We study the abelian sandpile growth model, where n particles are added at the origin on a stable background configuration in ℤd. 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 n1/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).
Abelian sandpile Bootstrap percolation Dimensional reduction Discrete Laplacian Growth model Least action principle