Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The one dimensional free Poincaré inequality

Authors: Michel Ledoux and Ionel Popescu
Journal: Trans. Amer. Math. Soc. 365 (2013), 4811-4849
MSC (2010): Primary 46L54; Secondary 60B20, 60E15, 33D45
Published electronically: April 24, 2013
MathSciNet review: 3066771
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we discuss the natural candidate for the one dimensional free Poincaré inequality. Two main strong points sustain this candidacy. One is the random matrix heuristic and the other the relations with the other free functional inequalities, namely, the free transportation and Log-Sobolev inequalities. As in the classical case the Poincaré is implied by the others. This investigation is driven by a nice lemma of Haagerup which relates logarithmic potentials and Chebyshev polynomials. The Poincaré inequality revolves around the counting number operator for the Chebyshev polynomials of the first kind with respect to the arcsine law on $ [-2,2]$. This counting number operator appears naturally in a representation of the minimum of the logarithmic energy with external fields discovered in Analyticity of the planar limit of a matrix model by S. Garoufalidis and the second author as well as in the perturbation of logarithmic energy with external fields, which is the essential connection between all these inequalities.

References [Enhancements On Off] (What's this?)

  • 1. D. Bakry.
    L'hypercontractivité et son utilisation en théorie des semigroupes.
    Ecole d'Eté de Probabilités de St-Flour. Lecture Notes in Math., 1581:1-114, 1994. MR 1307413 (95m:47075)
  • 2. Ph. Biane.
    Logarithmic Sobolev inequalities, matrix models and free entropy.
    Acta Math. Sin. (Engl. Ser.), 19(3):497-506, 2003. MR 2014030 (2004k:46113)
  • 3. Ph. Biane and D. Voiculescu.
    A free probability analogue of the Wasserstein metric on a trace-state space.
    GAFA - Geometric And Functional Analysis, 11:1125-1138, 2001. MR 1878316 (2003d:46087)
  • 4. S.G. Bobkov, I. Gentil, and M. Ledoux.
    Hypercontractivity of Hamilton-Jacobi equations.
    J. Math. Pures Appl. (9), 80(7):669-696, 2001. MR 1846020 (2003b:47073)
  • 5. T. Cabanal-Duvillard.
    Fluctuations de la loi empirique de grandes matrices aléatoires.
    Ann. Inst. H. Poincaré Probab. Statist., 37(3):373-402, 2001. MR 1831988 (2002c:15040)
  • 6. P. A. Deift.
    Orthogonal polynomials and random matrices: A Riemann-Hilbert approach, volume 3 of Courant Lecture Notes in Mathematics.
    New York University, Courant Institute of Mathematical Sciences, New York, 1999. MR 1677884 (2000g:47048)
  • 7. L. C. Evans.
    Partial differential equations, volume 19 of Graduate Studies in Mathematics.
    American Mathematical Society, Providence, RI, second edition, 2010. MR 2597943 (2011c:35002)
  • 8. S. Fang, F.-Yu Wang, and B. Wu.
    Transportation-cost inequality on path spaces with uniform distance.
    Stochastic Process. Appl., 118(12):2181-2197, 2008. MR 2474347 (2009m:60189)
  • 9. D. Feyel and A. S. Üstünel.
    Measure transport on Wiener space and the Girsanov theorem.
    C. R. Math. Acad. Sci. Paris, 334(11):1025-1028, 2002. MR 1913729 (2003h:60007)
  • 10. S. Garoufalidis and I. Popescu.
    Analyticity of the planar limit of a matrix model.
    arxiv:1010.0927, 2010.
  • 11. L. Gross.
    Logarithmic Sobolev inequalities.
    Amer. J. Math., 97(4):1061-1083, 1975. MR 0420249 (54:8263)
  • 12. U. Haagerup.
    Seminar notes on free probability.
  • 13. F. Hiai, D. Petz, and Y. Ueda.
    Free transportation cost inequalities via random matrix approximation.
    Probality Theory and Related Fields, 130:199-221, 2004. MR 2093762 (2005k:46177)
  • 14. E. P. Hsu.
    Logarithmic Sobolev inequalities on path spaces over Riemannian manifolds.
    Comm. Math. Phys., 189(1):9-16, 1997. MR 1478528 (98i:58249)
  • 15. M. Ledoux.
    A (one-dimensional) free Brunn-Minkowski inequality.
    C. R. Acad. Sciences, Paris, 340:301-304, 2005. MR 2121895 (2005m:60036)
  • 16. M. Ledoux and I. Popescu.
    Mass transportation proofs of free functional inequalities, and free Poincaré inequalities.
    J. Funct. Anal., 257(4):1175-1221, 2009. MR 2535467 (2011e:46106)
  • 17. F. Otto and C. Villani.
    Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality.
    Journal of Functional Analysis, 173(2):361-400, 2000. MR 1760620 (2001k:58076)
  • 18. K. T.-R. McLaughlin P. Deift, T. Kriecherbauer.
    New results for the equilibrium measure of logarithmic potentials with external fields obtained via the inverse spectral method.
    Journal of Approximation Theory, 95:388-475, 1998. MR 1657691 (2000j:31003)
  • 19. E. B. Saff and V. Totik.
    Logarithmic potentials with external fields, volume 316 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].
    Springer-Verlag, Berlin, 1997. MR 1485778 (99h:31001)
  • 20. C. Villani.
    Topics in optimal transportation, volume 58 of Graduate Studies in Mathematics.
    American Mathematical Society, Providence, RI, 2003. MR 1964483 (2004e:90003)
  • 21. F.-Y. Wang.
    Probability distance inequalities on Riemannian manifolds and path spaces.
    J. Funct. Anal., 206(1):167-190, 2004. MR 2024350 (2005a:58057)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 46L54, 60B20, 60E15, 33D45

Retrieve articles in all journals with MSC (2010): 46L54, 60B20, 60E15, 33D45

Additional Information

Michel Ledoux
Affiliation: Institut de Mathématiques de Toulouse, Université de Toulouse, F-31062 Toulouse, France – and – Institut Universitaire de France

Ionel Popescu
Affiliation: Department of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta Georgia 30332 – and – Institute of Mathematics of Romanian Academy, 21 Calea Grivitei Street, 010702-Bucharest, Sector 1, Romania

Received by editor(s): May 10, 2011
Received by editor(s) in revised form: October 21, 2011
Published electronically: April 24, 2013
Additional Notes: The second author was partially supported by Marie Curie Action grant no. 249200.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society