Growth Rates and Explosions in Sandpiles

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).

19 pages, 4 figures, to appear in Journal of Statistical Physics. v2 corrects the proof of the outer bound of Theorem 4.1 of arXiv:0704.0688

References in corpus (2)

• Limiting shapes for deterministic centrally seeded growth models
• Stabilizability and percolation in the infinite volume sandpile model

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• Abelian networks IV. Dynamics of nonhalting networks
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