Abelian Sandpile Models in Infinite Volume∗

Since its introduction by Bak,Tang and Wiesenfeld, 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 that review paper, we recall the needed results in finite volume, then explain how to deal with infinite volume when S = Z, S = T is an infinite tree, S = Z with d large, and when the dynamics is dissipative (i.e. sand grains may disappear at each toppling) .