Dates: ???
Each project consists of reading a research article or a book chapter, solving one or two related problems, and preparing a 50 minutes talk. Hand in the problem solutions by the time of your talk. Keep the talk simple, and don't try to cover everything that you have read about; rather, give just enough background and motiviation that makes it possible to appreciate your topic, and prove at least one non-trivial statement. (Could be just an interesting lemma that is important in the proof of the main theorem, but then you should have some idea about the outline of the entire proof.) The audience will be the participants of the course. I will help with the reading or the problem(s) if you come and ask.
1. For symmetric random walk on groups, zero speed is equivalent to the Liouville property for bounded harmonic functions. The only known proof goes through entropy and Poisson boundary, hence this is a project with several new notions --- but it is quite beautiful. It's better to focus on either entropy or Poisson boundary. The original main article is Kaimanovich-Vershik (1983), very nice, but it might be better to start with my PGG notes Sections 9.3, 9.4, 9.5. Part of the task is to solve two of Exercises 9.8, 9.9, 9.11, 9.13. A main part of your talk could be to give the solution of Exercise 9.11.
2. On any infinite transitive graph, SRW is at least diffusive. Lee-Peres (2009) Sections 1, 2, minus Subsection 2.1, plus solving Exercise 10.5. from my PGG notes. PGG Sections 10.1, 10.2 may contain explanations helping you understand the paper.
3. The evolving sets method to deduce heat kernel decay or mixing from the isoperimetric profile. Here is the original Morris-Peres paper (2003), but my PGG notes Sections 8.2 and 8.3 contain much more explanation (with some details missing). Plus, solve two of the PGG Exercises 8.1, 8.5, 8.7, or one of Exercises 8.4, 8.6.
4. Bottlenecks, burstiness, and fat tails regulate mixing times of non-Poissonian random walks. Paper by Delvenne, Lambiotte and Rocha (2013). There are probably some math details to fill in, but as an additional exercise, find the exact speed of convergence to stationarity on the complete graph with power-law waiting times with no cut-off. Plus, Proposition 7.9 from my PGG notes. [Alexey has taken this.]
5. The Benjamini-Schramm limit of bounded degree finite planar graphs is always recurrent. This is the original Benjamini-Schramm paper (2000) where they introduced this limit notion. The beautiful proof uses circle packings. Plus, solve two of the PGG Exercises 14.1, 14.13, 14.14.
6. David Wilson's algorithm to generate the uniform random spanning tree (UST) using loop-erased random walks (LERW). Lyons-Peres book Section 4.1, plus Exercise 4.29 using Wilson's algorithm.
7. Proof of the power-law degree distribution for preferential attachment models, using martingales. van der Hofstad book Sections 8.3, 8.4., 8.5. You may assume that m=1 and \delta=0. Plus, solve one or two of Exercises 8.14-17 from that book.
8. The concentration of the chromatic number of the Erdős-Rényi graph G(n,1/2), around the value 2\log_2 n, using martingales, due to Béla Bollobás (1988). The main source is the Alon-Spencer book Sections 4.5, 7.3, 10.3. As a background, you will need Sections 7.1, 7.2, too; for this, I also suggest my PGG notes Section 6.3, containing some explanation missing from Alon-Spencer. Plus, solve one of the PGG Exercises 6.12, 6.13.
9. Proof of the Harris-Kesten theorem p_c(\Z^2)=1/2 for bond percolation on the square grid. The Russo-Seymour-Welsh inequality may be assumed. PGG notes Section 12.3. Plus, one of the Exercises 12.36 and 12.38. See also Grimmett's book Sections 5.6, 5.8, in case you need more details.
10. Proof of the Edwars-Sokal coupling between the Potts(\beta,q) and the FK(p,q) models. For the square grid FK(p,2) model, finding the self-dual point. Second half of PGG Section 13.1, plus one of Exercises 13.4 and 13.5.
11. Mermin-Wagner theorem: in the O(n) spin model on \Z^2, due to recurrence, there is no phase transition. Marek Biskup's notes Theorem 3.6. Don't get frightened by the Lie-algebra in the proof: just take n=1, where every appearance of the Lie-algebra element R can simply be ignored. Plus one of the PGG Exercises 13.4 and 13.5.
12. Random-turn hex and other selection games: Peres-Schramm-Sheffield-Wilson paper in Amer Math Monthly. Plus, my PGG notes Exercise 12.35.