Large deviations of the extreme eigenvalues of random deformations of matrices

Consider a real diagonal deterministic matrix X (n) of size n with spectral measure converging to a compactly supported probability measure. We perturb this matrix by adding a random finite rank matrix, with delocalized eigenvectors. We show that the joint law of the extreme eigenvalues of the perturbed model satisfies a large deviation principle in the scale n, with a good rate function given by a variational formula. We tackle both cases when the extreme eigenvalues of X (n) converge to the edges of the support of the limiting measure and when we allow some eigenvalues of X (n) , that we call outliers, to converge out of the bulk. We can also generalise our results to the case when X (n) is random, with law proportional to e (-n Tr V(X)) dX, for V growing fast enough at infinity and any perturbation of finite rank.