Periodic Lorentz gas with small scatterers

The planar periodic Lorentz gas describes the motion of a billiard particle in a periodic arrangement of convex scatterers. The case of infinite horizon -- when the flight time between consecutive collisions is unbounded -- is a popular model of anomalous diffusion. For fixed scatterer size, Sz\asz and Varj\u proved a limit theorem for the displacement of the particle with a non-standard $\sqrt{n \log n}$ scaling. In my talk I would like to describe the asymptotics of this limit law in a setting when as time $n$ tends to infinity, the scatterer size may also tend to zero simultaneously at a sufficiently slow pace. This is joint work with Henk Bruin and Dalia Terhesiu.