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October 1999 Finite Size Scaling in Three-Dimensional Bootstrap Percolation
Raphaël Cerf, Emilio N. M. Cirillo
Ann. Probab. 27(4): 1837-1850 (October 1999). DOI: 10.1214/aop/1022874817


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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Raphaël Cerf. Emilio N. M. Cirillo. "Finite Size Scaling in Three-Dimensional Bootstrap Percolation." Ann. Probab. 27 (4) 1837 - 1850, October 1999.


Published: October 1999
First available in Project Euclid: 31 May 2002

zbMATH: 0960.60088
MathSciNet: MR1742890
Digital Object Identifier: 10.1214/aop/1022874817

Primary: 60K35
Secondary: 82B43

Keywords: Bootstrap percolation , cellular automata , critical length , finite size scaling

Rights: Copyright © 1999 Institute of Mathematical Statistics


Vol.27 • No. 4 • October 1999
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