Periodic Lorentz gas with small scatterers.

We prove limit laws for infinite horizon planar periodic Lorentz gases when, as time n tends to infinity, the scatterer size ρ 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 n log n scaling (i) for fixed infinite horizon configurations-letting first n → ∞ and then ρ → 0 -studied e.g. by Szász and Varjú (J Stat Phys 129(1):59-80, 2007) and (ii) Boltzmann-Grad type situations-letting first ρ → 0 and then n → ∞ -studied by Marklof and Tóth (Commun Math Phys 347(3):933-981, 2016) .