In Propp's "competitive erosion" model, each vertex of a finite graph is occupied by a red or blue particle. New red and blue particles are alternately emitted from their respective bases and perform random walk. On encountering a particle of the other 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. This part is joint work with Shirshendu Ganguly, Yuval Peres and James Propp: http://arxiv.org/abs/1501.03584

In the "oil and water" model, reds and blues exclude particles of the other color instead of killing them. The result is that the colors separate only microscopically, and the particles do not travel far: We prove that if $n$ red and $n$ blue particles start at one site of an infinite path, then most of them travel distance $O(n^{1/3})$. This part is joint work with Elisabetta Candellero, Shirshendu Ganguly and Christopher Hoffman: http://arxiv.org/abs/1408.0776