Open Access
1996 Percolation Times in Two-Dimensional Models For Excitable Media
Janko Gravner
Author Affiliations +
Electron. J. Probab. 1: 1-19 (1996). DOI: 10.1214/EJP.v1-12


The three-color Greenberg--Hastings model (GHM) is a simple cellular automaton model for an excitable medium. Each site on the lattice $Z^2$ 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 $1\to 2$ and $2\to 0$ 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.


Download Citation

Janko Gravner. "Percolation Times in Two-Dimensional Models For Excitable Media." Electron. J. Probab. 1 1 - 19, 1996.


Accepted: 10 October 1996; Published: 1996
First available in Project Euclid: 25 January 2016

zbMATH: 0888.60074
MathSciNet: MR1423465
Digital Object Identifier: 10.1214/EJP.v1-12

Primary: 60K35

Keywords: additive growth dynamics , Excitable media , Greenberg--Hastings model , percolation

Vol.1 • 1996
Back to Top