G. Anderson, A. Guionnet, and O. Zeitouni, An Introduction to Random Matrices, Cambridge Studies in Advanced Mathematics, vol.118, 2010.
DOI : 10.1017/CBO9780511801334

D. Bakry, F. Barthe, P. Cattiaux, and A. Guillin, A simple proof of the Poincar?? inequality for a large class of probability measures, Electronic Communications in Probability, vol.13, issue.0, pp.60-66, 2008.
DOI : 10.1214/ECP.v13-1352

D. Bakry and M. Emery, Diffusions hypercontractives, Séminaire de probabilités, pp.177-206, 1985.
DOI : 10.1007/BFb0075847

F. Barthe, Inégalités fonctionnelles et géométriques obtenues par transport de mesures, 1997.

F. Barthe, On a reverse form of the Brascamp-Lieb inequality, Inventiones Mathematicae, vol.134, issue.2, pp.335-361, 1998.
DOI : 10.1007/s002220050267
URL : https://hal.archives-ouvertes.fr/hal-00694256

F. Barthe, P. Cattiaux, and C. Roberto, Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry, Revista Matem??tica Iberoamericana, vol.22, issue.3, pp.993-1067, 2006.
DOI : 10.4171/RMI/482
URL : https://hal.archives-ouvertes.fr/hal-00014138

F. Barthe and A. V. Kolesnikov, Mass Transport and Variants of the Logarithmic Sobolev Inequality, Journal of Geometric Analysis, vol.22, issue.1, pp.921-979, 2008.
DOI : 10.1007/s12220-008-9039-6
URL : https://hal.archives-ouvertes.fr/hal-00634530

F. Barthe and C. Roberto, Sobolev inequalities for probability measures on the real line, Studia Mathematica, vol.159, issue.3, pp.481-497, 2003.
DOI : 10.4064/sm159-3-9
URL : https://hal.archives-ouvertes.fr/hal-00693117

F. Barthe and C. Roberto, Modified Logarithmic Sobolev Inequalities on $\mathbb{R}$, Potential Analysis, vol.21, issue.2, pp.167-193, 2008.
DOI : 10.1007/s11118-008-9093-5

W. Beckner, A generalized Poincaré inequality for Gaussian measures, Proc. Amer, pp.397-400, 1989.

M. Beiglböck and W. Schachermayer, Duality for Borel measurable cost functions. To appear in Trans

G. , B. Arous, and A. Guionnet, Large deviations for Wigner's law and Voiculescu's non-commutative entropy. Probab. Theory Related Fields, pp.517-542, 1997.

P. Biane, Logarithmic Sobolev Inequalities, Matrix Models and Free Entropy, International Workshop on Operator Algebra and Operator Theory, pp.497-506, 2001.
DOI : 10.1007/s10114-003-0271-5

P. Biane and D. Voiculescu, A free probability analogue of the Wasserstein metric on the trace-state space, Geom. Funct. Anal, vol.11, issue.6, pp.1125-1138, 2001.

S. Bobkov, I. Gentil, and M. Ledoux, Hypercontractivity of Hamilton???Jacobi equations, Journal de Math??matiques Pures et Appliqu??es, vol.80, issue.7, pp.669-696, 2001.
DOI : 10.1016/S0021-7824(01)01208-9

S. Bobkov and F. Götze, Exponential Integrability and Transportation Cost Related to Logarithmic Sobolev Inequalities, Journal of Functional Analysis, vol.163, issue.1, pp.1-28, 1999.
DOI : 10.1006/jfan.1998.3326

S. Bobkov and C. Houdré, Isoperimetric constants for product probability measures, The Annals of Probability, vol.25, issue.1, pp.184-205, 1997.
DOI : 10.1214/aop/1024404284

S. Bobkov and M. Ledoux, Poincaré's inequalities and Talagrand's concentration phenomenon for the exponential distribution. Probab. Theory Related Fields, pp.383-400, 1997.

S. Bobkov and M. Ledoux, From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities, Geometric and Functional Analysis, vol.10, issue.5, pp.1028-1052, 2000.
DOI : 10.1007/PL00001645

S. Bobkov and B. Zegarlinski, Entropy bounds and isoperimetry, Memoirs of the American Mathematical Society, vol.176, issue.829, p.69, 2005.
DOI : 10.1090/memo/0829

F. Bolley and C. Villani, Weighted Csisz??r-Kullback-Pinsker inequalities and applications to transportation inequalities, Annales de la facult?? des sciences de Toulouse Math??matiques, vol.14, issue.3, pp.331-352, 2005.
DOI : 10.5802/afst.1095

Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Communications on Pure and Applied Mathematics, vol.117, issue.4, pp.375-417, 1991.
DOI : 10.1002/cpa.3160440402

E. Carlen, Superadditivity of Fisher's information and logarithmic Sobolev inequalities, Journal of Functional Analysis, vol.101, issue.1, pp.194-211, 1991.
DOI : 10.1016/0022-1236(91)90155-X

P. Cattiaux and A. Guillin, Talagrand's like quadratic transportation cost inequalities, J. Math. Pures Appl, vol.86, issue.9, pp.341-361, 2006.
DOI : 10.1016/j.matpur.2006.06.003
URL : http://doi.org/10.1016/j.matpur.2006.06.003

P. Cattiaux, A. Guillin, and L. Wu, A note on Talagrand transportation inequality and logarithmic Sobolev inequality, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00629506

D. Cordero-erausquin, Some Applications of Mass Transport to Gaussian-Type Inequalities, Archive for Rational Mechanics and Analysis, vol.161, issue.3, pp.257-269, 2002.
DOI : 10.1007/s002050100185
URL : https://hal.archives-ouvertes.fr/hal-00693655

D. Cordero-erausquin, W. Gangbo, and C. Houdré, Inequalities for generalized entropy and optimal transportation, Contemp. Math, vol.353, pp.73-94, 2004.
DOI : 10.1090/conm/353/06433

D. Cordero-erausquin, R. Mccann, and M. Schmuckenschläger, A Riemannian interpolation inequality ?? la Borell, Brascamp and Lieb, Inventiones Mathematicae, vol.146, issue.2, pp.219-257, 2001.
DOI : 10.1007/s002220100160
URL : https://hal.archives-ouvertes.fr/hal-00693677

D. Cordero-erausquin, R. Mccann, and M. Schmuckenschläger, Pr??kopa???Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport, Annales de la facult?? des sciences de Toulouse Math??matiques, vol.15, issue.4, pp.613-635, 2006.
DOI : 10.5802/afst.1132

D. Cordero-erausquin, B. Nazaret, and C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo???Nirenberg inequalities, Advances in Mathematics, vol.182, issue.2, pp.307-332, 2004.
DOI : 10.1016/S0001-8708(03)00080-X
URL : https://hal.archives-ouvertes.fr/hal-00157943

I. Csiszár, Information-type measures of difference of probability distributions and indirect observations, Stud. Sci. Math. Hung, vol.2, pp.299-318, 1967.

A. Dembo, Information inequalities and concentration of measure, The Annals of Probability, vol.25, issue.2, pp.927-939, 1997.
DOI : 10.1214/aop/1024404424

A. Dembo and O. Zeitouni, Transportation Approach to Some Concentration Inequalities in Product Spaces, Electronic Communications in Probability, vol.1, issue.0, pp.83-90, 1996.
DOI : 10.1214/ECP.v1-979

A. Dembo and O. Zeitouni, Large deviations techniques and applications, 1998.
DOI : 10.1007/978-1-4612-5320-4

J. Deuschel and D. Stroock, Large Deviations, Pure and Applied Mathematics, vol.137, 1989.
DOI : 10.1090/chel/342

H. Djellout, A. Guillin, and L. Wu, Transportation cost-information inequalities and applications to random dynamical systems and diffusions, Ann. Probab, vol.32, issue.3B, pp.2702-2732, 2004.

R. Dobrushin, The Description of a Random Field by Means of Conditional Probabilities and Conditions of Its Regularity, Theory of Probability & Its Applications, vol.13, issue.2, pp.197-224, 1968.
DOI : 10.1137/1113026

R. Dobrushin, Prescribing a System of Random Variables by Conditional Distributions, Theory of Probability & Its Applications, vol.15, issue.3, pp.458-486, 1970.
DOI : 10.1137/1115049

M. Donsker and S. Varadhan, Asymptotic evaluation of certain markov process expectations for large time, I, Communications on Pure and Applied Mathematics, vol.XIX, issue.3, pp.1-47, 1975.
DOI : 10.1002/cpa.3160280102

R. Dudley, Real Analysis and Probability, volume 74 of Cambridge studies in advanced mathematics, 2002.

D. Feyel and A. S. , Monge-Kantorovitch measure transportation and Monge-Ampère equation on Wiener space. Probab. Theory Related Fields, pp.347-385, 2004.

H. Föllmer, Random fields and diffusion processes, in Ecole d'Eté de Probabilités de Saint-Flour XV- XVII-1985-87, Lecture Notes in Mathematics, vol.1362, 1988.

W. Gangbo and R. Mccann, The geometry of optimal transportation, Acta Mathematica, vol.177, issue.2, pp.113-161, 1996.
DOI : 10.1007/BF02392620

F. Gao, A. Guillin, and L. Wu, Bernstein type???s concentration inequalities for symmetric Markov processes, Teoriya Veroyatnostei i ee Primeneniya, vol.58, issue.3, 2009.
DOI : 10.4213/tvp4524

F. Gao and L. Wu, Transportation-information inequalities for Gibbs measures, 2007.

I. Gentil, A. Guillin, and L. Miclo, Modified logarithmic Sobolev inequalities and transportation inequalities . Probab. Theory Related Fields, pp.409-436, 2005.
DOI : 10.1007/s00440-005-0432-9
URL : https://hal.archives-ouvertes.fr/hal-00001609

I. Gentil and F. Malrieu, ??quations de Hamilton???Jacobi et in??galit??s entropiques g??n??ralis??es, Comptes Rendus Mathematique, vol.335, issue.5, pp.437-440, 2002.
DOI : 10.1016/S1631-073X(02)02513-X

N. Gozlan, Integral criteria for transportation cost inequalities, Electronic Communications in Probability, vol.11, issue.0, pp.64-77, 2006.
DOI : 10.1214/ECP.v11-1198
URL : https://hal.archives-ouvertes.fr/hal-00017036

N. Gozlan, Characterization of Talagrand's like transportation-cost inequalities on the real line, Journal of Functional Analysis, vol.250, issue.2, pp.400-425, 2007.
DOI : 10.1016/j.jfa.2007.05.025

N. Gozlan, Poincar?? inequalities and dimension free concentration of measure, Annales de l'Institut Henri Poincar??, Probabilit??s et Statistiques, vol.46, issue.3, 2008.
DOI : 10.1214/09-AIHP209

N. Gozlan, A characterization of dimension free concentration in terms of transportation inequalities, The Annals of Probability, vol.37, issue.6, pp.2480-2498, 2009.
DOI : 10.1214/09-AOP470
URL : https://hal.archives-ouvertes.fr/hal-00274548

N. Gozlan and C. Léonard, A large deviation approach to some transportation cost inequalities. Probab. Theory Related Fields, pp.235-283, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00012764

N. Gozlan, C. Roberto, and P. Samson, A refined version of Otto and Villani theorem, 2009.

M. Gromov and V. D. Milman, A Topological Application of the Isoperimetric Inequality, American Journal of Mathematics, vol.105, issue.4, pp.843-854, 1983.
DOI : 10.2307/2374298

A. Guillin, C. Léonard, F. Wang, and L. Wu, Transportation-information inequalities for markov processes (II) : relations with other functional inequalities, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00360854

A. Guillin, C. Léonard, L. Wu, and N. Yao, Transportation-information inequalities for Markov processes, Probability Theory and Related Fields, vol.128, issue.3, pp.669-695, 2009.
DOI : 10.1007/s00440-008-0159-5
URL : https://hal.archives-ouvertes.fr/hal-00158258

A. Guionnet, Large random matrices: lectures on macroscopic asymptotics, volume 1957 of Lecture Notes in Mathematics, Lectures from the 36th Probability Summer School held in Saint-Flour, 2006.

A. Guionnet and B. Zegarlinski, Lectures on Logarithmic Sobolev Inequalities, Lecture Notes in Mathematics, vol.1801, 2003.
DOI : 10.1007/978-3-540-36107-7_1
URL : http://archive.numdam.org/article/SPS_2002__36__1_0.pdf

F. Hiai, D. Petz, and Y. Ueda, Free transportation cost inequalities via random matrix approximation. Probab. Theory Related Fields, pp.199-221, 2004.
DOI : 10.1007/s00440-004-0351-1
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.62.1078

R. A. Horn and C. R. Johnson, Topics in matrix analysis, 1994.
DOI : 10.1017/CBO9780511840371

R. Kannan, L. Lovász, and M. Simonovits, Isoperimetric problems for convex bodies and a localization lemma, Discrete & Computational Geometry, vol.32, issue.312, pp.3-4541, 1995.
DOI : 10.1007/BF02574061

M. Knott and C. Smith, On the optimal mapping of distributions, Journal of Optimization Theory and Applications, vol.47, issue.1, pp.39-49, 1984.
DOI : 10.1007/BF00934745

S. Kullback, A lower bound for discrimination information in terms of variation, IEEE Trans. Inform. Theory, vol.4, pp.126-127, 1967.

R. Lataa-la and K. Oleszkiewicz, Between sobolev and poincar??, Lecture Notes in Math, vol.136, pp.147-168, 2000.
DOI : 10.1006/jfan.1996.0030

R. Lataa-la and J. Wojtaszczyk, On the infimum convolution inequality, Studia Mathematica, vol.189, issue.2, pp.147-187, 2008.
DOI : 10.4064/sm189-2-5

M. Ledoux, On Talagrand's deviation inequalities for product measures, ESAIM: Probability and Statistics, vol.1, pp.63-87, 1996.
DOI : 10.1051/ps:1997103

M. Ledoux, The concentration of measure phenomenon, volume 89 of Mathematical Surveys and Monographs, 2001.

M. Ledoux, A (one-dimensional) free Brunn???Minkowski inequality, Comptes Rendus Mathematique, vol.340, issue.4, pp.301-304, 2005.
DOI : 10.1016/j.crma.2004.12.017

M. Ledoux and I. Popescu, Mass transportation proofs of free functional inequalities, and free Poincar?? inequalities, Journal of Functional Analysis, vol.257, issue.4, pp.1175-1221, 2009.
DOI : 10.1016/j.jfa.2009.03.011

L. Leindler, On a certain converse of Hölder's inequality, Linear operators and approximations (Proc. Conf. Oberwolfach, pp.182-184, 1971.

C. Léonard, A saddle-point approach to the Monge-Kantorovich optimal transport problem Optimization and Calculus of Variations), ESAIM- COCV (Control

J. Lott and C. Villani, Hamilton???Jacobi semigroup on length spaces and applications, Journal de Math??matiques Pures et Appliqu??es, vol.88, issue.3, pp.219-229, 2007.
DOI : 10.1016/j.matpur.2007.06.003

Y. Ma, S. Shen, X. Wang, and L. Wu, Transportation inequalities: From Poisson to Gibbs measures, Proc. Appl
DOI : 10.3150/00-BEJ268

F. Martinelli, Lectures on Glauber Dynamics for Discrete Spin Models, Lecture Notes in Mathematics, vol.1717, 1997.
DOI : 10.1007/978-3-540-48115-7_2

K. Marton, A simple proof of the blowing-up lemma (Corresp.), IEEE Transactions on Information Theory, vol.32, issue.3, pp.445-446, 1986.
DOI : 10.1109/TIT.1986.1057176

K. Marton, Bounding $\bar{d}$-distance by informational divergence: a method to prove measure concentration, The Annals of Probability, vol.24, issue.2, pp.857-866, 1996.
DOI : 10.1214/aop/1039639365

K. Marton, A measure concentration inequality for contracting markov chains, Geometric and Functional Analysis, vol.81, issue.158, pp.556-571, 1996.
DOI : 10.1007/BF02249263

K. Marton, Measure concentration for a class of random processes. Probab. Theory Related Fields, pp.427-439, 1998.

K. Marton, Measure concentration for Euclidean distance in the case of dependent random variables, The Annals of Probability, vol.32, issue.3B, pp.2526-2544, 2004.
DOI : 10.1214/009117904000000702

P. Massart, Concentration inequalities and model selection Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, Lecture Notes in Mathematics, vol.1896, 2003.

B. Maurey, Some deviation inequalities, Geometric and Functional Analysis, vol.104, issue.2, pp.188-197, 1991.
DOI : 10.1007/BF01896377
URL : http://arxiv.org/abs/math/9201216

R. Mccann, A convexity theory for interacting gases and equilibrium crystals, 1994.

R. Mccann, Existence and uniqueness of monotone measure-preserving maps. Duke Math, J, vol.80, pp.309-323, 1995.

E. Milman, Isoperimetric and concentration inequalities -part i: Equivalence under curvature lower bound, 2009.
DOI : 10.1215/00127094-2010-038
URL : http://arxiv.org/abs/0902.1560

E. Milman, On the role of convexity in functional and isoperimetric inequalities, Proceedings of the London Mathematical Society, vol.99, issue.1, pp.32-66, 2009.
DOI : 10.1112/plms/pdn045

E. Milman, On the role of convexity in isoperimetry, spectral gap and??concentration, Inventiones mathematicae, vol.115, issue.9, pp.1-43, 2009.
DOI : 10.1007/s00222-009-0175-9

B. Muckenhoupt, Hardy's inequality with weights, Studia Math, vol.44, pp.31-38, 1972.

F. Otto and C. Villani, Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality, Journal of Functional Analysis, vol.173, issue.2, pp.361-400, 2000.
DOI : 10.1006/jfan.1999.3557

M. Pinsker, Information and information stability of random variables and processes, 1964.

I. Popescu, Talagrand Inequality for the Semicircular Law and Energy of the Eigenvalues of Beta Ensembles, Mathematical Research Letters, vol.14, issue.6, pp.1023-1032, 2007.
DOI : 10.4310/MRL.2007.v14.n6.a10

A. Prékopa, On logarithmic concave measures and functions, Acta Sci. Math. (Szeged), vol.34, pp.335-343, 1973.

G. Royer, Une initiation aux inégalités de Sobolev logarithmiques, volume 5 of Cours spécialisés, 1999.

P. Samson, Concentration of measure inequalities for Markov chains and $\Phi$-mixing processes, The Annals of Probability, vol.28, issue.1, pp.416-461, 2000.
DOI : 10.1214/aop/1019160125

P. Samson, Concentration Inequalities for Convex Functions on Product Spaces, Stochastic inequalities and applications, pp.33-52, 2003.
DOI : 10.1007/978-3-0348-8069-5_4

P. Samson, Infimum-convolution description of concentration properties of product probability measures, with applications, Annales de l'Institut Henri Poincare (B) Probability and Statistics, vol.43, issue.3, pp.321-338, 2007.
DOI : 10.1016/j.anihpb.2006.05.003
URL : https://hal.archives-ouvertes.fr/hal-00693095

D. Stroock and B. Zegarlinski, The equivalence of the logarithmic Sobolev inequality and the Dobrushin-Shlosman mixing condition, Communications in Mathematical Physics, vol.123, issue.1, pp.303-323, 1992.
DOI : 10.1007/BF02101094

D. Stroock and B. Zegarlinski, The logarithmic sobolev inequality for discrete spin systems on a lattice, Communications in Mathematical Physics, vol.133, issue.1, pp.175-193, 1992.
DOI : 10.1007/BF02096629

M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces. Publications Mathématiques de l'I, pp.73-203, 1995.

M. Talagrand, New concentration inequalities in product spaces, Inventiones Mathematicae, vol.126, issue.3, pp.505-563, 1996.
DOI : 10.1007/s002220050108

M. Talagrand, A new look at independence, The Annals of Probability, vol.24, issue.1, pp.1-34, 1996.
DOI : 10.1214/aop/1042644705

M. Talagrand, Transportation cost for Gaussian and other product measures, Geometric and Functional Analysis, vol.27, issue.3, pp.587-600, 1996.
DOI : 10.1007/BF02249265

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, vol.58, 2003.
DOI : 10.1090/gsm/058

C. Villani, Optimal transport, of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences, 2009.
DOI : 10.1007/978-3-540-71050-9
URL : https://hal.archives-ouvertes.fr/hal-00974787

F. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Related Fields, pp.417-424, 1997.
DOI : 10.1007/s004400050137

F. Wang, Functional Inequalities for Empty Essential Spectrum, Journal of Functional Analysis, vol.170, issue.1, pp.219-245, 2000.
DOI : 10.1006/jfan.1999.3516
URL : http://doi.org/10.1006/jfan.1999.3516

F. Wang, Probability distance inequalities on Riemannian manifolds and path spaces, Journal of Functional Analysis, vol.206, issue.1, pp.167-190, 2004.
DOI : 10.1016/S0022-1236(02)00100-3

F. Wang, A generalization of Poincaré and log-Sobolev inequalities. Potential Anal, pp.1-15, 2005.

F. Wang, From super Poincar?? to weighted log-Sobolev and entropy-cost inequalities, Journal de Math??matiques Pures et Appliqu??es, vol.90, issue.3, pp.270-285, 2008.
DOI : 10.1016/j.matpur.2008.06.004

F. Wang, Generalized transportation-cost inequalities and applications. Potential Anal, pp.321-334, 2008.
DOI : 10.1007/s11118-008-9079-3
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.619.749

L. Wu, Uniformly Integrable Operators and Large Deviations for Markov Processes, Journal of Functional Analysis, vol.172, issue.2, pp.301-376, 2000.
DOI : 10.1006/jfan.1999.3544

L. Wu, Poincar?? and transportation inequalities for Gibbs measures under the Dobrushin uniqueness condition, The Annals of Probability, vol.34, issue.5, pp.1960-1989, 2006.
DOI : 10.1214/009117906000000368

L. Wu, Transportation inequalities for stochastic differential equations of pure jumps, Annales de l'Institut Henri Poincar??, Probabilit??s et Statistiques, vol.46, issue.2, 2009.
DOI : 10.1214/09-AIHP320

B. Zegarlinski, Dobrushin uniqueness theorem and logarithmic Sobolev inequalities, Journal of Functional Analysis, vol.105, issue.1, pp.77-111, 1992.
DOI : 10.1016/0022-1236(92)90073-R
URL : http://doi.org/10.1016/0022-1236(92)90073-r

U. Laboratoire-d-'analyse-et-mathématiques-appliquées and . Cnrs, Université Paris Est. 5 bd Descartes, 77454 Marne la Vallée, France E-mail address: nathael.gozlan@univ-mlv.fr Modal-X, Bât, vol.8050