# Summer School on Markov Chains

Things to work out the details for:
- Approximate sampling gives approximate counting.
- Old mixing time arguments:
- What are coupling based arguments, and see for random graph coloring based?
- Cannonical path based examples.
- How to work with spectral gap, entropy contraction, MLSI, LSI.
- Understand matrix trickle-down theorem and local-to-global theorem, and Tali's new proof. [[Hopkins-Lovette Notes]](https://cseweb.ucsd.edu//classes/sp21/cse291-g/CSE_291_Expanders_and_HDX_Course_Notes.pdf), [[Tali Lecture Notes]](https://www.overleaf.com/project/62deffd91cf91e95036256d5), [[Tali Paper]](https://arxiv.org/pdf/2208.03241.pdf).
- Nima how to use trickle down by arguing about multi-partite matroid graph in the top layer. Based on [[A, Liu, Gharan, Vinzent]](https://arxiv.org/pdf/1811.01816.pdf).
- Capito approximate tensorization proof. [?]
- KuiKui proof of spectral independence and mixing time and relate it to HDX language. Understand the meaning of assumptions (A1, A2, A3). Based on [[Chen, Liu, Vigoda]](https://arxiv.org/abs/2011.02075).
- Understand how Heng Guo used entropy to improve over Nima, based on [[Cryan, Guo, Mousa]](https://homepages.inf.ed.ac.uk/hguo/papers/Log-Sob-matroid.pdf).
- Zhongchen Chen proof of zero-freeness implying spectral independence, and connections to complex analysis. Based on [[Chen, Liu, Vigoda]](https://arxiv.org/abs/2011.02075)
- Nima geometry of polynomials language to HDX proof, and how it translates to log-concave polynomials [[Fractionally Log-Concave Polynomial]](https://arxiv.org/pdf/2102.02708.pdf).
- [Jerrum and Sinclair](https://people.eecs.berkeley.edu/~sinclair/approx.pdf) paper on how intersection of matroid bases work, see how it applies to matchings in birartite graphs.
- Field dynamics markov chain [Chen et al.](https://arxiv.org/abs/2105.15005) is pretty powerful, and see how it relates to Eldan work on stochastic localization [Chen, Eldan](https://arxiv.org/abs/2203.04163).