Convergence of the spectral measure of non-normal matrices

We discuss regularization by noise of the spectrum of large random non-normal matrices. Under suitable conditions, we show that the regularization of a sequence of matrices that converges in *-moments to a regular element a by the addition of a polynomially vanishing Gaussian Ginibre matrix forces the empirical measure of eigenvalues to converge to the Brown measure of a.
France. Agence nationale de la recherche (Project ANR-08-BLAN-0311-01)