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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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The Galton board: Limit theorems and recurrence
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by N. Chernov and D. Dolgopyat PDF
J. Amer. Math. Soc. 22 (2009), 821-858


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
  • MR Author ID: 357840
  • Received by editor(s): December 12, 2007
  • Published electronically: October 21, 2008
  • © Copyright 2008 by the authors
  • Journal: J. Amer. Math. Soc. 22 (2009), 821-858
  • MSC (2000): Primary 37D50
  • DOI:
  • MathSciNet review: 2505302