| Duality and KPZ in Liouville Quantum Gravity (2009) | |||||||||
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 d\mu_\gamma=\epsilon^{\gamma^2/2} e^{\gamma h_\epsilon(z)}dz, where dz is Lebesgue measure on D, \gamma is a real parameter, 0\leq \gamma 2 is shown to be related to the quantum measure d\mu_{\gamma'}, \gamma' < 2, by the fundamental duality \gamma\gamma'=4.. Comment: 4 pages, 1 figure | |||||||||
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