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→∞ and p_n/n→0 as n→∞; (2). (n−p_n)/n→0 and (n−p_n)/(log⁡n)³→∞ as n→∞; (3). n−p_n→∞ and (n−p_n)/log⁡n→0 as n→∞ 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.