Breadcrumb
Tail asymptotics of free path lengths for the periodic Lorentz process. On Dettmann's geometric conjectures
Thu 21 February 2013, 15:00
Domokos Szasz
Technical University of Budapest
Ergodic Theory and Dynamical Systems
Organisers: Andrew Ferguson, Thomas Jordan
ABSTRACT
Consider a Z^d 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 prob-
ability that the so determined free flight (until the first hitting of
a ball) is larger than t >> 1 is ∼ C , where C is explicitly given t
by the geometry of the model. Dettmann’s second conjecture is related to the previous case with tangent balls (of radii 1/2). The conjectures are established in a general setup: for L-periodic con- figuration of - possibly intersecting - convex bodies with L being a non-degenerate lattice. These questions are related to Polya’s visibility problem (1918), to theories of Bourgain-Golse-Wennberg (1998-) and of Marklof-Strombergsson (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.