SOME HALTING PROBLEMS FOR ABELIAN SANDPILES ARE UNDECIDABLE IN DIMENSION THREE

The abelian sandpile model is a simple combinatorial model for critical behavior: a chip game on a graph with the abelian property that the order in which we make moves does not change the final outcome of the game. We prove the undecidability of three halting problems for the sandpile on the three-dimensional lattice. It is unknown whether or not the problems are decidable on a two-dimensional lattice or on periodic planar graphs.