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

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.

arXiv comment30 pages, 2 figures