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

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 \({\mathbb {Z}^d}\), \(d\ge 3\), and other transient graphs.