A Little Statistical Mechanics for the Graph Theorist

TL;DR

This survey introduces the Potts model from a graph theory perspective, highlighting its connections to graph invariants like the Tutte polynomial, and discusses its applications, computational complexity, and analysis methods in complex systems.

Contribution

It provides an accessible overview of the Potts model's relationship with graph invariants and explores its diverse applications and computational aspects for a general audience.

Findings

Equivalence of the Potts model partition function and the Tutte polynomial

Connection between chromatic polynomial and zero-temperature antiferromagnetic partition function

Applications of the Potts model in various scientific fields

Abstract

In this survey, we give a friendly introduction from a graph theory perspective to the q-state Potts model, an important statistical mechanics tool for analyzing complex systems in which nearest neighbor interactions determine the aggregate behavior of the system. We present the surprising equivalence of the Potts model partition function and one of the most renowned graph invariants, the Tutte polynomial, a relationship that has resulted in a remarkable synergy between the two fields of study. We highlight some of these interconnections, such as computational complexity results that have alternated between the two fields. The Potts model captures the effect of temperature on the system and plays an important role in the study of thermodynamic phase transitions. We discuss the equivalence of the chromatic polynomial and the zero-temperature antiferromagnetic partition function, and how this has led to the study of the complex zeros of these functions. We also briefly describe Monte Carlo simulations commonly used for Potts model analysis of complex systems. The Potts model has applications as widely varied as magnetism, tumor migration, foam behaviors, and social demographics, and we provide a sampling of these that also demonstrates some variations of the Potts model. We conclude with some current areas of investigation that emphasize graph theoretic approaches. This paper is an elementary general audience survey, intended to popularize the area and provide an accessible first point of entry for further exploration.