Diffusivity of the Lorentz mirror walk in high dimensions

TL;DR

This paper investigates the behavior of light trajectories in a high-dimensional Lorentz mirror walk, proving diffusive behavior in dimensions four and higher for small mirror densities over polynomial time scales.

Contribution

It establishes that in dimensions four and above, the trajectories are diffusive for small densities at polynomial time scales, advancing understanding of localization in high-dimensional Lorentz mirror walks.

Findings

Trajectories are diffusive in dimensions d≥4 for small p.

Trajectories do not close within polynomial time scales t≈p^{-M}.

Behavior is characterized at all polynomial time scales for small densities.

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.