Skip to main content

Euler characteristic and related measures for random geometric sets


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.

This is a preview of subscription content, access via your institution.


  1. B. L. Okun,J. Stat. Phys. 59:523 (1990).

    Google Scholar 

  2. S. B. Gray,IEEE Trans. Computers 20:551 (1971).

    Google Scholar 

  3. T. Hofsäss and H. Kleinert,J. Chem. Phys. 86:3565 (1987).

    Google Scholar 

  4. D. M. Anderson, H. T. Davis, and L. E. Scriven,J. Chem. Phys. 91:3246 (1989).

    Google Scholar 

  5. A. L. Melott,Phys. Rep. 193:1 (1990).

    Google Scholar 

  6. H. Hadwiger,Vorlesungen über Inhalt, Oberfläche und Isoperimetrie (Springer, 1957).

  7. P. Hall,Introduction to the Theory of Coverage Processes (Wiley, 1988).

  8. L. A. Santaló,Integral Geometry and Geometric Probability (Addison-Wesley, 1976).

  9. P. Davy,J. Appl. Prob. 13:714 (1976).

    Google Scholar 

  10. H. G. Kellerer,Z. Wahrsch. Verw. Geb. 67:63 (1984).

    Google Scholar 

  11. K. R. Mecke and H. Wagner, to be published.

  12. M. F. Sykes and J. W. Essam,J. Math. Phys. 5:1117 (1964).

    Google Scholar 

  13. H. Scher and R. Zallen,J. Chem. Phys. 53:3759 (1970).

    Google Scholar 

  14. G. E. Pike and C. H. Seager,Phys. Rev. B 10:1421 (1974).

    Google Scholar 

  15. I. Balberg,Phil. Mag. B 56:991 (1987).

    Google Scholar 

  16. E. Charlaix,J. Phys. A: Math. Gen. 19:L533 (1986).

    Google Scholar 

Download references

Author information

Authors and Affiliations


Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mecke, K.R., Wagner, H. Euler characteristic and related measures for random geometric sets. J Stat Phys 64, 843–850 (1991).

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI:

Key words

  • Random sets
  • integral geometry
  • Minkowski functionals
  • Euler characteristic
  • continuum percolation