Rotor-routing on Galton-Watson trees

TL;DR

This paper studies rotor-router walks, a deterministic process, on Galton-Watson trees with random initial configurations, providing a classification of when the walks are recurrent or transient.

Contribution

It offers the first classification of recurrence and transience for rotor-router walks on Galton-Watson trees with random initial rotor configurations.

Findings

Classification of recurrence and transience based on initial configurations

Analysis of rotor-router behavior on Galton-Watson trees

Extension of rotor-router theory to random tree structures

Abstract

A rotor-router walk on a graph is a deterministic process, in which each vertex is endowed with a rotor that points to one of the neighbors. A particle located at some vertex first rotates the rotor in a prescribed order, and then it is routed to the neighbor the rotor is now pointing at. In the current work we make a step toward in understanding the behavior of rotor-router walks on random trees. More precisely, we consider random i.i.d. initial configurations of rotors on Galton-Watson trees, i.e. on a family tree arising from a Galton-Watson process, and give a classification in recurrence and transience for rotor-router walks on these trees.