Four-point boundary connectivities in critical two-dimensional percolation from conformal invariance. (English) Zbl 1405.81130
Summary: 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 \(Q \rightarrow 1\) 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.
MSC:
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 62P35 | Applications of statistics to physics |
References:
| [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]. · Zbl 1005.60097 |
| [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]. · Zbl 0993.60083 |
| [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]. · Zbl 1006.60075 |
| [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]. · Zbl 0985.60090 |
| [5] | M. Bauer and D. Bernard, Conformal field theories of stochastic Loewner evolutions, Commun. Math. Phys.239 (2003) 493 [hep-th/0210015] [INSPIRE]. · Zbl 1046.81091 · doi:10.1007/s00220-003-0881-x |
| [6] | M. Bauer and D. Bernard, SLEκ growth processes and conformal field theories, Phys. Lett.B 543 (2002) 135. · Zbl 0997.60119 · doi:10.1016/S0370-2693(02)02423-1 |
| [7] | M. Bauer and D. Bernard, SLE martingales and the Virasoro algebra, Phys. Lett.B 557 (2003) 309. · Zbl 1009.81022 · doi:10.1016/S0370-2693(03)00189-8 |
| [8] | M. Bauer and D. Bernard, CFTs of SLEs: the Radial case, Phys. Lett.B 583 (2004) 324 [math-ph/0310032] [INSPIRE]. · Zbl 1246.81313 |
| [9] | M. Bauer and D. Bernard, Conformal transformations and the SLE partition function Martingale, Ann. Henri Poincaré5 (2004) 289 [math-ph/0305061] [INSPIRE]. · Zbl 1088.81083 |
| [10] | M. Bauer and D. Bernard, 2D growth processes: SLE and Loewner chains, Phys. Rept.432 (2006) 115 [math-ph/0602049] [INSPIRE]. |
| [11] | J.L. Cardy, SLE for theoretical physicists, Annals Phys.318 (2005) 81 [cond-mat/0503313] [INSPIRE]. · Zbl 1073.81068 |
| [12] | A. Belavin, A. Polyakov and A. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys.B 241 (1984) 333. · Zbl 0661.17013 · doi:10.1016/0550-3213(84)90052-X |
| [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). |
| [14] | J.L. Cardy, Critical percolation in finite geometries, J. Phys.A 25 (1992) L201. · Zbl 0965.82501 |
| [15] | J.L. Cardy, Conformal invariance and surface critical behavior, Nucl. Phys.B 240 (1984) 514. · doi:10.1016/0550-3213(84)90241-4 |
| [16] | J.L. Cardy, Boundary conditions, fusion rules and the Verlinde formula, Nucl. Phys.B 324 (1989) 581. · doi:10.1016/0550-3213(89)90521-X |
| [17] | P.P.R. Langlands and Y. Saint-Aubin, Conformal invariance in two-dimensional percolation, Bull. Amer. Math. Soc.30 (1994) 1 [math/9401222]. · Zbl 0794.60109 |
| [18] | V. Gurarie, Logarithmic operators in conformal field theory, Nucl. Phys.B 410 (1993) 535 [hep-th/9303160] [INSPIRE]. · Zbl 0990.81686 · doi:10.1016/0550-3213(93)90528-W |
| [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]. · doi:10.1103/PhysRevLett.56.742 |
| [20] | I. Affleck, Universal term in the free energy at a critical point and the conformal anomaly, Phys. Rev. Lett.56 (1986) 746 [INSPIRE]. · doi:10.1103/PhysRevLett.56.746 |
| [21] | L. Rozansky and H. Saleur, Quantum field theory for the multi-variable Alexander-Conway polynomial, Nucl. Phys.B 376 (1992) 461. · doi:10.1016/0550-3213(92)90118-U |
| [22] | H. Saleur, Polymers and percolation in two-dimensions and twisted N = 2 supersymmetry, Nucl. Phys.B 382 (1992) 486 [hep-th/9111007] [INSPIRE]. · doi:10.1016/0550-3213(92)90657-W |
| [23] | V. Gurarie and A.W.W. Ludwig, Conformal algebras of 2D disordered systems, J. Phys.A 35 (2002) L377 [cond-mat/9911392] [INSPIRE]. · Zbl 1066.81604 |
| [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]. · Zbl 1229.81271 · doi:10.1016/j.nuclphysb.2011.05.018 |
| [25] | T. Creutzig and D. Ridout, Logarithmic conformal field theory: beyond an introduction, J. Phys.A 46 (2013) 4006 [arXiv:1303.0847] [INSPIRE]. · Zbl 1280.81118 |
| [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] | 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]. · Zbl 1280.81116 |
| [28] | M. Hogervorst, M. Paulos and A. Vichi, The ABC (in any D) of Logarithmic CFT, JHEP10 (2017) 201 [arXiv:1605.03959] [INSPIRE]. · Zbl 1383.81230 · doi:10.1007/JHEP10(2017)201 |
| [29] | A. Gainutdinov, D. Ridout and I. Runkel, Logarithmic conformal field theory, J. Phys.A 46 (2013) 490301. · Zbl 1282.00021 |
| [30] | G.M.T. Watts, A crossing probability for critical percolation in two-dimensions, J. Phys.A 29 (1996) L363 [cond-mat/9603167] [INSPIRE]. · Zbl 0904.60078 |
| [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]. · Zbl 1314.35188 · doi:10.1007/s00220-014-2189-4 |
| [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]. · Zbl 1314.35189 · doi:10.1007/s00220-014-2185-8 |
| [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]. · Zbl 1311.35314 · doi:10.1007/s00220-014-2190-y |
| [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]. · Zbl 1314.35190 · doi:10.1007/s00220-014-2180-0 |
| [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]. · Zbl 1147.82335 |
| [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]. · Zbl 1294.82022 |
| [37] | G. Delfino and J. Viti, On three-point connectivity in two-dimensional percolation, J. Phys.A 44 (2011) 032001 [arXiv:1009.1314] [INSPIRE]. · Zbl 1208.82023 |
| [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]. · Zbl 1331.82017 · doi:10.1016/j.nuclphysb.2013.07.014 |
| [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]. · doi:10.1103/PhysRevLett.116.130601 |
| [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]. · doi:10.21468/SciPostPhys.1.1.009 |
| [41] | F.Y. Wu, The Potts model, Rev. Mod. Phys.54 (1982) 235. · doi:10.1103/RevModPhys.54.235 |
| [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]. · Zbl 1204.81154 · doi:10.1016/j.nuclphysb.2010.02.016 |
| [43] | R. Vasseur, J.L. Jacobsen and H. Saleur, Logarithmic observables in critical percolation, J. Stat. Mech.1207 (2012) L07001 [arXiv:1206.2312] [INSPIRE]. |
| [44] | P.A. Pearce, J. Rasmussen and J.-B. Zuber, Logarithmic minimal models, J. Stat. Mech.0611 (2006) P11017 [hep-th/0607232] [INSPIRE]. · Zbl 1456.81217 · doi:10.1088/1742-5468/2006/11/P11017 |
| [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]. · doi:10.1103/PhysRevLett.119.191601 |
| [46] | P. Kasteleyn and C. Fortuin, Phase transitions in lattice systems with random local properties, J. Phys. Soc. Jpn. Suppl.26 (1969) 11. |
| [47] | C.M. Fortuin and P.W. Kasteleyn, On the Random cluster model. 1. Introduction and relation to other models, Physica57 (1972) 536 [INSPIRE]. |
| [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]. · Zbl 1229.82032 · doi:10.1016/j.nuclphysb.2011.06.012 |
| [49] | F.R. Bernhart, Catalan, Motzkin, and Riordan numbers, Discrete Math.204 (1999) 73. · Zbl 0933.05002 · doi:10.1016/S0012-365X(99)00054-0 |
| [50] | G. Grimmett, The stochastic random-cluster process and the uniqueness of random-cluster measures, Ann. Prob. (1995) 1461. · Zbl 0852.60105 |
| [51] | I. Runkel, Boundary structure constants for the A series Virasoro minimal models, Nucl. Phys.B 549 (1999) 563 [hep-th/9811178] [INSPIRE]. · Zbl 0947.81102 · doi:10.1016/S0550-3213(99)00125-X |
| [52] | I. Runkel, Structure constants for the D series Virasoro minimal models, Nucl. Phys.B 579 (2000) 561 [hep-th/9908046] [INSPIRE]. · Zbl 1071.81571 · doi:10.1016/S0550-3213(99)00707-5 |
| [53] | V.S. Dotsenko and V.A. Fateev, Conformal algebra and multipoint correlation functions in 2D statistical models, Nucl. Phys.B 240 (1984) 312. · doi:10.1016/0550-3213(84)90269-4 |
| [54] | J. Cardy, Scaling and renormalization in statistical physics, Cambridge University Press, Cambridge U.K. (1996). · Zbl 0914.60002 · doi:10.1017/CBO9781316036440 |
| [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]. · Zbl 0945.82002 |
| [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] | P. Francesco, P. Mathieu and D. Senechal, Conformal field theory, Graduate texts in contemporary physics, Springer, Germany (1997). · Zbl 0869.53052 |
| [58] | A.B. Zamolodchikov, Conformal symmetry in two-dimensional space: on a recurrent representation of the conformal block, Teoret. Mat. Fiz.73 (1987) 103. |
| [59] | H. Saleur and B. Duplantier, Exact determination of the percolation hull exponent in two dimensions, Phys. Rev. Lett.58 (1987) 2325 [INSPIRE]. · doi:10.1103/PhysRevLett.58.2325 |
| [60] | J. Cardy, The number of incipient spanning clusters in two-dimensional percolation, J. Phys.A 31 (1998) L105 [cond-mat/9705137]. · Zbl 0973.82021 |
| [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]. · Zbl 1375.65097 · doi:10.1016/j.jsc.2016.11.014 |
| [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]. · Zbl 1285.81062 · doi:10.1016/j.nuclphysb.2014.02.022 |
| [63] | R.H. Swendsen and J.-S. Wang, Nonuniversal critical dynamics in Monte Carlo simulations, Phys. Rev. Lett.58 (1987) 86 [INSPIRE]. · doi:10.1103/PhysRevLett.58.86 |
| [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. · Zbl 0917.65005 · doi:10.1145/272991.272995 |
| [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].. · Zbl 1037.82020 |
| [66] | J.L. Cardy, M. Nauenberg and D. Scalapino, Scaling theory of the Potts-model multicritical point, Phys. Rev.B 22 (1980) 2560. · doi:10.1103/PhysRevB.22.2560 |
| [67] | M. Nauenberg and D.J. Scalapino, Singularities and scaling functions at the Potts model multicritical point, Phys. Rev. Lett.44 (1980) 837 [INSPIRE]. · doi:10.1103/PhysRevLett.44.837 |
| [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]. · doi:10.1103/PhysRevLett.99.055701 |
| [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. · Zbl 1084.82532 · doi:10.1007/BF01019724 |
| [70] | E.L. Ince, Ordinary differential equations, Dover Publications, New York U.S.A. (1944). · Zbl 0063.02971 |
| [71] | S. Ribault, Conformal field theory on the plane, arXiv:1406.4290 [INSPIRE]. · Zbl 1272.81178 |
| [72] | N. Javerzat, R. Santachiara and O. Foda, Notes on the solutions of Zamolodchikov-type recursion relations in Virasoro minimal models, JHEP08 (2018) 183 [arXiv:1806.02790] [INSPIRE]. · Zbl 1396.81177 · doi:10.1007/JHEP08(2018)183 |
| [73] | M. van Hoeij, private communication. |
| [74] | HypergeometricPFQ, http://functions.wolfram.com/HypergeometricFunctions/HypergeometricPFQ/06/01/05/02/0004/ |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
