## Electronic Journal of Probability

### Inequalities for critical exponents in $d$-dimensional sandpiles

#### Abstract

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.

#### Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 85, 51 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1507946760

Digital Object Identifier
doi:10.1214/17-EJP111

Mathematical Reviews number (MathSciNet)
MR3718713

Zentralblatt MATH identifier
06797895

#### Citation

Bhupatiraju, Sandeep; Hanson, Jack; Járai, Antal A. Inequalities for critical exponents in $d$-dimensional sandpiles. Electron. J. Probab. 22 (2017), paper no. 85, 51 pp. doi:10.1214/17-EJP111. https://projecteuclid.org/euclid.ejp/1507946760

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