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 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/lnlnL. We prove a conjecture proposed by
A.C.D. van Enter.Comment: 18 pages, LaTeX file, no figur