Limiting Spectral Radii of Circular Unitary Matrices under Light Truncation

Consider a truncated circular unitary matrix which is a $p_n$ by $p_n$ submatrix of an $n$ by $n$ circular unitary matrix after deleting the last $n-p_n$ columns and rows. Jiang and Qi \cite{JiangQi2017} and Gui and Qi \cite{GQ2018} study the limiting distributions of the maximum absolute value of the eigenvalues (known as spectral radius) of the truncated matrix. Some limiting distributions for the spectral radius for the truncated circular unitary matrix have been obtained under the following conditions: (1). $p_n/n$ is bounded away from $0$ and $1$; (2). $p_n\to\infty$ and $p_n/n\to 0$ as $n\to\infty$; (3). $(n-p_n)/n\to 0$ and $(n-p_n)/(\log n)^3\to\infty$ as $n\to\infty$; (4). $n-p_n\to\infty$ and $(n-p_n)/\log n\to 0$ as $n\to\infty$; and (5). $n-p_n=k\ge 1$ is a fixed integer. The spectral radius converges in distribution to the Gumbel distribution under the first four conditions and to a reversed Weibull distribution under the fifth condition. Apparently, the conditions above do not cover the case when $n-p_n$ is of order between $\log n$ and $(\log n)^3$. In this paper, we prove that the spectral radius converges in distribution to the Gumbel distribution as well in this case, as conjectured by Gui and Qi \cite{GQ2018}.