Résumé en anglais

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.