The Annals of Probability

A quantitative Burton–Keane estimate under strong FKG condition

Hugo Duminil-Copin, Dmitry Ioffe, and Yvan Velenik

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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

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

Received: October 2014
Revised: June 2015
First available in Project Euclid: 21 September 2016

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B43: Percolation [See also 60K35]

Reverse Poincaré inequality dependent percolation FK percolation random cluster model four-arms event Burton–Keane theorem negative association


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.

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