BULK EIGENVALUE FLUCTUATIONS OF SPARSE RANDOM MATRICES

• 成果类型: Article
• 署名作者: He, Yukun
• 署名单位: University of Zurich
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/20-AAP1575
• 发表日期: 2020
• 页码: 2846-2879
• 关键词: gaussian fluctuations; spectral statistics; UNIVERSALITY; delocalization; LAW

摘要

We consider a class of sparse random matrices, which includes the adjacency matrix of Erdos-Renyi graphs G(N, p) for p is an element of [N epsilon-1, N-epsilon]. We identify the joint limiting distributions of the eigenvalues away from 0 and the spectral edges. Our result indicates that unlike Wigner matrices, the eigenvalues of sparse matrices satisfy central limit theorems with normalization N root p. In addition, the eigenvalues fluctuate simultaneously: the correlation of two eigenvalues of the same/different sign is asymptotically 1/-1. We also prove CLTs for the eigenvalue counting function and trace of the resolvent at mesoscopic scales.