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


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.


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 



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.


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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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