Localization of eigenvectors of non-Hermitian banded noisy Toeplitz matrices

We prove localization with high probability on sets of size of order $N/\log N$ for the eigenvectors of non-Hermitian finitely banded $N\times N$ Toeplitz matrices $P_N$ subject to small random perturbations, in a very general setting. As perturbation we consider $N\times N$ random matrices with independent entries of zero mean, finite moments, and which satisfy an appropriate anti-concentration 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 $P_N-z$ 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.