Growth Rates and Explosions in Sandpiles

TL;DR

This paper analyzes the growth and explosion phenomena in the abelian sandpile model, establishing new bounds on the toppled set's diameter and conditions leading to infinite activity.

Contribution

It introduces a strong least action principle and background modification techniques to extend diameter bounds and identify conditions causing explosions in sandpiles.

Findings

Diameter of toppled sites scales as n^{1/d} for certain backgrounds.

Background modifications can lead to infinite, non-stabilizing sandpiles.

Extended bounds apply to backgrounds with high fractions of near-maximum height sites.

Abstract

We study the abelian sandpile growth model, where n particles are added at the origin on a stable background configuration in Z^d. Any site with at least 2d particles then topples by sending one particle to each neighbor. We find that with constant background height h <= 2d-2, the diameter of the set of sites that topple has order n^{1/d}. This was previously known only for h<d. Our proof uses a strong form of the least action principle for sandpiles, and a novel method of background modification. We can extend this diameter bound to certain backgrounds in which an arbitrarily high fraction of sites have height 2d-1. On the other hand, we show that if the background height 2d-2 is augmented by 1 at an arbitrarily small fraction of sites chosen independently at random, then adding finitely many particles creates an explosion (a sandpile that never stabilizes).