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A Course on Large Deviations with an Introduction to Gibbs Measures

Firas Rassoul-Agha University of Utah, Salt Lake City, UT
Available Formats:
Hardcover ISBN: 978-0-8218-7578-0
Product Code: GSM/162
318 pp
List Price: $84.00 MAA Member Price:$75.60
AMS Member Price: $67.20 Electronic ISBN: 978-1-4704-2222-6 Product Code: GSM/162.E 318 pp List Price:$79.00
MAA Member Price: $71.10 AMS Member Price:$63.20
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List Price: $126.00 MAA Member Price:$113.40
AMS Member Price: $100.80 Click above image for expanded view A Course on Large Deviations with an Introduction to Gibbs Measures Firas Rassoul-Agha University of Utah, Salt Lake City, UT Timo Seppäläinen University of Wisconsin–Madison, Madison, WI Available Formats:  Hardcover ISBN: 978-0-8218-7578-0 Product Code: GSM/162 318 pp  List Price:$84.00 MAA Member Price: $75.60 AMS Member Price:$67.20
 Electronic ISBN: 978-1-4704-2222-6 Product Code: GSM/162.E 318 pp
 List Price: $79.00 MAA Member Price:$71.10 AMS Member Price: $63.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version. List Price:$126.00
MAA Member Price: $113.40 AMS Member Price:$100.80
• Book Details

Volume: 1622015
MSC: 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 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.

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

• 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 Mathematiker-Vereinigung
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Volume: 1622015
MSC: 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 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.

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 Mathematiker-Vereinigung
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