The statistics of spectral shifts due to finite rank perturbations

Abstract

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 ⩽ t ⩽ 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.

The Statistics of Spectral Shifts due to Finite Rank Perturbations

“…Heuristic arguments supported by some estimates suggest that in this case the variance of the spectral shift due to interchanging or randomizing t diagonal elements grows linearly with τ . Numerical simulations confirm this intuition [26]. In other words, the Poissonian nature of the spectrum seems to lead to standard diffusion of the spectral shift.…”