Let G be the Cartesian product of a regular tree T and a finite connected transitive graph H. It is shown in  that the Free Uniform Spanning Forest () 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 exhibits a phase transition. For large enough w we show that the is connected, while for a wide family of H and T, the is disconnected when w is small (relying on ). Finally, we prove that when H is the graph of one edge, then for any w, the is a single tree, and we give an explicit formula for the distribution of the distance between two points within the tree.
The first three authors would like to thank the Rényi REU 2020 program for undergraduate research. The third author is partially supported by the ÚNKP-20-1 New National Excellence Program of the Ministry for Innovation and Technology from the source of the National Research, Development and Innovation Fund.
The last author was partially supported by the ERC Consolidator Grant 772466 “NOISE”, and by Icelandic Research Fund Grant 185233-051.
"Connectedness of the Free Uniform Spanning Forest as a function of edge weights." Electron. Commun. Probab. 27 1 - 12, 2022. https://doi.org/10.1214/22-ECP453