  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 $\ell$ among their $2d$ nearest neighbor sites are
occupied. When $d$ is fixed, the most interesting case is the one $\ell=d$:
this is a sort of threshold, in the sense that the critical probability $p_c$
for the dynamics on the infinite lattice ${\Bbb Z}^d$ switches from zero to one
when this limit is crossed. Finite size effects in the three-dimensional case
are already known in the cases $\ell\le 2$: in this paper we discuss the case
$\ell=3$ and we show that the finite size scaling function for this problem is
of the form $f(L)={\mathrm{const}}/\ln\ln L$. We prove a conjecture proposed by
A.C.D. van Enter.
