Power-law bounds for critical long-range percolation below the upper-critical dimension

Abstract

We study long-range Bernoulli percolation on ℤd in which each two vertices x and y are connected by an edge with probability 1−exp(−β‖x−y‖−d−α). It is a theorem of Noam Berger (CMP, 2002) that if 0<α<d then there is no infinite cluster at the critical parameter βc. We give a new, quantitative proof of this theorem establishing the power-law upper bound Pβc(|K|≥n)≤Cn−(d−α)/(2d+α) for every n≥1, where K is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli percolation model that is neither planar nor expected to exhibit mean-field critical behaviour. As part of the proof, we establish a universal inequality implying that the maximum size of a cluster in percolation on any finite graph is of the same order as its mean with high probability. We apply this inequality to derive a new rigorous hyperscaling inequality (2−η)(δ+1)≤d(δ−1) relating the cluster-volume exponent δ and two-point function exponent η.