The three-colour model with domain wall boundary conditions

TL;DR

This paper analyzes the three-colour model with domain wall boundary conditions, expressing its partition function via special polynomials, generalizing previous proofs, and deriving combinatorial results and conjectures on free energy.

Contribution

It introduces a recursive polynomial approach to the three-colour model, extending Kuperberg's method from the six-vertex to the eight-vertex-solid-on-solid model.

Findings

Partition function expressed in terms of special polynomials

Generalization of Kuperberg's proof to a new model

Conjecture on explicit free energy formula

Abstract

We study the partition function for the three-colour model with domain wall boundary conditions. We express it in terms of certain special polynomials, which can be constructed recursively. Our method generalizes Kuperberg's proof of the alternating sign matrix theorem, replacing the six-vertex model used by Kuperberg with the eight-vertex-solid-on-solid model. As applications, we obtain some combinatorial results on three-colourings. We also conjecture an explicit formula for the free energy of the model.