  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 $N$.
