Preface

A number of two-dimensional lattice models in statistical physics have contin-

uum limits that are conformally invariant. For example, the limit of simple random

walk is Brownian motion. This book will discuss the nature of conformally invari-

ant limits. Most of the processes discussed in this book are derived in one way

or another from Brownian motion. The exciting new development in this area is

the Schramm-Loewner evolution (SLE), which can be considered as a Brownian

motion on the space of conformal maps.

These notes arise from graduate courses at Cornell University given in 2002-

2003 on the mathematics behind conformally invariant processes. This may be

considered equal doses of probability and conformal mapping. It is assumed that the

reader knows the equivalent of first-year graduate courses in real analysis, complex

analysis, and probability.

Here is an outline of the book. We start with a quick introduction to some

discrete processes which have scaling limits that are conformally invariant. We only

present enough here to whet the appetite of the reader, and we will not use this

section later in the book. We will not prove any of the important results concerning

convergence of discrete processes. A good survey of some of these results is [83].

Chapter 1 gives the necessary facts about one-dimensional Brownian motion

and stochastic calculus. We have given an essentially self-contained treatment;

in order to do so, we only integrate with respect to continuous semimartingales

derived from Brownian motion and we only integrate adapted processes that are

continuous (or piecewise continuous). The latter assumption is more restrictive

than one generally wants for other applications of stochastic calculus, but it suffices

for our needs and avoids having to discuss certain technical aspects of stochastic

calculus. More detailed treatments can be found in many books, e.g., [5, 32, 72,

73]. Sections 1.10 and 1.11 discuss some particular stochastic differential equations

that arise in the analysis of SLE. The reader may wish to skip these sections until

Chapter 6 where these equations appear; however, since they discuss properties

of one-dimensional equations it logically makes sense to include them in the first

chapter.

The next chapter introduces the basics of two-dimensional (i.e., one complex

dimension) Brownian motion. It starts with the basic fact (dating back to Levy

[65] and implicit in earlier work on harmonic functions) that complex Brownian

motion is conformally invariant. Here we collect a number of standard facts about

harmonic functions and Green's function for complex Brownian motion. Because

much of this material is standard, a number of facts are labeled as exercises.

Conformal mapping is the topic of Chapter 3. The purpose is to present the

material about conformal mapping that is needed for SLE, especially material that

would not appear in a first course in complex variables. References for much of this

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