The Annals of Probability
- Ann. Probab.
- Volume 34, Number 5 (2006), 1665-1692.
Minimal spanning forests
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.
Ann. Probab., Volume 34, Number 5 (2006), 1665-1692.
First available in Project Euclid: 14 November 2006
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60B99: None of the above, but in this section 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Lyons,, Russell; Peres, Yuval; Schramm, Oded. Minimal spanning forests. Ann. Probab. 34 (2006), no. 5, 1665--1692. doi:10.1214/009117906000000269. https://projecteuclid.org/euclid.aop/1163517218