833 - Topics in Probability: Large deviations and Gibbs measures
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, available at http://www.math.wisc.edu/~seppalai/bookpage.html
Large deviation theory is an area of probability that quantifies chances of extremely rare behavior, the kind for example 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 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
(i) general large deviation theory and some convex analysis required for this
(ii) large deviations for independent random variables
(iii) Gibbs measures and their properties, especially their characterization through variational principles that involve entropy and a large deviations framework
(iv) the Ising model, which is the most important model of statistical physics, and its phase transition
(v) as time permits, further topics such as large deviations in Markov chains, and the related notion of moderate deviations.
Prerequisites for this course can be quite minimal. Measure theory and some advanced probability will be used, but prerequisites needed can be covered quickly at the beginning.