Open Access
2017 Inequalities for critical exponents in $d$-dimensional sandpiles
Sandeep Bhupatiraju, Jack Hanson, Antal A. Járai
Electron. J. Probab. 22: 1-51 (2017). DOI: 10.1214/17-EJP111


Consider the Abelian sandpile measure on $\mathbb{Z} ^d$, $d \ge 2$, obtained as the $L \to \infty $ limit of the stationary distribution of the sandpile on $[-L,L]^d \cap \mathbb{Z} ^d$. When adding a grain of sand at the origin, some region, called the avalanche cluster, topples during stabilization. We prove bounds on the behaviour of various avalanche characteristics: the probability that a given vertex topples, the radius of the toppled region, and the number of vertices toppled. Our results yield rigorous inequalities for the relevant critical exponents. In $d = 2$, we show that for any $1 \le k < \infty $, the last $k$ waves of the avalanche have an infinite volume limit, satisfying a power law upper bound on the tail of the radius distribution.


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Sandeep Bhupatiraju. Jack Hanson. Antal A. Járai. "Inequalities for critical exponents in $d$-dimensional sandpiles." Electron. J. Probab. 22 1 - 51, 2017.


Received: 21 February 2016; Accepted: 20 September 2017; Published: 2017
First available in Project Euclid: 14 October 2017

zbMATH: 06797895
MathSciNet: MR3718713
Digital Object Identifier: 10.1214/17-EJP111

Primary: 60K35
Secondary: 82B20

Keywords: Abelian sandpile , Critical exponent , Loop-erased random walk , Uniform spanning tree , wave

Vol.22 • 2017
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