Skip to main content
SpringerLink
Log in

KPZ in One Dimensional Random Geometry of Multiplicative Cascades

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

Abstract

We prove a formula relating the Hausdorff dimension of a subset of the unit interval and the Hausdorff dimension of the same set with respect to a random path matric on the interval, which is generated using a multiplicative cascade. When the random variables generating the cascade are exponentials of Gaussians, the well known KPZ formula of Knizhnik, Polyakov and Zamolodchikov from quantum gravity [KPZ88] appears. This note was inspired by the recent work of Duplantier and Sheffield [DS08] proving a somewhat different version of the KPZ formula for Liouville gravity. In contrast with the Liouville gravity setting, the one dimensional multiplicative cascade framework facilitates the determination of the Hausdorff dimension, rather than some expected box count dimension.

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. Duplantier, B., Sheffield, S.: In preparation, 2008

  2. Kolmogorov, A.N.: The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Translated from the Russian by V. Levin, Turbulence and stochastic processes: Kolmogorov’sideas 50 years on, Proc. Roy. Soc. London Ser. A 434(1890), 9–13 (1991)

  3. Kahane J.-P., Peyrière J.: Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22(2), 131–145 (1976)

    Article  MATH  Google Scholar 

  4. 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  ADS  MathSciNet  Google Scholar 

  5. Liu Q.: Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stoch. Proc. Appl. 95(1), 83–107 (2001)

    Article  MATH  Google Scholar 

  6. Mandelbrot B.: Intermittent turbulence in self similar cascades: divergence of high moments and dimension of carrier. J. Fluid Mech. 62, 331–333 (1974)

    Article  MATH  ADS  Google Scholar 

  7. Mattila, P.: Geometry of sets and measures in Euclidean spaces. Volume 44 of Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press, 1995

  8. Ossiander M., Waymire E.C.: Statistical estimation for multiplicative cascades. Ann. Stat. 28(6), 1533–1560 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Polyakov A.M.: Quantum gravity in two dimensions. Mod. Phys. Lett. A 2(11), 893–898 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  10. Yaglom A.M.: Effect of fluctuations in energy dissipation rate on the form of turbulence characteristics in the inertial subrange. Dokl. Akad. Nauk SSSR 166, 49–52 (1966)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Weizmann Institute of Science, Rehovot, 76100, Israel

    Itai Benjamini

  2. Microsoft Research, Redmond, WA, USA

    Itai Benjamini & Oded Schramm

Authors
  1. Itai Benjamini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oded Schramm
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Itai Benjamini.

Additional information

Communicated by M. Aizenman

Rights and permissions

Reprints and permissions

About this article

Cite this article

Benjamini, I., Schramm, O. KPZ in One Dimensional Random Geometry of Multiplicative Cascades. Commun. Math. Phys. 289, 653–662 (2009). https://doi.org/10.1007/s00220-009-0752-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-009-0752-1

Keywords

Search

Navigation