Competitive Erosion is Conformally Invariant

We study a graph-theoretic model of interface dynamics called $Competitive\, Erosion$. 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 bases and perform random walk. On encountering a particle of the opposite color they remove it and occupy its position. We consider competitive erosion on discretizations of smooth planar simply connected domains. The main result of this article shows that at stationarity, with high probability the blue and the red regions are separated by an orthogonal circular arc on the disc and by a suitable hyperbolic geodesic on a general `smooth' simply connected domain. This establishes $conformal\,invariance$ of the model.