Journal of Statistical Physics

, Volume 129, Issue 1, pp 59–80

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


DOI: 10.1007/s10955-007-9367-0

Cite this article as:
Szász, D. & Varjú, T. J Stat Phys (2007) 129: 59. doi:10.1007/s10955-007-9367-0


As Bleher (J. Stat. Phys. 66(1):315–373, 1992) observed the free flight vector of the planar, infinite horizon, periodic Lorentz process {Snn=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 Bálint and Gouëzel (Commun. Math. Phys. 263:461–512, 2006), helped us to essentially simplify the ideas of our earlier sketchy proof (Szász, D., Varjú, 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.


Lorentz processPeriodic configuration of scatterersInfinite horizonCorridorsNon-normal domain of attraction of the Gaussian lawLocal limit lawRecurrenceErgodicity

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Budapest University of Technology and Economics, Mathematical IntituteBudapestHungary