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Other version: http://www.ams.org/journals/tran/201236405/S000299472011055522/
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The sharp threshold for bootstrap percolation in all dimensions 

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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  
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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. https://archiveouverte.unige.ch/unige:30552 