LOCAL SINGLE RING THEOREM ON OPTIMAL SCALE

• 成果类型: Article
• 署名作者: Bao, Zhigang; Erdos, Laszlo; Schnelli, Kevin
• 署名单位: Hong Kong University of Science & Technology; Institute of Science & Technology - Austria; Royal Institute of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/18-AOP1284
• 发表日期: 2019
• 页码: 1270-1334
• 关键词: random matrices; Spectral Distribution; LAW; UNIVERSALITY; SUBORDINATION; CONVERGENCE; convolution

摘要

Let U and V be two independent N by N random matrices that are distributed according to Haar measure on U(N). Let Sigma be a nonnegative deterministic N by N matrix. The single ring theorem [Ann. of Math. (2) 174 (2011) 1189-1217] asserts that the empirical eigenvalue distribution of the matrix X : = U Sigma V* converges weakly, in the limit of large N, to a deterministic measure which is supported on a single ring centered at the origin in C. Within the bulk regime, that is, in the interior of the single ring, we establish the convergence of the empirical eigenvalue distribution on the optimal local scale of order N-1/2+epsilon and establish the optimal convergence rate. The same results hold true when U and V are Haar distributed on O(N).