CONCENTRATION INEQUALITIES FOR LOG-CONCAVE DISTRIBUTIONS WITH APPLICATIONS TO RANDOM SURFACE FLUCTUATIONS

We derive two concentration inequalities for linear functions of logconcave 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 (Formula presented) 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. Thisresultan-swers a question mentioned by Deuschel and Giacomin (Stochastic Process. Appl. 89 (2000) 333-354).