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Sampling from the Random Cluster Model on Random Regular Graphs at All Temperatures via Glauber Dynamics

Authors Andreas Galanis, Leslie Ann Goldberg, Paulina Smolarova



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Author Details

Andreas Galanis
  • Department of Computer Science, University of Oxford, UK
Leslie Ann Goldberg
  • Department of Computer Science, University of Oxford, UK
Paulina Smolarova
  • Department of Computer Science, University of Oxford, UK

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Andreas Galanis, Leslie Ann Goldberg, and Paulina Smolarova. Sampling from the Random Cluster Model on Random Regular Graphs at All Temperatures via Glauber Dynamics. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 64:1-64:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.64

Abstract

We consider the performance of Glauber dynamics for the random cluster model with real parameter q > 1 and temperature β > 0. Recent work by Helmuth, Jenssen and Perkins detailed the ordered/disordered transition of the model on random Δ-regular graphs for all sufficiently large q and obtained an efficient sampling algorithm for all temperatures β using cluster expansion methods. Despite this major progress, the performance of natural Markov chains, including Glauber dynamics, is not yet well understood on the random regular graph, partly because of the non-local nature of the model (especially at low temperatures) and partly because of severe bottleneck phenomena that emerge in a window around the ordered/disordered transition. Nevertheless, it is widely conjectured that the bottleneck phenomena that impede mixing from worst-case starting configurations can be avoided by initialising the chain more judiciously. Our main result establishes this conjecture for all sufficiently large q (with respect to Δ). Specifically, we consider the mixing time of Glauber dynamics initialised from the two extreme configurations, the all-in and all-out, and obtain a pair of fast mixing bounds which cover all temperatures β, including in particular the bottleneck window. Our result is inspired by the recent approach of Gheissari and Sinclair for the Ising model who obtained a similar-flavoured mixing-time bound on the random regular graph for sufficiently low temperatures. To cover all temperatures in the RC model, we refine appropriately the structural results of Helmuth, Jenssen and Perkins about the ordered/disordered transition and show spatial mixing properties "within the phase", which are then related to the evolution of the chain.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Gibbs sampling
  • Mathematics of computing → Random graphs
  • Mathematics of computing → Markov-chain Monte Carlo methods
Keywords
  • approximate counting
  • Glauber dynamics
  • random cluster model
  • approximate sampling
  • random regular graphs

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References

  1. Ferenc Bencs, Márton Borbényi, and Péter Csikvári. Random cluster model on regular graphs. Communications in Mathematical Physics, pages 1-46, 2022. Google Scholar
  2. Antonio Blanca, Zongchen Chen, Daniel Štefankovič, and Eric Vigoda. The Swendsen-Wang dynamics on trees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021), volume 207, pages 43:1-43:15, 2021. Google Scholar
  3. Antonio Blanca, Andreas Galanis, Leslie Ann Goldberg, Daniel Štefankovič, Eric Vigoda, and Kuan Yang. Sampling in uniqueness from the Potts and random-cluster models on random regular graphs. SIAM Journal on Discrete Mathematics, 34(1):742-793, 2020. Google Scholar
  4. Antonio Blanca and Reza Gheissari. Random-cluster dynamics on random regular graphs in tree uniqueness. Communications in Mathematical Physics, 386(2):1243-1287, 2021. Google Scholar
  5. Antonio Blanca and Reza Gheissari. On the tractability of sampling from the Potts model at low temperatures via Swendsen-Wang dynamics. CoRR, abs/2304.03182, 2023. Google Scholar
  6. Christian Borgs, Jennifer Chayes, Tyler Helmuth, Will Perkins, and Prasad Tetali. Efficient sampling and counting algorithms for the Potts model on ℤ^d at all temperatures. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, (STOC '20) , pages 738-751, 2020. Google Scholar
  7. Charlie Carlson, Ewan Davies, Nicolas Fraiman, Alexandra Kolla, Aditya Potukuchi, and Corrine Yap. Algorithms for the ferromagnetic Potts model on expanders. In 63rd Annual Symposium on Foundations of Computer Science (FOCS 2022), pages 344-355, 2022. Google Scholar
  8. Zongchen Chen, Andreas Galanis, Leslie A. Goldberg, Will Perkins, James Stewart, and Eric Vigoda. Fast algorithms at low temperatures via markov chains†. Random Structures & Algorithms, 58(2):294-321, 2021. Google Scholar
  9. Zongchen Chen, Andreas Galanis, Daniel Štefankovič, and Eric Vigoda. Sampling colorings and independent sets of random regular bipartite graphs in the non-uniqueness region. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms, (SODA '22), pages 2198-2207, 2022. Google Scholar
  10. 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. Google Scholar
  11. Matthew Coulson, Ewan Davies, Alexandra Kolla, Viresh Patel, and Guus Regts. Statistical physics approaches to unique games. In 35th Computational Complexity Conference, (CCC 2020), volume 169 of LIPIcs, pages 13:1-13:27, 2020. Google Scholar
  12. Charilaos Efthymiou. On sampling symmetric Gibbs distributions on sparse random graphs and hypergraphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022), pages 57:1-57:16, 2022. Google Scholar
  13. Weiming Feng, Heng Guo, and Jiaheng Wang. Sampling from the ferromagnetic Ising model with external fields. CoRR, abs/2205.01985, 2022. URL: https://arxiv.org/abs/2205.01985.
  14. Andreas Galanis, Leslie Ann Goldberg, and Paulina Smolarova. Sampling from the random cluster model on random regular graphs at all temperatures via Glauber dynamics, 2023. URL: https://arxiv.org/abs/2305.13239.
  15. Andreas Galanis, Daniel Štefankovic, Eric Vigoda, and Linji Yang. Ferromagnetic Potts model: Refined #bis-hardness and related results. SIAM Journal on Computing, 45(6):2004-2065, 2016. Google Scholar
  16. Reza Gheissari and Alistair Sinclair. Low-temperature Ising dynamics with random initializations. In 54th Annual ACM SIGACT Symposium on Theory of Computing, (STOC '22), pages 1445-1458, 2022. Google Scholar
  17. Reza Gheissari and Alistair Sinclair. Spatial mixing and the random-cluster dynamics on lattices. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms, (SODA '23) , pages 4606-4621, 2023. Google Scholar
  18. Leslie Ann Goldberg and Mark Jerrum. Approximating the partition function of the ferromagnetic Potts model. Journal of the ACM, 59(5):1-31, 2012. Google Scholar
  19. Geoffrey Grimmett. The random-cluster model, volume 333. Springer, 2006. Google Scholar
  20. Heng Guo and Mark Jerrum. Random cluster dynamics for the Ising model is rapidly mixing. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, (SODA 2017), pages 1818-1827, 2017. Google Scholar
  21. Olle Häggström. The random-cluster model on a homogeneous tree. Probability Theory and Related Fields, 104:231-253, 1996. Google Scholar
  22. Tyler Helmuth, Matthew Jenssen, and Will Perkins. Finite-size scaling, phase coexistence, and algorithms for the random cluster model on random graphs. Annales de l'Institut Henri Poincare (B) Probabilites et statistiques, 59(2):817-848, 2023. Google Scholar
  23. Tyler Helmuth, Will Perkins, and Guus Regts. Algorithmic Pirogov-Sinai theory. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, , (STOC '19) , pages 1009-1020, 2019. Google Scholar
  24. Johan Jonasson. The random cluster model on a general graph and a phase transition characterization of nonamenability. Stochastic Processes and their Applications, 79(2):335-354, 1999. Google Scholar
  25. Yuval Peres and Peter Winkler. Can extra updates delay mixing? Communications in Mathematical Physics, 323:1007-1016, 2013. Google Scholar
  26. Luca Trevisan. Lecture notes on graph partitioning, expanders and spectral methods, 2016. URL: https://lucatrevisan.github.io/books/expanders-2016.pdf.
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