Hardcover ISBN:  9780821875780 
Product Code:  GSM/162 
318 pp 
List Price:  $84.00 
MAA Member Price:  $75.60 
AMS Member Price:  $67.20 
Electronic ISBN:  9781470422226 
Product Code:  GSM/162.E 
318 pp 
List Price:  $79.00 
MAA Member Price:  $71.10 
AMS Member Price:  $63.20 

Book DetailsGraduate Studies in MathematicsVolume: 162; 2015MSC: Primary 60; 82;
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 GärtnerEllis 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.ReadershipGraduate students interested in probability, the theory of large deviations, and statistical mechanics.

Table of Contents

Part I. Large deviations: General theory and i.i.d. processes

Chapter 1. Introductory discussion

Chapter 2. The large deviation principle

Chapter 3. Large deviations and asymptotics of integrals

Chapter 4. Convex analysis in large deviation theory

Chapter 5. Relative entropy and large deviations for empirical measures

Chapter 6. Process level large deviations for i.i.d. fields

Part II. Statistical mechanics

Chapter 7. Formalism for classical lattice systems

Chapter 8. Large deviations and equilibrium statistical mechanics

Chapter 9. Phase transition in the Ising model

Chapter 10. Percolation approach to phase transition

Part II. Additional large deviation topics

Chapter 11. Further asymptotics for i.i.d. random variables

Chapter 12. Large deviations through the limiting generating function

Chapter 13. Large deviations for Markov chains

Chapter 14. Convexity criterion for large deviations

Chapter 15. Nonstationary independent variables

Chapter 16. Random walk in a dynamical random environment

Appendixes

Appendix A. Analysis

Appendix B. Probability

Appendix C. Inequalities from statistical mechanics

Appendix D. Nonnegative matrices


Additional Material

Reviews

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 MathematikerVereinigung


Request Review Copy

Get Permissions
 Book Details
 Table of Contents
 Additional Material
 Reviews

 Request Review Copy
 Get Permissions
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 GärtnerEllis 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.
Graduate students interested in probability, the theory of large deviations, and statistical mechanics.

Part I. Large deviations: General theory and i.i.d. processes

Chapter 1. Introductory discussion

Chapter 2. The large deviation principle

Chapter 3. Large deviations and asymptotics of integrals

Chapter 4. Convex analysis in large deviation theory

Chapter 5. Relative entropy and large deviations for empirical measures

Chapter 6. Process level large deviations for i.i.d. fields

Part II. Statistical mechanics

Chapter 7. Formalism for classical lattice systems

Chapter 8. Large deviations and equilibrium statistical mechanics

Chapter 9. Phase transition in the Ising model

Chapter 10. Percolation approach to phase transition

Part II. Additional large deviation topics

Chapter 11. Further asymptotics for i.i.d. random variables

Chapter 12. Large deviations through the limiting generating function

Chapter 13. Large deviations for Markov chains

Chapter 14. Convexity criterion for large deviations

Chapter 15. Nonstationary independent variables

Chapter 16. Random walk in a dynamical random environment

Appendixes

Appendix A. Analysis

Appendix B. Probability

Appendix C. Inequalities from statistical mechanics

Appendix D. Nonnegative matrices

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 MathematikerVereinigung