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Liouville quantum gravity and KPZ

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Abstract

Consider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy (2π)−1∫ D ∇h(z)⋅∇h(z)dz, and a constant 0≤γ<2. The Liouville quantum gravity measure on D is the weak limit as ε→0 of the measures

$$\varepsilon^{\gamma^2/2} e^{\gamma h_\varepsilon(z)}dz,$$

where dz is Lebesgue measure on D and h ε (z) denotes the mean value of h on the circle of radius ε centered at z. Given a random (or deterministic) subset X of D one can define the scaling dimension of X using either Lebesgue measure or this random measure. We derive a general quadratic relation between these two dimensions, which we view as a probabilistic formulation of the Knizhnik, Polyakov, Zamolodchikov (Mod. Phys. Lett. A, 3:819–826, 1988) relation from conformal field theory. We also present a boundary analog of KPZ (for subsets of ∂D). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity.

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Authors and Affiliations

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

    Bertrand Duplantier

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

    Scott Sheffield

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  1. Bertrand Duplantier
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  2. Scott Sheffield
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Correspondence to Scott Sheffield.

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B. Duplantier was partially supported by grant ANR-08-BLAN-0311-CSD5 and CNRS grant PEPS-PTI 2010.

S. Sheffield was partially supported by NSF grants DMS 0403182 and DMS 064558 and OISE 0730136.

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Duplantier, B., Sheffield, S. Liouville quantum gravity and KPZ. Invent. math. 185, 333–393 (2011). https://doi.org/10.1007/s00222-010-0308-1

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