LOCALIZATION OF EIGENVECTORS OF NONHERMITIAN BANDED NOISY TOEPLITZ MATRICES

Abstract

We prove localization with high probability on sets of size of order N/logN for the eigenvectors of nonhermitian finitely banded N × N Toeplitz matrices PN subject to small random perturbations, in a very general setting. As perturbation, we consider N × N random matrices with independent entries of zero mean, finite moments, and which satisfy an appropriate anticoncentration bound. We show via a Grushin problem that an eigenvector for a given eigenvalue z is well approximated by a random linear combination of the singular vectors of PN -z corresponding to its small singular values. We prove precise probabilistic bounds on the local distribution of the eigenvalues of the perturbed matrix and provide a detailed analysis of the singular vectors to conclude the localization result.