Minimal spanning forests

We study minimal spanning forests in infinite graphs, which are weak limits of minimal spanning trees from finite subgraphs corresponding to i.i.d. random labels on the edges. These limits can be taken with free or wired boundary conditions, and are denoted FMSF (free minimal spanning forest) and WMSF (wired minimal spanning forest), respectively. The WMSF is also the union of the trees that arise from invasion percolation started at all vertices. We show that on any Cayley graph where critical percolation has no infinite clusters, all the component trees in the WMSF have one end a.s. In Z this was proved by Alexander (1995), but a different method is needed for the nonamenable case. We show that on any connected graph, the union of the FMSF and independent Bernoulli percolation (with arbitrarily small parameter) is a.s. connected. In conjunction with a recent result of Gaboriau, this implies that in any Cayley graph, the expected degree of the FMSF is at least the expected degree of the FSF (the weak limit of uniform spanning trees). We show that on any graph, each component tree in the WMSF has pc = 1 a.s., where pc denotes the critical probability for having an infinite cluster in Bernoulli percolation. We show that the number of infinite clusters for Bernoulli(pu) percolation is at most the number of components of the FMSF, where pu denotes the critical probability for having a unique infinite cluster.