# Asymptotics of Hankel determinants with a one-cut regular potential and Fisher-Hartwig singularities

## TL;DR

This paper derives asymptotic formulas for large Hankel determinants with complex weights involving a regular potential and Fisher-Hartwig singularities, extending previous results and linking to eigenvalue gap probabilities in random matrix theory.

## Contribution

It generalizes existing asymptotic results for Hankel determinants to include multiple Fisher-Hartwig singularities and connects these to eigenvalue gap probabilities in thinned random matrix spectra.

## Key findings

- Derived asymptotics for Hankel determinants with multiple singularities
- Extended previous results to more general weights
- Linked determinants to eigenvalue gap probabilities

## Abstract

We obtain asymptotics of large Hankel determinants whose weight depends on a one-cut regular potential and any number of Fisher-Hartwig singularities. This generalises two results: 1) a result of Berestycki, Webb and Wong [5] for root-type singularities, and 2) a result of Its and Krasovsky [37] for a Gaussian weight with a single jump-type singularity. We show that when we apply a piecewise constant thinning on the eigenvalues of a random Hermitian matrix drawn from a one-cut regular ensemble, the gap probability in the thinned spectrum, as well as correlations of the characteristic polynomial of the associated conditional point process, can be expressed in terms of these determinants.