| The Schramm-Loewner evolutions (SLEs) are a one-parameter family of random curves, introduced by Oded Schramm, which are describing the scaling limit of many critical two-dimensional models arising from statistical physics. So far the existence of scaling limit has been shown for a number of these models (see Table 1.1). Since emerging SLE, the geometric properties of it has been studied extensively. In this thesis, first we review the SLE process, then we address three geometric properties of SLE curves. The first property, which we discuss in chapter 3, is the existence of Minkowski content for SLE curves at the critical dimension. Beffara computed the Hausdorff dimension of SLE curves. For definition and basic properties of Minkowski content see chapter 2. In a joint work with Greg Lawler, we prove its existence and show that it is a multiple of the natural parametrization for SLE curves. In chapter 4, we prove that the Hausdorff measure of SLE curves at the critical dimension is zero. Computing the measure at the critical dimension remained open. In chapter 4, by using the properties of natural parametrization, we prove that this measure is zero. In chapter 5, we prove up-to-constant bounds for two-point Green's function which is a central function that arises in this work. |
