Rotor-routing orbits in directed graphs and the Picard group

TL;DR

This paper provides new combinatorial proofs relating rotor-router orbits in directed graphs to the Picard group, connecting these concepts with chip-firing games and offering insights into their recurrence properties.

Contribution

It introduces purely combinatorial proofs for formulas linking rotor-router orbits to the Picard group, expanding understanding beyond linear algebraic methods.

Findings

Number of rotor-router unicycle-orbits equals the Picard group's order

Rotor-router orbits correspond to chip-firing game dynamics

New combinatorial proofs for recurrence formulas

Abstract

In [5], Holroyd, Levine, M\'esz\'aros, Peres, Propp and Wilson characterize recurrent chip-and-rotor configurations for strongly connected digraphs. However, the number of steps needed to recur, and the number of orbits is left open for general digraphs. Recently, these questions were answered by Pham [6], using linear algebraic methods. We give new, purely combinatorial proofs for these formulas. We also relate rotor-router orbits to the chip-firing game: The number of recurrent rotor-router unicycle-orbits equals the order of the Picard group of the graph, defined in the sense of [1], and during a period, the same chip-moves happen, as during firing the period vector in the chip-firing game.