Harmonic dynamics of the abelian sandpile

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 $\left(i,j\right)$ of the domain carries a nonnegative number $C_{ij}$ of particles (“grains of sand”), with $C$ referred to as the configuration of the sandpile. Starting from some initial configuration, particles are slowly dropped onto vertices chosen at random. When during this process the number of particles of any vertex exceeds three, this vertex becomes unstable and “topples,” decreasing the number of its particles by four and increasing the number of particles of each of its direct neighbors by one (Fig. 1A). 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).

Fig. 1.

Toppling of vertices and the sandpile identity. (A) If a vertex of the sandpile (here $3\times 3$) 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 $255\times 255$ 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.

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 $3-C_{ij}$ particles onto every vertex of a given configuration $C$, the “minimally stable configuration” where each vertex carries exactly three particles is necessarily recurrent (4). It follows that a given configuration is recurrent if and only if it can be reachable from the minimally stable configuration (4). The importance of the distinction between recurrent and transient configurations stems from the fact that the set of all recurrent configurations forms an abelian group, known as the sandpile group (4). The group operation thereby corresponds to the vertex-wise addition of particles and the subsequent relaxation of the sandpile (4).

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. 1B). 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. 1B) 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 $C=\left(C^u\right)°$ reached when relaxing an unstable configuration $C^u$, with $\left(\cdot \right)°$ the relaxation operator corresponding to a series of topplings resulting in a stable sandpile, can be expressed in terms of the toppling function $T$ [also referred to as the odometer function (16)], where $T_{ij}$ quantifies how often the vertex $\left(i,j\right)$ topples when relaxing $C^u$ (16, 17),

$$C=\left(C^u\right)°=C^u+\mathrm{\Delta }T,$$ [1]

with $\mathrm{\Delta }$ the discrete Laplace operator defined by

$${\left(\mathrm{\Delta }T\right)}_{ij}=T_{i+1,j}+T_{i-1,j}+T_{i,j+1}+T_{i,j-1}-4T_{i,j}.$$

Note that we adopt the convention to set $T_{ij}=0$ for any $\left(i,j\right)$ outside of the domain.

In general, it is nontrivial to find the toppling function $T$ corresponding to the relaxation of an unstable configuration $C^u$ without performing the relaxation itself. However, we can equivalently state the inverse problem of how to construct an unstable configuration $C^u$ such that the toppling function $T$ takes some predefined values. An interesting special case of this inverse problem arises when we require the toppling function to be harmonic, i.e., that ${\left(\mathrm{\Delta }T\right)}_{ij}=0$ for all vertices $\left(i,j\right)$ in the interior of the domain. Eq. 1 then implies that the relaxation of $C^u$ changes the particle numbers of vertices only at the boundary of the domain and that $C_{ij}=C_{ij}^u$ for all $\left(i,j\right)$ in the interior. Finally, if we assume that $C$ is recurrent, we can always require that $C^u$ should be the result of adding some “potential” $X$ to $C$ itself, i.e., that $C^u=C+X$ and, in consequence, that

$$C=\left(C+X\right)°=C+X+\mathrm{\Delta }T.$$ [2]

Clearly, the potential is given by $X=-\mathrm{\Delta }T$ and, since $T$ is harmonic, it has support only at the boundary; i.e., $X_{ij}=0$ for every $\left(i,j\right)$ in the interior. In other words, we add particles to the boundary of a given recurrent configuration such that we arrive at the same configuration again after relaxing the sandpile.

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 $C_{ij}^0=0$ (4). In this method, a single particle is dropped on each vertex at the sides of the rectangular domain and two particles on each vertex at the corners, and the sandpile is relaxed. These steps are repeated several times, and it turns out that the sandpile will eventually convergence to the identity. Notably, this corresponds to adding, in each step, the potential $X^0=-\mathrm{\Delta }{T}^0$, where $T^0$ denotes the constant harmonic toppling function $T_{ij}^0=1$. However, as will become clear in the following, this constant harmonic toppling function will not result in any nontrivial dynamics, and – to our knowledge – nobody yet analyzed the effect when adding particles one by one instead of all at once for higher-order harmonic functions.

Harmonic Dynamics of the Sandpile Identity.

Let $H$ be an integer-valued harmonic function on the standard square lattice ${\mathbb{Z}}^2$ and $H{\mid }_{\mathrm{\Gamma }}$ be its restriction to an $N\times M$ rectangular domain $\mathrm{\Gamma }$. Furthermore, let $X^H=-\mathrm{\Delta }H{\mid }_{\mathrm{\Gamma }}$, to which we refer as the potential of $H$ for the domain $\mathrm{\Gamma }$. Note that $X_{ij}^H=0$ for $\left(i,j\right)$ in the interior of $\mathrm{\Gamma }$.

For a given rectangular domain $\mathrm{\Gamma }$, we then define the dynamics $I^H\left(t\right)$ of the sandpile identity $I$ at time $t\ge 0$ induced by the harmonic field $H$ by

$$I^H\left(t\right)={\left(I+\lfloor t{X}^H\rfloor +k\left(t\right)X^0\right)}^{○},$$ [3]

with $\lfloor .\rfloor$ the floor function, $X^0$ the potential of the constant harmonic function $H_{ij}^0=1$, and $k:\left[0,\infty \right)\to {\mathbb{Z}}_{\ge 0}$ chosen such that $\lfloor t{X}^H\rfloor +k\left(t\right)X^0$ becomes nonnegative for all $t\ge 0$. We note that using the ceiling or round function instead of the floor function leads to negligible differences in the dynamics and that the definition (Eq. 3) can be trivially extended to superharmonic fields (SI Appendix, Figs. S1 and S2).

The term $k\left(t\right)X^0$ in Eq. 3 ensures that only valid ($C_{ij}\ge 0$) and recurrent configurations appear in the the sandpile dynamics induced by a given harmonic field $H$. The term is necessary because, in general, it is not guaranteed that the potential associated to a given harmonic field is nonnegative. In the following, however, we focus on harmonic fields with nonnegative potentials for which $k\left(t\right)X^0$ can be set to zero and use only the general formula when discussing the scaling limits of the sandpile dynamics on infinite domains. Note that $\left(C+X^0\right)°=C$ for every recurrent configuration $C$, such that any valid choice for $k\left(t\right)$ results in the same sandpile dynamics.

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

Lemma 1.

The sandpile identity dynamics $I^H\left(t\right)$ induced by an integer harmonic field $H$ have periodicity 1; i.e., $I^H\left(t+1\right)=I^H\left(t\right)$ for all $t\ge 0$.

Lemma 1 is a direct consequence of the standard identification of the sandpile group with the cokernel of the Laplacian ${\mathbb{Z}}^{\mathrm{\Gamma }}/\mathrm{\Delta }{\mathbb{Z}}^{\mathrm{\Gamma }}$ (17). The proof follows widely (4), which indeed directly provides the proof for the special case of the constant harmonic function $H_{ij}^0=1$.

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 $I+\lfloor t{X}^H\rfloor +k\left(t\right)X^0$ (Eq. 3) has to be harmonic, too. In general, this is, however, only the case for $t\in {\mathbb{Z}}_{\ge 0}$.

In the following, we focus our analysis on the sandpile identity dynamics on a $255\times 255$ square domain induced by a reasonably small and computationally feasible set of nine integer-valued harmonic fields forming the basis of the vector space of all harmonic functions of order four or less (Table 1). We used the freedom in the definition of an appropriate basis to ensure that the potential of each basis vector became nonnegative, which allows us to set $k\left(t\right)=0$ in Eq. 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 $H_{ij}^0=1$ does not lead to any nontrivial dynamics, we start the discussion with harmonic fields of order one. We refer to the various 2D patches of the sandpile identity by the names indicated in Fig. 1B.

Table 1.

Basis for discrete harmonic functions of order four or less

Name	Formula	$c^{\left(.\right)}$ for $255\times 255$	Effect
$H^0$	1	—	None
$H^{1a}$	$i+c^{1a}$	255	Horizontal transl. of TCs
$H^{1b}$	$j+c^{1b}$	255	Vertical transl. of TCs
$H^{2a}$	$ij+c^{2a}$	$32\prime 512$	Diagonal stretch.
$H^{2b}$	$i^2-j^2+c^{2b}$	$16\prime 384$	Vertical and horizontal stretch.
$H^{3a}$	$i^3-3i{j}^2+c^{3a}$	$8\prime 290\prime 305$	Horizontal transl. and stretch.
$H^{3b}$	$j^3-3j{i}^2+c^{3b}$	$8\prime 290\prime 305$	Vertical transl. and stretch.
$H^{4a}$	$\frac{1}{6}\left(i^4-6i^2{j}^2+j^4\right.$	$352\prime 332\prime 544$	Zooming
	$\left.-i^2-j^2\right)+c^{4a}$
$H^{4b}$	$\frac{1}{6}\left(i^{3}j-i{j}^3\right)+c^{4b}$	$17\prime 220\prime 096$	Zooming + rotation

For all harmonics and all $N\times M$ domains $\mathrm{\Gamma }$, the origin $\left(i,j\right)=\left(\mathrm{0,0}\right)$ is assumed to lie in the center of $\mathrm{\Gamma }$ (if both N and M are odd), respectively at the vertex closest to the center from the top left. For a given domain $\mathrm{\Gamma }$, the constants $c^{\left(.\right)}$ are chosen such that min$min\left(-\mathrm{\Delta }{H}^{\left(.\right)}{\mid }_{\mathrm{\Gamma }}\right)=0$. For $255\times 255$ domains, the respective values of$c^{\left(.\right)}$ are shown. TC, tropical curve; transl. translation; stretch., stretching.

The sandpile identity dynamics induced by the first-order harmonic field $H_{ij}^{1a}=i+c^{1a}$, with $c^{1a}=$ const, correspond to a smooth horizontal translation of the tropical curves from left to right, leading to the replacement of each tropical curve by its neighbor to the left after one full period (Fig. 2A and SI Appendix, Movie S1). Note that, while a tropical curve passes through a patch, the location and shape of the latter slightly change.

Fig. 2.

Sandpile identity dynamics induced by harmonic fields of order four or less on a $255\times 255$ square domain. For odd-order harmonics, only one of the two dynamics is shown since the other one is identical up to switching of axes. For each harmonic, the sandpile dynamics at five time points of specific interest are shown. The first frame ($t=0$) displayed for the dynamics induced by $H^{1a}$ corresponds to the sandpile identity, where the dynamics induced by all harmonics start $\left(t=0\right)$ and end $\left(t=1\right)$. (A–F)The six subfigures correspond to he harmonics: (A) $H_{ij}^{1a}=i+c^{1a}$; (B) $H_{ij}^{2a}=ij+c^{2a}$; (C) $H_{ij}^{2b}=i^2-j^2+c^{2b}$; (D) $H_{ij}^{3a}=i^3-3i{j}^2+c^{3a}$; (E) $H_{ij}^{4a}=\frac{1}{6}\left(i^4-6i^2{j}^2+j^4-i^2-j^2\right)+c^{4a}$, and (F) $H_{ij}^{4b}=\frac{1}{6}\left(i^{3}j-i{j}^3\right)+c^{4b}$ (F). In A–F, white, green, blue, and black pixels represent vertices carrying zero, one, two, or three particles, respectively.

The sandpile identity dynamics induced by $H_{ij}^{2a}=ij+c^{2a}$ (Fig. 2B 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 $H_{ij}^{2b}=i^2-j^2+c^{2b}$ (Fig. 2C 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 $H^{2a}$, the central square is “disassembled” in the horizontal direction into many tropical curves and thus becomes a rectangle of shrinking width and growing height. At $t=0.25$, the width of the rectangle approaches zero, leading to the fusion of the two background triangles which were originally to the left and the right of the central square. Subsequently, a rectangle is reestablished in the center of the domain by the “fusion” of many tropical curves entering the domain from the top and the bottom, reaching the shape of a square at $t=0.5$. While the configuration at $t=0.5$ is very similar to the sandpile identity, the tropical curves are at different positions. After $t=0.5$, the dynamics go through a similar cycle to that before and finally reach the sandpile identity at $t=1$. During each of these two half cycles, self-similar patches are smoothly transformed onto each other, enter the domain at its left and right boundaries, or leave it at its top and bottom boundaries.

The dynamics induced by $H_{ij}^{3a}=i^3-3i{j}^2+c^{3a}$ (Fig. 2D and SI Appendix, Movie S4) resemble a horizontal translation of the sandpile identity, overlaid by stretching dynamics similar to the ones induced by $H^{2b}$. New Sierpinski triangles enter the domain at its right boundary, leading to regular repetitive configurations of more and more, smaller and smaller Sierpinski triangles filling the whole domain. When the size of each individual Sierpinski triangle approaches one vertex, the dynamics enter extended periods of seemingly random configurations. However, at regular times corresponding to multiples of $1/12$ or other simple fractions, new fractal configurations emerge and subsequently disappear. At $t=1/3$ and $t=2/3$, these fractal configurations are similar, but not identical, to the sandpile identity, and, at $t=0.5$, the configuration is similar to the one observed for $H^{2a}$ at $t=0.5$ (compare Fig. 2B). Interestingly, the relative locations at the boundaries where new patches enter and leave the domain seem to be consistent with the dynamics induced by $H^{2b}$.

The dynamics induced by $H_{ij}^{4a}=\frac{1}{6}\left(i^4-6i^2{j}^2+j^4-i^2-j^2\right)+c^{4a}$ (Fig. 2E 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 $H^{2b}$ and $H^{3a}$. After some time, more and more regularly spaced fractal structures resembling the sandpile identity appear, each restricted to a region of the domain. Each of these fractals shrinks in size while converging to the center of the domain. When the size of each of these fractals approaches one vertex, the dynamics become rather random, except for a small region around the center of the domain. However, accentuated regular fractal structures filling the whole domain emerge at times corresponding to multiples of $1/12$ or other simple fractions. Shortly before these fractal structures emerge, small patches quickly expand and fill nearly the whole domain with a single, regular pattern. Subsequently, this pattern changes and the corresponding patch shrinks in size, giving rise to a fractal structure of which it becomes the center ($t=0.99995$ and $t=0.99997$ in Fig. 2E). Due to the observation that, in the dynamics induced by $H^{4a}$, local fractal structures appear which converge to the center of the domain and presumably form the regular configurations emerging at times corresponding to multiples of $1/12$ or other simple fractions, we refer to the latter configurations as hyperfractals.

The sandpile identity dynamics induced by $H_{ij}^{4b}=\frac{1}{6}\left(i^{3}j-i{j}^3\right)+c^{4b}$ (Fig. 2F and SI Appendix, Movie S6) show a similar zooming action to the ones induced by $H^{4a}$. Additionally, the central square slowly rotates, while all other patches become skewed. Similar to $H^{4a}$, local fractals resembling skewed copies of the sandpile identity enter the domain at its boundaries and slowly converge to its center while shrinking in size. The dynamics then become rather random, interrupted by regular hyperfractal configurations emerging at times corresponding to multiples of simple fractions. The hyperfractal configuration at $t=0.5$ is composed of isosceles triangles and reminiscent of sandpile identities on pseudo-Manhattan domains (ref. 18, figure 6). In contrast, the hyperfractal configurations at $t=1/3$ and $t=2/3$ are not composed of patches with clearly defined boundaries, but of patterns which smoothly “shade” into one another.

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 $H^{2a}$ to a configuration extracted from the dynamics induced by $H^{4a}$ at the time when multiple regularly spaced local fractal structures were visible, each of these local fractal structures showed similar dynamics to those of the sandpile identity under the harmonic field $H^{2a}$ (SI Appendix, Fig. S3C). Similarly, the hyperfractal configurations occurring in the dynamics of $H^{4a}$ showed dynamics consistent with those of the sandpile identity when induced by $H^{2a}$ or $H^{2b}$ (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 $H^{4a}$ on a rectangular or a circular domain show the emergence of a central square and surrounding patches typical of the sandpile identity on a square domain (Fig. 2E). When we further analyzed this effect, we observed that the dynamics induced by $H^{4a}$ are even similar on nonconvex domains (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 $H^{3a}$, but discuss other harmonic fields at the end of this section.

As shown in the previous section, accentuated fractal configurations emerge in the dynamics induced by $H^{3a}$ from seemingly random configurations at times corresponding to multiples of simple fractions. In the temporal vicinity of these fractal configurations, the dynamics seem to be “entrained” by these fractals, in the sense that new patches emerge at the boundaries of the domain, seemingly extending the fractal structure. To further analyze this effect, we compared the sandpile identity dynamics induced by $H^{3a}$ on $N\times N$ square domains for different domain sizes $N$. For different domain sizes, similar accentuated fractal configurations appeared at the same absolute times corresponding to multiples of $1/12$ or other simple fractions (Fig. 3A). However, the smaller the domain size and the “less simple” the fraction, the less visually pronounced these fractals became ($t=1/12$ or $t=3/12$ in Fig. 3A), and some weak fractal configurations were visually detectable only for large enough domain sizes ($t=9/10$, Fig. 3A).

Fig. 3.

Effect of scaling the domain size on the sandpile identity dynamics. (A) For different domain sizes (Top row, $63\times 63$; Bottom row, $255\times 255$), pronounced fractal configurations occur at the same times in the dynamics induced by $H_{ij}^{3a}=i^3-3i{j}^2+c^{3a}$ corresponding to multiples of simple fractions (e.g., of $1/12$). These fractals are easier to visually detect at larger domain sizes, and some are visually detectable only for sufficiently large domains ($t=0.9$). (B) Fractal structures seem to “entrain” the dynamics of patches in their temporal vicinity. To map the positions of such entrained patches onto one another for different domain sizes, time has to be scaled by a factor proportional to the domain size (here, by $255/63\approx 4$), which is in contrast to the absolute timescale at which the fractal configurations themselves appear. (C) Weak fractal structures can appear in the vicinity of stronger ones (here, at $t=1/30$ and $t=1/8$, i.e., close to the fractal structures at $t=0$ and $t=1/6$, respectively). (D) Qualitative model for the interplay between the different effects described in A–C. A global timescale determines when fractal configurations occur. These fractals impose a time distortion proportional to the domain size in their temporal vicinity. The absolute time periods during which these time distortions affect the dynamics are approximately constant and independent of the domain size. The entrainment by weak fractals in the vicinity of stronger ones might lead to a superposition of different local timescales. In A–C, white, green, blue, and black pixels represent vertices carrying zero, one, two, or three particles, respectively.

A different picture emerged when we compared the sandpile dynamics for different domain sizes $N$ in the temporal vicinity of strong fractal configurations (Fig. 3B). 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 $255/63\approx 4$ on a $63\times 63$ square domain than on a $255\times 255$ square domain for the positions and shapes of the patches to match.

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. 3C). In Fig. 3D, 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 $H^{3a}$ can also be observed for other harmonic fields. For $H^{1a}$, the sandpile identity dynamics describe the translation of tropical curves. When we compared these dynamics between different $N\times N$ domains with $N$ odd, the tropical curves seemed to have similar relative positions at the same absolute times (SI Appendix, Fig. S7A). 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 $H^{2a}$ and $H^{2b}$ had the same relative position and shapes at the same absolute times (SI Appendix, Fig. S7B). 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. S7C). 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 $d_c=1$ to tropical curves, $d_p=2$ to patches, $d_f=3$ to fractals, and $d_e=4$ to hyperfractals. A harmonic field of order $o$ acts on and transforms an object of dimension $d$ only if $o\ge d$, and the speed of the object in the sandpile dynamics then becomes proportional to $N^{o-d}$, with $N$ the domain size.

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 ${\left\{{\mathrm{\Gamma }}_N\right\}}_{N>0}$ of discrete domains ${\mathrm{\Gamma }}_N=\mathrm{\Omega }\cap \frac{1}{N}{\mathbb{Z}}^2$ approximating a given continuous and compact convex domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$, where $\frac{1}{N}{\mathbb{Z}}^2$ denotes the standard square lattice scaled by $\frac{1}{N}$. For example, when setting $\mathrm{\Omega }={\left[\mathrm{0,1}\right]}^2$, ${\mathrm{\Gamma }}_N$ corresponds to an $\left(N+1\right)\times \left(N+1\right)$ square domain and ${\left\{{\mathrm{\Gamma }}_N\right\}}_{N>0}$ to the sequence of square domains with increasing size. For every domain ${\mathrm{\Gamma }}_N$, we denote by $G_N$ the corresponding sandpile group and by $I_N\in G_N$ its identity. To compare configurations $C_N\in G_N$ belonging to different domains ${\mathrm{\Gamma }}_N$ in the same sequence, we denote by ${\tilde{C}}_N:\mathrm{\Omega }\to \mathbb{R}$, ${\tilde{C}}_N\left(p\right)=C_N\left(\lfloor Np\rfloor \right)$, the mathematical equivalence of the “scaled pixel images” in Fig. 3 A and B, with the floor function $\lfloor \cdot \rfloor$ operating coordinate-wise.

Given these definitions, a sequence ${\left\{{\tilde{C}}_N\right\}}_{N>0}$ weak-$*$ converges to $C_{\infty }:\mathrm{\Omega }\to \mathbb{R}$ if for any smooth test function $\psi :\mathrm{\Omega }\to \mathbb{R}$

$$\underset{N\to \infty }{lim}{\int }_{\mathrm{\Omega }}\psi \left(p\right){\tilde{C}}_N\left(p\right)dp={\int }_{\mathrm{\Omega }}\psi \left(p\right)C_{\infty }\left(p\right)dp.$$

By a weak abuse of notation, we drop the tilde in the following and say that a sequence of configurations ${\left\{C_N\right\}}_{N>0}$ weak-$*$ converges to $C_{\infty }$ whenever ${\left\{{\tilde{C}}_N\right\}}_{N>0}$ does.

Conjecture 1.

The sequence ${\left\{I_N\right\}}_{N>0}$ weak-$*$ converges to a piecewise smooth function $I_{\infty }$ on $\mathrm{\Omega }$.

Conjecture 1 states that the sandpile identity itself possesses a scaling limit, denoted by $I_{\infty }$. While being a common assumption in the field, the existence of such a limit has not yet been shown (Introduction). Even though 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 $I_N^H\left(t\right)$ denote the sandpile dynamics (Eq. 3) induced by a given harmonic field $H$ on the domain ${\mathrm{\Gamma }}_N$.

Conjecture 2.

For linear harmonic fields $H$, the sequence ${\left\{I_N^H\left(t\right)\right\}}_{N>0}$ weak-$*$ converges to $I_{\infty }$ for every $t\in \left[0,\infty \right)$.

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. 2A), we thus conjecture that the dynamics induced by them should be constant in the limit.

Conjecture 3.

For quadratic harmonic fields $H$, the sequence ${\left\{I_N^H\left(t\right)\right\}}_{N>0}$ weak-$*$converges to a piecewise smooth function $I_{\infty }^H\left(t\right)$ for every $t\in \left[0,\infty \right)$. Moreover, for every point in $\mathrm{\Omega }$, 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 $t\in \left[0,\infty \right)$, the sequence ${\left\{I_N^H\left(t{N}^{2-o}\right)\right\}}_{N>0}$ weak-$*$converges to a piecewise smooth function $I_{\infty }^H\left(t\right)$. Moreover, for every point in $\mathrm{\Omega }$, 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. 3B), time has to be scaled by a factor of $N^{2-o}$ to achieve scaling limits. Intuitively, the speedup of the harmonic dynamics occurs because the average number $\left|X_N^H\right|/\left|\partial {\mathrm{\Gamma }}_N\right|$ of particles dropped at boundary vertices increases approximately proportional to $N^o$. If we extend the definition of the sandpile identity dynamics (Eq. 3) to real valued harmonic fields, we can alternatively scale the harmonic field $H$ itself by $N^{-o}$ such that its restriction $N^{-o}H{\mid }_{{\mathrm{\Gamma }}_N}$ to the domain converges. Conjecture 4 then becomes equivalent to stating that the sequence ${\left\{I_N^{N^{-o}H}\left(t{N}^2\right)\right\}}_{N>0}$ weak-$*$ converges to a piecewise smooth function for all $t\in \left[0,\infty \right)$. Interestingly, this corresponds to a so-called diffusive scaling limit, a type of limit obtained by many models of physical processes (19). Due to the scaling of the harmonic field, all its lower-order terms have diminishing effects on the potential and thus on the diffusive scaling limit. It directly follows that the conjectured limit is invariant with respect to adding, to its inducing field, a harmonic field of lower order. Furthermore, since discrete polynomial harmonic fields differ from their continuous counterparts only in low-order terms (20), the dynamics are induced by continuous harmonic fields in the diffusive limit.

Conjecture 5.

For every polynomial harmonic field $H$ and every $t\in \left[0,\infty \right)\cap \mathbb{Q}$, the sequence ${\left\{I_N^H\left(t\right)\right\}}_{N>0}$ weak-$*$converges to a piecewise smooth function $I_{\infty }^H\left(t\right)$.

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. 3A). 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. S8B), even more scaling limits might exist.

Fig. 4.

(A) Topology of the usual and extended sandpile groups on a domain consisting of a single vertex: space of nonnegative configurations (Left) and the corresponding sandpile groups $G=\mathbb{Z}/4\mathbb{Z}\subset \tilde{G}=\mathbb{R}/4\mathbb{Z}$ (Right). (B) Space of legal configurations (Left) and the sandpile group (Right) on a domain consisting of two adjacent vertices. The extended sandpile group is a 2D torus obtained by gluing the opposite sides of the rhombus. The lattice points on $\tilde{G}$ form $G$. Moving along horizontal or vertical directions corresponds to increasing the amount of sand carried by the left or right vertex. (C) Harmonic functions $H$ serve as universal coordinates for configurations $\tilde{C}$ of the extended sandpile group and thus define natural renormalization maps between the extended sandpile groups on different domains [here from the two-vertices domain (B) to the one-vertex domain (A)]. A harmonic function $H$ identifying a given configuration $\tilde{C}$ can be determined by expressing it as a linear combination $H={\sum }_i{t}_i{H}^i$ of harmonic basis functions $H^i$, which results in a set of linear equations when inserted into the surjective homomorphism ${\tilde{C}}_{{\mathrm{\Gamma }}_2}=\left(-\mathrm{\Delta }\left(H{\mid }_{\mathrm{\Gamma }}+k\right)+I\right)°$ which can be solved for the coefficients $t_i$. The renormalization of the usual sandpile group corresponds to the floor of the renormalization of the extended one, restricted to integer-valued configurations. (D) Visual depiction of the result of the renormalization described in C. (E) Stochastic realization of the sandpile identity dynamics induced by $H_{ij}^{2a}=ij$. (F) Evolution of the normalized variation of information $\mathcal{V}\mathcal{I}\left(I;I\left(t\right)\right)=1-\frac{\mathcal{I}\left(I,I\left(t\right)\right)}{\mathcal{H}\left(I,I\left(t\right)\right)}$ between the stochastic identity dynamics in D and the sandpile identity over four periods, with $\mathcal{I}$ the mutual information and $\mathcal{H}$ the joint entropy. F, Inset shows the evolution at $t=\mathrm{1,2},\dots$ for 100 periods. (G) Avalanche size distribution over one full period of the stochastic identity dynamics induced by $H^{1a}$ (blue) and $H^{2a}$ (red). The dashed black lines show power-law distributions with critical coefficients −1.371 ($H^{1a}$) and −1.481 ($H^{2a}$), respectively. (H) Encoding of information (here the string “PNAS”) into the stochastic sandpile dynamics induced by $H^{3a}$. The encoded information is visually not detectable in intermediate configurations, while becoming clearly visible at multiples of t = $\frac{1}{3}$. In E and H, white, green, blue, and black pixels represent vertices carrying zero, one, two, and three particles, respectively.

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. 3A. 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 ($\left|G\right|=4$, Fig. 4A) and on a domain consisting of two adjacent vertices ($\left|G\right|=15$, Fig. 4B) 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 $\partial \mathrm{\Gamma }$ of the convex domain $\mathrm{\Gamma }$ is allowed to carry a nonnegative real number of particles, whereas each vertex in the interior of the domain can still carry only a nonnegative integer number of particles. Accordingly, we also allow to drop real numbers of particles at the boundaries but only integer numbers in the interior. The toppling rules for the extended sandpile model are the same as for the original one. The space of recurrent configurations of the extended sandpile model forms a topological abelian group $\tilde{G}$—a Lie group that we denote as the extended sandpile group—of which the original sandpile group $G$ is a discrete subgroup.

To understand the topology of the extended sandpile group, consider that, for each configuration $C$ of the usual sandpile group, $\tilde{G}$ contains all configurations which are equal to $C$ in the interior and differ only in the positions after the decimal point at the boundary of the domain. If we denote by $\mathbb{R}/\mathbb{Z}$ the circle of length 1, this implies that the quotient group $\tilde{G}/G$ is isomorphic to the torus ${\left(\mathbb{R}/\mathbb{Z}\right)}^{\partial \mathrm{\Gamma }}.$ In particular, there is a natural choice of the Riemannian metric on the extended sandpile group coming as a lift of the standard flat metric on ${\left(\mathbb{R}/\mathbb{Z}\right)}^{\partial \mathrm{\Gamma }}$, which allows us to discuss the volume of $\tilde{G}$. Since ${\left(\mathbb{R}/\mathbb{Z}\right)}^{\partial \mathrm{\Gamma }}$ is a torus of volume 1 and $\tilde{G}$ is its extension by $G$, the volume of $\tilde{G}$ is equal to $\left|G\right|$, the number of elements in the usual sandpile group. Furthermore, the extended sandpile group can be viewed to be constructed of cubes ${\left[\mathrm{0,1}\right]}^{\partial \mathrm{\Gamma }}$, one for each element of $G$, which are glued along their boundary faces. From this construction, it directly follows that each connected component of $\tilde{G}$ is a torus. To determine the number of such connected components, recall that $G$ is generated by dropping particles only onto vertices at the domain boundaries (4), which implies that $\tilde{G}$ is the quotient of ${\mathbb{R}}^{\partial \mathrm{\Gamma }}$. Since the continuous image of a connected space is connected, $\tilde{G}$ contains only one connected component, and thus the extended sandpile group is a single torus. We summarize these results in the following proposition.

Proposition 1.

The extended sandpile group $\tilde{G}$ is a torus of dimension $\left|\partial \mathrm{\Gamma }\right|$ and volume $\left|G\right|$.

The relationship between the extended sandpile group $\tilde{G}$ and its discrete subgroup $G$ is as follows: There exists a natural inclusion of $G$ into $\tilde{G}$. In the opposite direction, the floor function maps configurations from $\tilde{G}$ to $G$. The preimage of a recurrent configuration $C\in G$ under the floor function is a cube of volume 1. For example, in the case when $\mathrm{\Gamma }$ consists of just one vertex, all stable configurations are recurrent. Therefore, the usual sandpile group is $\mathbb{Z}/4\mathbb{Z}$ and the extended sandpile group is $\mathbb{R}/4\mathbb{Z}$, i.e., the circle of length 4 (Fig. 4A). If $\mathrm{\Gamma }$ consists of two adjacent vertices, 15 of the 16 stable configurations on $\mathrm{\Gamma }$ are recurrent, with the only nonrecurrent configuration being the configuration $C_{ij}^0=0$. The sandpile group $G$ is cyclic with the sandpile identity given by the configuration $C_{ij}^3=3$. Since $\left|\partial \mathrm{\Gamma }\right|=\left|\mathrm{\Gamma }\right|=2,$ the extended sandpile group $\tilde{G}$ is a 2D torus of area 15. It is isomorphic to ${\mathbb{R}}^{\mathrm{\Gamma }}/\mathrm{\Delta }{\mathbb{Z}}^{\mathrm{\Gamma }}$ (Fig. 4B), where ${\mathbb{R}}^{\mathrm{\Gamma }}/\mathrm{\Delta }{\mathbb{Z}}^{\mathrm{\Gamma }}$ denotes the quotient of ${\mathbb{R}}^{\mathrm{\Gamma }}$ by the image of the Laplacian $\mathrm{\Delta }{\mathbb{Z}}^{\mathrm{\Gamma }}\subset {\mathbb{Z}}^{\mathrm{\Gamma }}$.

Next, we analyze the relationship between the two sandpile groups and harmonic functions. For this, let $\mathcal{H}$ be the space of real-valued discrete harmonic functions on ${\mathbb{Z}}^2$ and denote by ${\mathcal{H}}_{\mathbb{Z}}\subset \mathcal{H}$ the subspace of integer-valued harmonics. We then define the homomorphism ${\eta }_{\mathrm{\Gamma }}:\mathcal{H}\to \tilde{G}$ mapping harmonic fields onto configurations of the extended sandpile group on $\mathrm{\Gamma }$ as follows: For a given harmonic function $H\in \mathcal{H}$, we first determine the minimal number $k\in {\mathbb{Z}}_{\ge 0}$ such that $-\mathrm{\Delta }\left(\tilde{H}{\mid }_{\mathrm{\Gamma }}+k\right)$ becomes nonnegative. We then define the homomorphism by $\eta \left(H\right)=\left(-\mathrm{\Delta }\left(H{\mid }_{\mathrm{\Gamma }}+k\right)+I\right)°$. Given this homomorphism, the sandpile identity dynamics of the usual sandpile group (Eq. 3) can be written as $I^H\left(t\right)=\eta \left(\lfloor tH\rfloor \right)$, and it is then natural to define the corresponding dynamics of the extended sandpile group by ${Ĩ}^H\left(t\right)=\eta \left(tH\right)$. The trajectories of these dynamics correspond to continuous geodesics on $\tilde{G}$ through the identity. If $H\in {\mathcal{H}}_{\mathbb{Z}}$, the geodesic is closed. The projection $I^H=\lfloor {Ĩ}^H\rfloor$ of this geodesic to $G$ via the floor function $\lfloor .\rfloor$ gives the cyclic trajectory of the harmonic sandpile dynamics on $G$. Thus, the usual and the extended sandpiles topple at the same time when induced by the same harmonic field, and their dynamics are indistinguishable in the interior of the domain. Again, recall that $G$ is generated by dropping particles only to boundary vertices (4), which also holds for $\tilde{G}$. Thus, the following lemma is a straightforward consequence of the existence of a solution to the Dirichlet problem on $\mathrm{\Gamma }$ and outside of $\mathrm{\Gamma }$ (21):

Lemma 2.

The homomorphism $\eta$ is surjective.

Now consider an exhausting injective family of domains ${\left\{{\mathrm{\Gamma }}_N\subset {\mathbb{Z}}^2\right\}}_{N\ge 1}$; i.e., ${\mathrm{\Gamma }}_N\subset {\mathrm{\Gamma }}_{N+1}$, where the union of all ${\mathrm{\Gamma }}_N$ is ${\bigcup }_N{\mathrm{\Gamma }}_N={\mathbb{Z}}^2.$ Let ${\tilde{G}}_N$ be the extended sandpile group of ${\mathrm{\Gamma }}_N$.

Theorem 1.

There are canonical surjective homomorphisms ${\tilde{G}}_{N+1}\to {\tilde{G}}_N$, with ${\tilde{G}}_{\infty }≔{proj\; lim}_{N\to \infty }{\tilde{G}}_N$, independent from the specific choice of the family ${\left\{{\mathrm{\Gamma }}_N\right\}}_{N\ge 1}$. There is a natural inclusion $\mathcal{H}/{\mathcal{H}}_{\mathbb{Z}}\to {\tilde{G}}_{\infty }$.

Proof:

Let $K_N$ be the kernel of ${\eta }_N:\mathcal{H}\to {\tilde{G}}_N$. This kernel is equal to the space of harmonic functions taking integer values on ${\mathrm{\Gamma }}_N.$ Therefore, $K_{N+1}\subset K_N$, which give projections between quotients $\mathcal{H}/K_{N+1}\to \mathcal{H}/K_N$. By Lemma 2, these quotients are canonically isomorphic to ${\tilde{G}}_{N+1}$ and ${\tilde{G}}_N$. Finally, we note that the intersection of all $K_N$ is ${\mathcal{H}}_{\mathbb{Z}}.$ $\square$.

Since $\mathcal{H}/{\mathcal{H}}_{\mathbb{Z}}$ projects onto all extended sandpile groups $\tilde{G}$ corresponding to finite domains $\mathrm{\Gamma }$, Theorem 1 provides a set of universal coordinates for the extended sandpile group. Since there exists a natural inclusion of $G$ into $\tilde{G}$, these coordinates, restricted to $G$, are also universal for the original sandpile group. The existence of these universal coordinates directly implies that the original as well as the extended sandpile group admits natural renormalizations. In more detail, for a pair of domains ${\mathrm{\Gamma }}_1\subset {\mathrm{\Gamma }}_2$, there is a canonical renormalization projection from ${\tilde{G}}_2$ to ${\tilde{G}}_1$ (Fig. 4 C and D). To obtain this projection for a given configuration ${\tilde{C}}_2\in {\tilde{G}}_2$, we first determine a harmonic function $H\in \mathcal{H}$ satisfying ${\tilde{C}}_2={\eta }_2\left(H\right)$ and thus serving as a universal coordinate for ${\tilde{C}}_2$ (Fig. 4C). Given $H$, the renormalization projection of ${\tilde{C}}_2$ is given by ${\tilde{C}}_1={\eta }_1\left(H\right)$ (Fig. 4D). Note that, while $\eta$ is not injective, for any $H_a,H_b\in \mathcal{H}$ with ${\eta }_2\left(H_a\right)={\eta }_2\left(H_b\right)$, we have ${\eta }_1\left(H_a\right)={\eta }_1\left(H_b\right)$ since ${\mathrm{\Gamma }}_1\subset {\mathrm{\Gamma }}_2$, and thus the renormalization projection is uniquely defined. By combining this renormalization projection for the extended sandpile group with the inclusion $G_2\to {\tilde{G}}_2$ and the floor function ${\tilde{G}}_1\to G_1$, we get the natural renormalization map $G_2\to G_1$ for the usual sandpile group.

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

Question 1.

Is the inclusion $\mathcal{H}/{\mathcal{H}}_{\mathbb{Z}}\to {\tilde{G}}_{\infty }$ 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 $X^H$ of a given integer harmonic field $H$, provided that this potential is nonnegative. After each particle drop, we can then relax the sandpile and associate the time $t=\frac{k}{\left|X^H\right|}$ to the resulting configuration, with $k=\mathrm{0,1,2},\dots$ the number of particles already dropped. This algorithm corresponds to a Markov process resulting in stochastic sandpile identity dynamics with the same expected time interval between two successive particle drops onto the same boundary vertex as the (fixed) time interval of its deterministic counterpart. Due to its simplicity, it might be possible to implement this Markov process experimentally in a physical system resembling the abelian sandpile.

In Fig. 4D, we depict the stochastic sandpile identity dynamics corresponding to a realization of this Markov process for the harmonic field $H^{2a}$. The stochastic dynamics closely resemble their deterministic counterparts, but are overlaid by noise. This noise is stronger toward the boundaries of the domain where particles are dropped, while in the interior of the domain the stochastic sandpile dynamics remain very close to their deterministic counterparts. After one period, nearly all details of the sandpile identity are reproduced ($t=1$ in Fig. 4D). Only after a significantly longer time does the structure of the identity become significantly disturbed ($t=10$ and $t=100$ in Fig. 4 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. 4G).

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. 4F). 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 $H^{4a}$ (Fig. 2E) 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 $H^{4a}$ on different domains (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 ($H^{2a}$, Fig. 2B) or into a large set of tropical curves ($H^{2b}$, Fig. 2C) suggests that all patches constituting the sandpile identity might be composed of tropical curves. In contrast, the dynamics induced by $H^{4a}$ (Fig. 2C) 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. 4G). 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.