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 (2017) 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). $p_n\to\infty$ and $p_n/n\to 0$ as $n\to\infty$; (2). $(n-p_n)/n\to 0$ and $(n-p_n)/(\log n)^3\to\infty$ as $n\to\infty$; (3). $n-p_n\to\infty$ and $(n-p_n)/\log n\to 0$ as $n\to\infty$ and (4). $n-p_n=k\ge 1$ 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.