Discrete Isoperimetric Inequalities via Curvature

Zongchen Chen (Georgia Tech) Discrete Isoperimetric Inequalities via Curvature Local to Global Principles Isoperimetric inequalities on the Boolean hypercube play a fundamental role in the analysis of Boolean functions and have broad applications in probability theory, combinatorics, and theoretical computer science. These inequalities have been established primarily for the uniform measure and for biased product measures, using Fourier-analytic or inductive arguments. In this talk, we extend the Kahn–Kalai–Linial inequality, Talagrand’s L^1–L^2 and variance–surface-area inequalities, and the Eldan–Gross inequality to weakly dependent measures. We show that a Dobrushin-type condition implies all four inequalities. In particular, our results apply to Ising models whose interaction matrix has 1-norm less than 1. Our approach uses a semigroup framework based on discrete Bakry–Émery theory and gradient estimates. Joint work with Zejia Chen and Xinyuan Zhang.