Probability Theory and Related Fields

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• KPZ line ensemble AI翻译
  作者：Corwin, Ivan; Hammond, Alan
  作者单位：Columbia University; Sorbonne Universite; University of California System; University of California Berkeley; University of California System; University of California Berkeley
  摘要：For each t >= 1we construct an -indexed ensemble of random continuous curves with three properties: the lowest indexed curve is distributed as the time t Hopf-Cole solution to the Kardar-Parisi-Zhang (KPZ) stochastic partial differential equation with narrow wedge initial data; the entire ensemble satisfies a resampling invariance which we call the -Brownian Gibbs property [with H(x) = e(x)]; increments of the lowest indexed curve, when centered by and scaled down vertically by and horizontall...
• Percolation and local isoperimetric inequalities AI翻译
  作者：Teixeira, Augusto
  作者单位：Instituto Nacional de Matematica Pura e Aplicada (IMPA)
  摘要：In this paper we establish some relations between percolation on a given graph G and its geometry. Our main result shows that, if G has polynomial growth and satisfies what we call the local isoperimetric inequality of dimension , then . This gives a partial answer to a question of Benjamini and Schramm (Electron Commun Probab 1(8):71-82 1996). As a consequence of this result and Haggstrom (Adv Appl Probab 32(1):39-66 2000) we derive, that these graphs also undergo a non-trivial phase transiti...
• Anomalous diffusion in fast cellular flows at intermediate time scales AI翻译
  作者：Iyer, Gautam; Novikov, Alexei
  作者单位：Carnegie Mellon University; Pennsylvania Commonwealth System of Higher Education (PCSHE); Pennsylvania State University; Pennsylvania State University - University Park
  摘要：It is well known that on long time scales the behaviour of tracer particles diffusing in a cellular flow is effectively that of a Brownian motion. This paper studies the behaviour on intermediate time scales before diffusion sets in. Various heuristics suggest that an anomalous diffusive behaviour should be observed. We prove that the variance on intermediate time scales grows like . Hence, on these time scales the effective behaviour can not be purely diffusive, and is consistent with an anom...
• Uniqueness vs. non-uniqueness for complete connections with modified majority rules AI翻译
  作者：Dias, J. C. A.; Friedli, S.
  作者单位：Universidade Federal de Minas Gerais
  摘要：We take a closer look at a class of chains with complete connections introduced by Berger, Hoffman and Sidoravicius. In particular, besides giving a sharper description of the uniqueness and non-uniqueness regimes, we show that if the pure majority rule used to fix the dependence on the past is replaced with a function that is Lipschitz at the origin, then uniqueness always holds, even with arbitrarily slow decaying variation.
• The continuum disordered pinning model AI翻译
  作者：Caravenna, Francesco; Sun, Rongfeng; Zygouras, Nikos
  作者单位：University of Milano-Bicocca; National University of Singapore; University of Warwick
  摘要：Any renewal processes on N-0 with a polynomial tail, with exponent a is an element of (0, 1), has a non-trivial scaling limit, known as the alpha-stable regenerative set. In this paper we consider Gibbs transformations of such renewal processes in an i. i. d. random environment, called disordered pinning models. We show that for alpha is an element of (1/2, 1) these models have a universal scaling limit, which we call the continuum disordered pinning model (CDPM). This is a random closed subse...
• Infinite dimensional stochastic differential equations for Dyson's model AI翻译
  作者：Tsai, Li-Cheng
  作者单位：Stanford University
  摘要：In this paper we show the strong existence and the pathwise uniqueness of an infinite-dimensional stochastic differential equation (SDE) corresponding to the bulk limit of Dyson's Brownian Motion, for all . Our construction applies to an explicit and general class of initial conditions, including the lattice configuration and the sine process. We further show the convergence of the finite to infinite-dimensional SDE. This convergence concludes the determinantal formula of Katori and Tanemura (...
• The hull process of the Brownian plane AI翻译
  作者：Curien, Nicolas; Le Gall, Jean-Francois
  作者单位：Universite Paris Saclay
  摘要：We study the random metric space called the Brownian plane, which is closely related to the Brownian map and is conjectured to be the universal scaling limit of many discrete random lattices such as the uniform infinite planar triangulation. We obtain a number of explicit distributions for the Brownian plane. In particular, we consider, for every , the hull of radius r, which is obtained by filling in the holes in the ball of radius r centered at the root. We introduce a quantity which is inte...
• Sharp asymptotics for the quasi-stationary distribution of birth-and-death processes AI翻译
  作者：Chazottes, J. -R.; Collet, P.; Meleard, S.
  作者单位：Institut Polytechnique de Paris; Ecole Polytechnique; Centre National de la Recherche Scientifique (CNRS); CNRS - Institute of Physics (INP); Institut Polytechnique de Paris; Ecole Polytechnique; ENSTA Paris; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
  摘要：We study a general class of birth-and-death processes with state space N that describes the size of a population going to extinction with probability one. This class contains the logistic case. The scale of the population is measured in terms of a 'carrying capacity' K. When K is large, the process is expected to stay close to its deterministic equilibrium during a long time but ultimately goes extinct. Our aim is to quantify the behavior of the process and the mean time to extinction in the q...
• Occupation times of long-range exclusion and connections to KPZ class exponents AI翻译
  作者：Bernardin, Cedric; Goncalves, Patricia; Sethuraman, Sunder
  作者单位：Universite Cote d'Azur; Centre National de la Recherche Scientifique (CNRS); Universidade do Minho; University of Arizona
  摘要：With respect to a class of long-range exclusion processes on , with single particle transition rates of order , starting under Bernoulli invariant measure with density , we consider the fluctuation behavior of occupation times at a vertex and more general additive functionals. Part of our motivation is to investigate the dependence on , d and with respect to the variance of these functionals and associated scaling limits. In the case the rates are symmetric, among other results, we find the sc...
• On the principal components of sample covariance matrices AI翻译
  作者：Bloemendal, Alex; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun
  作者单位：Harvard University; Swiss Federal Institutes of Technology Domain; ETH Zurich; University of Wisconsin System; University of Wisconsin Madison
  摘要：We introduce a class of M x M sample covariance matrices Q which subsumes and generalizes several previous models. The associated population covariance matrix Sigma = EQ is assumed to differ from the identity by a matrix of bounded rank. All quantities except the rank of Sigma - I-M may depend on M in an arbitrary fashion. We investigate the principal components, i. e. the top eigenvalues and eigenvectors, of Q. We derive precise large deviation estimates on the generalized components < w, xi(...