论文标题
极端特征值和高斯单位合奏的排名一号非弱点变形的新兴异常值
Extreme eigenvalues and the emerging outlier in rank-one non-Hermitian deformations of the Gaussian Unitary Ensemble
论文作者
论文摘要
随机矩阵的复杂特征值$ j = \ text {gue}+iγ\ diag（1，0，\ ldots，0）$提供了最简单的模型，用于通过单个开放的通道从量子混乱系统中研究波浪散射中的共振。众所周知，在大型矩阵尺寸的极限下，$ n \ gg 1 $ $ j $的特征值密度在$γ= 1 $下进行突然重组，这是一个关键的阈值，除此之外，单个特征值异常（````broad Resonance''出现。我们提供了对这种重组过渡的详细说明，包括与关键区域宽度的$ n $缩放，围绕较高的阈值$γ= 1 $以及对真实零件（``共振位置''）的相关缩放和假想的零件（``共振宽度''的eigenvalues of eigenvalues的范围（'resonance Widths''与eigenvalues的相比，相比是实际量相比。在临界机构中，我们确定了这种极端特征值的密度，并展示了离群值如何逐渐与其他极端特征值分开。最后，我们描述了与相关的大偏差函数有关的大而有限的$ n $的特征值离群值的高度波动。
Complex eigenvalues of random matrices $J=\text{GUE }+ iγ\diag (1, 0, \ldots, 0)$ provide the simplest model for studying resonances in wave scattering from a quantum chaotic system via a single open channel. It is known that in the limit of large matrix dimensions $N\gg 1$ the eigenvalue density of $J$ undergoes an abrupt restructuring at $γ= 1$, the critical threshold beyond which a single eigenvalue outlier (``broad resonance'') appears. We provide a detailed description of this restructuring transition, including the scaling with $N$ of the width of the critical region about the outlier threshold $γ=1$ and the associated scaling for the real parts (``resonance positions'') and imaginary parts (``resonance widths'') of the eigenvalues which are farthest away from the real axis. In the critical regime we determine the density of such extreme eigenvalues, and show how the outlier gradually separates itself from the rest of the extreme eigenvalues. Finally, we describe the fluctuations in the height of the eigenvalue outlier for large but finite $N$ in terms of the associated large deviation function.
