The problem of recurrence for Lorentz processes (Q1062691)

We prove that, for the planar Lorentz process with a periodic configuration of scatterers, the quasi-local CLT of the Gaussian \(\{\log^{\rho}n\}\) type holds for any \(\rho >1\). Consequently, for arbitrary \(\rho >3/2\), the probabilities that, at the moment of the nth reflection, this process lies in a square of size \(\log^{\rho}n\) are asymptotically Gaussian. This implies that these events occur for infinitely many values of n (i.e. a weaker form of recurrence).