00-81 Alexander, K. S.

The Asymmetric Random Cluster Model and Comparison of Ising and Potts Models (162K, AMS-LATeX 1.2) Feb 22, 00

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Abstract. 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 different parameter values; we give, for example, values $(\beta, h)$ for which the 0's configuration in the Potts lattice gas is dominated by the ``+'' configuration of the $(\beta,h)$ Ising model. The Potts model, with possibly an external field 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 $.571 \leq 1 - \exp(-\beta_{c}) \leq .600$ 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 effects of small annealed site dilution on the critical temperature of the Potts model.

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