The Annals of Probability

The growth of the infinite long-range percolation cluster

Pieter Trapman

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Abstract

We consider long-range percolation on ℤd, where the probability that two vertices at distance r are connected by an edge is given by p(r) = 1 − exp[−λ(r)] ∈ (0, 1) and the presence or absence of different edges are independent. Here, λ(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 → ∞, for different λ(r). We show that conditioned on the origin being in the (unique) infinite cluster, nonempty classes of nonincreasing regularly varying λ(r) exist, for which, respectively:

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

• there exist 1 < a1 < a2 < ∞ such that $\lim_{k\to \infty}\mathbb{P}(a_{1}\textless |\mathcal{B}_{k}|^{1/k}\textless a_{2})=1$;

$|\mathcal{B}_{k}|^{1/k}\to1$ almost surely.

This result can be applied to spatial SIR epidemics. In particular, regimes are identified for which the basic reproduction number, R0, which is an important quantity for epidemics in unstructured populations, has a useful counterpart in spatial epidemics.

Article information

Source
Ann. Probab. Volume 38, Number 4 (2010), 1583-1608.

Dates
First available: 8 July 2010

Permanent link to this document
http://projecteuclid.org/euclid.aop/1278593961

Digital Object Identifier
doi:10.1214/09-AOP517

Zentralblatt MATH identifier
05776091

Mathematical Reviews number (MathSciNet)
MR2663638

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 92D30: Epidemiology
Secondary: 82B28: Renormalization group methods [See also 81T17]

Keywords
Long-range percolation epidemics chemical distance

Citation

Trapman, Pieter. The growth of the infinite long-range percolation cluster. The Annals of Probability 38 (2010), no. 4, 1583--1608. doi:10.1214/09-AOP517. http://projecteuclid.org/euclid.aop/1278593961.


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