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

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 $\nabla \mathit{\phi }$ 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 $\mathit{U}(\mathit{x})=|\mathit{x}{|}^{\mathit{p}}$ with $\mathit{p}>1$ 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 $\mathit{U}(\mathit{x})=|\mathit{x}{|}^{\mathit{p}}+{\mathit{x}}^2$, $\mathit{p}>2$. This result answers a question mentioned by Deuschel and Giacomin (Stochastic Process. Appl. 89 (2000) 333–354).