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
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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