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The growth of the infinite long-range percolation cluster. (English) Zbl 1196.60171
Summary: We consider long-range percolation on $\Bbb Z^d$, 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, $|\cal B_k|$, that is, the number of vertices that are within graph-distance $k$ of the origin, for $k \rightarrow \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: $\bullet |\cal B_k|^{1/k} \to \infty$ almost surely; $\bullet $ there exist $1 < a_{1} < a_{2} < \infty $ such that $\lim_{k \to \infty} \Bbb P(a_1 < |\cal B_k|^{1/k} < a_2) = 1;$ $\bullet |\cal B_k|^{1/k}$ 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:
60K35Interacting random processes; statistical mechanics type models; percolation theory
92D30Epidemiology
82B28Renormalization group methods (equilibrium statistical mechanics)
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References:
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