Journal of High Energy Physics

, 2018:131 | Cite as

Four-point boundary connectivities in critical two-dimensional percolation from conformal invariance

  • Giacomo Gori
  • Jacopo VitiEmail author
Open Access
Regular Article - Theoretical Physics


We conjecture an exact form for an universal ratio of four-point cluster connectivities in the critical two-dimensional Q-color Potts model. We also provide analogous results for the limit Q → 1 that corresponds to percolation where the observable has a logarithmic singularity. Our conjectures are tested against Monte Carlo simulations showing excellent agreement for Q = 1, 2, 3.


Boundary Quantum Field Theory Conformal Field Theory Random Systems 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    G.F. Lawler, O. Schramm and W. Werner, Values of Brownian intersection exponents. I. Half-plane exponents, Acta Math. 187 (2001) 237 [math/9911084] [INSPIRE].
  2. [2]
    G.F. Lawler, O. Schramm and W. Werner, Values of Brownian intersection exponents. II. Plane exponents, Acta Math. 187 (2001) 275 [math/0003156] [INSPIRE].
  3. [3]
    G.F. Lawler, O. Schramm and W. Werner, Values of Brownian intersection exponents. III. Two-sided exponents, Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 109 [math/0005294] [INSPIRE].
  4. [4]
    S. Smirnov, Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001) 239 [arXiv:0909.4499] [arXiv:0909.4499].
  5. [5]
    M. Bauer and D. Bernard, Conformal field theories of stochastic Loewner evolutions, Commun. Math. Phys. 239 (2003) 493 [hep-th/0210015] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    M. Bauer and D. Bernard, SLEκ growth processes and conformal field theories, Phys. Lett. B 543 (2002) 135.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    M. Bauer and D. Bernard, SLE martingales and the Virasoro algebra, Phys. Lett. B 557 (2003) 309.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    M. Bauer and D. Bernard, CFTs of SLEs: the Radial case, Phys. Lett. B 583 (2004) 324 [math-ph/0310032] [INSPIRE].
  9. [9]
    M. Bauer and D. Bernard, Conformal transformations and the SLE partition function Martingale, Ann. Henri Poincaré 5 (2004) 289 [math-ph/0305061] [INSPIRE].
  10. [10]
    M. Bauer and D. Bernard, 2D growth processes: SLE and Loewner chains, Phys. Rept. 432 (2006) 115 [math-ph/0602049] [INSPIRE].
  11. [11]
    J.L. Cardy, SLE for theoretical physicists, Annals Phys. 318 (2005) 81 [cond-mat/0503313] [INSPIRE].
  12. [12]
    A. Belavin, A. Polyakov and A. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    B. Nienhuis, Coulomb gas formulation of 2-d phase transitions, in Phase transitions and critical phenomena, C. Domb and J.L. Lebowitz eds., Academic Press, New York U.S.A. (1987).Google Scholar
  14. [14]
    J.L. Cardy, Critical percolation in finite geometries, J. Phys. A 25 (1992) L201.MathSciNetzbMATHGoogle Scholar
  15. [15]
    J.L. Cardy, Conformal invariance and surface critical behavior, Nucl. Phys. B 240 (1984) 514.CrossRefGoogle Scholar
  16. [16]
    J.L. Cardy, Boundary conditions, fusion rules and the Verlinde formula, Nucl. Phys. B 324 (1989) 581.MathSciNetCrossRefGoogle Scholar
  17. [17]
    P.P.R. Langlands and Y. Saint-Aubin, Conformal invariance in two-dimensional percolation, Bull. Amer. Math. Soc. 30 (1994) 1 [math/9401222].
  18. [18]
    V. Gurarie, Logarithmic operators in conformal field theory, Nucl. Phys. B 410 (1993) 535 [hep-th/9303160] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    H.W.J. Bloete, J.L. Cardy and M.P. Nightingale, Conformal invariance, the central charge and universal finite size amplitudes at criticality, Phys. Rev. Lett. 56 (1986) 742 [INSPIRE].CrossRefGoogle Scholar
  20. [20]
    I. Affleck, Universal term in the free energy at a critical point and the conformal anomaly, Phys. Rev. Lett. 56 (1986) 746 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  21. [21]
    L. Rozansky and H. Saleur, Quantum field theory for the multi-variable Alexander-Conway polynomial, Nucl. Phys. B 376 (1992) 461.MathSciNetCrossRefGoogle Scholar
  22. [22]
    H. Saleur, Polymers and percolation in two-dimensions and twisted N = 2 supersymmetry, Nucl. Phys. B 382 (1992) 486 [hep-th/9111007] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  23. [23]
    V. Gurarie and A.W.W. Ludwig, Conformal algebras of 2D disordered systems, J. Phys. A 35 (2002) L377 [cond-mat/9911392] [INSPIRE].
  24. [24]
    R. Vasseur, J.L. Jacobsen and H. Saleur, Indecomposability parameters in chiral logarithmic conformal field theory, Nucl. Phys. B 851 (2011) 314 [arXiv:1103.3134] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    T. Creutzig and D. Ridout, Logarithmic conformal field theory: beyond an introduction, J. Phys. A 46 (2013) 4006 [arXiv:1303.0847] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  26. [26]
    .Cardy, The stress tensor in quenched random systems, in Statistical Field Theories (Proceedings of a NATO workshop, Como, June 2001), A. Cappelli and G. Mussardo eds., Springer, Germany (2002), cond-mat/0111031.
  27. [27]
    J. Cardy, Logarithmic conformal field theories as limits of ordinary CFTs and some physical applications, J. Phys. A 46 (2013) 494001 [arXiv:1302.4279] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  28. [28]
    M. Hogervorst, M. Paulos and A. Vichi, The ABC (in any D) of Logarithmic CFT, JHEP 10 (2017) 201 [arXiv:1605.03959] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    A. Gainutdinov, D. Ridout and I. Runkel, Logarithmic conformal field theory, J. Phys. A 46 (2013) 490301.MathSciNetzbMATHGoogle Scholar
  30. [30]
    G.M.T. Watts, A crossing probability for critical percolation in two-dimensions, J. Phys. A 29 (1996) L363 [cond-mat/9603167] [INSPIRE].
  31. [31]
    S.M. Flores and P. Kleban, A solution space for a system of null-state partial differential equations: part 1, Commun. Math. Phys. 333 (2015) 389 [arXiv:1212.2301] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    S.M. Flores and P. Kleban, A solution space for a system of null-state partial differential equations: part 2, Commun. Math. Phys. 333 (2015) 435 [arXiv:1404.0035] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    S.M. Flores and P. Kleban, A solution space for a system of null-state partial differential equations: part 3, Commun. Math. Phys. 333 (2015) 597 [arXiv:1303.7182] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    S.M. Flores and P. Kleban, A solution space for a system of null-state partial differential equations: part 4, Commun. Math. Phys. 333 (2015) 669 [arXiv:1405.2747] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    J.J.H. Simmons, P. Kleban and R.M. Ziff, Percolation crossing formulas and conformal field theory, J. Phys. A 40 (2007) F771 [arXiv:0705.1933] [INSPIRE].zbMATHGoogle Scholar
  36. [36]
    J.J.H. Simmons, Logarithmic operator intervals in the boundary theory of critical percolation, J. Phys. A 46 (2013) 494015 [arXiv:1311.5395] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  37. [37]
    G. Delfino and J. Viti, On three-point connectivity in two-dimensional percolation, J. Phys. A 44 (2011) 032001 [arXiv:1009.1314] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  38. [38]
    M. Picco, R. Santachiara, J. Viti and G. Delfino, Connectivities of Potts Fortuin-Kasteleyn clusters and time-like Liouville correlator, Nucl. Phys. B 875 (2013) 719 [arXiv:1304.6511] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    Y. Ikhlef, J.L. Jacobsen and H. Saleur, Three-point functions in c ≤ 1 Liouville theory and conformal loop ensembles, Phys. Rev. Lett. 116 (2016) 130601 [arXiv:1509.03538] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  40. [40]
    M. Picco, S. Ribault and R. Santachiara, A conformal bootstrap approach to critical percolation in two dimensions, SciPost Phys. 1 (2016) 009 [arXiv:1607.07224] [INSPIRE].CrossRefGoogle Scholar
  41. [41]
    F.Y. Wu, The Potts model, Rev. Mod. Phys. 54 (1982) 235.MathSciNetCrossRefGoogle Scholar
  42. [42]
    J. Dubail, J.L. Jacobsen and H. Saleur, Conformal field theory at central charge c = 0: a measure of the indecomposability (b) parameters, Nucl. Phys. B 834 (2010) 399 [arXiv:1001.1151] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    R. Vasseur, J.L. Jacobsen and H. Saleur, Logarithmic observables in critical percolation, J. Stat. Mech. 1207 (2012) L07001 [arXiv:1206.2312] [INSPIRE].Google Scholar
  44. [44]
    P.A. Pearce, J. Rasmussen and J.-B. Zuber, Logarithmic minimal models, J. Stat. Mech. 0611 (2006) P11017 [hep-th/0607232] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  45. [45]
    G. Gori and J. Viti, Exact logarithmic four-point functions in the critical two-dimensional Ising model, Phys. Rev. Lett. 119 (2017) 191601 [arXiv:1704.02893] [INSPIRE].CrossRefGoogle Scholar
  46. [46]
    P. Kasteleyn and C. Fortuin, Phase transitions in lattice systems with random local properties, J. Phys. Soc. Jpn. Suppl. 26 (1969) 11.Google Scholar
  47. [47]
    C.M. Fortuin and P.W. Kasteleyn, On the Random cluster model. 1. Introduction and relation to other models, Physica 57 (1972) 536 [INSPIRE].
  48. [48]
    G. Delfino and J. Viti, Potts q-color field theory and scaling random cluster model, Nucl. Phys. B 852 (2011) 149 [arXiv:1104.4323] [INSPIRE].CrossRefzbMATHGoogle Scholar
  49. [49]
    F.R. Bernhart, Catalan, Motzkin, and Riordan numbers, Discrete Math. 204 (1999) 73.MathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    G. Grimmett, The stochastic random-cluster process and the uniqueness of random-cluster measures, Ann. Prob. (1995) 1461.Google Scholar
  51. [51]
    I. Runkel, Boundary structure constants for the A series Virasoro minimal models, Nucl. Phys. B 549 (1999) 563 [hep-th/9811178] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    I. Runkel, Structure constants for the D series Virasoro minimal models, Nucl. Phys. B 579 (2000) 561 [hep-th/9908046] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    V.S. Dotsenko and V.A. Fateev, Conformal algebra and multipoint correlation functions in 2D statistical models, Nucl. Phys. B 240 (1984) 312.CrossRefGoogle Scholar
  54. [54]
    J. Cardy, Scaling and renormalization in statistical physics, Cambridge University Press, Cambridge U.K. (1996).CrossRefzbMATHGoogle Scholar
  55. [55]
    F. Wu and H. Huang, Sum rule identities and the duality relation for the Potts n-point boundary correlation function, Phys. Rev. Lett. 79 (1997) 4954, Phys. Rev. B 57 (1998) 3031 [cond-mat/9706250].
  56. [56]
    P. Kleban, J.J.H. Simmons and R.M. Ziff, Anchored critical percolation clusters and 2D electrostatics, Phys. Rev. Lett. 97 (2006) 115702 [cond-mat/0605120] [INSPIRE].
  57. [57]
    P. Francesco, P. Mathieu and D. Senechal, Conformal field theory, Graduate texts in contemporary physics, Springer, Germany (1997).Google Scholar
  58. [58]
    A.B. Zamolodchikov, Conformal symmetry in two-dimensional space: on a recurrent representation of the conformal block, Teoret. Mat. Fiz. 73 (1987) 103.MathSciNetGoogle Scholar
  59. [59]
    H. Saleur and B. Duplantier, Exact determination of the percolation hull exponent in two dimensions, Phys. Rev. Lett. 58 (1987) 2325 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  60. [60]
    J. Cardy, The number of incipient spanning clusters in two-dimensional percolation, J. Phys. A 31 (1998) L105 [cond-mat/9705137].
  61. [61]
    E. Imamoglu and M. van Hoeij, Computing hypergeometric solutions of second order linear differential equations using quotients of formal solutions and integral bases, J. Symb. Comput. 83 (2017) 254 [arXiv:1606.01576].MathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    R. Santachiara and J. Viti, Local logarithmic correlators as limits of Coulomb gas integrals, Nucl. Phys. B 882 (2014) 229 [arXiv:1311.2055] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    R.H. Swendsen and J.-S. Wang, Nonuniversal critical dynamics in Monte Carlo simulations, Phys. Rev. Lett. 58 (1987) 86 [INSPIRE].CrossRefGoogle Scholar
  64. [64]
    M. Matsumoto and T. Nishimura, Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator, ACM Trans. Model. Comput. Simul. 8 (1998) 3.CrossRefzbMATHGoogle Scholar
  65. [65]
    J. Cardy and R. M. Ziff, Exact results for the universal area distribution of clusters in percolation, Ising, and Potts models, J. Stat. Phys. 110 (2003) 1 [cond-mat/0205404]..
  66. [66]
    J.L. Cardy, M. Nauenberg and D. Scalapino, Scaling theory of the Potts-model multicritical point, Phys. Rev. B 22 (1980) 2560.MathSciNetCrossRefGoogle Scholar
  67. [67]
    M. Nauenberg and D.J. Scalapino, Singularities and scaling functions at the Potts model multicritical point, Phys. Rev. Lett. 44 (1980) 837 [INSPIRE].CrossRefGoogle Scholar
  68. [68]
    Y. Deng et al., Dynamic critical behavior of the Chayes-Machta-Swendsen-Wang algorithm, Phys. Rev. Lett. 99 (2007) 055701 [arXiv:0705.2751] [INSPIRE].CrossRefGoogle Scholar
  69. [69]
    C.J. Hamer, M.T. Batchelor and M.N. Barber, Logarithmic corrections to finite-size scaling in the four-state Potts model, J. Stat. Phys. 52 (1988) 679.MathSciNetCrossRefzbMATHGoogle Scholar
  70. [70]
    E.L. Ince, Ordinary differential equations, Dover Publications, New York U.S.A. (1944).zbMATHGoogle Scholar
  71. [71]
    S. Ribault, Conformal field theory on the plane, arXiv:1406.4290 [INSPIRE].
  72. [72]
    N. Javerzat, R. Santachiara and O. Foda, Notes on the solutions of Zamolodchikov-type recursion relations in Virasoro minimal models, JHEP 08 (2018) 183 [arXiv:1806.02790] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  73. [73]
    M. van Hoeij, private communication.Google Scholar
  74. [74]

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© The Author(s) 2018

Authors and Affiliations

  1. 1.Dipartimento di Fisica e Astronomia “Galileo Galilei”Università di PadovaPadovaItaly
  2. 2.CNR-IOMTriesteItaly
  3. 3.ECT & International Institute of PhysicsUFRNNatalBrazil

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