Journal Article10.1016/S0378-4371(03)00364-9
Sharp thresholds in Bootstrap percolation
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TL;DR: In this paper, the threshold function of the percolation is shown to be sharp and it is shown that the initial configuration percolates if at the end of the process every site is occupied, while an empty site becomes occupied if at least two of its neighbours are occupied.
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Abstract: In the standard bootstrap percolation on the d-dimensional grid G n d , in the initial position each of the nd sites is occupied with probability p and empty with probability 1−p, independently of the state of every other site. Once a site is occupied, it remains occupied for ever, while an empty site becomes occupied if at least two of its neighbours are occupied. If at the end of the process every site is occupied, we say that the (initial) configuration percolates. By making use of a theorem of Friedgut and Kalai (Proc. Amer. Math. Soc. 124 (1996) 2993), we shall show that the threshold function of the percolation is sharp. We shall prove similar results for three other models of bootstrap percolation as well.
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Citations
The sharp threshold for bootstrap percolation in all dimensions
TL;DR: In this paper, it was shown that there is a constant L(d,r) such that the density at which percolation becomes likely in any (fixed) number of dimensions.
Bootstrap percolation in three dimensions
TL;DR: In this paper, the exact threshold for percolation in the case of fixed d = r = 3 was shown to be (1+o(1))(1 + o(1)), where ρ is an iterated logarithm.
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The sharp threshold for bootstrap percolation in all dimensions
TL;DR: In this article, it was shown that there is a constant L(d,r) such that the density at which percolation becomes likely in any (fixed) number of dimensions.
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On the Behavior of Some Cellular Automata Related to Bootstrap Percolation
TL;DR: In this paper, the authors studied a family of models which has arisen in the interface between statistical mechanics and percolation, and showed that 0 < Pc = ir- < 1.