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THE GROWTH OF THE INFINITE LONG-RANGE PERCOLATION CLUSTERPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2010 (English)In: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 38, no 4, p. 1583-1608Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

2010. Vol. 38, no 4, p. 1583-1608
##### Keywords [en]

Long-range percolation, epidemics, chemical distance
##### National Category

Mathematics
##### Identifiers

URN: urn:nbn:se:su:diva-48969DOI: 10.1214/09-AOP517ISI: 000280387200010OAI: oai:DiVA.org:su-48969DiVA, id: diva2:376339
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##### Note

authorCount :1Available from: 2010-12-10 Created: 2010-12-10 Last updated: 2017-12-11Bibliographically approved

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(r) = 1 - exp[-lambda(r)] is an element of (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, vertical bar B-k vertical bar, that is, the number of vertices that ale within graphdistance k of the origin, for K -> infinity, for different lambda(r) We show that conditioned on the origin being in the (unique) infinite cluster, nonempty classes of nonmereasimi regularly varying lambda(r) exist, for which, respectively vertical bar B-k vertical bar(1/k) -> infinity almost surely, there exist 1 < a(1) < a(2) < infinity such that lim(k ->infinity)P(a(1) < vertical bar B-k vertical bar(1/k) < a(2)) = 1. vertical bar B-k vertical bar(1/k) -> 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.

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