Critical edge behavior in the modified Jacobi ensemble and Painlevé equations

Abstract

We study the Jacobi unitary ensemble perturbed by an algebraic singularity at t > 1. For fixed t, this is the modified Jacobi ensemble studied by Kuijlaars et al. The main focus here, however, is the case when the algebraic singularity approaches the hard edge, namely t → 1+.In the double scaling limit case when t − 1 is of the order of magnitude of 1/n2, n being the size of the matrix, the eigenvalue correlation kernel is shown to have a new limiting kernel at the hard edge 1, described by the ψ-functions for a certain second-order nonlinear equation. The equation is related to the Painlevé III equation by a Möbius transformation. It also furnishes a generalization of the Painlevé V equation, and can be reduced to a particular Painlevé V equation via the Bäcklund transformations in special cases. The transitions of the limiting kernel to Bessel kernels are also investigated, with n2(t − 1) being large or small.In the present paper, the approach is based on the Deift–Zhou nonlinear steepest descent analysis for Riemann–Hilbert problems.