Competitive erosion is conformally invariant

Опубликовано 22 июня 2016, 0:21

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. We establish conformal invariance of competitive erosion on discretizations of smooth planar simply connected 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. The talk will be based on joint works with Lionel Levine, Yuval Peres and James Propp.