Lorentz gas with small scatterers

Abstract

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).