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作者：Sarkar, Sourav; Roy, Parthanil
作者单位：University of California System; University of California Berkeley; Indian Statistical Institute; Indian Statistical Institute Bangalore
摘要：In this work, we investigate the extremal behaviour of left-stationary symmetric alpha-stable (S alpha S) random fields indexed by finitely generated free groups. We begin by studying the rate of growth of a sequence of partial maxima obtained by varying the indexing parameter of the field over balls of increasing size. This leads to a phase-transition that depends on the ergodic properties of the underlying nonsingular action of the free group but is different from what happens in the case of...
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作者：Barbour, A. D.; Luczak, M. J.; Xia, A.
作者单位：University of Zurich; University of London; Queen Mary University London; University of Melbourne
摘要：The paper applies the theory developed in Part I to the discrete normal approximation in total variation of random vectors in Z(d). We illustrate the use of the method for sums of independent integer valued random vectors, and for random vectors exhibiting an exchangeable pair. We conclude with an application to random colourings of regular graphs.
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作者：Collamore, Jeffrey F.; Mentemeier, Sebastian
作者单位：University of Copenhagen; Dortmund University of Technology; Dortmund University of Technology
摘要：We study the large exceedance probabilities and large exceedance paths of the recursive sequence V-n = MnVn-1 + Q(n), where {(M-n, Q(n))} is an i.i.d. sequence, and M-1 is a d x d random matrix and Q(1) is a random vector, both with nonnegative entries. We impose conditions which guarantee the existence of a unique stationary distribution for {V-n} and a Cramr-type condition for {M-n}. Under these assumptions, we characterize the distribution of the first passage time T-u(A) = inf{n : V-n is a...
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作者：Lyons, Russell
作者单位：Indiana University System; Indiana University Bloomington
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作者：Martinsson, Anders
作者单位：Chalmers University of Technology; University of Gothenburg
摘要：We consider first-passage percolation on the class of high-dimensional graphs that can be written as an iterated Cartesian product G square G square...square G of some base graph G as the number of factors tends to infinity. We propose a natural asymptotic lower bound on the first-passage time between (v, v,...,v) and (w, w,...,w) as n, the number of factors, tends to infinity, which we call the critical time t(G)*(v,w). Our main result characterizes when this lower bound is sharp as n ->infin...
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作者：Panchenko, Dmitry
作者单位：University of Toronto
摘要：Using the synchronization mechanism developed in the previous work on the Potts spin glass model, we obtain the analogue of the Parisi formula for the free energy in the mixed even p-spin models with vector spins, which include the Sherrington-Kirkpatrick model with vector spins interacting through their scalar product. As a special case, this also establishes the sharpness of Talagrand's upper bound for the free energy of multiple mixed p-spin systems coupled by constraining their overlaps.
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作者：Cook, Nicholas
作者单位：California State University System; California State University Los Angeles
摘要：We obtain lower tail estimates for the smallest singular value of random matrices with independent but nonidentically distributed entries. Specifically, we consider n x n matrices with complex entries of the form M = A o X + B = (a(ij) xi(ij) + b(ij)), where X = (xi(ij)) has i.i.d. centered entries of unit variance and A and B are fixed matrices. In our main result, we obtain polynomial bounds on the smallest singular value of M for the case that A has bounded (possibly zero) entries, and B = ...
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作者：Benjamini, Itai; Paquette, Elliot; Pfeffer, Joshua
作者单位：Weizmann Institute of Science; University System of Ohio; Ohio State University; Massachusetts Institute of Technology (MIT)
摘要：We show that a random walk on a stationary random graph with positive anchored expansion and exponential volume growth has positive speed. We also show that two families of random triangulations of the hyperbolic plane, the hyperbolic Poisson-Voronoi tessellation and the hyperbolic Poisson-Delaunay triangulation, have 1-skeletons with positive anchored expansion. As a consequence, we show that the simple random walks on these graphs have positive hyperbolic speed. Finally, we include a section...
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作者：Gorin, Vadim; Shkolnikov, Mykhaylo
作者单位：Massachusetts Institute of Technology (MIT); Kharkevich Institute for Information Transmission Problems of the RAS; Russian Academy of Sciences; Princeton University
摘要：We determine the operator limit for large powers of random symmetric tridiagonal matrices as the size of the matrix grows. The result provides a novel expression in terms of functionals of Brownian motions for the Laplace transform of the Airy beta process, which describes the largest eigen-values in the beta ensembles of random matrix theory. Another consequence is a Feynman-Kac formula for the stochastic Airy operator of Edelman-Sutton and Ramirez-Rider-Virag. As a side result, we find that ...
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作者：Cook, Nicholas; Goldstein, Larry; Johnson, Tobias
作者单位：Stanford University; University of Southern California
摘要：Let lambda be the second largest eigenvalue in absolute value of a uniform random d-regular graph on n vertices. It was famously conjectured by Alon and proved by Friedman that if d is fixed independent of n, then lambda = 2 root d - 1+ o(1) with high probability. In the present work, we show that lambda = O(root d) continues to hold with high probability as long as d = O(n(2/3)), making progress toward a conjecture of Vu that the bound holds for all 1 <= d <= n/2. Prior to this work the best ...
