On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures
Chafaï, Djalil; Lehec, Joseph (2018), On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures. https://basepub.dauphine.fr/handle/123456789/17979
TypeDocument de travail / Working paper
External document linkhttps://hal.archives-ouvertes.fr/hal-01781502
Series titleCahier de recherche CEREMADE, Université Paris-Dauphine
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Abstract (EN)This note, mostly expository, is devoted to Poincaré and logarithmic Sobolev inequalities for a class of singular Boltzmann-Gibbs measures. Such measures allow to model one-dimensional particles with confinement and singular pair interaction. The functional inequalities come from the convexity of confinement and interaction. We prove optimality in the case of quadratic confinement by using a factorization of the measure. This optimality phenomenon holds for all beta Hermite ensembles including the Gauss-ian unitary ensemble, a famous exactly solvable model of random matrix theory. We further explore exact solvability by reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics which admits the Hermite-Lassalle orthogonal polynomials as a complete set of eigenfunctions. We also discuss the McKean-Vlasov mean-field limit of the dynamics, as well as the consequence of the logarithmic Sobolev inequality in terms of concentration of measure for Lipschitz functions such as maxima and linear statistics.
Subjects / KeywordsBoltzmann–Gibbs measure; Gaussian unitary ensemble; Random matrix theory; Spectral analysis; Geometric functional analysis; Log-concave measure; Poincaré inequality; Logarithmic Sobolev inequality; Concentration of measure; Diffusion operator; Orthogonal polynomials
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