A quantitative Burton–Keane estimate under strong FKG condition

Autores: Hugo Duminil-Copin, Dmitry Ioffe, Yvan Velenik
Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 5, 2016, págs. 3335-3356
Idioma: inglés
DOI: 10.1214/15-AOP1049
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Resumen
- We consider translationally-invariant percolation models on ZdZd 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 nn (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≥1q≥1.