Concentration for Coulomb gases and Coulomb transport inequalities

Abstract

We study the non-asymptotic behavior of Coulomb gases in dimension two and more. Such gases are modeled by an exchangeable Boltzmann-Gibbs measure with a singular two-body interaction. We obtain concentration of measure inequalities for the empirical distribution of such gases around their equilibrium measure, with respect to bounded Lipschitz and Wasserstein distances. This implies macroscopic as well as mesoscopic convergence in such distances. In particular, we improve the concentration inequalities known for the empirical spectral distribution of Ginibre random matrices. Our approach is remarkably simple and bypasses the use of renormalized energy. It crucially relies on new inequalities between probability metrics, including Coulomb transport inequalities which can be of independent interest. Our work is inspired by the one of Ma{ï}da and Maurel-Segala, itself inspired by large deviations techniques. Our approach allows to recover, extend, and simplify previous results by Rougerie and Serfaty.

arXiv commentImprovement on an assumption, and minor modifications

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Cited by in corpus (18)

• A non-Hermitian generalisation of the Marchenko-Pastur distribution: from the circular law to multi-criticality
• The high temperature crossover for general 2D Coulomb gases
• Local incompressibility estimates for the Laughlin phase
• Dynamics of a planar Coulomb gas
• A localization theorem for the planar Coulomb gas in an external field
• Concentration for Coulomb gases on compact manifolds
• Simulating Coulomb gases and log-gases with hybrid Monte Carlo algorithms
• Rate of Convergence to the Circular Law via Smoothing Inequalities for Log-Potentials
• Zeros of random polynomials and its higher derivatives
• Macroscopic and edge behavior of a planar jellium
• Incomplete determinantal processes: from random matrix to Poisson statistics
• Coulomb gases under constraint: some theoretical and numerical results
• The Dyson and Coulomb games
• Lemniscate ensembles with spectral singularity
• Asymptotic properties of short-range interaction functionals
• Large Deviation Principle for local empirical measure of Coulomb gases at intermediate temperature regime
• Transport inequalities for random point measures
• On existence of measure with given marginals supported on a hyperplane