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 among their 2d nearest neighbor sites are occupied. When d is fixed, the most interesting case is the one = 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 2; in this paper we discuss the case = 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.