Harmonic dynamics of the abelian sandpile Moritz Langa,1 and Mikhail Shkolnikova aInstitute of Science and Technology Austria, 3400 Klosterneuburg, Austria Edited by Yuval Peres, Microsoft Research, Redmond, WA, and approved January 3, 2019 (received for review July 12, 2018) The abelian sandpile is a cellular automaton which serves as “avalanche.” Due to the loss of particles at the boundaries of the the archetypical model to study self-organized criticality, a phe- domain, this process eventually terminates (ref. 4, theorem 1), nomenon occurring in various biological, physical, and social pro- and the “relaxed” sandpile reaches a stable configuration which cesses. Its recurrent configurations form an abelian group, whose is independent of the order of topplings (ref. 3, p. 13). The distri- identity is a fractal composed of self-similar patches. Here, we ana- bution of avalanche sizes—the total number of topplings after a lyze the evolution of the sandpile identity under harmonic fields random particle drop—follows a power law (1) and is thus scale of different orders. We show that this evolution corresponds to invariant. However, the critical exponent for this power law is yet periodic cycles through the abelian group characterized by the unknown (5). smooth transformation and apparent conservation of the patches Soon after the introduction of the sandpile model (1), it was constituting the identity. The dynamics induced by second- and observed that the set of stable configurations can be divided third-order harmonics resemble smooth stretchings and transla- into two classes, recurrent and transient ones (6). Thereby, a tions, respectively,while the ones induced by fourth-order harmon- stable configuration is recurrent if it appears infinitely often ics resemble magnifications and rotations. Based on an extensive in the Markov process described above where the probability analysis of these sandpile dynamics on domains of different size, to drop a particle on any given vertex is nonzero (6). Tran- we conjecture the existence of several scaling limits for infinite sient configurations, in contrast, appear finitely often in the domains. Furthermore, we show that the space of harmonic func- same process (6). Equivalently, recurrent configurations can be tions provides a set of universal coordinates identifying configu- defined as those stable configurations which can be reached rations between different domains, which directly implies that the from any other configuration by dropping particles and relax- sandpile group admits a natural renormalization. Finally, we show ing the sandpile (4). Since it is always possible to drop 3 − APPLIED that the harmonic fields can be induced by simple Markov processes MATHEMATICS Cij particles onto every vertex of a given configuration C , the and that the corresponding stochastic dynamics show remarkable “minimally stable configuration” where each vertex carries robustness. Our results suggest that harmonic fields might split the exactly three particles is necessarily recurrent (4). It follows that sandpile group into subsets showing different critical coefficients a given configuration is recurrent if and only if it can be reach- and that it might be possible to extend the fractal structure of the able from the minimally stable configuration (4). The importance identity beyond the boundaries of its domain. of the distinction between recurrent and transient configurations stems from the fact that the set of all recurrent configurations abelian sandpile j identity j dynamics j criticality j forms an abelian group, known as the sandpile group (4). The group operation thereby corresponds to the vertex-wise addition he abelian sandpile is a mathematical model introduced by of particles and the subsequent relaxation of the sandpile (4). TBak, Tang, and Wiesenfeld in 1987 (1). The model describes The identity of this abelian group, the sandpile or Creutz the evolution of an idealized sandpile under random dropping identity—after Michael Creutz who first studied it in depth (addition) of grains of sand (1). The abelian sandpile was the first model which demonstrated the concept of self-organized critical- ity (SOC) (see ref. 2 for a recent review), the property of certain Significance dissipative systems driven by fluctuating forces to automatically converge into critical configurations which eventually become When slowly dropping sand on a sandpile, it automatically unstable and relax in processes referred to as avalanches, char- converges to a critical state where each additional grain of acterized by scale-free spatiotemporal correlations (1–3). Even sand may cause nothing at all or start an avalanche of any size. though more than 30 y passed since its initial introduction, the Since similar phenomena occur in many physical, biological, sandpile model remains an active field of research. Despite being and social processes, the abelian sandpile as the archetypical the first and archetypical model to study SOC (ref. 3, p. 1), this model for such “self-organized criticality” has been exten- continued interest of the scientific community can be explained sively studied for more than 30 y. Here, we demonstrate that by the various intriguing mathematical properties of the abelian the self-similar fractal structures arising in the abelian sand- sandpile, several of which are explained below. pile show smooth dynamics under harmonic fields, similar to The sandpile model is a cellular automaton defined on a rect- sand dunes which travel, transform, and merge, depending on angular domain of the standard square lattice. Each vertex (i, j ) the wind. These harmonic dynamics directly provide universal of the domain carries a nonnegative number Cij of particles coordinates for every possible sandpile configuration and can (“grains of sand”), with C referred to as the configuration of the thus help to explore scaling limits for infinitely big domains. sandpile. Starting from some initial configuration, particles are slowly dropped onto vertices chosen at random. When during Author contributions: M.L. designed research; M.L. and M.S. performed research; M.L. this process the number of particles of any vertex exceeds three, and M.S. analyzed data; and M.L. and M.S. wrote the paper.y this vertex becomes unstable and “topples,” decreasing the num- The authors declare no conflict of interest.y ber of its particles by four and increasing the number of particles This article is a PNAS Direct Submission.y of each of its direct neighbors by one (Fig. 1A). Thus, toppling of Published under the PNAS license.y vertices in the interior of the domain conserves the total number Data deposition: An open-source implementation of the algorithms to generate the of particles in the sandpile, whereas toppling of vertices at the sandpile identity dynamics is available at langmo.github.io/interpile/.y sides and the corners of the domain decreases the total number 1 To whom correspondence should be addressed. Email: [email protected] by one and two, respectively. The redistribution of particles due This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. to the toppling of a vertex can render other vertices unstable, 1073/pnas.1812015116/-/DCSupplemental.y resulting in subsequent topplings in a process referred to as an Published online February 6, 2019. www.pnas.org/cgi/doi/10.1073/pnas.1812015116 PNAS j February 19, 2019 j vol. 116 j no. 8 j 2821–2830 Downloaded by guest on September 30, 2021 A 0 3 0 B 3 42 Background Triangles 112 toppling Sierpinski 040 Triangles 403 Central 122 Square toppling Tropical 201 Curve 023 Tip Center Back 222 Fig. 1. Toppling of vertices and the sandpile identity. (A) If a vertex of the sandpile (here 3 × 3) carries four or more particles, it becomes unstable and topples, decreasing the number of its particles by four and increasing the number of particles carried by each of its (four or fewer) neighbors by one. The toppling of one vertex can render other, previously stable vertices unstable, resulting in an avalanche of subsequent topplings. (B) The sandpile identity on a 255 × 255 square domain. White, green, blue, and black pixels represent vertices carrying zero, one, two, or three particles, respectively. Orange arrows denote the three different patch types in the identity: (i) the central square, (ii) background triangles, and (iii) Sierpinski triangles (ref. 3, p. 109ff). The latter are composed of three different patterns (yellow arrows), which we refer to as the tips, centers, and backs of the Sierpinski triangles. Additionally, thin 1D tropical curves are visible in the sandpile identity, slightly disturbing the patches they cross. (4)—shows a remarkably complex self-similar fractal structure Results composed of patches covered with periodic patterns (Fig. 1B). Motivation. Recall that the stable configuration C = (C u )◦ Computational studies indicate that the structure of the iden- reached when relaxing an unstable configuration C u , with (·)◦ tity possesses a scaling limit for infinite domains. Indeed, for the relaxation operator corresponding to a series of topplings some configurations different from the sandpile identity, scal- resulting in a stable sandpile, can be expressed in terms of the ing limits for infinite domains were shown to exist, and the toppling function T [also referred to as the odometer function patches visible in these configurations as well as their robust- (16)], where Tij quantifies how often the vertex (i, j ) topples ness were analyzed (7–9). Nevertheless, corresponding results when relaxing C u (16, 17), for the sandpile identity—such as a closed formula for its construction—are still missing (ref. 3, p. 61), even though C = (C u )◦ = C u + ∆T , [1] recently a proof for the scaling limit of the sandpile identity was announced (10). At the time of writing, however, only rigorous with ∆ the discrete Laplace operator defined by proofs for some specific structural aspects of the sandpile iden- tity are available (11). For example, the thin (usually one pixel (∆T )ij = Ti+1,j + Ti−1,j + Ti,j +1 + Ti,j −1 − 4Ti,j : wide) “strings” or “curves” visible in the identity (Fig.