Limit Laws and Recurrence for the Planar Lorentz Process with Infinite Horizon

Abstract

As Bleher (J. Stat. Phys. 66(1):315–373, 1992) observed the free flight vector of the planar, infinite horizon, periodic Lorentz process {S n ∣n=0,1,2,…} belongs to the non-standard domain of attraction of the Gaussian law—actually with the $\sqrt{n\log n}$ scaling. Our first aim is to establish his conjecture that, indeed, $\frac{S_{n}}{\sqrt{n\log n}}$ converges in distribution to the Gaussian law (a Global Limit Theorem). Here the recent method of Balint and Gouezel (Commun. Math. Phys. 263:461–512, 2006), helped us to essentially simplify the ideas of our earlier sketchy proof (Szasz, D., Varju, T. in Modern dynamical systems and applications, pp. 433–445, 2004). Moreover, we can also derive (a) the local version of the Global Limit Theorem, (b) the recurrence of the planar, infinite horizon, periodic Lorentz process, and finally (c) the ergodicity of its infinite invariant measure.