Convergence of the spectral measure of non-normal matrices

Abstract

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. * UMPA, CNRS UMR 5669, ENS Lyon, 46 allée d'Italie,

Regularization of Non-Normal Matrices by Gaussian Noise

“…When Gaussian noise is added to T N , it is a consequence of [GWZ14] that T N + N −γ G N for γ > 1 /2 has empirical spectral measure converging weakly to the uniform distribution on the unit circle. One way of explaining why this limiting distribution appears is through the notion of * -moment convergence.…”

Large bidiagonal matrices and random perturbations

“…A recent result by A. Guionnet, P. Matched Wood and O. Zeitouni [10] implies that when the coupling constant is bounded from above and from below by (different) sufficiently negative powers of N , then the normalized counting measure of eigenvalues of the randomly perturbed Jordan block converges weakly in probability to the uniform measure on S 1 as the dimension of the matrix gets large. In [15], J. Sjöstrand obtained a probabilistic circular Weyl law for most of the eigenvalues of large Jordan block matrices subject to small random perturbations, and in [17], we obtained a precise asymptotic formula for the average density of the residual eigenvalues in the interior of a circle, where the result of Davies and Hager yielded a logarithmic upper bound on the number of eigenvalues.…”

Spectral statistics of non-selfadjoint operators subject to small random perturbations

“…Let α be a complex-valued random variables defined on (M, F, P) such that 14) where ε 0 > 0 is an arbitrarily small but fixed constant. As a consequence, the Markov inequality implies the following tail estimate: there exists a constant κ α > 0 such that…”

“…One can calculate the Lebesgue densities of the limiting 1-point measures µ 1 , which is given by 14) which is precise the average density of eigenvalues at z 0 which one would expect from the probabilistic Weyl law in Theorem 4, see also (2.1).…”

“…The case of large non-selfadjoint deterministic matrices perturbed by random matrices with coupling constants of order 1 where studied by Bordenave and Capitaine [4]. The regime of small coupling constants tending to 0 as the dimension of the matrix gets large, have been considered by Davies and Hager [8], Guionnet, Matchett-Wood and Zeitouni [14] and Sjöstrand and Vogel [28,27,29] in the case of large non-selfadjoint Toeplitz matrices.…”