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 ${\mathbb{Z}}^2$, 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 prove) a relationship between the two models: the threshold density of the fixed-energy sandpile is the point at which the driven-dissipative sandpile begins to lose a macroscopic amount of sand to the sink.

• Received 16 March 2010

DOI:https://doi.org/10.1103/PhysRevE.82.031121