On the Behavior of Some Cellular Automata Related to Bootstrap Percolation

Abstract

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.