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I have been looking at random (and periodic) Lorentz gas models recently in preparing a review which has just appeared at arxiv:1402.7010. As far as I know, there are rigorous results only for the low density (Boltzmann-Grad) limit, and for models where the scatterers are known not to overlap (for example placing a scatterer or empty space randomly at each site of a lattice). In the fixed density case there does not even appear to be a proof that "conditional on the scatterers not overlapping" converges. Choosing a point outside a scatterer at random is problematic since the measure is infinite; better to fix the point at the origin and then choose the scatterers conditional on not overlapping each other or the origin. The low density limit suggests that the velocity autocorrelation decays as $t^{-d/2-1}$, which is fast enough for the diffusion coefficient (its integral) to exist.