The time of bootstrap percolation for dense initial sets
Abstract
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. We study the distribution of the time t at which all vertices become infected. Given t = t(n) = o(log n/log log n), 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_n^d. 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.
Keywords
Cite
@article{arxiv.1205.3922,
title = {The time of bootstrap percolation for dense initial sets},
author = {Béla Bollobás and Cecilia Holmgren and Paul Smith and Andrew J. Uzzell},
journal= {arXiv preprint arXiv:1205.3922},
year = {2012}
}
Comments
32 pages, 2 figures. To appear in Annals of Probability