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Journal of Statistical Physics

, Volume 174, Issue 3, pp 692–714 | Cite as

Simulating Coulomb and Log-Gases with Hybrid Monte Carlo Algorithms

  • Djalil ChafaïEmail author
  • Grégoire Ferré
Article

Abstract

Coulomb and log-gases are exchangeable singular Boltzmann–Gibbs measures appearing in mathematical physics at many places, in particular in random matrix theory. We explore experimentally an efficient numerical method for simulating such gases. It is an instance of the Hybrid or Hamiltonian Monte Carlo algorithm, in other words a Metropolis–Hastings algorithm with proposals produced by a kinetic or underdamped Langevin dynamics. This algorithm has excellent numerical behavior despite the singular interaction, in particular when the number of particles gets large. It is more efficient than the well known overdamped version previously used for such problems, and allows new numerical explorations. It suggests for instance to conjecture a universality of the Gumbel fluctuation at the edge of beta Ginibre ensembles for all beta.

Keywords

Numerical simulation Random number generator Singular Stochastic differential equation Coulomb gas Monte Carlo adjusted Langevin Hybrid Monte Carlo Markov chain Monte Carlo Langevin dynamics Kinetic equation 

Mathematics Subject Classification

65C05 (Primary) 82C22 60G57 

Notes

Acknowledgements

We warmly thank Gabriel Stoltz for his encouragements and for very useful discussions on the theoretical and numerical sides of this work. We are also grateful to Thomas Leblé and Laure Dumaz for their comments on the first version.

References

  1. 1.
    Anderson, G.W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, vol. 118. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  2. 2.
    Bardenet, R., Hardy, A.: Monte Carlo with determinantal point processes. arXiv:1605.00361v1 (2016)
  3. 3.
    Berman, R.J.: On large deviations for Gibbs measures, mean energy and Gamma-convergence. arXiv:1610.08219v1 (2016)
  4. 4.
    Beskos, A., Pillai, N., Roberts, G., Sanz-Serna, J.-M., Stuart, A.: Optimal tuning of the hybrid Monte Carlo algorithm. Bernoulli 19(5A), 1501–1534 (2013)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Blanc, X., Lewin, M.: The crystallization conjecture: a review. EMS Surv. Math. Sci. 2(2), 225–306 (2015)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bolley, F., Chafï, D., Fontbona, J.: Dynamics of a planar Coulomb gas. Ann. Appl. Probab. arXiv:1706.08776v3 (2017)
  7. 7.
    Bou-Rabee, N., Eberle, A., Zimmer, R.: Coupling and convergence for Hamiltonian Monte Carlo. arXiv:1805.00452v1 (2018)
  8. 8.
    Bou-Rabee, N., Sanz-Serna, J.M.: Geometric integrators and the Hamiltonian Monte Carlo method. arXiv:1711.05337v1 (2017)
  9. 9.
    Bouchard-Côté, A., Vollmer, S.J., Doucet, A.: The bouncy particle sampler: a nonreversible rejection-free Markov chain Monte Carlo method. J. Am. Stat. Assoc. 113(522), 855–867 (2018)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Brooks, S., Gelman, A., Jones, G.L., Meng, X.L. (eds.): Handbook of Markov Chain Monte Carlo. Chapman Hall/CRC Handbooks of Modern Statistical Methods. CRC Press, Boca Raton (2011)Google Scholar
  11. 11.
    Brosse, N., Durmus, A., Moulines, É., Sabanis , S.: The tamed unadjusted Langevin algorithm arXiv:1710.05559v2 (2017)
  12. 12.
    Chafaï, D., Hardy, A., Maïda, M.: (2018) Concentration for Coulomb gases and Coulomb transport inequalities. J. Funct. Anal. arXiv:1610.00980v3
  13. 13.
    Chafaï, D., Lehec, J.: On Poincaré and logarithmic Sobolev inequalities for a class of singular Gibbs measures. arXiv:1805.00708v2 (2018)
  14. 14.
    Chafaï, D., Gozlan, N., Zitt, P.-A.: First-order global asymptotics for confined particles with singular pair repulsion. Ann. Appl. Probab. 24(6), 2371–2413 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Chafaï, D., Péché, S.: A note on the second order universality at the edge of Coulomb gases on the plane. J. Stat. Phys. 156(2), 368–383 (2014)ADSMathSciNetzbMATHGoogle Scholar
  16. 16.
    Chafaï, D., Saff, E.: Aspects of an Euclidean log-gas. Work in progress (2018)Google Scholar
  17. 17.
    Dalalyan, A., Riou-Durand, L.: On sampling from a log-concave density using kinetic Langevin diffusions. arXiv:1807.09382v1 (2018)
  18. 18.
    Decreusefond, L., Flint, I., Vergne, A.: Vergne: a note on the simulation of the Ginibre point process. J. Appl. Probab. 52(4), 1003–1012 (2015)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Duane, S., Kennedy, A., Pendleton, B.J., Roweth, D.: Hybrid Monte Carlo. Phys. Lett. B 195(2), 216–222 (1987)ADSGoogle Scholar
  20. 20.
    Dubach, G.: Powers of Ginibre Eigenvalues. arXiv:1711.03151v2 (2017)
  21. 21.
    Dumitriu, I., Edelman, A.: Matrix models for beta ensembles. J. Math. Phys. 43(11), 5830–5847 (2002)ADSMathSciNetzbMATHGoogle Scholar
  22. 22.
    Duncan, A.B., Lelièvre, T., Pavliotis, G.A.: Variance reduction using nonreversible Langevin samplers. J. Stat. Phys. 163, 457–491 (2016)ADSMathSciNetzbMATHGoogle Scholar
  23. 23.
    Durmus, A., Moulines, E., Saksman, E.: On the convergence of Hamiltonian Monte Carlo. arXiv:1705.00166v1 (2017)
  24. 24.
    Edelman, A., Rao, N.R.: Random matrix theory. Acta Numer. 14, 233–297 (2005)ADSMathSciNetzbMATHGoogle Scholar
  25. 25.
    Erdős, L., Yau, H.-T.: A dynamical approach to random matrix theory. Courant Lecture Notes in Mathematics, vol. 28. American Mathematical Society, Providence, RI (2017)Google Scholar
  26. 26.
    Ezawa, Z.E.: Quantum Hall Effects. Field Theoretical Approach and Related Topics, 2nd edn. World Scientific Publishing Co. Pt. Ltd., Hackensack, NJ (2008)zbMATHGoogle Scholar
  27. 27.
    Fathi, M., Homman, A.-A., Stoltz, G.: Error analysis of the transport properties of Metropolized schemes. ESAIM Proc. Surv. 48, 341–363 (2015)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Forrester, P .J.: Log-Gases and Random Matrices. London Mathematical Society Monographs Series, vol. 34. Princeton University Press, Princeton, NJ (2010)zbMATHGoogle Scholar
  29. 29.
    Forrester, P.J.: Analogies between random matrix ensembles and the one-component plasma in two-dimensions. Nuclear Phys. B 904, 253–281 (2016)ADSMathSciNetzbMATHGoogle Scholar
  30. 30.
    García-Zelada, D.: A large deviation principle for empirical measures on Polish spaces: application to singular Gibbs measures on manifolds. arXiv:1703.02680v2 (2017)
  31. 31.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, vol. 31. Springer Science Business Media, Berlin (2006)zbMATHGoogle Scholar
  32. 32.
    Hardy, A.: Polynomial ensembles and recurrence coefficients. arXiv:1709.01287v1 (2017)
  33. 33.
    Helms, L.L.: Potential Theory, 2nd edn. Springer, London (2014)zbMATHGoogle Scholar
  34. 34.
    Hoffman, M.D., Gelman, A.: The no-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo. J. Mach. Learn. Res. 15, 1593–1623 (2014)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Höft, T.A., Alpert, B.K.: Fast updating multipole Coulombic potential calculation. SIAM J. Sci. Comput. 39(3), A1038–A1061 (2017)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Horowitz, A.M.: A generalized guided Monte Carlo algorithm. Phys. Lett. B 268, 247–252 (1991)ADSGoogle Scholar
  37. 37.
    Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Determinantal processes and independence. Probab. Surv. 3, 206–229 (2006)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Hutzenthaler, M., Jentzen, A., Kloeden, P.E.: Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. Ann. Appl. Probab. 22(4), 1611–1641 (2012)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Jiang, T., Qi, Y.: Spectral radii of large non-Hermitian random matrices. J. Theoret. Probab. 30(1), 326–364 (2017)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Jones, A., Leimkuhler, B.: Adaptive stochastic methods for sampling driven molecular systems. J. Chem. Phys. 135(8), 084125 (2011)ADSGoogle Scholar
  41. 41.
    Kapfer, S.C., Krauth, W.: Cell-veto Monte Carlo algorithm for long-range systems. Phys. Rev. E 94, 031302 (2016)ADSGoogle Scholar
  42. 42.
    Kloeden, P.E., Platen, E., Schurz, H.: Numerical Solution of SDE Through Computer Experiments. Universitext. Springer, Berlin (1994)zbMATHGoogle Scholar
  43. 43.
    Krishnapur, M., Rider, B., Virág, B.: Universality of the stochastic Airy operator. Commun. Pure Appl. Math. 69(1), 145–199 (2016)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Landkof, N.S.: Foundations of Modern Potential Theory. Springer, New York (1972) (Translated from the Russian by A, p. 180. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band)Google Scholar
  45. 45.
    Lavancier, F., Møller, J., Rubak, E.: Determinantal point process models and statistical inference. J. R. Stat. Soc. Ser. B. Stat. Methodol 77(4), 853–877 (2015)MathSciNetGoogle Scholar
  46. 46.
    Ledoux, M.: Differential operators and spectral distributions of invariant ensembles from the classical orthogonal polynomials. The continuous case. Electron. J. Probab 9(7), 177–208 (2004)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Lee, Y.T., Vempala, S.S.: Convergence rate of Riemannian Hamiltonian Monte Carlo and faster polytope volume computation. arXiv:1710.06261v1 (2017)
  48. 48.
    Leimkuhler, B., Matthews, C., Stoltz, G.: The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics. IMA J. Numer. Anal. 36(1), 13–79 (2015)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Lelièvre, T., Nier, F., Pavliotis, G.A.: Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion. J. Stat. Phys. 152, 237–274 (2013)ADSMathSciNetzbMATHGoogle Scholar
  50. 50.
    Lelièvre, T., Rousset, M., Stoltz, G.: Free Energy Computations. A Mathematical Perspective. Imperial College Press, London (2010)zbMATHGoogle Scholar
  51. 51.
    Lelièvre, T., Rousset, M., Stoltz, G.: Langevin dynamics with constraints and computation of free energy differences. Math. Comput. 81(280), 2071–2125 (2012)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Lelièvre, T., Stoltz, G.: Partial differential equations and stochastic methods in molecular dynamics. Acta Numer. 25, 681–880 (2016)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Levesque, D., Verlet, L.: On the theory of classical fluids II. Physica 28(11), 1124–1142 (1962)ADSMathSciNetzbMATHGoogle Scholar
  54. 54.
    Levesque, D., Verlet, L.: Computer experiments on classical fluids. III. Time-dependent self-correlation functions. Phys. Rev. A 2, 2514 (1970)ADSGoogle Scholar
  55. 55.
    Li, X.H., Menon, G.: Numerical solution of Dyson Brownian motion and a sampling scheme for invariant matrix ensembles. J. Stat. Phys. 153(5), 801–812 (2013)ADSMathSciNetGoogle Scholar
  56. 56.
    Mattingly, J., Stuart, A., Higham, D.: Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stoch. Proc. Appl. 101(2), 185–232 (2002)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Mehta, M.L.: Random Matrices, 3rd edn. Pure and Applied Mathematics (Amsterdam), vol. 142. Elsevier/Academic Press, Amsterdam (2004)Google Scholar
  58. 58.
    Milstein, G .N., Tretyakov, M .V.: Stochastic Numerics for Mathematical Physics. Springer Science Business Media, Berlin (2013)zbMATHGoogle Scholar
  59. 59.
    Olver, S., Nadakuditi, R .R., Trogdon, T.: Sampling unitary ensembles. Random Matrices Theory Appl. 4(1), 1550002–22 (2015)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Rider, B.: A limit theorem at the edge of a non-Hermitian random matrix ensemble. Random matrix theory. J. Phys. A 36(12), 3401–3409 (2003)ADSMathSciNetzbMATHGoogle Scholar
  61. 61.
    Robert, C.P., Casella, G.: Monte Carlo Statistical Methods. Springer Texts in Statistics, 2nd edn. Springer, New York (2004)zbMATHGoogle Scholar
  62. 62.
    Roberts, G.O., Rosenthal, J.S.: Optimal scaling for various Metropolis-Hastings algorithms. Stat. Sci. 16(4), 351–367 (2001)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Roberts, G.O., Tweedie, R.L., et al.: Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli 2(4), 341–363 (1996)MathSciNetzbMATHGoogle Scholar
  64. 64.
    Rossky, P .H., Doll, J .D., Friedman, H .L.: Brownian dynamics as smart Monte Carlo simulation. J. Chem. Phys. 69, 4628 (1978)ADSGoogle Scholar
  65. 65.
    Saff, E.B, Totik, V.: Logarithmic Potentials with External Fields. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316, Springer, Berlin (1997) (Appendix B by Thomas Bloom)Google Scholar
  66. 66.
    Scardicchio, A., Zachary, C .E., Torquato, S.: Statistical properties of determinantal point processes in high-dimensional Euclidean spaces. Phys. Rev. E (3) 79(4), 041108 (2009). 19ADSMathSciNetGoogle Scholar
  67. 67.
    Serfaty, S.: Coulomb Gases and Ginzburg-Landau Vortices. Zurich Lectures in Advanced Mathematics. Euro. Math. Soc. (EMS), Zürich (2015)Google Scholar
  68. 68.
    Serfaty, S.: Systems of points with Coulomb interactions. arXiv:1712.04095v1 (2017)
  69. 69.
    Smale, S.: Mathematical problems for the next century. Math. Intell. 20(2), 7–15 (1998)MathSciNetzbMATHGoogle Scholar
  70. 70.
    Smale, S.: Mathematical problems for the next century. In: Arnold, V., Atiyah, M., Lax, P., Mazur, B. (eds.) Mathematics: Frontiers and Perspectives, pp. 271–294. Am. Math. Soc., Providence, RI (2000)Google Scholar
  71. 71.
    Stoltz, G., Trstanova, Z.: Stable and accurate schemes for Langevin dynamics with general kinetic energies. arXiv:1609.02891v1 (2016)
  72. 72.
    Vanetti, P., Bouchard-Côté, A., Deligiannidis, G., Doucet, A.: Piecewise-deterministic Markov Chain Monte Carlo. arXiv:1707.05296v1 (2018)
  73. 73.
    Verlet, L.: Computer experiments on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Phys. Rev. 159(98), 9 (1967)ADSGoogle Scholar

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Authors and Affiliations

  1. 1.Université Paris-Dauphine, PSL, CNRS, CEREMADEParisFrance
  2. 2.Université Paris-Est, CERMICS (ENPC), INRIAMarne-la-ValléeFrance

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