Growth Rates and Explosions in Sandpiles

We study the abelian sandpile growth model, where n particles are added at the origin on a stable background configuration in acurrency sign (d) . Any site with at least 2d particles then topples by sending one particle to each neighbor. We find that with constant background height ha parts per thousand currency sign2d-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).