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作者：Bauerschmidt, Roland; Helmuth, Tyler; Swan, Andrew
作者单位：University of Cambridge; University of Bristol
摘要：We prove the vertex-reinforced jump process (VRJP) is recurrent in two dimensions for any translation invariant finite-range initial rates. Our proof has two main ingredients. The first is a direct connection between the VRJP and sigma models whose target space is a hyperbolic space H-n or its super-symmetric counterpart H-2 broken vertical bar 2. These results are analogues of well-known relations between the Gaussian free field and the local times of simple random walk. The second ingredient...
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作者：Fan, Jianqing; Jiang, Tiefeng
作者单位：Princeton University; University of Minnesota System; University of Minnesota Twin Cities
摘要：The paper studies the limiting distribution of the largest off-diagonal entry of the sample correlation matrices of high-dimensional Gaussian populations with equi-correlation structure. Assume the entries of the population distribution have a common correlation coefficient rho > 0 and both the population dimension p and the sample size n tend to infinity with log p = o(n(1/3)). As 0 < rho < 1, we prove that the largest off-diagonal entry of the sample correlation matrix converges to a Gaussia...
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作者：He, Yukun; Knowles, Antti; Marcozzi, Matteo
作者单位：University of Zurich; University of Geneva
摘要：We prove a local law for the adjacency matrix of the Erdos-Renyi graph G(N, p) in the supercritical regime pN >= C logN where G(N, p) has with high probability no isolated vertices. In the same regime, we also prove the complete delocalization of the eigenvectors. Both results are false in the complementary subcritical regime. Our result improves the corresponding results from (Ann. Probab. 41 (2013) 2279-2375) by extending them all the way down to the critical scale pN = O(logN). A key ingred...
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作者：Patie, Pierre; Savov, Mladen; Zhao, Yixuan
作者单位：Cornell University; Bulgarian Academy of Sciences
摘要：In this paper, we start by showing that the intertwining relationship between two minimal Markov semigroups acting on Hilbert spaces implies that any recurrent extensions, in the sense of Ito, of these semigroups satisfy the same intertwining identity. Under mild additional assumptions on the intertwining operator, we prove that the converse also holds. This connection, which relies on the representation of excursion quantities as developed by Fitzsimmons and Getoor (Illinois J. Math. 50 (2006...
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作者：Soo, Terry; Wilkens, Amanda
作者单位：University of Kansas
摘要：As part of a general theory for the isomorphism problem for actions of amenable groups, Ornstein and Weiss (J. Anal. Math. 48 (1987) 1-141) proved that any two Poisson point processes are isomorphic as measure-preserving actions. We give an elementary construction of an isomorphism between Poisson point processes that is finitary.
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作者：Lambert, Gaultier; Ledoux, Michel; Webb, Christian
作者单位：University of Zurich; Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Institut Universitaire de France; Aalto University
摘要：We present a new approach, inspired by Stein's method, to prove a central limit theorem (CLT) for linear statistics of beta-ensembles in the one-cut regime. Compared with the previous proofs, our result requires less regularity on the potential and provides a rate of convergence in the quadratic Kantorovich or Wasserstein-2 distance. The rate depends both on the regularity of the potential and the test functions, and we prove that it is optimal in the case of the Gaussian Unitary Ensemble (GUE...
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作者：Rhodes, Remi; Vargas, Vincent
作者单位：Aix-Marseille Universite; Universite PSL; Ecole Normale Superieure (ENS); Universite PSL; Ecole Normale Superieure (ENS)
摘要：In this short note we derive a precise tail expansion for Gaussian multiplicative chaos (GMC) associated to the 2d Gaussian Free Fiedl (GFF) on the unit disk with zero average on the unit circle (and variants). More specifically, we show that to first order the tail is a constant times an inverse power with an explicit value for the tail exponent as well as an explicit value for the constant in front of the inverse power; we also provide a second order bound for the tail expansion. The main in...
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作者：Hutchcroft, Tom
作者单位：University of Cambridge
摘要：We study self-avoiding walk on graphs whose automorphism group has a transitive nonunimodular subgroup. We prove that self-avoiding walk is ballistic, that the bubble diagram converges at criticality, and that the critical two-point function decays exponentially in the distance from the origin. This implies that the critical exponent governing the susceptibility takes its mean-field value, and hence that the number of self-avoiding walks of length n is comparable to the nth power of the connec...
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作者：Alexander, Yotam; Eldan, Ronen
作者单位：Weizmann Institute of Science
摘要：We show that the convex hull of the path of Brownian motion in n-dimensions, up to time 1, is a smooth set. As a consequence we conclude that a Brownian motion in any dimension almost surely has no cone points for any cone whose dual cone is nontrivial.
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作者：Bae, Joohak; Kang, Jaehoon; Kim, Panki; Lee, Jaehun
作者单位：Seoul National University (SNU); University of Bielefeld; Seoul National University (SNU)
摘要：In this paper, we study the transition densities of pure-jump symmetric Markov processes in R-d, whose jumping kernels are comparable to radially symmetric functions with mixed polynomial growths. Under some mild assumptions on their scale functions, we establish sharp two-sided estimates of the transition densities (heat kernel estimates) for such processes. This is the first study on global heat kernel estimates of jump processes (including non-Levy processes) whose weak scaling index is not...
