Gaussian unitary ensemble with two jump discontinuities, PDEs and the coupled Painlevé II and IV systems

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 Painlevé 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 $n\to \infty$, the log derivative of the Hankel determinant in the scaled variables tends to the Hamiltonian of a coupled Painlevé 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 Painlevé II system.

arXiv comment21 pages

References in corpus (6)

• Critical edge behavior and the Bessel to Airy transition in the singularly perturbed Laguerre unitary ensemble
• A matrix model with a singular weight and Painleve' III
• Random Matrix Models, Double-Time Painlevé Equations, and Wireless Relaying
• Asymptotic Gap Probability Distributions of the Gaussian Unitary Ensembles and Jacobi Unitary Ensembles
• Linear Statistics of Random Matrix Ensembles at the Spectrum Edge Associated with the Airy Kernel
• Gaussian unitary ensemble with jump discontinuities and the coupled Painlevé II and IV systems

Cited by in corpus (1)

• Airy kernel determinant solutions to the KdV equation and integro-differential Painlevé equations