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The time of bootstrap percolation with dense initial sets
Cambridge University and Stockholms universitet, Matematiska institutionen.
2014 (English)In: Annals of Probability, ISSN 0091-1798, Vol. 42, no 4, 1337-1373 p.Article in journal (Refereed) Published
Abstract [en]

Let r∈N . In r -neighbour bootstrap percolation on the vertex set of a graph G , vertices are initially infected independently with some probability p . At each time step, the infected set expands by infecting all uninfected vertices that have at least r  infected neighbours. When p  is close to 1, we study the distribution of the time at which all vertices become infected. Given t=t(n)=o(logn/loglogn) , we prove a sharp threshold result for the probability that percolation occurs by time t  in d -neighbour bootstrap percolation on the d -dimensional discrete torus T d n  . Moreover, we show that for certain ranges of p=p(n) , the time at which percolation occurs is concentrated either on a single value or on two consecutive values. We also prove corresponding results for the modified d -neighbour rule

Place, publisher, year, edition, pages
2014. Vol. 42, no 4, 1337-1373 p.
National Category
URN: urn:nbn:se:uu:diva-260384DOI: 10.1214/12-AOP818OAI: oai:DiVA.org:uu-260384DiVA: diva2:846979
Swedish Research Council
Available from: 2015-02-26 Created: 2015-08-18 Last updated: 2015-08-19

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Holmgren, Cecilia
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