Simulating Coulomb gases and log-gases with hybrid Monte Carlo algorithms

TL;DR

This paper introduces an efficient hybrid Monte Carlo method for simulating Coulomb and log-gases, demonstrating superior performance and enabling new insights into their statistical properties, especially in large particle regimes.

Contribution

It presents a novel hybrid Monte Carlo algorithm tailored for Coulomb and log-gases, improving simulation efficiency over existing methods and facilitating new numerical investigations.

Findings

The hybrid algorithm exhibits excellent numerical stability despite singular interactions.

It outperforms the overdamped version in efficiency for large particle numbers.

Numerical experiments suggest a universality of Gumbel fluctuations at the edge of beta Ginibre ensembles.

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.