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 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.