Polynomial bound for the localization length of Lorentz mirror model on the 1D cylinder

TL;DR

This paper establishes a polynomial upper bound on the localization length of light trajectories in the Lorentz mirror and Manhattan models on a finite-width cylinder, demonstrating confinement within a region of size polynomial in the cylinder's width.

Contribution

It provides the first polynomial bound on the localization length for these models on a cylindrical geometry, advancing understanding of their long-term behavior.

Findings

Trajectories are confined within |x| ≤ O(n^{10}) for large n

High probability of confinement for large cylinder width

Polynomial bound on localization length

Abstract

We consider the Lorentz mirror model and the Manhattan model on the even-width cylinder $\mathbb{Z} \times (\mathbb{Z}/2n\mathbb{Z}) =\{(x,y):x,y\in \mathbb{Z}, 1\leq y\leq 2n\}$. For both models, we show that for large enough $n$, with high probability, any trajectory of light starting from the section $x=0$ is contained in the region $|x|\leq O(n^{10})$.