The devil's staircase for chip‐firing on random graphs and on graphons

Abstract We study the behavior of the activity of the parallel chip‐firing upon increasing the number of chips on an Erdős–Rényi random graph. We show that in various situations the resulting activity diagrams converge to a devil's staircase as we increase the number of vertices. Such a phenomenon was proved in an earlier paper by Levine for complete graphs, by relating the activity to the rotation number of a cycle map. Our method in this article is to generalize the parallel chip‐firing to graphons. Then we show that the earlier results on complete graphs generalize to constant graphons. Moreover, we prove a continuity result for the activity on graphons. These statements enable us to prove results on the activity of the parallel chip‐firing on large random graphs. We also address several problems concerning chip‐firing on graphons, and pose open problems. In particular, we show that the activity of a chip configuration on a graphon does not necessarily exist, but it does exist for every chip configuration on a large class of graphons.