2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/202018Since 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.11 pages, 8 figuresStatistical MechanicsHigh Energy Physics - LatticeMathematical PhysicsAdaptation and Self-Organizing Systemstext