Abelian Sandpiles and the Harmonic 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 T 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.

Sandpiles on the Square Lattice

“…The first equality in (9), which is the most delicate part to prove, is essentially a restatement of Theorem 2.4 in [35]. We provide a unified proof of all three parts of Theorem 4 in Section 3.…”

“…Schmidt and Verbitskiy [35] characterized the set of all functions in ℓ 1 pZ 2 q that are harmonic modulo 1. Their result can be stated using discrete derivatives of the Green's function on Z 2 .…”

“…The techniques use analysis of the space of functions on Z 2 which are harmonic modulo 1. In the course of our arguments, we characterize the harmonic modulo 1 functions in ℓ p pZ 2 q as linear combinations of certain discrete derivatives of Green's functions, extending a result of Schmidt and Verbitskiy [35].2010 Mathematics Subject Classification. Primary 82C20, 60B15, 60J10.…”

Discrete systems and abelian sandpiles

“…Before going into the mathematical details, we describe the physical meaning of this model (cf. [10]). Let Λ ⊂ Z 2 be non empty and finite, and that every location/site holds certain grains of sand.…”

“…In [10] a different approach has been proposed. Configurations of the infinite volume ASM are encoded as elements of a certain compact abelian group.…”