Diffusive motion and recurrence on an idealized Galton board.

We study a mechanical model known as a Galton board--a particle rolling on a tilted plane under gravitation and bouncing off a periodic array of rigid pegs. Incidentally, this model is identical to a periodic Lorentz gas where an electron is driven by a uniform electric field. 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 find exact limit distributions for the rescaled velocity t(-1/3)v(t) and position t(-2/3)x(t). In addition, we determine that the particle's motion is recurrent; i.e., the particle comes back to the top of the board with a probability of one.