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Inequalities for critical exponents in $d$-dimensional sandpiles
Probability · Submitted 20 Feb 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Our results yield rigorous inequalities for the relevant critical exponents.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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 $[-L,L]^d \cap \mathbb{Z}^d$. | background 0.80 | 0.14 | 0.03 | asserted 0.89 |
| s1 | When adding a grain of sand at the origin, some region, called the avalanche cluster, topples during stabilization. | background 1.00 | 0.07 | 0.03 | asserted 1.00 |
| s2 | 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. | contribution 0.87 | 0.95 | 0.20 | asserted 1.00 |
| s3 | Our results yield rigorous inequalities for the relevant critical exponents. | result 0.52 | 0.94 | 0.66 | asserted 1.00 |
| s4 | 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. | result 0.99 | 0.95 | 0.20 | asserted 1.00 |
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The paper
Original 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.
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How the literature uses it
2 arXiv papers cite this in 2 sentences
Papers citing it · 1 sentence without a confident label
Cites it as background 1 paper
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(Possibly Lemma [ref] could help resolve this question.) Some further studies of sandpile dynamics on $^d$ are [ref] .
Sandpiles on the square lattice · Historical review · 0.96
What it cites · 7 sentences without a confident label
Cites as background 2 papers
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Separation lemmas for a loop-erased walk and a simple random walk appeared in [ref] for $d = 2$ and in [ref] for $d = 3$.
Growth exponent for loop-erased random walk in three dimensions · Separation lemma · 0.92
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The surveys [ref] and [ref] collect some of this.
Sandpile models · Introduction · 1.00
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