Formation of an interface by competitive erosion

TL;DR

This paper studies a graph-theoretic model called competitive erosion, where particles of two colors perform random walks and interact, leading to the formation of a stable interface on a cylinder graph.

Contribution

It proves that an interface spontaneously forms and remains stable in a predictable position on the cylinder graph under the competitive erosion model.

Findings

Interface forms spontaneously on the cylinder graph

Interface remains stable and predictable over time

High probability of stable interface formation

Abstract

In 2006, the fourth author proposed a graph-theoretic model of interface dynamics called competitive erosion. Each vertex of the graph is occupied by a particle that can be either red or blue. New red and blue particles alternately get emitted from their respective bases and perform random walk. On encountering a particle of the opposite color they kill it and occupy its position. We prove that on the cylinder graph (the product of a path and a cycle) an interface spontaneously forms between red and blue and is maintained in a predictable position with high probability.