Modified logarithmic Sobolev inequalities in discrete settings

Journal of Theoretical Probability

Volumn 19, Issue 2, 2006, Pages 289-336

  Bobkov, Sergey G ^a   Tetali, Prasad ^b  

^a University of Minnesota   (United States)

^b Georgia Institute of Technology   (United States)

Author keywords

Entropy decay; Logarithmic Sobolev Inequalities; Spectral gap

Indexed keywords

EID: 33750501028     PISSN: 08949840     EISSN: 15729230     Source Type: Journal    
DOI: 10.1007/s10959-006-0016-3     Document Type: Article

References (37)

1. Alon, N., and Milman, V. D. (1985). λ1, isoperimetric inequalities for graphs and superconcentrators. J. Comb. Theory Ser. B 38, 73-88.
2. Bakry, D. (1994). L'hypercontractivité et son utilisation en théorie des semigroups. Ecole d'Été de Saint Flour, 1992. Lect. Notes in Math., Vol. 1581, Springer-Berlin.
3. Bakry, D., and Emery, M. (1994). Diffusions hypercontractive. Séminaire de Probabilités XIX, Lect. Notes in Math., Vol. 1123, Springer, Berlin, pp. 179-206.
4. Beckner, W. (1989). A generalized Poincaré inequality for Gaussian measures. Proc. of the AMS, 105(2), 397-400.
5. Bobkov, S. G., and Götze, F. (1999). Exponential integrability and transporation cost related to logarithmic transportation inequalities. J. Funct. Anal. 163, 1-28.
6. Bobkov, S. G., and Ledoux, M. (1997). Poincaré inequalities and Talagrand's measure concentration phenomenon for the exponential measure, Prob. Th. Relat. Fields 107, 383-400.
7. Bobkov, S. G., and Ledoux, M. (1998). On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156, 347-365.
8. Bobkov, S. G., and Tetali, P. (June 2003). Modified log-Sobolev inequalities, mixing and hypercontractivity. Proc. of Symposium on Theory of computing, San Diego, pp. 287-296.
9. Chen, G.-Y., and Shen, Y.-C. (2003). On the log-Sobolev constant for the simple random walk on the n-cycle: the even cases. J. Funct. Anal. 202, 473-485.
10. Chung, F. R. K. (1989). Diameters and Eigenvalues. J. Amer. Math. Soc. 2, 187-196.
11. Chung, F. R. K., Grigor'yan, A., and Yau, S.-T. (2000). Higher eigenvalues and isoperimetric inequalities on Riemannian manifolds and graphs. Comm. Anal. Geom. 8(5), 969-1026.
12. Cover, T. M., and Thomas, J. A. (1991). Elements of Information Theory, John Wiley & Sons, New York.
13. Dai Pra, P., Paganoni, A.M., and Posta, G. (2002). Entropy inequalities for unbounded spin systems. Ann. Probab. 30, 1959-1976.
14. Diaconis, P. (1988). Group representations in Probability and Statistics, IMS, Hayward, CA.
15. Diaconis, P., and Saloff-Coste, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6, 695-750.
16. Diaconis, P., and Shahshahani, M. (1981). Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57(2), 159-179.
17. Diaconis, P., and Shahshahani, M. (1987). Time to reach stationarity in the Bernoulli-Laplace diffusion model. SIAM J. Math. Anal. 18, 208-218.
18. Durrett, R. (1999). Essentials of Stochastic Processes. Springer Texts in Statistics, Springer-Verlag, New York Inc.
19. Frieze, A., and Kannan, R. (1998). Log-Sobolev Inequalities and Sampling from Log-Concave Distributions. Preprint.
20. Gao, F., and Quastel, J. (2003). Exponential decay of entropy in the Random Transposition and Bernoulli-Laplace models. Ann. Appl. Probab. 13, 1591-1600.
21. Goel, S. (2004). Modified logarithmic Sobolev inequalities for some models of random walk. Stoch. Proc. Their Appl. 114, 51-79.
22. Gross, L. (1975). Logarithmic Sobolev inequalities. Amer. J. Math. 97, 1060-1083.
23. Higuchi, Y., and Yoshida, N. (1995). Analytic conditions and phase transition for Ising models. Lect. Notes in Japanese, 1995.
24. Houdré, C., and Tetali, P. (2001). Concentration of measure for products of Markov kernels and graph products via functional inequalities. Comb. Probab. Comp. 10, 1-28.
25. Latala, R., and Oleszkiewicz, K. (2000). Between Sobolev and Poincaré. Geometric Aspects Function. Analy. Lect. Notes Math. 1745, 147-168.
26. Ledoux, M. (1996). On Talagrand's deviation inequalities for product measures. ESAIM Probab. Statist. 1, 63-87.
27. Ledoux, M. (1999). Concentration of measure and logarithmic Sobolev inequalities and the phenomenon. Séminaire de Probabilitiés XXXIII, Lecture Notes Math., 1709, 120-216.
28. Ledoux, M. (2001). The concentration of measure phenomenon. Mathematical Surveys and Monographs, 89. American Mathematical Society, Providence, RI.
29. Lee, T. Y., and Yau, H. T. (1998). Logarithmic Sobolev inequality for some models of random walks. Ann. Probab. 26(4), 1855-1873.
30. Lieb, E. H. (1975). Some convexity and subadditivity properties of entropy. Bull. Amer. Math. Soc. 81, 1-13.
31. Miclo, L. (1996). Sur les problémes de sortie discrets inhomogénes. Ann. Appl. Probab. 6(4), 1112-1156.
32. Mohar, B. (1991). Eigenvalues, diameter, and mean distance in graphs. Graphs Combin. 7, 53-64.
33. Rothaus, O. (1981). Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities. J. Funct. Anal. 42, 102-109.
34. Saloff-Coste, L. (1997). Lectures on finite Markov chains. Lect. Notes in Math. Vol. 1665, Springer, Berlin, pp. 301-413.
35. Sammer, M., and Tetali, P. (2004). Concentration and Transportation. Preprint (2004); conference version in SIAM Conf. on Discrete Math. (June 2004).
36. Stark, D. (2002). Information loss in top to random shuffling. Combin. Probab. & Comput. 11, 607-627.
37. Stroock, D. (1993). Logarithmic Sobolev inequalities for Gibbs measures. Lect. Notes in Math., Vol. 1563, Springer, Berlin.