Abelian sandpiles: An overview and results on certain transitive graphs

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Abstract

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.
Original languageEnglish
Pages (from-to)111-156
Number of pages44
JournalMarkov Processes and Related Fields
Volume18
Issue number1
Publication statusPublished - 2012

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