Uniform asymptotics of Toeplitz determinants with Fisher-Hartwig singularities

TL;DR

This paper derives uniform asymptotic formulas for large Toeplitz determinants with Fisher-Hartwig singularities, confirming a conjecture in random matrix theory and providing insights into quantum many-body systems.

Contribution

It presents a uniform asymptotic formula for Toeplitz determinants with Fisher-Hartwig singularities, extending previous results and confirming a conjecture related to random matrix theory.

Findings

Confirmed a conjecture by Fyodorov and Keating on moments of characteristic polynomials.

Derived asymptotics for the momentum of impenetrable bosons in one dimension.

Provided a uniform asymptotic formula valid for any fixed number of singularities.

Abstract

We obtain an asymptotic formula for $n\times n$ Toeplitz determinants as $n\to \infty$, for real valued symbols with any fixed number of Fisher-Hartwig singularities, which is uniform with respect to the location of the singularities. As an application, we prove a conjecture by Fyodorov and Keating regarding moments of averages of the characteristic polynomial of the Circular Unitary Ensemble. In addition, we obtain an asymptotic formula regarding the momentum of impenetrable bosons in one dimension with periodic boundary conditions.