Liouville Quantum Gravity and KPZ

Abstract

Consider a bounded planar domain D, an instance h of the Gaussian free field on D (with Dirichlet energy normalized by 1/(2π)), and a constant 0 < gamma < 2. The Liouville quantum gravity measure on D is the weak limit as epsilon tends to 0 of the measures ε^{γ^2/2} e^{γh_ε(z)}dz, where dz is Lebesgue measure on D and h_ε(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 KPZ relation from conformal field theory. We also present a boundary analog of KPZ (for subsets of the boundary of D). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity.

56 pages. Revised version contains more details. To appear in Inventiones

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• Holographic interpretation of two-dimensional O(N) models coupled to quantum gravity
• Multiple intersection exponents