Abstract.
By means of random matrix approximation procedure, we re-prove Biane and Voiculescu’s free analog of Talagrand’s transportation cost inequality for measures on R in a more general setup. Furthermore, we prove the free transportation cost inequality for measures on T as well by extending the method to special unitary random matrices.
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Bakry, D., Emery, M.: Diffusion hypercontractives, in Séminaire Probabilités XIX. Lecture Notes in Math. Vol. 1123, Springer-Verlag, 1985, pp. 177–206
Ben Arous, G., Guionnet, A.: Large deviation for Wigner’s law and Voiculescu’s noncommutative entropy. Probab. Theory Related Fields 108, 517–542 (1997)
Bhatia, R.: Matrix Analysis, Springer-Verlag, New York, 1996
Biane, Ph.: Logarithmic Sobolev inequalities, matrix models and free entropy. Acta Math. Sinica 19 (3), 1–11 (2003)
Biane, Ph., Speicher, R.: Free diffusions, free entropy and free Fisher information. Ann. Inst. H. Poincaré Probab. Statist. 37, 581–606 (2001)
Biane, Ph., Voiculescu, D.: A free probabilistic analogue of the Wasserstein metric on the trace-state space. Geom. Funct. Anal. 11, 1125–1138 (2001)
Deift, P.A.: Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes in Math., Vol. 3, New York Univ., Courant Inst. Math. Sci., New York, 1999
Dembo, A., Zeitouni, O.: Large deviations techniques and applications, Second edition. Applications of Mathematics, Vol. 38, Springer-Verlag, New York, 1998
Deuschel, J.D., Stroock, D.W.: Large deviations. Pure and Applied Mathematics, Vol. 137. Academic Press, Inc., Boston, MA, 1989
Donoghue, Jr., W.F.: Monotone Matrix Functions and Analytic Continuation. Springer-Verlag, Berlin-Heidelberg-New York, 1974
Hiai, F., Mizuo, M., Petz, D.: Free relative entropy for measures and a corresponding perturbation theory. J. Math. Soc. Japan 54, 679–718 (2002)
Hiai, F., Petz, D.: Properties of free entropy related to polar decomposition. Comm. Math. Phys. 202, 421–444 (1999)
Hiai, F., Petz, D.: A large deviation theorem for the empirical eigenvalue distribution of random unitary matrices. Ann. Inst. H. Poincaré Probab. Statist. 36, 71–85 (2000)
Hiai, F., Petz, D.: The Semicircle Law, Free Random Variables and Entropy. Mathematical Surveys and Monographs, Vol. 77, Amer. Math. Soc., Providence, 2000
Hiai, F., Petz, D., Ueda, Y.: In preparation
Knapp, A.W.: Representation Theory of Semisimple Groups, An Overview Based on Examples. Princeton University Press, Princeton, 1986
Koosis, P.: Introduction to H p Spaces, Second edition. Cambridge Tracts in Math., Vol. 115, Cambridge Univ. Press, Cambridge, 1998
Ledoux, M.: Concentration of measures and logarithmic Sobolev inequalities. In Séminaire de Probabilités XXXIII, Lecture Notes in Math. Vol. 1709, Springer-Verlag, 1999, pp. 120–216
Ledoux, M.: The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs, Vol. 89, Amer. Math. Soc., Providence, 2001
Mehta, M.L.: Random Matrices. Second edition, Academic Press, Boston, 1991
Milnor, J.: Curvature of left invariant metrics on Lie groups. Adv. Math. 21, 293–329 (1976)
Ohya, M., Petz, D.: Quantum Entropy and Its Use, Springer-Verlag, Berlin, 1993
Otto, F., Villani, C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173, 361–400 (2000)
Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields, Springer-Verlag, Berlin-Heidelberg-New York, 1997
Shlyakhtenko, D.: Free fisher information with respect to a complete positive map and cost of equivalence relations. Comm. Math. Phys. 218, 133–152 (2001)
Talagrand, M.: Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6, 587–600 (1996)
Villani, C.: Topics in Optimal Transportation, Grad. Studies in Math., Vol. 58, Amer. Math. Soc., Providence, 2003
Voiculescu, D.: The analogues of entropy and of Fisher’s information measure in free probability theory, I. Comm. Math. Phys. 155, 71–92 (1993)
Voiculescu, D.: The analogues of entropy and of Fisher’s information measure in free probability theory, II. Invent. Math. 118, 411–440 (1994)
Voiculescu, D.: The analogues of entropy and of Fisher’s information measure in free probability theory, V, Noncommutative Hilbert transforms. Invent. Math. 132, 189–227 (1998)
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, 101–166 (1999)
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Supported in part by Grant-in-Aid for Scientific Research (C)14540198 and by the program “R&D support scheme for funding selected IT proposals” of the Ministry of Public Management, Home Affairs, Posts and Telecommunications.
Supported in part by MTA-JSPS project (Quantum Probability and Information Theory) and by OTKA T032662.
Supported in part by Grant-in-Aid for Young Scientists (B)14740118.
Mathematics Subject Classification (2000): Primary: 46L54, 46L53; Secondary: 60F10, 15A52, 94A17.
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Hiai, F., Petz, D. & Ueda, Y. Free transportation cost inequalities via random matrix approximation. Probab. Theory Relat. Fields 130, 199–221 (2004). https://doi.org/10.1007/s00440-004-0351-1
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DOI: https://doi.org/10.1007/s00440-004-0351-1