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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

The Galton board: Limit theorems and recurrence


Authors: N. Chernov and D. Dolgopyat
Journal: J. Amer. Math. Soc. 22 (2009), 821-858
MSC (2000): Primary 37D50
Published electronically: October 21, 2008
MathSciNet review: 2505302
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Abstract: 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).


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Additional Information

N. Chernov
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294

D. Dolgopyat
Affiliation: Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742

DOI: http://dx.doi.org/10.1090/S0894-0347-08-00626-7
PII: S 0894-0347(08)00626-7
Received by editor(s): December 12, 2007
Published electronically: October 21, 2008
Article copyright: © Copyright 2008 by the authors