The time of bootstrap percolation with dense initial sets
2014 (English)In: Annals of Probability, ISSN 0091-1798, Vol. 42, no 4, 1337-1373 p.Article in journal (Refereed) Published
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.
IdentifiersURN: urn:nbn:se:su:diva-114308DOI: 10.1214/12-AOP818OAI: oai:DiVA.org:su-114308DiVA: diva2:791135