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Sandpiles, (un)decidability, and mod 1 harmonic functions

Lionel Levine, Cornell University
Tuesday, August 28, 2018 - 11:00pm
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 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 that certain avalanches go on forever.

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