´ Zt
+2 and taking expectation we obtain that
+E
+„
+Hte
+iWn´Wn,t
+vpnq
+` xWny8´xWnyt
+2vpnq2
+´ Zt
+2
+
+“ E
+”
+Hte´ Zt
+2
+ı
+(7.7)
+Hence we have (using that }Ht}8 ď }H}8)
+|E3pt, nq| ď E
+„
+|Ht|
+ˇˇˇˇei Wn
+vpnq ´ e
+ipWn´Wn,tq
+vpnq
+` xWny8´xWnyt
+2vpnq2
+´ Zt
+2
+ˇˇˇˇ
+
+ď }H}8E
+„ˇˇˇˇ1 ´ e
+´
+iWn,t
+vpnq ` xWny8´xWnyt
+2vpnq2
+´ Zt
+2
+ˇˇˇˇ
+
+,
+(7.8)
+From (3.22) we have the following convergence in probability for any fixed t
+lim
+nÑ8 ´iWn,t
+vpnq ` xWny8 ´ xWnyt
+2vpnq2
+´ Zt
+2 “ Z ´ Zt
+2
+.
+(7.9)
+Using assumption (7.2), taking the limit in the r.h.s. of (7.8) and using dominated con-
+vergence, we obtain that
+lim sup
+nÑ8 |E3pt, nq| ď }H}8E
+”ˇˇˇ1 ´ e
+Z´Zt
+2
+ˇˇˇ
+ı
+.
+(7.10)
+Since Zt converges to Z we can conclude that (7.4) also holds for i “ 3 using dominated
+convergence again (both variables are uniformly bounded).
+Let us now remove the boundedness assumption. Given A ą 0 we set
+TA,n :“ inftt : vpnq´2xWnyt “ Au
+and
+W A
+n :“ Wn,TA,n.
+Note that E
+“
+W A
+n | Ft
+‰
+“ Wt^TA,n so that (using the notation xW A
+n yt to denote the qua-
+dratic variation of this martingale) we have
+lim
+nÑ8 vpnq´2xW A
+n y8 “ Z ^ A.
+(7.11)
+Since we have proved (7.1) under the assumption (7.2) we know that for every A ą 0
+lim
+nÑ8 E
+„
+H
+ˆ
+ei ξW A
+n
+vpnq ´ e´ ξ2pZ^Aq
+2
+˙
+“ 0
+(7.12)
+From the convergence assumption, we have
+lim sup
+nÑ8 PrTA,n “ 8s ď P rZ ě As
+(7.13)
+and hence
+lim
+AÑ8 lim inf
+nÑ8 P
+“
+W A
+n “ Wn
+‰
+“ lim
+AÑ8 PrZ ^ A “ Zs “ 1.
+
+36
+HUBERT LACOIN
+As a consequence we can conclude using (7.12) that
+lim
+nÑ8 E
+„
+H
+ˆ
+eiξ Wn
+vpnq ´ e´ ξ2Z
+2
+˙
+“ lim
+AÑ8 lim
+nÑ8 E
+„
+H
+ˆ
+ei ξW A
+n
+vpnq ´ e´ ξ2pZ^Aq
+2
+˙
+“ 0.
+(7.14)
+□
+Acknowledgements: This work was supported by a productivity grant from CNPq and
+a JCNE grant from FAPERJ.
+Appendix A. Technical results and their proof
+A.1. Standard Gaussian tools. We first display two standard tools which are used
+throughout the proof. The first is the standard Cameron-Martin formula which describes
+how a Gaussian process is affected by an exponential tilt.
+Proposition A.1. Let pY pzqqzPZ be a centered Gaussian field indexed by a set Z. We
+let H denote its covariance and P denote its law. Given z0 P Z let us define rPz0 the
+probability obtained from P after a tilt by Y pz0q that is
+drPz0
+dP :“ eY pz0q´ 1
+2 Hpz0,z0q
+(A.1)
+Under rPz0, Y is a Gaussian field with covariance H, and mean rEz0rY pzqs “ Hpz, z0q.
+The second is a bound on the probability for a Brownian Motion to remain below a line.
+Both estimates can be proved directly using the reflexion principle.
+Lemma A.2. Let B be a standard Brownian Motion and let P denote its distribution,
+setting gtpaq :“
+şu`
+0
+e´ z2
+2t dz. we have
+P
+«
+sup
+sPr0,ts
+Bs ď a
+ff
+“
+c
+2π
+t gtpaq ď
+c
+2π
+t a.
+(A.2)
+Additionally for any a, b ą 0 there exists Ca,b such that f
+P
+«
+sup
+sPr0,ts
+pBs ` bsq ď a
+ff
+“
+1
+?
+2πt
+ż
+e´ u2
+2t p1 ´ e
+2apa`u´bsq`
+t
+qdu ď Ca,be´ b2t
+2 t´3{2.
+(A.3)
+A.2. Comparing exponentiated Gaussians. In or comparison of partition functions
+Lemma A.3. Consider pX1, X2, Y1, Y2q an R4 valued centered Gaussian vector and set
+X :“ X1 ` iX2 and Y “ Y1 ` iY2. Assuming that
+ErX2
+2s ď 1 and Er|X ´ Y |2s ď 1
+(A.4)
+then there exits a constant C such that
+E
+”ˇˇeX´ 1
+2ErX2s ´ eY ´ 1
+2ErY 2sˇˇ
+ı
+ď CE
+“
+|X ´ Y |2‰
+(A.5)
+Proof. We factorize eX´ 1
+2ErX2s, use the Cameron-Martin formula and rearrange the ex-
+pectation terms in the exponential, we obtain
+E
+„ˇˇeY ´ ErY 2s
+2
+´ eX´ ErX2s
+2
+ˇˇ
+
+“ E
+„
+eX1`
+ErX2
+2 s´ErX2
+1 s
+2
+ˇˇeY ´X` ErX2s´ErY 2s
+2
+´ 1
+ˇˇ
+
+“ e
+ErX2
+2 s
+2
+E
+„ˇˇeY ´X´ ErpX´Y q2s
+2
+´iErX2pY ´Xqs ´ 1
+ˇˇ
+
+.
+(A.6)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES37
+The prefactor is bounded (by assumption) by e1{2. For the rest, setting Z “ Y ´ X (and
+letting Z1 and Z2 denote the real and imaginary part) we have using the triangle inequality
+E
+„ˇˇeZ´ ErZ2s
+2
+´iErX2Zs ´ 1
+ˇˇ
+
+ď
+ˇˇeiErX2Zs ´ 1
+ˇˇE
+„ˇˇeZ´ ErZ2s
+2
+ˇˇ
+
+` E
+„ˇˇeZ´ ErZ2s
+2
+´ 1
+ˇˇ
+
+.
+(A.7)
+For the first term, using that |ErX2Zs| ď
+a
+ErX2
+2sEr|Z|2s ď
+a
+Er|Z|2s ď 1, and that
+|eu ´ 1| ď e|u| for u ď 1 and computing expectation, we obtain that
+ˇˇeiErX2Zs ´ 1
+ˇˇE
+„ˇˇeZ´ ErZ2s
+2
+ˇˇ
+
+ď e
+a
+Er|Z|2se
+ErZ2
+2 s
+2
+ď e3{2a
+Er|Z|2s.
+(A.8)
+For the second term we have (using again |eu ´ 1| ď e|u|)
+E
+„ˇˇeZ´ ErZ2s
+2
+´ 1
+ˇˇ
+
+ď
+d
+E
+„ˇˇeZ´ ErZ2s
+2
+´ 1
+ˇˇ2
+
+“
+a
+eEr|Z|2s ´ 1 ď e1{2a
+Er|Z|2s,
+(A.9)
+which yields the desired result for C “ e2 ` e.
+□
+A.3. Proof of Lemma 5.4. Let us first compute the order of magnitude of φptq. Let us
+set for practical purpose φptq :“
+ş
+Qtp0, zqe|γ|2Ktp0,zqdz. Using (3.12) (recall that |z| ď e´t
+on the integrand) and (3.13) we have
+φptq — ep|γ|2´dqt
+and
+φptq — t´1{2ep|γ|2´dqt
+(A.10)
+As a consequence when |γ|2 ą d most of the integral is carried by rt ´
+?
+t, ts and we have
+ż t
+0
+φpsqds “ p1 ` op1qq
+ż t
+t´
+?
+t
+φpsqds
+“ p1 ` op1qq
+ż t
+t´
+?
+t
+c
+s _ 1
+t
+φpsqds
+“ p1 ` op1qq
+c
+2
+πte|γ|2j
+ż t
+0
+φpsqds.
+(A.11)
+We observe that
+|γ|2Qsp0, zqe|γ|2Ksp0,zq “ Bs
+´
+e|γ|2Ksp0,zq¯
+.
+Using Fubini and integrating w.r.t. time and making a change of variable we have
+|γ|2
+ż t
+0
+φpsqds “
+ż
+Rd
+´
+e|γ|2Ktp0,zq ´ 1
+¯
+dz
+“ ep|γ|2´dqt
+ż
+Rd
+´
+e|γ|2pKtp0,e´tzq´tq ´ e´|γ|2t¯
+dz.
+(A.12)
+The integrand in the second line is bounded above by p|z| _ 1q´|γ|2. This is obvious for
+|z| ě et since the integrand vanishes, and when |z| ď et this can be obtainded from (3.11).
+Furthermore it converges to e|γ|2ℓpzq and we obtain using dominated convergence that
+lim
+tÑ8
+|γ|2 şt
+0 φpsqds
+ep|γ|2´dqt
+“
+ż
+Rd e|γ|2ℓpzqdz,
+(A.13)
+which combined with (A.11), proves the lemma in the case |γ|2 ą d. When |γ|2 “ d, we
+observe that using, as in (A.12), a change of variable and dominated convergence, we have
+lim
+sÑ8 φpsq “
+ż
+Rd κpe
+η1
+η2 zqedℓpzqdz.
+(A.14)
+
+38
+HUBERT LACOIN
+On the other hand we have from (A.12)
+ż t
+0
+φpsqds “
+ż
+Rd
+´
+edpKtp0,e´tzq´tq ´ e´dt¯
+dz
+(A.15)
+We have, for 1 ď |z| ď et,
+Ktp0, e´tzq “ Kp0, e´tzq “ Kp0, e´tzq ´ K0p0, e´tzq
+“ t ` log 1
+|z| ` pL ´ K0qp0, e´tzq.
+(A.16)
+Since pL ´ K0qp0, e´t|z|q “ ´j ` δ
+`
+e´t|z|
+˘
+where δpuq tends to zero when u Ñ 0 we can
+deduce that
+ż
+Rd
+´
+edpKtp0,e´tzq´tq ´ e´dt¯
+dz “ p1 ` op1qq
+ż
+1t1ď|z|ďetuedpKtp0,e´tzq´tqdz
+“ p1 ` op1qqe´dj
+ż
+1t1ď|z|ďetu|z|´ddz
+“ p1 ` op1qqe´djΣd´1t.
+(A.17)
+Since φpsq converges, we deduce that its limit equals its Cesaro limit and thus
+lim
+sÑ8 φpsq “ e´djΣd´1,
+(A.18)
+which implies in turn that
+ż t
+0
+d
+2
+πps ^ 1qedjφpsq “ p1 ` op1qq2
+c
+2t
+π Σd´1,
+(A.19)
+and concludes the proof of the lemma.
+□
+A.4. Proof of Lemma 6.4. Let us again start with the case |γ|2 ą d. As in the proof of
+Lemma 5.4, we can compute the asymptotic of φpt, εq (when t and ε goes to infinity and
+zero respectively)
+φpt, εq — t´1{2e|γ|2t´dt.
+(A.20)
+Since suptPr0,Ts
+εPp0,1q
+φpt, εq ă 8 for every finite T, this implies that the integral
+ş8
+0 φpt, εq is
+mostly carried by values of s around logp1{εq (say ˘
+a
+logp1{εq). For this reason, we can
+replace the term pt _ 1q´1{2 by plog 1{εq´1{2.
+|γ|2
+ż 8
+0
+φpt, εqdt “ p1 ` op1qq
+d
+2
+πplog 1{εqe|γ|2j
+ż 8
+0
+ż
+Rd |γ|2e|γ|2Kt,εp0,zqQt,εp0, zqdz (A.21)
+Using Fubini and integrating with respect to time as in (A.12) we have
+ż 8
+0
+ż
+Rd |γ|2e|γ|2Kt,εp0,zqQt,εp0, zqdz “
+ż
+Rd
+´
+e|γ|2Kεp0,zq ´ 1
+¯
+dz
+(A.22)
+We then perform a change of variable for z
+ż
+Rd
+´
+e|γ|2Kεp0,zq ´ 1
+¯
+dz “ εd´|γ|2 ż
+Rd
+´
+e|γ|2pKεp0,εzq`logpεqq ´ ε|γ|2¯
+dz
+(A.23)
+Next we observe that
+Kεp0, εzq ` logpεq “ ℓθpzq ` pLεp0, εzq ´ Kεp0, εzqq .
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES39
+Using dominated convergence as ε goes to zero (the integrand is bounded above by p|z| _
+1q´|γ|2 and recalling (3.9) we obtain that
+lim
+εÑ0
+ż
+Rd
+´
+e|γ|2pKεp0,εzq`logpεqq ´ ε|γ|2¯
+dz “ e´|γ|2j
+ż
+Rd e´|γ|2ℓθpzqdz.
+(A.24)
+The combination of (A.21)-(A.24) concludes the proof in the case |γ|2 ą d. For the case
+|γ|2 “ d, based on (A.20) we know that setting Tε “ logp1{εq ´
+a
+logp1{εq
+ż 8
+0
+φpt, εqdt “ p1 ` op1qq
+ż Tε
+0
+φpt, εqdt.
+(A.25)
+Now in this range for t it is tedious but not difficult to check that
+lim
+εÑ0 sup
+tPr0,Tεs
+ş
+Rd e|γ|2Kt,εp0,zqQt,εp0, zqdz
+ş
+Rd e|γ|2Ktp0,zqQtp0, zqdz
+“ 1.
+(A.26)
+From this we obtain that
+ż 8
+0
+φpt, εqdt “ p1 ` op1qq
+ż Tε
+0
+φpt, εqdt “ p1 ` op1qq
+ż Tε
+0
+φptqdt
+(A.27)
+and we can conclude using Lemma 5.4.
+Appendix B. The convergence of Mγ
+ε as a distribution
+We have chosen for simplicity, to present our convergence results as convergence of a
+collection of random variables Mγ
+ε pfq indexed by CcpRdq. We can go further and prove
+that Mγ
+ε p¨q converges as a distribution. For this purpose we need to recall the definition
+of local Sobolev/Bessel spaces.
+The Bessel space Hs,ppRkq, s P R and p P r1, 8s on Rk is defined by
+Hs,ppRkq :“ tϕ P D1pRkq : p1 ` |ξ|2qs{2 pϕpξq P LppRkqu
+(B.1)
+where D1pRkq is the space of distribution and pϕpξq is the Fourier transform of ϕ defined
+for ϕ P C8
+c pRkq by pϕpξq “
+ş
+Rk eiξxϕpxqdx. It is a Banach space when equiped with the
+norm
+}f}Hs,p “
+ż
+Rkp1 ` |ξ|2qps{2|pϕpξq|pdξ
+(B.2)
+For U Ă Rk open, the local Bessel space Hs,p
+locpUq denotes the set of distributions which
+belongs to Hs,ppUq after multiplication by an arbitrary smooth function with compact
+support
+Hs,p
+locpUq :“
+!
+ϕ P D1pUq | ρϕ P Hs,ppRdq for all ρ P C8
+c pUq
+)
+,
+(B.3)
+where above ρϕ is identified with its extension by zero on Rk. It is equiped with the
+topology generated by the family of seminorms rρ, ρ P C8
+c pUq defined by rρpϕq :“ }ϕρ}Hs,p.
+In the particular case where p “ 2 we write HspRkq :“ Hs,2pRkq which is a Hilbert space
+(and use the same convention for the local spaces). The convergence result for Mγ
+ε p¨q as a
+distribution for γ P PI{II is the following.
+Theorem B.1. If X is a centered Gaussian field whose covariance kernel K has an
+almost star-scale invariant part, γ P PI{II, p P r1,
+?
+2dαq and s ă ´ d
+p, then there exists
+Mγ
+8 P Hs,p
+locpRdq such that for every ρ P C8
+c pRdq
+lim
+εÑ0 E
+“
+}Mγ
+ε pρ ¨q ´ Mγ
+8pρ ¨q}p
+Hs,p
+‰
+“ 0.
+(B.4)
+
+40
+HUBERT LACOIN
+In particular Mγ
+ε converges to Mγ
+8 in probability in the Hs,p
+locpRdq topology.
+Similarly in P1
+II{III the convergence in law holds also for the distribution.
+Theorem B.2. Let X be a Gaussian random field with an almost star-scale covariance.
+Then given γ P P1
+II{III, and s ă ´ d
+2 the following joint convergence in law for the Hs
+locpRdq
+topology
+ˆ
+X,
+Mγ
+ε
+vpε, θ, γq
+˙
+εÑ0
+ñ pX, Mγq,
+(B.5)
+Remark B.3. In the proof of Theorems B.1 and B.2 presented below, we are going to
+assume that our probability space contains a martingale sequence pXtqtě0 of fields with co-
+variance (3.3) approximating X. For reasons analogous to the one exposed at the beginning
+of Section 4.2 this entails no loss of generality.
+B.1. The case of Theorem B.1. Let us fix ρ P C8
+c pRdq. We want to prove that Mγ
+ε pρ ¨q
+converges in Hs,ppRdq. We first define the limit point. We set (without underlying the
+dependence in ρ to keep the notation light)
+x
+Mγ
+ε pξq “
+ż
+Rd ρpxqeiξ.xeγXεpxq´ γ2
+2 Kεpxqdx,
+x
+Mγ
+8pξq “ lim
+εÑ0
+x
+Mγ
+ε pξq.
+(B.6)
+We let Mγ
+8pρ ¨q denote the random distribution whose Fourier transform is given by x
+Mγ
+8pξq.
+The proof of (B.4), implies that Mγ
+8pρ ¨q P Hs,ppRdq with probability one. To prove (B.4),
+note that we have
+E
+“
+}Mγ
+ε pρ ¨q ´ Mγ
+8pρ ¨q}p
+Hs,p
+‰
+“
+ż
+Rd E
+”
+|px
+Mγ
+ε ´ x
+Mγ
+8qpξq|pı
+p1 ` |ξ|2q
+ps
+2 dξ.
+(B.7)
+Hence with our assumption on s ă ´ d
+p it is sufficient to prove that
+lim sup
+εÑ0
+sup
+ξPRd E
+”
+|px
+Mγ
+ε ´ x
+Mγ
+8qpξq|pı
+ă 8,
+@ξ P Rd, lim
+εÑ0 E
+”
+|px
+Mγ
+ε ´ x
+Mγ
+8qpξq|pı
+“ 0.
+(B.8)
+and we can then conclude using dominated convergence (the first line yields the domina-
+tion). The second line is simply (2.2) with fpxq “ ρpxqeiξ.x. For the first line it sufficient
+to prove that
+lim sup
+εÑ0
+sup
+ξPRd E
+”
+|x
+Mγ
+ε pξq|pı
+ă 8,
+since the bound for x
+Mγ
+8pξq| can then be obtained by Fatou. We set V pεq
+t
+pξq “ Erx
+Mγ
+ε | Fts.
+Using the BDG inequality we have
+E
+”
+|x
+Mγ
+ε pξq|pı
+ď CE
+”
+|V pεq
+0
+pξq|p ` xV pεqpξqyp{2
+8
+ı
+.
+(B.9)
+Now we have (recall that p ă
+?
+2d{α ď 2)
+E
+”
+|V pεq
+0
+pξq|2ı
+“
+ż
+R2d ρpxqρpyqeiξ.px´yqe|γ|2K0,εpx,yqdxdy
+(B.10)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES41
+and we can conclude by replacing eiξ.px´yq by 1 and observing that since K is continuous
+K0,ε is uniformly bounded for x, y in the support of ρ and ε P p0, 1q. For the quadratic
+variation part, we have
+xV pεqy8 “ |γ|2
+ż 8
+0
+At,εpξqdt,
+(B.11)
+where,
+At,εpξq :“
+ż
+R2d ρpxqρpyqQt,εpx, yqeiξpx´yqeγXt,εpxq`γXt,εpyq´ γ2
+2 Kt,εpxq´ γ2
+2 Kt,εpyqdxdy
+ď
+ż
+R2d ρpxqρpyqQt,εpx, yqeαpXt,εpxq`Xt,εpyqq` β2´α2
+2
+pKt,ε`Kt,εpyqqdxdy “: At,ε,
+(B.12)
+the inequality being obtain by taking the modulus of the integrand. To conclude, we just
+need to prove that
+sup
+εPp0,1q
+E
+«ˆż 8
+0
+At,εdt
+˙p{2ff
+ă 8
+(B.13)
+For this part we can just repeat the computations made to prove (4.23) in Section 4.2.
+B.2. The case of Theorem B.2. Since the convergence of finite dimensional marginal
+has been established, we only need to prove tightness of the distribution of vpε, θ, γq´1Mγ
+ε pρ ¨q
+in HspRdq for every ρ. For this, we simply replicate the strategy presented in [16], with a
+minor twist. Since in our case, the Fourier transform in not in L2, we need to consider a
+restriction to the event Aq,R where R is such that the support ρ is contained in Bp0, Rq
+(recall (5.6)). Keeping the notation introduced in (B.6) for the Fourier transform, we are
+going to prove the following analogue of [16, Lemma B.2] (we use the notation vpεq for
+vpε, θ, γq for ease of reading)
+Lemma B.4. If the support of ρ is included in Bp0, Rq then the following holds for every
+a P Rd with a constant C which depends on ρ.
+sup
+εPp0,1q
+ξPRd
+E
+”
+vpεq´2|x
+Mγ
+ε pξq|21Aq,R
+ı
+ă 8,
+sup
+εPp0,1q
+ξPRd
+E
+”
+vpεq´2|x
+Mγ
+ε pξ ` aq ´ x
+Mγ
+ε pξq|21Aq,R
+ı
+ď C|a|2,
+(B.14)
+Proof. We introduce a martingale whose limit coincides with x
+Mγ
+ε pξq on the event Aq,R.
+Given x P Rd and q ą 0 we set
+Tqpxq :“ inftt ą 0 : Xtpxq “
+?
+2dt ` qu,
+(B.15)
+and define
+N pεq
+t
+pξq :“
+ż
+Rd ρpxqeiξ.xeγXt^Tqpxq,εpxq´ γ2
+2 Kt^Tqpxq,εpxqdx.
+(B.16)
+Since Tqpxq “ 8 for all x in the support of ρ on the event Aq,R, we have
+N pεq
+8 pξq1Aq,R “ x
+Mγ
+ε pξq1Aq,R.
+(B.17)
+
+42
+HUBERT LACOIN
+Hence it is sufficient to prove
+sup
+εPp0,1q
+ξPRd
+E
+”
+vpεq´2|N pεq
+8 pξq|2ı
+ă 8,
+sup
+εPp0,1q
+ξPRd
+E
+”
+vpεq´2|N pεq
+8 pξ ` aq ´ N pεq
+8 pξq|2ı
+ď C|a|2.
+(B.18)
+Let us prove the only second inequality, since the first one is only easier.
+We set for
+simplicity Wt :“ N pεq
+t
+pξ ` aq ´ N pεq
+t
+pξq. We have
+E
+”
+|N pεq
+8 pξ ` aq ´ N pεq
+8 pξq|2ı
+“ Er|W8|2s “ Er|W0|2s ` E rxWy8s .
+We are going to prove a bound for each of the term in the r.h.s. . We have
+Er|W0|2s “
+ż
+R2d ρpxqρpyq
+´
+eipξ`aq.x ´ eiξ.x¯ ´
+e´ipξ`aq.y ´ eiξ.y¯
+e|γ|2K0,εpx,yq
+ď C|a|2
+ż
+ρpxqρpyq|x||y|dxdy ď C1|a|2.
+(B.19)
+where in the second line have taken the modulus of the integrand, and used the fact
+that the complex exponential is Lipshitz. To bound the expected value of the quadratic
+variation, using Itˆo calculus, and observing that tTqpxq ă tu “ At,qpxq (recall (5.6)) we
+obtain that
+xWy8 “ |γ|2
+ż 8
+0
+Utdt.
+(B.20)
+where
+Ut :“
+ż
+R2d ρpxqρpyqQt,εpx, yq
+´
+eipξ`aq.x ´ eiξ.x¯ ´
+e´ipξ`aq.y ´ eiξ.y¯
+ˆ eγXt,εpxq`γXt,εpyq´ γ2
+2 Kt,εpxq´ γ2
+2 Kt,εpyq1At,qpxqXAt,qpyqdx.
+(B.21)
+Taking the modulus in the integrand value everywhere inside the integral and using the
+fact that the complex exponential is Lipshitz fwe obtain
+Ut ď |a|2
+ż
+R2d ρpxqρpyq|x||y|Qt,εpx, yq
+ˆ e
+?
+d{2pXt,εpxq`Xt,εpyqq` |γ|2´d
+2
+pKt,εpxq`Kt,εpyqq1At,qpxqXAt,qpyqdxdy
+ď C|a|2
+ż
+R2d ρpxq2|x|2Qt,εpx, yqe
+?
+2dXt,εpxq`p|γ|2´dqKt,εpxq1At,qpxqdxdy,
+(B.22)
+where the second line is obtained via the same step as (5.26) (ab ď a2`b2{2 and symmetry
+and in x and y). Now recalling (6.25) we have
+E
+”
+e
+?
+2dXt,εpxq´dKt,εpxq1At,qpxq
+ı
+ď Cpt _ 1q´1{2.
+(B.23)
+where to obtain the first inequality, we used (3.12) to show that Kspx, yq (and all similar
+terms) are well estimated by t for s P r0, ts. Now we have
+E rUts ď C|a|2e´dt
+?
+t _ 1
+ż
+R2d ρpxq2|x|2Qt,εpx, yqe|γ|2Kt,εpx,yqdx ď C1|a|2φpt, εq.
+(B.24)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES43
+After integrating with respect to t (recalling Lemma 6.4) we obtain that
+ErxWy8s ď C|a|2vpεq2,
+(B.25)
+for a constant C which is independent of ε and ξ and a, which combined with (B.19),
+concludes the proof.
+□
+Appendix C. Beyond star-scale invariance
+The assumption that the kernel can be written in the form (1.8) may be felt as unnec-
+essarily restrictive, since after all, given an open domain D Ă Rd and a positive definite
+Kernel kernel K :
+D2 Ñ p´8, 8s that admits a decomposition of the form (1.1), the
+mollified field Xε can be defined on
+Dε :“ tx P D : inf
+yPDA |x ´ y| ą 2εu.
+More precisely in that case the field X is indexed by CcpDq the set of functions with
+compact support on D (in (1.4), R2d is replaced by D2), and Xε remains defined by (1.5)
+(here θεpx ´ ¨q, which for x P Dε, has its support included in D, is identified with its
+restriction on D).
+It turns out that our results can be extended to the the general setup described above,
+only with an additional regularity assumption concerning the function L present in (1.1).
+Given U Ă D, we say that the restriction of K to U has an almost star-scale invariant
+part, if
+@x, y P U, Kpx, yq “ K0px, yq ` Kpx, yq
+(C.1)
+where K is an almost-star scale invariant Kernel, and K0 : U 2 Ñ R is positive definite
+and H¨older continuous.
+To extend the result we use the fact (proved in [12]) that if L is sufficiently regular
+then K is locally star-scale invariant in the sense defined above. We state this result as a
+proposition. It can be directly derived from [12, Theorem 4.5].
+Proposition C.1. If K is a positive definite kernel on D that can be written in the form
+(1.1) with L P Hs
+locpD2q with s ą d, then for every z P D, there exist δz ą 0 such that the
+restriction of K to Bpz, δzq has an almost star-scale invariant part.
+To extend Theorem 2.5 we require another technical result, which states that with the
+same assumption as above, and U an open set whose closure is included in D, K can be
+approximated by a kernel with an almost star-scale invariant part defined on U. This is
+the content of the following result, [16, Lemma 2.1]
+Proposition C.2. Given K a covariance kernel on D of the form (1.1) with L P Hs
+locpD2q
+for s ą d , U a bounded open set whose closure satisfies U Ă D and δ ą 0, then there
+exists a kernel Kpδq on U satisfying (C.1) such that
+(A) For all x, y P U,
+|Kpδqpx, yq ´ Kpx, yq| ď δ.
+(B) ∆pδqpx, yq “ Kpδqpx, yq ´ Kpx, yq is a positive definite kernel on U.
+Remark C.3. More precisely, [16, Lemma 2.1] states that one can chose η1 “ 0 (recall
+(1.8)) for the almost-star scale invariant part of Kpδq, but this refinement is not required
+for our purpose.
+
+44
+HUBERT LACOIN
+C.1. The case of Theorem 2.1. The extension of the result to the case of a general
+log-correlated field defined on a domain D is the following.
+Theorem C.4. If X is a centered Gaussian field defined on D whose covariance kernel
+K can be written in the form (1.1) with L P Hs
+locpD2q for s ą d, and f P CcpRdq, then
+there exists a complex valued random variable Mγ
+8pfq such that for any choice of mollifier
+θ the following convergence holds in Lp if p P
+”
+1,
+?
+2d{α
+¯
+.
+lim
+εÑ0 Mγ
+ε pfq “ Mγ
+8pfq.
+(C.2)
+Proof. This follows quite immediately via a localization argument using a partition of
+unity. Let f P CcpDq be fixed. Using Proposition C.1, we can cover the support of f (which
+is compact) by finitely many Euclidean balls Bpzi, εiq, i P I such that for every i P I the
+restriction of K to Bpzi, εiq has an almost star-scale invariant part. Using a partition of the
+unity, we can write f :“ ř
+iPI fi where fi is continuous with compact support included in
+Bpzi, εiq. Using Theorem 2.1 for K restricted to Bpzi, εiq, we obtain that Mγ
+ε pfiq converges
+in Lp for every fi and thus we obtain the convergence for Mγ
+ε pfq “ ř
+iPI Mγ
+ε pfiq.
+□
+C.2. The case of Theorem 2.5. To extend the result for γ P P1
+II{III it is sufficient to
+extend Proposition 2.8. In the statement below, we implicitely use the fact that the critical
+multiplicative chaos M1 is well defined under our assumptions (see [17, Theorem C.2] for
+a proof).
+Proposition C.5. If X is a centered Gaussian field defined on D whose covariance kernel
+K can be written in the form (1.1) with L P Hs
+locpD2q for s ą d, given ρ, f P CcpDq,
+ω P r0, 2πq, we have
+lim
+εÑ0 E
+„
+eixX,ρy`i Mγ
+ε pf,ωq
+vpε,θ,γq
+
+“ E
+„
+eixX,ρy´ 1
+2M1pe|γ|2L|f|2q
+
+.
+(C.3)
+Proof. Given a fixed f P CcpDq, and n ě 1, we chose U which contains the support of
+f and Kn : U 2 Ñ p´8, 8s satisfying the assumptions of Kpδq of Proposition C.2 with
+δ “ 1{n. We let Zn be a centered Gaussian field indexed by U, independent of X and with
+covariance ∆n “ Kn ´K and define Xn a field indexed by CcpUq by setting Xn “ X `Zn.
+Note that Xn has covariance Kn. We let Mγ,n
+ε
+and M
+1
+n denote the mollified GMC and
+critical GMC associated with Xn. For simplicity, we define all the pZnqně1 on the same
+probability space: the fields Zn form an independent sequence which is independent of
+X. We let P denote the corresponding probability. From Proposition 2.8 we have for each
+n ě 1
+lim
+εÑ0 E
+„
+eixXn,ρy`i Mγ,n
+ε
+pf,ωq
+vpε,θ,γq
+
+“ E
+„
+eixXn,ρy´ 1
+2 M
+1
+npe|γ|2Ln|f|2q
+
+,
+(C.4)
+where Ln :“ L ` ∆n. In order to conclude, we need to show that (for any choice of Kpδq)
+lim
+nÑ8 sup
+εPp0,1q
+ˇˇˇˇE
+„
+eixXn,ρy`i Mγ,n
+ε
+pf,ωq
+vpε,θ,γq
+
+´ E
+„
+eixX,ρy`i Mγ
+ε pf,ωq
+vpε,θ,γq
+ˇˇˇˇ “ 0
+(C.5)
+and that
+lim
+nÑ8 E
+„
+eixXn,ρy´ 1
+2M
+1
+npe|γ|2Ln|f|2q
+
+“ E
+„
+eixX,ρy´ 1
+2 M
+1pe|γ|2Lpδq|f|2q
+
+.
+(C.6)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES45
+Note that it is sufficient to show that the difference between the terms in the l.h.s. and
+the r.h.s. tend to zero in probability (uniformly in ε) that is
+lim
+nÑ8 E r|xXn, ρy ´ xX, ρy| _ 1s “ 0,
+lim
+nÑ8 E
+”ˇˇˇM
+1
+npe|γ|2Ln|f|2q ´ M
+1pe|γ|2L|f|2q
+ˇˇˇ _ 1
+ı
+“ 0,
+lim
+nÑ8 sup
+εPp0,1q
+E
+„ˇˇˇˇ
+Mγ,n
+ε
+pf, ωq
+vpε, θ, γq
+´ Mγ
+ε pf, ωq
+vpε, θ, γq
+ˇˇˇˇ _ 1
+
+“ 0.
+(C.7)
+The first line is immediate via the computation of the L2 norm (the convergence holds in
+L2). For the second line, we set gn “ e|γ|2Ln|f|2 and g “ e|γ|2L|f|2. Using the notational
+convention introduced in Section 3.1, we let Zn,ε denote the mollification of Zn and ∆n,ε
+its covariance. Letting e
+?
+2dZn,ε´d∆n,ε denote the function x ÞÑ e
+?
+2dZn,εpxq´d∆n,εpxq we have
+E
+„´
+M
+?
+2d,n
+ε
+pgnq ´ M
+?
+2d
+ε
+pgq
+¯2
+| X
+
+“ E
+”
+M
+?
+2d
+ε
+pe
+?
+2dZn,ε´d∆n,εgn ´ gq2 | X
+ı
+“
+ż
+U2
+´
+e2d∆n,εpx,yqgnpxqgnpyq ´ 2gnpxqgpyq ` gpxqgpyq
+¯
+M
+?
+2d
+ε
+pdxqM
+?
+2d
+ε
+pdyq.
+(C.8)
+From the assumption that |∆npx, yq| ď 1{n (and thus |Lnpxq ´ Lpxq| ď 1{n) we obtain
+that
+|e2d∆n,εpx,yqgnpxqgnpyq ´ 2gnpxqgpyq ` gpxqgpyq| ď Cgpxqgpyq
+n
+and hence
+E
+���´
+M
+?
+2d,n
+ε
+pgnq ´ M
+?
+2d
+ε
+pgq
+¯2
+| X
+
+ď C
+n M
+?
+2d
+ε
+pgq2
+(C.9)
+Using Fatou after renormalization we obtain that
+E
+”`
+M1
+npgnq ´ M1pgq
+˘2 | X
+ı
+ď C
+n M1pgq2
+(C.10)
+which implies the second line in (C.7). For the third line, we are going to Proposition
+(C.1). More precisely, we use a decomposition of f “ ř
+iPI fi where fi is continous with
+compact support included in Ui and the restriction of K to Ui has an almost star-scale
+invariant part. We are going to prove that for each i P I.
+lim
+nÑ8 sup
+εPp0,1q
+E
+„ˇˇˇˇ
+Mγ,n
+ε
+pfi, ωq
+vpε, θ, γq
+´ Mγ
+ε pfi, ωq
+vpε, θ, γq
+ˇˇˇˇ _ 1
+
+“ 0.
+(C.11)
+This operation shows that it is in fact sufficient to prove the third line of (C.7) assuming
+that K is an almost star-scale invariant Kernel. We can thus further our probability space
+contains pXtqtě0 a martingale sequence of fields with covariance Kt (we adopt the notation
+of Section 3.1) approximating X. We equip our space with the filtration
+Gt :“ σppXsqsPr0,ts, pZnqně1q.
+We recall the definition of Tqpxq in (B.15) and define
+W pn,εq
+t
+:“
+ż
+U
+peγZn,εpxq´ γ2
+2 ∆n,εpxq ´ 1qfpxqeγXt^Tqpxq,ε´ γ2
+2 Kt^Tq,εpxqdx
+(C.12)
+Setting
+Aq :“ t@x P Supppfq, @t ą 0, Xtpxq ď
+?
+2dt ` qu
+
+46
+HUBERT LACOIN
+We have from the definition
+W pn,εq
+8
+1Aq “ |Mγ,n
+ε
+pfq ´ Mγ
+ε pfq|1Aq
+Hence we have (for any ω P r0, 2πq since the projection on one axis reduces the modulus
+E
+“
+|Mγ,n
+ε
+pf, ωq ´ Mγ
+ε pf, ωq|21Aq
+‰
+ď Er|W pn,εq
+8
+|2s “ Er|W pn,εq
+0
+|2s ` ErxW pn,εqy8s.
+(C.13)
+Since from Lemma 5.2 we have limqÑ8 PrAqs “ 1, to prove the third line in (C.7), it is
+sufficient to show that for any q we have
+lim
+nÑ8 sup
+εPp0,1q
+vpε, θ, γq´2 ´
+Er|W pn,εq
+0
+|2s ` ErxW pn,εqy8s
+¯
+.
+(C.14)
+For the first term, we have
+Er|W pn,εq
+0
+|2s “
+ż
+U2 fpxqfpyqpe|γ|2∆n,εpx,yq´1qe|γ|2K0,εpx,yqdxdy ď Cn´1
+(C.15)
+where the inequality obtained taking the modulus of the integrand and using the fact that
+|∆npx, yq| ď 1{n and the other terms are uniformly bounded. The derivative of the bracket
+of W pn,εq is given by |γ|2 times (recall that by (5.6) we have At,qpxq “ tTqpxq ď tu)
+Dt :“
+ż
+R2d Qt,εpx, yqGn,εpxqGn,εpyq
+ˆ eγXt,ε`γXt,εpyq´ γ2
+2 Kt,εpxq´ γ2
+2 Kt,εpyq1At,qpxqXAt,qpyqdxdy.
+(C.16)
+with Gn,εpxq “ peγZn,εpxq´ γ2
+2 ∆n,εpxq ´ 1q. Repeating once more the computation in (5.26)
+we obtain that
+Dt ď
+ż
+R2d Qt,εpx, yq|Gn,εpxq|2e
+?
+2dXt,ε`p|γ|2´dqKt,εpxq1At,qpxqdxdy.
+(C.17)
+We define a martingale W
+pεq and W
+pεq
+t
+by setting
+W pεq
+t
+:“ E rMγ,n
+ε
+pf, ωq ´ Mγ
+ε pf, ωq | Gts
+(C.18)
+Now we have
+E
+“
+|Gn,εpxq|2‰
+“ e|γ|2∆n,εpxq ´ 1 ď Cn´1
+This the term is independent of the rest, thus using (B.23) we obtain that
+ErDts ď Cn´1
+ż
+R2d Qt,εpx, yqe|γ|2Kt,εpxqdxdy ď C1n´1φpt, εq.
+(C.19)
+Integrating against t we conclude that
+ErxW pn,εqy8s ď Cn´1vpε, θ, γq2
+and this concludes the proof of (C.14).
+□
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES47
+Appendix D. Proof of Lemma 4.1
+We use Kahane convexity inequality in order to compare Bn to the the partition function
+of a Gaussian branching random walk (or polymer on a 2d-adic tree). We assume without
+loss of generality that Supppfq Ă r0, 1sd. For x, y P r0, 1sd we let 2´kpx,yq be the sidelength
+of the smallest dyadic cube that contains x and y.
+kpx, yq :“ inf
+!
+n ě 0 : Dm P �0, 2n ´ 1�d, tx, yu Ă
+´
+2´nm ` r0, 2´nqd¯)
+.
+and set knpx, yq :“ kpx, yq ^ n. Note that kn defines a positive definite function and that
+kpx, yq ď log2
+´
+1
+|x´y|
+¯
+` C. Hence from Lemma 3.1 there exists a constant A ą 0 such
+that
+plog 2qknpx, yq ď Krn log 2spx, yq ` A.
+(D.1)
+Using Kahane’s convexity inequality (proved in [15] see also [27, Theorem 2.1]) which we
+introduce in a simplified setup
+Lemma D.1. If C1 and C2 are two bounded positive definite kernel on an arbitrary space
+X satisfying
+@x, y P X,
+C1px, yq ď C2px, yq
+µ is a finite measure on r0, 1sd and F : R` Ñ R is a concave function with at most
+polynomial growth at infinity and Y1 and Y2 are Gaussian fields with respective covariance
+C1 and C2 then we have for any θ P R
+E
+„
+F
+ˆż
+eθY1pxq´ θ2
+2 C1pxqµpdxq
+˙
+ď E
+„
+F
+ˆż
+eY2pxq´ θ2
+2 C2pxqµpdxq
+˙
+.
+(D.2)
+Hence if Zn denotes a field defined on r0, 1sd with covariance kn we can apply Lemma
+D.1 result for the fields ?log 2Zn and Xrn log 2s`
+?
+AN where N is an independent standard
+Gaussian (the fields have their resepective covariances given by the two sides of Equation
+(D.1)), µpdxq “ |fpxq|2dx and Fpuq “ up{2. Recalling (4.11) we have
+E
+”
+Bp{2
+rn log 2s
+ı
+ď CE
+»
+–
+˜
+2dn
+ż
+r0,1sd |fpxq|2e2α?log 2pZnpxq´?2d log 2nqdx
+¸p{2fi
+fl .
+(D.3)
+The constant C above takes care of the fact that the variance of ?log 2Znpxq and Xrn log 2s
+differ by a Op1q term, and also of the moment of the variable N. We can ignore the
+constant f at the cost of a prefactor }f}p
+8. To conclude we thus need a bound on the
+moment of order p{2 of the partition function of the Gaussian branching random walk
+Wn,ζ :“ 2dn
+ż
+r0,1sd eζpZnpxq´?2d log 2nqdx “
+ÿ
+mP�0,2n´1�d
+eζpZnpm2´nq´?2d log 2nq,
+(D.4)
+for ζ “ 2α?log 2. The following result is a particular case of [8, Theorem 1.6]. We present
+a shorter proof which is valid in our context for the sake of completeness.
+Lemma D.2. Given ζ ą ?2d log 2 and q ď
+?2d log 2
+ζ
+there exists positive constant C and
+b such that
+E rpWn,ζqqs ď Cn´
+3qζ
+2?2d log 2plog nq6
+
+48
+HUBERT LACOIN
+Proof. We split our integral in three parts. We set
+Bnpxq :“ tDm P �1, n�, Zmpxq ě
+a
+2d log 2 ` plog nq2u,
+Cnpxq :“ BA
+npxq X tZnpxq ď
+a
+2d log 2n ´ plog nq2u,
+Anpxq :“ BA
+npxq X CA
+npxq
+(D.5)
+We define Wn,ζpAq, Wn,ζpBq and Wn,ζpCq by setting, for I P tA, B, Cu
+Wn,ζpIq :“ 2dn
+ż
+r0,1sd eζpZnpxq´?2d log 2nq1Inpxqdx
+(D.6)
+Using subadditivity (4.9) we have
+E rpWn,ζqqs ď E rWn,ζpAqqs ` E rWn,ζpBqqs ` E rWn,ζpCqqs .
+(D.7)
+We are going to show that the two last terms in the r.h.s. decay faster than any negative
+power of n and then prove a bound of the right order of magnitude for E rpWn,ζqqs. Letting
+setting Bn :“ Ť
+xPr0,1s Bnpxq, and q1 “ ?2d log 2ζ´1 (q1 P rq, 1q) we have
+E rWn,ζpBqqs ď E rpWn,ζqq1Bns ď E
+”
+pWn,ζqq1ı q
+q1 P rBns1´ q
+q1 .
+(D.8)
+Using subadditivity (4.9) for the sum (D.4) with θ “ q1,
+E
+”
+pWn,ζqq1ı
+ď E
+“
+Wn,?2d log 2
+‰
+“ 1.
+(D.9)
+The inequality on the right comes from the fact that pWm,?2d log 2qmě1 is a martingale for
+the natural filtration associated with Zn. Using the optional stopping Theorem for this
+same martingale, we can obtain a bound for the probability of Bn,
+PrBns ď P
+”
+Dm, Wm,?2d log 2 ě e
+?2 log 2plog nq2ı
+ď e´?2 log 2plog nq2.
+(D.10)
+This yields a subpolynomial decay for ErWn,ζpBqqs. For Wn,ζpCq using the fact that Zn is a
+Gaussian of variance n, we obtain using Jensen’s inequality, the Cameron-Martin formula
+and Gaussian tail bounds
+E rpWn,ζpCqqqs1{q ď E rWn,ζpCqs “ 2dnE
+”
+eqpZn´?2d log 2nq1tZnď?2d log 2n´plog nq2u
+ı
+“ e
+ˆ
+d log 2` ζ2
+2 ´ζ?2d log 2
+˙
+n
+P
+”
+Znp0q ď p
+a
+2d log 2 ´ ζqn ´ plog nq2ı
+ď ep?2d log 2´ζqplog nq2,
+(D.11)
+also proving a subpolynomial decay. It remains to estimate the main part E rpWn,ζpAqqqs.
+Using first subaddivity (4.9) and then Jensen’s inequality
+E rpWn,ζpAqqqs ď E
+„
+Wn,?2d log 2pAq
+qζ
+?2d log 2
+
+ď E
+“
+Wn,?2d log 2pAq
+‰
+qζ
+?2d log 2 .
+(D.12)
+The Cameron-Martin formula directly expresses E
+“
+Wn,?2d log 2pAq
+‰
+as the probability con-
+cerning the Gaussian centered random walk pZmp0qqmě0,
+E
+“
+Wn,?2d log 2pAq
+‰
+“ P
+“
+@m P �1, n�, Zmp0q ď plog nq2 ; Znp0q ě ´plog nq2‰
+ď Cn´3{2plog nq6.
+(D.13)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES49
+The bound for the probability of the event above is valid for any random walk with IID
+centered increments with finite second moment (see for instance [1, Lemma A.3]) which
+concludes our proof.
+□
+References
+[1] Elie A¨ıd´ekon and Zhan Shi. Weak convergence for the minimal position in a branching random walk:
+a simple proof. Period. Math. Hungar., 61(1-2):43–54, 2010.
+[2] Nathana¨el Berestycki. An elementary approach to Gaussian multiplicative chaos. Electron. Commun.
+Probab., 22:Paper No. 27, 12, 2017.
+[3] B. Derrida, M. R. Evans, and E. R. Speer. Mean field theory of directed polymers with random
+complex weights. Comm. Math. Phys., 156(2):221–244, 1993.
+[4] Bertrand Duplantier, R´emi Rhodes, Scott Sheffield, and Vincent Vargas. Critical Gaussian multiplica-
+tive chaos: convergence of the derivative martingale. Ann. Probab., 42(5):1769–1808, 2014.
+[5] Bertrand Duplantier, R´emi Rhodes, Scott Sheffield, and Vincent Vargas. Renormalization of critical
+Gaussian multiplicative chaos and KPZ relation. Comm. Math. Phys., 330(1):283–330, 2014.
+[6] Lisa Hartung and Anton Klimovsky. The glassy phase of the complex branching Brownian motion
+energy model. Electron. Commun. Probab., 20:no. 78, 15, 2015.
+[7] Lisa Hartung and Anton Klimovsky. The phase diagram of the complex branching Brownian motion
+energy model. Electron. J. Probab., 23:Paper No. 127, 27, 2018.
+[8] Yueyun Hu and Zhan Shi. Minimal position and critical martingale convergence in branching random
+walks, and directed polymers on disordered trees. Ann. Probab., 37(2):742–789, 2009.
+[9] Jean Jacod and Albert N. Shiryaev. Limit theorems for stochastic processes, volume 288 of Grundlehren
+der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-
+Verlag, Berlin, second edition, 2003.
+[10] Janne Junnila and Eero Saksman. Uniqueness of critical Gaussian chaos. Electron. J. Probab., 22:Paper
+No. 11, 31, 2017.
+[11] Janne Junnila, Eero Saksman, and Lauri Viitasaari. On the regularity of complex multiplicative chaos.
+arXiv e-prints, page arXiv:1905.12027, May 2019.
+[12] Janne Junnila, Eero Saksman, and Christian Webb. Decompositions of log-correlated fields with ap-
+plications. Ann. Appl. Probab., 29(6):3786–3820, 2019.
+[13] Janne Junnila, Eero Saksman, and Christian Webb. Imaginary multiplicative chaos: moments, regu-
+larity and connections to the Ising model. Ann. Appl. Probab., 30(5):2099–2164, 2020.
+[14] Zakhar Kabluchko and Anton Klimovsky. Complex random energy model: zeros and fluctuations.
+Probab. Theory Related Fields, 158(1-2):159–196, 2014.
+[15] Jean-Pierre Kahane. Sur le chaos multiplicatif. (On multiplicative chaos). Ann. Sci. Math. Qu´e.,
+9:105–150, 1985.
+[16] Hubert Lacoin. Convergence in law for complex Gaussian multiplicative chaos in phase III. Ann.
+Probab., 50(3):950–983, 2022.
+[17] Hubert
+Lacoin.
+Critical
+Gaussian
+Multiplicative
+Chaos
+revisited.
+arXiv
+e-prints,
+page
+arXiv:2209.06683, September 2022.
+[18] Hubert Lacoin. A universality result for subcritical complex Gaussian multiplicative chaos. Ann. Appl.
+Probab., 32(1):269–293, 2022.
+[19] Hubert Lacoin, R´emi Rhodes, and Vincent Vargas. A probabilistic approach of ultraviolet renormali-
+sation in the boundary Sine-Gordon model. to appear in Probab. Theory Related Fields.
+[20] Hubert Lacoin, R´emi Rhodes, and Vincent Vargas. Complex Gaussian multiplicative chaos. Comm.
+Math. Phys., 337(2):569–632, 2015.
+[21] Jean-Fran¸cois Le Gall. Brownian motion, martingales, and stochastic calculus, volume 274 of Graduate
+Texts in Mathematics. Springer, 2016.
+[22] Thomas Madaule. Convergence in law for the branching random walk seen from its tip. J. Theor.
+Probab., 30:27–63, 2017.
+[23] Thomas Madaule, R´emi Rhodes, and Vincent Vargas. The glassy phase of complex branching Brow-
+nian motion. Comm. Math. Phys., 334(3):1157–1187, 2015.
+[24] Thomas Madaule, R´emi Rhodes, and Vincent Vargas. Glassy phase and freezing of log-correlated
+gaussian potentials. Ann. Appl. Probab., 26(2):643–690, 04 2016.
+
+50
+HUBERT LACOIN
+[25] Ellen Powell. Critical Gaussian multiplicative chaos: a review. arXiv e-prints, page arXiv:2006.13767,
+June 2020.
+[26] Ellen Powell. Critical Gaussian multiplicative chaos:
+a review. Markov Process. Related Fields,
+27(4):557–506, 2021.
+[27] R´emi Rhodes and Vincent Vargas. Gaussian multiplicative chaos and applications: a review. Probab.
+Surv., 11:315–392, 2014.
+IMPA, Institudo de Matem´atica Pura e Aplicada, Estrada Dona Castorina 110 Rio de
+Janeiro, CEP-22460-320, Brasil.
+