Topics in Probability -
Random Walks and Percolation
Ori Gurel-Gurevich and Gideon Amir (under supervision of Itai Benjamini).
This course deals with following topics:
- Simple Random walk on graphs, Recurrence and Transience, especially on Z^d.
- Harmonic functions and hitting probabilities.
- Electrical Networks.
- Other random walks: Self Interacting, Self Avoiding, Excited.
- Percolation, especially on Z^d and trees.
- Other kind of percolations: First Passage Percolation.
Time and Location
The course takes place at Ziskind 261 on Sundays 14-16
These are our lecture notes for the course. They are to be considered incomplete, erroneous and outdated. Use at you own Risk!
Part 1 - Random Walks basics (updated 14/11/05): ps, pdf, dvi.
Part 2 - Martingales and Harmonic functions (updated 15/11/05): ps, pdf, dvi.
Part 3 - Electric Networks (updated 07/12/05): ps, pdf, dvi.
Part 4 - Percolation on Trees: consult chapter 4 in Lyons book.
Part 5 - Percolation on Z^d (updated 17/01/06): ps, pdf, dvi.
Part 6 - P_c(Z^2)=1/2 (updated 17/01/06): ps, pdf, dvi.
Doyle & Snell is an excellent introduction to
Electric Networks and their connection to Random Walks.
Yuval Peres's homepage is a good place to
learn stuff about discrete probability.
Russell Lyons's book (with Yuval Peres) is a great reference for random walks, and many other topics.
Geoffrey Grimmett's St. Flour lectures are a good introduction to percolation.
Dana Randall's lecture notes can be useful, especially about the FKG and BK inequalities.