BK-type inequalities and generalized random-cluster representations

Recently, van den Berg and Jonasson gave the first substantial extension of the BK inequality for non-product measures: they proved that, for k -out-of- n measures, the probability that two increasing events occur disjointly is at most the product of the two individual probabilities. We show several other extensions and modifications of the BK inequality. In particular, we prove that the antiferromagnetic Ising Curie–Weiss model satisfies the BK inequality for all increasing events. We prove that this also holds for the Curie–Weiss model with three-body interactions under the so-called negative lattice condition. For the ferromagnetic Ising model we show that the probability that two events occur ‘cluster-disjointly’ is at most the product of the two individual probabilities, and we give a more abstract form of this result for arbitrary Gibbs measures. The above cases are derived from a general abstract theorem whose proof is based on an extension of the Fortuin–Kasteleyn random-cluster representation for all probability distributions and on a ‘folding procedure’ which generalizes an argument of Reimer.