# KPZ relation does not hold for the level lines and $$\hbox {SLE}_\kappa$$SLEκ flow lines of the Gaussian free field

@article{Aru2013KPZRD,
title={KPZ relation does not hold for the level lines and \$\$\hbox \{SLE\}\_\kappa \$\$SLE$\kappa$ flow lines of the Gaussian free field},
author={Juhan Aru},
journal={Probability Theory and Related Fields},
year={2013},
volume={163},
pages={465-526}
}
• Juhan Aru
• Published 4 December 2013
• Physics
• Probability Theory and Related Fields
In this paper we mingle the Gaussian free field, the Schramm–Loewner evolution (SLE) and the KPZ relation in a natural way, shedding new light on all of them. In particular, we describe the quantum fractal behaviour of the level lines and the $$\hbox {SLE}_\kappa$$SLEκ flow lines of the Gaussian free field by determining their quantum Minkowski dimensions. As a corollary we deduce that the usual KPZ relation is not satisfied. In order to determine the fractal dimensions, we have to make a…
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AbstractConsider 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