Theoretical Statistics and Mathematics Unit, ISI Delhi

Infinite volume limit for the stationary distribution of Abelian sandpile models

by 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]

Click here to return to Preprints Page