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Liouville quantum gravity and KPZ. (English) Zbl 1226.81241
Authors’ abstract: Consider a bounded planar domain \(D\), an instance \(h\) of the Gaussian free field on \(D\), with Dirichlet energy \((2\pi )^{-1}\int _{D }\nabla h(z)\cdot \nabla h(z)dz\), and a constant \(0\leq \gamma <2\). The Liouville quantum gravity measure on \(D\) is the weak limit as \(\varepsilon \rightarrow 0\) of the measures
\[ \varepsilon^{\gamma^2/2} e^{\gamma h_\varepsilon(z)}dz, \]
where \(dz\) is the Lebesgue measure on \(D\) and \(h _{\varepsilon }(z)\) denotes the mean value of \(h\) on the circle of radius \(\varepsilon\) centered at \(z\). Given a random (or deterministic) subset \(X\) of \(D\) one can define the scaling dimension of \(X\) using either the Lebesgue measure or this random measure. We derive a general quadratic relation between these two dimensions, which we view as a probabilistic formulation of the V. G. Knizhnik, A. M. Polyakov and A. B. Zamolodchikov [“Fractal structure of 2D-quantum gravity”, Mod. Phys. Lett. A 3, 819–826 (1988)] relation from conformal field theory. We also present a boundary analog of KPZ (for subsets of \(\partial D\)). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity.

MSC:
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
28A33 Spaces of measures, convergence of measures
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
60A10 Probabilistic measure theory
60K40 Other physical applications of random processes
83C80 Analogues of general relativity in lower dimensions
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[1] Ambjørn, J., Anagnostopoulos, K.N., Magnea, U., Thorleifsson, G.: Geometrical interpretation of the Knizhnik-Polyakov-Zamolodchikov exponents. Phys. Lett. B 388, 713–719 (1996). arXiv:hep-lat/9606012
[2] Alvarez-Gaumé, L., Barbón, J.L.F., Crnković, Č.: A proposal for strings at D&gt;1. Nucl. Phys. B 394, 383–422 (1993). arXiv:hep-th/9208026
[3] Anagnostopoulos, K., Bialas, P., Thorleifsson, G.: The Ising model on a quenched ensemble of c= gravity graphs. J. Stat. Phys. 94, 321–345 (1999). arXiv:cond-mat/9804137 · Zbl 0958.82021
[4] Aizenman, M., Duplantier, B., Aharony, A.: Path-crossing exponents and the external perimeter in 2D percolation. Phys. Rev. Lett. 83, 1359–1362 (1999). arXiv:cond-mat/9901018
[5] Ambjørn, J., Durhuus, B., Fröhlich, J.: Diseases of triangulated random surface models, and possible cures. Nucl. Phys. B 257, 433–449 (1985)
[6] Ambjørn, J., Durhuus, B., Jonsson, T.: A solvable 2d gravity model with {\(\gamma\)}&gt;0. Mod. Phys. Lett. A 9, 1221–1228 (1994) · Zbl 1022.81722
[7] Ambjørn, J., Durhuus, B., Jonsson, T.: Quantum Geometry, a Statistical Field Theory Approach. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1997) · Zbl 0993.82500
[8] Albeverio, S., Gallavotti, G., Høegh-Krohn, R.: Some results for the exponential interaction in two or more dimensions. Commun. Math. Phys. 70(2), 187–192 (1979) · Zbl 0433.60098
[9] Albeverio, S., Høegh-Krohn, R.: The Wightman axioms and the mass gap for strong interactions of exponential type in two-dimensional space-time. J. Funct. Anal. 16, 39–82 (1974) · Zbl 0279.60095
[10] Albeverio, S., Høegh-Krohn, R., Paycha, S., Scarlatti, S.: A global and stochastic analysis approach to bosonic strings and associated quantum fields. Acta Appl. Math. 26(2), 103–195 (1992) · Zbl 0755.60039
[11] Ambjørn, J., Jurkiewicz, J., Watabiki, Y.: On the fractal structure of two-dimensional quantum gravity. Nucl. Phys. B 454(1–2), 313–342 (1995). arXiv:hep-lat/9507014 · Zbl 0925.83006
[12] Angel, O., Schramm, O.: Uniform infinite planar triangulations. Commun. Math. Phys. 241(2–3), 191–213 (2003). arXiv:math/0207153 · Zbl 1098.60010
[13] Ambjørn, J., Watabiki, Y.: Scaling in quantum gravity. Nucl. Phys. B 445(1), 129–142 (1995). arXiv:hep-th/9501049 · Zbl 1006.83015
[14] Bauer, M.: Aspects de l’invariance conforme. Ph.D. Thesis, Université Paris 7 (1990)
[15] Bernardi, O., Bousquet-Mélou, M.: Counting colored planar maps: algebraicity results (2009). arXiv:0909.1695 · Zbl 1223.05123
[16] Bouttier, J., Di Francesco, P., Guitter, E.: Census of planar maps: from the one-matrix model solution to a combinatorial proof. Nucl. Phys. B 645(3), 477–499 (2002) · Zbl 0999.05052
[17] Bouttier, J., Di Francesco, P., Guitter, E.: Combinatorics of hard particles on planar graphs. Nucl. Phys. B 655(3), 313–341 (2003) · Zbl 1009.82005
[18] Bouttier, J., Di Francesco, P., Guitter, E.: Geodesic distance in planar graphs. Nucl. Phys. B 663(3), 535–567 (2003) · Zbl 1022.05022
[19] Bouttier, J., Di Francesco, P., Guitter, E.: Blocked edges on Eulerian maps and mobiles: application to spanning trees, hard particles and the Ising model. J. Phys. A Math. Theor. 40(27), 7411–7440 (2007) · Zbl 1147.82003
[20] Barbón, J.L.F., Demeterfi, K., Klebanov, I.R., Schmidhuber, C.: Correlation functions in matrix models modified by wormhole terms. Nucl. Phys. B 440, 189–214 (1995). arXiv:hep-th/9501058 · Zbl 0990.81703
[21] Bernardi, O.: Bijective counting of tree-rooted maps and shuffles of parenthesis systems. Electron. J. Comb. 14(1), #R9 (2007). arXiv:math/0601684 · Zbl 1115.05002
[22] Bernardi, O.: A characterization of the Tutte polynomial via combinatorial embeddings. Ann. Comb. 12(2), 139–153 (2008). arXiv:math/0608057 · Zbl 1169.05046
[23] Bernardi, O.: On triangulations with high vertex degree. Ann. Comb. 12(1), 17–44 (2008). arXiv:math/0601678 · Zbl 1154.05039
[24] Bernardi, O.: Tutte polynomial, subgraphs, orientations and sandpile model: new connections via embeddings. Electron. J. Comb. 15(1), #R109 (2008). arXiv:math/0612003 · Zbl 1179.05048
[25] Banderier, C., Flajolet, P., Schaeffer, G., Soria, M.: Random maps, coalescing saddles, singularity analysis, and Airy phenomena. Random Struct. Algorithms 19(3) (2001) · Zbl 1016.68179
[26] Bouttier, J., Guitter, E.: Statistics of geodesics in large quadrangulations. J. Phys. A Math. Theor. 41, 145001 (2008). arXiv:0712.2160 · Zbl 1183.82030
[27] Bouttier, J., Guitter, E.: The three-point function of planar quadrangulations. J. Stat. Mech. 7, P07020 (2008). arXiv:0805.2355 · Zbl 1183.82030
[28] Bouttier, J., Guitter, E.: Confluence of geodesic paths and separating loops in large planar quadrangulations. J. Stat. Mech., P03001 (2009). arXiv:0811.0509 · Zbl 1179.82069
[29] Brézin, E., Itzykson, C., Parisi, G., Zuber, J.B.: Planar diagrams. Commun. Math. Phys. 59, 35–51 (1978) · Zbl 0997.81548
[30] 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, 641–686 (1986) · Zbl 0968.81540
[31] Boulatov, D.V., Kazakov, V.A., Kostov, I.K., Migdal, A.A.: Possible types of critical behaviour and the mean size of dynamically triangulated random surfaces. Phys. Lett. B 174, 87–93 (1986) · Zbl 0968.81540
[32] Borodin, A.N., Salminen, P.: Handbook of Brownian Motion, p. 295, 2nd edn. Birkhäuser, Basel (2000). formulae 2.0.1–2.0.2 · Zbl 0859.60001
[33] Bousquet-Mélou, M., Schaeffer, G.: The degree distribution in bipartite planar maps: applications to the Ising model. In: Eriksson, K., Linusson, S. (eds.) Proceedings of FPSAC 03 (Formal Power Series and Algebraic Combinatorics), Vadstena, Sweden, June 2003, pp. 312–323 (2003). arXiv:math/0211070
[34] Benjamini, I., Schramm, O.: KPZ in one dimensional random geometry of multiplicative cascades. Commun. Math. Phys 289, 46–56 (2009). arXiv:0806.1347 · Zbl 1170.83006
[35] Chapuy, G.: Asymptotic enumeration of constellations and related families of maps on orientable surfaces. Comb. Probab. Comput. 18(4), 477–516 (2009) · Zbl 1221.05204
[36] Chapuy, G.: The structure of unicellular maps, and a connection between maps of positive genus and planar labelled trees. Probab. Theory Relat. Fields 147(3), 415–447 (2010) · Zbl 1195.60013
[37] Chern, S.: An elementary proof of the existence of isothermal parameters on a surface. Proc. Am. Math. Soc. 6, 771–782 (1955) · Zbl 0066.15402
[38] Chapuy, G., Marcus, M., Schaeffer, G.: A bijection for rooted maps on orientable surfaces. SIAM J. Discrete Math. 23(3), 1587–1611 (2009) · Zbl 1207.05087
[39] Daul, J.-M.: Q-States Potts model on a random planar lattice (1995). arXiv:hep-th/9502014 . Unpublished
[40] David, F.: Randomly triangulated surfaces in dimensions. Phys. Lett. B 159, 303–306 (1985)
[41] David, F.: Conformal field theories coupled to 2-D gravity in the conformal gauge. Mod. Phys. Lett. A 3(17), 1651–1656 (1988)
[42] David, F.: Random matrices and two-dimensional gravity. In: Fundamental Problems in Statistical Mechanics, VIII (Altenberg, 1993), pp. 105–126. North-Holland, Amsterdam (1994)
[43] David, F.: Simplicial quantum gravity and random lattices. In: Julia, B., Zinn-Justin, J. (eds.) Gravitation et quantifications (Les Houches, Session LVII, 1992), pp. 679–749. Elsevier B.V., Amsterdam (1995) · Zbl 0856.53069
[44] Duplantier, B., Binder, I.A.: Harmonic measure and winding of conformally invariant curves. Phys. Rev. Lett. 89, 264101 (2002). arXiv:cond-mat/0208045
[45] David, F., Bauer, M.: Another derivation of the geometrical KPZ relations. J. Stat. Mech. 3, P03004 (2009). arXiv:0810.2858
[46] Das, S.R., Dhar, A., Sengupta, A.M., Wadia, S.R.: New critical behavior in d=0 large-N matrix models. Mod. Phys. Lett. A 5, 1041–1056 (1990) · Zbl 1020.81740
[47] Di Francesco, P., Guitter, E.: Geometrically constrained statistical systems on regular and random lattices: from folding to meanders. Phys. Rep. 415(1), 1–88 (2005)
[48] Di Francesco, P., Golinelli, O., Guitter, E.: Meanders: exact asymptotics. Nucl. Phys. B 570(3), 699–712 (2000) · Zbl 0984.82024
[49] Di Francesco, P., Ginsparg, P., Zinn-Justin, J.: 2D gravity and random matrices. Phys. Rep. 254, 1–133 (1995)
[50] Duplantier, B., Kostov, I.K.: Conformal spectra of polymers on a random surface. Phys. Rev. Lett. 61, 1433–1437 (1988)
[51] Duplantier, B., Kwon, K.-H.: Conformal invariance and intersection of random walks. Phys. Rev. Lett. 61, 2514–2517 (1988)
[52] Distler, J., Kawai, H.: Conformal field theory and 2D quantum gravity. Nucl. Phys. B 321, 509–527 (1989)
[53] Duplantier, B., Kostov, I.K.: Geometrical critical phenomena on a random surface of arbitrary genus. Nucl. Phys. B 340, 491–541 (1990)
[54] Dai, J., Luo, W., Jin, M., Zeng, W., He, Y., Yau, S.-T., Gu, X.: Geometric accuracy analysis for discrete surface approximation. Comput. Aided Geom. Des. 24(6), 323–338 (2007) · Zbl 1171.65360
[55] Dorn, H., Otto, H.-J.: Two- and three-point functions in Liouville theory. Nucl. Phys. B 429, 375–388 (1994). arXiv:hep-th/9403141 · Zbl 1020.81770
[56] Duplantier, B., Sheffield, S.: Schramm-Loewner evolution and Liouville quantum gravity (in preparation) · Zbl 1226.81241
[57] Duplantier, B., Sheffield, S.: Duality and KPZ in Liouville quantum gravity. Phys. Rev. Lett. 102, 150603 (2009). arXiv:0901.0277 · Zbl 1226.81241
[58] Duplantier, B.: Random walks and quantum gravity in two dimensions. Phys. Rev. Lett. 81, 5489–5492 (1998) · Zbl 0949.83056
[59] Duplantier, B.: Harmonic measure exponents for two-dimensional percolation. Phys. Rev. Lett. 82, 3940–3943 (1999). arXiv:cond-mat/9901008 · Zbl 1042.82560
[60] Duplantier, B.: Random walks, polymers, percolation, and quantum gravity in two dimensions. Physica A 263(1–4), 452–465 (1999). STATPHYS 20 (Paris, 1998) · Zbl 0949.83056
[61] Duplantier, B.: Two-dimensional copolymers and exact conformal multifractality. Phys. Rev. Lett. 82, 880–883 (1999). arXiv:cond-mat/9812439
[62] Duplantier, B.: Conformally invariant fractals and potential theory. Phys. Rev. Lett. 84, 1363–1367 (2000). arXiv:cond-mat/9908314 · Zbl 1042.82577
[63] Duplantier, B.: Conformal fractal geometry &amp; boundary quantum gravity. In: Fractal Geometry and Applications: a Jubilee of Benoît Mandelbrot, Part 2. Proc. Sympos. Pure Math., vol. 72, pp. 365–482. Am. Math. Soc., Providence (2004). arXiv:math-ph/0303034 · Zbl 1068.60019
[64] Duplantier, B.: Conformal random geometry. In: Bovier, A., Dunlop, F., den Hollander, F., van Enter, A., Dalibard, J. (eds.) Mathematical Statistical Physics, Les Houches Summer School, Session LXXXIII, 2005, pp. 101–217. Elsevier B.V., Amsterdam (2006). arXiv:math-ph/0608053 · Zbl 1370.60013
[65] Durhuus, B.: Multi-spin systems on a randomly triangulated surface. Nucl. Phys. B 426, 203–222 (1994) · Zbl 1049.82521
[66] Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Springer, New York (1997) · Zbl 0793.60030
[67] Eynard, B., Bonnet, G.: The Potts-Q random matrix model: loop equations, critical exponents, and rational case. Phys. Lett. B 463, 273–279 (1999). arXiv:hep-th/9906130 · Zbl 1037.82522
[68] Eynard, B., Kristjansen, C.: Exact solution of the O(n) model on a random lattice. Nucl. Phys. B 455, 577–618 (1995). arXiv:hep-th/9506193 · Zbl 0925.81129
[69] Eynard, B., Orantin, N.: Invariants of algebraic curves and topological expansion. Commun. Number Theory Phys. 1(2), 347–452 (2007) · Zbl 1161.14026
[70] Eynard, B., Orantin, N.: Topological expansion and boundary conditions. J. High Energy Phys. 6, 37 (2008). arXiv:0710.0223 · Zbl 1134.81040
[71] Eynard, B.: Random matrices. Saclay Lectures in Theoretical Physics (2001). http://ipht.cea.fr/Docspht//search/article.php?IDA=257 , unpublished
[72] Eynard, B.: Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence. J. High Energy Phys. 3, 3 (2009). arXiv:0802.1788
[73] Eynard, B., Zinn-Justin, J.: The O(n) model on a random surface: critical points and large-order behaviour. Nucl. Phys. B 386, 558–591 (1992). arXiv:hep-th/9204082
[74] Farkas, H.M., Kra, I.: Riemann Surfaces, 2nd edn. Graduate Texts in Mathematics, vol. 71. Springer, New York (1992) · Zbl 0764.30001
[75] Flajolet, P., Salvy, B., Schaeffer, G.: Airy phenomena and analytic combinatorics of connected graphs. Electron. J. Comb. 11(1), #R34,1–30 (2004) · Zbl 1053.05064
[76] Fateev, V., Zamolodchikov, A.B., Zamolodchikov, Al.B.: Boundary Liouville field theory I. Boundary state and boundary two-point function (2000). arXiv:hep-th/0001012 . Unpublished · Zbl 0737.17014
[77] Gaudin, M., Kostov, I.K.: O(n) model on a fluctuating planar lattice. Some exact results. Phys. Lett. B 220, 200–206 (1989)
[78] Goulian, M., Li, M.: Correlation functions in Liouville theory. Phys. Rev. Lett. 66, 2051–2055 (1991)
[79] Ginsparg, P., Moore, G.: Lectures on 2d gravity and 2d string theory (TASI 1992). In: Harvey, J., Polchinski, J. (eds.) Recent Direction in Particle Theory, Proceedings of the 1992 TASI. World Scientific, Singapore (1993)
[80] Gu, X., Wang, Y., Yau, S.-T.: Geometric compression using Riemann surface structure. Commun. Inf. Syst. 3(3), 171–182 (2004) · Zbl 1112.68478
[81] Gu, X., Yau, S.-T.: Computing conformal structures of surfaces. Commun. Inf. Syst. 2(2), 121–145 (2002) · Zbl 1092.14514
[82] Høegh-Krohn, R.: A general class of quantum fields without cut-offs in two space-time dimensions. Commun. Math. Phys. 21, 244–255 (1971)
[83] Hu, X., Miller, J., Peres, Y.: Thick points of the Gaussian free field. Ann. Probab. 38(2), 896–926 (2010). arXiv:0902.3842 · Zbl 1201.60047
[84] Hosomichi, K.: Bulk-boundary propagator in Liouville theory on a disc. J. High Energy Phys. 11, 44 (2001). arXiv:hep-th/0108093
[85] Janke, W., Johnston, D.A.: The wrong kind of gravity. Phys. Lett. B 460, 271–275 (1999)
[86] Jain, S., Mathur, S.D.: World-sheet geometry and baby universes in 2D quantum gravity. Phys. Lett. B 286, 239–246 (1992)
[87] Jin, M., Wang, Y., Gu, X., Yau, S.-T.: Optimal global conformal surface parameterization for visualization. Commun. Inf. Syst. 4(2), 117–134 (2005) · Zbl 1092.14515
[88] Kazakov, V.A.: Ising model on a dynamical planar random lattice: Exact solution. Phys. Lett. A 119, 140–144 (1986)
[89] Klebanov, I.R., Hashimoto, A.: Non-perturbative solution of matrix models modified by trace-squared terms. Nucl. Phys. B 434, 264–282 (1995) · Zbl 1020.81751
[90] Klebanov, I.R., Hashimoto, A.: Wormholes, matrix models, and Liouville gravity. Nucl. Phys. B Proc. Suppl. 45, 135–148 (1996) · Zbl 0991.81582
[91] Kazakov, V.A., Kostov, I.K.: Loop gas model for open strings. Nucl. Phys. B 386, 520–557 (1992)
[92] Kazakov, V.A., Kostov, I.K., Migdal, A.A.: Critical properties of randomly triangulated planar random surfaces. Phys. Lett. B 157, 295–300 (1985)
[93] Klebanov, I.R.: Touching random surfaces and Liouville gravity. Phys. Rev. D 51, 1836–1841 (1995)
[94] Korchemsky, G.P.: Loops in the curvature matrix model. Phys. Lett. B 296, 323–334 (1992)
[95] Korchemsky, G.P.: Matrix model perturbed by higher order curvature terms. Mod. Phys. Lett. A 7, 3081–3100 (1992) · Zbl 1021.81839
[96] Kostov, I.K.: O(n) vector model on a planar random lattice: Spectrum of anomalous dimensions. Mod. Phys. Lett. A 4, 217–226 (1989)
[97] Kostov, I.K.: The ADE face models on a fluctuating planar lattice. Nucl. Phys. B 326, 583–612 (1989)
[98] Kostov, I.K.: Exact solution of the six-vertex model on a random lattice. Nucl. Phys. B 575(3), 513–534 (2000) · Zbl 1037.82509
[99] Kostov, I.K.: Boundary correlators in 2D quantum gravity: Liouville versus discrete approach. Nucl. Phys. B 658, 397–416 (2003). arXiv:hep-th/0212194 · Zbl 1017.83012
[100] Kostov, I.K.: Boundary loop models and 2D quantum gravity. J. Stat. Mech. 08, P08023 (2007). arXiv:hep-th/0703221 · Zbl 1219.83003
[101] Kostov, I.K.: Boundary O(n) models and 2D quantum gravity. In: Ouvry, S., Jacobsen, J., Pasquier, V., Serban, D., Cugliandolo, L. (eds.) Exact Methods in Low-Dimensional Statistical Physics and Quantum Theory, Les Houches Summer School, Session LXXXIX, 2008. Oxford University Press, London (2009)
[102] Kostov, I.K., Ponsot, B., Serban, D.: Boundary Liouville theory and 2D quantum gravity. Nucl. Phys. B 683, 309–362 (2004) · Zbl 1107.83309
[103] Knizhnik, V.G., Polyakov, A.M., Zamolodchikov, A.B.: Fractal structure of 2D-quantum gravity. Mod. Phys. Lett. A 3, 819–826 (1988)
[104] Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Graduate Texts in Mathematics, vol. 113. Springer, New York (1991) · Zbl 0734.60060
[105] Kazakov, V.A., Zinn-Justin, P.: Two-matrix model with ABAB interaction. Nucl. Phys. B 546(3), 647–668 (1999) · Zbl 0958.81026
[106] Le Gall, J.-F.: The topological structure of scaling limits of large planar maps. Invent. Math. 169, 621–670 (2007) · Zbl 1132.60013
[107] Le Gall, J.-F.: Geodesics in large planar maps and the Brownian map (2008). arXiv:0804.3012 [math.PR]. To appear in Acta Math. doi: 10.1007/s11511-010-0056-5
[108] Le Gall, J.-F., Miermont, G.: Scaling limits of random planar maps with large faces (2009). arXiv:0907.3262 . To appear in Ann. Probab. · Zbl 1204.05088
[109] Lawler, G.F., Schramm, O., Werner, W.: Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math. 187(2), 237–273 (2001). arXiv:math.PR/9911084 · Zbl 1005.60097
[110] Lawler, G.F., Schramm, O., Werner, W.: Values of Brownian intersection exponents. II. Plane exponents. Acta Math. 187(2), 275–308 (2001). arXiv:math.PR/0003156 · Zbl 0993.60083
[111] Lawler, G.F., Schramm, O., Werner, W.: Values of Brownian intersection exponents. III. Two-sided exponents. Ann. Inst. Henri Poincaré Probab. Stat. 38(1), 109–123 (2002). arXiv:math.PR/0005294 · Zbl 1006.60075
[112] Lawler, G.F., Werner, W.: Intersection exponents for planar Brownian motion. Ann. Probab. 27(4), 1601–1642 (1999) · Zbl 0965.60071
[113] Miermont, G.: On the sphericity of scaling limits of random planar quadrangulations. Electron. Commun. Probab. 13, 248–257 (2008) · Zbl 1193.60016
[114] Miermont, G.: Tessellations of random maps of arbitrary genus. Ann. Sci. Éc. Norm. Supér. (4) 42(5), 725–781 (2009) · Zbl 1228.05118
[115] Marckert, J.-F., Miermont, G.: Invariance principles for random bipartite planar maps. Ann. Probab. 35(5), 1642–1705 (2007) · Zbl 1208.05135
[116] Moore, G., Seiberg, N., Staudacher, M.: From loops to states in two-dimensional quantum gravity. Nucl. Phys. B 362, 665–709 (1991)
[117] Miermont, G., Weill, M.: Radius and profile of random planar maps with faces of arbitrary degrees. Electron. J. Probab. 13(4), 79–106 (2008) · Zbl 1190.60024
[118] Nakayama, Y.: Liouville field theory. Int. J. Mod. Phys. A 19, 2771–2930 (2004) · Zbl 1080.81056
[119] Peres, Y., Mörters, P.: Brownian motion. Unpublished draft (2006). http://www.stat.berkeley.edu/\(\sim\)peres/bmbook.pdf · Zbl 1243.60002
[120] Polyakov, A.M.: From quarks to strings. In: Cappelli, A., Castellani, E., Colomo, F., Di Vecchia, P. (eds.) The Birth of String Theory. Cambridge University Press, Cambridge (to appear November 2011). arXiv:0812.0183
[121] Polyakov, A.M.: Quantum geometry of bosonic strings. Phys. Lett. B 103(3), 207–210 (1981)
[122] Polyakov, A.M.: Quantum geometry of fermionic strings. Phys. Lett. B 103(3), 211–213 (1981)
[123] Polyakov, A.M.: Gauge Fields and Strings. Harwood Academic Publishers, Chur (1987)
[124] Polyakov, A.M.: Quantum gravity in two-dimensions. Mod. Phys. Lett. A 2, 893 (1987)
[125] Polyakov, A.M.: Two-dimensional quantum gravity. Superconductivity at high T C . In: Fields, Strings and Critical Phenomena, Les Houches, Session XLIX, 1988, pp. 305–368. North-Holland, Amsterdam (1989)
[126] Ponsot, B., Teschner, J.: Boundary Liouville field theory: boundary three-point function. Nucl. Phys. B 622, 309–327 (2002) · Zbl 0988.81068
[127] Rhodes, R., Vargas, V.: KPZ formula for log-infinitely divisible multifractal random measures (2008). arXiv:0807.1036 [math.PR]. To appear in ESAIM P&amp;S. doi: 10.1051/ps/2010007
[128] Schaeffer, G.: Conjugaison d’arbres et cartes combinatoires aléatoires. Ph.D. Thesis, Univ. Bordeaux I, Talence (1998)
[129] Saleur, H., Duplantier, B.: Exact determination of the percolation hull exponent in two dimensions. Phys. Rev. Lett. 58, 2325–2328 (1987)
[130] Seiberg, N.: Notes on quantum Liouville theory and quantum gravity. Prog. Theor. Phys. Suppl. 102, 319–349 (1990) · Zbl 0790.53059
[131] Sheffield, S.: Conformal weldings of random surfaces: SLE and the quantum gravity zipper. arXiv:1012.4797 · Zbl 1388.60144
[132] Sheffield, S.: Gaussian free fields for mathematicians. Probab. Theory Relat. Fields 139, 521–541 (2007) · Zbl 1132.60072
[133] Simon, B.: The P({\(\Phi\)})2 Euclidean (Quantum) Field Theory. Princeton University Press, Princeton (1974) · Zbl 1175.81146
[134] Smirnov, S., Werner, W.: Critical exponents for two-dimensional percolation. Math. Res. Lett. (2001). arXiv:math.PR/0109120 · Zbl 1009.60087
[135] Takhtajan, L.A.: Liouville Theory: Quantum geometry of Riemann surfaces. Mod. Phys. Lett. A 8, 3529–3535 (1993) · Zbl 1021.81891
[136] Teschner, J.: On the Liouville three-point function. Phys. Lett. B 363, 65–70 (1995). arXiv:hep-th/9507109
[137] Teschner, J.: Liouville theory revisited. Class. Quantum Gravity 18, R153–R222 (2001) · Zbl 1022.81047
[138] Teschner, J.: From Liouville theory to the quantum geometry of Riemann surfaces. In: Prospects in Mathematical Physics. Contemp. Math., vol. 437, pp. 231–246. Am. Math. Soc., Providence (2007) · Zbl 1127.81331
[139] Takhtajan, L.A., Teo, L.-P.: Quantum Liouville theory in the background field formalism I. Compact Riemann surfaces. Commun. Math. Phys. 268, 135–197 (2006). arXiv:hep-th/0508188 · Zbl 1114.32007
[140] Wang, Y., Gu, X., Yau, S.-T.: Surface segmentation using global conformal structure. Commun. Inf. Syst. 4(2), 165–179 (2005) · Zbl 1092.14516
[141] Zamolodchikov, Al.B.: Higher equations of motion in Liouville field theory. Int. J. Mod. Phys. A 19, 510–523 (2004) · Zbl 1080.81062
[142] Zamolodchikov, A.B., Zamolodchikov, Al.B.: Structure constants and conformal bootstrap in Liouville field theory. Nucl. Phys. B 477, 577–605 (1996). arXiv:hep-th/9506136 · Zbl 0925.81301
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