Sandpile models

Járai, Antal

doi:10.1214/14-ps228

Cited by 27 publications

(23 citation statements)

References 77 publications

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“…Dhar's formula [24] states that the expected number of times u topples when we add a grain of sand at v is given by the Greens function. That is, See [21,38] for detailed discussions of these properties. We now apply these relations to deduce Theorems 1.7-1.9 from the analogous results concerning the WUSF and v-WUSF.…”

Section: Spectral Dimension Anomalous Diffusionmentioning

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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Section: Spectral Dimension Anomalous Diffusionmentioning

confidence: 99%

Universality of high-dimensional spanning forests and sandpiles

Hutchcroft

2019

Probab. Theory Relat. Fields

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“…Our proof of (21) involves the asymptotics of the generating function E z |T π t | as z → 1 (see Section 5) and thus it is analytic in nature. Let us therefore provide a non-rigorous probabilistic explanation of (21). Given an age-critical distribution π t and h ∈ R + , let us define the distribution π t,h by f (x)dπ t,h (x) = f (t + h)dπ t (x).…”

Section: Theorem 219 (Convergence In Probability Of Empirical Age Distribution)mentioning

confidence: 99%

Age evolution in the mean field forest fire model via multitype branching processes

2021

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We study the distribution of ages in the mean field forest fire model introduced by Ráth and Tóth. This model is an evolving random graph whose dynamics combine Erdős-Rényi edge-addition with a Poisson rain of lightning strikes. All edges in a connected component are deleted when any of its vertices is struck by lightning. We consider the asymptotic regime of lightning rates for which the model displays self-organized criticality. The age of a vertex increases at unit rate, but it is reset to zero at each burning time. We show that the empirical age distribution converges as a process to a deterministic solution of an autonomous measure-valued differential equation. The main technique is to observe that, conditioned on the vertex ages, the graph is an inhomogeneous random graph in the sense of Bollobás, Janson and Riordan. We then study the evolution of the ages via the multitype Galton-Watson trees that arise as the limit in law of the component of an identified vertex at any fixed time. These trees are critical from the gelation time onwards.

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“…We first define the model in finite volume and then give its infinite volume construction. Standard references are [14,15,21] and [23].…”

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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Sandpile models

Cited by 27 publications

References 77 publications

Universality of high-dimensional spanning forests and sandpiles

Universality of high-dimensional spanning forests and sandpiles

Age evolution in the mean field forest fire model via multitype branching processes

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

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