Loop models, random matrices and planar algebras

TL;DR

This paper introduces matrix models that approximate the generating functions of various loop models and applies these to compute the Potts model's generating functions on random planar maps.

Contribution

It develops a unified matrix model framework for loop models with complex fugacity sets and connects these to non-commutative laws in subfactor planar algebras.

Findings

Matrix models converge to loop model generating functions.

Application to Potts model on random planar maps.

Provides a new computational approach for complex loop models.

Abstract

We define matrix models that converge to the generating functions of a wide variety of loop models with fugacity taken in sets with an accumulation point. The latter can also be seen as moments of a non-commutative law on a subfactor planar algebra. We apply this construction to compute the generating functions of the Potts model on a random planar map.