Spectral radii of truncated circular unitary matrices

Consider a truncated circular unitary matrix which is a pn by pn submatrix of an n by n circular unitary matrix by deleting the last n−pn 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 pn/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). pn→∞ and pn/n→0 as n→∞; (2). (n−pn)/n→0 and (n−pn)/(log⁡n)3→∞ as n→∞; (3). n−pn→∞ and (n−pn)/log⁡n→0 as n→∞ and (4). n−pn=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.