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作者：Fyodorov, Yan V.; Muirhead, Stephen
作者单位：University of London; King's College London; University of Melbourne
摘要：We study a random permutation of a lattice box in which each permutation is given a Boltzmann weight with energy equal to the total Euclidean displacement. Our main result establishes the band structure of the model as the box-size N tends to infinity and the inverse temperature beta tends to zero; in particular, we show that the mean displacement is of order min{1/beta,N}. In one dimension our results are more precise, specifying leading-order constants and giving bounds on the rates of conve...
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作者：Gamarnik, David; Jagannath, Aukosh; Sen, Subhabrata
作者单位：Massachusetts Institute of Technology (MIT); University of Waterloo; University of Waterloo; Harvard University
摘要：We study support recovery for a k x k principal submatrix with elevated mean lambda/N, hidden in an N x N symmetric mean zero Gaussian matrix. Here lambda > 0 is a universal constant, and we assume k = N rho for some constant rho is an element of (0, 1). We establish that there exists a constant C > 0 such that the MLE recovers a constant proportion of the hidden submatrix if lambda >= C root 1/rho log 1/rho, while such recovery is information theoretically impossible if lambda = o(root 1/rho ...
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作者：Basak, Anirban; Rudelson, Mark
作者单位：Tata Institute of Fundamental Research (TIFR); International Centre for Theoretical Sciences, Bengaluru; Weizmann Institute of Science; University of Michigan System; University of Michigan
摘要：We consider three models of sparse random graphs: undirected and directed Erdos-Renyi graphs and random bipartite graph with two equal parts. For such graphs, we show that if the edge connectivity probability p satisfies np >= log n + k(n) with k(n) -> infinity as n -> infinity, then the adjacency matrix is invertible with probability approaching one (n is the number of vertices in the two former cases and the same for each part in the latter case). For np <= log n -k(n) these matrices are inv...
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作者：Eberhard, Sean; Varju, Peter P.
作者单位：University of Cambridge
摘要：Define (X-n) on Z/qZ by Xn+1 = 2X(n) + b(n), where the steps b(n) are chosen independently at random from -1, 0,+1. The mixing time of this random walk is known to be at most 1.02 log(2) q for almost all odd q (Chung, Diaconis, Graham in Ann Probab 15(3):1148-1165, 1987), and at least 1.004 log(2) q (Hildebrand in Proc Am Math Soc 137(4):1479-1487, 2009). We identify a constant c = 1.01136 ... such that the mixing time is (c + o(1)) log(2) q for almost all odd q. In general, the mixing time of...
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作者：Duminil-Copin, Hugo; Grimmett, Geoffrey R.
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作者：Haas, Benedicte; Stephenson, Robin
作者单位：Universite Paris 13; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); University of Sheffield; University of Sheffield
摘要：We introduce multi-type Markov Branching trees, which are simple random population tree models where individuals are characterized by their size and their type and give rise to (size,type)-children in a Galton-Watson fashion, with the rule that the size of any individual is at least the sum of the sizes of its children. Assuming that the macroscopic size-splittings are rare, we describe the scaling limits of multi-type Markov Branching trees in terms of multi-type self-similar fragmentation tr...
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作者：Friz, Peter K.; Tran, Huy; Yuan, Yizheng
作者单位：Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics; Technical University of Berlin
摘要：Schramm-Loewner evolution (SLE kappa) is classically studied via Loewner evolution with half-plane capacity parametrization, driven by root kappa times Brownian motion. This yields a (half-plane) valued random field gamma = gamma (t, kappa;omega). (Holder) regularity of in gamma (center dot, kappa; omega), a.k.a. SLE trace, has been considered by many authors, starting with Rohde and Schramm (Ann Math (2) 161(2):883-924, 2005). Subsequently, Johansson Viklund et al. (Probab Theory Relat Fields...
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作者：Garban, Christophe; Holden, Nina; Sepulveda, Avelio; Sun, Xin
作者单位：Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet; Institut Universitaire de France; Swiss Federal Institutes of Technology Domain; ETH Zurich; Universidad de Chile; University of Pennsylvania
摘要：We construct and analyze a continuum dynamical percolation process which evolves in a random environment given by a gamma-Liouville measure. The homogeneous counterpart of this process describes the scaling limit of discrete dynamical percolation on the rescaled triangular lattice. Our focus here is to study the same limiting dynamics, but where the speed of microscopic updates is highly inhomogeneous in space and is driven by the gamma-Liouville measure associated with a two-dimensional log-c...
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作者：Augeri, Fanny
摘要：We develop transportation-entropy inequalities which are saturated by measures such that their log-density with respect to the background measure is an affine function, in the setting of the uniform measure on the discrete hypercube and the exponential measure. In this sense, this extends the well-known result of Talagrand in the Gaussian case. By duality, these transportation-entropy inequalities imply a strong integrability inequality for Bernoulli and exponential processes. As a result, we ...
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作者：Angel, Omer; Hutchcroft, Tom; Jarai, Antal
作者单位：University of British Columbia; University of Cambridge; University of Bath
摘要：Consider a critical branching random walk on Z(d), d >= 1, started with a single particle at the origin, and let L(x) be the total number of particles that ever visit a vertex x. We study the tail of L(x) under suitable conditions on the offspring distribution. In particular, our results hold if the offspring distribution has an exponential moment.
