On the thermodynamic limit for a one-dimensional sandpile process

TL;DR

This paper constructs and analyzes the thermodynamic limit of a one-dimensional abelian sandpile model, demonstrating convergence to a unique invariant measure despite the non-local dynamics.

Contribution

It introduces a method to define an infinite volume Markov process for the 1D sandpile model and proves convergence to equilibrium.

Findings

Existence of a well-defined infinite volume process.

Convergence to a unique invariant measure.

Identification of 'good' configurations for local dynamics.

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.