Asymptotic height distribution in high-dimensional sandpiles

TL;DR

This paper derives an asymptotic formula for the distribution of heights in high-dimensional Abelian sandpiles, revealing how the distribution approaches a Poisson(1) model as the dimension increases, with explicit error bounds.

Contribution

It provides the first asymptotic analysis of the height distribution in high-dimensional sandpiles, connecting it to Poisson probabilities and including precise error estimates.

Findings

Height distribution converges to Poisson(1) as dimension increases

Explicit error bounds for the asymptotic approximation

Asymptotic formula applicable for large dimensions

Abstract

We give an asymptotic formula for the single site height distribution of Abelian sandpiles on $\mathbb{Z}^d$ as $d \to \infty$, in terms of $\mathsf{Poisson}(1)$ probabilities. We provide error estimates.