The statistics of particle trajectories in the homogeneous Sinai problem for a two-dimensional lattice

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 → 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).