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

Consider the domain Z(epsilon) = {x is an element of R-n/dist(x, epsilon Z(n)) > epsilon(gamma)}, and let the free path length be defined as tau(epsilon)(x, omega) = inf{t > 0 \x - t omega is an element of Z(epsilon)}. The distribution of values of tau(epsilon) is studied in the limit as epsilon --> 0 for all gamma greater than or equal to 1. It is shown that the value gamma(c) = n/n-1 is critical for this problem: in other words, the limiting behavior of tau(epsilon) depends only on whether gamma is larger or smaller than gamma(c).