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

TL;DR

This paper establishes that in abelian sandpiles on infinite Galton-Watson trees, the avalanche size distribution follows a power law decay with an exponent of 1/2, under certain conditions, extending previous conductance martingale methods.

Contribution

It proves the mean-field avalanche size exponent of 1/2 for sandpiles on Galton-Watson trees using conductance martingale analysis, both quenched and annealed.

Findings

Avalanche size probability decays as t^{-1/2}.

Results hold under suitable moment conditions.

Method extends conductance martingale techniques to 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 $\mathbb{Z}^d$, $d\geq 3$, and other transient graphs.