Quadratic differentials and Loewner evolutions

Abstract

Oded Schramm's discovery of stochastic Loewner evolution (SLE) in 1999 as the scaling limit of many important 2-dimensional random processes on lattices has opened up an exciting area of research. One of the central ideas in his theory is the use of a classic tool in function theory, the Loewner differential equation (LDE), to study random paths growing in simply-connected domains. In this thesis, we develop a method using quadratic differentials to study paths on 2-dimensional lattices. We will then use this method to derive properties of the LDE. In particular, we will be able to derive formulae for the driving function of the LDE for paths on certain lattices. We will also show how these formulae can be applied numerically. This provides some insight into what happens when we take scaling limits. We also use quadratic differentials to derive a generalized version of the LDE on simply-connected domains as well as a version of the LDE for paths . on Riemann surfaces. We can then use this to define SLE on Riemann surfaces and derive some of its properties.