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

Abstract

We study the orthogonal polynomials and the Hankel determinants associated with Gaussian weight with two jump discontinuities. When the degree $n$ is finite, the orthogonal polynomials and the Hankel determinants are shown to be connected to the coupled Painlevé IV system. In the double scaling limit as the jump discontinuities tend to the edge of the spectrum and the degree $n$ grows to infinity, we establish the asymptotic expansions for the Hankel determinants and the orthogonal polynomials, which are expressed in terms of solutions of the coupled Painlevé II system. As applications, we re-derive the recently found Tracy-Widom type expressions for the gap probability of there being no eigenvalues in a finite interval near the the extreme eigenvalue of large Gaussian unitary ensemble and the limiting conditional distribution of the largest eigenvalue in Gaussian unitary ensemble by considering a thinned process.

34 pages, 2 figures

References in corpus (3)

• Randomly incomplete spectra and intermediate statistics
• Large gap asymptotics for Airy kernel determinants with discontinuities
• Four-dimensional Painlevé-type equations associated with ramified linear equations II: Sasano systems

Cited by in corpus (4)

• Airy kernel determinant solutions to the KdV equation and integro-differential Painlevé equations
• Exponential moments for disk counting statistics of random normal matrices in the critical regime
• Laguerre Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé V System
• Gaussian unitary ensemble with two jump discontinuities, PDEs and the coupled Painlevé II and IV systems