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PreprintarXiv licencearXiv

Discrete Isoperimetric Inequalities via Curvature

  • Zejia Chen
  • Zongchen Chen
  • Xinyuan Zhang
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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 L1--L2 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.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026discrete,
  title = {{Discrete Isoperimetric Inequalities via Curvature}},
  author = {Zejia Chen and Zongchen Chen and Xinyuan Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24687},
  eprint = {2609.24687},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24687},
}
RIS
TY  - UNPB
TI  - Discrete Isoperimetric Inequalities via Curvature
AU  - Zejia Chen
AU  - Zongchen Chen
AU  - Xinyuan Zhang
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24687
UR  - https://arxiv.org/abs/2609.24687
AB  - 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.
ER  -