Uniform spanning forests

We study uniform spanning forest measures on infinite graphs. which are wea k limits of uniform spanning tree measures from finite subgraphs. These lim its can be taken with free (FSF) or wired (WSF) boundary conditions. Pemant le proved that the free and wired spanning forests coincide in Z(d) and tha t they give a single tree iff d less than or equal to 4. In the present work, we extend Pemantle's alternative to general graphs and exhibit further connections of uniform spanning forests to random walks, p otential theory, invariant percolation and amenability. The uniform spannin g forest model is related to random cluster models in statistical physics, but, because of the preceding connections, its analysis can be carried furt her. Among our results are the following: The FSF and WSF in a graph G coincide iff all harmonic Dirichlet functions on G are constant. The tail sigma -fields of the WSF and the FSF are trivial on any graph. On any Cayley graph that is not a finite extension of Z, all component tree s of the WSF have one end; this is new in Z(d) for d greater than or equal to 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 measur es on lattices in hyperbolic space H-d is analyzed. A Cayley graph is amenable iff for all epsilon > 0, the union of the WSF an d Bernoulli percolation with parameter E is connected. Harmonic measure from infinity is shown to exist on any recurrent proper pl anar graph with finite codegrees. We also present numerous open problems and conjectures.