The Asymmetric Random Cluster Model and Comparison of Ising and Potts Models

We introduce the asymmetric random cluster (or ARC) model, which is a graphical representation of the Potts lattice gas, and establish its basic properties. The ARC model allows a rich variety of comparisons (in the FKG sense) between models with diierent parameter values; we give, for example, values (; h) for which the 0's connguration in the Potts lattice gas is dominated by the \+" connguration of the (; h) Ising model. The Potts model, with possibly an external eld applied to one of the spins, is a special case of the Potts lattice gas, which allows our comparisons to yield rigorous bounds on the critical temperatures of Potts models. For example, we obtain :571 1 ? exp(? c) :600 for the 9-state Potts model on the hexagonal lattice. Another comparison bounds the movement of the critical line when a small Potts interaction is added to a lattice gas which otherwise has only interparticle attraction. ARC models can also be compared to related models such as the partial FK model, obtained by deleting a fraction of the nonsingleton clusters from a realization of the Fortuin-Kasteleyn random cluster model. This comparison leads to bounds on the eeects of small annealed site dilution on the critical temperature of the Potts model. 1. Introduction Random cluster models, or graphical representations, have become an increasingly important tool in the study of lattice models. Most prominently, the Fortuin-Kasteleyn random cluster model (or simply, the FK model), introduced in 17], 15] and 16], has been used to analyze aspects of the Potts and Ising models, including critical behavior 31], long-range versions 3], mean-eld behavior in high dimensions 28], covariance structure 6], mixing properties 4] and eecient simulation 38]. Wiseman and Domany 40] and PPster and Ve-lenik 35] considered graphical representations of the Ashkin-Teller model, and graphical representations for large classes of models have been considered in the contexts of eecient simulation ((11],,12]) and conditions for Gibbs uniqueness 5]. A principal advantage of random cluster models is that the connguration space, typically