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 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 with 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 , . This result answers a question mentioned by Deuschel and Giacomin (Stochastic Process. Appl. 89 (2000) 333–354).
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).
We are grateful to Gady Kozma for letting us know of the work  and for sharing with us the question of understanding the typical maximal value of the random surface with potential on the three-dimensional grid . 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 . Lastly, we thank Shangjie Yang and an anonymous referee for helpful comments on the manuscript.
"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