Infinite volume limit for the stationary distribution of Abelian sandpile models

Abstract: We study the stationary distribution of the standard Abelian sandpile model in the box $Lam_n = [-n, n]^d cap d$ for $d ge 2$. We show that as $n o infty$, the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in $d$. 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 Majumdar and Dhar, and the existence of the wired uniform spanning forest measure on $d$. In the case $d > 4$, we also make use of Wilson's method.