The Tutte-Potts connection in the presence of an external magnetic field

TL;DR

This paper introduces the V-polynomial, extending the classical Tutte-Potts relationship to include external magnetic fields, thereby unifying and broadening the applicability of combinatorial methods in statistical mechanics.

Contribution

The paper defines the V-polynomial, generalizing the Tutte polynomial to incorporate external magnetic fields in the Potts model, and proves its relation to the partition function.

Findings

V-polynomial unifies Tutte polynomial and Potts model with magnetic fields.

Partition function is an evaluation of the V-polynomial.

Enables application of combinatorial techniques to a wider class of models.

Abstract

The classical relationship between the Tutte polynomial of graph theory and the Potts model of statistical mechanics has resulted in valuable interactions between the disciplines. Unfortunately, it does not include the external magnetic fields that appear in most Potts model applications. Here we define the V-polynomial, which lifts the classical relationship between the Tutte polynomial and the zero field Potts model to encompass external magnetic fields. The V-polynomial generalizes Nobel and Welsh's W-polynomial, which extends the Tutte polynomial by incorporating vertex weights and adapting contraction to accommodate them. We prove that the variable field Potts model partition function (with its many specializations) is an evaluation of the V-polynomial, and hence a polynomial with deletion-contraction reduction and Fortuin-Kasteleyn type representation. This unifies an important segment of Potts model theory and brings previously successful combinatorial machinery, including complexity results, to bear on a wider range of statistical mechanics models.