The local circular law III: general case

• 成果类型: Article
• 署名作者: Yin, Jun
• 署名单位: University of Wisconsin System; University of Wisconsin Madison
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-013-0539-3
• 发表日期: 2014
• 页码: 679-732
• 关键词: random matrices; INVERTIBILITY; UNIVERSALITY; eigenvalues

摘要

In the first part (Bourgade et al., Local circular law for random matrices, preprint, arXiv: 1206.1449, 2012) of this article series, Bourgade, Yau and the author of this paper proved a local version of the circular law up to the finest scale N-1/2+epsilon for non-Hermitian random matrices at any point z is an element of C with parallel to z vertical bar-1 vertical bar > c for any constant c > 0 independent of the size of the matrix. In the second part (Bourgade et al., The local circular law II: the edge case, preprint, arXiv: 1206.3187, 2012), they extended this result to include the edge case vertical bar z vertical bar-1 = o(1), under themain assumption that the third moments of the matrix elements vanish. (Without the vanishing third moment assumption, they proved that the circular law is valid near the spectral edge vertical bar z vertical bar - 1 = o(1) up to scale N-1/4+epsilon.) In this paper, we will remove this assumption, i.e. we prove a local version of the circular law up to the finest scale N-1/2+epsilon for non-Hermitian random matrices at any point z is an element of C.