The growth of the infinite long-range percolation cluster

Abstract

We consider long-range percolation on Z{double-struck}d, where the probability that two vertices at distance r are connected by an edge is given by p(r) = 1 - exp[-λ(r)] ∈ (0, 1) and the presence or absence of different edges are independent. Here, λ(r) is a strictly positive, nonincreasing, regularly varying function. We investigate the asymptotic growth of the size of the k-ball around the origin, |B{script}k|, that is, the number of vertices that are within graphdistance k of the origin, for k→∞, for different λ(r). We show that conditioned on the origin being in the (unique) infinite cluster, nonempty classes of nonincreasing regularly varying λ(r) exist, for which, respectively: • |B{script}k|1/k→∞almost surely; • there exist 1 a1 < a2 < ∞ such that limk→∞P{double-struck}(a1