The growth of the infinite long-range percolation cluster

Abstract

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 ${\lim }_{k\to \infty }\mathbb{P}(a_1<|{\mathcal{B}}_k{|}^{1/k}<a_2)=1$; $\bullet$ $|{\mathcal{B}}_k{|}^{1/k}\to 1$ 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.

Published in at http://dx.doi.org/10.1214/09-AOP517 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• On the scaling of the chemical distance in long-range percolation models
• A lower bound for the chemical distance in sparse long-range percolation models

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• Stopping explosion by penalising transmission to hubs in scale-free spatial random graphs
• The longest edge in discrete and continuous long-range percolation