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 in [4].