Rotor walks on transient graphs and the wired spanning forest

TL;DR

This paper investigates rotor walks on transient graphs with initial configurations from the wired spanning forest, showing they are transient and relating their behavior to simple random walks, with convergence results for vertex-transitive graphs.

Contribution

It establishes that rotor walks with OWUSF initial configurations are transient and their visit counts relate to simple random walks, answering a question from prior research.

Findings

Rotor walks are transient on transient graphs with OWUSF initial configurations.

Expected visits by rotor walks are bounded by those of simple random walks.

Average visits converge to the Green function in vertex-transitive graphs.

Abstract

We study rotor walks on transient graphs with initial rotor configuration sampled from the oriented wired uniform spanning forest (OWUSF) measure. We show that the expected number of visits to any vertex by the rotor walk is at most equal to the expected number of visits by the simple random walk. In particular, this implies that this walk is transient. When these two numbers coincide, we show that the rotor configuration at the end of the process also has the law of OWUSF. Furthermore, if the graph is vertex-transitive, we show that the average number of visits by $n$ consecutive rotor walks converges to the Green function of the simple random walk as $n$ tends to infinity. This answers a question posed by Florescu, Ganguly, Levine, and Peres (2014).