# Spectrum of random perturbations of Toeplitz matrices with finite symbols

## TL;DR

This paper studies how the eigenvalues of Toeplitz matrices with finite symbols are affected by small random perturbations, showing they converge to a distribution determined by the symbol evaluated on the unit circle.

## Contribution

It extends previous results to non-triangular Toeplitz matrices with more general noise, confirming predictions about eigenvalue distributions under perturbations.

## Key findings

- Eigenvalue empirical measure converges to the law of the symbol on the unit circle.
- Results apply to non-triangular matrices and non-Gaussian noise.
- Confirms pseudo-spectrum predictions for eigenvalue behavior.

## Abstract

Let $T_N$ denote an $N\times N$ Toeplitz matrix with finite, $N$ independent symbol ${\bf a}$. For $E_N$ a noise matrix satisfying mild assumptions (ensuring, in particular, that $N^{-1/2}\|E_N\|_{{\rm HS}}\to_{N\to\infty} 0$ at a polynomial rate), we prove that the empirical measure of eigenvalues of $T_N+E_N$ converges to the law of ${\bf a}(U)$, where $U$ is uniformly distributed on the unit circle in the complex plane. This extends results from arXiv:1712.00042 to the non-triangular setup and non complex Gaussian noise, and confirms predictions obtained in Reichel and Trefethen (1992) using the notion of pseudo-spectrum.