A quantitative Burton-Keane estimate under strong FKG condition

TL;DR

This paper establishes explicit upper bounds on cluster probabilities in percolation models satisfying FKG and finite energy conditions, extending Burton-Keane estimates and analyzing four-arm events in planar models.

Contribution

It provides a quantitative Burton-Keane estimate under strong FKG conditions and generalizes a reverse Poincaré inequality for percolation models.

Findings

Upper bounds on two-cluster connection probabilities

Bounds on four-arm event probabilities in planar models

Extension of Burton-Keane argument to finite size scenarios

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 [Comm. Math. Phys. 121 (1989) 501-505] argument proving uniqueness of the infinite-cluster). The proof is based on the generalization of a reverse Poincar\'{e} 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 $q\ge1$.