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September 2016 A quantitative Burton–Keane estimate under strong FKG condition
Hugo Duminil-Copin, Dmitry Ioffe, Yvan Velenik
Ann. Probab. 44(5): 3335-3356 (September 2016). DOI: 10.1214/15-AOP1049


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 $q\ge1$.


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Hugo Duminil-Copin. Dmitry Ioffe. Yvan Velenik. "A quantitative Burton–Keane estimate under strong FKG condition." Ann. Probab. 44 (5) 3335 - 3356, September 2016.


Received: 1 October 2014; Revised: 1 June 2015; Published: September 2016
First available in Project Euclid: 21 September 2016

zbMATH: 1357.60109
MathSciNet: MR3551198
Digital Object Identifier: 10.1214/15-AOP1049

Primary: 60K35
Secondary: 82B20 , 82B43

Keywords: Burton–Keane theorem , Dependent percolation , FK percolation , four-arms event , negative association , random cluster model , Reverse Poincaré inequality

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.44 • No. 5 • September 2016
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