Some Exact Results on the Potts Model Partition Function in a Magnetic Field

TL;DR

This paper investigates the Potts model in a magnetic field on arbitrary graphs, deriving properties of its partition function, relating it to a generalized Tutte polynomial, and exploring implications for graph coloring and specific graph structures.

Contribution

It provides new exact results on the Potts model partition function, including factorization, monotonicity, zeros, and a generalized Tutte polynomial, extending understanding of graph-based statistical models.

Findings

Partition function $Z$ exhibits specific factorization and monotonicity properties.

Zeros of the partition function are characterized under certain conditions.

A generalized Tutte polynomial corresponding to the Potts model is introduced.

Abstract

We consider the Potts model in a magnetic field on an arbitrary graph $G$. Using a formula of F. Y. Wu for the partition function $Z$ of this model as a sum over spanning subgraphs of $G$, we prove some properties of $Z$ concerning factorization, monotonicity, and zeros. A generalization of the Tutte polynomial is presented that corresponds to this partition function. In this context we formulate and discuss two weighted graph-coloring problems. We also give a general structural result for $Z$ for cyclic strip graphs.