Toeplitz band matrices with small random perturbations

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