## The Annals of Probability

### A quantitative Burton–Keane estimate under strong FKG condition

#### Abstract

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 [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 $q\ge1$.

#### Article information

Source
Ann. Probab., Volume 44, Number 5 (2016), 3335-3356.

Dates
Revised: June 2015
First available in Project Euclid: 21 September 2016

https://projecteuclid.org/euclid.aop/1474462099

Digital Object Identifier
doi:10.1214/15-AOP1049

Mathematical Reviews number (MathSciNet)
MR3551198

Zentralblatt MATH identifier
1357.60109

#### Citation

Duminil-Copin, Hugo; Ioffe, Dmitry; Velenik, Yvan. A quantitative Burton–Keane estimate under strong FKG condition. Ann. Probab. 44 (2016), no. 5, 3335--3356. doi:10.1214/15-AOP1049. https://projecteuclid.org/euclid.aop/1474462099

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