Co-author: Shirshendu Ganguly (University of Washington)

We study a graph-theoretic model of interface dynamics called {\bf competitive erosion}. Each vertex of the graph is occupied by a particle, which can be either red or blue. New red and blue particles are emitted alternately from their respective sources and perform random walk. On encountering a particle of the opposite color they remove it and occupy its position. This is a finite competitive version of the celebrated Internal DLA growth model first analyzed by Lawler, Bramson and Griffeath in 1992. We establish conformal invariance of competitive erosion on discretizations of smooth, simply connected planar domains. This is done by showing that at stationarity, with high probability the blue and the red regions are separated by an orthogonal circular arc on the disc and more generally by a hyperbolic geodesic. (Joint work with Shirshendu Ganguly, available at http://arxiv.org/abs/1503.06989 ).