Asymptotics for a singularly perturbed GUE, Painlevé III, double-confluent Heun equations, and small eigenvalues

In this talk, we discuss the recurrence coefficients of the three-term recurrence relation for the orthogonal polynomials with a singularly perturbed Gaussian weight w(z)=|z|^{\alpha}\exp({-z^{2}-t/z^{2}}), z\in R, t>0, \alpha>1. Based on the ladder operator approach, two auxiliary quantities are defined. We show that the auxiliary quantities and the recurrence coefficients satisfy some equations with the aid of three compatibility conditions, which will be used to derive the Riccati equations and Painlevé III. We show that the Hankel determinant has an integral representation involving a particular \sigma-form of Painlevé III. We study the asymptotics of the Hankel determinant under a suitable double scaling, i.e. n\rightarrow\infty and t \rightarrow 0 such that s=(2n+1+\lambda)t is fixed, where \lambda is a parameter with \lambda:=(\alpha\mp1)/2. The asymptotic behaviors of the Hankel determinant for large s and small s are obtained and the Dyson's constant is recovered here. They have generalized the results in the literature [C. Min et al, Nucl. Phys. B,  936 (2018) 169-188] where \alpha=0. Together the Coulomb fluid method with the orthogonality principle, we obtain the asymptotic expansions of the recurrence coefficients, which are applied to derive the relationship between second order differential equations satisfied by our monic orthogonal polynomials and the double-confluent Heun equations, as well as to calculate the smallest eigenvalue of the large Hankel matrices generated by the above weight.