Cyclic Cellular Automata and Greenberg-Hastings Models on Regular Trees

We study the cyclic cellular automaton (CCA) and the Greenberg-Hastings model (GHM) with $\kappa\ge 3$ colors and contact threshold $\theta\ge 2$ on the infinite $(d+1)$-regular tree, $T_d$. When the initial state has the uniform product distribution, we show that these dynamical systems exhibit at least two distinct phases. For sufficiently large $d$, we show that if $\kappa(\theta-1) \le d - O(\sqrt{d\kappa \ln(d)})$, then every vertex almost surely changes its color infinitely often, while if $\kappa\theta \ge d + O(\kappa\sqrt{d\ln(d)})$, then every vertex almost surely changes its color only finitely many times. Roughly, this implies that as $d\to \infty$, there is a phase transition where $\kappa\theta/d = 1$. For the GHM dynamics, in the scenario where every vertex changes color finitely many times, we moreover give an exponential tail bound for the distribution of the time of the last color change at a given vertex.