ON DETTMANN'S 'HORIZON' CONJECTURES

In the simplest case consider a Z d periodic (d 3) arrangement of balls of radii < 1=2, and select a random direc- tion and point (outside the balls ). According to Dettmann's rst conjecture the probability that the so determined free ight (until the rst 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: forL periodic conguration of convex bodies withL being a non-degenerate lattice. These questions are related to P olya's visibility problem (1918), to the results of Bourgain- Golse (1998-) and of Marklof-Strombergsson (2010-). The results, joint with P. N andori and T. Varj u, also provide the asymptotic covariance of the periodic Lorentz process assuming it has a limit in the super-diusively scaling, a fact if d = 2 and the horizon is innite.