We show that in Abelian sandpiles on infinite Galton–Watson trees, the probability that the total avalanche has more than t topplings decays as $$t^{-1/2}$$ 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 $${\mathbb {Z}^d}$$ Z d , $$d\ge 3$$ d ≥ 3 , and other transient graphs.

中文翻译：

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Galton-Watson 树上沙堆的平均场雪崩大小指数

我们表明，在无限高尔顿-沃森树上的阿贝尔沙堆中，总雪崩超过 t 次倾覆的概率衰减为 $$t^{-1/2}$$ t - 1 / 2 。我们在合适的力矩条件下证明了淬火和退火边界。我们的证明基于对 Morris 电导鞅的分析（Probab Theory Relat Fields 125:259–265, 2003），该分析之前由 Lyons 等人使用。(Electron J Probab 13(58):1702–1725, 2008) 在 $${\mathbb {Z}^d}$$Z d , $$d\ge 3$$ d ≥ 3 上研究均匀跨越森林其他瞬态图。