Lionel Levine, Cornell University

PDL C-401

In the abelian sandpile model on the d-dimensional cubic lattice Z^d, each site that has at least 2d particles gives one particle to each of its 2d nearest neighbors. An "avalanche" is what happens when you iterate this move. In https://arxiv.org/abs/1508.00161 Hannah Cairns proved that for d=3, questions of the form "Will this avalanche ever stop?" are algorithmically undecidable: they can be as hard as the halting problem! This infinite unclimbable peak is surrounded by appealing finite peaks: What about d=2? What if the initial configuration of sand is random? I’ll tell you about the “mod 1 harmonic functions” Bob Hough and Daniel Jerison and I used to prove in https://arxiv.org/abs/1703.00827 that certain avalanches go on forever.