Document Open Access Logo

Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs

Authors Aiya Kuchukova , Marcus Pappik , Will Perkins , Corrine Yap

PDF

File

Thumbnail PDF
PDF
LIPIcs.APPROX-RANDOM.2024.56.pdf
  • Filesize: 1.13 MB
  • 24 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Markov-chain Monte Carlo methods
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • ferromagnetic Ising model
  • fixed-magnetization Ising model
  • Kawasaki dynamics
  • Glauber dynamics
  • mixing time

Metrics

Abstract

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.

Cite As Get BibTex

Aiya Kuchukova, Marcus Pappik, Will Perkins, and Corrine Yap. Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 56:1-56:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.56

Author Details

Aiya Kuchukova
  • School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA
Marcus Pappik
  • Hasso Plattner Institute, University of Potsdam, Germany
Will Perkins
  • School of Computer Science, Georgia Institute of Technology, Atlanta, GA, USA
Corrine Yap
  • School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA

Funding

  • Pappik, Marcus: supported by the HPI Research School on Data Science and Engineering.
  • Perkins, Will: supported in part by NSF grant CCF-2309708.
  • Yap, Corrine: supported in part by NIH grant R01GM126554.

Acknowledgements

The authors thank Zongchen Chen and Marcus Michelen for very helpful discussions.

References

  1. Vedat Levi Alev and Lap Chi Lau. Improved analysis of higher order random walks and applications. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 1198-1211, 2020. URL: https://doi.org/10.1145/3357713.3384317.
  2. Nima Anari, Kuikui Liu, and Shayan Oveis Gharan. Spectral independence in high-dimensional expanders and applications to the hardcore model. SIAM Journal on Computing, 0(0):FOCS20-1, 2021. URL: https://doi.org/10.1137/20M1367696.
  3. Victor Bapst and Amin Coja-Oghlan. Harnessing the Bethe free energy. Random structures & algorithms, 49(4):694-741, 2016. URL: https://doi.org/10.1002/rsa.20692.
  4. Roland Bauerschmidt, Thierry Bodineau, and Benoit Dagallier. Kawasaki dynamics beyond the uniqueness threshold. arXiv preprint arXiv:2310.04609, 2023. URL: https://doi.org/10.48550/arXiv.2310.04609.
  5. Roland Bauerschmidt, Thierry Bodineau, and Benoit Dagallier. Stochastic dynamics and the Polchinski equation: an introduction. arXiv preprint arXiv:2307.07619, 2023. URL: https://doi.org/10.48550/arXiv.2307.07619.
  6. Rodney J Baxter. Exactly solved models in statistical mechanics. Elsevier, 2016. Google Scholar
  7. Russ Bubley and Martin Dyer. Path coupling: A technique for proving rapid mixing in Markov chains. In Proceedings 38th Annual Symposium on Foundations of Computer Science, pages 223-231. IEEE, 1997. URL: https://doi.org/10.1109/SFCS.1997.646111.
  8. Van Hao Can, Remco van der Hofstad, and Takashi Kumagai. Glauber dynamics for Ising models on random regular graphs: cut-off and metastability. ALEA, 18(1):1441-1482, 2021. URL: https://doi.org/10.30757/ALEA.v18-52.
  9. N Cancrini and F Martinelli. On the spectral gap of Kawasaki dynamics under a mixing condition revisited. Journal of Mathematical Physics, 41(3):1391-1423, 2000. URL: https://doi.org/10.1063/1.533192.
  10. N Cancrini, F Martinelli, and C Roberto. The logarithmic Sobolev constant of Kawasaki dynamics under a mixing condition revisited. Annales de l'Institut Henri Poincare (B) Probability and Statistics, 38(4):385-436, 2002. URL: https://doi.org/10.1016/S0246-0203(01)01096-2.
  11. Nicoletta Cancrini, F Cesi, and F Martinelli. The spectral gap for the Kawasaki dynamics at low temperature. Journal of statistical physics, 95:215-271, 1999. URL: https://doi.org/10.1023/A:1004581512343.
  12. Charlie Carlson, Ewan Davies, Alexandra Kolla, and Will Perkins. Computational thresholds for the fixed-magnetization Ising model. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 1459-1472, 2022. URL: https://doi.org/10.1145/3519935.3520003.
  13. Raphaël Cerf and Ágoston Pisztora. On the Wulff crystal in the Ising model. Annals of probability, pages 947-1017, 2000. URL: https://doi.org/10.1214/aop/1019160324.
  14. Yuansi Chen and Ronen Eldan. Localization schemes: A framework for proving mixing bounds for Markov chains. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 110-122. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00018.
  15. Zongchen Chen, Kuikui Liu, and Eric Vigoda. Optimal mixing of Glauber dynamics: Entropy factorization via high-dimensional expansion. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 1537-1550, 2021. URL: https://doi.org/10.1145/3406325.3451035.
  16. Zongchen Chen, Kuikui Liu, and Eric Vigoda. Spectral independence via stability and applications to holant-type problems. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 149-160. IEEE, 2022. URL: https://doi.org/10.1109/FOCS52979.2021.00023.
  17. Zongchen Chen, Kuikui Liu, and Eric Vigoda. Rapid mixing of Glauber dynamics up to uniqueness via contraction. SIAM Journal on Computing, 52(1):196-237, 2023. URL: https://doi.org/10.1137/20M136685X.
  18. Amin Coja-Oghlan, Charilaos Efthymiou, and Samuel Hetterich. On the chromatic number of random regular graphs. Journal of Combinatorial Theory, Series B, 116:367-439, 2016. URL: https://doi.org/10.1016/j.jctb.2015.09.006.
  19. Amin Coja-Oghlan, Andreas Galanis, Leslie Ann Goldberg, Jean Bernoulli Ravelomanana, Daniel Štefankovič, and Eric Vigoda. Metastability of the Potts ferromagnet on random regular graphs. Communications in Mathematical Physics, pages 1-41, 2023. URL: https://doi.org/10.1007/s00220-023-04644-6.
  20. Amin Coja-Oghlan, Florent Krzakala, Will Perkins, and Lenka Zdeborová. Information-theoretic thresholds from the cavity method. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 146-157, 2017. URL: https://doi.org/10.1145/3055399.3055420.
  21. Amin Coja-Oghlan, Philipp Loick, Balázs F Mezei, and Gregory B Sorkin. The Ising antiferromagnet and max cut on random regular graphs. SIAM Journal on Discrete Mathematics, 36(2):1306-1342, 2022. URL: https://doi.org/10.1137/20M137999X.
  22. Ewan Davies and Will Perkins. Approximately counting independent sets of a given size in bounded-degree graphs. SIAM Journal on Computing, 52(2):618-640, 2023. URL: https://doi.org/10.1137/21M1466220.
  23. Amir Dembo and Andrea Montanari. Ising models on locally tree-like graphs. Ann. Appl. Probab., 20(1):565-592, 2010. URL: https://doi.org/10.1214/09-AAP627.
  24. Roland Lvovich Dobrushin, Roman Koteckỳ, and Senya Shlosman. Wulff construction: a global shape from local interaction, volume 104. American Mathematical Society Providence, 1992. Google Scholar
  25. Martin Dyer, Alan Frieze, and Mark Jerrum. On counting independent sets in sparse graphs. SIAM Journal on Computing, 31(5):1527-1541, 2002. URL: https://doi.org/10.1137/S0097539701383844.
  26. Martin Dyer, Leslie Ann Goldberg, Mark Jerrum, and Russell Martin. Markov chain comparison. Probability Surveys, 3(none):89-111, 2006. URL: https://doi.org/10.1214/154957806000000041.
  27. Andreas Galanis, Qi Ge, Daniel Štefankovič, Eric Vigoda, and Linji Yang. Improved inapproximability results for counting independent sets in the hard-core model. Random Structures & Algorithms, 45(1):78-110, 2014. URL: https://doi.org/10.1002/rsa.20479.
  28. 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(4):500-559, 2016. URL: https://doi.org/10.1017/S0963548315000401.
  29. Andreas Galanis, Daniel Štefankovič, Eric Vigoda, and Linji Yang. Ferromagnetic Potts model: Refined #BIS-hardness and related results. SIAM Journal on Computing, 45(6):2004-2065, 2016. URL: https://doi.org/10.1137/140997580.
  30. Hans-Otto Georgii. Gibbs measures and phase transitions, volume 9. Walter de Gruyter, 2011. Google Scholar
  31. Antoine Gerschenfeld and Andrea Montanari. Reconstruction for models on random graphs. In 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07), pages 194-204. IEEE, 2007. URL: https://doi.org/10.1109/FOCS.2007.58.
  32. Heng Guo and Pinyan Lu. Uniqueness, spatial mixing, and approximation for ferromagnetic 2-spin systems. ACM Transactions on Computation Theory (TOCT), 10(4):1-25, 2018. URL: https://doi.org/10.1145/3265025.
  33. Vishesh Jain, Marcus Michelen, Huy Tuan Pham, and Thuy-Duong Vuong. Optimal mixing of the down-up walk on independent sets of a given size. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 1665-1681. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00101.
  34. Svante Janson, Tomasz Łuczak, and Andrzej Rucinski. Random graphs. John Wiley & Sons, 2011. Google Scholar
  35. Mark Jerrum and Alistair Sinclair. Polynomial-time approximation algorithms for the Ising model. SIAM Journal on computing, 22(5):1087-1116, 1993. URL: https://doi.org/10.1137/0222066.
  36. Aiya Kuchukova, Marcus Pappik, Will Perkins, and Corrine Yap. Fast and slow mixing of the kawasaki dynamics on bounded-degree graphs. arXiv preprint arXiv:2405.06209, 2024. URL: https://doi.org/10.48550/arXiv.2405.06209.
  37. David A Levin and Yuval Peres. Markov chains and mixing times, volume 107. American Mathematical Soc., 2017. Google Scholar
  38. Sheng Lin Lu and Horng-Tzer Yau. Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. Communications in Mathematical Physics, 156(2):399-433, 1993. URL: https://doi.org/10.1007/BF02098489.
  39. Fabio Martinelli, Alistair Sinclair, and Dror Weitz. Fast mixing for independent sets, colorings, and other models on trees. Random Structures & Algorithms, 31(2):134-172, 2007. URL: https://doi.org/10.1002/rsa.20132.
  40. Elchanan Mossel and Allan Sly. Exact thresholds for Ising-Gibbs samplers on general graphs. The Annals of Probability, 41(1):294-328, 2013. URL: https://doi.org/10.1214/11-AOP737.
  41. Elchanan Mossel, Dror Weitz, and Nicholas Wormald. On the hardness of sampling independent sets beyond the tree threshold. Probability Theory and Related Fields, 143(3-4):401-439, 2009. URL: https://doi.org/10.1007/s00440-007-0131-9.
  42. Dana Randall and David Wilson. Sampling spin configurations of an Ising system. In Symposium on Discrete Algorithms: Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms, pages 959-960, 1999. URL: https://doi.org/10.1145/314500.314945.
  43. Guus Regts. Absence of zeros implies strong spatial mixing. Probability Theory and Related Fields, pages 1-21, 2023. URL: https://doi.org/10.1007/s00440-023-01190-z.
  44. Shuai Shao and Yuxin Sun. Contraction: A unified perspective of correlation decay and zero-freeness of 2-spin systems. Journal of Statistical Physics, 185:1-25, 2021. URL: https://doi.org/10.1007/s10955-021-02831-0.
  45. Shuai Shao and Xiaowei Ye. From zero-freeness to strong spatial mixing via a christoffel-darboux type identity. arXiv preprint arXiv:2401.09317, 2024. URL: https://doi.org/10.48550/arXiv.2401.09317.
  46. Allan Sly. Computational transition at the uniqueness threshold. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, pages 287-296. IEEE, 2010. URL: https://doi.org/10.1109/FOCS.2010.34.
  47. Allan Sly and Nike Sun. Counting in two-spin models on d-regular graphs. Annals of Probability, 42(6):2383-2416, 2014. URL: https://doi.org/10.1214/13-AOP888.
  48. Dror Weitz. Counting independent sets up to the tree threshold. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pages 140-149, 2006. URL: https://doi.org/10.1145/1132516.1132538.
  49. Lawrence Zalcman. Normal families: new perspectives. Bulletin of the American Mathematical Society, 35(3):215-230, 1998. URL: https://doi.org/10.1090/S0273-0979-98-00755-1.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact