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

Peter Nandori, Domokos Szasz, Tamas Varju

Published 2012-10-08, updated 2013-05-06Version 3

In the simplest case, consider a $\mathbb{Z}^d$-periodic ($d \geq 3$) 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 P\'olya's visibility problem (1918), to theories of Bourgain-Golse-Wennberg (1998-) and of Marklof-Str\"{o}mbergsson (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.