Color War: Cellular Automata with Majority-Rule

Abstract

Consider a graph $G=(V,E)$ and a random initial vertex-coloring such that each vertex is blue independently with probability $p_{b}\le 1/2$, and red otherwise. In each step, all vertices change their current color synchronously to the most frequent color in their neighborhood (in the case of a tie, a vertex conserves its current color). We are interested in the behavior of this very natural process, especially in 2-dimensional grids and tori (cellular automata with majority rule). In the present paper, as a main result we prove that a grid $G_{n,n}$ or a torus $T_{n,n}$ with 4-neighborhood (8-neighborhood) exhibits a threshold behavior: with high probability, it reaches a red monochromatic configuration in a constant number of steps if $p_b\ll n^{-\frac{1}{2}}$ ($p_b\ll n^{-\frac{1}{6}}$), but $p_b\gg n^{-\frac{1}{2}}$ ($p_b\gg n^{-\frac{1}{6}}$) results in a bichromatic period of configurations of length one or two, after at most $2n^2$ ($4n^2$) steps with high probability.