New correlation functions for random matrices and integrals over supergroups

Abstract

The averages of ratios of characteristic polynomials det(lambda - X) of N x N random matrices X, are investigated in the large N limit for the GUE, GOE and GSE ensemble. The density of states and the two-point correlation function are derived from these ratios. The method relies on an extension of the Harish-Chandra-Itzykson-Zuber integrals to the GOE ensemble and to supergroups, which are explicitly evaluated as solutions of heat kernel differential equations. An external matrix source, linearly coupled to the random matrices, may also be added to the Gaussian distribution, and allows for a discussion of universality of the GOE results in the large N limit.