Harmonic dynamics of the abelian sandpile

The abelian sandpile is a cellular automaton which serves as the archetypical model to study self-organized criticality, a phenomenon occurring in various biological, physical, and social processes. Its recurrent configurations form an abelian group, whose identity is a fractal composed of self-similar patches. 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 and translations, respectively, while the ones induced by fourth-order harmonics resemble magnifications and rotations. Based on an extensive analysis of these sandpile dynamics on domains of different size, we conjecture the existence of several scaling limits for infinite domains. Furthermore, we show that the space of harmonic functions provides a set of universal coordinates identifying configurations between different domains, which directly implies that the sandpile group admits a natural renormalization. Finally, we show that the harmonic fields can be induced by simple Markov processes and that the corresponding stochastic dynamics show remarkable robustness. Our results suggest that harmonic fields might split the sandpile group into subsets showing different critical coefficients and that it might be possible to extend the fractal structure of the identity beyond the boundaries of its domain.