Rotor-Router Walk on a Semi-infinite Cylinder

TL;DR

This paper investigates the rotor-router walk on a semi-infinite cylinder, analyzing cluster growth, internal structure, and emergent geometric patterns, revealing KPZ-like scaling and helical label arrangements.

Contribution

It introduces a novel analysis of rotor-router walks on cylindrical geometries, highlighting KPZ scaling behavior and the formation of helical label structures.

Findings

Cluster surface width follows a KPZ-like scaling law.

Labels form an average ordered helix structure.

Growth dynamics exhibit stationary scaling parameters.

Abstract

We study the rotor-router walk with the clockwise ordering of outgoing edges on the semi-infinite cylinder. Imposing uniform conditions on the boundary of the cylinder, we consider growth of the cluster of visited sites and its internal structure. The average width of the surface region of the cluster evolves with time to the stationary value by a scaling law whose parameters are close to the standard KPZ exponents. We introduce characteristic labels corresponding to closed clockwise contours formed by rotors and show that the sequence of labels has in average an ordered helix structure.