The Galton board: Limit theorems and recurrence

Abstract

We study a particle moving in R2 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 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 t1/3 and its coordinate x(t) as t2/3. We prove these conjectures rigorously; we also find limit distributions for the rescaled velocity t−1/3v(t) and position t−2/3x(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).