  We provide a perturbative expansion for the empirical spectral distribution
of a Hermitian matrix with large size perturbed by a random matrix with small
operator norm whose entries in the eigenvector basis of the first one are
independent with a variance profile. We prove that, depending on the order of
magnitude of the perturbation, several regimes can appear, called perturbative
and semi-perturbative regimes. Depending on the regime, the leading terms of
the expansion are either related to the one-dimensional Gaussian free field or
to free probability theory.
