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Critical Percolation on Non-Amenable Groups

  • Itai Benjamini
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2100)

Abstract

For a given graph G, let \(\theta _{G}(p) = P_{p}(0 \leftrightarrow \infty )\) (or just θ(p) when G is clear from the context). From the definition of p c we know that θ(p) = 0 for any p < p c , and θ(p) > 0 whenever p > p c . A major and natural question that arises is: Does θ(p c )= 0 or θ(p c ) > 0?.

References

  1. [BLPS99b]
    I. Benjamini, R. Lyons, Y. Peres, O. Schramm, Critical percolation on any nonamenable group has no infinite clusters. Ann. Probab. 27(3), 1347–1356 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [Dek91]
    F.M. Dekking, Branching processes that grow faster than binary splitting. Am. Math. Mon. 98(8), 728–731 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [Gri99]
    G. Grimmett, Percolation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321, 2nd edn. (Springer, Berlin, 1999)Google Scholar
  4. [HM09]
    O. Häggström, P. Mester, Some two-dimensional finite energy percolation processes. Electron. Commun. Probab. 14, 42–54 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [HS90]
    T. Hara, G. Slade, Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys. 128(2), 333–391 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Itai Benjamini
    • 1
  1. 1.Department of MathematicsThe Weizmann Institute of ScienceRehovotIsrael

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