Explicit characterization of the identity configuration in an Abelian Sandpile Model

TL;DR

This paper provides an explicit characterization of the identity configuration in the Abelian Sandpile Model across various geometries, revealing exact and self-similar structures, especially in cylindrical and directed variants.

Contribution

It offers a new explicit description of the identity configuration in ASM, including for directed and cylindrical geometries, advancing understanding of its algebraic structure.

Findings

Cylinders enable easy determination of the identity configuration.

Directed ASM on square lattice exhibits an exact, asymptotically self-similar structure.

The identity configuration is a homogeneous function in cylindrical geometries.

Abstract

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.