Random matrices: tail bounds for gaps between eigenvalues

• 成果类型: Article
• 署名作者: Hoi Nguyen; Tao, Terence; Van Vu
• 署名单位: University System of Ohio; Ohio State University; University of California System; University of California Los Angeles; Yale University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-016-0693-5
• 发表日期: 2017
• 页码: 777-816
• 关键词: littlewood-offord theorems; smallest singular-value; geometric diffusions; structure definition; harmonic-analysis; condition number; UNIVERSALITY; statistics; INVERTIBILITY; graphs

摘要

Gaps (or spacings) between consecutive eigenvalues are a central topic in random matrix theory. The goal of this paper is to study the tail distribution of these gaps in various random matrix models. We give the first repulsion bound for random matrices with discrete entries and the first super-polynomial bound on the probability that a random graph has simple spectrum, along with several applications.