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The statistics of spectral shifts due to finite rank perturbations
Journal of Physics A: Mathematical and Theoretical ( IF 2.1 ) Pub Date : 2020-12-09 , DOI: 10.1088/1751-8121/abc9da
Barbara Dietz 1 , Holger Schanz 2 , Uzy Smilansky 3 , Hans Weidenmller 4
Affiliation  

This article is dedicated to the following class of problems. Start with an N 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 ⩽ tN, 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.



中文翻译:

有限秩扰动引起的谱位移统计

本文致力于以下类别的问题。从一个N N Hermitian矩阵开始,该矩阵是从一个矩阵集合(参考矩阵)中随机选择的。施加秩扰动给它,用取的值1⩽Ñ,我们研究了扰动的谱和参考矩阵作为的函数之间的差和其上的底层普遍性类随机的依赖性矩阵合奏。我们考虑两种较弱的扰动,它们t个 对角元素进行排列或随机化,而较强的扰动t 行和列依次进行随机化。在第一种情况下,我们推导出缩放参数τ = t / N的通用表达式,以期望频谱偏移函数的方差,选择戴森的三个高斯合奏作为随机矩阵合奏。在第二种情况下,我们发现对矩阵大小N的附加依赖性。

更新日期:2020-12-09
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