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.
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