Infinite-step stationarity of rotor walk and the wired spanning forest

TL;DR

This paper establishes that on certain infinite graphs, the final rotor configuration of a rotor walk aligns with the wired uniform spanning forest measure, answering a previously posed question.

Contribution

It proves the law of the final rotor configuration matches the wired uniform spanning forest measure for graphs with single-ended trees, extending understanding of rotor walk behavior.

Findings

Final rotor configuration follows the wired uniform spanning forest measure.

Results apply to all graphs with single-ended trees in the wired spanning forest.

Answers a previously open question about rotor walk configurations.

Abstract

We study rotor walk, a deterministic counterpart of the simple random walk, on infinite transient graphs. We show that the final rotor configuration of the rotor walk follows the law of the wired uniform spanning forest oriented toward infinity (OWUSF) measure when the initial rotor configuration is sampled from OWUSF. This result holds for all graphs for which each tree in the wired spanning forest has one single end almost surely. This answers a question posed in a previous work of the author (Chan 2018).