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

Thomas Joachim Bothner, Purdue University

Abstract

We study the one-parameter family of determinants det( I – γKPII), γ ∈[special characters omitted] 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 Painlevé 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 –γ KPII) for all values of the real parameter γ.

Degree

Ph.D.

Advisors

Its, Purdue University.

Subject Area

Mathematics

COinS