References$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_upper_j_idt147",{id:"formSmash:upper:j_idt147",widgetVar:"widget_formSmash_upper_j_idt147",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:upper:referencesLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_upper_j_idt148_j_idt150",{id:"formSmash:upper:j_idt148:j_idt150",widgetVar:"widget_formSmash_upper_j_idt148_j_idt150",target:"formSmash:upper:j_idt148:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});

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});
function selectAll()
{
var panelSome = $(PrimeFaces.escapeClientId("formSmash:some"));
var panelAll = $(PrimeFaces.escapeClientId("formSmash:all"));
panelAll.toggle();
toggleList(panelSome.get(0).childNodes, panelAll);
toggleList(panelAll.get(0).childNodes, panelAll);
}
/*Toggling the list of authorPanel nodes according to the toggling of the closeable second panel */
function toggleList(childList, panel)
{
var panelWasOpen = (panel.get(0).style.display == 'none');
// console.log('panel was open ' + panelWasOpen);
for (var c = 0; c < childList.length; c++) {
if (childList[c].classList.contains('authorPanel')) {
clickNode(panelWasOpen, childList[c]);
}
}
}
/*nodes have styleClass ui-corner-top if they are expanded and ui-corner-all if they are collapsed */
function clickNode(collapse, child)
{
if (collapse && child.classList.contains('ui-corner-top')) {
// console.log('collapse');
child.click();
}
if (!collapse && child.classList.contains('ui-corner-all')) {
// console.log('expand');
child.click();
}
}
PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2010 (English)In: Annals of Probability, ISSN 0091-1798, Vol. 38, no 4, 1583-1608 p.Article in journal (Refereed) Published
##### Abstract [en]

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

2010. Vol. 38, no 4, 1583-1608 p.
##### Keyword [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: diva2:376339
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt388",{id:"formSmash:j_idt388",widgetVar:"widget_formSmash_j_idt388",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt394",{id:"formSmash:j_idt394",widgetVar:"widget_formSmash_j_idt394",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt400",{id:"formSmash:j_idt400",widgetVar:"widget_formSmash_j_idt400",multiple:true});
##### Note

authorCount :1Available from: 2010-12-10 Created: 2010-12-10 Last updated: 2010-12-10Bibliographically 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.

References$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_lower_j_idt1106",{id:"formSmash:lower:j_idt1106",widgetVar:"widget_formSmash_lower_j_idt1106",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:lower:referencesLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_lower_j_idt1107_j_idt1109",{id:"formSmash:lower:j_idt1107:j_idt1109",widgetVar:"widget_formSmash_lower_j_idt1107_j_idt1109",target:"formSmash:lower:j_idt1107:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});