Hostname: page-component-5db58dd55d-mhzq2 Total loading time: 0 Render date: 2026-05-25T08:11:59.467Z Has data issue: false hasContentIssue false
  • English
  • Français

On a continuum percolation model

Published online by Cambridge University Press:  01 July 2016

Mathew D. Penrose
Mathew D. Penrose*
Affiliation:
University of California, Santa Barbara
*
∗ Postal address: Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Consider particles placed in space by a Poisson process. Pairs of particles are bonded together, independently of other pairs, with a probability that depends on their separation, leading to the formation of clusters of particles. We prove the existence of a non-trivial critical intensity at which percolation occurs (that is, an infinite cluster forms). We then prove the continuity of the cluster density, or free energy. Also, we derive a formula for the probability that an arbitrary Poisson particle lies in a cluster consisting of k particles (or equivalently, a formula for the density of such clusters), and show that at high Poisson intensity, the probability that an arbitrary Poisson particle is isolated, given that it lies in a finite cluster, approaches 1.

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 3 , September 1991 , pp. 536 - 556
Copyright
Copyright © Applied Probability Trust 1991 

Footnotes

POISSON PROCESS; CLUSTER DENSITY; LARGE DEVIATIONS AT HIGH DENSITY

References

Aizenman, M., Kesten, H. and Newman, C. M. (1987) Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Commun. Math. Physics. 111, 505–531.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (1988) An Introduction to the Theory of Point Processes . Springer-Verlag, New York.Google Scholar
Dunford, N. and Schwartz, J. T. (1958) Linear Operators , part I. Wiley-Interscience, New York.Google Scholar
Gilbert, E. N. (1961) Random plane networks. J. Soc. Ind. Appl. Math. 9, 533–543.CrossRefGoogle Scholar
Grimmett, G. R. (1976) On the number of clusters in the percolation model. J. London Math. Soc. (2) 13, 346–350.CrossRefGoogle Scholar
Grimmett, G. R. (1989) Percolation. Springer-Verlag, New York.CrossRefGoogle Scholar
Hall, P. (1985) On continuum percolation. Ann. Prob. 13, 1250–1266.CrossRefGoogle Scholar
Hall, P. (1986) Clump counts in a mosaic. Ann. Prob. 14, 424–458.CrossRefGoogle Scholar
Hall, , (1988) Introduction to the Theory of Coverage Processes. Wiley, New York.Google Scholar
Kesten, H. (1987) Percolation theory and first-passage percolation. Ann. Prob. 15, 1231–1271.CrossRefGoogle Scholar
Mack, C. (1954) The expected number of clumps when convex laminae are placed at random and with random orientation on a plane area. Proc. Camb. Phil. Soc. 50, 581–585.CrossRefGoogle Scholar
Mack, C. (1956) On clumps formed when convex laminae are placed at random in two or three dimensions. Proc. Camb. Phil. Soc. 52, 246–256.CrossRefGoogle Scholar
Resnick, S. I. (1987) Extreme Values, Regular Variation and Point Processes. Springer-Verlag, New York.CrossRefGoogle Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1987) Stochastic Geometry and its Applications. Wiley, Chichester.Google Scholar
Sykes, M. F. and Essam, J. W. (1964) Exact critical probabilities for site and bond problems in two dimensions. J. Math. Phys. 5, 1117–1127.CrossRefGoogle Scholar
Zuev, S. A. and Sidorenko, A. F. (1985a) Continuous models of percolation theory I. Theoret. Math. Phys. 62, 76–86 (551–553 in translation from Russian).Google Scholar
Zuev, S. A. and Sidorenko, A. F. (1985b) Continuous models of percolation theory II. Theoret. Math. Phys. 62, 253–262 (171–177 in translation from Russian).Google Scholar