The Galton board: Limit theorems and recurrence

Chernov, N.¹ ; Dolgopyat, D.²

¹ Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294
² Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742

Journal of the American Mathematical Society, Tome 22 (2009) no. 3, pp. 821-858

Résumé

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).

Détail

DOI : 10.1090/S0894-0347-08-00626-7

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