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The sharp threshold for bootstrap percolation in all dimensions

Balogh, József
Bollobás, Béla
Morris, Robert
Published in Transactions of the American Mathematical Society. 2012, vol. 364, p. 2667-2701
Abstract In r-neighbour 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 already-infected 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 d-dimensional 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 $p_c([n]^d,r) = [(L(d,r) + o(1)) / log_(r-1) (n)]^{d-r+1}$ as $n o infty$, where $log_r$ denotes an r-times 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).
arXiv: 1010.3326
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BALOGH, József et al. The sharp threshold for bootstrap percolation in all dimensions. In: Transactions of the American Mathematical Society, 2012, vol. 364, p. 2667-2701. doi: 10.1090/s0002-9947-2011-05552-2 https://archive-ouverte.unige.ch/unige:30552

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Deposited on : 2013-10-21

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