POTTS MODELS AND RANDOM-CLUSTER PROCESSES WITH MANY-BODY INTERACTIONS

Known differential inequalities for certain ferromagnetic Potts models with pair interactions may be extended to Potts models with many-body interactions. As a major application of such differential inequalitie s, we obtain necessary and sufficient conditions on the set of interac tions of such a Potts model in order that its critical point be a stri ctly monotonic function of the strengths of interactions. The method y ields some ancillary information concerning the equality of certain cr itical exponents for Potts models; this amounts to a small amount of r igorous universality. These results are achieved in the context of a ' 'Fortuin-Kasteleyn representation'' of Potts models with many-body int eractions. For such a Potts model, the corresponding random-cluster pr ocess is a (random) hypergraph.