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 $Z_{n,N}^{-1}|\det M|^{2\alpha}e^{-N\mathrm{TrV(M)}}dM$, with α > − 1/2, defined on n × 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 → ∞ 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 |x|^2αe^−NV(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.