  The dynamics of a point particle in a periodic array of spherical scatterers
converges, in the limit of small scatterer size, to a random flight process,
whose paths are piecewise linear curves generated by a Markov process with
memory two. The corresponding transport equation is distinctly different from
the linear Boltzmann equation observed in the case of a random configuration of
scatterers. In the present paper we provide asymptotic estimates for the
transition probabilities of this Markov process. Our results in particular
sharpen previous upper and lower bounds on the distribution of free path
lengths obtained by Bourgain, Golse and Wennberg.
