Spiral Structures in the Rotor-Router Walk

We study the rotor-router walk on the infinite square lattice with the outgoing edges at each lattice site ordered clockwise. In the previous paper [J.Phys.A: Math. Theor. 48, 285203 (2015)], we have considered the loops created by rotors and labeled sites where the loops become closed. The sequence of labels in the rotor-router walk was conjectured to form a spiral structure obeying asymptotically an Archimedean property. In the present paper, we select a subset of labels called "nodes" and consider spirals formed by nodes. The new spirals are directly related to tree-like structures which represent the evolution of the cluster of vertices visited by the walk. We show that the average number of visits to the origin $\left<n_0(t)\right>$ by the moment $t\gg 1$ is $\left<n_0(t)\right> = 4 \left<n(t)\right> + O(1)$ where $\left<n(t)\right>$ is the average number of rotations of the spiral.