  We study the behavior of the activity of the parallel chip-firing upon
increasing the number of chips on an Erd\H{o}s--R\'enyi 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. Our method is to
generalize the parallel chip-firing to graphons, and to prove a continuity
result for the activity. We also 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.
