Structured perturbations of tridiagonal twisted Toeplitz matrices

TL;DR

This paper investigates the eigenvalue distribution of perturbed non-Hermitian twisted Toeplitz matrices, revealing a unique limiting measure different from classical predictions, with potential extensions to banded matrices.

Contribution

It introduces the limiting eigenvalue distribution for non-Hermitian twisted Toeplitz matrices under small random perturbations, extending understanding beyond classical Toeplitz cases.

Findings

Eigenvalues follow a specific two-dimensional limiting distribution.

The distribution differs from the push-forward of Lebesgue measure by the symbol.

Results suggest possible extensions to banded non-Hermitian twisted Toeplitz matrices.

Abstract

Twisted Toeplitz matrices constitute a generalization of Toeplitz matrices in the sense that the entries on each diagonal no longer need to be constant, but are given by the values of a continuous function on a partition of $[0,1]$. We study the limiting statistical distribution of the eigenvalues of matrices of the form $R_n(a) = T_n(a) + \sigma_n X_n$, where $T_n(a)$ is a sequence of non-Hermitian tridiagonal twisted Toeplitz matrices, $X_n$ is a sequence of tridiagonal random matrices whose entries have mean $0$ and finite variance, and $\sigma_n\to0$. The limiting distribution turns out to be a two-dimensional measure which is in general different from the push-forward of the Lebesgue measure by the symbol. We also explain how the results could extend to banded non-Hermitian twisted Toeplitz matrices.