Critical edge behavior in unitary random matrix ensembles and the thirty fourth Painlevé transcendent

We describe a new universality class for unitary invariant random matrix ensembles. It arises in the double scaling limit of ensembles of random n × n Hermitian matrices Z −1 n,N | det M | 2α e −N Tr V (M) dM with α > −1/2, where the factor | det M | 2α induces critical eigenvalue behavior near the origin. Under the assumption that the limiting mean eigenvalue density associated with V 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 → ∞ 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 on the line orthogonal with respect to the weight |x| 2α e −N V (x). Our main attention is on the construction of a local parametrix near the origin by means of the ψ-functions associated with a distinguished solution of the Painlevé XXXIV equation. This solution is related to a particular solution of the Painlevé II equation, which however is different from the usual Hastings-McLeod solution.