Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
G.F. Lawler, O. Schramm and W. Werner, Values of Brownian intersection exponents. II. Plane exponents, Acta Math. 187 (2001) 275 [math/0003156] [INSPIRE].
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].
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].
M. Bauer and D. Bernard, Conformal field theories of stochastic Loewner evolutions, Commun. Math. Phys. 239 (2003) 493 [hep-th/0210015] [INSPIRE].
M. Bauer and D. Bernard, SLEκ growth processes and conformal field theories, Phys. Lett. B 543 (2002) 135.
M. Bauer and D. Bernard, SLE martingales and the Virasoro algebra, Phys. Lett. B 557 (2003) 309.
M. Bauer and D. Bernard, CFTs of SLEs: the Radial case, Phys. Lett. B 583 (2004) 324 [math-ph/0310032] [INSPIRE].
M. Bauer and D. Bernard, Conformal transformations and the SLE partition function Martingale, Ann. Henri Poincaré 5 (2004) 289 [math-ph/0305061] [INSPIRE].
M. Bauer and D. Bernard, 2D growth processes: SLE and Loewner chains, Phys. Rept. 432 (2006) 115 [math-ph/0602049] [INSPIRE].
J.L. Cardy, SLE for theoretical physicists, Annals Phys. 318 (2005) 81 [cond-mat/0503313] [INSPIRE].
A. Belavin, A. Polyakov and A. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333.
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).
J.L. Cardy, Critical percolation in finite geometries, J. Phys. A 25 (1992) L201.
J.L. Cardy, Conformal invariance and surface critical behavior, Nucl. Phys. B 240 (1984) 514.
J.L. Cardy, Boundary conditions, fusion rules and the Verlinde formula, Nucl. Phys. B 324 (1989) 581.
P.P.R. Langlands and Y. Saint-Aubin, Conformal invariance in two-dimensional percolation, Bull. Amer. Math. Soc. 30 (1994) 1 [math/9401222].
V. Gurarie, Logarithmic operators in conformal field theory, Nucl. Phys. B 410 (1993) 535 [hep-th/9303160] [INSPIRE].
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].
I. Affleck, Universal term in the free energy at a critical point and the conformal anomaly, Phys. Rev. Lett. 56 (1986) 746 [INSPIRE].
L. Rozansky and H. Saleur, Quantum field theory for the multi-variable Alexander-Conway polynomial, Nucl. Phys. B 376 (1992) 461.
H. Saleur, Polymers and percolation in two-dimensions and twisted N = 2 supersymmetry, Nucl. Phys. B 382 (1992) 486 [hep-th/9111007] [INSPIRE].
V. Gurarie and A.W.W. Ludwig, Conformal algebras of 2D disordered systems, J. Phys. A 35 (2002) L377 [cond-mat/9911392] [INSPIRE].
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].
T. Creutzig and D. Ridout, Logarithmic conformal field theory: beyond an introduction, J. Phys. A 46 (2013) 4006 [arXiv:1303.0847] [INSPIRE].
.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.
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].
M. Hogervorst, M. Paulos and A. Vichi, The ABC (in any D) of Logarithmic CFT, JHEP 10 (2017) 201 [arXiv:1605.03959] [INSPIRE].
A. Gainutdinov, D. Ridout and I. Runkel, Logarithmic conformal field theory, J. Phys. A 46 (2013) 490301.
G.M.T. Watts, A crossing probability for critical percolation in two-dimensions, J. Phys. A 29 (1996) L363 [cond-mat/9603167] [INSPIRE].
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].
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].
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].
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].
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].
J.J.H. Simmons, Logarithmic operator intervals in the boundary theory of critical percolation, J. Phys. A 46 (2013) 494015 [arXiv:1311.5395] [INSPIRE].
G. Delfino and J. Viti, On three-point connectivity in two-dimensional percolation, J. Phys. A 44 (2011) 032001 [arXiv:1009.1314] [INSPIRE].
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].
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].
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].
F.Y. Wu, The Potts model, Rev. Mod. Phys. 54 (1982) 235.
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].
R. Vasseur, J.L. Jacobsen and H. Saleur, Logarithmic observables in critical percolation, J. Stat. Mech. 1207 (2012) L07001 [arXiv:1206.2312] [INSPIRE].
P.A. Pearce, J. Rasmussen and J.-B. Zuber, Logarithmic minimal models, J. Stat. Mech. 0611 (2006) P11017 [hep-th/0607232] [INSPIRE].
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].
P. Kasteleyn and C. Fortuin, Phase transitions in lattice systems with random local properties, J. Phys. Soc. Jpn. Suppl. 26 (1969) 11.
C.M. Fortuin and P.W. Kasteleyn, On the Random cluster model. 1. Introduction and relation to other models, Physica 57 (1972) 536 [INSPIRE].
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].
F.R. Bernhart, Catalan, Motzkin, and Riordan numbers, Discrete Math. 204 (1999) 73.
G. Grimmett, The stochastic random-cluster process and the uniqueness of random-cluster measures, Ann. Prob. (1995) 1461.
I. Runkel, Boundary structure constants for the A series Virasoro minimal models, Nucl. Phys. B 549 (1999) 563 [hep-th/9811178] [INSPIRE].
I. Runkel, Structure constants for the D series Virasoro minimal models, Nucl. Phys. B 579 (2000) 561 [hep-th/9908046] [INSPIRE].
V.S. Dotsenko and V.A. Fateev, Conformal algebra and multipoint correlation functions in 2D statistical models, Nucl. Phys. B 240 (1984) 312.
J. Cardy, Scaling and renormalization in statistical physics, Cambridge University Press, Cambridge U.K. (1996).
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].
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].
P. Francesco, P. Mathieu and D. Senechal, Conformal field theory, Graduate texts in contemporary physics, Springer, Germany (1997).
A.B. Zamolodchikov, Conformal symmetry in two-dimensional space: on a recurrent representation of the conformal block, Teoret. Mat. Fiz. 73 (1987) 103.
H. Saleur and B. Duplantier, Exact determination of the percolation hull exponent in two dimensions, Phys. Rev. Lett. 58 (1987) 2325 [INSPIRE].
J. Cardy, The number of incipient spanning clusters in two-dimensional percolation, J. Phys. A 31 (1998) L105 [cond-mat/9705137].
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].
R. Santachiara and J. Viti, Local logarithmic correlators as limits of Coulomb gas integrals, Nucl. Phys. B 882 (2014) 229 [arXiv:1311.2055] [INSPIRE].
R.H. Swendsen and J.-S. Wang, Nonuniversal critical dynamics in Monte Carlo simulations, Phys. Rev. Lett. 58 (1987) 86 [INSPIRE].
M. Matsumoto and T. Nishimura, Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator, ACM Trans. Model. Comput. Simul. 8 (1998) 3.
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]..
J.L. Cardy, M. Nauenberg and D. Scalapino, Scaling theory of the Potts-model multicritical point, Phys. Rev. B 22 (1980) 2560.
M. Nauenberg and D.J. Scalapino, Singularities and scaling functions at the Potts model multicritical point, Phys. Rev. Lett. 44 (1980) 837 [INSPIRE].
Y. Deng et al., Dynamic critical behavior of the Chayes-Machta-Swendsen-Wang algorithm, Phys. Rev. Lett. 99 (2007) 055701 [arXiv:0705.2751] [INSPIRE].
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.
E.L. Ince, Ordinary differential equations, Dover Publications, New York U.S.A. (1944).
S. Ribault, Conformal field theory on the plane, arXiv:1406.4290 [INSPIRE].
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].
M. van Hoeij, private communication.
HypergeometricPFQ, http://functions.wolfram.com/HypergeometricFunctions/HypergeometricPFQ/06/01/05/02/0004/
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1806.02330
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Gori, G., Viti, J. Four-point boundary connectivities in critical two-dimensional percolation from conformal invariance. J. High Energ. Phys. 2018, 131 (2018). https://doi.org/10.1007/JHEP12(2018)131
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2018)131