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 $a$. 
Alice Guionnet, Philip Matchett Wood, Ofer Zeitouni
math.PR