Perfect Sampling for Hard Spheres from Strong Spatial Mixing

Authors Konrad Anand, Andreas Göbel, Marcus Pappik, Will Perkins

Thumbnail PDF


  • Filesize: 0.76 MB
  • 18 pages

Document Identifiers

Author Details

Konrad Anand
  • Queen Mary, University of London, UK
Andreas Göbel
  • Hasso Plattner Institute, University of Potsdam, Germany
Marcus Pappik
  • Hasso Plattner Institute, University of Potsdam, Germany
Will Perkins
  • School of Computer Science, Georgia Institute of Technology, Atlanta, GA, USA


We thank Mark Jerrum for very helpful discussions on this topic.

Cite AsGet BibTex

Konrad Anand, Andreas Göbel, Marcus Pappik, and Will Perkins. Perfect Sampling for Hard Spheres from Strong Spatial Mixing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 38:1-38:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We provide a perfect sampling algorithm for the hard-sphere model on subsets of R^d with expected running time linear in the volume under the assumption of strong spatial mixing. A large number of perfect and approximate sampling algorithms have been devised to sample from the hard-sphere model, and our perfect sampling algorithm is efficient for a range of parameters for which only efficient approximate samplers were previously known and is faster than these known approximate approaches. Our methods also extend to the more general setting of Gibbs point processes interacting via finite-range, repulsive potentials.

Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
  • perfect sampling
  • hard-sphere model
  • Gibbs point processes


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


  1. Michael Aizenman and Richard Holley. Rapid convergence to equilibrium of stochastic Ising models in the Dobrushin Shlosman regime. Percolation theory and ergodic theory of infinite particle systems, pages 1-11, 1987. Google Scholar
  2. Berni Julian Alder and Thomas Everett Wainwright. Phase transition for a hard sphere system. The Journal of Chemical Physics, 27(5):1208-1209, 1957. Google Scholar
  3. Konrad Anand, Andreas Göbel, Marcus Pappik, and Will Perkins. Perfect sampling for hard spheres from strong spatial mixing. arXiv preprint, 2023. URL:
  4. Konrad Anand and Mark Jerrum. Perfect sampling in infinite spin systems via strong spatial mixing. SIAM Journal on Computing, 51(4):1280-1295, 2022. Google Scholar
  5. Søren Asmussen, Peter W Glynn, and Hermann Thorisson. Stationarity detection in the initial transient problem. ACM Transactions on Modeling and Computer Simulation (TOMACS), 2(2):130-157, 1992. Google Scholar
  6. Etienne P Bernard and Werner Krauth. Two-step melting in two dimensions: first-order liquid-hexatic transition. Physical Review Letters, 107(15):155704, 2011. Google Scholar
  7. Etienne P Bernard, Werner Krauth, and David B Wilson. Event-chain Monte Carlo algorithms for hard-sphere systems. Physical Review E, 80(5):056704, 2009. Google Scholar
  8. Steffen Betsch and Günter Last. On the uniqueness of Gibbs distributions with a non-negative and subcritical pair potential. In Annales de l'Institut Henri Poincare (B) Probabilites et statistiques, volume 59(2), pages 706-725. Institut Henri Poincaré, 2023. Google Scholar
  9. Siddharth Bhandari and Sayantan Chakraborty. Improved bounds for perfect sampling of k-colorings in graphs. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 631-642, 2020. Google Scholar
  10. Zongchen Chen, Kuikui Liu, Nitya Mani, and Ankur Moitra. Strong spatial mixing for colorings on trees and its algorithmic applications. arXiv preprint, 2023. URL:
  11. Hofer Temmel Christoph. Disagreement percolation for the hard-sphere model. Electronic Journal of Probability, 24:1-22, 2019. Google Scholar
  12. David Dereudre. Introduction to the theory of Gibbs point processes. In Stochastic Geometry, pages 181-229. Springer, 2019. Google Scholar
  13. Persi Diaconis. The Markov Chain Monte Carlo revolution. Bulletin of the American Mathematical Society, 46(2):179-205, 2009. Google Scholar
  14. Shaddin Dughmi, Jason Hartline, Robert D Kleinberg, and Rad Niazadeh. Bernoulli factories and black-box reductions in mechanism design. Journal of the ACM (JACM), 68(2):1-30, 2021. Google Scholar
  15. Martin Dyer, Alistair Sinclair, Eric Vigoda, and Dror Weitz. Mixing in time and space for lattice spin systems: A combinatorial view. Random Structures & Algorithms, 24(4):461-479, 2004. Google Scholar
  16. Michael Engel, Joshua A Anderson, Sharon C Glotzer, Masaharu Isobe, Etienne P Bernard, and Werner Krauth. Hard-disk equation of state: First-order liquid-hexatic transition in two dimensions with three simulation methods. Physical Review E, 87(4):042134, 2013. Google Scholar
  17. Stefan Felsner and Lorenz Wernisch. Markov chains for linear extensions, the two-dimensional case. In SODA, pages 239-247, 1997. Google Scholar
  18. Weiming Feng, Heng Guo, and Yitong Yin. Perfect sampling from spatial mixing. Random Structures & Algorithms, 61(4):678-709, 2022. Google Scholar
  19. Weiming Feng and Yitong Yin. On local distributed sampling and counting. In Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, pages 189-198, 2018. Google Scholar
  20. Roberto Fernández, Aldo Procacci, and Benedetto Scoppola. The analyticity region of the hard sphere gas. Improved bounds. J. Stat. Phys., 5:1139-1143, 2007. Google Scholar
  21. Pablo A Ferrari, Roberto Fernández, and Nancy L Garcia. Perfect simulation for interacting point processes, loss networks and Ising models. Stochastic Processes and their Applications, 102(1):63-88, 2002. Google Scholar
  22. David Gamarnik, Dmitriy Katz, and Sidhant Misra. Strong spatial mixing of list coloring of graphs. Random Structures & Algorithms, 46(4):599-613, 2015. Google Scholar
  23. Nancy L Garcia. Perfect simulation of spatial processes. Resenhas do Instituto de Matemática e Estatística da Universidade de São Paulo, 4(3):283-325, 2000. Google Scholar
  24. J Groeneveld. Two theorems on classical many-particle systems. Phys. Letters, 3, 1962. Google Scholar
  25. Heng Guo and Mark Jerrum. Perfect simulation of the hard disks model by partial rejection sampling. Annales de l’Institut Henri Poincaré D, 8(2):159-177, 2021. Google Scholar
  26. Heng Guo, Mark Jerrum, and Jingcheng Liu. Uniform sampling through the Lovász local lemma. Journal of the ACM (JACM), 66(3):1-31, 2019. Google Scholar
  27. Olle Häggström and Karin Nelander. Exact sampling from anti-monotone systems. Statistica Neerlandica, 52(3):360-380, 1998. Google Scholar
  28. Olle Häggström, Marie-Colette N.M. van Lieshout, and Jesper Møller. Characterization results and Markov chain Monte Carlo algorithms including exact simulation for some spatial point processes. Bernoulli, 5(4):641-658, 1999. Google Scholar
  29. Thomas P Hayes and Cristopher Moore. Lower bounds on the critical density in the hard disk model via optimized metrics. arXiv preprint, 2014. URL:
  30. Kun He, Xiaoming Sun, and Kewen Wu. Perfect sampling for (atomic) Lovász Local Lemma. arXiv preprint, 2021. URL:
  31. Kun He, Chunyang Wang, and Yitong Yin. Sampling Lovász Local Lemma for general constraint satisfaction solutions in near-linear time. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 147-158. IEEE, 2022. Google Scholar
  32. Kun He, Kewen Wu, and Kuan Yang. Improved bounds for sampling solutions of random CNF formulas. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3330-3361. SIAM, 2023. Google Scholar
  33. Tyler Helmuth, Will Perkins, and Samantha Petti. Correlation decay for hard spheres via Markov chains. The Annals of Applied Probability, 32(3):2063-2082, 2022. Google Scholar
  34. Christoph Hofer-Temmel and Pierre Houdebert. Disagreement percolation for Gibbs ball models. Stochastic Processes and their Applications, 129(10):3922-3940, 2019. Google Scholar
  35. Richard Holley. Possible rates of convergence in finite range, attractive spin systems. Part. Syst. Random Media Large Deviat., 41:215, 1985. Google Scholar
  36. Mark Huber. Spatial birth-death swap chains. Bernoulli, 18(3):1031-1041, 2012. Google Scholar
  37. Mark Huber. Nearly optimal Bernoulli factories for linear functions. Combin. Probab. Comput., 25(4):577-591, 2016. Google Scholar
  38. Mark Huber, Elise Villella, Daniel Rozenfeld, and Jason Xu. Bounds on the artificial phase transition for perfect simulation of hard core Gibbs processes. Involve, a Journal of Mathematics, 5(3):247-255, 2013. Google Scholar
  39. Masaharu Isobe. Hard sphere simulation in statistical physics—methodologies and applications. Molecular Simulation, 42(16):1317-1329, 2016. Google Scholar
  40. Vishesh Jain, Ashwin Sah, and Mehtaab Sawhney. Perfectly sampling k ≥ (8/3+ o (1)) Δ-colorings in graphs. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 1589-1600, 2021. Google Scholar
  41. Matthew Jenssen, Marcus Michelen, and Mohan Ravichandran. Quasipolynomial-time algorithms for repulsive Gibbs point processes. arXiv preprint, 2022. URL:
  42. Mark Jerrum and Alistair Sinclair. The Markov chain Monte Carlo method: an approach to approximate counting and integration. Approximation algorithms for NP-hard problems, pages 482-520, 1996. Google Scholar
  43. Ravi Kannan, Michael W. Mahoney, and Ravi Montenegro. Rapid mixing of several Markov chains for a hard-core model. In Algorithms and computation, volume 2906 of Lecture Notes in Comput. Sci., pages 663-675. Springer, Berlin, 2003. Google Scholar
  44. Frank P Kelly and Brian D Ripley. A note on Strauss’s model for clustering. Biometrika, pages 357-360, 1976. Google Scholar
  45. Wilfrid S Kendall. Perfect simulation for the area-interaction point process. In Probability towards 2000, pages 218-234. Springer, 1998. Google Scholar
  46. 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, pages 844-865, 2000. Google Scholar
  47. Botao Li, Yoshihiko Nishikawa, Philipp Höllmer, Louis Carillo, AC Maggs, and Werner Krauth. Hard-disk pressure computations - A historic perspective. The Journal of Chemical Physics, 157(23):234111, 2022. Google Scholar
  48. Jingcheng Liu, Alistair Sinclair, and Piyush Srivastava. Correlation decay and partition function zeros: Algorithms and phase transitions. SIAM Journal on Computing, pages FOCS19-200, 2022. Google Scholar
  49. Laszlo Lovasz and Peter Winkler. Exact mixing in an unknown Markov chain. The Electronic Journal of Combinatorics, pages R15-R15, 1995. Google Scholar
  50. Hartmut Löwen. Fun with hard spheres. In Statistical physics and spatial statistics, volume 554, pages 295-331. Springer, 2000. Google Scholar
  51. Pinyan Lu and Yitong Yin. Improved FPTAS for multi-spin systems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques: 16th International Workshop, APPROX 2013, and 17th International Workshop, RANDOM 2013, Berkeley, CA, USA, August 21-23, 2013. Proceedings, pages 639-654. Springer, 2013. Google Scholar
  52. Fabio Martinelli. Lectures on Glauber dynamics for discrete spin models. In Lectures on probability theory and statistics, pages 93-191. Springer, 1999. Google Scholar
  53. 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. Google Scholar
  54. Marcus Michelen and Will Perkins. Potential-weighted connective constants and uniqueness of Gibbs measures. arXiv preprint, 2021. URL:
  55. Marcus Michelen and Will Perkins. Analyticity for classical gasses via recursion. Communications in Mathematical Physics, pages 1-22, 2022. Google Scholar
  56. Marcus Michelen and Will Perkins. Strong spatial mixing for repulsive point processes. Journal of Statistical Physics, 189(1):9, 2022. Google Scholar
  57. Sarat B Moka, Dirk P Kroese, et al. Perfect sampling for Gibbs point processes using partial rejection sampling. Bernoulli, 26(3):2082-2104, 2020. Google Scholar
  58. Jesper Møller. A review of perfect simulation in stochastic geometry. Lecture Notes-Monograph Series, pages 333-355, 2001. Google Scholar
  59. Jesper Møller and Rasmus Plenge Waagepetersen. Statistical inference and simulation for spatial point processes. CRC Press, 2003. Google Scholar
  60. Duncan J Murdoch and Peter J Green. Exact sampling from a continuous state space. Scandinavian Journal of Statistics, 25(3):483-502, 1998. Google Scholar
  61. Serban Nacu and Yuval Peres. Fast simulation of new coins from old. The Annals of Applied Probability, 15(1A):93-115, 2005. Google Scholar
  62. Oliver Penrose. Convergence of fugacity expansions for fluids and lattice gases. Journal of Mathematical Physics, 4(10):1312-1320, 1963. Google Scholar
  63. James Gary Propp and David Bruce Wilson. Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures & Algorithms, 9(1-2):223-252, 1996. Google Scholar
  64. James Gary Propp and David Bruce Wilson. How to get a perfectly random sample from a generic Markov chain and generate a random spanning tree of a directed graph. Journal of Algorithms, 27(2):170-217, 1998. Google Scholar
  65. Dana Randall. Rapidly mixing Markov chains with applications in computer science and physics. Computing in Science & Engineering, 8(2):30-41, 2006. Google Scholar
  66. Guus Regts. Absence of zeros implies strong spatial mixing. Probability Theory and Related Fields, pages 1-21, 2023. Google Scholar
  67. David Ruelle. Correlation functions of classical gases. Annals of Physics, 25:109-120, 1963. Google Scholar
  68. David Ruelle. Statistical mechanics: Rigorous results. World Scientific, 1999. Google Scholar
  69. 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. Google Scholar
  70. Yinon Spinka. Finitary codings for spatial mixing Markov random fields. Ann. Probab., 48(3):1557-1591, 2020. Google Scholar
  71. David J Strauss. A model for clustering. Biometrika, 62(2):467-475, 1975. Google Scholar
  72. Daniel W Stroock and Boguslaw Zegarlinski. The logarithmic Sobolev inequality for discrete spin systems on a lattice. Communications in Mathematical Physics, 149(1):175-193, 1992. Google Scholar
  73. Marie-Colette N.M. van Lieshout. Markov Point Processes and Their Applications. Imperial College Press, 2000. Google Scholar
  74. Dror Weitz. Counting independent sets up to the tree threshold. In Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, STOC 2006, pages 140-149. ACM, 2006. Google Scholar
  75. William W. Wood, F. Raymond Parker, and Jack David Jacobson. Recent monte carlo calculations of the equation of state of lenard-jones and hard sphere molecules. Il Nuovo Cimento (1955-1965), 9:133-143, 1958. Google Scholar
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