Random growth models with polygonal shapes

Abstract

We consider discrete-time random perturbations of monotone cellular automata (CA) in two dimensions. Under general conditions, we prove the existence of half-space velocities, and then establish the validity of the Wulff construction for asymptotic shapes arising from finite initial seeds. Such a shape converges to the polygonal invariant shape of the corresponding deterministic model as the perturbation decreases. In many cases, exact stability is observed. That is, for small perturbations, the shapes of the deterministic and random processes agree exactly. We give a complete characterization of such cases, and show that they are prevalent among threshold growth CA with box neighborhood. We also design a nontrivial family of CA in which the shape is exactly computable for all values of its probability parameter.