Bulk universality for generalized Wigner matrices with few moments

• 成果类型: Article
• 署名作者: Aggarwal, Amol
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-018-0836-y
• 发表日期: 2019
• 页码: 375-432
• 关键词: local semicircle law; spectral statistics; delocalization; CONVERGENCE; eigenvalues; energy

摘要

In this paper we consider NxN real generalized Wigner matrices whose entries are only assumed to have finite (2+epsilon)th moment for some fixed, but arbitrarily small, epsilon>0. We show that the Stieltjes transforms mN(z) of these matrices satisfy a weak local semicircle law on the nearly smallest possible scale, when =I(z) is almost of order N-1. As a consequence, we establish bulk universality for local spectral statistics of these matrices at fixed energy levels, both in terms of eigenvalue gap distributions and correlation functions, meaning that these statistics converge to those of the Gaussian orthogonal ensemble in the large N limit.