Liouville quantum gravity and KPZ

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]γ0 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.French National Research Agency (ANR-08-BLAN-0311-CSD5)Centre National de la Recherche Scientifique (CNRS grant PEPS-PTI 2010)National Science Foundation (U.S.) (NSF grant DMS 0403182)National Science Foundation (U.S.) (grant DMS 064558)National Science Foundation (U.S.) (grant OISE 0730136