Lectures: Tuesdays 12:15-13:45, H306
The final grade will be composed of 25% for each of two HW sets, and 50% for a final presentation.
Here are the final projects.
Here is the first problem set.
And here is the second one.
And here is the third one.
1.) Feb 15. Survival of Galton-Watson branching processes with two different methods: generating function fixed point, and 1st and 2nd Moment Methods. Different definitions of the percolation critical density, Kolmogorov's 0/1 law. [PGG Section 12.1, LyP Section 5.1]
2.) Feb 22. Upper and lower growth rates gr(T) and branching number br(T) of general trees. Relationship to Hausdorff and Minkowski dimensions of the boundary. Starting the connection between percolation and branching number. [LyP Sections 1.2, 5.2]
3.) March 1. GW 0/1 law for hereditary tree properties. p_c(T_\infty)=br(T_\infty)=\mu for GW trees conditioned to survive. p_c(T)=1/br(T) for any infinite tree, using the Max Flow Min Cut theorem and a weighted 2nd MM. Mentioning capacity. [LyP Sections 5.1-5.3]
4.) March 8. Reversible Markov chains, and the self-adjointness of the Markov operator. Flows, harmonic functions, hitting probabilities.[PGG Section 6.1, Durrett Section 4.8.1]
5.) March 22. Flows, energy, harmonic functions. Electric network interpretations, hitting probabilities, Dirichlet's and Thompson's principles. [PGG Sections 6.1, 6.2, LyP Chp 2.]
6.) March 29. Dirichlet's and Thompson's principles, recurrence vs transience, Pólya's theorem, Nash-Williams criterion. [PGG Sections 6.1, 6.2, LyP Chp 2.]
7.) April 5. Transitive graphs, Cayley graphs of groups, such as the free group and the modular group PSL(2,\Z). Ping-pong lemma. [PGG Sections 2.1, 3.1, or Lyons-Peres Section 3.4.]
8.) April 12. The basics of nilpotent and solvable groups, semidirect products, lamplighter groups [PGG Sections 4.1, 4.2, 5.1] Growth and isoperimetric inequalities, amenability. [PGG Sections 5.1, 5.2, 5.3]
April 19 Spring Break. April 26 I was sick.
9.) May 3 (plan). Isoperimetric profile for finite and infinite graphs. Kazhdan groups and expander graphs. Some basic examples of return probabilities and mixing times. Evolving sets.
Rick Durrett. Probability: theory and examples. 5th edition. Cambridge University Press, 2019. https://services.math.duke.edu/~rtd/PTE/PTE5_011119.pdf.
Geoffrey Grimmett. Probability on graphs. Cambridge University Press, 2010. http://www.statslab.cam.ac.uk/~grg/books/pgs.html.
Russ Lyons and Yuval Peres. Probability on trees and networks. Cambridge University Press, 2016. http://mypage.iu.edu/%7Erdlyons/prbtree/prbtree.html
Gábor Pete. Probability and geometry on groups. Book in preparation. PGG.pdf