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作者：Rath, Balazs; Swart, Jan M.; Terpai, Tamas
作者单位：Budapest University of Technology & Economics; MTA-BME Stochastics Research Group; Czech Academy of Sciences; Institute of Information Theory & Automation of the Czech Academy of Sciences; Eotvos Lorand University
摘要：In frozen percolation, i.i.d. uniformly distributed activation times are assigned to the edges of a graph. At its assigned time an edge opens provided neither of its end vertices is part of an infinite open cluster; in the opposite case it freezes. Aldous (Math. Proc. Cambridge Philos. Soc. 128 (2000) 465477) showed that such a process can be constructed on the infinite 3-regular tree and asked whether the event that a given edge freezes is a measurable function of the activation times assigne...
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作者：Alt, Johannes; Ducatez, Raphael; Knowles, Antti
作者单位：University of Geneva
摘要：We complete the analysis of the extremal eigenvalues of the adjacency matrix A of the Erdos-Renyi graph G(N, d/N) in the critical regime d asymptotic to log N of the transition uncovered in (Ann. Inst. Henri Poincare Probab. Stat. 56 (2020) 2141-2161; Ann. Probab. 47 (2019) 1653-1676), where the regimes d >> log N and d << log N were studied. We establish a one-to-one correspondence between vertices of degree at least 2d and nontrivial (excluding the trivial top eigenvalue) eigenvalues of A/ro...
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作者：Coupier, David; Saha, Kumarjit; Sarkar, Anish; Viet Chi Tran
作者单位：Universite de Lille; IMT - Institut Mines-Telecom; IMT Nord Europe; Ashoka University; Indian Statistical Institute; Indian Statistical Institute Bangalore; Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Centre National de la Recherche Scientifique (CNRS); Universite Gustave-Eiffel
摘要：The two-dimensional directed spanning forest (DSF) introduced by Baccelli and Bordenave is a planar directed forest whose vertex set is given by a homogeneous Poisson point process N on R-2. If the DSF has direction -e(y), the ancestor h(u) of a vertex u is an element of N is the nearest Poisson point (in the L-2 distance) having strictly larger y-coordinate. This construction induces complex geometrical dependencies. In this paper, we show that the collection of DSF paths, properly scaled, co...
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作者：Arguin, Louis-Pierre; Ouimet, Frederic; Radziwill, Maksym
作者单位：City University of New York (CUNY) System; Baruch College (CUNY); City University of New York (CUNY) System; California Institute of Technology
摘要：We show that as T -> infinity, for all t is an element of [T, 2T] outside of a set of measure o(T), integral(log theta T)(-log theta T)vertical bar zeta(1/2 + it +ih)vertical bar(beta) dh = (log T)(f)((beta)+)(o(1))(theta), for some explicit exponent f(theta)(beta), where theta > -1 and beta > 0. This proves an extended version of a conjecture of Fyodorov and Keating (Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014) 20120503, 32). In particular, it shows that, for all theta...
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作者：Xu, Changji
作者单位：Harvard University
摘要：Consider the discrete cube {-1, 1}(N) and a random collection of half spaces which includes each half space H(x) := {y is an element of {-1, 1}(N) : x center dot y >=kappa root N} for x. {-1, 1}(N) independently with probability p. Is the intersection of these half spaces empty? This is called the Ising perceptron model under Bernoulli disorder. We prove that this event has a sharp threshold, that is, the probability that the intersection is empty increases quickly from is an element of to 1 -...
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作者：Roeckner, Michael; Xie, Longjie
作者单位：University of Bielefeld; Jiangsu Normal University; Jiangsu Normal University
摘要：We consider a Poisson equation in R-d for the elliptic operator corresponding to an ergodic diffusion process. Optimal regularity and smoothness with respect to the parameter are obtained under mild conditions on the coefficients. The result is then applied to establish a general diffusion approximation for fully coupled multitime scales stochastic differential equations with only Holder continuous coefficients. Four different averaged equations as well as rates of convergence are obtained. Mo...
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作者：Livshyts, Galyna, V; Tikhomirov, Konstantin; Vershynin, Roman
作者单位：University System of Georgia; Georgia Institute of Technology; University of California System; University of California Irvine
摘要：We show that, for an n x n random matrix A with independent uniformly anticoncentrated entries such that E parallel to A parallel to(2)(HS) <= Kn(2), the smallest singular value sigma(n)(A) of A satisfies P{sigma(n)(A) <= epsilon/root n} <= C epsilon + 2e(-cn), epsilon >= 0. This extends earlier results (Adv. Math. 218 (2008) 600-633; Israel J. Math. 227 (2018) 507-544) by removing the assumption of mean zero and identical distribution of the entries across the matrix as well as the recent res...
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作者：Lee, James R.
作者单位：University of Washington; University of Washington Seattle
摘要：For a unimodular random graph (G, rho), we consider deformations of its intrinsic path metric by a (random) weighting of its vertices. This leads to the notion of the conformal growth exponent of (G, rho), which is the best asymptotic degree of volume growth of balls that can be achieved by such a reweighting. Under moment conditions on the degree of the root, we show that the conformal growth exponent of a unimodular random graph bounds its almost sure spectral dimension. This has interesting...
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作者：Dauvergne, Duncan; Virag, Balint
作者单位：Princeton University; University of Toronto
摘要：The Airy line ensemble is a central object in random matrix theory and last passage percolation defined by a determinantal formula. The goal of this paper is to provide a set of tools, which allow for precise probabilistic analysis of the Airy line ensemble. The two main theorems are a representation in terms of independent Brownian bridges connecting a fine grid of points, and a modulus of continuity result for all lines. Along the way, we give tail bounds and moduli of continuity for noninte...
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作者：Bakhtin, Yuri; Chen, Hong-Bin
作者单位：New York University
摘要：We consider exit problems for small, white noise perturbations of a dynamical system generated by a vector field and a domain containing a critical point with all positive eigenvalues of linearization. We prove that, in the vanishing noise limit, the probability of exit through a generic set on the boundary is asymptotically polynomial in the noise strength with exponent depending on the mutual position of the set and the flag of the invariant manifolds associated with the top eigenvalues. Fur...
