曼哈顿格上镜像模型中的边界奇偶性
Boundary parity in the mirror model on the Manhattan lattice
- ESSEC（法国高等经济商业学院）
- Polytechnic Institute of Paris（巴黎理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究提出有限尺度判据与三色张量表示，证明曼哈顿格镜像模型中边界奇偶概率非单调，未达约束阈值，为平面局域化提供严格数值方法。
AI中文摘要：
我们研究了曼哈顿格上镜像模型的一个有限域事件：每条边界到边界的轨迹穿过一个标记连接器的次数为偶数，而内部环不受限制。我们的主要定理给出了一个有限尺度约束判据：如果该事件的概率在一个偶数二方格矩形中达到一个普适阈值，那么平面上几乎所有的轨迹都是周期的。我们构造了一个精确归一化的三色张量表示，并利用有向延续的符号部分置换结构证明了近似收缩的确定性界。这些工具将数值评估转化为严格的有限体积陈述。在镜像密度\(p=2/5\)时，我们证明了边界奇偶概率在矩形宽度上非单调，先下降后经认证的上升。这些值仍低于充分约束阈值，并未解决该密度下的平面局域化问题。进一步的实验追踪了测量的有限尺寸最小值随密度的变化，比较了基于匹配和逐项误差界，并展示了局部镜像变化对端点敏感和协同的响应。
英文摘要：
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.
