Discrete Isoperimetric Inequalities via Curvature

Abstract

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--\'Emery theory and gradient estimates and can also be extended to measures on Hamming slices or hypergrids.