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

TL;DR

This paper explores Poincaré and log-Sobolev inequalities for singular Gibbs measures, highlighting their optimality, connections to random matrix models, and implications for measure concentration.

Contribution

It provides a characterization of optimality for these inequalities in quadratic confinement and relates them to Hermite ensembles and Dyson-Ornstein-Uhlenbeck dynamics.

Findings

Optimality of inequalities for quadratic confinement

Connection to Hermite ensembles and random matrix theory

Implications for measure concentration and Lipschitz functions

Abstract

This note, mostly expository, is devoted to Poincar{\'e} 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.