M 393C Gibbs Measures & Random Graphs (54495)

Dr. Lewis Bowen

MWF 12 - 1pm 

RLM 10.176

 

Course Description:   

This course is on random processes (such as random walks, percolation, uniform spanning trees, the Ising model) on graphs, usually Cayley graphs of groups. The emphasis is on how geometry (or geometric group theory) plays a role in the qualitative aspects of the process. For example, amenable Cayley graphs admit at most one infinite Bernoulli percolation cluster, but non-amenable Cayley graphs can admit infinitely many. 

Part of the course will consist of student lectures. I will provide additional materials and coaching to help students with their lectures.

Some useful (but not required) references include: 
2) N. Alon and J. Spencer, The probabilistic method, 3rd Edition, Wiley, 2008. 

3) Itai Benjamini and Oded Schramm. Percolation beyond Zd, many questions and a few answers [mr1423907]. In Selected works of Oded Schramm. Volume 1, 2, Sel. Works Probab. Stat., pages 679–690. Springer, New York, 2011. 

4) Itai Benjamini, Russell Lyons, Yuval Peres, and Oded Schramm. Uniform spanning forests. Ann. Probab., 29(1):1–65, 2001. 

5) N. C. Wormald. Models of random regular graphs. In Surveys in combinatorics, 1999 (Canterbury), volume 267 of London Math. Soc. Lecture Note Ser., pages 239–298. Cambridge Univ. Press, Cambridge, 1999. 

Basic knowledge of probability theory is necessary. No knowledge of amenability or geometric group theory will be assumed.