The random minimal spanning tree in high dimensions

Annals of Probability

Volumn 24, Issue 4, 1996, Pages 1903-1925

  Penrose, Mathew D ^a  

^a DURHAM UNIVERSITY   (United Kingdom)

Author keywords

Continuum percolation; Geometric probability; Invasion percolation; Minimal spanning tree; Vertex degrees

Indexed keywords

EID: 0030353886     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/aop/1041903210     Document Type: Article

References (23)

1. ALDOUS, D. (1990). A random tree model associated with random graphs. Random Structures and Algorithms 1 383-402.
2. ALDOUS, D. and STEELE, J. M. (1992). Asymptotics for Euclidean minimal spanning trees on random points. Probab. Theory Related Fields 92 247-258.
3. ALEXANDER, K. (1995). Percolation and minimal spanning forests in infinite graphs. Ann. Probab. 23 87-104.
4. AVRAM, F. and BERTSIMAS, D. (1992). The minimum spanning tree constant in geometrical probability and under the independent model: a unified approach. Ann. Appl. Probab. 2 113-130.
5. BENTLEY, J. L. and FRIEDMAN, J. H. (1978). Fast algorithms for constructing minimal spanning trees in coordinate spaces. IEEE Trans. Comput. 27 97-105.
6. BERTSIMAS, D. J. and VAN RYZIN, G. (1990). An asymptotic determination of the minimum spanning tree and minimum matching constants in geometrical probability. Oper. Res. Lett. 9 223-231.
7. DURRETT, R. (1988). Lecture Notes on Particle Systems and Percolation. Wadsworth and Brooks Cole, Pacific Grove, CA.
8. DUSSERT, C., RASIGNI, M., PALMARI, J., RASIGNI, G., LLEBARIA, A. and MARTY, F. (1987). Minimal spanning tree analysis of biological structures. J. Theoret. Biol. 125 317-323.
9. FRIEDMAN, J. H. and RAFSKY, L. C. (1979). Multivariate generalizations of the Wolfowitz and Smirnov two-sample tests. Ann. Statist. 7 697-717.
10. FRIEDMAN, J. H. and RAFSKY, L. C. (1983). Graph-theoretic measures of multivariate association and prediction. Ann. Statist. 11 377-391.
11. 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.
12. KANG, A. N. C., LEE, R. C. T., CHANG, C.-L. and CHANG, S.-K. (1977). Storage reduction through minimal spanning trees and spanning forests. IEEE Trans. Comput. 26 425-434.
13. KESTEN, H. and LEE, S. (1996). The central limit theorem for weighted minimal spanning trees on random points. Ann. Appl. Probab. 6 495-527.
14. MEESTER, R. W. J. and ROY, R. (1995). Continuum Percolation. Cambridge Univ. Press.
15. NEWMAN, C. M., RINOTT, Y. and TVERSKY, A. (1983). Nearest neighbors and Voronoi regions in certain point processes. Adv. in Appl. Probab. 15 726-751.
16. PENROSE, M. D. (1993). On the spread-out limit for bond and continuum percolation. Ann. Appl. Probab. 3 253-256.
17. PENROSE, M. D. (1996). Continuum percolation and Euclidean minimal spanning trees in high dimensions. Ann. Appl. Probab. 6 528-544.
18. RESNICK, S. I. (1987). Extreme Values, Point Processes and Regular Variation. Springer, New York.
19. ROHLF, F. J. (1975). Generalization of the gap test for the detection of multivariate outliers. Biometrics 31 93-101.
20. RUDIN, W. (1987). Real and Complex Analysis, 3rd ed. McGraw-Hill, New York.
21. SCHILLING, M. F. (1986). Mutual and shared neighbor probabilities: finite- and infinite-dimensional results. Adv. in Appl. Probab. 18 388-405.
22. STEELE, J. M. (1993). Probability and problems in Euclidean combinatorial optimization. Statist. Sci. 8 48-56.
23. STEELE, J. M., SHEPP, L. A. and EDDY, W. F. (1987). On the number of leaves of a Euclidean minimal spanning tree. J. Appl. Probab. 24 809-826.