Synopsis for C10.1b: Brownian Motion and Conformal Invariance

Level: M-level Method of Assessment: Written examination. Weight: Half-unit (OSS paper code 2B83).

Recommended Prerequisites

Essential Prerequisites: Part A Analysis and knowledge of Brownian Motion. Recommended Prerequisites: Part A Differential Equations, B10a Martingales through Measure and C10.1a Stochastic Differential Equations.

Overview

This course is devoted to connections between two-dimensional Brownian Motion, Complex Analysis and Lattice Models. An important example is the standard random walk on a square lattice. It is known, that after proper rescaling, the random walk converges to Brownian Motion (this is a corollary of the Central Limit Theorem). It was observed by Lévy that Brownian Motion is conformally invariant i.e. the image of the Brownian trajectory under an analytic map looks like a Brownian trajectory. In particular, this implies that the limiting object has more symmetry than the original random walk on the lattice. There is a big class of other lattice models which are conjectured to have scaling limits and these limits are conjectured to be conformally invariant.

This course will consist of two parts: in the first part we will discuss the conformal invariance of Brownian Motion. In the second part we will learn about the very recent and exciting theory of Schramm-Loewner Evolution (SLE). This theory will provide the necessary tools to study conformally invariant limits of various lattice models.

Learning Outcomes

The students will develop an understanding of the role the Brownian Motion plays in different areas of mathematics and physics. They will be familiar with basic ideas and techniques of Schramm-Loewner Evolution.

Synopsis

Brief introduction to Brownian Motion, continuous martingales, and Ito formula. Conformal invariance of Brownian Motion. Brief introduction to conformal maps and Loewner Evolution. Lattice models: percolation, Ising, Loop-erased Random Walk etc. Schramm's principle and the introduction of SLE (Schramm-Loewner Evolution or Stochastic Loewner Evolution). Properties of SLE. [Convergence of percolation to SLE(6)]

This is a very young and actively developing area of research. Unfortunately this means that there are very few books, in fact, there is only one proper book and a couple of lecture notes.

1. G. Lawler, Conformally Invariant Processes in the Plane, Mathematical Surveys and Monographs, Vol 114 (2005)
2. W. Werner, Random planar curves and Schramm-Loewner evolutions
   http://arxiv.org/abs/math/0303354