The Annals of Probability

Minimal spanning forests

Russell Lyons,, Yuval Peres, and Oded Schramm

Full-text: Open access


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

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]

Spanning trees Cayley graphs amenability percolation


Lyons,, Russell; Peres, Yuval; Schramm, Oded. Minimal spanning forests. Ann. Probab. 34 (2006), no. 5, 1665--1692. doi:10.1214/009117906000000269.

Export citation


  • Adams, S. R. and Spatzier, R. J. (1990). Kazhdan groups, cocycles and trees. Amer. J. Math. 112 271--287.
  • Aizenman, M., Burchard, A., Newman, C. M. and Wilson, D. B. (1999). Scaling limits for minimal and random spanning trees in two dimensions. Random Structures Algorithms 15 319--367.
  • Aldous, D. J. and Steele, J. M. (1992). Asymptotics for Euclidean minimal spanning trees on random points. Probab. Theory Related Fields 92 247--258.
  • Aldous, D. J. and Steele, J. M. (2004). The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures. Encyclopaedia Math. Sci. (H. Kesten, ed.) 110 1--72. Springer, Berlin.
  • Alexander, K. S. (1995). Percolation and minimal spanning forests in infinite graphs. Ann. Probab. 23 87--104.
  • Alexander, K. S. and Molchanov, S. A. (1994). Percolation of level sets for two-dimensional random fields with lattice symmetry. J. Statist. Phys. 77 627--643.
  • Barsky, D. J., Grimmett, G. R. and Newman, C. M. (1991). Percolation in half-spaces: Equality of critical densities and continuity of the percolation probability. Probab. Theory Related Fields 90 111--148.
  • Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (1999). Critical percolation on any nonamenable group has no infinite clusters. Ann. Probab. 27 1347--1356.
  • Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (1999). Group-invariant percolation on graphs. Geom. Funct. Anal. 9 29--66.
  • Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (2001). Uniform spanning forests. Ann. Probab. 29 1--65.
  • Benjamini, I. and Schramm, O. (1996). Percolation beyond $\mathbfZ^d$, many questions and a few answers. Electron. Comm. Probab. 1 71--82 (electronic).
  • Benjamini, I. and Schramm, O. (1998). Personal communication.
  • Benjamini, I. and Schramm, O. (2001). Percolation in the hyperbolic plane. J. Amer. Math. Soc. 14 487--507.
  • Burton, R. M. and Keane, M. (1989). Density and uniqueness in percolation. Comm. Math. Phys. 121 501--505.
  • Doyle, P. G. and Snell, J. L. (1984). Random Walks and Electric Networks. Mathematical Association of America, Washington, DC. Available at
  • Gaboriau, D. (2002). Invariants $l^2$ de relations d'équivalence et de groupes. Publ. Math. Inst. Hautes Études Sci. 95 93--150.
  • Gaboriau, D. (2006). Invariant percolation and harmonic Dirichlet functions. GAFA. To appear.
  • Grimmett, G. (1999). Percolation, 2nd ed. Springer, Berlin.
  • Grimmett, G. R. and Marstrand, J. M. (1990). The supercritical phase of percolation is well behaved. Proc. Roy. Soc. London Ser. A 430 439--457.
  • Häggström, O. (1995). Random-cluster measures and uniform spanning trees. Stochastic Process. Appl. 59 267--275.
  • Häggström, O. (1997). Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab. 25 1423--1436.
  • Häggström, O. (1998). Uniform and minimal essential spanning forests on trees. Random Structures Algorithms 12 27--50.
  • Häggström, O. and Peres, Y. (1999). Monotonicity of uniqueness for percolation on Cayley graphs: All infinite clusters are born simultaneously. Probab. Theory Related Fields 113 273--285.
  • Häggström, O., Peres, Y. and Schonmann, R. H. (1999). Percolation on transitive graphs as a coalescent process: Relentless merging followed by simultaneous uniqueness. In Perplexing Problems in Probability (M. Bramson and R. Durrett, eds.) 69--90. Birkhäuser, Boston.
  • Hara, T. and Slade, G. (1994). Mean-field behaviour and the lace expansion. In Probability and Phase Transition (G. Grimmett, ed.) 87--122. Kluwer, Dordrecht.
  • Harris, T. E. (1960). A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc. 56 13--20.
  • Kesten, H. (1980). The critical probability of bond percolation on the square lattice equals $1\over 2$. Comm. Math. Phys. 74 41--59.
  • Kesten, H. (1982). Percolation Theory for Mathematicians. Birkhäuser, Boston.
  • Lyons, R. (2000). Phase transitions on nonamenable graphs. J. Math. Phys. 41 1099--1126.
  • Lyons, R. (2005). Random complexes and $\ell^2$-Betti numbers. To appear.
  • Lyons, R. with Peres, Y. (2006). Probability on Trees and Networks. Cambridge Univ. Press. Available at\string rdlyons/.
  • Lyons, R. and Schramm, O. (1999). Indistinguishability of percolation clusters. Ann. Probab. 27 1809--1836.
  • Morris, B. (2003). The components of the wired spanning forest are recurrent. Probab. Theory Related Fields 125 259--265.
  • Newman, C. M. (1997). Topics in Disordered Systems. Birkhäuser, Basel.
  • Newman, C. M. and Stein, D. L. (1996). Ground-state structure in a highly disordered spin-glass model. J. Statist. Phys. 82 1113--1132.
  • Pemantle, R. (1991). Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19 1559--1574.
  • Pemantle, R. and Peres, Y. (2000). Nonamenable products are not treeable. Israel J. Math. 118 147--155.
  • Peres, Y. and Steif, J. E. (1998). The number of infinite clusters in dynamical percolation. Probab. Theory Related Fields 111 141--165.
  • Schonmann, R. H. (1999). Percolation in $\infty+1$ dimensions at the uniqueness threshold. In Perplexing Problems in Probability (M. Bramson and R. Durrett, eds.) 53--67. Birkhäuser, Boston.
  • Schonmann, R. H. (1999). Stability of infinite clusters in supercritical percolation. Probab. Theory Related Fields 113 287--300.
  • Timár, A. (2006). Ends in free minimal spanning forests. Ann. Probab. 34 865--869.
  • van den Berg, J. and Kesten, H. (1985). Inequalities with applications to percolation and reliability. J. Appl. Probab. 22 556--569.