III Schramm--Loewner Evolutions

0Introduction

0 Introduction

Schramm–Loewner evolution is a random curve in a complex domain

(which

we will often take to be

for convenience), parametrized by a positive real

number

. This was introduced by Schramm in 1999 to describe various scaling

limits that arise in probability theory and statistical physics.

Recall that if we take a random walk on the integer lattice

, and take the

scaling limit as the grid size tends to 0, we converge towards a Brownian motion.

There are other discrete models that admit a scaling limit, and the limit is often

something that is not Brownian motion.

Schramm showed that if the scaling limit satisfies certain conformal invariance

properties, which many models do, then it must be

SLE

for some

. If we can

identify exactly which

it belongs to, then this will completely determine what

the scaling limit is, and often conveys a lot of extra information such as the

critical exponents. This has been done for several discrete models:

– The scaling limit of loop-erased random walk is SLE

– The scaling limit of hexagonal percolation is SLE

(see chapter 5)

–

It is conjectured that the scaling limit of self-avoiding random walk is

SLE

8/3

(see chapter 6)

– The “level sets” of a Gaussian free field are SLE

’s (see chapter7).

In this course, we will prove some basic properties of

SLE

, and then establish

the last three results/conjectures.