Explicit isoperimetric constants, phase transitions in the random-cluster and Potts models, and Bernoullicity

The random-cluster model is a dependent percolation model that has applications in the study of Ising and Potts models. In this paper, several new results for the random-cluster model with cluster parameter q ≥ 1 are obtained. These include an explicit pointwise dynamical construction of random-cluster measures for arbitrary graphs, and for unimodular transitive graphs, lack of percolation for the free random-cluster measure at the lower critical value on nonamenable graphs, and a number of inequalities for the critical values. Some of these inequalities lead to considerations of isoperimetric constants in certain hyperbolic graphs, and the first nontrivial explicit calculations of such constants are obtained. Applications to the Potts model include Bernoullicity in the Z case at all temperatures, and non-robust phase transition in the case of nonamenable regular graphs.