# A quantitative Burton-Keane estimate under strong FKG condition

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Hugo Duminil-Copin, Dmitry Ioffe, Yvan Velenik

Hugo Duminil-Copin, Dmitry Ioffe, Yvan Velenik

**18/9/14**Published in : arXiv:1409.5199

We consider translationally-invariant percolation models on 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≥1.