The Stochastic Random-Cluster Process and the Uniqueness of Random-Cluster Measures

The random-cluster model is a generalization of percolation and ferromagnetic Potts models, due to Fortuin and Kasteleyn. Not only is the random-cluster model a worthwhile topic for study in its own right, but also it provides much information about phase transitions in the associated physical models. This paper serves two functions. First, we introduce and survey random-cluster measures from the probabilist's point of view, giving clear statements of some of the many open problems. Second, we present new results for such measures, as follows. We discuss the relationship between weak limits of random-cluster measures and measures satisfying a suitable DLR condition. Using an argument based on the convexity of pressure, we prove the uniqueness of random-cluster measures for all but (at most) countably many values of the parameter $p$. Related results concerning phase transition in two or more dimensions are included, together with various stimulating conjectures. The uniqueness of the infinite cluster is employed in an intrinsic way in part of these arguments. In the second part of this paper is constructed a Markov process whose level sets are reversible Markov processes with random-cluster measures as unique equilibrium measures. This construction enables a coupling of random-cluster measures for all values of $p$. Furthermore, it leads to a proof of the semicontinuity of the percolation probability and provides a heuristic probabilistic justification for the widely held belief that there is a first-order phase transition if and only if the cluster-weighting factor $q$ is sufficiently large.