Comparison of bootstrap percolation models

Abstract

New simulations involving up to nearly 10⁹ sites for square lattice bootstrap percolation give a percolation threshold compatible with unity. In this model sites in a square lattice are initially occupied randomly, and then are removed if they have less than three occupied neighbours. The results are compared with those of similar models, such as that studied by Frobose (1989), and the earlier bootstrap and diffusion percolation models, in two, three and four dimensions, of Adler and Aharony (1986-8). In the latter models, the effective threshold approaches its asymptotic limit (one or zero) as ( lambda /ln L)^d-1, for samples of size L^d, in agreement with theoretical predictions. The coefficient lambda decreases monotonically with the parameters which facilitate the removal (addition) of sites.