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Russo's Formula, Uniqueness of the Infinite Cluster, and Continuous Differentiability of Free Energy for Continuum Percolation

Part of: Special processes Stochastic processes Equilibrium statistical mechanics

Published online by Cambridge University Press:  14 July 2016

Jianping Jiang ,
Sanguo Zhang  and
Tiande Guo
Jianping Jiang*
Affiliation:
Graduate University of Chinese Academy of Sciences
Sanguo Zhang*
Affiliation:
Graduate University of Chinese Academy of Sciences
Tiande Guo*
Affiliation:
Graduate University of Chinese Academy of Sciences
*
∗ Current address: Department of Mathematics, University of Arizona, Tucson, Arizona 85721, USA. Email address: jiangjianping11@gmail.com
∗∗ Postal address: School of Mathematical Sciences, Graduate University of Chinese Academy of Sciences, 100049, Beijing, P. R. China.
∗∗ Postal address: School of Mathematical Sciences, Graduate University of Chinese Academy of Sciences, 100049, Beijing, P. R. China.
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Abstract

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A new formula for continuum percolation on the Euclidean space R d (d ≥ 2), which is analogous to Russo's formula for bond or site percolation, is proved. Using this formula, we prove the equivalence between uniqueness of the infinite cluster and continuous differentiability of the mean number of clusters per Poisson point (or free energy). This yields a new proof for uniqueness of the infinite cluster since the continuous differentiability of free energy has been proved by Bezuidenhout, Grimmett and Löffler (1998); a consequence of this new proof gives the continuity of connectivity functions.

Keywords

Continuum percolationRusso's formulauniqueness of the infinite clusterfree energyconnectivity function

MSC classification

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B43: Percolation 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 3 , September 2011 , pp. 597 - 610
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] 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. Phys. 111, 505–531.Google Scholar
[2] Bezuidenhout, C. E., Grimmett, G. and Löffler, A. (1998). Percolation and minimal spanning trees. J. Statist. Phys. 92, 1–34.Google Scholar
[3] Durrett, R. (2007). Probability: Theory and Examples, 3rd edn. Duxbury Press, Belmont, CA.Google Scholar
[4] Franceschetti, M., Penrose, M. D. and Rosoman, T. (2010). Strict inequalities of critical probabilities on Gilbert's continuum percolation graph. Preprint. Available at http://arxiv.org/abs/1004.1596v2.Google Scholar
[5] Grimmett, G. (1999). Percolation, 2nd edn. Springer, Berlin.Google Scholar
[6] Grimmett, G. (2010). Probability on Graphs. Cambridge University Press.Google Scholar
[7] Margulis, G. A. (1974). Probabilistic characteristics of graphs with large connectivity. Probl. Peredachi Inf. 10, 101–108.Google Scholar
[8] Meester, R. and Roy, R. (1994). Uniqueness of unbounded occupied and vacant components in Boolean models. Ann. Appl. Prob. 4, 933–951.Google Scholar
[9] Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press.Google Scholar
[10] Penrose, M. D. (2003). Random Geometric Graphs. Oxford University Press.Google Scholar
[11] Rockafellar, R. T. (1970). Convex Analysis. Princeton University Press.Google Scholar
[12] Rudin, W. (1976). Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York.Google Scholar
[13] Russo, L. (1981). On the critical percolation probabilities. Z. Wahrscheinlichkeitsth. 56, 229–237.Google Scholar
[14] Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
[15] Zuev, S. A. (1993). Russo's formula for Poisson point fields and its application. Discrete Math. Appl. 3, 63–73.Google Scholar