# Asymptotics of Hankel determinants with a Laguerre-type or Jacobi-type potential and Fisher-Hartwig singularities

## TL;DR

This paper derives large n asymptotics for Hankel determinants with Laguerre or Jacobi-type potentials and Fisher-Hartwig singularities, with applications to random matrix theory and related fields.

## Contribution

It provides the first detailed asymptotic analysis of Hankel determinants with these specific singularities and edge configurations, extending previous results in the field.

## Key findings

- Asymptotics for partition functions with singularities
- Central limit theorems for related ensembles
- Correlation and gap probability results

## Abstract

We obtain large $n$ asymptotics of $n \times n$ Hankel determinants whose weight has a one-cut regular potential and Fisher-Hartwig singularities. We restrict our attention to the case where the associated equilibrium measure possesses either one soft edge and one hard edge (Laguerre-type) or two hard edges (Jacobi-type). We also present some applications in the theory of random matrices. In particular, we can deduce from our results asymptotics for partition functions with singularities, central limit theorems, correlations of the characteristic polynomials, and gap probabilities for (piecewise constant) thinned Laguerre and Jacobi-type ensembles. Finally, we mention some links with the topics of rigidity and Gaussian multiplicative chaos.