Fractal and Multifractal Properties of Sle

This is a slightly expanded version of my lectures at the 2010 Clay Mathematics Institute summer (winter) school in Buzios, Brazil. The theme is the fine properties of Schramm-Loewner evolution (SLE) curves with an emphasis on recent work I have done with a number of co-authors on fractal and multifractal properties. I assume basic knowledge of SLE at the level of foundational course presented by Vincent Beffara. I will try to discuss both results and ideas of proofs. Although discrete models motivate SLE, I will focus only on SLE itself and will not discuss convergence of discrete models. The basic theme tying the results together is the SLE curve. Fine analysis of the curve requires estimates of moments of the derivatives, and in turn leads to studying martingales and local martingales. In the process, I will discuss existence of the curve, Hausdorff dimension of the curve, and a number or more recent results that I have obtained with a number of co-authors. The five sections correspond roughly to the five lectures that I gave. Here is a quick summary. • Section 1 proves a basic result of Rohde and Schramm [15] on the existence of the SLE curve for κ 6= 8. Many small steps are left to the reader; one can treat this as an exercise in the deterministic Loewner equation and classical properties of univalent functions such as the distortion theorem. Two main ingredients go into the proof: the modulus of continuity of Brownian motion and an estimate of the moments of the derivative of the reverse map. By computing the moment, we can determine the optimal Hölder exponent and see why κ = 8 is the delicate case. The estimation of the moment is left to the next section. • Section 2 discusses how to use the reverse Loewner flow to estimate the exponent. This was the tool in [15] to get their estimate. Here we expand signficantly on their work because finer analysis is needed to derive “two-point” or “second moment” estimates which are required to establish fractal and multifractal behavior with probability one. Although there is a fair amount of calculation involved, there are a few general tools: