Inequalities for critical exponents in $d$-dimensional sandpiles

TL;DR

This paper establishes rigorous bounds and inequalities for critical exponents related to avalanche behavior in the Abelian sandpile model on $ olinebreak Z^d$, including power law bounds in two dimensions.

Contribution

It provides the first rigorous bounds on avalanche critical exponents and demonstrates the existence of infinite volume limits for the last waves in two dimensions.

Findings

Bounds on probability of vertex toppling

Power law upper bounds on avalanche radius distribution in 2D

Existence of infinite volume limits for last avalanche waves in 2D

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.