A pr 1 99 7 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. The introduction by Fortuin and Kasteleyn [FK, F1, F2] of the random–cluster model in the late 60s has given rise to numerous important results. First it provided a unified representation of several famous models, including the Ising, Potts and percolation models, thus allowing the comparison between them. It also brought a whole class of models interpolating between the latter ones. The random–cluster representation has been used in many recent proofs in statistical mechanics, for example in large deviations theory [I, Pi]. The fact is that this model has several nice properties, as FKG and comparison inequalities, allowing to derive non–perturbative results for the original models. One of the properties which has also often been used is that the two-dimensional random–cluster model is self–dual, and that this duality commutes with the duality of the original models; this has been used for example in the study of the decay of the connectivity in the Ising model [CCS]. Other applications of this representation have been found in numerical studies, in particular Random–cluster representation of Ashkin–Teller model 2 the Swendsen-Wang algorithm is based on it. It would then be interesting to be able to extend this representation to a wider class of models, while keeping most of its properties. This appears to be possible. We show that the Ashkin–Teller model (and a class of models generalizing the Ashkin– Teller model, and containing the partially-symmetric Potts models) admits a similar representation, which in fact generalizes the usual one. The nice point is that it is still possible to prove FKG inequalities, comparison inequalities and commutativity of the dual transformations for this new representation. Such a representation has already been considered in [WD, SS]. The main goal in these papers is to develop a Swendsen-Wang type algorithm for the Ashkin-Teller model. A closely related representation has also appeared in the study of partially symmetric Potts models [LMaR]. Their representation appears as a special case of the one studied here. Nevertheless, properties of the measure were not studied in these …