Existence of a planar sandpile avalanche-size exponent
Let $\nu$ be the Abelian sandpile measure on $\mathbb{Z}^2$: the weak limit, as $n \to \infty$, of the uniform measures on recurrent stable configurations of the box $[-n, n]^2$ with wired boundary, in which the exterior is identified with a sink; stable heights are $0, 1, 2, 3$, and the limit exists by Athreya and Járai. Sample a configuration from $\nu$, add one grain at the origin and stabilise, toppling any site with at least four grains by sending one grain to each of its four neighbours. Let $S$ be the total number of topplings, counted with multiplicity, and $\mathrm{Av}$ the set of sites that topple. Bou-Rabee (2026) proved that $\nu$-almost surely the configuration stabilises after finitely many topplings, so that $S < \infty$ almost surely. By Dhar's formula $\mathbb{E}_\nu S = \infty$ (see Járai). Bhupatiraju, Hanson and Járai proved $\nu(|\mathrm{Av}| \ge k) \ge k^{-3/8 + o(1)}$ as $k \to \infty$, and $S \ge |\mathrm{Av}|$. On $\mathbb{Z}^d$ with $d \ge 5$, Hutchcroft proved that the probability of at least $k$ topplings is of order $k^{-1/2}$.
Problem. Does there exist $a \in [0, \infty)$ such that $$\lim_{k \to \infty} \frac{\log \nu(S \ge k)}{\log k} = -a?$$ Only the existence of the exponent is asked, not its value.
Source: Járai (2018), Conjecture 3.8, for every d ≥ 2 and for both the number of topplings and the number of toppled sites; AIM — Open Applied Problems states the case d = 2 for the number of topplings.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- Abelian sandpiles, AIM Open Applied Problems, critical exponents, self-organized criticality
References
@article{jarai2018,
author = {J{\'a}rai, Antal A.},
title = {Sandpile models},
journal = {Probab. Surv.},
volume = {15},
year = {2018},
pages = {243--306},
doi = {10.1214/14-PS228},
eprint = {1401.0354},
archivePrefix = {arXiv},
}
@article{athreya2004,
author = {Athreya, Siva R. and J{\'a}rai, Antal A.},
title = {Infinite volume limit for the stationary distribution of {A}belian sandpile models},
journal = {Comm. Math. Phys.},
volume = {249},
number = {1},
year = {2004},
pages = {197--213},
doi = {10.1007/s00220-004-1080-0},
}
@misc{bourabee2026,
author = {Bou-Rabee, Ahmed},
title = {Sandpile avalanches in two dimensions are finite},
year = {2026},
eprint = {2610.13940},
archivePrefix = {arXiv},
}
@article{bhupatiraju2017,
author = {Bhupatiraju, Sandeep and Hanson, Jack and J{\'a}rai, Antal A.},
title = {Inequalities for critical exponents in $d$-dimensional sandpiles},
journal = {Electron. J. Probab.},
volume = {22},
year = {2017},
pages = {1--51},
note = {Paper no. 85},
doi = {10.1214/17-EJP111},
eprint = {1602.06475},
archivePrefix = {arXiv},
}
@article{hutchcroft2019,
author = {Hutchcroft, Tom},
title = {Universality of high-dimensional spanning forests and sandpiles},
journal = {Probab. Theory Related Fields},
volume = {176},
number = {1-2},
year = {2020},
pages = {533--597},
doi = {10.1007/s00440-019-00923-3},
eprint = {1804.04120},
archivePrefix = {arXiv},
}
@misc{aim-planar-sandpile-size-exponent,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Existence of a planar sandpile avalanche-size exponent}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/281-planar-sandpile-size-exponent.md},
note = {Entry 281 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00459, "Existence of a planar sandpile avalanche-size exponent", clopen.solutions.