Uniform and minimal essential spanning forests on trees

Random Structures and Algorithms

Volumn 12, Issue 1, 1998, Pages 27-50

  Häggström, Olle ^a,b  

^a UTRECHT UNIVERSITY   (Netherlands)

^b CHALMERS UNIVERSITY OF TECHNOLOGY   (Sweden)

Author keywords

Minimal spanning tree; Random spanning forest; Random walk; Random cluster model; Uniform spanning tree

Indexed keywords

EID: 0040759189     PISSN: 10429832     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1098-2418(199801)12:1<27::AID-RSA2>3.0.CO;2-V     Document Type: Article

References (26)

1. D. Aldous, The random walk construction of uniform spanning trees and uniform labeled trees, SIAM J. Discrete Math., 3, 450-465 (1990).
2. D. Aldous, A random tree model associated with random graphs, Random Structures Algorithms, 1, 383-402 (1990).
3. D. Aldous, Asymptotic fringe distributions for general families of random trees, Ann. Appl. Probab., 1, 228-266 (1991).
4. D. Aldous and J. M. Steele, Asymptotics for Euclidean minimal spanning trees on random points, Probab. Theory Related Fields, 92, 247-258 (1992).
5. K. Alexander, Percolation and minimal spanning forests in infinite graphs, Ann. Probab. 23, 87-104 (1995).
6. I. Benjamini, R. Lyons, Y. Peres, and O. Schramm, Uniform spanning forests, in preparation.
7. A. Broder, Generating random spanning trees, Symposium on the Foundations of Computer Science, IEEE, New York, 1989, pp. 442-447.
8. R. Burton and R. Pemantle, Local characteristics, entropy and limit theorems for spanning trees and domino tiltings via transfer impedances, Ann. Probab., 21, 1329-1371 (1993).
9. J. T. Chayes, L. Chayes, and C. M. Newman, The stochastic geometry of invasion percolation, Commun. Math. Phys., 101, 383-407 (1985).
10. P. Doyle and J. L. Snell, Random Walks and Electric Networks, Mathematical Monograph Vol. 22, Mathematical Association of America, Washington, DC, 1984.
11. C. M. Fortuin and P. W. Kasteleyn, On the random-cluster model I. Introduction and relation to other models, Physica, 57, 536-564 (1972).
12. G. Grimmett, Percolative problems, in Probability and Phase Transition, G. Grimmett, Ed., Kluwer, Dordrecht, 1994, pp. 69-86.
13. H. Haase, On the incipient cluster of the binary tree, Arch. Math., 63, 465-471 (1994).
14. O. Häggström, Random-cluster measures and uniform spanning trees, Stochastic Process. Appl., 59, 267-275 (1995).
15. O. Häggström, The random-cluster model on a homogeneous tree, Probab. Theory Related Fields, 104, 231-253 (1996).
16. O. Häggström, Almost sure quasilocality fails for the random-cluster model on a tree, J. Statist. Phys., 84, 1351-1361 (1996).
17. O. Häggström, Infinite clusters in dependent automorphism invariant percolation on trees, Ann. Probab. (1997), to appear.
18. O. Häggström, Random-cluster representations in the study of phase transitions, preprint, 1997.
19. H. Kesten, Subdiffusive behavior of random walk on a random cluster, Ann. Inst. H. Poincaré Probab. Statist., 22, 425-487 (1986).
20. R. Lyons, Random walks and percolation on trees, Ann. Probab., 18, 931-958 (1990).
21. R. Lyons, Random walks, capacity and percolation on trees, Ann. Probab., 20, 2043-2088 (1992).
22. R. Lyons, R. Pemantle, and Y. Peres, Conceptual proofs of L log L criteria for mean behavior of branching processes, Ann. Probab., 23, 1125-1138 (1995).
23. J. Machta, Y. S. Choi, A. Lucke, T. Schweizer, and L. Chayes, Invaded cluster algorithms for equilibrium critical points, Phys. Rev. Lett., 75, 2792-2795 (1995).
24. C. M. Newman and D. L. Stein, Ground-state structure in a highly disordered spin-glass model, J. Statist. Phys., 82, 1113-1132 (1996).
25. B. Nickel and D. Wilkinson, Invasion percolation on the Caley tree: exact solution of a modified percolation model, Phys. Rev. Lett., 51, 71-74 (1983).
26. R. Pemantle, Choosing a spanning tree for the integer lattice uniformly, Ann. Probab., 19, 1559-1574 (1991).