Gaussian unitary ensembles 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 Min and Chen [Math. Methods Appl Sci. 2019;42:301-321] where a second-order partial differential equation (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 Wu and Xu [arXiv: 2002.11240v2] by a study of the Riemann-Hilbert problem for orthogonal polynomials. Under double scaling, we show that, as → ∞, 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.

Exponential moments for disk counting statistics of random normal matrices in the critical regime

“…In the aforementioned works, the discontinuities are merging in the bulk. Hankel determinants with merging discontinuities at the edge are also related to the Painlevé theory, see [45,35] for soft edges, [36] for a hard edge, and e.g. [16,13,12] for studies on related Fredholm determinants.…”

Laguerre Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé V System

“…The two-jump case (m = 2) was investigated in [25] via the ladder operator approach and a second order PDE was deduced for the logarithmic derivative of the Hankel determinant. By making use of the finite dimensional results therein, the first and the second author of the present paper reproduce the aforementioned coupled P IV and P II system of [29] with m = 2 [20]. See also [6,11,30] for the applications of coupled P II system in the studies Airy kernel determinants with discontinuities and the Painlevé-type kernel determinants.…”

Exponential moments for disk counting statistics of random normal matrices in the critical regime

“…In the aforementioned works, the discontinuities are merging in the bulk. Hankel determinants with merging discontinuities at the edge are also related to the Painlevé theory, see [41,51] for soft edges, [42] for a hard edge, and e.g. [14,15,18] for studies on related Fredholm determinants.…”