TR-2020-05

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

Viktor Kiss, Lionel Levine, Lilla Tóthmérész



Abstract

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:

@techreport{egres-20-05,
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 www.cs.elte.hu/egres}},
INSTITUTION = {Egerv{\'a}ry Research Group, Budapest},
YEAR = {2020},
NUMBER = {TR-2020-05}
}


Last modification: 29.12.2020. Please email your comments to Tamás Király!