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作者：Tao, Terence; Vu, Van
作者单位：University of California System; University of California Los Angeles; Rutgers University System; Rutgers University New Brunswick
摘要：We study the eigenvalues of the covariance matrix 1/n M*M of a large rectangular matrix M = M-n,M-p = (zeta(ij))(1 <= i <= p;1 <= j <= n) whose entries are i.i.d. random variables of mean zero, variance one, and having finite C(0)th moment for some sufficiently large constant C-0. The main result of this paper is a Four Moment theorem for i.i.d. covariance matrices (analogous to the Four Moment theorem for Wigner matrices established by the authors in [Acta Math. (2011) Random matrices: Univer...
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作者：Carraro, Laurent; El Karoui, Nicole; Obloj, Jan
作者单位：Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Sorbonne Universite; Universite Paris Cite; University of Oxford
摘要：We study the class of Azema-Yor processes defined from a general semi-martingale with a continuous running maximum process. We show that they arise as unique strong solutions of the Bachelier stochastic differential equation which we prove is equivalent to the drawdown equation. Solutions of the latter have the drawdown property: they always stay above a given function of their past maximum. We then show that any process which satisfies the drawdown property is in fact an Azema-Yor process. Th...
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作者：Pitman, Jim; Bravo, Geronimo Uribe
作者单位：University of California System; University of California Berkeley
摘要：We offer a unified approach to the theory of convex minorants of Levy processes with continuous distributions. New results include simple explicit constructions of the convex minorant of a Levy process on both finite and infinite time intervals, and of a Poisson point process of excursions above the convex minorant up to an independent exponential time. The Poisson Dirichlet distribution of parameter 1 is shown to be the universal law of ranked lengths of excursions of a Levy process with cont...
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作者：Damron, Michael; Sapozhnikov, Artem
作者单位：Princeton University; Swiss Federal Institutes of Technology Domain; ETH Zurich
摘要：We prove limit theorems and variance estimates for quantities related to ponds and outlets for 2D invasion percolation. We first exhibit several properties of a sequence (O(n)) of outlet variables, the nth of which gives the number of outlets in the box centered at the origin of side length 2(n). The most important of these properties describes the sequence's renewal structure and exponentially fast mixing behavior. We use these to prove a central limit theorem and strong law of large numbers ...
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作者：Ioffe, Dmitry; Velenik, Yvan
作者单位：Technion Israel Institute of Technology; University of Geneva
摘要：We consider a model of a polymer in Z(d+1), constrained to join 0 and a hyperplane at distance N. The polymer is subject to a quenched nonnegative random environment. Alternatively, the model describes crossing random walks in a random potential (see Zerner [Ann Appl. Probab. 8 (1998) 246-280] or Chapter 5 of Sznitman [Brownian Motion, Obstacles and Random Media (1998) Springer] for the original Brownian motion formulation). It was recently shown [Ann. Probab. 36 (2008) 1528-1583; Probab. Theo...
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作者：Haas, Benedicte; Miermont, Gregory
作者单位：Universite PSL; Universite Paris-Dauphine; Universite Paris Saclay
摘要：We consider a family of random trees satisfying a Markov branching property. Roughly, this property says that the subtrees above some given height are independent with a law that depends only on their total size, the latter being either the number of leaves or vertices. Such families are parameterized by sequences of distributions on partitions of the integers that determine how the size of a tree is distributed in its different subtrees. Under some natural assumption on these distributions, s...
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作者：Alsmeyer, Gerold; Biggins, J. D.; Meiners, Matthias
作者单位：University of Munster; University of Sheffield; Uppsala University
摘要：Given a sequence T = (T-i)(i >= 1) of nonnegative random variables, a function f on the positive halfline can be transformed to E Pi(i >= 1) f (tT(i)). We study the fixed points of this transform within the class of decreasing functions. By exploiting the intimate relationship with general branching processes, a full description of the set of solutions is established without the moment conditions that figure in earlier studies. Since the class of functions under consideration contains all Lapl...
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作者：Hairer, Martin; Maas, Jan
作者单位：University of Warwick; University of Bonn
摘要：We consider a class of stochastic PDEs of Burgers type in spatial dimension 1, driven by space-time white noise. Even though it is well known that these equations are well posed, it turns out that if one performs a spatial discretization of the nonlinearity in the wrong way, then the sequence of approximate equations does converge to a limit, but this limit exhibits an additional correction term. This correction term is proportional to the local quadratic cross-variation (in space) of the grad...
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作者：Menshikov, Mikhail; Popov, Serguei; Ramirez, Alejandro F.; Vachkovskaia, Marina
作者单位：Durham University; Universidade Estadual de Campinas; Pontificia Universidad Catolica de Chile
摘要：In this paper we study a substantial generalization of the model of excited random walk introduced in [Electron. Commun. Probab. 8 (2003) 86-92] by Benjamini and Wilson. We consider a discrete-time stochastic process (X-n, n = 0, 1, 2, ...) taking values on Z(d), d >= 2, described as follows: when the particle visits a site for the first time, it has a uniformly-positive drift in a given direction l; when the particle is at a site which was already visited before, it has zero drift. Assuming u...
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作者：O'Connell, Neil
作者单位：University of Warwick
摘要：We characterize the law of the partition function of a Brownian directed polymer model in terms of a diffusion process associated with the quantum Toda lattice. The proof is via a multidimensional generalization of a theorem of Matsumoto and Yor concerning exponential functionals of Brownian motion. It is based on a mapping which can be regarded as a geometric variant of the RSK correspondence.
