Conservation laws for strings in the Abelian Sandpile Model

TL;DR

This paper studies the structure and interactions of strings in the Abelian Sandpile Model, revealing their classification, a dispersion-like relation, and the role of the modular group SL(2,Z) in their conservation laws.

Contribution

It provides a complete classification of strings in the Abelian Sandpile Model and uncovers their interaction rules and underlying mathematical symmetries.

Findings

Strings are classified by their principal periodic vector k.

A dispersion relation E=k^2 relates string density to momentum.

String interactions conserve momentum and involve merging and splitting.

Abstract

The Abelian Sandpile generates complex and beautiful patterns and seems to display allometry. On the plane, beyond patches, patterns periodic in both dimensions, we remark the presence of structures periodic in one dimension, that we call strings. We classify completely their constituents in terms of their principal periodic vector k, that we call momentum. We derive a simple relation between the momentum of a string and its density of particles, E, which is reminiscent of a dispersion relation, E=k^2. Strings interact: they can merge and split and within these processes momentum is conserved. We reveal the role of the modular group SL(2,Z) behind these laws.