# Asymptotics of eigenvalues of large symmetric Toeplitz matrices with smooth simple-loop symbols

## TL;DR

This paper investigates the asymptotic distribution of eigenvalues for large symmetric Toeplitz matrices with smooth symbols forming a simple loop, providing uniform expansions as matrix size grows.

## Contribution

It offers a comprehensive asymptotic description of all eigenvalues for symmetric Toeplitz matrices with smooth simple-loop symbols, extending understanding of their spectral behavior.

## Key findings

- Eigenvalues follow a specific asymptotic pattern
- Uniform expansion valid for all eigenvalues
- Results applicable to large matrix dimensions

## Abstract

This paper is devoted to the asymptotic behavior of all eigenvalues of Symmetric (in general non Hermitian) Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the complex plane line as the dimension of the matrices increases to infinity. The main result describes the asymptotic structure of all eigenvalues. The constructed expansion is uniform with respect to the number of eigenvalues. Keywords: Toeplitz matrices, eigenvalues, asymptotic expansions