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

Abstract

Suppose that $η$ is a whole-plane space-filling SLE$_κ$ for $κ\in (4,8)$ from $\infty$ to $\infty$ parameterized by Lebesgue measure and normalized so that $η(0) = 0$. For each $T > 0$ and $κ\in (4,8)$ we let $μ_{κ,T}$ denote the law of $η|_{[0,T]}$. We show for each $ν, T > 0$ that the family of laws $μ_{κ,T}$ for $κ\in [4+ν,8)$ 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 $κ\uparrow 8$), 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.