Painlevé III\('\) and the Hankel Determinant Generated by a Singularly Perturbed Gaussian Weight

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 der...

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Bibliographic Details
Published in:arXiv.org 2018-07
Main Authors: Chao, Min, Lyu, Shulin, Chen, Yang
Format: Article
Language:English
Subjects:
Asymptotic series
Difference equations
Differential equations
Operators (mathematics)
Polynomials
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1807.05961