Inequalities for critical exponents in $d$-dimensional sandpiles

Bhupatiraju, Sandeep; Hanson, Jack; Járai, Antal

doi:10.1214/17-ejp111

Cited by 16 publications

(22 citation statements)

References 37 publications

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“…In comparison to the intrinsic diameter, our methods to study the extrinsic diameter are more delicate and require stronger assumptions on the graph in order to derive sharp estimates. Our first result on the extrinsic diameter concerns Z d , and improves upon the results of Bhupatiraju, Hanson, and Járai [21] by removing the polylogarithmic errors present in their results. Theorem 1.4 (Mean-field Euclidean extrinsic diameter).…”

Section: Extrinsic Exponentssupporting

confidence: 76%

“…• Also in the case of Z d for d ≥ 5, Bhupatiraju, Hanson, and Járai [21] followed the strategy of an unpublished proof of Lyons, Morris, and Schramm [53] to prove that the probability that the past reaches extrinsic distance n is n −2 log O (1) n and that the probability that the past has volume n is n −1/2 log O(1) n. Our results improve upon theirs in this case by reducing the error from polylogarithmic to constant order. Moreover, their proof relies heavily on transitivity and cannot be used to derive universal results of the kind we prove here.…”

Section: Relation To Other Workmentioning

confidence: 99%

“…(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].) Bhupatiraju, Hanson, and Járai [21] built upon these methods to prove that, when d ≥ 5, the probability that the diameter of the avalanche is at least n scales as n −2 log O(1) n and the probability that the total number of topplings in the avalanche is at least n is between cn −1/2 and n −2/5+o (1) . Using the combinatorial tools that they developed, the following theorem, which improves upon theirs, follows straightforwardly from our results concerning the WUSF.…”

Section: Applications To the Abelian Sandpile Modelmentioning

confidence: 99%

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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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Section: Extrinsic Exponentssupporting

confidence: 76%

Section: Relation To Other Workmentioning

confidence: 99%

Section: Applications To the Abelian Sandpile Modelmentioning

confidence: 99%

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Universality of high-dimensional spanning forests and sandpiles

Hutchcroft

2019

Probab. Theory Relat. Fields

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“…Their result can be stated using discrete derivatives of the Green's function on Z 2 . Let (6) ν :" 1 4`δ p´1,0q`δp1,0q`δp0,´1q`δp0,1qb e the measure that drives simple random walk on Z 2 , and let ν˚n be its n-th convolution power, so that ν˚npxq is the probability that a random walker started from the origin is at site x after n steps. The Green's function is defined by (7) G Z 2 pxq :" 1 4 8 ÿ n"0 rν˚npxq´ν˚np0, 0qs .…”

Section: 2mentioning

confidence: 99%

Sandpiles on the Square Lattice

Hough

Jerison

Levine

2019

Commun. Math. Phys.

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We give a non-trivial upper bound for the critical density when stabilizing i.i.d. distributed sandpiles on the lattice Z 2 . We also determine the asymptotic spectral gap, asymptotic mixing time and prove a cutoff phenomenon for the recurrent state abelian sandpile model on the torus pZ{mZq 2 . The techniques use analysis of the space of functions on Z 2 which are harmonic modulo 1. In the course of our arguments, we characterize the harmonic modulo 1 functions in ℓ p pZ 2 q as linear combinations of certain discrete derivatives of Green's functions, extending a result of Schmidt and Verbitskiy [35].2010 Mathematics Subject Classification. Primary 82C20, 60B15, 60J10. Key words and phrases. Abelian sandpile model, random walk on a group, spectral gap, cutoff phenomenon, critical density, harmonic modulo 1 functions.

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“…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 [11]. Figure 1.…”

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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Inequalities for critical exponents in $d$-dimensional sandpiles

Cited by 16 publications

References 37 publications

Universality of high-dimensional spanning forests and sandpiles

Universality of high-dimensional spanning forests and sandpiles

Sandpiles on the Square Lattice

Harmonic dynamics of the abelian sandpile

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