  The Greenberg-Hastings Model (GHM) is a family of multitype cellular automata
that emulate excitable media, exhibiting the nucleation and spiral formation
characteristic of such complex systems. In this paper we study the asymptotic
frequency of nucleation in 2-d GHM dynamics as the number k of types, or
colors, becomes large. Starting from uniform product measure over colors, and
assuming that the excitation threshold is not too large, the box size needed
for formation of a spiral core is shown to grow exponentially in k. By
exploiting connections with percolation theory we find that a box of size
exp{(.23+-.06)k} is required in the nearest neighbor, threshold 1 case. In
contrast, GHM rules obey power law nucleation scaling when started from a
suitable non-uniform product measure over the k colors; this effect is driven
by critical percolation. Finally, we present some analogous results for a
Random GHM, an interacting Markovian system closely related to the epidemic
with regrowth of Durrett and Neuhauser.
