Carl Dettmann (Bristol University)
New horizons in multidimensional diffusion: The Lorentz gas and the 
Riemann
Hypothesis

<br><br>
Abstract: The Lorentz gas is a billiard model involving a point particle
diffusing deterministically in a periodic array of convex scatterers. In 
the
two dimensional finite horizon case, in which all trajectories involve
collisions with the scatterers, displacements scaled by the usual 
diffusive
factor sqrt(t) are normally distributed, as shown by Bunimovich and Sinai 
in
1981. In the infinite horizon case, motion is superdiffusive, however the
normal distribution is recovered when scaling by sqrt(t ln t), with an
explicit formula for its variance. Here we explore the infinite horizon 
case
in arbitrary dimensions, giving explicit formulas for the mean square
displacement, arguing that it differs from the variance of the limiting
distribution, making connections with the Riemann Hypothesis in the small
scatterer limit, and providing evidence for a critical dimension d=6 
beyond
which correlation decay exhibits fractional powers. The results are
conditional on a number of conjectures, and are corroborated by numerical
simulations in up to ten dimensions.