In memory of Roland L. Dobrushin DECAY OF CORRELATIONS IN SUBCRITICAL POTTS AND RANDOM-CLUSTER MODELS

We prove exponential decay for the tail of the radius R of the cluster at the origin, for subcritical random-cluster models, under an assumption slightly weaker than that E(R d1) < 1 (here, d is the number of dimensions). Speciically, if E(R d1) < 1 throughout the subcritical phase, then P(R n) exp(n) for some > 0. This implies the exponential decay of the two-point correlation function of subcritical Potts models, subject to a hypothesis of (at least) polynomial decay of this function. Similar results are known already for percolation and Ising models, and for Potts models when the number q of available states is suuciently large; indeed the hypothesis of polynomial decay has been proved rigorously for these cases. In two dimensions, the hypothesis that E(R) < 1 is weaker than requiring that the susceptibility be nite, i.e., that the two-point function be summable. The principal new technique is a form of Russo's formula for random-cluster models reported by Bezuidenhout, Grimmett, and Kesten. For the current application, this leads to an analysis of a rst-passage problem for random-cluster models, and a proof that the associated time constant is strictly positive if and only if the tail of R decays exponentially.