Negative association in uniform forests and connected graphs

Random Structures and Algorithms

Volumn 24, Issue 4, 2004, Pages 444-460

  Grimmett, G R ^a   Winkler, S N ^a  

^a UNIVERSITY OF CAMBRIDGE   (United Kingdom)

Author keywords

Negative association; Uniform connected graph; Uniform forest; Uniform spanning tree

Indexed keywords

EID: 11144283653     PISSN: 10429832     EISSN: None     Source Type: Journal    
DOI: 10.1002/rsa.20012     Document Type: Article

References (21)

1. N. Alon, A. Frieze, and D. Welsh, Polynomial time randomized approximation schemes for Tutte-Gröthendieck invariants: The dense case, Random Structures Algorithms 6 (1995), 459-478.
2. J. D. Annan, A randomised approximation algorithm for counting the number of forests in dense graphs, Combin Probab Comput 3 (1994), 273-283.
3. I. Benjamini, R. Lyons, Y. Peres, and O. Schramm, Uniform spanning forests, Ann Probab 29 (2001), 1-65.
4. B. Bollobás, Graph theory, an introductory course, Springer-Verlag, Berlin, 1979.
5. B. Bollobás, Random graphs, Academic Press, London, 1985.
6. P. G. Doyle and E. L. Snell, Random walks and electric networks, Carus Mathematical Monograph No. 22, American Mathematical Association, Washington, DC, 1984.
7. T. Feder and M. Mihail, Balanced matroids, Proc 24th ACM Symp Theory of Computing, 1992, pp. 26-38.
8. C. M. Fortuin and P. W. Kasteleyn, On the random-cluster model. I. Introduction and relation to other models, Physica 57 (1972), 536-564.
9. G. R. Grimmett, Percolation, 2nd edition, Springer-Verlag, Berlin, 1999.
10. G. R. Grimmett, The random-cluster model, Probability on discrete structures, Encyclopedia of Mathematical Sciences, Volume 110, H. Kesten (Editor), Springer-Verlag, Berlin, 2003, pp. 73-123.
11. G. R. Grimmett, The random-cluster model (2003), in preparation.
12. O. Häggström, Random-cluster measures and uniform spanning trees, Stochastic Process Appl 59(1995), 267-275.
13. S. Janson, T. Łuczak, and A. Ruciński, Random graphs, Wiley, New York, 2000.
14. M. Jerrum, Counting trees in a graph is #P-complete, Inform Process Lett 51 (1994), 111-116.
15. J. Kahn, A normal law for matchings, Combinatorica 20 (2000), 339-391.
16. G. Kirchhoff, Uber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Strome gefuhrt wird, Anna Phys Chem 72 (1847), 497-508.
17. R. Lyons and Y. Peres, Probability on trees and networks, http://www.math.gatech.edu/~rdlyons/prbtree/prbtree.html, Cambridge University Press, Cambridge, to appear.
18. B. McKay, Practical graph isomorphism, Congr Numer 30 (1981), 45-87.
19. R. Pemantle, Towards a theory of negative dependence, J Math Phys 41 (2000), 1371-1390.
20. J. H. van Lint and R. M. Wilson, A course in combinatorics, 2nd edition. Cambridge University Press, Cambridge, 2001.
21. D. L. Vertigan and D. J. A. Welsh, The computational complexity of the Tutte plane: The bipartite case, Combin Probab Comput 1 (1992), 181-187.