Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity

TL;DR

This paper derives large-size asymptotics for Hankel determinants with Hermite weights featuring a Fisher-Hartwig singularity, revealing a critical transition related to jump size variations, relevant in random matrix theory and Painlevé equations.

Contribution

It provides the first detailed asymptotic analysis of Hankel determinants with varying Fisher-Hartwig singularities in the Hermite case, highlighting a phase transition.

Findings

Asymptotic formulas for Hankel determinants with singularities

Identification of a critical transition in jump size effects

Connections to Gaussian ensembles and Painlevé IV solutions

Abstract

We study $n\times n$ Hankel determinants constructed with moments of a Hermite weight with a Fisher-Hartwig singularity on the real line. We consider the case when the singularity is in the bulk and is both of root-type and jump-type. We obtain large $n$ asymptotics for these Hankel determinants, and we observe a critical transition when the size of the jumps varies with $n$. These determinants arise in the thinning of the generalised Gaussian unitary ensembles and in the construction of special function solutions of the Painlev\'e IV equation.