On the distribution of free path lengths for the periodic Lorentz gas III

For r is an element of (0, 1), let Z(r) = {x is an element of R-2} \ dist(x, Z(2)) > r/2} and define tau(r)(x, v) = inf {t > 0\x + tv is an element of partial derivative Z(r)}. Let Phi(r)(t) be the probability that tau(r)(x, v) greater than or equal to t for x and v uniformly distributed in Z(r) and S-1 respectively. We prove in this paper that [GRAPHICS] [GRAPHICS] as t --> +infinity. This result improves upon the bounds on Phi(r) in Bourgain-Golse-Wennberg [Commun. Math. Phys. 190, 491-508 (1998)]. We also discuss the applications of this result in the context of kinetic theory.