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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2011.12904 |
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Table of Contents:
- Let $G$ be the Cartesian product of a regular tree $T$ and a finite connected transitive graph $H$. It is shown in arXiv:2006.06387 that the Free Uniform Spanning Forest ($\mathsf{FSF}$) of this graph may not be connected, but the dependence of this connectedness on $H$ remains somewhat mysterious. We study the case when a positive weight $w$ is put on the edges of the $H$-copies in $G$, and conjecture that the connectedness of the $\mathsf{FSF}$ exhibits a phase transition. For large enough $w$ we show that the $\mathsf{FSF}$ is connected, while for a large family of $H$ and $T$, the $\mathsf{FSF}$ is disconnected when $w$ is small (relying on arXiv:2006.06387). Finally, we prove that when $H$ is the graph of one edge, then for any $w$, the $\mathsf{FSF}$ is a single tree, and we give an explicit formula for the distribution of the distance between two points within the tree.