Dynamics of a planar Coulomb gas

Abstract

We study the long-time behavior of the dynamics of interacting planar Brow-nian particles, confined by an external field and subject to a singular pair repulsion. The invariant law is an exchangeable Boltzmann -- Gibbs measure. For a special inverse temperature, it matches the Coulomb gas known as the complex Ginibre ensemble. The difficulty comes from the interaction which is not convex, in contrast with the case of one-dimensional log-gases associated with the Dyson Brownian Motion. Despite the fact that the invariant law is neither product nor log-concave, we show that the system is well-posed for any inverse temperature and that Poincar{é} inequalities are available. Moreover the second moment dynamics turns out to be a nice Cox -- Ingersoll -- Ross process in which the dependency over the number of particles leads to identify two natural regimes related to the behavior of the noise and the speed of the dynamics.

arXiv commentMinor revision for Annals of Applied Probability

References in corpus (9)

• Around the circular law
• Invariant $\beta$-ensembles and the Gauss-Wigner crossover
• First order global asymptotics for confined particles with singular pair repulsion
• Propagation of chaos for the 2D viscous vortex model
• The distribution of overlaps between eigenvectors of Ginibre matrices
• Concentration for Coulomb gases and Coulomb transport inequalities
• Dynamics of a planar Coulomb gas
• Generalized Dyson Brownian motion, McKean-Vlasov equation and eigenvalues of random matrices
• Ornstein-Uhlenbeck diffusion of hermitian and non-hermitian matrices - unexpected links

Cited by in corpus (12)

• Mean Field Limit for Coulomb-Type Flows
• The distribution of overlaps between eigenvectors of Ginibre matrices
• On the geometry of Stein variational gradient descent
• Geometric ergodicity of Langevin dynamics with Coulomb interactions
• The high temperature crossover for general 2D Coulomb gases
• Full Dysonian dynamics of the complex Ginibre ensemble
• Dynamics of a planar Coulomb gas
• Simulating Coulomb gases and log-gases with hybrid Monte Carlo algorithms
• On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures
• Universal cutoff for Dyson Ornstein Uhlenbeck process
• The Dyson and Coulomb games
• A Spectral Dominance Approach to Large Random Matrices