Outliers of perturbations of banded Toeplitz matrices

TL;DR

This paper investigates how the eigenvalues of banded Toeplitz matrices are affected by small random perturbations, revealing the structure of outliers and providing new theoretical insights into spectral sensitivity.

Contribution

It introduces a novel analysis of outlier eigenvalues in perturbed banded Toeplitz matrices, including new proofs and a functional central limit theorem for related trace functions.

Findings

Outlier eigenvalues are governed by a Gaussian analytic matrix field.

The spectral sensitivity of Toeplitz matrices is characterized by explicit deterministic and random components.

New variations of Szego's strong limit theorem are presented.

Abstract

Toeplitz matrices form a rich class of possibly non-normal matrices whose asymptotic spectral analysis in high dimension is well-understood. The spectra of these matrices are notoriously highly sensitive to small perturbations. In this work, we analyze the spectrum of a banded Toeplitz matrix perturbed by a random matrix with iid entries of variance $\sigma_n^2 / n$ in the asymptotic of high dimension and $\sigma_n$ converging to $\sigma \geq 0$. Our results complement and provide new proofs on recent progresses in the case $\sigma = 0$. For any $\sigma \geq 0$, we show that the point process of outlier eigenvalues is governed by a low-dimensional random analytic matrix field, typically Gaussian, alongside an explicit deterministic matrix that captures the algebraic structure of the resonances responsible for the outlier eigenvalues. On our way, we prove a new functional central limit theorem for trace of polynomials in deterministic and random matrices and present new variations around Szego's strong limit theorem.