Explicit characterization of the identity configuration in an Abelian Sandpile Model

Understand

Since the work of Creutz, identifying the group identities for the Abelian Sandpile Model (ASM) on a given lattice is a puzzling issue: on rectangular portions of Z^2 complex quasi-self-similar structures arise.

• We study the ASM on the square lattice, in different geometries, and a variant with directed edges.
• Cylinders, through their extra symmetry, allow an easy determination of the identity, which is a homogeneous function.
• The directed variant on square geometry shows a remarkable exact structure, asymptotically self-similar.