Random clusters in the Villain and XY models

TL;DR

This paper explores geometric cluster representations for the Villain and XY models, extending the concept from Ising and Potts models, and discusses their interpretations, extensions, and open problems.

Contribution

It introduces cluster representations for the Villain and XY models, providing new geometric insights and extending existing frameworks from Ising and Potts models.

Findings

Cluster representations for Villain and XY models are developed.

Interpretation via cable system extension is provided.

Open problems in the cluster representations are listed.

Abstract

In the Ising and Potts model, random cluster representations provide a geometric interpretation to spin correlations. We discuss similar constructions for the Villain and XY models, where spins take values in the circle, as well as extensions to models with different single site spin spaces. In the Villain case, we highlight natural interpretation in terms of the cable system extension of the model. We also list questions and open problems on the cluster representations obtained in this fashion.