{
  "abstract": "Ultra-log-concave distributions are ubiquitous in probability, combinatorics, and statistical mechanics and have been studied extensively. In this paper, we introduce a quantitative high-dimensional extension of this notion, called $δ$-ultra-log-concavity, for probability measures on $\\mathbb{N}^d$ with downward closed support. When $δ= 1$, this notion coincides with the class studied by Gurvits (2009) via strongly log-concave generating functions, and with the class defined by Anari, Oveis Gharan, and Vinzant (2021) via completely log-concave generating functions; in one dimension, it reduces to classical ultra-log-concavity. We establish several functional inequalities, including a weighted Poincaré inequality, a discrete Brascamp--Lieb inequality, and a weighted Wu-type modified log-Sobolev inequality. Our approach combines integrated Bakry--Émery calculus for a canonical birth-death chain with Poisson stochastic localization, which arises as the time reversal of coordinatewise binomial thinning. We further establish concentration of measure, maximum-entropy principles, and several closure properties for ultra-log-concave measures, and develop applications to queueing models, polymatroids, antiferromagnetic Potts models, and hardcore models. Finally, a lattice scaling limit of the discrete theory yields Poincaré and Brascamp--Lieb inequalities for Laguerre diffusions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23994",
  "author_total": 2,
  "authors": [
    {
      "name": "Zongchen Chen"
    },
    {
      "name": "Sihan Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ek6ak4x9tvbsw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23994",
  "fulltext_url": null,
  "id": "ek6ak4x9tvbsw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23994",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214052269",
  "pdf_url": "https://arxiv.org/pdf/2609.23994",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ek6ak4x9tvbsw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.23994",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23994"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "High-Dimensional Ultra-Log-Concave Distributions",
  "title_text": "High-Dimensional Ultra-Log-Concave Distributions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/ek6ak4x9tvbsw",
  "venue": "arXiv",
  "versions": []
}