LOCAL SINGLE RING THEOREM

• 成果类型: Article
• 署名作者: Benaych-Georges, Florent
• 署名单位: Universite Paris Cite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/16-AOP1151
• 发表日期: 2017
• 页码: 3850-3885
• 关键词: free additive convolution; RANDOM MATRICES; circular law; SUBORDINATION; UNIVERSALITY; probability

摘要

The single ring theorem, by Guionnet, Krishnapur and Zeitouni in Ann. of Math. (2) 174 (2011) 1189-1217, describes the empirical eigenvalue distribution of a large generic matrix with prescribed singular values, that is, an N x N matrix of the form A = UTV, with U, V some independent Haardistributed unitary matrices and T a deterministic matrix whose singular values are the ones prescribed. In this text, we give a local version of this result, proving that it remains true at the microscopic scale (logN)(-1/4). On our way to prove it, we prove a matrix subordination result for singular values of sums of non-Hermitian matrices, as Kargin did in Ann. Probab. 43 (2015) 2119-2150 for Hermitian matrices. This allows to prove a local law for the singular values of the sum of two non-Hermitian matrices and a delocalization result for singular vectors.