The Annals of Probability

Finite Size Scaling in Three-Dimensional Bootstrap Percolation

Raphaël Cerf and Emilio N. M. Cirillo

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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.

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Ann. Probab., Volume 27, Number 4 (1999), 1837-1850.

First available in Project Euclid: 31 May 2002

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35]

Cellular automata bootstrap percolation finite size scaling critical length


Cerf, Raphaël; Cirillo, Emilio N. M. Finite Size Scaling in Three-Dimensional Bootstrap Percolation. Ann. Probab. 27 (1999), no. 4, 1837--1850. doi:10.1214/aop/1022874817.

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