The Free Uniform Spanning Forest is disconnected in some virtually free groups, depending on the generating set
Gábor Pete (Budapest)
Abstract: The uniform random spanning tree of a finite graph is a classical object in probability and combinatorics. In an infinite graph, one can take any 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. Taking a finitely generated group, 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 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 but has infinitely many trees in the other. Since our example is a virtually free group, this is also a counterexample to the general expectation that such "tree-like" graphs would have connected FUSF. Many open questions are inspired by the results.
combinatorics
Audience: researchers in the topic
Series comments: This is the online combinatorics seminar at Warwick.
Organizers: | Jan Grebik, Oleg Pikhurko |
Curator: | Hong Liu* |
*contact for this listing |