Rotor-Router Aggregation on the Layered Square Lattice

Abstract

In rotor-router aggregation on the square lattice $\mathbb{Z}^2$, 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.