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The growth of the infinite long-range percolation cluster. (English) Zbl 1196.60171

Summary: 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 $p\left(r\right)=1-exp\left[-\lambda \left(r\right)\right]\in \left(0,1\right)$ 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, $|{ℬ}_{k}|$, 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:

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

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

$•|{ℬ}_{k}{|}^{1/k}$ 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.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 92D30 Epidemiology 82B28 Renormalization group methods (equilibrium statistical mechanics)
##### Keywords:
long-range percolation; epidemics; chemical distance