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 $t^{-1\mathrm{/}2}$. We prove both quenched and annealed bounds, under suitable moment conditions. Our proofs are based on an analysis of the conductance martingale of Morris (2003), that was previously used by Lyons, Morris and Schramm (2008) to study uniform spanning forests on ${\mathbb{Z}}^d$, $d\ge 3$, and other transient graphs.