Abelian Sandpiles and the Harmonic Model

TL;DR

This paper constructs entropy-preserving maps between abelian sandpile models and harmonic models, establishing key properties like uniqueness and Bernoulli nature of the maximal entropy measure.

Contribution

It introduces a novel construction of entropy-preserving maps linking sandpile models to harmonic models, proving measure uniqueness and Bernoulli properties.

Findings

Constructed entropy-preserving equivariant surjective maps

Proved uniqueness of the measure of maximal entropy

Established Bernoulli property for the dissipative model

Abstract

We present a construction of an entropy-preserving equivariant surjective map from the $d$-dimensional critical sandpile model to a certain closed, shift-invariant subgroup of $\mathbb{T}^{\mathbb{Z}^d}$ (the `harmonic model'). A similar map is constructed for the dissipative abelian sandpile model and is used to prove uniqueness and the Bernoulli property of the measure of maximal entropy for that model.