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January, 1992 On the Behavior of Some Cellular Automata Related to Bootstrap Percolation
Roberto H. Schonmann
Ann. Probab. 20(1): 174-193 (January, 1992). DOI: 10.1214/aop/1176989923


We consider some deterministic cellular automata on the state space $\{0, 1\}^{\mathbb{Z}^d}$ evolving in discrete time, starting from product measures. Basic features of the dynamics include: 1's do not change, translation invariance, attractiveness and nearest neighbor interaction. The class of models which is studied generalizes the bootstrap percolation rules, in which a 0 changes to a 1 when it has at least $l$ neighbors which are 1. Our main concern is with critical phenomena occurring with these models. In particular, we define two critical points: $p_c$, the threshold of the initial density for convergence to total occupancy, and $\pi_c$, the threshold for this convergence to occur exponentially fast. We locate these critical points for all the bootstrap percolation models, showing that they are both 0 when $l \leq d$ and both 1 when $l > d$. For certain rules in which the orientation is important, we show that $0 < p_c = \pi_c < 1$, by relating these systems to oriented site percolation. Finally, these oriented models are used to obtain an estimate for a critical exponent of these models.


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Roberto H. Schonmann. "On the Behavior of Some Cellular Automata Related to Bootstrap Percolation." Ann. Probab. 20 (1) 174 - 193, January, 1992.


Published: January, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0742.60109
MathSciNet: MR1143417
Digital Object Identifier: 10.1214/aop/1176989923

Primary: 60K35

Keywords: Bootstrap percolation , cellular automata , Critical behavior , critical points

Rights: Copyright © 1992 Institute of Mathematical Statistics


Vol.20 • No. 1 • January, 1992
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