A Spectral Independence View on Hard Spheres via Block Dynamics

Authors Tobias Friedrich , Andreas Göbel , Martin S. Krejca , Marcus Pappik

Thumbnail PDF


  • Filesize: 0.67 MB
  • 15 pages

Document Identifiers

Author Details

Tobias Friedrich
  • Hasso Plattner Institute, University of Potsdam, Germany
Andreas Göbel
  • Hasso Plattner Institute, University of Potsdam, Germany
Martin S. Krejca
  • Sorbonne University, CNRS, LIP6, Paris, France
Marcus Pappik
  • Hasso Plattner Institute, University of Potsdam, Germany

Cite AsGet BibTex

Tobias Friedrich, Andreas Göbel, Martin S. Krejca, and Marcus Pappik. A Spectral Independence View on Hard Spheres via Block Dynamics. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 66:1-66:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


The hard-sphere model is one of the most extensively studied models in statistical physics. It describes the continuous distribution of spherical particles, governed by hard-core interactions. An important quantity of this model is the normalizing factor of this distribution, called the partition function. We propose a Markov chain Monte Carlo algorithm for approximating the grand-canonical partition function of the hard-sphere model in d dimensions. Up to a fugacity of λ < e/2^d, the runtime of our algorithm is polynomial in the volume of the system. This covers the entire known real-valued regime for the uniqueness of the Gibbs measure. Key to our approach is to define a discretization that closely approximates the partition function of the continuous model. This results in a discrete hard-core instance that is exponential in the size of the initial hard-sphere model. Our approximation bound follows directly from the correlation decay threshold of an infinite regular tree with degree equal to the maximum degree of our discretization. To cope with the exponential blow-up of the discrete instance we use clique dynamics, a Markov chain that was recently introduced in the setting of abstract polymer models. We prove rapid mixing of clique dynamics up to the tree threshold of the univariate hard-core model. This is achieved by relating clique dynamics to block dynamics and adapting the spectral expansion method, which was recently used to bound the mixing time of Glauber dynamics within the same parameter regime.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
  • Hard-sphere Model
  • Markov Chain
  • Partition Function
  • Gibbs Distribution
  • Approximate Counting
  • Spectral Independence


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Vedat Levi Alev and Lap Chi Lau. Improved analysis of higher order random walks and applications. In Proc. of STOC'20, 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. CoRR, abs/2001.00303, 2020. URL: http://arxiv.org/abs/2001.00303.
  3. Ivona Bezáková, Andreas Galanis, Leslie Ann Goldberg, and Daniel Štefankovič. Inapproximability of the independent set polynomial in the complex plane. In Proc. of STOC'18, page 1234–1240, 2018. URL: https://doi.org/10.1145/3188745.3188788.
  4. Antonio Blanca, Pietro Caputo, Alistair Sinclair, and Eric Vigoda. Spatial mixing and nonlocal Markov chains. Random Structures & Algorithms, 55(3):584-614, 2019. URL: https://doi.org/10.1002/rsa.20844.
  5. Antonio Blanca, Zongchen Chen, and Eric Vigoda. Swendsen-Wang dynamics for general graphs in the tree uniqueness region. Random Structures & Algorithms, 56(2):373-400, 2020. URL: https://doi.org/10.1002/rsa.20858.
  6. Christian Borgs, Jennifer T. Chayes, Tyler Helmuth, Will Perkins, and Prasad Tetali. Efficient sampling and counting algorithms for the Potts model on 𝕫^d at all temperatures. In Proc. of STOC'20, pages 738-751, 2020. URL: https://doi.org/10.1145/3357713.3384271.
  7. Tomáš Boublik, Ivo Nezbeda, and Karel Hlavaty. Statistical thermodynamics of simple liquids and their mixtures. Fundamental Studies in Engineering. Elsevier, 1980. Google Scholar
  8. Sarah Cannon and Will Perkins. Counting independent sets in unbalanced bipartite graphs. In Proc. of SODA'20, pages 1456-1466, 2020. URL: https://doi.org/10.1137/1.9781611975994.88.
  9. Katrin Casel, Philipp Fischbeck, Tobias Friedrich, Andreas Göbel, and J. A. Gregor Lagodzinski. Zeros and approximations of holant polynomials on the complex plane. CoRR, abs/1905.03194, 2019. URL: http://arxiv.org/abs/1905.03194.
  10. Zongchen Chen, Andreas Galanis, Leslie Ann Goldberg, Will Perkins, James Stewart, and Eric Vigoda. Fast algorithms at low temperatures via Markov chains. In Proc. of APPROX/RANDOM'19, pages 41:1-41:14, 2019. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.41.
  11. Zongchen Chen, Andreas Galanis, Daniel Stefankovic, and Eric Vigoda. Rapid mixing for colorings via spectral independence. CoRR, abs/2007.08058, 2020. URL: http://arxiv.org/abs/2007.08058.
  12. Zongchen Chen, Kuikui Liu, and Eric Vigoda. Optimal mixing of Glauber dynamics: Entropy factorization via high-dimensional expansion. CoRR, abs/2011.02075, 2020. URL: http://arxiv.org/abs/2011.02075.
  13. Zongchen Chen, Kuikui Liu, and Eric Vigoda. Rapid mixing of Glauber dynamics up to uniqueness via contraction. CoRR, abs/2004.09083, 2020. URL: http://arxiv.org/abs/2004.09083.
  14. Hofer-Temmel Christoph et al. Disagreement percolation for the hard-sphere model. Electronic Journal of Probability, 24, 2019. Google Scholar
  15. Persi Diaconis and Laurent Saloff-Coste. Comparison theorems for reversible Markov chains. The Annals of Applied Probability, pages 696-730, 1993. Google Scholar
  16. Martin Dyer, Abraham D. Flaxman, Alan M. Frieze, and Eric Vigoda. Randomly coloring sparse random graphs with fewer colors than the maximum degree. Random Structures & Algorithms, 29(4):450–465, 2006. URL: https://doi.org/10.1002/rsa.20129.
  17. 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.
  18. Charilaos Efthymiou. MCMC sampling colourings and independent sets of g(n, d/n) near uniqueness threshold. In Proc. of SODA'14, page 305–316, 2014. URL: https://doi.org/10.1137/1.9781611973402.22.
  19. Charilaos Efthymiou, Thomas P. Hayes, Daniel Štefankovič, and Eric Vigoda. Sampling random colorings of sparse random graphs. In Proc. of SODA'18, page 1759–1771, 2018. URL: https://doi.org/10.1137/1.9781611975031.115.
  20. Pep Espanol. Statistical mechanics of coarse-graining. In Novel Methods in Soft Matter Simulations, pages 69-115. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-39895-0_3.
  21. Weiming Feng, Heng Guo, Yitong Yin, and Chihao Zhang. Rapid mixing from spectral independence beyond the Boolean domain. CoRR, abs/2007.08091, 2020. URL: http://arxiv.org/abs/2007.08091.
  22. Roberto Fernández, Aldo Procacci, and Benedetto Scoppola. The analyticity region of the hard sphere gas. Improved bounds. Journal of Statistical Physics, 128(5):1139-1143, 2007. URL: https://doi.org/10.1007/s10955-007-9352-7.
  23. Tobias Friedrich, Andreas Göbel, Martin S. Krejca, and Marcus Pappik. Polymer dynamics via cliques: New conditions for approximations. CoRR, abs/2007.08293, 2020. URL: http://arxiv.org/abs/2007.08293.
  24. Andreas Galanis, Qi Ge, Daniel Stefankovic, 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.
  25. Andreas Galanis, Leslie Ann Goldberg, and Daniel Stefankovic. Inapproximability of the independent set polynomial below the Shearer threshold. CoRR, abs/1612.05832, 2016. URL: http://arxiv.org/abs/1612.05832.
  26. Andreas Galanis, Leslie Ann Goldberg, and James Stewart. Fast algorithms for general spin systems on bipartite expanders. In Proc. of MFCS'20, 2020. To appear. URL: http://arxiv.org/abs/2004.13442.
  27. David Galvin and Prasad Tetali. Slow mixing of Glauber dynamics for the hard-core model on regular bipartite graphs. Random Structures & Algorithms, 28(4):427-443, 2006. URL: https://doi.org/10.1002/rsa.20094.
  28. Heng Guo and Mark Jerrum. Perfect simulation of the hard disks model by partial rejection sampling. CoRR, abs/1801.07342, 2018. URL: http://arxiv.org/abs/1801.07342v2.
  29. Jean-Pierre Hansen and Ian R. McDonald. Theory of Simple Liquids. Academic Press, fourth edition edition, 2013. URL: https://doi.org/10.1016/B978-0-12-387032-2.00013-1.
  30. Nicholas J.A. Harvey, Piyush Srivastava, and Jan Vondrák. Computing the independence polynomial: From the tree threshold down to the roots. In Proc. of SODA'18, pages 1557-1576, 2018. URL: https://doi.org/10.1137/1.9781611975031.102.
  31. Thomas P. Hayes and Cristopher Moore. Lower bounds on the critical density in the hard disk model via optimized metrics. CoRR, abs/1407.1930, 2014. URL: http://arxiv.org/abs/abs/1407.1930.
  32. Tyler Helmuth, Will Perkins, and Samantha Petti. Correlation decay for hard spheres via Markov chains. CoRR, abs/2001.05323, 2020. URL: http://arxiv.org/abs/2001.05323.
  33. Tyler Helmuth, Will Perkins, and Guus Regts. Algorithmic Pirogov-Sinai theory. In Proc. of STOC'19, pages 1009-1020, 2019. URL: https://doi.org/10.1145/3313276.3316305.
  34. Matthew Jenssen, Felix Joos, and Will Perkins. On the hard sphere model and sphere packings in high dimensions. Forum of Mathematics, Sigma, 7:e1, 2019. URL: https://doi.org/10.1017/fms.2018.25.
  35. Matthew Jenssen, Peter Keevash, and Will Perkins. Algorithms for #bis-hard problems on expander graphs. In Proc. of SODA'19, pages 2235-2247, 2019. URL: https://doi.org/10.1137/1.9781611975482.135.
  36. Ravi Kannan, Michael W. Mahoney, and Ravi Montenegro. Rapid mixing of several Markov chains for a hard-core model. In Proc. of ISAAC'03, pages 663-675, 2003. URL: https://doi.org/10.1007/978-3-540-24587-2_68.
  37. Wilfrid S. Kendall. Perfect simulation for the area-interaction point process. In Probability Towards 2000, pages 218-234. Springer, 1998. URL: https://doi.org/10.1007/978-1-4612-2224-8_13.
  38. Wilfrid S. Kendall and Jesper Møller. Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Advances in Applied Probability, 32(3):844–865, 2000. URL: https://doi.org/10.1239/aap/1013540247.
  39. Liang Li, Pinyan Lu, and Yitong Yin. Correlation decay up to uniqueness in spin systems. In Proc. of SODA'13, pages 67-84, 2013. URL: https://doi.org/10.1137/1.9781611973105.5.
  40. Fabio Martinelli. Lectures on Glauber dynamics for discrete spin models. In Lectures on probability theory and statistics, pages 93-191. Springer, 1999. URL: https://doi.org/10.1007/978-3-540-48115-7_2.
  41. Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N. Rosenbluth, Augusta H. Teller, and Edward Teller. Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21(6):1087-1092, 1953. URL: https://doi.org/10.1063/1.1699114.
  42. Marcus Michelen and Will Perkins. Analyticity for classical gasses via recursion. CoRR, abs/2008.00972, 2020. URL: http://arxiv.org/abs/2008.00972.
  43. Sarat Babu Moka, Sandeep Juneja, and Michel R. H. Mandjes. Analysis of perfect sampling methods for hard-sphere models. SIGMETRICS Performance Evaluation Review, 45(3):69–75, 2018. URL: https://doi.org/10.1145/3199524.3199536.
  44. Elchanan Mossel and Allan Sly. Gibbs rapidly samples colorings of g (n, d/n). Probability Theory and Related Fields, 148(1-2):37-69, 2010. URL: https://doi.org/10.1007/s00440-009-0222-x.
  45. 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.
  46. Viresh Patel and Guus Regts. Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials. SIAM Journal on Computing, 46(6):1893-1919, 2017. URL: https://doi.org/10.1137/16M1101003.
  47. Mathew D Penrose. Self-avoiding walks and trees in spread-out lattices. Journal of Statistical Physics, 77(1-2):3-15, 1994. URL: https://doi.org/10.1007/BF02186829.
  48. Ricardo Restrepo, Jinwoo Shin, Prasad Tetali, Eric Vigoda, and Linji Yang. Improved mixing condition on the grid for counting and sampling independent sets. Probability Theory and Related Fields, 156(1-2):75-99, 2013. URL: https://doi.org/10.1007/s00440-012-0421-8.
  49. Alistair Sinclair, Piyush Srivastava, Daniel Štefankovič, and Yitong Yin. Spatial mixing and the connective constant: Optimal bounds. Probability Theory and Related Fields, 168(1-2):153-197, 2017. URL: https://doi.org/10.1007/s00440-016-0708-2.
  50. Allan Sly. Computational transition at the uniqueness threshold. In Proc. of FOCS'10, pages 287-296, 2010. URL: https://doi.org/10.1109/FOCS.2010.34.
  51. J. van den Berg and R. Brouwer. Random sampling for the monomer–dimer model on a lattice. Journal of Mathematical Physics, 41(3):1585-1597, 2000. URL: https://doi.org/10.1063/1.533198.
  52. Juan C Vera, Eric Vigoda, and Linji Yang. Improved bounds on the phase transition for the hard-core model in 2-dimensions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 699-713. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-40328-6_48.
  53. Dror Weitz. Combinatorial criteria for uniqueness of Gibbs measures. Random Structures & Algorithms, 27(4):445-475, 2005. Google Scholar
  54. Dror Weitz. Counting independent sets up to the tree threshold. In Proc. of STOC'06, pages 140-149, 2006. URL: https://doi.org/10.1145/1132516.1132538.
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail