Mean-field avalanche size exponent for sandpiles on Galton–Watson trees
Antal A. Járai (University of Bath)
Wioletta M. Ruszel (TU Delft - Applied Probability, Universiteit Utrecht)
Ellen Saada (Université Paris Descartes)
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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 Zd, d≥ 3 , and other transient graphs.
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