Communications in Mathematical Physics

, Volume 330, Issue 1, pp 283–330 | Cite as

Renormalization of Critical Gaussian Multiplicative Chaos and KPZ Relation

  • Bertrand Duplantier
  • Rémi RhodesEmail author
  • Scott Sheffield
  • Vincent Vargas


Gaussian Multiplicative Chaos is a way to produce a measure on \({\mathbb{R}^d}\) (or subdomain of \({\mathbb{R}^d}\)) of the form \({e^{\gamma X(x)} dx}\), where X is a log-correlated Gaussian field and \({\gamma \in [0, \sqrt{2d})}\) is a fixed constant. A renormalization procedure is needed to make this precise, since X oscillates between −∞ and ∞ and is not a function in the usual sense. This procedure yields the zero measure when \({\gamma = \sqrt{2d}}\).

Two methods have been proposed to produce a non-trivial measure when \({\gamma = \sqrt{2d}}\). The first involves taking a derivative at \({\gamma = \sqrt{2d}}\) (and was studied in an earlier paper by the current authors), while the second involves a modified renormalization scheme. We show here that the two constructions are equivalent and use this fact to deduce several quantitative properties of the random measure. In particular, we complete the study of the moments of the derivative multiplicative chaos, which allows us to establish the KPZ formula at criticality. The case of two-dimensional (massless or massive) Gaussian free fields is also covered.


Hausdorff Dimension Random Measure Bessel Process Covariance Kernel Liouville Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Bertrand Duplantier
    • 1
  • Rémi Rhodes
    • 2
    Email author
  • Scott Sheffield
    • 3
  • Vincent Vargas
    • 2
  1. 1.Institut de Physique ThéoriqueGif-sur-Yvette CedexFrance
  2. 2.Université Paris-Dauphine, Ceremade, UMR 7564Paris CedexFrance
  3. 3.Department of MathematicsMassachusetts Institute for TechnologyCambridgeUSA

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