Rotor-router aggregation on the layered square lattice

TL;DR

This paper studies a deterministic growth model on a layered square lattice, showing that a small change in routing causes the cluster to form a perfect diamond shape instead of a disk.

Contribution

It introduces a modified rotor-router model on the layered square lattice that results in a diamond-shaped limiting cluster, a novel shape for this type of aggregation.

Findings

Cluster forms a perfect diamond shape with specific initial rotors.

Small routing modifications significantly alter the limiting shape.

The model demonstrates deterministic shape formation in rotor-router aggregation.

Abstract

In rotor-router aggregation on the square lattice Z^2, particles starting at the origin perform deterministic analogues of random walks until reaching an unoccupied site. The limiting shape of the cluster of occupied sites is a disk. We consider a small change to the routing mechanism for sites on the x- and y-axes, resulting in a limiting shape which is a diamond instead of a disk. We show that for a certain choice of initial rotors, the occupied cluster grows as a perfect diamond.