Concentration inequalities for log-concave distributions with applications to random surface fluctuations

TL;DR

This paper develops new concentration inequalities for log-concave distributions and applies them to analyze fluctuations in random surface models, extending classical bounds to broader potential functions and providing new tail probability estimates.

Contribution

It introduces enhanced concentration inequalities for log-concave distributions and extends fluctuation bounds to non-uniformly convex potentials in statistical mechanics.

Findings

Extended Brascamp--Lieb inequalities to convex potentials with degenerate second derivatives.

Derived new tail bounds for potentials of the form |x|^p + x^2, p>2.

Answered longstanding questions on fluctuation bounds in random surface models.

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 $\nabla\varphi$ 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 (1975). Additionally, new tail probability bounds are obtained for the family of potentials $U(x) = |x|^p+x^2$, $p>2$. This result answers a question mentioned by Deuschel and Giacomin (2000).