Eigenvector distribution of Wigner matrices

• 成果类型: Article
• 署名作者: Knowles, Antti; Yin, Jun
• 署名单位: Harvard University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-011-0407-y
• 发表日期: 2013
• 页码: 543-582
• 关键词: semicircle law; Orthogonal polynomials; UNIVERSALITY; asymptotics; delocalization; eigenvalues; spectrum; Respect; edge

摘要

We consider N x N Hermitian or symmetric random matrices with independent entries. The distribution of the (i, j)-th matrix element is given by a probability measure nu (ij) whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution nu (ij) coincide with those of the corresponding Gaussian ensemble. More generally, we prove that the joint eigenvector-eigenvalue distributions near the spectral edge of two generalized Wigner ensembles agree, provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate. If in addition the first four moments match then this result holds also in the bulk.