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 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 root n log n scaling (i) for fixed infinite horizon configurations & mdash;letting first n -> infinity and then rho -> 0 & mdash;studied e.g. by Sz & aacute;sz and Varj & uacute; (J Stat Phys 129(1):59-80, 2007) and (ii) Boltzmann-Grad type situations & mdash; letting first rho -> 0 and then n -> infinity & mdash;studied by Marklof and T & oacute;th (Commun Math Phys 347(3):933-981, 2016) .

Automated summary

We prove limit laws for infinite horizon planar periodic Lorentz gases when both the time n and scatterer size rho tend to infinity and zero simultaneously at a sufficiently slow pace. Our results include a non-standard Central Limit Theorem and a Local Limit Theorem for the displacement function. These results are the first in an intermediate case between two well-studied regimes: (i) fixed infinite horizon configurations with superdiffusive root n log n scaling, and (ii) Boltzmann-Grad type situations.