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作者：Lyons, Russell; White, Graham
作者单位：Indiana University System; Indiana University Bloomington
摘要：Consider continuous-time random walks on Cayley graphs where the rate assigned to each edge depends only on the corresponding generator. We show that the limiting speed is monotone increasing in the rates for infinite Cay-ley graphs that arise from Coxeter systems but not for all Cayley graphs. On finite Cayley graphs, we show that the distance-in various senses-to stationarity is monotone decreasing in the rates for Coxeter systems and for abelian groups but not for all Cayley graphs. We also...
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作者：Alt, Johannes; Ducatez, Raphael; Knowles, Antti
作者单位：University of Geneva; New York University; Ecole Normale Superieure de Lyon (ENS de LYON)
摘要：We consider the adjacency matrix A of the Erdos-Renyi graph on N ver-tices with edge probability d/N. For (log log N)4 << d < log N, we prove that the eigenvalues near the spectral edge form asymptotically a Poisson point process and the associated eigenvectors are exponentially localized. As a corollary, at the critical scale d kappa log N, the limiting distribution of the largest nontrivial eigenvalue does not match with any previously known dis-tribution. Together with (Comm. Math. Phys. 38...
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作者：Halberstam, Noah; Hutchcroft, Tom
作者单位：University of Cambridge; California Institute of Technology
摘要：We prove that if (X-n)(n >= 0) is a random walk on a transient graph such that the Green's function decays at least polynomially along the random walk, then (X-n)(n >= 0) has infinitely many cut times almost surely. This condition applies in particular to any graph of spectral dimension strictly larger than 2. In fact, our proof applies to general (possibly nonreversible) Markov chains satisfying a similar decay condition for the Green's function that is sharp for birth-death chains. We deduce...
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作者：Najnudel, Joseph; Paquette, Elliot; Simm, Nick
作者单位：University of Bristol; McGill University; University of Sussex
摘要：We study the secular coefficients of N x N random unitary matrices UN drawn from the Circular-Ensemble which are defined as the coefficients of {zn} in the characteristic polynomial det(1 - zU*N). When > 4, we obtain a new class of limiting distributions that arise when both n and N tend to infinity simultaneously. We solve an open problem of Diaconis and Gamburd (Electron. J. Combin. 11 (2004/06) 2) by showing that, for = 2, the mid-dle coefficient of degree n = N2 j tends to zero as N -oo. W...
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作者：Das, Sayan; Ghosal, Promit
作者单位：Columbia University; Massachusetts Institute of Technology (MIT)
摘要：We consider the Cole-Hopf solution of the (1 + 1)-dimensional KPZ equation started from the narrow wedge initial condition. In this article we ask how the peaks and valleys of the KPZ height function (centered by time/24) at any spatial point grow as time increases. Our first main result is about the law of iterated logarithms for the KPZ equation. As time variable t goes to infinity, we show that the limsup of the KPZ height function with the scaling by root t1/3(loglogt)2/3 is almost surely ...
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作者：Chong, Carsten; Kevei, Peter
作者单位：Columbia University; Szeged University
摘要：We show that the spatial profile of the solution to the stochastic heat equation features multiple layers of intermittency islands if the driving noise is non-Gaussian. On the one hand, as expected, if the noise is sufficiently heavy-tailed, the largest peaks of the solution will be taller under multiplica-tive than under additive noise. On the other hand, surprisingly, as soon as the noise has a finite moment of order d2, where dis the spatial dimension, the largest peaks will be of the same ...
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作者：Diaconu, Simona
作者单位：Stanford University
摘要：The Tracy-Widom distributions are among the most famous laws in probability theory, partly due to their connection with Wigner matrices. In particular, for A = 1/root n(a(ij))(1 <= i,j <= n) is an element of R-nxn symmetric with (a(ij))(1 <= i <= j <= n) i.i.d. standard normal, the fluctuations of its largest eigenvalue lambda(1)(A) are asymptotically described by a real-valued Tracy-Widom distribution TW1 : n(2/3)(lambda(1)(A) - 2) double right arrow TW1. As it often happens, Gaussianity can ...
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作者：Kosygina, Elena; Mountford, Thomas; Peterson, Jonathon
作者单位：City University of New York (CUNY) System; Baruch College (CUNY); Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Purdue University System; Purdue University
摘要：We use generalized Ray-Knight theorems, introduced by B. Toth in 1996, together with techniques developed for excited random walks as main tools for establishing positive and negative results concerning convergence of some classes of diffusively scaled self-interacting random walks (SIRW) to Brownian motions perturbed at extrema (BMPE). Toth's work studied two classes of SIRWs: asymptotically free and polynomially self-repelling walks. For both classes Toth has shown, in particular, that the d...
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作者：Sjostrand, Jonas
作者单位：Malardalen University
摘要：A locally uniform random permutation is generated by sampling n points independently from some absolutely continuous distribution p on the plane and interpreting them as a permutation by the rule that i maps to j if the ith point from the left is the jth point from below. As n tends to infinity, decreas-ing subsequences in the permutation will appear as curves in the plane, and by interpreting these as level curves, a union of decreasing subsequences give rise to a surface. We show that, under...
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作者：Salez, Justin
作者单位：Universite PSL; Universite Paris-Dauphine
摘要：We consider the exclusion process with reservoirs on arbitrary networks. We characterize the spectral gap, mixing time, and mixing window of the pro-cess, in terms of certain simple spectral statistics of the underlying network. Among other consequences we establish a nonconservative analogue of Al-dous's spectral gap conjecture, and we show that cutoff occurs if and only if the product condition is satisfied. We illustrate this by providing explicit cutoffs on discrete lattices of arbitrary d...
