THE LARGEST EIGENVALUES OF FINITE RANK DEFORMATION OF LARGE WIGNER MATRICES: CONVERGENCE AND NONUNIVERSALITY OF THE FLUCTUATIONS

• 成果类型: Article
• 署名作者: Capitaine, Mireille; Donati-Martin, Catherine; Feral, Delphine
• 署名单位: Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Sorbonne Universite; Sorbonne Universite; Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Centre National de la Recherche Scientifique (CNRS); Inria; Universite de Bordeaux
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/08-AOP394
• 发表日期: 2009
• 页码: 1-47
• 关键词: limiting spectral distribution; distributions; UNIVERSALITY; edge

摘要

In this paper, we investigate the asymptotic spectrum of complex or real Deformed Wigner matrices (M-N)(N) defined by M-N = W-N/root N + A(N) where W-N is all N x N Hermitian (resp., symmetric) Wigner matrix whose entries have a symmetric law satisfying a Poincare inequality. The matrix A(N) is Hermitian (resp., symmetric) and deterministic with all but finitely many eigenvalues equal to zero. We first show that, as soon as the first largest or last smallest eigenvalues of A(N) are sufficiently far from zero, the corresponding eigenvalues of M-N almost surely exit the limiting semicircle compact support as the size N becomes large. The corresponding limits are universal in the sense that they only involve the variance of the entries of W-N. On the other hand, when A(N) is diagonal with a sole simple nonnull eigenvalue large enough, we prove that the fluctuations of the largest eigenvalue are not universal and vary with the particular distribution of the entries of W-N.