Fine regularity properties of SLE_4 and SLE_8

Abstract

The Schramm-Loewner evolution (SLE$_{\kappa}$) is a one parameter family ($\kappa$>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 $\kappa$ = 4 and $\kappa$ = 8. The value $\kappa$ = 4 is special because it is the critical value at or below which SLE${\kappa}$ curves are simple and above which they are not. In particular, SLE${4}$ curves are almost self-intersecting in the sense that the harmonic measure of a ball of radius $\epsilon$ centred at a point on the curve can decay as $\epsilon$ $\rightarrow$ 0 faster than any power of $\epsilon$ reflecting the fact that the uniformizing map is not Holder continuous. We show that it can decay as quickly as exp(-$\epsilon$$^{-3+o(1)}$) as $\epsilon$ $\rightarrow$ 0 and deduce that the modulus of continuity of the SLE${4}$ uniformizing map is given by (log($\delta$$^{-1}$))$^{-1/3 + o(1)}$ as $\delta$ $\rightarrow$ 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 SLE${4}$. In addition to the above results, we show that the modulus of continuity for SLE${8}$ with the capacity parameterization is given by (log($\delta$$^{-1}$))$^{-1/4 + o(1)}$ as $\delta$ $\rightarrow$ 0, proving a conjecture of Alvisio and Lawler. Note that the value $\kappa$ = 8 is special because it is the value at or above which SLE${\kappa}$ is space-filling while for $\kappa$<8 it is not. This is reflected to the fact that the left and right sides of the outer boundary of the curve drawn up until capacity time t are almost intersecting. In particular, we show that the harmonic measure of a ball of radius $\epsilon$ centred at the tip of the curve can decay to 0 as $\epsilon$ $\rightarrow$ 0 as quickly as exp(-$\epsilon$$^{-4 + o(1)}$). The second part of the thesis focuses on showing that the range of an SLE${4}$ curve is almost surely conformally removable, answering a question of Sheffield. Note that SLE${4}$ arises as the conformal welding of a pair of independent critical ($\gamma$ = 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 SLE${4}$, in order to establish our result we give a new sufficient condition for a set K $\subset$ 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 SLE${\kappa}$ curves and the GFF.