Advertisement

Hypercontractive Measures, Talagrand’s Inequality, and Influences

  • Dario Cordero-Erausquin
  • Michel Ledoux
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2050)

Abstract

We survey several Talagrand type inequalities and their application to influences with the tool of hypercontractivity for both discrete and continuous, and product and non-product models. The approach covers similarly by a simple interpolation the framework of geometric influences recently developed by N. Keller, E. Mossel and A. Sen. Geometric Brascamp-Lieb decompositions are also considered in this context.

Keywords

Cayley Graph Dirichlet Form Logarithmic Sobolev Inequality Markov Operator Markov Semigroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

We thank F. Barthe and P. Cattiaux for their help with the bound (26), and R. Rossignol for pointing out to us the references [20, 21]. We also thank J. van den Berg and D. Kiss for pointing out that the techniques developed here cover the example of the complete graph and for letting us know about [14].

References

  1. 1.
    C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, G. Scheffer, Sur les inégalités de Sobolev logarithmiques. (French. Frech summary) [Logarithmic Sobolev inequalities] Panoramas et Synthèses [Panoramas and Syntheses] vol. 10 (Société Mathématique de France, Paris, 2000)Google Scholar
  2. 2.
    D. Bakry, in L’hypercontractivité et son utilisation en théorie des semigroupes. Ecole d’Eté de Probabilités de Saint-Flour, Lecture Notes in Math., vol. 1581 (Springer, Berlin, 1994), pp. 1–114Google Scholar
  3. 3.
    D. Bakry, M. Ledoux, Lévy-Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator. Invent. Math. 123, 259–281 (1996)Google Scholar
  4. 4.
    D. Bakry, I. Gentil, M. Ledoux, Forthcoming monograph (2012)Google Scholar
  5. 5.
    F. Barthe, P. Cattiaux, C. Roberto, Interpolated inequalities between exponential and gaussian Orlicz hypercontractivity and isoperimetry. Revista Mat. Iberoamericana 22, 993–1067 (2006)Google Scholar
  6. 6.
    F. Barthe, D. Cordero-Erausquin, M. Ledoux, B. Maurey, Correlation and Brascamp-Lieb inequalities for Markov semigroups. Int. Math. Res. Not. IMRN 10, 2177–2216 (2011)Google Scholar
  7. 7.
    W. Beckner, Inequalities in Fourier analysis. Ann. Math. 102, 159–182 (1975)Google Scholar
  8. 8.
    S. Bobkov, C. Houdré, A converse Gaussian Poincaré-type inequality for convex functions. Statist. Probab. Lett. 44, 281–290 (1999)Google Scholar
  9. 9.
    A. Bonami, Étude des coefficients de Fourier des fonctions de Lp(G). Ann. Inst. Fourier 20, 335–402 (1971)Google Scholar
  10. 10.
    P. Diaconis, L. Saloff-Coste, Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Prob. 6, 695–750 (1996)Google Scholar
  11. 11.
    D. Falik, A. Samorodnitsky, Edge-isoperimetric inequalities and influences. Comb. Probab. Comp. 16, 693–712 (2007)Google Scholar
  12. 12.
    J. Kahn, G. Kalai, N. Linial, The Influence of Variables on Boolean Functions. Foundations of Computer Science, IEEE Annual Symposium, 29th Annual Symposium on Foundations of Computer Science (FOCS, White Planes, 1988), pp. 68–80Google Scholar
  13. 13.
    N. Keller, E. Mossel, A. Sen, Geometric influences. Ann. Probab. 40(3), 1135–1166 (2012)Google Scholar
  14. 14.
    D. Kiss, A note on Talagrand’s variance bound in terms of influences. Preprint (2011)Google Scholar
  15. 15.
    M. Ledoux, The geometry of Markov diffusion generators. Ann. Fac. Sci. Toulouse IX, 305–366 (2000)Google Scholar
  16. 16.
    M. Ledoux, in The Concentration of Measure Phenomenon. Math. Surveys and Monographs, vol. 89 (Amer. Math. Soc., Providence, 2001)Google Scholar
  17. 17.
    T.Y. Lee, H.-T. Yau, Logarithmic Sobolev inequality for some models of random walks. Ann. Probab. 26, 1855–1873 (1998)Google Scholar
  18. 18.
    E. Milman, On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177, 1–43 (2009)Google Scholar
  19. 19.
    E. Milman, Isoperimetric and concentration inequalities – Equivalence under curvature lower bound. Duke Math. J. 154, 207–239 (2010)Google Scholar
  20. 20.
    R. O’Donnell, K. Wimmer, KKL, Kruskal-Katona, and Monotone Nets. 2009 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009) (IEEE Computer Soc., Los Alamitos, 2009), pp. 725–734Google Scholar
  21. 21.
    R. O’Donnell, K. Wimmer, Sharpness of KKL on Schreier graphs. Preprint (2011)Google Scholar
  22. 22.
    R. Rossignol, Threshold for monotone symmetric properties through a logarithmic Sobolev inequality. Ann. Probab. 34, 1707–1725 (2006)Google Scholar
  23. 23.
    G. Royer, An initiation to logarithmic Sobolev inequalities. Translated from the 1999 French original by Donald Babbitt. SMF/AMS Texts and Monographs, vol. 14 (American Mathematical Society, Providence, RI (Société Mathématique de France, Paris, 2007)Google Scholar
  24. 24.
    M. Talagrand, On Russo’s approximate zero-one law. Ann. Probab. 22, 1576–1587 (1994)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institut de MathématiquesUniversité Pierre et Marie Curie (Paris 6 – Jussieu)ParisFrance
  2. 2.Institut Universitaire de FranceParisFrance
  3. 3.Institut de Mathématiques de ToulouseUniversité de ToulouseToulouseFrance

Personalised recommendations