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作者：El Karoui, Noureddine
作者单位：University of California System; University of California Berkeley
摘要：It has been recently shown that if X is an n x N matrix whose entries are i.i.d. standard complex Gaussian and l(1) is the largest eigenvalue of X*X, there exist sequences m(n,N) and s(n,N) such that (l(1) - m(n,N))/s(n,N) converges in distribution to W-2, the Tracy-Widom law appearing in the study of the Gaussian unitary ensemble. This probability law has a density which is known and computable. The cumulative distribution function of W2 is denoted F-2. In this paper we show that, under the a...
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作者：Chatterjee, Sourav
作者单位：University of California System; University of California Berkeley
摘要：We generalize Lindeberg's proof of the central limit theorem to an invariance principle for arbitrary smooth functions of independent and weakly dependent random variables. The result is applied to get a similar theorem for smooth functions of exchangeable random variables. This theorem allows us to identify, for the first time, the limiting spectral distributions of Wigner matrices with exchangeable entries.
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作者：Gnedin, Alexander V.; Yakubovich, Yuri
作者单位：Utrecht University
摘要：A class of random discrete distributions P is introduced by means of a recursive splitting of unity. Assuming supercritical branching, we show that for partitions induced by sampling from such P a power growth of the number of blocks is typical. Some known and some new partition structures appear when P is induced by a Dirichlet splitting.
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作者：Marckert, Jean-Francois; Mokkadem, Abdelkader
作者单位：Universite Paris Saclay
摘要：Consider q(n) a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with n faces. In this paper we show that, when n goes to +infinity, qn suitably normalized converges weakly in a certain sense to a random limit object, which is continuous and compact, and that we name the Brownian map. The same result is shown for a model of rooted quadrangulations and for some models of rooted quadrangulations with random edge lengths. A metric space of rooted (resp. poin...
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作者：Bally, Vlad
作者单位：Universite Gustave-Eiffel; Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
摘要：We give lower bounds for the density p(T) (x, y) of the law of X-t, the solution of dX(t) = sigma(X-t)dB(t) +b(X-t)dt, X-0 = x, under the following local ellipticity hypothesis: there exists a deterministic differentiable curve x(t), 0 <= t <= T, such that x(0) = x, x(T) = y and sigma sigma* (x(t)) > 0, for all t is an element of [0, T]. The lower bound is expressed in terms of a distance related to the skeleton of the diffusion process. This distance appears when we optimize over all the curv...
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作者：Gaertner, J.; den Hollander, F.
作者单位：Technical University of Berlin; Leiden University; Leiden University - Excl LUMC
摘要：In this paper, we study intermittency for the parabolic Anderson equation partial derivative u/partial derivative t = K Delta u + xi u, where u: Z(d) x [0, infinity) -> R, K is the diffusion constant, Delta is the discrete Laplacian and xi : Z(d) X [0, infinity) -> R is a space-time random medium. We focus on the case where xi is gamma times the random medium that is obtained by running independent simple random walks with diffusion constant rho starting from a Poisson random field with intens...
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作者：Rider, B.; Silverstein, Jack W.
作者单位：University of Colorado System; University of Colorado Boulder; North Carolina State University
摘要：Consider an ensemble of N x N non-Hermitian matrices in which all entries are independent identically distributed complex random variables of mean zero and absolute mean-square one. If the entry distributions also possess bounded densities and finite (4 + epsilon) moments, then Z. D. Bai [Ann. Probab. 25 (1997) 494-529] has shown the ensemble to satisfy the circular law: after scaling by a factor of 1/root N and letting N -> infinity, the empirical measure Of the eigenvalues converges weakly t...
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作者：Timar, Adam
作者单位：Indiana University System; Indiana University Bloomington
摘要：We consider Bernoulli percolation on a locally finite quasi-transitive unimodular graph and prove that two infinite clusters cannot have infinitely many pairs of vertices at distance I from one another or, in other words, that such graphs exhibit cluster repulsion. This partially answers a question of Haggstrom, Peres and Schonmann.
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作者：Timar, Adam
作者单位：Indiana University System; Indiana University Bloomington
摘要：We extend some of the fundamental results about percolation on unimodular nonamenable graphs to nonunimodular graphs. We show that they cannot have infinitely many infinite clusters at critical Bernoulli percolation. In the case of heavy clusters, this result has already been established, but it also follows from one of our results. We give a general necessary condition for nonunimodular graphs to have a phase with infinitely many heavy clusters. We present an invariant spanning tree with p(c)...
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作者：Burdzy, Krzysztof; Quastel, Jeremy
作者单位：University of Washington; University of Washington Seattle; University of Toronto
摘要：We consider two species of particles performing random walks in a domain in R-d with reflecting boundary conditions, which annihilate on contact. In addition, there is a conservation law so that the total number of particles of each type is preserved: When the two particles of different species annihilate each other, particles of each species, chosen at random, give birth. We assume initially equal numbers of each species and show that the system has a diffusive scaling limit in which the dens...
