Open Access
September 2006 Minimal spanning forests
Russell Lyons,, Yuval Peres, Oded Schramm
Ann. Probab. 34(5): 1665-1692 (September 2006). DOI: 10.1214/009117906000000269


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 ℤd 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 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 also 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. Finally, an example is given to show that the minimal spanning tree measure does not have negative associations.


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Russell Lyons,. Yuval Peres. Oded Schramm. "Minimal spanning forests." Ann. Probab. 34 (5) 1665 - 1692, September 2006.


Published: September 2006
First available in Project Euclid: 14 November 2006

zbMATH: 1142.60065
MathSciNet: MR2271476
Digital Object Identifier: 10.1214/009117906000000269

Primary: 60B99 , 60K35 , 82B43
Secondary: 20F65 , 60D05

Keywords: amenability , Cayley graphs , percolation , Spanning trees

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.34 • No. 5 • September 2006
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