Mean-field avalanche size exponent for sandpiles on Galton–Watson trees

Járai, Antal; Ruszel, Wioletta M.; Saada, Ellen

doi:10.1007/s00440-019-00951-z

Cited by 2 publications

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References 31 publications

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“…As a consequence, because not all vertices have maximal degree with positive probability, there is a positive density of dissipative sites outside the boundary, which intuitively should lead to a non-critical model. Note that this set-up, similar to what was done in [24], is in contrast with the set-up of [18], where the random toppling matrix depends on the realization of the tree in such a way that the only dissipative sites are the boundary sites, so that the model there is critical.…”

Section: The Sandpile Model On the Random Binomial Treementioning

confidence: 99%

Non-criticality criteria for Abelian sandpile models with sources and sinks

Redig

Ruszel

Saada

2018

Journal of Mathematical Physics

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We prove that the Abelian sandpile model on a random binary and binomial tree, as introduced in Redig, Ruszel, and Saada [J. Stat. Phys. 147, 653-677 (2012)], is not critical for all branching probabilities p < 1; by estimating the tail of the annealed survival time of a random walk on the binary tree with randomly placed traps, we obtain some more information about the exponential tail of the avalanche radius. Next we study the sandpile model on Z d with some additional dissipative sites: we provide examples and sufficient conditions for non-criticality; we also make a connection with the parabolic Anderson model. Finally we initiate the study of the sandpile model with both sources and sinks and give a sufficient condition for non-criticality in the presence of a finite number of sources, using a connection with the homogeneous pinning model. Published by AIP Publishing. https://doi

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Section: The Sandpile Model On the Random Binomial Treementioning

confidence: 99%

Non-criticality criteria for Abelian sandpile models with sources and sinks

Redig

Ruszel

Saada

2018

Journal of Mathematical Physics

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Sandpile models

Járai¹

2018

Probab. Surveys

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This survey is an extended version of lectures given at the Cornell Probability Summer School 2013. The fundamental facts about the Abelian sandpile model on a finite graph and its connections to related models are presented. We discuss exactly computable results via Majumdar and Dhar's method. The main ideas of Priezzhev's computation of the height probabilities in 2D are also presented, including explicit error estimates involved in passing to the limit of the infinite lattice. We also discuss various questions arising on infinite graphs, such as convergence to a sandpile measure, and stabilizability of infinite configurations.

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Mean-field avalanche size exponent for sandpiles on Galton–Watson trees

Cited by 2 publications

References 31 publications

Non-criticality criteria for Abelian sandpile models with sources and sinks

Non-criticality criteria for Abelian sandpile models with sources and sinks

Sandpile models

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