paper

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

arXiv:1203.1477 · doi:10.1214/ECP.v17-2096

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.

14 pages, 3 figures; The previous version contains an error in the proof of Corollary 3.8, which is essential for the transience part of the main result. This is now corrected by constructing a new rotor-router process, which we call the frontier process

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