Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-09T01:37:28.322Z Has data issue: false hasContentIssue false
  • English
  • Français

Critical intensities of Boolean models with different underlying convex shapes

Part of: Special processes

Published online by Cambridge University Press:  19 February 2016

Rahul Roy  and
Hideki Tanemura
Rahul Roy*
Affiliation:
Indian Statistical Institute, New Delhi
Hideki Tanemura*
Affiliation:
Chiba University
*
∗ Postal address: Indian Statistical Institute, 7 S. J. S. Sansanwal Marg, New Delhi 110016, India. Email address: [email protected]
∗∗ Postal address: Department of Mathematics and Informatics, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the Poisson Boolean model of percolation where the percolating shapes are convex regions. By an enhancement argument we strengthen a result of Jonasson (2000) to show that the critical intensity of percolation in two dimensions is minimized among the class of convex shapes of unit area when the percolating shapes are triangles, and, for any other shape, the critical intensity is strictly larger than this minimum value. We also obtain a partial generalization to higher dimensions. In particular, for three dimensions, the critical intensity of percolation is minimized among the class of regular polytopes of unit volume when the percolating shapes are tetrahedrons. Moreover, for any other regular polytope, the critical intensity is strictly larger than this minimum value.

Keywords

Poisson processBoolean modelpercolationcritical intensity

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 34 , Issue 1 , March 2002 , pp. 48 - 57
Copyright
Copyright © Applied Probability Trust 2002 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aizenman, M. and Grimmett, G. (1991). Strict monotonicity for critical points in percolation and ferromagnetic models. J. Statist. Phys. 63, 817–835.Google Scholar
Coxeter, H. S. M. (1989). Introduction to Geometry, 2nd edn. John Wiley, New York.Google Scholar
Hall, P. (1985). On continuum percolation. Ann. Prob. 13, 1250–1266.CrossRefGoogle Scholar
Hall, P. (1988). Introduction to the Theory of Coverage Processes. John Wiley, New York.Google Scholar
Jonasson, J. (2001). Optimization of shape in continuum percolation. To appear in Ann. Prob. Available at http://www.math.chalmers.se/jonasson/recent.html. Ann. Prob. 29, 624–635.CrossRefGoogle Scholar
Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press.Google Scholar
Meester, R., Roy, R. and Sarkar, A. (1994). Non-universality and continuity of the critical covered volume fraction. J. Statist. Phys. 75, 123–134.CrossRefGoogle Scholar
Sarkar, A. (1997). Co-existence of the occupied and vacant phase in Boolean models in three or more dimensions. Adv. Appl. Prob. 29, 878–889.Google Scholar
Tanemura, H. (2000). Scaling and power laws. In Percolation and Particle Systems, ed. Roy, R., Universities Press, Hyderabad, pp. 59–72.Google Scholar