Toeplitz band matrices with small random perturbations

Abstract We study the spectra of N × N Toeplitz band matrices perturbed by small complex Gaussian random matrices, in the regime N ≫ 1 . We prove a probabilistic Weyl law, which provides a precise asymptotic formula for the number of eigenvalues in certain domains, which may depend on N , with probability sub-exponentially (in N ) close to 1. We show that most eigenvalues of the perturbed Toeplitz matrix are at a distance of at most O ( N − 1 + e ) , for all e > 0 , to the curve in the complex plane given by the symbol of the unperturbed Toeplitz matrix.