Random rotor walks and i.i.d. sandpiles on Sierpinski graphs
Probability
2024-02-27 v3 Combinatorics
Abstract
We prove that, on the infinite Sierpinski gasket graph SG, rotor walk with random initial configuration of rotors is recurrent. We also give a necessary condition for an i.i.d. sandpile to stabilize. In particular, we prove that an i.i.d. sandpile with expected number of chips per site greater or equal to three does not stabilize almost surely. Furthermore, the proof also applies to divisible sandpiles and shows that divisible sandpile at critical density one does not stabilize almost surely on SG.
Cite
@article{arxiv.2210.00810,
title = {Random rotor walks and i.i.d. sandpiles on Sierpinski graphs},
author = {Robin Kaiser and Ecaterina Sava-Huss},
journal= {arXiv preprint arXiv:2210.00810},
year = {2024}
}
Comments
13 pages, 3 figures; referee's suggestions implemented; to appear in Statistics and Probability Letters (2024)