Convergence of the Abelian sandpile

Pegden, Wesley; Smart, Charles K.

doi:10.1215/00127094-2079677

Cited by 52 publications

(63 citation statements)

References 13 publications

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“…This local rule allows (and in fact forces) huge masses to accumulate on the free boundary of the moving front. In the case of a single source mass, the numerics indicates that shapes generated by this model do not converge to a sphere under a scaling limit, but to a shape somewhat reminiscent of the classical Abelian sandpile (see [2] for definition, and [13] for the scaling limit).…”

Section: Introductionmentioning

confidence: 95%

Perturbed Divisible Sandpiles and Quadrature Surfaces

Aleksanyan

Shahgholian

2018

Potential Anal

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The main purpose of the present paper is to establish a link between quadrature surfaces (potential theoretic concept) and sandpile dynamics (Laplacian growth models). For this aim, we introduce a new model of Laplacian growth on the lattice Z d (d ≥ 2) which continuously deforms occupied regions of the divisible sandpile model of Levine and Peres (J. Anal. Math. 111(1), 151-219 2010), by redistributing the total mass of the system onto 1 msub-level sets of the odometer which is a function counting total emissions of mass from lattice vertices. In free boundary terminology this goes in parallel with singular perturbation, which is known to converge to a Bernoulli type free boundary. We prove that models, generated from a single source, have a scaling limit, if the threshold m is fixed. Moreover, this limit is a ball, and the entire mass of the system is being redistributed onto an annular ring of thickness 1 m . By compactness argument we show that when m tends to infinity sufficiently slowly with respect to the scale of the model, then in this case also there is scaling limit which is a ball, with the mass of the system being uniformly distributed onto the boundary of that ball, and hence we recover a quadrature surface in this case. Depending on the speed of decay of 1/m, the visited set of the sandpile interpolates between spherical and polygonal shapes. Finding a precise characterisation of this shape-transition phenomenon seems to be a considerable challenge, which we cannot address at this moment.

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Section: Introductionmentioning

confidence: 95%

Perturbed Divisible Sandpiles and Quadrature Surfaces

Aleksanyan

Shahgholian

2018

Potential Anal

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“…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%

Harmonic dynamics of the abelian sandpile

Lang

Shkolnikov

2019

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

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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.

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“…Various special cases of the least action principle to particular abelian networks have enabled a flurry of recent progress: bounds on the growth rate of sandpiles [FLP10], exact shape theorems for rotor aggregation [KL10,HS11], proof of a phase transition for activated random walkers [RS12], and a fast simulation algorithm for growth models [FL13]. The least action principle was also the starting point for the recent breakthrough by Pegden and Smart [PS13] showing existence of the abelian sandpile scaling limit.…”

Section: Least Action Principlementioning

confidence: 99%

Abelian Networks I. Foundations and Examples

Bond¹,

Levine²

2016

SIAM J. Discrete Math.

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Abstract. In Deepak Dhar's model of abelian distributed processors, automata occupy the vertices of a graph and communicate via the edges. We show that two simple axioms ensure that the final output does not depend on the order in which the automata process their inputs. A collection of automata obeying these axioms is called an abelian network. We prove a least action principle for abelian networks. As an application, we show how abelian networks can solve certain linear and nonlinear integer programs asynchronously. In most previously studied abelian networks, the input alphabet of each automaton consists of a single letter; in contrast, we propose two non-unary examples of abelian networks: oil and water and abelian mobile agents.

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Convergence of the Abelian sandpile

Cited by 52 publications

References 13 publications

Perturbed Divisible Sandpiles and Quadrature Surfaces

Perturbed Divisible Sandpiles and Quadrature Surfaces

Harmonic dynamics of the abelian sandpile

Abelian Networks I. Foundations and Examples

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