Asymptotics for Toeplitz determinants: perturbation of symbols with a gap

TL;DR

This paper investigates the asymptotic behavior of Toeplitz determinants with symbols having two jump discontinuities and vanishing on an arc, extending Widom's expansion and applying results to random matrix theory and Fredholm determinants.

Contribution

It generalizes Widom's asymptotic expansion for Toeplitz determinants to symbols with gaps and applies findings to the Circular Unitary Ensemble and sine kernel Fredholm determinants.

Findings

Extended asymptotic expansion for Toeplitz determinants with symbols having gaps.

Applied results to Circular Unitary Ensemble analysis.

Connected Toeplitz asymptotics to Fredholm determinants of the sine kernel.

Abstract

We study the determinants of Toeplitz matrices as the size of the matrices tends to infinity, in the particular case where the symbol has two jump discontinuities and tends to zero on an arc of the unit circle at a sufficiently fast rate. We generalize an asymptotic expansion by Widom [22], which was known for symbols supported on an arc. We highlight applications of our results in the Circular Unitary Ensemble and in the study of Fredholm determinants associated to the sine kernel.