A continuous proof of the existence of the SLE$_8$ curve

TL;DR

This paper proves the existence of the SLE$_8$ curve by establishing the tightness of a family of laws for space-filling SLE$_{κ}$ curves as κ approaches 8, providing a new continuous proof independent of discrete models.

Contribution

It introduces a novel continuous approach to prove the existence of SLE$_8$, avoiding reliance on discrete uniform spanning tree limits.

Findings

Demonstrates tightness of laws for space-filling SLE$_{κ}$ as κ approaches 8

Provides a new proof of SLE$_8$ existence without discrete model dependence

Establishes compactness in the space of continuous curves for the family of laws

Abstract

Suppose that $\eta$ is a whole-plane space-filling SLE$_\kappa$ for $\kappa \in (4,8)$ from $\infty$ to $\infty$ parameterized by Lebesgue measure and normalized so that $\eta(0) = 0$. For each $T > 0$ and $\kappa \in (4,8)$ we let $\mu_{\kappa,T}$ denote the law of $\eta|_{[0,T]}$. We show for each $\nu, T > 0$ that the family of laws $\mu_{\kappa,T}$ for $\kappa \in [4+\nu,8)$ is compact in the weak topology associated with the space of probability measures on continuous curves $[0,T] \to {\mathbf C}$ equipped with the uniform distance. As a direct byproduct of this tightness result (taking a limit as $\kappa \uparrow 8$), we obtain a new proof of the existence of the SLE$_8$ curve which does not build on the discrete uniform spanning tree scaling limit of Lawler-Schramm-Werner.