Gaussian Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlev\'{e} IV System

TL;DR

This paper investigates Hankel determinants with jump discontinuities in Gaussian weights, deriving PDEs linked to Painlevé IV equations and establishing connections to coupled Painlevé IV systems via Riemann-Hilbert analysis.

Contribution

It introduces a novel analysis of Hankel determinants with jump discontinuities, deriving PDEs and relating them to coupled Painlevé IV systems using ladder operators and Riemann-Hilbert methods.

Findings

Logarithmic derivative of Hankel determinant satisfies a PDE reducible to Painlevé IV.

Established relationships between auxiliary quantities and coupled Painlevé IV solutions.

Derived explicit connections between jump discontinuities and integrable systems.

Abstract

We study the Hankel determinant generated by the Gaussian weight with jump discontinuities at $t_1,\cdots,t_m$. By making use of a pair of ladder operators satisfied by the associated monic orthogonal polynomials and three supplementary conditions, we show that the logarithmic derivative of the Hankel determinant satisfies a second order partial differential equation which is reduced to the $\sigma$-form of a Painlev\'{e} IV equation when $m=1$. Moreover, under the assumption that $t_k-t_1$ is fixed for $k=2,\cdots,m$, by considering the Riemann-Hilbert problem for the orthogonal polynomials, we construct direct relationships between the auxiliary quantities introduced in the ladder operators and solutions of a coupled Painlev\'{e} IV system.