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

Antal A. Jarai, Wioletta M. Ruszel, Ellen Saada

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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 Zd, d≥ 3 , and other transient graphs.

Original languageEnglish
Pages (from-to)1-28
Number of pages28
JournalProbability Theory and Related Fields
Publication statusE-pub ahead of print - 2019


  • Abelian sandpile
  • Conductance martingale
  • Uniform spanning tree
  • Wired spanning forest

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