Character Expansion Methods for Matrix Models of Dually Weighted Graphs

Abstract

We consider generalized one-matrix models in which external fields allow control over the coordination numbers on both the original and dual lattices. We rederive in a simple fashion a character expansion formula for these models originally due to Itzykson and Di Francesco, and then demonstrate how to take the large N limit of this expansion. The relationship to the usual matrix model resolvent is elucidated. Our methods give as a by-product an extremely simple derivation of the Migdal integral equation describing the large $N$ limit of the Itzykson-Zuber formula. We illustrate and check our methods by analyzing a number of models solvable by traditional means. We then proceed to solve a new model: a sum over planar graphs possessing even coordination numbers on both the original and the dual lattice. We conclude by formulating equations for the case of arbitrary sets of even, self-dual coupling constants. This opens the way for studying the deep problem of phase transitions from random to flat lattices.