Rare events and breakdown of simple scaling in the Abelian sandpile model

Menech, Mario De; Stella, Attilio L.; Tebaldi, Claudio

doi:10.1103/physreve.58.r2677

Cited by 114 publications

(140 citation statements)

References 21 publications

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“…Recently, multifractal scaling was observed in the avalanche size distribution of the BTW model [29,56,57]. This indicates that a finite size scaling analysis of the form of Eq.…”

Section: Discussionmentioning

confidence: 99%

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Sandpile models and random walkers on finite lattices

Shilo

Biham

2003

Phys. Rev. E

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Abelian sandpile models, both deterministic, such as the Bak, Tang, Wiesenfeld (BTW) model [P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987) L. The first few moments of this distribution are evaluated numerically and their dependence on the system size is examined. The sandpile models are conservative in the sense that grains are conserved in the bulk and can leave the system only through the boundaries. It is shown that the conservation law provides an interesting connection between sandpile models and random walk models. Using this connection, it is shown that the average avalanche sizes, n L , for the BTW and the Manna models are equal to each other, and both are equal to the average path-length of a random walker starting from a random initial site on the same lattice of size L. This is in spite of the fact that sandpile models with deterministic (BTW) and stochastic (Manna) toppling rules exhibit different critical exponents, indicating that they belong to different universality classes.

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“…Recently, multifractal scaling was observed in the avalanche size distribution of the BTW model [29,56,57]. This indicates that a finite size scaling analysis of the form of Eq.…”

Section: Discussionmentioning

confidence: 99%

“…This indicates that a finite size scaling analysis of the form of Eq. (24) is not sufficient for describing the scaling behavior of the BTW model, although it was found to apply in the case of the Manna model [29,56,57].…”

Section: Discussionmentioning

confidence: 99%

Sandpile models and random walkers on finite lattices

Shilo

Biham

2003

Phys. Rev. E

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“…In a recent paper De Menech et al [22] have argued that avalanches that reach the boundary scale differently than those that don't. This leads to a breakdown of the simple finite-size scaling theory usually assumed in deriving scaling relations, and in data analysis.…”

Section: Recent Developmentsmentioning

confidence: 99%

The Abelian sandpile and related models

Dhar

1999

Physica A: Statistical Mechanics and its Applications

296

291

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The Abelian sandpile model is the simplest analytically tractable model of self-organized criticality. This paper presents a brief review of known results about the model. The abelian group structure of the algebra of operators allows an exact calculation of many of its properties. In particular, when there is a preferred direction, one can calculate all the critical exponents characterizing the distribution of avalanche-sizes in all dimensions. For the undirected case, the model is related to q → 0 state Potts model. This enables exact calculation of some exponents in two dimensions, and there are some conjectures about others. We also discuss a generalization of the model to a network of communicating reactive processors. This includes sandpile models with stochastic toppling rules as a special case. We also consider a non-abelian stochastic variant, which lies in a different universality class, related to directed percolation.

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“…A powerful method to test these scaling laws, extracting τ and D, is provided by the moment analysis [39]. We compute the qth moment of the distribution M q ≡ s q and plot it as a function of N .…”

Section: Avalanchesmentioning

confidence: 99%

Crack roughness and avalanche precursors in the random fuse model

2005

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We analyze the scaling of the crack roughness and of avalanche precursors in the two dimensional random fuse model by numerical simulations, employing large system sizes and extensive sample averaging. We find that the crack roughness exhibits anomalous scaling, as recently observed in experiments. The roughness exponents (ζ, ζ loc ) and the global width distributions are found to be universal with respect to the lattice geometry. Failure is preceded by avalanche precursors whose distribution follows a power law up to a cutoff size. While the characteristic avalanche size scales as s0 ∼ L D , with a universal fractal dimension D, the distribution exponent τ differs slightly for triangular and diamond lattices and, in both cases, it is larger than the mean-field (fiber bundle) value τ = 5/2.

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Rare events and breakdown of simple scaling in the Abelian sandpile model

Cited by 114 publications

References 21 publications

Sandpile models and random walkers on finite lattices

Sandpile models and random walkers on finite lattices

The Abelian sandpile and related models

Crack roughness and avalanche precursors in the random fuse model

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