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

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 (2003), that was previously used by Lyons, Morris and Schramm (2008) to study uniform spanning forests on Z d , d ≥ 3, and other transient graphs.

Non-criticality criteria for Abelian sandpile models with sources and sinks

“…As a consequence, because not all vertices have maximal degree with positive probability, there is a positive density of dissipative sites outside the boundary, which intuitively should lead to a non-critical model. Note that this set-up, similar to what was done in [24], is in contrast with the set-up of [18], where the random toppling matrix depends on the realization of the tree in such a way that the only dissipative sites are the boundary sites, so that the model there is critical.…”