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. 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.

References in corpus (5)

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• Parallel Chip-Firing on the Complete Graph: Devil's Staircase and Poincare Rotation Number
• Komlós's tiling theorem via graphon covers
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