A QUANTITATIVE BURTON-KEANE ESTIMATE UNDER STRONG FKG CONDITION

• 成果类型: Article
• 署名作者: Duminil-Copin, Hugo; Ioffe, Dmitry; Velenik, Yvan
• 署名单位: University of Geneva; Technion Israel Institute of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1049
• 发表日期: 2016
• 页码: 3335-3356
• 关键词: random-cluster model; infinite cluster; uniqueness; percolation; continuity

摘要

We consider translationally-invariant percolation models on 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 end-points 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 Poincare 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 >= 1.