Formation of an interface by competitive erosion
نویسندگان
چکیده
In 2006, the fourth author of this paper proposed a graph-theoretic model of interface dynamics called competitive erosion. Each vertex of the graph is occupied by a particle that can be either red or blue. New red and blue particles alternately get emitted from their respective bases and perform random walk. On encountering a particle of the opposite color they kill it and occupy its position. We prove that on the cylinder graph (the product of a path and a cycle) an interface spontaneously forms between red and blue and is maintained in a predictable position with high probability. t = 0 t = 125 t = 325 Figure 1: Competitive erosion spontaneously forms an interface. In the initial state S(0) each vertex of the 30 × 30 cylinder graph is independently colored blue with probability .33 and red otherwise. After 325 time steps of competitive erosion, an interface has formed between red and blue. ∗[email protected] www.math.cornell.edu/~levine. Supported by NSF grant DMS-1243606 and a Sloan Fellowship. ‡[email protected] §www.jamespropp.org. Partially supported by NSF grant DMS-1001905.
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