  We prove limit laws for infinite horizon planar periodic Lorentz gases when,
as time $n$ tends to infinity, the scatterer size $\rho$ may also tend to zero
simultaneously at a sufficiently slow pace. In particular we obtain a
non-standard Central Limit Theorem as well as a Local Limit Theorem for the
displacement function. To the best of our knowledge, these are the first
results on an intermediate case between the two well-studied regimes with
superdiffusive $\sqrt{n\log n}$ scaling (i) for fixed infinite horizon
configurations -- letting first $n\to \infty$ and then $\rho \to 0$ -- studied
e.g.~by Sz\'asz \& Varj\'u (2007) and (ii) Boltzmann-Grad type situations --
letting first $\rho \to 0$ and then $n \to \infty$ -- studied by Marklof \&
T\'oth (2016).
