STAT C206A / MATH C223A : Percolation theory (Fall 2008; Listed as "Stochastic Processes")
Instructor: Sourav Chatterjee
Class time: TuTh 2:00 - 3:30
Location: 330 Evans
Percolation is one of the deepest and most well-researched branches of probability theory, and yet a repository of a large number of important open questions. The purpose of this course is to serve as an introduction to this fascinating and beautiful area of mathematics to advanced graduate students.
Special attraction: Professor Yuval Peres will co-teach a part of the course.
Course Topics
The following is a possible list of topics for this course. Ideally we would like to cover all of them in the order in which they are listed, but adjustments may have to be made because of time constraints.
- Basic definitions, critical exponents.
- Exact evaluation of critical exponents for percolation on trees.
- Basic inequalities: FKG, BK. Russo's formula.
- Percolation on the square lattice: Results about size and density of open clusters in the subcritical phase.
- Menshikov's theorem.
- Evaluation of the critical probability in the square lattice in 2D (the Harris-Kesten theorem).
- Percolation on the complete graph: The Erdos-Renyi random graph model.
- Critical random graphs. Aldous's theorem about the sizes of large components.
- Percolation on non-amenable graphs. Cayley graphs.
- Uniqueness transitions in non-amenable graphs.
- Critical exponents in non-amenable graphs.
- The Russo-Seymour-Welsh theorem.
- Smirnov's proof of Cardy's formula.
- First and last passage percolation.