Date: Probably April 11 Wed afternoon.
1. Ising models as factor of iid processes. Here are two papers. Consult me if you need advice what to cover. J. E. Steif and J. van den Berg. On the Existence and Nonexistence of Finitary Codings for a Class of Random Fields, Ann. Probab. (1999). Olle Haggstrom, Johan Jonasson, Russell Lyons. Coupling and Bernoullicity in random-cluster and Potts models, Bernoulli (2002).
2. Non-existence of atoms at the spectral radius of groups. The key step is from this book: Wolfgang Woess: Random Walks on Infinite Graphs and Groups, Cambridge Univ Press, 2000. Section 7B
3. The Trotter-Virág proof of Wigner's semicircle law for the limiting eigenvalue distribution of large random symmetric Gaussian matrices (the GOE ensemble), using Benjamini-Schramm convergence of graphs. Watch Virág's ICM 2014 talk up to time 21:16, and fill in the details.
4. Karpas' partial result on Frankl's conjecture on union-closed families, via discrete Fourier analysis. Blogpost by Gil Kalai. The preprint by Karpas.
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. Follow-up: Ori Gurel-Gurevich and Asaf Nachmias. Recurrence of planar graph limits, Ann Math (2013).
6. On any infinite transitive graph, SRW is at least diffusive. Lee-Peres (2009) Sections 1, 2, minus Subsection 2.1, plus try to solve Exercise 10.5 from my PGG notes. PGG Sections 10.1, 10.2 may contain explanations helping you understand the paper.
7. Emergence of giant cycles in random transpositions. Originally, this was proved by Oded Schramm, but here is a simple proof by Nathanael Berestycki, in Electr. J. Probab. (2011)
8. Something about the Abelian sandpile model. For an introduction, see Wikipedia page, and/or this AMS Notices article. And here is a survey by Antal Járai. If you are interested in the topic, I'll figure out the exact task.
9. The statement of the Szemerédi Regularity Lemma, and its application to testing triangle-freeness and to the simplest case of Szemerédi's Abel-prize winning theorem (proved first by Roth, 1953): every positive density subset of the integers contains a 3-term arithmetic progression. See the Alon-Spencer book Section 17.4, then Section 17.6, Exercise 3, and the Tao-Vu book Section 10.6.
10. 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.