1998/12/04 by Raphael Cerf, Emilio N. M. Cirillo · 2 citations
Physics and Astronomy · #cond-mat.stat-mech
published as The Annals of Probability, 27, 1833-1850 (1999) · 18 pages, LaTeX file, no figure
arxiv created 1998/12/04 · arxiv updated 2009/11/30
We consider the problem of bootstrap percolation on a three dimensional lattice and we study its finite size scaling behavior. Bootstrap percolation is an example of Cellular Automata defined on the d-dimensional lattice \1,2,...,L\d in which each site can be empty or occupied by a single particle; in the starting configuration each site is occupied with probability p, occupied sites remain occupied for ever, while empty sites are occupied by a particle if at least ℓ among their 2d nearest neighbor sites are occupied. When d is fixed, the most interesting case is the one ℓ=d: this is a sort of threshold, in the sense that the critical probability pc for the dynamics on the infinite lattice \Bbb Zd switches from zero to one when this limit is crossed. Finite size effects in the three-dimensional case are already known in the cases ℓ≤ 2: in this paper we discuss the case ℓ=3 and we show that the finite size scaling function for this problem is of the form f(L)=const/lnln L. We prove a conjecture proposed by A.C.D. van Enter.