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作者：Lawler, Gregory; Sun, Xin; Wu, Wei
作者单位：University of Chicago; Columbia University; University of Warwick
摘要：The loop-erased random walk (LERW) in Z(4) is the process obtained by erasing loops chronologically for a simple random walk. We prove that the escape probability of the LERW renormalized by (log n) 1/3 converges almost surely and in L-p for all p > 0. Along the way, we extend previous results by the first author building on slowly recurrent sets. We provide two applications for the escape probability. We construct the two-sided LERW, and we construct a +/- 1 spin model coupled with the wired ...
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作者：Tang, Pengfei
作者单位：Indiana University System; Indiana University Bloomington
摘要：We prove that the heavy clusters are indistinguishable for Bernoulli percolation on quasi-transitive nonunimodular graphs. As an application, we show that the uniqueness threshold of any quasi-transitive graph is also the threshold for connectivity decay. This resolves a question of Lyons and Schramm (Ann. Probab.27 (1999) 1809-1836) in the Bernoulli percolation case and confirms a conjecture of Schonmann (Comm. Math. Phys. 219 (2001) 271-322). We also prove that every infinite cluster of Bern...
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作者：Martineau, Sebastien; Severo, Franco
作者单位：Universite Paris Saclay; Universite Paris Saclay
摘要：We answer a question of Benjamini and Schramm by proving that under reasonable conditions, quotienting a graph strictly increases the value of its percolation critical parameter p(c). More precisely, let G = (V, E) be a quasi-transitive graph with p(c) (G) < 1, and let G be a nontrivial group that acts freely on V by graph automorphisms. Assume that H := G/G is quasi-transitive. Then one has p(c) (G) < P-c (H). We provide results beyond this setting: we treat the case of general covering maps ...
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作者：Bally, Vlad; Caramellino, Lucia
作者单位：Universite Gustave-Eiffel; Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Centre National de la Recherche Scientifique (CNRS); University of Rome Tor Vergata; University of Rome Tor Vergata
摘要：The goal of this paper is to estimate the total variation distance between two general stochastic polynomials. As a consequence, one obtains an invariance principle for such polynomials. This generalizes known results concerning the total variation distance between two multiple stochastic integrals on one hand, and invariance principles in Kolmogorov distance for multilinear stochastic polynomials on the other hand. As an application, we first discuss the asymptotic behavior of U-statistics as...
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作者：Miller, Jason; Sheffield, Scott
作者单位：University of Cambridge; Massachusetts Institute of Technology (MIT)
摘要：Let h be an instance of the GFF. Fix kappa is an element of (0, 4) and chi = 2//root kappa - root kappa/2. Recall that an imaginary geometry ray is a flow line of e(i(h/chi+theta)) that looks locally like SEE kappa. The light cone with parameter theta is an element of [0, pi] is the set of points reachable from the origin by a sequence of rays with angles in [-theta/2, theta/2]. It is known that when theta = 0, the light cone looks like SLE kappa, and when = theta = pi looks like the range of ...
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作者：Senizergues, Delphin
作者单位：Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris 13
摘要：We construct random metric spaces by gluing together an infinite sequence of pointed metric spaces that we call blocks. At each step, we glue the next block to the structure constructed so far by randomly choosing a point on the structure and then identifying it with the distinguished point of the block. The random object that we study is the completion of the structure that we obtain after an infinite number of steps. In (Ann. Inst. Fourier (Grenoble) 67 (2017) 1963-2001), Curien and Haas stu...
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作者：Berger, Quentin; Torri, Niccolo
作者单位：Sorbonne Universite
摘要：We study the directed polymer model in dimension 1 + 1 when the environment is heavy-tailed, with a decay exponent alpha is an element of (0, 2). We give all possible scaling limits of the model in the weak-coupling regime, that is, when the inverse temperature temperature beta = beta(n) vanishes as the size of the system n goes to infinity. When alpha is an element of (1/2, 2), we show that all possible transversal fluctuations root n <= h(n) <= n can be achieved by tuning properly beta(n), a...
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作者：Rhee, Chang-Han; Blanchet, Jose; Zwart, Bert
作者单位：Northwestern University
摘要：Let X be a Levy process with regularly varying Levy measure v. We obtain sample-path large deviations for scaled processes (X) over bar (n) (t) (sic) X (nt)/n and obtain a similar result for random walks with regularly varying increments. Our results yield detailed asymptotic estimates in scenarios where multiple big jumps in the increment are required to make a rare event happen; we illustrate this through detailed conditional limit theorems. In addition, we investigate connections with the c...
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作者：Borodin, Alexei; Gorin, Vadim
作者单位：Massachusetts Institute of Technology (MIT); Kharkevich Institute for Information Transmission Problems of the RAS
摘要：A stochastic telegraph equation is defined by adding a random inhomogeneity to the classical (second-order linear hyperbolic) telegraph differential equation. The inhomogeneities we consider are proportional to the two-dimensional white noise, and solutions to our equation are two-dimensional random Gaussian fields. We show that such fields arise naturally as asymptotic fluctuations of the height function in a certain limit regime of the stochastic six-vertex model in a quadrant. The correspon...
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作者：Ganguly, Shirshendu; Levine, Lionel; Sarkar, Sourav
作者单位：University of California System; University of California Berkeley; Cornell University
摘要：We study the following one-dimensional model of annihilating particles. Beginning with all sites of Z uncolored, a blue particle performs simple random walk from 0 until it reaches a nonzero red or uncolored site, and turns that site blue; then a red particle performs simple random walk from 0 until it reaches a nonzero blue or uncolored site, and turns that site red. We prove that after n blue and n red particles alternately perform such walks, the total number of colored sites is of order n(...
