Critical edge behavior in the singularly perturbed Pollaczek-Jacobi type unitary ensemble

We investigate the orthogonal polynomials associated with a singularly perturbed Pollaczek-Jacobi type weight wPJ(2)(x,t; alpha,beta) = x(alpha)(1 - x)(beta)e-t/x(1-x), where t is an element of [0,infinity), alpha > 0, beta > 0 and 0 < x < 1. Based on our observation, we find that this weight includes the symmetric constraint wPJ2(x,t; alpha,beta) = wPJ2(1 - x,t; beta,alpha). Our main results obtained here include two aspects: (1) Strong asymptotics: we deduce strong asymptotics of monic orthogonal polynomials with respect to the above weight function in different regions in the complex plane when the polynomial degree n goes to infinity. Because of the effect of t/x(1-x) for varying t, the asymptotic behavior in a neighborhood of point 1 is described in terms of the Airy function as zeta = 2n(2)t ->infinity,n ->infinity, but the Bessel function as zeta -> 0,n ->infinity. Due to symmetry, the similar local asymptotic behavior near the singular point x = 0 can be derived. (2) Limiting eigenvalue correlation kernels: We calculate the limit of the eigenvalue correlation kernel of the corresponding unitary random matrix ensemble in the bulk of the spectrum described by the sine kernel, and at both sides of hard edge, expressed as a Painleve III kernel. Our analysis is based on the Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems.