Liouville quantum gravity and KPZ

Consider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy (2π)−1 ∫ D ∇h(z) · ∇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 2/2eγhε(z)dz, where dz is Lebesgue measure on D and hε(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 (KPZ, 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. e-mail: [email protected]. Partially supported by grant ANR-08-BLAN-0311-CSD5 and CNRS grant PEPS-PTI 2010. e-mail: [email protected]. Partially supported by NSF grants DMS 0403182 and DMS 064558 and OISE 0730136.