Vertices from replica in a random matrix theory

Kontsevich's work on Airy matrix integrals has led to explicit results for the intersection numbers of the moduli space of curves. In a subsequent work Okounkov rederived these results from the edge behavior of a Gaussian matrix integral. In our work we consider the correlation functions of vertices in a Gaussian random matrix theory, with an external matrix source. We deal with operator products of the form <Πi=1n1/N tr Mki>, in a 1/N expansion. For large values of the powers ki, in an appropriate scaling limit relating large k's to large N, universal scaling functions are derived. Furthermore, we show that the replica method applied to characteristic polynomials of the random matrices, together with a duality exchanging N and the number of points, provides a new way to recover Kontsevich's results on these intersection numbers