Spectral radii of truncated circular unitary matrices

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.