Boundary parity in the mirror model on the Manhattan lattice
Abstract
We study a finite-domain event for the mirror model on the Manhattan lattice: every boundary-to-boundary trajectory crosses a marked connector an even number of times, while internal cycles are unrestricted. Our main theorem gives a finite-scale confinement criterion: if this event's probability reaches a universal threshold in one even two-square rectangle, then almost surely every trajectory in the plane is periodic. We construct an exactly normalized three-color tensor representation and prove deterministic bounds for approximate contractions using the signed partial-permutation structure of directed continuations. These tools turn numerical evaluations into rigorous finite-volume statements. At mirror density p=2/5, we prove that the boundary-parity probability is nonmonotone in the rectangle width, with a decrease followed by a certified increase. These values remain below the sufficient confinement threshold and do not resolve planar localization at this density. Further experiments track how the measured finite-size minimum varies with density, compare matching-based and entrywise error bounds, and exhibit endpoint-sensitive and cooperative responses to local mirror changes.
Related
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BibTeX
@misc{gu2026boundary,
title = {{Boundary parity in the mirror model on the Manhattan lattice}},
author = {Jian Gu and Qin Hao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19498},
eprint = {2609.19498},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19498},
}
RIS
TY - UNPB
TI - Boundary parity in the mirror model on the Manhattan lattice
AU - Jian Gu
AU - Qin Hao
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.19498
UR - https://arxiv.org/abs/2609.19498
AB - We study a finite-domain event for the mirror model on the Manhattan lattice: every boundary-to-boundary trajectory crosses a marked connector an even number of times, while internal cycles are unrestricted. Our main theorem gives a finite-scale confinement criterion: if this event's probability reaches a universal threshold in one even two-square rectangle, then almost surely every trajectory in the plane is periodic. We construct an exactly normalized three-color tensor representation and prove deterministic bounds for approximate contractions using the signed partial-permutation structure of directed continuations. These tools turn numerical evaluations into rigorous finite-volume statements. At mirror density p=2/5, we prove that the boundary-parity probability is nonmonotone in the rectangle width, with a decrease followed by a certified increase. These values remain below the sufficient confinement threshold and do not resolve planar localization at this density. Further experiments track how the measured finite-size minimum varies with density, compare matching-based and entrywise error bounds, and exhibit endpoint-sensitive and cooperative responses to local mirror changes.
ER -