Finite size scaling in three-dimensional bootstrap percolation

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 forever, 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 p(c) for the dynamics on the infinite lattice Z(d) 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 less than or equal to 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/ In In L. We prove a conjecture proposed by A. C. D. van Enter.