Liouville quantum gravity and KPZ

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

Alvarez-Gaumé, L., Barbón, J.L.F., Crnković, Č.: A proposal for strings at D>1. Nucl. Phys. B 394, 383–422 (1993). arXiv:hep-th/9208026

Anagnostopoulos, K., Bialas, P., Thorleifsson, G.: The Ising model on a quenched ensemble of c=−5 gravity graphs. J. Stat. Phys. 94, 321–345 (1999). arXiv:cond-mat/9804137

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

Ambjørn, J., Durhuus, B., Fröhlich, J.: Diseases of triangulated random surface models, and possible cures. Nucl. Phys. B 257, 433–449 (1985)

Ambjørn, J., Durhuus, B., Jonsson, T.: A solvable 2d gravity model with γ>0. Mod. Phys. Lett. A 9, 1221–1228 (1994)

Ambjørn, J., Durhuus, B., Jonsson, T.: Quantum Geometry, a Statistical Field Theory Approach. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1997)

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)

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)

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)

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

Angel, O., Schramm, O.: Uniform infinite planar triangulations. Commun. Math. Phys. 241(2–3), 191–213 (2003). arXiv:math/0207153

Ambjørn, J., Watabiki, Y.: Scaling in quantum gravity. Nucl. Phys. B 445(1), 129–142 (1995). arXiv:hep-th/9501049

Bauer, M.: Aspects de l’invariance conforme. Ph.D. Thesis, Université Paris 7 (1990)

Bernardi, O., Bousquet-Mélou, M.: Counting colored planar maps: algebraicity results (2009). arXiv:0909.1695

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)

Bouttier, J., Di Francesco, P., Guitter, E.: Combinatorics of hard particles on planar graphs. Nucl. Phys. B 655(3), 313–341 (2003)

Bouttier, J., Di Francesco, P., Guitter, E.: Geodesic distance in planar graphs. Nucl. Phys. B 663(3), 535–567 (2003)

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)

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

Bernardi, O.: Bijective counting of tree-rooted maps and shuffles of parenthesis systems. Electron. J. Comb. 14(1), #R9 (2007). arXiv:math/0601684

Bernardi, O.: A characterization of the Tutte polynomial via combinatorial embeddings. Ann. Comb. 12(2), 139–153 (2008). arXiv:math/0608057

Bernardi, O.: On triangulations with high vertex degree. Ann. Comb. 12(1), 17–44 (2008). arXiv:math/0601678

Bernardi, O.: Tutte polynomial, subgraphs, orientations and sandpile model: new connections via embeddings. Electron. J. Comb. 15(1), #R109 (2008). arXiv:math/0612003

Banderier, C., Flajolet, P., Schaeffer, G., Soria, M.: Random maps, coalescing saddles, singularity analysis, and Airy phenomena. Random Struct. Algorithms 19(3) (2001)

Bouttier, J., Guitter, E.: Statistics of geodesics in large quadrangulations. J. Phys. A Math. Theor. 41, 145001 (2008). arXiv:0712.2160

Bouttier, J., Guitter, E.: The three-point function of planar quadrangulations. J. Stat. Mech. 7, P07020 (2008). arXiv:0805.2355

Bouttier, J., Guitter, E.: Confluence of geodesic paths and separating loops in large planar quadrangulations. J. Stat. Mech., P03001 (2009). arXiv:0811.0509

Brézin, E., Itzykson, C., Parisi, G., Zuber, J.B.: Planar diagrams. Commun. Math. Phys. 59, 35–51 (1978)

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)

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)

Borodin, A.N., Salminen, P.: Handbook of Brownian Motion, p. 295, 2nd edn. Birkhäuser, Basel (2000). formulae 2.0.1–2.0.2

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

Benjamini, I., Schramm, O.: KPZ in one dimensional random geometry of multiplicative cascades. Commun. Math. Phys 289, 46–56 (2009). arXiv:0806.1347

Chapuy, G.: Asymptotic enumeration of constellations and related families of maps on orientable surfaces. Comb. Probab. Comput. 18(4), 477–516 (2009)

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)

Chern, S.: An elementary proof of the existence of isothermal parameters on a surface. Proc. Am. Math. Soc. 6, 771–782 (1955)

Chapuy, G., Marcus, M., Schaeffer, G.: A bijection for rooted maps on orientable surfaces. SIAM J. Discrete Math. 23(3), 1587–1611 (2009)

Daul, J.-M.: Q-States Potts model on a random planar lattice (1995). arXiv:hep-th/9502014 . Unpublished

David, F.: Randomly triangulated surfaces in −2 dimensions. Phys. Lett. B 159, 303–306 (1985)

David, F.: Conformal field theories coupled to 2-D gravity in the conformal gauge. Mod. Phys. Lett. A 3(17), 1651–1656 (1988)

David, F.: Random matrices and two-dimensional gravity. In: Fundamental Problems in Statistical Mechanics, VIII (Altenberg, 1993), pp. 105–126. North-Holland, Amsterdam (1994)

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)

Duplantier, B., Binder, I.A.: Harmonic measure and winding of conformally invariant curves. Phys. Rev. Lett. 89, 264101 (2002). arXiv:cond-mat/0208045

David, F., Bauer, M.: Another derivation of the geometrical KPZ relations. J. Stat. Mech. 3, P03004 (2009). arXiv:0810.2858

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)

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)

Di Francesco, P., Golinelli, O., Guitter, E.: Meanders: exact asymptotics. Nucl. Phys. B 570(3), 699–712 (2000)

Di Francesco, P., Ginsparg, P., Zinn-Justin, J.: 2D gravity and random matrices. Phys. Rep. 254, 1–133 (1995)

Duplantier, B., Kostov, I.K.: Conformal spectra of polymers on a random surface. Phys. Rev. Lett. 61, 1433–1437 (1988)

Duplantier, B., Kwon, K.-H.: Conformal invariance and intersection of random walks. Phys. Rev. Lett. 61, 2514–2517 (1988)

Distler, J., Kawai, H.: Conformal field theory and 2D quantum gravity. Nucl. Phys. B 321, 509–527 (1989)

Duplantier, B., Kostov, I.K.: Geometrical critical phenomena on a random surface of arbitrary genus. Nucl. Phys. B 340, 491–541 (1990)

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)

Dorn, H., Otto, H.-J.: Two- and three-point functions in Liouville theory. Nucl. Phys. B 429, 375–388 (1994). arXiv:hep-th/9403141

Duplantier, B., Sheffield, S.: Schramm-Loewner evolution and Liouville quantum gravity (in preparation)

Duplantier, B., Sheffield, S.: Duality and KPZ in Liouville quantum gravity. Phys. Rev. Lett. 102, 150603 (2009). arXiv:0901.0277

Duplantier, B.: Random walks and quantum gravity in two dimensions. Phys. Rev. Lett. 81, 5489–5492 (1998)

Duplantier, B.: Harmonic measure exponents for two-dimensional percolation. Phys. Rev. Lett. 82, 3940–3943 (1999). arXiv:cond-mat/9901008

Duplantier, B.: Random walks, polymers, percolation, and quantum gravity in two dimensions. Physica A 263(1–4), 452–465 (1999). STATPHYS 20 (Paris, 1998)

Duplantier, B.: Two-dimensional copolymers and exact conformal multifractality. Phys. Rev. Lett. 82, 880–883 (1999). arXiv:cond-mat/9812439

Duplantier, B.: Conformally invariant fractals and potential theory. Phys. Rev. Lett. 84, 1363–1367 (2000). arXiv:cond-mat/9908314

Duplantier, B.: Conformal fractal geometry & 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

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

Durhuus, B.: Multi-spin systems on a randomly triangulated surface. Nucl. Phys. B 426, 203–222 (1994)

Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Springer, New York (1997)

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

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

Eynard, B., Orantin, N.: Invariants of algebraic curves and topological expansion. Commun. Number Theory Phys. 1(2), 347–452 (2007)

Eynard, B., Orantin, N.: Topological expansion and boundary conditions. J. High Energy Phys. 6, 37 (2008). arXiv:0710.0223

Eynard, B.: Random matrices. Saclay Lectures in Theoretical Physics (2001). http://ipht.cea.fr/Docspht//search/article.php?IDA=257 , unpublished

Eynard, B.: Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence. J. High Energy Phys. 3, 3 (2009). arXiv:0802.1788

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

Farkas, H.M., Kra, I.: Riemann Surfaces, 2nd edn. Graduate Texts in Mathematics, vol. 71. Springer, New York (1992)

Flajolet, P., Salvy, B., Schaeffer, G.: Airy phenomena and analytic combinatorics of connected graphs. Electron. J. Comb. 11(1), #R34,1–30 (2004)

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

Gaudin, M., Kostov, I.K.: O(n) model on a fluctuating planar lattice. Some exact results. Phys. Lett. B 220, 200–206 (1989)

Goulian, M., Li, M.: Correlation functions in Liouville theory. Phys. Rev. Lett. 66, 2051–2055 (1991)

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)

Gu, X., Wang, Y., Yau, S.-T.: Geometric compression using Riemann surface structure. Commun. Inf. Syst. 3(3), 171–182 (2004)

Gu, X., Yau, S.-T.: Computing conformal structures of surfaces. Commun. Inf. Syst. 2(2), 121–145 (2002)

Høegh-Krohn, R.: A general class of quantum fields without cut-offs in two space-time dimensions. Commun. Math. Phys. 21, 244–255 (1971)

Hu, X., Miller, J., Peres, Y.: Thick points of the Gaussian free field. Ann. Probab. 38(2), 896–926 (2010). arXiv:0902.3842

Hosomichi, K.: Bulk-boundary propagator in Liouville theory on a disc. J. High Energy Phys. 11, 44 (2001). arXiv:hep-th/0108093

Janke, W., Johnston, D.A.: The wrong kind of gravity. Phys. Lett. B 460, 271–275 (1999)

Jain, S., Mathur, S.D.: World-sheet geometry and baby universes in 2D quantum gravity. Phys. Lett. B 286, 239–246 (1992)

Jin, M., Wang, Y., Gu, X., Yau, S.-T.: Optimal global conformal surface parameterization for visualization. Commun. Inf. Syst. 4(2), 117–134 (2005)

Kazakov, V.A.: Ising model on a dynamical planar random lattice: Exact solution. Phys. Lett. A 119, 140–144 (1986)

Klebanov, I.R., Hashimoto, A.: Non-perturbative solution of matrix models modified by trace-squared terms. Nucl. Phys. B 434, 264–282 (1995)

Klebanov, I.R., Hashimoto, A.: Wormholes, matrix models, and Liouville gravity. Nucl. Phys. B Proc. Suppl. 45, 135–148 (1996)

Kazakov, V.A., Kostov, I.K.: Loop gas model for open strings. Nucl. Phys. B 386, 520–557 (1992)

Kazakov, V.A., Kostov, I.K., Migdal, A.A.: Critical properties of randomly triangulated planar random surfaces. Phys. Lett. B 157, 295–300 (1985)

Klebanov, I.R.: Touching random surfaces and Liouville gravity. Phys. Rev. D 51, 1836–1841 (1995)

Korchemsky, G.P.: Loops in the curvature matrix model. Phys. Lett. B 296, 323–334 (1992)

Korchemsky, G.P.: Matrix model perturbed by higher order curvature terms. Mod. Phys. Lett. A 7, 3081–3100 (1992)

Kostov, I.K.: O(n) vector model on a planar random lattice: Spectrum of anomalous dimensions. Mod. Phys. Lett. A 4, 217–226 (1989)

Kostov, I.K.: The ADE face models on a fluctuating planar lattice. Nucl. Phys. B 326, 583–612 (1989)

Kostov, I.K.: Exact solution of the six-vertex model on a random lattice. Nucl. Phys. B 575(3), 513–534 (2000)

Kostov, I.K.: Boundary correlators in 2D quantum gravity: Liouville versus discrete approach. Nucl. Phys. B 658, 397–416 (2003). arXiv:hep-th/0212194

Kostov, I.K.: Boundary loop models and 2D quantum gravity. J. Stat. Mech. 08, P08023 (2007). arXiv:hep-th/0703221

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)

Kostov, I.K., Ponsot, B., Serban, D.: Boundary Liouville theory and 2D quantum gravity. Nucl. Phys. B 683, 309–362 (2004)

Knizhnik, V.G., Polyakov, A.M., Zamolodchikov, A.B.: Fractal structure of 2D-quantum gravity. Mod. Phys. Lett. A 3, 819–826 (1988)

Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Graduate Texts in Mathematics, vol. 113. Springer, New York (1991)

Kazakov, V.A., Zinn-Justin, P.: Two-matrix model with ABAB interaction. Nucl. Phys. B 546(3), 647–668 (1999)

Le Gall, J.-F.: The topological structure of scaling limits of large planar maps. Invent. Math. 169, 621–670 (2007)

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

Le Gall, J.-F., Miermont, G.: Scaling limits of random planar maps with large faces (2009). arXiv:0907.3262 . To appear in Ann. Probab.

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

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

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

Lawler, G.F., Werner, W.: Intersection exponents for planar Brownian motion. Ann. Probab. 27(4), 1601–1642 (1999)

Miermont, G.: On the sphericity of scaling limits of random planar quadrangulations. Electron. Commun. Probab. 13, 248–257 (2008)

Miermont, G.: Tessellations of random maps of arbitrary genus. Ann. Sci. Éc. Norm. Supér. (4) 42(5), 725–781 (2009)

Marckert, J.-F., Miermont, G.: Invariance principles for random bipartite planar maps. Ann. Probab. 35(5), 1642–1705 (2007)

Moore, G., Seiberg, N., Staudacher, M.: From loops to states in two-dimensional quantum gravity. Nucl. Phys. B 362, 665–709 (1991)

Miermont, G., Weill, M.: Radius and profile of random planar maps with faces of arbitrary degrees. Electron. J. Probab. 13(4), 79–106 (2008)

Nakayama, Y.: Liouville field theory. Int. J. Mod. Phys. A 19, 2771–2930 (2004)

Peres, Y., Mörters, P.: Brownian motion. Unpublished draft (2006). http://www.stat.berkeley.edu/~peres/bmbook.pdf

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

Polyakov, A.M.: Quantum geometry of bosonic strings. Phys. Lett. B 103(3), 207–210 (1981)

Polyakov, A.M.: Quantum geometry of fermionic strings. Phys. Lett. B 103(3), 211–213 (1981)

Polyakov, A.M.: Gauge Fields and Strings. Harwood Academic Publishers, Chur (1987)

Polyakov, A.M.: Quantum gravity in two-dimensions. Mod. Phys. Lett. A 2, 893 (1987)

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)

Ponsot, B., Teschner, J.: Boundary Liouville field theory: boundary three-point function. Nucl. Phys. B 622, 309–327 (2002)

Rhodes, R., Vargas, V.: KPZ formula for log-infinitely divisible multifractal random measures (2008). arXiv:0807.1036 [math.PR]. To appear in ESAIM P&S. doi: 10.1051/ps/2010007

Schaeffer, G.: Conjugaison d’arbres et cartes combinatoires aléatoires. Ph.D. Thesis, Univ. Bordeaux I, Talence (1998)

Saleur, H., Duplantier, B.: Exact determination of the percolation hull exponent in two dimensions. Phys. Rev. Lett. 58, 2325–2328 (1987)

Seiberg, N.: Notes on quantum Liouville theory and quantum gravity. Prog. Theor. Phys. Suppl. 102, 319–349 (1990)

Sheffield, S.: Conformal weldings of random surfaces: SLE and the quantum gravity zipper. arXiv:1012.4797

Sheffield, S.: Gaussian free fields for mathematicians. Probab. Theory Relat. Fields 139, 521–541 (2007)

Simon, B.: The P(Φ)2 Euclidean (Quantum) Field Theory. Princeton University Press, Princeton (1974)

Smirnov, S., Werner, W.: Critical exponents for two-dimensional percolation. Math. Res. Lett. (2001). arXiv:math.PR/0109120

Takhtajan, L.A.: Liouville Theory: Quantum geometry of Riemann surfaces. Mod. Phys. Lett. A 8, 3529–3535 (1993)

Teschner, J.: On the Liouville three-point function. Phys. Lett. B 363, 65–70 (1995). arXiv:hep-th/9507109

Teschner, J.: Liouville theory revisited. Class. Quantum Gravity 18, R153–R222 (2001)

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)

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

Wang, Y., Gu, X., Yau, S.-T.: Surface segmentation using global conformal structure. Commun. Inf. Syst. 4(2), 165–179 (2005)

Zamolodchikov, Al.B.: Higher equations of motion in Liouville field theory. Int. J. Mod. Phys. A 19, 510–523 (2004)

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