Results and conjectures on the Sandpile Identity on a lattice

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.