Character Expansion Methods for Matrix Models of Dually Weighted Graphs

TL;DR

This paper develops character expansion techniques for matrix models of dually weighted graphs, enabling analysis of large N limits and phase transitions in models with controlled coordination numbers.

Contribution

It introduces a simplified derivation of character expansions for generalized matrix models and applies these methods to solve new models with even coordination numbers.

Findings

Derived a simple character expansion formula for matrix models.

Solved a new model of planar graphs with even coordination numbers.

Formulated equations for models with arbitrary self-dual coupling constants.

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.