Painlevé III${}^{\prime }$ 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 weight $$w(x,t)={\mathrm{e}}^{-x^2-\frac{t}{x^2}},x\in (-\infty ,\infty ),t>0.$$ 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${}^{\prime }$. Furthermore, we consider the asymptotics of the Hankel determinant under a double scaling, i.e. $n\to \infty$ and $t\to 0$ 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.

22 pages