The local circular law II: the edge case

• 成果类型: Article
• 署名作者: Bourgade, Paul; Yau, Horng-Tzer; Yin, Jun
• 署名单位: Harvard University; University of Wisconsin System; University of Wisconsin Madison
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-013-0516-x
• 发表日期: 2014
• 页码: 619-660
• 关键词: random matrices; gaussian fluctuation; eigenvalues; INVERTIBILITY; UNIVERSALITY

摘要

In the first part of this article (Bourgade et al. arXiv:1206.1449, 2012), we proved a local version of the circular law up to the finest scale for non-Hermitian random matrices at any point with for any independent of the size of the matrix. Under the main assumption that the first three moments of the matrix elements match those of a standard Gaussian random variable after proper rescaling, we extend this result to include the edge case . Without the vanishing third moment assumption, we prove that the circular law is valid near the spectral edge up to scale .