Spectral radii of truncated circular unitary matrices

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:458

Article

Gui, Wenhao¹  Qi, Yongcheng² 

[1] Beijing Jiaotong Univ, Dept Math, Beijing 100044, Peoples R China

[2] Univ Minnesota, Dept Math & Stat, Duluth, MN 55812 USA

关键词: Spectral radius;    Eigenvalue;    Limiting distribution;    Extreme value;    Circular unitary matrix;   

DOI  :  10.1016/j.jmaa.2017.09.030

来源: Elsevier

【 摘 要 】

Consider a truncated circular unitary matrix which is a p(n) by p(n) submatrix of an n by n. circular unitary matrix by deleting the last n - p(n) columns and rows. Jiang and Qi [11] proved that the maximum absolute value of the eigenvalues (known as spectral radius) of the truncated matrix, after properly normalized, converges in distribution to the Gumbel distribution if p(n)/n is bounded away from 0 and 1. In this paper we investigate the limiting distribution of the spectral radius under one of the following four conditions: (1). p(n) -> infinity and p(n)/n -> 0 as n -> infinity; (2). (n - p(n))/n -> 0 and (n - p(n))/(logn)(3) -> infinity as n -> infinity; (3). n - p(n) -> infinity and (n - p(n))/log n -> 0 as n -> infinity and (4). n p(n) = k >= 1 is a fixed integer. We prove that the spectral radius converges in distribution to the Gumbel distribution under the first three conditions and to a reversed Weibull distribution under the fourth condition. (C) 2017 Elsevier Inc. All rights reserved.

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