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 the Gaussian weight with two jump discontinuities. When the degree n is finite, the orthogonal polynomials and the Hankel determinants are shown to be connected with the coupled Painlevé IV system. Using this connection, we obtain a sequence of special function solutions 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 extreme eigenvalue of large Hermitian matrix from the Gaussian unitary ensemble (GUE) and the limiting conditional distribution of the largest eigenvalue in the GUE by considering a thinned process.