Hyperbolic Radial Spanning Tree

径向双曲测度：壳几何、金字塔极限与高斯相变

Radial Hyperbolic Measures: Shell Geometry, Pyramid Limits, and Gaussian Phase Transitions

该研究探讨高维双曲空间中径向测度的壳几何、金字塔极限与高斯相变，明确有效半径的作用，揭示两类高斯的临界阶差异及对应收敛性质。

双曲双圆填充中的完美匹配与本质生成森林

Perfect Matchings and Essential Spanning Forests in Hyperbolic Double Circle Packings

本文利用双曲图的双圆填充方法，证明了调和 Dirichlet 函数与均匀生成森林极值无限体积测度的存在性，并给出完美匹配柱事件概率的显式公式，进而证明特定边界条件下高度差方差有限，与二维欧氏可容图的 $\log n$ 增长形成对比。

Hyperbolic and Parabolic Unimodular Random Maps

We show that for infinite planar unimodular random rooted maps, many global geometric and probabilistic properties are equivalent, and are determined by a natural, local notion of average curvature. This dichotomy includes properties relating to amenability, conformal geometry, random walks, uniform and minimal spanning forests, and Bernoulli bond percolation. We also prove that every simply connected unimodular random rooted map is sofic, that is, a Benjamini-Schramm limit of finite maps.

Random spanning forests and hyperbolic symmetry

We study (unrooted) random forests on a graph where the probability of a forest is multiplicatively weighted by a parameter $β>0$ per edge. This is called the arboreal gas model, and the special case when $β=1$ is the uniform forest model. The arboreal gas can equivalently be defined to be Bernoulli bond percolation with parameter $p=β/(1+β)$ conditioned to be acyclic, or as the limit $q\to 0$ with $p=βq$ of the random cluster model. It is known that on the complete graph $K_{N}$ with $β=α/N$ there is a phase transition similar to that of the Erdős--Rényi random graph: a giant tree percolates …

Long Brownian bridges in hyperbolic spaces converge to Brownian trees

We show that the range of a long Brownian bridge in the hyperbolic space converges after suitable renormalisation to the Brownian continuum random tree. This result is a relatively elementary consequence of $\bullet$ A theorem by Bougerol and Jeulin, stating that the rescaled radial process converges to the normalized Brownian excursion, $\bullet$ A property of invariance under re-rooting, $\bullet$ The hyperbolicity of the ambient space in the sense of Gromov. A similar result is obtained for the rescaled infinite Brownian loop in hyperbolic space.

The Matsumoto and Yor process and infinite dimensional hyperbolic space

The Matsumoto\,--Yor process is $\int\_0^t \exp(2B\_s-B\_t)\, ds$, where $(B\_t)$ is a Brownian motion. It is shown that it is the limit of the radial part of the Brownian motion at the bottom of the spectrum on the hyperbolic space of dimension $q$, when $q$ tends to infinity. Analogous processes on infinite series of non compact symmetric spaces and on regular trees are described.

关于双曲布朗运动的渐近性

On the asymptotic behavior of the hyperbolic Brownian motion

本文的主要结果涉及Poincaré半空间上$n$维双曲布朗运动（对于$n\geq 2$）径向分量的大偏差和中等偏差.当双曲布朗运动的起点距离$η$趋于无穷大时，我们还研究了半径为$η_1$的球的命中概率$P_η(T_{η_1}^{(n)}<\infty)$的渐近行为。

径向生成树在所有维度中都是直线的

The Radial Spanning Tree is straight in all dimensions

本文证明了径向生成树在所有维度中几乎必然直线，解决了 RST 研究中的关键障碍，揭示了无限分支行为的性质。

均匀生成树中的三脚架和三边径向SLE$_2$

Tripod in uniform spanning tree and three-sided radial SLE$_2$

研究均匀生成树中三叉点的缩放极限，证明其分布绝对连续于勒贝格测度且具有显式密度，并给出三脚架的条件分布为三边径向SLE$_2$。

径向泊松网的标度极限

Scaling limit of the radial Poissonian web

我们考虑一个变种的径向生成树引入Baccelli和Bordenave。与原始模型一样，我们的模型是一棵以原点为根的树，建立在平面泊松点过程的实现上。与之不同的是，我们模型的路径具有独立的跳跃。我们表明，我们的扩散重新缩放的树，被视为连接其网站的根的路径的集合，收敛分布的布朗桥网络，这是粗略地说，从一个平面带垂直于时间轴的所有点开始的合并布朗桥的集合，并在原点结束。

平面环擦除随机游动与一致生成树的共形不变性

Conformal invariance of planar loop-erased random walks and uniform spanning trees

证明了单连通区域$D$中环擦除随机游动的标度极限等于$D$中的径向SLE（2）路.特别地，极限存在并且是共形不变的。证明了Jordan域上的一致生成树的标度极限是存在的，并且是共形不变的。假设域的边界是一条$C^1$简单闭曲线，同样的方法被用来证明均匀生成树Peano曲线的缩放极限是连接$A$端点的$D$闭包中的弦SLE（8）路径，其中树沿着边界上的适当弧$A$连接。这个结果的一个副产品是SLE（8）几乎必然是由一条连续路生成的。