Skip to main content
SpringerLink
Log in

Gaussian Multiplicative Chaos and KPZ Duality

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

This paper is concerned with the construction of atomic Gaussian multiplicative chaos and the KPZ formula in Liouville quantum gravity. On the first hand, we construct purely atomic random measures corresponding to values of the parameter γ 2 beyond the transition phase (i.e. γ 2 > 2d) and check the duality relation with sub-critical Gaussian multiplicative chaos. On the other hand, we give a simplified proof of the classical KPZ formula as well as the dual KPZ formula for atomic Gaussian multiplicative chaos. In particular, this framework allows to construct singular Liouville measures and to understand the duality relation in Liouville quantum gravity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aïdékon, E., Shi, Z.: The Seneta-Heyde scaling for the branching random walk. http://arxiv.org/abs/1102.0217v2 [math.PR], 2012

  2. Allez R., Rhodes R., Vargas V.: Lognormal \({\star}\)-scale invariant random measures. Prob. Th. Rel. Fields 155(3-4), 751–788 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bacry E., Muzy J.F.: Log-infinitely divisible multifractal processes. Commun. Math. Phys. 236(3), 449–475 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Barral J., Mandelbrot B.B.: Multifractal products of cylindrical pulses. Prob. Th. Rel. Fields 124, 409–430 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barral J., Rhodes R., Vargas V.: Limiting laws of supercritical branching random walks. C. R. Acad. Sci. Paris, Ser. I 350, 535–538 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Barral J., Seuret S.: The singularity spectrum of Lévy processes in multifractal time. Adv. Math. 214, 437–468 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Benjamini I., Schramm O.: KPZ in one dimensional random geometry of multiplicative cascades. Commun. Math. Phys. 289(2), 653–662 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Carpentier D., Le Doussal P.: Glass transition of a particle in a random potential, front selection in nonlinear RG and entropic phenomena in Liouville and SinhGordon models. Phys. Rev. E 63, 026110 (2001)

    Article  ADS  Google Scholar 

  9. Castaing B., Gagne Y., Hopfinger E.J.: Velocity probability density-functions of high Reynolds-number turbulence. Physica D 46(2), 177–200 (1990)

    Article  ADS  MATH  Google Scholar 

  10. Castaing B., Gagne Y., Marchand M.: Conditional velocity pdf in 3-D turbulence. J. Phys. II France 4, 1–8 (1994)

    Article  Google Scholar 

  11. David F.: Conformal Field Theories Coupled to 2-D Gravity in the Conformal Gauge. Mod. Phys. Lett. A 3, 1651 (1988)

    Article  ADS  Google Scholar 

  12. Daley, D.J., Vere-Jones, D.: An introduction to the theory of point processes. Volume 2, Probability and its applications, 2nd edition, Berlin-Heidelberg-New York: Springer, 2007

  13. Duplantier, B.: A rigorous perspective on Liouville quantum gravity and KPZ. In: Exact Methods in Low-dimensional Statistical Physics and Quantum Computing, J. Jacobsen, S. Ouvry, V. Pasquier, D. Serban, L.F. Cugliandolo, eds., Lecture Notes of the Les Houches Summer School: Volume 89, (July 2008) Oxford: Oxford University Press Clarendon, 2010

  14. Duplantier B., Sheffield S.: Liouville Quantum Gravity and KPZ. Invent. Math. 185(2), 333–393 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Duplantier B., Sheffield S.: Duality and KPZ in Liouville Quantum Gravity. Phys. Rev. Lett. 102, 150603 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  16. Duplantier, B., Rhodes, R., Sheffield, S., Vargas, V.: Critical Gaussian Multiplicative Chaos: convergence of the derivative martingale. http://arvix.org/abs/1206.1671v2 [math.PR], 2012

  17. Duplantier, B., Rhodes, R., Sheffield, S., Vargas, V.: Renormalization of critical Gaussian multiplicative chaos and KPZ formula. http://arvix.org/abs/1212.0529v1 [math.PR], 2012

  18. Durrett R., Liggett T.M.: Fixed points of the smoothing transformation. Prob. Th. Rel. Fields 64(3), 275–301 (1983)

    MathSciNet  MATH  Google Scholar 

  19. Falconer, K.J.: The geometry of fractal sets. Cambridge: Cambridge University Press, 1985

  20. Falconer, K.J.: Fractal geometry. Hoboken, NJ: John Wiley & Sons Inc., Second edition, 2003

  21. Fan A.H.: Sur le chaos de Lévy d’indice \({0 < \alpha < 1}\). Ann. Sci. Math. Québec 21(1), 53–66 (1997)

    MATH  Google Scholar 

  22. Frisch U.: Turbulence. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  23. Garban, C.: Quantum gravity and the KPZ formula. Séminaire Bourbaki, 64e année, 2011-2012, no 1052

  24. Garban, C., Rhodes, R., Vargas, V.: Liouville Brownian motion. http://arxiv.org/abs/1301.2876v1 [math.PR], 2013

  25. Kahane J.-P.: Sur le modèle de turbulence de Benoît Mandelbrot. C.R. Acad. Sci. Paris 278, 567–569 (1974)

    Google Scholar 

  26. Kahane J.-P.: Sur le chaos multiplicatif. Ann. Sci. Math. Québec 9(2), 105–150 (1985)

    MathSciNet  MATH  Google Scholar 

  27. Kahane J.-P., Peyrière J.: Sur certaines martingales de B. Mandelbrot. Adv. Math. 22, 131–145 (1976)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  29. Klebanov I.R., Hashimoto A.: Non-perturbative Solution of Matrix Models Modified by Trace-squared Terms. Nucl. Phys. B 434, 264–282 (1995)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. Klebanov I.R., Hashimoto A.: Wormholes, Matrix Models, and Liouville Gravity. Nucl. Phys. (Proc. Suppl). 45B(C), 135–148C (1996)

    Article  MathSciNet  ADS  Google Scholar 

  31. Knizhnik V.G., Polyakov A.M., Zamolodchikov A.B.: Fractal structure of 2D-quantum gravity. Mod. Phys. Lett A 3(8), 819–826 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  32. Lawler, G.: Conformally Invariant Processes in the Plane. Providence, RI: Amer. Math. Soc., 2005

  33. Liu Q.: On generalized multiplicative cascades. Stoch. Proc. Appl. 86, 263–286 (2000)

    Article  MATH  Google Scholar 

  34. Madaule, T.: Convergence in law for the branching random walk seen from its tip. http://arxiv.org/abs/1107.2543v2 [math.PR], 2012

  35. Mandelbrot, B.B.: Possible refinement of the lognormal hypothesis concerning the distribution of energy in intermittent turbulence Statistical Models and Turbulence. In: Rosenblatt, M. Atta, C.V. ed., Lectures Notes in Physics. 12, New York: Springer–Verlag, 1972, pp. 333–351

  36. Mandelbrot B.B.: Intermittent turbulence in self-similar cascades, divergence of high moments and dimension of the carrier. J. Fluid. Mech. 62, 331–358 (1974)

    Article  ADS  MATH  Google Scholar 

  37. Mandelbrot, B.B.: Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Berlin-Heidelberg-New York: Springer, 1997

  38. Pasenchenko O.Y.: Sufficient conditions for the characteristic function of a two-dimensional isotropic distribution. Th. Prob. Math. Stat. 53, 149–152 (1996)

    MathSciNet  Google Scholar 

  39. Peyrière J.: Turbulence et dimension de Hausdorff. C. R. Acad. Sc. Paris 278, 567–569 (1974)

    MATH  Google Scholar 

  40. Rajput B., Rosinski J.: Spectral representations of infinitely divisible processes. Prob. Th. Rel. Fields 82, 451–487 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  41. Robert R., Vargas V.: Hydrodynamic Turbulence and Intermittent Random Fields. Commun. Math. Phys. 284(3), 649–673 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. Robert R., Vargas V.: Gaussian Multiplicative Chaos revisited. Ann. Prob. 38(2), 605–631 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  43. Rhodes R., Vargas V.: KPZ formula for log-infinitely divisible multifractal random measures. ESAIM Prob. and Stat. 15, 358–371 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  44. Rhodes R., Vargas V.: Multidimensional multifractal random measures. Electronic J. Prob. 15, 241–258 (2010)

    MathSciNet  MATH  Google Scholar 

  45. Rhodes, R., Sohier, J., Vargas, V.: \({\star}\)-scale invariant random measures. http://arxiv.org/abs/1201.5219v1 [math.PR], 2012

  46. Schmitt F., Lavallee D., Schertzer D., Lovejoy S.: Empirical determination of universal multifractal exponents in turbulent velocity fields. Phys. Rev. Lett. 68, 305–308 (1992)

    Article  ADS  Google Scholar 

  47. Sheffield S.: Gaussian free fields for mathematicians. Prob. Th. Rel. Fields 139, 521–541 (1989)

    Article  MathSciNet  Google Scholar 

  48. Stolovitzky G., Kailasnath P., Sreenivasan K.R.: Kolmogorov’s Refined Similarity Hypotheses. Phys. Rev. Lett. 69(8), 1178–1181 (1992)

    Article  ADS  Google Scholar 

  49. Villani, C.: Optimal Transport. Old and new. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338., Berlin: Springer-Verlag, 2009

  50. Webb C.: Exact asymptotics of the freezing transition of a logarithmically correlated random energy model. J. Stat. Phys. 145, 1595–1619 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université Paris 13, Institut Galilée, LAGA, UMR CNRS 7539, 99 rue Jean-Baptiste Clément, 93430, Villetaneuse, France

    Julien Barral

  2. University of St Andrews, Mathematics Institute, North Haugh, St Andrews, KY16 9SS, Scotland

    Xiong Jin

  3. Université Paris-Dauphine, Ceremade, UMR 7534, Place du maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Rémi Rhodes & Vincent Vargas

Authors
  1. Julien Barral
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiong Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rémi Rhodes
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Vincent Vargas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rémi Rhodes.

Additional information

Communicated by M. Aizenman and S. Smirnov

Xiong Jin: is supported by a Royal Society Newton International Fellowship.

Rémi Rhodes, Vincent Vargas: are partially supported by the CHAMU ANR project.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barral, J., Jin, X., Rhodes, R. et al. Gaussian Multiplicative Chaos and KPZ Duality. Commun. Math. Phys. 323, 451–485 (2013). https://doi.org/10.1007/s00220-013-1769-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-013-1769-z

Keywords

Search

Navigation