-
作者：Lygkonis, Dimitris; Zygouras, Nikos
作者单位：University of Warwick
摘要：The Erdos-Taylor theorem (Acta Math Acad Sci Hungar, 1960) states that if L(N )is the local time at zero, up to time 2N, of a two-dimensional simple, symmetric random walk, then pi/logN L(N )converges in distribution to an exponential random variable with parameter one. This can be equivalently stated in terms of the total collision time of two independent simple random walks on the plane. More precisely, if L-N((1-2)) = Sigma(N)(n=1) 1({Sn(1)=Sn(2)}), then pi/logN L-N((1, 2)) converges in dis...
-
作者：Qi, Weiwei; Shen, Zhongwei; Yi, Yingfei
作者单位：University of Alberta; Jilin University
摘要：The present paper is devoted to the investigation of an important family of absorbed singular diffusion processes exhibiting long transient dynamics, namely, interesting and important dynamical behaviours over long but finite time scales. We explore the multiscale dynamics by establishing the asymptotic distribution of the normalized extinction time, the asymptotic reciprocal relationship between the mean extinction time and the principal eigenvalue of the generator, and a sophisticated multis...
-
作者：Cipolloni, Giorgio; Erdos, Laszlo; Schroder, Dominik
作者单位：Princeton University; Institute of Science & Technology - Austria; Swiss Federal Institutes of Technology Domain; ETH Zurich
摘要：We prove that the mesoscopic linear statistics Sigma(i)f (n(a)(sigma(i) - z(0))) of the eigenvalues {sigma(i)}(i) of large nxn non-Hermitian random matrices with complex centred i.i.d. entries are asymptotically Gaussian for any H-0(2) -functions f around any point z0 in the bulk of the spectrum on any mesoscopic scale 0 < a < 1/2. This extends our previous result (Cipolloni et al. in Commun Pure Appl Math, 2019. arXiv:1912.04100), that was valid on the macroscopic scale, a = 0, to cover the e...
-
作者：Neufeld, Leonie
作者单位：University of Bielefeld
摘要：We study weighted sums of free identically distributed self-adjoint random variables with weights chosen randomly from the unit sphere and show that the Kolmogorov distance between the distribution of such a weighted sum and Wigner's semicircle law is of order n(-1/2) with high probability. Replacing the Kolmogorov distance by a weaker pseudometric, we obtain a rate of convergence of order n(-1), thus providing a free analog of the Klartag-Sodin result in classical probability theory. Moreover...
-
作者：Gouezel, Sebastien; Rousseau, Jerome; Stadlbauer, Manuel
作者单位：Universite de Rennes; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universidade Federal da Bahia
摘要：We study the minimal distance between two orbit segments of length n, in a random dynamical system with sufficiently good mixing properties. This problem has already been solved in non-random dynamical system, and on average in random dynamical systems (the so-called annealed version of the problem): it is known that the asymptotic behavior for this question is given by a dimension-like quantity associated to the invariant measure, called correlation dimension (or R & eacute;nyi entropy). We s...
-
作者：Deuschel, Jean-Dominique; Rodriguez, Pierre-Francois
作者单位：Technical University of Berlin; Imperial College London
摘要：We introduce a natural measure on bi-infinite random walk trajectories evolving in a time-dependent environment driven by the Langevin dynamics associated to a gradient Gibbs measure with convex potential. We derive an identity relating the occupation times of the Poissonian cloud induced by this measure to the square of the corresponding gradient field, which - generically - is not Gaussian. In the quadratic case, we recover a well-known generalization of the second Ray-Knight theorem. We fur...
-
作者：Kwasnicki, Mateusz
作者单位：Wroclaw University of Science & Technology
摘要：We prove a non-extinction result for Fleming-Viot-type systems of two particles with dynamics described by an arbitrary symmetric Hunt process under the assumption that the reference measure is finite. Additionally, we describe an invariant measure for the system, we discuss its ergodicity, and we prove that the reference measure is a stationary measure for the embedded Markov chain of positions of the surviving particle at successive branching times.
-
作者：Boedihardjo, H.; Geng, X.
作者单位：University of Warwick; University of Melbourne
摘要：We show that the tail probability of the rough line integral f(0)(1) phi(X-t)dY(t), where (X, Y) is a 2D fractional Brownian motion with Hurst parameter H is an element of (1/4, 1/2) and phi is a C-b(infinity)-function satisfying a mild non-degeneracy condition on its derivative, cannot decay faster than a gamma-Weibull tail with any exponent gamma > 2H + 1. In particular, this produces a simple class of examples of differential equations driven by fBM, whose solutions fail to have Gaussian ta...
-
作者：Huang, Jiaoyang; Mcswiggen, Colin
作者单位：University of Pennsylvania; New York University
摘要：This paper studies the asymptotic behavior of several central objects in Dunkl theory as the dimension of the underlying space grows large. Our starting point is the observation that a recent result from the random matrix theory literature implies a large deviations principle for the hydrodynamic limit of radial Dunkl processes. Using this fact, we prove a variational formula for the large-N asymptotics of generalized Bessel functions, as well as a large deviations principle for the more gener...
-
作者：Bazaes, Rodrigo; Lammers, Isabel; Mukherjee, Chiranjib
作者单位：University of Munster
摘要：We construct and study properties of an infinite dimensional analog of Kahane's theory of Gaussian multiplicative chaos (Kahane in Ann Sci Math Quebec 9(2):105-150, 1985). Namely, if H-T (omega) is a random field defined w.r.t. space-time white noise (B) over dot and integrated w.r.t. Brownian paths in d >= 3, we consider the renormalized exponential mu(gamma),T, weighted w.r.t. the Wiener measure P-0(d omega). We construct the almost sure limit mu(gamma) = lim(T ->) (infinity) (mu gamma,T) in...
