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

Abstract

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 Ψ-function associated with the Hastings–McLeod solution of the second Painleve 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) .