Abelian Sandpile Models in Infinite Volume

摘要

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 = Z,S = 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).