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Abstract

This article is dedicated to the following class of problems. Start with
an N x 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 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.