  The Lorentz gas is a billiard model involving a point particle diffusing
deterministically in a periodic array of convex scatterers. In the two
dimensional finite horizon case, in which all trajectories involve collisions
with the scatterers, displacements scaled by the usual diffusive factor
$\sqrt{t}$ are normally distributed, as shown by Bunimovich and Sinai in 1981.
In the infinite horizon case, motion is superdiffusive, however the normal
distribution is recovered when scaling by $\sqrt{t\ln t}$, with an explicit
formula for its variance. Here we explore the infinite horizon case in
arbitrary dimensions, giving explicit formulas for the mean square
displacement, arguing that it differs from the variance of the limiting
distribution, making connections with the Riemann Hypothesis in the small
scatterer limit, and providing evidence for a critical dimension $d=6$ beyond
which correlation decay exhibits fractional powers. The results are conditional
on a number of conjectures, and are corroborated by numerical simulations in up
to ten dimensions.
