Duality and KPZ in Liouville Quantum Gravity

Abstract

We present a (mathematically rigorous) probabilistic and geometrical proof of the KPZ relation between scaling exponents in a Euclidean planar domain D and in Liouville quantum gravity. It uses the properly regularized quantum area measure dμ_γ=ε^{γ^2/2} e^{γh_ε(z)}dz, where dz is Lebesgue measure on D, γis a real parameter, 0\leq γ<2, and h_ε(z) denotes the mean value on the circle of radius εcentered at z of an instance h of the Gaussian free field on D. The proof extends to the boundary geometry. The singular case γ>2 is shown to be related to the quantum measure dμ_{γ'}, γ' < 2, by the fundamental duality γγ'=4.

4 pages, 1 figure