THE STOCHASTIC RANDOM-CLUSTER PROCESS AND THE UNIQUENESS OF RANDOM-CLUSTER MEASURES

The random-cluster model is a generalization of percolation and ferrom agnetic Potts models, due to Fortuin and Kasteleyn. Not only is the ra ndom-cluster model a worthwhile topic for study in its own right, but also it provides much information about phase transitions in the assoc iated physical models. This paper serves two functions. First, we intr oduce and survey random-cluster measures from the probabilist's point of view, giving clear statements of some of the many open problems. Se cond, we present new results for such measures, as follows. We discuss the relationship between weak limits of random-cluster measures and m easures satisfying a suitable DLR condition. Using an argument based o n 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 dimensi ons are included, together with various stimulating conjectures. The u niqueness of the infinite cluster is employed in an intrinsic way in p art of these arguments. In the second part of this paper is constructe d a Markov process whose level sets are reversible Markov processes wi th random-cluster measures as unique equilibrium measures. This constr uction enables a coupling of random-cluster measures for all values of p. Furthermore, it leads to a proof of the semicontinuity of the perc olation probability and provides a heuristic probabilistic justificati on for the widely held belief that there is a first-order phase transi tion if and only if the cluster-weighting factor q is sufficiently lar ge.