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arXiv · 1807.05961

Painlevé III$'$ and the Hankel Determinant Generated by a Singularly Perturbed Gaussian Weight

Chao Min·Shulin Lyu·Yang Chen
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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$'$. Furthermore, we consider the asymptotics of the Hankel determinant under a double scaling, i.e. $n\rightarrow\infty$ and $t\rightarrow 0$ 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.

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Chao Min, Shulin Lyu, Yang Chen. 2018-07-16. Painlevé III$'$ and the Hankel Determinant Generated by a Singularly Perturbed Gaussian Weight. https://doi.org/10.1016/j.nuclphysb.2018.09.016

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