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arXiv · 2003.04548

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

Omer Angel·David A. Croydon·Sarai Hernandez-Torres·Daisuke Shiraishi
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Abstract

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).

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Omer Angel, David A. Croydon, Sarai Hernandez-Torres, Daisuke Shiraishi. 2020-03-10. The number of spanning clusters of the uniform spanning tree in three dimensions. https://arxiv.org/abs/2003.04548

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