Transience and recurrence of rotor-router walks on directed covers of graphs

TL;DR

This paper extends the understanding of rotor-router walks' recurrence and transience from homogeneous trees to directed covers of finite graphs, showing how embedding affects walk behavior.

Contribution

It generalizes previous results to directed covers of finite graphs and demonstrates the impact of planar embedding on walk recurrence or transience.

Findings

Rotor-router walks can be recurrent or transient depending on the embedding.

Extension of Angel and Holroyd's results to directed covers of graphs.

Identification of conditions influencing walk behavior on periodic trees.

Abstract

The aim of this note is to extend the result of Angel and Holroyd concerning the transience and the recurrence of transfinite rotor-router walks, for random initial configuration of rotors on homogeneous trees. We address the same question on directed covers of finite graphs, which are also called trees with finitely many cone types or periodic trees. Furthermore, we provide an example of a directed cover such that the rotor-router walk can be either recurrent or transient, depending only on the planar embedding of the periodic tree.