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 inequalities, we obtain necessary and sufficient conditions on the set of interactions of such a Potts model in order that its critical point be astrictly monotonic function of the strengths of interactions. The method yields some ancillary information concerning the equality of certain critical exponents for Potts models; this amounts to a small amount of rigorous universality. These results are achieved in the context of a “Fortuin-Kasteleyn representation” of Potts models with many-body interactions. For such a Potts model, the corresponding random-cluster process is a (random) hypergraph.