A Free Logarithmic Sobolev Inequality on the Circle

Hiai, Fumio ; Petz, Dénes ; Ueda, Yoshimichi

Canadian mathematical bulletin, Tome 49 (2006) no. 3, pp. 389-406

Résumé

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.

Détail

DOI : 10.4153/CMB-2006-039-7

Mots-clés : 46L54, 60E15, 94A17

Bibliographie

[1] [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] [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] [3] Biane, Ph., Logarithmic Sobolev inequalities, matrix models and free entropy. Acta Math. Sin. 19(2003), no. 3, 497–506. Google Scholar

[4] [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] [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] [6] Gross, L., Logarithmic Sobolev inequalities. Amer. J. Math. 97(1975), no. 4, 1061–1083. Google Scholar

[7] [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] [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] [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] [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] [11] Hiai, F., Petz, D., and Ueda, Y., Inequalities related to free entropy derived from random matrix approximation, Unpublished notes, 2003, math.OA/0310453.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] [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] [13] Koosis, P., Introduction to Hp Spaces. Second edition. Cambridge Tracts in Mathematics 115, Cambridge University Press, Cambridge, 1998. Google Scholar

[14] [14] Milnor, J., Curvature of left invariant metrics on Lie groups. Advances in Math. 21(1976), no. 3, 293–329. Google Scholar

[15] [15] Saff, E. B. and Totik, V., Logarithmic Potentials with External Fields. Grundlehren der Mathematischen Wissenschaften 316, Springer-Verlag, Berlin, 1997. Google Scholar

[16] [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] [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] [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] [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