(Biased) Majority Rule Cellular Automata
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Consider a graph G=(V,E) and a random initial vertex-coloring, where each vertex is blue independently with probability p_b, and red with probability p_r=1-p_b. 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 conserves its current color; this model is called majority model. If in case of a tie a vertex always chooses blue color, it is called biased majority model. We are interested in the behavior of these deterministic processes, especially in a two-dimensional torus (i.e., cellular automaton with (biased) majority rule). In the present paper, as a main result we prove both majority and biased majority cellular automata exhibit a threshold behavior with two phase transitions. More precisely, it is shown that for a two-dimensional torus T_n,n, there are two thresholds 0≤ p_1, p_2≤ 1 such that p_b ≪ p_1, p_1 ≪ p_b ≪ p_2, and p_2 ≪ p_b result in monochromatic configuration by red, stable coexistence of both colors, and monochromatic configuration by blue, respectively in O(n^2) number of steps
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