Math/Stat 833 Topics in Probability Spring 2013: Large Deviations and Gibbs Measures

Meetings: MWF 2:25-3:15, Van Vleck B129

Instructor: Timo Seppäläinen

Office: 419 Van Vleck

Telephone: 263-2812

E-mail: seppalai at math dot wisc dot edu

Lecture notes

Homework

• Homework 1 due Monday, February 11. Exercises 2.16, 2.25, 2.34(a), 3.9. Some solutions.
• Homework 2 due Friday, March 8. Exercises 5.13, 5.15, 5.19 (but only the case of bounded continuous φ), 5.20, 5.22. Exercise 5.15 is important. Please organize your exposition well. If it gets too messy I may have to give up on reading it. If you are not familiar with measure theoretic conditional probability, do the finite case for partial credit.
• Homework 3 due Wednesday, April 3. Exercises 7.5, 7.11, 7.13.

Spring 2013 Schedule

• Week 1. First examples, tentative definition of a large deviation principle.
• Week 2. LDP, weak LDP, discussion about Crámer's theorem.
• Week 3. Contraction principle, Varadhan's theorem, Curie-Weiss model.
• Week 4. Convex analysis. Multidimensional Crámer's theorem.
• Week 5. Relative entropy and Sanov's theorem.
• Week 6. Classes rescheduled.
• Week 7. Maximum entropy principle. Specific entropy, pressure, and process level large deviations.
• Week 8. Lower bound for IID process level LDP. Begin Gibbs measures.
• Week 9. Further basics of Gibbs measures. Finite Markov chain as Gibbs measure. Dobrushin's uniqueness theorem.
• Spring Break March 25-29.
• Week 10. Extreme Gibbs measures, large deviations.
• Week 11. DLR variational principle. Phase transition in the Ising model: 1 dimension, Peierls argument in 2 dimensions.
• Week 12. Stochastic monotonicity, its application to the Ising model.
• Week 13. Completion of the phase transition in the Ising model. Percolation, random cluster measures.
• Week 14. Proof of the Ising phase transition via random cluster measures. Essentially smooth convex functions, exposed points.
• Week 15. Gärtner-Ellis theorem. No class on Friday, May 10.

Further recommended materials

• Amir Dembo and Ofer Zeitouni, Large Deviations Techniques and Applications.
• Frank den Hollander, Large Deviations.
• Jean-Dominique Deuschel and Daniel Stroock, Large Deviations.

Description of the course

Large deviation theory is an area of probability that seeks to quantify chances of extremely rare behavior, for example the kind that falls outside the central limit theorem. It has its origins in early probability and statistical mechanics. A unified formulation emerged in the 1960's especially through the work of S. Varadhan, who received the Abel Prize ("Nobel Prize of mathematics") for this and other achievements in 2007.

There are applications in various fields such as engineering, especially queueing theory, statistics and economics. Notions of entropy appear naturally in descriptions of these small probabilities.

Gibbs measures are probability measures that describe random systems in multidimensional space and possess a natural spatial Markovian property. They arose in the context of statistical physics.

This course intends to cover

• General large deviation theory, including the role of convex analysis.
• Large deviation theory for independent random variables, including Sanov's theorem and relative entropy.
• Gibbs measures and their properties, especially their characterization through variational principles that involve entropy and a large deviations framework.
• The Ising model, which is the most important model of statistical physics, and its phase transition.
• As time permits, further topics such as large deviations for Markov chains, more refined estimates for independent variables, moderate deviations.

Prerequisites for this course can be quite minimal. Measure theory and some advanced probability will be used, but the prerequisites needed can be covered quickly at the beginning.

There are a number of books on these topics. No textbook purchase is required. The instructor will use his own manuscript for an introductory text.