Regularity of the SLE$_4$ uniformizing map and the SLE$_8$ trace

TL;DR

This paper analyzes the modulus of continuity for SLE$_4$ and SLE$_8$ maps, revealing new regularity properties and disproving a conformal removability condition for SLE$_4$, while confirming a conjecture for SLE$_8$.

Contribution

It establishes precise logarithmic modulus of continuity estimates for SLE$_4$ and SLE$_8$, and shows the Jones-Smirnov condition fails for SLE$_4$, advancing understanding of SLE regularity.

Findings

SLE$_4$ uniformizing map modulus of continuity is $(rac{1}{ ext{log} rac{1}{ ext{delta}}})^{1/3+o(1)}$

Jones-Smirnov condition does not hold for SLE$_4$

SLE$_8$ modulus of continuity is $(rac{1}{ ext{log} rac{1}{ ext{delta}}})^{1/4+o(1)}$, confirming a conjecture

Abstract

We show that the modulus of continuity of the SLE$_4$ uniformizing map is given by $(\log \delta^{-1})^{-1/3+o(1)}$ as $\delta \to 0$. As a consequence of our analysis, we show that the Jones-Smirnov condition for conformal removability (with quasihyperbolic geodesics) does not hold for SLE$_4$. We also show that the modulus of continuity for SLE$_8$ with the capacity time parameterization is given by $(\log \delta^{-1})^{-1/4+o(1)}$ as $\delta \to 0$, proving a conjecture of Alvisio and Lawler.