Recurrent Rotor-Router Configurations

TL;DR

This paper establishes the existence of recurrent configurations for rotor walks on various graphs and shows that recurrence or transience is unaffected by initial conditions or starting points.

Contribution

It proves the existence of recurrent configurations on many graphs and demonstrates invariance of recurrence and transience under certain modifications.

Findings

Recurrent configurations exist on Z^d and planar graphs.

Recurrence and transience are invariant under initial configuration changes.

Rotor walk behavior is robust to starting point variations.

Abstract

We prove the existence of recurrent initial configurations for the rotor walk on many graphs, including Z^d, and planar graphs with locally finite embeddings. We also prove that recurrence and transience of rotor walks are invariant under changes in the starting vertex and finite changes in the initial configuration.