The Annals of Applied Probability

Subcritical regimes in some models of continuum percolation

Jean-Baptiste Gouéré

Full-text: Open access

Abstract

We consider some continuum percolation models. We are mainly interested in giving some sufficient conditions for the absence of percolation. We give some general conditions and then focus on two examples. The first one is a multiscale percolation model based on the Boolean model. It was introduced by Meester and Roy and subsequently studied by Menshikov, Popov and Vachkovskaia. The second one is based on the stable marriage of Poisson and Lebesgue introduced by Hoffman, Holroyd and Peres and whose percolation properties have been studied by Freire, Popov and Vachkovskaia.

Article information

Source
Ann. Appl. Probab., Volume 19, Number 4 (2009), 1292-1318.

Dates
First available in Project Euclid: 27 July 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1248700618

Digital Object Identifier
doi:10.1214/08-AAP575

Mathematical Reviews number (MathSciNet)
MR2538071

Zentralblatt MATH identifier
1172.60335

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35] 60G55: Point processes

Keywords
Continuum percolation Boolean percolation multiscale percolation marriage of Poisson and Lebesgue

Citation

Gouéré, Jean-Baptiste. Subcritical regimes in some models of continuum percolation. Ann. Appl. Probab. 19 (2009), no. 4, 1292--1318. doi:10.1214/08-AAP575. https://projecteuclid.org/euclid.aoap/1248700618


Export citation

References

  • [1] Freire, M. V., Popov, S. and Vachkovskaia, M. (2007). Percolation for the stable marriage of Poisson and Lebesgue. Stochastic Process. Appl. 117 514–525.
  • [2] Gale, D. and Shapley, L. S. (1962). College admissions and the stability of marriage. Amer. Math. Monthly 69 9–15.
  • [3] Gouéré, J.-B. (2008). Subcritical regimes in the Poisson Boolean model of continuum percolation. Ann. Probab. 36 1209–1220.
  • [4] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin.
  • [5] Hall, P. (1985). On continuum percolation. Ann. Probab. 13 1250–1266.
  • [6] Hall, P. (1988). Introduction to the Theory of Coverage Processes. Wiley, New York.
  • [7] Hoffman, C., Holroyd, A. E. and Peres, Y. (2006). A stable marriage of Poisson and Lebesgue. Ann. Probab. 34 1241–1272.
  • [8] Hoffman, C., Holroyd, A. E. and Peres, Y. (2008). Tail bounds for the stable marriage of Poisson and Lebesgue. Canad. J. Math. To appear. Available at arXiv:math/0507324.
  • [9] Kallenberg, O. (1986). Random Measures, 4th ed. Akademie-Verlag, Berlin.
  • [10] Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge Tracts in Mathematics 119. Cambridge Univ. Press, Cambridge.
  • [11] Meester, R., Roy, R. and Sarkar, A. (1994). Nonuniversality and continuity of the critical covered volume fraction in continuum percolation. J. Statist. Phys. 75 123–134.
  • [12] Menshikov, M. V., Popov, S. Y. and Vachkovskaia, M. (2001). On the connectivity properties of the complementary set in fractal percolation models. Probab. Theory Related Fields 119 176–186.
  • [13] Menshikov, M. V., Popov, S. Y. and Vachkovskaia, M. (2003). On a multiscale continuous percolation model with unbounded defects. Bull. Braz. Math. Soc. (N.S.) 34 417–435.
  • [14] Menshikov, M. V. and Sidorenko, A. F. (1987). Coincidence of critical points in Poisson percolation models. Teor. Veroyatnost. i Primenen. 32 603–606.
  • [15] Møller, J. (1994). Lectures on Random Voronoĭ Tessellations. Lecture Notes in Statistics 87. Springer, New York.
  • [16] Neveu, J. (1977). Processus ponctuels. In École D’Été de Probabilités de Saint-Flour, VI—1976. Lecture Notes in Math. 598 249–445. Springer, Berlin.
  • [17] Zuev, S. A. and Sidorenko, A. F. (1985). Continuous models of percolation theory. I. Teoret. Mat. Fiz. 62 76–86.
  • [18] Zuev, S. A. and Sidorenko, A. F. (1985). Continuous models of percolation theory. II. Teoret. Mat. Fiz. 62 253–262.