SLE as a mating of trees in Euclidean geometry

TL;DR

This paper explores the mating of trees approach to SLE in Euclidean geometry, establishing regularity properties of the contour function and demonstrating convergence of the uniform spanning tree to SLE_8.

Contribution

It extends the mating of trees framework from Liouville quantum gravity to Euclidean geometry, providing new regularity results and convergence proofs.

Findings

Proves regularity properties of the contour function.

Shows the contour function encodes all information about the SLE curve.

Demonstrates convergence of the uniform spanning tree to SLE_8.

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-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 $\eta$. We prove regularity properties of the contour function and show that (as in the LQG case) it encodes all the information about the curve $\eta$. We also prove that the uniform spanning tree on $\mathbb Z^2$ converges to $\mathrm{SLE}_8$ in the natural topology associated with the mating of trees approach.