A Loop Reversibility and Subdiffusion of the Rotor-Router Walk

TL;DR

This paper proves a property of the rotor-router walk on planar graphs involving contour reversibility and uses this to analyze the walk's subdiffusive behavior, extending previous results to more general rotor configurations.

Contribution

It introduces a new property of rotor-router walks related to contour reversibility and applies it to analyze subdiffusive behavior, broadening the understanding of deterministic walks.

Findings

Proved that rotor configurations forming a closed contour can reverse orientation through the walk.

Demonstrated the property extends previous theorems to more general rotor configurations.

Used the property to analyze and explain the subdiffusive behavior of the rotor-router walk.

Abstract

The rotor-router model on a graph describes a discrete-time walk accompanied by the deterministic evolution of configurations of rotors randomly placed on vertices of the graph. We prove the following property: if at some moment of time, the rotors form a closed clockwise contour on the planar graph, then the clockwise rotations of rotors generate a walk which enters into the contour at some vertex $v$, performs a number of steps inside the contour so that the contour formed by rotors becomes anti-clockwise, and then leaves the contour at the same vertex $v$. This property generalizes the previously proved theorem for the case when the rotor configuration inside the contour is a cycle-rooted spanning tree, and all rotors inside the contour perform a full rotation. We use the proven property for an analysis of the sub-diffusive behavior of the rotor-router walk.