Truncated Wiener-Hopf operators with Fisher Hartwig singularities

TL;DR

This paper analyzes the asymptotic behavior of determinants of truncated Wiener-Hopf operators with Fisher-Hartwig singularities using Riemann-Hilbert problem techniques, also providing new derivations for Toeplitz determinants with similar singularities.

Contribution

It introduces a systematic Riemann-Hilbert problem approach to derive asymptotics of Wiener-Hopf and Toeplitz determinants with Fisher-Hartwig singularities, including sub-leading terms.

Findings

Derived asymptotics for truncated Wiener-Hopf determinants with Fisher-Hartwig singularities.

Provided a new derivation for the asymptotics of Toeplitz determinants with Fisher-Hartwig singularities.

Developed a method to compute sub-leading asymptotics systematically.

Abstract

We derive the asymptotic behavior of determinants of truncated Wiener-Hopf operators generated by symbols having Fisher-Hartwig singularities. This task is achieved thanks to an asymptotic resolution of the Riemann-Hilbert problem associated to some generalized sine kernel. As a byproduct, we give yet another derivation of the asymptotic behavior of Toeplitz determinants having Fisher-Hartwig singularities. The Riemann-Hilbert problem approach to these asymptotics yields a systematic although quickly cumbersome way to compute their sub-leading asymptotics.