Title: Universality of a Double Scaling Limit near Singular Edge Points in Random Matrix Models
Abstract: We consider unitary random matrix ensembles $$Z_{n,s,t}^{-1}e^{-n tr V_{s,t}(M)}dM$$ on the space of Hermitian n × n matrices M, where the confining potential V s,t is such that the limiting mean density of eigenvalues (as n→∞ and s,t→ 0) vanishes like a power 5/2 at a (singular) endpoint of its support. The main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double scaling limit. The limiting kernel is built out of functions associated with a special solution of the P 2 equation, which is a fourth order analogue of the Painlevé I equation. In order to prove our result, we use the well-known connection between the eigenvalue correlation kernel and the Riemann-Hilbert (RH) problem for orthogonal polynomials, together with the Deift/Zhou steepest descent method to analyze the RH problem asymptotically. The key step in the asymptotic analysis will be the construction of a parametrix near the singular endpoint, for which we use the model RH problem for the special solution of the P 2 equation. In addition, the RH method allows us to determine the asymptotics (in a double scaling limit) of the recurrence coefficients of the orthogonal polynomials with respect to the varying weights $$e^{-nV_{s,t}}$$ on $${\mathbb{R}}$$ . The special solution of the P 2 equation pops up in the n −2/7-term of the asymptotics.