| MINIMAL SPANNING FORESTS (2009) | |||||||||||||||
Abstract | |||||||||||||||
| Minimal spanning forests on infinite graphs are weak limits of minimal spanning trees from finite subgraphs. 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 d this was proved by Alexander [Ann. Probab. 23 (1995) 87–104], but a different method is needed for the nonamenable case. We also prove that the WMSF components are “thin ” in a different sense, namely, 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. On the other hand, the FMSF is shown to be “thick”: 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 | |||||||||||||||
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