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作者：Tran, Hong Quan
作者单位：Universite PSL; Universite Paris-Dauphine
摘要：We consider the mean-field zero-range process in the regime where the potential function r is increasing to infinity at sublinear speed, and the den-sity of particles is bounded. We determine the mixing time of the system, and establish cutoff. We also prove that the Poincare constant is bounded away from zero and infinity. This mean-field estimate extends to arbitrary geome-tries via a comparison argument. Our proof uses the path-coupling method of Bubley and Dyer and stochastic calculus.
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作者：Genovese, Giuseppe
作者单位：University of Zurich
摘要：We study the free energy of a most used deep architecture for restricted Boltzmann machines, where the layers are disposed in series. Assuming inde-pendent Gaussian distributed random weights, we show that the error term in the so-called replica symmetric sum rule can be optimised as a saddle point. This leads us to conjecture that in the replica symmetric approximation the free energy is given by a min max formula, which parallels the one achieved for case.
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作者：Damron, Michael; Hanson, Jack; Lam, Wai -Kit
作者单位：University System of Georgia; Georgia Institute of Technology; City University of New York (CUNY) System; City College of New York (CUNY); National Taiwan University
摘要：We consider first-passage percolation (FPP) on the triangular lattice with vertex weights (t(v)) whose common distribution function F satisfies F(0) = 1/2. This is known as the critical case of FPP because large (critical) zeroweight clusters allow travel between distant points in time which is sublinear in the distance. Denoting by T (0, partial derivative B(n)) the first-passage time from 0 to {x : parallel to x parallel to(infinity) = n}, we show existence of a time constant and find its ex...
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作者：Heydecker, Daniel
作者单位：Max Planck Society
摘要：We consider the dynamic large deviation behaviour of Kac's collisional process for a range of initial conditions including equilibrium. We prove an upper bound with a rate function of the type which has previously been found for kinetic large deviation problems, and a matching lower bound restricted to a class of sufficiently good paths. However, we are able to show by an explicit counterexample that the predicted rate function does not extend to a global lower bound: even though the particle ...
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作者：Cosco, Clement; Nakajima, Shuta
作者单位：Weizmann Institute of Science; Meiji University
摘要：Consider the first passage percolation on the d-dimensional lattice Z(d) with identical and independent weight distributions and the first passage time T. In this paper, we study the upper tail large deviations P(T(0, nx) > n(mu + xi)), for xi > 0 and x not equal 0 with a time constant mu, for weights that sat-isfy a tail assumption P(tau e > t) asymptotic to beta exp (-alpha t(r)). When r <= 1 (this includes the well-known Eden growth model), we show that the upper tail large deviation decays...
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作者：Chernozhukov, Victor; Chetverikov, Denis; Koike, Yuta
作者单位：Massachusetts Institute of Technology (MIT); Massachusetts Institute of Technology (MIT); University of California System; University of California Los Angeles; University of Tokyo; University of Tokyo
摘要：In this paper, we derive new, nearly optimal bounds for the Gaussian ap-proximation to scaled averages of n independent high-dimensional centered random vectors X1, . .. , Xn over the class of rectangles in the case when the covariance matrix of the scaled average is nondegenerate. In the case of bounded Xi's, the implied bound for the Kolmogorov distance between the distribution of the scaled average and the Gaussian vector takes the form C(Bn2 log3 d/n)1/2log n, where d is the dimension of t...
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作者：Buckdahn, Rainer; Li, Juan; Ma, Jin
作者单位：Universite de Bretagne Occidentale; Shandong University; Shandong University; University of Southern California
摘要：In this paper we consider a class of conditional McKean Vlasov SDES (CMVSDE for short). Such an SDE can be considered as an extended version of McKean-Vlasov SDEs with common noises, as well as the general version of the so-called conditional mean-field SDEs (CMFSDE) studied previously by the authors (Ann. Appl. Probab. 27 (2017) 3201-3245; SIAM J. Control Optim. 56 (2018) 1154-1180), but with some fundamental differences. In particular, due to the lack of compactness of the iterated condition...
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作者：Chenavier, Nicolas; Coupier, David; Rousselle, Arnaud
作者单位：Centre National de la Recherche Scientifique (CNRS)
摘要：We investigate three types of internal diffusion limited aggregation (IDLA) models. These models are based on simple random walks on Z2 with infinitely many sources that are the points of the vertical axis I (infinity) = {0}x Z. Various properties are provided, such as stationarity, mixing, stabilization and shape theorems. Our results allow us to define a new directed (w.r.t. the hor-izontal direction) random forest spanning Z2, based on an IDLA protocol, which is invariant in distribution w....
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作者：Klimek, Aleksander; Rosati, Tommaso Cornelis
作者单位：University of Edinburgh; University of Warwick
摘要：We study the large scale behaviour of a population consisting of two types which evolve in dimension d = 1, 2 according to a spatial Lambda-Fleming-Viot process subject to random time-independent selection. If one of the two types is rare compared to the other, we prove that its evolution can be approximated by a super-Brownian motion in a random (and singular) en-vironment. Without the sparsity assumption, a diffusion approximation leads to a Fisher-KPP equation in a random potential. The pro...
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作者：Deb, Nabarun; Mukherjee, Sumit
作者单位：Columbia University
摘要：In this paper, we study the fluctuations of the average magnetization in an Ising model on an approximately dN regular graph GN on N vertices. In particular, if GN satisfies a spectral gap condition, we show that when-ever dN >> ./N, the fluctuations are universal and the same as that of the Curie-Weiss model in the entire ferromagnetic parameter regime. We give a counterexample to demonstrate that the condition dN >> ./N is tight, in the sense that the limiting distribution changes if dN simi...
