论文标题
统一树的跨越簇的数量在三个维度
The number of spanning clusters of the uniform spanning tree in three dimensions
论文作者
论文摘要
令$ {\ Mathcal U}_δ$为$δ\ Mathbb {z}^{3} $的统一跨越树。 $ {\ Mathcal U}_δ$的跨度群集是$ {\ Mathcal U}_δ$限制的一个连接的组件，即连接左face $ \ \ \ {0 \} \} \ {0 \} \ times [0,1] $ [0,1] $^{2} $ {2 {2} $ {2 {2} $ {2 [0,1]^{2} $。在本说明中，我们将证明跨越簇的数量紧张为$δ\至0 $，这解决了Benjamini（1999）提出的一个空旷的问题。
Let ${\mathcal U}_δ$ be the uniform spanning tree on $δ\mathbb{Z}^{3}$. A spanning cluster of ${\mathcal U}_δ$ is a connected component of the restriction of ${\mathcal U}_δ$ to the unit cube $[0,1]^{3}$ that connects the left face $\{ 0 \} \times [0,1]^{2}$ to the right face $\{ 1 \} \times [0,1]^{2}$. In this note, we will prove that the number of the spanning clusters is tight as $δ\to 0$, which resolves an open question raised by Benjamini (1999).
