This article is cited in 6 papers

The Statistics of Particle Trajectories in the Homogeneous Sinai Problem for a Two-Dimensional Lattice

V. A. Bykovskii, A. V. Ustinov

Institute for Applied Mathematics, Khabarovsk Division, Far-Eastern Branch of the Russian Academy of Sciences

Abstract: In this paper, we generalize and refine some results by F. P. Boca, R. N. Gologan, and A. Zaharescu on the asymptotic behavior as $h\to 0$ of the statistics of the free path length until the first hit of the $h$-neighborhood (a disk of radius $h$) of a nonzero integer for a particle issuing from the origin. The established facts imply that the limit distribution function for the free path length and for the sighting parameter (the distance from the trajectory to the integer point in question) does not depend on the particle escape direction (the property of isotropy).

Keywords: integer lattice, continued fraction, Kloosterman's sum.

UDC: 511.336+511.9+517.987.5

Received: 24.01.2007

DOI: 10.4213/faa2909

 English version: Functional Analysis and Its Applications, 2008, 42:3, 169–179

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