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

Part of: Equilibrium statistical mechanics Special processes Stochastic processes

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: [email protected]
∗∗ 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 

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