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On Poincaré and Logarithmic Sobolev Inequalities for a Class of Singular Gibbs Measures

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Geometric Aspects of Functional Analysis

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2256))

Abstract

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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Acknowledgements

This work is linked with the French research project ANR-17-CE40-0030 - EFI - Entropy, flows, inequalities. A significant part was carried out during a stay at the Institute for Computational and Experimental Research in Mathematics (ICERM), during the 2018 Semester Program on “Point Configurations in Geometry, Physics and Computer Science”, thanks to the kind invitation by Edward Saff and Sylvia Serfaty. We thank also Sergio Andraus, François Bolley, Nizar Demni, Peter Forrester, Nathaël Gozlan, Michel Ledoux, Mylène Maïda, and Elizabeth Meckes for useful discussions and some help to locate references. We would also like to thank the anonymous referee for his careful reading of the manuscript and his suggestion to use [27, 29, 30] to prove Lemma 10.3.1 and Theorem 10.1.3. This note takes its roots in the blog post [23].

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Authors and Affiliations

  1. CEREMADE, CNRS UMR 7534, Université Paris-Dauphine, PSL, Paris, France

    Djalil Chafaï & Joseph Lehec

  2. DMA, CNRS UMR 8553, École Normale Supérieure, Paris, France

    Joseph Lehec

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  1. Djalil Chafaï
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Correspondence to Joseph Lehec .

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Editors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel, Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel

    Bo'az Klartag

  2. Department of Mathematics, Technion – Israel Institute of Technology, Haifa, Israel

    Emanuel Milman

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Chafaï, D., Lehec, J. (2020). On Poincaré and Logarithmic Sobolev Inequalities for a Class of Singular Gibbs Measures. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2256. Springer, Cham. https://doi.org/10.1007/978-3-030-36020-7_10

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