Instructor: Anirban Basak

Email: anirban.basak@icts.res.in

Office location: L14, IISc Mathematics Department.

Office hours: By appointment.

Class time and location: TuTh 2.003.30 PM, LH3, IISc Mathematics department. First class is on August 01.

Prerequisite: This is a graduate level topics course in probability theory. Graduate
level measure theoretic probability will be useful, but not a requirement. The course will be
accessible to advanced undergraduates who have had sufficient exposure to probability.

Course outline: Large deviations provide quantitative estimates of the probabilities of rare events in (highdimensional) stochastic systems.
The course will begin with general foundations of the theory of large deviations and will cover classical large deviations techniques such as Sanov's Theorem, Cramér's Theorem, GärtnerEllis Theorem, Contraction principle, Varadhan's Integral Lemma, and Bryc's Inverse Varadhan Lemma.
The latter half of the course will focus on some recent developments. Topics include large deviations in the context of random matrices, large deviations for subgraphs counts in random graphs (both dense and sparse regime), nonlinear large deviations, and low complexity of Gibbs measures. If time allows selected topics from Spin Glasses will be introduced.

Suggested books: 1. Amir Dembo and Ofer Zeitouni, Large Deviations Techniques and Applications.
2. Firas RassoulAgha and Timo Seppäläinen, A Course on Large Deviations with an Introduction to Gibbs Measures.
3. Marc Mézard and Andrea Montanari, Information, Physics, and Computation.
4. Sourav Chatterjee, Large Deviations for Random Graphs.

Tentative Weekly Schedule:
Aug 01: Course outline and examples.
Aug 06 and 08: Examples (continued), Sanov's Theorem for finite alphabet, properties of lower semicontinuous functions.
Aug 13: Sanov's Theorem for finite alphabet (continued).
Aug 20 and 22: Cramér's Theorem in \(\mathbb{R}\), large deviations for sums of i.i.d. random variables beyond the scope of Cramér's Theorem.
Aug 27 and 29: Cramér's Theorem in \(\mathbb{R}\) (continued), weak large deviations principle, Cramér's Theorem in \(\mathbb{R}^d\).
Sep 03 and 05: Weak large deviations principle and Cramér's Theorem in \(\mathbb{R}^d\) (continued), GärtnerEllis Theorem.
Sep 12: GärtnerEllis Theorem (continued), large deviations for the flow of current.
Sep 17 and 19: Large deviations for the flow of current (continued), exponential equivalence, contraction and inverse contraction principles.
Sep 24 and 26: No lectures.
Oct 01 and 03: Varadhan's Integral Lemma, Bryc's Inverse Varadhan Lemma, CurieWeiss Ising spin model: free energy.
Oct 10: CurieWeiss Ising spin model (continued): large deviations of the magnetization and analysis of the fixed point equation.
Oct 15 and 17: Examples of nonlinear large deviations, large deviations of the upper tail of nonlinear functions of i.i.d. random variables.
Oct 22 and 24: Large deviations of the upper tail of triangle counts in sparse ErdősRényi graphs, naïve MeanField approximation for partition functions.
Oct 29 and 31: Naïve MeanField approximation for Ising on general graphs.
Nov 05 and 07: Local weak convergence, belief propagation and Bethe free entropy for factor models on finite graphs and trees.
Nov 12 and 14: No lectures.
Nov 19 and 21: Asymptotic tightness of Bethe free energy prediction for ferromagnetic Ising on locally treelike graphs
