Harmonic dynamics of the Abelian sandpile

TL;DR

This paper explores how harmonic fields influence the evolution of the abelian sandpile's identity, revealing periodic, smooth, and complex dynamics, and introduces a universal coordinate system for configurations.

Contribution

It demonstrates the effects of harmonic fields of various orders on sandpile identities, linking harmonic functions to the sandpile group and revealing new universal coordinates.

Findings

Harmonic fields induce periodic and smooth transformations of the sandpile identity.

The extended sandpile group provides a universal coordinate system for configurations.

Harmonic dynamics are robust and can encode and decode information efficiently.

Abstract

The abelian sandpile serves as a model to study self-organized criticality, a phenomenon occurring in biological, physical and social processes. The identity of the abelian group is a fractal composed of self-similar patches, and its limit is subject of extensive collaborative research. Here, we analyze the evolution of the sandpile identity under harmonic fields of different orders. We show that this evolution corresponds to periodic cycles through the abelian group characterized by the smooth transformation and apparent conservation of the patches constituting the identity. The dynamics induced by second and third order harmonics resemble smooth stretchings, respectively translations, of the identity, while the ones induced by fourth order harmonics resemble magnifications and rotations. Starting with order three, the dynamics pass through extended regions of seemingly random configurations which spontaneously reassemble into accentuated patterns. We show that the space of harmonic functions projects to the extended analogue of the sandpile group, thus providing a set of universal coordinates identifying configurations between different domains. Since the original sandpile group is a subgroup of the extended one, this directly implies that it admits a natural renormalization. Furthermore, we show that the harmonic fields can be induced by simple Markov processes, and that the corresponding stochastic dynamics show remarkable robustness over hundreds of periods. Finally, we encode information into seemingly random configurations, and decode this information with an algorithm requiring minimal prior knowledge. Our results suggest that harmonic fields might split the sandpile group into sub-sets showing different critical coefficients, and that it might be possible to extend the fractal structure of the identity beyond the boundaries of its domain.