The Rotor-Router Model on Regular Trees

TL;DR

This paper studies the rotor-router model on regular trees, proving that the occupied set forms a perfect ball under certain conditions and exploring recurrence, transience, and escape sequences.

Contribution

It establishes the isomorphism between the rotor-router group and the sandpile group and characterizes escape sequences on the ternary tree.

Findings

Occupied sites form perfect balls on regular trees with acyclic initial configurations

Rotor-router group is isomorphic to the sandpile group

Characterization of escape sequences for the ternary tree

Abstract

The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We show that the set of occupied sites for this model on an infinite regular tree is a perfect ball whenever it can be, provided the initial rotor configuration is acyclic (that is, no two neighboring vertices have rotors pointing to one another). This is proved by defining the rotor-router group of a graph, which we show is isomorphic to the sandpile group. We also address the question of recurrence and transience: We give two rotor configurations on the infinite ternary tree, one for which chips exactly alternate escaping to infinity with returning to the origin, and one for which every chip returns to the origin. Further, we characterize the possible "escape sequences" for the ternary tree, that is, binary words a_1 ... a_n for which there exists a rotor configuration so that the k-th chip escapes to infinity if and only if a_k=1.