Quantitative results for banded Toeplitz matrices subject to random and deterministic perturbations

TL;DR

This paper analyzes how the eigenvalues of banded Toeplitz matrices are affected by small random or deterministic perturbations, providing non-asymptotic spectral results and eigenvalue concentration properties.

Contribution

It introduces new non-asymptotic eigenvalue bounds, a rigidity result, and auxiliary tools for spectral analysis of perturbed non-normal matrices.

Findings

Eigenvalues concentrate around classical locations

Established a local law and convergence rate in Wasserstein distance

Provided bounds for singular values in adversarial models

Abstract

We consider the eigenvalues of a fixed, non-normal matrix subject to a small additive perturbation. In particular, we consider the case when the fixed matrix is a banded Toeplitz matrix, where the bandwidth is allowed to grow slowly with the dimension, and the perturbation matrix is drawn from one of several different random matrix ensembles. We establish a number of non-asymptotic results for the eigenvalues of this model, including a local law and a rate of convergence in Wasserstein distance of the empirical spectral measure to its limiting distribution. In addition, we define the classical locations of the eigenvalues and prove a rigidity result showing that, on average, the eigenvalues concentrate closely around their classical locations. While proving these results we also establish a number of auxiliary results that may be of independent interest, including a quantitative version of the Tao--Vu replacement principle, a general least singular value bound that applies to adversarial models, and a description of the limiting empirical spectral measure for random multiplicative perturbations.