The single ring theorem

• 成果类型: Article
• 署名作者: Guionnet, Alice; Krishnapur, Manjunath; Zeitouni, Ofer
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2011.174.2.10
• 发表日期: 2011
• 页码: 1189-1217
• 关键词: random matrices; circular law; UNIVERSALITY

摘要

We study the empirical measure LA of the eigenvalues of nonnormal square matrices of the form A(n) = UnTnVn, with U-n, V-n independent Haar distributed on the unitary group and T-n diagonal. We show that when the empirical measure of the eigenyalues of T-n converges, and T-n satisfies some technical conditions, L-An converges towards a rotationally invariant measure mu on the complex plane whose support is a single ring. In particular, we provide a complete proof of the Feinberg-Zee single ring theorem [6]. We also consider the case where U-n, V-n are independently Haar distributed on the orthogonal group.