Liouville quantum gravity and KPZ

Consider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy (2 pi)(-1) integral(D) del h(z) . del h(z)dz, and a constant 0 <= gamma < 2. The Liouville quantum gravity measure on D is the weak limit as epsilon -> 0 of the measures epsilon(gamma 2/2)e(gamma h epsilon(z)) dz, 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 delta D). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity.