C. Maes, F. Redig, E. Saada, A. Van Moffaert
On the thermodynamic limit for a one-dimensional sandpile process
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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. 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. We prove that every configuration
converges in a finite time to the unique invariant measure.