Jens Marklof (University of Bristol)

The distribution of visible lattice points and collision times in the periodic Lorentz gas.

Abstract. I will discuss two closely related problems.

(1) Take integer lattice points in a large ball of radius $R$ and project them onto the unit sphere $S$ centered at $x$.   It is well known that the sequence of projected points becomes uniformly distributed on $S$ as $R$ becomes large. I will show that, for every fixed $x$,  the statistical correlation functions of this sequence have limiting distributions with some remarkable properties. The proof uses Ratner's classification of ergodic measures invariant under unipotent flows.

(2) The periodic Lorentz gas describes a point particle moving in a periodic array of spherical scatterers of radius $r$.   I'll explain why the probability for a particle to hit the first scatterer after time $T$ has a limiting distribution in the small scatterer limit $r\to 0$ and discuss some of its properties. This particular question was raised by Sinai in the early 1980s. This is joint work with A. Strombergsson.