Random-Cluster Representation of the Ashkin-Teller Model

TL;DR

This paper introduces a generalized random-cluster representation for the Ashkin-Teller model, extending key properties like FKG inequalities and duality, thereby enhancing understanding of its probabilistic structure.

Contribution

It develops a generalized random-cluster model for the Ashkin-Teller model, preserving fundamental properties and analyzing duality transformations.

Findings

The GRC representation admits FKG and comparison inequalities.

Duality transformations commute in both spin and GRC models.

Elementary consequences of the GRC model are derived.

Abstract

We show that a class of spin models, containing the Ashkin-Teller model, admits a generalized random-cluster (GRC) representation. Moreover we show that basic properties of the usual representation, such as FKG inequalities and comparison inequalities, still hold for this generalized random-cluster model. Some elementary consequences are given. We also consider the duality transformations in the spin representation and in the GRC model and show that they commute.