Random–Cluster Representation of the Ashkin–Teller Model

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.