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

Abstract

We describe a new universality class for unitary invariant random matrix ensembles. It arises in the double scaling limit of ensembles of random \(n \times n\) Hermitian matrices \(Z_{n,N}^{-1} |\det M|^{2\alpha} e^{-N \Tr V(M)} dM\) with \(\alpha > -1/2\), where the factor \(|\det M|^{2\alpha}\) 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 \(n, N \to \infty\) 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 \(|x|^{2\alpha} e^{-NV(x)}\). 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. This solution is related to a particular solution of the Painleve II equation, which however is different from the usual Hastings-McLeod solution.