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 \(\left\) by the moment \(t\gg 1\) is \(\left = 4 \left + O(1)\) where \(\left\) is the average number of rotations of the spiral.