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We survey the current state of affairs in the study of thresholds and sharp thresholds in random structures on the occasion of the recent proof of the Kahn–Kalai conjecture by Park and Pham and the fairly recent proof of the satisfiability conjecture for large k by Ding, Sly, and Sun. Random discrete structures appear as fundamental objects of study in many scientific and mathematical fields including statistical physics, combinatorics, algorithms and complexity, social choice theory, coding theory, and statistics. While the models and properties of interest in these fields vary widely, much progress has been made through the development of general tools applicable to large families of models and properties all at once. Historically, these tools originated to solve or make progress on specific difficult conjectures in the areas mentioned above. We will survey recent progress on some of these hard problems and describe some challenges for the future. This survey was prepared in conjunction with a talk for the Current Events Bulletin at the 2024 Joint Mathematics Meetings (JMM) in San Francisco, California. 

We prove a hardness of sampling result for the anti-ferromagnetic Ising model on random graphs of average degree d for large constant d, proving that when the normalized inverse temperature satisfies β > 1 (asymptotically corresponding to the condensation threshold), then w.h.p. over the random graph there is no stable sampling algorithm that can output a sample close in W₂ distance to the Gibbs measure. The results also apply to a fixed-magnetization version of the model, showing that there are no stable sampling algorithms for low but positive temperature max and min bisection distributions. These results show a gap in the tractability of search and sampling problems: while there are efficient algorithms to find near optimizers, stable sampling algorithms cannot access the Gibbs distribution concentrated on such solutions. Our techniques involve extensions of the interpolation technique relating behavior of the mean field Sherrington-Kirkpatrick model to behavior of Ising models on random graphs of average degree d for large d. While previous interpolation arguments compared the free energies of the two models, our argument compares the average energies and average overlaps in the two models. 

We study the worst-case mixing time of the global Kawasaki dynamics for the fixed-magnetization Ising model on the class of graphs of maximum degree Δ. Proving a conjecture of Carlson, Davies, Kolla, and Perkins, we show that below the tree uniqueness threshold, the Kawasaki dynamics mix rapidly for all magnetizations. Disproving a conjecture of Carlson, Davies, Kolla, and Perkins, we show that the regime of fast mixing does not extend throughout the regime of tractability for this model: there is a range of parameters for which there exist efficient sampling algorithms for the fixed-magnetization Ising model on max-degree Δ graphs, but the Kawasaki dynamics can take exponential time to mix. Our techniques involve showing spectral independence in the fixed-magnetization Ising model and proving a sharp threshold for the existence of multiple metastable states in the Ising model with external field on random regular graphs. 

We give an efficient perfect sampling algorithm for weighted, connected induced subgraphs (orgraphlets) of rooted, bounded degree graphs. Our algorithm utilizes a vertex-percolation process with a carefully chosen rejection filter and works under a percolation subcriticality condition. We show that this condition is optimal in the sense that the task of (approximately) sampling weighted rooted graphlets becomes impossible in finite expected time for infinite graphs and intractable for finite graphs when the condition does not hold. We apply our sampling algorithm as a subroutine to give near linear-time perfect sampling algorithms for polymer models and weighted non-rooted graphlets in finite graphs, two widely studied yet very different problems. This new perfect sampling algorithm for polymer models gives improved sampling algorithms for spin systems at low temperatures on expander graphs and unbalanced bipartite graphs, among other applications. 

Abstract We study the locations of complex zeroes of independence polynomials of bounded-degree hypergraphs. For graphs, this is a long-studied subject with applications to statistical physics, algorithms, and combinatorics. Results on zero-free regions for bounded-degree graphs include Shearer’s result on the optimal zero-free disc, along with several recent results on other zero-free regions. Much less is known for hypergraphs. We make some steps towards an understanding of zero-free regions for bounded-degree hypergaphs by proving that all hypergraphs of maximum degree$$\Delta$$have a zero-free disc almost as large as the optimal disc for graphs of maximum degree$$\Delta$$established by Shearer (of radius$$\sim 1/(e \Delta )$$). Up to logarithmic factors in$$\Delta$$this is optimal, even for hypergraphs with all edge sizes strictly greater than$$2$$. We conjecture that for$$k\ge 3$$,$$k$$-uniformlinearhypergraphs have a much larger zero-free disc of radius$$\Omega (\Delta ^{- \frac{1}{k-1}} )$$. We establish this in the case of linear hypertrees.