{
  "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--Émery theory and gradient estimates and can also be extended to measures on Hamming slices or hypergrids.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24687",
  "author_total": 3,
  "authors": [
    {
      "name": "Zejia Chen"
    },
    {
      "name": "Zongchen Chen"
    },
    {
      "name": "Xinyuan Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dagf196twtmn0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24687",
  "fulltext_url": null,
  "id": "dagf196twtmn0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24687",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214050513",
  "pdf_url": "https://arxiv.org/pdf/2609.24687",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dagf196twtmn0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24687",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24687"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Discrete Isoperimetric Inequalities via Curvature",
  "title_text": "Discrete Isoperimetric Inequalities via Curvature",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/dagf196twtmn0",
  "venue": "arXiv",
  "versions": []
}