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作者：Perkowski, Nicolas; Rosati, Tommaso
作者单位：Free University of Berlin
摘要：We study the scaling limit of a branching random walk in static random environment in dimension d = 1, 2 and show that it is given by a superBrownian motion in a white noise potential. In dimension 1 we characterize the limit as the unique weak solution to the stochastic PDE partial derivative t mu=(Delta+xi)mu+root 2 nu mu xi for independent space white noise xi and space-time white noise (xi) over tilde. In dimension 2 the study requires paracontrolled theory and the limit process is describ...
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作者：Cannizzaro, Giuseppe; Erhard, Dirk; Schonbauer, Philipp
作者单位：University of Warwick; Universidade Federal da Bahia; Imperial College London
摘要：In this work we focus on the two-dimensional anisotropic KPZ (aKPZ) equation, which is formally given by partial derivative(t)h = v/2 Delta h + lambda(partial derivative(1)h)(2) - (partial derivative(2)h)(2)) + v(1/2)xi, where xi denotes a noise which is white in both space and time, and lambda and nu are positive constants. Due to the wild oscillations of the noise and the quadratic nonlinearity, the previous equation is classically ill posed. It is not possible to linearise it via the Cole-H...
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作者：Cryan, Mary; Guo, Heng; Mousa, Giorgos
作者单位：University of Edinburgh
摘要：We show that the modified log-Sobolev constant for a natural Markov chain which converges to an r -homogeneous strongly log-concave distribution is at least 1/r. Applications include a sharp mixing time bound for the bases-exchange walk for matroids, and a concentration bound for Lipschitz functions over these distributions.
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作者：Feldheim, Naomi; Feldheim, Ohad; Nitzan, Shahaf
作者单位：Bar Ilan University; Hebrew University of Jerusalem; University System of Georgia; Georgia Institute of Technology
摘要：We study the persistence probability of a centered stationary Gaussian process on Z or R, that is, its probability to remain positive for a long time. We describe the delicate interplay between this probability and the behavior of the spectral measure of the process near zero and infinity.
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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：Bailleul, Ismael; Catellier, Remi; Delarue, Francois
作者单位：Centre National de la Recherche Scientifique (CNRS); Universite de Rennes; Centre National de la Recherche Scientifique (CNRS); Universite Cote d'Azur
摘要：We address propagation of chaos for large systems of rough differential equations associated with random rough differential equations of mean field type dX(t) = V(X-t, L(X-t)) dt + F(X-t, L(X-t)) dW(t), where W is a random rough path and L(X-t) is the law of X-t. We prove propagation of chaos, and provide also an explicit optimal convergence rate. The analysis is based upon the tools we developed in our companion paper (Electron. J. Probab. 25 (2020) 21) for solving mean field rough differenti...
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作者：Addario-Berry, Louigi; Corsini, Benoit
作者单位：McGill University
摘要：Random binary search trees are obtained by recursively inserting the elements sigma(1), sigma(2),..., sigma(n) of a uniformly random permutation s of [n] = {1,..., n} into a binary search tree data structure. Devroye (J. Assoc. Comput. Mach. 33 (1986) 489-498) proved that the height of such trees is asymptotically of order c* log n, where c* = 4.311... is the unique solution of c log((2e)/c) = 1 with c >= 2. In this paper, we study the structure of binary search trees T-n,T- q built from Mallo...
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作者：Bhattacharya, Bhaswar B.; Bhattacharya, Sohom; Ganguly, Shirshendu
作者单位：University of Pennsylvania; Stanford University; University of California System; University of California Berkeley
摘要：In this paper, we consider the problem of estimating the joint upper and lower tail large deviations of the edge eigenvalues of an Erdos-Renyi random graph G(n,p), in the regime of p where the edge of the spectrum is no longer governed by global observables, such as the number of edges, but rather by localized statistics, such as high degree vertices. Going beyond the recent developments in mean-field approximations of related problems, this paper provides a comprehensive treatment of the larg...
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作者：Sarkar, Sourav; Virag, Balint
作者单位：University of Toronto; University of Toronto
摘要：We show that the law of the KPZ fixed point starting from arbitrary initial condition is absolutely continuous with respect to the law of Brownian motion B on every compact interval. In particular, the Airy(1) process is absolutely continuous with respect to B on any compact interval.
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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...
