The growth of the infinite long-range percolation cluster

Details The growth of the infinite long-range percolation cluster The growth of the infinite long-range percolation cluster

2009-01-06

arxiv.org/abs/0901.0661v3

Annals of Probability 2010, Vol. 38, No. 4, 1583-1608

Mathematics

Probability

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

Scientific paper

10.1214/09-AOP517

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

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