**Graduate Studies in Mathematics**

Volume: 162;
2015;
318 pp;
Hardcover

MSC: Primary 60; 82;

Print ISBN: 978-0-8218-7578-0

Product Code: GSM/162

List Price: $79.00

AMS Member Price: $63.20

MAA Member Price: $71.10

**Electronic ISBN: 978-1-4704-2222-6
Product Code: GSM/162.E**

List Price: $79.00

AMS Member Price: $63.20

MAA Member Price: $71.10

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#### Supplemental Materials

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

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*Firas Rassoul-Agha; Timo Seppäläinen*

This is an introductory course on the methods of computing asymptotics
of probabilities of rare events: the theory of large deviations. The
book combines large deviation theory with basic statistical mechanics,
namely Gibbs measures with their variational characterization and the
phase transition of the Ising model, in a text intended for a one
semester or quarter course.

The book begins with a straightforward approach to the key ideas
and results of large deviation theory in the context of independent
identically distributed random variables. This includes Cramér's
theorem, relative entropy, Sanov's theorem, process level large
deviations, convex duality, and change of measure arguments.

Dependence is introduced through the interactions potentials of
equilibrium statistical mechanics. The phase transition of the Ising
model is proved in two different ways: first in the classical way with
the Peierls argument, Dobrushin's uniqueness condition, and
correlation inequalities and then a second time through the
percolation approach.

Beyond the large deviations of independent variables and Gibbs
measures, later parts of the book treat large deviations of Markov
chains, the Gärtner-Ellis theorem, and a large deviation theorem of
Baxter and Jain that is then applied to a nonstationary process and a
random walk in a dynamical random environment.

The book has been used with students from mathematics, statistics,
engineering, and the sciences and has been written for a broad
audience with advanced technical training. Appendixes review basic
material from analysis and probability theory and also prove some of
the technical results used in the text.

#### Readership

Graduate students interested in probability, the theory of large deviations, and statistical mechanics.

#### Reviews & Endorsements

It possesses a great value as an introduction for more and more people (students and experienced researchers) to these beautiful and highly active theories, as it is written in a very motivating and fresh style...I think the authors did a very good job to provide a text that can be taken as a base for an interesting and useful lecture without much preparation or as a quick but thorough introduction to this subject.

-- Wolfgang König, Jahresbericht der Deutschen Mathematiker-Vereinigung

#### Table of Contents

# Table of Contents

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

- Cover Cover11
- Title page iii4
- Dedication v6
- Contents vii8
- Preface xi12
- Part I. Large deviations: General theory and i.i.d. processes 116
- Chapter 1. Introductory discussion 318
- Chapter 2. The large deviation principle 1732
- Chapter 3. Large deviations and asymptotics of integrals 3550
- Chapter 4. Convex analysis in large deviation theory 4964
- Chapter 5. Relative entropy and large deviations for empirical measures 6782
- Chapter 6. Process level large deviations for i.i.d. fields 8398
- Part II. Statistical mechanics 97112
- Chapter 7. Formalism for classical lattice systems 99114
- Chapter 8. Large deviations and equilibrium statistical mechanics 121136
- Chapter 9. Phase transition in the Ising model 133148
- Chapter 10. Percolation approach to phase transition 149164
- Part II. Additional large deviation topics 159174
- Chapter 11. Further asymptotics for i.i.d. random variables 161176
- Chapter 12. Large deviations through the limiting generating function 167182
- Chapter 13. Large deviations for Markov chains 187202
- Chapter 14. Convexity criterion for large deviations 213228
- Chapter 15. Nonstationary independent variables 221236
- Chapter 16. Random walk in a dynamical random environment 233248
- Appendixes 257272
- Appendix A. Analysis 259274
- Appendix B. Probability 273288
- Appendix C. Inequalities from statistical mechanics 293308
- Appendix D. Nonnegative matrices 297312
- Bibliography 299314
- Notation index 305320
- Author index 311326
- General index 313328
- Back Cover Back Cover1335