Abstract: The uniform measure on the set of all spanning trees of a finite graph is a classical object in probability. In an infinite graph, one can take an exhaustion by finite subgraphs, with some boundary conditions, and take the limit measure. The Free Uniform Spanning Forest (FUSF) is one of the natural limits, but it is less understood than the wired version, the WUSF. If we take a finitely generated group, then several properties of WUSF and FUSF have been known to be independent of the chosen Cayley graph of the group: the average degree in WUSF and in FUSF; the number of ends in the components of the WUSF and of the FUSF; the number of trees in the WUSF. Lyons and Peres asked if this latter should also be the case for the FUSF.
In a joint work with Gábor Pete we give two different Cayley graphs of the same group such that the FUSF is connected in one of them and it has infinitely many trees in the other. Furthermore, since our example is a virtually free group, we obtained a counterexample to the general expectation, that such "tree-like" graphs would have connected FUSF. Several open questions are inspired by the results. We also present some preliminary results and conjectures on phase transition phenomena that happen if we put conductances on the edges of the underlying graph.
