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

Abstract

This note, mostly expository, is devoted to Poincar{é} 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.

Minor improvements. To appear in Geometric Aspects of Functional Analysis -- Israel Seminar (GAFA) 2017-2019", Lecture Notes in Mathematics 2256

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Cited by in corpus (4)

• Universal cutoff for Dyson Ornstein Uhlenbeck process
• Coulomb gases under constraint: some theoretical and numerical results
• On cutoff via rigidity for high dimensional curved diffusions
• Aspects of Coulomb gases