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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