A quantitative Burton-Keane estimate under strong FKG condition. (English) Zbl 1357.60109

Authors’ abstract: 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 argument [R. M. Burton and M. Keane, Comm. Math. Phys. 121, No. 3, 501–505 (1989; Zbl 0662.60113)] proving uniqueness of the infinite-cluster). The proof is based on the generalization of a reverse Poincaré inequality proved in [S. Chatterjee and S. Sen, “Minimal spanning trees and Stein’s method”, Ann. App. Probab. (to appear); arXiv:1307.1661]. 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 \(q \geq 1\).


60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics


Zbl 0662.60113
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