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

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.

96 pages, 3 figures. Final accepted version to Proceedings of the LMS

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