Article (590 Kb)  Free access
Other version: http://www.ams.org/journals/tran/201236405/S000299472011055522/
Highlights
Title 
The sharp threshold for bootstrap percolation in all dimensions 

Authors  
Published in  Transactions of the American Mathematical Society. 2012, vol. 364, p. 26672701  
Abstract  In rneighbour bootstrap percolation on a graph G, a (typically random) set A of initially 'infected' vertices spreads by infecting (at each time step) vertices with at least r alreadyinfected neighbours. This process may be viewed as a monotone version of the Glauber dynamics of the Ising model, and has been extensively studied on the ddimensional grid $[n]^d$. The elements of the set A are usually chosen independently, with some density p, and the main question is to determine $p_c([n]^d,r)$, the density at which percolation (infection of the entire vertex set) becomes likely. In this paper we prove, for every pair $d ge r ge 2$, that there is a constant L(d,r) such that $p_c([n]^d,r) = [(L(d,r) + o(1)) / log_(r1) (n)]^{dr+1}$ as $n o infty$, where $log_r$ denotes an rtimes iterated logarithm. We thus prove the existence of a sharp threshold for percolation in any (fixed) number of dimensions. Moreover, we determine L(d,r) for every pair (d,r).  
Identifiers  arXiv: 1010.3326  
Full text  
Structures  
Citation (ISO format)  BALOGH, József et al. The sharp threshold for bootstrap percolation in all dimensions. In: Transactions of the American Mathematical Society, 2012, vol. 364, p. 26672701. doi: 10.1090/s000299472011055522 https://archiveouverte.unige.ch/unige:30552 