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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：Goldstein, Larry; Xia, Aihua
作者单位：University of Southern California; University of Melbourne
摘要：We introduce a new family of distributions to approximate P(W epsilon A) for A subset of {...-2, -1, 0,1, 2...} and W a sum of independent integer-valued random variables xi(1), xi(2), ..., xi(n) with finite second moments, where, with large probability, W is not concentrated on a lattice of span greater than 1. The well-known Berry-Esseen theorem states that, for Z a normal random variable with mean E(W) and variance Var(W), P(Z epsilon A) provides a good approximation to P(W epsilon A) for A...
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作者：Gine, Evarist; Koltchinskii, Vladimir
作者单位：University of Connecticut; University of New Mexico
摘要：Let F be a class of measurable functions on a measurable space (S, S) with values in [0, 1] and let [GRAPHICS] be the empirical measure based on an i.i.d. sample (X-1,..., X-n) from a probability distribution P on (S, S). We study the behavior of suprema of the following type: [GRAPHICS] where sigma(p)f >= Var(p)(1/2) f and phi is a continuous, strictly increasing function with phi(0) = 0. Using Talagrand's concentration inequality for empirical processes, we establish concentration inequaliti...
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作者：Ben Arous, Gerard; Voiculescu, Dan Virgil
作者单位：New York University; University of California System; University of California Berkeley
摘要：Free probability analogs of the basics of extreme-value theory are obtained, based on Ando's spectral order. This includes classification of freely max-stable laws and their domains of attraction, using free extremal convolutions on the distributions. These laws coincide with the limit laws in the classical peaks-over-threshold approach. A free extremal projection-valued process over a measure-space is constructed, which is related to the free Poisson point process.
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作者：Lyons, Russell; Peres, Yuval; Schramm, Oded
作者单位：Indiana University System; Indiana University Bloomington; University of California System; University of California Berkeley; Microsoft
摘要：Minimal spanning forests on infinite graphs are weak limits of minimal spanning trees from finite subgraphs. These limits can be taken with free or wired boundary conditions and are denoted FMSF (free minimal spanning forest) and WMSF (wired minimal spanning forest), respectively. The WMSF is also the union of the trees that arise from invasion percolation started at all vertices. We show that on any Cayley graph where critical percolation has no infinite clusters, all the component trees in t...
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作者：Gravner, J; Griffeath, D
作者单位：University of California System; University of California Davis; University of Wisconsin System; University of Wisconsin Madison
摘要：We consider discrete-time random perturbations of monotone cellular automata (CA) in two dimensions. Under general conditions, we prove the existence of half-space velocities, and then establish the validity of the Wulff construction for asymptotic shapes arising from finite initial seeds. Such a shape converges to the polygonal invariant shape of the corresponding deterministic model as the perturbation decreases. In many cases, exact stability is observed. That is, for small perturbations, t...
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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：Asselah, Amine; Ferrari, Pablo A.
作者单位：Aix-Marseille Universite; Universidade de Sao Paulo
摘要：We consider a system of asymmetric independent random walks on Z(d), denoted by {eta(t), t is an element of R}, stationary under the product Poisson measure v(rho) of marginal density p > 0. We fix a pattern A, an increasing local event, and denote by tau the hitting time of A. By using a loss network representation of our system, at small density, we obtain a coupling between the laws of eta(t) conditioned on (tau > t) for all times t. When d >= 3, this provides bounds on the rate of converge...
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作者：Conger, Mark; Viswanath, D.
作者单位：University of Michigan System; University of Michigan
摘要：By a well-known result of Bayer and Diaconis, the maximum entropy model of the common riffle shuffle implies that the number of riffle shuffles necessary to mix a standard deck of 52 cards is either 7 or 11-with the former number applying when the metric used to define mixing is the total variation distance and the latter when it is the separation distance. This and other related results assume all 52 cards in the deck to be distinct and require all 52! permutations of the deck to be almost eq...
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作者：Jiang, Tiefeng
作者单位：University of Minnesota System; University of Minnesota Twin Cities
摘要：We solve an open problem of Diaconis that asks what are the largest orders of p(n) and q(n) such that Z(n), the p(n) x q(n) upper left block of a random matrix Gamma(n) which is uniformly distributed on the orthogonal group O(n), can be approximated by independent standard normals? This problem is solved by two different approximation methods. First, we show that the variation distance between the joint distribution of entries of Z(n) and that of p(n)q(n) independent standard normals goes to z...
