A loop reversibility and subdiffusion of the rotor-router walk

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...

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Published inJournal of physics. A, Mathematical and theoretical Vol. 48; no. 28; pp. 285203 - 11
Main Authors Papoyan, Vl V, Poghosyan, V S, Priezzhev, V B
Format Journal Article
LanguageEnglish
Published IOP Publishing 17.07.2015
Subjects
Euler walk
Evolution
Graph theory
Graphs
Mathematical models
rotor-router walk
Rotors
Shape
spanning tree
Theorems
unicycle
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Summary: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.
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ISSN:1751-8113
1751-8121
DOI:10.1088/1751-8113/48/28/285203