Infinite volume limits of high-dimensional sandpile models

We study the Abelian sandpile model on Z. In d ≥ 3 we prove existence of the infinite volume addition operator, almost surely with respect to the infinite volume limit μ of the uniform measures on recurrent configurations. We prove the existence of a Markov process with stationary measure μ, and study ergodic properties of this process. The main techniques we use are a connection between the statistics of waves and uniform two-component spanning trees and results on the uniform spanning tree measure on Z. Key-words: Abelian sandpile model, wave, addition operator, uniform spanning tree, two-component spanning tree, loop-erased random walk, tail triviality.