Advertisement

Journal of Statistical Physics

, Volume 64, Issue 3–4, pp 843–850 | Cite as

Euler characteristic and related measures for random geometric sets

  • K. R. Mecke
  • H. Wagner
Short Communications

Abstract

By an elementary calculation we obtain the exact mean values of Minkowksi functionals for a standard model of percolating sets. In particular, a recurrence theorem for the mean Euler characteristic recently put forward is shown to be incorrect. Related previous mathematical work is mentioned. We also conjecture bounds for the threshold density of continuum percolation, which are associated with the Euler characteristic.

Key words

Random sets integral geometry Minkowski functionals Euler characteristic continuum percolation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    B. L. Okun,J. Stat. Phys. 59:523 (1990).Google Scholar
  2. 2.
    S. B. Gray,IEEE Trans. Computers 20:551 (1971).Google Scholar
  3. 3.
    T. Hofsäss and H. Kleinert,J. Chem. Phys. 86:3565 (1987).Google Scholar
  4. 4.
    D. M. Anderson, H. T. Davis, and L. E. Scriven,J. Chem. Phys. 91:3246 (1989).Google Scholar
  5. 5.
    A. L. Melott,Phys. Rep. 193:1 (1990).Google Scholar
  6. 6.
    H. Hadwiger,Vorlesungen über Inhalt, Oberfläche und Isoperimetrie (Springer, 1957).Google Scholar
  7. 7.
    P. Hall,Introduction to the Theory of Coverage Processes (Wiley, 1988).Google Scholar
  8. 8.
    L. A. Santaló,Integral Geometry and Geometric Probability (Addison-Wesley, 1976).Google Scholar
  9. 9.
    P. Davy,J. Appl. Prob. 13:714 (1976).Google Scholar
  10. 10.
    H. G. Kellerer,Z. Wahrsch. Verw. Geb. 67:63 (1984).Google Scholar
  11. 11.
    K. R. Mecke and H. Wagner, to be published.Google Scholar
  12. 12.
    M. F. Sykes and J. W. Essam,J. Math. Phys. 5:1117 (1964).Google Scholar
  13. 13.
    H. Scher and R. Zallen,J. Chem. Phys. 53:3759 (1970).Google Scholar
  14. 14.
    G. E. Pike and C. H. Seager,Phys. Rev. B 10:1421 (1974).Google Scholar
  15. 15.
    I. Balberg,Phil. Mag. B 56:991 (1987).Google Scholar
  16. 16.
    E. Charlaix,J. Phys. A: Math. Gen. 19:L533 (1986).Google Scholar

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • K. R. Mecke
    • 1
  • H. Wagner
    • 1
  1. 1.Sektion Physik der Universität MünchenMünchen 2Germany

Personalised recommendations