# Abstract

A Quantitative Burton-Keane Estimate Under Strong FKG Condition
H. Duminil-Copin, D. Ioffe, Y. Velenik
Ann. Probab.
44,
3335-3356
(2016).
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 Poincaré 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 $q\ge1$.

**Key words:**Reverse Poincaré inequality, pivotal, dependent percolation, FK percolation, random cluster model, four-arms, Burton-Keane theorem, negative association.**Files:**PDF file, Published version, bibtex