Sankhya: The Indian Journal of Statistics

2005, Volume 67, Pt. 4, 634--661

Abelian Sandpile Models in Infinite Volume

C. Maes, Instituut voor Theoretische Fysica, Belgium
F. Redig, Technische Universiteit Eindhoven, The Netherlands
E. Saada, CNRS, Universit\'{e} de Rouen, France

SUMMARY. 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 $S= \mathbb{Z},\,S= \mathbb{T}$ is an infinite tree, $S=\Z^d$ with $d$ large, and when the dynamics is dissipative (i.e. sand grains may disappear at each toppling).

AMS (2000) subject classification. Primary 82C22; secondary 60K35.

Key words and phrases. Sandpile dynamics, nonlocal interactions, interacting particle systems, thermodynamic limit, spanning trees.

Full paper (PDF)