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On the Mixing Time of Glauber Dynamics for the Hard-Core and Related Models on G(n,d/n)

Authors Charilaos Efthymiou, Weiming Feng

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Charilaos Efthymiou
  • Computer Science, University of Warwick, Coventry, UK
Weiming Feng
  • School of Informatics, University of Edinburgh, Edinburgh, UK


Weiming Feng would like to thank Heng Guo for the helpful discussions.

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Charilaos Efthymiou and Weiming Feng. On the Mixing Time of Glauber Dynamics for the Hard-Core and Related Models on G(n,d/n). In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 54:1-54:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


We study the single-site Glauber dynamics for the fugacity λ, Hard-Core model on the random graph G(n, d/n). We show that for the typical instances of the random graph G(n,d/n) and for fugacity λ < {d^d} / {(d-1)^(d+1)}, the mixing time of Glauber dynamics is n^{1 + O(1/log log n)}. Our result improves on the recent elegant algorithm in [Bezáková, Galanis, Goldberg and Štefankovič; ICALP'22]. The algorithm there is an MCMC-based sampling algorithm, but it is not the Glauber dynamics. Our algorithm here is simpler, as we use the classic Glauber dynamics. Furthermore, the bounds on mixing time we prove are smaller than those in Bezáková et al. paper, hence our algorithm is also faster. The main challenge in our proof is handling vertices with unbounded degrees. We provide stronger results with regard the spectral independence via branching values and show that the our Gibbs distributions satisfy the approximate tensorisation of the entropy. We conjecture that the bounds we have here are optimal for G(n,d/n). As corollary of our analysis for the Hard-Core model, we also get bounds on the mixing time of the Glauber dynamics for the Monomer-Dimer model on G(n,d/n). The bounds we get for this model are slightly better than those we have for the Hard-Core model

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Randomness, geometry and discrete structures
  • Mathematics of computing → Discrete mathematics
  • spin-system
  • spin-glass
  • sparse random (hyper)graph
  • approximate sampling
  • efficient algorithm


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  1. Nima Anari, Kuikui Liu, and Shayan Oveis Gharan. Spectral independence in high-dimensional expanders and applications to the hardcore model. In FOCS, pages 1319-1330, 2020. Google Scholar
  2. Jean Barbier, Florent Krzakala, Lenka Zdeborová, and Pan Zhang. The hard-core model on random graphs revisited. Journal of Physics: Conference Series, 473(1):012021, December 2013. Google Scholar
  3. Ivona Bezáková, Andreas Galanis, Leslie Ann Goldberg, and Daniel Štefankovič. Fast sampling via spectral independence beyond bounded-degree graphs. In ICALP, volume 229 of LIPIcs, pages 21:1-21:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  4. Pietro Caputo, Georg Menz, and Prasad Tetali. Approximate tensorization of entropy at high temperature. Ann. Fac. Sci. Toulouse Math. (6), 24(4):691-716, 2015. Google Scholar
  5. Pietro Caputo and Daniel Parisi. Block factorization of the relative entropy via spatial mixing. arXiv preprint, 2020. URL:
  6. Filippo Cesi. Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields. Probab. Theory Related Fields, 120(4):569-584, 2001. Google Scholar
  7. Xiaoyu Chen, Weiming Feng, Yitong Yin, and Xinyuan Zhang. Rapid mixing of glauber dynamics via spectral independence for all degrees. In FOCS, pages 137-148. IEEE, 2021. Google Scholar
  8. Xiaoyu Chen, Weiming Feng, Yitong Yin, and Xinyuan Zhang. Optimal mixing for two-state anti-ferromagnetic spin systems. In FOCS, pages 588-599. IEEE, 2022. Google Scholar
  9. Zongchen Chen, Kuikui Liu, and Eric Vigoda. Optimal mixing of Glauber dynamics: Entropy factorization via high-dimensional expansion. In STOC, 2021. arXiv:2011.02075. URL:
  10. Amin Coja-Oghlan and Charilaos Efthymiou. On independent sets in random graphs. Random Struct. Algorithms, 47(3):436-486, 2015. Google Scholar
  11. Varsha Dani and Cristopher Moore. Independent sets in random graphs from the weighted second moment method. In RANDOM, volume 6845 of Lecture Notes in Computer Science, pages 472-482. Springer, 2011. Google Scholar
  12. Charilaos Efthymiou. Spectral independence beyond uniqueness using the topological method. CoRR, abs/2211.03753, 2022. URL:
  13. Charilaos Efthymiou, Thomas P. Hayes, Daniel Stefankovic, and Eric Vigoda. Sampling random colorings of sparse random graphs. In SODA, pages 1759-1771, 2018. Google Scholar
  14. Weiming Feng, Heng Guo, Yitong Yin, and Chihao Zhang. Fast sampling and counting k-SAT solutions in the local lemma regime. J. ACM, 68(6):40:1-40:42, 2021. Google Scholar
  15. Weiming Feng, Heng Guo, Yitong Yin, and Chihao Zhang. Rapid mixing from spectral independence beyond the boolean domain. In SODA, pages 1558-1577, 2021. Google Scholar
  16. Weiming Feng, Kun He, and Yitong Yin. Sampling constraint satisfaction solutions in the local lemma regime. In STOC, pages 1565-1578. ACM, 2021. Google Scholar
  17. Alan M. Frieze. On the independence number of random graphs. Discret. Math., 81(2):171-175, 1990. Google Scholar
  18. Andreas Galanis, Leslie Ann Goldberg, Heng Guo, and Kuan Yang. Counting solutions to random CNF formulas. SIAM J. Comput., 50(6):1701-1738, 2021. Google Scholar
  19. Andreas Galanis, Daniel Štefankovič, and Eric Vigoda. Inapproximability of the partition function for the antiferromagnetic Ising and hard-core models. Combinatorics, Probability and Computing, 25(04):500-559, 2016. Google Scholar
  20. David Gamarnik and Madhu Sudan. Limits of local algorithms over sparse random graphs. In ITCS, pages 369-376. ACM, 2014. Google Scholar
  21. Vishesh Jain, Huy Tuan Pham, and Thuy-Duong Vuong. On the sampling Lovász local lemma for atomic constraint satisfaction problems. arXiv preprint, 2021. URL:
  22. Vishesh Jain, Huy Tuan Pham, and Thuy Duong Vuong. Spectral independence, coupling with the stationary distribution, and the spectral gap of the Glauber dynamics. arXiv preprint, 2021. URL:
  23. Mark Jerrum and Alistair Sinclair. Approximating the permanent. SIAM J. Comput., 18(6):1149-1178, 1989. Google Scholar
  24. F. P. Kelly. Stochastic models of computer communication systems. Journal of the Royal Statistical Society. Series B (Methodological), 47:379-395, 1985. Google Scholar
  25. Florent Krzakala, Andrea Montanari, Federico Ricci-Tersenghi, Guilhem Semerjian, and Lenka Zdeborová. Gibbs states and the set of solutions of random constraint satisfaction problems. Proc. Natl. Acad. Sci. USA, 104(25):10318-10323, 2007. Google Scholar
  26. Ankur Moitra. Approximate counting, the Lovász local lemma, and inference in graphical models. J. ACM, 66(2):10:1-10:25, 2019. Google Scholar
  27. Jesús Salas and Alan D Sokal. Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem. J. Stat. Phys., 86(3):551-579, 1997. Google Scholar
  28. Alistair Sinclair, Piyush Srivastava, Daniel Štefankovič, and Yitong Yin. Spatial mixing and the connective constant: optimal bounds. Probab. Theory Related Fields, 168(1-2):153-197, 2017. Google Scholar
  29. Allan Sly. Computational transition at the uniqueness threshold. In FOCS, pages 287-296, 2010. Google Scholar
  30. Allan Sly and Nike Sun. Counting in two-spin models on d-regular graphs. Ann. Probab., 42(6):2383-2416, 2014. Google Scholar
  31. Dror Weitz. Counting independent sets up to the tree threshold. In STOC, pages 140-149, 2006. Google Scholar
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