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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
D. Bakry, I. Gentil, M. Ledoux, Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften, vol. 348 (Springer, Berlin, 2014)
F. Baudoin, J. Wang, Curvature dimension inequalities and subelliptic heat kernel gradient bounds on contact manifolds. Potential Anal. 40, 163–193 (2014)
S. Boucheron, M. Thomas, Concentration inequalities for order statistics. Electron. Commun. Probab. 17, 1–12 (2012)
S. Boucheron, G. Lugosi, P. Massart, Concentration Inequalities: A Nonasymptotic Theory of Independance (Oxford University Press, Oxford, 2013)
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)
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)
S. Chatterjee, Superconcentration and Related Topics (Springer, Berlin, 2014)
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.169–189
M. Ledoux, The geometry of Markov diffusions operators. Ann. Fac. Sci. Toulouse Math. 6 9(2), 305–366 (2000)
M. Ledoux, The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs, vol. 89 (American Mathematical Society, Providence, 2001)
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)
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)
K. Tanguy, Talagrand’s inequality at higher order and application to Boolean analysis (2019). https://arxiv.org/pdf/1801.08931.pdf
K. Tanguy, Some superconcentration inequalities for extrema of stationary Gaussian processes. Stat. Probab. Lett. 106, 239–246 (2015)
K. Tanguy, Non asymptotic variance bounds and deviation inequalities by optimal transport (2017). Preprint, http://arxiv.org/abs/1708.08620
K. Tanguy, Quelques inégalités de concentration: théorie et applications (in French). Ph.D. thesis, Institute of Mathematics of Toulouse, 2017
P. Valettas, On the tightness of Gaussian concentration for convex functions. J. Anal. Math. (to appear)
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Tanguy, K. (2019). Remarks on Superconcentration and Gamma Calculus: Applications to Spin Glasses. In: Gozlan, N., Latała, R., Lounici, K., Madiman, M. (eds) High Dimensional Probability VIII. Progress in Probability, vol 74. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-26391-1_10
Download citation
DOI: https://doi.org/10.1007/978-3-030-26391-1_10
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-26390-4
Online ISBN: 978-3-030-26391-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)