The devil's staircase for chip-firing on random graphs and on graphons
Combinatorics
2024-08-16 v1 Dynamical Systems
Probability
Abstract
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.
Cite
@article{arxiv.2004.13104,
title = {The devil's staircase for chip-firing on random graphs and on graphons},
author = {Viktor Kiss and Lionel Levine and Lilla Tóthmérész},
journal= {arXiv preprint arXiv:2004.13104},
year = {2024}
}