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Critical percolation: the expected number of clusters in a rectangle
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  • Published: 03 August 2010

Critical percolation: the expected number of clusters in a rectangle

  • Clément Hongler1 &
  • Stanislav Smirnov1 

Probability Theory and Related Fields volume 151, pages 735–756 (2011)Cite this article

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  • 9 Citations

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Abstract

We show that for critical site percolation on the triangular lattice two new observables have conformally invariant scaling limits. In particular the expected number of clusters separating two pairs of points converges to an explicit conformal invariant. Our proof is independent of earlier results and SLE techniques, and might provide a new approach to establishing conformal invariance of percolation.

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References

  1. Aizenman M., Duplantier B., Aharony A.: Path-crossing exponents and the external perimeter in 2D percolation. Phys. Rev. Lett. 83, 1359–1362 (1999)

    Article  Google Scholar 

  2. Beffara, V.: Cardy’s formula on the triangular lattice, the easy way. In: Universality and Renormalization, pp. 39–45. Fields Inst. Commun., vol. 50. Amer. Math. Soc., Providence (2007)

  3. Bollobás B., Riordan O.: Percolation. Cambridge University Press, Cambridge (2006)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Cardy, J.: Conformal Invariance and Percolation. arXiv:math-ph/0103018

  6. Camia F., Newman C.: Critical percolation exploration path and SLE 6: a proof of convergence. Probab. Theory Relat. Fields 139(3), 473–519 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dubédat J.: Excursion decompositions for SLE and Watts’ crossing formula. Probab. Theory Relat. Fields 3, 453–488 (2006)

    Article  Google Scholar 

  8. Grimmett G.: Percolation, 2nd edn. Springer-Verlag, Berlin (1999)

    MATH  Google Scholar 

  9. Kesten H.: Percolation Theory for Mathematicians. Birkäuser, Boston (1982)

    MATH  Google Scholar 

  10. Lawler, G.F.: Conformally Invariant Processes in the Plane, vol. 114. Mathematical Surveys and Monographs (2005)

  11. Maier R.S.: On crossing event formulas in critical two-dimensional percolation. J. Stat. Phys. 111, 1027–1048 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Sheffield, S., Wilson, D.B.: Schramm’s proof of Watts’ formula. arXiv:math-ph/1003.3271

  13. Simmons J.H., Kleban P., Ziff R.M.: Percolation crossing formulae and conformal field theory. J. Phys. A 40(31), F771–F784 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  15. Smirnov, S.: Critical percolation in the plane. Preprint. arXiv:0909.4490 (2001)

  16. Smirnov, S.: Critical percolation and conformal invariance. In: XIVth International Congress on Mathematical Physics (Lissbon, 2003), pp. 99–112. World Scientific Publication, Hackensack (2003)

  17. Smirnov, S.: Towards conformal invariance of 2D lattice models. In: Sanz-Sole, M., et al. (eds.) Proceedings of the International Congress of Mathematicians (ICM), Madrid, Spain, 22–30 August 2006. Volume II: Invited Lectures, pp. 1421–1451. European Mathematical Society (EMS), Zurich (2006)

  18. Smirnov S., Werner W.: Critical exponents for two-dimensional percolation. Math. Res. Lett. 8(5–6), 729–744 (2001)

    MathSciNet  MATH  Google Scholar 

  19. Watts G.M.T.: A crossing probability for critical percolation in two dimensions. J. Phys. A29, L363 (1996)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, Case postale 64, 1211, Geneva 4, Switzerland

    Clément Hongler & Stanislav Smirnov

Authors
  1. Clément Hongler
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  2. Stanislav Smirnov
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Correspondence to Clément Hongler.

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Cite this article

Hongler, C., Smirnov, S. Critical percolation: the expected number of clusters in a rectangle. Probab. Theory Relat. Fields 151, 735–756 (2011). https://doi.org/10.1007/s00440-010-0313-8

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  • Received: 07 August 2009

  • Revised: 14 June 2010

  • Published: 03 August 2010

  • Issue Date: December 2011

  • DOI: https://doi.org/10.1007/s00440-010-0313-8

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Mathematics Subject Classification (2000)

  • Primary 60K35
  • Secondary 30C35
  • 81T40
  • 82B43
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