The growth of the infinite long-range percolation cluster

Abstract

We consider long-range percolation on ℤd, 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, $|\mathcal{B}_{k}|$, that is, the number of vertices that are within graph-distance 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: • $|\mathcal{B}_{k}|^{1/k}\to\infty$> almost surely; • there exist 1 < a1 < a2 < ∞ such that $\lim_{k\to \infty}\mathbb{P}(a_{1}< |\mathcal{B}_{k}|^{1/k}< a_{2})=1$; • $|\mathcal{B}_{k}|^{1/k}\to1$ almost surely. This result can be applied to spatial SIR epidemics. In particular, regimes are identified for which the basic reproduction number, R0, which is an important quantity for epidemics in unstructured populations, has a useful counterpart in spatial epidemics.