Mathematics Colloquium

On Dettmann's 'Horizon' Conjectures

Speaker: Domokos Szász, Budapest University of Technology

Location: Warren Weaver Hall 1302

Date: Monday, November 12, 2012, 3:45 p.m.


In the simplest case consider a \(\mathbb{Z}^{d}\)−periodic \(\left ( d \geq 3 \right )\) 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 \(t \gg 1\) 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 convex bodies with \(\mathcal{L}\) being a non-degenerate lattice. These questions are related to Pólya's visibility problem (1918), to the results of Bourgain-Golse (1998-) and of Marklof-Strömbergsson (2010-). The results, joint with P. Nándori and T. Varjú, also provide the asymptotic covariance of the periodic Lorentz process assuming it has a limit in the super-diffusively scaling, a fact if \(d = 2\) and the horizon is infinite.