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

Alexander Magazinov, Ron Peled
<span title="2021-11-22">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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
more &raquo; ... ever 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).
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