Range and speed of rotor walks on trees

TL;DR

This paper establishes a law of large numbers for the range of rotor walks on trees, demonstrating linear growth and the existence of speed even in recurrent cases, advancing understanding of rotor walk behavior.

Contribution

It proves a law of large numbers and the existence of linear speed for rotor walks on regular and Galton-Watson trees, including recurrent cases.

Findings

Range grows linearly on the considered trees.

Existence of a well-defined speed for rotor walks.

Linear growth holds even in recurrent scenarios.

Abstract

We prove a law of large numbers for the range of rotor walks with random initial configuration on regular trees and on Galton-Watson trees. More precisely, we show that on the classes of trees under consideration, even in the case when the rotor walk is recurrent, the range grows at linear speed. We also show the existence of the speed for such rotor walks.