Universality of a double scaling limit near singular edge points in random matrix models

We consider unitary random matrix ensembles Z n,s,te −n tr s,tdM on the space of Hermitian n × n matrices M , where the confining potential Vs,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 I equation, which is a fourth order analogue of the Painlevé I equation. In order to prove our result, we use the wellknown 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 I 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 es,t on R. The special solution of the P 2 I equation pops up in the n -term of the asymptotics.