SPECTRUM OF LARGE RANDOM REVERSIBLE MARKOV CHAINS: HEAVY-TAILED WEIGHTS ON THE COMPLETE GRAPH

• 成果类型: Article
• 署名作者: Bordenave, Charles; Caputo, Pietro; Chafai, Djalil
• 署名单位: Universite Federale Toulouse Midi-Pyrenees (ComUE); Universite de Toulouse; Institut National des Sciences Appliquees de Toulouse; Universite Toulouse III - Paul Sabatier; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Roma Tre University; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Universite Gustave-Eiffel
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP587
• 发表日期: 2011
• 页码: 1544-1590
• 关键词: random matrices; LARGEST EIGENVALUES; trap model; CONVERGENCE; statistics; LAW

摘要

We consider the random reversible Markov kernel K obtained by assigning i.i.d. nonnegative weights to the edges of the complete graph over n vertices and normalizing by the corresponding row sum. The weights are assumed to be in the domain of attraction of an a-stable law, alpha is an element of (0, 2). When 1 <= alpha < 2, we show that for a suitable regularly varying sequence kappa(n) of index 1 - 1/alpha, the limiting spectral distribution mu(alpha) of kappa(n) K coincides with the one of the random symmetric matrix of the un-normalized weights (Levy matrix with i.i.d. entries). In contrast, when 0 < alpha < 1, we show that the empirical spectral distribution of K converges without rescaling to a nontrivial law (mu) over tilde (alpha) supported on [-1, 1], whose moments are the return probabilities of the random walk on the Poisson weighted infinite tree (PWIT) introduced by Aldous. The limiting spectral distributions are given by the expected value of the random spectral measure at the root of suitable self-adjoint operators defined on the PWIT. This characterization is used together with recursive relations on the tree to derive some properties of mu a and (mu) over tilde (alpha). We also study the limiting behavior of the invariant probability measure of K.