无限光线与具有无限多个枢轴的无限簇
Infinite light rays and infinite clusters with infinitely many pivots
- University of Tübingen（蒂宾根大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对$\text{Z}^2$上洛伦兹镜像模型的无限轨迹相关问题，构造了含无限轨迹的平面图，证明其无限簇顶点具无限多枢轴边，并刻画了键渗流、位点渗流及镜像的对应性质。
AI中文摘要：
在$\boldsymbol{\text{Z}^2}$上的洛伦兹镜像模型中，所有轨迹均为有限的这一猜想已被提出。Quas证明，几乎必然地，任意无限轨迹在某条边处被分割后的两个半段，会在无限多个顶点处相遇，且每个这样的顶点都是枢轴：改变其状态可使该轨迹变为有限。我们构造了存在无限轨迹的平面图，且所有此类轨迹均具备该性质。给定$p \boldsymbol{\text{∈}}(0,1)$，我们将$\boldsymbol{\text{Z}^2}$的随机单端子树的每条边替换为若干平行边，其数量取决于$p$，且不超过对应后代树高度的对数。所得平面多重图的中间图是该树的一个因子，对于均匀生成树，其顶点强度有限。在该中间图上的镜像模型中，若原边处镜像的概率为$p$，对偶边处镜像的任意概率为$r \boldsymbol{\text{∈}}(0,1-p]$，则存在无限轨迹，且每条无限轨迹的两个半段会无限次相遇。该构造基于伯努利渗流：每个随机单端局部有限树均可按此方式装饰，使得参数为$p$的键渗流是临界的且发生渗流，且无限簇的每个顶点都有无限多个枢轴边。我们针对键渗流、位点渗流及镜像对该性质进行了刻画。
英文摘要：
It is conjectured that in the Lorentz mirror model on $\Z^2$ all trajectories are finite. Quas showed that, almost surely, the two halves of any infinite trajectory, cut at an edge, meet at infinitely many vertices, each of which is pivotal: changing its state can make the trajectory finite. We construct plane graphs on which infinite trajectories exist, and all of them have this property. Given $p\in(0,1)$, we replace each edge of a random one-ended subtree of $\Z^2$ by a number of parallel edges, depending on $p$, that is at most logarithmic in the height of the corresponding descendant tree. The medial graph of the resulting plane multigraph is a factor of the tree, and for the uniform spanning tree it has finite vertex intensity. In the mirror model on this medial graph, with probability $p$ for mirrors along primal edges and any probability $r\in(0,1-p]$ for mirrors along dual edges, infinite trajectories exist, and the two halves of each of them meet infinitely often. The construction rests on Bernoulli percolation: every random one-ended locally finite tree can be decorated in this way such that bond percolation with parameter $p$ is critical and percolates, and every vertex of the infinite cluster has infinitely many pivotal edges. We characterize this property for bond and site percolation and for mirrors.
