On the Behavior of Some Cellular Automata Related to Bootstrap Percolation

We consider some deterministic cellular automata on the state space \\0, 1\\^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: pc, the threshold of the initial density for convergence to total occupancy, and \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 \ d and both 1 when l > d. For certain rules in which the orientation is important, we show that 0 < pc = \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.