Tail asymptotics of free path lengths for the periodic Lorentz process. On Dettmann's geometric conjectures

Abstract

In the simplest case, consider a ${\mathbb{Z}}^d$-periodic ($d\ge 3$) 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 (1918), to theories of Bourgain-Golse-Wennberg (1998-) and of Marklof-Strömbergsson (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.

References in corpus (4)

• Free path lengths in quasi crystals
• Normal diffusion in crystal structures and higher-dimensional billiard models with gaps
• Deterministic Diffusion in Periodic Billiard Models
• Periodizing quasicrystals: Anomalous diffusion in quasiperiodic systems

Cited by in corpus (6)

• The low-density limit of the Lorentz gas: periodic, aperiodic and random
• Horizons and free path distributions in quasiperiodic Lorentz gases
• Random walks and Lorentz processes
• Efficient algorithms for general periodic Lorentz gases in two and three dimensions
• Escape Dynamics of Many Hard Disks
• Invariance principle for the periodic Lorentz gas in the Boltzmann-Grad limit