Communications in Mathematical Physics

, Volume 331, Issue 1, pp 111–137

Tail Asymptotics of Free Path Lengths for the Periodic Lorentz Process: On Dettmann’s Geometric Conjectures

Article

DOI: 10.1007/s00220-014-2086-x

Cite this article as:
Nándori, P., Szász, D. & Varjú, T. Commun. Math. Phys. (2014) 331: 111. doi:10.1007/s00220-014-2086-x

Abstract

In the simplest case, consider a \({\mathbb{Z}^d}\)-periodic (d ≥ 3) arrangement of balls of radii < 1/2, and select a random direction and point (outside the balls). According to Dettmann’s first conjecture, the probability that the so determined free flight (until the first hitting of a ball) is larger than t >  > 1 is \({\sim\frac{C}{t}}\), where C is explicitly given by the geometry of the model. In its simplest form, Dettmann’s second conjecture is related to the previous case with tangent balls (of radii 1/2). The conjectures are established in a more general setup: for \({\mathcal{L}}\)-periodic configuration of—possibly intersecting—convex bodies with \({\mathcal{L}}\) being a non-degenerate lattice. These questions are related to Pólya’s visibility problem (Arch Math Phys Ser 2:135–142, 1918), to theories of Bourgain et al. (Commun Math Phys 190:491–508,1998), and of Marklof–Strömbergsson (Ann Math 172:1949–2033,2010). The results also provide the asymptotic covariance of the periodic Lorentz process assuming it has a limit in the super-diffusive scaling, a fact if d = 2 and the horizon is infinite.

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Péter Nándori
    • 1
    • 2
  • Domokos Szász
    • 1
  • Tamás Varjú
    • 1
  1. 1.Institute of MathematicsBudapest University of Technology and EconomicsBudapestHungary
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA