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VOL. 87 | 2021 The number of spanning clusters of the uniform spanning tree in three dimensions
Omer Angel, David A. Croydon, Sarai Hernandez-Torres, Daisuke Shiraishi

Editor(s) Yuzuru Inahama, Hirofumi Osada, Tomoyuki Shirai

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].

Information

Published: 1 January 2021
First available in Project Euclid: 20 January 2022

Digital Object Identifier: 10.2969/aspm/08710403

Subjects:
Primary: 60D05
Secondary: 05C80

Keywords: spanning clusters , Uniform spanning tree

Rights: Copyright © 2021 Mathematical Society of Japan

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