度量树上的环汤渗流与边界维数
Loop soup percolation and boundary dimension on metric trees
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中文总结 AI 辅助
该研究确定了具有适当电缆度量的暂态局部有限加权树上布朗环汤的渗流阈值，给出了伯努利渗流 Lyons 公式的类似物，并将分类扩展至超临界 Galton-Watson 树及相关实树。
中文摘要 AI 辅助
我们确定了具有适当电缆度量且无杀死的暂态局部有限加权树上的布朗环汤的渗流阈值，该阈值等于 $1-D_G/2$，其中 $D_G\in[0,2]$ 是归一化格林函数诱导度量下边界的豪斯多夫维数。这给出了伯努利渗流的 Lyons 分支数公式的类似物，其中格林距离取代了图距离。具有单位电导和分支数为 2 的球对称树实现了 $[1/2,1]$ 中的每个阈值，而几何电导给出了低于二分之一的显式正阈值，细分二叉树的阈值为零。在 Drewitz、Prévost 和 Rodriguez 的容量假设下，我们无需最小度假设即可通过有限能量条件表征阈值二分之一，随后根据他们的定理和 Lupu 的耦合得到二分之一处的非渗流结果。对于具有单位电导的超临界 Galton-Watson 树，在以存活为条件时，边界维数几乎必然为 1，阈值为二分之一，且在临界点处非渗流，无需后代矩假设，允许存在叶子和有一个孩子的顶点。该分类也固有地适用于由其无限射线张成的适当实树。
英文摘要
We determine the percolation threshold of the Brownian loop soup on a transient locally finite weighted tree with proper cable metric and no killing. It equals $1-D_G/2$, where $D_G\in[0,2]$ is the Hausdorff dimension of the boundary in the metric induced by the normalized Green function. This gives an analogue of Lyons' branching-number formula for Bernoulli percolation, with Green distance replacing graph distance. Spherically symmetric trees with unit conductances and branching number two realize every threshold in $[1/2,1]$, while geometric conductances give explicit positive thresholds below one-half and a subdivided binary tree has threshold zero. Under the capacity assumption of Drewitz, Prévost and Rodriguez, we characterize the threshold one-half by a finite-energy condition, without a minimum-degree assumption. Non-percolation at one-half then follows from their theorem and Lupu's coupling. For supercritical Galton-Watson trees with unit conductances, conditioned on survival, the boundary dimension is almost surely one and the threshold is one-half, with non-percolation at criticality. No offspring moment assumption is required, and leaves and vertices with one child are allowed. The classification also applies intrinsically to proper real trees spanned by their infinite rays.
