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

  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},
  • 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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