Finite Size Scaling in Three-Dimensional Bootstrap Percolation

Abstract

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 ${1,2,\ldots,L^d}$ 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 $\ell$ among their 2$d$ 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 $\mathbb{Z}^d$ switches from zero to one when this limit is crossed. Finite size effects in the three-dimensional case are already known in the cases $\ell\leq2$; 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 $f(L) =const/lnlnL$.We prove a conjecture proposed by A.C.D. van Enter.