Critical edge behavior in the modified Jacobi ensemble and Painlevé equations

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/ n 2 , 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 n 2 ( 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.