The computational complexity of sandpiles

Given an initial distribution of sand in an Abelian sandpile, what final state does it relax to after all possible avalanches have taken place? In d greater than or equal to 3, we show that this problem is P-complete, so that explicit simulation of the system is almost certainly necessary. We also show that the problem of determining whether a sandpile state is recurrent is P-complete in d greater than or equal to 3, and briefly discuss the problem of constructing the identity. In d = 1, we give two algorithms for predicting the sandpile on a lattice of size n, both faster than explicit simulation: a serial one that runs in time O(n log n), and a parallel one that runs in time O(log(3) n), i.e., the class NC3. The latter is based on a more general problem we call additive ranked generability. This leaves the two-dimensional case as an interesting open problem.