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作者：Chen, Xia; Deya, Aurelien; Ouyang, Cheng; Tindel, Samy
作者单位：University of Tennessee System; University of Tennessee Knoxville; Universite de Lorraine; University of Illinois System; University of Illinois Chicago; University of Illinois Chicago Hospital; Purdue University System; Purdue University
摘要：The theory of regularity structures enables the definition of the following parabolic Anderson model in a very rough environment: partial derivative(t)u(t) (x) = 1/2 Delta u(t) (x) + u(t) (x). (W) over dot (x), for t is an element of R+ and x is an element of R-d, where (W) over dot (x) is a Gaussian noise whose space time covariance function is singular. In this rough context we shall give some information about the moments of u(t) (x) when the stochastic heat equation is interpreted in the S...
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作者：Au, Benson; Cebron, Guillaume; Dahlqvist, Antoine; Gabriel, Franck; Male, Camille
作者单位：University of California System; University of California San Diego; Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Universite Federale Toulouse Midi-Pyrenees (ComUE); Institut National des Sciences Appliquees de Toulouse; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Centre National de la Recherche Scientifique (CNRS); University of Sussex; Universite de Bordeaux
摘要：We prove that independent families of permutation invariant random matrices are asymptotically free with amalgamation over the diagonal, both in expectation and in probability, under a uniform boundedness assumption on the operator norm. We can relax the operator norm assumption to an estimate on sums associated to graphs of matrices, further extending the range of applications (e.g., to Wigner matrices with exploding moments and the sparse regime of the Erdos-Renyi model). The result still ho...
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作者：Feng, Renjie; Wei, Dongyi
作者单位：Chinese Academy of Sciences; University of Science & Technology of China, CAS; Peking University
摘要：In this article, we study the smallest gaps of the circular beta-ensemble (C beta E) on the unit circle, where beta is any positive integer. The main result is that the smallest gaps, after being normalized by n(beta+2/beta+1), will converge in distribution to a Poisson point process with some explicit intensity. And thus one can derive the limiting density of the kth smallest gap, which is proportional to x(k(beta+1)-1)e(-x beta+1). In particular, the results apply to the classical COE, CUE a...
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作者：Ahlberg, Daniel; Griffiths, Simon; Janson, Svante
作者单位：Stockholm University; Pontificia Universidade Catolica do Rio de Janeiro; Uppsala University
摘要：We study a model of competition between two types evolving as branching random walks on Z(d). The two types are represented by red and blue balls, respectively, with the rule that balls of different colour annihilate upon contact. We consider initial configurations in which the sites of Z(d) contain one ball each which are independently coloured red with probability p and blue otherwise. We address the question of fixation, referring to the sites and eventually settling for a given colour or n...
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作者：Dalang, Robert C.; Mueller, Carl; Xiao, Yimin
作者单位：Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; University of Rochester; Michigan State University
摘要：We study vector-valued solutions u(t, x) is an element of R-d to systems of nonlinear stochastic heat equations with multiplicative noise, partial derivative/partial derivative t u(t, x) = partial derivative(2)/partial derivative x(2) u(t, x) + sigma (u(t, x)(W) over dot (t, x). Here, t >= 0, x is an element of R and (W) over dot (t, x) is an R-d-valued space-time white noise. We say that a point z is an element of R-d is polar if P{u(t, x) = z for some t > 0 and x is an element of R} = 0. We ...
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作者：Miller, Jason; Sheffield, Scott
作者单位：University of Cambridge; Massachusetts Institute of Technology (MIT)
摘要：We endow the root 8/3-Liouville quantum gravity sphere with a metric space structure and show that the resulting metric measure space agrees in law with the Brownian map. Recall that a Liouville quantum gravity sphere is a priori naturally parameterized by the Euclidean sphere S-2. Previous work in this series used quantum Loewner evolution (QLE) to construct a metric d(Q) on a countable dense subset of S-2. Here, we show that d(Q) a.s. extends uniquely and continuously to a metric (d) over ba...
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作者：Banerjee, Sayan; Burdzy, Krzysztof
作者单位：University of North Carolina; University of North Carolina Chapel Hill; University of Washington; University of Washington Seattle
摘要：We analyze the local and global smoothing rates of the smoothing process and obtain convergence rates to stationarity for the dual process known as the potlatch process. For general finite graphs we connect the smoothing and convergence rates to the spectral gap of the associated Markov chain. We perform a more detailed analysis of these processes on the torus. Polynomial corrections to the smoothing rates are obtained. They show that local smoothing happens faster than global smoothing. These...
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作者：Vanneuville, Hugo
作者单位：Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet
摘要：In this paper, we introduce and study the annealed spectral sample of Voronoi percolation, which is a continuous and finite point process in R-2 whose definition is mostly inspired by the spectral sample of Bernoulli percolation introduced in (Acta Math. 205 (2010) 19-104) by Garban, Pete and Schramm. We show a clustering effect as well as estimates on the full lower tail of this spectral object. Our main motivation is the study of two models of dynamical critical Voronoi percolation in the pl...
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作者：Rossignol, Raphael
作者单位：Centre National de la Recherche Scientifique (CNRS); Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA)
摘要：Consider a critical Erdos-Renyi random graph: n is the number of vertices, each one of the ((n)(2)) possible edges is kept in the graph independently from the others with probability n(-1) + lambda n (-4/3), lambda being a fixed real number. When n goes to infinity, Addario-Berry, Broutin and Goldschmidt (Probab. Theory Related Fields 152 (2012) 367-406) have shown that the collection of connected components, viewed as suitably normalized measured compact metric spaces, converges in distributi...
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作者：Berggren, Tomas
作者单位：University of Michigan System; University of Michigan
摘要：In this paper we consider domino tilings of the Aztec diamond with doubly periodic weightings. In particular, a family of models which, for any k is an element of N, includes models with k smooth regions is analyzed as the size of the Aztec diamond tends to infinity. We use a nonintersecting paths formulation and give a double integral formula for the correlation kernel of the Aztec diamond of finite size. By a classical steepest descent analysis of the correlation kernel, we obtain the local ...
