SHARP NONASYMPTOTIC BOUNDS ON THE NORM OF RANDOM MATRICES WITH INDEPENDENT ENTRIES

• 成果类型: Article
• 署名作者: Bandeira, Afonso S.; van Handel, Ramon
• 署名单位: Princeton University; Princeton University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1025
• 发表日期: 2016
• 页码: 2479-2506
• 关键词: INEQUALITIES

摘要

This bound is optimal in the sense that a matching lower bound holds under mild assumptions, and the constants are sufficiently sharp that we can often capture the precise edge of the spectrum. Analogous results are obtained for rectangular matrices and for more general sub-Gaussian or heavy-tailed distributions of the entries, and we derive tail bounds in addition to bounds on the expected norm. The proofs are based on a combination of the moment method and geometric functional analysis techniques. As an application, we show that our bounds immediately yield the correct phase transition behavior of the spectral edge of random band matrices and of sparse Wigner matrices. We also recover a result of Seginer on the norm of Rademacher matrices.