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Improved Mixing of Critical Hardcore Model

Authors Zongchen Chen , Tianhui Jiang

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Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
  • Mathematics of computing → Markov processes
Keywords
  • Hardcore model
  • Phase transition
  • Glauber dynamics
  • Spectral independence
  • Online decision making
  • Site percolation

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Abstract

The hardcore model is one of the most classic and widely studied examples of undirected graphical models. Given a graph G, the hardcore model describes a Gibbs distribution of λ-weighted independent sets of G. In the last two decades, a beautiful computational phase transition has been established at a precise threshold λ_c(Δ) where Δ denotes the maximum degree, where the task of sampling independent sets transitions from polynomial-time solvable to computationally intractable. We study the critical hardcore model where λ = λ_c(Δ) and show that the Glauber dynamics, a simple yet popular Markov chain algorithm, mixes in Õ(n^{7.44 + O(1/Δ)}) time on any n-vertex graph of maximum degree Δ ≥ 3, significantly improving the previous upper bound Õ(n^{12.88 + O(1/Δ)}) by the recent work [Chen et al., 2024]. The core property we establish in this work is that the critical hardcore model is O(√n)-spectrally independent, improving the trivial bound of n and matching the critical behavior of the Ising model. Our proof approach utilizes an online decision-making framework to study a site percolation model on the infinite (Δ-1)-ary tree, which can be interesting by itself.

Cite As Get BibTex

Zongchen Chen and Tianhui Jiang. Improved Mixing of Critical Hardcore Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 51:1-51:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.51

Author Details

Zongchen Chen
  • School of Computer Science, Georgia Institute of Technology, Atlanta, GA, USA
Tianhui Jiang
  • Zhiyuan College, Shanghai Jiao Tong University, China

References

  1. Nima Anari, Kuikui Liu, and Shayan Oveis Gharan. Spectral independence in high-dimensional expanders and applications to the hardcore model. In Proceedings of the Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 1319-1330, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00125.
  2. Alexander Barvinok. Combinatorics and complexity of partition functions, volume 30 of Algorithms and Combinatorics. Springer, Cham, 2016. URL: https://doi.org/10.1007/978-3-319-51829-9.
  3. Xiaoyu Chen, Zongchen Chen, Yitong Yin, and Xinyuan Zhang. Rapid mixing at the uniqueness threshold. arXiv preprint arXiv:2411.03413, 2024. URL: https://doi.org/10.48550/arXiv.2411.03413.
  4. Xiaoyu Chen and Weiming Feng. Rapid mixing via coupling independence for spin systems with unbounded degree. arXiv preprint arXiv:2407.04672, 2024. URL: https://doi.org/10.48550/arXiv.2407.04672.
  5. Xiaoyu Chen, Weiming Feng, Yitong Yin, and Xinyuan Zhang. Optimal mixing for two-state anti-ferromagnetic spin systems. In Proceedings of the Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 588-599, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00062.
  6. Xiaoyu Chen and Xinyuan Zhang. A near-linear time sampler for the ising model with external field. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 4478-4503, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH170.
  7. Yuansi Chen and Ronen Eldan. Localization schemes: A framework for proving mixing bounds for Markov chains. In Proceedings of the Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 110-122, 2022. Google Scholar
  8. Zongchen Chen and Tianhui Jiang. Improved mixing of critical hardcore model. Full version, 2025. URL: https://arxiv.org/abs/2505.07515.
  9. Zongchen Chen, Kuikui Liu, and Eric Vigoda. Optimal mixing of Glauber dynamics: entropy factorization via high-dimensional expansion. In Proceedings of the Annual ACM Symposium on Theory of Computing (STOC), pages 1537-1550, 2021. URL: https://doi.org/10.1145/3406325.3451035.
  10. Zongchen Chen, Kuikui Liu, and Eric Vigoda. Rapid mixing of Glauber dynamics up to uniqueness via contraction. SIAM J. Comput., 52(1):196-237, 2023. URL: https://doi.org/10.1137/20M136685X.
  11. John Newton Darroch. On the distribution of the number of successes in independent trials. Ann. Math. Stat., 35(3):1317-1321, 1964. Google Scholar
  12. Sudhakar Dharmadhikari and Kumar Joag-dev. Unimodality, Convexity, and Applications. Probability and Mathematical Statistics. Academic Press, San Diego, 1988. Google Scholar
  13. Weiming Feng, Heng Guo, Yitong Yin, and Chihao Zhang. Rapid mixing from spectral independence beyond the boolean domain. ACM Trans. Algorithms, 18(3):1-32, 2022. URL: https://doi.org/10.1145/3531008.
  14. Andreas Galanis, Daniel Štefankovič, and Eric Vigoda. Inapproximability of the partition function for the antiferromagnetic Ising and hard-core models. Combin. Probab. Comput., 25(4):500-559, 2016. URL: https://doi.org/10.1017/S0963548315000401.
  15. Yuanying Guan, Muqiao Huang, and Ruodu Wang. A new characterization of second-order stochastic dominance. Insur. Math. Econ., 119:261-267, 2024. Google Scholar
  16. Haim Levy. Stochastic dominance and expected utility: Survey and analysis. Manag. Sci., 38(4):555-593, 1992. Google Scholar
  17. Liang Li, Pinyan Lu, and Yitong Yin. Correlation decay up to uniqueness in spin systems. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 67-84, 2013. URL: https://doi.org/10.1137/1.9781611973105.5.
  18. Kuikui Liu. Spectral Independence: A New Tool to Analyze Markov Chains. PhD thesis, University of Washington, 2023. Google Scholar
  19. Elchanan Mossel, Dror Weitz, and Nicholas Wormald. On the hardness of sampling independent sets beyond the tree threshold. Probab. Theory Related Fields, 143(3-4):401-439, 2009. Google Scholar
  20. Alfred Müller and Dietrich Stoyan. Comparison of Stochastic Models and Risks, volume 389 of Wiley Series in Probability and Statistics. John Wiley & Sons, 2002. Google Scholar
  21. Han Peters and Guus Regts. On a conjecture of Sokal concerning roots of the independence polynomial. Michigan Math. J., 68(1):33-55, 2019. Google Scholar
  22. Michael Rothschild and Joseph E. Stiglitz. Increasing risk: I. a definition. J. Econ. Theory, 2(3):225-243, 1970. Google Scholar
  23. Allan Sly. Computational transition at the uniqueness threshold. In Proceedings of the Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 287-296, 2010. URL: https://doi.org/10.1109/FOCS.2010.34.
  24. Allan Sly and Nike Sun. The computational hardness of counting in two-spin models on d-regular graphs. In Proceedings of the Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 361-369, 2012. URL: https://doi.org/10.1109/FOCS.2012.56.
  25. Daniel Štefankovič and Eric Vigoda. Lecture notes on spectral independence and bases of a matroid: Local-to-global and trickle-down from a Markov chain perspective. arXiv preprint, 2023. URL: https://arxiv.org/abs/2307.13826.
  26. Wenpin Tang and Fengmin Tang. The poisson binomial distribution - Old & new. Statist. Sci., 38(1):108-119, 2023. Google Scholar
  27. Remco van der Hofstad and Michael Keane. An elementary proof of the hitting time theorem. Amer. Math. Monthly, 115(8):753-756, 2008. URL: http://www.jstor.org/stable/27642587.
  28. Dror Weitz. Counting independent sets up to the tree threshold. In Proceedings of the Annual ACM Symposium on Theory of Computing (STOC), pages 140-149, 2006. URL: https://doi.org/10.1145/1132516.1132538.
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