Critical Edge Behavior in Unitary Random Matrix Ensembles and the Thirty-Fourth Painlevé Transcendent

Abstract

We describe a new universality class for unitary invariant random matrix ensembles. It arises in the double scaling limit of ensembles Z(n,N)(-1)vertical bar det M vertical bar (2 alpha)e(-NTrV(M)) dM, with alpha > -1/2, defined on n x n Hermitian matrices M. Assuming that the limiting mean eigenvalue density is regular and that the origin is a right endpoint of its support, we compute the limiting eigenvalue correlation kernel in the double scaling limit as n, N -> infinity such that n(2/3)(n/N-1) = O(1). We use the Deift-Zhou steepest descent method for the Riemann-Hilbert problem for polynomials orthogonal with respect to the weight vertical bar x vertical bar(2 alpha)e(-NV(x)). Our main attention is on the construction of a local parametrix near the origin by means of the psi-functions associated with a distinguished solution of the Painleve XXXIV equation.