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• W2964202470 startingPage "2667" @default.
• W2964202470 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 $d,r in mathbb {N}$ with $d geqslant r geqslant 2$, that [ p_cbig ( [n]^d,r big ) = left ( frac {lambda (d,r) + o(1)}{log _{(r-1)} (n)} right )^{d-r+1}] as $n to infty$, for some constant $lambda (d,r) > 0$, and thus prove the existence of a sharp threshold for percolation in any (fixed) number of dimensions. We moreover determine $lambda (d,r)$ for every $d geqslant r geqslant 2$." @default.
• W2964202470 created "2019-07-30" @default.
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• W2964202470 date "2012-05-01" @default.
• W2964202470 modified "2023-10-03" @default.
• W2964202470 title "The sharp threshold for bootstrap percolation in all dimensions" @default.
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• W2964202470 doi "https://doi.org/10.1090/s0002-9947-2011-05552-2" @default.
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