Skip to main content
SpringerLink
Log in

Two-Dimensional Critical Percolation: The Full Scaling Limit

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We use SLE 6 paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice – that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aizenman, M.: The geometry of critical percolation and conformal invariance. In: Bai-lin, H. (ed.) STATPHYS 19, Proceeding Xiamen 1995, Singapore: World Scientific, 1995

  2. Aizenman M., (1998) Scaling limit for the incipient spanning clusters. In: Golden K., Grimmett G., James R., Milton G., Sen P. (eds) Mathematics of Multiscale Materials; the IMA Volumes in Mathematics and its Applications. Berlin-Heidelberg New York, Springer

    Google Scholar 

  3. Aizenman M., Burchard A. (1999) Hölder regularity and dimension bounds for random curves. Duke Math. J. 99, 419–453

    Article  MATH  MathSciNet  Google Scholar 

  4. Aizenman M., Burchard A., Newman C.M., Wilson D.B. (1999) Scaling limits for minimal and random spanning trees in two dimensions. Ran. Structures Alg. 15, 316–367

    MathSciNet  Google Scholar 

  5. Aizenman M., Duplantier B., Aharony A. (1999) Connectivity exponents and the external perimeter in 2D independent percolation. Phys. Rev. Lett. 83, 1359–1362

    Article  ADS  Google Scholar 

  6. Belavin A.A., Polyakov A.M., Zamolodchikov A.B. (1984) Infinite conformal symmetry of critical fluctuations in two dimensions. J. Stat. Phys. 34, 763–774

    Article  MathSciNet  ADS  Google Scholar 

  7. Belavin A.A., Polyakov A.M., Zamolodchikov A.B. (1984) Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241, 333–380

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. Benjamini I., Schramm O.(1998) Conformal invariance of Voronoi percolation. Commun. Math. Phys. 197, 75–107

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. Billingsley, P.: Weak Convergence of Measures: Applications in Probability. Section 3, Philadelphia, PA: Society for Industrial and Applied Mathematics, 1971

  10. Camia F., Newman C.M. (2004) Continuum Nonsimple Loops and 2D Critical Percolation. J. Stat. Phys. 116, 157–173

    Article  MathSciNet  MATH  ADS  Google Scholar 

  11. Camia, F., Newman, C.M.: The Full Scaling Limit of Two-Dimensional Critical Percolation (original preprint version of this paper and reference [cn2]), available at http://arxiv:org/list/math.PR/0504036, 2005

  12. Camia, F., Newman, C.M.: Critical Percolation Exploration Path and SLE 6: a Proof of Convergence. available at http://arxiv:org/list/math.PR/0604487, 2006

  13. Cardy J.L. (1992) Critical percolation in finite geometries. J. Phys. A 25, L201–L206

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Cardy, J.: Lectures on Conformal Invariance and Percolation, available at http://arxiv:org/list/math-ph/0103018, 2001

  15. Durrett R., (2004) Probability: Theory and Examples Third Edition. Belmont CA, Duxbury Advanced Series

    Google Scholar 

  16. Grimmett G.R., (1999) Percolation Second Edition. Berlin, Springer

    MATH  Google Scholar 

  17. Kager W., Nienhuis B. (2004) A Guide to Stochastic Löwner Evolution and Its Applications. J. Phys. A 115, 1149–1229

    MathSciNet  MATH  Google Scholar 

  18. Kenyon R. (2000) Long-range properties of spanning trees. J. Math. Phys. 41, 1338–1363

    Article  MATH  MathSciNet  ADS  Google Scholar 

  19. Kenyon R. (2000) Conformal invariance of domino tiling. Ann. Probab. 28, 759–795

    Article  MATH  MathSciNet  Google Scholar 

  20. Kesten H., (1982) Percolation Theory for Mathematicians. Boston, Birkhäuser

    MATH  Google Scholar 

  21. Kesten, H., Sidoravicius, V., Zhang, Y.: Almost all words are seen in critical site percolation on the triangular lattice. Electr. J. Probab. 3(10) (1998)

  22. Langlands R., Pouliot P., Saint-Aubin Y. (1994) Conformal invariance for two-dimensional percolation. Bull. Am. Math. Soc. 30, 1–61

    MATH  MathSciNet  Google Scholar 

  23. Lawler, G.: Conformally Invariant Processes in the Plane. In: Lecture notes for the 2002 ICTP School and Conference on Probability, ICTP Lecture Notes Series, Vol. XVII, available at http://users:ictp.it/~pub_off/lectures/vol17.html, 2004

  24. Lawler, G.F.: Conformally Invariant Processes in the Plane, Mathematical Surveys and Monographs, 114, Providence, RI: American Mathematical Society, 2005

  25. Lawler G., Schramm O., Werner W. (2001) Values of Brownian intersection exponents I: Half-plane exponents. Acta Math. 187, 237–273

    Article  MATH  MathSciNet  Google Scholar 

  26. Lawler, G., Schramm, O., Werner, W.: One arm exponent for critical 2D percolation. Electronic J. Probab. 7(2) (2002)

  27. Polyakov A.M. (1970) Conformal symmetry of critical fluctuations. JETP Letters 12, 381–383

    ADS  Google Scholar 

  28. Pommerenke Ch., (1992) Boundary Behaviour of Conformal Maps. Berlin, Springer-Verlag

    MATH  Google Scholar 

  29. Radó T. (1923) Sur la représentation conforme de domaines variables. Acta Sci. Math. (Szeged) 1, 180–186

    MATH  Google Scholar 

  30. Rohde S., Schramm O. (2005) Basic properties of SLE. Ann. Math. 161, 883–924

    Article  MATH  MathSciNet  Google Scholar 

  31. Russo L. (1978) A note on percolation. Z. Wahrsch. Ver. Geb. 43, 39–48

    Article  MATH  Google Scholar 

  32. Schramm O. (2000) Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221–288

    MATH  MathSciNet  Google Scholar 

  33. Schramm, O., Sheffield, S.: Contour lines of the two-dimensional discrete Gaussian free field. Available at http://arxiv.org/list/math.PR/0605337, 2006

  34. Seymour P.D., Welsh D.J.A. (1978), Percolation probabilities on the square lattice. In: Bollobás B. (ed) Advances in Graph Theory Annals of Discrete Mathematics 3. Amsterdam, North-Holland, pp. 227–245

    Google Scholar 

  35. Sheffield, S., Werner, W.: In preparation

  36. Smirnov S. (2001) Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris 333, 239–244

    MATH  ADS  Google Scholar 

  37. Smirnov, S.: Critical percolation in the plane. I. Conformal invariance and Cardy’s formula. II. Continuum scaling limit. (long version of [36], dated Nov. 15, 2001), available at http://www.math. kth.se/~stas/papers/index.html

  38. Smirnov, S.: In preparation

  39. Smirnov, S.: Private communication

  40. Smirnov S., Werner W. (2001) Critical exponents for two-dimensional percolation. Math. Rev. Lett. 8, 729–744

    MATH  MathSciNet  Google Scholar 

  41. Tsirelson, B.: Percolation, boundary, noise: an experiment, available at http://arxiv.org/list/math.PR/ 0506269, 2005

  42. Werner W. (2003) SLEs as boundaries of clusters of Brownian loops, C. R. Math. Acad. Sci. Paris 337, 481–486

    MATH  MathSciNet  Google Scholar 

  43. Werner, W.: Random planar curves and Schramm-Loewner Evolutions. In: Lectures on probability theory and statistics, Lecture Notes in Math., Vol. 1840, Berlin: Springer, 2004, pp. 107–195

  44. Werner, W.: Some recent aspects of random conformally invariant systems. Lecture notes available at http://arxiv.org/list/math.PR/0511268, 2005

  45. Werner, W.: The conformally invariant measure on self-avoiding loops. Available at http://arxiv.org/ list/math.PR/0511605, 2005

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Vrije Universiteit, 1081, Amsterdam, HV, The Netherlands

    Federico Camia

  2. Courant Inst. of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA

    Charles M. Newman

Authors
  1. Federico Camia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Charles M. Newman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Federico Camia.

Additional information

Communicated by M. Aizenman

Research partially supported by a Marie Curie Intra-European Fellowship under contract MEIF-CT-2003-500740 and by a Veni grant of the Dutch Organization for Scientific Research (NWO).

Research partially supported by the U.S. NSF under grant DMS-01-04278.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Camia, F., Newman, C.M. Two-Dimensional Critical Percolation: The Full Scaling Limit. Commun. Math. Phys. 268, 1–38 (2006). https://doi.org/10.1007/s00220-006-0086-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-006-0086-1

Keywords

Search

Navigation