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MATH 7880: Large Deviations and Gibbs Measures (Spring '18)

The law of large numbers says that if we toss a fair coin a large number of times we will get about 50% heads. In fact, getting 50% heads is the most probable event. The central limit theorem, on the other hand, describes fluctuations from this event. It says that if we draw the histogram of a large number of tosses we will see the famous bell curve.

These basic limit theorems in probability theory allow us to understand the typical behavior of a stochastic system. However, a better understanding of the system can only be achieved when we also understand how atypical events can occur. This is particularly important when rare events have a high cost associated with them (e.g. earthquakes and other natural disasters).

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.R.S. Varadhan, who received the 2007 Abel Prize ("Nobel Prize of mathematics") for this and other achievements.

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, etc.

If you need to brush up on some standard material in probability theory, Davar's book and Varadhan's notes are a good resource.

For more large deviations (after having completed this course):

Notes on Large Deviations (Cramer's theorem, Sanov's theorem, Schilder's theorem, LIL, small noise, exit problem, Markov chains, random walk among Bernoulli traps, hydrodynamic limits)

More notes on Large Deviations (more advanced, leading to hydrodynamic scaling of exclusion processes)

1. Work out the development below (1.1), up to (and including) Exercise 1.2.

2. Solve Exercises 2.2, 2.4, 2.5, 2.15, 2.16, 2.24, 2.29, 2.30.

**Homework 2 (due Thursday Feb 15):**

1. Solve Exercises 2.37, 2.38, 3.5, 3.15, 3.17, 3.19, 4.12, 4.20, 4.22, 4.23.

2. Read and sort out the metrizability discussion in Remark 4.4. (No need to submit this part)

3. Prove Proposition 4.6.

4. Look up (and digest!) the Hahn-Banach separation theorems (Section A.4). (No need to submit this part.)

5. Read the proof of Lemma 4.16. (No need to submit this part.)

**Homework 3 (due Tuesday Mar 13):**

1. Solve Exercises 4.27, 4.29, 5.11, 5.12, 5.17, 5.21.

2. Read the proof of Lemma A.11 and Prohorov's Theorem (page 280).

3. Read Theorem B.5 and the paragraph following it.

Suggested (but no need to submit):

Solve Exercise 5.16.

**Homework 4 (due Thursday Apr 5):**

1. Solve Exercises 6.1, 6.2, B.20, 6.10, 6.11, 6.17.

2. Recall the proof of Kolmogorov's 0-1 law (Exercise B.22) (no need to submit)

Suggested (but no need to submit):

Solve Exercises 6.3, B.23, B.24

Longer (but useful): Solve Exercises B.30 and B.31

**Homework 5 (due Thursday May 3):**

Solve Exercises 7.4, 7.7, 7.16, 7.35.