Uniform spanning forests

Abstract

We study uniform spanning forest measures on infinite graphs, which are weak limits of uniform spanning tree measures from finite subgraphs. These limits can be taken with free (FSF )or wired (WSF ) boundary conditions. Pemantle proved that the free and wired spanning forests coincide in $\mathbb{Z}^d$ and that they give a single tree iff $d </4$.

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 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 f $\mathbb{Z}$ all component trees of the WSF have one end; this is new in $\mathbb{Z}^d$ for $d \ge 5$.

• 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 $\mathbb{H}^d$ is analyzed.

• A Cayley graph is amenable iff for all $\epsilon > 0$, the union of the WSF and Bernoulli percolation with parameter $\epsilon$ is connected.

• Harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees.

We also present numerous open problems and conjectures.