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

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...

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Veröffentlicht in:Journal of applied probability 2011-09, Vol.48 (3), p.597-610
Hauptverfasser: Jiang, Jianping, Zhang, Sanguo, Guo, Tiande
Format: Artikel
Sprache:eng
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Zusammenfassung: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.
ISSN:0021-9002
1475-6072
DOI:10.1017/S0021900200008184