Curvature Matrix Models for Dynamical Triangulations and the Itzykson-DiFrancesco Formula

TL;DR

This paper analyzes matrix models for dynamical triangulations using character expansions, revealing a correspondence with models of random surfaces and providing explicit solutions for specific vertex weightings.

Contribution

It introduces a novel approach to solve matrix models for random surfaces via character expansion and connects these models to even coordination number vertex models.

Findings

Large-N limit leads to a correspondence with even coordination number models

Explicit solutions obtained for simple vertex weightings

Reproduction of random surface features through matrix models

Abstract

We study the large-N limit of a class of matrix models for dually weighted triangulated random surfaces using character expansion techniques. We show that for various choices of the weights of vertices of the dynamical triangulation the model can be solved by resumming the Itzykson-Di Francesco formula over congruence classes of Young tableau weights modulo three. From this we show that the large-N limit implies a non-trivial correspondence with models of random surfaces weighted with only even coordination number vertices. We examine the critical behaviour and evaluation of observables and discuss their interrelationships in all models. We obtain explicit solutions of the model for simple choices of vertex weightings and use them to show how the matrix model reproduces features of the random surface sum. We also discuss some general properties of the large-N character expansion approach as well as potential physical applications of our results.