Gaussian unitary ensemble with two jump discontinuities, PDEs and the coupled Painlev\'{e} II and IV systems

TL;DR

This paper investigates the asymptotic behavior of Hankel determinants with jump discontinuities in the Gaussian weight, linking them to coupled Painlevé systems and PDEs, and deriving new results on recurrence coefficients.

Contribution

It derives a coupled Painlevé IV system from Hankel determinants with jump discontinuities and connects it to coupled Painlevé II systems under double scaling, extending previous work.

Findings

Asymptotic behavior of Hankel determinants is characterized by coupled Painlevé systems.

The log derivative of the Hankel determinant satisfies a second order PDE.

Recurrence coefficients are asymptotically connected to solutions of coupled Painlevé II.

Abstract

We consider the Hankel determinant generated by the Gaussian weight with two jump discontinuities. Utilizing the results of [C. Min and Y. Chen, Math. Meth. Appl. Sci. {\bf 42} (2019), 301--321] where a second order PDE was deduced for the log derivative of the Hankel determinant by using the ladder operators adapted to orthogonal polynomials, we derive the coupled Painlev\'{e} IV system which was established in [X. Wu and S. Xu, arXiv: 2002.11240v2] by a study of the Riemann-Hilbert problem for orthogonal polynomials. Under double scaling, we show that, as $n\rightarrow\infty$, the log derivative of the Hankel determinant in the scaled variables tends to the Hamiltonian of a coupled Painlev\'{e} II system and it satisfies a second order PDE. In addition, we obtain the asymptotics for the recurrence coefficients of orthogonal polynomials, which are connected with the solutions of the coupled Painlev\'{e} II system.