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Renormalization of Critical Gaussian Multiplicative Chaos and KPZ Relation

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Abstract

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.

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Authors and Affiliations

  1. Institut de Physique Théorique, CEA/Saclay, 91191, Gif-sur-Yvette Cedex, France

    Bertrand Duplantier

  2. Université Paris-Dauphine, Ceremade, UMR 7564, Paris Cedex, France

    Rémi Rhodes & Vincent Vargas

  3. Department of Mathematics, Massachusetts Institute for Technology, Cambridge, MA, 02139, USA

    Scott Sheffield

Authors
  1. Bertrand Duplantier
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  2. Rémi Rhodes
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  3. Scott Sheffield
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  4. Vincent Vargas
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Corresponding author

Correspondence to Rémi Rhodes.

Additional information

Communicated by F. Toninelli

R. Duplantier was partially supported by grant ANR-08-BLAN-0311-CSD5 and by the MISTI MIT-France Seed Fund.

R. Rhodes was partially supported by grant ANR-11-JCJC.

S. Sheffield was partially supported by NSF grants DMS 064558, OISE 0730136 and DMS 1209044, and by the MISTI MIT-France Seed Fund.

V. Vargas was partially supported by grant ANR-11-JCJC.

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Duplantier, B., Rhodes, R., Sheffield, S. et al. Renormalization of Critical Gaussian Multiplicative Chaos and KPZ Relation. Commun. Math. Phys. 330, 283–330 (2014). https://doi.org/10.1007/s00220-014-2000-6

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