On the asymptotics of some large Hankel determinants generated by Fisher-Hartwig symbols defined on the real line

TL;DR

This paper analyzes the asymptotic behavior of large Hankel determinants generated by Fisher-Hartwig symbols on the real line, with applications in random matrix theory and quantum many-body systems.

Contribution

It provides a rigorous asymptotic analysis of Hankel determinants with Fisher-Hartwig symbols, verifying a recent conjecture for specific cases using a duality formula.

Findings

Asymptotic formulas for Hankel determinants with Fisher-Hartwig symbols.

Verification of a conjecture by Forrester and Frankel for specific parameter cases.

Application of duality formulas to compute large N asymptotics.

Abstract

We investigate the asymptotics of the determinant of N by N Hankel matrices generated by Fisher-Hartwig symbols defined on the real line, as N becomes large. Such objects are natural analogues of Toeplitz determinants generated by Fisher-Hartwig symbols, and arise in random matrix theory in the investigation of certain expectations involving random characteristic polynomials. The reduced density matrices of certain one-dimensional systems of impenetrable bosons can also be expressed in terms of Hankel determinants of this form. We focus on the specific cases of scaled Hermite and Laguerre weights. We compute the asymptotics using a duality formula expressing the N by N Hankel determinant as a 2|q|-fold integral, where q is a fixed vector, which is valid when each component of q is natural.We thus verify, for such q, a recent conjecture of Forrester and Frankel derived using a log-gas argument.