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


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.


Triangular Lattice Harmonic Measure Percolation Cluster Factorization Formula Critical Percolation 
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  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. [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. [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

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