Sandpile probabilities on triangular and hexagonal lattices

Poncelet, Adrien; Ruelle, Philippe

doi:10.1088/1751-8121/aa9255

Cited by 4 publications

(3 citation statements)

References 39 publications

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“…The results have been extended to higher heights on the honeycomb lattice, and to all heights on the triangular lattice [PR18]. Interestingly, these two regular lattices have coordination numbers different from the square lattice, with the consequence that the height variables take in each case a different number of values : four for the square lattice, three for the honeycomb lattice and six for the triangular lattice.…”

Section: Aspects Of Universalitymentioning

confidence: 94%

Sandpile models in the large

Ruelle

2021

Preprint

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This contribution is a review of the deep and powerful connection between the large scale properties of critical systems and their description in terms of a field theory. Although largely applicable to many other models, the details of this connection are illustrated in the class of two-dimensional Abelian sandpile models. Bulk and boundary height variables, spanning tree related observables, boundary conditions and dissipation are all discussed in this context and found to have a proper match in the field theoretic description.

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Section: Aspects Of Universalitymentioning

confidence: 94%

Sandpile models in the large

Ruelle

2021

Preprint

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“…The results have been extended to higher heights on the honeycomb lattice and to all heights on the triangular lattice [61]. Interestingly, these two regular lattices have coordination numbers different from the square lattice, with the consequence that the height variables take in each case a different number of values: four for the square lattice, three for the honeycomb lattice and six for the triangular lattice.…”

Section: Aspects Of Universalitymentioning

confidence: 94%

Sandpile Models in the Large

Ruelle

2021

Front. Phys.

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This contribution is a review of the deep and powerful connection between the large-scale properties of critical systems and their description in terms of a field theory. Although largely applicable to many other models, the details of this connection are illustrated in the class of two-dimensional Abelian sandpile models. Bulk and boundary height variables, spanning tree–related observables, boundary conditions, and dissipation are all discussed in this context and found to have a proper match in the field theoretic description.

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“…§ 3. The order of growth of the domain occupied by sand: a lower bound Lemma 3.1 (see [34]- [36]). There is a discrete Green's function (a function with the properties ∆g(x) = δ 0 and g(0, 0) = 0) for the triangular lattice g : △ → R that has the following asymptotic expansion for some constant C 0 :…”

Section: Introduction and Notationmentioning

confidence: 99%

Convergence of a sandpile on a triangular lattice under rescaling

Aliev,

Kalinin

2023

Sb. Math.

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We present a survey of results on convergence in sandpile models. For a sandpile model on a triangular lattice we prove results similar to the ones known for a square lattice. Namely, consider the sandpile model on the integer points of the plane and put $n$ grains of sand at the origin. Let us begin the process of relaxation: if the number of grains of sand at some vertex $z$ is not less than its valency (in this case we say that the vertex $z$ is unstable), then we move a grain of sand from $z$ to each adjacent vertex, and then repeat this operation as long as there are unstable vertices. We prove that the support of the state $(n\delta_0)^\circ$ in which the process stabilizes grows at a rate of $\sqrt n$ and, after rescaling with coefficient $\sqrt n$, $(n\delta_0)^\circ$ has a limit in the weak-$^*$ topology. This result was established by Pegden and Smart for the square lattice (where every vertex is connected with four nearest neighbours); we extend it to a triangular lattice (where every vertex is connected with six neighbours). Bibliography: 39 titles.

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Sandpile probabilities on triangular and hexagonal lattices

Abstract: We consider the Abelian sandpile model on triangular and hexagonal lattices. We compute several height probabilities on the full plane and on half-planes, and discuss some properties of the universality of the model.

Cited by 4 publications

References 39 publications

Sandpile models in the large

Sandpile models in the large

Sandpile Models in the Large

Convergence of a sandpile on a triangular lattice under rescaling

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