Probabilistic Weyl Law for Twisted Toeplitz Matrices with Rough Symbols

TL;DR

This paper proves that the spectral distribution of twisted Toeplitz matrices with rough symbols converges to a predictable measure, extending classical results to less regular symbols under small random perturbations.

Contribution

It establishes a probabilistic Weyl law for twisted Toeplitz matrices with piecewise Hölder symbols, including discontinuities, under small random perturbations.

Findings

Spectral measure converges weakly in probability to the push-forward of Lebesgue measure.

Convergence holds for symbols that are smooth in frequency and piecewise Hölder in position.

Results extend classical spectral distribution laws to rougher symbol classes.

Abstract

In this article, we study the convergence of the empirical spectral measure of twisted Toeplitz matrices subject to small random perturbations. We show that the empirical spectral measure converges weakly in probability to the push-forward of the Lebesgue measure by the symbol. The symbol of the twisted Toeplitz matrices is assumed to be smooth in frequency, and only piecewise H{\"o}lder continuous with respect to the position variable with discontinuities of jump type.