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# Harmonic dynamics of the abelian sandpile

Edited by Yuval Peres, Microsoft Research, Redmond, WA, and approved January 3, 2019 (received for review July 12, 2018)

## Significance

When slowly dropping sand on a sandpile, it automatically converges to a critical state where each additional grain of sand may cause nothing at all or start an avalanche of any size. Since similar phenomena occur in many physical, biological, and social processes, the abelian sandpile as the archetypical model for such “self-organized criticality” has been extensively studied for more than 30 y. Here, we demonstrate that the self-similar fractal structures arising in the abelian sandpile show smooth dynamics under harmonic fields, similar to sand dunes which travel, transform, and merge, depending on the wind. These harmonic dynamics directly provide universal coordinates for every possible sandpile configuration and can thus help to explore scaling limits for infinitely big domains.

## Abstract

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.

The abelian sandpile is a mathematical model introduced by Bak, Tang, and Wiesenfeld in 1987 (1). The model describes the evolution of an idealized sandpile under random dropping (addition) of grains of sand (1). The abelian sandpile was the first model which demonstrated the concept of self-organized criticality (SOC) (see ref. 2 for a recent review), the property of certain dissipative systems driven by fluctuating forces to automatically converge into critical configurations which eventually become unstable and relax in processes referred to as avalanches, characterized by scale-free spatiotemporal correlations (1⇓–3). Even though more than 30 y passed since its initial introduction, the sandpile model remains an active field of research. Despite being the first and archetypical model to study SOC (ref. 3, p. 1), this continued interest of the scientific community can be explained by the various intriguing mathematical properties of the abelian sandpile, several of which are explained below.

The sandpile model is a cellular automaton defined on a rectangular domain of the standard square lattice. Each vertex *A*). Thus, toppling of vertices in the interior of the domain conserves the total number of particles in the sandpile, whereas toppling of vertices at the sides and the corners of the domain decreases the total number by one and two, respectively. The redistribution of particles due to the toppling of a vertex can render other vertices unstable, resulting in subsequent topplings in a process referred to as an “avalanche.” Due to the loss of particles at the boundaries of the domain, this process eventually terminates (ref. 4, theorem 1), and the “relaxed” sandpile reaches a stable configuration which is independent of the order of topplings (ref. 3, p. 13). The distribution of avalanche sizes—the total number of topplings after a random particle drop—follows a power law (1) and is thus scale invariant. However, the critical exponent for this power law is yet unknown (5).

Soon after the introduction of the sandpile model (1), it was observed that the set of stable configurations can be divided into two classes, recurrent and transient ones (6). Thereby, a stable configuration is recurrent if it appears infinitely often in the Markov process described above where the probability to drop a particle on any given vertex is nonzero (6). Transient configurations, in contrast, appear finitely often in the same process (6). Equivalently, recurrent configurations can be defined as those stable configurations which can be reached from any other configuration by dropping particles and relaxing the sandpile (4). Since it is always possible to drop

The identity of this abelian group, the sandpile or Creutz identity—after Michael Creutz who first studied it in depth (4)—shows a remarkably complex self-similar fractal structure composed of patches covered with periodic patterns (Fig. 1*B*). Computational studies indicate that the structure of the identity possesses a scaling limit for infinite domains. Indeed, for some configurations different from the sandpile identity, scaling limits for infinite domains were shown to exist, and the patches visible in these configurations as well as their robustness were analyzed (7⇓–9). Nevertheless, corresponding results for the sandpile identity—such as a closed formula for its construction—are still missing (ref. 3, p. 61), even though recently a proof for the scaling limit of the sandpile identity was announced (10). At the time of writing, however, only rigorous proofs for some specific structural aspects of the sandpile identity are available (11). For example, the thin (usually one pixel wide) “strings” or “curves” visible in the identity (Fig. 1*B*) were recently identified as tropical curves (12⇓⇓–15), structures from tropical geometry arising, e.g., in string theory and statistical physics.

In this article we study a yet unknown property of the sandpile model: the evolution of the sandpile identity under harmonic fields externally imposed by deterministically or stochastically dropping particles on boundary vertices of the domain. We show that such harmonic fields induce cyclic dynamics of the sandpile identity through the abelian group, smoothly transforming individual patches and tropical curves, mapping them onto one another or merging them into different objects. The dynamics induced by the same harmonic field on domains of different size show remarkable similarities, strongly indicating that not only the sandpile identity, but also sandpile dynamics possess scaling limits. To mathematically interpret these observations, we introduce an extended analogue of the sandpile model where each vertex at the domain boundary is allowed to carry a real number of particles. The set of recurrent configurations for this extended sandpile model forms a connected Lie group, and we show that, on this group, the harmonic fields define closed geodesics and thus provide universal coordinates allowing us to map configurations between the sandpile groups on different domains. Since there exists a natural inclusion of the usual sandpile group into the extended one, the former can be interpreted as a discretization of the latter, and the existence of universal coordinates thus directly implies that the usual sandpile group admits a natural renormalization.

## Results

### Motivation.

Recall that the stable configuration

In general, it is nontrivial to find the toppling function T corresponding to the relaxation of an unstable configuration **1** then implies that the relaxation of

In the process described above, instead of adding all particles at once, we can equivalently add one particle after the other and relax the sandpile after each step. Since the addition of particles and the toppling operator commute (4), we will still finally arrive at the same configuration with which we started. However, we will also observe a series of intermediate, stable configurations while executing this algorithm. Intuitively, these intermediate configurations will first become more and more dissimilar from the initial configuration, before finally converging back to it. The dynamic properties of these “oscillations,” and in which order the particles have to be added such that these dynamics become meaningful, is the focus of this article.

Before describing our approach in more detail, we note that the procedure described above is reminiscent of a method proposed nearly 30 y ago by Michael Creutz to construct the sandpile identity starting from the empty configuration

### Harmonic Dynamics of the Sandpile Identity.

Let H be an integer-valued harmonic function on the standard square lattice

For a given rectangular domain Γ, we then define the dynamics **3**) can be trivially extended to superharmonic fields (*SI Appendix*, Figs. S1 and S2).

The term **3** ensures that only valid (

Before discussing the sandpile dynamics induced by specific harmonic fields, let us first state the following lemma:

### Lemma 1.

*The sandpile identity dynamics* *induced by an integer harmonic field* H *have periodicity* 1; *i.e*., *for all*

*Lemma 1* is a direct consequence of the standard identification of the sandpile group with the cokernel of the Laplacian

### Remark 1.

*Due to the similarities between* *Eqs.***2** *and* **3**, *it might be tempting to assume that*, *because* H *is harmonic*, *the toppling function associated to the relaxation of* *Eq.***3**) *has to be harmonic*, *too. In general*, *this is*, *however*, *only the case for*

In the following, we focus our analysis on the sandpile identity dynamics on a **3** and thus simplifies numeric computations. Since on a square domain, each pair of odd-order harmonic functions induces equivalent dynamics up to reflection along the diagonal, we discuss only the dynamics induced by one of them. Because the constant harmonic field *B*.

The sandpile identity dynamics induced by the first-order harmonic field *A* and *SI Appendix*, Movie S1). Note that, while a tropical curve passes through a patch, the location and shape of the latter slightly change.

The sandpile identity dynamics induced by *B* and *SI Appendix*, Movie S2) correspond to a smooth “stretching” of the central square in the direction of one diagonal and to a compression in the direction of the other. During this process, the central square gradually changes its pattern by the subsequent action of tropical curves. Thereafter, the central square splits into the tips of two Sierpinski triangles which continue traveling on the diagonal to the corners of the domain. On their way, these tips combine with the rest of the patterns of the Sierpinski triangle which resolved from the innermost Sierpinski triangles on the other diagonal. The tips of the latter continue moving to the center of the domain and eventually form a new central square. Thus, in the course of one period, the self-similar patches of the sandpile identity are smoothly transformed onto each other.

Also the dynamics induced by *C* and *SI Appendix*, Movie S3) resemble stretching actions, however, along the horizontal and vertical axes instead of the diagonal ones. Different from the dynamics induced by

The dynamics induced by *D* and *SI Appendix*, Movie S4) resemble a horizontal translation of the sandpile identity, overlaid by stretching dynamics similar to the ones induced by *B*). Interestingly, the relative locations at the boundaries where new patches enter and leave the domain seem to be consistent with the dynamics induced by

The dynamics induced by *E* and *SI Appendix*, Movie S5) initially seem to “zoom out” from the sandpile identity. During this process, patches enter the domain at relative positions seemingly consistent with the ones observed for *E*). Due to the observation that, in the dynamics induced by

The sandpile identity dynamics induced by *F* and *SI Appendix*, Movie S6) show a similar zooming action to the ones induced by

Due to the floor function in Eq. **3**, the sandpile identity dynamics are not linear in the harmonic fields inducing them. Nevertheless, when we analyzed the sandpile identity dynamics induced by linear combinations of two harmonic fields, we observed that their combined actions are well described by the sum of their individual ones (*SI Appendix*, Fig. S3 *A* and *B*). Furthermore, when we applied the harmonic *SI Appendix*, Fig. S3*C*). Similarly, the hyperfractal configurations occurring in the dynamics of *SI Appendix*, Fig. S3 *D* and *E*). This indicates that the action induced by the sum of two harmonics is well described by the sum of their individual actions.

The sandpile identity dynamics on nonsquare domains closely resemble those on square domains (*SI Appendix*, Figs. S4 and S5). Interestingly, the dynamics induced by *E*). When we further analyzed this effect, we observed that the dynamics induced by *SI Appendix*, Fig. S6), in the sense that for all tested domains regularly spaced local fractal structures emerged, resembling the structure of the sandpile identity on a square domain.

### Effect of Scaling the Domain Size.

In the previous section, we discussed the sandpile identity dynamics on a given domain induced by harmonic fields of different orders. In this section, we analyze how the dynamics induced by a given harmonic field change when scaling the domain size. We focus our analysis on the dynamics induced by

As shown in the previous section, accentuated fractal configurations emerge in the dynamics induced by *A*). However, the smaller the domain size and the “less simple” the fraction, the less visually pronounced these fractals became (*A*), and some weak fractal configurations were visually detectable only for large enough domain sizes (*A*).

A different picture emerged when we compared the sandpile dynamics for different domain sizes N in the temporal vicinity of strong fractal configurations (Fig. 3*B*). Here, the configurations at the same absolute times were clearly different. However, when we scaled time by a factor inversely proportional to the respective domain size, the dynamics in the vicinity of the strong fractal configurations coincided. For example, time had to run faster by a factor of

To interpret this “time distortion” in the vicinity of strong fractal configurations, recall that the sandpile dynamics can enter periods characterized by seemingly random configurations only when the areas of the individual patches approach one vertex. Without any time distortion in the vicinity of strong fractal configurations, we would thus expect, for only large enough domains, to see regular configurations composed of small patches extending over the whole sandpile identity dynamics. In contrast, when we estimated the time period during which the entrainment by a strong fractal configuration was observable, this period had similar absolute lengths for different domain sizes. The local time distortion in the proximity of strong fractals thus seems to compensate for the effect of the “space distortion” caused by scaling the domain size, ensuring that periods of seemingly random configurations always start at approximately the same absolute times. Interestingly, some weak fractal configurations emerge in the temporal vicinity of stronger ones (Fig. 3*C*). In Fig. 3*D*, we depict a model incorporating all effects discussed above, which proposes a superposition of different time lines in such situations.

Similar effects to those for *SI Appendix*, Fig. S7*A*). Note that already in the identity the positions of the tropical curves are completely different for odd- and even-sized domains and that they “fluctuate” when increasing the domain size. Similarly, the patches in the dynamics induced by *SI Appendix*, Fig. S7*B*). However, only when we scaled time by a factor inversely proportional to the domain size, did the positions of the tropical curves become comparable (*SI Appendix*, Fig. S7*C*). Finally, for fourth-order harmonic fields, hyperfractal configurations appeared at the same absolute times, while the speed of the regularly spaced local fractal structures scaled with the domain size, and the speed of individual patches scaled with the square of the domain size (*SI Appendix*, Fig. S8). Given these observations, we propose to assign the dimensions

In the following, we formalize several aspects of our observations in terms of mathematical conjectures about the scaling limits of configurations appearing in the sandpile identity dynamics. These limits are defined with respect to sequences *A* and *B*, with the floor function

Given these definitions, a sequence

### Conjecture 1.

*The sequence* *weak*-* *converges to a piecewise smooth function* *on* Ω.

*Conjecture 1* states that the sandpile identity itself possesses a scaling limit, denoted by *Conjecture 1* can thus be considered folklore, we included it here because all of the following original conjectures extend from it. For these conjectures, let **3**) induced by a given harmonic field H on the domain

### Conjecture 2.

*For linear harmonic fields* H, *the sequence* *weak*-* *converges to* *for every*

The tropical curves in the identity are commonly considered to represent 1D “defects” whose influence converges to zero when taking the limit. Since our simulations indicate that first-order harmonics affect only the positions of tropical curves (Fig. 2*A*), we thus conjecture that the dynamics induced by them should be constant in the limit.

### Conjecture 3.

*For quadratic harmonic fields* H, *the sequence* *weak*-**converges to a piecewise smooth function* *for every* *Moreover*, *for every point in* Ω, *the values of this function are piecewise smooth on* t.

*Conjecture 3* formalizes our intuition that the dynamics induced by second-order harmonics (Fig. 2 *B* and *C*) approach a piecewise smooth limit.

### Conjecture 4.

*For every polynomial harmonic field* H *of order* o *and every* *the sequence* *weak*-**converges to a piecewise smooth function* *Moreover*, *for every point in* Ω, *the values of this function are piecewise smooth on* t.

*Conjecture 4* generalizes *Conjecture 3* to higher-order harmonic fields. Since, for such fields, the sandpile dynamics become faster when increasing the domain size N (Fig. 3*B*), time has to be scaled by a factor of **3**) to real valued harmonic fields, we can alternatively scale the harmonic field H itself by *Conjecture 4* then becomes equivalent to stating that the sequence

### Conjecture 5.

*For every polynomial harmonic field* H *and every* *the sequence* *weak*-**converges to a piecewise smooth function*

As described above, in the sandpile dynamics induced by higher-order harmonic fields, regular fractal configurations appear at times corresponding to simple fractions (Fig. 2 *D*–*F*). Furthermore, when increasing the domain size, more and more of such regular configurations occur at times corresponding to less and less simple fractions (Fig. 3*A*). *Conjecture 5* states that, in the limit, such regular fractal configurations should appear at all rational times. Together with *Conjecture 4*, this means that we expect that there exist at least two different scaling limits for the sandpile identity dynamics. However, when we consider that local fractal structures appear in the dynamics of fourth-order harmonics at predictable times and positions when scaling the domain size (Fig. 2 *E* and *F* and *SI Appendix*, Fig. S8*B*), even more scaling limits might exist.

### The Topology and Renormalization of the Abelian Sandpile Group.

In the previous section, we analyzed the relationship between the harmonic sandpile dynamics on domains of different size and stated several conjectures concerning their scaling limits. However, already on sufficiently big but finite domains, the apparent “smoothness” of the harmonic dynamics suggests that they approximate some continuous trajectories. In the following, we formalize this intuition by showing that the usual sandpile group G can be interpreted as a discretization of a Lie group and that the harmonic dynamics approximate continuous geodesics of this Lie group. Furthermore, we show that the harmonic fields inducing these dynamics directly provide universal coordinates allowing us to map configurations between domains of different size—similar to the mapping indicated in Fig. 3*A*. This immediately implies that the sandpile group admits a natural renormalization. The existence of a natural renormalization might seem surprising at first glance since it is easy to see that, in general, any nontrivial renormalization cannot respect the underlying group structure of the abelian sandpile. For example, the orders of the sandpile groups on a domain consisting of just one vertex (*A*) and on a domain consisting of two adjacent vertices (*B*) are coprime, and thus any group homomorphism between them has to be trivial. However, as we will show, the natural renormalization of the sandpile group approximates a group homomorphism of its underlying Lie group, which explains its existence.

We start our analysis by first introducing an extended sandpile model where each vertex at the boundary

To understand the topology of the extended sandpile group, consider that, for each configuration C of the usual sandpile group,

### Proposition 1.

*The extended sandpile group* *is a torus of dimension* *and volume*

The relationship between the extended sandpile group *A*). If Γ consists of two adjacent vertices, 15 of the 16 stable configurations on Γ are recurrent, with the only nonrecurrent configuration being the configuration *B*), where

Next, we analyze the relationship between the two sandpile groups and harmonic functions. For this, let H be the space of real-valued discrete harmonic functions on **3**) can be written as

### Lemma 2.

*The homomorphism* η *is surjective.*

Now consider an exhausting injective family of domains

### Theorem 1.

*There are canonical surjective homomorphisms* *with* *independent from the specific choice of the family* *There is a natural inclusion*

#### Proof:

Let *Lemma 2*, these quotients are canonically isomorphic to

Since *Theorem 1* provides a set of universal coordinates for the extended sandpile group. Since there exists a natural inclusion of G into *C* and *D*). To obtain this projection for a given configuration *C*). Given H, the renormalization projection of *D*). Note that, while η is not injective, for any

Finally, note that *Theorem 1* directly poses the following question, which we hope to answer in our future research:

### Question 1.

*Is the inclusion* *an isomorphism?*

### Stochastic Realizations of Harmonic Potentials.

Instead of deterministically dropping particles at boundary vertices (Eq. **3**), we can alternatively drop them stochastically according to a probability distribution proportional to the potential

In Fig. 4*D*, we depict the stochastic sandpile identity dynamics corresponding to a realization of this Markov process for the harmonic field *D*). Only after a significantly longer time does the structure of the identity become significantly disturbed (*D* and *E*).

This robustness is remarkable when considering that every recurrent configuration can be reached by dropping particles only onto boundary vertices of the domain (4), which implies that the stochastic sandpile dynamics induced by most harmonic fields are ergodic. This result indicates that even hundreds of billions of random particle drops might not be sufficient to explore the whole sandpile group, which partially explains the discrepancy between experimentally obtained values for the critical coefficient of the avalanche size distribution in different studies. As a consequence of this robustness, it furthermore becomes possible to encode information into seemingly random configurations by utilizing that higher-order sandpile dynamics show extended periods of noise before returning to the neighborhood of their initial configurations (Fig. 4*G*).

Finally, we note that different from the deterministic version of our algorithm to generate the sandpile dynamics (Eq. **3**), only one particle is dropped at every time step of the stochastic algorithm. This allows us to determine and compare the critical exponents of the avalanche size distributions for the stochastic sandpile identity dynamics induced by harmonic fields of different order (Fig. 4*F*). Our results show that the critical exponent depends on the specific harmonic field, indicating that there might exist different regions of the sandpile group where the critical coefficient has different values or that the critical coefficient might depend on probability distribution determining on which vertices particles are dropped.

## Discussion

We expect that our results showing the existence of smooth sandpile identity dynamics induced by harmonic fields will provide important guidelines for future studies of the abelian sandpile. For example, it might be possible to prove the conjectured limits for the sandpile identity dynamics (*Conjectures 2–5*) using well-established techniques from statistical mechanics (19). Specifically, if one can determine the universal coordinates of particularly simple configurations, like the minimally stable one where each vertex carries three particles (*SI Appendix*, Fig. S9), it might become possible to use these configurations as initial conditions for a coarse-grained model describing the limiting dynamics and, by simulating back in time, indirectly prove the existence of a scaling limit for the identity. As another example, the emergence of new patches at boundary positions seemingly consistent between the dynamics induced by different harmonic fields (Fig. 2) suggests that it might be possible to extend the fractal structure of the sandpile identity beyond the boundaries of the domain, similar to the continuation of analytic functions in the complex plane. Provided that the dynamics induced by the harmonic *E*) are indeed similar to zooming actions, this extension might result in a (potentially infinite) wallpaper-like structure composed of regularly spaced, locally similar fractal structures. The similarities between the sandpile identity dynamics induced by *SI Appendix*, Fig. S6) furthermore suggest that there might exist only one fundamental extension of the identity for each domain topology.

That second-order harmonics transform the central square of the identity either into the tips of two Sierpinski triangles (*B*) or into a large set of tropical curves (*C*) suggests that all patches constituting the sandpile identity might be composed of tropical curves. In contrast, the dynamics induced by *C*) suggest that the sandpile identity itself might be composed of fractals. These two interpretations do not necessarily have to contradict each other, e.g., if we allowed tropical curves to be composed of fine strings of fractals. We note that such interpretations are somewhat reminiscent of discussions in string theory, where strings (tropical curves) and branes of different dimensions (patches, fractals, hyperfractals, and so on) occur.

The close relationship between the usual and the extended sandpile group (Fig. 4 *A*–*D*) indicates that we can extend our knowledge of the former by studying the latter. For example, *Question 1* proposes a concrete scaling limit for the extended sandpile group. Based on the observation that, when induced by the same harmonic field, both sandpile models topple at identical times while the time interval between subsequent topplings quickly decreases when increasing the domain size, one might speculate whether the limit of the usual sandpile group might actually be the same as the limit of the extended one. Furthermore, given that the harmonic fields themselves generate the sandpile group, our observation that regular fractal configurations occur at times corresponding to multiples of simple fractions has an interesting explanation: These simple fractions correspond to simple roots of the identity with respect to the respective generator/harmonic.

Finally, our results indicate that harmonic fields might divide the sandpile group into different regions, each showing scale-free spatiotemporal relationships, but with different critical exponents (Fig. 4*G*). If one could determine the critical exponents corresponding to a basis for the harmonic fields with high confidence using the periodicity of the sandpile identity dynamics, it might be possible to reconstruct the critical exponent of the whole sandpile group by taking an adequately weighted mean.

## Materials and Methods

An open-source implementation of the algorithms to generate the sandpile identity dynamics is available at langmo.github.io/interpile/ (22, 23). This website also contains additional movies for other domains, harmonics, and initial configurations.

## Acknowledgments

M.L. is grateful to the members of the C Guet and G Tkačik groups for valuable comments and support. M.S. is grateful to Nikita Kalinin for inspiring communications.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: moritz.lang{at}ist.ac.at.

Author contributions: M.L. designed research; M.L. and M.S. performed research; M.L. and M.S. analyzed data; and M.L. and M.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Data deposition: An open-source implementation of the algorithms to generate the sandpile identity dynamics is available at langmo.github.io/interpile/.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1812015116/-/DCSupplemental.

Published under the PNAS license.

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