Infinite excursions of rotor walks on regular trees

TL;DR

This paper investigates the behavior of rotor walks on regular trees with random configurations, revealing that they generally mimic simple random walks unless they deviate significantly from the standard rotor-router model.

Contribution

It provides new insights into the conditions under which rotor walks on regular trees behave like simple random walks, especially regarding infinite excursions.

Findings

Rotor walks behave like simple random walks unless far from the standard rotor-router model.

The study characterizes when infinite excursions occur in random rotor configurations.

Results suggest a threshold behavior depending on the configuration's proximity to the standard model.

Abstract

A rotor configuration on a graph contains in every vertex an infinite ordered sequence of rotors, each is pointing to a neighbor of the vertex. After sampling a configuration according to some probability measure, a rotor walk is a deterministic process: at each step it chooses the next unused rotor in its current location, and uses it to jump to the neighboring vertex to which it points. Rotor walks capture many aspects of the expected behavior of simple random walks. However, this similarity breaks down for the property of having an infinite excursion. In this paper we study that question for natural random configuration models on regular trees. Our results suggest that in this context the rotor model behaves like the simple random walk unless it is not "close to" the standard rotor-router model.