Perturbed Hankel determinant, correlation functions and Painlev\'e equations

TL;DR

This paper investigates the asymptotic behavior of Hankel determinants generated by a Pollaczek-Jacobi weight, revealing their connections to Painlevé equations under various double scaling limits.

Contribution

It establishes new links between scaled Hankel determinants and Painlevé equations, providing integral representations and asymptotic expansions in different scaling regimes.

Findings

Double scaling limits relate Hankel determinants to Painlevé III' and V equations.

Asymptotic expansions for small and large scaling parameters are derived.

Integral representations connect Hankel determinants with special Painlevé functions.

Abstract

We continue with the study of the Hankel determinant, $$ D_{n}(t,\alpha,\beta):=\det\left(\int_{0}^{1}x^{j+k}w(x;t,\alpha,\beta)dx\right)_{j,k=0}^{n-1}, $$ generated by a Pollaczek-Jacobi type weight, $$ w(x;t,\alpha,\beta):=x^{\alpha}(1-x)^{\beta}{\rm e}^{-t/x}, \quad x\in [0,1], \quad \alpha>0, \quad \beta>0, \quad t\geq 0. $$ This reduces to the "pure" Jacobi weight at $t=0.$ We may take $\alpha\in \mathbb{R}$, in the situation while $t$ is strictly greater than $0.$ It was shown in Chen and Dai (2010), that the logarithmic derivative of this Hankel determinant satisfies a Jimbo-Miwa-Okamoto $\sigma$-form of Painlev\'e \uppercase\expandafter{\romannumeral5} (${\rm P_{\uppercase\expandafter{\romannumeral5}}}$). In fact the logarithmic of the Hankel determinant has an integral representation in terms of a particular ${\rm P_{\uppercase\expandafter{\romannumeral5}}}.$ \\ In this paper, we show that, under a double scaling, where $n$ the dimension of the Hankel matrix tends to $\infty$, and $t$ tends to $0^{+},$ such that $s:=2n^2t$ is finite, the double scaled Hankel determinant (effectively an operator determinant) has an integral representation in terms of a particular ${\rm P_{\uppercase\expandafter{\romannumeral3}'}}.$ Expansions of the scaled Hankel determinant for small and large $s$ are found. A further double scaling with $\alpha=-2n+\lambda,$ where $n\rightarrow \infty$ and $t,$ tends to $0^{+},$ such that $s:=nt$ is finite. In this situation the scaled Hankel determinant has an integral representation in terms of a particular ${\rm P_{\uppercase\expandafter{\romannumeral5}}},$ %which can be degenerate to a particular ${\rm P_{\uppercase\expandafter{\romannumeral3}}}$ and its small and large $s$ asymptotic expansions are also found.