On the Thermodynamic Limit for a One-Dimensional Sandpile Process

C. Maes, F. Redig, E. Saada, A. van Moffaert

2000, v.6, Issue 1, 1-21

ABSTRACT

Considering the standard Abelian sandpile model in one dimension, we construct an infinite volume Markov process corresponding to its thermodynamic (infinite volume) limit. The main difficulty we overcome is the strong non-locality of the dynamics by which, depending on the sand configuration, changes in the height variables far away can still influence the origin in a single updating. However, using similar ideas as in recent extensions of the standard Gibbs formalism for lattice spin systems, we can identify a set of `good' configurations on which the dynamics is effectively local. Finally, we prove that every configuration converges in a finite time to the unique invariant measure. Up to this time, the expected height increases linearly in time.

Keywords: sandpile model,non-Feller process,thermodynamic limit,interactingparticle systems

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