This talk is a sequel to the one I gave last year, but I will remind you about the original setting and results.

The model is simple: consider a translation invariant measure on arrow configurations on the usual nearest-neighbor lattice in d dimensions. There is an arrow on every point on Z^d, and following arrows produces a semi-infinite trajectory. Last time, I showed you an striking dichotomy theorem: either walks from every pair of points coalesce with probability one, or they form bi-infinite trajectories. We believe that when trajectories are random enough, bi-infinite trajectories do not exist. One such measure of randomness is entropy.

So we consider various entropic properties of these systems. We show that in systems with completely positive entropy, bi-infinite trajectories must carry entropy. In some specialized situations, we show that positive entropy guarantees that bi-infinite trajectories do not exist. Many classical models fall into our simple framework: we construct a stationary discrete-time symmetric exclusion process whose particle trajectories form bi-infinite trajectories carrying entropy.