*(English)*Zbl 1196.60171

Summary: 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 $p\left(r\right)=1-exp[-\lambda (r\left)\right]\in (0,1)$ and the presence or absence of different edges are independent. Here, $\lambda \left(r\right)$ 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 $\lambda \left(r\right)$. We show that conditioned on the origin being in the (unique) infinite cluster, nonempty classes of nonincreasing regularly varying $\lambda \left(r\right)$ exist, for which, respectively:

$\u2022|{\mathcal{B}}_{k}{|}^{1/k}\to \infty $ almost surely;

$\u2022$ there exist $1<{a}_{1}<{a}_{2}<\infty $ such that ${lim}_{k\to \infty}\mathbb{P}({a}_{1}<|{\mathcal{B}}_{k}{|}^{1/k}<{a}_{2})=1;$

$\u2022|{\mathcal{B}}_{k}{|}^{1/k}$ 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.