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

Abstract

In the simplest case, consider a Zd-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 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 L-periodic configuration of - possibly intersecting - convex bodies with 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¨ombergsson (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.