Diffusivity of the Lorentz mirror walk in high dimensions

Abstract

In the Lorentz mirror walk in dimension $d\geq 2$, mirrors are randomly placed on the vertices of $\mathbb{Z}^d$ at density $p\in[0,1]$. A light ray is then shot from the origin and deflected through the various mirrors in space. The object of study is the random trajectory obtained in this way, and it is of upmost interest to determine whether these trajectories are localized (finite) or delocalized (infinite). A folklore conjecture states that for $d=2$ these trajectories are finite for any density $p>0$, while in dimensions $d\geq 3$ and for $p>0$ small enough some trajectories are infinite. In this paper we prove that for all dimensions $d\geq 4$ and any small density $p$, the trajectories behave diffusively at all polynomial time scales $t\approx p^{-M}$ with $M>1$, and in particular, they do not close by this time.