diff --git "a/1tE4T4oBgHgl3EQfzg0x/content/tmp_files/load_file.txt" "b/1tE4T4oBgHgl3EQfzg0x/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/1tE4T4oBgHgl3EQfzg0x/content/tmp_files/load_file.txt" @@ -0,0 +1,1913 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf,len=1912 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='05274v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='PR] 12 Jan 2023 CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES HUBERT LACOIN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The correlation function of X is of the form Kpx, yq “ log 1 |x ´ y| ` Lpx, yq, where L is a continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' |α| ą |β| ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' |α| ` |β| “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2du, and P1 II{III :“ tα ` iβ : α, β P R ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' |α| “ a d{2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' |β| ą ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The limit does not depend on the mollification kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' When γ P PI{II, the convergence holds in probability and in Lp for some value of p P r1, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d{αq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' When γ P P1 II{III the convergence holds only in law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' In this latter case, the limit can be described a complex Gaussian white noise with a random intensity given by a critical real GMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The regions PI{II and P1 II{III correspond to phase boundary between the three different regions of the complex GMC phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2010 Mathematics Subject Classification: 60F99, 60G15, 82B99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Keywords: Random distributions, log-correlated fields, Gaussian Multiplicative Chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Introduction 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Main results 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The martingale approximation for GMC 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proof of convergence results on for γ P PI{II 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5 18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8 27 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6 34 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Technical results and their proof 36 Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The convergence of Mγ ε as a distribution 39 Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Beyond star-scale invariance 43 Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1 47 References 49 1 2 HUBERT LACOIN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1) (with the convention logp1{0q “ 8) where L is a continuous function on R2d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' A kernel K is positive definite if for ρ P CcpRdq (ρ continuous with compact support) ż R2d Kpx, yqρpxqρpyqdxdy ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3) A difficulty comes up when trying to give an interpretation to the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' A field X with a covariance given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1) can be defined only as a random distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' For a fixed x P Rd it is not possible to make sense of Xpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The problem of providing a mathematical construction of Mγ that gives a meaning to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The case of γ P C was considered only more recently, see for instance [11, 12, 13, 16, 18, 19, 20] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' In the present paper we present convergence results for the random distribution eγXεpxqdx and eγXtpxqdx and in a certain range of parameter γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Before describing our results in more details and provide some motivation, we first rigorously introduce the setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The mollification of a log-correlated field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Log-correlated fields defined as distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Since K is infinite on the diagonal, it is not possible to define a Gaussian field indexed by Rd with covariance function K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We consider instead a process indexed by test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We define pK, a bilinear form on CcpRdq (the set of compactly supported continuous functions) by pKpρ, ρ1q “ ż R2d Kpx, yqρpxqρ1pyqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3) in the appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' For this reason (and although we will not use this fact) we refer to X as a random distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES 3 Approximation of X via mollification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5) From (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4), the field Xεp¨q has covariance Kεpx, yq :“ ErXεpxqXεpyqs “ ż R2d θεpx ´ z1qθεpy ´ z2qKpz1, z2qdz1dz2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6) We set Kεpxq :“ Kεpx, xq and extend this convention to other functions of two variables in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Since Kε is infinitely differentiable - thus in particular is H¨older continuous - by Kolmogorov’s Continuity Theorem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' [21, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9]) there exists a continuous modification of Xεp¨q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' In the remainder of the paper, we always consider the continuous modification of a process when it exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' This ensures that integrals such as the one appearing in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='7) are well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We define the distribution Mγ ε by setting for f P CcpRdq Mγ ε pfq :“ ż Rd fpxqeγXεpxq´ γ2 2 Kεpxqdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='7) The question of interest in the present paper is the convergence of Mγ ε when ε Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' An important feature of our results is that the limits obtained for Mγ ε pfq do not depend on θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Star-scale invariance and our assumption on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' On top of assuming that K admits a decomposition like (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1), we also assume that it has an almost star-scale invariant part (see the definition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1) is sufficiently regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9) where Kpx, yq is an almost star-scale invariant kernel and K0 is H¨older continuous on R2d and positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 4 HUBERT LACOIN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Phase transitions and phase diagrams for GMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Our main results concerns the asymptotic behavior of Mγ ε in the specific range of γ given in the abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Phase transition at |α| “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d for the real valued GMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The obtained limit in that case crucially depends on α: when |α| ă ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' When |α| ě ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d, we have limεÑ0 Mα ε pfq “ 0 and a rescaling procedure is needed in order to obtain a non-trivial limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The phenomenology is however different according to whether |α| “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d (α critical) or |α| ą ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d (α supercritical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' In the critical case (α “ ˘ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' When |α| ą ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d, the results are less complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' So far the convergence has not been proved for Mα ε but only for an approximating martingale sequence Mα t (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6)) in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Besides this technical point, the most important differences with the case |α| ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d concerns the type of the convergence and the nature of limiting object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Phase diagram for complex GMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' When γ is allowed to assume complex value, the phase diagram becomes more intricate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The complex plane can be divided in three open regions with intersecting boundaries PI :“ ␣ α ` iβ : α2 ` β2 ă d ( Y !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' α ` iβ : α P p a d{2, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dq ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' |α| ` |β| ă ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d ) , PII :“ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' α ` iβ : |α| ` |β| ą ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' |α| ą a d{2 ) , PIII :“ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' α ` iβ : α2 ` β2 ą d ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' |α| ă a d{2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The region PI corresponds to the subcritical phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' This conjecture is supported by rigorous results obtained in the case of Complex Branching Brownian Motion [6, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES 5 PSfrag replacements β α ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' d a d{2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d ´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d PI PII PIII Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The phase diagram of the complex GMC in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' In the present paper, we prove results concerning the asymptotic behavior on frontier of PI Y PIII with PII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Results concerning convergence in PI Y PIII where proved in [12, 18] (for PI) and [16] (for PIII Y PI{III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The convergence in the region PII remains a challenging conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Finally the region PIII corresponds to yet another asymptotic behavior for Mγ ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Like in PII, Mγ ε - properly rescaled - only converges in law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' A similar convergence result holds on the boundary between PII and PIII that is PII{III :“ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' α ` iβ : α2 ` β2 “ d ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' |α| ă a d{2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' These convergence statements for γ P PIII Y PI{III are proved in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The present contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' In each case, the limit obtained does not depend on the regularization kernel θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We leave as an open problem the challenging task of proving a convergence result in the frozen phase PII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The boundary between phase I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' α ` β “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d{α ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' lim εÑ0 Mγ ε pfq “ Mγ 8pfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2) The above result extends [18, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2] which established convergence for γ P PI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The method which we use to prove it however, completely differs from the one employed in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' In fact the method of proof that we employ in Section 4 balso provides an alternative and much shorter proof of [18, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2], with the additional benefit of establishing convergence in Lp for an optimal range of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We have chosen to denote the limit by Mγ 8 rather than Mγ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We include the argument in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The boundary between phase II and III, and the triple point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3) In that case Mγ ε needs to be rescaled in order to obtain a non-trivial limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The convergence holds only in law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' To describe the limit we need to introduce two notions: Critical Gaussian Multiplicative Chaos, and Gaussian White Noise with a random intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Critical GMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' As explained in the introduction critical Gaussian Multiplicative Chaos is obtained as the limit of Mα ε when α “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The value ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d represent a threshold for the convergence of Mα ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4] which establish that the limit is the same for the exponential of the mollified field Mα ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Alternative concise proofs of these results have been recently given in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Let X be a Gaussian random field with an almost-star scale invariant kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d ε pfq “ M1pfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Note that we have set different conventions and that our M1 differs from that in [5, Theorem 5] by a factor b 2 π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES 7 Complex white noise with random intensity given by a Real GMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' , ρm, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' , 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6) and set (recall (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='7) where Σd is the volume of the d ´ 1 dimensional sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Note that limεÑ0 vpε, θ, γq “ 8 in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Let X be a Gaussian random field with an almost star-scale covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The convergence in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8) implies that vpε, θ, γq´1Mγ ε does not converge in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' These fluctuations are produced by high frequencies in the Fourier spectrum of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The set of frequencies that produce the fluctuations diverges to infinity when ε Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' This means that the randomness that produces the white noise become asymptotically independent of X in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The convergence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9) 8 HUBERT LACOIN The convergence can also be shown to hold in a space of distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='10) Hence Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5 can be reduced to the proof of the following statement Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Under the assumption of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 \uf6be “ E „ eixX,ρy´ 1 2M1pe|γ|2L|f|2q \uf6be .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='11) The r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='11) of corresponds to the Fourier transform of pX, Mγq (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5)) E „ eixX,ρy´ 1 2 M1pe|γ|2L|f|2q \uf6be “ E ” eixX,ρy`iMγpfqı , and the convergence of the Fourier transform implies that of finite dimensional marginals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' More detailed justifications are exposed in [16, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5: the martingale decomposition of the field X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Extending the probability space, one can define a martingale sequence of smooth fields pXtqtě0 that converges to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' This allows for another approach to the construction of GMC, considering the exponen- tial of the martingale approximation of X (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='7)) which we call Mγ t (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2 concerning the conflict of notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Convergence results for Mγ t which are a analogous to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5 are also presented in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' In section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5 we introduce an important technical tool which is used to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The martingale decomposition of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Given K with an almost star-scale invariant part, and using the decomposition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2) CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES 9 Note that we have limtÑ8 Ktpx, yq “ Kpx, yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3) Since ps, t, x, yq ÞÑ Ks^tpx, yq is H¨older continuous, the field admits a continuous modifi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We let Ft :“ σ ´ pXspxqqxPRd,sPr0,ts ¯ denote the natural filtration associated with X¨p¨q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We also define X¨ :“ X¨ ´ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Recalling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2) we have ErXtpxqXspyqs “ Ks^tpx, yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We also introduce the field Xt,ε which is the mollification of Xt, that is Xt,εpxq :“ ż Rd θεpx ´ zqXtpzqdz “ E rXεpxq | Fts .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The martingale approximation for the GMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We define the distribution Mγ t by setting for f P CcpRdq Mγ t pfq :“ ż Rd fpxqeγXtpxq´ γ2 2 Ktpxqdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6) Using the independence of the increments of X, it is elementary to check that Mtpfq is an pFtq-martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We also define Mγ t,εpfq :“ ż Rd fpxqeγXt,εpxq´ γ2 2 Kt,εpxqdx “ E rMγ ε pfq | Fts .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='7) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' A few properties of the covariance kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We introduce some technical nota- tion and estimates that are going to be of use throughout the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Indeed, if K satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8) then the function L defined for x ‰ y by Lpx, yq :“ Kpx, yq ` log |x ´ y|, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8) can be extended to a continuous function on R2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Note that we have Lpxq “ lim yÑx pKpx, yq ` log |x ´ y|q “ K0pxq ´ j (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' converges to 1 ´ rκpe η1 η2 ´sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='11) where in the integral in s, s1 is related to s via (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Note that ℓpzq is a continuous negative function and that for any |z| ě e´ η1 η2 we have ℓpzq “ log 1 |z| ´ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' To conclude this subsection, we gather in a a technical lemma a couple of useful estimates concerning Kt, Qt and their variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) The bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) remains valid with Kt,ε replaced by Kt (with ε “ 0), Kt,ε,0, Kt,ε etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We also have ż Rd Qtpx, yq “ ż Rd Qt,εpx, yqdy “ ż Rd Qt,ε,0px, yqdy ď Ce´dt, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='13) and 0 ď t ´ Kt,εpx, yq ď C ` etp|x ´ y| ` εq ˘2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='14) The estimates above can be proved rather directly from the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' A detailed proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) is provided in [17, Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='13) follows directly from the definition of Qt given above (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1) and the fact that |t´t1| is uniformy bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The upper bound in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' There is an obvious conflict of notation between Kt introduced above and Kε introduced in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6) and the same can be said about Xt and Mγ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Convergence results for the martingale approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' An intermediate step to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5 is to show that similar results hold for the mar- tingale approximation Mγ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' These results present of course an interest in their own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES11 The case of γ P PI{II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' When γ P PI{II, the martingale Mγ t pfq is bounded Lp for p P r1, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d{|α|q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' As a consequence the limit lim tÑ8 Mγ t pfq “: Mγ 8pfq (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='15) exists almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The convergence holds in Lp and the limit is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The martingale limit in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='15) is the same as the limit of Mγ ε appearing in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1 (this is the reason why we use the same notation), we have lim tÑ8 Mγ t pfq “ lim εÑ0 Mγ ε pfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='16) This observation is important, since it establishes that the limit in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2) does not depend on the choice of the mollifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The case γ P P1 II{III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' In order to state the convergence in law result for Mγ t , we need to introduce a normalization factor vpt, γq (analogous to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='7) for the mollified case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='17) and define Mγ t pf, ωq :“ Re ` e´iωMγ t pfq ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='18) The following analogue of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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 \uf6be “ E „ eixX,ρy´ 1 2 M1pe|γ|2L|f|2q \uf6be .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='20) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' CLT towards a Gaussian with random variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We provide the result and its proof in a reasonably high level of generality since it may find application in other contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Consider pFtqtě0 a filtration and pWnqně1 a sequence of real valued random variables in L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We introduce for each n ě 1 the martingale Wn,t :“ E rWn | Fts .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The result generalizes a similar CLT established for a single mar- tingale process (see [9, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='50, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' VIII-Section 5c] or [16, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 12 HUBERT LACOIN Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 \uf6be (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='23) This is equivalent to saying that for any F8 random variable Y we have the following convergence in law pY, Wnq ùñ pY, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' ZNq where N is a standard Gaussian which is independent of Z and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Since we have no application in that setup, we restricted ourselves to the continuous case where the proof is technically simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' In Section 6 we apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' These setups are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Organization of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The remainder of the paper is organized as follows ‚ In Section 4 we prove all the statements concerning convergence in PI{II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1 is devoted to the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The more technical proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1, which uses Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3 as in imput is displayed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' ‚ The statements concerning γ P P1 II{III, namely Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8, while relying on relatively simple ideas, require a certain amount of technical computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' In Section 5 we prove Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5, in Section 6 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' ‚ In Section 7, we present the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' A significant amount of material is presented in appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' ‚ In Appendix A, we prove a couple of auxilliary results used in Section 5/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' ‚ 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' ‚ 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' ‚ In Appendix D, we present a relatively short proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1 for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' It the same as the one presented in [20, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='15], except that we include a short proof of Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES13 A comment on notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Throughout the paper, we use the letter C for a generic positive constant when we need to compare two quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' It may depend on some parameters (for instance on γ or on the kernel K) but never on the variable t or ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The value of C might change from one equation to the other withing the same proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We use C1 and C2 if we need several constants in the same display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proof of convergence results on for γ P PI{II In this section we prove Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' This is performed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We rely on the Burkeholder-Davis-Gundy (BDG) inequality, compute the quadratic variation of the martingale and studying its moment of order p{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2, we adapt the same method to estimate the Lp norm of Mγ ε ´ Mγ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Recalling that ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d{α ą 1, we are going to prove that Mγ t pfq is bounded in Lp for p P ´?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 8d{p3αq _ 1, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d{α ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' It is the predictable process such that |Mt|2 ´ xMyt is a local martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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 ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2) We have E “ |Mγ 0 pfq|2‰ “ ż R2d e|γ|2K0px,yqfpxqfpyqdxdy ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3) Since p ă 2 by assumption, Jensen’s inequality implies that E r|Mγ 0 pfq|ps ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Using Itˆo calculus, we obtain an explicit expression for the quadratic variation xMγpfqy8 “ |γ|2 ż 8 0 Atdt (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4) where At :“ ż R2d fpxqfpyqQtpx, yqeγXtpxq`γXtpyq´ γ2 2 Ktpxq´ γ2 2 Ktpyqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5) Note that At is real and positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2), we deduce that Mγ t pfq is bounded in Lp if E “ p ş8 0 Atdtqp{2‰ ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' To bound At from above, we take the modulus of the integrand in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5) and using the assumption that β “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d ´ α (γ P PI{II) we obtain that At ď ż R2d |fpxqfpyq|Qtpx, yqeαpXtpxq`Xtpyqq` 2d´2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dα 2 pKtpxq`Ktpyqqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6) Then using the inequality ab ď a2 2 ` b2 2 with a “ |fpxq|eαXtpxq` 2d´2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dα 2 Ktpxq and b “ |fpyq|eαXtpyq` 2d´2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dα 2 Ktpyq 14 HUBERT LACOIN and symmetry in x and y, we have At ď ż R2d |fpxq|2Qtpx, yqe2αXtpxq`p2d´2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dαqKtpxqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='7) We use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='13) to integrate over y and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) to replace replace Ktpxq by t (at the cost of multiplicative constant) and we have At ď Cedt ż Rd |fpxq|2e2αpXtpxq´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dtqdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8) Now, as α ą a d{2, we have by p{2 ă 1 by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9) with q “ p{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' In the remainder of the paper, we simply say “by subadditivity” when using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='10) Averaging with respect to pXs ´ Xnq we obtain from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8) ż n`1 n E rAs | Fns ď Cedn ż R2d |fpxq|2e2αpXnpxq´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dnqdx “: CBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='11) As p ą ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 8d{3α by assumption, we can conclude using the estimate in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1 below for the fractional moments of Bn (the assumption on p makes the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) summable in n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' More precisely, we deduce from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='10),(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='11) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) that E ”`ş8 0 Atdt ˘p{2ı ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' For α ą a d{2 and p ă ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d{α we have E ” Bp{2 n ı ď Cn´ 3αp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 8d plog nq6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) This result is a weaker version of [20, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We provide, for the commodity of the reader a self-contained of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1 in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1 in the setup where our probability space contains a martingale approximation pXtqtě0 of the field X with covariance 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' More precisely we show that Mγ ε pfq converges to the same limit as Mγ t pfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Given γ P PI{II and p P r1, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d{αq we have lim εÑ0 sup tą0 E “ |pMγ t ´ Mγ t,εqpfq|p‰ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='13) As a consequence the following convergence holds in Lp lim εÑ0 Mγ ε pfq “ Mγ 8pfq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='14) CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Let us first show indicate how (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='14) follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='15) Since limtÑ8 Kt,εpx, yq “ Kεpx, yq, using dominated convergence the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' tends to 0 when t Ñ 8 and thus limtÑ8 Mγ t,εpfq “ Mγ ε pfq in L2, and hence also in Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3 we thus have the following convergence in Lp lim tÑ8pMγ t ´ Mγ t,εqpfq “ pMγ 8 ´ Mγ ε qpfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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‰ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='16) and we conclude using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' To prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='13), we assume that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Then using the BDG inequality (we omit the dependence in f for ease of reading).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='18) Hence in view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='17)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='18), to prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='13) we need to show that lim εÑ0 ErxMγ ´ Mγ ¨,εyp{2 8 s “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='20) where, At is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8), and recalling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='21) We are going to reduce the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='22) lim rÑ8 sup εPp0,1s E «ˆż 8 r |Apiq t,ε|dt ˙p{2ff “ 0 for i P t1, 2u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='23) Before proving (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='22)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='23) let us explain how (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='19) is deduced from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Note that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='23) is also valid for At (this can be extracted from the proof in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Given δ ą 0, 16 HUBERT LACOIN using subadditivity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='24) Now using first Jensen’s inequality and then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='25) Using subadditivity again we deduce from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='24)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='26) which (recalling (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='20)) concludes the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Let us now prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='22)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The proof (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='22) follows from a rather pedestrian but rather cumbersome computation of the L2 norm of pAt ´ Apiq t,εq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The following lemma summarizes the key points of this computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Consider the following: ‚ Let pX, µq be a measured space and T be a set of indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' ‚ Let Zt,εp¨q, t P T , ε P p0, 1s be a collection of complex valued Gaussian processes defined on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We set Gt,εpx, yq :“ ErZt,εpxqZt,εpyqs and Ht,εpx, yq :“ ErZt,εpxqZt,εpyqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='27) ‚ Let Zt be defined on the same probability space in such a way that pZt, Zt,εq is jointly Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We let Gt and Ht be defined as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='27) and set Ht,ε,0px, yq :“ ErZt,εpxqZtpyqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' ‚ Let gt,ε and gt be deterministic functions X Ñ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We have lim εÑ0 sup tPT E “ |Wε ´ Wt,ε|2‰ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='29) Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The proof is actually much shorter than the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='30) and using our assumptions we can apply dominated convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' □ Proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We consider the case i “ 2 but the other one is identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Then the assumptions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3 are immediate to check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' □ We now provide the details for the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='23) i “ 2 (the case i “ 1 is similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Let us set n0pεq “ rlogp1{εqs and assume (without loss of generality) that r is an integer and is smaller than n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Using - as in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2 - subadditivity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='31) Proceeding as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1 to replace the covariance Kn,εpxq by n) E ” |Ap2q t,ε | | Fn ı ď C ż R2d |fpxq|2Qs,εpx, yqe2αpXn,εpxq´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dnq`2dndxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='32) Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='13) to integrate over y and setting Bp2q n,ε :“ ż Rd |fpxq|2e2αpXn,εpxq´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dnq`dndx (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dnq`dndy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1, that E ” pBp2q n,εqp{2ı ď Cn´ 3αp ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 8d for a constant which does not depend on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Recalling that p ą ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 8d{3α (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1)) we obtain combining(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='32), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='34) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='35) that E «ˆż 8 r |Ap2q t,ε |ds ˙p{2ff ď Cr1´ 3αp 2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='36) This concludes the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='23), and thus of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Reduction to a statement concerning the total variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Using [16, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5] (which is a simpler version of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6 displayed above) we can reduce the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='19) to the following convergence statement about the quadratic variation of the martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We have the following lim tÑ8 vpt, γq´2xMγpf, ωqyt “ M1pe|γ|2L|f|2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1) Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5 from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We simply apply [16, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5] to the martingale Mγ t pf, ωq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' □ Setting, for notational simplicity Wt :“ Mγ t pfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Recall that for a complex value mar- tingale such as Wt we use the notation xWyt for the bracket between W and its conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Using bilinearity of the martingale brackets we have xMγpf, ωqyt “ 1 2 ` xWyt ` Repe´2iωxW, Wytq ˘ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2) Hence to prove (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1), it is sufficient to prove that following convergences hold in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' lim tÑ8 vpt, γq´2xWyt “ 2M1pe|γ|2L|f|2q, lim tÑ8 vpt, γq´2xW, Wyt “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3) The expression for the bracket of Wt can be obtained by using Itˆo calculus (recall (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4)) More precisely we have xWyt “ |γ|2 ż t 0 Asds and xW, Wyt “ γ2 ż t 0 Bsds, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4) CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES19 where At is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5) and Bt :“ ż R2d fpxqfpyqQtpx, yqeγpXtpxq`Xtpyqq´ γ2 2 pKtpxq`Ktpyqqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5) Now using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4) our first idea is to deduce (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3) from a convergence statement concerning At and Bt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' To bypass this problem, restrict ourselves to likely family of event and prove L1 convergence for the restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Recalling the definition of X (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 ´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dsq ă q , Aq,R :“ # sup sě0,|x|ďR pXspxq ´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dtq ă q + “ č xPBp0,Rq tě0 At,qpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6) A very important fact, which is a direct consequence of [4, Proposition 19] (see also [17, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4] for a concise proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We have for any fixed R ą 0 lim qÑ8 P rAq,Rs “ 1 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8) which plays the role of a rescaling function for At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9) lim tÑ8 E “ |Bt{φptq| 1Aq,R ‰ “ 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='10) and the above quantities are finite for every t ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' To show that Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3 implies the convergence stated in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' This is the purpose of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We have for any |γ| ě d lim tÑ8 |γ|2 şt 0 φpsqds 2vpt, γq2 “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='11) 20 HUBERT LACOIN The proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4 is presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1 and hence Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5 are valid with v replaced by vpt, γq :“ |γ| b p şt 0 φpsqdsq{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' As we have seen, it is sufficient to prove (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We provide the details concerning the convergence of xWyt (the first line in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3)) but that of xW, Wyt can be obtained exactly in the same manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) Observing that ş8 0 φpsqds “ 8, the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) is simply a weighted Cesaro mean and thus we deduce from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3 and more precisely from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9) that lim tÑ8 E «ˇˇˇˇˇ xWyt |γ|2 şt 0 φpsqds ´ M1pe|γ|2L|f|2q ˇˇˇˇˇ 1Aq,R ff “ 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2) is the desired conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Restricted convergence in L2 for the critical GMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Before starting the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3, we recall a result which play a key role in the proof, the L2 convergence of M ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d t pgq towards M1pgq when considering the restriction to the event Aq,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' This also implies convergence in L1 which is what we require for the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The result can be deduced from the L2 convergence of the truncated version of M ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d t pgq which is proved in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d t pgq ´ M1pgq ˇˇˇˇˇ 2 1Aq,R fi fl “ 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='14) and E “ |M1pgq|21Aq,R ‰ ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The fact that E “ |M1pgq|21Aq,R ‰ ă 8 is a simple consequence of the convergence since for any fixed t, Er|M ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d t pgq|2s ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We set (recall (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6)) M ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d,pqq t pgq :“ ż gpxqe ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXtpxq´dKtpxq1At,qpxqdx, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='15) CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES21 From [17, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1], there exists an L2 variable D pqq 8 pgq such that lim tÑ8 E » – ˇˇˇˇˇ c πt 2 M ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d,pqq t pgq ´ D pqq 8 pgq ˇˇˇˇˇ 2fi fl “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='16) It satisfies D pqq 8 pgq “ M1pgq on the event Aq,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' More precisely [17, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' On the event Aq,R we have M ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d,pqq t pgq “ M ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d t pgq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Hence lim sup tÑ8 E » – ˇˇˇˇˇ c πt 2 M ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d t pgq ´ M1pgq ˇˇˇˇˇ 2 1Aq,R fi fl “ lim sup tÑ8 E » – ˇˇˇˇˇ c πt 2 M ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d,pqq t pfq ´ D pqq 8 pgq ˇˇˇˇˇ 2 1Aq,R fi fl “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='17) where the last equality follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Organizing the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The two convergences rely on similar ideas, we focus on (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9) which is the more delicate of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The quantity obtained after this replacement is, up to a multiplicative factor, of the form M ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d t pgq (recall that γ ` γ “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d) for some function g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Hence we should be able to conclude the proof of the convergence statement using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' While this idea is relatively simple, it requires several steps to be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We set K˚ t px, yq :“ K0pxq ` Ktpx, yq and r “ rptq :“ t ´ log log t (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='18) (we are assuming that t ą e so that 0 ď r ď t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXrpxq´dKrpxqdxdy “ ˆż Rd Qtp0, zqe|γ|2Ktp0,zqdz ˙ ż Rd e|γ|2K0ptq|fpxq|2e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXrpxq´dKrpxqdx “ φptq c πt 2 ż Rd e|γ|2Lpxq|fpxq|2e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXrpxq´dKrpxqdx “ φptq c πt 2 M ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d r pe|γ|2L|f|2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='19) As a direct consequence of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5 (since r “ t ´ optq the presence of ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' t instead of ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='r does not affect the convergence), we have lim tÑ8 E «ˇˇˇˇˇ rAt φptq ´ M1pe|γ|2L|f|2q ˇˇˇˇˇ 1Aq,R ff “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='20) 22 HUBERT LACOIN With this observation the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9) reduces to showing that lim tÑ0 1 φptqE ” |At ´ rAt|1Aq,R ı “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='21) This requires some care but before going in the depth of the proof, let us explain the heuristic behind (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Note that rAt is obtained from At with two simple modifications: ‚ We have replaced fpxqfpyq by |fpxq|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' ‚ In the exponential, we have replaced γXtpxq`γXtpyq by ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXrpxq “ pγ`γqXrpxq and ´ γ2 2 Ktpxq ´ γ2 2 Ktpyq by ´dKrpxq ` |γ|2K˚ t px, yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The first modification is rather straightfoward, we are integrating close to the diagonal so that fpyq is close to fpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' In our case we have E „ eγXtpxq`γXtpyq´ γ2 2 Ktpxq´ γ2 2 Ktpyq \uf6be “ e|γ|2Ktpx,yq, E ” e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXrpxq´dKrpxq`|γ|2K˚ t px,yqı “ e|γ|2K˚ t px,yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='21) requires three distinct steps which are detailed in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Step 1: Changing the deterministic prefactor in the integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The integrand of At and rAt have different expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Our first step aims to fix this by replacing fpyq by fpxq in At and doing a small modification in the exponential factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We set Ap1q t :“ ż R2d |fpxq|2Qtpx, yqeγXtpxq`γXtpyq` γ2 2 Ktpxq` γ2 2 Ktpyq`|γ|2pK0pxq´K0px,yqqdxdy (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='23) We are going to prove that lim tÑ8 φptq´1E ” |At ´ Ap1q t |1Aq,R ı “ 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXtpxq`p|γ|2´dqKtpxqdxdy (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Then we observe that (λ denotes the Lebesgue measure) since At,qpxq Ă Aq,R we have E ” e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXtpxq´dKtpxq1Aq,R ı ď E ” e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXpxq´dKtpxq1At,qpxq ı “ Pr@s P r0, ts, Bs ď qs ď c 2 πtq (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='27) where in the last line, we used Cameron-Martin formula (see Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1 in the ap- pendix) and the fact that pXtpxqqtě0 is a standard Brownian Motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The last inequality is simply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='26) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='27) and the fact that K0 is bounded, we have E ” |At ´ Ap1q t |1Aq,R ı ď Cδptq ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' t ż Bp0,Rq2 Qtpx, yqe|γ|2Ktpxqdxdy ď C1δptqφptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='28) □ Step 2: Taking conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Recalling the definition of rptq (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='18) we set Ap2q t :“ ErAp1q t | Frs (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='29) For this step of the proof (and only this step), we are going to assume that K0 ” 0 (and hence X0 ” 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' This assumption allows to replace Ktpxq and Ktpyq by t, and we get the following simplification for the expression of Ap1q t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Ap1q t :“ ż R2d |fpxq|2Qtpx, yqeγXtpxq`γXtpyq`p|γ|2´dqtdxdy (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='30) Then we have Ap2q t “ ż R2d |fpxq|2Qtpx, yqeγXrpxq`γXrpyq`p|γ|2´dqr`|γ|2Krr,tspx,yqdxdy, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We are going to show that lim tÑ8 φptq´1E ” |Ap1q t ´ Ap2q t |1Aq,R ı “ 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='32) Recalling (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='34) and thus we can prove that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='32) holds by showing that lim tÑ8 φptq´2E ” pA p2q t ´ A p1q t q2ı “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='35) To bound ErpA p2q t ´ A p1q t q2s we expand the square, making it an integral on R4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We set ξpx, yq :“ |fpxq|2Qtpx, yqe´p|γ|2´dqt ´ eγXspxq`γXspyq ´ E ” eγXspxq`γXspyqq | Fr ı¯ 1Ar,qpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We have E ” pA p2q t ´ A p1q t q2ı “ ż R4d E “ ξpx1, y1qξpx2, y2q ‰ dx1dy1dx2dy2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='37) Hence we only need to integrate the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='36) on the set |x1 ´ x2| ď 3e´r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' In that case we use E “ ξpx1, y1qξpx2, y2q ‰ ď E “ |ξpx1, y1q|2‰1{2 E “ |ξpx2, y2q|2‰1{2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='38) and E “ |ξpx, yq|2‰ “ |fpxq|4Qtpx, yq2e2p|γ|2´dqtE ” e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dpXtpxq`Xtpyqq1Ar,qpxq ı (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='39) Using Cameron-Martin formula and the fact that pXtpxqqtě0 is a standard Brownian motion we have E ” e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dpXtpxq`Xtpyqq1Ar,qpxq ı “ e2dpt`Ktpx,yqqP r@u P r0, rs, Bu ď q ´ Kupx, yqs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) (and then Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2) we obtain for a constant q1 ą q e´4dtE ” e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dpXtpxq`Xtpyqq1Aq,rpxq ı ď P ” @u P r0, rs, Bu ď q1 ´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2du ı ď Cr´3{2e´dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The third inequality is a consequence of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='10) (see the computation in the Appendix, while the last inequality follows from the the fact that with our choice of parameters (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='18) we have t ´ r “ oplog tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Step 3: Comparing Ap2q t and rAt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Finally, we show that lim tÑ8 φptq´1E ” |Ap2q t ´ rAt|1Aq,R ı “ 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='41) which together with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='24)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='32), concludes the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We introduce another smaller time parameter, namely r “ t{2 and define p Xpx, yq “ Xrpxq ` Xrr,rspyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We want to replace Xrpyq by Xrpxq in the exponential with an intermediate steps, so we set Z1pxq :“ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXrpxq, Z2px, yq :“ γXrpxq ` γ p Xpx, yq, Z3px, yq :“ γXrpxq ` γXrpyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='yq´ 1 2ErZ3px,yqsdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='43) In order to prove (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 \uf6be ˆ ż R2d |fpxq|2Qtpx, yqe|γ|2K˚ t px,yqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='44) Since the integral is of order ep|γ|2´dqt (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12)), which is the same order as φptq ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' t (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='10)), the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='41) boilds down to proving lim tÑ8 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' t max |x|ďR |x´y|ďe´t E „ˇˇˇeZ1pxq´ ErZ2 1 s 2 ´ eZ3´ ErZ2 3 s 2 ˇˇˇ1Aq,R \uf6be “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='45) To prove (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='45) we start with the decomposition E „ˇˇˇeZ1pxq´ ErZ2 1 s 2 ´ eZ3´ ErZ2 3 s 2 ˇˇˇ1Aq,R \uf6be ď E „ˇˇˇeZ1´ ErZ2 1 s 2 ´ eZ2´ ErZ2s 2 ˇˇˇ1Ar,qpxq \uf6be ` E „ˇˇˇeZ3´ ErZ2 3 s 2 ´ eZ2 2´ ErZ2 2 s 2 ˇˇˇ \uf6be (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We start with the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' From Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3, we have E „ eZ3´ ErZ2 3 s 2 ´ eZ2 2´ ErZ2 2 s 2 | \uf6be ď C a Er|Z3 ´ Z2|2s “ C|γ| a ErpXrpxq ´ Xrpyqq2s ď C1e´ct, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='47) where we have used that ErpXrpxq ´ Xrpyqq2s “ 2pr ´ Krpx, yqq ` pK0pxq ` K0pyq ´ 2K0px, yqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='14)), both are exponentially small in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' For the first term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 \uf6be “ E ” e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXrpxq´dKrpxq1Aq,rpxq ı E „ |eZ1 1´ ErpZ1 1q2s 2 ´ eZ1 2´ ErpZ1q2 2s 2 | \uf6be , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='48) where Z1 i “ Zi ´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXrpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Using Cameron-Martin formula and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2, we have E ” e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXrpxq´dKrpxq1Ar,qpxq ı “ P r@s P r0, rs, Bs ď qs ď c 2 rπq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='49) The factor r´1{2 is sufficient to cancel the ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' t in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='45) and we just have to show that the second factor in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='48) is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' From Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3 we have E „ |eZ1 1´ ErpZ1 1q2s 2 ´ eZ1 2´ ErpZ1 2q2s 2 | \uf6be 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='50) where the penultimate inequality can be deduced from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The combination of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='47)- (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='49) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='50) concludes the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' □ Bonus step: the case of Bt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' To conclude let us sketch rapidly the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We can repeat the argument of step 2 to show that lim tÑ8 φptq´2Er|Bt ´ ErBt | Frs|2s “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='51) Then it is rather direct to check that lim tÑ8 φptq´1E “ |ErBt | Frs|1Aq,R ‰ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='52) More precisely we have ErBt | Frs “ ż R2d fpxqfpyqQtpx, yqeγpXrpxq`Xrpyqq` γ2 2 p2Krr,tspx,yq´Krpxq´Krpyqqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES27 Taking the absolute value of the integrand, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) to evaluate Kt and Krr,ts, then the inequality ab ď pa2 ` b2q{2 and symmetry, and finally (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='13) |ErBt | Frs| ď Cep|γ2|´dqp2r´tq ż R2d |fpxqfpyq|Qtpx, yqe ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' d{2pXrpxq`Xrpyqqdxdy ď Cep|γ2|´dqp2r´tq ż R2d |fpxq|2Qtpx, yqe ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXrpxqdxdy ď C1ep|γ2|´dqp2r´tq´dt ż R2d |fpxq|2e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXrpxqdx (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='53) Hence we have E “ |ErBt | Frs|1Aq,R ‰ ď ep|γ2|´dqp2r´tq´dt ż R2d |fpxq|2E ” e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXrpxq1Ar,qpxq ı dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='54) Using Cameron Martin formula, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2 (recall that r „ t) we obtain that E ” e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXrpxq1Ar,qpxq ı ď Ct´1{2edr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='55) Overall using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='10) we have φptq´1E “ |ErBt | Frs|1Aq,R ‰ ď Ce´|γ2|pt´rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Organization of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Like for the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1, we assume that our probability space contains a martingale approximation sequence pXtqtě0 of the field X, with covariance given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' For the same reason as the one exposed at the beginning of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2 this entails no loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The main idea is to apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Hence need to check that the martingale Mγ t,εpf, ωq :“ E rMγ ε pf, ωq | Fts satisfy all the requirements in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Setting W pεq t :“ Mγ t,ε (recall (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='7)), and using the bilinearity of the martingale bracket like in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2) we obtain xMγ ¨,εpf, ωqyt “ 1 2 ´ xW pεqyt ` Repe´2iωxW pεq, W pεqytq ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1) The requirements concerning the quadratic variation of Mγ t,εpf, ωq can be obtained as consequences of the following, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2) Furthermore we have for any fixed t we have sup εPp0,1q ErxW pεqyts ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3) Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1 is proved in the next subsection, let us first show how our main results can be deduced from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 28 HUBERT LACOIN Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We must check that the three requirements in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='22) are satis- fied since the result follows then from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The sequences are in fact uniformly bounded in L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We have sup εPp0,1q Er|Mγ 0,εpf, ωq|s ď sup εPp0,1q Er|Mγ 0,εpfq|s ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4) Indeed taking the absolute value of the integrand, we have Er|Mγ 0,εpfq|s ď ż Rd E „ fpxqe ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' d{2X0,εpxq` β2´pd{2q 2 K0,εpxq \uf6be dx “ ż Rd fpxqeβ2K0,εpxqdx, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5) and the uniform bound follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' From (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1) we have xMγ ¨,εpf, ωqyt ď xW pεqyt and thus the uniform boundedness in L1 is consequence of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Let us now turn to the first and main requirement in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The convergences in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6) Using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2 and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1), we conclude that lim εÑ0 vpε, θ, γq´2xMγ ¨,εpf, ωqy8 “ M1pe|γ|2L|f|2q in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' □ As another preliminary step to our proof, we reduce the convergence statement in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1 to a convergence of the derivative of the martingale brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='7) Similarly to what has been done in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' To this end we introduce a couple of parameters (recall (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5)) tpt, εq :“ t ^ logp1{εq φpt, εq :“ d 2 πpt _ 1qe|γ|2j ˆż Rd e|γ|2Kt,εp0,zqQt,εp0, zqdz ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8) The quantity r will on Our aim is to prove the following CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES29 Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' lim εÑ0 tÑ8 E „ˇˇˇˇ At,ε φpt, εq ´ M1pe|γ|2L|f|2q ˇˇˇˇ 1Aq,R \uf6be “ 0, lim εÑ0 tÑ8 E „ˇˇˇˇ Bt,ε φpt, εq ˇˇˇˇ 1Aq,R \uf6be “ 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9) and for any T ă 8 sup tPr0,Ts εPp0,1q E r|At,ε|s ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='10) Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' This is a stronger statement than both limεÑ0 limtÑ8 Fpt, εq “ 0 or limtÑ8 limεÑ0 Fpt, εq “ 0 Clearly (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='10) implies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' To deduce (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2) from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9), we need to check that renormal- izing factor 2vpε, θ, γq2 corresponds to the integral of φpt, εq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' This is the content of the following lemma whose proof is presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We have for any |γ| ě d lim εÑ0 |γ|2 ş8 0 φpt, εqdt 2vpε, θ, γq2 “ 1 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='11) We can now complete the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1 using Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2 Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' From Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 \uf6be dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 \uf6be ď δ 2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='13) In the integral we can distinguish the contribution from r0, Ts from the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We have from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 \uf6be ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 \uf6be dt ď δ 2, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='16) and thus conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Let us start with the proof of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Noting that At,ε is positive, we have E rAt,εs “ ż R2d fpxqfpyqQt,εpx, yqe|γ|2Kt,εpx,yqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='17) We can just use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) and bound Qt,εpx, yq by 1 and Kt,εpx, yq by T `C to conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' For the proof of the convergence of At,ε we proceed exactly as for the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We assume that tpt, εq ą e (recall (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8)) and set r “ rpt, εq :“ t ´ log log t, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXrpxq´2dKrpxqdxdy “ φpt, εqM ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d r pe|γ|2L|f|2q (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='19) Since lim εÑ0 tÑ8 rpt, εq “ 8, we obtain, as a direct consequence of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5 that lim εÑ0 tÑ8 E «ˇˇˇˇˇ rAt,ε φpt, εq ´ M1pe|γ|2L|f|2q ˇˇˇˇˇ 1Aq,R ff “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='20) Our task is thus to prove that lim εÑ0 tÑ8 φpt, εq´1E ” | rAt,ε ´ At,ε|1Aq,R ı “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='21) Like for the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='21) in the previous section, we proceed in three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Step 1: Changing the deterministic prefactor in the integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='23) CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES31 we have lim εÑ0 tÑ8 δpε, tq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Since Qt,εpx, yq “ 0 when |x ´ y| ě et ` 2ε repeating the computation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='26) and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='13) we get |Ap1q t,ε ´ At,ε| ď δpε, tq ż Bp0,Rq2 Qt,εpx, yqe ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXt,εpxq`p|γ|2´dqKt,εpxqdxdy ď Ce´dtδpε, tq ż Bp0,Rq e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXt,εpxq`p|γ|2´dqKt,εpxqdx (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXt,εpxq´dKt,εpxq1Aq,R ı ď E ” e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXt,εpxq`p|γ|2´dqKt,εpxq1At,qpxq ı “ P ” @s P r0, ts, Bs ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dps ´ Ks,ε,0pxqq ` q ı ď Pp@s P r0, ts, Bs ď q1q ď Cpt _ 1q´1{2et|γ|2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='25) where in the second inequality we have used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) to estimate covariances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Hence, using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='20) we deduce from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='24) that E ” |Ap1q t,ε ´ At,ε|1Aq,R ı ď Cδpε, tqt´1{2et|γ|2´dt ď C1δpε, tqφpt, εq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' This concludes the proof of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' □ Step 2: Taking conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We set Ap2q t,ε :“ E ” Ap1q t,ε | Fr ı (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='26) and we are going to prove lim εÑ0 tÑ8 φpt, εq´2E ” |Ap2q t,ε ´ Ap1q t,ε |21Aq,R ı “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' In that case note that since Kt,εpxq “ Kt,εpyq “ Kt,εpxq, we can factorize the term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='30) Now we set ξt,εpx, yq :“ |fpxq|2Qt,εpx, yq pζpx, yq ´ Erζpx, yq |Frsq 1Ar,qpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8)) then E “ ξpx1, y1qξpx2, y2q | Fr ‰ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='33) When |x1 ´ x2| ď 2e´r we can use Cauchy-Schwarz to bound the covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We have E “ |ξpx, yq|2‰ ď |fpxq|4 pQt,εpx, yqq2 Er|ζpx, yq|21Ar,qpxqs (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dps ´ Ks,ε,0pxq ´ Ks,ε,0py, xqq ` q ¯ ď Cep|γ|2`dqtP ´ @s P r0, rs, Bs ď ´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2ds ` q1¯ ď C1etp2|γ|2`dqr´3{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' where in the first inequality we used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='35) We conclude the proof of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='27) by observing (recall (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8)) that lim εÑ0 tÑ8 edpt´rqr´1{2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='36) □ Step 3: Final comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Finally to conclude we need to show that lim εÑ0 tÑ8 φpt, εq´2E ” |Ap2q t,ε ´ rAt,ε|21Aq,R ı “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='37) We set (recall that Xrs,ts,ε “ Xt,ε ´ Xs,ε´) Z1pxq :“ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXrpxq, Z2px, yq :“ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXrpxq ` γXrr,rs,εpxq ` γXrr,rs,εpyq, Z3px, yq :“ γXr,εpxq ` γXr,εpyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='38) CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES33 The reader can check (using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='26) rather than (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='39) In order to prove (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 \uf6be “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='40) The restriction for x and y comes from the support of f and Qt,ε respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' To prove (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='45) we start with the decomposition E „ˇˇˇeZ1pxq´ ErZ2 1 s 2 ´ eZ3´ ErZ2 3 s 2 ˇˇˇ1Aq,R \uf6be ď E „ˇˇˇeZ1´ ErZ2 1 s 2 ´ eZ2´ ErZ2s 2 ˇˇˇ1Ar,qpxq \uf6be ` E „ˇˇˇeZ3´ ErZ2 3 s 2 ´ eZ2 2´ ErZ2 2 s 2 ˇˇˇ \uf6be (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Let us start with the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3, we have E „ eZ3´ ErZ2 3 s 2 ´ eZ2 2´ ErZ2 2 s 2 | \uf6be ď C a Er|Z3 ´ Z2|2s “ C b Er| ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='42) where to obtain the last line we have used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='14) and the H¨older continuity of K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' For the first term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='41) we factorize the Fr-measureable part and use independence to obtain E „ |eZ1´ ErZ2 1 s 2 ´ eZ2´ ErZ2s 2 |1Ar,qpxq \uf6be “ E ” e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXrpxq´dKrpxq1Ar,qpxq ı E „ |eZ1 1´ ErpZ1 1q2s 2 ´ eZ1 2´ ErpZ1q2 2s 2 | \uf6be , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='43) where Z1 i “ Zi ´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXrpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We have from Cameron-Martin Formula and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2 E ” e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXrpxq´dKrpxq1Ar,qpxq ı “ P r@s P r0, rs, Bs ď qs ď Cr´1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='44) 34 HUBERT LACOIN This is obviously Opt´1{2q so to conclude we only need to show that the other factor in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='43) goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We also have from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='14) E „ |eZ1 1´ ErpZ1 1q2s 2 ´ eZ1 2´ ErpZ1q2 2s 2 | \uf6be 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='45) This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' □ The convergence of Bt,ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' To prove the second convergence in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9), it is sufficient again to show first that lim tÑ8 εÑ0 ϕpt, εq´2E “ |Bt,ε ´ ErBt,ε | Frs|21Aq,R ‰ “ 0 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='46) repeating the computation of step two, and then prove that lim tÑ8 εÑ0 Er|ErBt,ε | Frs|1Aq,Rs “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We leave this part to the reader, since this is very similar to the computation performed at the end of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 ˙\uf6be “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2) Note that this implies also that P rZ ě Ms “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We assume, to simplify notation that ξ “ 1 (this entails no loss of generality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We set Ht :“ E rH | Fts and Zt :“ E rZ | Fts we have E „ H ˆ ei ξWn vpnq ´ e´ ξ2Z 2 ˙\uf6be “ 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3) We prove the convergence (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1) by showing that for i “ 1, 2, 3 lim tÑ8 lim sup nÑ8 |Eipt, nq| “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5) CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES35 Since Ht and Zt converge respectively to H and Z in L1, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4) holds for i “ 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' In particular we have E „ e iWn´Wn,t vpnq ` xWny8´xWnyt 2vpnq2 | Ft \uf6be “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6) Multiplying by Hte´ Zt 2 and taking expectation we obtain that E „ Hte iWn´Wn,t vpnq ` xWny8´xWnyt 2vpnq2 ´ Zt 2 \uf6be “ E ” Hte´ Zt 2 ı (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 \uf6be ď }H}8E „ˇˇˇˇ1 ´ e ´ iWn,t vpnq ` xWny8´xWnyt 2vpnq2 ´ Zt 2 ˇˇˇˇ \uf6be , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8) From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9) Using assumption (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2), taking the limit in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8) and using dominated con- vergence, we obtain that lim sup nÑ8 |E3pt, nq| ď }H}8E ”ˇˇˇ1 ´ e Z´Zt 2 ˇˇˇ ı .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='10) Since Zt converges to Z we can conclude that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4) also holds for i “ 3 using dominated convergence again (both variables are uniformly bounded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Let us now remove the boundedness assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Given A ą 0 we set TA,n :“ inftt : vpnq´2xWnyt “ Au and W A n :“ Wn,TA,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='11) Since we have proved (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1) under the assumption (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2) we know that for every A ą 0 lim nÑ8 E „ H ˆ ei ξW A n vpnq ´ e´ ξ2pZ^Aq 2 ˙\uf6be “ 0 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) From the convergence assumption, we have lim sup nÑ8 PrTA,n “ 8s ď P rZ ě As (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='13) and hence lim AÑ8 lim inf nÑ8 P “ W A n “ Wn ‰ “ lim AÑ8 PrZ ^ A “ Zs “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 36 HUBERT LACOIN As a consequence we can conclude using (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) that lim nÑ8 E „ H ˆ eiξ Wn vpnq ´ e´ ξ2Z 2 ˙\uf6be “ lim AÑ8 lim nÑ8 E „ H ˆ ei ξW A n vpnq ´ e´ ξ2pZ^Aq 2 ˙\uf6be “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='14) □ Acknowledgements: This work was supported by a productivity grant from CNPq and a JCNE grant from FAPERJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Technical results and their proof A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Standard Gaussian tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We first display two standard tools which are used throughout the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The first is the standard Cameron-Martin formula which describes how a Gaussian process is affected by an exponential tilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Let pY pzqqzPZ be a centered Gaussian field indexed by a set Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We let H denote its covariance and P denote its law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1) Under rPz0, Y is a Gaussian field with covariance H, and mean rEz0rY pzqs “ Hpz, z0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The second is a bound on the probability for a Brownian Motion to remain below a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Both estimates can be proved directly using the reflexion principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Let B be a standard Brownian Motion and let P denote its distribution, setting gtpaq :“ şu` 0 e´ z2 2t dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' we have P « sup sPr0,ts Bs ď a ff “ c 2π t gtpaq ď c 2π t a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2πt ż e´ u2 2t p1 ´ e 2apa`u´bsq` t qdu ď Ca,be´ b2t 2 t´3{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Comparing exponentiated Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' In or comparison of partition functions Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Consider pX1, X2, Y1, Y2q an R4 valued centered Gaussian vector and set X :“ X1 ` iX2 and Y “ Y1 ` iY2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Assuming that ErX2 2s ď 1 and Er|X ´ Y |2s ď 1 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4) then there exits a constant C such that E ”ˇˇeX´ 1 2ErX2s ´ eY ´ 1 2ErY 2sˇˇ ı ď CE “ |X ´ Y |2‰ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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 \uf6be “ E „ eX1` ErX2 2 s´ErX2 1 s 2 ˇˇeY ´X` ErX2s´ErY 2s 2 ´ 1 ˇˇ \uf6be “ e ErX2 2 s 2 E „ˇˇeY ´X´ ErpX´Y q2s 2 ´iErX2pY ´Xqs ´ 1 ˇˇ \uf6be .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6) CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES37 The prefactor is bounded (by assumption) by e1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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 \uf6be ď ˇˇeiErX2Zs ´ 1 ˇˇE „ˇˇeZ´ ErZ2s 2 ˇˇ \uf6be ` E „ˇˇeZ´ ErZ2s 2 ´ 1 ˇˇ \uf6be .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 \uf6be ď e a Er|Z|2se ErZ2 2 s 2 ď e3{2a Er|Z|2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8) For the second term we have (using again |eu ´ 1| ď e|u|) E „ˇˇeZ´ ErZ2s 2 ´ 1 ˇˇ \uf6be ď d E „ˇˇeZ´ ErZ2s 2 ´ 1 ˇˇ2 \uf6be “ a eEr|Z|2s ´ 1 ď e1{2a Er|Z|2s, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9) which yields the desired result for C “ e2 ` e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' □ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Let us first compute the order of magnitude of φptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Let us set for practical purpose φptq :“ ş Qtp0, zqe|γ|2Ktp0,zqdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) (recall that |z| ď e´t on the integrand) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='13) we have φptq — ep|γ|2´dqt and φptq — t´1{2ep|γ|2´dqt (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='10) As a consequence when |γ|2 ą d most of the integral is carried by rt ´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' t, ts and we have ż t 0 φpsqds “ p1 ` op1qq ż t t´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' t φpsqds “ p1 ` op1qq ż t t´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' t c s _ 1 t φpsqds “ p1 ` op1qq c 2 πte|γ|2j ż t 0 φpsqds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='11) We observe that |γ|2Qsp0, zqe|γ|2Ksp0,zq “ Bs ´ e|γ|2Ksp0,zq¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Using Fubini and integrating w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) The integrand in the second line is bounded above by p|z| _ 1q´|γ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' This is obvious for |z| ě et since the integrand vanishes, and when |z| ď et this can be obtainded from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='13) which combined with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='11), proves the lemma in the case |γ|2 ą d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' When |γ|2 “ d, we observe that using, as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12), a change of variable and dominated convergence, we have lim sÑ8 φpsq “ ż Rd κpe η1 η2 zqedℓpzqdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='14) 38 HUBERT LACOIN On the other hand we have from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) ż t 0 φpsqds “ ż Rd ´ edpKtp0,e´tzq´tq ´ e´dt¯ dz (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='17) Since φpsq converges, we deduce that its limit equals its Cesaro limit and thus lim sÑ8 φpsq “ e´djΣd´1, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='18) which implies in turn that ż t 0 d 2 πps ^ 1qedjφpsq “ p1 ` op1qq2 c 2t π Σd´1, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='19) and concludes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' □ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Let us again start with the case |γ|2 ą d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' As in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' For this reason, we can replace the term pt _ 1q´1{2 by plog 1{εq´1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' |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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='21) Using Fubini and integrating with respect to time as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='23) Next we observe that Kεp0, εzq ` logpεq “ ℓθpzq ` pLεp0, εzq ´ Kεp0, εzqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='24) The combination of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='21)-(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='24) concludes the proof in the case |γ|2 ą d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' For the case |γ|2 “ d, based on (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='20) we know that setting Tε “ logp1{εq ´ a logp1{εq ż 8 0 φpt, εqdt “ p1 ` op1qq ż Tε 0 φpt, εqdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='27) and we can conclude using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We can go further and prove that Mγ ε p¨q converges as a distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' For this purpose we need to recall the definition of local Sobolev/Bessel spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' It is a Banach space when equiped with the norm }f}Hs,p “ ż Rkp1 ` |ξ|2qps{2|pϕpξq|pdξ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 :“ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' ϕ P D1pUq | ρϕ P Hs,ppRdq for all ρ P C8 c pUq ) , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3) where above ρϕ is identified with its extension by zero on Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' It is equiped with the topology generated by the family of seminorms rρ, ρ P C8 c pUq defined by rρpϕq :“ }ϕρ}Hs,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The convergence result for Mγ ε p¨q as a distribution for γ P PI{II is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4) 40 HUBERT LACOIN In particular Mγ ε converges to Mγ 8 in probability in the Hs,p locpRdq topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Similarly in P1 II{III the convergence in law holds also for the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Let X be a Gaussian random field with an almost star-scale covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5) Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' In the proof of Theorems B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2 presented below, we are going to assume that our probability space contains a martingale sequence pXtqtě0 of fields with co- variance (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3) approximating X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' For reasons analogous to the one exposed at the beginning of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2 this entails no loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The case of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Let us fix ρ P C8 c pRdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We want to prove that Mγ ε pρ ¨q converges in Hs,ppRdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We first define the limit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We set (without underlying the dependence in ρ to keep the notation light) x Mγ ε pξq “ ż Rd ρpxqeiξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='xeγXεpxq´ γ2 2 Kεpxqdx, x Mγ 8pξq “ lim εÑ0 x Mγ ε pξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6) We let Mγ 8pρ ¨q denote the random distribution whose Fourier transform is given by x Mγ 8pξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The proof of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4), implies that Mγ 8pρ ¨q P Hs,ppRdq with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' To prove (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8) and we can then conclude using dominated convergence (the first line yields the domina- tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The second line is simply (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2) with fpxq “ ρpxqeiξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We set V pεq t pξq “ Erx Mγ ε | Fts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Using the BDG inequality we have E ” |x Mγ ε pξq|pı ď CE ” |V pεq 0 pξq|p ` xV pεqpξqyp{2 8 ı .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9) Now we have (recall that p ă ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d{α ď 2) E ” |V pεq 0 pξq|2ı “ ż R2d ρpxqρpyqeiξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='px´yqe|γ|2K0,εpx,yqdxdy (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='10) CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES41 and we can conclude by replacing eiξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' For the quadratic variation part, we have xV pεqy8 “ |γ|2 ż 8 0 At,εpξqdt, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) the inequality being obtain by taking the modulus of the integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' To conclude, we just need to prove that sup εPp0,1q E «ˆż 8 0 At,εdt ˙p{2ff ă 8 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='13) For this part we can just repeat the computations made to prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='23) in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The case of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' For this, we simply replicate the strategy presented in [16], with a minor twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Keeping the notation introduced in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6) for the Fourier transform, we are going to prove the following analogue of [16, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2] (we use the notation vpεq for vpε, θ, γq for ease of reading) Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='14) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We introduce a martingale whose limit coincides with x Mγ ε pξq on the event Aq,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Given x P Rd and q ą 0 we set Tqpxq :“ inftt ą 0 : Xtpxq “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dt ` qu, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='15) and define N pεq t pξq :“ ż Rd ρpxqeiξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='xeγXt^Tqpxq,εpxq´ γ2 2 Kt^Tqpxq,εpxqdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='18) Let us prove the only second inequality, since the first one is only easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We set for simplicity Wt :“ N pεq t pξ ` aq ´ N pεq t pξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We have E ” |N pεq 8 pξ ` aq ´ N pεq 8 pξq|2ı “ Er|W8|2s “ Er|W0|2s ` E rxWy8s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We are going to prove a bound for each of the term in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We have Er|W0|2s “ ż R2d ρpxqρpyq ´ eipξ`aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='x ´ eiξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='x¯ ´ e´ipξ`aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='y ´ eiξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='y¯ e|γ|2K0,εpx,yq ď C|a|2 ż ρpxqρpyq|x||y|dxdy ď C1|a|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='19) where in the second line have taken the modulus of the integrand, and used the fact that the complex exponential is Lipshitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' To bound the expected value of the quadratic variation, using Itˆo calculus, and observing that tTqpxq ă tu “ At,qpxq (recall (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6)) we obtain that xWy8 “ |γ|2 ż 8 0 Utdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='20) where Ut :“ ż R2d ρpxqρpyqQt,εpx, yq ´ eipξ`aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='x ´ eiξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='x¯ ´ e´ipξ`aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='y ´ eiξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='y¯ ˆ eγXt,εpxq`γXt,εpyq´ γ2 2 Kt,εpxq´ γ2 2 Kt,εpyq1At,qpxqXAt,qpyqdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXt,εpxq`p|γ|2´dqKt,εpxq1At,qpxqdxdy, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='22) where the second line is obtained via the same step as (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='26) (ab ď a2`b2{2 and symmetry and in x and y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Now recalling (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='25) we have E ” e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXt,εpxq´dKt,εpxq1At,qpxq ı ď Cpt _ 1q´1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='23) where to obtain the first inequality, we used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) to show that Kspx, yq (and all similar terms) are well estimated by t for s P r0, ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Now we have E rUts ď C|a|2e´dt ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' t _ 1 ż R2d ρpxq2|x|2Qt,εpx, yqe|γ|2Kt,εpx,yqdx ď C1|a|2φpt, εq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='24) CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES43 After integrating with respect to t (recalling Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4) we obtain that ErxWy8s ď C|a|2vpεq2, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='25) for a constant C which is independent of ε and ξ and a, which combined with (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='19), concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' □ Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Beyond star-scale invariance The assumption that the kernel can be written in the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1), the mollified field Xε can be defined on Dε :“ tx P D : inf yPDA |x ´ y| ą 2εu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' More precisely in that case the field X is indexed by CcpDq the set of functions with compact support on D (in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4), R2d is replaced by D2), and Xε remains defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5) (here θεpx ´ ¨q, which for x P Dε, has its support included in D, is identified with its restriction on D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1) where K is an almost-star scale invariant Kernel, and K0 : U 2 Ñ R is positive definite and H¨older continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We state this result as a proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' It can be directly derived from [12, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' If K is a positive definite kernel on D that can be written in the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' To extend Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' This is the content of the following result, [16, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1] Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Given K a covariance kernel on D of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1) such that (A) For all x, y P U, |Kpδqpx, yq ´ Kpx, yq| ď δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (B) ∆pδqpx, yq “ Kpδqpx, yq ´ Kpx, yq is a positive definite kernel on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Remark C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' More precisely, [16, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1] states that one can chose η1 “ 0 (recall (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8)) for the almost-star scale invariant part of Kpδq, but this refinement is not required for our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 44 HUBERT LACOIN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The case of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The extension of the result to the case of a general log-correlated field defined on a domain D is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' If X is a centered Gaussian field defined on D whose covariance kernel K can be written in the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d{α ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' lim εÑ0 Mγ ε pfq “ Mγ 8pfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' This follows quite immediately via a localization argument using a partition of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Let f P CcpDq be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Using Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Using a partition of the unity, we can write f :“ ř iPI fi where fi is continuous with compact support included in Bpzi, εiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' □ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The case of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' To extend the result for γ P P1 II{III it is sufficient to extend Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2] for a proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' If X is a centered Gaussian field defined on D whose covariance kernel K can be written in the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 \uf6be “ E „ eixX,ρy´ 1 2M1pe|γ|2L|f|2q \uf6be .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2 with δ “ 1{n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Note that Xn has covariance Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We let Mγ,n ε and M 1 n denote the mollified GMC and critical GMC associated with Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' For simplicity, we define all the pZnqně1 on the same probability space: the fields Zn form an independent sequence which is independent of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We let P denote the corresponding probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' From Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8 we have for each n ě 1 lim εÑ0 E „ eixXn,ρy`i Mγ,n ε pf,ωq vpε,θ,γq \uf6be “ E „ eixXn,ρy´ 1 2 M 1 npe|γ|2Ln|f|2q \uf6be , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4) where Ln :“ L ` ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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 \uf6be ´ E „ eixX,ρy`i Mγ ε pf,ωq vpε,θ,γq \uf6beˇˇˇˇ “ 0 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='5) and that lim nÑ8 E „ eixXn,ρy´ 1 2M 1 npe|γ|2Ln|f|2q \uf6be “ E „ eixX,ρy´ 1 2 M 1pe|γ|2Lpδq|f|2q \uf6be .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' and the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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 \uf6be “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='7) The first line is immediate via the computation of the L2 norm (the convergence holds in L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' For the second line, we set gn “ e|γ|2Ln|f|2 and g “ e|γ|2L|f|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Using the notational convention introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1, we let Zn,ε denote the mollification of Zn and ∆n,ε its covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Letting e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dZn,ε´d∆n,ε denote the function x ÞÑ e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dZn,εpxq´d∆n,εpxq we have E „´ M ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d,n ε pgnq ´ M ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d ε pgq ¯2 | X \uf6be “ E ” M ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d ε pe ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dZn,ε´d∆n,εgn ´ gq2 | X ı “ ż U2 ´ e2d∆n,εpx,yqgnpxqgnpyq ´ 2gnpxqgpyq ` gpxqgpyq ¯ M ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d ε pdxqM ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d ε pdyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d,n ε pgnq ´ M ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d ε pgq ¯2 | X \uf6be ď C n M ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2d ε pgq2 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9) Using Fatou after renormalization we obtain that E ”` M1 npgnq ´ M1pgq ˘2 | X ı ď C n M1pgq2 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='10) which implies the second line in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' For the third line, we are going to Proposition (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We are going to prove that for each i P I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' lim nÑ8 sup εPp0,1q E „ˇˇˇˇ Mγ,n ε pfi, ωq vpε, θ, γq ´ Mγ ε pfi, ωq vpε, θ, γq ˇˇˇˇ _ 1 \uf6be “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='11) This operation shows that it is in fact sufficient to prove the third line of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='7) assuming that K is an almost star-scale invariant Kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We can thus further our probability space contains pXtqtě0 a martingale sequence of fields with covariance Kt (we adopt the notation of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1) approximating X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We equip our space with the filtration Gt :“ σppXsqsPr0,ts, pZnqně1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We recall the definition of Tqpxq in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='15) and define W pn,εq t :“ ż U peγZn,εpxq´ γ2 2 ∆n,εpxq ´ 1qfpxqeγXt^Tqpxq,ε´ γ2 2 Kt^Tq,εpxqdx (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) Setting Aq :“ t@x P Supppfq, @t ą 0, Xtpxq ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='13) Since from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2 we have limqÑ8 PrAqs “ 1, to prove the third line in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The derivative of the bracket of W pn,εq is given by |γ|2 times (recall that by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='16) with Gn,εpxq “ peγZn,εpxq´ γ2 2 ∆n,εpxq ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Repeating once more the computation in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='26) we obtain that Dt ď ż R2d Qt,εpx, yq|Gn,εpxq|2e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 2dXt,ε`p|γ|2´dqKt,εpxq1At,qpxqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='23) we obtain that ErDts ď Cn´1 ż R2d Qt,εpx, yqe|γ|2Kt,εpxqdxdy ď C1n´1φpt, εq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='19) Integrating against t we conclude that ErxW pn,εqy8s ď Cn´1vpε, θ, γq2 and this concludes the proof of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' □ CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES47 Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We assume without loss of generality that Supppfq Ă r0, 1sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' For x, y P r0, 1sd we let 2´kpx,yq be the sidelength of the smallest dyadic cube that contains x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' kpx, yq :“ inf !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' n ě 0 : Dm P �0, 2n ´ 1�d, tx, yu Ă ´ 2´nm ` r0, 2´nqd¯) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' and set knpx, yq :“ kpx, yq ^ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Note that kn defines a positive definite function and that kpx, yq ď log2 ´ 1 |x´y| ¯ ` C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Hence from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1 there exists a constant A ą 0 such that plog 2qknpx, yq ď Krn log 2spx, yq ` A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1) Using Kahane’s convexity inequality (proved in [15] see also [27, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1]) which we introduce in a simplified setup Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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 ˙\uf6be ď E „ F ˆż eY2pxq´ θ2 2 C2pxqµpdxq ˙\uf6be .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2) Hence if Zn denotes a field defined on r0, 1sd with covariance kn we can apply Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1 result for the fields ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='log 2Zn and Xrn log 2s` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' AN where N is an independent standard Gaussian (the fields have their resepective covariances given by the two sides of Equation (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='1)), µpdxq “ |fpxq|2dx and Fpuq “ up{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Recalling (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='11) we have E ” Bp{2 rn log 2s ı ď CE » – ˜ 2dn ż r0,1sd |fpxq|2e2α?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='log 2pZnpxq´?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2d log 2nqdx ¸p{2fi fl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3) The constant C above takes care of the fact that the variance of ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='log 2Znpxq and Xrn log 2s differ by a Op1q term, and also of the moment of the variable N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We can ignore the constant f at the cost of a prefactor }f}p 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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´?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2d log 2nqdx “ ÿ mP�0,2n´1�d eζpZnpm2´nq´?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2d log 2nq, (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4) for ζ “ 2α?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The following result is a particular case of [8, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We present a shorter proof which is valid in our context for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Given ζ ą ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2d log 2 and q ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2d log 2 ζ there exists positive constant C and b such that E rpWn,ζqqs ď Cn´ 3qζ 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2d log 2plog nq6 48 HUBERT LACOIN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' We split our integral in three parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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´?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2d log 2nq1Inpxqdx (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='6) Using subadditivity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9) we have E rpWn,ζqqs ď E rWn,ζpAqqs ` E rWn,ζpBqqs ` E rWn,ζpCqqs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='7) We are going to show that the two last terms in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' decay faster than any negative power of n and then prove a bound of the right order of magnitude for E rpWn,ζqqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Letting setting Bn :“ Ť xPr0,1s Bnpxq, and q1 “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='8) Using subadditivity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9) for the sum (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='4) with θ “ q1, E ” pWn,ζqq1ı ď E “ Wn,?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2d log 2 ‰ “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9) The inequality on the right comes from the fact that pWm,?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2d log 2qmě1 is a martingale for the natural filtration associated with Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Using the optional stopping Theorem for this same martingale, we can obtain a bound for the probability of Bn, PrBns ď P ” Dm, Wm,?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2d log 2 ě e ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2 log 2plog nq2ı ď e´?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2 log 2plog nq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='10) This yields a subpolynomial decay for ErWn,ζpBqqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' 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´?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2d log 2nq1tZnď?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2d log 2n´plog nq2u ı “ e ˆ d log 2` ζ2 2 ´ζ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2d log 2 ˙ n P ” Znp0q ď p a 2d log 2 ´ ζqn ´ plog nq2ı ď ep?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2d log 2´ζqplog nq2, (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='11) also proving a subpolynomial decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' It remains to estimate the main part E rpWn,ζpAqqqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Using first subaddivity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='9) and then Jensen’s inequality E rpWn,ζpAqqqs ď E „ Wn,?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2d log 2pAq qζ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2d log 2 \uf6be ď E “ Wn,?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2d log 2pAq ‰ qζ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2d log 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='12) The Cameron-Martin formula directly expresses E “ Wn,?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2d log 2pAq ‰ as the probability con- cerning the Gaussian centered random walk pZmp0qqmě0, E “ Wn,?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='2d log 2pAq ‰ “ P “ @m P �1, n�, Zmp0q ď plog nq2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Znp0q ě ´plog nq2‰ ď Cn´3{2plog nq6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content='3]) which concludes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' □ References [1] Elie A¨ıd´ekon and Zhan Shi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Weak convergence for the minimal position in a branching random walk: a simple proof.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Evans, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Speer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Mean field theory of directed polymers with random complex weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=', 156(2):221–244, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' [4] Bertrand Duplantier, R´emi Rhodes, Scott Sheffield, and Vincent Vargas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Critical Gaussian multiplica- tive chaos: convergence of the derivative martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=', 42(5):1769–1808, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' [5] Bertrand Duplantier, R´emi Rhodes, Scott Sheffield, and Vincent Vargas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Renormalization of critical Gaussian multiplicative chaos and KPZ relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=', 330(1):283–330, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' [6] Lisa Hartung and Anton Klimovsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' The glassy phase of the complex branching Brownian motion energy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'} +page_content=' Commun.' metadata={'source': 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