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).

A popular theory of self-organized criticality relates driven dissipative systems to systems with conservation. This theory predicts that the stationary density of the Abelian sandpile model equals the threshold density of the fixed-energy sandpile. We refute this prediction for a wide variety of underlying graphs, including the square grid. Driven dissipative sandpiles continue to evolve even after reaching criticality. This result casts doubt on the validity of using fixed-energy sandpiles to explore the critical behavior of the Abelian sandpile model at stationarity.

We study the sandpile model in infinite volume on $\mathbb{Z}^d$. In particular, we are interested in the question whether or not initial configurations, chosen according to a stationary measure $\mu$, are $\mu$-almost surely stabilizable. We prove that stabilizability does not depend on the particular procedure of stabilization we adopt. In $d=1$ and $\mu$ a product measure with density $\rho=1$ (the known critical value for stabilizability in $d=1$) with a positive density of empty sites, we prove that $\mu$ is not stabilizable. Furthermore, we study, for values of $\rho$ such that $\mu$ is stabilizable, percolation of toppled sites. We find that for $\rho>0$ small enough, there is a subcritical regime where the distribution of a cluster of toppled sites has an exponential tail, as is the case in the subcritical regime for ordinary percolation.Comment: Published in at http://dx.doi.org/10.1214/08-AOP415 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

In this paper we analyze several anisotropic bootstrap percolation models in three dimensions. We present the order of magnitude for the metastability thresholds for a fairly general class of models. In our proofs, we use an adaptation of the technique of dimensional reduction. We find that the order of the metastability threshold is generally determined by the 'easiest growth direction' in the model. In contrast to anisotropic bootstrap percolation in two dimensions, in three dimensions the order of the metastability threshold for anisotropic bootstrap percolation can be equal to that of isotropic bootstrap percolation.

A popular theory of self-organized criticality predicts that the stationary density of the Abelian sandpile model equals the threshold density of the corresponding fixed-energy sandpile. We recently announced that this "density conjecture" is false when the underlying graph is any of Z 2 , the complete graph K n , the Cayley tree, the ladder graph, the bracelet graph, or the flower graph. In this paper, we substantiate this claim by rigorous proof and extensive simulations. We show that driven-dissipative sandpiles continue to evolve even after a constant fraction of the sand has been lost at the sink. Nevertheless, we do find ͑and prove͒ a relationship between the two models: the threshold density of the fixed-energy sandpile is the point at which the drivendissipative sandpile begins to lose a macroscopic amount of sand to the sink.