Competitive Erosion is Conformally Invariant

TL;DR

This paper proves that the interface dynamics of the competitive erosion model on planar domains are conformally invariant, with regions separated by level curves of the Green function at stationarity.

Contribution

It demonstrates the conformal invariance of the competitive erosion model and characterizes the stationary interface as level curves of the Green function.

Findings

Interfaces align with Green function level curves at stationarity.

The model exhibits conformal invariance on smooth planar domains.

Red and blue regions are separated by orthogonal circular arcs or hyperbolic geodesics.

Abstract

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 the level curves of the Green function, with Neumann boundary conditions, which are orthogonal circular arcs on the disc and hyperbolic geodesics on a general simply connected domain. This establishes $conformal\,invariance$ of the model.