Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Uniqueness of infinite rigid components in percolation models: the case of nonplanar lattices
Download PDF
Download PDF
  • Published: 04 November 2003

Uniqueness of infinite rigid components in percolation models: the case of nonplanar lattices

  • Olle Häggström1 

Probability Theory and Related Fields volume 127, pages 513–534 (2003)Cite this article

  • 73 Accesses

  • Metrics details

Abstract.

We prove uniqueness of the infinite rigid component for standard bond percolation on periodic lattices in d-dimensional Euclidean space for arbitrary d, and more generally when the lattice is a quasi-transitive and amenable graph. Our approach to uniqueness of the infinite rigid component improves earlier ones, that were confined to planar settings.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Aizenman, M., Kesten, H., Newman, C.M.: Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Comm. Math. Phys. 111, 505–531 (1987)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. van den Berg, J., Keane, M.: On the continuity of the percolation function. In Conference in modern analysis and probability, R. Beals et al. (eds.), American Mathematical Society, Providence, RI, 1984, pp. 61–65

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

    MathSciNet  MATH  Google Scholar 

  6. Graver, J., Servatius, B., Servatius, H.: Combinatorial Rigidity. Am. Math. Soc., Providence, RI, 1993

  7. Grimmett, G.R.: The stochastic random-cluster process, and the uniqueness of random-cluster measures. Ann. Probab. 23, 1461–1510 (1995)

    MathSciNet  MATH  Google Scholar 

  8. Grimmett, G.R.: Percolation, 2nd ed. Springer, New York, 1999

  9. Häggström, O.: Uniqueness in two-dimensional rigidity percolation. Math. Proc. Cambridge Phil. Soc. 130, 175–188 (2001)

    Article  MathSciNet  Google Scholar 

  10. Häggström, O.: Uniqueness of the infinite entangled component in three-dimensional bond percolation. Ann. Probab. 29, 127–136 (2001)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Harris, T.E.: A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Phil. Soc. 56, 13–20 (1960)

    MATH  Google Scholar 

  13. Holroyd, A.E.: Existence and uniqueness of infinite components in generic rigidity percolation. Ann. Appl. Probab. 8, 944–973 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Holroyd, A.E.: Percolation Beyond Connectivity. Ph.D. thesis, University of Cambridge, 1999

  15. Holroyd, A.E.: Rigidity percolation and boundary conditions. Ann. Appl. Probab. 11, 1063–1078 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jacobs, D.J., Thorpe, M.F.: Generic rigidity percolation in two dimensions. Phys. Rev. E 53, 3682–3693 (1996)

    Article  Google Scholar 

  17. Liggett, T.M., Schonmann, R.H., Stacey, A.M.: Domination by product measures. Ann. Probab. 25, 71–95 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Newman, C.M., Schulman, L.S.: Infinite clusters in percolation models. J. Statist. Phys. 26, 613–628 (1981)

    MathSciNet  MATH  Google Scholar 

  19. Schonmann, R.H.: Stability of infinite clusters in supercritical percolation. Probab. Th. Rel. Fields 113, 287–300 (1999)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Statistics, Chalmers University of Technology, 412 96, Göteborg, Sweden

    Olle Häggström

Authors
  1. Olle Häggström
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Olle Häggström.

Additional information

Research supported by the Swedish Research Council

Mathematics Subject Classification (2000): 60K35, 82B43

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Häggström, O. Uniqueness of infinite rigid components in percolation models: the case of nonplanar lattices. Probab. Theory Relat. Fields 127, 513–534 (2003). https://doi.org/10.1007/s00440-003-0290-2

Download citation

  • Received: 08 October 2002

  • Revised: 06 June 2003

  • Published: 04 November 2003

  • Issue Date: December 2003

  • DOI: https://doi.org/10.1007/s00440-003-0290-2

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Euclidean Space
  • Periodic Lattice
  • Percolation Model
  • Planar Setting
  • Bond Percolation
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature