Mathematics – Probability
Scientific paper
2011-06-27
Markov Process. Related Fields, Vol. 18, 111-156 (2012)
Mathematics
Probability
44 pages. Version 2 incorporates some smaller changes. To appear in Markov Processes and Related Fields in the proceedings of
Scientific paper
We review the Majumdar-Dhar bijection between recurrent states of the Abelian sandpile model and spanning trees. We generalize earlier results of Athreya and Jarai on the infinite volume limit of the stationary distribution of the sandpile model on Z^d, d >= 2, to a large class of graphs. This includes: (i) graphs on which the wired spanning forest is connected and has one end; (ii) transitive graphs with volume growth at least c n^5 on which all bounded harmonic functions are constant. We also extend a result of Maes, Redig and Saada on the stationary distribution of sandpiles on infinite regular trees, to arbitrary exhaustions.
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