Improved geometric and recollision estimates for the invariance principle of the random Lorentz gas

TL;DR

This paper enhances geometric recollision estimates for the random Lorentz gas, extending the invariance principle timescale and providing insights into longer recollision patterns using convexity and dispersive estimates.

Contribution

It improves the timescale for the invariance principle in the Lorentz gas by refining geometric estimates and leveraging convexity for better dispersive bounds.

Findings

Extended the invariance principle timescale from r^2|log(r)|^2 to r^2|log(r)|

Developed better dispersive estimates using convexity of scatterers

Potential to reach timescales close to r^{-2} with advanced coupling methods

Abstract

By improving geometric recollision estimates for a random Lorentz gas, we extend the timescale $T(r)$ of the invariance principle for a Lorentz gas with particle size $r$ obtained by Lutsko and Toth (2020) from $\lim_{r \rightarrow 0} T(r) r^{2} |\log(r)|^2 =0$ to $\lim_{r \rightarrow 0} T(r) r^{2} |\log(r)| =0$. We show that this is the maximal reachable timescale with the coupling of stochastic processes introduced in the original result. In our improved geometric estimates we make use of the convexity of scatterers to obtain better dispersive estimates for the associated billiard map. We provide additional estimates which potentially open the possibility to reach, with a more elaborate coupling argument, up to timescales just below of $T(r)\sim r^{-2}$ when recollision patterns of arbitrary length occur.