Asymptotics of a Fredholm Determinant Corresponding to the First Bulk Critical Universality Class in Random Matrix Models

We study the determinant det(I − KPII) of an integrable Fredholm operator KPII acting on the interval (−s,s) whose kernel is constructed out of the ﰪ-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 − KPII).