Article

Journal of Statistical Physics

, Volume 129, Issue 1, pp 59-80

First online:

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

  • Domokos SzászAffiliated withBudapest University of Technology and Economics, Mathematical Intitute Email author 
  • , Tamás VarjúAffiliated withBudapest University of Technology and Economics, Mathematical Intitute

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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 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.

Keywords

Lorentz process Periodic configuration of scatterers Infinite horizon Corridors Non-normal domain of attraction of the Gaussian law Local limit law Recurrence Ergodicity