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

Abstract

We show that the modulus of continuity of the SLE4 uniformizing map is given by (log δ −1 ) −1/3+o(1) as δ → 0. As a consequence of our analysis, we show that the Jones-Smirnov condition for conformal removability (with quasihyperbolic geodesics) does not hold for SLE4. We also show that the modulus of continuity for SLE8 with the capacity time parameterization is given by (log δ −1 ) −1/4+o(1) as δ → 0, proving a conjecture of Alvisio and Lawler.

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

“…The reason that the proof given in [28] is restricted to the case that κ = 8 is related to the regularity of SLE 8 . In particular, it turns out that the SLE κ curves are Hölder continuous for κ = 8 [7,18] but have modulus of continuity (log δ −1 ) −1/4+o (1) as δ → 0 for κ = 8 [9], which was previously conjectured by Alvisio and Lawler [1].…”

“…In view of (1.2) we have that the amount of capacity time elapsed between the times t and τ , i.e., τ − t, is related to the harmonic measure of B(η(t), ) in H \ η([0, t]). It was shown in [9] that this harmonic measure can decay as fast as exp(− −4+o (1) ) as → 0 which is in contrast to the case κ = 8 where it can only decay as fast as a power of [8].…”