Concentration for Coulomb gases and Coulomb transport inequalities
Copyright policy )Concentration for Coulomb gases and Coulomb transport inequalities
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.
Improvement on an assumption, and minor modifications
[MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Random matrices (algebraic aspects), FOS: Physical sciences, Set functions and measures on topological spaces (regularity of measures, etc.), Ginibre ensemble, 510, Measures and integrals in product spaces, astrophysique), FOS: Mathematics, Inequalities; stochastic orderings, Wasserstein distance, [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph], Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics, Mathematical Physics, Coulomb gas, Probability (math.PR), [MATH.MATH-FA] Mathematics [math]/Functional Analysis [math.FA], Kinetic theory of gases in time-dependent statistical mechanics, Inequalities involving derivatives and differential and integral operators, Random matrix, Mathematical Physics (math-ph), Talagrand inequality, 520, concentration of measure, Statistical mechanics of gases, Functional Analysis (math.FA), Mathematics - Functional Analysis, Transport of measure, Kantorovich distance, Sciences connexes (physique, Transport inequality, Concentration of measure, Mathematics - Probability
[MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Random matrices (algebraic aspects), FOS: Physical sciences, Set functions and measures on topological spaces (regularity of measures, etc.), Ginibre ensemble, 510, Measures and integrals in product spaces, astrophysique), FOS: Mathematics, Inequalities; stochastic orderings, Wasserstein distance, [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph], Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics, Mathematical Physics, Coulomb gas, Probability (math.PR), [MATH.MATH-FA] Mathematics [math]/Functional Analysis [math.FA], Kinetic theory of gases in time-dependent statistical mechanics, Inequalities involving derivatives and differential and integral operators, Random matrix, Mathematical Physics (math-ph), Talagrand inequality, 520, concentration of measure, Statistical mechanics of gases, Functional Analysis (math.FA), Mathematics - Functional Analysis, Transport of measure, Kantorovich distance, Sciences connexes (physique, Transport inequality, Concentration of measure, Mathematics - Probability
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