More topics about SLE are discussed in Chapter 7, most particularly, the
technical problems of showing that SLE generates a random curve and determining
the dimension of the path. Many of the results in this chapter first appeared in
[74]. The techniques are similar to those used in the previous chapter. The main
difference is that one considers the Bessel process in the upper half-plane HI rather
than just on the real line.
The next chapter gives an application of SLE to Brownian motion. The "in-
tersection exponents" for Brownian motion are examples of critical exponents that
give the fractal dimension of certain exceptional sets on the path. They are also
nontrivial exponents for a lattice model, simple random walk. The close relationship
and Brownian motion can be used to derive the values of the Brownian
exponents from the
crossing exponents.
The Schramm-Loewner evolution gives a one-parameter family of conformally
invariant measures. There is another important one-parameter family of measures
called restriction measures. Roughly speaking, the restriction measure with pa-
rameter a corresponds to the union of a Brownian motions (we actually allow a to
be any positive real). In Chapter 9 we show the relationship between SLE and
restriction measures.
Needless to say, this book would not exist if it were not for Oded Schramm
and Wendelin Werner with whom I have had the great opportunity to collaborate.
Their ideas permeate this entire book. There are a number of other people who have
helped by answering questions or commenting on earlier versions. These include:
Christian Benes, Nathanael Berestycki, Zhen-Qing Chen, Keith Crank, Rick Dur-
rett, Clifford Earle, Christophe Garban, Lee Gibson, Pavel Gyrya, John Hubbard,
Harry Kesten, Evgueni Klebanov, Ming Kou, Michael Kozdron, Robin Pemantle,
Melanie Pivarski, Jose Ramirez, Luke Rogers, Jason Schweinsberg, John Thacker,
Jose Trujillo Ferreras, Brigitta Vermesi. Figures 0.4 and 0.5 were produced by
Vincent Beffara and Geoffrey Grimmett, respectively.
During the preparation of this book I have enjoyed extended visits at the
Mittag-LefHer Institute, l'lnstitut Henri Poincare, the Issac Newton Institute for
the Mathematical Sciences, and the Pacific Institute for the Mathematical Sciences
at the University of British Columbia, and I have received support from the Na-
tional Science Foundation.
Finally, and most importantly, I thank Marcia for all her understanding, pa-
tience, and support.
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