Scaling Laws for a Class of Critical Cellular Automaton Growth Rules

Assume that a cellular automaton (CA) rule enlarges subsets of Z 2 , and does so in such a way that a larger set results in a larger outcome. Such models are called monotone solidi cation CA. In the critical case, these dynamics cannot cover the lattice starting from any nite set, but are able to do so from any set with nite complement. We assume that the initial set is a product measure with small density p, and address various scaling laws for the rst passage time to the origin, emergence of shapes, and the ability of the dynamics to overcome pollution of space. 1991 Mathematics Subject Classi cation. Primary 60K35. Secondary 52A10.