Journal of Statistical Physics

, Volume 75, Issue 1–2, pp 123–134 | Cite as

Nonuniversality and continuity of the critical covered volume fraction in continuum percolation

  • Ronald Meester
  • Rahul Roy
  • Anish Sarkar
Articles

Abstract

We establish, using mathematically rigorous methods, that the critical covered volume fraction (CVF) for a continuum percolation model with overlapping balls of random sizes is not a universal constant independent of the distribution of the size of the balls. In addition, we show that the critical CVF is a continuous function of the distribution of the radius random variable, in the sense that if a sequence of random variables converges weakly to some random variable, then the critical CVF based on these random variables converges to the critical CVF of the limiting random variable.

Key Words

Poisson point process continuum percolation critical intensity covered volume fraction 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. T. Gawlinski and S. Redner, Monte Carlo renormalisation group for continuum percolation with excluded volume interactions,J. Phys. A: Math. Gen. 16:1063–1071 (1983).CrossRefGoogle Scholar
  2. 2.
    P. Hall, On continuum percolation,Ann. Prob. 13:1250–1260 (1985).Google Scholar
  3. 3.
    P. Hall,Introduction to the Theory of Coverage Processes (Wiley, New York, 1988).Google Scholar
  4. 4.
    J. Kertesz and T. Vicsek, Monte Carlo renormalisation group study of the percolation problem of discs with a distribution of radii,Z. Physik B 45:345–350 (1982).CrossRefGoogle Scholar
  5. 5.
    H. Kesten,Percolation Theory for Mathematicians (Birkhäuser, Boston, 1982).Google Scholar
  6. 6.
    M. V. Menshikov, Coincidence of critical points in percolation problems,Sov. Math. Dokl. 33:856–859 (1986).Google Scholar
  7. 7.
    M.K. Phani and D. Dhar, Continuum percolation with discs having a distribution of adii.J. Phys. A: Math. Gen. 17:L645-L649 (1984).CrossRefGoogle Scholar
  8. 8.
    G. E. Pike and C. H. Seager, Percolation and conductivity: A computer study I,Phys. Rev. B 10:1421–1446 (1974).CrossRefGoogle Scholar
  9. 9.
    R. Roy, The Russo-Seymour-Welsh theorem and the equality of critical densities and the ‘dual’ critical densities for continuum percolation on ℝ2,Ann. Prob. 18:1563–1575 (1990).Google Scholar
  10. 10.
    H. Scher and R. Zallen, Critical density in percolation processes,J. Chem. Phys. 53:3759–3761 (1970).CrossRefGoogle Scholar
  11. 11.
    S. A. Zuev and A. F. Sidorenko, Continuous models of percolation theory I,Theor. Math. Phys. 62:76–88 (1985).Google Scholar

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • Ronald Meester
    • 1
  • Rahul Roy
    • 2
  • Anish Sarkar
    • 2
  1. 1.Department of MathematicsUniversity of UtrechtUtrechtThe Netherlands
  2. 2.Indian Statistical InstituteNew DelhiIndia

Personalised recommendations