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Scientific article
English

A quantitative Burton-Keane estimate under strong FKG condition

Published inAnnals of probability, vol. 44, no. 5, p. 3335-3356
Publication date2016
Abstract

We consider translationally-invariant percolation models on the d-dimensional cubic lattice, 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 larger or equal to 1.

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Keywords
  • Reverse Poincaré inequality
  • Pivotal
  • Dependent percolation
  • FK percolation
  • Random cluster model
  • Four-arms
  • Burton-Keane theorem
  • Negative association
Citation (ISO format)
DUMINIL-COPIN, Hugo, IOFFE, Dmitry, VELENIK, Yvan. A quantitative Burton-Keane estimate under strong FKG condition. In: Annals of probability, 2016, vol. 44, n° 5, p. 3335–3356. doi: 10.1214/15-aop1049
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Article (Submitted version)
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Identifiers
ISSN of the journal0091-1798
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