RANDOM COVARIANCE MATRICES: UNIVERSALITY OF LOCAL STATISTICS OF EIGENVALUES

• 成果类型: Article
• 署名作者: Tao, Terence; Vu, Van
• 署名单位: University of California System; University of California Los Angeles; Rutgers University System; Rutgers University New Brunswick
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP648
• 发表日期: 2012
• 页码: 1285-1315
• 关键词: bulk universality; semicircle law; delocalization; CONVERGENCE; ensembles

摘要

We study the eigenvalues of the covariance matrix 1/n M*M of a large rectangular matrix M = M-n,M-p = (zeta(ij))(1 <= i <= p;1 <= j <= n) whose entries are i.i.d. random variables of mean zero, variance one, and having finite C(0)th moment for some sufficiently large constant C-0. The main result of this paper is a Four Moment theorem for i.i.d. covariance matrices (analogous to the Four Moment theorem for Wigner matrices established by the authors in [Acta Math. (2011) Random matrices: Universality of local eigenvalue statistics] (see also [Comm. Math. Phys. 298 (2010) 549-572])). We can use this theorem together with existing results to establish universality of local statistics of eigenvalues under mild conditions. As a byproduct of our arguments, we also extend our previous results on random Hermitian matrices to the case in which the entries have finite C(0)th moment rather than exponential decay.