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

Abstract

We investigate the orthogonal polynomials associated with a singularly perturbed Pollaczek–Jacobi type weight [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. Based on our observation, we find that this weight includes the symmetric constraint [Formula: see text]. 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 [Formula: see text] goes to infinity. Because of the effect of [Formula: see text] for varying [Formula: see text], the asymptotic behavior in a neighborhood of point [Formula: see text] is described in terms of the Airy function as [Formula: see text], but the Bessel function as [Formula: see text]. Due to symmetry, the similar local asymptotic behavior near the singular point [Formula: see text] 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 Painlevé III kernel. Our analysis is based on the Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems.