SLE as a mating of trees in Euclidean geometry

Understand

The mating of trees approach to Schramm-Loewner evolution (SLE) in the random geometry of Liouville quantum gravity (LQG) has been recently developed by Duplantier-Miller-Sheffield (2014).

• In this paper we consider the mating of trees approach to SLE in Euclidean geometry.
• Let $\eta$ be a whole-plane space-filling SLE with parameter $\kappa>4$, parameterized by Lebesgue measure.
• The main observable in the mating of trees approach is the contour function, a two-dimensional continuous process describing the evolution of the Minkowski content of the left and right frontier of $\eta$.