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.