A Loop Reversibility and Subdiffusion 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.

15 pages, 5 figures

References in corpus (2)

• Return probability for the loop-erased random walk and mean height in sandpile : a proof
• Transience and recurrence of rotor-router walks on directed covers of graphs

Cited by in corpus (1)

• Perfect boundaries in rotor-router aggregation on cylinders