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.
Ann. Probab.
27(4):
1837-1850
(October 1999).
DOI: 10.1214/aop/1022874817
[1] Adler,J. and Aharony,A. (1988). Diffusion percolation I. Infinite time limit and bootstrap percolation. J. Phys. A: Math. Gen. 21 1387-1404. MR939745 10.1088/0305-4470/21/6/015[1] Adler,J. and Aharony,A. (1988). Diffusion percolation I. Infinite time limit and bootstrap percolation. J. Phys. A: Math. Gen. 21 1387-1404. MR939745 10.1088/0305-4470/21/6/015
[3] Aizenman,M. and Lebowitz,J. L. (1988). Metastability effects in bootstrap percolation. J. Phys. A: Math. Gen. 21 3801-3813. MR90e:82047 0656.60106 10.1088/0305-4470/21/19/017[3] Aizenman,M. and Lebowitz,J. L. (1988). Metastability effects in bootstrap percolation. J. Phys. A: Math. Gen. 21 3801-3813. MR90e:82047 0656.60106 10.1088/0305-4470/21/19/017
[4] Branco,N. S.,Dos Santos,R. R. and de Queiroz,S. L. A. (1984). Bootstrap percolation: a renormalization group approach. J. Phys. C 17 L373-L377; Khan, M. A., Gould, H. and [4] Branco,N. S.,Dos Santos,R. R. and de Queiroz,S. L. A. (1984). Bootstrap percolation: a renormalization group approach. J. Phys. C 17 L373-L377; Khan, M. A., Gould, H. and
Chalupa, J. (1985). Monte Carlo renormalization group study of bootstrap percolation. J. Phys. C 18 L223-L228; Branco, N. S., de Queiroz, S. L. A. and Dos Santos, R. R. Chalupa, J. (1985). Monte Carlo renormalization group study of bootstrap percolation. J. Phys. C 18 L223-L228; Branco, N. S., de Queiroz, S. L. A. and Dos Santos, R. R.
[9] Mountford,T. S. (1995). Critical length for semi-oriented bootstrap percolation. Stochastic Process. Appl. 56 185-205. MR96b:60254 0821.60092 10.1016/0304-4149(94)00061-W[9] Mountford,T. S. (1995). Critical length for semi-oriented bootstrap percolation. Stochastic Process. Appl. 56 185-205. MR96b:60254 0821.60092 10.1016/0304-4149(94)00061-W
[10] Schonmann,R. H. (1990). Critical Points of two-dimensional bootstrap percolation-like cellular automata. J. Statist. Phys. 58 1239-1244. MR91f:82038 0712.68071 10.1007/BF01026574[10] Schonmann,R. H. (1990). Critical Points of two-dimensional bootstrap percolation-like cellular automata. J. Statist. Phys. 58 1239-1244. MR91f:82038 0712.68071 10.1007/BF01026574
[11] Schonmann,R. H. (1990). Finite size scaling behavior of a biased majority rule cellular automaton. Phys. A 167 619-627. MR91m:82087 10.1016/0378-4371(90)90280-6[11] Schonmann,R. H. (1990). Finite size scaling behavior of a biased majority rule cellular automaton. Phys. A 167 619-627. MR91m:82087 10.1016/0378-4371(90)90280-6
[12] Schonmann,R. H. (1992). On the behavior of some cellular automata related to bootstrap percolation. Ann. Probab. 20 174-193. MR93b:60231 0742.60109 10.1214/aop/1176989923 euclid.aop/1176989923
[12] Schonmann,R. H. (1992). On the behavior of some cellular automata related to bootstrap percolation. Ann. Probab. 20 174-193. MR93b:60231 0742.60109 10.1214/aop/1176989923 euclid.aop/1176989923
[13] Toffoli,T. and Margolus,N. (1987). Cellular Automata Machines. A New Environment for Modeling. MIT Press. 0655.68055[13] Toffoli,T. and Margolus,N. (1987). Cellular Automata Machines. A New Environment for Modeling. MIT Press. 0655.68055
[14] Ulam,S. (1950). Random processes and transformations. Proc. Internat. Congr. Math. 264- 275. MR45334 0049.09511[14] Ulam,S. (1950). Random processes and transformations. Proc. Internat. Congr. Math. 264- 275. MR45334 0049.09511
[15] van Enter,A. C. D. (1987). Proof of Straley's argument for bootstrap percolation. J. Statist. Phys. 48 943-945. MR88j:82024 1084.82548 10.1007/BF01019705[15] van Enter,A. C. D. (1987). Proof of Straley's argument for bootstrap percolation. J. Statist. Phys. 48 943-945. MR88j:82024 1084.82548 10.1007/BF01019705
[16] van Enter,A. C. D.,Adler,J. and Duarte,J. A. M. S. (1990). Finite-size effects for some bootstrap percolation models. J. Statist. Phys. 60 323-332. MR92c:82054a 10.1007/BF01314923[16] van Enter,A. C. D.,Adler,J. and Duarte,J. A. M. S. (1990). Finite-size effects for some bootstrap percolation models. J. Statist. Phys. 60 323-332. MR92c:82054a 10.1007/BF01314923
[17] van Enter,A. C. D.,Adler,J. and Duarte,J. A. M. S. (1991). Addendum: Finite-size effects for some bootstrap percolation models. J. Statist. Phys. 62 505-506. MR92c:82054b 10.1007/BF01020891 63.1006.01[17] van Enter,A. C. D.,Adler,J. and Duarte,J. A. M. S. (1991). Addendum: Finite-size effects for some bootstrap percolation models. J. Statist. Phys. 62 505-506. MR92c:82054b 10.1007/BF01020891 63.1006.01
[18] Vichniac,G. Y. (1984). Simulating physics with cellular automata. Phys. D 10 96-116. MR85h:00018 10.1016/0167-2789(84)90253-7 0563.68053[18] Vichniac,G. Y. (1984). Simulating physics with cellular automata. Phys. D 10 96-116. MR85h:00018 10.1016/0167-2789(84)90253-7 0563.68053
[20] Wolfram,S. (1983). Statistical mechanics of cellular automata. Rev. Modern Phys. 55, 601- MR85d:68057 1174.82319 10.1103/RevModPhys.55.601[20] Wolfram,S. (1983). Statistical mechanics of cellular automata. Rev. Modern Phys. 55, 601- MR85d:68057 1174.82319 10.1103/RevModPhys.55.601
644; Wolfram,S. (1986). Theory and Applications of Cellular Automata. World Scientific, Singapore. MR87j:68007 0609.68043644; Wolfram,S. (1986). Theory and Applications of Cellular Automata. World Scientific, Singapore. MR87j:68007 0609.68043