CS294-180 Partition Functions: Project Suggestions

Below is a list of suggestions for project topics. Please read the detailed guidelines here first. You are also free to propose your own topic, provided you send me details and get it approved first. Note that such proposals must be in line with the themes and spirit of this class.

Once you have found a partner and made your selections, please email me your first choice plus at least one, and preferably two, alternatives. Projects marked XXX have already been selected by someone. Deadline for selecting a project: 31 October (Halloween)!

Note: Most of the suggestions contain multiple papers. These should be viewed as a guide, and are maximal for the scope. Most likely you will end up covering only one of the papers, and/or parts of the papers. Please discuss your proposed scope with me before finalizing your presentation (but after you've got your topic).

Hammersley-Clifford for hard constraints. [Moussouris, Journal of Statistical Physics, 1974; Gandolfi & Lenarda, Mathematics & Mechanics of Complex Systems, 2016.]
This is intended as an open-ended project: find out what is known (in addition to these papers), conduct a critical review of it, and ideally propose new approaches.
XXX Group representations and mixing times. [Diaconis & Shashahani, Zeitschrift fuer Wahrscheinlichkeitstheorie, 1981; Diaconis, Group Representations in Probability & Statistics, 1988.]
For those with a strong interest in group theory; explore the use of group representations to analyze mixing times of card-shuffling and other Markov chains.
An approximation algorithm for network reliability. [Guo & Jerrum SIAM Journal on Computing, 2019; Guo, Jerrum & Liu, ACM STOC, 2017.]
Complements Karger's algorithm from Lecture 5; the algorithm builds on the earlier 2017 paper, which is related to the algorithmic Lovasz Local Lemma.
XXX Generating random spanning trees. [Wilson, ACM STOC, 1996; Madry, Straszak & Tarnawski, ACM-SIAM SODA, 2015; Anari, Liu & Vuong, IEEE FOCS, 2022.]
Various approaches, of increasing sophistication, to the important problem of generating random spanning trees (faster than the obvious reduction to counting via the matrix tree theorem). The last paper makes use of entropy methods.
The cutoff phenomenon for mixing times. [Levin, Luczak & Peres, Probability Theory & Related Fields, 2010.]
A nice example (Ising model on the complete graph) that exhibits a cutoff phenomenon and can be cleanly analyzed. (There are many other interesting papers on cutoff.)
The Bipartite Independent Set problem (BIS). [Dyer, Goldberg, Greenhill & Jerrum, Algorithmica, 2003; Goldberg & Jerrum, JACM, 2012.]
A class of approximate counting problems, all equivalent to counting independent sets in bipartite graphs, whose status is still unresolved.
More on coupling for colorings. [Vigoda, IEEE FOCS, 1999; Hayes/Vigoda, ACM-SIAM SODA, 2005; Chen, Delcourt, Moitra, Perarnau & Postle, ACM-SIAM SODA, 2019.]
Vigoda's 11/6 Δ algorithm and a recent improvement, plus better algorithms with stronger assumptions.
An almost-optimal algorithm for colorings. [Chen, Liu, Mani & Moitra, arXiv 2304.01954]
This recent breakthrough proves rapid mixing (as well as strong spatial mixing) for the colorings dynamics all the way down to Δ + 3 colors, under a girth condition.
XXX Coupling from the Past. [Propp & Wilson, Random Structures & Algorithms, 1996; Fill, Annals of Applied Prob., 1998; Fill, Machida, Murdoch & Rosenthal, Random Structures & Algorithms, 2000.]
An interesting twist on coupling that eliminates all bias from the samples.
XXX Approximating the volume of a convex body. [Lovasz & Vempala, IEEE FOCS, 2003; Cousins & Vempala, ACM STOC, 2015; Cousins & Vempala, Math Programming C, 2016.]
Another fundamental application of MCMC that we didn't have time for in the class; the latter papers are the state of the art.
Kawasaki dynamics beyond the uniqueness threshold. [Bauerschmidt, Bodineau & Dagallier, arXiv https://arxiv.org/abs/2310.04609, 2024.]\br A nice paper that shows how a simple modification of the Glauber dynamics can overcome the exponential slowdown at the tree uniqueness threshold on random d-regular graphs.
Matroid bases via high-dimensional expanders. [Anari, Liu, Oveis Gharan & Vinzant, IEEE FOCS, 2018; Alev & Lau, ACM STOC, 2020.]
The original paper on this result, which replaces the log-Sobolev approach of Cryan et al. we saw in class by a spectral analysis. The second paper simplifies and generalizes the approach.
Spectral independence and optimal mixing times for Glauber dynamics. [Anari, Liu & Oveis Gharan, IEEE FOCS 2020; Chen, Liu & Vigoda, ACM STOC, 2021; Anari, Jain, Koehler, Pham & Vuong, ACM STOC, 2022.]
A line of work that develops the idea of spectral independence for mixing times of Glauber dynamics, culminating eventually (when applied to entropy) in optimal O(n log n) mixing times in almost all cases where polynomial mixing time is known.
Inapproximability of antiferromagnetic two-spin systems in the non-uniqueness region. [Sly & Sun, Annals of Probability, 2014; Galanis, Stefankovic & Vigoda, Combinatorics, Probability & Computing , 2016; Galanis, Stefankovic & Vigoda JACM, 2015.]
Two alternative approaches to proving hardness of approximation for the partition function of the anti-ferromagnetic Ising model in the non-uniqueness region, generalizing Sly's original result, and an extension to multi-spin systems.
XXX Approximate counting via the cluster expansion. [Helmuth, Perkins & Regts, Probability Theory & Related Fields, 2020.]
Use of the cluster expansion from statistical physics to obtain algorithms for spin systems at low temperatures.
XXX Connections to quantum computing. [Bakshi, Liu, Moitra & Tang, IEEE FOCS, 2024; Bakshi, Liu, Moitra & Tang, arXiv https://arxiv.org/abs/2510.08542, 2025; Gilyen, Chen, Doriguello, Kastoryano, arXiv https://arxiv.org/abs/2405.20322, 2024]
The first two papers aim to develop quantum analogs of some of the mixing time theory in this class. The third paper explores appropriate models of quantum dynamics.