Fine regularity properties of SLE₄ and SLE

• Main Author: Kavvadias, Konstantinos
• Format: Dissertation
• Language: English
• Published: University of Cambridge 2022
• Subjects: Conformal removability of SLE4; Uniformizing map of SLE4
• DOI: 10.17863/CAM.91577

The Schramm-Loewner evolution (SLEκ) is a one parameter family (κ > 0) of curves which connect two boundary points of a simply connected domain. It was introduced by Schramm in 1999 as a candidate to describe the scaling limits of the interfaces in statistical mechanics models on two-dimensional lattices at criticality, such as loop erased random walk and the percolation model. It has also been the subject of intensive study as it has deep connections to the Gaussian free field (GFF) and Liouville quantum gravity (LQG). The first part of the thesis treats the values κ = 4 and κ = 8. The value κ = 4 is special because it is the critical value at or below which SLEκ curves are simple and above which they are not. In particular, SLE₄ curves are almost self-intersecting in the sense that the harmonic measure of a ball of radius ε centred at a point on the curve can decay to 0 as ε→0 faster than any power of ε reflecting the fact that the uniformizing map is not Holder continuous. We show that it can decay as quickly as exp(-ε⁻³⁺⁰⁽¹⁾) as ε→0 and deduce that the modulus of continuity of the SLE₄ uniformizing map is given by (log(δ⁻¹))⁻¹/³⁺⁰⁽¹⁾ as δ→0. As a consequence of our analysis, we also show that the Jones-Smirnov condition for conformal removability (with quasi hyperbolic geodesics) does not hold for SLE₄. In addition to the above results, we show that the modulus of continuity for SLE₈ with the capacity parameterization is given by (log(δ⁻¹))⁻¹/⁴⁺⁰⁽¹⁾ as δ→0, proving a conjecture of Alvisio and Lawler. Note that the value κ = 8 is special because it is the value at or above which SLEκ is space-filling while for κ < 8 it is not. This is reflected to the fact that the left and right sides of the outer boundary of the curve drawn up until capacity time t are almost intersecting. In particular, we show that the harmonic measure of a ball of radius ε centred at the tip of the curve can decay to 0 as ε→0 as quickly as exp(-ε⁻⁴⁺⁰⁽¹⁾). The second part of the thesis focuses on showing that the range of an SLE₄ curve is almost surely conformally removable, answering a question of Sheffield. Note that SLE₄ arises as the conformal welding of a pair of independent critical (γ = 2) Liouville quantum gravity (LQG) surfaces along their boundaries and our result implies that this conformal welding is unique. As the Jones-Smirnov condition for conformal removability does not hold for SLE₄, in order to establish our result we give a new sufficient condition for a set K ⊂ C to be conformally removable which applies in the case that K is not the boundary of a simply connected domain. Our proof makes uses of the connections recently established between SLEκ curves and the GFF.