We prove localization with high probability on sets of size of order
for the eigenvectors of nonhermitian finitely banded
Toeplitz
matrices
subject to small random perturbations, in a very general setting. As perturbation, we
consider
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
is
well approximated by a random linear combination of the singular vectors of
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.
Keywords
spectral theory, nonselfadjoint operators, random
perturbations