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Chapitre D'ouvrage Année : 2020

On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures

Résumé

This note, mostly expository, is devoted to Poincaré and log-Sobolev inequalities for a class of Boltzmann-Gibbs measures with singular interaction. Such measures allow to model one-dimensional particles with confinement and singular pair interaction. The functional inequalities come from convexity. We prove and characterize optimality in the case of quadratic confinement via a factorization of the measure. This optimality phenomenon holds for all beta Hermite ensembles including the Gaussian 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 admitting the Hermite-Lassalle orthogonal polynomials as a complete set of eigenfunctions. We also discuss the consequence of the log-Sobolev inequality in terms of concentration of measure for Lipschitz functions such as maxima and linear statistics.
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Dates et versions

hal-01781502 , version 1 (30-04-2018)
hal-01781502 , version 2 (05-06-2018)
hal-01781502 , version 3 (15-11-2019)

Identifiants

Citer

Djalil Chafai, Joseph Lehec. On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures. Geometric aspects of functional analysis. Israel seminar (GAFA) 2017–2019. Volume 1., pp.219-246, 2020, Lecture Notes in Mathematics 2256, 978-3-030-36019-1. ⟨10.1007/978-3-030-36020-7_10⟩. ⟨hal-01781502v3⟩
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