Abelian sandpile models in infinite volume

C. Maes, F.H.J. Redig, E. Saada

Research output: Contribution to journalArticleAcademicpeer-review

5 Citations (Scopus)


Since its introduction by Bak, Tang and Wiessenfeld, the abelian sandpile dynamics has been studied extensively in finite volume. There are many problems posed by the existence of a sandpile dynamics in an infinite volume S: its invariant distribution should be the thermodynamic limit (does the latter exist?) of the invariant measure for the finite volume dynamics; the extension of the sand grains addition operator to infinite volume is related to the boundary effects of the dynamics in finite volume; finally, the crucial difficulty of the definition of a Markov process in infinite volume is that, due to sand avalanches, the interaction is long range, so that no use of the Hille-Yosida theorem is possible. In this review paper, we recall the needed results in finite volume, then explain how to deal with infinite volume when $S={\Bbb Z},S={\Bbb T}$ is an infinite tree, $S={\Bbb Z}^{d}$ with d large, and when the dynamics is dissipative (i.e. sand grains may disappear at each toppling).
Original languageEnglish
Pages (from-to)634-661
JournalSankhya : the Indian journal of statistics
Issue number4
Publication statusPublished - 2005


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