The Potts and random-cluster models
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Potts Models and Random-cluster Processes with Many-body Interactions
Geoffrey Grimmett Abstra t. Known differential inequalities for certain ferromagnetic Potts models with pairinteractions may be extended to Potts models with many-body interactions. As a major application of such differential inequalities, we obtain necessary and sufficient conditions on the set of interactions of such a Potts model in order that its critical point be a strictly monotonic funct...
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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 \+" conngura...
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The class of random-cluster models is a unification of a variety of stochastic processes of significance for probability and statistical physics, including percolation, Ising, and Potts models; in addition, their study has impact on the theory of certain random combinatorial structures, and of electrical networks. Much (but not all) of the physical theory of Ising/Potts models is best implement...
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An explicit coupling construction of random-cluster measures is presented. As one of the applications of the construction, the Potts model on amenable Cayley graphs is shown to exhibit at every temperature the mixing property known as Bernoullicity.
متن کاملIn memory of Roland L. Dobrushin DECAY OF CORRELATIONS IN SUBCRITICAL POTTS AND RANDOM-CLUSTER MODELS
We prove exponential decay for the tail of the radius R of the cluster at the origin, for subcritical random-cluster models, under an assumption slightly weaker than that E(R d1) < 1 (here, d is the number of dimensions). Speciically, if E(R d1) < 1 throughout the subcritical phase, then P(R n) exp(n) for some > 0. This implies the exponential decay of the two-point correlation function of subc...
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