The Annals of Applied Probability

Metastability in the Greenberg-Hastings Model

Robert Fisch, Janko Gravner, and David Griffeath

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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 GHM dynamics on $\mathbb{Z}^2$ as the number of types, or colors, becomes large. Starting from uniform product measure over $\kappa$ colors, and assuming that the excitation threshold $\theta$ is not too large, the box size $L_\kappa$ needed for formation of a spiral core is shown to grow exponentially: $L_\kappa \approx \exp\{C_\kappa\}$ as $\kappa \rightarrow \infty$. By exploiting connections with percolation theory, we find that $C = 0.23 \pm 0.06$ in the nearest neighbor, $\theta = 1$ case. In contrast, GHM rules obey power law nucleation scaling when started from a suitable nonuniform product measure over the $\kappa$ 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.

Article information

Ann. Appl. Probab., Volume 3, Number 4 (1993), 935-967.

First available in Project Euclid: 19 April 2007

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

metastability nucleation self-organization excitable medium cellular automation


Fisch, Robert; Gravner, Janko; Griffeath, David. Metastability in the Greenberg-Hastings Model. Ann. Appl. Probab. 3 (1993), no. 4, 935--967. doi:10.1214/aoap/1177005268.

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