发表机构
Georgia Institute of Technology; Shanghai Jiao Tong University（佐治亚理工学院; 上海交通大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出高维超对数凹分布的定量推广（$\delta$-超对数凹性），建立函数不等式、集中性与最大熵原理，并应用于多个模型。
AI 中文摘要
超对数凹分布在概率论、组合学和统计力学中普遍存在，并已被广泛研究。在本文中，我们针对定义在具有向下闭支撑的 $\mathbb{N}^d$ 上的概率测度，引入了这一概念的定量高维推广，称为 $\delta$-超对数凹性。当 $\delta = 1$ 时，该概念与 Gurvits (2009) 通过强对数凹生成函数研究的类以及 Anari、Oveis Gharan 和 Vinzant (2021) 通过完全对数凹生成函数定义的类一致；在一维情形下，它退化为经典的超对数凹性。我们建立了若干函数不等式，包括加权 Poincaré 不等式、离散 Brascamp--Lieb 不等式以及加权 Wu 型修正对数 Sobolev 不等式。我们的方法将典范生灭过程的集成 Bakry--Émery 演算与 Poisson 随机局域化相结合，后者作为坐标二项式稀疏化的时间反转出现。我们进一步建立了超对数凹测度的集中不等式、最大熵原理以及若干闭包性质，并发展了在排队模型、多拟阵、反铁磁 Potts 模型和硬核模型中的应用。最后，离散理论的格点标度极限导出了 Laguerre 扩散的 Poincaré 和 Brascamp--Lieb 不等式。
英文摘要
Ultra-log-concave distributions are ubiquitous in probability, combinatorics, and statistical mechanics and have been studied extensively. In this paper, we introduce a quantitative high-dimensional extension of this notion, called $δ$-ultra-log-concavity, for probability measures on $\mathbb{N}^d$ with downward closed support. When $δ= 1$, this notion coincides with the class studied by Gurvits (2009) via strongly log-concave generating functions, and with the class defined by Anari, Oveis Gharan, and Vinzant (2021) via completely log-concave generating functions; in one dimension, it reduces to classical ultra-log-concavity. We establish several functional inequalities, including a weighted Poincaré inequality, a discrete Brascamp--Lieb inequality, and a weighted Wu-type modified log-Sobolev inequality. Our approach combines integrated Bakry--Émery calculus for a canonical birth-death chain with Poisson stochastic localization, which arises as the time reversal of coordinatewise binomial thinning. We further establish concentration of measure, maximum-entropy principles, and several closure properties for ultra-log-concave measures, and develop applications to queueing models, polymatroids, antiferromagnetic Potts models, and hardcore models. Finally, a lattice scaling limit of the discrete theory yields Poincaré and Brascamp--Lieb inequalities for Laguerre diffusions.
