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March 2022 Concentration inequalities for log-concave distributions with applications to random surface fluctuations
Alexander Magazinov, Ron Peled
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Ann. Probab. 50(2): 735-770 (March 2022). DOI: 10.1214/21-AOP1545

Abstract

We derive two concentration inequalities for linear functions of log-concave distributions: an enhanced version of the classical Brascamp–Lieb concentration inequality and an inequality quantifying log-concavity of marginals in a manner suitable for obtaining variance and tail probability bounds.

These inequalities are applied to the statistical mechanics problem of estimating the fluctuations of random surfaces of the φ type. The classical Brascamp–Lieb inequality bounds the fluctuations whenever the interaction potential is uniformly convex. We extend these bounds to the case of convex potentials whose second derivative vanishes only on a zero measure set, when the underlying graph is a d-dimensional discrete torus. The result applies, in particular, to potentials of the form U(x)=|x|p with p>1 and answers a question discussed by Brascamp–Lieb–Lebowitz (In Statistical Mechanics (1975) 379–390, Springer). Additionally, new tail probability bounds are obtained for the family of potentials U(x)=|x|p+x2, p>2. This result answers a question mentioned by Deuschel and Giacomin (Stochastic Process. Appl. 89 (2000) 333–354).

Funding Statement

The work of AM was supported, in part, by Israel Science Foundation Grant 861/15, the European Research Council Starting Grant 678520 (LocalOrder) and the Russian Science Foundation Grant 20-41-09009.
The work of RP was supported in part by Israel Science Foundation Grants 861/15 and 1971/19 and by the European Research Council Starting Grant 678520 (LocalOrder).

Acknowledgments

We are grateful to Gady Kozma for letting us know of the work [4] and for sharing with us the question of understanding the typical maximal value of the random surface with potential U(x)=x4+x2 on the three-dimensional grid V(ΛLd):={1,,L}3. We thank Ronen Eldan, Emanuel Milman and Sasha Sodin for fruitful discussions of the presented results and related questions. We thank Jean-Dominique Deuschel for a discussion of the results of [21]. Lastly, we thank Shangjie Yang and an anonymous referee for helpful comments on the manuscript.

Citation

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Alexander Magazinov. Ron Peled. "Concentration inequalities for log-concave distributions with applications to random surface fluctuations." Ann. Probab. 50 (2) 735 - 770, March 2022. https://doi.org/10.1214/21-AOP1545

Information

Received: 1 November 2020; Revised: 1 July 2021; Published: March 2022
First available in Project Euclid: 24 March 2022

Digital Object Identifier: 10.1214/21-AOP1545

Subjects:
Primary: 60E15 , 60K35 , 82C24
Secondary: 82B20

Keywords: Brascamp–Lieb concentration inequality , effective interface models , Localization , log-concave distributions , random surfaces of the ∇φ type , tail probability bounds

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.50 • No. 2 • March 2022
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