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 Electronic Journal of Probability > Vol. 1 (1996) > Paper 12 open journal systems 


Percolation Times in Two-Dimensional Models For Excitable Media

Janko Gravner, University of California, Davis


Abstract
The three-color Greenberg--Hastings model (GHM) is a simple cellular automaton model for an excitable medium. Each site on the lattice $bZ^2$ is initially assigned one of the states 0, 1 or 2. At each tick of a discrete--time clock, the configuration changes according to the following synchronous rule: changes $1to 2$ and $2to 0$ are automatic, while an $x$ in state 0 may either stay in the same state or change to 1, the latter possibility occurring iff there is at least one representative of state 1 in the local neighborhood of $x$. Starting from a product measure with just 1's and 0's such dynamics quickly die out (turn into 0's), but not before 1's manage to form infinite connected sets. A very precise description of this ``transient percolation'' phenomenon can be obtained when the neighborhood of $x$ consists of 8 nearest points, the case first investigated by S. Fraser and R. Kapral. In addition, first percolation times for related monotone models are addressed.


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Pages: 1-19

Published on: October 10, 1996
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Electronic Journal of Probability. ISSN: 1083-6489