Skip to main content

Uniqueness of the Infinite Percolation Cluster

  • 1567 Accesses

Part of the Lecture Notes in Mathematics book series (LNMECOLE,volume 2100)

Abstract

Since we know that for p > p c there is an infinite percolation cluster a.s., it is natural to ask whether it is unique or not. As it turns out the answer to this question leads to a rather rich landscape with applications in group theory and many still open problems. In this section we study the question of the number of infinite clusters in percolation configurations in the regime p > p c .

Keywords

  • Unique Infinite Cluster
  • Percolation Configuration
  • Bernoulli Percolation
  • Infinite Component
  • Uniform Spanning Tree

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   49.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. A. Ancona, Positive harmonic functions and hyperbolicity, in Potential Theory—Surveys and Problems, Prague, 1987. Lecture Notes in Mathematics, vol. 1344 (Springer, Berlin, 1988), pp. 1–23

    Google Scholar 

  2. L. Babai, The growth rate of vertex-transitive planar graphs, in Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, New Orleans, 1997 (ACM, New York, 1997), pp. 564–573

    Google Scholar 

  3. R.M. Burton, M. Keane, Density and uniqueness in percolation. Commun. Math. Phys. 121(3), 501–505 (1989)

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. I. Benjamini, G. Kozma, Uniqueness of percolation on products with z. Lat. Am. J. Probab. Math. Stat. 10, 15–25 (2013)

    MATH  MathSciNet  Google Scholar 

  5. I. Benjamini, R. Lyons, Y. Peres, O. Schramm, Group-invariant percolation on graphs. Geom. Funct. Anal. 9(1), 29–66 (1999)

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. 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)

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. I. Benjamini, R. Lyons, Y. Peres, O. Schramm, Uniform spanning forests. Ann. Probab. 29(1), 1–65 (2001)

    MathSciNet  MATH  Google Scholar 

  8. I. Benjamini, O. Schramm, Percolation beyond Z d, many questions and a few answers. Electron. Commun. Probab. 1(8), 71–82 (1996) (electronic)

    Google Scholar 

  9. I. Benjamini, O. Schramm, Percolation in the hyperbolic plane. J. Am. Math. Soc. 14(2), 487–507 (2001) (electronic)

    Google Scholar 

  10. N. Gantert, S. Müller, The critical branching Markov chain is transient. Markov Process. Relat. Fields 12(4), 805–814 (2006)

    MATH  Google Scholar 

  11. G.R. Grimmett, C.M. Newman, Percolation in + 1 dimensions, in Disorder in Physical Systems (Oxford Science Publication/Oxford University Press, New York, 1990), pp. 167–190

    Google Scholar 

  12. G. Grimmett, Percolation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321, 2nd edn. (Springer, Berlin, 1999)

    Google Scholar 

  13. C. Houdayer, Invariant percolation and measured theory of nonamenable groups. Arxiv preprint arXiv:1106.5337 (2011)

    Google Scholar 

  14. O. Häggström, Y. Peres, Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously. Probab. Theory Relat. Fields 113(2), 273–285 (1999)

    CrossRef  MATH  Google Scholar 

  15. O. Häggström, Y. Peres, R.H. Schonmann, Percolation on transitive graphs as a coalescent process: relentless merging followed by simultaneous uniqueness, in Perplexing Problems in Probability. Progress in Probability, vol. 44 (Birkhäuser, Boston, 1999), pp. 69–90

    Google Scholar 

  16. W. Imrich, On Whitney’s theorem on the unique embeddability of 3-connected planar graphs, in Recent Advances in Graph Theory (Proceedings of Second Czechoslovak Symposium), Prague, 1974 (Academia, Prague, 1975), pp. 303–306 (loose errata)

    Google Scholar 

  17. H. Kesten, Asymptotics in high dimensions for percolation. In Disorder in Physical Systems (Oxford Science Publication/Oxford University Press, New York, 1990), pp. 219–240

    Google Scholar 

  18. R. Lyons, Y. Peres, Probability on Trees and Networks (2009). (in preparation)

    Google Scholar 

  19. B. Mohar, Isoperimetric inequalities, growth, and the spectrum of graphs. Linear Algebra Appl. 103, 119–131 (1988)

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. G. Pete, Probability and Geometry on Groups (2009) (preprint)

    Google Scholar 

  21. L.S. Pontryagin, Selected Works. Classics of Soviet Mathematics, vol. 2, 3rd edn. (Gordon & Breach Science, New York, 1986). Topological groups, Edited and with a preface by R.V. Gamkrelidze, Translated from the Russian and with a preface by Arlen Brown, With additional material translated by P.S.V. Naidu

    Google Scholar 

  22. I. Pak, T. Smirnova-Nagnibeda, On non-uniqueness of percolation on nonamenable Cayley graphs. C. R. Acad. Sci. Paris Sér. I Math. 330(6), 495–500 (2000)

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. R.H. Schonmann, Percolation in + 1 dimensions at the uniqueness threshold, in Perplexing Problems in Probability. Progress in Probability, vol. 44 (Birkhäuser, Boston, 1999), pp. 53–67

    Google Scholar 

  24. R.H. Schonmann, Multiplicity of phase transitions and mean-field criticality on highly non-amenable graphs. Commun. Math. Phys. 219(2), 271–322 (2001)

    CrossRef  MathSciNet  MATH  Google Scholar 

  25. M.E. Watkins, Connectivity of transitive graphs. J. Comb. Theory 8, 23–29 (1970)

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 2013 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Benjamini, I. (2013). Uniqueness of the Infinite Percolation Cluster. In: Coarse Geometry and Randomness. Lecture Notes in Mathematics(), vol 2100. Springer, Cham. https://doi.org/10.1007/978-3-319-02576-6_9

Download citation