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作者：Capitaine, Mireille; Peche, Sandrine
作者单位：Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Centre National de la Recherche Scientifique (CNRS); Universite Paris Cite
摘要：We consider a full rank deformation of the GUE where is a full rank Hermitian matrix of size and is a GUE. The empirical eigenvalue distribution of converges to a probability distribution . We identify all the possible limiting eigenvalue statistics at the edges of the spectrum, including outliers, edges and merging points of connected components of the limiting spectrum. The results are stated in terms of a deterministic equivalent of the empirical eigenvalue distribution of , namely the free...
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作者：Herdade, Simo; Vu, Van
作者单位：Rutgers University System; Rutgers University New Brunswick; Yale University
摘要：As an extension of Polya's classical result on random walks on the square grids (), we consider a random walk where the steps, while still have unit length, point to different directions. We show that in dimensions at least 4, the returning probability after n steps is at most , which is sharp. The real surprise is in dimensions 2 and 3. In dimension 2, where the traditional grid walk is recurrent, our upper bound is , which is much worse than in higher dimensions. In dimension 3, we prove an ...
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作者：De Santis, Emilio; Maffei, Andrea
作者单位：Sapienza University Rome; University of Pisa
摘要：In this article we create a new algorithm for the perfect simulation of the infinite random cluster model for a sufficiently small or a sufficiently high value of the parameters. This implies the simulation of the Ising and Potts models with free boundary conditions.
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作者：Marahrens, Daniel; Otto, Felix
作者单位：Max Planck Society
摘要：We consider a random, uniformly elliptic coefficient field on the -dimensional integer lattice . We are interested in the spatial decay of the quenched elliptic Green function . Next to stationarity, we assume that the spatial correlation of the coefficient field decays sufficiently fast to the effect that a logarithmic Sobolev inequality holds for the ensemble . We prove that all stochastic moments of the first and second mixed derivatives of the Green function, that is, and , have the same d...
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作者：Gorin, Vadim; Shkolnikov, Mykhaylo
作者单位：Massachusetts Institute of Technology (MIT); Russian Academy of Sciences; Kharkevich Institute for Information Transmission Problems of the RAS; University of California System; University of California Berkeley
摘要：We introduce multilevel versions of Dyson Brownian motions of arbitrary parameter , generalizing the interlacing reflected Brownian motions of Warren for . Such processes unify corners processes and Dyson Brownian motions in a single object. Our approach is based on the approximation by certain multilevel discrete Markov chains of independent interest, which are defined by means of Jack symmetric polynomials. In particular, this approach allows to show that the levels in a multilevel Dyson Bro...
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作者：Mossel, Elchanan; Neeman, Joe; Sly, Allan
作者单位：University of California System; University of California Berkeley; University of California System; University of California Berkeley; Australian National University
摘要：The planted partition model (also known as the stochastic blockmodel) is a classical cluster-exhibiting random graph model that has been extensively studied in statistics, physics, and computer science. In its simplest form, the planted partition model is a model for random graphs on nodes with two equal-sized clusters, with an between-class edge probability of and a within-class edge probability of . Although most of the literature on this model has focused on the case of increasing degrees (...
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作者：Aimino, Romain; Nicol, Matthew; Vaienti, Sandro
作者单位：Centre National de la Recherche Scientifique (CNRS); CNRS - Institute of Physics (INP); Aix-Marseille Universite; Aix-Marseille Universite; Centre National de la Recherche Scientifique (CNRS); CNRS - Institute of Physics (INP); Universite de Toulon; University of Houston System; University of Houston
摘要：In this paper, we investigate annealed and quenched limit theorems for random expanding dynamical systems. Making use of functional analytic techniques and more probabilistic arguments with martingales, we prove annealed versions of a central limit theorem, a large deviation principle, a local limit theorem, and an almost sure invariance principle. We also discuss the quenched central limit theorem, dynamical Borel-Cantelli lemmas, Erdos-R,nyi laws and concentration inequalities.
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作者：Le Gall, Jean-Francois
作者单位：Universite Paris Saclay
摘要：We study properties of the random metric space called the Brownian map. For every , we consider the connected components of the complement of the open ball of radius centered at the root, and we let be the number of those connected components that intersect the complement of the ball of radius . We then prove that converges as to a constant times the density at of the profile of distances from the root. In terms of the Brownian cactus, this gives asymptotics for the number of points at height ...
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作者：Kiss, Demeter
作者单位：University of Cambridge; Tohoku University
摘要：Aldous (Math Proc Camb Philos Soc 128:465-477, 2000) introduced a modification of the bond percolation process on the binary tree where clusters stop growing (freeze) as soon as they become infinite. We investigate the site version of this process on the triangular lattice where clusters freeze as soon as they reach diameter at least for some parameter . We show, informally speaking, that in the limit , the clusters only freeze in the critical window of site percolation on the triangular latti...
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作者：Damron, Michael; Hanson, Jack; Sosoe, Philippe
作者单位：Princeton University; Princeton University
摘要：We prove that the variance of the passage time from the origin to a point in first-passage percolation on is sublinear in the distance to when , obeying the bound , under minimal assumptions on the edge-weight distribution. The proof applies equally to absolutely continuous, discrete and singular continuous distributions and mixtures thereof, and requires only moments. The main result extends work of Benjamini-Kalai-Schramm (Ann Prob 31, 2003) and Benaim-Rossignol (Ann Inst Henri Poincar, Prob...
