Connectedness of the Free Uniform Spanning Forest as a function of edge weights
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866917787013742592 |
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| author | Alexy, Marcell Borbényi, Márton Imolay, András Timár, Ádám |
| author_facet | Alexy, Marcell Borbényi, Márton Imolay, András Timár, Ádám |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_12904 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Connectedness of the Free Uniform Spanning Forest as a function of edge weights Alexy, Marcell Borbényi, Márton Imolay, András Timár, Ádám Probability Combinatorics 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. |
| title | Connectedness of the Free Uniform Spanning Forest as a function of edge weights |
| topic | Probability Combinatorics |
| url | https://arxiv.org/abs/2011.12904 |