Appears in collection : Les probabilités de demain 2017

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): the leading terms of the expansion are either related to free probability theory or to the one-dimensional Gaussian free field.