# Outliers of random perturbations of Toeplitz matrices with finite symbols

## TL;DR

This paper investigates the behavior of outlier eigenvalues in large Toeplitz matrices with finite symbols under random perturbations, showing they do not appear outside the spectrum of the limiting operator and characterizing their distribution inside.

## Contribution

It provides a detailed analysis of outliers in perturbed Toeplitz matrices, establishing their absence outside the spectrum and describing their convergence to zeros of certain random analytic functions.

## Key findings

- No outliers outside the spectrum of the limiting Toeplitz operator.
- Outliers inside the spectrum converge to zeros of random analytic functions.
- The distribution of outliers depends on roots of a specific polynomial and Young tableaux.

## Abstract

Consider an $N\times N$ Toeplitz matrix $T_N$ with symbol ${a }(\lambda) := \sum_{\ell=-d_2}^{d_1} a_\ell \lambda^\ell$, perturbed by an additive noise matrix $N^{-\gamma} E_N$, where the entries of $E_N$ are centered i.i.d.~random variables of unit variance and $\gamma>1/2$. It is known that the empirical measure of eigenvalues of the perturbed matrix converges weakly, as $N\to\infty$, to the law of ${a}(U)$, where $U$ is distributed uniformly on $\mathbb{S}^1$. In this paper, we consider the outliers, i.e. eigenvalues that are at a positive ($N$-independent) distance from ${a}(\mathbb{S}^1)$. We prove that there are no outliers outside ${\rm spec} \, T({a})$, the spectrum of the limiting Toeplitz operator, with probability approaching one, as $N \to \infty$. {In contrast,} in ${\rm spec}\, T({a})\setminus {a}({\mathbb S}^1)$ the process of outliers converges to the point process described by the zero set of certain random {analytic} functions. The limiting random {analytic} functions can be expressed as linear combinations of the determinants of finite sub-matrices of an infinite dimensional matrix, whose entries are i.i.d.~having the same law as that of $E_N$. The coefficients in the linear combination depend on the roots of the polynomial $P_{z, {a}}(\lambda):= ({a}(\lambda) -z)\lambda^{d_2}=0$ and semi-standard Young Tableaux with shapes determined by the number of roots of $P_{z,{a}}(\lambda)=0$ that are greater than one in moduli.