Open Access
December 2018 Limiting behavior of 3-color excitable media on arbitrary graphs
Janko Gravner, Hanbaek Lyu, David Sivakoff
Ann. Appl. Probab. 28(6): 3324-3357 (December 2018). DOI: 10.1214/17-AAP1350


Fix a simple graph $G=(V,E)$ and choose a random initial 3-coloring of vertices drawn from a uniform product measure. The 3-color cycle cellular automaton is a process in which at each discrete time step in parallel, every vertex with color $i$ advances to the successor color $(i+1)\bmod 3$ if in contact with a neighbor with the successor color, and otherwise retains the same color. In the Greenberg–Hastings model, the same update rule applies only to color 0, while other two colors automatically advance. The limiting behavior of these processes has been studied mainly on the integer lattices. In this paper, we introduce a monotone comparison process defined on the universal covering space of the underlying graph, and characterize the limiting behavior of these processes on arbitrary connected graphs. In particular, we establish a phase transition on the Erdős–Rényi random graph. On infinite trees, we connect the rate of color change to the cloud speed of an associated tree-indexed walk. We give estimates of the cloud speed by generalizing known results to trees with leaves.


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Janko Gravner. Hanbaek Lyu. David Sivakoff. "Limiting behavior of 3-color excitable media on arbitrary graphs." Ann. Appl. Probab. 28 (6) 3324 - 3357, December 2018.


Received: 1 December 2016; Revised: 1 June 2017; Published: December 2018
First available in Project Euclid: 8 October 2018

zbMATH: 06994395
MathSciNet: MR3861815
Digital Object Identifier: 10.1214/17-AAP1350

Primary: 60K35 , 82B43

Keywords: cellular automaton , Excitable media , tournament expansion , tree-indexed random walk

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.28 • No. 6 • December 2018
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