PreprintarXiv licencearXiv
Sandpile avalanches in two dimensions are finite
Abstract
In the abelian sandpile model on ℤd, a site with at least 2d grains topples by sending one grain to each neighbor. In a box [−N,N]d whose boundary absorbs grains, adding grains at random sites and toppling defines a Markov chain. Athreya and Járai (2004) constructed the stationary state on ℤd as the limit of its stationary distributions as N→∞. They asked whether, when a grain is added at the origin of a sandpile in the stationary state, the resulting avalanche of topplings is finite. Járai and Redig (2008) proved that the answer is yes for d≥ 3, using the transience of random walk. We prove, using a different method, that avalanches are also finite for d=2.
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BibTeX
@misc{bourabee2026sandpile,
title = {{Sandpile avalanches in two dimensions are finite}},
author = {Ahmed Bou-Rabee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13940},
eprint = {2610.13940},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13940},
}
RIS
TY - UNPB
TI - Sandpile avalanches in two dimensions are finite
AU - Ahmed Bou-Rabee
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.13940
UR - https://arxiv.org/abs/2610.13940
AB - In the abelian sandpile model on ℤ^d, a site with at least 2d grains topples by sending one grain to each neighbor. In a box [−N,N]^d whose boundary absorbs grains, adding grains at random sites and toppling defines a Markov chain. Athreya and Járai (2004) constructed the stationary state on ℤ^d as the limit of its stationary distributions as N→∞. They asked whether, when a grain is added at the origin of a sandpile in the stationary state, the resulting avalanche of topplings is finite. Járai and Redig (2008) proved that the answer is yes for d≥ 3, using the transience of random walk. We prove, using a different method, that avalanches are also finite for d=2.
ER -