Asymptotics of a Fredholm determinant involving the second Painlevé transcendent

Abstract

We study the determinant $\det (I-K_{\text{PII}})$ of an integrable Fredholm operator $K_{\text{PII}}$ acting on the interval $(-s,s)$ whose kernel is constructed out of the $\Psi$-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 $\det (I-K_{\text{PII}})$.

49 pages, 12 figures