Electronic Journal of Probability

Inequalities for critical exponents in $d$-dimensional sandpiles

Sandeep Bhupatiraju, Jack Hanson, and Antal A. Járai

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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

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

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

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Abelian sandpile critical exponent wave uniform spanning tree loop-erased random walk

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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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