The Galton board: Limit theorems and recurrence

We study a particle moving in $\mathbb{R}^2$ under a constant (external) force and bouncing off a periodic array of convex domains (scatterers); the latter must satisfy a standard `finite horizon' condition to prevent `ballistic' (collision-free) motion. This model is known to physicists as the Galton board (it is also identical to a periodic Lorentz gas). Previous heuristic and experimental studies have suggested that the particle's speed $v(t)$ should grow as $t^{1/3}$ and its coordinate $x(t)$ as $t^{2/3}$. We prove these conjectures rigorously; we also find limit distributions for the rescaled velocity $t^{-1/3} v(t)$ and position $t^{-2/3} x(t)$. In addition, quite unexpectedly, we discover that the particle's motion is recurrent. That means that a ball dropped on an idealized Galton board will roll down, but from time to time it should bounce all the way back up (with probability one).