## The Annals of Probability

### Uniform spanning forests

#### Abstract

We study uniform spanning forest measures on infinite graphs, which are weak limits of uniform spanning tree measures from finite subgraphs. These limits can be taken with free (FSF )or wired (WSF ) boundary conditions. Pemantle proved that the free and wired spanning forests coincide in $\mathbb{Z}^d$ and that they give a single tree iff $d </4$.

In the present work, we extend Pemantle’s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks, potential theory, invariant percolation and amenability. The uniform spanning forest model is related to random cluster models in statistical physics, but, because of the preceding connections, its analysis can be carried further. Among our results are the following:

The FSF and WSF in a graph G coincide iff all harmonic Dirichlet functions on G are constant.

The tail $\sigma$-fields of the WSF and the FSF are trivial on any graph.

On any Cayley graph that is not a finite extension of f $\mathbbf{Z}$ all component trees of the WSF have one end; this is new in $\mathbb{Z}^d$ for $d \ge 5. On any tree, as well as on any graph with spectral radius less than 1, a.s. all components of the WSF are recurrent. The basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space$\mathbb{H}^d$is analyzed. A Cayley graph is amenable iff for all$\epsilon > 0$, the union of the WSF and Bernoulli percolation with parameter$\epsilon\$ is connected.

Harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees.

We also present numerous open problems and conjectures.

#### Article information

Source
Ann. Probab., Volume 29, Number 1 (2001), 1-65.

Dates
First available in Project Euclid: 21 December 2001

https://projecteuclid.org/euclid.aop/1008956321

Digital Object Identifier
doi:10.1214/aop/1008956321

Mathematical Reviews number (MathSciNet)
MR1825141

Zentralblatt MATH identifier
1016.60009

#### Citation

Benjamini, Itai; Lyons, Russell; Peres, Yuval; Schramm, Oded. Uniform spanning forests. Ann. Probab. 29 (2001), no. 1, 1--65. doi:10.1214/aop/1008956321. https://projecteuclid.org/euclid.aop/1008956321

#### References

• Aizenman, M., Burchard, A., Newman, C. and Wilson, D. (1999). Scaling limits for minimal and random spanning trees in two dimensions. Random Structures Algorithms 15 319-367.
• Aldous, D. (1990). The random walk construction of uniform random spanning trees and uniform labelled trees. SIAM J. Disc. Math. 3 450-465.
• Alexander, K. S. (1995). Percolation and minimal spanning forests in infinite graphs. Ann. Probab. 23 87-104.
• Beckenbach, E. F. and Bellman, R. (1965). Inequalities (revised). Springer, NewYork.
• Bekka, M. E. B. and Valette, A. (1997). Group cohomology, harmonic functions and the first L2-Betti number. Potential Anal. 6 313-326.
• Benjamini, I., Kesten, H., Peres, Y. and Schramm, O. (1998). Geometry of the uniform spanning forest: transitions in dimensions 4, 8, 12,. Unpublished manuscript.
• 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. and Schramm, O. (1999). Percolation perturbations in potential theory and random walks. In Random Walks and Discrete Potential Theory (M. Picardello and W. Woess, eds.) 56-84. Cambridge Univ. Press.
• Benjamini, I. and Schramm, O. (1996a). Harmonic functions on planar and almost planar graphs and manifolds, via circle packings. Invent. Math. 126 565-587.
• Benjamini, I. and Schramm, O. (1996b). Random walks and harmonic functions on infinite planar graphs using square tilings. Ann. Probab. 24 1219-1238.
• Benjamini, I. and Schramm, O. (1996c). Percolation beyond d, many questions and a few answers. Electronic Comm. Probab. 1 71-82.
• Benjamini, I. and Schramm, O. (1998). Conformal invariance of Voronoi percolation. Comm. Math. Phys. 197 75-107.
• Broder, A. (1989). Generating random spanning trees. In Thirtieth Annual Symposium Foundations Computer Sci. 442-447. IEEE, NewYork.
• Brooks, R. L., Smith, C. A. B., Stone, A. H. and Tutte, W. T. (1940). The dissection of rectangles into squares. Duke Math. J. 7 312-340.
• Burton, R. M. and Keane, M. (1989). Density and uniqueness in percolation. Comm. Math. Phys. 121 501-505.
• Burton, R. and Pemantle, R. (1993). Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances. Ann. Probab. 21 1329-1371.
• Chaboud, T. and Kenyon, C. (1996). Planar Cayley graphs with regular dual. Internat. J. Algebra Comput. 6 553-561.
• Dodziuk, J. (1979). L2 harmonic forms on rotationally symmetric Riemannian manifolds. Proc. Amer. Math. Soc. 77 395-400.
• Dodziuk, J. (1984). Difference equations, isoperimetric inequality, and transience of certain random walks. Trans. Amer. Math. Soc. 284 787-794.
• Doyle, P. G. (1988). Electric currents in infinite networks. Unpublished manuscript. Available at www.math.dartmouth.edu/ doyle.
• Doyle, P. G. and Snell, J. L. (1984). Random Walks and Electric Networks. Mathematical Assoc. of America, Washington, D.C.
• Duplantier, B. (1992). Loop-erased self-avoiding walks in 2D. Phys. A 191 516-522.
• Feder, T. and Mihail, M. (1992). Balanced matroids. In Proceedings 24th Annual ACM Symposium. Theory Computing 26-38. ACM Press, NewYork.
• Furstenberg, H. (1981). Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton Univ. Press.
• Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. de Gruyter, Berlin.
• Gerl, P. (1988). Random walks on graphs with a strong isoperimetric property. J. Theoret. Probab. 1 171-187.
• Grimmett, G. R. (1995). The stochastic random-cluster process and the uniqueness of randomcluster measures. Ann. Probab. 23 1461-1510.
• Gromov, M. (1981). Groups of polynomial growth and expanding maps. Publ. Math. I.H.E.S. 53 53-73.
• H¨aggstr ¨om, O. (1995). Random-cluster measures and uniform spanning trees. Stochastic Process. Appl. 59 267-275.
• H¨aggstr ¨om, O. (1997). Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab. 25 1423-1436.
• H¨aggstr ¨om, O. (1998). Uniform and minimal essential spanning forests on trees. Random Structures Algebras 12 27-50.
• Hebisch, W. and Saloff-Coste, L. (1993). Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. 21 673-709.
• Holopainen, I. and Soardi, P. M. (1997). p-harmonic functions on graphs and manifolds. Manuscripta Math. 94 95-110.
• Kanai, M. (1986). Rough isometries and the parabolicity of riemannian manifolds, J. Math. Soc. Japan 38 227-238.
• Kenyon, R. (1997). Conformal invariance of domino tiling. Ann. Probab. To appear.
• Kenyon, R. (2000). Long-range properties of spanning trees in 2. J. Math. Phys. 41 1338-1363.
• Kesten, H. (1959). Symmetric random walks on groups. Trans. Amer. Math. Soc. 92 336-354.
• Kesten, H. (1986). Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincar´e Probab. Statist. 22 425-487.
• Kirchhoff, G. (1847). Ueber die Aufl¨osung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Str¨ome gef ¨uhrt wird. Ann. Phys. Chem. 72 497-508.
• Langlands, R., Pouliot, P. and Saint-Aubin, Y. (1994). Conformal invariance in twodimensional percolation. Bull. Amer. Math. Soc. (N.S.) 30 1-61.
• Lawler, G. (1988). Loop-erased self-avoiding random walk in two and three dimensions. J. Statist. Phys. 50 91-108.
• Lawler, G. (1991). Intersections of Random Walks. Birkh¨auser, Boston.
• Lawler, G. (1999). A lower bound on the growth exponent for loop-erased random walk in two dimensions. ESAIM Probab. Statist. 3 1-21.
• Lyons, R. (1990). Random walks and percolation on trees. Ann. Probab. 18 931-958.
• Lyons, R. (1992). Random walks, capacity, and percolation on trees. Ann. Probab. 20 2043-2088.
• Lyons, R. (1998). A bird's-eye viewof uniform spanning trees and forests. In Microsurveys in Discrete Probability (D. Aldous and J. Propp eds.) 135-162. Amer. Math. Soc., Providence, RI.
• Lyons, R. and Peres, Y. (1997). Probability on Trees and Networks, Cambridge Univ. Press. To appear. Current version available at php.indiana.edu/ rdlyons/.
• Lyons, R., Peres, Y. and Schramm, O. (1998). Intersections of Markov chains and their looperasures. Unpublished manuscript.
• Lyons, R. and Schramm, O. (1999). Indistinguishability of percolation clusters. Ann. Probab. 27 1809-1836.
• Majumdar, S. N. (1992). Exact fractal dimension of the loop-erased self-avoiding random walk in two dimensions. Phys. Rev. Lett. 68 2329-2331.
• Medolla, G. and Soardi, P. M. (1995). Extension of Foster's averaging formula to infinite networks with moderate growth. Math.219 171-185.
• Mohar, B. (1991). Some relations between analytic and geometric properties of infinite graphs. Discrete Math. 95 193-219.
• Pemantle, R. (1991). Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19 1559-1574.
• Pemantle, R. (2000). Towards a theory of negative dependence. J. Math. Phys. 41 1371-1390.
• Pemantle, R. and Peres, Y. (1995). Critical random walk in random environment on trees. Ann. Probab. 23 105-140.
• Propp, J. G. (1997). Continuum spanning trees. Available at math.wisc.edu/ propp/continuum.ps.
• Propp, J. G. and Wilson, D. B. (1998). Howto get a perfectly random sample from a generic Markov chain and generate a random spanning tree of a directed graph. J. Algorithms 27 170-217.
• Sario, L., Nakai, M., Wang, C. and Chung, L. O. (1977). Classification Theory of Riemannian Manifolds. Springer, Berlin.
• Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221-288.
• Soardi, P. M. (1993). Rough isometries and Dirichlet finite harmonic functions on graphs. Proc. Amer. Math. Soc. 119 1239-1248.
• Soardi, P. M. (1994). Potential Theory on Infinite Networks. Springer, Berlin.
• Solomyak, R. (1999). Essential spanning forests and electrical networks on groups. J. Theoret. Probab. 12 523-548.
• Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist. 36 423-439.
• Thomassen, C. (1989). Transient random walks, harmonic functions, and electrical currents in infinite electrical networks. Mat-Report 1989-07, Technical Univ. Denmark.
• Thomassen, C. (1990). Resistances and currents in infinite electrical networks. J. Combin. Theory Ser. B 49 87-102.
• T ´oth, B. and Werner, W. (1998). The true self-repelling motion. Probab. Theory Related Fields 111 375-452.
• van den Berg, J. and Kesten, H. (1985). Inequalities with applications to percolation and reliability. J. Appl. Probab. 22 556-569.
• Varopoulos, N. Th. (1985). Isoperimetric inequalities and Markov chains. J. Funct. Anal. 63 215-239.
• Wilson, D. B. (1996). Generating random spanning trees more quickly than the cover time. In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing 296-303. ACM, NewYork.