A LARGE DEVIATION PRINCIPLE FOR WIGNER MATRICES WITHOUT GAUSSIAN TAILS

• 成果类型: Article
• 署名作者: Bordenave, Charles; Caputo, Pietro
• 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); Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Roma Tre University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP866
• 发表日期: 2014
• 页码: 2454-2496
• 关键词: semicircular distribution; free convolution; Random graphs

摘要

We consider n x n Hermitian matrices with i.i.d. entries X-ij whose tail probabilities P(vertical bar X-ij vertical bar >= t) behave like e(-at alpha) for some a > 0 and alpha is an element of (0, 2). We establish a large deviation principle for the empirical spectral measure of X/root n with speed n(1+alpha/2) with a good rate function J(mu) that is finite only if mu is of the form mu = mu(sc) boxed plus nu for some probability measure nu on R, where boxed plus denotes the free convolution and mu(sc) is Wigner's semicircle law. We obtain explicit expressions for J(mu(sc) boxed plus nu) in terms of the alpha th moment of nu. The proof is based on the analysis of large deviations for the empirical distribution of very sparse random rooted networks.