The Statistics of Spectral Shifts due to Finite Rank Perturbations

TL;DR

This paper investigates how finite-rank perturbations affect the spectra of Hermitian random matrices, deriving universal formulas for spectral shifts and analyzing their dependence on perturbation strength, matrix size, and ensemble class.

Contribution

It provides new universal expressions for spectral shift variances under finite-rank perturbations across different random matrix ensembles.

Findings

Universal formulas for spectral shift variance in Dyson's ensembles

Dependence of spectral shifts on perturbation rank and matrix size

Analysis of different types of finite-rank perturbations

Abstract

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 $\tau=t/N$ 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 $N$.