EIGENVECTOR STATISTICS OF LEVY MATRICES

• 成果类型: Article
• 署名作者: Aggarwal, Amol; Lopatto, Patrick; Marcinek, Jake
• 署名单位: Columbia University; Institute for Advanced Study - USA; Harvard University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/20-AOP1493
• 发表日期: 2021
• 页码: 1778-1846
• 关键词: spectral statistics; bulk universality; LARGEST EIGENVALUES; local statistics; wigner matrices; band; delocalization; localization; CONVERGENCE; energy

摘要

We analyze statistics for eigenvector entries of heavy-tailed random symmetric matrices (also called Levy matrices) whose associated eigenvalues are sufficiently small. We show that the limiting law of any such entry is nonGaussian, given by the product of a normal distribution with another random variable that depends on the location of the corresponding eigenvalue. Although the latter random variable is typically nonexplicit, for the median eigenvector it is given by the inverse of a one-sided stable law. Moreover, we show that different entries of the same eigenvector are asymptotically independent, but that there are nontrivial correlations between eigenvectors with nearby eigenvalues. Our findings contrast sharply with the known eigenvector behavior for Wigner matrices and sparse random graphs.