Apollonian structure in the Abelian sandpile

Levine, Lionel; Pegden, Wesley; Smart, Charles K.

doi:10.1007/s00039-016-0358-7

Cited by 46 publications

(39 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, there exists some "hints" in the literature: it was shown that, while 1-dimensional defects of the background are tropical curves, i.e. governed by piecewise-linear functions, patches correspond to polynomials of order two [14,15]. These results are in agreement with our observation that first order harmonic fields seem to only transform tropical curves, while second order harmonic fields also transform patches.…”

Section: Discussionsupporting

confidence: 91%

“…The identity of this group, the sandpile or Creutz identity -after Michael Creutz who first studied it in depth [10] -shows a remarkably complex self-similar fractal structure composed of patches covered with periodic patterns ("textures", see Figure 1B) which is still not completely understood. For some configurations different to the sandpile identity, scaling limits for infinite domains were shown to exist, and the patches visible in these configurations as well as their robustness was analyzed [13,14,15]. Corresponding results for the sandpile identity -like a closed formula for its construction -are still missing [3, p. 61], even though recently a proof for the scaling limit of the sandpile identity was announced [16].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Harmonic dynamics of the abelian sandpile

Lang

Shkolnikov

2019

Proc. Natl. Acad. Sci. U.S.A.

View full text Add to dashboard Cite

The abelian sandpile serves as a simple model system to study self-organized criticality, a phenomenon occurring in many important biological, physical and social processes. The identity element of the abelian group is a fractal composed of self-similar patches with periodic patterns, and the limit of this identity is subject of extensive collaborative research. Here, we analyze the evolution of the sandpile identity under harmonic fields of different orders. We show that this evolution corresponds to periodic cycles through the abelian group characterized -to a large extend -by the smooth transformation and apparent conservation of the patches constituting the identity. The dynamics induced by second and third order harmonic fields resemble smooth stretchings, respectively translations, of the identity, while the ones induced by fourth order harmonic fields resemble magnifications and rotations. Starting with order three, the dynamics pass through extended regions of seemingly random configurations which spontaneously reassemble into accentuated patterns. We show that the space of harmonic functions projects to the extended analogue of the sandpile group, thus providing a set of universal coordinates identifying configurations between different domains. Since the original sandpile group is a subgroup of the extended one, this directly implies that the sandpile group admits a natural renormalization. Furthermore, we show that the harmonic fields can be induced by simple Markov processes, and that the corresponding stochastic sandpile identity dynamics show remarkable robustness over dozens or hundreds of periods. Finally, we encode information into seemingly random sandpile configurations, and decode this information with an algorithm requiring minimal prior knowledge. Our results suggest that harmonic fields might split the sandpile group into sub-sets showing different critical coefficients, and that it might be possible to extend the fractal structure of the sandpile identity beyond the boundaries of its finite domain.

show abstract

Section: Discussionsupporting

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Harmonic dynamics of the abelian sandpile

Lang

Shkolnikov

2019

Proc. Natl. Acad. Sci. U.S.A.

View full text Add to dashboard Cite

show abstract

“…Without reopening that particular debate, we view the stabilizing/exploding dichotomy as a topic with its own intrinsic mathematical interest. An example of its importance can be seen in the partial differential equation for the scaling limit of the abelian sandpile on Z 2 , which relies on a classification of certain 'quadratic' sandpiles according to whether they are stabilizing or exploding [LPS12].…”

Section: Proof Ideasmentioning

confidence: 99%

The Divisible Sandpile at Critical Density

Levine

Murugan

Peres

et al. 2015

Ann. Henri Poincaré

Self Cite

View full text Add to dashboard Cite

Abstract. The divisible sandpile starts with i.i.d. random variables ("masses") at the vertices of an infinite, vertex-transitive graph, and redistributes mass by a local toppling rule in an attempt to make all masses ≤ 1. The process stabilizes almost surely if m < 1 and it almost surely does not stabilize if m > 1, where m is the mean mass per vertex. The main result of this paper is that in the critical case m = 1, if the initial masses have finite variance, then the process almost surely does not stabilize. To give quantitative estimates on a finite graph, we relate the number of topplings to a discrete bi-Laplacian Gaussian field.

show abstract

“…The shapes of the limiting patches are known in many cases. Exact solutions for some other choices of domain are constructed by Levine and the authors [8]; the key point is that the notion of convergence used in this previous work ignores smallscale structure, and thus does not address the appearance of patterns. The ansatz of Sportiello [14] can be used to adapt these methods to the square with cutoff 3, which yields the continuum limit of the sandpile identity on the square.…”

Section: Introductionmentioning

confidence: 99%