Abstract
We prove a precise relationship between the threshold state of the fixed-energy sandpile and the stationary state of Dhar’s abelian sandpile: in the limit as the initial condition s 0 tends to \({-\infty}\) , the former is obtained by size-biasing the latter according to burst size, an avalanche statistic. The question of whether and how these two states are related has been a subject of some controversy since 2000.
The size-biasing in our result arises as an instance of a Markov renewal theorem, and implies that the threshold and stationary distributions are not equal even in the \({s_0 \to -\infty}\) limit. We prove that, nevertheless, in this limit the total amount of sand in the threshold state converges in distribution to the total amount of sand in the stationary state, confirming a conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle.
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Communicated by H. Spohn
The author was partly supported by NSF grant DMS-1243606.
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Levine, L. Threshold State and a Conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle. Commun. Math. Phys. 335, 1003–1017 (2015). https://doi.org/10.1007/s00220-014-2216-5
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DOI: https://doi.org/10.1007/s00220-014-2216-5