A Course on Large Deviations with an Introduction to Gibbs Measures.

xii Preface (and to the reviewers)
theorem, Sanov’s theorem, and the process level LDP for i.i.d. variables
indexed by a multidimensional square lattice.
Part II introduces Gibbs measures and proves the Dobrushin-Lanford-
Ruelle variational principle that characterizes translation-invariant Gibbs
measures in terms of the vanishing of a large deviation rate function and
in terms of a minimization problem. After this we proceed to study the
phase transition of the Ising model. We also plan to add a section on the
Fortyin-Kasteleyn random cluster model and cover the modern percolation
approach to phase transitions.
Part III develops the large deviation themes of Part I in several directions.
Large deviations of i.i.d. variables are complemented with moderate
deviations and with more precise large deviation asymptotics. From i.i.d.
processes we generalize to Markov chains and to independent but nonstationary
processes. The latter topic gives us the opportunity to introduce
an entirely different way of proving large deviation principles, namely the
Baxter-Jain theorem. This material has not previously appeared in textbooks.
We are also planning a section on large deviations for some flavor of
random walk in random environment.
The reader should note that starred exercises are used in the text later
on. Also, Dembo and Zeitouni contain thorough historical notes and references,
hence we have not attempted to present the historical record.
The ideal background for reading this book would be some familiarity
with the language of measure-theoretic probability, and with analysis
in general so that the reader is comfortable working with notions such as
lower semicontinuity or with linear spaces. In reality our courses have been
populated by students with diverse backgrounds, many with less than ideal
knowledge of analysis and probability. To make the material accessible to
these students we plan to enhance the appendixes to include an overview
of the main theorems of measure theory and probability. (And once we do
this the frequent references to basic probability textbooks that now interrupt
the flow of the prose will disappear.) The actual technical needs for
following the book are not deep and it should be possible for an instructor
to accommodate students with quick lectures on analytic technicalities
whenever needed. We have found that there is great interest in probability
theory among students of economics, engineering and sciences. This interest
should be encouraged and nurtured with accessible courses.
We are of course greatly indebted to the existing books on the subject,
especially those by Amir Dembo and Ofer Zeitouni, Jean-Dominique Deuchel
and Daniel Stroock, Frank den Hollander, and Richard Ellis. We thank Jeff
Steif for lecture notes that helped shape the proof of Theorem 10.2, Jim
Kuelbs for the material for Sections 12.1 and 12.2, and Chuck Newman for