The Free Uniform Spanning Forest is disconnected in some virtually free groups, depending on the generating set
19th June 2020, 3:30 pm – 4:30 pm
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 Cayley graph of the group: whether WUSF=FUSF; the average degree in WUSF and in FUSF; the number of trees in the WUSF. Lyons and Peres asked if this should also be the case for the FUSF.
In recent joint work with Ádám Timár, 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.