Sharp thresholds in Bootstrap percolation


In the standard bootstrap percolation on the d-dimensional grid double-struck G signnd, 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. © 2003 Published by Elsevier B.V.

Publication Title

Physica A: Statistical Mechanics and its Applications