Duminil-copin, Hugo et al., A quantitative Burton.Keane estimate under strong FKG condition, Annals of probability (Online) , 44(5), 2016, pp. 3335-3356
We consider translationally-invariant percolation models on Zd satisfying the finite energy and the FKG properties. We provide explicit upper bounds on the probability of having two distinct clusters going from the endpoints of an edge to distance n (this corresponds to a finite size version of the celebrated Burton.Keane [Comm. Math. Phys. 121 (1989) 501.505] argument proving uniqueness of the infinite-cluster). The proof is based on the generalization of a reverse Poincaré inequality proved in Chatterjee and Sen (2013). As a consequence, we obtain upper bounds on the probability of the so-called four-arm event for planar random-cluster models with cluster-weight q.1.