Special invited paper uniform spanning forests

Annals of Probability

Volumn 29, Issue 1, 2001, Pages 1-65

  Benjamini, Itai ^a   Lyons, Russell ^b   Peres, Yuval ^c,d   Schramm, Oded ^a  

^a WEIZMANN INSTITUTE OF SCIENCE   (Israel)

^b INDIANA UNIVERSITY   (United States)

^c HEBREW UNIVERSITY OF JERUSALEM   (Israel)

^d UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Amenability; Cayley graphs; Electrical networks; Harmonic dirichlet functions; Loop erased walk; Percolation; Spanning trees

Indexed keywords

EID: 0040434878     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/aop/1008956321     Document Type: Article

References (69)

1. 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.
2. ALDOUS, D. (1990). The random walk construction of uniform random spanning trees and uniform labelled trees. SIAM J. Disc. Math. 3 450-465.
3. ALEXANDER, K. S. (1995). Percolation and minimal spanning forests in infinite graphs. Ann. Probab. 23 87-104.
4. BECKENBACH, E. F. and BELLMAN, R. (1965). Inequalities (revised). Springer, New York.
5. 2-Betti number. Potential Anal. 6 313-326.
6. BENJAMINI, I., KESTEN, H., PERES, Y. and SCHRAMM, O. (1998). Geometry of the uniform spanning forest: transitions in dimensions 4, 8, 12, . . . . Unpublished manuscript.
7. BENJAMINI, I., LYONS, R., PERES, Y. and SCHRAMM, O. (1999). Group-invariant percolation on graphs. Geom. Funct. Anal. 9 29-66.
8. 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.
9. BENJAMINI, I. and SCHRAMM, O. (1996a). Harmonic functions on planar and almost planar graphs and manifolds, via circle packings. Invent. Math. 126 565-587.
10. BENJAMINI, I. and SCHRAMM, O. (1996b). Random walks and harmonic functions on infinite planar graphs using square tilings. Ann. Probab. 24 1219-1238.
11. d, many questions and a few answers. Electronic Comm. Probab. 1 71-82.
12. BENJAMINI, I. and SCHRAMM, O. (1998). Conformal invariance of Voronoi percolation. Comm. Math. Phys. 197 75-107.
13. BRODER, A. (1989). Generating random spanning trees. In Thirtieth Annual Symposium Foundations Computer Sci. 442-447. IEEE, New York.
14. 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.
15. BURTON, R. M. and KEANE, M. (1989). Density and uniqueness in percolation. Comm. Math. Phys. 121 501-505.
16. 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.
17. CHABOUD, T. and KENYON, C. (1996). Planar Cayley graphs with regular dual. Internat. J. Algebra Comput. 6 553-561.
18. 2 harmonic forms on rotationally symmetric Riemannian manifolds. Proc. Amer. Math. Soc. 77 395-400.
19. DODZIUK, J. (1984). Difference equations, isoperimetric inequality, and transience of certain random walks. Trans. Amer. Math. Soc. 284 787-794.
20. DOYLE, P. G. (1988). Electric currents in infinite networks. Unpublished manuscript. Available at www.math.dartmouth.edu/~doyle.
21. DOYLE, P. G. and SNELL, J. L. (1984). Random Walks and Electric Networks. Mathematical Assoc. of America, Washington, D.C.
22. DUPLANTIER, B. (1992). Loop-erased self-avoiding walks in 2D. Phys. A 191 516-522.
23. FEDER, T. and MIHAIL, M. (1992). Balanced matroids. In Proceedings 24th Annual ACM Symposium. Theory Computing 26-38. ACM Press, New York.
24. FURSTENBERG, H. (1981). Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton Univ. Press.
25. GEORGII, H.-O. (1988). Gibbs Measures and Phase Transitions. de Gruyter, Berlin.
26. GERL, P. (1988). Random walks on graphs with a strong isoperimetric property. J. Theoret. Probab. 1 171-187.
27. GRIMMETT, G. R. (1995). The stochastic random-cluster process and the uniqueness of random-cluster measures. Ann. Probab. 23 1461-1510.
28. GROMOV, M. (1981). Groups of polynomial growth and expanding maps. Publ. Math. I.H.E.S. 53 53-73.
29. HÄGGSTRÖM, O. ( 1995). Random-cluster measures and uniform spanning trees. Stochastic Process. Appl. 59 267-275.
30. HÄGGSTRÖM, O. (1997). Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab. 25 1423-1436.
31. HÄGGSTRÖM, O. (1998). Uniform and minimal essential spanning forests on trees. Random Structures Algebras 12 27-50.
32. HEBISCH, W. and SALOFF-COSTE, L. (1993). Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. 21 673-709.
33. HOLOPAINEN, I. and SOARDI, P. M. (1997). p-harmonic functions on graphs and manifolds. Manuscripta Math. 94 95-110.
34. KANAI, M. (1986). Rough isometries and the parabolicity of riemannian manifolds, J. Math. Soc. Japan 38 227-238.
35. KENYON, R. (1997). Conformal invariance of domino tiling. Ann. Probab. To appear.
36. 2. J. Math. Phys. 41 1338-1363.
37. KESTEN, H. (1959). Symmetric random walks on groups. Trans. Amer. Math. Soc. 92 336-354.
38. KESTEN, H. (1986). Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Probab. Statist. 22 425-487.
39. KIRCHHOFF, G. (1847). Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird. Ann. Phys. Chem. 72 497-508.
40. LANGLANDS, R., POULIOT, P. and SAINT-AUBIN, Y. (1994). Conformal invariance in two-dimensional percolation. Bull. Amer. Math. Soc. (N.S.) 30 1-61.
41. LAWLER, G. ( 1988). Loop-erased self-avoiding random walk in two and three dimensions. J. Statist. Phys. 50 91-108.
42. LAWLER, G. (1991). Intersections of Random Walks. Birkhäuser, Boston.
43. LAWLER, G. (1999). A lower bound on the growth exponent for loop-erased random walk in two dimensions. ESAIM Probab. Statist. 3 1-21.
44. LYONS, R. (1990). Random walks and percolation on trees. Ann. Probab. 18 931-958.
45. LYONS, R. (1992). Random walks, capacity, and percolation on trees. Ann. Probab. 20 2043-2088.
46. LYONS, R. (1998). A bird's-eye view of uniform spanning trees and forests. In Microsurveys in Discrete Probability (D. Aldous and J. Propp eds.) 135-162. Amer. Math. Soc., Providence, RI.
47. LYONS, R. and PERES, Y. (1997). Probability on Trees and Networks, Cambridge Univ. Press. To appear. Current version available at php.indiana.edu/~rdlyons/.
48. LYONS, R., PERES, Y. and SCHRAMM, O. (1998). Intersections of Markov chains and their loop-erasures. Unpublished manuscript.
49. LYONS, R. and SCHRAMM, O. (1999). Indistinguishability of percolation clusters. Ann. Probab. 27 1809-1836.
50. MAJUMDAR, S. N. (1992). Exact fractal dimension of the loop-erased self-avoiding random walk in two dimensions. Phys. Rev. Lett. 68 2329-2331.
51. MEDOLLA, G. and SOARDI, P. M. (1995). Extension of Foster's averaging formula to infinite networks with moderate growth. Math. Z. 219 171-185.
52. MOHAR, B. (1991). Some relations between analytic and geometric properties of infinite graphs. Discrete Math. 95 193-219.
53. PEMANTLE, R. (1991). Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19 1559-1574.
54. PEMANTLE, R. (2000). Towards a theory of negative dependence. J. Math. Phys. 41 1371-1390.
55. PEMANTLE, R. and PERES, Y. (1995). Critical random walk in random environment on trees. Ann. Probab. 23 105-140.
56. PROPP, J. G. (1997). Continuum spanning trees. Available at math.wisc.edu/'propp/continuum.ps.
57. PROPP, J. G. and WILSON, D. B. (1998). How to 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.
58. SARIO, L., NAKAI, M., WANG, C. and CHUNG, L. O. (1977). Classification Theory of Riemannian Manifolds. Springer, Berlin.
59. SCHRAMM, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221-288.
60. SOARDI, P. M. (1993). Rough isometries and Dirichlet finite harmonic functions on graphs. Proc. Amer. Math. Soc. 119 1239-1248.
61. SOARDI, P. M. (1994). Potential Theory on Infinite Networks. Springer, Berlin.
62. SOLOMYAK, R. (1999). Essential spanning forests and electrical networks on groups. J. Theoret. Probab. 12 523-548.
63. STRASSEN, V. ( 1965). The existence of probability measures with given marginals. Ann. Math. Statist. 36 423-439.
64. THOMASSEN, C. (1989). Transient random walks, harmonic functions, and electrical currents in infinite electrical networks. Mat-Report 1989-07, Technical Univ. Denmark.
65. THOMASSEN, C. (1990). Resistances and currents in infinite electrical networks. J. Combin. Theory Ser. B 49 87-102.
66. TÓTH, B. and WERNER, W. (1998). The true self-repelling motion. Probab. Theory Related Fields 111 375-452.
67. VAN DEN BERG, J. and KESTEN, H. (1985). Inequalities with applications to percolation and reliability. J. Appl. Probab. 22 556-569.
68. VAROPOULOS, N. TH. (1985). Isoperimetric inequalities and Markov chains. J. Funct. Anal. 63 215-239.
69. 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, New York.