Coloring Random Triangulations

TL;DR

This paper introduces a two-matrix model for vertex tri-coloring in random triangulations, providing three solutions and revealing its universality class in 2D quantum gravity.

Contribution

It presents a novel two-matrix model for vertex tri-coloring and offers three distinct solution methods, advancing understanding of related combinatorial and physical models.

Findings

Model lies in the universality class of pure 2D quantum gravity

Three solution methods demonstrated: orthogonal polynomials, Hirota equation, direct expansion

Model's potential is non-polynomial but still exhibits universal behavior

Abstract

We introduce and solve a two-matrix model for the tri-coloring problem of the vertices of a random triangulation. We present three different solutions: (i) by orthogonal polynomial techniques (ii) by use of a discrete Hirota bilinear equation (iii) by direct expansion. The model is found to lie in the universality class of pure two-dimensional quantum gravity, despite the non-polynomiality of its potential.