论文标题
双曲线径向生成树
Hyperbolic Radial Spanning Tree
论文作者
论文摘要
我们定义并分析了Baccelli和Bordenave在二维欧几里得空间中引入的径向生成树（RST）的$ D $维二线空间的扩展。特别是，我们将重点关注树的无限分支的描述。二维欧几里得rST的性质在每个维度上都扩展到双曲线的情况：几乎可以肯定的是，每个无限分支都接收一个渐近方向，并且至少一个无限分支达到了每个渐近方向。此外，分支聚集到任何确定性渐近方向几乎是唯一的。为了获得任何维度的结果，这里考虑了一种全新的方法。 \ tvc {我们的策略主要涉及以下两种成分，这些成分依赖于在Flammant中引入和研究的双曲线定向跨越森林（DSF）（2019年）。}首先，双曲线指标使我们能够很好地控制分支机构在不使用平面力参数的超纤维DSF中分支机构的波动。然后，我们将双曲线rst与双曲线DSF息息结束，并结束。
We define and analyze an extension to the $d$-dimensional hyperbolic space of the Radial Spanning Tree (RST) introduced by Baccelli and Bordenave in the two-dimensional Euclidean space (2007). In particular, we will focus on the description of the infinite branches of the tree. The properties of the two-dimensional Euclidean RST are extended to the hyperbolic case in every dimension: almost surely, every infinite branch admits an asymptotic direction and each asymptotic direction is reached by at least one infinite branch. Moreover, the branch converging to any deterministic asymptotic direction is unique almost surely. To obtain results for any dimension, a completely new approach is considered here. \tvc{Our strategy mainly involves the two following ingredients, that rely on the hyperbolic Directed Spanning Forest (DSF) introduced and studied in Flammant (2019).} First, the hyperbolic metric allows us to obtain fine control of the branches' fluctuations in the hyperbolic DSF without using planarity arguments. Then, we couple the hyperbolic RST with the hyperbolic DSF and conclude.
