Parallel Chip Firing on Digraphs

Given some multidigraph , a state is any dist ribut ion of some chips on its vert ices. We transform t his initial st ate step by step. Every vert ex checks whet her it is able to send one chip through every outgoing arc. If it can, it does; otherwise it does not send any chip. All vert ices check and send in parallel. Finally, at every vert ex all incoming chips are added to the remaining chips. This tr ansformation on the set of states is iterated. If the digraph and the total numb er of chips are finite, then we finally arrive at some periodic configurat ion. Here we investigate how t hese periodic configurations depend on the digraph and the total number of chips. There is a sharp cont rast in the behavior for Eulerian digraphs (where the in-degree of each vert ex equals its out-degree) and non-Eulerian digraphs.