Perfect boundaries in rotor-router aggregation on cylinders

TL;DR

This paper proves that rotor-router aggregation on a semi-infinite cylinder has boundary fluctuations of constant width, independent of the cluster size, improving understanding of deterministic growth models.

Contribution

It establishes that boundary fluctuations in rotor-router internal DLA on cylinders are bounded by a constant, regardless of the cluster radius, which was previously unknown.

Findings

Boundary fluctuations are bounded by a constant width.

Fluctuations do not grow with the cluster radius.

Improves understanding of deterministic aggregation models.

Abstract

We study a rotor-router version of the internal diffusion-limited aggregation introduced by J.Propp. The existing estimations of boundary fluctuations of the aggregation cluster show that they grow not faster than $O(\log r)$ with the cluster radius $r$. We consider the rotor-router internal DLA on the semi-infinite cylinder and prove a constant width of boundary fluctuations not depending on the radius of the cylinder.