Localization of light rays in the planar random mirror model
Fix $p \in (0, 1]$. Independently at each vertex of $\mathbb{Z}^2$, place no mirror with probability $1 - p$, a two-sided mirror of slope $+1$ with probability $p/2$, and a two-sided mirror of slope $-1$ with probability $p/2$. A ray of light travels along the edges of $\mathbb{Z}^2$: at a vertex without a mirror it continues straight, and at a vertex with a mirror it is reflected by it. The ray starts at the origin heading east, and the mirror at the origin, if any, acts whenever the ray returns there. This is the Lorentz mirror model. For $p = 1$ the trajectory is almost surely periodic: the question reduces to bond percolation on $\mathbb{Z}^2$ at parameter $1/2$ (see Grimmett, §13.3), which has no infinite open path (Harris). Kozma and Sidoravicius proved that for every $p \in (0, 1]$ the ray leaves the box $[-n, n]^2$ with probability at least $1/(2n + 1)$. In dimensions $d \ge 4$ and at small mirror density, Elboim, Gloria and Hernández proved that trajectories behave diffusively up to all polynomial time scales in $1/p$.
Problem. Is it true that for every $p \in (0, 1]$ the set of vertices visited by the ray is almost surely finite, that is, that the trajectory almost surely closes into a finite periodic orbit?
Source: Folklore conjecture (see Elboim, Gloria and Hernández (2025), abstract), asked for every mirror density p ∈ (0, 1] by Kozma and Sidoravicius (2015); the mirror model goes back to Ruijgrok and Cohen (1988).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Lorentz mirror model, random environments
References
@article{kozma2015,
author = {Kozma, Gady and Sidoravicius, Vladas},
title = {Lower bound for the escape probability in the {L}orentz mirror model on {$\mathbb{Z}^2$}},
journal = {Israel J. Math.},
volume = {209},
number = {2},
year = {2015},
pages = {683--685},
doi = {10.1007/s11856-015-1233-1},
eprint = {1311.7437},
archivePrefix = {arXiv},
}
@misc{elboim2025,
author = {Elboim, Dor and Gloria, Antoine and Hern{\'a}ndez, Felipe},
title = {Diffusivity of the {L}orentz mirror walk in high dimensions},
year = {2025},
eprint = {2505.01341},
archivePrefix = {arXiv},
}
@article{ruijgrok1988,
author = {Ruijgrok, Th. W. and Cohen, E. G. D.},
title = {Deterministic lattice gas models},
journal = {Phys. Lett. A},
volume = {133},
number = {7-8},
year = {1988},
pages = {415--418},
doi = {10.1016/0375-9601(88)90927-9},
}
@book{grimmett1999,
author = {Grimmett, Geoffrey},
title = {Percolation},
edition = {Second},
series = {Grundlehren der mathematischen Wissenschaften},
volume = {321},
publisher = {Springer},
address = {Berlin},
year = {1999},
doi = {10.1007/978-3-662-03981-6},
}
@article{harris1960,
author = {Harris, T. E.},
title = {A lower bound for the critical probability in a certain percolation process},
journal = {Proc. Cambridge Philos. Soc.},
volume = {56},
number = {1},
year = {1960},
pages = {13--20},
doi = {10.1017/S0305004100034241},
}
@misc{aim-planar-lorentz-mirrors,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Localization of light rays in the planar random mirror model}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/280-planar-lorentz-mirrors.md},
note = {Entry 280 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00458, "Localization of light rays in the planar random mirror model", clopen.solutions.