Uniform Spanning

We study uniform spanning forest measures on innnite graphs, which are weak limits of uniform spanning tree measures from nite subgraphs. These limits can be taken with free (FSF) or wired (WSF) boundary conditions. Pemantle (1991) proved that the free and wired spanning forests coincide in Z d and that they give a single tree ii d 6 4. In the present work, we extend Pemantle's alternative to general graphs and exhibit further connections of uniform spanning forests to random walks, potential theory, invariant percolation, and amenability. The uniform spanning forest model is related to random cluster models in statistical physics, but, because of the preceding connections, its analysis can be carried further. Among our results are the following: The FSF and WSF in a graph G coincide ii all harmonic Dirichlet functions on G are constant. The tail-eld of the WSF and the FSF is trivial on any graph. On any Cayley graph which is not a nite extension of Z, all component trees of the WSF have one end; this is new in Z d for d > 5. On any tree, as well as on any graph with spectral radius less than 1, a.s. all components of the WSF are recurrent. The basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space H d is analyzed. A Cayley graph is amenable ii for all > 0, the union of the WSF and Bernoulli percolation with parameter is connected. Harmonic measure from innnity is shown to exist on any recurrent proper planar graph with nite co-degrees. We also present numerous open problems and conjectures.