The Annals of Probability

On the Behavior of Some Cellular Automata Related to Bootstrap Percolation

Roberto H. Schonmann

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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.

Article information

Ann. Probab., Volume 20, Number 1 (1992), 174-193.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


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

Cellular automata bootstrap percolation critical points critical behavior


Schonmann, Roberto H. On the Behavior of Some Cellular Automata Related to Bootstrap Percolation. Ann. Probab. 20 (1992), no. 1, 174--193. doi:10.1214/aop/1176989923.

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