Majority rule cellular automata

Abstract

Consider a graph G=(V,E) and a random initial coloring where each vertex is black independently with probability pb, and white with probability pw=1−pb. In each step, all vertices change their current color synchronously to the most frequent color in their neighborhood and in case of a tie, a vertex keeps its current color. This model is called the majority model. If in case of a tie a vertex always selects black color, it is called the biased majority model. We are interested in the behavior of these two processes, especially when the underlying graph is a two-dimensional torus (cellular automaton with (biased) majority rule). In the present paper, as our main result we prove that both majority and biased majority cellular automata exhibit a threshold behavior with two phase transitions. More precisely, we prove for a two-dimensional torus Tn,n, there are two threshold values 0≤p1,p2≤1 such that pb≪p1, p1≪pb≪p2, and p2≪pb result in final complete occupancy by white, stable coexistence of both colors, and final complete occupancy by black, respectively in O(n2) number of steps. (For two functions f(n) and g(n), we shortly write f(n)≪g(n) instead of f(n)∈o(g(n)).) We finally argue that our proof techniques can be used to prove a similar threshold behavior for a larger class of models.