Some halting problems for abelian sandpiles are undecidable in dimension three

Abstract

The abelian sandpile model is a simple combinatorial model for critical behaviour which has the "abelian property" that the order in which we make moves does not change the final outcome of the game. This might seem to restrict the model's computational ability, but we show that, given three dimensions to work with, the sandpile is able to simulate a Turing machine. We use that to prove the undecidability of three halting problems.