Rotor-Router Aggregation on the Comb

TL;DR

This paper proves a shape theorem for rotor-router aggregation on the comb, analyzes harmonic measure, and identifies conditions for uniform harmonic measure, revealing growth patterns and measure properties of the cluster.

Contribution

It establishes the first shape theorem for rotor-router aggregation on the comb with specific configurations and explores harmonic measure properties of the resulting shapes.

Findings

Shape theorem for rotor-router aggregation on the comb.

Identification of shapes with uniform harmonic measure.

First example of non-uniform harmonic measure in rotor-router clusters.

Abstract

We prove a shape theorem for rotor-router aggregation on the comb, for a specific initial rotor configuration and clockwise rotor sequence for all vertices. Furthermore, as an application of rotor-router walks, we describe the harmonic measure of the rotor-router aggregate and related shapes, which is useful in the study of other growth models on the comb. We also identify the shape for which the harmonic measure is uniform. This gives the first known example where the rotor-router cluster has non-uniform harmonic measure, and grows with different speeds in different directions.