The Random Cluster Model on A

The random cluster model on a general innnite graph with bounded degree wired at innnity is considered and a \ghost vertex" method is introduced in order to explicitly construct random cluster measures satisfying the Dobrushin-Lanford-Ruelle condition for q 1. It is proved that on a regular nonamenable graph there is a q 0 such that for q q 0 there is a phase transition for an entire interval of values of p, whereas on a quasi-transitive amenable graph there is a phase transition for at most a countable number of values of p. In particular, a transitive graph is nonamenable if and only if there is a phase transition for an entire interval of p-values for large enough q. It is also observed that these results have a Potts model interpretation. In particular a transitive graph is nonamenable if and only if the q-state Potts model on that graph has the property that for q large enough there is an entire interval of temperatures for which the free Gibbs state is not a convex combination of the q Gibbs states obtained from one-spin boundary conditions. It is also proved that on the regular tree, T n , with q 1 and p close enough to 1, there is unique random cluster measure despite the presence of more than one innnite cluster. This partly proves Conjecture 1.9 of 11].