Explicit characterization of the identity configuration in an Abelian sandpile model

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.