The growth of the infinite long-range percolation cluster

We consider long-range percolation on $\mathbb{Z}^d$, where the probability that two vertices at distance $r$ are connected by an edge is given by $p(r)=1-\exp[-\lambda(r)]\in(0,1)$ and the presence or absence of different edges are independent. Here, $\lambda(r)$ 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 $k\to\infty$, for different $\lambda(r)$. We show that conditioned on the origin being in the (unique) infinite cluster, nonempty classes of nonincreasing regularly varying $\lambda(r)$ exist, for which, respectively: $\bullet$ $|\mathcal{B}_k|^{1/k}\to\infty$ almost surely; $\bullet$ there exist $1<a_1<a_2<\infty$ such that $\lim_{k\to \infty}\mathbb{P}(a_1<|\mathcal{B}_k|^{1/k}<a_2)=1$; $\bullet$ $|\mathcal{B}_k|^{1/k}\to1$ almost surely. This result can be applied to spatial SIR epidemics. In particular, regimes are identified for which the basic reproduction number, $R_0$, which is an important quantity for epidemics in unstructured populations, has a useful counterpart in spatial epidemics.