Sharp thresholds in Bootstrap percolation

József Balogh and Béla Bollobás

Physica A: Statistical Mechanics and its Applications, 2003, vol. 326, issue 3, 305-312

Abstract: In the standard bootstrap percolation on the d-dimensional grid Gnd, 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.

Keywords: Percolation; Bootstrap percolation; Cellular automata; Threshold; Sharp threshold (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:326:y:2003:i:3:p:305-312

DOI: 10.1016/S0378-4371(03)00364-9

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