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Logarithmic Sobolev Inequalities, Matrix Models and Free Entropy

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Abstract

We give two applications of logarithmic Sobolev inequalities to matrix models and free probability. We also provide a new characterization of semi-circular systems through a Poincaré-type inequality.

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References

  1. Voiculescu, D. V.: The analogues of entropy and of Fisher’s information measure in free probability theory. II. Invent. Math., 118(3), 411–440 (1994)

    Article  MathSciNet  Google Scholar 

  2. Voiculescu, D. V.: The analogues of entropy and of Fisher’s information measure in free probability theory. V. Noncommutative Hilbert transforms. Invent. Math., 132(1), 189–227 (1998)

    Article  MathSciNet  Google Scholar 

  3. Ledoux, M.: Concentration of measure and logarithmic Sobolev inequalities. Séminaire de Probabilités, XXXIII, Lecture Notes in Mathematics, 1709, Springer, 120–216 (1999)

  4. Bakry, D., Émery, M.: Diffusions hypercontractives. Séminaire de Probabilités, XIX, 1983/84, 177–206, Lecture Notes in Math., 1123, Springer, Berlin, 1985

  5. Biane, P., Speicher, R.: Free diffusions, free entropy and free fisher information. Ann. Inst. H. Poincaré, Probabilités et Statistiques, PR, 37, 581–606 (2001)

    Article  MathSciNet  Google Scholar 

  6. Biane, P., Voiculescu, D.: A free probability analogue of the Wasserstein distance on the trace-state space. Geometric and Functional Analysis, 11(6), 1125–1138 (2001)

    Article  MathSciNet  Google Scholar 

  7. Saff, E. B., Totik, V.: Logarithmic potentials with external fields. Appendix B by T. Bloom. Grundlehren der Mathematischen Wissenschaften, 316. Springer-Verlag, Berlin (1997)

  8. Deift, P. A.: Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. Courant Lecture Notes in Mathematics, 3. New York University, Courant Institute of Mathematical Sciences, New York (1999)

  9. Abraham, R., Marsden, J. E.: Foundations of mechanics. Second edition. Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass. (1978)

  10. Voiculescu, D. V.: Cyclomorphy. MSRI preprint # 2001–019.

  11. Voiculescu, D. V.: The analogues of entropy and of Fisher’s information measure in free probability theory IV. Maximum entropy and freeness. Free probability theory (Waterloo, ON, 1995), 293–302, Fields Inst. Commun., 12, Amer. Math. Soc., Providence, RI (1997)

  12. Sauvageot, J. L.: Quantum Dirichlet forms, differential calculus and semigroups. Quantum probability and applications, V (Heidelberg, 1988), 334–346, Lecture Notes in Math., 1442, Springer, Berlin (1990)

  13. Davies, E. B., Lindsay, J. M.: Noncommutative symmetric Markov semigroups. Math. Z., 210(3), 379–411 (1992)

    MathSciNet  Google Scholar 

  14. Nica, A., Shlyakhtenko, D., Speicher, R.: Some minimization problems for the free analogue of the Fisher information. Adv. Math., 141(2), 282–321 (1999)

    Article  MathSciNet  Google Scholar 

  15. Borovkov, A. A., Utev, S. A.: An inequality and a characterization of the normal distribution connected with it. Teor. Veroyatnost. i Primenen., 28(2), 209–218 (1983)

    MathSciNet  Google Scholar 

  16. Biane, P.: Free hypercontractivity. Communications in Mathematical Physics, 184, 457–474 (1997)

    Article  MathSciNet  Google Scholar 

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  1. CNRS, Départment de Mathématiques et Applications, École Normale Supérieure, 45, rue d'Ulm, 75005, Paris, France

    Philippe Biane

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  1. Philippe Biane
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Correspondence to Philippe Biane.

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Biane, P. Logarithmic Sobolev Inequalities, Matrix Models and Free Entropy. Acta Math Sinica 19, 497–506 (2003). https://doi.org/10.1007/s10114-003-0271-5

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  • DOI: https://doi.org/10.1007/s10114-003-0271-5

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