Harmonic dynamics of the abelian sandpile

Abstract

The abelian sandpile serves as a simple model system to study self-organized criticality, a phenomenon occurring in many important biological, physical and social processes. The identity element of the abelian group is a fractal composed of self-similar patches with periodic patterns, and the limit of this identity 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 -to a large extend -by the smooth transformation and apparent conservation of the patches constituting the identity. The dynamics induced by second and third order harmonic fields resemble smooth stretchings, respectively translations, of the identity, while the ones induced by fourth order harmonic fields 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 the sandpile group admits a natural renormalization. Furthermore, we show that the harmonic fields can be induced by simple Markov processes, and that the corresponding stochastic sandpile identity dynamics show remarkable robustness over dozens or hundreds of periods. Finally, we encode information into seemingly random sandpile 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 sandpile identity beyond the boundaries of its finite domain.

Pattern Formation and Tropical Geometry

“…It turns out that we can associate the "energy of the particle" with each world line such that total energy is conserved in these collisions. As it was recently shown (experimentally), quadratic patches may be thought of as a limit of many line-shaped patterns coming together during a relaxation [38].…”

Sandpile monomorphisms and limits

“…For example, the order of the sandpile group on a 3 × 3 square domain is 2 11 7 2 , while the one on a 5 × 5 domain is 2 18 3 5 5 2 11 2 13 2 . Recently, we have shown that the sandpile group can be considered as a discretization of a |∂Γ|-dimensional torus, to which we refer to as the extended sandpile group [35]. We have then derived epimorphisms from the extended sandpile group on a given domain to the corresponding group on a subdomain.…”

“…We have then derived epimorphisms from the extended sandpile group on a given domain to the corresponding group on a subdomain. On the level of the (usual) sandpile group, due to the discretization, this renormalization is however defined in the category of sets and in general only "approximates" group homeomorphism for sufficiently large domains [35]. Under which conditions these "approximations" can be lifted to true group homeomorphisms, if possible at all, is yet unknown.…”

“…This lack of known relationships in terms of homeomorphism between the sandpile groups on different domains of Z 2 is in stark contrast to the role of the sandpile model as the archetypical example for self-organized criticality, given that the concept of criticality itself is based on the notion of scaling. Progress in this subject might also provide means by which the existence of the conjectured scaling limits of the sandpile identity [28] and of other recurrent configurations [35] might be proven.…”

Criticality and sampling in neural networks

“…Colors correspond to number of grains: white is 0, green is 1, blue is 2 and black is 3. Figure generated with the Interpile toolbox [57]. 12 Figure 2.3 Directed Percolation.…”