Rotor-routing on Galton-Watson trees

Wilfried Huss,Sebastian Müller,Ecaterina Sava-Huss

doi:10.1214/ecp.v20-4000

Wilfried Huss, Sebastian Müller + Show 1 more

Open Access

https://doi.org/10.1214/ecp.v20-4000

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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 T, 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.

Highlights

A rotor-router walk on a graph is a deterministic process in which the exits from each vertex follow a prescribed periodic sequence
Rotor-router walks capture in many aspects the expected behavior of simple random walks, but with significantly reduced fluctuations compared to a typical random walk trajectory; for more details see [CS06, FL11, HP10]
We say that a rotorrouter walk which started at the origin with initial rotor configuration r is recurrent if it

Summary

Introduction

A rotor-router walk on a graph is a deterministic process in which the exits from each vertex follow a prescribed periodic sequence. A key result of Schramm states that for any choice of the initial rotor configuration, the rotor-router walk is in a certain sense no more transient than the random walk. In this note we investigate the recurrence and transience properties of rotor-router walks with random initial rotor configuration ρ on Galton-Watson trees. The main result Theorem 3.2 gives a criterion for recurrence and transience of rotor-router walks. If we run n rotor walks starting from the root and record whether each walk returns to the root or escapes to infinity, we show that in the transient regime the relative density of escapes of the rotor-router walk equals almost surely the return probability of the simple random walk on Galton-Watson trees. For GW-almost all trees T , the rotor-router walk on T is recurrent if and only if m ≤ 2

Galton-Watson trees

Rotor-router walks

Random initial rotor configuration

Recurrent part

The frontier process

Transient part