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

We discuss the recurrence coefficients of the three-term recurrence relation for the orthogonal polynomials with a singularly perturbed Gaussian weight w(z)=|z|α⁡exp−z2−t/z2,z∈R,t>0,α>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 σ-form of Painlevé III and to calculate the asymptotics of the Hankel determinant under a suitable double scaling, i.e., n → ∞ and t → 0 such that s = (2n + 1 + λ)t is fixed, where λ is a parameter with λ ≔ (α ∓ 1)/2. The asymptotic behaviors of the Hankel determinant for large s and small s are obtained, and Dyson’s constant is recovered here. They have generalized the results in the literature [Min et al., Nucl. Phys. B 936, 169–188 (2018)] where α = 0. By combining 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. In particular, when α = t = 0, the asymptotic behavior of the smallest eigenvalue for the classical Gaussian weight exp(−z2) [Szegö, Trans. Am. Math. Soc. 40, 450–461 (1936)] is recovered.