Connectedness of the Free Uniform Spanning Forest as a function of edge weights

Abstract

Let G be the Cartesian product of a regular tree T and a finite connected transitive graph H. It is shown in [4] 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 wide family of H and T, the $\mathsf{FSF}$ is disconnected when w is small (relying on [4]). 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.