×

Gaussian multiplicative chaos and applications: a review. (English) Zbl 1316.60073

This rich and interesting survey begins with the seminal work by J.-P. Kahane [Ann. Sci. Math. Qué. 9, 105–150 (1985; Zbl 0596.60041)], which rigorously introduced the standard Gaussian multiplicative chaos, i.e., the random measure \(M_\gamma\) on \(D \subset\mathbb R^d\) defined (at least formally) by: \[ M_\gamma (A) :=\int_A e^{\gamma X(x)-\frac12 \gamma^2\mathbb E[X(x)^2]}\sigma dx, \] where \(\sigma\) is a fixed measure on \(D\), and \(X = (X(x), x \in D)\) is a centred Gaussian field, with correlation \(\mathbb E[X(x)X(y)] \sim \log^+ (\frac{1}{|x-y|})\) as \(|x-y|\to 0\); moreover, \(\gamma < \gamma_c = \sqrt{2d}\) if \(\sigma\) is the Lebesgue measure.
Kahane defined \(M_\gamma\) through a limiting procedure, considering an always slighter cut-off of the field \(X\). Then other regularization procedures have been considered, mainly the convolution of \(X\) by a kernel going to the Dirac mass \(\delta_0\).
The following related questions are also discussed in this survey:
Does the limiting measure \(M_\gamma\) depend on the chosen cut-off procedure?
What are the geometrical and statistical properties of the measure \(M_\gamma\)?
What are the regularity properties (multifractal analysis) of the measure \(M_\gamma\)?
How can we characterize the measure \(M_\gamma\)?
What happens at \(\gamma = \gamma_c\)? And what about \(\gamma > \gamma_c\)?
The case of \(X\) being a Gaussian free field has also been extensively analyzed, and is considered in detail in this survey. A central application of this important case concerns the \(2d\)-Liouville quantum gravity, an attempt to construct a canonical “Riemannian” random metric on the sphere (related to the so-called Brownian map).
As a rigorous construction of this random metric seems to lay beyond the actual possibilities of the theory, instead of the metric itself, the Gaussian multiplicative chaos theory allows to rigorously construct the associated volume form, namely, the so-called (random) Liouville measure.
Thence, on the one hand, important geometrical KPZ formulae are derived, and on the other hand, the whole procedure is generalized to other types of Gaussian fields \(X\).
The authors carefully expose all this matter, and moreover, they show that the Liouville measure can be approached by discrete versions of it on triangular graphs.
Finally, they investigate the case \(\gamma^2\geq \gamma_c^2 = 2d\), too, leading to “critical Gaussian multiplicative chaos” and to “atomic Gaussian multiplicative chaos”, and then in particular to their relationships to duality in \(2d\)-Liouville quantum gravity.

MSC:

60G57 Random measures
60G15 Gaussian processes
60G60 Random fields
28A80 Fractals
60-02 Research exposition (monographs, survey articles) pertaining to probability theory

Citations:

Zbl 0596.60041
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] Aidekon, E. and Shi, Z. (2014). The Seneta-Heyde scaling for the branching random walk. Ann. Probab. 42 , 3, 959-993. · Zbl 1304.60092 · doi:10.1214/12-AOP809
[2] Albeverio, S. and Høegh-Krohn, R. (1974). The Wightman axioms and the mass gap for strong interactions of exponential type in two-dimensional space-time. J. Functional Analysis 16 , 39-82. · Zbl 0279.60095 · doi:10.1016/0022-1236(74)90070-6
[3] Albeverio, S., Gallavotti, G., and Høegh-Krohn, R. (1979). Some results for the exponential interaction in two or more dimensions. Comm. Math. Phys. 70 , 2, 187-192. · Zbl 0433.60098 · doi:10.1007/BF01982355
[4] Aïdékon, E. (2013). Convergence in law of the minimum of a branching random walk. Ann. Probab. 41 , 3A, 1362-1426. · Zbl 1285.60086 · doi:10.1214/12-AOP750
[5] Allez, R., Rhodes, R., and Vargas, V. (2013). Lognormal \(\star\)-scale invariant random measures. Probab. Theory Related Fields 155 , 3-4, 751-788. · Zbl 1278.60083 · doi:10.1007/s00440-012-0412-9
[6] Ambjørn, J., Nielsen, J. L., Rolf, J., Boulatov, D., and Watabiki, Y. (1998). The spectral dimension of \(2\)D quantum gravity. J. High Energy Phys. 2, Paper 10, 8 pp. (electronic). · Zbl 0955.83005 · doi:10.1088/1126-6708/1998/02/010
[7] Ambjørn, J., Anagnostopoulos, K. N., Jensen, L., Ichihara, T., and Watabiki, Y. (1998). Quantum geometry and diffusion. J. High Energy Phys. 11, Paper 22, 16 pp. (electronic). · Zbl 0951.83013 · doi:10.1088/1126-6708/1998/11/022
[8] Arguin, L.-P. and Zindy, O. (2014). Poisson-Dirichlet statistics for the extremes of a log-correlated Gaussian field. Ann. Appl. Probab. 24 , 4, 1446-1481. · Zbl 1301.60042 · doi:10.1214/13-AAP952
[9] Alvarez-Gaumé, L., Barbón, J. L. F., and Crnković, Č. (1993). A proposal for strings at \(D&gt;1\). Nuclear Phys. B 394 , 2, 383-422.
[10] Bacry, E., Delour, J., and Muzy, J. F. (2001). Multifractal random walks. Phys. Rev. E 64 , 026103-026106. · Zbl 0974.91045
[11] Bacry, E., Kozhemyak, A., and Muzy, J.-F. (2008). Continuous cascade models for asset returns. J. Econom. Dynam. Control 32 , 1, 156-199. · Zbl 1181.91338 · doi:10.1016/j.jedc.2007.01.024
[12] Bacry, E. and Muzy, J. F. (2003). Log-infinitely divisible multifractal processes. Comm. Math. Phys. 236 , 3, 449-475. · Zbl 1032.60046 · doi:10.1007/s00220-003-0827-3
[13] Bailleul, I., KPZ in a multidimensional random geometry of multiplicative cascades, available at . · Zbl 1170.83006 · doi:10.1007/s00220-009-0752-1
[14] Barral, J. (1999). Moments, continuité, et analyse multifractale des martingales de Mandelbrot. Probab. Theory Related Fields 113 , 4, 535-569. · Zbl 0936.60045 · doi:10.1007/s004400050217
[15] Barral, J. (2000). Continuity of the multifractal spectrum of a random statistically self-similar measure. J. Theoret. Probab. 13 , 4, 1027-1060. · Zbl 0977.37024 · doi:10.1023/A:1007866024819
[16] Barral, J., Fan, A. H., and Peyrière, J. (2010). Mesures engendrées par multiplications. In Quelques interactions entre analyse, probabilités et fractals . Panor. Synthèses, Vol. 32 . Soc. Math. France, Paris, pp. 57-189. · Zbl 1238.28004
[17] Barral, J., Jin, X., Rhodes, R., and Vargas, V. (2013). Gaussian multiplicative chaos and KPZ duality. Comm. Math. Phys. 323 , 2, 451-485. · Zbl 1287.83019 · doi:10.1007/s00220-013-1769-z
[18] Barral, J. and Jin, X., On exact scaling log-infinitely divisible cascades, to appear in Probability Theory and Related Fields , arXiv: · Zbl 1323.60069 · doi:10.1007/s00440-013-0534-8
[19] Barral, J., Kupiainen, A., Nikula, M., Saksman, E., and Webb, C. (2014). Critical Mandelbrot cascades. Comm. Math. Phys. 325 , 2, 685-711. · Zbl 1302.60065 · doi:10.1007/s00220-013-1829-4
[20] Barral, J., Kupiainen, A., Nikula, M., Saksman, E., and Webb, C., Basic properties of critical lognormal multiplicative chaos, arXiv: · Zbl 1302.60065
[21] Barral, J. and Mandelbrot, B. B. (2002). Multifractal products of cylindrical pulses. Probab. Theory Related Fields 124 , 3, 409-430. · Zbl 1014.60042 · doi:10.1007/s004400200220
[22] Barral, J. and Mandelbrot, B. B. (2004). Introduction to infinite products of independent random functions (Random multiplicative multifractal measures. I). In Fractal geometry and applications: A jubilee of Benoît Mandelbrot, Part 2 . Proc. Sympos. Pure Math., Vol. 72 . Amer. Math. Soc., Providence, RI, pp. 3-16. · Zbl 1088.28004
[23] Barral, J., Rhodes, R., and Vargas, V. (2012). Limiting laws of supercritical branching random walks. C. R. Math. Acad. Sci. Paris 350 , 9-10, 535-538. · Zbl 1260.60173 · doi:10.1016/j.crma.2012.05.013
[24] Barral, J. and Seuret, S. (2007). The singularity spectrum of Lévy processes in multifractal time. Adv. Math. 214 , 1, 437-468. · Zbl 1131.60039 · doi:10.1016/j.aim.2007.02.007
[25] Bauer, M., Bernard, D., and Cantini, L. (2009). Off-critical \(\mathrm{SLE}(2)\) and \(\mathrm{SLE}(4)\): A field theory approach. J. Stat. Mech. Theory Exp. 7, P07037, 32.
[26] Benjamini, I. and Curien, N. (2013). Simple random walk on the uniform infinite planar quadrangulation: Subdiffusivity via pioneer points. Geom. Funct. Anal. 23 , 2, 501-531. · Zbl 1274.60143 · doi:10.1007/s00039-013-0212-0
[27] Benjamini, I. and Schramm, O. (2009). KPZ in one dimensional random geometry of multiplicative cascades. Comm. Math. Phys. 289 , 2, 653-662. · Zbl 1170.83006 · doi:10.1007/s00220-009-0752-1
[28] Biggins, J. D. and Kyprianou, A. E. (2004). Measure change in multitype branching. Adv. in Appl. Probab. 36 , 2, 544-581. · Zbl 1056.60082 · doi:10.1239/aap/1086957585
[29] Biggins, J. D. and Kyprianou, A. E. (2005). Fixed points of the smoothing transform: The boundary case. Electron. J. Probab. 10 , 17, 609-631. · Zbl 1110.60081 · doi:10.1214/EJP.v10-255
[30] Biskup, M. and Louidor, O., Extreme local extrema of the two-dimensional discrete Gaussian free field, arXiv: · Zbl 1347.82007
[31] Boulatov, D. V. and Kazakov, V. A. (1987). Phys. Lett. 184B , 247.
[32] Bramson, M., Ding, J., and Zeitouni, O., Convergence in law of the maximum of the two-dimensional discrete Gaussian Free Field, arXiv: · Zbl 1237.60041
[33] Brézin, É., Kazakov, V. A., and Zamolodchikov, A. B. (1990). Scaling violation in a field theory of closed strings in one physical dimension. Nuclear Phys. B 338 , 3, 673-688. · doi:10.1016/0550-3213(90)90647-V
[34] Carpentier, D. and Le Doussal, P. (2001). Glass transition of a particle in a random potential, front selection in nonlinear RG and entropic phenomena in Liouville and Sinh-Gordon models. Phys. Rev. E 63 , 026110.
[35] Castaing, B., Gagne, Y., and Hopfinger, E. J. (1990). Velocity probability density-functions of high Reynolds-number turbulence. Physica D 46 , 2, 177-200. · Zbl 0718.60097 · doi:10.1016/0167-2789(90)90035-N
[36] Castaing, B., Gagne, Y., and Marchand, M. (1994). Conditional velocity pdf in 3-D turbulence. J. Phys. II France 4 , 1-8.
[37] Chainais, P. (2006). Multidimensional infinitely divisible cascades. Application to the modelling of intermittency in turbulence. European Physical Journal B 51 , 2, 229-243.
[38] Chelkak, D. and Smirnov, S. (2011). Discrete complex analysis on isoradial graphs. Adv. Math. 228 , 3, 1590-1630. · Zbl 1227.31011 · doi:10.1016/j.aim.2011.06.025
[39] Chen, L. and Jakobson, D. (2014). Gaussian free fields and KPZ relation in \(\mathbb{R}^{4}\). Ann. Henri Poincaré 15 , 7, 1245-1283. · Zbl 1298.81341 · doi:10.1007/s00023-013-0277-1
[40] Chevillard, L., Rhodes, R., and Vargas, V. (2013). Gaussian multiplicative chaos for symmetric isotropic matrices. J. Stat. Phys. 150 , 4, 678-703. · Zbl 1264.82008 · doi:10.1007/s10955-013-0697-9
[41] Chevillard L., Robert R., and Vargas V. (2010). A stochastic representation of the local structure of turbulence. EPL , 89 , 54002.
[42] Cizeau, P., Gopikrishnan, P., Liu, Y., Meyer, M., Peng, C. K., and Stanley, E. (1999). Statistical properties of the volatility of price fluctuations. Physical Review E 60 , 2, 1390-1400.
[43] Cont, R. (2001). Empirical properties of asset returns: Stylized facts and statistical issues. Quantitative Finance 1 , 2, 223-236.
[44] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes . Springer Series in Statistics. Springer-Verlag, New York. · Zbl 0657.60069
[45] Das, S. R., Dhar, A., Sengupta, A. M., Wadia, S. R. (1990). Mod. Phys. Lett. A5, 1041.
[46] David, F. (1985). Planar diagrams, two-dimensional lattice gravity and surface models. Nuclear Phys. B 257 , 1, 45-58. · doi:10.1016/0550-3213(85)90335-9
[47] David, F. (1985). A model of random surfaces with nontrivial critical behaviour. Nuclear Phys. B 257 , 4, 543-576. · doi:10.1016/0550-3213(85)90363-3
[48] David, F. (1988). Conformal field theories coupled to \(2\)-D gravity in the conformal gauge. Modern Phys. Lett. A 3 , 17, 1651-1656. · doi:10.1142/S0217732388001975
[49] Distler, J. and Kawai, H. (1989). Conformal field theory and \(2\)D quantum gravity. Nuclear Phys. B 321 , 2, 509-527. · doi:10.1016/0550-3213(89)90354-4
[50] Duchon, J., Robert, R., and Vargas, V. (2012). Forecasting volatility with the multifractal random walk model. Math. Finance 22 , 1, 83-108. · Zbl 1279.60051 · doi:10.1111/j.1467-9965.2010.00458.x
[51] Duplantier, B. (2010). A rigorous perspective on Liouville quantum gravity and the KPZ relation. In Exact Methods in Low-Dimensional Statistical Physics and Quantum Computing . Oxford Univ. Press, Oxford, pp. 529-561. · Zbl 1208.83040
[52] Duplantier, B. (1998). Random walks and quantum gravity in two dimensions. Phys. Rev. Lett. 81 , 25, 5489-5492. · Zbl 0949.83056 · doi:10.1103/PhysRevLett.81.5489
[53] Duplantier, B. (2004). Conformal fractal geometry & boundary quantum gravity. In Fractal geometry and applications: A jubilee of Benoît Mandelbrot, Part 2 . Proc. Sympos. Pure Math., Vol. 72 . Amer. Math. Soc., Providence, RI, pp. 365-482. · Zbl 1068.60019
[54] Duplantier, B. and Kostov, I. (1988). Conformal spectra of polymers on a random surface. Phys. Rev. Lett. 61 , 13, 1433-1437. · doi:10.1103/PhysRevLett.61.1433
[55] Duplantier, D. and Kwon, K.-H. (1988). Conformal invariance and intersection of random walks. Phys. Rev. Lett. 61, 2514-2517.
[56] Duplantier, B. and Sheffield, S. (2009). Duality and the Knizhnik-Polyakov-Zamolodchikov relation in Liouville quantum gravity. Phys. Rev. Lett. 102 , 15, 150603, 4. · doi:10.1103/PhysRevLett.102.150603
[57] Duplantier, B. and Sheffield, S. (2011). Liouville quantum gravity and KPZ. Invent. Math. 185 , 2, 333-393. · Zbl 1226.81241 · doi:10.1007/s00222-010-0308-1
[58] Duplantier, B., Rhodes, R., Sheffield, S., and Vargas, V. (2014). Critical Gaussian multiplicative chaos: Convergence of the derivative martingale. Ann. Probab. 42 , 5, 1769-1808. · Zbl 1306.60055 · doi:10.1214/13-AOP890
[59] Duplantier, B., Rhodes, R., Sheffield, S., and Vargas, V., Renormalization of Critical Gaussian Multiplicative Chaos and KPZ, to appear in Communications in Mathematical Physics , arXiv. · Zbl 1297.60033
[60] Duplantier, B., Rhodes, R., Sheffield, S., and Vargas, V., Log-correlated Gaussian fields: An overview, arXiv: · Zbl 1297.60033
[61] Durrett, R. and Liggett, T. M. (1983). Fixed points of the smoothing transformation. Z. Wahrsch. Verw. Gebiete 64 , 3, 275-301. · Zbl 0506.60097 · doi:10.1007/BF00532962
[62] Falconer, K. J. (1986). The Geometry of Fractal Sets . Cambridge Tracts in Mathematics, Vol. 85 . Cambridge University Press, Cambridge. · Zbl 0599.28009
[63] Fan, A. H. (2004). Some topics in the theory of multiplicative chaos. In Fractal Geometry and Stochastics III . Progr. Probab., Vol. 57 . Birkhäuser, Basel, pp. 119-134. · Zbl 1062.60050 · doi:10.1007/978-3-0348-7891-3_8
[64] Fan, A. H. (1997). Sur les chaos de Lévy stables d’indice \(0&lt;\alpha&lt;1\). Ann. Sci. Math. Québec 21 , 1, 53-66. · Zbl 0884.60040
[65] Foias, C., Manley, O., Rosa, R., and Temam, R. (2001). Navier-Stokes Equations and Turbulence . Encyclopedia of Mathematics and Its Applications, Vol. 83 . Cambridge University Press, Cambridge. · Zbl 0994.35002
[66] Frisch, U. (1995). Turbulence . Cambridge University Press, Cambridge. The legacy of A. N. Kolmogorov. · Zbl 0832.76001
[67] Garrido, L. (Ed.) (1985). Applications of Field Theory to Statistical Mechanics . Lecture Notes in Physics, Vol. 216 . Springer-Verlag, Berlin.
[68] Fyodorov, Y. V. and Bouchaud, J.-P. (2008). Freezing and extreme-value statistics in a random energy model with logarithmically correlated potential. J. Phys. A 41 , 37, 372001, 12. · Zbl 1214.82016 · doi:10.1088/1751-8113/41/37/372001
[69] Fyodorov, Y., Le Doussal, P., and Rosso A. (2009). Statistical mechanics of logarithmic REM: Duality, freezing and extreme value statistics of \(1/f\) noises generated by Gaussian free fields. J. Stat. Mech. , P10005.
[70] Fyodorov, Y., Le Doussal, P., and Rosso, A. (2010). Freezing transition in decaying Burgers turbulence and random matrix dualities. Europhysics Letters , 90, 60004.
[71] Garban, C. (2011-2012). Quantum gravity and the KPZ formula. Séminaire Bourbaki , 64e année, 1052.
[72] Garban, C., Rhodes, R., and Vargas, V., Liouville Brownian motion, arXiv:
[73] Garban, C., Rhodes, R., and Vargas, V., On the heat kernel and the Dirichlet form of Liouville Brownian motion, to appear in Electronic Journal of Probability , arXiv: · Zbl 1334.60175
[74] Ginsparg, P. and Moore, G. (1993). Lectures on 2D gravity and 2D string theory. in Recent Direction in Particle Theory . Proceedings of the 1992 TASI, edited by J. Harvey and J. Polchinski, World Scientific, Singapore.
[75] Ginsparg, P. and Zinn-Justin, J. (1990). \(2\)D gravity \({}+1\)D matter. Phys. Lett. B 240 , 3-4, 333-340. · doi:10.1016/0370-2693(90)91108-N
[76] Glimm, J. and Jaffe, A. (1981). Quantum Physics . Springer-Verlag, New York-Berlin. A functional integral point of view. · Zbl 0461.46051
[77] Gneiting, T. (2001). Criteria of Pólya type for radial positive definite functions. Proc. Amer. Math. Soc. 129 , 8, 2309-2318 (electronic). · Zbl 1008.42012 · doi:10.1090/S0002-9939-01-05839-7
[78] Gray, A., Mathews, G. B., and Macrobert, T. M. (1922). Bessel Functions , Macmillan and Co. · JFM 48.0423.05
[79] Gross, D. J. and Klebanov, I. (1990). One-dimensional string theory on a circle. Nuclear Phys. B 344 , 3, 475-498. · doi:10.1016/0550-3213(90)90667-3
[80] Guerra, F. and Toninelli, F. L. (2002). The thermodynamic limit in mean field spin glass models. Comm. Math. Phys. 230 , 1, 71-79. · Zbl 1004.82004 · doi:10.1007/s00220-002-0699-y
[81] Høegh-Krohn, R. (1971). A general class of quantum fields without cut-offs in two space-time dimensions. Comm. Math. Phys. 21 , 244-255. · doi:10.1007/BF01647122
[82] Hu, X., Miller, J., and Peres, Y. (2010). Thick points of the Gaussian free field. Ann. Probab. 38 , 2, 896-926. · Zbl 1201.60047 · doi:10.1214/09-AOP498
[83] Hu, Y. and Shi, Z. (2009). Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab. 37 , 2, 742-789. · Zbl 1169.60021 · doi:10.1214/08-AOP419
[84] Janson, S. (1997). Gaussian Hilbert Spaces . Cambridge Tracts in Mathematics, Vol. 129 . Cambridge University Press, Cambridge. · Zbl 0887.60009 · doi:10.1017/CBO9780511526169
[85] Kahane, J.-P. (1985). Sur le chaos multiplicatif. Ann. Sci. Math. Québec 9 , 2, 105-150. · Zbl 0596.60041
[86] Kahane, J.-P. (1974). Sur le modèle de turbulence de Benoît Mandelbrot. C.R. Acad. Sci. Paris 278 , 567-569.
[87] Kahane, J.-P. and Peyrière, J. (1976). Sur certaines martingales de Benoit Mandelbrot. Advances in Math. 22 , 2, 131-145. · Zbl 0349.60051 · doi:10.1016/0001-8708(76)90151-1
[88] Kazakov, V., Kostov, I., and Kutasov, D., A Matrix Model for the 2d Black Hole. In Nonperturbative Quantum Effects 2000 , JHEP Proceedings. · Zbl 0988.81099
[89] Kazakov, V. A. (1986). Ising model on a dynamical planar random lattice: Exact solution. Phys. Lett. A 119 , 3, 140-144. · doi:10.1016/0375-9601(86)90433-0
[90] Kazakov, V. A. (1988). Exactly solvable Potts models, bond- and tree-like percolation on dynamical (random) planar lattice. Nuclear Phys. B Proc. Suppl. 4 , 93-97. Field theory on the lattice (Seillac, 1987). · Zbl 0957.82503 · doi:10.1016/0920-5632(88)90089-8
[91] Klebanov, I. R. (1995). Touching random surfaces and Liouville gravity. Phys. Rev. D (3) 51 , 4, 1836-1841. · doi:10.1103/PhysRevD.51.1836
[92] Klebanov, I. R. and Hashimoto, A. (1995). Non-perturbative solution of matrix models modified by trace-squared terms. Nuclear Phys. B 434 , 1-2, 264-282. · Zbl 1020.81751 · doi:10.1016/0550-3213(94)00518-J
[93] Klebanov, I. R. and Hashimoto, A. (1996). Wormholes, matrix models, and Liouville gravity. Nuclear Phys. B Proc. Suppl. 45BC , 135-148. String theory, gauge theory and quantum gravity (Trieste, 1995). · Zbl 0991.81582 · doi:10.1016/0920-5632(95)00631-1
[94] Knizhnik, V. G., Polyakov, A. M., and Zamolodchikov, A. B. (1988). Fractal structure of \(2\)D-quantum gravity. Modern Phys. Lett. A 3 , 8, 819-826. · doi:10.1142/S0217732388000982
[95] Kolmogorov, A. N. (1991). The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Proc. Roy. Soc. London Ser. A 434 , 1890, 9-13. Translated from the Russian by V. Levin, Turbulence and stochastic processes: Kolmogorov’s ideas 50 years on. · Zbl 1142.76389 · doi:10.1098/rspa.1991.0075
[96] Kolmogorov, A. N. (1962). A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13 , 82-85. · Zbl 0112.42003 · doi:10.1017/S0022112062000518
[97] Korchemsky, G. P. (1992). Mod. Phys. Lett. , A7, 3081; (1992). Phys. Lett. 296B, 323. · Zbl 1021.81839 · doi:10.1142/S0217732392002470
[98] Kusuoka, S. (1988). Hoegh-Krohn’s model of quantum fields and the absolute continuity of measures. Ideas and Methods in Quantum and Statistical Physics , Oslo.
[99] Kyprianou, A. E. (1998). Slow variation and uniqueness of solutions to the functional equation in the branching random walk. J. Appl. Probab. 35 , 4, 795-801. · Zbl 0930.60066 · doi:10.1239/jap/1032438375
[100] Lalley, S. P. and Sellke, T. (1987). A conditional limit theorem for the frontier of a branching Brownian motion. Ann. Probab. 15 , 3, 1052-1061. · Zbl 0622.60085 · doi:10.1214/aop/1176992080
[101] Lawler, G. and Limic, V., Random Walk, a Modern Introduction. In Cambridge Studies in Advanced Mathematics , ISBN: 9780521519182. · Zbl 1210.60002 · doi:10.1017/CBO9780511750854
[102] Lawler, G. F., Schramm, O., and Werner, W. (2001). Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math. 187 , 2, 237-273. · Zbl 0993.60083 · doi:10.1007/BF02392619
[103] Lawler, G. F., Schramm, O., and Werner, W. (2001). Values of Brownian intersection exponents. II. Plane exponents. Acta Math. 187 , 2, 275-308. · Zbl 0993.60083 · doi:10.1007/BF02392619
[104] Lawler, G. F., Schramm, O., and Werner, W. (2002). Values of Brownian intersection exponents. III. Two-sided exponents. Ann. Inst. H. Poincaré Probab. Statist. 38 , 1, 109-123. · Zbl 1006.60075 · doi:10.1016/S0246-0203(01)01089-5
[105] Ledoux, M. and Talagrand, M. (1991). Probability in Banach spaces . Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Vol. 23 . Springer-Verlag, Berlin. Isoperimetry and processes. · Zbl 0748.60004
[106] Le Gall, J.-F. (2007). The topological structure of scaling limits of large planar maps. Invent. Math. 169 , 3, 621-670. · Zbl 1132.60013 · doi:10.1007/s00222-007-0059-9
[107] Le Gall, J.-F. (2013). Uniqueness and universality of the Brownian map. Ann. Probab. 41 , 4, 2880-2960. · Zbl 1282.60014 · doi:10.1214/12-AOP792
[108] Le Gall, J. F. and Miermont, G. (2011). Scaling limits of random trees and planar maps, arXiv: · Zbl 1204.05088
[109] Liu, Q. (1998). Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. in Appl. Probab. 30 , 1, 85-112. · Zbl 0909.60075 · doi:10.1239/aap/1035227993
[110] Madaule, T., Convergence in law for the branching random walk seen from its tip, arXiv: · Zbl 1384.60093
[111] Madaule, T., Maximum of a log-correlated Gaussian field, to appear in Annales de l’IHP , arXiv: · Zbl 1329.60138
[112] Madaule, T., Rhodes, R., and Vargas, V., Glassy phase and freezing of log-correlated Gaussian potentials, arXiv: · Zbl 1322.60177
[113] Makarov, N. and Smirnov, S. (2010). Off-critical lattice models and massive SLEs. In XVIth International Congress on Mathematical Physics . World Sci. Publ., Hackensack, NJ, pp. 362-371. · Zbl 1205.82055
[114] Mandelbrot, B. B. (1972). A possible refinement of the lognormal hypothesis concerning the distribution of energy in intermittent turbulence. In Statistical Models and Turbulence , La Jolla, CA, Lecture Notes in Phys., Vol. 12 , Springer, pp. 333-351. · Zbl 0227.76081
[115] Mandelbrot, B. B. (1974). Intermittent turbulence in self-similar cascades, divergence of high moments and dimension of the carrier. J. Fluid. Mech. 62 , 331-358. · Zbl 0289.76031 · doi:10.1017/S0022112074000711
[116] Maruyama, G. (1970). Infinitely divisible processes. Teor. Verojatnost. i Primenen. 15 , 3-23. · Zbl 0268.60036
[117] Miermont, G. (2013). The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. 210 , 2, 319-401. · Zbl 1278.60124 · doi:10.1007/s11511-013-0096-8
[118] Molchan, G. M. (1996). Scaling exponents and multifractal dimensions for independent random cascades. Comm. Math. Phys. 179 , 3, 681-702. · Zbl 0853.76032 · doi:10.1007/BF02100103
[119] Neveu, J. (1988). Multiplicative martingales for spatial branching processes. In Seminar on Stochastic Processes, 1987 (Princeton, NJ, 1987) . Progr. Probab. Statist., Vol. 15 . Birkhäuser Boston, Boston, MA, pp. 223-242. · Zbl 0652.60089 · doi:10.1007/978-1-4684-0550-7_10
[120] Nienhuis, B. (1987). Coulomb gas formulation of two-dimensional phase transitions. In Phase Transitions and Critical Phenomena, Vol. 11 . Academic Press, London, pp. 1-53.
[121] Oboukhov, A. M. (1962). Some specific features of atmospheric turbulence. J. Fluid Mech. 13 , 77-81. · Zbl 0112.42002 · doi:10.1017/S0022112062000506
[122] Parisi, G. (1990). On the one-dimensional discretized string. Phys. Lett. B 238 , 2-4, 209-212. · Zbl 1332.81185 · doi:10.1016/0370-2693(90)91722-N
[123] Parisi, G. and Frisch, U. (1983). On the singularity structure of fully developed turbulence. Proceed. Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics , M. Ghil, R. Benzi and G. Parisi, eds, 1985, Ed. Varenna, pp. 84-87.
[124] Pasenchenko, O. Yu. (1996). Sufficient conditions for the characteristic function of a two-dimensional isotropic distribution. Theory Probab. Math. Statist. 53 , 149-152. · Zbl 0941.60034
[125] Peyrière, J. (1974). Turbulence et dimension de Hausdorff. C. R. Acad. Sci. Paris Sér. A 278 , 567-569. · Zbl 0278.60082
[126] Polchinski, J. (1990). Critical behavior of random surfaces in one dimension. Nuclear Phys. B 346 , 2-3, 253-263. · doi:10.1016/0550-3213(90)90280-Q
[127] Polchinski, J. (1998). String theory. Vol. I . Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge. An Introduction to the Bosonic String. · Zbl 1006.81521
[128] Polyakov, A. M. (1981). Quantum geometry of bosonic strings. Phys. Lett. B 103 , 3, 207-210. · doi:10.1016/0370-2693(81)90743-7
[129] Rhodes, R., Sohier, J., and Vargas, V. (2014). Levy multiplicative chaos and star scale invariant random measures. Ann. Probab. 42 , 2, 689-724. · Zbl 1295.60064 · doi:10.1214/12-AOP810
[130] Rhodes, R. and Vargas, V. (2010). Multidimensional multifractal random measures. Electron. J. Probab. 15 , 9, 241-258. · Zbl 1201.60046
[131] Rhodes, R. and Vargas, V. (2011). KPZ formula for log-infinitely divisible multifractal random measures. ESAIM Probab. Stat. 15 , 358-371. · Zbl 1268.60070 · doi:10.1051/ps/2010007
[132] Rhodes, R. and Vargas, V. (2013). Optimal transportation for multifractal random measures and applications. Ann. Inst. Henri Poincaré Probab. Stat. 49 , 1, 119-137. · Zbl 1296.60130 · doi:10.1214/11-AIHP443
[133] Rhodes, R. and Vargas, V., Spectral dimension of Liouville quantum gravity, to appear in Annales Henri Poincaré , arxiv. · Zbl 1305.83036 · doi:10.1007/s00023-013-0308-y
[134] Robert, R. and Vargas, V. (2008). Hydrodynamic turbulence and intermittent random fields. Comm. Math. Phys. 284 , 3, 649-673. · Zbl 1157.60322 · doi:10.1007/s00220-008-0642-y
[135] Robert, R. and Vargas, V. (2010). Gaussian multiplicative chaos revisited. Ann. Probab. 38 , 2, 605-631. · Zbl 1191.60066 · doi:10.1214/09-AOP490
[136] Schmitt, F., Lavallee, D., Schertzer, D., and Lovejoy, S. (1992). Empirical determination of universal multifractal exponents in turbulent velocity fields. Phys. Rev. Lett. 68 , 305-308.
[137] Sheffield, S. (2007). Gaussian free fields for mathematicians. Probab. Theory Related Fields 139 , 3-4, 521-541. · Zbl 1132.60072 · doi:10.1007/s00440-006-0050-1
[138] Sheffield, S. (2005). Random surfaces. Astérisque 304, vi+175. · Zbl 1104.60002
[139] Simon, B. (1974). The \(P(\phi)_{2}\) Euclidean (quantum) field theory . Princeton University Press, Princeton, N.J., Princeton Series in Physics. · Zbl 1175.81146
[140] Smirnov, S. (2010). Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. of Math. (2) 172 , 2, 1435-1467. · Zbl 1200.82011 · doi:10.4007/annals.2010.172.1441
[141] Stolovitzky, G., Kailasnath, P., and Sreenivasan, K. R. (1992). Kolmogorov’s refined similarity hypotheses. Phys. Rev. Lett. 69 , 8, 1178-1181. · Zbl 0849.76031
[142] Webb, C. (2011). Exact asymptotics of the freezing transition of a logarithmically correlated random energy model. J. Stat. Phys. 145 , 6, 1595-1619. · Zbl 1231.82091 · doi:10.1007/s10955-011-0359-8
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.