Advertisement

Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach

  • Ewain Gwynne
  • Nina HoldenEmail author
  • Joshua Pfeffer
  • Guillaume Remy
Article
  • 29 Downloads

Abstract

There is a substantial literature concerning Liouville quantum gravity (LQG) in two dimensions with conformal matter field of central charge \({{\mathbf {c}}}_{\mathrm M} \in (-\infty ,1]\). Via the DDK ansatz, LQG can equivalently be described as the random geometry obtained by exponentiating \(\gamma \) times a variant of the planar Gaussian free field, where \(\gamma \in (0,2]\) satisfies \({\mathbf {c}}_{\mathrm M} = 25 - 6(2/\gamma + \gamma /2)^2\). Physics considerations suggest that LQG should also make sense in the regime when \({\mathbf {c}}_{\mathrm M} > 1\). However, the behavior in this regime is rather mysterious in part because the corresponding value of \(\gamma \) is complex, so analytic continuations of various formulas give complex answers which are difficult to interpret in a probabilistic setting. We introduce and study a discretization of LQG which makes sense for all values of \({\mathbf {c}}_{\mathrm M} \in (-\infty ,25)\). Our discretization consists of a random planar map, defined as the adjacency graph of a tiling of the plane by dyadic squares which all have approximately the same “LQG size" with respect to the Gaussian free field. We prove that several formulas for dimension-related quantities are still valid for \(\mathbf{c}_{\mathrm M} \in (1,25)\), with the caveat that the dimension is infinite when the formulas give a complex answer. In particular, we prove an extension of the (geometric) KPZ formula for \(\mathbf{c}_{\mathrm M} \in (1,25)\), which gives a finite quantum dimension if and only if the Euclidean dimension is at most \((25-\mathbf{c}_{\mathrm M} )/12\). We also show that the graph distance between typical points with respect to our discrete model grows polynomially whereas the cardinality of a graph distance ball of radius r grows faster than any power of r (which suggests that the Hausdorff dimension of LQG in the case when \({\mathbf {c}}_{\mathrm M} \in (1,25)\) is infinite). We include a substantial list of open problems.

Notes

Acknowledgements

We are grateful to several individuals for helpful discussions, including Timothy Budd, Jian Ding, Bertrand Duplantier, Antti Kupiainen, Greg Lawler, Eveliina Peltola, Rémi Rhodes, Scott Sheffield, Xin Sun, and Vincent Vargas. We thank Scott Sheffield for suggesting the idea of using square subdivisions to approximate LQG for \({{\mathbf {c}}_{\mathrm M}}\in (1,25)\). We also thank the anonymous referee for numerous helpful suggestions and comments. E.G. was partially supported by a Herchel Smith fellowship and a Trinity College junior research fellowship. N.H. was supported by Dr. Max Rössler, the Walter Haefner Foundation, and the ETH Zürich Foundation. G.R. was partially supported by a National Science Foundation mathematical sciences postdoctoral research fellowship. J.P. was partially supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1122374.

References

  1. [AB14]
    Ambjörn, J., Budd, T.G.: Geodesic distances in Liouville quantum gravity. Nucl. Phys. B 889, 676–691 (2014). arXiv:1405.3424 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  2. [ADF86]
    Ambjørn, J., Durhuus, B., Fröhlich, J.: The appearance of critical dimensions in regulated string theories. II. Nucl. Phys. B 275(2), 161–184 (1986)ADSMathSciNetCrossRefGoogle Scholar
  3. [ADH13]
    Abraham, R., Delmas, J.-F., Hoscheit, P.: A note on the Gromov–Hausdorff–Prokhorov distance between (locally) compact metric measure spaces. Electron. J. Probab. 18 14, 21 (2013). arXiv:1202.5464 zbMATHGoogle Scholar
  4. [ADJT93]
    Ambjørn, J., Durhuus, B., Jónsson, T., Thorleifsson, G.: Matter fields with \(c>1\) coupled to 2d gravity. Nucl. Phys. B 398, 568–592 (1993). arXiv:hep-th/9208030 ADSCrossRefGoogle Scholar
  5. [Ald91a]
    Aldous, D.: The continuum random tree. I. Ann. Probab. 19(1), 1–28 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  6. [Ald91b]
    Aldous, D.: The continuum random tree. II. An overview. In Stochastic analysis (Durham, 1990), London Mathematical Society Lecture Note Series, vol. 167, pp. 23–70. Cambridge University Press, Cambridge (1991)CrossRefGoogle Scholar
  7. [Ald93]
    Aldous, D.: The continuum random tree. III. Ann. Probab. 21(1), 248–289 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  8. [Amb94]
    Ambjørn, J.: Remarks about c \(>\) 1 and D \(>\) 2. Teoret. Mat. Fiz. 98(3), 326–336 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  9. [Ang19]
    Ang, M.: Comparison of discrete and continuum Liouville first passage percolation. ArXiv e-prints, Apr (2019). arXiv:1904.09285
  10. [Aru15]
    Aru, J.: KPZ relation does not hold for the level lines and \(\text{ SLE }_\kappa \) flow lines of the Gaussian free field. Probab. Theory Relat. Fields 163(3–4), 465–526 (2015). arXiv:1312.1324 MathSciNetzbMATHCrossRefGoogle Scholar
  11. [Aru17]
    Aru, J.: Gaussian multiplicative chaos through the lens of the 2D Gaussian free field. ArXiv e-prints, Sept (2017). arXiv:1709.04355
  12. [BB19]
    Barkley, J., Budd, T.: Precision measurements of Hausdorff dimensions in two-dimensional quantum gravity. ArXiv e-prints, Aug (2019) arXiv:1908.09469 ADSCrossRefGoogle Scholar
  13. [BD86]
    Billoire, A., David, F.: Scaling properties of randomly triangulated planar random surfaces: a numerical study. Nucl. Phys. B 275(4), 617–640 (1986)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  14. [Bef08]
    Beffara, V.: The dimension of the SLE curves. Ann. Probab. 36(4), 1421–1452 (2008). arXiv:math/0211322 MathSciNetzbMATHCrossRefGoogle Scholar
  15. [Ber17]
    Berestycki, N.: An elementary approach to Gaussian multiplicative chaos. Electron. Commun. Probab. 22(27), 12 (2017). arXiv:1506.09113 MathSciNetzbMATHGoogle Scholar
  16. [BGRV16]
    Berestycki, N., Garban, C., Rhodes, R., Vargas, V.: KPZ formula derived from Liouville heat kernel. J. Lond. Math. Soc. (2) 94(1), 186–208 (2016). arXiv:1406.7280 MathSciNetzbMATHCrossRefGoogle Scholar
  17. [BH92]
    Brézin, E., Hikami, S.: A naive matrix-model approach to 2D quantum gravity coupled to matter of arbitrary central charge. Phys. Lett. B 283, 203–208 (1992). arXiv:hep-th/9204018 ADSMathSciNetCrossRefGoogle Scholar
  18. [BJ92]
    Baillie, C.F., Johnston, D.A.: A numerical test of Kpz scaling: Potts models coupled to two-dimensional quantum gravity. Mod. Phys. Lett. A 7, 1519–1533 (1992). arXiv:hep-lat/9204002 ADSCrossRefGoogle Scholar
  19. [BJM14]
    Bettinelli, J., Jacob, E., Miermont, G.: The scaling limit of uniform random plane maps, via the Ambjørn–Budd bijection. Electron. J. Probab. 19(74), 16 (2014). arXiv:1312.5842 zbMATHGoogle Scholar
  20. [BJRV13]
    Barral, J., Jin, X., Rhodes, R., Vargas, V.: Gaussian multiplicative chaos and KPZ duality. Commun. Math. Phys. 323(2), 451–485 (2013). arXiv:1202.5296 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  21. [BKKM86]
    Boulatov, D.V., Kazakov, V.A., Kostov, I.K., Migdal, A.A.: Analytical and numerical study of a model of dynamically triangulated random surfaces. Nucl. Phys. B 275(4), 641–686 (1986)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  22. [BS09]
    Benjamini, I., Schramm, O.: KPZ in one dimensional random geometry of multiplicative cascades. Commun. Math. Phys. 289(2), 653–662 (2009). arXiv:0806.1347 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  23. [Cat88]
    Cates, M.E.: The Liouville field theory of random surfaces: when is the bosonic string a branched polymer? EPL (Europhys. Lett.) 7, 719 (1988)ADSCrossRefGoogle Scholar
  24. [CKR92]
    Catterall, S., Kogut, J., Renken, R.: Numerical study of c \(>\) 1 matter coupled to quantum gravity. Phys. Lett. B 292, 277–282 (1992)ADSzbMATHCrossRefGoogle Scholar
  25. [Cur16]
    Curien, N.: Peeling random planar maps. Notes du cours Peccot. Available at https://www.math.u-psud.fr/~curien/cours/peccot.pdf (2016). Accessed Mar 2019
  26. [Dav88]
    David, F.: Conformal field theories coupled to 2-D gravity in the conformal gauge. Mod. Phys. Lett. A 3, 1651–1656 (1988)ADSMathSciNetCrossRefGoogle Scholar
  27. [Dav97]
    David, F.: A scenario for the c \(>\) 1 barrier in non-critical bosonic strings. Nucl. Phys. B 487, 633–649 (1997). arXiv:hep-th/9610037 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  28. [DD18]
    Ding, J., Dunlap, A.: Subsequential scaling limits for Liouville graph distance. ArXiv e-prints, Dec (2018). arXiv:1812.06921
  29. [DDDF19]
    Ding, J., Dubédat, J., Dunlap, A., Falconet, H.: Tightness of Liouville first passage percolation for \(\gamma \in (0,2)\). ArXiv e-prints, Apr (2019). arXiv:1904.08021
  30. [DFG+19]
    Dubédat, J., Falconet, H., Gwynne, E., Pfeffer, J., Sun, X.: Weak LQG metrics and Liouville first passage percolation. ArXiv e-prints, May (2019). arXiv:1905.00380
  31. [DFJ84]
    Durhuus, B., Frohlich, J., Jonsson, T.: Critical behavior in a model of planar random surfaces. Nucl. Phys. B 240, 453 (1984). [Phys. Lett.137B,93(1984)]ADSCrossRefGoogle Scholar
  32. [DG16]
    Ding, J., Goswami, S.: Upper bounds on Liouville first passage percolation and Watabiki’s prediction. Commun. Pure Appl. Math., to appear (2016). arXiv:1610.09998
  33. [DG18]
    Ding, J., Gwynne, E.: The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds. Commun. Math. Phys., to appear (2018). arXiv:1807.01072
  34. [DJKP87]
    David, F., Jurkiewicz, J., Krzywicki, A., Petersson, B.: Critical exponents in a model of dynamically triangulated random surfaces. Nucl. Phys. B 290, 218–230 (1987)ADSCrossRefGoogle Scholar
  35. [DK89]
    Distler, J., Kawai, H.: Conformal field theory and 2D quantum gravity. Nucl. Phys. B 321, 509–527 (1989)ADSCrossRefGoogle Scholar
  36. [DKRV16]
    David, F., Kupiainen, A., Rhodes, R., Vargas, V.: Liouville quantum gravity on the Riemann sphere. Commun. Math. Phys. 342(3), 869–907 (2016). arXiv:1410.7318 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  37. [DL18]
    Ding, J., Li, L.: Chemical distances for percolation of planar Gaussian free fields and critical random walk loop soups. Commun. Math. Phys. 360(2), 523–553 (2018). arXiv:1605.04449 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  38. [DMS14]
    Duplantier, B., Miller, J., Sheffield, S.: Liouville quantum gravity as a mating of trees. ArXiv e-prints, Sept (2014). arXiv:1409.7055
  39. [DO94]
    Dorn, H., Otto, H.-J.: Two- and three-point functions in Liouville theory. Nucl. Phys. B 429, 375–388 (1994). arXiv:hep-th/9403141 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  40. [DP86]
    D’Hoker, E., Phong, D.H.: Multiloop amplitudes for the bosonic Polyakov string. Nucl. Phys. B 269(1), 205–234 (1986)ADSMathSciNetCrossRefGoogle Scholar
  41. [DRSV14a]
    Duplantier, B., Rhodes, R., Sheffield, S., Vargas, V.: Critical Gaussian multiplicative chaos: convergence of the derivative martingale. Ann. Probab. 42(5), 1769–1808 (2014). arXiv:1206.1671 MathSciNetzbMATHCrossRefGoogle Scholar
  42. [DRSV14b]
    Duplantier, B., Rhodes, R., Sheffield, S., Vargas, V.: Renormalization of critical Gaussian multiplicative chaos and KPZ relation. Commun. Math. Phys. 330(1), 283–330 (2014). arXiv:1212.0529 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  43. [DRV16]
    David, F., Rhodes, R., Vargas, V.: Liouville quantum gravity on complex tori. J. Math. Phys. 57(2), 022302 (2016). arXiv:1504.00625 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  44. [DS11]
    Duplantier, B., Sheffield, S.: Liouville quantum gravity and KPZ. Invent. Math. 185(2), 333–393 (2011). arXiv:1206.0212 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  45. [DS19]
    Dubédat, J., Shen, H.: Stochastic Ricci flow on compact surfaces. ArXiv e-prints, Apr (2019). arXiv:1904.10909
  46. [Dup10]
    Duplantier, B.: A rigorous perspective on Liouville quantum gravity and the KPZ relation. In: Jacobsen, J., Ouvry, S., Pasquier, V., Serban, D., Cugliandolo, L. (eds.) Exact Methods in Low-Dimensional Statistical Physics and Quantum Computing, pp. 529–561. Oxford University Press, Oxford (2010)zbMATHGoogle Scholar
  47. [DZZ18]
    Ding, J., Zeitouni, O., Zhang, F.: Heat kernel for Liouville Brownian motion and Liouville graph distance. Commun. Math. Phys., to appear (2018). arXiv:1807.00422
  48. [FK02]
    Faddeev, L.D., Kashaev, R.M.: Strongly coupled quantum discrete Liouville theory: II. Geometric interpretation of the evolution operator. J. Phys. Math. Gen. 35, 4043–4048 (2002). arXiv:hep-th/0201049 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  49. [FKV01]
    Faddeev, L.D., Kashaev, R.M., Volkov, A.Y.: Strongly coupled quantum discrete Liouville theory. I: algebraic approach and duality. Commun. Math. Phys. 219, 199–219 (2001). arXiv:hep-th/0006156 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  50. [Gar18]
    Garban, C.: Dynamical Liouville. ArXiv e-prints, May (2018). arXiv:1805.04507
  51. [GGN13]
    Gurel-Gurevich, O., Nachmias, A.: Recurrence of planar graph limits. Ann. Math. (2) 177(2), 761–781 (2013). arXiv:1206.0707 MathSciNetzbMATHCrossRefGoogle Scholar
  52. [GHM15]
    Gwynne, E., Holden, N., Miller, J.: An almost sure KPZ relation for SLE and Brownian motion. Ann. Probab., to appear (2015). arXiv:1512.01223
  53. [GHS17]
    Gwynne, E., Holden, N., Sun, X.: A mating-of-trees approach for graph distances in random planar maps. ArXiv e-prints, Nov (2017). arXiv:1711.00723
  54. [GHS19]
    Gwynne, E., Holden, N., Sun, X.: Mating of trees for random planar maps and Liouville quantum gravity: a survey. ArXiv e-prints, Oct (2019). arXiv:1910.04713
  55. [GM17]
    Gwynne, E., Miller, J.: Characterizations of \(\text{ SLE }_{\kappa }\) for \(\kappa \in (4,8)\) on Liouville quantum gravity. ArXiv e-prints, Jan (2017). arXiv:1701.05174
  56. [GM19a]
    Gwynne, E., Miller, J.: Conformal covariance of the Liouville quantum gravity metric for \(\gamma \in (0,2)\). ArXiv e-prints, May (2019). arXiv:1905.00384
  57. [GM19b]
    Gwynne, E., Miller, J.: Existence and uniqueness of the Liouville quantum gravity metric for \(\gamma \in (0,2)\). ArXiv e-prints, May (2019). arXiv:1905.00383
  58. [GMS17]
    Gwynne, E., Miller, J., Sheffield, S.: The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity. ArXiv e-prints, May (2017). arXiv:1705.11161
  59. [GMS18]
    Gwynne, E., Miller, J., Sheffield, S.: An invariance principle for ergodic scale-free random environments. ArXiv e-prints, July (2018). arXiv:1807.07515
  60. [GP19a]
    Gwynne, E., Pfeffer, J.: Bounds for distances and geodesic dimension in Liouville first passage percolation. Electron. Commun. Probab. 24(56), 12 (2019). arXiv:1903.09561 MathSciNetzbMATHGoogle Scholar
  61. [GP19b]
    Gwynne, E., Pfeffer, J.: KPZ formulas for the Liouville quantum gravity metric. ArXiv e-prints, May (2019). arXiv:1905.11790
  62. [GRV16]
    Guillarmou, C., Rhodes, R., Vargas, V.: Polyakov’s formulation of 2d bosonic string theory. ArXiv e-prints, July (2016). arXiv:1607.08467
  63. [Gwy19]
    Gwynne, E.: The dimension of the boundary of a Liouville quantum gravity metric ball. arXiv e-prints, Sept (2019). arXiv:1909.08588
  64. [HMP10]
    Hu, X., Miller, J., Peres, Y.: Thick points of the Gaussian free field. Ann. Probab. 38(2), 896–926 (2010). arXiv:0902.3842 MathSciNetzbMATHCrossRefGoogle Scholar
  65. [HRV18]
    Huang, Y., Rhodes, R., Vargas, V.: Liouville quantum gravity on the unit disk. Ann. Inst. Henri Poincaré Probab. Stat. 54(3), 1694–1730 (2018). arXiv:1502.04343 MathSciNetzbMATHCrossRefGoogle Scholar
  66. [HS19]
    Holden, N., Sun, X.: Convergence of uniform triangulations under the Cardy embedding. ArXiv e-prints, May (2019). arXiv:1905.13207
  67. [Hua18]
    Huang, Y.: Path integral approach to analytic continuation of Liouville theory: the pencil region. ArXiv e-prints, Sept (2018). arXiv:1809.08650
  68. [IJS16]
    Ikhlef, Y., Jacobsen, J.L., Saleur, H.: Three-point functions in \(c\le 1\) Liouville theory and conformal loop ensembles. Phys. Rev. Lett. 116, 130601 (2016)ADSMathSciNetCrossRefGoogle Scholar
  69. [JSW18a]
    Junnila, J., Saksman, E., Webb, C.: Imaginary multiplicative chaos: moments, regularity and connections to the Ising model. ArXiv e-prints, June (2018). arXiv:1806.02118
  70. [JSW18b]
    Junnila, J., Saksman, E., Webb, C.: Decompositions of log-correlated fields with applications. ArXiv e-prints, Aug (2018). arXiv:1808.06838
  71. [Kah85]
    Kahane, J.-P.: Sur le chaos multiplicatif. Ann. Sci. Math. Québec 9(2), 105–150 (1985)MathSciNetzbMATHGoogle Scholar
  72. [KL13]
    Kennedy, T., Lawler, G.F.: Lattice effects in the scaling limit of the two-dimensional self-avoiding walk. In: Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics. II. Fractals in Applied Mathematics, volume 601 of Contemporary Mathematics, pp. 195–210. American Mathematical Society, Providence, RI (2013). arXiv:1109.3091
  73. [Kle95]
    Klebanov, I.R.: Touching random surfaces and Liouville gravity. Phys. Rev. D 51, 1836–1841 (1995). arXiv:hep-th/9407167 ADSMathSciNetCrossRefGoogle Scholar
  74. [KPZ88]
    Knizhnik, V., Polyakov, A., Zamolodchikov, A.: Fractal structure of 2D-quantum gravity. Mod. Phys. Lett A 3(8), 819–826 (1988)ADSCrossRefGoogle Scholar
  75. [KRV15]
    Kupiainen, A., Rhodes, R., Vargas, V.: Local conformal structure of Liouville quantum gravity. ArXiv e-prints, Dec (2015). arXiv:1512.01802
  76. [KRV17]
    Kupiainen, A., Rhodes, R., Vargas, V.: Integrability of Liouville theory: proof of the DOZZ Formula. Ann. Math., to appear (2017). arXiv:1707.08785
  77. [Le13]
    Le Gall, J.-F.: Uniqueness and universality of the Brownian map. Ann. Probab. 41(4), 2880–2960 (2013). arXiv:1105.4842 MathSciNetzbMATHCrossRefGoogle Scholar
  78. [Le14]
    Le Gall, J.-F.: Random geometry on the sphere. In: Proceedings of the ICM (2014). arXiv:1403.7943
  79. [LR15]
    Lawler, G.F., Rezaei, M.A.: Minkowski content and natural parameterization for the Schramm–Loewner evolution. Ann. Probab. 43(3), 1082–1120 (2015). arXiv:1211.4146 MathSciNetzbMATHCrossRefGoogle Scholar
  80. [LRV13]
    Lacoin, H., Rhodes, R., Vargas, V.: Complex Gaussian multiplicative chaos. ArXiv e-prints, July (2013). arXiv:1307.6117
  81. [LRV19]
    Lacoin, H., Rhodes, R., Vargas, V.: A probabilistic approach of ultraviolet renormalisation in the boundary Sine–Gordon model. ArXiv e-prints, Mar (2019). arXiv:1903.01394
  82. [Mie13]
    Miermont, G.: The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. 210(2), 319–401 (2013). arXiv:1104.1606 MathSciNetzbMATHCrossRefGoogle Scholar
  83. [Mie14]
    Miermont, G.: Aspects of random maps. St. Flour Lecture Notes. Available at http://perso.ens-lyon.fr/gregory.miermont/coursSaint-Flour.pdf (2014). Accessed Mar 2019
  84. [MS15]
    Miller, J., Sheffield, S.: Liouville quantum gravity and the Brownian map I: the QLE(8/3,0) metric. Invent. Math., to appear (2015). arXiv:1507.00719
  85. [MS16a]
    Miller, J., Sheffield, S.: Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding. ArXiv e-prints, May (2016). arXiv:1605.03563
  86. [MS16b]
    Miller, J., Sheffield, S.: Liouville quantum gravity and the Brownian map III: the conformal structure is determined. ArXiv e-prints, Aug (2016). arXiv:1608.05391
  87. [MS16c]
    Miller, J., Sheffield, S.: Imaginary geometry I: interacting SLEs. Probab. Theory Relat. Fields 164(3–4), 553–705 (2016). arXiv:1201.1496 MathSciNetzbMATHCrossRefGoogle Scholar
  88. [MS17]
    Miller, J., Sheffield, S.: Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees. Probab. Theory Relat. Fields 169(3–4), 729–869 (2017). arXiv:1302.4738 MathSciNetzbMATHCrossRefGoogle Scholar
  89. [MS19]
    Miller, J., Sheffield, S.: Liouville quantum gravity spheres as matings of finite-diameter trees. Ann. Inst. Henri Poincaré Probab. Stat. 55(3), 1712–1750 (2019). arXiv:1506.03804 MathSciNetzbMATHCrossRefGoogle Scholar
  90. [Pol81]
    Polyakov, A.M.: Quantum geometry of bosonic strings. Phys. Lett. B 103(3), 207–210 (1981)ADSMathSciNetCrossRefGoogle Scholar
  91. [Rem18]
    Remy, G.: Liouville quantum gravity on the annulus. J. Math. Phys. 59(8), 082303 (2018). arXiv:1711.06547 ADSMathSciNetzbMATHCrossRefGoogle Scholar
  92. [Rib14]
    Ribault, S.: Conformal field theory on the plane. ArXiv e-prints, June (2014). arXiv:1406.4290
  93. [Rib18]
    Ribault S (2018) Minimal lectures on two-dimensional conformal field theory. SciPost Phys. Lect. Notes 1.  https://doi.org/10.21468/SciPostPhysLectNotes.1, https://scipost.org/10.21468/SciPostPhysLectNotes.1
  94. [RS15]
    Ribault, S., Santachiara, R.: Liouville theory with a central charge less than one. J. High Energy Phys. 2015(8), 109 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  95. [RV11]
    Rhodes, R., Vargas, V.: KPZ formula for log-infinitely divisible multifractal random measures. ESAIM Probab. Stat. 15, 358–371 (2011). arXiv:0807.1036 MathSciNetzbMATHCrossRefGoogle Scholar
  96. [RV14]
    Rhodes, R., Vargas, V.: Gaussian multiplicative chaos and applications: a review. Probab. Surv. 11, 315–392 (2014). arXiv:1305.6221 MathSciNetzbMATHCrossRefGoogle Scholar
  97. [She07]
    Sheffield, S.: Gaussian free fields for mathematicians. Probab. Theory Relat. Fields 139(3–4), 521–541 (2007). arXiv:math/0312099 MathSciNetzbMATHCrossRefGoogle Scholar
  98. [She16]
    Sheffield, S.: Conformal weldings of random surfaces: SLE and the quantum gravity zipper. Ann. Probab. 44(5), 3474–3545 (2016). arXiv:1012.4797 MathSciNetzbMATHCrossRefGoogle Scholar
  99. [SS13]
    Schramm, O., Sheffield, S.: A contour line of the continuum Gaussian free field. Probab. Theory Relat. Fields 157(1–2), 47–80 (2013). arXiv:math/0605337 MathSciNetzbMATHCrossRefGoogle Scholar
  100. [Suz97]
    Suzuki, T.: A note on quantum liouville theory via the quantum group an approach to strong coupling liouville theory. Nucl. Phys. B 492(3), 717–742 (1997)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  101. [Tes04]
    Teschner, J.: A lecture on the Liouville vertex operators. Int. J. Mod. Phys. A 19(supp02), 436–458 (2004)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  102. [Wat93]
    Watabiki, Y.: Analytic study of fractal structure of quantized surface in two-dimensional quantum gravity. Prog. Theor. Phys. Suppl. 114, 1–17 (1993). Quantum gravity (Kyoto, 1992)ADSMathSciNetCrossRefGoogle Scholar
  103. [Zam05]
    Zamolodchikov, A.B.: Three-point function in the minimal Liouville gravity. Theor. Math. Phys. 142(2), 183–196 (2005)zbMATHCrossRefGoogle Scholar
  104. [ZZ96]
    Zamolodchikov, A., Zamolodchikov, A.: Conformal bootstrap in Liouville field theory. Nucl. Phys. B 477, 577–605 (1996). arXiv:hep-th/9506136 ADSMathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2020

Authors and Affiliations

  1. 1.University of CambridgeCambridgeUK
  2. 2.ETH ZürichZürichSwitzerland
  3. 3.MITCambridgeUSA
  4. 4.Columbia UniversityNew York CityUSA

Personalised recommendations