SLE as a Mating of Trees in Euclidean Geometry

Abstract

Abstract 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 et al. (Liouville quantum gravity as a mating of trees, 2014. arXiv:1409.7055

). In this paper we consider the mating of trees approach to SLE in Euclidean geometry. Let

$${\eta}$$

η

be a whole-plane space-filling SLE with parameter

$${\kappa > 4}$$

κ > 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

$${\eta}$$

η

. We prove regularity properties of the contour function and show that (as in the LQG case) it encodes all the information about the curve

$${\eta}$$

η

. We also prove that the uniform spanning tree on

$${\mathbb{Z}^2}$$

Z

2

converges to SLE8 in the natural topology associated with the mating of trees approach.