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

Antal Jarai; Wioletta Magdalena Ruszel; Ellen Saada

doi:10.1007/s00440-019-00951-z

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

Antal Jarai, Wioletta Magdalena Ruszel, Ellen Saada

Research output: Contribution to journal › Article › peer-review

Abstract

We show that in Abelian sandpiles on infinite Galton–Watson trees, the probability that the total avalanche has more than t topplings decays as t ^{- 1 / 2}. We prove both quenched and annealed bounds, under suitable moment conditions. Our proofs are based on an analysis of the conductance martingale of Morris (Probab Theory Relat Fields 125:259–265, 2003), that was previously used by Lyons et al. (Electron J Probab 13(58):1702–1725, 2008) to study uniform spanning forests on Z ^d, d≥ 3 , and other transient graphs.

Original language	English
Pages (from-to)	369-396
Number of pages	28
Journal	Probability Theory and Related Fields
Volume	177
Early online date	3 Nov 2019
DOIs	https://doi.org/10.1007/s00440-019-00951-z
Publication status	Published - 1 Jun 2020

Keywords

Abelian sandpile
uniform spanning tree
conductance martingale
Wired spanning forest
Conductance martingale
Uniform spanning tree

ASJC Scopus subject areas

Analysis
Statistics and Probability
Statistics, Probability and Uncertainty

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10.1007/s00440-019-00951-zLicence: CC BY

https://arxiv.org/abs/1807.01809

Cite this

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title = "Mean-field avalanche size exponent for sandpiles on Galton-Watson trees",

abstract = "We show that in Abelian sandpiles on infinite Galton–Watson trees, the probability that the total avalanche has more than t topplings decays as t - 1 / 2. We prove both quenched and annealed bounds, under suitable moment conditions. Our proofs are based on an analysis of the conductance martingale of Morris (Probab Theory Relat Fields 125:259–265, 2003), that was previously used by Lyons et al. (Electron J Probab 13(58):1702–1725, 2008) to study uniform spanning forests on Z d, d≥ 3 , and other transient graphs. ",

keywords = "Abelian sandpile, uniform spanning tree, conductance martingale, Wired spanning forest, Conductance martingale, Uniform spanning tree",

author = "Antal Jarai and Ruszel, {Wioletta Magdalena} and Ellen Saada",

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doi = "10.1007/s00440-019-00951-z",

language = "English",

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journal = "Probability Theory and Related Fields",

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T1 - Mean-field avalanche size exponent for sandpiles on Galton-Watson trees

AU - Jarai, Antal

AU - Ruszel, Wioletta Magdalena

AU - Saada, Ellen

PY - 2020/6/1

Y1 - 2020/6/1

N2 - We show that in Abelian sandpiles on infinite Galton–Watson trees, the probability that the total avalanche has more than t topplings decays as t - 1 / 2. We prove both quenched and annealed bounds, under suitable moment conditions. Our proofs are based on an analysis of the conductance martingale of Morris (Probab Theory Relat Fields 125:259–265, 2003), that was previously used by Lyons et al. (Electron J Probab 13(58):1702–1725, 2008) to study uniform spanning forests on Z d, d≥ 3 , and other transient graphs.

AB - We show that in Abelian sandpiles on infinite Galton–Watson trees, the probability that the total avalanche has more than t topplings decays as t - 1 / 2. We prove both quenched and annealed bounds, under suitable moment conditions. Our proofs are based on an analysis of the conductance martingale of Morris (Probab Theory Relat Fields 125:259–265, 2003), that was previously used by Lyons et al. (Electron J Probab 13(58):1702–1725, 2008) to study uniform spanning forests on Z d, d≥ 3 , and other transient graphs.

KW - Abelian sandpile

KW - uniform spanning tree

KW - conductance martingale

KW - Wired spanning forest

KW - Conductance martingale

KW - Uniform spanning tree

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U2 - 10.1007/s00440-019-00951-z

DO - 10.1007/s00440-019-00951-z

M3 - Article

SN - 0178-8051

VL - 177

SP - 369

EP - 396

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

ER -

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

Abstract

Keywords

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Antal Jarai

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

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

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Antal Jarai

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