Toeplitz band matrices with small random perturbations

We study the spectra of N x 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+epsilon), for all epsilon > 0, to the curve in the complex plane given by the symbol of the unperturbed Toeplitz matrix. (C) 2020 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.