Joint Dynamical Systems / Probability Seminar
In this talk I’ll take a dynamical view on certain questions in probability.
Perhaps the simplest question about any dynamical system is whether it converges to a fixed point. A classic example in probability is whether a branching process dies out.
For a spatially infinite system, we can ask about *local* fixation: does every bounded subset eventually stop changing? A classic example is whether a random walk is transient or recurrent.
In the case of sandpile models (I’ll define them precisely in the talk) we can ask whether an avalanche will stop or go on forever. In https://arxiv.org/abs/1508.00161 Hannah Cairns proved for 3-dimensional abelian sandpiles that this question is algorithmically undecidable: it is as hard as the halting problem! But this infinite unclimbable peak is surrounded by appealing finite peaks: What about 2 dimensions? What if the initial configuration of sand is i.i.d.? Deciding local fixation of i.i.d. abelian sandpiles is still open, even on the square grid Z^2. 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 avalanches go on forever if the density of sand is high enough.
What about dimension 1? There the abelian sandpile is too simple (Z has too few harmonic functions). An interesting contrast is the model of "activated random walks", where the critical density for local fixation is universal by a recent result of Rolla, Sidoravicius, and Zindy https://arxiv.org/abs/1707.06081 , but the actual value of the critical density is still unknown even on Z.
This is a joint talk with the Dynamical Systems Seminar.