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arXiv · 1209.5415

Asymptotics of a Fredholm determinant involving the second Painlevé transcendent

Thomas Bothner·Alexander Its
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Abstract

We study the determinant $\det(I-K_{\textnormal{PII}})$ of an integrable Fredholm operator $K_{\textnormal{PII}}$ acting on the interval $(-s,s)$ whose kernel is constructed out of the $Ψ$-function associated with the Hastings-McLeod solution of the second Painlevé equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random Hermitian matrix chosen from the Unitary Ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density. Using the Riemann-Hilbert method, we evaluate the large $s$-asymptotics of $\det(I-K_{\textnormal{PII}})$.

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Thomas Bothner, Alexander Its. 2012-11-02. Asymptotics of a Fredholm determinant involving the second Painlevé transcendent. https://arxiv.org/abs/1209.5415

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