Skip to main content Accessibility help
×
Home

A Free Logarithmic Sobolev Inequality on the Circle

  • Fumio Hiai (a1), Dénes Petz (a2) and Yoshimichi Ueda (a3)

Abstract

Free analogues of the logarithmic Sobolev inequality compare the relative free Fisher information with the relative free entropy. In the present paper such an inequality is obtained for measures on the circle. The method is based on a random matrix approximation procedure, and a large deviation result concerning the eigenvalue distribution of special unitary matrices is applied and discussed.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      A Free Logarithmic Sobolev Inequality on the Circle
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      A Free Logarithmic Sobolev Inequality on the Circle
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      A Free Logarithmic Sobolev Inequality on the Circle
      Available formats
      ×

Copyright

References

Hide All
[1] Bakry, D. and Emery, M., Diffusion hypercontractives. In: Séminaire de probabilités XIX, Lecture Notes in Math. 1123, Springer, 1985, pp. 177206.
[2] Arous, G. Ben and Guionnet, A., Large deviations for Wigner's law and Voiculescu's noncommutative entropy. Probab. Theory Related Fields 108(1997), no. 4, 517542.
[3] Biane, Ph., Logarithmic Sobolev inequalities, matrix models and free entropy. Acta Math. Sin. 19(2003), no. 3, 497506.
[4] Biane, Ph. and Speicher, R., Free diffusions, free entropy and free Fisher information. Ann. Inst. H. Poincaré Probab. Statist. 37(2001), no. 5, 581606.
[5] Edwards, R. E., Fourier Series, A Modern Introduction, 2. Second edition, Graduate Texts in Mathematics 85, Springer-Verlag, New York, 1982.
[6] Gross, L., Logarithmic Sobolev inequalities. Amer. J. Math. 97(1975), no. 4, 10611083.
[7] Hiai, F., Mizuo, M. and Petz, D., Free relative entropy for measures and a corresponding perturbation theory. J. Math. Soc. Japan 54(2002), no. 3, 679718.
[8] Hiai, F. and Petz, D., Properties of free entropy related to polar decomposition. Comm. Math. Phys. 202(1999), no. 2, 421444.
[9] Hiai, F. and Petz, D., A large deviation theorem for the empirical eigenvalue distribution of random unitary matrices. Ann. Inst. H. Poincaré Probab. Statist. 36(2000), no. 1, 7185.
[10] Hiai, F. and Petz, D., The Semicircle Law, Free Random Variables and Entropy. Mathematical Surveys and Monographs 77, American Mathematical Society, Providence, RI, 2000.
[11] Hiai, F., Petz, D., and Ueda, Y., Inequalities related to free entropy derived from random matrix approximation, Unpublished notes, 2003, math.OA/0310453.
[12] Hiai, F., Petz, D., and Ueda, Y., Free transportation cost inequalities via random matrix approximation. Probab. Theory Related Fields 130(2004), no. 2, 199221.
[13] Koosis, P., Introduction to Hp Spaces. Second edition. Cambridge Tracts in Mathematics 115, Cambridge University Press, Cambridge, 1998.
[14] Milnor, J., Curvature of left invariant metrics on Lie groups. Advances in Math. 21(1976), no. 3, 293329.
[15] Saff, E. B. and Totik, V., Logarithmic Potentials with External Fields. Grundlehren der Mathematischen Wissenschaften 316, Springer-Verlag, Berlin, 1997.
[16] Voiculescu, D., The analogues of entropy and of Fisher's information measure in free probability theory. I. Comm. Math. Phys. 155(1993), no. 1, 7192.
[17] Voiculescu, D., The analogues of entropy and of Fisher's information measure in free probability theory. II. Invent. Math. 118(1994), no. 3, 411440.
[18] Voiculescu, D., The analogues of entropy and of Fisher's information measure in free probability theory. V. Noncommutative Hilbert transforms. Invent. Math. 132(1998), no. 1, 189227.
[19] Voiculescu, D., The analogue of entropy and of Fisher's information measure in free probability theory. VI. Liberation and mutual free information. Adv. Math. 146(1999), no. 2, 101166.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

Related content

Powered by UNSILO

A Free Logarithmic Sobolev Inequality on the Circle

  • Fumio Hiai (a1), Dénes Petz (a2) and Yoshimichi Ueda (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.