论文标题
由于有限等级扰动而引起的光谱变化的统计数据
The Statistics of Spectral Shifts due to Finite Rank Perturbations
论文作者
论文摘要
本文致力于以下问题类。从从矩阵集合 - 参考矩阵中随机选择的$ n \ times n $ hermitian矩阵开始。在其上应用等级$ t $扰动，$ t $将值$ 1 \ le t \ le n $，我们研究了扰动矩阵的光谱与参考矩阵的差异，这是$ t $的函数及其对随机Matrix Ensemble的基本普遍性类别的依赖。我们认为，这两者都较弱的扰动，它可以将$ t $对角线元素排列或随机化，并且更强的扰动随机化$ t $行和列。在第一种情况下，我们在缩放参数$τ= t/n $中得出通用表达式，以期期望光谱移位功能的方差，选择为随机 - 矩阵合奏Dyson的三个高斯共e剂。在第二种情况下，我们发现对矩阵尺寸$ n $的额外依赖性。
This article is dedicated to the following class of problems. Start with an $N\times N$ Hermitian matrix randomly picked from a matrix ensemble - the reference matrix. Applying a rank-$t$ perturbation to it, with $t$ taking the values $1\le t \le N$, we study the difference between the spectra of the perturbed and the reference matrices as a function of $t$ and its dependence on the underlying universality class of the random matrix ensemble. We consider both, the weaker kind of perturbation which either permutes or randomizes $t$ diagonal elements and a stronger perturbation randomizing successively $t$ rows and columns. In the first case we derive universal expressions in the scaled parameter $τ=t/N$ for the expectation of the variance of the spectral shift functions, choosing as random-matrix ensembles Dyson's three Gaussian ensembles. In the second case we find an additional dependence on the matrix size $N$.
