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 2 d particles then topples by sending one particle to each neighbor. We find that with constant background height h≤2 d2, 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 2 d1. On the other hand, we show that if the background height 2 d2 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).
 Publication:

Journal of Statistical Physics
 Pub Date:
 February 2010
 DOI:
 10.1007/s1095500998996
 arXiv:
 arXiv:0901.3805
 Bibcode:
 2010JSP...138..143F
 Keywords:

 Abelian sandpile;
 Bootstrap percolation;
 Dimensional reduction;
 Discrete Laplacian;
 Growth model;
 Least action principle;
 Mathematics  Combinatorics;
 Condensed Matter  Statistical Mechanics;
 Mathematics  Probability;
 60K35
 EPrint:
 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