离散等周不等式：基于曲率的方法
Discrete Isoperimetric Inequalities via Curvature
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中文总结 AI 辅助
本文提出基于曲率的半群框架，在弱依赖坐标测度下证明多种离散等周不等式，适用于Dobrushin唯一性区域的Ising模型。
中文摘要 AI 辅助
布尔超立方体上的等周不等式在布尔函数分析中起着基础性作用。这些不等式主要针对均匀测度和有偏乘积测度建立，通常通过傅里叶分析或归纳论证。我们开发了一个基于曲率的半群框架，用于建立具有弱依赖坐标的测度的等周不等式。特别地，我们证明了一个Dobrushin型条件，连同分布的边际有界性，蕴含局部Bobkov不等式、Talagrand的$L^1$--$L^2$和方差--表面积不等式、Kahn--Kalai--Linial不等式以及Eldan--Gross不等式。我们的结果适用于相互作用矩阵为$J$的零场Ising模型，在整个Dobrushin唯一性区域$\u001b|J\u001b|_1<1$内。该框架建立在离散Bakry--Émery理论和梯度估计的基础上，也可扩展到Hamming切片或超网格上的测度。
英文摘要
Isoperimetric inequalities on the Boolean hypercube play a fundamental role in the analysis of Boolean functions. These inequalities have been established primarily for the uniform measure and for biased product measures, often through Fourier-analytic or inductive arguments. We develop a curvature-based semigroup framework to establish isoperimetric inequalities for measures with weakly dependent coordinates. In particular, we show that a Dobrushin-type condition, together with marginal boundedness of the distribution, implies the local Bobkov inequality, Talagrand's $L^1$--$L^2$ and variance--surface-area inequalities, the Kahn--Kalai--Linial inequality, and the Eldan--Gross inequality. Our results apply to zero-field Ising models with interaction matrix $J$ throughout the Dobrushin uniqueness regime $\|J\|_1<1$. The framework builds on discrete Bakry--Émery theory and gradient estimates and can also be extended to measures on Hamming slices or hypergrids.
发表机构
- Georgia Institute of Technology（佐治亚理工学院）
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