随机最近邻图：平移不变情形
Random nearest neighbor graphs: the translation invariant case
摘要：
如果$(ω(e))$是分配给$\mathbb{Z}^d$的边的随机变量（权重）族，则最近邻图是由所有边$\langle x,y \rangle$诱导的有向图，使得$ω(\{x,y\})$在$x$的所有邻居$y$中最小。也就是说，如果权重被视为边长，则每个顶点指向其最近的邻居。Nanda-Newman在权值为独立同分布时引入了最近邻图。并不断地分发，并证明了a.s.，无向图的所有分量都是有限的。我们研究了平移不变，不同的权重的情况下，并证明了最近邻图不包含双无限有向路。与i.i.d.的情况下，我们表明，在这种平稳的情况下，图可以包含一个或两个无限的组件（但不是更多）在二维，和$k$无限组件的任何$k \in [1,\infty]$在尺寸$\geq 3$。后者的结构使用一个通用的程序来表现出一定类别的有向图作为具有不同权重的最近邻图，从而表征所有的平移不变最近邻图。我们还讨论了测地线图的第一次通过渗流和影响的Chaika-Krishnan的聚结行走模型的关系。
英文摘要：
If $(ω(e))$ is a family of random variables (weights) assigned to the edges of $\mathbb{Z}^d$, the nearest neighbor graph is the directed graph induced by all edges $\langle x,y \rangle$ such that $ω(\{x,y\})$ is minimal among all neighbors $y$ of $x$. That is, each vertex points to its closest neighbor, if the weights are viewed as edge-lengths. Nanda-Newman introduced nearest neighbor graphs when the weights are i.i.d. and continuously distributed and proved that a.s., all components of the undirected version of the graph are finite. We study the case of translation invariant, distinct weights, and prove that nearest neighbor graphs do not contain doubly-infinite directed paths. In contrast to the i.i.d. case, we show that in this stationary case, the graphs can contain either one or two infinite components (but not more) in dimension two, and $k$ infinite components for any $k \in [1,\infty]$ in dimension $\geq 3$. The latter constructions use a general procedure to exhibit a certain class of directed graphs as nearest neighbor graphs with distinct weights, and thereby characterize all translation invariant nearest neighbor graphs. We also discuss relations to geodesic graphs from first-passage percolation and implications for the coalescing walk model of Chaika-Krishnan.
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