## Asymptotics of a Fredholm determinant involving the second Painlevé transcendent    [PDF]

Thomas Bothner, Alexander Its
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 \$\Psi\$-function associated with the Hastings-McLeod solution of the second Painlev\'e 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}})\$.
View original: http://arxiv.org/abs/1209.5415