Regularity of the SLE(4) uniformizing map and the SLE(8) trace
The Schramm-Loewner evolution (SLE($\kappa$)) is a one-parameter family of random curves which live in a planar domain and describe the scaling limits of the interfaces for different models in statistical mechanics in two dimensions. The parameter $\kappa$ determines the roughness of the curve. For example, SLE($\kappa$) curves are simple for $\kappa \leq 4$, self-intersecting for $\kappa \in (4,8)$, and space-filling for $\kappa \geq 8$. The optimal Holder exponent for an SLE$(\kappa)$ curve for $\kappa \neq 8$ was determined in works of Lind and Lawler-Viklund and the optimal Holder exponent for a conformal map which uniformizes a component in the complement of an SLE$(\kappa)$ curve for $\kappa \neq 4$ was determined by Gwynne-M.-Sun. In this talk, I will describe recent joint work with Kavvadias and Schoug which determines the continuity properties of SLE$(\kappa)$ at the critical values $\kappa=4, 8$. In particular, we show that the modulus of continuity of SLE$(8)$ curve is given by $(\log \delta^{-1})^{-1/4+o(1)}$ as $\delta \to 0$, proving a conjecture of Alvisio and Lawler. We also show that the modulus of continuity of a conformal map which uniformizes component in the complement of an SLE$(4)$ curve is given by $(\log \delta^{-1})^{-1/3+o(1)}$ as $\delta \to 0$. As a consequence of the analysis to prove this, we show that the Jones-Smirnov condition for conformal removability (with quasihyperbolic geodesics) does not hold for SLE$(4)$.