论文标题
在光截断下圆形单位矩阵的限制光谱半径
Limiting Spectral Radii of Circular Unitary Matrices under Light Truncation
论文作者
论文摘要
考虑一个截断的圆形统一矩阵，该矩阵是$ p_n $ by $ p_n $ subsatrix of $ n $ by $ n $ cundular unimare矩阵后，删除了最后的$ n-p_n $ columns and lows。 Jiang和Qi \ Cite {Jiangqi2017}以及GUI和Qi \ cite {gq2018}研究了截短基质的特征值（称为光谱半径）的最大绝对值的限制分布。在以下条件下，获得了截短的圆形统一基质的光谱半径的一些限制分布：（1）。 $ p_n/n $远离$ 0 $和$ 1 $； （2）。 $ p_n \ to \ to \ infty $和$ p_n/n \ to 0 $ as $ n \ to \ infty $; （3）。 $（n-p_n）/n \ to 0 $和$（n-p_n）/（\ log n）^3 \ to \ infty $ as $ n \ to \ infty $; （4）。 $ n-p_n \ to \ infty $和$（n-p_n）/\ log n \ to 0 $ as $ n \ to \ to \ infty $; （5）。 $ n-p_n = k \ ge 1 $是固定整数。光谱半径在前四个条件下分布与牙龈分布收敛，并在第五条件下逆转了瓦布尔分布。显然，当$ n-p_n $属于$ \ log n $和$（\ log n）^3 $之间的订单时，以上条件不涵盖案例。在本文中，我们证明，在这种情况下，光谱半径也以GUI和QI \ cite {GQ2018}的猜想收敛到牙龈分布。
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}.
