Rotor walks on transient graphs and the wired spanning forest

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).