Finite rank perturbations of random matrices and free probability theory

Abstract : The question adressed in this talk is the following one : how are the extreme eigenvalues of a matrix X moved by a small rank perturbation P of X ?
We shall consider this question in its generic apporach, i.e. when the matrices X and P are chosen at random independently and in isotropic ways.
We shall give a general answer, uncovering a remarkable phase transition phenomenon: the limit of the extreme eigenvalues of the perturbed matrix differs from the original matrix if and only if the eigenvalues of the perturbing matrix are above a certain critical threshold. We also examine the consequences of this eigenvalue phase transition on the associated eigenvectors and generalize our results to examine the case of multiplicative perturbations or of additive perturbations for the singular values of rectangular matrices.