Rotor-routing orbits in directed graphs and the Picard group

In [5], Holroyd, Levine, Mészáros, 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 unicycleorbits equals to the order of the Picard group of the graph, defined in the sense of [1]. Moreover, during a period of the rotor-router process, the same chip-moves happen, as during firing the period vector in the chip-firing game.