论文标题
对数孔循环分布的浓度不平等，并应用于随机表面波动
Concentration inequalities for log-concave distributions with applications to random surface fluctuations
论文作者
论文摘要
我们得出了两个浓度不平等的对数孔分布的线性函数：经典的brascamp- lieb浓度不平等的增强版本，以及一种不平等的边际量子concovity，以一种适合获取方差和尾巴概率界限的方式来量化边缘的对数。 这些不等式应用于估计$ \nablaφ$类型的随机表面的波动的统计力学问题。每当相互作用势均匀凸电势时，经典的布拉斯帕姆 - lieb不等式就会界定波动。我们将这些边界扩展到凸电势的情况，当基础图为$ d $维离散的圆环时，其第二个衍生物仅在零度量集中消失。结果尤其适用于$ u（x）= | x |^p $带有$ p> 1 $的潜力，并回答了Brascamp-lieb-lebowitz（1975）讨论的问题。此外，还为$ u（x）= | x |^p+x^2 $，$ p> 2 $的家族获得了新的尾巴概率界限。该结果回答了Deuschel和Giacomin（2000）提到的一个问题。
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 $\nablaφ$ 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).
