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We study the behavior of the activity of the parallel chip-firing upon increasing the number of chips on an Erd\H os--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.
Bibtex entry:
| AUTHOR | = | {Kiss, Viktor and Levine, Lionel and T{\'o}thm{\'e}r{\'e}sz, Lilla}, |
| TITLE | = | {The devil's staircase for chip-firing on random graphs and on graphons}, |
| NOTE | = | {{\tt egres.elte.hu}}, |
| INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
| YEAR | = | {2020}, |
| NUMBER | = | {TR-2020-05} |