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Journal of Statistical Physics

, Volume 138, Issue 1–3, pp 143–159 | Cite as

Growth Rates and Explosions in Sandpiles

  • Anne Fey
  • Lionel Levine
  • Yuval Peres
Open Access
Article

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).

Abelian sandpile Bootstrap percolation Dimensional reduction Discrete Laplacian Growth model Least action principle 

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Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Delft Institute of Applied MathematicsDelft University of TechnologyDelftThe Netherlands
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  3. 3.Theory GroupMicrosoft ResearchRedmondUSA

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