×

zbMATH — the first resource for mathematics

Nonuniversality and continuity of the critical covered volume fraction in continuum percolation. (English) Zbl 0828.60083
Summary: 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.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
PDF BibTeX XML Cite
Full Text: DOI
References:
[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).
[2] P. Hall, On continuum percolation,Ann. Prob. 13:1250–1260 (1985). · Zbl 0588.60096
[3] P. Hall,Introduction to the Theory of Coverage Processes (Wiley, New York, 1988). · Zbl 0659.60024
[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). · Zbl 1342.60170
[5] H. Kesten,Percolation Theory for Mathematicians (Birkhäuser, Boston, 1982). · Zbl 0522.60097
[6] M. V. Menshikov, Coincidence of critical points in percolation problems,Sov. Math. Dokl. 33:856–859 (1986). · Zbl 0615.60096
[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).
[8] G. E. Pike and C. H. Seager, Percolation and conductivity: A computer study I,Phys. Rev. B 10:1421–1446 (1974).
[9] R. Roy, The Russo-Seymour-Welsh theorem and the equality of critical densities and the ’dual’ critical densities for continuum percolation on \(\mathbb{R}\)2,Ann. Prob. 18:1563–1575 (1990). · Zbl 0719.60119
[10] H. Scher and R. Zallen, Critical density in percolation processes,J. Chem. Phys. 53:3759–3761 (1970).
[11] S. A. Zuev and A. F. Sidorenko, Continuous models of percolation theory I,Theor. Math. Phys. 62:76–88 (1985).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.