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 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 red and the blue regions are separated by an orthogonal circular arc on the disc and more generally by a hyperbolic geodesic. The proof relies on convergence of solutions of the discrete Poisson problem with Neumann boundary conditions to their continuous counterparts and robust IDLA estimates.

This is joint work with Yuval Peres, available at arxiv.org/abs/1503.06989.

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