Painlev\'e III$'$ and the Hankel Determinant Generated by a Singularly Perturbed Gaussian Weight

TL;DR

This paper investigates the properties and asymptotics of a Hankel determinant generated by a singularly perturbed Gaussian weight, revealing connections to Painlevé III' equations and Dyson's constant through advanced analytical techniques.

Contribution

It introduces a novel analysis of the Hankel determinant with a singular perturbation, deriving differential and difference equations, and providing asymptotic expansions in a double scaling limit.

Findings

Hankel determinant satisfies a Painlevé III' equation.

Asymptotic expansions are obtained for large and small scaling parameters.

Dyson's constant appears in the asymptotic analysis.

Abstract

In this paper, we study the Hankel determinant generated by a singularly perturbed Gaussian weight $$ w(x,t)=\mathrm{e}^{-x^{2}-\frac{t}{x^{2}}},\;\;x\in(-\infty, \infty),\;\;t>0. $$ By using the ladder operator approach associated with the orthogonal polynomials, we show that the logarithmic derivative of the Hankel determinant satisfies both a non-linear second order difference equation and a non-linear second order differential equation. The Hankel determinant also admits an integral representation involving a Painlev\'e III$'$. Furthermore, we consider the asymptotics of the Hankel determinant under a double scaling, i.e. $n\rightarrow\infty$ and $t\rightarrow 0$ such that $s=(2n+1)t$ is fixed. The asymptotic expansions of the scaled Hankel determinant for large $s$ and small $s$ are established, from which Dyson's constant appears.