Infinite volume limit of the Abelian sandpile model in dimensions d ≥  3

Járai, Antal; Redig, Frank

doi:10.1007/s00440-007-0083-0

Cited by 25 publications

(43 citation statements)

References 23 publications

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“…From the definition of topological entropy we obtain that 13) which completes the proof of the theorem. L the equidistributed probability measure on the set S…”

Section: Surjectivity Of the Maps ξmentioning

confidence: 56%

“…1 In [10], Dhar showed that the topological entropy of the shift-action σ R ∞ on R ∞ is also given by (3.4), which implies that every shift-invariant measure µ of maximal entropy on R ∞ has entropy (1.1). Shift-invariant measures on R ∞ were studied in some detail by Athreya and Jarai in [1,2], Jarai and Redig in [13]; however, the question of uniqueness of the measure of maximal entropy is still unresolved. Spanning trees of finite graphs are classical objects in combinatorics and graph theory.…”

Section: Introductionmentioning

confidence: 99%

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Abelian Sandpiles and the Harmonic Model

Schmidt

Verbitskiy

2009

Commun. Math. Phys.

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“…From the definition of topological entropy we obtain that 13) which completes the proof of the theorem. L the equidistributed probability measure on the set S…”

Section: Surjectivity Of the Maps ξmentioning

confidence: 56%

Section: Introductionmentioning

confidence: 99%

Abelian Sandpiles and the Harmonic Model

Schmidt

Verbitskiy

2009

Commun. Math. Phys.

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“…The following immediate corollary will be useful in controlling avalanches. A version of part (i) was proved in [15,Lemma 7.5].…”

Section: Intermediate Configurationsmentioning

confidence: 97%

“…All our results have analogues in large finite V (L), or indeed are derived therefrom. Passing to the limit of Z d is not too difficult when d ≥ 3, due to the result of Járai and Redig [15,Theorem 3.11] showing that ν(|S| < ∞) = 1. When d = 2, this is not known.…”

Section: Introductionmentioning

confidence: 99%

Inequalities for critical exponents in $d$-dimensional sandpiles

Bhupatiraju¹,

Hanson²,

Járai³

2017

Electron. J. Probab.

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Consider the Abelian sandpile measure on Z d , d ≥ 2, obtained as the L → ∞ limit of the stationary distribution of the sandpile on [−L, L] d ∩ Z d . When adding a grain of sand at the origin, some region, called the avalanche cluster, topples during stabilization. We prove bounds on the behaviour of various avalanche characteristics: the probability that a given vertex topples, the radius of the toppled region, and the number of vertices toppled. Our results yield rigorous inequalities for the relevant critical exponents. In d = 2, we show that for any 1 ≤ k < ∞, the last k waves of the avalanche have an infinite volume limit, satisfying a power law upper bound on the tail of the radius distribution.

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“…Járai and Redig [39] showed that the burning bijection allows one to relate avalanches to the past of the WUSF, which allowed them to prove that avalanches in Z d satisfy P(v ∈ AvC 0 (H)) v −d+2 for d ≥ 5. (The fact that the expected number of times v topples scales this way is an immediate consequence of Dhar's formula, see [38, Section 3.3.1].)…”

Section: Applications To the Abelian Sandpile Modelmentioning

confidence: 99%

Universality of high-dimensional spanning forests and sandpiles

Hutchcroft

2019

Probab. Theory Relat. Fields

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We prove that the wired uniform spanning forest exhibits mean-field behaviour on a very large class of graphs, including every transitive graph of at least quintic volume growth and every bounded degree nonamenable graph. Several of our results are new even in the case of Z d , d ≥ 5. In particular, we prove that every tree in the forest has spectral dimension 4/3 and walk dimension 3 almost surely, and that the critical exponents governing the intrinsic diameter and volume of the past of a vertex in the forest are 1 and 1/2 respectively. (The past of a vertex in the uniform spanning forest is the union of the vertex and the finite components that are disconnected from infinity when that vertex is deleted from the forest.) We obtain as a corollary that the critical exponent governing the extrinsic diameter of the past is 2 on any transitive graph of at least five dimensional polynomial growth, and is 1 on any bounded degree nonamenable graph. We deduce that the critical exponents describing the diameter and total number of topplings in an avalanche in the Abelian sandpile model are 2 and 1/2 respectively for any transitive graph with polynomial growth of dimension at least five, and are 1 and 1/2 respectively for any bounded degree nonamenable graph.In the case of Z d , d ≥ 5, some of our results regarding critical exponents recover earlier results of Bhupatiraju, Hanson, and Járai (Electron. J. Probab. Volume 22 (2017), no. 85 ). In this case, we improve upon their results by showing that the tail probabilities in question are described by the appropriate power laws to within constant-order multiplicative errors, rather than the polylogarithmic-order multiplicative errors present in that work. * Statslab, DPMMS, University of Cambridge

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Infinite volume limit of the Abelian sandpile model in dimensions d ≥ 3

Cited by 25 publications

References 23 publications

Abelian Sandpiles and the Harmonic Model

Abelian Sandpiles and the Harmonic Model

Inequalities for critical exponents in $d$-dimensional sandpiles

Universality of high-dimensional spanning forests and sandpiles

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