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 Λn = [-n, n]^d ∩ Z^d for d≥ 2. We show that as n→ ∞, the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in 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 Majumdar and Dhar, and the existence of the wired uniform spanning forest measure on Z^d. In the case d > 4, we also make use of Wilson’s method.