The number of spanning clusters of the uniform spanning tree in three dimensions

Abstract

Let ${\mathcal{U}}_{\delta }$ be the uniform spanning tree on $\delta {\mathbb{Z}}^3$. A spanning cluster of ${\mathcal{U}}_{\delta }$ is a connected component of the restriction of ${\mathcal{U}}_{\delta }$ 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 $\delta \to 0$, which resolves an open question raised by Benjamini (1999).

8 pages, 1 figure