Limiting behavior of 3-color excitable media on arbitrary graphs

TL;DR

This paper investigates the long-term behavior of 3-color excitable media on arbitrary graphs, introducing a comparison process and analyzing phase transitions and dynamics on various graph structures.

Contribution

It introduces a monotone comparison process on the universal cover of graphs and characterizes the limiting behavior of excitable media on arbitrary connected graphs.

Findings

Established a phase transition on Erdős-Rényi graphs.

Connected color change rate to cloud speed on infinite trees.

Generalized estimates of cloud speed for trees with leaves.

Abstract

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 $(i+1)$ mod 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 Erd\"os-R\'enyi 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.