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arXiv · 0901.0277

Duality and KPZ in Liouville Quantum Gravity

Bertrand Duplantier·Scott Sheffield
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Abstract

We present a (mathematically rigorous) probabilistic and geometrical proof of the KPZ relation between scaling exponents in a Euclidean planar domain D and in Liouville quantum gravity. It uses the properly regularized quantum area measure dμ_γ=ε^{γ^2/2} e^{γh_ε(z)}dz, where dz is Lebesgue measure on D, γis a real parameter, 0\leq γ<2, and h_ε(z) denotes the mean value on the circle of radius εcentered at z of an instance h of the Gaussian free field on D. The proof extends to the boundary geometry. The singular case γ>2 is shown to be related to the quantum measure dμ_{γ'}, γ' < 2, by the fundamental duality γγ'=4.

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Bertrand Duplantier, Scott Sheffield. 2009-01-02. Duality and KPZ in Liouville Quantum Gravity. https://doi.org/10.1103/physrevlett.102.150603

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