%0 Journal Article
%T A quantitative Burton-Keane estimate under strong FKG condition
%A Hugo Duminil-Copin
%A Dmitry Ioffe
%A Yvan Velenik
%J Mathematics
%D 2014
%I arXiv
%X We consider translationally-invariant percolation models on $\mathbb Z^d$ 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 argument proving uniqueness of the infinite-cluster). The proof is based on the generalization of a reverse Poincar\'e inequality proved by Chatterjee and Sen. 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 $q\ge 1$.
%U http://arxiv.org/abs/1409.5199v4