The number of link and cluster states: the core of the 2D $q$ state Potts model

TL;DR

This paper introduces a method to compute the combinatorial factor in the random cluster representation of the q-state Potts model using Monte Carlo simulations, enhancing understanding of its graph-based structure.

Contribution

The paper presents a novel approach to calculate the combinatorial factor in the random cluster representation of the Potts model from Monte Carlo data.

Findings

Method successfully computes the combinatorial factor

Enhances analysis of the Potts model's graph structure

Provides insights into cluster and link configurations

Abstract

Due to Fortuin and Kastelyin the $q$ state Potts model has a representation as a sum over random graphs, generalizing the Potts model to arbitrary $q$ is based on this representation. A key element of the Random Cluster representation is the combinatorial factor $\Gamma_{\Graph{G}}(\Clusters,\Edges)$, which is the number of ways to form $\Clusters$ distinct clusters, consisting of totally $\Edges$ edges. We have devised a method to calculate $\Gamma_{\Graph{G}}(\Clusters,\Edges)$ from Monte Carlo simulations.