Matrix Model Combinatorics: Applications to Folding and Coloring

TL;DR

This paper explores the combinatorial aspects of matrix integrals and applies these methods to complex folding and coloring problems in physics, including polymer folding and membrane triangulations, providing new insights into their enumeration.

Contribution

It introduces a detailed combinatorial interpretation of matrix integrals and applies it to solve folding and coloring problems on random surfaces, expanding the understanding of these models.

Findings

Enumerated topologically distinct polymer foldings.

Counted vertex-tricolored triangulations of arbitrary genus.

Connected matrix models to physical folding problems.

Abstract

We present a detailed study of the combinatorial interpretation of matrix integrals, including the examples of tessellations of arbitrary genera, and loop models on random surfaces. After reviewing their methods of solution, we apply these to the study of various folding problems arising from physics, including: the meander (or polymer folding) problem ``enumeration of all topologically inequivalent closed non-intersecting plane curves intersecting a line through a given number of points" and a fluid membrane folding problem reformulated as that of ``enumerating all vertex-tricolored triangulations of arbitrary genus, with given numbers of vertices of either color".