Advertisement

Communications in Mathematical Physics

, Volume 310, Issue 3, pp 611–623 | Cite as

A Proof of Factorization Formula for Critical Percolation

  • Dmitri BeliaevEmail author
  • Konstantin Izyurov
Article

Abstract

We give mathematical proofs to a number of statements which appeared in the series of papers by Simmons et al. (Phys Rev E 76(4):041106, 2007; J Stat Mech Theory Exp 2009(2):P02067, 33, 2009) where they computed the probabilities of several percolation events.

Keywords

Triangular Lattice Harmonic Measure Percolation Cluster Factorization Formula Critical Percolation 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Dubédat J.: Sle(κ, ρ) martingales and duality. Ann. Prob. 33(1), 223–243 (2005)zbMATHCrossRefGoogle Scholar
  2. 2.
    Dubédat J.: Excursion decompositions for SLE and Watts’ crossing formula. Probab. Th. Rel. Fields 134(3), 453–488 (2006)zbMATHCrossRefGoogle Scholar
  3. 3.
    Hongler, C., Smirnov, S.: Critical percolation: the expected number of clusters in a rectangle. http://arxiv.org/abs/0909.4490v1 [math.PR], 2009
  4. 4.
    Lawler, G.: Conformally Invariant Processes in the Plane. Volume 114 of Mathematical Surveys and Monographs. Providence, RI: Amer. Math. Soc., 2005Google Scholar
  5. 5.
    Lawler G., Schramm O., Werner W.: Values of brownian intersection exponents, i: Half-plane exponents. Acta Math. 187(2), 237–273 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Lawler G.F., Schramm O., Werner W.: One-arm exponent for critical 2D percolation. Electron. J. Probab. 7(2), 13 (2002) (electronic)MathSciNetGoogle Scholar
  7. 7.
    Naimark M.A.: Linear Differential Operators. George G. Harrap and Co, LTD, London (1968)zbMATHGoogle Scholar
  8. 8.
    Rohde S., Schramm O.: Basic properties of SLE. Ann. of Math. (2) 161(2), 883–924 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Sheffield, S., Wilson, D.: Schramm’s proof of Watts’ formula. http://arxiv.org/abs/1003.3271v3 [math.PR], 2010
  10. 10.
    Simmons J.J.H., Kleban P., Ziff R.M.: Exact factorization of correlation functions in two-dimensional critical percolation. Phys. Rev. E 76(4), 041106 (2007)ADSCrossRefGoogle Scholar
  11. 11.
    Simmons, J.J.H., Ziff, R.M., Kleban, P.: Factorization of percolation density correlation functions for clusters touching the sides of a rectangle. J. Stat. Mech. Theory Exp. 2009(2), P02067, 33 (2009)Google Scholar
  12. 12.
    Smirnov S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333(3), 239–244 (2001)ADSzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Section de MathématiquesUniversité de GenèveGenève 4Switzerland
  3. 3.Chebyshev LaboratorySaint-Petersburg State UniversitySt. PetersburgRussia

Personalised recommendations