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

Let G be the Cartesian product of a regular tree T and finite connected transitive graph H. It is shown in [4] that Free Uniform Spanning Forest (FSF) this may not connected, but dependence connectedness on H remains somewhat mysterious. We study case when positive weight w put edges H-copies G, conjecture FSF exhibits phase transition. For large enough we show while for wide family T, disconnected small (relying [4]). Finally, prove one edge, then any w, single tree, give an explicit formula distribution distance between two points within tree.