A Course on Large Deviations with an Introduction to Gibbs Measures
About this Title
Firas Rassoul-Agha, University of Utah, Salt Lake City, UT and Timo Seppäläinen, University of Wisconsin–Madison, Madison, WI
Publication: Graduate Studies in Mathematics
Publication Year: 2015; Volume 162
ISBNs: 978-0-8218-7578-0 (print); 978-1-4704-2222-6 (online)
MathSciNet review: MR3309619
MSC: Primary 60-01; Secondary 60F10, 60J10, 60K35, 60K37, 82B20
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.
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
- Appendix A. Analysis
- Appendix B. Probability
- Appendix C. Inequalities from statistical mechanics
- Appendix D. Nonnegative matrices