Liouville quantum gravity and KPZ
Author(s)Duplantier, Bertrand; Sheffield, Scott Roger
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Consider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy ... and a constant 0[less than or equal to]γ<2. The Liouville quantum gravity measure on D is the weak limit as epsilon-->0 of the measures ... where dz is Lebesgue measure on D and h epsilon (z) denotes the mean value of h on the circle of radius epsilon 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.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Duplantier, Bertrand, and Scott Sheffield. “Liouville Quantum Gravity and KPZ.” Inventiones mathematicae 185.2 (2010): 333–393.
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