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作者：Fox, Jacob; Kwan, Matthew; Sauermann, Lisa
作者单位：Stanford University; Institute for Advanced Study - USA
摘要：Fix a graph H and some p is an element of (0, 1), and let X-H be the number of copies of H in a random graph G(n, p). Random variables of this form have been intensively studied since the foundational work of Erdos and Renyi. There has been a great deal of progress over the years on the large-scale behaviour of X-H, but the more challenging problem of understanding the small-ball probabilities has remained poorly understood until now. More precisely, how likely can it be that X-H falls in some...
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作者：Holmes, Mark; Salisbury, Thomas S.
作者单位：University of Melbourne; York University - Canada
摘要：We study a particular model of a random medium, called the orthant model, in general dimensions d >= 2. Each site x is an element of Z(d) independently has arrows pointing to its positive neighbours x + e(i), i = 1, ..., d with probability p and, otherwise, to its negative neighbours x - e(i), i = 1, ..., d (with probability 1 - p). We prove a shape theorem for the set of sites reachable by following arrows, starting from the origin, when p is large. The argument uses subadditivity, as would b...
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作者：Deuschel, Jean-Dominique; Orenshtein, Tal; Perkowski, Nicolas
作者单位：Technical University of Berlin; Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics; Free University of Berlin
摘要：We consider additive functionals of stationary Markov processes and show that under Kipnis-Varadhan type conditions they converge in rough path topology to a Stratonovich Brownian motion, with a correction to the Levy area that can be described in terms of the asymmetry (nonreversibility) of the underlying Markov process. We apply this abstract result to three model problems: First, we study random walks with random conductances under the annealed law. If we consider the Ito rough path, then w...
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作者：Timar, Adam
作者单位：HUN-REN; HUN-REN Alfred Renyi Institute of Mathematics
摘要：Answering a question of Benjamini, we present an isometry-invariant random partition of the Euclidean space R-d, d >= 3, into infinite connected indistinguishable pieces, such that the adjacency graph defined on the pieces is the 3-regular infinite tree. Along the way, it is proved that any finitely generated one-ended amenable Cayley graph can be represented in R-d as an isometry-invariant random partition of R-d to bounded polyhedra, and also as an isometry-invariant random partition of R-d ...
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作者：Gwynne, Ewain; Miller, Jason
作者单位：University of Cambridge
摘要：We study simple random walk on the class of random planar maps which can be encoded by a two-dimensional random walk with i.i.d.. increments or a two-dimensional Brownian motion via a mating-of-trees type bijection. This class includes the uniform infinite planar triangulation (UIPT), the infinite-volume limits of random planar maps weighted by the number of spanning trees, bipolar orientations, or Schnyder woods they admit, and the gamma-mated-CRT map for gamma is an element of (0, 2). For ea...
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作者：Buraczewski, Dariusz; Iksanov, Alexander; Mallein, Bastien
作者单位：University of Wroclaw; Ministry of Education & Science of Ukraine; Taras Shevchenko National University of Kyiv
摘要：We work under the Aidekon-Chen conditions which ensure that the derivative martingale in a supercritical branching random walk on the line converges almost surely to a nondegenerate nonnegative random variable that we denote by Z. It is shown that EZ 1({Z <= x}) = log x + o(log x) as x -> infinity. Also, we provide necessary and sufficient conditions under which EZ 1({Z <= x}) = log x + const + o(1) as x -> infinity. This more precise asymptotics is a key tool for proving distributional limit ...
