Connectedness of the free uniform spanning forest as a function of edge weights

DOI10.1214/22-ECP453zbMATH Open1492.60018arXiv2011.12904MaRDI QIDQ2113269

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Publication date: 11 March 2022

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Abstract: 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 ($mathsfFSF$) 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 $mathsfFSF$ exhibits a phase transition. For large enough $w$ we show that the $mathsfFSF$ is connected, while for a large family of $H$ and $T$, the $mathsfFSF$ 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 $mathsfFSF$ is a single tree, and we give an explicit formula for the distribution of the distance between two points within the tree.

Full work available at URL: https://arxiv.org/abs/2011.12904

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