A quantitative Burton–Keane estimate under strong FKG condition

Abstract

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 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 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 ≥ 1.

Minimal spanning trees and Stein’s method

“…In [23], Cerf improves the bound given in Lemma 5.2 but only at p = p c . In the recent preprint [28], the authors prove a bound of the form O(1/n) for bond percolation in Z 2 (in fact, their result is true for the more general random cluster model), and Kozma and Nachmias [37] prove a bound of the form O(1/n 4 ) for bond percolation in Z d when d ≥ 19 but again, these bounds hold only at p = p c . For site percolation on the triangular lattice, a bound of the form O(n −5/4+o (1) ) is known to hold at criticality [54], but an analogous result is not known for the square lattice Z 2 .…”

“…Thus the technique used in the proofs of Lemma 5.1 and Lemma 5.7 is expected to have wider applicability in other contexts, where the Burton-Keane argument works but the AKN argument does not. As mentioned earlier, using a generalization of the arguments used in the proof of Lemma 5.7, Duminil-Copin, Ioffe and Velenik [28] have recently obtained bounds on the probability of two-arm events in a broad class of translation-invariant percolation models on Z d . Due to this recent development, we have included a brief sketch of the proof of Lemma 5.7 in Appendix A even though in the proof of Theorem 2.4 we will use Lemma 5.2 which gives a sharper bound.…”

“…Secondly, we present a way of quantifying the Burton-Keane argument for the uniqueness of the infinite open cluster. The latter is interesting in its own right and based on a generalization of our technique, Duminil-Copin, Ioffe and Velenik [28] have recently obtained bounds on probability of two-arm events in a broad class of translation-invariant percolation models.…”