Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-23T02:40:45.187Z Has data issue: false hasContentIssue false
  • English
  • Français

A Free Logarithmic Sobolev Inequality on the Circle

Published online by Cambridge University Press:  20 November 2018

Fumio Hiai ,
Dénes Petz  and
Yoshimichi Ueda
Fumio Hiai
Affiliation:
Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-8579, Japan e-mail: hiai@math.is.tohoku.ac.jp
Dénes Petz
Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, H-1053 Budapest, Reáltanoda u. 13-15, Hungary e-mail: petz@renyi.hu
Yoshimichi Ueda
Affiliation:
Graduate School of Mathematics, Kyushu University, Fukuoka 810-8560, Japan e-mail: ueda@math.kyushu-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

46L5460E1594A17
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 49 , Issue 3 , 01 September 2006 , pp. 389 - 406
Copyright
Copyright © Canadian Mathematical Society 2006

References

[1] Bakry, D. and Emery, M., Diffusion hypercontractives. In: Séminaire de probabilités XIX, Lecture Notes in Math. 1123, Springer, 1985, pp. 177–206.Google Scholar
[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, 517–542.Google Scholar
[3] Biane, Ph., Logarithmic Sobolev inequalities, matrix models and free entropy. Acta Math. Sin. 19(2003), no. 3, 497–506.Google Scholar
[4] Biane, Ph. and Speicher, R., Free diffusions, free entropy and free Fisher information. Ann. Inst. H. Poincaré Probab. Statist. 37(2001), no. 5, 581–606.Google Scholar
[5] Edwards, R. E., Fourier Series, A Modern Introduction, 2. Second edition, Graduate Texts in Mathematics 85, Springer-Verlag, New York, 1982.Google Scholar
[6] Gross, L., Logarithmic Sobolev inequalities. Amer. J. Math. 97(1975), no. 4, 1061–1083.Google Scholar
[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, 679–718.Google Scholar
[8] Hiai, F. and Petz, D., Properties of free entropy related to polar decomposition. Comm. Math. Phys. 202(1999), no. 2, 421–444.Google Scholar
[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, 71–85.Google Scholar
[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.Google Scholar
[11] Hiai, F., Petz, D., and Ueda, Y., Inequalities related to free entropy derived from random matrix approximation, Unpublished notes, 2003, math.OA/0310453.Google Scholar
[12] Hiai, F., Petz, D., and Ueda, Y., Free transportation cost inequalities via random matrix approximation. Probab. Theory Related Fields 130(2004), no. 2, 199–221.Google Scholar
[13] Koosis, P., Introduction to Hp Spaces. Second edition. Cambridge Tracts in Mathematics 115, Cambridge University Press, Cambridge, 1998.Google Scholar
[14] Milnor, J., Curvature of left invariant metrics on Lie groups. Advances in Math. 21(1976), no. 3, 293–329.Google Scholar
[15] Saff, E. B. and Totik, V., Logarithmic Potentials with External Fields. Grundlehren der Mathematischen Wissenschaften 316, Springer-Verlag, Berlin, 1997.Google Scholar
[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, 71–92.Google Scholar
[17] Voiculescu, D., The analogues of entropy and of Fisher's information measure in free probability theory. II. Invent. Math. 118(1994), no. 3, 411–440.Google Scholar
[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, 189–227.Google Scholar
[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, 101–166.Google Scholar