Combinatorics of Hard Particles on Planar Graphs

TL;DR

This paper explores the combinatorial structure of hard particles on planar tetravalent graphs, connecting recent diagram techniques to matrix models, and providing a new combinatorial perspective on the problem.

Contribution

It introduces a combinatorial approach to analyze hard particles on planar graphs, linking diagram techniques to matrix model solutions.

Findings

Recovered the two-matrix model solution using combinatorial methods

Established a connection between planar diagrams and decorated trees

Provided a new combinatorial framework for the problem

Abstract

We revisit the problem of hard particles on planar random tetravalent graphs in view of recent combinatorial techniques relating planar diagrams to decorated trees. We show how to recover the two-matrix model solution to this problem in this purely combinatorial language.