On the growth of the supercritical long-range percolation cluster on $\mathbb{Z}^d$ and an application for spatial epidemics

We consider long-range percolation on $\mathbb{Z}^d$ in which the measure on the configuration of edges is a product measure and the probability that two vertices at distance $r$ share an edge is given by $p(r) = 1-\exp[-\lambda(r)] \in (0,1)$. Here $\lambda(r)$ is a strictly positive, non-increasing regularly varying function. We investigate the asymptotic growth of the size of the $k$-ball around the origin, $|\mathcal{B}_k|$, i.e. 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 infinite component, non-empty classes of non-increasing regularly varying $\lambda(r)$ exist for which respectively $|\mathcal{B}_k|^{1/k} \to \infty$ almost surely, there exist $1 almost surely. This result can be applied to spatial $SIR$ epidemics. In particular, we show that it is possible to construct a distribution of long-range contacts between individuals only depending on their distance, such that the number of infectious individuals in the $k$-th infection generation stochastically dominates an exponentially growing function.