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1993 Asymptotic behavior of excitable cellular automata
Richard Durrett, David Griffeath
Experiment. Math. 2(3): 183-208 (1993).


We study two families of excitable cellular automata known as the Greenberg-Hastings model and the cyclic cellular automaton. Each family consists of local deterministic oscillating lattice dynamics, with parallel discrete-time updating, parametrized by the range $\rho$ of interaction, $\ell^p$ shape of its neighbor set, threshold $\theta$ for contact updating, and number $\kappa$ of possible states per site. These models are mathematically tractable prototypes for the spatially distributed periodic wave activity of so-called excitable media observed in diverse disciplines of experimental science.

Fisch, Gravner and Griffeath studied experimentally the ergodic behavior of these models on $\Bbb Z^2$, started from random initial states. Among other phenomena, they noted the emergence of asymptotic phase diagrams (and dynamics on $\Bbb R^2$) in the threshold-range scaling limit as $\rho,\theta\to \infty$ with $\theta/\rho^2$ constant.

Here we present several rigorous results and some experimental findings concerning various phase transitions in the asymptotic diagrams. Our efforts focus on evaluating bend$(p)$, the limiting threshold cutoff for existence of the spirals that characterize many excitable media. Our main results are formulated in terms of spo$(p)$, the cutoff for existence of stable periodic objects that arise as spiral cores. Some subtle consequences of anisotropic neighbor sets $(p\neq2)$ are also discussed; the case of box neighborhoods $(p=\infty)$ is examined in detail.


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Richard Durrett. David Griffeath. "Asymptotic behavior of excitable cellular automata." Experiment. Math. 2 (3) 183 - 208, 1993.


Published: 1993
First available in Project Euclid: 3 September 2003

zbMATH: 0796.60100
MathSciNet: MR1273408

Primary: 58F08
Secondary: 68Q80 , 82C44

Rights: Copyright © 1993 A K Peters, Ltd.

Vol.2 • No. 3 • 1993
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