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Remarks on Superconcentration and Gamma Calculus: Applications to Spin Glasses

  • Kevin TanguyEmail author
Conference paper
Part of the Progress in Probability book series (PRPR, volume 74)

Abstract

This note is concerned with the so-called superconcentration phenomenon. It shows that the Bakry-Émery’s Gamma calculus can provide relevant bound on the variance of function satisfying a inverse, integrated, curvature criterion. As an illustration, we present some variance bounds for the Free Energy in different models from Spin Glasses Theory.

Notes

Acknowledgements

I thank M. Ledoux for fruitful discussions on this topic. I also warmly thank the referee for helpful comments in improving the exposition and the simplification of the proof of Lemma 10.4.1.

References

  1. 1.
    D. Bakry, I. Gentil, M. Ledoux, Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften, vol. 348 (Springer, Berlin, 2014)Google Scholar
  2. 2.
    F. Baudoin, J. Wang, Curvature dimension inequalities and subelliptic heat kernel gradient bounds on contact manifolds. Potential Anal. 40, 163–193 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    S. Boucheron, M. Thomas, Concentration inequalities for order statistics. Electron. Commun. Probab. 17, 1–12 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    S. Boucheron, G. Lugosi, P. Massart, Concentration Inequalities: A Nonasymptotic Theory of Independance (Oxford University Press, Oxford, 2013)CrossRefGoogle Scholar
  5. 5.
    A. Bovier, Gaussian Processes on Trees: From Spin Glasses to Branching Brownian Motion. Cambridge Studies in Advanced Mathematics, vol. 163 (Cambridge University Press, Cambridge, 2016)Google Scholar
  6. 6.
    A. Bovier, I. Kurkova, M. Löwe, Fluctuations of the free energy in the REM and the p-spin SK models. Ann. Probab. 30(2), 605–651 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    S. Chatterjee, Superconcentration and Related Topics (Springer, Berlin, 2014)CrossRefGoogle Scholar
  8. 8.
    D. Cordero-Erausquin, M. Ledoux, Hypercontractive measures, Talagrand’s inequality, and influences, in Geometric Aspects of Functional Analysis. Lectures Notes in Mathematics, vol. 2050 (Springer, Berlin, 2012), pp.169–189Google Scholar
  9. 9.
    M. Ledoux, The geometry of Markov diffusions operators. Ann. Fac. Sci. Toulouse Math. 6 9(2), 305–366 (2000)Google Scholar
  10. 10.
    M. Ledoux, The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs, vol. 89 (American Mathematical Society, Providence, 2001)Google Scholar
  11. 11.
    M. Talagrand, Mean Field Models for Spin Glasses. Volume I. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics (Springer, Berlin, 2011)Google Scholar
  12. 12.
    M. Talagrand, Mean Field Models for Spin Glasses. Volume II: Advanced Replica-Symmetry and Low Temperature, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 55 (Springer, Heidelberg, 2011)Google Scholar
  13. 13.
    K. Tanguy, Talagrand’s inequality at higher order and application to Boolean analysis (2019). https://arxiv.org/pdf/1801.08931.pdf
  14. 14.
    K. Tanguy, Some superconcentration inequalities for extrema of stationary Gaussian processes. Stat. Probab. Lett. 106, 239–246 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    K. Tanguy, Non asymptotic variance bounds and deviation inequalities by optimal transport (2017). Preprint, http://arxiv.org/abs/1708.08620
  16. 16.
    K. Tanguy, Quelques inégalités de concentration: théorie et applications (in French). Ph.D. thesis, Institute of Mathematics of Toulouse, 2017Google Scholar
  17. 17.
    P. Valettas, On the tightness of Gaussian concentration for convex functions. J. Anal. Math. (to appear)Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of AngersAngersFrance

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