Rotor-router aggregation on the layered square lattice (Q612935)

Summary: 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.