Metastability in the Greenberg-Hastings Model

Details Metastability in the Greenberg-Hastings Model Metastability in the Greenberg-Hastings Model

1993-03-23

arxiv.org/abs/patt-sol/9303005v1

Nonlinear Sciences

Adaptation and Self-Organizing Systems

31 pages. To appear as a Special Invited Paper in the Annals of Applied Probability. arXiv admin note, 17Jun2003: original PCL

Scientific paper

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.

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