The growth of the infinite long-range percolation cluster

Abstract

We consider long-range percolation on ℤ^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, $|\mathcal{B}_{k}|$, 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.