Publications and Preprints

Infinite volume limit for the stationary distribution of Abelian sandpile models
Siva R. Athreya and Antal A. J\'arai
We study the stationary distribution of the standard Abelian sandpile model in the box $\Lambda_n = [−n, n]^d \cap \mathbb{Z}^d$ for $d \ge 2$. We show that as $n \rightarrow \infty$, the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in $\mathbb{Z}^d$ . This allows us to define infinite volume versions of the avalanche-size distribution and related quantities. The proof is based on a mapping of the sandpile model to the uniform spanning tree due to Dhar and Majumdar, and the existence of the wired uniform spanning forest measure on $\mathbb{Z}^d$ . In the case $d > 4$ we also make use of Wilson’s method.

isid/ms/2003/16 [fulltext]

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