Asymptotics of the Fredholm determinant corresponding to the first bulk critical universality class in random matrix models

Abstract

We study the one-parameter family of determinants det(I−γKPII),γ∈R 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 Painlev'e equation. In case γ=1, 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−γKPII) for all values of the real parameter γ.