The Annals of Probability

The random minimal spanning tree in high dimensions

Mathew D. Penrose

Full-text: Open access

Abstract

For the minimal spanning tree on n independent uniform points in the d-dimensional unit cube, the proportionate number of points of degree k is known to converge to a limit $\alpha_{k,d}$ as $n \to \infty$. We show that $\alpha_{k,d}$ converges to a limit $\alpha_k$ as $d \to \infty$ for each k. The limit $\alpha_k$ arose in earlier work by Aldous, as the asymptotic proportionate number of vertices of degree k in the minimum-weight spanning tree on k vertices, when the edge weights are taken to be independent, identically distributed random variables. We give a graphical alternative to Aldous's characterization of the $\alpha_k$.

Article information

Source
Ann. Probab., Volume 24, Number 4 (1996), 1903-1925.

Dates
First available in Project Euclid: 6 January 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1041903210

Digital Object Identifier
doi:10.1214/aop/1041903210

Mathematical Reviews number (MathSciNet)
MR1415233

Zentralblatt MATH identifier
0866.60021

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 05C05: Trees 90C27: Combinatorial optimization

Keywords
Geometric probability minimal spanning tree vertex degrees continuum percolation invasion percolation

Citation

Penrose, Mathew D. The random minimal spanning tree in high dimensions. Ann. Probab. 24 (1996), no. 4, 1903--1925. doi:10.1214/aop/1041903210. https://projecteuclid.org/euclid.aop/1041903210


Export citation

References

  • 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. Asy mptotics 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 Ry ZIN, G. 1990. An asy mptotic 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 Sy stems 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 Analy sis, 3rd ed. McGraw-Hill, New York.
  • 21 SCHILLING, M. F. 1986. Mutual and shared neighbor probabilities: finiteand infinitedimensional 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.