Consider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy (2π)~(-1)∫_D?h(z){dot operator}?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 ε~(γ2/2)e~(γhε(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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