The Annals of Probability

Minimal spanning forests

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 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.

Article information

Source
Ann. Probab., Volume 34, Number 5 (2006), 1665-1692.

Dates
First available in Project Euclid: 14 November 2006

https://projecteuclid.org/euclid.aop/1163517218

Digital Object Identifier
doi:10.1214/009117906000000269

Mathematical Reviews number (MathSciNet)
MR2271476

Zentralblatt MATH identifier
1142.60065

Citation

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

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