Results and conjectures on the Sandpile Identity on a lattice

Arnaud Dartois (1) , Clémence Magnien (1)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

In this paper we study the identity of the Abelian Sandpile Model on a rectangular lattice.This configuration can be computed with the burning algorithm, which, starting from the empty lattice, computes a sequence of configurations, the last of which is the identity.We extend this algorithm to an infinite lattice, which allows us to prove that the first steps of the algorithm on a finite lattice are the same whatever its size.Finally we introduce a new configuration, which shares the intriguing properties of the identity, but is easier to study.

Mots clés: Abelian sandpile; Identity; Burning algorithm; Infinite lattice; Toppling

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]; Automates cellulaires et gaz sur réseau [nlin.CG]

Arnaud Dartois, Clémence Magnien. Results and conjectures on the Sandpile Identity on a lattice. Discrete Models for Complex Systems, DMCS'03, 2003, Lyon, France. pp.89-102, ⟨10.46298/dmtcs.2308⟩. ⟨hal-01183316⟩