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

In this paper, we study the strong asymptotic for the orthogonal polynomials and universality associated with singularly perturbed Pollaczek-Jacobi type weight $$w_{p_J2}(x,t)=e^{-\frac{t}{x(1-x)}}x^\alpha(1-x)^\beta, $$ where $t \ge 0$, $\alpha >0$, $\beta >0$ and $x \in [0,1].$ Our main results obtained here include two aspects: { I. Strong asymptotics:} We obtain the strong asymptotic expansions for the monic Pollaczek-Jacobi type orthogonal polynomials in different interval $(0,1)$ and outside of interval $\mathbb{C}\backslash (0,1)$, respectively; Due to the effect of $\frac{t}{x(1-x)}$ for varying $t$, different asymptotic behaviors at the hard edge $0$ and $1$ were found with different scaling schemes. Specifically, the uniform asymptotic behavior can be expressed as a Airy function in the neighborhood of point $1$ as $\zeta= 2n^2t \to \infty, n\to \infty$, while it is given by a Bessel function as $\zeta \to 0, n \to \infty$. { II. Universality:} We respectively calculate the limit of the eigenvalue correlation kernel in the bulk of the spectrum and at the both side of hard edge, which will involve a $\psi$-functions associated with a particular Painlev$\acute{e}$ \uppercase\expandafter{\romannumeral3} equation near $x=\pm 1$. Further, we also prove the $\psi$-funcation can be approximated by a Bessel kernel as $\zeta \to 0$ compared with a Airy kernel as $\zeta \to \infty$. Our analysis is based on the Deift-Zhou nonlinear steepest descent method for the Riemann-Hilbert problems.