Localization of eigenvectors of non-Hermitian banded noisy Toeplitz matrices

TL;DR

This paper demonstrates that eigenvectors of non-Hermitian banded Toeplitz matrices with small random perturbations are highly localized, with high probability, on sets of size roughly N divided by log N, using probabilistic and spectral analysis techniques.

Contribution

It establishes the localization of eigenvectors for perturbed non-Hermitian Toeplitz matrices in a very general setting, extending understanding of eigenvector behavior under randomness.

Findings

Eigenvectors are localized on sets of size approximately N/log N.

Probabilistic bounds on eigenvalue distribution are established.

Singular vectors of the shifted matrix approximate eigenvectors well.

Abstract

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.