Inequalities for critical exponents in $d$-dimensional sandpiles

Abstract

Consider the Abelian sandpile measure on $\mathbb{Z} ^d$, $d \ge 2$, obtained as the $L \to \infty $ limit of the stationary distribution of the sandpile on $[-L,L]^d \cap \mathbb{Z} ^d$. When adding a grain of sand at the origin, some region, called the avalanche cluster, topples during stabilization. We prove bounds on the behaviour of various avalanche characteristics: the probability that a given vertex topples, the radius of the toppled region, and the number of vertices toppled. Our results yield rigorous inequalities for the relevant critical exponents. In $d = 2$, we show that for any $1 \le k < \infty $, the last $k$ waves of the avalanche have an infinite volume limit, satisfying a power law upper bound on the tail of the radius distribution.