A Contribution in the Rotor-router model

TL;DR

This paper explores the Rotor-router model as part of a broader family of similar models, aiming to deepen understanding of the J.Propp model through generalizations and explicit algorithms, especially focusing on the Abelian version.

Contribution

It introduces a generalized approach to the Rotor-router model, including explicit algorithms for the Abelian version and connections to symmetrical models, enhancing theoretical understanding.

Findings

Generalized Rotor-router models with different geometric shapes.

Explicit iterative algorithms for the Abelian Rotor-router.

Relationships established between J.Propp model and symmetrical models.

Abstract

In this paper I propose to approach the Rotor-router problem by considering it as one example of a big family of many other similar models. The study of some specific samples of them may contribute, in my opinion, at a more understanding of the J.Propp model. In fact we can easily generalize the Rotor-router to many other models, with different regular geometric shapes, by slightly changing the ants' displacements rules. The two directions Rotor-Router (RR2) is particularly interesting because in it's Abelian version it is generated by an easy mathematical explicit scheme. Moreover we can also generate the J.Propp Rotor-Router using a similar iterative explicit algorithm. The study of RR2 establishes also a relationship between the J.Propp model and a family of symmetrical models which generates the same round forms.