Spiral Structures in the Rotor-Router Walk

TL;DR

This paper investigates the spiral structures formed by rotor-router walks on an infinite square lattice, revealing their relation to cluster growth and providing asymptotic behavior of visits to the origin.

Contribution

It introduces the concept of node-based spirals in rotor-router walks and links these structures to tree-like growth models, advancing understanding of their geometric properties.

Findings

Average visits to the origin scale linearly with the number of spiral rotations.

Spirals formed by nodes exhibit properties related to Archimedean spirals.

The study connects spiral structures to cluster evolution in rotor-router walks.

Abstract

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.