Painlevé III′ and the Hankel determinant generated by a singularly perturbed Gaussian weight

Abstract

In this paper, we study the Hankel determinant generated by a singularly perturbed Gaussian weightw(x,t)=e−x2−tx2,x∈(−∞,∞),t>0. By using the ladder operator approach associated with the orthogonal polynomials, we show that the logarithmic derivative of the Hankel determinant satisfies both a non-linear second order difference equation and a non-linear second order differential equation. The Hankel determinant also admits an integral representation involving a Painlevé III′. Furthermore, we consider the asymptotics of the Hankel determinant under a double scaling, i.e. n→∞ and t→0 such that s=(2n+1)t is fixed. The asymptotic expansions of the scaled Hankel determinant for large s and small s are established, from which Dyson's constant appears.