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Behavior of the supercritical phase of a continuum percolation model on ℝd

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Hideki Tanemura
Hideki Tanemura*
Affiliation:
Chiba University
*
∗ Postal address: Department of Mathematics, Faculty of Science, Chiba University, Chiba 263, Japan.
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Abstract

A continuum percolation model on is considered. Using a renormalization technique developed by Grimmett and Marstrand, we show a continuum analogue of their results. We prove the critical value of the percolation equals the limit of the critical value of a slice as the thickness of the slice tends to infinity. We also prove that the effective conductivity in the model is bounded from below by a positive constant in the supercritical case.

Keywords

CRITICAL VALUEEFFECTIVE CONDUCTIVITYPOISSON BLOB MODELPOISSON DISTRIBUTIONRENORMALIZATION TECHNIQUE

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 2 , June 1993 , pp. 382 - 396
Copyright
Copyright © Applied Probability Trust 1993 

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References

[1] Barlow, M. T. and Bass, R. F. (1990) On the resistance of the Sierpinski carpet. Proc. R. Soc. London A 431, 345–360.Google Scholar
[2] Bass, R. F. and Hsu, P. (1991) Some potential theory for reflecting Brownian motion in Holder and Lipschitz domains. Ann. Prob. 19, 486–508.CrossRefGoogle Scholar
[3] Chayes, J. T. and Chayes, L. (1986) Bulk transport properties and exponent inequalities for random resistor and flow networks. Comm. Math. Phys. 105, 133–152.CrossRefGoogle Scholar
[4] Doyle, P. G. and Snell, J. L. (1984) Random Walks and Electrical Networks. Mathematical Association of America, Washington, DC.CrossRefGoogle Scholar
[5] Fitzpatrick, J. P., Malt, R. B. and Spaepen, F. (1974) Percolation theory and the conductivity of random close packed mixtures of hard spheres. Phys. Lett. A 47, 207–208.CrossRefGoogle Scholar
[6] Grimmett, G. R. (1989) Percolation. Springer-Verlag, Berlin.CrossRefGoogle Scholar
[7] Grimmett, G. R. and Kesten, H. (1984) First passage percolation, network flows and electrical resistances. Z. Wahrscheinlichkeitsth. 66, 335–366.CrossRefGoogle Scholar
[8] Grimmett, G. R. and Marstrand, J. M. (1990) The supercritical phase of percolation is well behaved. Proc. R. Soc. London A 430, 439–457.Google Scholar
[9] Hall, P. (1985) On continuum percolation. Ann. Prob. 13, 1250–1266.CrossRefGoogle Scholar
[10] Halperin, B. L., Feng, S. and Sen, P. N. (1985) Differences between lattice and continuum percolation transport exponents. Phys. Rev. Lett. 54, 2391–2394.CrossRefGoogle ScholarPubMed
[11] Men'Shikov, M. V., Molchanov, S. A. and Sidorenko, A. F. (1986) Percolation theory and some applications. Itogi Nauki. Tekh. 24, 53–110.Google Scholar
[12] Roy, R. (1990) The Russo-Welsh theorem and the equality of critical densities and the dual critical densities for continuum percolation on ℝ d . Ann. Prob. 18, 1563–1575.CrossRefGoogle Scholar
[13] Sidorenko, A. F. and Zuev, S. A. (1985) Continuous models of percolation theory, I, II. Theoret. Math. Phys. 62, 76–88; 253-262.Google Scholar